Subgroups — #30839073

Order 17152.

606 conjugacy classes of subgroups.

The subgroup lattice has 298 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#22571157	C₂	1	2	yes	maximal
2	#22569143	C₂	1	1	yes	maximal, characteristic
3	#22559724	C₂	1	4	yes	maximal, complemented
4	#22559600	C₂	1	4	yes	maximal, complemented
5	#22558957	C₂	1	1	yes	maximal, characteristic
6	#22558894	C₂	1	2	yes	maximal
7	C₂×#500×C₆₇	C₂	1	1	yes	maximal, characteristic
8	#16863953	C₂²	1	1	yes	characteristic
9	#16863775	—	2	2	no	—
10	#16863746	—	2	2	no	—
11	#16863603	C₂²	1	8	yes	—
12	#16863552	—	2	4	no	—
13	#16863526	—	2	1	no	—
14	#16863526	C₂²	1	2	yes	—
15	C₈×Dic₁₃₄	—	2	1	no	—
16	C₈×Dic₁₃₄	C₂²	1	2	yes	—
17	#16863449	C₂²	1	2	yes	—
18	#16863422	—	2	1	no	—
19	#16863420	C₂²	1	1	yes	characteristic
20	#16863351	—	2	1	no	—
21	#16863149	C₂²	1	2	yes	—
22	#16863147	C₂²	1	2	yes	—
23	#16863142	C₂²	1	8	yes	—
24	C₂×C₄⋊Q₈×C₆₇	C₂²	1	2	yes	—
25	#500×C₆₇	—	2	1	no	—
26	#500×C₆₇	—	2	1	no	—
27	#500×C₆₇	C₂²	1	2	yes	—
28	#500×C₆₇	C₂²	1	2	yes	—
29	C₂×#107×C₆₇	—	2	1	no	—
30	C₂×#107×C₆₇	—	2	1	no	—
31	C₂×C₄×C₈×C₆₇	C₂²	1	1	yes	characteristic
32	C₂×#11761618	—	2	8	no	—
33	C₂×#11761612	—	2	1	no	—
34	#14524043	D₄	1	1	yes	characteristic
35	#14524043	D₄	1	1	yes	characteristic
36	#14524035	—	2	1	no	—
37	#14524022	—	2	1	no	—
38	#14524022	D₄	1	2	yes	—
39	C₄×Dic₁₃₄	—	2	1	no	—
40	C₄×Dic₁₃₄	C₂³	1	2	yes	—
41	#14524002	—	2	1	no	—
42	C₄⋊(C₆₇⋊C₈)	—	2	1	no	—
43	C₄⋊(C₆₇⋊C₈)	C₂³	1	2	yes	—
44	C₄⋊(C₆₇⋊C₈)	C₂³	1	4	yes	—
45	Dic₆₇⋊C₈	—	2	2	no	—
46	Dic₆₇⋊C₈	—	2	2	no	—
47	#14523967	—	2	8	no	—
48	C₂²⋊(C₆₇⋊C₈)	—	2	2	no	—
49	#14523953	—	2	2	no	—
50	C₄×C₆₇⋊C₈	—	2	2	no	—
51	C₈×Dic₆₇	—	2	2	no	—
52	C₂²×Q₈×C₆₇	—	2	2	no	—
53	C₂²×C₈×C₆₇	D₄	1	1	yes	characteristic
54	C₂²×C₈×C₆₇	D₄	1	1	yes	characteristic
55	C₄⋊Q₈×C₆₇	—	2	2	no	—
56	C₄⋊Q₈×C₆₇	—	2	2	no	—
57	C₄⋊Q₈×C₆₇	—	2	2	no	—
58	C₄⋊Q₈×C₆₇	C₂³	1	4	yes	—
