Subgroups — C₆₇×#34383 #30713634

Order 17152.

2190 conjugacy classes of subgroups.

The subgroup lattice has 732 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×#529×C₆₇	C₂	1	1	yes	maximal, characteristic
2	C₆₇×#2376	C₂	1	4	yes	maximal, complemented
3	C₆₇×#2370	C₂	1	2	yes	maximal, complemented
4	C₆₇×#2367	C₂	1	2	yes	maximal, complemented
5	C₆₇×#2355	C₂	1	1	yes	maximal, characteristic, complemented
6	C₆₇×#2354	C₂	1	2	yes	maximal, complemented
7	C₆₇×#2354	C₂	1	1	yes	maximal, characteristic, complemented
8	C₆₇×#2352	C₂	1	2	yes	maximal, complemented
9	Q₈×C₄⋊C₄×C₆₇	C₂	1	1	yes	maximal, characteristic, complemented
10	D₄×C₄⋊C₄×C₆₇	C₂	1	1	yes	maximal, characteristic, complemented
11	C₆₇×#2334	C₂	1	4	yes	maximal, complemented
12	C₆₇×#2324	C₂	1	2	yes	maximal
13	C₆₇×#2323	C₂	1	2	yes	maximal, complemented
14	C₆₇×#2317	C₂	1	2	yes	maximal, complemented
15	C₆₇×#2314	C₂	1	2	yes	maximal, complemented
16	C₆₇×#2285	C₂	1	1	yes	maximal, characteristic, complemented
17	C₆₇×#2285	C₂	1	1	yes	maximal, characteristic, complemented
18	#529×C₆₇	C₄	1	16	yes	complemented
19	C₂×#126×C₆₇	C₂²	1	2	yes	—
20	C₂×#125×C₆₇	C₂²	1	1	yes	characteristic
21	C₂×#124×C₆₇	C₂²	1	1	yes	characteristic
22	C₂×#123×C₆₇	C₂²	1	2	yes	—
23	C₂×C₂²⋊Q₈×C₆₇	C₂²	1	1	yes	characteristic
24	C₂×#121×C₆₇	C₂²	1	1	yes	characteristic
25	C₂×C₄×Q₈×C₆₇	C₂²	1	2	yes	—
26	C₂×C₄×Q₈×C₆₇	C₂²	1	1	yes	characteristic
27	C₂×C₄×D₄×C₆₇	C₂²	1	2	yes	—
28	C₂×C₄×D₄×C₆₇	C₂²	1	1	yes	characteristic
29	C₂×C₄×D₄×C₆₇	C₂²	1	2	yes	—
30	C₂×#117×C₆₇	C₂²	1	2	yes	—
31	C₂×#117×C₆₇	C₂²	1	2	yes	—
32	C₂×#117×C₆₇	C₂²	1	2	yes	—
33	C₂×#117×C₆₇	C₂²	1	2	yes	—
34	C₂²×C₄⋊C₄×C₆₇	C₂²	1	2	yes	—
35	C₂²×C₄²×C₆₇	C₂²	1	1	yes	characteristic
36	#391×C₆₇	C₂²	1	2	yes	—
37	#390×C₆₇	C₂²	1	2	yes	—
38	#390×C₆₇	C₂²	1	4	yes	—
39	#389×C₆₇	C₂²	1	4	yes	—
40	#389×C₆₇	C₂²	1	4	yes	—
41	#389×C₆₇	C₂²	1	2	yes	—
42	#389×C₆₇	C₂²	1	2	yes	—
43	#389×C₆₇	C₂²	1	4	yes	—
44	#388×C₆₇	C₂²	1	4	yes	—
45	#388×C₆₇	C₂²	1	4	yes	—
46	#388×C₆₇	C₂²	1	4	yes	—
47	#388×C₆₇	C₂²	1	4	yes	—
48	#388×C₆₇	C₂²	1	4	yes	—
49	#388×C₆₇	C₂²	1	4	yes	—
50	#387×C₆₇	C₂²	1	4	yes	—
51	#387×C₆₇	C₂²	1	4	yes	—
52	#387×C₆₇	C₂²	1	4	yes	—
53	#387×C₆₇	C₂²	1	4	yes	—
54	#386×C₆₇	C₂²	1	4	yes	—
55	#385×C₆₇	C₂²	1	4	yes	—
56	#385×C₆₇	C₂²	1	4	yes	—
57	#385×C₆₇	C₂²	1	4	yes	—
58	#384×C₆₇	C₂²	1	1	yes	characteristic
59	#384×C₆₇	C₂²	1	1	yes	characteristic
60	#384×C₆₇	C₂²	1	1	yes	characteristic
61	#384×C₆₇	C₂²	1	1	yes	characteristic
62	#384×C₆₇	C₂²	1	2	yes	—
63	#382×C₆₇	C₂²	1	2	yes	—
64	#382×C₆₇	C₂²	1	2	yes	—
65	#382×C₆₇	C₂²	1	2	yes	—
66	#380×C₆₇	C₂²	1	2	yes	—
67	#380×C₆₇	C₂²	1	1	yes	characteristic
68	#380×C₆₇	C₂²	1	1	yes	characteristic
69	#380×C₆₇	C₂²	1	2	yes	—
70	#380×C₆₇	C₂²	1	2	yes	—
71	#380×C₆₇	C₂²	1	2	yes	—
72	#380×C₆₇	C₂²	1	2	yes	—
73	C₄×C₄⋊C₄×C₆₇	C₂²	1	2	yes	—
74	C₄×C₄⋊C₄×C₆₇	C₂²	1	1	yes	characteristic
75	C₄×C₄⋊C₄×C₆₇	C₂²	1	1	yes	characteristic
76	C₄×C₄⋊C₄×C₆₇	C₂²	1	2	yes	—
77	C₄×C₄⋊C₄×C₆₇	C₂²	1	2	yes	—
78	C₄×C₄⋊C₄×C₆₇	C₂²	1	2	yes	—
79	C₄×C₄⋊C₄×C₆₇	C₂²	1	2	yes	—
80	C₄×C₄⋊C₄×C₆₇	C₂²	1	2	yes	—
81	C₄×C₄⋊C₄×C₆₇	C₂²	1	2	yes	—
