Subgroups — C₆₇×#59321 #30738572

Order 17152.

1758 conjugacy classes of subgroups.

The subgroup lattice has 672 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₆₇×#2791	C₂	1	4	yes	maximal, complemented
2	C₆₇×#2582	C₂	1	1	yes	maximal, characteristic, complemented
3	C₆₇×#2565	C₂	1	2	yes	maximal, complemented
4	C₆₇×#2552	C₂	1	2	yes	maximal, complemented
5	C₆₇×#2552	C₂	1	1	yes	maximal, characteristic, complemented
6	C₆₇×#2511	C₂	1	2	yes	maximal, complemented
7	C₆₇×#2506	C₂	1	2	yes	maximal, complemented
8	C₆₇×#2502	C₂	1	2	yes	maximal
9	C₆₇×#2473	C₂	1	4	yes	maximal
10	C₆₇×#2442	C₂	1	2	yes	maximal
11	C₆₇×#2360	C₂	1	2	yes	maximal
12	C₆₇×#2355	C₂	1	1	yes	maximal, characteristic, complemented
13	Q₈×C₄⋊C₄×C₆₇	C₂	1	1	yes	maximal, characteristic, complemented
14	C₆₇×#2345	C₂	1	2	yes	maximal
15	C₄×C₂²⋊Q₈×C₆₇	C₂	1	1	yes	maximal, characteristic
16	C₄×C₂²⋊Q₈×C₆₇	C₂	1	1	yes	maximal, characteristic
17	C₂×#384×C₆₇	C₂	1	1	yes	maximal, characteristic
18	C₂×#125×C₆₇	C₂²	1	4	yes	—
19	C₂×#125×C₆₇	C₂²	1	2	yes	—
20	C₂×C₂²⋊Q₈×C₆₇	C₂²	1	1	yes	characteristic
21	C₂×C₂²⋊Q₈×C₆₇	C₂²	1	1	yes	characteristic
22	C₂×C₄×Q₈×C₆₇	C₂²	1	1	yes	characteristic
23	C₂×C₄×Q₈×C₆₇	C₂²	1	1	yes	characteristic
24	C₂×C₄×Q₈×C₆₇	C₂²	1	2	yes	—
25	C₂²×C₄⋊C₄×C₆₇	C₂²	1	2	yes	—
26	C₂²×C₄⋊C₄×C₆₇	C₂²	1	2	yes	—
27	C₂²×C₄⋊C₄×C₆₇	C₂²	1	2	yes	—
28	C₂²×C₄²×C₆₇	C₂²	1	1	yes	characteristic
29	#400×C₆₇	C₂²	1	4	yes	—
30	#400×C₆₇	C₂²	1	4	yes	—
31	#398×C₆₇	C₂²	1	4	yes	—
32	#398×C₆₇	C₂²	1	4	yes	—
33	#398×C₆₇	C₂²	1	4	yes	—
34	#398×C₆₇	C₂²	1	4	yes	—
35	#398×C₆₇	C₂²	1	4	yes	—
36	#398×C₆₇	C₂²	1	4	yes	—
37	#391×C₆₇	C₂²	1	2	yes	—
38	#391×C₆₇	C₂²	1	2	yes	—
39	#391×C₆₇	C₂²	1	1	yes	characteristic
40	#391×C₆₇	C₂²	1	1	yes	characteristic
41	#391×C₆₇	C₂²	1	2	yes	—
42	#389×C₆₇	C₂²	1	1	yes	characteristic
43	#389×C₆₇	C₂²	1	2	yes	—
44	#389×C₆₇	C₂²	1	2	yes	—
45	#389×C₆₇	C₂²	1	2	yes	—
46	#389×C₆₇	C₂²	1	2	yes	—
47	#389×C₆₇	C₂²	1	4	yes	—
48	#389×C₆₇	C₂²	1	4	yes	—
49	#389×C₆₇	C₂²	1	1	yes	characteristic
50	#389×C₆₇	C₂²	1	2	yes	—
51	#387×C₆₇	C₂²	1	4	yes	—
52	#387×C₆₇	C₂²	1	4	yes	—
53	#387×C₆₇	C₂²	1	4	yes	—
54	#387×C₆₇	C₂²	1	2	yes	—
55	#387×C₆₇	C₂²	1	2	yes	—
56	#387×C₆₇	C₂²	1	4	yes	—
57	#387×C₆₇	C₂²	1	4	yes	—
58	#385×C₆₇	C₂²	1	4	yes	—
59	#385×C₆₇	C₂²	1	4	yes	—
60	#385×C₆₇	C₂²	1	2	yes	—
61	#385×C₆₇	C₂²	1	2	yes	—
62	#384×C₆₇	C₂²	1	2	yes	—
63	#384×C₆₇	C₂²	1	2	yes	—
64	#384×C₆₇	C₂²	1	2	yes	—
65	#384×C₆₇	C₂²	1	2	yes	—
66	#384×C₆₇	C₂²	1	2	yes	—
67	#384×C₆₇	C₂²	1	1	yes	characteristic
68	#384×C₆₇	C₂²	1	1	yes	characteristic
69	#382×C₆₇	C₂²	1	2	yes	—
70	#380×C₆₇	C₂²	1	1	yes	characteristic
71	#380×C₆₇	C₂²	1	1	yes	characteristic
72	#380×C₆₇	C₂²	1	4	yes	—
