Subgroups — #30597630

Order 15616.

498 conjugacy classes of subgroups.

The subgroup lattice has 342 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#21462625	C₂	1	1	yes	maximal, characteristic
2	C₄×#14519842	C₂	1	1	yes	maximal, characteristic
3	#21461532	C₂	1	1	yes	maximal, characteristic, complemented
4	#21461021	C₂	1	1	yes	maximal, characteristic, complemented
5	#21460651	C₂	1	1	yes	maximal, characteristic, complemented
6	#21460646	C₂	1	1	yes	maximal, characteristic, complemented
7	#21444200	C₂	1	1	yes	maximal, characteristic
8	Q₈×#92420	C₄	1	2	yes	complemented
9	#16840790	—	2	2	no	—
10	#16840790	C₄	1	2	yes	complemented
11	#16840781	—	2	1	no	—
12	#16840781	C₂²	1	1	yes	characteristic
13	C₂×#14519842	C₄	1	2	yes	—
14	C₄×#11758213	C₂²	1	1	yes	characteristic
15	C₄×D₆₁⋊C₈	—	2	1	no	—
16	C₄×D₆₁⋊C₈	C₂²	1	1	yes	characteristic
17	C₄²×#6923	C₂²	1	1	yes	characteristic
18	#16840632	C₄	1	2	yes	—
19	(C₂×#6923)⋊C₈	—	2	2	no	—
20	#92420⋊C₈	—	2	1	no	—
21	C₈×#92420	—	2	1	no	—
22	C₄×#11758226	C₂²	1	1	yes	characteristic
23	#16839814	C₂²	1	1	yes	characteristic
24	C₄×C₈×D₆₁	C₂²	1	1	yes	characteristic
25	Q₈⋊#92418	C₄	1	1	yes	characteristic, complemented
26	#16839491	C₄	1	1	yes	characteristic, complemented
27	#16839489	C₄	1	1	yes	characteristic, complemented
28	C₆₁×Q₈⋊C₈	C₄	1	1	yes	characteristic, complemented
29	C₂×#11758213	C₂×C₄	1	2	yes	—
30	C₂×D₆₁⋊C₈	—	2	2	no	—
31	C₂×D₆₁⋊C₈	D₄	1	2	yes	—
32	C₂×C₄×#6923	—	2	1	no	—
33	C₂×C₄×#6923	—	2	1	no	—
34	C₂×C₄×#6923	C₂×C₄	1	2	yes	—
35	C₄⋊#92420	—	2	2	no	—
36	C₄⋊#92420	C₂×C₄	1	2	yes	—
37	#14519842	C₈	1	8	yes	complemented
38	#14519834	—	2	1	no	—
39	#14519834	—	2	2	no	—
40	#14519832	—	2	1	no	—
41	C₄×#92420	—	2	2	no	—
42	C₄×#92420	—	2	1	no	—
43	C₄×#92420	—	2	1	no	—
44	C₄×#92420	C₂×C₄	1	2	yes	—
45	C₄×#92420	D₄	1	1	yes	characteristic
46	C₄×#92420	D₄	1	1	yes	characteristic
47	C₂×#11758226	C₂×C₄	1	1	yes	characteristic
48	C₂×C₈×D₆₁	—	2	1	no	—
49	Q₈×Dic₆₁	C₂×C₄	1	1	yes	characteristic
50	C₄×D₂₄₄	—	2	1	no	—
51	C₄×D₂₄₄	C₂×C₄	1	1	yes	characteristic
52	#14519739	—	2	1	no	—
53	#14519739	C₂×C₄	1	1	yes	characteristic
54	#14519735	—	2	1	no	—
55	#14519735	C₂×C₄	1	1	yes	characteristic
56	C₄²×D₆₁	—	2	1	no	—
57	C₄²×D₆₁	C₂³	1	1	yes	characteristic
58	C₄⋊#92418	C₂×C₄	1	1	yes	characteristic
59	#14519689	—	2	1	no	—
60	C₄×#92418	C₂×C₄	1	1	yes	characteristic
61	C₈×Dic₆₁	—	2	1	no	—
62	C₈×Dic₆₁	—	2	1	no	—
63	C₄×Q₈×C₆₁	C₂×C₄	1	1	yes	characteristic
