Subgroups — #30955584

Order 17152.

710 conjugacy classes of subgroups.

The subgroup lattice has 362 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#22568572	C₂	1	1	yes	maximal, characteristic
2	#22561766	C₂	1	2	yes	maximal
3	#22561452	C₂	1	2	yes	maximal
4	#22560902	C₂	1	4	yes	maximal
5	#22560214	C₂	1	2	yes	maximal
6	#22560214	C₂	1	2	yes	maximal
7	#22560062	C₂	1	1	yes	maximal, characteristic
8	C₄×C₄⋊Q₈×C₆₇	C₂	1	1	yes	maximal, characteristic
9	C₂×#14524035	—	2	1	no	—
10	C₂×C₄⋊(C₆₇⋊C₈)	C₂²	1	2	yes	—
11	C₂×#14523967	C₂²	1	4	yes	—
12	C₂×#14523967	C₂²	1	4	yes	—
13	C₂×#14523954	C₂²	1	2	yes	—
14	#16863294	—	2	1	no	—
15	#16863294	—	2	1	no	—
16	#16863294	C₂²	1	2	yes	—
17	#16863294	C₂²	1	2	yes	—
18	#16863266	C₂²	1	1	yes	characteristic
19	#16863266	C₂²	1	1	yes	characteristic
20	#16863262	—	2	1	no	—
21	#16863158	C₂²	1	4	yes	—
22	#16863158	C₂²	1	4	yes	—
23	#16863151	C₂²	1	2	yes	—
24	C₂×C₄⋊Q₈×C₆₇	C₂²	1	1	yes	characteristic
25	C₂×C₄×Q₈×C₆₇	C₂²	1	2	yes	—
26	#391×C₆₇	C₂²	1	2	yes	—
27	#389×C₆₇	—	2	1	no	—
28	#389×C₆₇	—	2	1	no	—
29	#384×C₆₇	C₂²	1	1	yes	characteristic
30	C₄×C₄⋊C₄×C₆₇	—	2	1	no	—
31	C₄×C₄⋊C₄×C₆₇	—	2	1	no	—
32	C₄³×C₆₇	C₂²	1	1	yes	characteristic
33	C₂²×C₆₇⋊C₈	C₂³	1	4	yes	—
34	#14524035	—	4	4	no	—
35	C₂×C₄⋊Dic₆₇	—	2	1	no	—
36	C₂×C₄⋊Dic₆₇	—	2	1	no	—
37	C₂×C₄⋊Dic₆₇	—	2	1	no	—
38	C₂×C₄⋊Dic₆₇	—	2	1	no	—
39	C₂×C₄⋊Dic₆₇	C₂³	1	2	yes	—
40	C₂×C₄⋊Dic₆₇	C₂³	1	2	yes	—
41	C₂×Dic₆₇⋊C₄	—	2	2	no	—
42	C₂×Dic₆₇⋊C₄	—	2	2	no	—
43	C₄⋊(C₆₇⋊C₈)	—	2	4	no	—
44	#14523967	—	2	8	no	—
45	#14523967	—	4	4	no	—
46	#14523954	—	4	2	no	—
47	C₆₇⋊#95	—	2	2	no	—
48	C₆₇⋊#95	—	2	2	no	—
49	C₂²×Q₈×C₆₇	C₂³	1	2	yes	—
50	C₄⋊Q₈×C₆₇	—	2	4	no	—
51	C₄×Q₈×C₆₇	—	2	4	no	—
52	C₄×Q₈×C₆₇	—	2	4	no	—
53	C₂×C₄⋊C₄×C₆₇	—	2	1	no	—
54	C₂×C₄⋊C₄×C₆₇	—	2	1	no	—
55	C₂×C₄⋊C₄×C₆₇	—	2	2	no	—
56	C₂×C₄⋊C₄×C₆₇	—	2	1	no	—
