Subgroups — #30875744

Order 17152.

736 conjugacy classes of subgroups.

The subgroup lattice has 494 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#22561150	C₂	1	2	yes	maximal
2	#22560775	C₂	1	2	yes	maximal
3	#22560768	C₂	1	2	yes	maximal
4	#22560361	C₂	1	1	yes	maximal, characteristic
5	#22560326	C₂	1	1	yes	maximal, characteristic
6	#22560326	C₂	1	1	yes	maximal, characteristic
7	#22560084	C₂	1	2	yes	maximal
8	#22560062	C₂	1	1	yes	maximal, characteristic
9	#22560062	C₂	1	1	yes	maximal, characteristic
10	C₄×C₄⋊(C₆₇⋊C₈)	C₂	1	1	yes	maximal, characteristic
11	C₄²×Q₈×C₆₇	C₂	1	1	yes	maximal, characteristic
12	C₂×C₄⋊(C₆₇⋊C₈)	C₂²	1	1	yes	characteristic
13	C₂×C₄⋊(C₆₇⋊C₈)	C₂²	1	1	yes	characteristic
14	C₂×C₄⋊(C₆₇⋊C₈)	C₂²	1	1	yes	characteristic
15	C₂×C₄⋊(C₆₇⋊C₈)	C₂²	1	2	yes	—
16	C₂×C₄⋊(C₆₇⋊C₈)	C₂²	1	2	yes	—
17	C₂×C₄⋊(C₆₇⋊C₈)	C₂²	1	1	yes	characteristic
18	C₂×#14523954	C₂²	1	2	yes	—
19	C₂×#14523954	C₂²	1	2	yes	—
20	C₂×C₄×C₆₇⋊C₈	C₂²	1	2	yes	—
21	#16863151	C₂²	1	2	yes	—
22	#16863151	C₂²	1	2	yes	—
23	#16863151	C₂²	1	2	yes	—
24	#16863151	C₂²	1	2	yes	—
25	#16863151	C₂²	1	2	yes	—
26	#16863151	C₂²	1	2	yes	—
27	#16863151	C₂²	1	2	yes	—
28	C₂×C₄×Q₈×C₆₇	C₄	1	1	yes	characteristic
29	C₂×C₄×Q₈×C₆₇	C₄	1	1	yes	characteristic
30	C₂×C₄×Q₈×C₆₇	C₂²	1	1	yes	characteristic
31	C₄×C₄⋊C₄×C₆₇	C₄	1	1	yes	characteristic
32	C₄×C₄⋊C₄×C₆₇	C₄	1	1	yes	characteristic
33	C₄×C₄⋊C₄×C₆₇	C₄	1	1	yes	characteristic
34	C₄×C₄⋊C₄×C₆₇	C₄	1	1	yes	characteristic
35	C₄×C₄⋊C₄×C₆₇	C₄	1	1	yes	characteristic
36	C₄×C₄⋊C₄×C₆₇	C₄	1	1	yes	characteristic
37	C₄×C₄⋊C₄×C₆₇	C₂²	1	1	yes	characteristic
38	C₄×C₄⋊C₄×C₆₇	C₂²	1	1	yes	characteristic
39	C₄×C₄⋊C₄×C₆₇	C₂²	1	1	yes	characteristic
40	C₄³×C₆₇	C₂²	1	1	yes	characteristic
41	C₄³×C₆₇	C₂²	1	1	yes	characteristic
42	C₄³×C₆₇	C₂²	1	1	yes	characteristic
43	C₂²×C₆₇⋊C₈	C₂³	1	2	yes	—
44	C₂²×C₆₇⋊C₈	C₂³	1	2	yes	—
45	C₂²×C₆₇⋊C₈	C₂³	1	2	yes	—
46	C₂²×C₆₇⋊C₈	C₂³	1	2	yes	—
47	C₄⋊(C₆₇⋊C₈)	—	2	4	no	—
48	C₄⋊(C₆₇⋊C₈)	—	2	2	no	—
49	C₄⋊(C₆₇⋊C₈)	—	2	2	no	—
50	C₄⋊(C₆₇⋊C₈)	—	2	2	no	—
51	C₄⋊(C₆₇⋊C₈)	—	4	2	no	—