64	C₄⋊C₈×C₆₁	C₂×C₄	1	1	yes	characteristic
65	C₄×C₈×C₆₁	C₂×C₄	1	1	yes	characteristic
66	C₂²×#6923	—	2	2	no	—
67	C₂×Q₈×C₆₁	C₄²	1	1	yes	characteristic
68	#11758226	—	2	2	no	—
69	#11758226	C₂×C₈	1	4	yes	—
70	C₂×D₂₄₄	—	2	1	no	—
71	C₂×D₂₄₄	C₄²	1	1	yes	characteristic
72	C₂×C₄×D₆₁	—	2	1	no	—
73	C₂×C₄×D₆₁	—	2	1	no	—
74	C₂×C₄×D₆₁	—	2	1	no	—
75	C₂×C₄×D₆₁	C₂²×C₄	1	1	yes	characteristic
76	C₂×C₄×D₆₁	C₂×D₄	1	1	yes	characteristic
77	C₂×C₄×D₆₁	#41	1	1	yes	characteristic
78	C₂×#92420	—	2	2	no	—
79	C₂×#92420	—	2	2	no	—
80	C₂×#92420	C₂²⋊C₄	1	2	yes	—
81	C₂×#92420	C₂²⋊C₄	1	2	yes	—
82	C₄×#6923	—	2	2	no	—
83	C₄×#6923	—	4	1	no	—
84	C₄×#6923	—	4	1	no	—
85	C₄×#6923	—	4	2	no	—
86	C₄×#6923	C₂×C₈	1	4	yes	complemented
87	#11758213	—	2	2	no	—
88	#11758213	C₂×C₈	1	4	yes	complemented
89	D₆₁⋊C₈	—	4	2	no	—
90	D₆₁⋊C₈	—	4	4	no	—
91	C₂×C₈×C₆₁	—	2	1	no	—
92	C₂×C₈×C₆₁	—	2	1	no	—
93	C₄⋊C₄×C₆₁	—	2	1	no	—
94	C₄⋊C₄×C₆₁	C₄²	1	1	yes	characteristic
95	C₄²×C₆₁	—	2	1	no	—
96	C₄²×C₆₁	C₂²×C₄	1	1	yes	characteristic
97	#11758198	—	2	1	no	—
98	#11758198	—	2	1	no	—
99	#11758198	—	2	1	no	—
100	C₄⋊Dic₆₁	—	2	1	no	—
101	C₄⋊Dic₆₁	C₄²	1	1	yes	characteristic
102	C₄×Dic₆₁	—	2	1	no	—
103	C₄×Dic₆₁	—	2	1	no	—
104	C₄×Dic₆₁	C₂²×C₄	1	1	yes	characteristic
105	C₄×Dic₆₁	C₂×D₄	1	1	yes	characteristic
106	C₄×Dic₆₁	#41	1	1	yes	characteristic
107	C₂×#92418	—	2	1	no	—
108	C₂×#92418	—	2	1	no	—
109	C₈×D₆₁	—	4	2	no	—
110	C₂²×D₆₁	—	2	1	no	—
111	C₂²×D₆₁	C₄×D₄	1	1	yes	characteristic
112	C₂×#6923	—	2	4	no	—
113	C₂×#6923	—	4	2	no	—
114	C₂×#6923	—	4	2	no	—
115	Q₈×C₆₁	—	2	1	no	—
116	Q₈×C₆₁	C₄×C₈	1	2	yes	—
117	C₂×C₄×C₆₁	—	2	1	no	—
118	C₂×C₄×C₆₁	—	2	1	no	—
119	C₂×C₄×C₆₁	C₂×C₄²	1	1	yes	characteristic
120	C₂×C₄×C₆₁	C₄×D₄	1	1	yes	characteristic
121	C₂×C₄×C₆₁	C₄×D₄	1	1	yes	characteristic
122	C₂×Dic₆₁	—	2	1	no	—
123	C₂×Dic₆₁	C₂×C₂²⋊C₄	1	1	yes	characteristic
124	C₂×Dic₆₁	#117	1	1	yes	characteristic
125	C₂×Dic₆₁	C₄×D₄	1	1	yes	characteristic
126	D₂₄₄	—	2	4	no	—
127	D₂₄₄	—	2	1	no	—
128	D₂₄₄	C₄×C₈	1	2	yes	—
129	C₄×D₆₁	—	2	1	no	—
130	C₄×D₆₁	—	2	1	no	—
131	C₄×D₆₁	—	2	2	no	—
132	C₄×D₆₁	—	2	4	no	—
133	C₄×D₆₁	—	4	1	no	—
134	C₄×D₆₁	—	4	1	no	—
135	C₄×D₆₁	—	4	2	no	—
136	C₄×D₆₁	C₂²×C₈	1	2	yes	—
137	C₄×D₆₁	#131	1	2	yes	—
138	#92420	—	2	2	no	—
139	#92420	—	2	2	no	—
140	#92420	—	2	4	no	—
141	#92420	—	4	2	no	—
142	C₈×C₆₁	—	4	1	no	—
143	C₈×C₆₁	—	4	1	no	—
144	#92418	—	4	1	no	—
145	#92418	—	4	1	no	—
146	#8191	—	61	1	no	maximal, Sylow
147	C₂²×C₆₁	C₄×C₂²⋊C₄	1	1	yes	characteristic
