Subgroups — #30749145

Order 17152.

384 conjugacy classes of subgroups.

The subgroup lattice has 306 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₄×#14523975	C₂	1	1	yes	maximal, characteristic, complemented
2	#22558972	C₂	1	1	yes	maximal, characteristic, complemented
3	#22558734	C₂	1	2	yes	maximal
4	#22558066	C₂	1	2	yes	maximal, complemented
5	C₆₇×#2966	C₂	1	1	yes	maximal, characteristic
6	#16863453	C₂²	1	2	yes	—
7	C₂×#14523975	—	2	1	no	—
8	C₄×C₄⋊Dic₆₇	C₂²	1	2	yes	—
9	#16863159	—	2	1	no	—
10	#16863149	—	2	1	no	—
11	#16863147	—	2	1	no	—
12	#16863133	—	2	2	no	—
13	#16863130	—	2	2	no	—
14	#529×C₆₇	C₂²	1	1	yes	characteristic
15	#446×C₆₇	C₄	1	1	yes	characteristic, complemented
16	C₈×Q₈×C₆₇	C₄	1	1	yes	characteristic, complemented
17	#436×C₆₇	C₄	1	1	yes	characteristic, complemented
18	#435×C₆₇	—	2	1	no	—
19	C₈×D₄×C₆₇	—	2	1	no	—
20	C₈×D₄×C₆₇	C₄	1	1	yes	characteristic, complemented
21	#433×C₆₇	—	2	1	no	—
22	#432×C₆₇	C₂²	1	1	yes	characteristic
23	#405×C₆₇	—	2	1	no	—
24	C₂×C₄×C₈×C₆₇	C₂²	1	1	yes	characteristic
25	C₂×C₄⋊Dic₆₇	—	2	2	no	—
26	C₂×C₄×Dic₆₇	—	2	2	no	—
27	#14523975	—	2	4	no	—
28	C₄⋊(C₆₇⋊C₈)	—	2	2	no	—
29	C₂²⋊(C₆₇⋊C₈)	—	2	2	no	—
30	#14523954	—	2	2	no	—
31	C₆₇⋊#95	—	2	2	no	—
32	C₂×M₁₆×C₆₇	—	2	1	no	—
33	C₂²×C₈×C₆₇	—	2	1	no	—
34	C₂²×C₈×C₆₇	D₄	1	1	yes	characteristic
35	C₂²×C₈×C₆₇	D₄	1	1	yes	characteristic
36	#126×C₆₇	—	2	1	no	—
37	#125×C₆₇	Q₈	1	1	yes	characteristic
38	#124×C₆₇	Q₈	1	1	yes	characteristic
39	#123×C₆₇	—	2	1	no	—
40	C₂²⋊Q₈×C₆₇	D₄	1	1	yes	characteristic
41	#121×C₆₇	D₄	1	1	yes	characteristic
42	C₄×Q₈×C₆₇	C₂×C₄	1	1	yes	characteristic
43	C₄×D₄×C₆₇	—	2	1	no	—
44	C₄×D₄×C₆₇	C₂×C₄	1	1	yes	characteristic
45	#117×C₆₇	—	2	1	no	—
46	C₂×C₄²×C₆₇	C₂³	1	1	yes	characteristic
47	C₄⋊C₈×C₆₇	—	2	1	no	—
48	C₄⋊C₈×C₆₇	—	2	1	no	—
49	C₄⋊C₈×C₆₇	C₂×C₄	1	1	yes	characteristic
50	C₄⋊C₈×C₆₇	C₂×C₄	1	1	yes	characteristic
51	C₂²⋊C₈×C₆₇	—	2	1	no	—
52	C₂²⋊C₈×C₆₇	—	2	1	no	—
53	C₂²⋊C₈×C₆₇	—	2	1	no	—
54	#97×C₆₇	—	2	1	no	—
55	#97×C₆₇	—	2	1	no	—
56	C₄×C₈×C₆₇	—	2	1	no	—
57	C₄×C₈×C₆₇	C₂×C₄	1	1	yes	characteristic
58	C₄×C₈×C₆₇	C₂×C₄	1	1	yes	characteristic
59	C₄×C₈×C₆₇	D₄	1	1	yes	characteristic
60	C₄×C₈×C₆₇	D₄	1	1	yes	characteristic
61	C₂²×Dic₆₇	—	4	2	no	—
62	C₂×C₆₇⋊C₈	—	4	2	no	—
63	C₄⋊Dic₆₇	—	2	4	no	—
64	C₄⋊Dic₆₇	—	4	2	no	—
65	C₄×Dic₆₇	—	8	2	no	—
66	C₂×Q₈×C₆₇	C₄⋊C₄	1	1	yes	characteristic