57	C₂×C₄⋊C₄×C₆₇	—	2	1	no	—
58	C₂×C₄⋊C₄×C₆₇	C₂³	1	2	yes	—
59	C₂×C₄²×C₆₇	—	2	2	no	—
60	C₂×C₄²×C₆₇	D₄	1	1	yes	characteristic
61	C₂×C₄²×C₆₇	D₄	1	1	yes	characteristic
62	C₂×C₄²×C₆₇	D₄	1	1	yes	characteristic
63	C₂×C₄²×C₆₇	D₄	1	1	yes	characteristic
64	C₂×C₄²×C₆₇	C₂³	1	1	yes	characteristic
65	C₂×C₄²×C₆₇	C₂³	1	1	yes	characteristic
66	C₂×C₄²×C₆₇	C₂³	1	1	yes	characteristic
67	#95×C₆₇	—	2	2	no	—
68	#95×C₆₇	—	2	2	no	—
69	C₂²×Dic₆₇	—	2	2	no	—
70	C₂²×Dic₆₇	—	2	2	no	—
71	C₂×C₆₇⋊C₈	—	2	4	no	—
72	C₂×C₆₇⋊C₈	—	2	4	no	—
73	C₂×C₆₇⋊C₈	—	2	4	no	—
74	C₄⋊Dic₆₇	—	2	4	no	—
75	C₄⋊Dic₆₇	—	2	2	no	—
76	C₄⋊Dic₆₇	—	2	2	no	—
77	C₄⋊Dic₆₇	—	4	2	no	—
78	C₄⋊Dic₆₇	—	4	2	no	—
79	C₄⋊Dic₆₇	—	4	2	no	—
80	C₄⋊Dic₆₇	—	4	2	no	—
81	Dic₆₇⋊C₄	—	4	4	no	—
82	Dic₆₇⋊C₄	—	4	4	no	—
83	C₂×Q₈×C₆₇	—	2	2	no	—
84	C₂×Q₈×C₆₇	—	2	2	no	—
85	C₂×Q₈×C₆₇	—	2	2	no	—
86	C₂×Q₈×C₆₇	—	2	2	no	—
87	C₂×Q₈×C₆₇	—	2	2	no	—
88	C₂×Q₈×C₆₇	#41	1	4	yes	—
89	C₂²×C₄×C₆₇	—	2	2	no	—
90	C₂²×C₄×C₆₇	—	2	2	no	—
91	C₂²×C₄×C₆₇	C₂×D₄	1	1	yes	characteristic
92	C₂²×C₄×C₆₇	C₂×D₄	1	1	yes	characteristic
93	C₂²×C₄×C₆₇	C₂×D₄	1	1	yes	characteristic
94	C₂²×C₄×C₆₇	C₂×D₄	1	1	yes	characteristic
95	C₂²×C₄×C₆₇	C₂×D₄	1	1	yes	characteristic
96	C₂²×C₄×C₆₇	C₂×D₄	1	1	yes	characteristic
97	C₂²×C₄×C₆₇	C₂⁴	1	1	yes	characteristic
98	C₄⋊C₄×C₆₇	—	2	4	no	—
99	C₄⋊C₄×C₆₇	—	2	4	no	—
100	C₄⋊C₄×C₆₇	—	2	2	no	—
101	C₄⋊C₄×C₆₇	—	2	2	no	—
102	C₄⋊C₄×C₆₇	—	4	4	no	—
103	C₄⋊C₄×C₆₇	—	4	2	no	—
104	C₄⋊C₄×C₆₇	—	4	2	no	—
105	C₄²×C₆₇	—	2	2	no	—
106	C₄²×C₆₇	—	2	2	no	—
107	C₄²×C₆₇	—	2	2	no	—
108	C₄²×C₆₇	—	2	2	no	—
109	C₄²×C₆₇	—	2	2	no	—
110	C₄²×C₆₇	—	2	1	no	—
111	C₄²×C₆₇	—	2	1	no	—
112	C₄²×C₆₇	—	4	4	no	—
113	C₄²×C₆₇	#41	1	4	yes	—
114	C₂³×C₆₇	C₂²×D₄	1	1	yes	characteristic
115	Q₈×C₆₇	—	2	4	no	—
116	Q₈×C₆₇	—	2	4	no	—
117	Q₈×C₆₇	—	4	2	no	—
118	Q₈×C₆₇	—	4	2	no	—
119	C₂×C₄×C₆₇	—	2	1	no	—
120	C₂×C₄×C₆₇	—	2	1	no	—