82	C₄×C₄⋊C₄×C₆₇	C₂²	1	4	yes	—
83	C₄×C₂²⋊C₄×C₆₇	C₂²	1	4	yes	—
84	C₂²×Q₈×C₆₇	C₂³	1	1	yes	characteristic
85	C₂²×D₄×C₆₇	C₂³	1	1	yes	characteristic
86	C₂²×D₄×C₆₇	C₂³	1	2	yes	—
87	C₂³×C₄×C₆₇	C₂³	1	1	yes	characteristic
88	C₂³×C₄×C₆₇	C₂³	1	1	yes	characteristic
89	C₂³×C₄×C₆₇	C₂³	1	2	yes	—
90	C₂³×C₄×C₆₇	C₂³	1	4	yes	—
91	C₂³×C₄×C₆₇	C₂³	1	1	yes	characteristic
92	#126×C₆₇	C₂×C₄	1	16	yes	complemented
93	#125×C₆₇	C₂×C₄	1	8	yes	complemented
94	#124×C₆₇	C₂×C₄	1	8	yes	—
95	#123×C₆₇	C₂×C₄	1	16	yes	complemented
96	C₂²⋊Q₈×C₆₇	C₂×C₄	1	8	yes	complemented
97	#121×C₆₇	C₂×C₄	1	8	yes	—
98	C₄×Q₈×C₆₇	—	4	4	no	—
99	C₄×Q₈×C₆₇	C₂×C₄	1	8	yes	complemented
100	C₄×D₄×C₆₇	—	4	4	no	—
101	C₄×D₄×C₆₇	C₂×C₄	1	8	yes	—
102	C₄×D₄×C₆₇	C₂×C₄	1	16	yes	complemented
103	#117×C₆₇	—	4	4	no	—
104	#117×C₆₇	—	4	4	no	—
105	#117×C₆₇	—	4	4	no	—
106	#117×C₆₇	C₂×C₄	1	16	yes	complemented
107	C₂×C₄⋊C₄×C₆₇	—	4	4	no	—
108	C₂×C₄⋊C₄×C₆₇	C₂³	1	1	yes	characteristic
109	C₂×C₄⋊C₄×C₆₇	C₂³	1	2	yes	—
110	C₂×C₄⋊C₄×C₆₇	C₂³	1	2	yes	—
111	C₂×C₄⋊C₄×C₆₇	C₂³	1	1	yes	characteristic
112	C₂×C₄⋊C₄×C₆₇	C₂³	1	2	yes	—
113	C₂×C₄⋊C₄×C₆₇	C₂³	1	1	yes	characteristic
114	C₂×C₄⋊C₄×C₆₇	C₂³	1	1	yes	characteristic
115	C₂×C₄⋊C₄×C₆₇	C₂³	1	2	yes	—
116	C₂×C₄⋊C₄×C₆₇	C₂³	1	4	yes	—
117	C₂×C₄⋊C₄×C₆₇	C₂³	1	2	yes	—
118	C₂×C₄⋊C₄×C₆₇	C₂³	1	2	yes	—
119	C₂×C₄⋊C₄×C₆₇	C₂³	1	4	yes	—
120	C₂×C₄⋊C₄×C₆₇	C₂³	1	2	yes	—
121	C₂×C₄⋊C₄×C₆₇	C₂³	1	2	yes	—
122	C₂×C₄⋊C₄×C₆₇	C₂³	1	2	yes	—
123	C₂×C₄⋊C₄×C₆₇	C₂³	1	2	yes	—
124	C₂×C₄⋊C₄×C₆₇	C₂³	1	4	yes	—
125	C₂×C₄⋊C₄×C₆₇	C₂³	1	2	yes	—
126	C₂×C₄⋊C₄×C₆₇	C₂³	1	2	yes	—
127	C₂×C₄⋊C₄×C₆₇	C₂³	1	1	yes	characteristic
128	C₂×C₄⋊C₄×C₆₇	C₂³	1	2	yes	—
129	C₂×C₄⋊C₄×C₆₇	C₂³	1	1	yes	characteristic
130	C₂×C₄⋊C₄×C₆₇	C₂³	1	2	yes	—
131	C₂×C₂²⋊C₄×C₆₇	C₂³	1	2	yes	—
132	C₂×C₂²⋊C₄×C₆₇	C₂³	1	4	yes	—
133	C₂×C₂²⋊C₄×C₆₇	C₂³	1	2	yes	—
134	C₂×C₂²⋊C₄×C₆₇	C₂³	1	4	yes	—
135	C₂×C₂²⋊C₄×C₆₇	C₂³	1	4	yes	—
136	C₂×C₂²⋊C₄×C₆₇	C₂³	1	4	yes	—
137	C₂×C₂²⋊C₄×C₆₇	C₂³	1	2	yes	—
138	C₂×C₂²⋊C₄×C₆₇	C₂³	1	2	yes	—
139	C₂×C₂²⋊C₄×C₆₇	C₂³	1	2	yes	—
140	C₂×C₂²⋊C₄×C₆₇	C₂³	1	2	yes	—
141	C₂×C₂²⋊C₄×C₆₇	C₂³	1	2	yes	—
142	C₂×C₄²×C₆₇	C₂×C₄	1	8	yes	complemented
143	C₂×C₄²×C₆₇	C₂³	1	2	yes	—
144	C₂×C₄²×C₆₇	C₂³	1	2	yes	—
145	C₂×C₄²×C₆₇	C₂³	1	4	yes	—
146	C₂×C₄²×C₆₇	C₂³	1	4	yes	—
147	C₂×C₄²×C₆₇	C₂³	1	4	yes	—
148	C₂×C₄²×C₆₇	C₂³	1	4	yes	—
149	C₂×C₄²×C₆₇	C₂³	1	1	yes	characteristic
150	C₂×C₄²×C₆₇	C₂³	1	1	yes	characteristic
151	C₂×C₄²×C₆₇	C₂³	1	1	yes	characteristic
152	C₂×C₄²×C₆₇	C₂³	1	1	yes	characteristic
153	C₂×C₄²×C₆₇	C₂³	1	2	yes	—
154	#95×C₆₇	C₂³	1	2	yes	—
155	#95×C₆₇	C₂³	1	4	yes	—
156	#95×C₆₇	C₂³	1	2	yes	—
157	#95×C₆₇	C₂³	1	2	yes	—
158	#95×C₆₇	C₂³	1	4	yes	—
159	#95×C₆₇	C₂³	1	4	yes	—
160	#95×C₆₇	C₂³	1	2	yes	—
161	#95×C₆₇	C₂³	1	4	yes	—
162	#95×C₆₇	C₂³	1	4	yes	—
163	#95×C₆₇	C₂³	1	4	yes	—
164	#95×C₆₇	C₂³	1	4	yes	—
165	#95×C₆₇	C₂³	1	4	yes	—
166	C₂⁴×C₆₇	C₂⁴	1	1	yes	characteristic
167	C₂⁴×C₆₇	C₂⁴	1	2	yes	—
168	C₂×Q₈×C₆₇	—	2	2	no	—
169	C₂×Q₈×C₆₇	—	2	2	no	—
170	C₂×Q₈×C₆₇	C₂²×C₄	1	4	yes	—
171	C₂×D₄×C₆₇	—	2	2	no	—
172	C₂×D₄×C₆₇	—	2	2	no	—
173	C₂×D₄×C₆₇	—	4	2	no	—
174	C₂×D₄×C₆₇	—	4	2	no	—
175	C₂×D₄×C₆₇	C₂²×C₄	1	4	yes	—