148	C₂×D₆₁	—	2	1	no	—
149	C₂×D₆₁	—	2	1	no	—
150	C₂×D₆₁	—	2	2	no	—
151	C₂×D₆₁	—	4	1	no	—
152	C₂×D₆₁	—	4	1	no	—
153	C₂×D₆₁	C₈×D₄	1	2	yes	—
154	#6923	—	4	4	no	—
155	C₄×C₆₁	—	2	2	no	—
156	C₄×C₆₁	—	2	1	no	—
157	C₄×C₆₁	—	4	1	no	—
158	C₄×C₆₁	C₂×C₄×C₈	1	1	yes	characteristic
159	C₄×C₆₁	#405	1	1	yes	characteristic
160	C₄×C₆₁	#444	1	1	yes	characteristic
161	C₄×C₆₁	#444	1	1	yes	characteristic
162	Dic₆₁	—	2	1	no	—
163	Dic₆₁	—	4	1	no	—
164	Dic₆₁	#421	1	2	yes	—
165	Dic₆₁	C₈×D₄	1	2	yes	—
166	C₈×#41	—	61	1	no	—
167	C₄×#104	—	61	1	no	—
168	#1744	—	61	1	no	—
169	#1476	—	61	1	no	—
170	C₈×M₁₆	—	61	1	no	—
171	#1278	—	61	1	no	—
172	#1276	—	61	1	no	—
173	C₂×C₆₁	C₈×C₂²⋊C₄	1	1	yes	characteristic
174	C₂×C₆₁	#1759	1	1	yes	characteristic
175	C₂×C₆₁	#1765	1	1	yes	characteristic
176	D₆₁	—	2	2	no	—
177	D₆₁	—	4	2	no	—
178	C₄×#41	—	61	1	no	—
179	C₈×Q₈	—	61	2	no	—
180	C₈×D₄	—	61	2	no	—
181	C₈×D₄	—	122	2	no	—
182	#431	—	61	1	no	—
183	#431	—	61	1	no	—
184	#431	—	122	1	no	—
185	C₂×#104	—	61	2	no	—
186	C₄×M₁₆	—	61	1	no	—
187	C₂×C₄×C₈	—	61	1	no	—
188	C₂×C₄×C₈	—	61	1	no	—
189	C₂×C₄×C₈	—	122	1	no	—
190	C₄³	—	61	1	no	—
191	#339	—	61	2	no	—
192	#336	—	122	2	no	—
193	Q₈⋊C₈	—	61	1	no	—
194	Q₈⋊C₈	#6923	1	1	yes	characteristic, complemented
195	D₄⋊C₈	—	61	1	no	—
196	D₄⋊C₈	—	61	1	no	—
197	#322	—	122	1	no	—
198	C₈²	—	122	1	no	—
199	C₆₁	#8191	1	1	yes	characteristic, complemented, Sylow
200	C₂×#41	—	61	1	no	—
201	C₂×M₁₆	—	61	2	no	—
202	C₂²×C₈	—	61	2	no	—
203	C₂²×C₈	—	122	1	no	—
204	C₂²×C₈	—	122	2	no	—
205	C₄×Q₈	—	61	1	no	—
206	C₄×Q₈	C₂×#6923	1	1	yes	characteristic
207	C₄×D₄	—	61	1	no	—
208	C₄×D₄	—	61	1	no	—
209	C₄×D₄	—	122	1	no	—
210	C₄×D₄	—	122	1	no	—
211	#117	—	61	1	no	—
212	#117	—	122	1	no	—
213	C₂×C₄²	—	61	1	no	—
214	C₂×C₄²	—	61	2	no	—
215	C₂×C₄²	—	122	1	no	—
216	C₂×C₄²	—	122	1	no	—
217	C₂×C₄²	—	122	1	no	—
218	C₄⋊C₈	—	61	2	no	—
219	C₄⋊C₈	—	61	1	no	—
220	C₄⋊C₈	—	122	2	no	—
221	C₄⋊C₈	C₂×#6923	1	1	yes	characteristic
222	#104	—	61	8	no	—
223	C₂²⋊C₈	—	122	1	no	—
224	C₂²⋊C₈	—	122	2	no	—
225	C₂²⋊C₈	—	122	1	no	—
226	#97	—	122	1	no	—
227	C₄×C₈	—	61	2	no	—
228	C₄×C₈	—	61	1	no	—
229	C₄×C₈	—	61	1	no	—
230	C₄×C₈	—	61	1	no	—
231	C₄×C₈	—	122	1	no	—
232	C₄×C₈	—	122	1	no	—
233	C₄×C₈	—	122	1	no	—
234	C₄×C₈	—	122	1	no	—
235	C₄×C₈	—	122	2	no	—
236	C₄×C₈	C₂×#6923	1	1	yes	characteristic
237	#41	—	61	4	no	—
238	#41	—	122	2	no	—
239	C₂×Q₈	C₄×#6923	1	1	yes	characteristic
240	C₂×D₄	—	61	1	no	—
241	C₂×D₄	—	122	1	no	—
