Subgroups — C₆₇×#34401 #30713652

Order 17152.

2118 conjugacy classes of subgroups.

The subgroup lattice has 620 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×#2383	C₂	1	4	yes	maximal, complemented
3	C₆₇×#2359	C₂	1	4	yes	maximal, complemented
4	C₆₇×#2353	C₂	1	2	yes	maximal, complemented
5	C₆₇×#2349	C₂	1	2	yes	maximal, complemented
6	C₆₇×#2347	C₂	1	2	yes	maximal, complemented
7	C₆₇×#2343	C₂	1	2	yes	maximal, complemented
8	C₆₇×#2337	C₂	1	2	yes	maximal, complemented
9	C₆₇×#2336	C₂	1	2	yes	maximal, complemented
10	C₆₇×#2329	C₂	1	2	yes	maximal, complemented
11	C₆₇×#2325	C₂	1	2	yes	maximal
12	C₆₇×#2320	C₂	1	2	yes	maximal, complemented
13	C₆₇×#2318	C₂	1	2	yes	maximal, complemented
14	C₆₇×#2285	C₂	1	1	yes	maximal, characteristic, complemented
15	C₂×#383×C₆₇	C₂	1	1	yes	maximal, characteristic, complemented
16	#529×C₆₇	C₄	1	16	yes	complemented
17	C₂×#126×C₆₇	C₂²	1	2	yes	—
18	C₂×#125×C₆₇	C₂²	1	1	yes	characteristic
19	C₂×#124×C₆₇	C₂²	1	1	yes	characteristic
20	C₂×#123×C₆₇	C₂²	1	2	yes	—
21	C₂×C₂²⋊Q₈×C₆₇	C₂²	1	1	yes	characteristic
22	C₂×#121×C₆₇	C₂²	1	1	yes	characteristic
23	C₂×C₄×Q₈×C₆₇	C₂²	1	1	yes	characteristic
24	C₂×C₄×D₄×C₆₇	C₂²	1	4	yes	—
25	C₂×C₄×D₄×C₆₇	C₂²	1	1	yes	characteristic
26	C₂×C₄×D₄×C₆₇	C₂²	1	2	yes	—
27	C₂×#117×C₆₇	C₂²	1	2	yes	—
28	C₂×#117×C₆₇	C₂²	1	2	yes	—
29	C₂²×C₄²×C₆₇	C₂²	1	1	yes	characteristic
30	#391×C₆₇	C₂²	1	4	yes	—
31	#389×C₆₇	C₂²	1	4	yes	—
32	#389×C₆₇	C₂²	1	4	yes	—
33	#389×C₆₇	C₂²	1	4	yes	—
34	#388×C₆₇	C₂²	1	4	yes	—
35	#388×C₆₇	C₂²	1	4	yes	—
36	#388×C₆₇	C₂²	1	4	yes	—
37	#388×C₆₇	C₂²	1	4	yes	—
38	#387×C₆₇	C₂²	1	4	yes	—
39	#387×C₆₇	C₂²	1	4	yes	—
40	#387×C₆₇	C₂²	1	4	yes	—
41	#387×C₆₇	C₂²	1	4	yes	—
42	#387×C₆₇	C₂²	1	4	yes	—
43	#387×C₆₇	C₂²	1	4	yes	—
44	#386×C₆₇	C₂²	1	4	yes	—
45	#386×C₆₇	C₂²	1	4	yes	—
46	#385×C₆₇	C₂²	1	4	yes	—
47	#385×C₆₇	C₂²	1	4	yes	—
48	#385×C₆₇	C₂²	1	4	yes	—
49	#385×C₆₇	C₂²	1	4	yes	—
50	#384×C₆₇	C₂²	1	2	yes	—
51	#383×C₆₇	C₂²	1	2	yes	—
52	#383×C₆₇	C₂²	1	2	yes	—
53	#383×C₆₇	C₂²	1	2	yes	—
54	#383×C₆₇	C₂²	1	2	yes	—
55	#383×C₆₇	C₂²	1	2	yes	—
56	#382×C₆₇	C₂²	1	2	yes	—
57	#382×C₆₇	C₂²	1	2	yes	—
58	#381×C₆₇	C₂²	1	2	yes	—
59	#380×C₆₇	C₂²	1	2	yes	—
60	#380×C₆₇	C₂²	1	2	yes	—
61	#380×C₆₇	C₂²	1	2	yes	—
62	C₄×C₄⋊C₄×C₆₇	C₂²	1	2	yes	—
63	C₄×C₄⋊C₄×C₆₇	C₂²	1	1	yes	characteristic
64	C₄×C₄⋊C₄×C₆₇	C₂²	1	1	yes	characteristic
65	C₄×C₄⋊C₄×C₆₇	C₂²	1	2	yes	—
66	C₄×C₄⋊C₄×C₆₇	C₂²	1	4	yes	—
67	C₄×C₂²⋊C₄×C₆₇	C₂²	1	2	yes	—
68	C₄×C₂²⋊C₄×C₆₇	C₂²	1	4	yes	—
69	C₄×C₂²⋊C₄×C₆₇	C₂²	1	4	yes	—
70	#376×C₆₇	C₂²	1	2	yes	—
71	C₂×#95×C₆₇	C₂²	1	2	yes	—
72	C₂×#95×C₆₇	C₂²	1	1	yes	characteristic