176	C₂×D₄×C₆₇	C₂²×C₄	1	8	yes	—
177	C₂²×C₄×C₆₇	—	2	4	no	—
178	C₂²×C₄×C₆₇	—	2	2	no	—
179	C₂²×C₄×C₆₇	—	2	2	no	—
180	C₂²×C₄×C₆₇	—	2	2	no	—
181	C₂²×C₄×C₆₇	—	2	4	no	—
182	C₂²×C₄×C₆₇	—	2	2	no	—
183	C₂²×C₄×C₆₇	—	4	1	no	—
184	C₂²×C₄×C₆₇	—	4	1	no	—
185	C₂²×C₄×C₆₇	—	4	4	no	—
186	C₂²×C₄×C₆₇	—	4	4	no	—
187	C₂²×C₄×C₆₇	—	4	4	no	—
188	C₂²×C₄×C₆₇	C₂²×C₄	1	4	yes	—
189	C₂²×C₄×C₆₇	C₂²×C₄	1	4	yes	—
190	C₂²×C₄×C₆₇	C₂²×C₄	1	8	yes	—
191	C₂²×C₄×C₆₇	C₂²×C₄	1	4	yes	—
192	C₂²×C₄×C₆₇	C₂⁴	1	2	yes	—
193	C₂²×C₄×C₆₇	C₂⁴	1	2	yes	—
194	C₂²×C₄×C₆₇	C₂⁴	1	2	yes	—
195	C₂²×C₄×C₆₇	C₂⁴	1	4	yes	—
196	C₂²×C₄×C₆₇	C₂⁴	1	4	yes	—
197	C₂²×C₄×C₆₇	C₂⁴	1	2	yes	—
198	C₂²×C₄×C₆₇	C₂⁴	1	1	yes	characteristic
199	C₂²×C₄×C₆₇	C₂⁴	1	2	yes	—
200	C₂²×C₄×C₆₇	C₂⁴	1	2	yes	—
201	C₂²×C₄×C₆₇	C₂⁴	1	2	yes	—
202	C₂²×C₄×C₆₇	C₂⁴	1	1	yes	characteristic
203	C₂²×C₄×C₆₇	C₂⁴	1	1	yes	characteristic
204	C₂²×C₄×C₆₇	C₂⁴	1	1	yes	characteristic
205	C₂²×C₄×C₆₇	C₂⁴	1	1	yes	characteristic
206	C₂²×C₄×C₆₇	C₂⁴	1	1	yes	characteristic
207	C₄⋊C₄×C₆₇	—	2	2	no	—
208	C₄⋊C₄×C₆₇	—	2	4	no	—
209	C₄⋊C₄×C₆₇	—	2	4	no	—
210	C₄⋊C₄×C₆₇	—	2	2	no	—
211	C₄⋊C₄×C₆₇	—	2	4	no	—
212	C₄⋊C₄×C₆₇	—	2	2	no	—
213	C₄⋊C₄×C₆₇	—	2	2	no	—
214	C₄⋊C₄×C₆₇	—	2	4	no	—
215	C₄⋊C₄×C₆₇	—	4	4	no	—
216	C₄⋊C₄×C₆₇	—	4	2	no	—
217	C₄⋊C₄×C₆₇	—	4	2	no	—
218	C₄⋊C₄×C₆₇	—	4	2	no	—
219	C₄⋊C₄×C₆₇	—	4	2	no	—
220	C₄⋊C₄×C₆₇	—	4	4	no	—
221	C₄⋊C₄×C₆₇	—	4	2	no	—
222	C₄⋊C₄×C₆₇	—	4	2	no	—
223	C₄⋊C₄×C₆₇	—	4	4	no	—
224	C₄⋊C₄×C₆₇	C₂²×C₄	1	8	yes	—
225	C₄⋊C₄×C₆₇	C₂²×C₄	1	8	yes	—
226	C₄⋊C₄×C₆₇	C₂²×C₄	1	4	yes	—
227	C₄⋊C₄×C₆₇	C₂²×C₄	1	8	yes	—
228	C₄⋊C₄×C₆₇	C₂²×C₄	1	4	yes	—
229	C₄⋊C₄×C₆₇	C₂²×C₄	1	8	yes	—
230	C₂²⋊C₄×C₆₇	—	4	4	no	—
231	C₂²⋊C₄×C₆₇	—	4	2	no	—
232	C₂²⋊C₄×C₆₇	—	4	2	no	—
233	C₂²⋊C₄×C₆₇	—	4	4	no	—
234	C₂²⋊C₄×C₆₇	—	4	4	no	—
235	C₂²⋊C₄×C₆₇	—	4	4	no	—
236	C₂²⋊C₄×C₆₇	C₂²×C₄	1	8	yes	—
237	C₂²⋊C₄×C₆₇	C₂²×C₄	1	8	yes	—
238	C₂²⋊C₄×C₆₇	C₂²×C₄	1	8	yes	—
239	C₂²⋊C₄×C₆₇	C₂²×C₄	1	8	yes	—
240	C₂²⋊C₄×C₆₇	C₂²×C₄	1	8	yes	—
241	C₄²×C₆₇	—	2	4	no	—
242	C₄²×C₆₇	—	2	4	no	—
243	C₄²×C₆₇	—	4	4	no	—
244	C₄²×C₆₇	—	4	4	no	—
245	C₄²×C₆₇	—	4	4	no	—
246	C₄²×C₆₇	—	4	4	no	—
247	C₄²×C₆₇	C₂²×C₄	1	4	yes	—
248	C₄²×C₆₇	C₂²×C₄	1	4	yes	—
249	C₄²×C₆₇	C₂²×C₄	1	4	yes	—
250	C₄²×C₆₇	C₂²×C₄	1	4	yes	—
251	C₄²×C₆₇	C₂²×C₄	1	8	yes	—
252	C₂³×C₆₇	—	2	1	no	—
253	C₂³×C₆₇	—	2	1	no	—
254	C₂³×C₆₇	—	2	2	no	—
255	C₂³×C₆₇	—	2	2	no	—
256	C₂³×C₆₇	—	2	1	no	—
257	C₂³×C₆₇	—	2	2	no	—
258	C₂³×C₆₇	—	2	1	no	—
259	C₂³×C₆₇	—	2	2	no	—
260	C₂³×C₆₇	—	2	2	no	—
261	C₂³×C₆₇	—	2	2	no	—
262	C₂³×C₆₇	—	2	1	no	—
263	C₂³×C₆₇	—	2	1	no	—
264	C₂³×C₆₇	C₂³×C₄	1	4	yes	—
265	C₂³×C₆₇	C₂³×C₄	1	2	yes	—
266	C₂³×C₆₇	C₂⁵	1	1	yes	characteristic
267	Q₈×C₆₇	—	2	8	no	—
268	D₄×C₆₇	—	2	8	no	—
269	D₄×C₆₇	—	4	8	no	—
270	C₂×C₄×C₆₇	—	2	1	no	—
271	C₂×C₄×C₆₇	—	2	2	no	—
272	C₂×C₄×C₆₇	—	2	2	no	—
273	C₂×C₄×C₆₇	—	2	1	no	—
274	C₂×C₄×C₆₇	—	2	1	no	—
275	C₂×C₄×C₆₇	—	2	2	no	—
276	C₂×C₄×C₆₇	—	2	1	no	—
277	C₂×C₄×C₆₇	—	2	2	no	—
278	C₂×C₄×C₆₇	—	2	2	no	—
279	C₂×C₄×C₆₇	—	2	1	no	—
280	C₂×C₄×C₆₇	—	2	1	no	—
281	C₂×C₄×C₆₇	—	2	1	no	—
282	C₂×C₄×C₆₇	—	2	1	no	—
283	C₂×C₄×C₆₇	—	2	2	no	—
284	C₂×C₄×C₆₇	—	2	16	no	—
285	C₂×C₄×C₆₇	—	2	8	no	—