121	C₂×C₄×C₆₇	—	2	1	no	—
122	C₂×C₄×C₆₇	—	2	1	no	—
123	C₂×C₄×C₆₇	—	2	1	no	—
124	C₂×C₄×C₆₇	—	2	1	no	—
125	C₂×C₄×C₆₇	—	2	4	no	—
126	C₂×C₄×C₆₇	—	2	1	no	—
127	C₂×C₄×C₆₇	—	2	1	no	—
128	C₂×C₄×C₆₇	—	2	1	no	—
129	C₂×C₄×C₆₇	—	2	1	no	—
130	C₂×C₄×C₆₇	—	2	1	no	—
131	C₂×C₄×C₆₇	—	2	1	no	—
132	C₂×C₄×C₆₇	—	4	2	no	—
133	C₂×C₄×C₆₇	—	4	2	no	—
134	C₂×C₄×C₆₇	—	4	2	no	—
135	C₂×C₄×C₆₇	—	4	2	no	—
136	C₂×C₄×C₆₇	—	4	2	no	—
137	C₂×C₄×C₆₇	#124	1	2	yes	—
138	C₂×C₄×C₆₇	#124	1	2	yes	—
139	C₂×C₄×C₆₇	AGL(1,ℤ/8)	1	2	yes	—
140	C₂×C₄×C₆₇	#137	1	2	yes	—
141	C₂×C₄×C₆₇	#137	1	2	yes	—
142	C₂×C₄×C₆₇	#137	1	2	yes	—
143	C₂×C₄×C₆₇	C₂×#41	1	2	yes	—
144	C₂×C₄×C₆₇	C₂×#41	1	2	yes	—
145	C₂×C₄×C₆₇	H(5,𝔽₂)	1	1	yes	characteristic
146	C₂×C₄×C₆₇	2₋¹⁺⁴	1	1	yes	characteristic
147	C₂×Dic₆₇	—	4	2	no	—
148	C₂×Dic₆₇	—	4	2	no	—
149	C₂×Dic₆₇	—	4	2	no	—
150	C₂×Dic₆₇	—	4	2	no	—
151	C₂×Dic₆₇	—	4	2	no	—
152	C₂×Dic₆₇	—	4	2	no	—
153	C₆₇⋊C₈	—	4	4	no	—
154	C₂²×C₆₇	#525	1	1	yes	characteristic
155	C₂²×C₆₇	C₂×#124	1	1	yes	characteristic
156	C₂²×C₆₇	#537	1	1	yes	characteristic
157	C₂²×C₆₇	C₂×AGL(1,ℤ/8)	1	1	yes	characteristic
158	C₂²×C₆₇	C₂×#137	1	1	yes	characteristic
159	C₂²×C₆₇	C₂×#137	1	1	yes	characteristic
160	C₂²×C₆₇	C₂×#137	1	1	yes	characteristic
161	C₄×C₆₇	—	2	2	no	—
162	C₄×C₆₇	—	2	2	no	—
163	C₄×C₆₇	—	2	2	no	—
164	C₄×C₆₇	—	4	4	no	—
165	C₄×C₆₇	—	4	1	no	—
166	C₄×C₆₇	—	4	1	no	—
167	C₄×C₆₇	—	8	2	no	—
168	C₄×C₆₇	#491	1	4	yes	—
169	C₄×C₆₇	#552	1	2	yes	—
170	C₄×C₆₇	#553	1	2	yes	—
171	Dic₆₇	—	8	2	no	—
172	Dic₆₇	—	8	2	no	—
173	#29132	—	67	1	no	maximal, Sylow
174	C₂×C₆₇	#2562	1	1	yes	characteristic
175	C₂×C₆₇	#3112	1	1	yes	characteristic
176	C₂×C₆₇	#3113	1	1	yes	characteristic
177	C₂×C₆₇	C₂×#491	1	1	yes	characteristic
178	C₂×C₆₇	#3141	1	1	yes	characteristic
179	C₂×C₆₇	#3254	1	1	yes	characteristic
180	C₂×C₆₇	#3255	1	1	yes	characteristic