52	C₄⋊(C₆₇⋊C₈)	Q₈	1	4	yes	complemented
53	#14523954	—	2	4	no	—
54	#14523954	—	4	2	no	—
55	C₄×C₆₇⋊C₈	—	4	2	no	—
56	C₂²×Q₈×C₆₇	C₂×C₄	1	1	yes	characteristic
57	C₄×Q₈×C₆₇	—	2	4	no	—
58	C₄×Q₈×C₆₇	—	2	4	no	—
59	C₄×Q₈×C₆₇	—	2	2	no	—
60	C₄×Q₈×C₆₇	—	2	2	no	—
61	C₂×C₄⋊C₄×C₆₇	C₂×C₄	1	1	yes	characteristic
62	C₂×C₄⋊C₄×C₆₇	C₂×C₄	1	1	yes	characteristic
63	C₂×C₄⋊C₄×C₆₇	C₂×C₄	1	1	yes	characteristic
64	C₂×C₄⋊C₄×C₆₇	C₂×C₄	1	1	yes	characteristic
65	C₂×C₄⋊C₄×C₆₇	C₂×C₄	1	1	yes	characteristic
66	C₂×C₄⋊C₄×C₆₇	C₂×C₄	1	1	yes	characteristic
67	C₂×C₄⋊C₄×C₆₇	C₂×C₄	1	1	yes	characteristic
68	C₂×C₄⋊C₄×C₆₇	C₂×C₄	1	1	yes	characteristic
69	C₂×C₄⋊C₄×C₆₇	C₂×C₄	1	1	yes	characteristic
70	C₂×C₄²×C₆₇	C₂×C₄	1	1	yes	characteristic
71	C₂×C₄²×C₆₇	C₂×C₄	1	1	yes	characteristic
72	C₂×C₄²×C₆₇	C₂×C₄	1	1	yes	characteristic
73	C₂×C₄²×C₆₇	C₂×C₄	1	1	yes	characteristic
74	C₂×C₄²×C₆₇	C₂×C₄	1	1	yes	characteristic
75	C₂×C₄²×C₆₇	C₂×C₄	1	1	yes	characteristic
76	C₂×C₄²×C₆₇	C₂×C₄	1	1	yes	characteristic
77	C₂×C₄²×C₆₇	C₂×C₄	1	1	yes	characteristic
78	C₂×C₄²×C₆₇	C₂×C₄	1	1	yes	characteristic
79	C₂×C₄²×C₆₇	C₂×C₄	1	1	yes	characteristic
80	C₂×C₄²×C₆₇	C₂×C₄	1	1	yes	characteristic
81	C₂×C₄²×C₆₇	C₂×C₄	1	1	yes	characteristic
82	C₂×C₄²×C₆₇	C₂³	1	1	yes	characteristic
83	C₂×C₄²×C₆₇	C₂³	1	1	yes	characteristic
84	C₂×C₄²×C₆₇	C₂³	1	1	yes	characteristic
85	C₂×C₄²×C₆₇	C₂³	1	1	yes	characteristic
86	C₂×C₄²×C₆₇	C₂³	1	1	yes	characteristic
87	C₂×C₄²×C₆₇	C₂³	1	1	yes	characteristic
88	C₂×C₄²×C₆₇	C₂³	1	1	yes	characteristic
89	#95×C₆₇	C₂×C₄	1	1	yes	characteristic
90	#95×C₆₇	C₂×C₄	1	1	yes	characteristic
91	#95×C₆₇	C₂×C₄	1	1	yes	characteristic
92	#95×C₆₇	C₂×C₄	1	1	yes	characteristic
93	#95×C₆₇	C₂×C₄	1	1	yes	characteristic
94	#95×C₆₇	C₂×C₄	1	1	yes	characteristic
95	C₂×C₆₇⋊C₈	—	2	2	no	—
96	C₂×C₆₇⋊C₈	—	2	2	no	—
97	C₂×C₆₇⋊C₈	—	2	2	no	—
98	C₂×C₆₇⋊C₈	—	2	2	no	—
99	C₂×C₆₇⋊C₈	—	2	2	no	—
100	C₂×C₆₇⋊C₈	—	2	2	no	—
101	C₂×C₆₇⋊C₈	—	2	2	no	—
102	C₂×C₆₇⋊C₈	—	2	2	no	—
103	C₂×C₆₇⋊C₈	—	2	2	no	—
104	C₂×C₆₇⋊C₈	—	2	2	no	—
105	C₂×C₆₇⋊C₈	—	2	2	no	—
106	C₂×C₆₇⋊C₈	C₂×Q₈	1	4	yes	—
107	C₂×Q₈×C₆₇	—	2	2	no	—
