Subgroups — #30843534

Order 17152.

706 conjugacy classes of subgroups.

The subgroup lattice has 336 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#22570796	C₂	1	1	yes	maximal, characteristic
2	#22570794	C₂	1	2	yes	maximal
3	#22570793	C₂	1	1	yes	maximal, characteristic
4	#22570718	C₂	1	2	yes	maximal
5	#22559826	C₂	1	1	yes	maximal, characteristic, complemented
6	#22559825	C₂	1	1	yes	maximal, characteristic, complemented
7	#22559817	C₂	1	2	yes	maximal, complemented
8	#22559806	C₂	1	1	yes	maximal, characteristic, complemented
9	#22559804	C₂	1	2	yes	maximal, complemented
10	C₈²×D₆₇	C₂	1	1	yes	maximal, characteristic
11	C₆₇×#1707	C₂	1	1	yes	maximal, characteristic, complemented
12	#16863951	C₂²	1	2	yes	—
13	#16863951	C₂²	1	2	yes	—
14	#16863801	C₂²	1	1	yes	characteristic
15	#16863799	C₂²	1	2	yes	—
16	#16863799	C₂²	1	2	yes	—
17	#16863799	C₂²	1	2	yes	—
18	#16863796	—	2	4	no	—
19	#16863794	C₂²	1	2	yes	—
20	#16863776	C₂²	1	1	yes	characteristic
21	#16863775	C₂²	1	2	yes	—
22	#16863748	C₂²	1	2	yes	—
23	#16863747	C₂²	1	2	yes	—
24	#16863746	C₂²	1	2	yes	—
25	#16863638	—	2	2	no	—
26	#16863636	—	2	4	no	—
27	#16863634	—	2	2	no	—
28	Q₁₆×Dic₆₇	—	2	2	no	—
29	#16863490	—	2	2	no	—
30	#16863488	—	2	4	no	—
31	#16863431	—	2	4	no	—
32	C₄×C₈×D₆₇	C₂²	1	1	yes	characteristic
33	C₄×C₈×D₆₇	C₂²	1	1	yes	characteristic
34	C₄×C₈×D₆₇	C₂²	1	1	yes	characteristic
35	C₈×C₆₇⋊C₈	C₂²	1	1	yes	characteristic
36	C₈×C₆₇⋊C₈	C₂²	1	1	yes	characteristic
37	C₈×C₆₇⋊C₈	C₂²	1	1	yes	characteristic
38	#500×C₆₇	C₂²	1	1	yes	characteristic
39	#498×C₆₇	C₂²	1	2	yes	—
40	#494×C₆₇	C₂²	1	1	yes	characteristic
41	#487×C₆₇	C₂²	1	2	yes	—
42	C₈²×C₆₇	C₂²	1	1	yes	characteristic
43	C₂×Dic₂₆₈	—	2	2	no	—
44	C₂×#11761618	—	2	4	no	—
45	C₂×C₈×D₆₇	D₄	1	2	yes	—
46	C₂×C₈×D₆₇	D₄	1	2	yes	—
47	C₂×C₈×D₆₇	D₄	1	2	yes	—
48	#14524043	C₂³	1	2	yes	—
49	#14524043	C₂³	1	2	yes	—
50	#14524034	—	2	2	no	—
51	#14524034	—	2	2	no	—
52	#14524023	—	2	2	no	—
53	#14524023	—	2	2	no	—
54	#14524019	—	2	2	no	—
55	#14524019	—	2	2	no	—
56	Q₈×Dic₆₇	—	2	2	no	—
57	Q₈×Dic₆₇	—	2	2	no	—
58	#14524008	—	2	2	no	—
59	#14524008	—	2	2	no	—
60	#14524001	—	2	2	no	—
61	#14524001	—	2	2	no	—
62	C₄²×D₆₇	C₂³	1	1	yes	characteristic
63	#14523977	—	2	2	no	—
64	#14523976	—	2	4	no	—
65	#14523976	—	2	4	no	—
66	#14523976	—	2	4	no	—
67	#14523975	—	2	4	no	—
68	#14523968	—	2	4	no	—