73	C₂×#95×C₆₇	C₂²	1	1	yes	characteristic
74	C₂×#95×C₆₇	C₂²	1	2	yes	—
75	C₂×#95×C₆₇	C₂²	1	2	yes	—
76	C₂²×Q₈×C₆₇	C₂³	1	1	yes	characteristic
77	C₂²×D₄×C₆₇	C₂³	1	2	yes	—
78	C₂²×D₄×C₆₇	C₂³	1	1	yes	characteristic
79	C₂³×C₄×C₆₇	C₂³	1	1	yes	characteristic
80	C₂³×C₄×C₆₇	C₂³	1	2	yes	—
81	C₂³×C₄×C₆₇	C₂³	1	2	yes	—
82	C₂³×C₄×C₆₇	C₂³	1	4	yes	—
83	C₂³×C₄×C₆₇	C₂³	1	1	yes	characteristic
84	C₂³×C₄×C₆₇	C₂³	1	1	yes	characteristic
85	C₂³×C₄×C₆₇	C₂³	1	2	yes	—
86	#126×C₆₇	C₂×C₄	1	16	yes	complemented
87	#125×C₆₇	C₂×C₄	1	8	yes	complemented
88	#124×C₆₇	C₂×C₄	1	8	yes	complemented
89	#123×C₆₇	C₂×C₄	1	16	yes	complemented
90	C₂²⋊Q₈×C₆₇	C₂×C₄	1	8	yes	complemented
91	#121×C₆₇	C₂×C₄	1	8	yes	—
92	C₄×Q₈×C₆₇	C₂×C₄	1	8	yes	complemented
93	C₄×D₄×C₆₇	—	4	8	no	—
94	C₄×D₄×C₆₇	C₂×C₄	1	8	yes	complemented
95	C₄×D₄×C₆₇	C₂×C₄	1	16	yes	complemented
96	#117×C₆₇	—	4	4	no	—
97	#117×C₆₇	C₂×C₄	1	16	yes	complemented
98	C₂×C₄⋊C₄×C₆₇	C₂³	1	4	yes	—
99	C₂×C₄⋊C₄×C₆₇	C₂³	1	4	yes	—
100	C₂×C₄⋊C₄×C₆₇	C₂³	1	2	yes	—
101	C₂×C₄⋊C₄×C₆₇	C₂³	1	4	yes	—
102	C₂×C₄⋊C₄×C₆₇	C₂³	1	2	yes	—
103	C₂×C₄⋊C₄×C₆₇	C₂³	1	4	yes	—
104	C₂×C₄⋊C₄×C₆₇	C₂³	1	4	yes	—
105	C₂×C₄⋊C₄×C₆₇	C₂³	1	2	yes	—
106	C₂×C₄⋊C₄×C₆₇	C₂³	1	2	yes	—
107	C₂×C₄⋊C₄×C₆₇	C₂³	1	1	yes	characteristic
108	C₂×C₄⋊C₄×C₆₇	C₂³	1	2	yes	—
109	C₂×C₄⋊C₄×C₆₇	C₂³	1	1	yes	characteristic
110	C₂×C₄⋊C₄×C₆₇	C₂³	1	2	yes	—
111	C₂×C₂²⋊C₄×C₆₇	C₂³	1	4	yes	—
112	C₂×C₂²⋊C₄×C₆₇	C₂³	1	4	yes	—
113	C₂×C₂²⋊C₄×C₆₇	C₂³	1	4	yes	—
114	C₂×C₂²⋊C₄×C₆₇	C₂³	1	4	yes	—
115	C₂×C₂²⋊C₄×C₆₇	C₂³	1	2	yes	—
116	C₂×C₂²⋊C₄×C₆₇	C₂³	1	2	yes	—
117	C₂×C₂²⋊C₄×C₆₇	C₂³	1	2	yes	—
118	C₂×C₂²⋊C₄×C₆₇	C₂³	1	2	yes	—
119	C₂×C₂²⋊C₄×C₆₇	C₂³	1	2	yes	—
120	C₂×C₄²×C₆₇	C₂×C₄	1	8	yes	complemented
121	C₂×C₄²×C₆₇	C₂³	1	2	yes	—
122	C₂×C₄²×C₆₇	C₂³	1	2	yes	—
123	C₂×C₄²×C₆₇	C₂³	1	4	yes	—
124	C₂×C₄²×C₆₇	C₂³	1	4	yes	—
125	C₂×C₄²×C₆₇	C₂³	1	4	yes	—
126	C₂×C₄²×C₆₇	C₂³	1	1	yes	characteristic
127	C₂×C₄²×C₆₇	C₂³	1	1	yes	characteristic
128	C₂×C₄²×C₆₇	C₂³	1	1	yes	characteristic
129	C₂×C₄²×C₆₇	C₂³	1	1	yes	characteristic
130	C₂×C₄²×C₆₇	C₂³	1	2	yes	—
131	#95×C₆₇	C₂³	1	2	yes	—
132	#95×C₆₇	C₂³	1	2	yes	—
133	#95×C₆₇	C₂³	1	2	yes	—
134	#95×C₆₇	C₂³	1	4	yes	—
135	#95×C₆₇	C₂³	1	2	yes	—
136	#95×C₆₇	C₂³	1	4	yes	—
137	#95×C₆₇	C₂³	1	4	yes	—
138	#95×C₆₇	C₂³	1	4	yes	—
139	#95×C₆₇	C₂³	1	2	yes	—
140	#95×C₆₇	C₂³	1	4	yes	—
141	#95×C₆₇	C₂³	1	4	yes	—
142	#95×C₆₇	C₂³	1	2	yes	—
143	#95×C₆₇	C₂³	1	4	yes	—
144	#95×C₆₇	C₂³	1	4	yes	—
145	#95×C₆₇	C₂³	1	4	yes	—
146	#95×C₆₇	C₂³	1	4	yes	—
147	#95×C₆₇	C₂³	1	4	yes	—
148	C₂⁴×C₆₇	C₂⁴	1	2	yes	—
149	C₂⁴×C₆₇	C₂⁴	1	1	yes	characteristic
150	C₂×Q₈×C₆₇	—	4	2	no	—