181	#2859	—	67	1	no	—
182	C₄×C₄⋊Q₈	D₆₇	1	1	yes	characteristic
183	#2099	—	67	2	no	—
184	#2043	—	67	2	no	—
185	#1935	—	67	4	no	—
186	#1803	—	67	2	no	—
187	#1803	—	67	2	no	—
188	#1767	—	67	1	no	—
189	C₆₇	#29132	1	1	yes	characteristic, complemented, Sylow
190	C₂×C₄⋊Q₈	C₂×D₆₇	1	1	yes	characteristic
191	C₂×#125	—	134	1	no	—
192	C₂×C₄×Q₈	C₂×D₆₇	1	2	yes	—
193	C₂×C₄⋊C₈	—	67	2	no	—
194	C₂×#103	—	67	4	no	—
195	C₂×#103	—	67	4	no	—
196	C₂×#97	—	67	2	no	—
197	#391	C₂×D₆₇	1	2	yes	—
198	#389	—	2	1	no	—
199	#389	—	2	1	no	—
200	#389	—	67	2	no	—
201	#389	—	67	2	no	—
202	#389	—	134	1	no	—
203	#389	—	134	1	no	—
204	#384	—	67	1	no	—
205	#384	—	67	1	no	—
206	#384	C₂×D₆₇	1	1	yes	characteristic
207	#382	—	134	1	no	—
208	C₄×C₄⋊C₄	—	2	1	no	—
209	C₄×C₄⋊C₄	—	2	1	no	—
210	C₄³	C₂×D₆₇	1	1	yes	characteristic
211	#340	—	67	4	no	—
212	#340	—	67	4	no	—
213	#336	—	67	2	no	—
214	C₂²×Q₈	C₂²×D₆₇	1	2	yes	—
215	C₂²×C₈	—	67	4	no	—
216	C₄⋊Q₈	—	2	4	no	—
217	#125	—	268	4	no	—
218	C₄×Q₈	—	2	4	no	—
219	C₄×Q₈	—	2	4	no	—
220	C₂×C₄⋊C₄	—	2	1	no	—
221	C₂×C₄⋊C₄	—	2	1	no	—
222	C₂×C₄⋊C₄	—	2	2	no	—
223	C₂×C₄⋊C₄	—	2	1	no	—
224	C₂×C₄⋊C₄	—	2	1	no	—
225	C₂×C₄⋊C₄	—	67	2	no	—
226	C₂×C₄⋊C₄	—	67	2	no	—
227	C₂×C₄⋊C₄	—	134	2	no	—
228	C₂×C₄⋊C₄	—	134	2	no	—
229	C₂×C₄⋊C₄	—	134	1	no	—
230	C₂×C₄⋊C₄	—	134	1	no	—
231	C₂×C₄⋊C₄	—	134	1	no	—
232	C₂×C₄⋊C₄	—	134	1	no	—
233	C₂×C₄⋊C₄	C₂²×D₆₇	1	2	yes	—
234	C₂×C₄²	—	2	2	no	—
235	C₂×C₄²	C₂²⋊D₆₇	1	1	yes	characteristic
236	C₂×C₄²	C₂²⋊D₆₇	1	1	yes	characteristic
237	C₂×C₄²	C₂²⋊D₆₇	1	1	yes	characteristic
238	C₂×C₄²	C₂²⋊D₆₇	1	1	yes	characteristic
239	C₂×C₄²	C₂²×D₆₇	1	1	yes	characteristic
240	C₂×C₄²	C₂²×D₆₇	1	1	yes	characteristic
241	C₂×C₄²	C₂²×D₆₇	1	1	yes	characteristic
242	C₄⋊C₈	—	134	4	no	—
243	#103	—	268	4	no	—
244	#103	—	134	8	no	—
245	#97	—	268	2	no	—
246	#95	—	2	2	no	—
247	#95	—	2	2	no	—