69	#14523967	—	2	4	no	—
70	#14523966	—	2	4	no	—
71	C₄×C₆₇⋊C₈	C₂³	1	1	yes	characteristic
72	C₄×C₆₇⋊C₈	C₂³	1	1	yes	characteristic
73	C₄×C₆₇⋊C₈	C₂³	1	1	yes	characteristic
74	C₈×Dic₆₇	D₄	1	2	yes	—
75	C₈×Dic₆₇	D₄	1	2	yes	—
76	C₈×Dic₆₇	D₄	1	2	yes	—
77	C₂×Q₁₆×C₆₇	—	2	2	no	—
78	C₄⋊Q₈×C₆₇	C₂³	1	2	yes	—
79	C₄⋊Q₈×C₆₇	C₂³	1	2	yes	—
80	#107×C₆₇	—	2	2	no	—
81	#106×C₆₇	—	2	4	no	—
82	#103×C₆₇	—	2	4	no	—
83	C₄×C₈×C₆₇	C₂³	1	1	yes	characteristic
84	C₄×C₈×C₆₇	C₂³	1	1	yes	characteristic
85	C₄×C₈×C₆₇	C₂³	1	1	yes	characteristic
86	C₂×Dic₁₃₄	—	2	2	no	—
87	C₂×Dic₁₃₄	—	2	2	no	—
88	C₂×C₄×D₆₇	C₂×D₄	1	1	yes	characteristic
89	C₂×C₄×D₆₇	C₂×D₄	1	1	yes	characteristic
90	C₂×C₄×D₆₇	C₂×D₄	1	1	yes	characteristic
91	Dic₂₆₈	—	4	4	no	—
92	#11761618	—	4	8	no	—
93	C₂×C₆₇⋊C₈	C₂×D₄	1	2	yes	—
94	C₂×C₆₇⋊C₈	C₂×D₄	1	2	yes	—
95	C₂×C₆₇⋊C₈	C₂×D₄	1	2	yes	—
96	C₈×D₆₇	—	2	4	no	—
97	C₈×D₆₇	—	2	4	no	—
98	C₈×D₆₇	Q₁₆	1	8	yes	complemented
99	C₄⋊Dic₆₇	—	2	2	no	—
100	C₄⋊Dic₆₇	—	2	2	no	—
101	C₄⋊Dic₆₇	—	2	2	no	—
102	C₄⋊Dic₆₇	—	2	2	no	—
103	C₄⋊Dic₆₇	—	4	2	no	—
104	C₄⋊Dic₆₇	—	4	2	no	—
105	Dic₆₇⋊C₄	—	4	2	no	—
106	Dic₆₇⋊C₄	—	4	2	no	—
107	(C₂×D₆₇)⋊C₄	—	4	2	no	—
108	(C₂×D₆₇)⋊C₄	—	4	2	no	—
109	C₄×Dic₆₇	—	4	2	no	—
110	C₄×Dic₆₇	—	4	2	no	—
111	C₄×Dic₆₇	C₂×D₄	1	1	yes	characteristic
112	C₄×Dic₆₇	C₂×D₄	1	1	yes	characteristic
113	C₄×Dic₆₇	C₂×D₄	1	1	yes	characteristic
114	C₂×Q₈×C₆₇	—	2	2	no	—
115	C₂×Q₈×C₆₇	—	2	2	no	—
116	Q₁₆×C₆₇	—	4	4	no	—
117	C₂×C₈×C₆₇	C₂×D₄	1	2	yes	—
118	C₂×C₈×C₆₇	C₂×D₄	1	2	yes	—
119	C₂×C₈×C₆₇	C₂×D₄	1	2	yes	—
120	C₄⋊C₄×C₆₇	—	2	2	no	—
121	C₄⋊C₄×C₆₇	—	2	2	no	—
122	C₄⋊C₄×C₆₇	—	2	2	no	—
123	C₄⋊C₄×C₆₇	—	2	2	no	—
124	C₄²×C₆₇	C₂⁴	1	1	yes	characteristic
125	C₂²×D₆₇	#127	1	1	yes	characteristic
126	Q₈×C₆₇	—	4	4	no	—
127	Q₈×C₆₇	—	4	4	no	—
128	C₂×C₄×C₆₇	—	4	2	no	—
129	C₂×C₄×C₆₇	—	4	2	no	—
130	C₂×C₄×C₆₇	C₂²×D₄	1	1	yes	characteristic
131	C₂×C₄×C₆₇	C₂²×D₄	1	1	yes	characteristic
132	C₂×C₄×C₆₇	C₂²×D₄	1	1	yes	characteristic
133	C₂×Dic₆₇	—	4	2	no	—
134	C₂×Dic₆₇	—	4	2	no	—
135	C₂×Dic₆₇	#127	1	1	yes	characteristic
136	C₂×Dic₆₇	#127	1	1	yes	characteristic
137	C₂×Dic₆₇	#127	1	1	yes	characteristic
138	C₄×D₆₇	—	2	2	no	—
139	C₄×D₆₇	—	2	2	no	—
140	C₄×D₆₇	C₂×Q₁₆	1	4	yes	—
141	Dic₁₃₄	—	4	4	no	—
142	Dic₁₃₄	—	4	4	no	—
143	C₈×C₆₇	—	2	2	no	—
144	C₈×C₆₇	C₂×Q₁₆	1	4	yes	—
145	C₈×C₆₇	#135	1	4	yes	—
146	C₆₇⋊C₈	—	2	2	no	—
147	C₆₇⋊C₈	C₂×Q₁₆	1	4	yes	—