59	C₂×C₄⋊C₄×C₆₇	—	2	2	no	—
60	C₂×C₄⋊C₄×C₆₇	—	2	2	no	—
61	C₂×C₄²×C₆₇	C₂³	1	1	yes	characteristic
62	#107×C₆₇	—	2	2	no	—
63	#107×C₆₇	—	2	2	no	—
64	#107×C₆₇	—	2	2	no	—
65	#107×C₆₇	—	2	2	no	—
66	C₄×C₈×C₆₇	D₄	1	1	yes	characteristic
67	C₄×C₈×C₆₇	D₄	1	1	yes	characteristic
68	C₄×C₈×C₆₇	C₂³	1	1	yes	characteristic
69	C₄×C₈×C₆₇	C₂³	1	1	yes	characteristic
70	C₂×Dic₁₃₄	—	2	1	no	—
71	C₂×Dic₁₃₄	C₂×D₄	1	2	yes	—
72	#11761618	—	8	8	no	—
73	#11761612	—	4	2	no	—
74	C₂×C₆₇⋊C₈	—	2	2	no	—
75	C₂×C₆₇⋊C₈	—	2	4	no	—
76	C₄⋊Dic₆₇	—	2	1	no	—
77	C₄⋊Dic₆₇	—	2	1	no	—
78	C₄⋊Dic₆₇	—	2	1	no	—
79	C₄⋊Dic₆₇	C₂×D₄	1	2	yes	—
80	Dic₆₇⋊C₄	—	2	2	no	—
81	Dic₆₇⋊C₄	—	2	2	no	—
82	C₂²⋊Dic₆₇	—	2	2	no	—
83	C₄×Dic₆₇	—	2	2	no	—
84	C₂×Q₈×C₆₇	—	2	4	no	—
85	C₂×Q₈×C₆₇	—	4	2	no	—
86	C₂×Q₈×C₆₇	—	8	4	no	—
87	C₂²×C₄×C₆₇	—	4	2	no	—
88	C₂²×C₄×C₆₇	C₂×D₄	1	1	yes	characteristic
89	C₂²×C₄×C₆₇	C₂×D₄	1	1	yes	characteristic
90	C₂²×C₄×C₆₇	#41	1	1	yes	characteristic
91	C₂×C₈×C₆₇	—	2	2	no	—
92	C₂×C₈×C₆₇	Q₁₆	1	4	yes	complemented
93	C₂×C₈×C₆₇	C₂×D₄	1	1	yes	characteristic
94	C₂×C₈×C₆₇	C₂×D₄	1	1	yes	characteristic
95	C₂×C₈×C₆₇	C₂×D₄	1	1	yes	characteristic
96	C₂×C₈×C₆₇	C₂×D₄	1	1	yes	characteristic
97	C₄⋊C₄×C₆₇	—	2	4	no	—
98	C₄⋊C₄×C₆₇	—	2	4	no	—
99	C₄⋊C₄×C₆₇	—	4	2	no	—
100	C₄⋊C₄×C₆₇	—	4	2	no	—
101	C₄²×C₆₇	C₂×D₄	1	1	yes	characteristic
102	C₄²×C₆₇	C₂×D₄	1	1	yes	characteristic
103	C₄²×C₆₇	#41	1	1	yes	characteristic
104	C₄²×C₆₇	C₂⁴	1	1	yes	characteristic
105	C₂³×C₆₇	#121	1	1	yes	characteristic
106	Q₈×C₆₇	—	4	8	no	—
107	Q₈×C₆₇	—	8	4	no	—
108	C₂×C₄×C₆₇	—	2	2	no	—
109	C₂×C₄×C₆₇	—	2	2	no	—
110	C₂×C₄×C₆₇	—	4	4	no	—
111	C₂×C₄×C₆₇	—	8	4	no	—
112	C₂×C₄×C₆₇	#121	1	1	yes	characteristic
113	C₂×C₄×C₆₇	#121	1	1	yes	characteristic
114	C₂×C₄×C₆₇	#121	1	1	yes	characteristic
115	C₂×C₄×C₆₇	C₂×Q₁₆	1	2	yes	—
116	C₂×C₄×C₆₇	#137	1	2	yes	—
117	C₂×C₄×C₆₇	C₂²×D₄	1	1	yes	characteristic
118	C₂×C₄×C₆₇	C₂²×D₄	1	1	yes	characteristic