67	C₂×D₄×C₆₇	—	2	1	no	—
68	C₂×D₄×C₆₇	C₄⋊C₄	1	1	yes	characteristic
69	C₂²×C₄×C₆₇	—	2	1	no	—
70	C₂²×C₄×C₆₇	C₂×D₄	1	1	yes	characteristic
71	C₂²×C₄×C₆₇	C₂×D₄	1	1	yes	characteristic
72	C₂²×C₄×C₆₇	#41	1	1	yes	characteristic
73	M₁₆×C₆₇	—	4	2	no	—
74	C₂×C₈×C₆₇	—	2	2	no	—
75	C₂×C₈×C₆₇	—	2	1	no	—
76	C₂×C₈×C₆₇	—	2	1	no	—
77	C₂×C₈×C₆₇	—	4	1	no	—
78	C₂×C₈×C₆₇	—	4	2	no	—
79	C₂×C₈×C₆₇	C₂²⋊C₄	1	1	yes	characteristic
80	C₂×C₈×C₆₇	C₂²⋊C₄	1	1	yes	characteristic
81	C₂×C₈×C₆₇	C₂²⋊C₄	1	1	yes	characteristic
82	C₂×C₈×C₆₇	C₂²⋊C₄	1	1	yes	characteristic
83	C₄⋊C₄×C₆₇	—	2	1	no	—
84	C₄⋊C₄×C₆₇	—	2	1	no	—
85	C₄⋊C₄×C₆₇	—	2	1	no	—
86	C₄⋊C₄×C₆₇	—	2	1	no	—
87	C₄⋊C₄×C₆₇	C₄⋊C₄	1	1	yes	characteristic
88	C₄⋊C₄×C₆₇	C₄⋊C₄	1	1	yes	characteristic
89	C₂²⋊C₄×C₆₇	—	2	1	no	—
90	C₂²⋊C₄×C₆₇	—	2	1	no	—
91	C₂²⋊C₄×C₆₇	—	2	1	no	—
92	C₂²⋊C₄×C₆₇	—	2	1	no	—
93	C₂²⋊C₄×C₆₇	—	2	1	no	—
94	C₄²×C₆₇	—	2	1	no	—
95	C₄²×C₆₇	C₂²×C₄	1	1	yes	characteristic
96	C₄²×C₆₇	C₂×D₄	1	1	yes	characteristic
97	C₄²×C₆₇	C₂×Q₈	1	1	yes	characteristic
98	C₄²×C₆₇	#41	1	1	yes	characteristic
99	C₂³×C₆₇	—	2	1	no	—
100	C₂³×C₆₇	#121	1	1	yes	characteristic
101	Q₈×C₆₇	—	4	1	no	—
102	D₄×C₆₇	—	4	1	no	—
103	D₄×C₆₇	—	4	2	no	—
104	C₂×C₄×C₆₇	—	2	2	no	—
105	C₂×C₄×C₆₇	—	2	1	no	—
106	C₂×C₄×C₆₇	—	2	1	no	—
107	C₂×C₄×C₆₇	—	2	1	no	—
108	C₂×C₄×C₆₇	—	2	1	no	—
109	C₂×C₄×C₆₇	—	2	1	no	—
110	C₂×C₄×C₆₇	—	4	1	no	—
111	C₂×C₄×C₆₇	—	4	1	no	—
112	C₂×C₄×C₆₇	—	4	1	no	—
113	C₂×C₄×C₆₇	C₂×C₂²⋊C₄	1	1	yes	characteristic
114	C₂×C₄×C₆₇	C₂×C₄⋊C₄	1	1	yes	characteristic
115	C₂×C₄×C₆₇	#117	1	1	yes	characteristic
116	C₂×C₄×C₆₇	#121	1	1	yes	characteristic
117	C₂×C₄×C₆₇	C₂²⋊Q₈	1	1	yes	characteristic
118	C₂×C₄×C₆₇	C₂²⋊Q₈	1	1	yes	characteristic
119	C₂×Dic₆₇	—	4	4	no	—
120	C₂×Dic₆₇	—	8	2	no	—
121	C₂×Dic₆₇	—	8	2	no	—
122	C₈×C₆₇	—	2	1	no	—
123	C₈×C₆₇	—	2	1	no	—
124	C₈×C₆₇	—	2	1	no	—
125	C₈×C₆₇	—	4	1	no	—
126	C₈×C₆₇	—	4	1	no	—
127	C₈×C₆₇	#104	1	2	yes	—
128	C₆₇⋊C₈	—	8	2	no	—
129	C₂²×C₆₇	—	2	1	no	—
130	C₂²×C₆₇	—	2	1	no	—
131	C₂²×C₆₇	—	4	1	no	—
132	C₂²×C₆₇	—	4	1	no	—
133	C₂²×C₆₇	—	4	1	no	—
134	C₂²×C₆₇	#380	1	1	yes	characteristic
135	C₂²×C₆₇	#471	1	2	yes	—
136	C₄×C₆₇	—	2	1	no	—
137	C₄×C₆₇	—	2	1	no	—
138	C₄×C₆₇	—	2	1	no	—
139	C₄×C₆₇	—	2	1	no	—
140	C₄×C₆₇	—	2	1	no	—
141	C₄×C₆₇	—	4	1	no	—
142	C₄×C₆₇	—	4	1	no	—
143	C₄×C₆₇	—	4	1	no	—
144	C₄×C₆₇	C₂×#104	1	1	yes	characteristic
