Subgroups — #29588965

Order 15616.

842 conjugacy classes of subgroups.

The subgroup lattice has 314 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#21459224	C₂	1	1	yes	maximal, characteristic
2	#21445558	C₂	1	1	yes	maximal, characteristic
3	#21445557	C₂	1	2	yes	maximal
4	#21445515	C₂	1	2	yes	maximal
5	C₆₁×#1729	C₂	1	1	yes	maximal, characteristic
6	C₂²×#11758194	C₂²	1	2	yes	—
7	#16840393	—	2	2	no	—
8	#16840390	—	2	4	no	—
9	#16840388	C₄	1	4	yes	—
10	#16840319	—	2	2	no	—
11	#16840315	—	2	4	no	—
12	C₂×#14519736	C₂²	1	2	yes	—
13	#16839531	—	2	2	no	—
14	#16839530	—	2	4	no	—
15	#16839516	—	2	4	no	—
16	C₂²×M₁₆×C₆₁	C₂²	1	1	yes	characteristic
17	C₂³×C₈×C₆₁	C₂²	1	1	yes	characteristic
18	C₂×#117×C₆₁	C₂²	1	1	yes	characteristic
19	C₆₁×#336	—	2	2	no	—
20	C₂×#11758194	—	2	2	no	—
21	C₂×#11758194	—	2	2	no	—
22	C₂×#11758194	—	2	2	no	—
23	C₂×#11758194	—	2	2	no	—
24	C₂×#11758194	—	2	2	no	—
25	C₂×#11758194	Q₈	1	4	yes	complemented
26	C₂²×#92418	—	2	2	no	—
27	#14519777	—	2	4	no	—
28	#14519776	—	2	8	no	—
29	#14519769	—	2	4	no	—
30	#14519768	—	2	8	no	—
31	#14519756	—	2	8	no	—
32	#14519754	—	2	16	no	—
33	C₄×Dic₁₂₂	—	2	8	no	—
34	#14519736	—	2	2	no	—
35	#14519736	—	2	2	no	—
36	#14519736	—	2	2	no	—
37	#14519736	—	2	2	no	—
38	C₂×C₄⋊Dic₆₁	D₄	1	4	yes	—
39	C₂×C₂²⋊Dic₆₁	C₂×C₄	1	4	yes	—
40	C₂×C₄×Dic₆₁	D₄	1	4	yes	—
41	C₂³×C₄×C₆₁	C₂³	1	1	yes	characteristic
42	C₂×M₁₆×C₆₁	—	2	1	no	—
43	C₂×M₁₆×C₆₁	—	2	2	no	—
44	C₂×M₁₆×C₆₁	—	2	2	no	—
45	C₂×M₁₆×C₆₁	Q₈	1	2	yes	complemented
46	C₂²×C₈×C₆₁	—	2	1	no	—
47	C₂²×C₈×C₆₁	—	2	2	no	—
48	C₂²×C₈×C₆₁	—	2	1	no	—
49	C₂²×C₈×C₆₁	—	2	1	no	—
50	C₂²×C₈×C₆₁	—	2	2	no	—
51	C₂²×C₈×C₆₁	Q₈	1	2	yes	complemented
52	#117×C₆₁	—	2	2	no	—
53	#117×C₆₁	—	2	2	no	—
54	C₂×C₄⋊C₄×C₆₁	D₄	1	2	yes	—
55	C₂×C₂²⋊C₄×C₆₁	C₂×C₄	1	2	yes	—
56	C₂×C₄²×C₆₁	D₄	1	2	yes	—