248	#95	—	134	2	no	—
249	#95	—	134	2	no	—
250	C₂×Q₈	—	2	2	no	—
251	C₂×Q₈	—	2	2	no	—
252	C₂×Q₈	—	2	2	no	—
253	C₂×Q₈	—	2	2	no	—
254	C₂×Q₈	—	2	2	no	—
255	C₂×Q₈	#11761628	1	4	yes	—
256	C₂²×C₄	—	2	2	no	—
257	C₂²×C₄	—	2	2	no	—
258	C₂²×C₄	—	134	2	no	—
259	C₂²×C₄	—	134	2	no	—
260	C₂²×C₄	C₂×C₂²⋊D₆₇	1	1	yes	characteristic
261	C₂²×C₄	C₂×C₂²⋊D₆₇	1	1	yes	characteristic
262	C₂²×C₄	C₂×C₂²⋊D₆₇	1	1	yes	characteristic
263	C₂²×C₄	C₂×C₂²⋊D₆₇	1	1	yes	characteristic
264	C₂²×C₄	C₂×C₂²⋊D₆₇	1	1	yes	characteristic
265	C₂²×C₄	C₂×C₂²⋊D₆₇	1	1	yes	characteristic
266	C₂²×C₄	C₂³×D₆₇	1	1	yes	characteristic
267	C₂×C₈	—	134	4	no	—
268	C₂×C₈	—	134	4	no	—
269	C₂×C₈	—	134	4	no	—
270	C₄⋊C₄	—	2	4	no	—
271	C₄⋊C₄	—	2	4	no	—
272	C₄⋊C₄	—	2	2	no	—
273	C₄⋊C₄	—	2	2	no	—
274	C₄⋊C₄	—	4	4	no	—
275	C₄⋊C₄	—	4	2	no	—
276	C₄⋊C₄	—	4	2	no	—
277	C₄⋊C₄	—	268	4	no	—
278	C₄⋊C₄	—	268	2	no	—
279	C₄⋊C₄	—	268	2	no	—
280	C₄⋊C₄	—	268	4	no	—
281	C₄⋊C₄	—	268	2	no	—
282	C₄⋊C₄	—	268	2	no	—
283	C₄⋊C₄	—	134	4	no	—
284	C₄⋊C₄	—	134	2	no	—
285	C₄⋊C₄	—	134	2	no	—
286	C₄²	—	2	2	no	—
287	C₄²	—	2	2	no	—
288	C₄²	—	2	2	no	—
289	C₄²	—	2	2	no	—
290	C₄²	—	2	2	no	—
291	C₄²	—	2	1	no	—
292	C₄²	—	2	1	no	—
293	C₄²	—	4	4	no	—
294	C₄²	#11761627	1	4	yes	—
295	C₂³	C₂²×C₂²⋊D₆₇	1	1	yes	characteristic
296	Q₈	—	2	4	no	—
297	Q₈	—	2	4	no	—
298	Q₈	—	4	2	no	—
299	Q₈	—	4	2	no	—
300	C₂×C₄	—	2	1	no	—
301	C₂×C₄	—	2	1	no	—
302	C₂×C₄	—	2	1	no	—
303	C₂×C₄	—	2	1	no	—
304	C₂×C₄	—	2	1	no	—
305	C₂×C₄	—	2	1	no	—
306	C₂×C₄	—	2	4	no	—
307	C₂×C₄	—	2	1	no	—
308	C₂×C₄	—	2	1	no	—
309	C₂×C₄	—	2	1	no	—
310	C₂×C₄	—	2	1	no	—
311	C₂×C₄	—	2	1	no	—
312	C₂×C₄	—	2	1	no	—
313	C₂×C₄	—	4	2	no	—
314	C₂×C₄	—	4	2	no	—
315	C₂×C₄	—	4	2	no	—
316	C₂×C₄	—	4	2	no	—
317	C₂×C₄	—	4	2	no	—
318	C₂×C₄	—	268	2	no	—
319	C₂×C₄	—	268	2	no	—
320	C₂×C₄	—	268	2	no	—
321	C₂×C₄	—	268	2	no	—