73	C₄×C₄⋊C₄×C₆₇	C₂²	1	2	yes	—
74	C₄×C₄⋊C₄×C₆₇	C₂²	1	2	yes	—
75	C₄×C₄⋊C₄×C₆₇	C₂²	1	2	yes	—
76	C₄×C₄⋊C₄×C₆₇	C₂²	1	2	yes	—
77	C₄×C₄⋊C₄×C₆₇	C₂²	1	1	yes	characteristic
78	C₄×C₄⋊C₄×C₆₇	C₂²	1	1	yes	characteristic
79	C₄×C₄⋊C₄×C₆₇	C₂²	1	2	yes	—
80	C₄×C₂²⋊C₄×C₆₇	C₂²	1	2	yes	—
81	C₄×C₂²⋊C₄×C₆₇	C₂²	1	2	yes	—
82	C₄×C₂²⋊C₄×C₆₇	C₂²	1	4	yes	—
83	C₂²×Q₈×C₆₇	C₂³	1	1	yes	characteristic
84	C₂²×Q₈×C₆₇	C₂³	1	1	yes	characteristic
85	C₂³×C₄×C₆₇	C₂³	1	1	yes	characteristic
86	C₂³×C₄×C₆₇	C₂³	1	1	yes	characteristic
87	C₂³×C₄×C₆₇	C₂³	1	1	yes	characteristic
88	C₂³×C₄×C₆₇	C₂³	1	4	yes	—
89	#125×C₆₇	—	4	4	no	—
90	#125×C₆₇	—	4	8	no	—
91	C₂²⋊Q₈×C₆₇	—	2	4	no	—
92	C₂²⋊Q₈×C₆₇	Q₈	1	8	yes	complemented
93	C₄×Q₈×C₆₇	—	2	4	no	—
94	C₄×Q₈×C₆₇	—	2	4	no	—
95	C₄×Q₈×C₆₇	—	4	4	no	—
96	C₂×C₄⋊C₄×C₆₇	—	2	8	no	—
97	C₂×C₄⋊C₄×C₆₇	—	4	4	no	—
98	C₂×C₄⋊C₄×C₆₇	—	4	4	no	—
99	C₂×C₄⋊C₄×C₆₇	C₂³	1	2	yes	—
100	C₂×C₄⋊C₄×C₆₇	C₂³	1	4	yes	—
101	C₂×C₄⋊C₄×C₆₇	C₂³	1	2	yes	—
102	C₂×C₄⋊C₄×C₆₇	C₂³	1	2	yes	—
103	C₂×C₄⋊C₄×C₆₇	C₂³	1	2	yes	—
104	C₂×C₄⋊C₄×C₆₇	C₂³	1	4	yes	—
105	C₂×C₄⋊C₄×C₆₇	C₂³	1	1	yes	characteristic
106	C₂×C₄⋊C₄×C₆₇	C₂³	1	1	yes	characteristic
107	C₂×C₄⋊C₄×C₆₇	C₂³	1	2	yes	—
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×C₄⋊C₄×C₆₇	C₂³	1	1	yes	characteristic
112	C₂×C₄⋊C₄×C₆₇	C₂³	1	2	yes	—
113	C₂×C₄⋊C₄×C₆₇	C₂³	1	2	yes	—
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	4	yes	—
119	C₂×C₄⋊C₄×C₆₇	C₂³	1	4	yes	—
120	C₂×C₄⋊C₄×C₆₇	C₂³	1	4	yes	—
121	C₂×C₄⋊C₄×C₆₇	C₂³	1	4	yes	—
122	C₂×C₄⋊C₄×C₆₇	C₂³	1	4	yes	—
123	C₂×C₄⋊C₄×C₆₇	C₂³	1	4	yes	—
124	C₂×C₄⋊C₄×C₆₇	C₂³	1	1	yes	characteristic
125	C₂×C₄⋊C₄×C₆₇	C₂³	1	2	yes	—
126	C₂×C₂²⋊C₄×C₆₇	C₂³	1	2	yes	—
127	C₂×C₂²⋊C₄×C₆₇	C₂³	1	2	yes	—
128	C₂×C₂²⋊C₄×C₆₇	C₂³	1	2	yes	—
129	C₂×C₂²⋊C₄×C₆₇	C₂³	1	2	yes	—
130	C₂×C₄²×C₆₇	—	2	4	no	—
131	C₂×C₄²×C₆₇	C₂³	1	1	yes	characteristic
132	C₂×C₄²×C₆₇	C₂³	1	1	yes	characteristic
133	C₂×C₄²×C₆₇	C₂³	1	1	yes	characteristic
134	C₂×C₄²×C₆₇	C₂³	1	2	yes	—
135	C₂×C₄²×C₆₇	C₂³	1	2	yes	—
136	C₂×C₄²×C₆₇	C₂³	1	2	yes	—
137	C₂×C₄²×C₆₇	C₂³	1	2	yes	—
138	C₂×C₄²×C₆₇	C₂³	1	4	yes	—
139	C₂×C₄²×C₆₇	C₂³	1	4	yes	—
140	C₂×C₄²×C₆₇	C₂³	1	1	yes	characteristic
141	#95×C₆₇	C₂³	1	2	yes	—
142	#95×C₆₇	C₂³	1	2	yes	—
143	#95×C₆₇	C₂³	1	2	yes	—
144	#95×C₆₇	C₂³	1	4	yes	—
145	#95×C₆₇	C₂³	1	2	yes	—
146	#95×C₆₇	C₂³	1	4	yes	—
147	#95×C₆₇	C₂³	1	4	yes	—
148	#95×C₆₇	C₂³	1	2	yes	—
149	#95×C₆₇	C₂³	1	4	yes	—
150	#95×C₆₇	C₂³	1	2	yes	—
151	#95×C₆₇	C₂³	1	2	yes	—
152	#95×C₆₇	C₂³	1	2	yes	—
153	#95×C₆₇	C₂³	1	4	yes	—
154	#95×C₆₇	C₂³	1	2	yes	—
155	#95×C₆₇	C₂³	1	2	yes	—
156	#95×C₆₇	C₂³	1	4	yes	—
157	#95×C₆₇	C₂³	1	4	yes	—
158	#95×C₆₇	C₂³	1	4	yes	—
159	C₂⁴×C₆₇	C₂⁴	1	1	yes	characteristic
160	C₂×Q₈×C₆₇	—	2	2	no	—