151	C₂×Q₈×C₆₇	C₂²×C₄	1	4	yes	—
152	C₂×D₄×C₆₇	—	2	4	no	—
153	C₂×D₄×C₆₇	—	2	4	no	—
154	C₂×D₄×C₆₇	—	4	2	no	—
155	C₂×D₄×C₆₇	C₂²×C₄	1	8	yes	—
156	C₂×D₄×C₆₇	C₂²×C₄	1	4	yes	—
157	C₂²×C₄×C₆₇	—	2	2	no	—
158	C₂²×C₄×C₆₇	—	2	2	no	—
159	C₂²×C₄×C₆₇	—	2	4	no	—
160	C₂²×C₄×C₆₇	—	2	4	no	—
161	C₂²×C₄×C₆₇	—	4	4	no	—
162	C₂²×C₄×C₆₇	—	4	2	no	—
163	C₂²×C₄×C₆₇	—	4	2	no	—
164	C₂²×C₄×C₆₇	—	4	4	no	—
165	C₂²×C₄×C₆₇	—	4	2	no	—
166	C₂²×C₄×C₆₇	—	4	2	no	—
167	C₂²×C₄×C₆₇	—	4	4	no	—
168	C₂²×C₄×C₆₇	—	4	4	no	—
169	C₂²×C₄×C₆₇	—	4	2	no	—
170	C₂²×C₄×C₆₇	—	4	2	no	—
171	C₂²×C₄×C₆₇	C₂²×C₄	1	4	yes	—
172	C₂²×C₄×C₆₇	C₂²×C₄	1	4	yes	—
173	C₂²×C₄×C₆₇	C₂²×C₄	1	4	yes	—
174	C₂²×C₄×C₆₇	C₂²×C₄	1	8	yes	—
175	C₂²×C₄×C₆₇	C₂⁴	1	4	yes	—
176	C₂²×C₄×C₆₇	C₂⁴	1	4	yes	—
177	C₂²×C₄×C₆₇	C₂⁴	1	4	yes	—
178	C₂²×C₄×C₆₇	C₂⁴	1	4	yes	—
179	C₂²×C₄×C₆₇	C₂⁴	1	1	yes	characteristic
180	C₂²×C₄×C₆₇	C₂⁴	1	1	yes	characteristic
181	C₂²×C₄×C₆₇	C₂⁴	1	2	yes	—
182	C₂²×C₄×C₆₇	C₂⁴	1	2	yes	—
183	C₂²×C₄×C₆₇	C₂⁴	1	1	yes	characteristic
184	C₂²×C₄×C₆₇	C₂⁴	1	2	yes	—
185	C₂²×C₄×C₆₇	C₂⁴	1	1	yes	characteristic
186	C₂²×C₄×C₆₇	C₂⁴	1	1	yes	characteristic
187	C₂²×C₄×C₆₇	C₂⁴	1	1	yes	characteristic
188	C₄⋊C₄×C₆₇	—	4	4	no	—
189	C₄⋊C₄×C₆₇	—	4	4	no	—
190	C₄⋊C₄×C₆₇	—	4	4	no	—
191	C₄⋊C₄×C₆₇	—	4	2	no	—
192	C₄⋊C₄×C₆₇	—	4	2	no	—
193	C₄⋊C₄×C₆₇	—	4	4	no	—
194	C₄⋊C₄×C₆₇	—	4	4	no	—
195	C₄⋊C₄×C₆₇	C₂²×C₄	1	8	yes	—
196	C₄⋊C₄×C₆₇	C₂²×C₄	1	8	yes	—
197	C₄⋊C₄×C₆₇	C₂²×C₄	1	4	yes	—
198	C₄⋊C₄×C₆₇	C₂²×C₄	1	8	yes	—
199	C₄⋊C₄×C₆₇	C₂²×C₄	1	4	yes	—
200	C₄⋊C₄×C₆₇	C₂²×C₄	1	8	yes	—
201	C₂²⋊C₄×C₆₇	—	2	8	no	—
202	C₂²⋊C₄×C₆₇	—	4	4	no	—
203	C₂²⋊C₄×C₆₇	—	4	4	no	—
204	C₂²⋊C₄×C₆₇	—	4	4	no	—
205	C₂²⋊C₄×C₆₇	C₂²×C₄	1	8	yes	—
206	C₂²⋊C₄×C₆₇	C₂²×C₄	1	8	yes	—
207	C₂²⋊C₄×C₆₇	C₂²×C₄	1	8	yes	—
208	C₂²⋊C₄×C₆₇	C₂²×C₄	1	8	yes	—
209	C₂²⋊C₄×C₆₇	C₂²×C₄	1	8	yes	—
210	C₄²×C₆₇	—	4	4	no	—
211	C₄²×C₆₇	—	4	4	no	—
212	C₄²×C₆₇	—	4	2	no	—
213	C₄²×C₆₇	—	4	2	no	—
214	C₄²×C₆₇	—	4	4	no	—
215	C₄²×C₆₇	C₂²×C₄	1	4	yes	—
216	C₄²×C₆₇	C₂²×C₄	1	4	yes	—
217	C₄²×C₆₇	C₂²×C₄	1	4	yes	—
218	C₄²×C₆₇	C₂²×C₄	1	4	yes	—
219	C₄²×C₆₇	C₂²×C₄	1	8	yes	—
220	C₂³×C₆₇	—	2	2	no	—
221	C₂³×C₆₇	—	2	2	no	—
222	C₂³×C₆₇	—	2	2	no	—
223	C₂³×C₆₇	—	2	2	no	—
224	C₂³×C₆₇	—	2	4	no	—
225	C₂³×C₆₇	—	2	4	no	—
226	C₂³×C₆₇	C₂³×C₄	1	4	yes	—
227	C₂³×C₆₇	C₂³×C₄	1	2	yes	—
228	C₂³×C₆₇	H(5,𝔽₂)	1	2	yes	—
229	C₂³×C₆₇	2₋¹⁺⁴	1	2	yes	—
230	C₂³×C₆₇	C₂⁵	1	1	yes	characteristic
231	Q₈×C₆₇	—	4	4	no	—
232	D₄×C₆₇	—	2	16	no	—
233	D₄×C₆₇	—	4	4	no	—
234	C₂×C₄×C₆₇	—	2	8	no	—
235	C₂×C₄×C₆₇	—	2	1	no	—
236	C₂×C₄×C₆₇	—	2	2	no	—
237	C₂×C₄×C₆₇	—	2	1	no	—
238	C₂×C₄×C₆₇	—	2	1	no	—
239	C₂×C₄×C₆₇	—	2	1	no	—
240	C₂×C₄×C₆₇	—	2	2	no	—