242	C₂²×C₄	—	61	1	no	—
243	C₂²×C₄	—	61	1	no	—
244	C₂²×C₄	—	61	1	no	—
245	C₂²×C₄	—	122	1	no	—
246	C₂²×C₄	—	122	1	no	—
247	C₂²×C₄	—	122	2	no	—
248	C₂²×C₄	—	122	1	no	—
249	M₁₆	—	61	4	no	—
250	M₁₆	—	122	2	no	—
251	C₂×C₈	—	2	1	no	—
252	C₂×C₈	—	2	1	no	—
253	C₂×C₈	—	61	2	no	—
254	C₂×C₈	—	61	2	no	—
255	C₂×C₈	—	122	2	no	—
256	C₂×C₈	—	122	2	no	—
257	C₂×C₈	—	122	1	no	—
258	C₂×C₈	—	122	1	no	—
259	C₂×C₈	—	244	2	no	—
260	C₂×C₈	—	244	2	no	—
261	C₂×C₈	—	244	4	no	—
262	C₄⋊C₄	—	2	1	no	—
263	C₄⋊C₄	—	61	1	no	—
264	C₄⋊C₄	—	122	1	no	—
265	C₄⋊C₄	C₄×#6923	1	1	yes	characteristic
266	C₂²⋊C₄	—	122	1	no	—
267	C₂²⋊C₄	—	122	1	no	—
268	C₂²⋊C₄	—	122	1	no	—
269	C₄²	—	2	1	no	—
270	C₄²	—	61	1	no	—
271	C₄²	—	61	1	no	—
272	C₄²	—	61	1	no	—
273	C₄²	—	61	4	no	—
274	C₄²	—	122	1	no	—
275	C₄²	—	122	1	no	—
276	C₄²	—	122	2	no	—
277	C₄²	—	244	2	no	—
278	C₄²	—	244	1	no	—
279	C₄²	—	244	1	no	—
280	C₄²	C₂²×#6923	1	1	yes	characteristic
281	C₂³	—	61	1	no	—
282	C₂³	—	122	1	no	—
283	Q₈	—	2	1	no	—
284	Q₈	C₈×#6923	1	2	yes	—
285	D₄	—	61	2	no	—
286	D₄	—	122	4	no	—
287	D₄	—	122	1	no	—
288	C₂×C₄	—	2	1	no	—
289	C₂×C₄	—	2	1	no	—
290	C₂×C₄	—	61	1	no	—
291	C₂×C₄	—	61	1	no	—
292	C₂×C₄	—	61	2	no	—
293	C₂×C₄	—	61	2	no	—
294	C₂×C₄	—	61	1	no	—
295	C₂×C₄	—	122	2	no	—
296	C₂×C₄	—	122	4	no	—
297	C₂×C₄	—	122	1	no	—
298	C₂×C₄	—	122	1	no	—
299	C₂×C₄	—	122	4	no	—
300	C₂×C₄	—	122	1	no	—
301	C₂×C₄	—	244	2	no	—
302	C₂×C₄	—	244	2	no	—
303	C₂×C₄	—	244	2	no	—
304	C₂×C₄	—	244	1	no	—
305	C₂×C₄	—	244	1	no	—
306	C₂×C₄	C₂×C₄×#6923	1	1	yes	characteristic
307	C₂×C₄	D₄×#6923	1	1	yes	characteristic
308	C₂×C₄	D₄×#6923	1	1	yes	characteristic
309	C₈	—	4	1	no	—
310	C₈	—	4	1	no	—
311	C₈	—	122	2	no	—
312	C₈	—	122	2	no	—
313	C₈	—	122	4	no	—
314	C₈	—	244	1	no	—
315	C₈	—	244	2	no	—
316	C₈	—	244	1	no	—
317	C₂²	—	61	2	no	—
318	C₂²	—	122	1	no	—
319	C₂²	—	122	1	no	—
320	C₂²	—	122	2	no	—
321	C₂²	—	244	1	no	—
322	C₂²	—	244	1	no	—
323	C₂²	C₂²⋊C₄×#6923	1	1	yes	characteristic
324	C₄	—	2	2	no	—
325	C₄	—	2	1	no	—
326	C₄	—	4	1	no	—
327	C₄	—	61	2	no	—
328	C₄	—	61	2	no	—
329	C₄	—	122	1	no	—
330	C₄	—	244	4	no	—
331	C₄	—	244	1	no	—
332	C₄	C₂×C₈×#6923	1	1	yes	characteristic
333	C₄	#16840728	1	1	yes	characteristic
334	C₄	#16840807	1	1	yes	characteristic
335	C₄	#16840808	1	1	yes	characteristic
336	C₂	—	122	2	no	—
337	C₂	—	244	2	no	—
338	C₂	#21461554	1	1	yes	characteristic
339	C₂	#21461584	1	1	yes	characteristic
340	C₂	#21461600	1	1	yes	characteristic