119	C₂×C₄×C₆₇	C₂×#41	1	1	yes	characteristic
120	C₂×Dic₆₇	—	2	2	no	—
121	C₂×Dic₆₇	—	2	2	no	—
122	Dic₁₃₄	—	2	4	no	—
123	Dic₁₃₄	—	4	2	no	—
124	C₈×C₆₇	—	2	1	no	—
125	C₈×C₆₇	—	2	1	no	—
126	C₈×C₆₇	C₂×Q₁₆	1	2	yes	—
127	C₈×C₆₇	C₂×Q₁₆	1	2	yes	—
128	C₆₇⋊C₈	—	4	2	no	—
129	C₆₇⋊C₈	—	4	4	no	—
130	C₂²×C₆₇	—	2	1	no	—
131	C₂²×C₆₇	—	2	1	no	—
132	C₂²×C₆₇	#462	1	2	yes	—
133	C₂²×C₆₇	C₂×#121	1	1	yes	characteristic
134	C₄×C₆₇	—	2	1	no	—
135	C₄×C₆₇	—	2	1	no	—
136	C₄×C₆₇	—	8	4	no	—
137	C₄×C₆₇	#462	1	2	yes	—
138	C₄×C₆₇	#473	1	2	yes	—
139	C₄×C₆₇	C₂²×Q₁₆	1	1	yes	characteristic
140	C₄×C₆₇	C₂×#137	1	1	yes	characteristic
141	C₄×C₆₇	#576	1	2	yes	—
142	Dic₆₇	—	4	2	no	—
143	Dic₆₇	—	4	2	no	—
144	#19208	—	67	1	no	maximal, Sylow
145	C₂×C₆₇	—	2	2	no	—
146	C₂×C₆₇	C₂×#462	1	1	yes	characteristic
147	C₂×C₆₇	#3058	1	1	yes	characteristic
148	C₂×C₆₇	C₂×#473	1	1	yes	characteristic
149	#3209	—	67	2	no	—
150	C₂×#500	D₆₇	1	1	yes	characteristic
151	#2960	—	67	1	no	—
152	#1694	—	67	4	no	—
153	#1674	—	67	4	no	—
154	#1564	—	67	1	no	—
155	#1549	—	67	2	no	—
156	C₆₇	#19208	1	1	yes	characteristic, complemented, Sylow
157	#533	—	67	1	no	—
158	C₂×C₄⋊Q₈	C₂×D₆₇	1	2	yes	—
159	#500	—	2	1	no	—
160	#500	—	2	1	no	—
161	#500	C₂×D₆₇	1	2	yes	—
162	#500	C₂×D₆₇	1	2	yes	—
163	#494	—	134	2	no	—
164	#487	—	134	2	no	—
165	#462	—	67	8	no	—
166	#451	—	134	4	no	—
167	#446	—	67	2	no	—
168	#446	—	134	1	no	—
169	C₈×Q₈	—	67	2	no	—
170	C₈×Q₈	—	134	1	no	—
171	#431	—	67	2	no	—
172	C₂×#107	—	2	1	no	—
173	C₂×#107	—	2	1	no	—
174	#424	—	134	1	no	—
175	#423	—	67	1	no	—
176	#405	—	134	1	no	—
177	C₂×C₄×C₈	C₂×D₆₇	1	1	yes	characteristic
178	#335	—	67	2	no	—
179	#334	—	67	2	no	—
180	#332	—	67	8	no	—
181	C₂²×Q₈	—	2	2	no	—
182	C₂×Q₁₆	—	134	8	no	—
183	C₂×M₁₆	—	134	1	no	—
184	C₂²×C₈	C₂²⋊D₆₇	1	1	yes	characteristic
185	C₂²×C₈	C₂²⋊D₆₇	1	1	yes	characteristic
186	C₄⋊Q₈	—	2	2	no	—
187	C₄⋊Q₈	—	2	2	no	—
188	C₄⋊Q₈	—	2	2	no	—