57	C₂⁴×C₆₁	C₂²×C₄	1	1	yes	characteristic
58	C₂²×C₄×C₆₁	C₂²⋊C₄	1	4	yes	—
59	C₂²×C₄×C₆₁	C₂×D₄	1	1	yes	characteristic
60	C₂²×C₄×C₆₁	C₂×D₄	1	2	yes	—
61	C₂²×C₄×C₆₁	C₂×D₄	1	2	yes	—
62	C₂²×C₄×C₆₁	C₂×D₄	1	1	yes	characteristic
63	C₂²×C₄×C₆₁	C₂×Q₈	1	1	yes	characteristic
64	C₂²×C₄×C₆₁	C₂×Q₈	1	1	yes	characteristic
65	C₂²×C₄×C₆₁	C₂×Q₈	1	2	yes	—
66	C₂²×C₄×C₆₁	#41	1	1	yes	characteristic
67	C₂²×C₄×C₆₁	#41	1	1	yes	characteristic
68	C₂²×C₄×C₆₁	#41	1	2	yes	—
69	C₂²×Dic₆₁	C₂²⋊C₄	1	8	yes	—
70	C₂×Dic₁₂₂	—	2	8	no	—
71	M₁₆×C₆₁	—	2	4	no	—
72	M₁₆×C₆₁	—	4	2	no	—
73	C₂×C₈×C₆₁	—	2	4	no	—
74	C₂×C₈×C₆₁	—	2	2	no	—
75	C₂×C₈×C₆₁	—	2	2	no	—
76	C₂×C₈×C₆₁	—	4	1	no	—
77	C₂×C₈×C₆₁	—	4	1	no	—
78	C₂×C₈×C₆₁	—	4	2	no	—
79	C₂×C₈×C₆₁	—	4	1	no	—
80	C₂×C₈×C₆₁	—	4	1	no	—
81	C₄⋊C₄×C₆₁	—	2	4	no	—
82	C₂²⋊C₄×C₆₁	—	2	4	no	—
83	C₄²×C₆₁	—	2	2	no	—
84	C₄²×C₆₁	—	2	2	no	—
85	C₂²⋊Dic₆₁	—	2	4	no	—
86	C₂²⋊Dic₆₁	—	2	4	no	—
87	C₄⋊Dic₆₁	—	2	8	no	—
88	C₄⋊Dic₆₁	—	2	4	no	—
89	C₄⋊Dic₆₁	—	2	4	no	—
90	#11758196	—	2	16	no	—
91	C₄×Dic₆₁	—	2	4	no	—
92	C₄×Dic₆₁	—	2	4	no	—
93	#11758194	—	2	4	no	—
94	#11758194	—	2	4	no	—
95	#11758194	—	4	2	no	—
96	#11758194	—	4	2	no	—
97	C₂×#92418	—	2	4	no	—
98	C₂×#92418	—	4	2	no	—
99	C₂×#92418	—	4	2	no	—
100	C₂³×C₆₁	—	4	2	no	—
101	C₂³×C₆₁	C₂×C₂²⋊C₄	1	2	yes	—
102	C₂³×C₆₁	C₂×C₂²⋊C₄	1	1	yes	characteristic
103	C₂³×C₆₁	C₄×Q₈	1	2	yes	—
104	C₂³×C₆₁	C₄×Q₈	1	1	yes	characteristic
105	C₂³×C₆₁	C₄×Q₈	1	1	yes	characteristic
106	C₂×C₄×C₆₁	—	2	4	no	—
107	C₂×C₄×C₆₁	—	4	4	no	—
108	C₂×C₄×C₆₁	#120	1	1	yes	characteristic
109	C₂×C₄×C₆₁	#120	1	1	yes	characteristic
110	C₂×C₄×C₆₁	#120	1	2	yes	—
111	C₂×C₄×C₆₁	C₂²⋊Q₈	1	2	yes	—
112	C₂×C₄×C₆₁	C₂²⋊Q₈	1	2	yes	—
113	C₂×C₄×C₆₁	C₂²⋊Q₈	1	2	yes	—
114	C₂×C₄×C₆₁	C₂²⋊Q₈	1	2	yes	—
115	C₂×C₄×C₆₁	C₂²⋊Q₈	1	1	yes	characteristic
116	C₂×C₄×C₆₁	C₂²⋊Q₈	1	1	yes	characteristic