286	C₂×C₄×C₆₇	—	2	8	no	—
287	C₂×C₄×C₆₇	—	2	2	no	—
288	C₂×C₄×C₆₇	—	2	2	no	—
289	C₂×C₄×C₆₇	—	2	2	no	—
290	C₂×C₄×C₆₇	—	2	2	no	—
291	C₂×C₄×C₆₇	—	2	2	no	—
292	C₂×C₄×C₆₇	—	2	4	no	—
293	C₂×C₄×C₆₇	—	2	4	no	—
294	C₂×C₄×C₆₇	—	2	4	no	—
295	C₂×C₄×C₆₇	—	2	2	no	—
296	C₂×C₄×C₆₇	—	2	2	no	—
297	C₂×C₄×C₆₇	—	2	2	no	—
298	C₂×C₄×C₆₇	—	2	4	no	—
299	C₂×C₄×C₆₇	—	2	2	no	—
300	C₂×C₄×C₆₇	—	2	4	no	—
301	C₂×C₄×C₆₇	—	2	2	no	—
302	C₂×C₄×C₆₇	—	2	2	no	—
303	C₂×C₄×C₆₇	—	2	4	no	—
304	C₂×C₄×C₆₇	—	2	2	no	—
305	C₂×C₄×C₆₇	—	4	4	no	—
306	C₂×C₄×C₆₇	—	8	8	no	—
307	C₂×C₄×C₆₇	C₂³×C₄	1	2	yes	—
308	C₂×C₄×C₆₇	C₂³×C₄	1	2	yes	—
309	C₂×C₄×C₆₇	C₂³×C₄	1	2	yes	—
310	C₂×C₄×C₆₇	C₂³×C₄	1	2	yes	—
311	C₂×C₄×C₆₇	C₂³×C₄	1	4	yes	—
312	C₂×C₄×C₆₇	C₂³×C₄	1	4	yes	—
313	C₂×C₄×C₆₇	C₂³×C₄	1	4	yes	—
314	C₂×C₄×C₆₇	C₂³×C₄	1	2	yes	—
315	C₂×C₄×C₆₇	C₂³×C₄	1	2	yes	—
316	C₂×C₄×C₆₇	H(5,𝔽₂)	1	2	yes	—
317	C₂×C₄×C₆₇	H(5,𝔽₂)	1	2	yes	—
318	C₂×C₄×C₆₇	2₋¹⁺⁴	1	2	yes	—
319	C₂×C₄×C₆₇	2₋¹⁺⁴	1	2	yes	—
320	C₂²×C₆₇	—	2	2	no	—
321	C₂²×C₆₇	—	2	2	no	—
322	C₂²×C₆₇	—	2	4	no	—
323	C₂²×C₆₇	—	2	4	no	—
324	C₂²×C₆₇	—	2	2	no	—
325	C₂²×C₆₇	—	2	4	no	—
326	C₂²×C₆₇	—	4	2	no	—
327	C₂²×C₆₇	—	4	1	no	—
328	C₂²×C₆₇	—	4	1	no	—
329	C₂²×C₆₇	—	4	2	no	—
330	C₂²×C₆₇	—	4	1	no	—
331	C₂²×C₆₇	—	4	2	no	—
332	C₂²×C₆₇	—	4	1	no	—
333	C₂²×C₆₇	—	4	2	no	—
334	C₂²×C₆₇	C₂⁴×C₄	1	1	yes	characteristic
335	C₂²×C₆₇	C₂×H(5,𝔽₂)	1	1	yes	characteristic
336	C₂²×C₆₇	C₂×H(5,𝔽₂)	1	1	yes	characteristic
337	C₂²×C₆₇	C₂×2₋¹⁺⁴	1	1	yes	characteristic
338	C₂²×C₆₇	C₂×2₋¹⁺⁴	1	1	yes	characteristic
339	C₂²×C₆₇	#585	1	1	yes	characteristic
340	C₂²×C₆₇	#585	1	1	yes	characteristic
341	C₄×C₆₇	—	2	2	no	—
342	C₄×C₆₇	—	2	4	no	—
343	C₄×C₆₇	—	2	2	no	—
344	C₄×C₆₇	—	2	4	no	—
345	C₄×C₆₇	—	2	2	no	—
346	C₄×C₆₇	—	2	2	no	—
347	C₄×C₆₇	—	2	4	no	—
348	C₄×C₆₇	—	4	2	no	—
349	C₄×C₆₇	—	4	4	no	—
350	C₄×C₆₇	—	4	2	no	—
351	C₄×C₆₇	—	4	2	no	—
352	C₄×C₆₇	—	4	2	no	—
353	C₄×C₆₇	—	4	4	no	—
354	C₄×C₆₇	#520	1	4	yes	—
355	C₄×C₆₇	#520	1	4	yes	—
356	#34383	C₆₇	1	1	yes	maximal, characteristic, direct product factor, Sylow
357	C₂×C₆₇	—	4	2	no	—
358	C₂×C₆₇	—	4	4	no	—
359	C₂×C₆₇	C₂×#520	1	1	yes	characteristic
360	C₂×C₆₇	C₂×#520	1	1	yes	characteristic
361	C₂×C₆₇	#3428	1	1	yes	characteristic
362	C₂×C₆₇	#3546	1	1	yes	characteristic
363	C₂×C₆₇	#3547	1	1	yes	characteristic
364	C₂×C₆₇	#3554	1	1	yes	characteristic
365	C₂×C₆₇	#3555	1	1	yes	characteristic
366	C₂×#529	C₂×C₆₇	1	1	yes	characteristic
367	#2376	C₂×C₆₇	1	4	yes	complemented
368	#2370	C₂×C₆₇	1	2	yes	complemented
369	#2367	C₂×C₆₇	1	2	yes	complemented
370	#2355	C₂×C₆₇	1	1	yes	characteristic, complemented
371	#2354	C₂×C₆₇	1	1	yes	characteristic, complemented
372	#2354	C₂×C₆₇	1	2	yes	complemented
373	#2352	C₂×C₆₇	1	2	yes	complemented
374	Q₈×C₄⋊C₄	C₂×C₆₇	1	1	yes	characteristic, complemented
375	D₄×C₄⋊C₄	C₂×C₆₇	1	1	yes	characteristic, complemented
376	#2334	C₂×C₆₇	1	4	yes	complemented
377	#2324	C₂×C₆₇	1	2	yes	—
378	#2323	C₂×C₆₇	1	2	yes	complemented
379	#2317	C₂×C₆₇	1	2	yes	complemented
380	#2314	C₂×C₆₇	1	2	yes	complemented
381	#2285	C₂×C₆₇	1	1	yes	characteristic, complemented
382	#2285	C₂×C₆₇	1	1	yes	characteristic, complemented