148	C₆₇⋊C₈	#135	1	4	yes	—
149	C₂²×C₆₇	C₂×#127	1	1	yes	characteristic
150	C₂×D₆₇	—	2	1	no	—
151	C₂×D₆₇	—	2	1	no	—
152	C₂×D₆₇	#494	1	2	yes	—
153	C₄×C₆₇	—	8	2	no	—
154	C₄×C₆₇	—	8	2	no	—
155	C₄×C₆₇	C₂²×Q₁₆	1	2	yes	—
156	C₄×C₆₇	C₂×#135	1	2	yes	—
157	C₄×C₆₇	C₂×#135	1	2	yes	—
158	Dic₆₇	—	2	1	no	—
159	Dic₆₇	—	2	1	no	—
160	Dic₆₇	—	8	2	no	—
161	Dic₆₇	—	8	2	no	—
162	Dic₆₇	#494	1	2	yes	—
163	#19604	—	67	1	no	maximal, Sylow
164	C₂×C₆₇	C₂×#494	1	1	yes	characteristic
165	C₂×C₆₇	#3147	1	1	yes	characteristic
166	C₂×C₆₇	#3147	1	1	yes	characteristic
167	D₆₇	—	2	2	no	—
168	#3170	—	67	1	no	—
169	#3170	—	67	1	no	—
170	#3170	—	67	2	no	—
171	#3160	—	67	2	no	—
172	#1713	—	67	1	no	—
173	#1713	—	67	1	no	—
174	#1710	—	67	2	no	—
175	#1707	—	67	2	no	—
176	#1707	—	67	1	no	—
177	#1707	D₆₇	1	1	yes	characteristic, complemented
178	C₂×C₈²	—	67	1	no	—
179	C₆₇	#19604	1	1	yes	characteristic, complemented, Sylow
180	#533	—	67	2	no	—
181	#533	—	67	2	no	—
182	#500	—	67	2	no	—
183	#500	—	67	2	no	—
184	#500	—	67	2	no	—
185	#500	—	67	1	no	—
186	#500	C₂×D₆₇	1	1	yes	characteristic
187	#499	—	134	4	no	—
188	#498	—	67	2	no	—
189	#498	C₂×D₆₇	1	2	yes	—
190	#494	—	67	2	no	—
191	#494	—	67	1	no	—
192	#494	C₂×D₆₇	1	1	yes	characteristic
193	#487	—	67	2	no	—
194	#487	—	67	2	no	—
195	#487	—	67	2	no	—
196	#487	C₂×D₆₇	1	2	yes	—
197	#467	—	134	4	no	—
198	#467	—	134	2	no	—
199	#467	—	134	2	no	—
200	C₄×Q₁₆	—	134	4	no	—
201	C₄×Q₁₆	—	134	2	no	—
202	C₄×Q₁₆	—	134	2	no	—
203	#427	—	134	4	no	—
204	C₂×C₄×C₈	—	67	1	no	—
205	C₂×C₄×C₈	—	67	1	no	—
206	C₂×C₄×C₈	—	67	1	no	—
207	C₈²	—	67	1	no	—
208	C₈²	—	67	1	no	—
209	C₈²	—	67	1	no	—
210	C₈²	C₂×D₆₇	1	1	yes	characteristic
211	C₂×Q₁₆	—	2	2	no	—
212	C₂×Q₁₆	—	134	4	no	—
213	C₂×Q₁₆	—	134	2	no	—
214	C₂²×C₈	—	67	2	no	—
215	C₂²×C₈	—	67	2	no	—
216	C₂²×C₈	—	67	2	no	—
217	C₄⋊Q₈	—	67	2	no	—
218	C₄⋊Q₈	—	67	2	no	—
219	C₄⋊Q₈	C₂²×D₆₇	1	2	yes	—
220	C₄⋊Q₈	C₂²×D₆₇	1	2	yes	—
221	#125	—	134	2	no	—
222	#125	—	134	2	no	—
223	C₂²⋊Q₈	—	134	2	no	—
224	C₂²⋊Q₈	—	134	2	no	—
225	C₂²⋊Q₈	—	134	2	no	—
226	C₂²⋊Q₈	—	134	2	no	—
227	C₄×Q₈	—	134	2	no	—
228	C₄×Q₈	—	134	2	no	—
229	C₄×Q₈	—	134	2	no	—
230	C₄×Q₈	—	134	2	no	—
231	#117	—	134	2	no	—
232	#117	—	134	2	no	—
233	C₂×C₄²	—	67	1	no	—
234	#107	—	2	2	no	—
235	#107	—	134	4	no	—
236	#107	—	134	4	no	—
237	#107	—	134	4	no	—
238	#107	—	134	2	no	—
239	#106	—	2	4	no	—