161	C₂×Q₈×C₆₇	—	2	2	no	—
162	C₂×Q₈×C₆₇	—	2	2	no	—
163	C₂×Q₈×C₆₇	—	2	2	no	—
164	C₂×Q₈×C₆₇	—	2	2	no	—
165	C₂×Q₈×C₆₇	C₂×Q₈	1	4	yes	—
166	C₂²×C₄×C₆₇	—	2	2	no	—
167	C₂²×C₄×C₆₇	—	2	2	no	—
168	C₂²×C₄×C₆₇	—	2	2	no	—
169	C₂²×C₄×C₆₇	—	2	2	no	—
170	C₂²×C₄×C₆₇	—	2	2	no	—
171	C₂²×C₄×C₆₇	—	2	2	no	—
172	C₂²×C₄×C₆₇	—	2	2	no	—
173	C₂²×C₄×C₆₇	—	2	8	no	—
174	C₂²×C₄×C₆₇	—	4	4	no	—
175	C₂²×C₄×C₆₇	—	4	4	no	—
176	C₂²×C₄×C₆₇	C₂×Q₈	1	4	yes	—
177	C₂²×C₄×C₆₇	#41	1	4	yes	—
178	C₂²×C₄×C₆₇	C₂⁴	1	2	yes	—
179	C₂²×C₄×C₆₇	C₂⁴	1	2	yes	—
180	C₂²×C₄×C₆₇	C₂⁴	1	4	yes	—
181	C₂²×C₄×C₆₇	C₂⁴	1	2	yes	—
182	C₂²×C₄×C₆₇	C₂⁴	1	2	yes	—
183	C₂²×C₄×C₆₇	C₂⁴	1	2	yes	—
184	C₂²×C₄×C₆₇	C₂⁴	1	2	yes	—
185	C₂²×C₄×C₆₇	C₂⁴	1	4	yes	—
186	C₂²×C₄×C₆₇	C₂⁴	1	2	yes	—
187	C₂²×C₄×C₆₇	C₂⁴	1	2	yes	—
188	C₂²×C₄×C₆₇	C₂⁴	1	1	yes	characteristic
189	C₂²×C₄×C₆₇	C₂⁴	1	1	yes	characteristic
190	C₂²×C₄×C₆₇	C₂⁴	1	1	yes	characteristic
191	C₂²×C₄×C₆₇	C₂⁴	1	1	yes	characteristic
192	C₂²×C₄×C₆₇	C₂⁴	1	1	yes	characteristic
193	C₂²×C₄×C₆₇	C₂⁴	1	1	yes	characteristic
194	C₄⋊C₄×C₆₇	—	2	8	no	—
195	C₄⋊C₄×C₆₇	—	2	4	no	—
196	C₄⋊C₄×C₆₇	—	2	4	no	—
197	C₄⋊C₄×C₆₇	—	2	4	no	—
198	C₄⋊C₄×C₆₇	—	2	2	no	—
199	C₄⋊C₄×C₆₇	—	2	2	no	—
200	C₄⋊C₄×C₆₇	—	2	4	no	—
201	C₄⋊C₄×C₆₇	—	2	2	no	—
202	C₄⋊C₄×C₆₇	—	2	4	no	—
203	C₄⋊C₄×C₆₇	—	2	4	no	—
204	C₄⋊C₄×C₆₇	—	2	8	no	—
205	C₄⋊C₄×C₆₇	—	2	4	no	—
206	C₄⋊C₄×C₆₇	—	2	4	no	—
207	C₄⋊C₄×C₆₇	—	2	4	no	—
208	C₄⋊C₄×C₆₇	—	2	2	no	—
209	C₄⋊C₄×C₆₇	—	2	2	no	—
210	C₄⋊C₄×C₆₇	—	4	2	no	—
211	C₄⋊C₄×C₆₇	—	4	4	no	—
212	C₄⋊C₄×C₆₇	—	4	2	no	—
213	C₄⋊C₄×C₆₇	—	4	4	no	—
214	C₄⋊C₄×C₆₇	—	4	4	no	—
215	C₄⋊C₄×C₆₇	—	4	4	no	—
216	C₄⋊C₄×C₆₇	—	4	4	no	—
217	C₄⋊C₄×C₆₇	—	4	4	no	—
218	C₄⋊C₄×C₆₇	—	4	4	no	—
219	C₄⋊C₄×C₆₇	C₂×Q₈	1	4	yes	—
220	C₄⋊C₄×C₆₇	C₂×Q₈	1	8	yes	—
221	C₂²⋊C₄×C₆₇	—	2	4	no	—
222	C₂²⋊C₄×C₆₇	—	2	4	no	—
223	C₂²⋊C₄×C₆₇	—	2	4	no	—
224	C₂²⋊C₄×C₆₇	C₂×Q₈	1	8	yes	—
225	C₄²×C₆₇	—	2	2	no	—
226	C₄²×C₆₇	—	2	2	no	—
227	C₄²×C₆₇	—	2	2	no	—
228	C₄²×C₆₇	—	2	4	no	—
229	C₄²×C₆₇	—	2	4	no	—
230	C₄²×C₆₇	—	2	4	no	—
231	C₄²×C₆₇	—	2	4	no	—
232	C₄²×C₆₇	—	4	4	no	—
233	C₄²×C₆₇	—	4	4	no	—
234	C₄²×C₆₇	#41	1	4	yes	—
235	C₂³×C₆₇	—	2	1	no	—
236	C₂³×C₆₇	—	2	1	no	—
237	C₂³×C₆₇	—	2	1	no	—
238	C₂³×C₆₇	—	2	1	no	—
239	C₂³×C₆₇	C₂²×Q₈	1	2	yes	—
240	C₂³×C₆₇	C₂×#41	1	2	yes	—
241	C₂³×C₆₇	2₋¹⁺⁴	1	2	yes	—
242	C₂³×C₆₇	C₂⁵	1	1	yes	characteristic
243	Q₈×C₆₇	—	2	8	no	—
244	Q₈×C₆₇	—	4	4	no	—
245	C₂×C₄×C₆₇	—	2	8	no	—
246	C₂×C₄×C₆₇	—	2	1	no	—
247	C₂×C₄×C₆₇	—	2	1	no	—
248	C₂×C₄×C₆₇	—	2	1	no	—
249	C₂×C₄×C₆₇	—	2	2	no	—
250	C₂×C₄×C₆₇	—	2	2	no	—
251	C₂×C₄×C₆₇	—	2	2	no	—
252	C₂×C₄×C₆₇	—	2	1	no	—
253	C₂×C₄×C₆₇	—	2	1	no	—
254	C₂×C₄×C₆₇	—	2	2	no	—
255	C₂×C₄×C₆₇	—	2	1	no	—
256	C₂×C₄×C₆₇	—	2	1	no	—