145	C₄×C₆₇	#421	1	1	yes	characteristic
146	Dic₆₇	—	8	4	no	—
147	#8696	—	67	1	no	maximal, Sylow
148	C₂×C₆₇	—	2	2	no	—
149	C₂×C₆₇	—	8	1	no	—
150	C₂×C₆₇	#1806	1	1	yes	characteristic
151	C₂×C₆₇	#1812	1	1	yes	characteristic
152	C₂×C₆₇	#1815	1	1	yes	characteristic
153	#2966	D₆₇	1	1	yes	characteristic
154	C₄×#106	—	67	1	no	—
155	#1568	—	67	1	no	—
156	#1509	—	67	2	no	—
157	#1285	—	67	2	no	—
158	C₆₇	#8696	1	1	yes	characteristic, complemented, Sylow
159	#529	C₂×D₆₇	1	1	yes	characteristic
160	#446	Dic₆₇	1	1	yes	characteristic, complemented
161	C₈×Q₈	Dic₆₇	1	1	yes	characteristic, complemented
162	#436	Dic₆₇	1	1	yes	characteristic, complemented
163	#435	—	2	1	no	—
164	C₈×D₄	—	2	1	no	—
165	C₈×D₄	Dic₆₇	1	1	yes	characteristic, complemented
166	#433	—	2	1	no	—
167	#432	—	67	2	no	—
168	#432	C₂×D₆₇	1	1	yes	characteristic
169	C₂×#106	—	134	1	no	—
170	#405	—	2	1	no	—
171	C₂×C₄×C₈	C₂×D₆₇	1	1	yes	characteristic
172	C₄×C₄⋊C₄	—	67	2	no	—
173	#340	—	134	1	no	—
174	#335	—	134	1	no	—
175	#334	—	134	1	no	—
176	#328	—	134	2	no	—
177	#327	—	134	2	no	—
178	C₂×M₁₆	—	2	1	no	—
179	C₂²×C₈	—	2	1	no	—
180	C₂²×C₈	C₂²⋊D₆₇	1	1	yes	characteristic
181	C₂²×C₈	C₂²⋊D₆₇	1	1	yes	characteristic
182	#126	—	2	1	no	—
183	#125	Dic₁₃₄	1	1	yes	characteristic
184	#124	Dic₁₃₄	1	1	yes	characteristic
185	#123	—	2	1	no	—
186	C₂²⋊Q₈	D₂₆₈	1	1	yes	characteristic
187	#121	D₂₆₈	1	1	yes	characteristic
188	C₄×Q₈	C₂×Dic₆₇	1	1	yes	characteristic
189	C₄×D₄	—	2	1	no	—
190	C₄×D₄	C₂×Dic₆₇	1	1	yes	characteristic
191	#117	—	2	1	no	—
192	C₂×C₄⋊C₄	—	134	2	no	—
193	C₂×C₄²	—	134	2	no	—
194	C₂×C₄²	C₂²×D₆₇	1	1	yes	characteristic
195	#106	—	134	4	no	—
196	C₄⋊C₈	—	2	1	no	—
197	C₄⋊C₈	—	2	1	no	—
198	C₄⋊C₈	—	134	2	no	—
199	C₄⋊C₈	C₂×Dic₆₇	1	1	yes	characteristic
200	C₄⋊C₈	C₂×Dic₆₇	1	1	yes	characteristic
201	C₂²⋊C₈	—	2	1	no	—
202	C₂²⋊C₈	—	2	1	no	—
203	C₂²⋊C₈	—	2	1	no	—
204	C₂²⋊C₈	—	134	2	no	—
205	#97	—	2	1	no	—
206	#97	—	2	1	no	—
207	#97	—	134	2	no	—
208	C₄×C₈	—	2	1	no	—
209	C₄×C₈	C₂×Dic₆₇	1	1	yes	characteristic
210	C₄×C₈	C₂×Dic₆₇	1	1	yes	characteristic
211	C₄×C₈	C₂²⋊D₆₇	1	1	yes	characteristic
212	C₄×C₈	C₂²⋊D₆₇	1	1	yes	characteristic
213	#95	—	134	2	no	—
214	C₂×Q₈	C₄⋊Dic₆₇	1	1	yes	characteristic
215	C₂×D₄	—	2	1	no	—
216	C₂×D₄	C₄⋊Dic₆₇	1	1	yes	characteristic
217	C₂²×C₄	—	2	1	no	—
218	C₂²×C₄	—	268	2	no	—
219	C₂²×C₄	C₂×C₂²⋊D₆₇	1	1	yes	characteristic