241	C₂×C₄×C₆₇	—	2	4	no	—
242	C₂×C₄×C₆₇	—	2	4	no	—
243	C₂×C₄×C₆₇	—	2	2	no	—
244	C₂×C₄×C₆₇	—	2	2	no	—
245	C₂×C₄×C₆₇	—	2	2	no	—
246	C₂×C₄×C₆₇	—	2	2	no	—
247	C₂×C₄×C₆₇	—	2	4	no	—
248	C₂×C₄×C₆₇	—	2	4	no	—
249	C₂×C₄×C₆₇	—	2	4	no	—
250	C₂×C₄×C₆₇	—	2	4	no	—
251	C₂×C₄×C₆₇	—	2	4	no	—
252	C₂×C₄×C₆₇	—	2	4	no	—
253	C₂×C₄×C₆₇	—	2	4	no	—
254	C₂×C₄×C₆₇	—	2	4	no	—
255	C₂×C₄×C₆₇	—	2	4	no	—
256	C₂×C₄×C₆₇	—	2	4	no	—
257	C₂×C₄×C₆₇	—	2	4	no	—
258	C₂×C₄×C₆₇	—	2	4	no	—
259	C₂×C₄×C₆₇	—	4	4	no	—
260	C₂×C₄×C₆₇	—	4	8	no	—
261	C₂×C₄×C₆₇	—	4	4	no	—
262	C₂×C₄×C₆₇	—	8	4	no	—
263	C₂×C₄×C₆₇	—	8	4	no	—
264	C₂×C₄×C₆₇	—	8	8	no	—
265	C₂×C₄×C₆₇	C₂³×C₄	1	2	yes	—
266	C₂×C₄×C₆₇	C₂³×C₄	1	2	yes	—
267	C₂×C₄×C₆₇	C₂³×C₄	1	2	yes	—
268	C₂×C₄×C₆₇	C₂³×C₄	1	2	yes	—
269	C₂×C₄×C₆₇	C₂³×C₄	1	2	yes	—
270	C₂×C₄×C₆₇	C₂³×C₄	1	2	yes	—
271	C₂×C₄×C₆₇	C₂³×C₄	1	4	yes	—
272	C₂×C₄×C₆₇	C₂³×C₄	1	4	yes	—
273	C₂×C₄×C₆₇	C₂³×C₄	1	4	yes	—
274	C₂²×C₆₇	—	2	2	no	—
275	C₂²×C₆₇	—	2	2	no	—
276	C₂²×C₆₇	—	2	4	no	—
277	C₂²×C₆₇	—	2	4	no	—
278	C₂²×C₆₇	—	2	4	no	—
279	C₂²×C₆₇	—	2	4	no	—
280	C₂²×C₆₇	—	2	4	no	—
281	C₂²×C₆₇	—	4	4	no	—
282	C₂²×C₆₇	—	4	4	no	—
283	C₂²×C₆₇	#520	1	4	yes	—
284	C₂²×C₆₇	C₂⁴×C₄	1	1	yes	characteristic
285	C₂²×C₆₇	C₂×H(5,𝔽₂)	1	1	yes	characteristic
286	C₂²×C₆₇	C₂×2₋¹⁺⁴	1	1	yes	characteristic
287	C₂²×C₆₇	#585	1	2	yes	—
288	C₂²×C₆₇	#585	1	2	yes	—
289	C₄×C₆₇	—	2	2	no	—
290	C₄×C₆₇	—	2	2	no	—
291	C₄×C₆₇	—	2	4	no	—
292	C₄×C₆₇	—	2	2	no	—
293	C₄×C₆₇	—	2	4	no	—
294	C₄×C₆₇	—	2	2	no	—
295	C₄×C₆₇	—	2	2	no	—
296	C₄×C₆₇	—	2	2	no	—
297	C₄×C₆₇	—	2	4	no	—
298	C₄×C₆₇	—	4	4	no	—
299	C₄×C₆₇	—	4	4	no	—
300	C₄×C₆₇	—	4	4	no	—
301	C₄×C₆₇	—	4	4	no	—
302	#34401	C₆₇	1	1	yes	maximal, characteristic, direct product factor, Sylow
303	C₂×C₆₇	—	2	4	no	—
304	C₂×C₆₇	—	4	4	no	—
305	C₂×C₆₇	C₂×#520	1	1	yes	characteristic
306	C₂×C₆₇	#3428	1	1	yes	characteristic
307	C₂×C₆₇	#3428	1	1	yes	characteristic
308	C₂×C₆₇	#3560	1	2	yes	—
309	C₂×C₆₇	#3564	1	2	yes	—
310	C₂×#529	C₂×C₆₇	1	1	yes	characteristic
311	#2383	C₂×C₆₇	1	4	yes	complemented
312	#2359	C₂×C₆₇	1	4	yes	complemented
313	#2353	C₂×C₆₇	1	2	yes	complemented
314	#2349	C₂×C₆₇	1	2	yes	complemented
315	#2347	C₂×C₆₇	1	2	yes	complemented
316	#2343	C₂×C₆₇	1	2	yes	complemented
317	#2337	C₂×C₆₇	1	2	yes	complemented
318	#2336	C₂×C₆₇	1	2	yes	complemented
319	#2329	C₂×C₆₇	1	2	yes	complemented
320	#2325	C₂×C₆₇	1	2	yes	—
321	#2320	C₂×C₆₇	1	2	yes	complemented
322	#2318	C₂×C₆₇	1	2	yes	complemented
323	#2285	C₂×C₆₇	1	1	yes	characteristic, complemented
324	C₂×#383	C₂×C₆₇	1	1	yes	characteristic, complemented
325	C₆₇	#34401	1	1	yes	characteristic, direct product factor, Sylow