257	C₂×C₄×C₆₇	—	2	1	no	—
258	C₂×C₄×C₆₇	—	2	8	no	—
259	C₂×C₄×C₆₇	—	2	8	no	—
260	C₂×C₄×C₆₇	—	2	2	no	—
261	C₂×C₄×C₆₇	—	2	2	no	—
262	C₂×C₄×C₆₇	—	2	2	no	—
263	C₂×C₄×C₆₇	—	2	2	no	—
264	C₂×C₄×C₆₇	—	2	2	no	—
265	C₂×C₄×C₆₇	—	2	4	no	—
266	C₂×C₄×C₆₇	—	2	4	no	—
267	C₂×C₄×C₆₇	—	2	2	no	—
268	C₂×C₄×C₆₇	—	2	2	no	—
269	C₂×C₄×C₆₇	—	2	2	no	—
270	C₂×C₄×C₆₇	—	2	4	no	—
271	C₂×C₄×C₆₇	—	2	4	no	—
272	C₂×C₄×C₆₇	—	2	2	no	—
273	C₂×C₄×C₆₇	—	2	2	no	—
274	C₂×C₄×C₆₇	—	2	2	no	—
275	C₂×C₄×C₆₇	—	2	4	no	—
276	C₂×C₄×C₆₇	—	2	4	no	—
277	C₂×C₄×C₆₇	—	2	2	no	—
278	C₂×C₄×C₆₇	—	2	2	no	—
279	C₂×C₄×C₆₇	—	2	2	no	—
280	C₂×C₄×C₆₇	—	2	2	no	—
281	C₂×C₄×C₆₇	—	2	2	no	—
282	C₂×C₄×C₆₇	—	2	2	no	—
283	C₂×C₄×C₆₇	—	2	2	no	—
284	C₂×C₄×C₆₇	—	8	8	no	—
285	C₂×C₄×C₆₇	C₂²×Q₈	1	4	yes	—
286	C₂×C₄×C₆₇	C₂²×Q₈	1	4	yes	—
287	C₂×C₄×C₆₇	C₂²×Q₈	1	2	yes	—
288	C₂×C₄×C₆₇	C₂²×Q₈	1	2	yes	—
289	C₂×C₄×C₆₇	C₂×#41	1	2	yes	—
290	C₂×C₄×C₆₇	C₂×#41	1	2	yes	—
291	C₂×C₄×C₆₇	C₂×#41	1	2	yes	—
292	C₂×C₄×C₆₇	C₂×#41	1	2	yes	—
293	C₂×C₄×C₆₇	C₂×#41	1	2	yes	—
294	C₂×C₄×C₆₇	H(5,𝔽₂)	1	2	yes	—
295	C₂×C₄×C₆₇	H(5,𝔽₂)	1	2	yes	—
296	C₂×C₄×C₆₇	2₋¹⁺⁴	1	2	yes	—
297	C₂²×C₆₇	—	2	2	no	—
298	C₂²×C₆₇	—	2	2	no	—
299	C₂²×C₆₇	—	2	2	no	—
300	C₂²×C₆₇	—	2	2	no	—
301	C₂²×C₆₇	—	2	2	no	—
302	C₂²×C₆₇	—	2	2	no	—
303	C₂²×C₆₇	#557	1	4	yes	—
304	C₂²×C₆₇	C₂³×Q₈	1	1	yes	characteristic
305	C₂²×C₆₇	C₂²×#41	1	1	yes	characteristic
306	C₂²×C₆₇	C₂²×#41	1	1	yes	characteristic
307	C₂²×C₆₇	C₂×H(5,𝔽₂)	1	1	yes	characteristic
308	C₂²×C₆₇	C₂×H(5,𝔽₂)	1	1	yes	characteristic
309	C₂²×C₆₇	C₂×2₋¹⁺⁴	1	1	yes	characteristic
310	C₂²×C₆₇	C₂×2₋¹⁺⁴	1	1	yes	characteristic
311	C₄×C₆₇	—	2	2	no	—
312	C₄×C₆₇	—	2	2	no	—
313	C₄×C₆₇	—	2	4	no	—
314	C₄×C₆₇	—	2	2	no	—
315	C₄×C₆₇	—	2	2	no	—
316	C₄×C₆₇	—	2	4	no	—
317	C₄×C₆₇	—	4	2	no	—
318	C₄×C₆₇	—	4	2	no	—
319	C₄×C₆₇	—	4	2	no	—
320	C₄×C₆₇	—	4	2	no	—
321	C₄×C₆₇	—	4	4	no	—
322	C₄×C₆₇	—	4	4	no	—
323	C₄×C₆₇	—	4	2	no	—
324	C₄×C₆₇	—	4	2	no	—
325	C₄×C₆₇	#552	1	4	yes	—
326	C₄×C₆₇	#554	1	4	yes	—
327	#59321	C₆₇	1	1	yes	maximal, characteristic, direct product factor, Sylow
328	C₂×C₆₇	—	2	4	no	—
329	C₂×C₆₇	C₂×#552	1	1	yes	characteristic
330	C₂×C₆₇	C₂×#554	1	1	yes	characteristic
331	C₂×C₆₇	C₂×#557	1	1	yes	characteristic
332	C₂×C₆₇	#3492	1	1	yes	characteristic
333	C₂×C₆₇	#3503	1	1	yes	characteristic
334	C₂×C₆₇	#3508	1	1	yes	characteristic
335	C₂×C₆₇	#3552	1	1	yes	characteristic
336	#2791	C₂×C₆₇	1	4	yes	complemented
337	#2582	C₂×C₆₇	1	1	yes	characteristic, complemented
338	#2565	C₂×C₆₇	1	2	yes	complemented
339	#2552	C₂×C₆₇	1	1	yes	characteristic, complemented
340	#2552	C₂×C₆₇	1	2	yes	complemented
341	#2511	C₂×C₆₇	1	2	yes	complemented
342	#2506	C₂×C₆₇	1	2	yes	complemented