108	C₂×Q₈×C₆₇	—	2	2	no	—
109	C₂×Q₈×C₆₇	—	2	2	no	—
110	C₂²×C₄×C₆₇	C₂²×C₄	1	1	yes	characteristic
111	C₂²×C₄×C₆₇	C₂²×C₄	1	1	yes	characteristic
112	C₂²×C₄×C₆₇	C₂²×C₄	1	1	yes	characteristic
113	C₂²×C₄×C₆₇	C₂²×C₄	1	1	yes	characteristic
114	C₂²×C₄×C₆₇	C₂²×C₄	1	1	yes	characteristic
115	C₂²×C₄×C₆₇	C₂²×C₄	1	1	yes	characteristic
116	C₂²×C₄×C₆₇	C₂²×C₄	1	1	yes	characteristic
117	C₂²×C₄×C₆₇	C₂²×C₄	1	1	yes	characteristic
118	C₂²×C₄×C₆₇	C₂²×C₄	1	1	yes	characteristic
119	C₂²×C₄×C₆₇	C₂²×C₄	1	1	yes	characteristic
120	C₂²×C₄×C₆₇	C₂²×C₄	1	1	yes	characteristic
121	C₂²×C₄×C₆₇	C₂²×C₄	1	1	yes	characteristic
122	C₂²×C₄×C₆₇	C₂²×C₄	1	1	yes	characteristic
123	C₂²×C₄×C₆₇	C₂²×C₄	1	1	yes	characteristic
124	C₂²×C₄×C₆₇	C₂⁴	1	1	yes	characteristic
125	C₄⋊C₄×C₆₇	—	2	2	no	—
126	C₄⋊C₄×C₆₇	—	2	2	no	—
127	C₄⋊C₄×C₆₇	—	2	2	no	—
128	C₄⋊C₄×C₆₇	—	2	2	no	—
129	C₄⋊C₄×C₆₇	—	2	2	no	—
130	C₄⋊C₄×C₆₇	—	2	2	no	—
131	C₄⋊C₄×C₆₇	—	2	2	no	—
132	C₄⋊C₄×C₆₇	—	2	2	no	—
133	C₄⋊C₄×C₆₇	M₁₆	1	4	yes	—
134	C₄²×C₆₇	—	2	2	no	—
135	C₄²×C₆₇	—	2	2	no	—
136	C₄²×C₆₇	—	2	2	no	—
137	C₄²×C₆₇	—	2	2	no	—
138	C₄²×C₆₇	—	2	2	no	—
139	C₄²×C₆₇	—	2	2	no	—
140	C₄²×C₆₇	—	2	2	no	—
141	C₄²×C₆₇	—	2	2	no	—
142	C₄²×C₆₇	—	2	1	no	—
143	C₄²×C₆₇	—	2	1	no	—
144	C₄²×C₆₇	—	2	1	no	—
145	C₄²×C₆₇	—	2	1	no	—
146	C₄²×C₆₇	—	2	1	no	—
147	C₄²×C₆₇	—	2	1	no	—
148	C₄²×C₆₇	—	2	1	no	—
149	C₄²×C₆₇	—	2	1	no	—
150	C₄²×C₆₇	—	2	1	no	—
151	C₄²×C₆₇	—	2	1	no	—
152	C₄²×C₆₇	—	4	1	no	—
153	C₄²×C₆₇	—	4	1	no	—
154	C₄²×C₆₇	—	4	1	no	—
155	C₄²×C₆₇	M₁₆	1	4	yes	—
156	C₄²×C₆₇	C₂×Q₈	1	2	yes	—
157	C₄²×C₆₇	#41	1	2	yes	—
158	C₄²×C₆₇	#41	1	2	yes	—
159	C₄²×C₆₇	#41	1	2	yes	—
160	C₂³×C₆₇	C₂³×C₄	1	1	yes	characteristic
161	Q₈×C₆₇	—	2	8	no	—
162	C₂×C₄×C₆₇	—	2	1	no	—
163	C₂×C₄×C₆₇	—	2	1	no	—
164	C₂×C₄×C₆₇	—	2	1	no	—
165	C₂×C₄×C₆₇	—	2	1	no	—
166	C₂×C₄×C₆₇	—	2	1	no	—
167	C₂×C₄×C₆₇	—	2	1	no	—
168	C₂×C₄×C₆₇	—	2	1	no	—
169	C₂×C₄×C₆₇	—	2	1	no	—
170	C₂×C₄×C₆₇	—	2	1	no	—
171	C₂×C₄×C₆₇	—	2	1	no	—
172	C₂×C₄×C₆₇	—	2	1	no	—
173	C₂×C₄×C₆₇	—	2	1	no	—
174	C₂×C₄×C₆₇	—	2	1	no	—
175	C₂×C₄×C₆₇	—	2	1	no	—
176	C₂×C₄×C₆₇	—	2	1	no	—
177	C₂×C₄×C₆₇	—	2	1	no	—
