Subgroups — #30886710

Order 17152.

616 conjugacy classes of subgroups.

The subgroup lattice has 360 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#22569113	C₂	1	1	yes	maximal, characteristic
2	#22561789	C₂	1	1	yes	maximal, characteristic, complemented
3	#22561784	C₂	1	1	yes	maximal, characteristic, complemented
4	#22561660	C₂	1	2	yes	maximal, complemented
5	#22561515	C₂	1	1	yes	maximal, characteristic
6	#22561496	C₂	1	1	yes	maximal, characteristic
7	#22561378	C₂	1	2	yes	maximal, complemented
8	#22560046	C₂	1	2	yes	maximal, complemented
9	#22560010	C₂	1	2	yes	maximal, complemented
10	#22559964	C₂	1	1	yes	maximal, characteristic, complemented
11	C₆₇×#2960	C₂	1	1	yes	maximal, characteristic, complemented
12	#16864196	C₂²	1	1	yes	characteristic
13	C₂×#14524049	C₂²	1	2	yes	—
14	#16863519	—	2	2	no	—
15	#16863514	—	2	2	no	—
16	#16863445	C₂²	1	1	yes	characteristic
17	#16863414	C₂²	1	2	yes	—
18	C₂×#14523971	C₂²	1	2	yes	—
19	#16863393	C₂²	1	2	yes	—
20	#16863378	C₂²	1	2	yes	—
21	#16863369	C₂²	1	2	yes	—
22	#16863351	C₂²	1	2	yes	—
23	#16863168	C₂²	1	2	yes	—
24	#16863166	C₂²	1	2	yes	—
25	#16863161	C₂²	1	2	yes	—
26	#16863156	C₂²	1	2	yes	—
27	C₆₇⋊#338	C₂²	1	1	yes	characteristic
28	#16863153	C₂²	1	1	yes	characteristic
29	#16863152	C₂²	1	2	yes	—
30	#533×C₆₇	C₂²	1	1	yes	characteristic
31	#446×C₆₇	—	2	2	no	—
32	C₈×Q₈×C₆₇	—	2	2	no	—
33	#424×C₆₇	C₂²	1	1	yes	characteristic
34	#424×C₆₇	C₂²	1	1	yes	characteristic
35	#423×C₆₇	C₂²	1	1	yes	characteristic
36	#405×C₆₇	C₂²	1	1	yes	characteristic
37	#405×C₆₇	C₂²	1	1	yes	characteristic
38	C₂×C₄×C₈×C₆₇	C₂²	1	1	yes	characteristic
39	C₂×#11761628	C₂³	1	2	yes	—
40	#14524076	—	2	2	no	—
41	#14524071	—	2	2	no	—
42	#14524065	—	2	2	no	—
43	#14524049	—	2	2	no	—
44	#14524049	—	2	2	no	—
45	#14524049	—	2	2	no	—
46	#14524049	—	2	2	no	—
47	C₂×#11761612	C₂³	1	2	yes	—
48	C₂×#11761612	C₂³	1	2	yes	—
49	C₂×#11761611	—	2	2	no	—
50	C₂×C₈×D₆₇	—	2	2	no	—
51	#14524002	C₂³	1	2	yes	—
52	#14523979	—	2	2	no	—
53	#14523971	—	2	4	no	—
54	#14523971	—	2	2	no	—
55	#14523971	—	2	2	no	—
56	#14523968	—	2	2	no	—
57	#14523965	—	2	2	no	—
58	#14523961	—	2	2	no	—
59	#14523959	—	2	2	no	—
60	#14523957	—	2	4	no	—
61	#14523953	—	2	2	no	—
62	C₈×Dic₆₇	—	2	2	no	—
63	C₂×M₁₆×C₆₇	C₂³	1	1	yes	characteristic
64	C₂×M₁₆×C₆₇	C₂³	1	1	yes	characteristic
65	C₂²×C₈×C₆₇	C₂³	1	1	yes	characteristic
66	C₂²×C₈×C₆₇	C₂³	1	1	yes	characteristic
67	C₄⋊Q₈×C₆₇	—	2	1	no	—
68	#125×C₆₇	—	2	1	no	—
69	C₂²⋊Q₈×C₆₇	—	2	2	no	—
70	C₄×Q₈×C₆₇	—	2	2	no	—
71	#117×C₆₇	C₂³	1	1	yes	characteristic