343	#2502	C₂×C₆₇	1	2	yes	—
344	#2473	C₂×C₆₇	1	4	yes	—
345	#2442	C₂×C₆₇	1	2	yes	—
346	#2360	C₂×C₆₇	1	2	yes	—
347	#2355	C₂×C₆₇	1	1	yes	characteristic, complemented
348	Q₈×C₄⋊C₄	C₂×C₆₇	1	1	yes	characteristic, complemented
349	#2345	C₂×C₆₇	1	2	yes	—
350	C₄×C₂²⋊Q₈	C₂×C₆₇	1	1	yes	characteristic
351	C₄×C₂²⋊Q₈	C₂×C₆₇	1	1	yes	characteristic
352	C₂×#384	C₂×C₆₇	1	1	yes	characteristic
353	C₆₇	#59321	1	1	yes	characteristic, direct product factor, Sylow
354	C₂×#125	C₂²×C₆₇	1	2	yes	—
355	C₂×#125	C₂²×C₆₇	1	4	yes	—
356	C₂×C₂²⋊Q₈	C₂²×C₆₇	1	1	yes	characteristic
357	C₂×C₂²⋊Q₈	C₂²×C₆₇	1	1	yes	characteristic
358	C₂×C₄×Q₈	C₂²×C₆₇	1	1	yes	characteristic
359	C₂×C₄×Q₈	C₂²×C₆₇	1	1	yes	characteristic
360	C₂×C₄×Q₈	C₂²×C₆₇	1	2	yes	—
361	C₂²×C₄⋊C₄	C₂²×C₆₇	1	2	yes	—
362	C₂²×C₄⋊C₄	C₂²×C₆₇	1	2	yes	—
363	C₂²×C₄⋊C₄	C₂²×C₆₇	1	2	yes	—
364	C₂²×C₄²	C₂²×C₆₇	1	1	yes	characteristic
365	#400	C₂²×C₆₇	1	4	yes	—
366	#400	C₂²×C₆₇	1	4	yes	—
367	#398	C₂²×C₆₇	1	4	yes	—
368	#398	C₂²×C₆₇	1	4	yes	—
369	#398	C₂²×C₆₇	1	4	yes	—
370	#398	C₂²×C₆₇	1	4	yes	—
371	#398	C₂²×C₆₇	1	4	yes	—
372	#398	C₂²×C₆₇	1	4	yes	—
373	#391	C₂²×C₆₇	1	1	yes	characteristic
374	#391	C₂²×C₆₇	1	2	yes	—
375	#391	C₂²×C₆₇	1	2	yes	—
376	#391	C₂²×C₆₇	1	2	yes	—
377	#391	C₂²×C₆₇	1	1	yes	characteristic
378	#389	C₂²×C₆₇	1	2	yes	—
379	#389	C₂²×C₆₇	1	2	yes	—
380	#389	C₂²×C₆₇	1	2	yes	—
381	#389	C₂²×C₆₇	1	2	yes	—
382	#389	C₂²×C₆₇	1	4	yes	—
383	#389	C₂²×C₆₇	1	2	yes	—
384	#389	C₂²×C₆₇	1	4	yes	—
385	#389	C₂²×C₆₇	1	1	yes	characteristic
386	#389	C₂²×C₆₇	1	1	yes	characteristic
387	#387	C₂²×C₆₇	1	4	yes	—
388	#387	C₂²×C₆₇	1	4	yes	—
389	#387	C₂²×C₆₇	1	4	yes	—
390	#387	C₂²×C₆₇	1	4	yes	—
391	#387	C₂²×C₆₇	1	4	yes	—
392	#387	C₂²×C₆₇	1	2	yes	—
393	#387	C₂²×C₆₇	1	2	yes	—
394	#385	C₂²×C₆₇	1	4	yes	—
395	#385	C₂²×C₆₇	1	4	yes	—
396	#385	C₂²×C₆₇	1	2	yes	—
397	#385	C₂²×C₆₇	1	2	yes	—
398	#384	C₂²×C₆₇	1	2	yes	—
399	#384	C₂²×C₆₇	1	1	yes	characteristic
400	#384	C₂²×C₆₇	1	1	yes	characteristic
401	#384	C₂²×C₆₇	1	2	yes	—
402	#384	C₂²×C₆₇	1	2	yes	—
403	#384	C₂²×C₆₇	1	2	yes	—
404	#384	C₂²×C₆₇	1	2	yes	—
405	#382	C₂²×C₆₇	1	2	yes	—
406	#380	C₂²×C₆₇	1	1	yes	characteristic
407	#380	C₂²×C₆₇	1	4	yes	—
408	#380	C₂²×C₆₇	1	1	yes	characteristic
409	C₄×C₄⋊C₄	C₂²×C₆₇	1	2	yes	—
410	C₄×C₄⋊C₄	C₂²×C₆₇	1	2	yes	—
411	C₄×C₄⋊C₄	C₂²×C₆₇	1	2	yes	—
412	C₄×C₄⋊C₄	C₂²×C₆₇	1	1	yes	characteristic
413	C₄×C₄⋊C₄	C₂²×C₆₇	1	2	yes	—
414	C₄×C₄⋊C₄	C₂²×C₆₇	1	1	yes	characteristic
415	C₄×C₄⋊C₄	C₂²×C₆₇	1	2	yes	—
416	C₄×C₂²⋊C₄	C₂²×C₆₇	1	4	yes	—
417	C₄×C₂²⋊C₄	C₂²×C₆₇	1	2	yes	—
418	C₄×C₂²⋊C₄	C₂²×C₆₇	1	2	yes	—
419	C₂²×Q₈	C₂³×C₆₇	1	1	yes	characteristic
420	C₂²×Q₈	C₂³×C₆₇	1	1	yes	characteristic
421	C₂³×C₄	C₂³×C₆₇	1	1	yes	characteristic