178	C₂×C₄×C₆₇	—	2	1	no	—
179	C₂×C₄×C₆₇	—	2	1	no	—
180	C₂×C₄×C₆₇	—	2	1	no	—
181	C₂×C₄×C₆₇	—	2	1	no	—
182	C₂×C₄×C₆₇	—	2	1	no	—
183	C₂×C₄×C₆₇	—	2	1	no	—
184	C₂×C₄×C₆₇	—	2	1	no	—
185	C₂×C₄×C₆₇	—	2	1	no	—
186	C₂×C₄×C₆₇	—	2	1	no	—
187	C₂×C₄×C₆₇	—	2	1	no	—
188	C₂×C₄×C₆₇	—	2	1	no	—
189	C₂×C₄×C₆₇	—	2	1	no	—
190	C₂×C₄×C₆₇	—	2	1	no	—
191	C₂×C₄×C₆₇	—	2	1	no	—
192	C₂×C₄×C₆₇	—	2	1	no	—
193	C₂×C₄×C₆₇	#117	1	2	yes	—
194	C₂×C₄×C₆₇	#117	1	2	yes	—
195	C₂×C₄×C₆₇	C₄×Q₈	1	2	yes	—
196	C₂×C₄×C₆₇	C₄×Q₈	1	2	yes	—
197	C₂×C₄×C₆₇	C₂×M₁₆	1	2	yes	—
198	C₂×C₄×C₆₇	C₂×M₁₆	1	2	yes	—
199	C₂×C₄×C₆₇	C₂×M₁₆	1	2	yes	—
200	C₂×C₄×C₆₇	C₂×M₁₆	1	2	yes	—
201	C₂×C₄×C₆₇	C₂×M₁₆	1	2	yes	—
202	C₂×C₄×C₆₇	C₂×M₁₆	1	2	yes	—
203	C₂×C₄×C₆₇	#131	1	2	yes	—
204	C₂×C₄×C₆₇	C₂²×Q₈	1	1	yes	characteristic
205	C₂×C₄×C₆₇	C₂×#41	1	1	yes	characteristic
206	C₂×C₄×C₆₇	C₂×#41	1	1	yes	characteristic
207	C₂×C₄×C₆₇	C₂×#41	1	1	yes	characteristic
208	C₂×C₄×C₆₇	H(5,𝔽₂)	1	1	yes	characteristic
209	C₂×C₄×C₆₇	2₋¹⁺⁴	1	1	yes	characteristic
210	C₆₇⋊C₈	—	2	4	no	—
211	C₆₇⋊C₈	—	4	2	no	—
212	C₆₇⋊C₈	—	4	2	no	—
213	C₆₇⋊C₈	—	4	2	no	—
214	C₂²×C₆₇	C₂×#117	1	1	yes	characteristic
215	C₂²×C₆₇	C₂×C₄×Q₈	1	1	yes	characteristic
216	C₂²×C₆₇	#520	1	1	yes	characteristic
217	C₂²×C₆₇	C₂²×M₁₆	1	1	yes	characteristic
218	C₂²×C₆₇	C₂²×M₁₆	1	1	yes	characteristic
219	C₂²×C₆₇	C₂×#131	1	1	yes	characteristic
220	C₂²×C₆₇	#568	1	1	yes	characteristic
221	C₄×C₆₇	—	2	2	no	—
222	C₄×C₆₇	—	2	2	no	—
223	C₄×C₆₇	—	2	2	no	—
224	C₄×C₆₇	—	2	2	no	—
225	C₄×C₆₇	—	2	2	no	—
226	C₄×C₆₇	—	2	2	no	—
227	C₄×C₆₇	—	2	2	no	—
228	C₄×C₆₇	—	2	2	no	—
229	C₄×C₆₇	—	4	1	no	—
230	C₄×C₆₇	—	4	1	no	—
231	C₄×C₆₇	—	4	1	no	—
232	C₄×C₆₇	—	4	1	no	—
233	C₄×C₆₇	—	4	1	no	—
234	C₄×C₆₇	#446	1	4	yes	—
235	C₄×C₆₇	#554	1	1	yes	characteristic
236	C₄×C₆₇	#556	1	1	yes	characteristic
237	C₄×C₆₇	#557	1	1	yes	characteristic
238	C₄×C₆₇	#559	1	1	yes	characteristic
239	#22823	—	67	1	no	maximal, Sylow
240	C₂×C₆₇	#2355	1	1	yes	characteristic
241	C₂×C₆₇	#2920	1	1	yes	characteristic
242	C₂×C₆₇	#2925	1	1	yes	characteristic
243	C₂×C₆₇	C₂×#446	1	1	yes	characteristic