	subgroup	G/H
center	C₂×C₄	D₄×#6923
commutator	C₄×C₆₁	C₂×C₄×C₈
Frattini	C₄²	C₂²×#6923
Fitting	C₆₁×Q₈⋊C₈	C₄
socle	C₂²×C₆₁	C₄×C₂²⋊C₄
prime radical	C₂×#11758226	C₂×C₄
cosocle	C₄²×D₆₁	C₂³
hypercenter	Q₈⋊C₈	#6923
hypocenter	C₆₁	#8191
LCS	G ▷ C₄×C₆₁ ▷ C₂×C₆₁ ▷ C₆₁
UCS	C₁ ◁ C₂×C₄ ◁ C₄² ◁ Q₈⋊C₈

Complements

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

N	H ≅ G/N	type
3. #21461532	C₂	semidirect
4. #21461021	C₂	semidirect
5. #21460651	C₂	semidirect
6. #21460646	C₂	semidirect
8. Q₈×#92420	C₄	semidirect
10. #16840790	C₄	semidirect
25. Q₈⋊#92418	C₄	semidirect
26. #16839491	C₄	semidirect
27. #16839489	C₄	semidirect
28. C₆₁×Q₈⋊C₈	C₄	semidirect
37. #14519842	C₈	semidirect
86. C₄×#6923	C₂×C₈	semidirect
88. #11758213	C₂×C₈	semidirect
194. Q₈⋊C₈	#6923	semidirect
199. C₆₁	#8191	semidirect

Maximal subgroups

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

subgroup	index
#21462625	2
C₄×#14519842	2
#21461532	2
#21461021	2
#21460651	2
#21460646	2
#21444200	2
#8191	61