422	C₂³×C₄	C₂³×C₆₇	1	1	yes	characteristic
423	C₂³×C₄	C₂³×C₆₇	1	1	yes	characteristic
424	C₂³×C₄	C₂³×C₆₇	1	4	yes	—
425	#125	—	4	4	no	—
426	#125	—	4	8	no	—
427	C₂²⋊Q₈	—	2	4	no	—
428	C₂²⋊Q₈	Q₈×C₆₇	1	8	yes	complemented
429	C₄×Q₈	—	2	4	no	—
430	C₄×Q₈	—	2	4	no	—
431	C₄×Q₈	—	4	4	no	—
432	C₂×C₄⋊C₄	—	2	8	no	—
433	C₂×C₄⋊C₄	—	4	4	no	—
434	C₂×C₄⋊C₄	—	4	4	no	—
435	C₂×C₄⋊C₄	C₂³×C₆₇	1	2	yes	—
436	C₂×C₄⋊C₄	C₂³×C₆₇	1	4	yes	—
437	C₂×C₄⋊C₄	C₂³×C₆₇	1	2	yes	—
438	C₂×C₄⋊C₄	C₂³×C₆₇	1	2	yes	—
439	C₂×C₄⋊C₄	C₂³×C₆₇	1	2	yes	—
440	C₂×C₄⋊C₄	C₂³×C₆₇	1	4	yes	—
441	C₂×C₄⋊C₄	C₂³×C₆₇	1	1	yes	characteristic
442	C₂×C₄⋊C₄	C₂³×C₆₇	1	1	yes	characteristic
443	C₂×C₄⋊C₄	C₂³×C₆₇	1	2	yes	—
444	C₂×C₄⋊C₄	C₂³×C₆₇	1	2	yes	—
445	C₂×C₄⋊C₄	C₂³×C₆₇	1	1	yes	characteristic
446	C₂×C₄⋊C₄	C₂³×C₆₇	1	2	yes	—
447	C₂×C₄⋊C₄	C₂³×C₆₇	1	1	yes	characteristic
448	C₂×C₄⋊C₄	C₂³×C₆₇	1	2	yes	—
449	C₂×C₄⋊C₄	C₂³×C₆₇	1	2	yes	—
450	C₂×C₄⋊C₄	C₂³×C₆₇	1	1	yes	characteristic
451	C₂×C₄⋊C₄	C₂³×C₆₇	1	2	yes	—
452	C₂×C₄⋊C₄	C₂³×C₆₇	1	4	yes	—
453	C₂×C₄⋊C₄	C₂³×C₆₇	1	2	yes	—
454	C₂×C₄⋊C₄	C₂³×C₆₇	1	4	yes	—
455	C₂×C₄⋊C₄	C₂³×C₆₇	1	4	yes	—
456	C₂×C₄⋊C₄	C₂³×C₆₇	1	4	yes	—
457	C₂×C₄⋊C₄	C₂³×C₆₇	1	4	yes	—
458	C₂×C₄⋊C₄	C₂³×C₆₇	1	4	yes	—
459	C₂×C₄⋊C₄	C₂³×C₆₇	1	4	yes	—
460	C₂×C₄⋊C₄	C₂³×C₆₇	1	1	yes	characteristic
461	C₂×C₄⋊C₄	C₂³×C₆₇	1	2	yes	—
462	C₂×C₂²⋊C₄	C₂³×C₆₇	1	2	yes	—
463	C₂×C₂²⋊C₄	C₂³×C₆₇	1	2	yes	—
464	C₂×C₂²⋊C₄	C₂³×C₆₇	1	2	yes	—
465	C₂×C₂²⋊C₄	C₂³×C₆₇	1	2	yes	—
466	C₂×C₄²	—	2	4	no	—
467	C₂×C₄²	C₂³×C₆₇	1	2	yes	—
468	C₂×C₄²	C₂³×C₆₇	1	2	yes	—
469	C₂×C₄²	C₂³×C₆₇	1	2	yes	—
470	C₂×C₄²	C₂³×C₆₇	1	2	yes	—
471	C₂×C₄²	C₂³×C₆₇	1	1	yes	characteristic
472	C₂×C₄²	C₂³×C₆₇	1	1	yes	characteristic
473	C₂×C₄²	C₂³×C₆₇	1	1	yes	characteristic
474	C₂×C₄²	C₂³×C₆₇	1	4	yes	—
475	C₂×C₄²	C₂³×C₆₇	1	4	yes	—
476	C₂×C₄²	C₂³×C₆₇	1	1	yes	characteristic
477	#95	C₂³×C₆₇	1	2	yes	—
478	#95	C₂³×C₆₇	1	2	yes	—
479	#95	C₂³×C₆₇	1	2	yes	—
480	#95	C₂³×C₆₇	1	4	yes	—
481	#95	C₂³×C₆₇	1	2	yes	—
482	#95	C₂³×C₆₇	1	4	yes	—
483	#95	C₂³×C₆₇	1	4	yes	—
484	#95	C₂³×C₆₇	1	2	yes	—
485	#95	C₂³×C₆₇	1	4	yes	—
486	#95	C₂³×C₆₇	1	2	yes	—
487	#95	C₂³×C₆₇	1	2	yes	—
488	#95	C₂³×C₆₇	1	2	yes	—
489	#95	C₂³×C₆₇	1	4	yes	—
490	#95	C₂³×C₆₇	1	2	yes	—
491	#95	C₂³×C₆₇	1	2	yes	—
492	#95	C₂³×C₆₇	1	4	yes	—
493	#95	C₂³×C₆₇	1	4	yes	—
494	#95	C₂³×C₆₇	1	4	yes	—
495	C₂⁴	C₂⁴×C₆₇	1	1	yes	characteristic
496	C₂×Q₈	—	2	2	no	—
497	C₂×Q₈	—	2	2	no	—
498	C₂×Q₈	—	2	2	no	—
499	C₂×Q₈	—	2	2	no	—
500	C₂×Q₈	—	2	2	no	—
501	C₂×Q₈	C₂×Q₈×C₆₇	1	4	yes	—
502	C₂²×C₄	—	2	2	no	—
503	C₂²×C₄	—	2	2	no	—
504	C₂²×C₄	—	2	2	no	—
505	C₂²×C₄	—	2	2	no	—
506	C₂²×C₄	—	2	2	no	—
507	C₂²×C₄	—	2	2	no	—
508	C₂²×C₄	—	2	2	no	—
509	C₂²×C₄	—	2	8	no	—
510	C₂²×C₄	—	4	4	no	—