244	C₂×C₆₇	Q₈×M₁₆	1	1	yes	characteristic
245	C₂×C₆₇	#2989	1	1	yes	characteristic
246	C₂×C₆₇	#2990	1	1	yes	characteristic
247	C₄²×Q₈	D₆₇	1	1	yes	characteristic
248	#1985	—	67	2	no	—
249	#1917	—	67	2	no	—
250	#1916	—	67	2	no	—
251	#1842	—	67	1	no	—
252	#1831	—	67	1	no	—
253	#1831	—	67	1	no	—
254	#1771	—	67	2	no	—
255	#1767	—	67	1	no	—
256	#1767	—	67	1	no	—
257	C₄×C₄⋊C₈	—	67	1	no	—
258	C₆₇	#22823	1	1	yes	characteristic, complemented, Sylow
259	C₂×C₄×Q₈	Dic₆₇	1	1	yes	characteristic
260	C₂×C₄×Q₈	Dic₆₇	1	1	yes	characteristic
261	C₂×C₄×Q₈	C₂×D₆₇	1	1	yes	characteristic
262	C₂×C₄⋊C₈	—	67	2	no	—
263	C₂×C₄⋊C₈	—	67	1	no	—
264	C₂×C₄⋊C₈	—	67	1	no	—
265	C₂×C₄⋊C₈	—	67	1	no	—
266	C₂×C₄⋊C₈	—	67	1	no	—
267	C₂×C₄⋊C₈	—	67	2	no	—
268	C₂×#97	—	67	2	no	—
269	C₂×#97	—	67	2	no	—
270	C₂×C₄×C₈	—	67	2	no	—
271	C₄×C₄⋊C₄	Dic₆₇	1	1	yes	characteristic
272	C₄×C₄⋊C₄	Dic₆₇	1	1	yes	characteristic
273	C₄×C₄⋊C₄	Dic₆₇	1	1	yes	characteristic
274	C₄×C₄⋊C₄	Dic₆₇	1	1	yes	characteristic
275	C₄×C₄⋊C₄	Dic₆₇	1	1	yes	characteristic
276	C₄×C₄⋊C₄	Dic₆₇	1	1	yes	characteristic
277	C₄×C₄⋊C₄	C₂×D₆₇	1	1	yes	characteristic
278	C₄×C₄⋊C₄	C₂×D₆₇	1	1	yes	characteristic
279	C₄×C₄⋊C₄	C₂×D₆₇	1	1	yes	characteristic
280	C₄³	C₂×D₆₇	1	1	yes	characteristic
281	C₄³	C₂×D₆₇	1	1	yes	characteristic
282	C₄³	C₂×D₆₇	1	1	yes	characteristic
283	#336	—	67	2	no	—
284	#336	—	67	2	no	—
285	#336	—	67	2	no	—
286	#336	—	67	2	no	—
287	#336	—	67	2	no	—
288	#336	—	67	2	no	—
289	#336	—	67	2	no	—
290	C₂²×Q₈	C₂×Dic₆₇	1	1	yes	characteristic
291	C₂²×C₈	—	67	2	no	—
292	C₂²×C₈	—	67	2	no	—
293	C₂²×C₈	—	67	2	no	—
294	C₂²×C₈	—	67	2	no	—
295	C₄×Q₈	—	2	4	no	—
296	C₄×Q₈	—	2	4	no	—
297	C₄×Q₈	—	2	2	no	—
298	C₄×Q₈	—	2	2	no	—
299	C₂×C₄⋊C₄	C₂×Dic₆₇	1	1	yes	characteristic
300	C₂×C₄⋊C₄	C₂×Dic₆₇	1	1	yes	characteristic
301	C₂×C₄⋊C₄	C₂×Dic₆₇	1	1	yes	characteristic
302	C₂×C₄⋊C₄	C₂×Dic₆₇	1	1	yes	characteristic
303	C₂×C₄⋊C₄	C₂×Dic₆₇	1	1	yes	characteristic
304	C₂×C₄⋊C₄	C₂×Dic₆₇	1	1	yes	characteristic
305	C₂×C₄⋊C₄	C₂×Dic₆₇	1	1	yes	characteristic
306	C₂×C₄⋊C₄	C₂×Dic₆₇	1	1	yes	characteristic