383	C₆₇	#34383	1	1	yes	characteristic, direct product factor, Sylow
384	#529	C₄×C₆₇	1	16	yes	complemented
385	C₂×#126	C₂²×C₆₇	1	2	yes	—
386	C₂×#125	C₂²×C₆₇	1	1	yes	characteristic
387	C₂×#124	C₂²×C₆₇	1	1	yes	characteristic
388	C₂×#123	C₂²×C₆₇	1	2	yes	—
389	C₂×C₂²⋊Q₈	C₂²×C₆₇	1	1	yes	characteristic
390	C₂×#121	C₂²×C₆₇	1	1	yes	characteristic
391	C₂×C₄×Q₈	C₂²×C₆₇	1	2	yes	—
392	C₂×C₄×Q₈	C₂²×C₆₇	1	1	yes	characteristic
393	C₂×C₄×D₄	C₂²×C₆₇	1	2	yes	—
394	C₂×C₄×D₄	C₂²×C₆₇	1	1	yes	characteristic
395	C₂×C₄×D₄	C₂²×C₆₇	1	2	yes	—
396	C₂×#117	C₂²×C₆₇	1	2	yes	—
397	C₂×#117	C₂²×C₆₇	1	2	yes	—
398	C₂×#117	C₂²×C₆₇	1	2	yes	—
399	C₂×#117	C₂²×C₆₇	1	2	yes	—
400	C₂²×C₄⋊C₄	C₂²×C₆₇	1	2	yes	—
401	C₂²×C₄²	C₂²×C₆₇	1	1	yes	characteristic
402	#391	C₂²×C₆₇	1	2	yes	—
403	#390	C₂²×C₆₇	1	2	yes	—
404	#390	C₂²×C₆₇	1	4	yes	—
405	#389	C₂²×C₆₇	1	4	yes	—
406	#389	C₂²×C₆₇	1	4	yes	—
407	#389	C₂²×C₆₇	1	2	yes	—
408	#389	C₂²×C₆₇	1	2	yes	—
409	#389	C₂²×C₆₇	1	4	yes	—
410	#388	C₂²×C₆₇	1	4	yes	—
411	#388	C₂²×C₆₇	1	4	yes	—
412	#388	C₂²×C₆₇	1	4	yes	—
413	#388	C₂²×C₆₇	1	4	yes	—
414	#388	C₂²×C₆₇	1	4	yes	—
415	#388	C₂²×C₆₇	1	4	yes	—
416	#387	C₂²×C₆₇	1	4	yes	—
417	#387	C₂²×C₆₇	1	4	yes	—
418	#387	C₂²×C₆₇	1	4	yes	—
419	#387	C₂²×C₆₇	1	4	yes	—
420	#386	C₂²×C₆₇	1	4	yes	—
421	#385	C₂²×C₆₇	1	4	yes	—
422	#385	C₂²×C₆₇	1	4	yes	—
423	#385	C₂²×C₆₇	1	4	yes	—
424	#384	C₂²×C₆₇	1	1	yes	characteristic
425	#384	C₂²×C₆₇	1	1	yes	characteristic
426	#384	C₂²×C₆₇	1	2	yes	—
427	#384	C₂²×C₆₇	1	1	yes	characteristic
428	#384	C₂²×C₆₇	1	1	yes	characteristic
429	#382	C₂²×C₆₇	1	2	yes	—
430	#382	C₂²×C₆₇	1	2	yes	—
431	#382	C₂²×C₆₇	1	2	yes	—
432	#380	C₂²×C₆₇	1	2	yes	—
433	#380	C₂²×C₆₇	1	2	yes	—
434	#380	C₂²×C₆₇	1	2	yes	—
435	#380	C₂²×C₆₇	1	1	yes	characteristic
436	#380	C₂²×C₆₇	1	1	yes	characteristic
437	#380	C₂²×C₆₇	1	2	yes	—
438	#380	C₂²×C₆₇	1	2	yes	—
439	C₄×C₄⋊C₄	C₂²×C₆₇	1	2	yes	—
440	C₄×C₄⋊C₄	C₂²×C₆₇	1	2	yes	—
441	C₄×C₄⋊C₄	C₂²×C₆₇	1	2	yes	—
442	C₄×C₄⋊C₄	C₂²×C₆₇	1	2	yes	—
443	C₄×C₄⋊C₄	C₂²×C₆₇	1	2	yes	—
444	C₄×C₄⋊C₄	C₂²×C₆₇	1	4	yes	—
445	C₄×C₄⋊C₄	C₂²×C₆₇	1	1	yes	characteristic
446	C₄×C₄⋊C₄	C₂²×C₆₇	1	1	yes	characteristic
447	C₄×C₄⋊C₄	C₂²×C₆₇	1	2	yes	—
448	C₄×C₄⋊C₄	C₂²×C₆₇	1	2	yes	—
449	C₄×C₂²⋊C₄	C₂²×C₆₇	1	4	yes	—
450	C₂²×Q₈	C₂³×C₆₇	1	1	yes	characteristic
451	C₂²×D₄	C₂³×C₆₇	1	1	yes	characteristic
452	C₂²×D₄	C₂³×C₆₇	1	2	yes	—
453	C₂³×C₄	C₂³×C₆₇	1	2	yes	—
454	C₂³×C₄	C₂³×C₆₇	1	1	yes	characteristic
455	C₂³×C₄	C₂³×C₆₇	1	1	yes	characteristic
456	C₂³×C₄	C₂³×C₆₇	1	4	yes	—
457	C₂³×C₄	C₂³×C₆₇	1	1	yes	characteristic
458	#126	C₂×C₄×C₆₇	1	16	yes	complemented
459	#125	C₂×C₄×C₆₇	1	8	yes	complemented
460	#124	C₂×C₄×C₆₇	1	8	yes	—
461	#123	C₂×C₄×C₆₇	1	16	yes	complemented
462	C₂²⋊Q₈	C₂×C₄×C₆₇	1	8	yes	complemented
463	#121	C₂×C₄×C₆₇	1	8	yes	—
464	C₄×Q₈	—	4	4	no	—
465	C₄×Q₈	C₂×C₄×C₆₇	1	8	yes	complemented
466	C₄×D₄	—	4	4	no	—
467	C₄×D₄	C₂×C₄×C₆₇	1	8	yes	—
468	C₄×D₄	C₂×C₄×C₆₇	1	16	yes	complemented
469	#117	—	4	4	no	—
470	#117	—	4	4	no	—
471	#117	—	4	4	no	—
472	#117	C₂×C₄×C₆₇	1	16	yes	complemented
473	C₂×C₄⋊C₄	—	4	4	no	—
474	C₂×C₄⋊C₄	C₂³×C₆₇	1	1	yes	characteristic