220	C₂²×C₄	C₂×D₂₆₈	1	1	yes	characteristic
221	C₂²×C₄	#11761628	1	1	yes	characteristic
222	M₁₆	—	4	2	no	—
223	C₂×C₈	—	2	2	no	—
224	C₂×C₈	—	2	1	no	—
225	C₂×C₈	—	2	1	no	—
226	C₂×C₈	—	4	1	no	—
227	C₂×C₈	—	4	2	no	—
228	C₂×C₈	—	268	2	no	—
229	C₂×C₈	C₂²⋊Dic₆₇	1	1	yes	characteristic
230	C₂×C₈	C₂²⋊Dic₆₇	1	1	yes	characteristic
231	C₂×C₈	C₂²⋊Dic₆₇	1	1	yes	characteristic
232	C₂×C₈	C₂²⋊Dic₆₇	1	1	yes	characteristic
233	C₄⋊C₄	—	2	1	no	—
234	C₄⋊C₄	—	2	1	no	—
235	C₄⋊C₄	—	2	1	no	—
236	C₄⋊C₄	—	2	1	no	—
237	C₄⋊C₄	—	268	2	no	—
238	C₄⋊C₄	—	134	4	no	—
239	C₄⋊C₄	C₄⋊Dic₆₇	1	1	yes	characteristic
240	C₄⋊C₄	C₄⋊Dic₆₇	1	1	yes	characteristic
241	C₂²⋊C₄	—	2	1	no	—
242	C₂²⋊C₄	—	2	1	no	—
243	C₂²⋊C₄	—	2	1	no	—
244	C₂²⋊C₄	—	2	1	no	—
245	C₂²⋊C₄	—	2	1	no	—
246	C₄²	—	2	1	no	—
247	C₄²	—	536	2	no	—
248	C₄²	C₂²×Dic₆₇	1	1	yes	characteristic
249	C₄²	C₂×C₂²⋊D₆₇	1	1	yes	characteristic
250	C₄²	C₂×Dic₁₃₄	1	1	yes	characteristic
251	C₄²	#11761628	1	1	yes	characteristic
252	C₂³	—	2	1	no	—
253	C₂³	#14524017	1	1	yes	characteristic
254	Q₈	—	4	1	no	—
255	D₄	—	4	1	no	—
256	D₄	—	4	2	no	—
257	C₂×C₄	—	2	2	no	—
258	C₂×C₄	—	2	1	no	—
259	C₂×C₄	—	2	1	no	—
260	C₂×C₄	—	2	1	no	—
261	C₂×C₄	—	2	1	no	—
262	C₂×C₄	—	2	1	no	—
263	C₂×C₄	—	4	1	no	—
264	C₂×C₄	—	4	1	no	—
265	C₂×C₄	—	4	1	no	—
266	C₂×C₄	—	268	4	no	—
267	C₂×C₄	—	536	2	no	—
268	C₂×C₄	—	536	2	no	—
269	C₂×C₄	C₂×C₂²⋊Dic₆₇	1	1	yes	characteristic
270	C₂×C₄	C₂×C₄⋊Dic₆₇	1	1	yes	characteristic
271	C₂×C₄	#14524002	1	1	yes	characteristic
272	C₂×C₄	#14524017	1	1	yes	characteristic
273	C₂×C₄	#14524022	1	1	yes	characteristic
274	C₂×C₄	#14524022	1	1	yes	characteristic
275	C₈	—	2	1	no	—
276	C₈	—	2	1	no	—
277	C₈	—	2	1	no	—
278	C₈	—	4	1	no	—
279	C₈	—	4	1	no	—
280	C₈	—	536	2	no	—
281	C₈	#14523970	1	2	yes	—
282	C₂²	—	2	1	no	—
283	C₂²	—	2	1	no	—
284	C₂²	—	4	1	no	—
285	C₂²	—	4	1	no	—
286	C₂²	—	4	1	no	—
287	C₂²	#16863256	1	1	yes	characteristic
288	C₂²	#16863659	1	2	yes	—
289	C₄	—	2	1	no	—
290	C₄	—	2	1	no	—
291	C₄	—	2	1	no	—
292	C₄	—	2	1	no	—
293	C₄	—	2	1	no	—
294	C₄	—	4	1	no	—
295	C₄	—	4	1	no	—
296	C₄	—	4	1	no	—
297	C₄	—	536	4	no	—
298	C₄	C₂×#14523970	1	1	yes	characteristic
299	C₄	#16863413	1	1	yes	characteristic
300	C₂	—	2	2	no	—
301	C₂	—	8	1	no	—
302	C₂	#22560229	1	1	yes	characteristic
303	C₂	#22560255	1	1	yes	characteristic
304	C₂	#22560268	1	1	yes	characteristic