307	C₂×C₄⋊C₄	C₂×Dic₆₇	1	1	yes	characteristic
308	C₂×C₄²	C₂×Dic₆₇	1	1	yes	characteristic
309	C₂×C₄²	C₂×Dic₆₇	1	1	yes	characteristic
310	C₂×C₄²	C₂×Dic₆₇	1	1	yes	characteristic
311	C₂×C₄²	C₂×Dic₆₇	1	1	yes	characteristic
312	C₂×C₄²	C₂×Dic₆₇	1	1	yes	characteristic
313	C₂×C₄²	C₂×Dic₆₇	1	1	yes	characteristic
314	C₂×C₄²	C₂×Dic₆₇	1	1	yes	characteristic
315	C₂×C₄²	C₂×Dic₆₇	1	1	yes	characteristic
316	C₂×C₄²	C₂×Dic₆₇	1	1	yes	characteristic
317	C₂×C₄²	C₂×Dic₆₇	1	1	yes	characteristic
318	C₂×C₄²	C₂×Dic₆₇	1	1	yes	characteristic
319	C₂×C₄²	C₂×Dic₆₇	1	1	yes	characteristic
320	C₂×C₄²	C₂²×D₆₇	1	1	yes	characteristic
321	C₂×C₄²	C₂²×D₆₇	1	1	yes	characteristic
322	C₂×C₄²	C₂²×D₆₇	1	1	yes	characteristic
323	C₂×C₄²	C₂²×D₆₇	1	1	yes	characteristic
324	C₂×C₄²	C₂²×D₆₇	1	1	yes	characteristic
325	C₂×C₄²	C₂²×D₆₇	1	1	yes	characteristic
326	C₂×C₄²	C₂²×D₆₇	1	1	yes	characteristic
327	C₄⋊C₈	—	268	2	no	—
328	C₄⋊C₈	—	67	4	no	—
329	C₄⋊C₈	—	134	4	no	—
330	C₄⋊C₈	—	134	2	no	—
331	C₄⋊C₈	—	134	2	no	—
332	C₄⋊C₈	—	134	2	no	—
333	#97	—	268	2	no	—
334	#97	—	134	4	no	—
335	C₄×C₈	—	268	2	no	—
336	#95	C₂×Dic₆₇	1	1	yes	characteristic
337	#95	C₂×Dic₆₇	1	1	yes	characteristic
338	#95	C₂×Dic₆₇	1	1	yes	characteristic
339	#95	C₂×Dic₆₇	1	1	yes	characteristic
340	#95	C₂×Dic₆₇	1	1	yes	characteristic
341	#95	C₂×Dic₆₇	1	1	yes	characteristic
342	C₂×Q₈	—	2	2	no	—
343	C₂×Q₈	—	2	2	no	—
344	C₂×Q₈	—	2	2	no	—
345	C₂²×C₄	C₂²×Dic₆₇	1	1	yes	characteristic
346	C₂²×C₄	C₂²×Dic₆₇	1	1	yes	characteristic
347	C₂²×C₄	C₂²×Dic₆₇	1	1	yes	characteristic
348	C₂²×C₄	C₂²×Dic₆₇	1	1	yes	characteristic
349	C₂²×C₄	C₂²×Dic₆₇	1	1	yes	characteristic
350	C₂²×C₄	C₂²×Dic₆₇	1	1	yes	characteristic
351	C₂²×C₄	C₂²×Dic₆₇	1	1	yes	characteristic
352	C₂²×C₄	C₂²×Dic₆₇	1	1	yes	characteristic
353	C₂²×C₄	C₂²×Dic₆₇	1	1	yes	characteristic
354	C₂²×C₄	C₂²×Dic₆₇	1	1	yes	characteristic
355	C₂²×C₄	C₂²×Dic₆₇	1	1	yes	characteristic
356	C₂²×C₄	C₂²×Dic₆₇	1	1	yes	characteristic
357	C₂²×C₄	C₂²×Dic₆₇	1	1	yes	characteristic
358	C₂²×C₄	C₂²×Dic₆₇	1	1	yes	characteristic
359	C₂²×C₄	C₂³×D₆₇	1	1	yes	characteristic