72	#117×C₆₇	C₂³	1	1	yes	characteristic
73	C₂×C₄²×C₆₇	C₂³	1	1	yes	characteristic
74	C₄⋊C₈×C₆₇	—	2	2	no	—
75	C₄⋊C₈×C₆₇	—	2	2	no	—
76	C₄⋊C₈×C₆₇	—	2	2	no	—
77	#97×C₆₇	—	2	1	no	—
78	#97×C₆₇	D₄	1	2	yes	—
79	C₄×C₈×C₆₇	—	2	1	no	—
80	C₄×C₈×C₆₇	—	2	1	no	—
81	C₄×C₈×C₆₇	—	2	1	no	—
82	C₄×C₈×C₆₇	D₄	1	2	yes	—
83	#11761628	—	2	2	no	—
84	#11761628	—	2	2	no	—
85	#11761628	—	2	2	no	—
86	#11761628	—	2	2	no	—
87	C₂×Dic₁₃₄	#41	1	2	yes	—
88	C₂×D₂₆₈	#41	1	2	yes	—
89	C₂×C₂²⋊D₆₇	—	2	2	no	—
90	C₂×C₄×D₆₇	—	2	2	no	—
91	Dic₂₆₈	—	2	2	no	—
92	#11761617	—	2	4	no	—
93	D₅₃₆	—	2	2	no	—
94	#11761612	—	2	2	no	—
95	#11761612	—	2	2	no	—
96	#11761612	—	2	2	no	—
97	#11761612	—	2	2	no	—
98	#11761611	—	4	2	no	—
99	#11761611	—	4	2	no	—
100	C₂×C₆₇⋊C₈	—	2	2	no	—
101	C₂×C₆₇⋊C₈	—	2	2	no	—
102	C₈×D₆₇	—	4	2	no	—
103	C₈×D₆₇	—	4	2	no	—
104	C₄⋊Dic₆₇	—	2	2	no	—
105	C₂²⋊Dic₆₇	—	2	2	no	—
106	C₄×Dic₆₇	—	2	2	no	—
107	C₂×Q₈×C₆₇	—	2	2	no	—
108	C₂²×C₄×C₆₇	—	2	1	no	—
109	C₂²×C₄×C₆₇	C₂⁴	1	1	yes	characteristic
110	M₁₆×C₆₇	—	2	2	no	—
111	M₁₆×C₆₇	—	2	2	no	—
112	C₂×C₈×C₆₇	—	2	1	no	—
113	C₂×C₈×C₆₇	—	2	1	no	—
114	C₂×C₈×C₆₇	—	2	2	no	—
115	C₂×C₈×C₆₇	C₂×D₄	1	2	yes	—
116	C₂×C₈×C₆₇	C₂×D₄	1	1	yes	characteristic
117	C₂×C₈×C₆₇	C₂×D₄	1	1	yes	characteristic
118	C₂×C₈×C₆₇	#41	1	2	yes	—
119	C₂×C₈×C₆₇	#41	1	2	yes	—
120	C₄⋊C₄×C₆₇	—	2	4	no	—
121	C₄⋊C₄×C₆₇	—	2	2	no	—
122	C₄⋊C₄×C₆₇	—	2	1	no	—
123	C₄⋊C₄×C₆₇	—	2	1	no	—
124	C₂²⋊C₄×C₆₇	—	2	1	no	—
125	C₂²⋊C₄×C₆₇	#41	1	2	yes	—
126	C₄²×C₆₇	—	2	1	no	—
127	C₄²×C₆₇	—	2	1	no	—
128	C₄²×C₆₇	C₂×D₄	1	2	yes	—
129	C₄²×C₆₇	#41	1	1	yes	characteristic
130	C₄²×C₆₇	#41	1	1	yes	characteristic
131	C₂³×C₆₇	C₂×#41	1	1	yes	characteristic
132	C₂²×D₆₇	—	2	2	no	—
133	Q₈×C₆₇	—	4	4	no	—
134	C₂×C₄×C₆₇	—	2	1	no	—
135	C₂×C₄×C₆₇	—	2	1	no	—
136	C₂×C₄×C₆₇	—	2	2	no	—
137	C₂×C₄×C₆₇	—	2	2	no	—
138	C₂×C₄×C₆₇	—	4	2	no	—
139	C₂×C₄×C₆₇	C₂²×D₄	1	1	yes	characteristic
140	C₂×C₄×C₆₇	C₂×#41	1	2	yes	—
141	C₂×C₄×C₆₇	C₂×#41	1	1	yes	characteristic
142	C₂×C₄×C₆₇	C₂×#41	1	2	yes	—
143	C₂²⋊D₆₇	—	4	2	no	—
144	C₂²⋊D₆₇	—	4	2	no	—
145	C₂×Dic₆₇	—	2	2	no	—
146	C₂×Dic₆₇	—	4	2	no	—
147	D₂₆₈	—	2	2	no	—
148	D₂₆₈	—	2	2	no	—
149	C₄×D₆₇	—	4	2	no	—
150	C₄×D₆₇	—	4	2	no	—
151	Dic₁₃₄	—	2	2	no	—
152	Dic₁₃₄	—	2	2	no	—
153	C₈×C₆₇	—	2	2	no	—
154	C₈×C₆₇	—	4	1	no	—
155	C₈×C₆₇	—	4	1	no	—
156	C₈×C₆₇	#124	1	2	yes	—
157	C₈×C₆₇	#124	1	1	yes	characteristic
158	C₈×C₆₇	#124	1	1	yes	characteristic
159	C₆₇⋊C₈	—	4	2	no	—
160	C₆₇⋊C₈	—	4	2	no	—
161	C₂²×C₆₇	—	4	1	no	—
162	C₂²×C₆₇	#525	1	1	yes	characteristic
163	C₂²×C₆₇	#529	1	2	yes	—