511	C₂²×C₄	—	4	4	no	—
512	C₂²×C₄	C₂×Q₈×C₆₇	1	4	yes	—
513	C₂²×C₄	#41×C₆₇	1	4	yes	—
514	C₂²×C₄	C₂⁴×C₆₇	1	2	yes	—
515	C₂²×C₄	C₂⁴×C₆₇	1	2	yes	—
516	C₂²×C₄	C₂⁴×C₆₇	1	2	yes	—
517	C₂²×C₄	C₂⁴×C₆₇	1	2	yes	—
518	C₂²×C₄	C₂⁴×C₆₇	1	2	yes	—
519	C₂²×C₄	C₂⁴×C₆₇	1	4	yes	—
520	C₂²×C₄	C₂⁴×C₆₇	1	4	yes	—
521	C₂²×C₄	C₂⁴×C₆₇	1	2	yes	—
522	C₂²×C₄	C₂⁴×C₆₇	1	2	yes	—
523	C₂²×C₄	C₂⁴×C₆₇	1	2	yes	—
524	C₂²×C₄	C₂⁴×C₆₇	1	1	yes	characteristic
525	C₂²×C₄	C₂⁴×C₆₇	1	1	yes	characteristic
526	C₂²×C₄	C₂⁴×C₆₇	1	1	yes	characteristic
527	C₂²×C₄	C₂⁴×C₆₇	1	1	yes	characteristic
528	C₂²×C₄	C₂⁴×C₆₇	1	1	yes	characteristic
529	C₂²×C₄	C₂⁴×C₆₇	1	1	yes	characteristic
530	C₄⋊C₄	—	2	8	no	—
531	C₄⋊C₄	—	2	4	no	—
532	C₄⋊C₄	—	2	4	no	—
533	C₄⋊C₄	—	2	4	no	—
534	C₄⋊C₄	—	2	2	no	—
535	C₄⋊C₄	—	2	2	no	—
536	C₄⋊C₄	—	2	4	no	—
537	C₄⋊C₄	—	2	2	no	—
538	C₄⋊C₄	—	2	4	no	—
539	C₄⋊C₄	—	2	4	no	—
540	C₄⋊C₄	—	2	8	no	—
541	C₄⋊C₄	—	2	4	no	—
542	C₄⋊C₄	—	2	4	no	—
543	C₄⋊C₄	—	2	4	no	—
544	C₄⋊C₄	—	2	2	no	—
545	C₄⋊C₄	—	2	2	no	—
546	C₄⋊C₄	—	4	2	no	—
547	C₄⋊C₄	—	4	4	no	—
548	C₄⋊C₄	—	4	2	no	—
549	C₄⋊C₄	—	4	4	no	—
550	C₄⋊C₄	—	4	4	no	—
551	C₄⋊C₄	—	4	4	no	—
552	C₄⋊C₄	—	4	4	no	—
553	C₄⋊C₄	—	4	4	no	—
554	C₄⋊C₄	—	4	4	no	—
555	C₄⋊C₄	C₂×Q₈×C₆₇	1	4	yes	—
556	C₄⋊C₄	C₂×Q₈×C₆₇	1	8	yes	—
557	C₂²⋊C₄	—	2	4	no	—
558	C₂²⋊C₄	—	2	4	no	—
559	C₂²⋊C₄	—	2	4	no	—
560	C₂²⋊C₄	C₂×Q₈×C₆₇	1	8	yes	—
561	C₄²	—	2	2	no	—
562	C₄²	—	2	2	no	—
563	C₄²	—	2	2	no	—
564	C₄²	—	2	4	no	—
565	C₄²	—	2	4	no	—
566	C₄²	—	2	4	no	—
567	C₄²	—	2	4	no	—
568	C₄²	—	4	4	no	—
569	C₄²	—	4	4	no	—
570	C₄²	#41×C₆₇	1	4	yes	—
571	C₂³	—	2	1	no	—
572	C₂³	—	2	1	no	—
573	C₂³	—	2	1	no	—
574	C₂³	—	2	1	no	—
575	C₂³	C₂²×Q₈×C₆₇	1	2	yes	—
576	C₂³	C₂×#41×C₆₇	1	2	yes	—
577	C₂³	2₋¹⁺⁴×C₆₇	1	2	yes	—
578	C₂³	C₂⁵×C₆₇	1	1	yes	characteristic
579	Q₈	—	2	8	no	—
580	Q₈	—	4	4	no	—
581	C₂×C₄	—	2	8	no	—
582	C₂×C₄	—	2	1	no	—
583	C₂×C₄	—	2	1	no	—
584	C₂×C₄	—	2	1	no	—
585	C₂×C₄	—	2	2	no	—
586	C₂×C₄	—	2	2	no	—
587	C₂×C₄	—	2	2	no	—
588	C₂×C₄	—	2	1	no	—
589	C₂×C₄	—	2	1	no	—
590	C₂×C₄	—	2	2	no	—
591	C₂×C₄	—	2	1	no	—
592	C₂×C₄	—	2	1	no	—
593	C₂×C₄	—	2	1	no	—
594	C₂×C₄	—	2	8	no	—
595	C₂×C₄	—	2	8	no	—
596	C₂×C₄	—	2	2	no	—
597	C₂×C₄	—	2	2	no	—
598	C₂×C₄	—	2	2	no	—
599	C₂×C₄	—	2	2	no	—
600	C₂×C₄	—	2	2	no	—
601	C₂×C₄	—	2	4	no	—
602	C₂×C₄	—	2	4	no	—
603	C₂×C₄	—	2	2	no	—
604	C₂×C₄	—	2	2	no	—
605	C₂×C₄	—	2	2	no	—
606	C₂×C₄	—	2	4	no	—
607	C₂×C₄	—	2	4	no	—
608	C₂×C₄	—	2	2	no	—
609	C₂×C₄	—	2	2	no	—
610	C₂×C₄	—	2	2	no	—
611	C₂×C₄	—	2	4	no	—
612	C₂×C₄	—	2	4	no	—
613	C₂×C₄	—	2	2	no	—
614	C₂×C₄	—	2	2	no	—
615	C₂×C₄	—	2	2	no	—
616	C₂×C₄	—	2	2	no	—
617	C₂×C₄	—	2	2	no	—
618	C₂×C₄	—	2	2	no	—
619	C₂×C₄	—	2	2	no	—
620	C₂×C₄	—	8	8	no	—