322	C₂×C₄	—	268	2	no	—
323	C₂×C₄	—	268	2	no	—
324	C₂×C₄	#14524030	1	2	yes	—
325	C₂×C₄	#14524030	1	2	yes	—
326	C₂×C₄	#14524068	1	2	yes	—
327	C₂×C₄	#14524073	1	2	yes	—
328	C₂×C₄	#14524074	1	2	yes	—
329	C₂×C₄	#14524074	1	2	yes	—
330	C₂×C₄	C₂×#11761627	1	2	yes	—
331	C₂×C₄	C₂×#11761628	1	2	yes	—
332	C₂×C₄	#14524089	1	1	yes	characteristic
333	C₂×C₄	#14524090	1	1	yes	characteristic
334	C₈	—	268	4	no	—
335	C₂²	#16863907	1	1	yes	characteristic
336	C₂²	C₂×#14524030	1	1	yes	characteristic
337	C₂²	#16863974	1	1	yes	characteristic
338	C₂²	C₂×#14524068	1	1	yes	characteristic
339	C₂²	C₂×#14524073	1	1	yes	characteristic
340	C₂²	C₂×#14524074	1	1	yes	characteristic
341	C₂²	C₂×#14524074	1	1	yes	characteristic
342	C₄	—	2	2	no	—
343	C₄	—	2	2	no	—
344	C₄	—	2	2	no	—
345	C₄	—	4	1	no	—
346	C₄	—	4	1	no	—
347	C₄	—	4	4	no	—
348	C₄	—	8	2	no	—
349	C₄	—	536	2	no	—
350	C₄	—	536	2	no	—
351	C₄	#16863764	1	4	yes	—
352	C₄	#16864086	1	2	yes	—
353	C₄	#16864092	1	2	yes	—
354	C₂	#22565375	1	1	yes	characteristic
355	C₂	#22570331	1	1	yes	characteristic
356	C₂	#22570339	1	1	yes	characteristic
357	C₂	C₂×#16863764	1	1	yes	characteristic
358	C₂	#22570602	1	1	yes	characteristic
359	C₂	#22571592	1	1	yes	characteristic
360	C₂	#22571600	1	1	yes	characteristic

	subgroup	G/H
center	C₂³	C₂²×C₂²⋊D₆₇
commutator, cosocle	C₂²×C₄×C₆₇	C₂⁴
Frattini	C₂²×C₄	C₂³×D₆₇
Fitting	C₄×C₄⋊Q₈×C₆₇	C₂
socle, prime radical	C₂³×C₆₇	C₂²×D₄
hypercenter	C₄×C₄⋊Q₈	D₆₇
hypocenter	C₆₇	#29132
LCS	G ▷ C₂²×C₄×C₆₇ ▷ C₂×C₆₇ ▷ C₆₇
UCS	C₁ ◁ C₂³ ◁ C₄³ ◁ C₄×C₄⋊Q₈

Complements

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

N	H ≅ G/N	type
189. C₆₇	#29132	semidirect

Maximal subgroups

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

subgroup	index
#22568572	2
#22561766	2
#22561452	2
#22560902	2
#22560214	2
#22560062	2
C₄×C₄⋊Q₈×C₆₇	2
#29132	67