164	C₂×D₆₇	—	4	2	no	—
165	C₂×D₆₇	—	8	2	no	—
166	C₄×C₆₇	—	2	2	no	—
167	C₄×C₆₇	—	4	1	no	—
168	C₄×C₆₇	—	4	2	no	—
169	C₄×C₆₇	—	4	2	no	—
170	C₄×C₆₇	#525	1	1	yes	characteristic
171	C₄×C₆₇	C₂×#124	1	1	yes	characteristic
172	C₄×C₆₇	#529	1	2	yes	—
173	Dic₆₇	—	4	2	no	—
174	Dic₆₇	—	8	2	no	—
175	#23833	—	67	1	no	maximal, Sylow
176	C₂×C₆₇	—	2	2	no	—
177	C₂×C₆₇	—	2	1	no	—
178	C₂×C₆₇	#2398	1	1	yes	characteristic
179	D₆₇	—	8	2	no	—
180	#2960	D₆₇	1	1	yes	characteristic, complemented
181	#2955	—	67	1	no	—
182	#2103	—	67	1	no	—
183	#2102	—	67	1	no	—
184	#2080	—	67	2	no	—
185	#2053	—	67	1	no	—
186	#2050	—	67	1	no	—
187	#2031	—	67	2	no	—
188	#1764	—	67	2	no	—
189	#1757	—	67	2	no	—
190	#1747	—	67	1	no	—
191	C₆₇	#23833	1	1	yes	characteristic, complemented, Sylow
192	#575	—	67	1	no	—
193	C₂×#131	—	67	2	no	—
194	#533	C₂×D₆₇	1	1	yes	characteristic
195	#446	—	2	2	no	—
196	C₈×Q₈	—	2	2	no	—
197	#444	—	134	2	no	—
198	#443	—	134	2	no	—
199	#430	—	67	1	no	—
200	#424	C₂×D₆₇	1	1	yes	characteristic
201	#424	C₂×D₆₇	1	1	yes	characteristic
202	#423	C₂×D₆₇	1	1	yes	characteristic
203	#421	—	67	2	no	—
204	C₂×#104	—	67	2	no	—
205	#416	—	67	2	no	—
206	#413	—	67	2	no	—
207	#410	—	67	2	no	—
208	#405	—	67	2	no	—
209	#405	C₂×D₆₇	1	1	yes	characteristic
210	#405	C₂×D₆₇	1	1	yes	characteristic
211	C₂×C₄×C₈	C₂×D₆₇	1	1	yes	characteristic
212	#344	—	67	2	no	—
213	#344	—	67	2	no	—
214	#341	—	67	2	no	—
215	#339	—	67	2	no	—
216	#338	—	67	1	no	—
217	H(3,ℤ/4)	—	67	1	no	—
218	H(3,ℤ/4)	—	67	2	no	—
219	C₂×#41	—	67	2	no	—
220	#137	—	134	2	no	—
221	AGL(1,ℤ/8)	—	134	2	no	—
222	#135	—	134	2	no	—
223	#131	—	134	2	no	—
224	#131	—	134	2	no	—
225	#131	—	134	2	no	—
226	#131	—	134	2	no	—
227	C₂×M₁₆	—	67	2	no	—
228	C₂×M₁₆	—	67	2	no	—
229	C₂×M₁₆	—	134	2	no	—
230	C₂×M₁₆	C₂²×D₆₇	1	1	yes	characteristic
231	C₂×M₁₆	C₂²×D₆₇	1	1	yes	characteristic
232	C₂²×C₈	—	134	2	no	—
233	C₂²×C₈	C₂²×D₆₇	1	1	yes	characteristic
234	C₂²×C₈	C₂²×D₆₇	1	1	yes	characteristic
235	C₄⋊Q₈	—	2	1	no	—
236	#125	—	2	1	no	—
237	C₂²⋊Q₈	—	2	2	no	—
238	C₄×Q₈	—	2	2	no	—
239	#117	—	67	2	no	—
240	#117	C₂²×D₆₇	1	1	yes	characteristic
241	#117	C₂²×D₆₇	1	1	yes	characteristic
242	C₂×C₄²	C₂²×D₆₇	1	1	yes	characteristic
243	#108	—	134	2	no	—
244	C₄⋊C₈	—	2	2	no	—
245	C₄⋊C₈	—	2	2	no	—
246	C₄⋊C₈	—	2	2	no	—
247	#104	—	134	4	no	—
248	#104	—	134	2	no	—
249	#104	—	134	2	no	—
250	#103	—	134	2	no	—
251	#102	—	134	2	no	—
252	#101	—	134	2	no	—
253	#100	—	134	2	no	—
254	#99	—	134	4	no	—
255	#97	—	2	1	no	—
256	#97	—	134	2	no	—
257	#97	C₂²⋊D₆₇	1	2	yes	—
258	C₄×C₈	—	2	1	no	—