360	C₂×C₈	—	67	4	no	—
361	C₂×C₈	—	134	2	no	—
362	C₂×C₈	—	134	2	no	—
363	C₂×C₈	—	134	2	no	—
364	C₂×C₈	—	134	2	no	—
365	C₂×C₈	—	134	2	no	—
366	C₂×C₈	—	134	2	no	—
367	C₂×C₈	—	134	2	no	—
368	C₂×C₈	—	134	2	no	—
369	C₂×C₈	—	134	2	no	—
370	C₂×C₈	—	134	2	no	—
371	C₂×C₈	—	134	2	no	—
372	C₄⋊C₄	—	2	2	no	—
373	C₄⋊C₄	—	2	2	no	—
374	C₄⋊C₄	—	2	2	no	—
375	C₄⋊C₄	—	2	2	no	—
376	C₄⋊C₄	—	2	2	no	—
377	C₄⋊C₄	—	2	2	no	—
378	C₄⋊C₄	—	2	2	no	—
379	C₄⋊C₄	—	2	2	no	—
380	C₄⋊C₄	#11761612	1	4	yes	—
381	C₄²	—	2	2	no	—
382	C₄²	—	2	2	no	—
383	C₄²	—	2	2	no	—
384	C₄²	—	2	2	no	—
385	C₄²	—	2	2	no	—
386	C₄²	—	2	2	no	—
387	C₄²	—	2	2	no	—
388	C₄²	—	2	2	no	—
389	C₄²	—	2	1	no	—
390	C₄²	—	2	1	no	—
391	C₄²	—	2	1	no	—
392	C₄²	—	2	1	no	—
393	C₄²	—	2	1	no	—
394	C₄²	—	2	1	no	—
395	C₄²	—	2	1	no	—
396	C₄²	—	2	1	no	—
397	C₄²	—	2	1	no	—
398	C₄²	—	2	1	no	—
399	C₄²	—	4	1	no	—
400	C₄²	—	4	1	no	—
401	C₄²	—	4	1	no	—
402	C₄²	#11761612	1	4	yes	—
403	C₄²	Q₈×D₆₇	1	2	yes	—
404	C₄²	#11761628	1	2	yes	—
405	C₄²	#11761628	1	2	yes	—
406	C₄²	#11761629	1	2	yes	—
407	C₂³	C₂³×Dic₆₇	1	1	yes	characteristic
408	Q₈	—	2	8	no	—
409	C₂×C₄	—	2	1	no	—
410	C₂×C₄	—	2	1	no	—
411	C₂×C₄	—	2	1	no	—
412	C₂×C₄	—	2	1	no	—
413	C₂×C₄	—	2	1	no	—
414	C₂×C₄	—	2	1	no	—
415	C₂×C₄	—	2	1	no	—
416	C₂×C₄	—	2	1	no	—
417	C₂×C₄	—	2	1	no	—
418	C₂×C₄	—	2	1	no	—
419	C₂×C₄	—	2	1	no	—
420	C₂×C₄	—	2	1	no	—
421	C₂×C₄	—	2	1	no	—
422	C₂×C₄	—	2	1	no	—
423	C₂×C₄	—	2	1	no	—
424	C₂×C₄	—	2	1	no	—
425	C₂×C₄	—	2	1	no	—
426	C₂×C₄	—	2	1	no	—
427	C₂×C₄	—	2	1	no	—
428	C₂×C₄	—	2	1	no	—
429	C₂×C₄	—	2	1	no	—
430	C₂×C₄	—	2	1	no	—
431	C₂×C₄	—	2	1	no	—
432	C₂×C₄	—	2	1	no	—
433	C₂×C₄	—	2	1	no	—
434	C₂×C₄	—	2	1	no	—
435	C₂×C₄	—	2	1	no	—
436	C₂×C₄	—	2	1	no	—
437	C₂×C₄	—	2	1	no	—
438	C₂×C₄	—	2	1	no	—
439	C₂×C₄	—	2	1	no	—
440	C₂×C₄	#14524002	1	2	yes	—
441	C₂×C₄	#14524002	1	2	yes	—
442	C₂×C₄	Q₈×Dic₆₇	1	2	yes	—
443	C₂×C₄	Q₈×Dic₆₇	1	2	yes	—
444	C₂×C₄	C₂×#11761612	1	2	yes	—
445	C₂×C₄	C₂×#11761612	1	2	yes	—
446	C₂×C₄	C₂×#11761612	1	2	yes	—
447	C₂×C₄	C₂×#11761612	1	2	yes	—
448	C₂×C₄	C₂×#11761612	1	2	yes	—