326	#529	C₄×C₆₇	1	16	yes	complemented
327	C₂×#126	C₂²×C₆₇	1	2	yes	—
328	C₂×#125	C₂²×C₆₇	1	1	yes	characteristic
329	C₂×#124	C₂²×C₆₇	1	1	yes	characteristic
330	C₂×#123	C₂²×C₆₇	1	2	yes	—
331	C₂×C₂²⋊Q₈	C₂²×C₆₇	1	1	yes	characteristic
332	C₂×#121	C₂²×C₆₇	1	1	yes	characteristic
333	C₂×C₄×Q₈	C₂²×C₆₇	1	1	yes	characteristic
334	C₂×C₄×D₄	C₂²×C₆₇	1	4	yes	—
335	C₂×C₄×D₄	C₂²×C₆₇	1	1	yes	characteristic
336	C₂×C₄×D₄	C₂²×C₆₇	1	2	yes	—
337	C₂×#117	C₂²×C₆₇	1	2	yes	—
338	C₂×#117	C₂²×C₆₇	1	2	yes	—
339	C₂²×C₄²	C₂²×C₆₇	1	1	yes	characteristic
340	#391	C₂²×C₆₇	1	4	yes	—
341	#389	C₂²×C₆₇	1	4	yes	—
342	#389	C₂²×C₆₇	1	4	yes	—
343	#389	C₂²×C₆₇	1	4	yes	—
344	#388	C₂²×C₆₇	1	4	yes	—
345	#388	C₂²×C₆₇	1	4	yes	—
346	#388	C₂²×C₆₇	1	4	yes	—
347	#388	C₂²×C₆₇	1	4	yes	—
348	#387	C₂²×C₆₇	1	4	yes	—
349	#387	C₂²×C₆₇	1	4	yes	—
350	#387	C₂²×C₆₇	1	4	yes	—
351	#387	C₂²×C₆₇	1	4	yes	—
352	#387	C₂²×C₆₇	1	4	yes	—
353	#387	C₂²×C₆₇	1	4	yes	—
354	#386	C₂²×C₆₇	1	4	yes	—
355	#386	C₂²×C₆₇	1	4	yes	—
356	#385	C₂²×C₆₇	1	4	yes	—
357	#385	C₂²×C₆₇	1	4	yes	—
358	#385	C₂²×C₆₇	1	4	yes	—
359	#385	C₂²×C₆₇	1	4	yes	—
360	#384	C₂²×C₆₇	1	2	yes	—
361	#383	C₂²×C₆₇	1	2	yes	—
362	#383	C₂²×C₆₇	1	2	yes	—
363	#383	C₂²×C₆₇	1	2	yes	—
364	#383	C₂²×C₆₇	1	2	yes	—
365	#383	C₂²×C₆₇	1	2	yes	—
366	#382	C₂²×C₆₇	1	2	yes	—
367	#382	C₂²×C₆₇	1	2	yes	—
368	#381	C₂²×C₆₇	1	2	yes	—
369	#380	C₂²×C₆₇	1	2	yes	—
370	#380	C₂²×C₆₇	1	2	yes	—
371	#380	C₂²×C₆₇	1	2	yes	—
372	C₄×C₄⋊C₄	C₂²×C₆₇	1	2	yes	—
373	C₄×C₄⋊C₄	C₂²×C₆₇	1	2	yes	—
374	C₄×C₄⋊C₄	C₂²×C₆₇	1	4	yes	—
375	C₄×C₄⋊C₄	C₂²×C₆₇	1	1	yes	characteristic
376	C₄×C₄⋊C₄	C₂²×C₆₇	1	1	yes	characteristic
377	C₄×C₂²⋊C₄	C₂²×C₆₇	1	2	yes	—
378	C₄×C₂²⋊C₄	C₂²×C₆₇	1	4	yes	—
379	C₄×C₂²⋊C₄	C₂²×C₆₇	1	4	yes	—
380	#376	C₂²×C₆₇	1	2	yes	—
381	C₂×#95	C₂²×C₆₇	1	2	yes	—
382	C₂×#95	C₂²×C₆₇	1	2	yes	—
383	C₂×#95	C₂²×C₆₇	1	1	yes	characteristic
384	C₂×#95	C₂²×C₆₇	1	1	yes	characteristic
385	C₂×#95	C₂²×C₆₇	1	2	yes	—
386	C₂²×Q₈	C₂³×C₆₇	1	1	yes	characteristic
387	C₂²×D₄	C₂³×C₆₇	1	2	yes	—
388	C₂²×D₄	C₂³×C₆₇	1	1	yes	characteristic
389	C₂³×C₄	C₂³×C₆₇	1	1	yes	characteristic
390	C₂³×C₄	C₂³×C₆₇	1	2	yes	—
391	C₂³×C₄	C₂³×C₆₇	1	2	yes	—
392	C₂³×C₄	C₂³×C₆₇	1	4	yes	—
393	C₂³×C₄	C₂³×C₆₇	1	2	yes	—
394	C₂³×C₄	C₂³×C₆₇	1	1	yes	characteristic
395	C₂³×C₄	C₂³×C₆₇	1	1	yes	characteristic
396	#126	C₂×C₄×C₆₇	1	16	yes	complemented
397	#125	C₂×C₄×C₆₇	1	8	yes	complemented
398	#124	C₂×C₄×C₆₇	1	8	yes	complemented
399	#123	C₂×C₄×C₆₇	1	16	yes	complemented
400	C₂²⋊Q₈	C₂×C₄×C₆₇	1	8	yes	complemented
401	#121	C₂×C₄×C₆₇	1	8	yes	—
402	C₄×Q₈	C₂×C₄×C₆₇	1	8	yes	complemented
403	C₄×D₄	—	4	8	no	—