189	C₄⋊Q₈	—	67	1	no	—
190	C₄⋊Q₈	—	67	1	no	—
191	C₄⋊Q₈	C₂²×D₆₇	1	4	yes	—
192	#125	—	134	1	no	—
193	C₂²⋊Q₈	—	67	2	no	—
194	C₂²⋊Q₈	—	134	1	no	—
195	C₄×Q₈	—	67	2	no	—
196	C₄×Q₈	—	134	1	no	—
197	#117	—	134	1	no	—
198	C₂×C₄⋊C₄	—	2	2	no	—
199	C₂×C₄⋊C₄	—	2	2	no	—
200	C₂×C₄²	C₂²×D₆₇	1	1	yes	characteristic
201	#107	—	2	2	no	—
202	#107	—	2	2	no	—
203	#107	—	2	2	no	—
204	#107	—	2	2	no	—
205	C₄⋊C₈	—	67	2	no	—
206	C₄⋊C₈	—	67	4	no	—
207	C₄⋊C₈	—	134	2	no	—
208	C₄⋊C₈	—	134	2	no	—
209	C₄⋊C₈	—	134	1	no	—
210	#103	—	134	8	no	—
211	C₂²⋊C₈	—	134	2	no	—
212	#97	—	134	2	no	—
213	C₄×C₈	—	134	2	no	—
214	C₄×C₈	—	134	2	no	—
215	C₄×C₈	C₂²⋊D₆₇	1	1	yes	characteristic
216	C₄×C₈	C₂²⋊D₆₇	1	1	yes	characteristic
217	C₄×C₈	C₂²×D₆₇	1	1	yes	characteristic
218	C₄×C₈	C₂²×D₆₇	1	1	yes	characteristic
219	C₂×Q₈	—	2	4	no	—
220	C₂×Q₈	—	4	2	no	—
221	C₂×Q₈	—	8	4	no	—
222	C₂×Q₈	—	67	2	no	—
223	C₂×Q₈	—	134	1	no	—
224	C₂²×C₄	—	4	2	no	—
225	C₂²×C₄	D₄×D₆₇	1	1	yes	characteristic
226	C₂²×C₄	C₂×C₂²⋊D₆₇	1	1	yes	characteristic
227	C₂²×C₄	#11761627	1	1	yes	characteristic
228	Q₁₆	—	536	8	no	—
229	M₁₆	—	268	2	no	—
230	C₂×C₈	—	2	2	no	—
231	C₂×C₈	—	134	4	no	—
232	C₂×C₈	—	134	2	no	—
233	C₂×C₈	#11761618	1	4	yes	complemented
234	C₂×C₈	C₂×C₂²⋊D₆₇	1	1	yes	characteristic
235	C₂×C₈	C₂×C₂²⋊D₆₇	1	1	yes	characteristic
236	C₂×C₈	C₂×C₂²⋊D₆₇	1	1	yes	characteristic
237	C₂×C₈	C₂×C₂²⋊D₆₇	1	1	yes	characteristic
238	C₄⋊C₄	—	2	4	no	—
239	C₄⋊C₄	—	2	4	no	—
240	C₄⋊C₄	—	4	2	no	—
241	C₄⋊C₄	—	4	2	no	—
242	C₄⋊C₄	—	67	2	no	—
243	C₄⋊C₄	—	134	1	no	—
244	C₄⋊C₄	—	134	1	no	—
245	C₄⋊C₄	—	134	2	no	—
246	C₄⋊C₄	—	134	1	no	—
247	C₄⋊C₄	—	134	2	no	—
248	C₂²⋊C₄	—	134	2	no	—
249	C₄²	—	134	2	no	—
250	C₄²	D₄×D₆₇	1	1	yes	characteristic
251	C₄²	C₂×C₂²⋊D₆₇	1	1	yes	characteristic
252	C₄²	#11761627	1	1	yes	characteristic
253	C₄²	C₂³×D₆₇	1	1	yes	characteristic
254	C₂³	#14524014	1	1	yes	characteristic
255	Q₈	—	4	8	no	—