	subgroup	G/H
center	C₂²	#16863256
commutator	C₄²×C₆₇	C₂²×C₄
Frattini	C₂×C₄²	C₂²×D₆₇
Fitting	C₆₇×#2966	C₂
socle	C₂²×C₆₇	#380
prime radical	#121×C₆₇	D₄
cosocle	C₂×C₄²×C₆₇	C₂³
hypercenter	#2966	D₆₇
hypocenter	C₆₇	#8696
LCS	G ▷ C₄²×C₆₇ ▷ C₂²×C₆₇ ▷ C₆₇
UCS	C₁ ◁ C₂² ◁ C₂×C₄² ◁ #2966

Complements

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

N	H ≅ G/N	type
1. C₄×#14523975	C₂	semidirect
2. #22558972	C₂	semidirect
4. #22558066	C₂	semidirect
15. #446×C₆₇	C₄	semidirect
16. C₈×Q₈×C₆₇	C₄	semidirect
17. #436×C₆₇	C₄	semidirect
20. C₈×D₄×C₆₇	C₄	semidirect
158. C₆₇	#8696	semidirect
160. #446	Dic₆₇	semidirect
161. C₈×Q₈	Dic₆₇	semidirect
162. #436	Dic₆₇	semidirect
165. C₈×D₄	Dic₆₇	semidirect

Maximal subgroups

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

subgroup	index
C₄×#14523975	2
#22558972	2
#22558734	2
#22558066	2
C₆₇×#2966	2
#8696	67