475	C₂×C₄⋊C₄	C₂³×C₆₇	1	2	yes	—
476	C₂×C₄⋊C₄	C₂³×C₆₇	1	2	yes	—
477	C₂×C₄⋊C₄	C₂³×C₆₇	1	1	yes	characteristic
478	C₂×C₄⋊C₄	C₂³×C₆₇	1	2	yes	—
479	C₂×C₄⋊C₄	C₂³×C₆₇	1	1	yes	characteristic
480	C₂×C₄⋊C₄	C₂³×C₆₇	1	1	yes	characteristic
481	C₂×C₄⋊C₄	C₂³×C₆₇	1	2	yes	—
482	C₂×C₄⋊C₄	C₂³×C₆₇	1	4	yes	—
483	C₂×C₄⋊C₄	C₂³×C₆₇	1	2	yes	—
484	C₂×C₄⋊C₄	C₂³×C₆₇	1	2	yes	—
485	C₂×C₄⋊C₄	C₂³×C₆₇	1	4	yes	—
486	C₂×C₄⋊C₄	C₂³×C₆₇	1	2	yes	—
487	C₂×C₄⋊C₄	C₂³×C₆₇	1	2	yes	—
488	C₂×C₄⋊C₄	C₂³×C₆₇	1	2	yes	—
489	C₂×C₄⋊C₄	C₂³×C₆₇	1	2	yes	—
490	C₂×C₄⋊C₄	C₂³×C₆₇	1	4	yes	—
491	C₂×C₄⋊C₄	C₂³×C₆₇	1	2	yes	—
492	C₂×C₄⋊C₄	C₂³×C₆₇	1	2	yes	—
493	C₂×C₄⋊C₄	C₂³×C₆₇	1	1	yes	characteristic
494	C₂×C₄⋊C₄	C₂³×C₆₇	1	2	yes	—
495	C₂×C₄⋊C₄	C₂³×C₆₇	1	1	yes	characteristic
496	C₂×C₄⋊C₄	C₂³×C₆₇	1	2	yes	—
497	C₂×C₂²⋊C₄	C₂³×C₆₇	1	2	yes	—
498	C₂×C₂²⋊C₄	C₂³×C₆₇	1	4	yes	—
499	C₂×C₂²⋊C₄	C₂³×C₆₇	1	2	yes	—
500	C₂×C₂²⋊C₄	C₂³×C₆₇	1	4	yes	—
501	C₂×C₂²⋊C₄	C₂³×C₆₇	1	4	yes	—
502	C₂×C₂²⋊C₄	C₂³×C₆₇	1	4	yes	—
503	C₂×C₂²⋊C₄	C₂³×C₆₇	1	2	yes	—
504	C₂×C₂²⋊C₄	C₂³×C₆₇	1	2	yes	—
505	C₂×C₂²⋊C₄	C₂³×C₆₇	1	2	yes	—
506	C₂×C₂²⋊C₄	C₂³×C₆₇	1	2	yes	—
507	C₂×C₂²⋊C₄	C₂³×C₆₇	1	2	yes	—
508	C₂×C₄²	C₂×C₄×C₆₇	1	8	yes	complemented
509	C₂×C₄²	C₂³×C₆₇	1	2	yes	—
510	C₂×C₄²	C₂³×C₆₇	1	2	yes	—
511	C₂×C₄²	C₂³×C₆₇	1	4	yes	—
512	C₂×C₄²	C₂³×C₆₇	1	4	yes	—
513	C₂×C₄²	C₂³×C₆₇	1	4	yes	—
514	C₂×C₄²	C₂³×C₆₇	1	4	yes	—
515	C₂×C₄²	C₂³×C₆₇	1	1	yes	characteristic
516	C₂×C₄²	C₂³×C₆₇	1	2	yes	—
517	C₂×C₄²	C₂³×C₆₇	1	1	yes	characteristic
518	C₂×C₄²	C₂³×C₆₇	1	1	yes	characteristic
519	C₂×C₄²	C₂³×C₆₇	1	1	yes	characteristic
520	#95	C₂³×C₆₇	1	2	yes	—
521	#95	C₂³×C₆₇	1	4	yes	—
522	#95	C₂³×C₆₇	1	2	yes	—
523	#95	C₂³×C₆₇	1	2	yes	—
524	#95	C₂³×C₆₇	1	4	yes	—
525	#95	C₂³×C₆₇	1	4	yes	—
526	#95	C₂³×C₆₇	1	2	yes	—
527	#95	C₂³×C₆₇	1	4	yes	—
528	#95	C₂³×C₆₇	1	4	yes	—
529	#95	C₂³×C₆₇	1	4	yes	—
530	#95	C₂³×C₆₇	1	4	yes	—
531	#95	C₂³×C₆₇	1	4	yes	—
532	C₂⁴	C₂⁴×C₆₇	1	1	yes	characteristic
533	C₂⁴	C₂⁴×C₆₇	1	2	yes	—
534	C₂×Q₈	—	2	2	no	—
535	C₂×Q₈	—	2	2	no	—
536	C₂×Q₈	C₂²×C₄×C₆₇	1	4	yes	—
537	C₂×D₄	—	2	2	no	—
538	C₂×D₄	—	2	2	no	—
539	C₂×D₄	—	4	2	no	—
540	C₂×D₄	—	4	2	no	—
541	C₂×D₄	C₂²×C₄×C₆₇	1	4	yes	—
542	C₂×D₄	C₂²×C₄×C₆₇	1	8	yes	—
543	C₂²×C₄	—	2	4	no	—
544	C₂²×C₄	—	2	2	no	—
545	C₂²×C₄	—	2	2	no	—
546	C₂²×C₄	—	2	4	no	—
547	C₂²×C₄	—	2	2	no	—
548	C₂²×C₄	—	2	2	no	—
549	C₂²×C₄	—	4	1	no	—
550	C₂²×C₄	—	4	1	no	—
551	C₂²×C₄	—	4	4	no	—
552	C₂²×C₄	—	4	4	no	—
553	C₂²×C₄	—	4	4	no	—
554	C₂²×C₄	C₂²×C₄×C₆₇	1	4	yes	—
555	C₂²×C₄	C₂²×C₄×C₆₇	1	4	yes	—
556	C₂²×C₄	C₂²×C₄×C₆₇	1	8	yes	—
557	C₂²×C₄	C₂²×C₄×C₆₇	1	4	yes	—
558	C₂²×C₄	C₂⁴×C₆₇	1	2	yes	—
559	C₂²×C₄	C₂⁴×C₆₇	1	2	yes	—
560	C₂²×C₄	C₂⁴×C₆₇	1	4	yes	—
561	C₂²×C₄	C₂⁴×C₆₇	1	2	yes	—
562	C₂²×C₄	C₂⁴×C₆₇	1	4	yes	—
563	C₂²×C₄	C₂⁴×C₆₇	1	2	yes	—
564	C₂²×C₄	C₂⁴×C₆₇	1	1	yes	characteristic
565	C₂²×C₄	C₂⁴×C₆₇	1	2	yes	—
566	C₂²×C₄	C₂⁴×C₆₇	1	2	yes	—
567	C₂²×C₄	C₂⁴×C₆₇	1	2	yes	—
568	C₂²×C₄	C₂⁴×C₆₇	1	1	yes	characteristic
569	C₂²×C₄	C₂⁴×C₆₇	1	1	yes	characteristic
570	C₂²×C₄	C₂⁴×C₆₇	1	1	yes	characteristic
571	C₂²×C₄	C₂⁴×C₆₇	1	1	yes	characteristic
572	C₂²×C₄	C₂⁴×C₆₇	1	1	yes	characteristic