449	C₂×C₄	C₂×#11761612	1	2	yes	—
450	C₂×C₄	#14524050	1	2	yes	—
451	C₂×C₄	C₂×Q₈×D₆₇	1	1	yes	characteristic
452	C₂×C₄	C₂×#11761628	1	1	yes	characteristic
453	C₂×C₄	C₂×#11761628	1	1	yes	characteristic
454	C₂×C₄	C₂×#11761629	1	1	yes	characteristic
455	C₂×C₄	#14524089	1	1	yes	characteristic
456	C₂×C₄	#14524090	1	1	yes	characteristic
457	C₈	—	268	2	no	—
458	C₈	—	268	2	no	—
459	C₈	—	268	2	no	—
460	C₈	—	134	4	no	—
461	C₂²	C₂×#14524002	1	1	yes	characteristic
462	C₂²	C₂×Q₈×Dic₆₇	1	1	yes	characteristic
463	C₂²	#16863879	1	1	yes	characteristic
464	C₂²	C₂²×#11761612	1	1	yes	characteristic
465	C₂²	C₂²×#11761612	1	1	yes	characteristic
466	C₂²	C₂×#14524050	1	1	yes	characteristic
467	C₂²	#16864163	1	1	yes	characteristic
468	C₄	—	2	2	no	—
469	C₄	—	2	2	no	—
470	C₄	—	2	2	no	—
471	C₄	—	2	2	no	—
472	C₄	—	2	2	no	—
473	C₄	—	2	2	no	—
474	C₄	—	2	2	no	—
475	C₄	—	2	2	no	—
476	C₄	—	4	1	no	—
477	C₄	—	4	1	no	—
478	C₄	—	4	1	no	—
479	C₄	—	4	1	no	—
480	C₄	—	4	1	no	—
481	C₄	#16863527	1	4	yes	—
482	C₄	#16864106	1	1	yes	characteristic
483	C₄	#16864119	1	1	yes	characteristic
484	C₄	#16864124	1	1	yes	characteristic
485	C₄	#16864131	1	1	yes	characteristic
486	C₂	#22563166	1	1	yes	characteristic
487	C₂	#22568842	1	1	yes	characteristic
488	C₂	#22568875	1	1	yes	characteristic
489	C₂	C₂×#16863527	1	1	yes	characteristic
490	C₂	Q₈×#11761612	1	1	yes	characteristic
491	C₂	#22569326	1	1	yes	characteristic
492	C₂	#22569334	1	1	yes	characteristic

	subgroup	G/H
center, Frattini	C₂²×C₄	C₂³×D₆₇
commutator, socle, prime radical	C₂³×C₆₇	C₂³×C₄
Fitting	C₄²×Q₈×C₆₇	C₂
cosocle	C₂²×C₄×C₆₇	C₂⁴
hypercenter	C₄²×Q₈	D₆₇
hypocenter	C₆₇	#22823
LCS	G ▷ C₂³×C₆₇ ▷ C₆₇
UCS	C₁ ◁ C₂²×C₄ ◁ C₄²×Q₈

Complements

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

N	H ≅ G/N	type
52. C₄⋊(C₆₇⋊C₈)	Q₈	semidirect
258. C₆₇	#22823	semidirect

Maximal subgroups

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

subgroup	index
#22561150	2
#22560775	2
#22560768	2
#22560361	2
#22560326	2
#22560084	2
#22560062	2
C₄×C₄⋊(C₆₇⋊C₈)	2
C₄²×Q₈×C₆₇	2
#22823	67