404	C₄×D₄	C₂×C₄×C₆₇	1	8	yes	complemented
405	C₄×D₄	C₂×C₄×C₆₇	1	16	yes	complemented
406	#117	—	4	4	no	—
407	#117	C₂×C₄×C₆₇	1	16	yes	complemented
408	C₂×C₄⋊C₄	C₂³×C₆₇	1	4	yes	—
409	C₂×C₄⋊C₄	C₂³×C₆₇	1	4	yes	—
410	C₂×C₄⋊C₄	C₂³×C₆₇	1	2	yes	—
411	C₂×C₄⋊C₄	C₂³×C₆₇	1	4	yes	—
412	C₂×C₄⋊C₄	C₂³×C₆₇	1	2	yes	—
413	C₂×C₄⋊C₄	C₂³×C₆₇	1	4	yes	—
414	C₂×C₄⋊C₄	C₂³×C₆₇	1	4	yes	—
415	C₂×C₄⋊C₄	C₂³×C₆₇	1	2	yes	—
416	C₂×C₄⋊C₄	C₂³×C₆₇	1	2	yes	—
417	C₂×C₄⋊C₄	C₂³×C₆₇	1	1	yes	characteristic
418	C₂×C₄⋊C₄	C₂³×C₆₇	1	2	yes	—
419	C₂×C₄⋊C₄	C₂³×C₆₇	1	1	yes	characteristic
420	C₂×C₄⋊C₄	C₂³×C₆₇	1	2	yes	—
421	C₂×C₂²⋊C₄	C₂³×C₆₇	1	4	yes	—
422	C₂×C₂²⋊C₄	C₂³×C₆₇	1	4	yes	—
423	C₂×C₂²⋊C₄	C₂³×C₆₇	1	4	yes	—
424	C₂×C₂²⋊C₄	C₂³×C₆₇	1	4	yes	—
425	C₂×C₂²⋊C₄	C₂³×C₆₇	1	2	yes	—
426	C₂×C₂²⋊C₄	C₂³×C₆₇	1	2	yes	—
427	C₂×C₂²⋊C₄	C₂³×C₆₇	1	2	yes	—
428	C₂×C₂²⋊C₄	C₂³×C₆₇	1	2	yes	—
429	C₂×C₂²⋊C₄	C₂³×C₆₇	1	2	yes	—
430	C₂×C₄²	C₂×C₄×C₆₇	1	8	yes	complemented
431	C₂×C₄²	C₂³×C₆₇	1	2	yes	—
432	C₂×C₄²	C₂³×C₆₇	1	2	yes	—
433	C₂×C₄²	C₂³×C₆₇	1	4	yes	—
434	C₂×C₄²	C₂³×C₆₇	1	4	yes	—
435	C₂×C₄²	C₂³×C₆₇	1	4	yes	—
436	C₂×C₄²	C₂³×C₆₇	1	1	yes	characteristic
437	C₂×C₄²	C₂³×C₆₇	1	2	yes	—
438	C₂×C₄²	C₂³×C₆₇	1	1	yes	characteristic
439	C₂×C₄²	C₂³×C₆₇	1	1	yes	characteristic
440	C₂×C₄²	C₂³×C₆₇	1	1	yes	characteristic
441	#95	C₂³×C₆₇	1	2	yes	—
442	#95	C₂³×C₆₇	1	2	yes	—
443	#95	C₂³×C₆₇	1	2	yes	—
444	#95	C₂³×C₆₇	1	4	yes	—
445	#95	C₂³×C₆₇	1	2	yes	—
446	#95	C₂³×C₆₇	1	4	yes	—
447	#95	C₂³×C₆₇	1	4	yes	—
448	#95	C₂³×C₆₇	1	4	yes	—
449	#95	C₂³×C₆₇	1	2	yes	—
450	#95	C₂³×C₆₇	1	4	yes	—
451	#95	C₂³×C₆₇	1	4	yes	—
452	#95	C₂³×C₆₇	1	2	yes	—
453	#95	C₂³×C₆₇	1	4	yes	—
454	#95	C₂³×C₆₇	1	4	yes	—
455	#95	C₂³×C₆₇	1	4	yes	—
456	#95	C₂³×C₆₇	1	4	yes	—
457	#95	C₂³×C₆₇	1	4	yes	—
458	C₂⁴	C₂⁴×C₆₇	1	2	yes	—
459	C₂⁴	C₂⁴×C₆₇	1	1	yes	characteristic
460	C₂×Q₈	—	4	2	no	—
461	C₂×Q₈	C₂²×C₄×C₆₇	1	4	yes	—
462	C₂×D₄	—	2	4	no	—
463	C₂×D₄	—	2	4	no	—
464	C₂×D₄	—	4	2	no	—
465	C₂×D₄	C₂²×C₄×C₆₇	1	8	yes	—
466	C₂×D₄	C₂²×C₄×C₆₇	1	4	yes	—
467	C₂²×C₄	—	2	2	no	—
468	C₂²×C₄	—	2	2	no	—
469	C₂²×C₄	—	2	4	no	—
470	C₂²×C₄	—	2	4	no	—
471	C₂²×C₄	—	4	4	no	—
472	C₂²×C₄	—	4	2	no	—
473	C₂²×C₄	—	4	2	no	—
474	C₂²×C₄	—	4	4	no	—
475	C₂²×C₄	—	4	2	no	—
476	C₂²×C₄	—	4	2	no	—
477	C₂²×C₄	—	4	4	no	—
478	C₂²×C₄	—	4	4	no	—
479	C₂²×C₄	—	4	2	no	—
480	C₂²×C₄	—	4	2	no	—
481	C₂²×C₄	C₂²×C₄×C₆₇	1	4	yes	—
482	C₂²×C₄	C₂²×C₄×C₆₇	1	4	yes	—
483	C₂²×C₄	C₂²×C₄×C₆₇	1	4	yes	—
484	C₂²×C₄	C₂²×C₄×C₆₇	1	8	yes	—
485	C₂²×C₄	C₂⁴×C₆₇	1	4	yes	—
486	C₂²×C₄	C₂⁴×C₆₇	1	4	yes	—
487	C₂²×C₄	C₂⁴×C₆₇	1	4	yes	—
488	C₂²×C₄	C₂⁴×C₆₇	1	4	yes	—
489	C₂²×C₄	C₂⁴×C₆₇	1	1	yes	characteristic