240	#106	—	134	4	no	—
241	#103	—	2	4	no	—
242	#103	—	134	4	no	—
243	#103	—	134	4	no	—
244	#103	—	134	4	no	—
245	C₄×C₈	—	67	2	no	—
246	C₄×C₈	—	67	2	no	—
247	C₄×C₈	—	67	2	no	—
248	C₄×C₈	—	67	1	no	—
249	C₄×C₈	—	67	1	no	—
250	C₄×C₈	—	67	1	no	—
251	C₄×C₈	C₂²×D₆₇	1	1	yes	characteristic
252	C₄×C₈	C₂²×D₆₇	1	1	yes	characteristic
253	C₄×C₈	C₂²×D₆₇	1	1	yes	characteristic
254	C₂×Q₈	—	2	2	no	—
255	C₂×Q₈	—	2	2	no	—
256	C₂×Q₈	—	134	2	no	—
257	C₂×Q₈	—	134	2	no	—
258	C₂²×C₄	—	67	1	no	—
259	C₂²×C₄	—	67	1	no	—
260	C₂²×C₄	—	67	1	no	—
261	Q₁₆	—	4	4	no	—
262	Q₁₆	—	268	8	no	—
263	Q₁₆	—	268	4	no	—
264	C₂×C₈	—	67	2	no	—
265	C₂×C₈	—	67	2	no	—
266	C₂×C₈	—	67	8	no	—
267	C₂×C₈	—	67	2	no	—
268	C₂×C₈	—	134	4	no	—
269	C₂×C₈	—	134	4	no	—
270	C₂×C₈	D₄×D₆₇	1	2	yes	—
271	C₂×C₈	D₄×D₆₇	1	2	yes	—
272	C₂×C₈	D₄×D₆₇	1	2	yes	—
273	C₄⋊C₄	—	2	2	no	—
274	C₄⋊C₄	—	2	2	no	—
275	C₄⋊C₄	—	2	2	no	—
276	C₄⋊C₄	—	2	2	no	—
277	C₄⋊C₄	—	268	2	no	—
278	C₄⋊C₄	—	268	2	no	—
279	C₄⋊C₄	—	268	2	no	—
280	C₄⋊C₄	—	268	2	no	—
281	C₄⋊C₄	—	134	2	no	—
282	C₄⋊C₄	—	134	2	no	—
283	C₄⋊C₄	—	134	2	no	—
284	C₄⋊C₄	—	134	2	no	—
285	C₂²⋊C₄	—	268	2	no	—
286	C₂²⋊C₄	—	268	2	no	—
287	C₄²	—	268	2	no	—
288	C₄²	—	268	2	no	—
289	C₄²	—	67	1	no	—
290	C₄²	—	67	1	no	—
291	C₄²	—	67	1	no	—
292	C₄²	C₂³×D₆₇	1	1	yes	characteristic
293	C₂³	—	67	1	no	—
294	Q₈	—	4	4	no	—
295	Q₈	—	4	4	no	—
296	Q₈	—	268	4	no	—
297	Q₈	—	268	4	no	—
298	C₂×C₄	—	4	2	no	—
299	C₂×C₄	—	4	2	no	—
300	C₂×C₄	—	268	2	no	—
301	C₂×C₄	—	268	2	no	—
302	C₂×C₄	—	67	1	no	—
303	C₂×C₄	—	67	4	no	—
304	C₂×C₄	—	67	1	no	—
305	C₂×C₄	—	67	1	no	—
306	C₂×C₄	—	134	2	no	—
307	C₂×C₄	—	134	2	no	—
308	C₂×C₄	C₂×D₄×D₆₇	1	1	yes	characteristic
309	C₂×C₄	C₂×D₄×D₆₇	1	1	yes	characteristic
310	C₂×C₄	C₂×D₄×D₆₇	1	1	yes	characteristic
311	C₈	—	2	2	no	—
312	C₈	—	67	4	no	—
313	C₈	—	67	4	no	—
314	C₈	—	134	2	no	—
315	C₈	Q₁₆×D₆₇	1	4	yes	—
316	C₈	#14524064	1	4	yes	—
317	C₂²	—	67	2	no	—
318	C₂²	—	134	1	no	—
319	C₂²	—	134	1	no	—
320	C₂²	#127×D₆₇	1	1	yes	characteristic
321	C₄	—	8	2	no	—
322	C₄	—	8	2	no	—
323	C₄	—	536	2	no	—
324	C₄	—	536	2	no	—
325	C₄	—	67	2	no	—
326	C₄	—	134	1	no	—
327	C₄	—	134	1	no	—
328	C₄	C₂×Q₁₆×D₆₇	1	2	yes	—
329	C₄	#135×D₆₇	1	2	yes	—
330	C₄	C₂×#14524064	1	2	yes	—
331	C₂	—	134	2	no	—
332	C₂	#494×D₆₇	1	1	yes	characteristic
333	C₂	#22570632	1	1	yes	characteristic
334	C₂	#22570634	1	1	yes	characteristic