259	C₄×C₈	—	2	1	no	—
260	C₄×C₈	—	2	1	no	—
261	C₄×C₈	—	134	2	no	—
262	C₄×C₈	C₂²⋊D₆₇	1	2	yes	—
263	#41	—	134	2	no	—
264	#41	—	134	2	no	—
265	#41	—	134	2	no	—
266	#41	—	134	2	no	—
267	C₂×Q₈	—	2	2	no	—
268	C₂×Q₈	—	67	2	no	—
269	C₂×D₄	—	67	2	no	—
270	C₂×D₄	—	134	2	no	—
271	C₂²×C₄	—	2	1	no	—
272	C₂²×C₄	—	134	2	no	—
273	C₂²×C₄	C₂³×D₆₇	1	1	yes	characteristic
274	Q₁₆	—	134	2	no	—
275	SD₁₆	—	134	4	no	—
276	D₈	—	134	2	no	—
277	M₁₆	—	2	2	no	—
278	M₁₆	—	2	2	no	—
279	M₁₆	—	268	2	no	—
280	M₁₆	—	268	2	no	—
281	M₁₆	—	134	2	no	—
282	M₁₆	—	134	2	no	—
283	M₁₆	—	134	2	no	—
284	M₁₆	—	134	2	no	—
285	C₂×C₈	—	2	1	no	—
286	C₂×C₈	—	2	1	no	—
287	C₂×C₈	—	2	2	no	—
288	C₂×C₈	—	268	2	no	—
289	C₂×C₈	—	268	2	no	—
290	C₂×C₈	—	134	2	no	—
291	C₂×C₈	—	134	2	no	—
292	C₂×C₈	C₂×C₂²⋊D₆₇	1	2	yes	—
293	C₂×C₈	C₂×C₂²⋊D₆₇	1	1	yes	characteristic
294	C₂×C₈	C₂×C₂²⋊D₆₇	1	1	yes	characteristic
295	C₂×C₈	#11761629	1	2	yes	—
296	C₂×C₈	#11761629	1	2	yes	—
297	C₄⋊C₄	—	2	4	no	—
298	C₄⋊C₄	—	2	2	no	—
299	C₄⋊C₄	—	2	1	no	—
300	C₄⋊C₄	—	2	1	no	—
301	C₄⋊C₄	—	134	2	no	—
302	C₂²⋊C₄	—	2	1	no	—
303	C₂²⋊C₄	—	134	2	no	—
304	C₂²⋊C₄	#11761628	1	2	yes	—
305	C₄²	—	2	1	no	—
306	C₄²	—	2	1	no	—
307	C₄²	—	134	2	no	—
308	C₄²	C₂×C₂²⋊D₆₇	1	2	yes	—
309	C₄²	#11761628	1	1	yes	characteristic
310	C₄²	#11761628	1	1	yes	characteristic
311	C₂³	—	134	2	no	—
312	C₂³	C₂×#11761628	1	1	yes	characteristic
313	Q₈	—	4	4	no	—
314	Q₈	—	134	2	no	—
315	Q₈	—	134	2	no	—
316	D₄	—	268	2	no	—
317	D₄	—	268	2	no	—
318	D₄	—	134	2	no	—
319	D₄	—	134	2	no	—
320	C₂×C₄	—	2	1	no	—
321	C₂×C₄	—	2	1	no	—
322	C₂×C₄	—	2	2	no	—
323	C₂×C₄	—	2	2	no	—
324	C₂×C₄	—	4	2	no	—
325	C₂×C₄	—	268	2	no	—
326	C₂×C₄	—	268	2	no	—
327	C₂×C₄	—	268	2	no	—
328	C₂×C₄	—	134	2	no	—
329	C₂×C₄	C₂²×C₂²⋊D₆₇	1	1	yes	characteristic
330	C₂×C₄	#41×D₆₇	1	2	yes	—
331	C₂×C₄	C₂×#11761628	1	1	yes	characteristic
332	C₂×C₄	C₂×#11761629	1	2	yes	—
333	C₈	—	2	2	no	—
334	C₈	—	4	1	no	—
335	C₈	—	4	1	no	—
336	C₈	—	268	2	no	—
337	C₈	—	268	2	no	—
338	C₈	#14524032	1	2	yes	—
339	C₈	#14524032	1	1	yes	characteristic
340	C₈	#14524032	1	1	yes	characteristic
341	C₂²	—	4	1	no	—
342	C₂²	—	268	2	no	—
343	C₂²	—	536	2	no	—
344	C₂²	#16863907	1	1	yes	characteristic
345	C₂²	#16863933	1	2	yes	—
346	C₄	—	2	2	no	—
347	C₄	—	4	1	no	—
348	C₄	—	4	2	no	—
349	C₄	—	4	2	no	—
350	C₄	—	268	2	no	—
351	C₄	—	536	2	no	—
352	C₄	#16863903	1	1	yes	characteristic
353	C₄	C₂×#14524032	1	1	yes	characteristic
354	C₄	#16863933	1	2	yes	—
355	C₂	—	2	1	no	—
356	C₂	—	2	2	no	—
357	C₂	—	536	2	no	—
358	C₂	#22563492	1	1	yes	characteristic