256	Q₈	—	8	4	no	—
257	Q₈	—	268	2	no	—
258	Q₈	—	134	4	no	—
259	C₂×C₄	—	2	2	no	—
260	C₂×C₄	—	2	2	no	—
261	C₂×C₄	—	4	4	no	—
262	C₂×C₄	—	8	4	no	—
263	C₂×C₄	—	134	2	no	—
264	C₂×C₄	—	134	2	no	—
265	C₂×C₄	#14524014	1	1	yes	characteristic
266	C₂×C₄	#14524014	1	1	yes	characteristic
267	C₂×C₄	#14524014	1	1	yes	characteristic
268	C₂×C₄	C₂×#11761618	1	2	yes	—
269	C₂×C₄	#14524074	1	2	yes	—
270	C₂×C₄	C₂×D₄×D₆₇	1	1	yes	characteristic
271	C₂×C₄	C₂²×C₂²⋊D₆₇	1	1	yes	characteristic
272	C₂×C₄	C₂×#11761627	1	1	yes	characteristic
273	C₈	—	2	1	no	—
274	C₈	—	2	1	no	—
275	C₈	—	268	2	no	—
276	C₈	—	268	4	no	—
277	C₈	C₂×#11761618	1	2	yes	—
278	C₈	C₂×#11761618	1	2	yes	—
279	C₂²	—	2	1	no	—
280	C₂²	—	2	1	no	—
281	C₂²	#16863603	1	2	yes	—
282	C₂²	C₂×#14524014	1	1	yes	characteristic
283	C₄	—	2	1	no	—
284	C₄	—	2	1	no	—
285	C₄	—	8	4	no	—
286	C₄	—	268	2	no	—
287	C₄	—	268	2	no	—
288	C₄	#16863603	1	2	yes	—
289	C₄	#16863668	1	2	yes	—
290	C₄	C₂²×#11761618	1	1	yes	characteristic
291	C₄	C₂×#14524074	1	1	yes	characteristic
292	C₄	#16864197	1	2	yes	—
293	C₂	—	2	2	no	—
294	C₂	C₂×#16863603	1	1	yes	characteristic
295	C₂	#22569853	1	1	yes	characteristic
296	C₂	C₂×#16863668	1	1	yes	characteristic

	subgroup	G/H
center	C₂²	C₂×#14524014
commutator, cosocle	C₄²×C₆₇	C₂⁴
Frattini	C₄²	C₂³×D₆₇
Fitting	C₂×#500×C₆₇	C₂
socle	C₂²×C₆₇	C₂×#121
prime radical	C₂³×C₆₇	#121
hypercenter	C₂×#500	D₆₇
hypocenter	C₆₇	#19208
LCS	G ▷ C₄²×C₆₇ ▷ C₂²×C₆₇ ▷ C₆₇
UCS	C₁ ◁ C₂² ◁ C₂×C₄² ◁ C₂×#500

Complements

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

N	H ≅ G/N	type
3. #22559724	C₂	semidirect
4. #22559600	C₂	semidirect
92. C₂×C₈×C₆₇	Q₁₆	semidirect
156. C₆₇	#19208	semidirect
233. C₂×C₈	#11761618	semidirect

Maximal subgroups

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

subgroup	index
#22571157	2
#22569143	2
#22559724	2
#22559600	2
#22558957	2
#22558894	2
C₂×#500×C₆₇	2
#19208	67