490	C₂²×C₄	C₂⁴×C₆₇	1	1	yes	characteristic
491	C₂²×C₄	C₂⁴×C₆₇	1	1	yes	characteristic
492	C₂²×C₄	C₂⁴×C₆₇	1	2	yes	—
493	C₂²×C₄	C₂⁴×C₆₇	1	2	yes	—
494	C₂²×C₄	C₂⁴×C₆₇	1	1	yes	characteristic
495	C₂²×C₄	C₂⁴×C₆₇	1	2	yes	—
496	C₂²×C₄	C₂⁴×C₆₇	1	1	yes	characteristic
497	C₂²×C₄	C₂⁴×C₆₇	1	1	yes	characteristic
498	C₄⋊C₄	—	4	4	no	—
499	C₄⋊C₄	—	4	4	no	—
500	C₄⋊C₄	—	4	4	no	—
501	C₄⋊C₄	—	4	4	no	—
502	C₄⋊C₄	—	4	4	no	—
503	C₄⋊C₄	—	4	2	no	—
504	C₄⋊C₄	—	4	2	no	—
505	C₄⋊C₄	C₂²×C₄×C₆₇	1	8	yes	—
506	C₄⋊C₄	C₂²×C₄×C₆₇	1	8	yes	—
507	C₄⋊C₄	C₂²×C₄×C₆₇	1	4	yes	—
508	C₄⋊C₄	C₂²×C₄×C₆₇	1	8	yes	—
509	C₄⋊C₄	C₂²×C₄×C₆₇	1	4	yes	—
510	C₄⋊C₄	C₂²×C₄×C₆₇	1	8	yes	—
511	C₂²⋊C₄	—	2	8	no	—
512	C₂²⋊C₄	—	4	4	no	—
513	C₂²⋊C₄	—	4	4	no	—
514	C₂²⋊C₄	—	4	4	no	—
515	C₂²⋊C₄	C₂²×C₄×C₆₇	1	8	yes	—
516	C₂²⋊C₄	C₂²×C₄×C₆₇	1	8	yes	—
517	C₂²⋊C₄	C₂²×C₄×C₆₇	1	8	yes	—
518	C₂²⋊C₄	C₂²×C₄×C₆₇	1	8	yes	—
519	C₂²⋊C₄	C₂²×C₄×C₆₇	1	8	yes	—
520	C₄²	—	4	2	no	—
521	C₄²	—	4	2	no	—
522	C₄²	—	4	4	no	—
523	C₄²	—	4	4	no	—
524	C₄²	—	4	4	no	—
525	C₄²	C₂²×C₄×C₆₇	1	4	yes	—
526	C₄²	C₂²×C₄×C₆₇	1	4	yes	—
527	C₄²	C₂²×C₄×C₆₇	1	4	yes	—
528	C₄²	C₂²×C₄×C₆₇	1	4	yes	—
529	C₄²	C₂²×C₄×C₆₇	1	8	yes	—
530	C₂³	—	2	2	no	—
531	C₂³	—	2	2	no	—
532	C₂³	—	2	2	no	—
533	C₂³	—	2	2	no	—
534	C₂³	—	2	4	no	—
535	C₂³	—	2	4	no	—
536	C₂³	C₂³×C₄×C₆₇	1	4	yes	—
537	C₂³	C₂³×C₄×C₆₇	1	2	yes	—
538	C₂³	H(5,𝔽₂)×C₆₇	1	2	yes	—
539	C₂³	2₋¹⁺⁴×C₆₇	1	2	yes	—
540	C₂³	C₂⁵×C₆₇	1	1	yes	characteristic
541	Q₈	—	4	4	no	—
542	D₄	—	2	16	no	—
543	D₄	—	4	4	no	—
544	C₂×C₄	—	2	8	no	—
545	C₂×C₄	—	2	1	no	—
546	C₂×C₄	—	2	2	no	—
547	C₂×C₄	—	2	1	no	—
548	C₂×C₄	—	2	1	no	—
549	C₂×C₄	—	2	1	no	—
550	C₂×C₄	—	2	2	no	—
551	C₂×C₄	—	2	4	no	—
552	C₂×C₄	—	2	4	no	—
553	C₂×C₄	—	2	2	no	—
554	C₂×C₄	—	2	2	no	—
555	C₂×C₄	—	2	2	no	—
556	C₂×C₄	—	2	2	no	—
557	C₂×C₄	—	2	4	no	—
558	C₂×C₄	—	2	4	no	—
559	C₂×C₄	—	2	4	no	—
560	C₂×C₄	—	2	4	no	—
561	C₂×C₄	—	2	4	no	—
562	C₂×C₄	—	2	4	no	—
563	C₂×C₄	—	2	4	no	—
564	C₂×C₄	—	2	4	no	—
565	C₂×C₄	—	2	4	no	—
566	C₂×C₄	—	2	4	no	—
567	C₂×C₄	—	2	4	no	—
568	C₂×C₄	—	2	4	no	—
569	C₂×C₄	—	4	4	no	—
570	C₂×C₄	—	4	8	no	—
571	C₂×C₄	—	4	4	no	—
572	C₂×C₄	—	8	4	no	—
573	C₂×C₄	—	8	4	no	—
574	C₂×C₄	—	8	8	no	—
575	C₂×C₄	C₂³×C₄×C₆₇	1	2	yes	—
576	C₂×C₄	C₂³×C₄×C₆₇	1	2	yes	—
577	C₂×C₄	C₂³×C₄×C₆₇	1	2	yes	—
578	C₂×C₄	C₂³×C₄×C₆₇	1	2	yes	—
579	C₂×C₄	C₂³×C₄×C₆₇	1	2	yes	—
580	C₂×C₄	C₂³×C₄×C₆₇	1	2	yes	—
581	C₂×C₄	C₂³×C₄×C₆₇	1	4	yes	—
582	C₂×C₄	C₂³×C₄×C₆₇	1	4	yes	—
583	C₂×C₄	C₂³×C₄×C₆₇	1	4	yes	—
584	C₂²	—	2	4	no	—
585	C₂²	—	2	4	no	—
586	C₂²	—	2	2	no	—
587	C₂²	—	2	2	no	—
588	C₂²	—	2	4	no	—
589	C₂²	—	2	4	no	—
590	C₂²	—	2	4	no	—
591	C₂²	—	4	4	no	—