573	C₄⋊C₄	—	2	2	no	—
574	C₄⋊C₄	—	2	4	no	—
575	C₄⋊C₄	—	2	4	no	—
576	C₄⋊C₄	—	2	2	no	—
577	C₄⋊C₄	—	2	4	no	—
578	C₄⋊C₄	—	2	2	no	—
579	C₄⋊C₄	—	2	2	no	—
580	C₄⋊C₄	—	2	4	no	—
581	C₄⋊C₄	—	4	4	no	—
582	C₄⋊C₄	—	4	2	no	—
583	C₄⋊C₄	—	4	2	no	—
584	C₄⋊C₄	—	4	4	no	—
585	C₄⋊C₄	—	4	2	no	—
586	C₄⋊C₄	—	4	2	no	—
587	C₄⋊C₄	—	4	2	no	—
588	C₄⋊C₄	—	4	2	no	—
589	C₄⋊C₄	—	4	4	no	—
590	C₄⋊C₄	C₂²×C₄×C₆₇	1	8	yes	—
591	C₄⋊C₄	C₂²×C₄×C₆₇	1	8	yes	—
592	C₄⋊C₄	C₂²×C₄×C₆₇	1	4	yes	—
593	C₄⋊C₄	C₂²×C₄×C₆₇	1	8	yes	—
594	C₄⋊C₄	C₂²×C₄×C₆₇	1	4	yes	—
595	C₄⋊C₄	C₂²×C₄×C₆₇	1	8	yes	—
596	C₂²⋊C₄	—	4	2	no	—
597	C₂²⋊C₄	—	4	2	no	—
598	C₂²⋊C₄	—	4	4	no	—
599	C₂²⋊C₄	—	4	4	no	—
600	C₂²⋊C₄	—	4	4	no	—
601	C₂²⋊C₄	—	4	4	no	—
602	C₂²⋊C₄	C₂²×C₄×C₆₇	1	8	yes	—
603	C₂²⋊C₄	C₂²×C₄×C₆₇	1	8	yes	—
604	C₂²⋊C₄	C₂²×C₄×C₆₇	1	8	yes	—
605	C₂²⋊C₄	C₂²×C₄×C₆₇	1	8	yes	—
606	C₂²⋊C₄	C₂²×C₄×C₆₇	1	8	yes	—
607	C₄²	—	2	4	no	—
608	C₄²	—	2	4	no	—
609	C₄²	—	4	4	no	—
610	C₄²	—	4	4	no	—
611	C₄²	—	4	4	no	—
612	C₄²	—	4	4	no	—
613	C₄²	C₂²×C₄×C₆₇	1	4	yes	—
614	C₄²	C₂²×C₄×C₆₇	1	4	yes	—
615	C₄²	C₂²×C₄×C₆₇	1	4	yes	—
616	C₄²	C₂²×C₄×C₆₇	1	4	yes	—
617	C₄²	C₂²×C₄×C₆₇	1	8	yes	—
618	C₂³	—	2	1	no	—
619	C₂³	—	2	1	no	—
620	C₂³	—	2	2	no	—
621	C₂³	—	2	2	no	—
622	C₂³	—	2	1	no	—
623	C₂³	—	2	2	no	—
624	C₂³	—	2	1	no	—
625	C₂³	—	2	2	no	—
626	C₂³	—	2	2	no	—
627	C₂³	—	2	2	no	—
628	C₂³	—	2	1	no	—
629	C₂³	—	2	1	no	—
630	C₂³	C₂³×C₄×C₆₇	1	4	yes	—
631	C₂³	C₂³×C₄×C₆₇	1	2	yes	—
632	C₂³	C₂⁵×C₆₇	1	1	yes	characteristic
633	Q₈	—	2	8	no	—
634	D₄	—	2	8	no	—
635	D₄	—	4	8	no	—
636	C₂×C₄	—	2	1	no	—
637	C₂×C₄	—	2	2	no	—
638	C₂×C₄	—	2	2	no	—
639	C₂×C₄	—	2	1	no	—
640	C₂×C₄	—	2	1	no	—
641	C₂×C₄	—	2	2	no	—
642	C₂×C₄	—	2	1	no	—
643	C₂×C₄	—	2	2	no	—
644	C₂×C₄	—	2	2	no	—
645	C₂×C₄	—	2	1	no	—
646	C₂×C₄	—	2	1	no	—
647	C₂×C₄	—	2	1	no	—
648	C₂×C₄	—	2	1	no	—
649	C₂×C₄	—	2	2	no	—
650	C₂×C₄	—	2	16	no	—
651	C₂×C₄	—	2	8	no	—
652	C₂×C₄	—	2	8	no	—
653	C₂×C₄	—	2	2	no	—
654	C₂×C₄	—	2	2	no	—
655	C₂×C₄	—	2	2	no	—
656	C₂×C₄	—	2	2	no	—
657	C₂×C₄	—	2	2	no	—
658	C₂×C₄	—	2	4	no	—
659	C₂×C₄	—	2	4	no	—
660	C₂×C₄	—	2	4	no	—
661	C₂×C₄	—	2	2	no	—
662	C₂×C₄	—	2	2	no	—
663	C₂×C₄	—	2	2	no	—
664	C₂×C₄	—	2	4	no	—
665	C₂×C₄	—	2	2	no	—
666	C₂×C₄	—	2	4	no	—
667	C₂×C₄	—	2	2	no	—
668	C₂×C₄	—	2	2	no	—
669	C₂×C₄	—	2	4	no	—
670	C₂×C₄	—	2	2	no	—
671	C₂×C₄	—	4	4	no	—
672	C₂×C₄	—	8	8	no	—
673	C₂×C₄	C₂³×C₄×C₆₇	1	2	yes	—
674	C₂×C₄	C₂³×C₄×C₆₇	1	2	yes	—
675	C₂×C₄	C₂³×C₄×C₆₇	1	2	yes	—
676	C₂×C₄	C₂³×C₄×C₆₇	1	2	yes	—
677	C₂×C₄	C₂³×C₄×C₆₇	1	4	yes	—
678	C₂×C₄	C₂³×C₄×C₆₇	1	4	yes	—
679	C₂×C₄	C₂³×C₄×C₆₇	1	4	yes	—
680	C₂×C₄	C₂³×C₄×C₆₇	1	2	yes	—
681	C₂×C₄	C₂³×C₄×C₆₇	1	2	yes	—
682	C₂×C₄	H(5,𝔽₂)×C₆₇	1	2	yes	—
683	C₂×C₄	H(5,𝔽₂)×C₆₇	1	2	yes	—
684	C₂×C₄	2₋¹⁺⁴×C₆₇	1	2	yes	—
685	C₂×C₄	2₋¹⁺⁴×C₆₇	1	2	yes	—
686	C₂²	—	2	2	no	—
687	C₂²	—	2	2	no	—
688	C₂²	—	2	4	no	—
689	C₂²	—	2	4	no	—
690	C₂²	—	2	2	no	—
691	C₂²	—	2	4	no	—
692	C₂²	—	4	2	no	—
693	C₂²	—	4	1	no	—
694	C₂²	—	4	1	no	—
695	C₂²	—	4	2	no	—
696	C₂²	—	4	1	no	—
697	C₂²	—	4	2	no	—
698	C₂²	—	4	1	no	—
699	C₂²	—	4	2	no	—
700	C₂²	C₂⁴×C₄×C₆₇	1	1	yes	characteristic
701	C₂²	C₂×H(5,𝔽₂)×C₆₇	1	1	yes	characteristic