621	C₂×C₄	C₂²×Q₈×C₆₇	1	4	yes	—
622	C₂×C₄	C₂²×Q₈×C₆₇	1	4	yes	—
623	C₂×C₄	C₂²×Q₈×C₆₇	1	2	yes	—
624	C₂×C₄	C₂²×Q₈×C₆₇	1	2	yes	—
625	C₂×C₄	C₂×#41×C₆₇	1	2	yes	—
626	C₂×C₄	C₂×#41×C₆₇	1	2	yes	—
627	C₂×C₄	C₂×#41×C₆₇	1	2	yes	—
628	C₂×C₄	C₂×#41×C₆₇	1	2	yes	—
629	C₂×C₄	C₂×#41×C₆₇	1	2	yes	—
630	C₂×C₄	H(5,𝔽₂)×C₆₇	1	2	yes	—
631	C₂×C₄	H(5,𝔽₂)×C₆₇	1	2	yes	—
632	C₂×C₄	2₋¹⁺⁴×C₆₇	1	2	yes	—
633	C₂²	—	2	2	no	—
634	C₂²	—	2	2	no	—
635	C₂²	—	2	2	no	—
636	C₂²	—	2	2	no	—
637	C₂²	—	2	2	no	—
638	C₂²	—	2	2	no	—
639	C₂²	#557×C₆₇	1	4	yes	—
640	C₂²	C₂³×Q₈×C₆₇	1	1	yes	characteristic
641	C₂²	C₂²×#41×C₆₇	1	1	yes	characteristic
642	C₂²	C₂²×#41×C₆₇	1	1	yes	characteristic
643	C₂²	C₂×H(5,𝔽₂)×C₆₇	1	1	yes	characteristic
644	C₂²	C₂×H(5,𝔽₂)×C₆₇	1	1	yes	characteristic
645	C₂²	C₂×2₋¹⁺⁴×C₆₇	1	1	yes	characteristic
646	C₂²	C₂×2₋¹⁺⁴×C₆₇	1	1	yes	characteristic
647	C₄	—	2	2	no	—
648	C₄	—	2	2	no	—
649	C₄	—	2	4	no	—
650	C₄	—	2	2	no	—
651	C₄	—	2	2	no	—
652	C₄	—	2	4	no	—
653	C₄	—	4	2	no	—
654	C₄	—	4	2	no	—
655	C₄	—	4	2	no	—
656	C₄	—	4	2	no	—
657	C₄	—	4	2	no	—
658	C₄	—	4	4	no	—
659	C₄	—	4	4	no	—
660	C₄	—	4	2	no	—
661	C₄	#552×C₆₇	1	4	yes	—
662	C₄	#554×C₆₇	1	4	yes	—
663	C₂	—	2	4	no	—
664	C₂	C₂×#552×C₆₇	1	1	yes	characteristic
665	C₂	C₂×#554×C₆₇	1	1	yes	characteristic
666	C₂	C₂×#557×C₆₇	1	1	yes	characteristic
667	C₂	C₆₇×#3492	1	1	yes	characteristic
668	C₂	C₆₇×#3503	1	1	yes	characteristic
669	C₂	C₆₇×#3508	1	1	yes	characteristic
670	C₂	C₆₇×#3552	1	1	yes	characteristic

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

Complements

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

N	H ≅ G/N	type
1. C₆₇×#2791	C₂	semidirect
2. C₆₇×#2582	C₂	semidirect
3. C₆₇×#2565	C₂	semidirect
4. C₆₇×#2552	C₂	semidirect
5. C₆₇×#2552	C₂	semidirect
6. C₆₇×#2511	C₂	semidirect
7. C₆₇×#2506	C₂	semidirect
12. C₆₇×#2355	C₂	semidirect
13. Q₈×C₄⋊C₄×C₆₇	C₂	semidirect
92. C₂²⋊Q₈×C₆₇	Q₈	semidirect
327. #59321	C₆₇	direct (N × H)
336. #2791	C₂×C₆₇	semidirect
337. #2582	C₂×C₆₇	semidirect
338. #2565	C₂×C₆₇	semidirect
339. #2552	C₂×C₆₇	semidirect
340. #2552	C₂×C₆₇	semidirect
341. #2511	C₂×C₆₇	semidirect
342. #2506	C₂×C₆₇	semidirect
347. #2355	C₂×C₆₇	semidirect
348. Q₈×C₄⋊C₄	C₂×C₆₇	semidirect
353. C₆₇	#59321	direct (N × H)
428. C₂²⋊Q₈	Q₈×C₆₇	semidirect

Maximal subgroups

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

subgroup	index
C₆₇×#2791	2
C₆₇×#2582	2
C₆₇×#2565	2
C₆₇×#2552	2
C₆₇×#2511	2
C₆₇×#2506	2
C₆₇×#2502	2
C₆₇×#2473	2
C₆₇×#2442	2
C₆₇×#2360	2
C₆₇×#2355	2
Q₈×C₄⋊C₄×C₆₇	2
C₆₇×#2345	2
C₄×C₂²⋊Q₈×C₆₇	2
C₂×#384×C₆₇	2
#59321	67

Subgroups — C67×#59321 #30738572

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₆₇×#59321 #30738572