592	C₂²	—	4	4	no	—
593	C₂²	#520×C₆₇	1	4	yes	—
594	C₂²	C₂⁴×C₄×C₆₇	1	1	yes	characteristic
595	C₂²	C₂×H(5,𝔽₂)×C₆₇	1	1	yes	characteristic
596	C₂²	C₂×2₋¹⁺⁴×C₆₇	1	1	yes	characteristic
597	C₂²	#585×C₆₇	1	2	yes	—
598	C₂²	#585×C₆₇	1	2	yes	—
599	C₄	—	2	2	no	—
600	C₄	—	2	2	no	—
601	C₄	—	2	4	no	—
602	C₄	—	2	2	no	—
603	C₄	—	2	2	no	—
604	C₄	—	2	4	no	—
605	C₄	—	2	2	no	—
606	C₄	—	2	2	no	—
607	C₄	—	2	4	no	—
608	C₄	—	4	4	no	—
609	C₄	—	4	4	no	—
610	C₄	—	4	4	no	—
611	C₄	—	4	4	no	—
612	C₂	—	2	4	no	—
613	C₂	—	4	4	no	—
614	C₂	C₂×#520×C₆₇	1	1	yes	characteristic
615	C₂	C₆₇×#3428	1	1	yes	characteristic
616	C₂	C₆₇×#3428	1	1	yes	characteristic
617	C₂	C₆₇×#3560	1	2	yes	—
618	C₂	C₆₇×#3564	1	2	yes	—

	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×#2383	C₂	semidirect
3. C₆₇×#2359	C₂	semidirect
4. C₆₇×#2353	C₂	semidirect
5. C₆₇×#2349	C₂	semidirect
6. C₆₇×#2347	C₂	semidirect
7. C₆₇×#2343	C₂	semidirect
8. C₆₇×#2337	C₂	semidirect
9. C₆₇×#2336	C₂	semidirect
10. C₆₇×#2329	C₂	semidirect
12. C₆₇×#2320	C₂	semidirect
13. C₆₇×#2318	C₂	semidirect
14. C₆₇×#2285	C₂	semidirect
15. C₂×#383×C₆₇	C₂	semidirect
16. #529×C₆₇	C₄	semidirect
86. #126×C₆₇	C₂×C₄	semidirect
87. #125×C₆₇	C₂×C₄	semidirect
88. #124×C₆₇	C₂×C₄	semidirect
89. #123×C₆₇	C₂×C₄	semidirect
90. C₂²⋊Q₈×C₆₇	C₂×C₄	semidirect
92. C₄×Q₈×C₆₇	C₂×C₄	semidirect
94. C₄×D₄×C₆₇	C₂×C₄	semidirect
95. C₄×D₄×C₆₇	C₂×C₄	semidirect
97. #117×C₆₇	C₂×C₄	semidirect
120. C₂×C₄²×C₆₇	C₂×C₄	semidirect
302. #34401	C₆₇	direct (N × H)
311. #2383	C₂×C₆₇	semidirect
312. #2359	C₂×C₆₇	semidirect
313. #2353	C₂×C₆₇	semidirect
314. #2349	C₂×C₆₇	semidirect
315. #2347	C₂×C₆₇	semidirect
316. #2343	C₂×C₆₇	semidirect
317. #2337	C₂×C₆₇	semidirect
318. #2336	C₂×C₆₇	semidirect
319. #2329	C₂×C₆₇	semidirect
321. #2320	C₂×C₆₇	semidirect
322. #2318	C₂×C₆₇	semidirect
323. #2285	C₂×C₆₇	semidirect
324. C₂×#383	C₂×C₆₇	semidirect
325. C₆₇	#34401	direct (N × H)
326. #529	C₄×C₆₇	semidirect
396. #126	C₂×C₄×C₆₇	semidirect
397. #125	C₂×C₄×C₆₇	semidirect
398. #124	C₂×C₄×C₆₇	semidirect
399. #123	C₂×C₄×C₆₇	semidirect
400. C₂²⋊Q₈	C₂×C₄×C₆₇	semidirect
402. C₄×Q₈	C₂×C₄×C₆₇	semidirect
404. C₄×D₄	C₂×C₄×C₆₇	semidirect
405. C₄×D₄	C₂×C₄×C₆₇	semidirect
407. #117	C₂×C₄×C₆₇	semidirect
430. 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×#2383	2
C₆₇×#2359	2
C₆₇×#2353	2
C₆₇×#2349	2
C₆₇×#2347	2
C₆₇×#2343	2
C₆₇×#2337	2
C₆₇×#2336	2
C₆₇×#2329	2
C₆₇×#2325	2
C₆₇×#2320	2
C₆₇×#2318	2
C₆₇×#2285	2
C₂×#383×C₆₇	2
#34401	67

Subgroups — C67×#34401 #30713652

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₆₇×#34401 #30713652