	subgroup	G/H
center	C₈	#14524032
commutator, cosocle	C₂²×C₄×C₆₇	C₂⁴
Frattini	C₂²×C₄	C₂³×D₆₇
Fitting	C₆₇×#2960	C₂
socle	C₂×C₆₇	#2398
prime radical	#16864196	C₂²
hypercenter	#2960	D₆₇
hypocenter	C₆₇	#23833
LCS	G ▷ C₂²×C₄×C₆₇ ▷ C₂×C₆₇ ▷ C₆₇
UCS	C₁ ◁ C₈ ◁ C₂²×C₈ ◁ #2960

Complements

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

N	H ≅ G/N	type
2. #22561789	C₂	semidirect
3. #22561784	C₂	semidirect
4. #22561660	C₂	semidirect
7. #22561378	C₂	semidirect
8. #22560046	C₂	semidirect
9. #22560010	C₂	semidirect
10. #22559964	C₂	semidirect
11. C₆₇×#2960	C₂	semidirect
180. #2960	D₆₇	semidirect
191. C₆₇	#23833	semidirect

Maximal subgroups

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

subgroup	index
#22569113	2
#22561789	2
#22561784	2
#22561660	2
#22561515	2
#22561496	2
#22561378	2
#22560046	2
#22560010	2
#22559964	2
C₆₇×#2960	2
#23833	67