	subgroup	G/H
center	C₂²	#127×D₆₇
commutator, cosocle	C₄²×C₆₇	C₂⁴
Frattini	C₄²	C₂³×D₆₇
Fitting	C₆₇×#1707	C₂
socle	C₂²×C₆₇	C₂×#127
prime radical	C₂²×D₆₇	#127
hypercenter	#1707	D₆₇
hypocenter	C₆₇	#19604
LCS	G ▷ C₄²×C₆₇ ▷ C₂²×C₆₇ ▷ C₆₇
UCS	C₁ ◁ C₂² ◁ C₄² ◁ #1707

Complements

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

N	H ≅ G/N	type
5. #22559826	C₂	semidirect
6. #22559825	C₂	semidirect
7. #22559817	C₂	semidirect
8. #22559806	C₂	semidirect
9. #22559804	C₂	semidirect
11. C₆₇×#1707	C₂	semidirect
98. C₈×D₆₇	Q₁₆	semidirect
177. #1707	D₆₇	semidirect
179. C₆₇	#19604	semidirect

Maximal subgroups

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

subgroup	index
#22570796	2
#22570794	2
#22570793	2
#22570718	2
#22559826	2
#22559825	2
#22559817	2
#22559806	2
#22559804	2
C₈²×D₆₇	2
C₆₇×#1707	2
#19604	67