702	C₂²	C₂×H(5,𝔽₂)×C₆₇	1	1	yes	characteristic
703	C₂²	C₂×2₋¹⁺⁴×C₆₇	1	1	yes	characteristic
704	C₂²	C₂×2₋¹⁺⁴×C₆₇	1	1	yes	characteristic
705	C₂²	#585×C₆₇	1	1	yes	characteristic
706	C₂²	#585×C₆₇	1	1	yes	characteristic
707	C₄	—	2	2	no	—
708	C₄	—	2	4	no	—
709	C₄	—	2	2	no	—
710	C₄	—	2	4	no	—
711	C₄	—	2	2	no	—
712	C₄	—	2	2	no	—
713	C₄	—	2	4	no	—
714	C₄	—	4	2	no	—
715	C₄	—	4	4	no	—
716	C₄	—	4	2	no	—
717	C₄	—	4	2	no	—
718	C₄	—	4	2	no	—
719	C₄	—	4	4	no	—
720	C₄	#520×C₆₇	1	4	yes	—
721	C₄	#520×C₆₇	1	4	yes	—
722	C₂	—	4	2	no	—
723	C₂	—	4	4	no	—
724	C₂	C₂×#520×C₆₇	1	1	yes	characteristic
725	C₂	C₂×#520×C₆₇	1	1	yes	characteristic
726	C₂	C₆₇×#3428	1	1	yes	characteristic
727	C₂	C₆₇×#3546	1	1	yes	characteristic
728	C₂	C₆₇×#3547	1	1	yes	characteristic
729	C₂	C₆₇×#3554	1	1	yes	characteristic
730	C₂	C₆₇×#3555	1	1	yes	characteristic

	subgroup	G/H
center, socle	C₂³×C₆₇	C₂⁵
commutator	C₂²	C₂⁴×C₄×C₆₇
Frattini, cosocle	C₂³	C₂⁵×C₆₇
prime radical	C₂×#121×C₆₇	C₂²
LCS	G ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂³×C₆₇ ◁ G

Complements

Splittings G = N ⋊ H with N normal; H ≅ G/N.

N	H ≅ G/N	type
2. C₆₇×#2376	C₂	semidirect
3. C₆₇×#2370	C₂	semidirect
4. C₆₇×#2367	C₂	semidirect
5. C₆₇×#2355	C₂	semidirect
6. C₆₇×#2354	C₂	semidirect
7. C₆₇×#2354	C₂	semidirect
8. C₆₇×#2352	C₂	semidirect
9. Q₈×C₄⋊C₄×C₆₇	C₂	semidirect
10. D₄×C₄⋊C₄×C₆₇	C₂	semidirect
11. C₆₇×#2334	C₂	semidirect
13. C₆₇×#2323	C₂	semidirect
14. C₆₇×#2317	C₂	semidirect
15. C₆₇×#2314	C₂	semidirect
16. C₆₇×#2285	C₂	semidirect
17. C₆₇×#2285	C₂	semidirect
18. #529×C₆₇	C₄	semidirect
92. #126×C₆₇	C₂×C₄	semidirect
93. #125×C₆₇	C₂×C₄	semidirect
95. #123×C₆₇	C₂×C₄	semidirect
96. C₂²⋊Q₈×C₆₇	C₂×C₄	semidirect
99. C₄×Q₈×C₆₇	C₂×C₄	semidirect
102. C₄×D₄×C₆₇	C₂×C₄	semidirect
106. #117×C₆₇	C₂×C₄	semidirect
142. C₂×C₄²×C₆₇	C₂×C₄	semidirect
356. #34383	C₆₇	direct (N × H)
367. #2376	C₂×C₆₇	semidirect
368. #2370	C₂×C₆₇	semidirect
369. #2367	C₂×C₆₇	semidirect
370. #2355	C₂×C₆₇	semidirect
371. #2354	C₂×C₆₇	semidirect
372. #2354	C₂×C₆₇	semidirect
373. #2352	C₂×C₆₇	semidirect
374. Q₈×C₄⋊C₄	C₂×C₆₇	semidirect
375. D₄×C₄⋊C₄	C₂×C₆₇	semidirect
376. #2334	C₂×C₆₇	semidirect
378. #2323	C₂×C₆₇	semidirect
379. #2317	C₂×C₆₇	semidirect
380. #2314	C₂×C₆₇	semidirect
381. #2285	C₂×C₆₇	semidirect
382. #2285	C₂×C₆₇	semidirect
383. C₆₇	#34383	direct (N × H)
384. #529	C₄×C₆₇	semidirect
458. #126	C₂×C₄×C₆₇	semidirect
459. #125	C₂×C₄×C₆₇	semidirect
461. #123	C₂×C₄×C₆₇	semidirect
462. C₂²⋊Q₈	C₂×C₄×C₆₇	semidirect
465. C₄×Q₈	C₂×C₄×C₆₇	semidirect
468. C₄×D₄	C₂×C₄×C₆₇	semidirect
472. #117	C₂×C₄×C₆₇	semidirect
508. C₂×C₄²	C₂×C₄×C₆₇	semidirect

Maximal subgroups

By isomorphism type — maximal subgroups of the same type appear once, whether or not they are normal.

subgroup	index
C₂×#529×C₆₇	2
C₆₇×#2376	2
C₆₇×#2370	2
C₆₇×#2367	2
C₆₇×#2355	2
C₆₇×#2354	2
C₆₇×#2352	2
Q₈×C₄⋊C₄×C₆₇	2
D₄×C₄⋊C₄×C₆₇	2
C₆₇×#2334	2
C₆₇×#2324	2
C₆₇×#2323	2
C₆₇×#2317	2
C₆₇×#2314	2
C₆₇×#2285	2
#34383	67

Subgroups — C67×#34383 #30713634

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₆₇×#34383 #30713634