117	C₂×C₄×C₆₁	C₂²⋊Q₈	1	2	yes	—
118	C₂×C₄×C₆₁	#124	1	2	yes	—
119	C₂×C₄×C₆₁	#124	1	2	yes	—
120	C₂×C₄×C₆₁	#124	1	1	yes	characteristic
121	C₂×C₄×C₆₁	#124	1	1	yes	characteristic
122	C₂×C₄×C₆₁	C₄⋊Q₈	1	1	yes	characteristic
123	C₂×C₄×C₆₁	C₄⋊Q₈	1	2	yes	—
124	C₂×C₄×C₆₁	C₄⋊Q₈	1	2	yes	—
125	C₂×C₄×C₆₁	C₄⋊Q₈	1	1	yes	characteristic
126	C₂×Dic₆₁	—	2	8	no	—
127	C₂×Dic₆₁	—	4	8	no	—
128	Dic₁₂₂	—	8	8	no	—
129	C₈×C₆₁	—	4	2	no	—
130	C₈×C₆₁	—	4	2	no	—
131	#92418	—	4	4	no	—
132	#11391	—	61	1	no	maximal, Sylow
133	C₂²×C₆₁	—	2	4	no	—
134	C₂²×C₆₁	—	2	2	no	—
135	C₂²×C₆₁	—	4	2	no	—
136	C₂²×C₆₁	—	4	1	no	—
137	C₂²×C₆₁	—	4	1	no	—
138	C₂²×C₆₁	#379	1	1	yes	characteristic
139	C₂²×C₆₁	#391	1	2	yes	—
140	C₂²×C₆₁	#391	1	2	yes	—
141	C₂²×C₆₁	#391	1	1	yes	characteristic
142	C₂²×C₆₁	#391	1	1	yes	characteristic
143	C₄×C₆₁	—	4	4	no	—
144	C₄×C₆₁	#393	1	1	yes	characteristic
145	C₄×C₆₁	#393	1	1	yes	characteristic
146	C₄×C₆₁	#393	1	2	yes	—
147	C₄×C₆₁	#395	1	1	yes	characteristic
148	C₄×C₆₁	#395	1	1	yes	characteristic
149	C₄×C₆₁	#395	1	2	yes	—
150	Dic₆₁	—	4	8	no	—
151	#3458	—	61	1	no	—
152	#1742	—	61	2	no	—
153	#1742	—	61	1	no	—
154	#1729	—	61	2	no	—
155	#1729	D₆₁	1	1	yes	characteristic
156	C₂×C₆₁	—	2	2	no	—
157	C₂×C₆₁	—	2	2	no	—
158	C₂×C₆₁	—	2	1	no	—
159	C₂×C₆₁	—	2	1	no	—
160	C₂×C₆₁	#1446	1	1	yes	characteristic
161	C₂×C₆₁	#1791	1	2	yes	—
162	C₂²×M₁₆	—	61	2	no	—
163	C₂²×M₁₆	C₂×D₆₁	1	1	yes	characteristic
164	C₂³×C₈	C₂×D₆₁	1	1	yes	characteristic
165	#544	—	122	4	no	—
166	#544	—	122	2	no	—
167	#543	—	61	4	no	—
168	#533	—	122	4	no	—
169	#533	—	122	2	no	—
170	C₂×#117	—	61	2	no	—
171	C₂×#117	C₂×D₆₁	1	1	yes	characteristic
172	#343	—	122	4	no	—
173	#343	—	122	2	no	—
174	#336	—	2	2	no	—
175	#336	—	122	4	no	—
176	C₆₁	#11391	1	1	yes	characteristic, complemented, Sylow
177	C₂³×C₄	C₂²×D₆₁	1	1	yes	characteristic
178	C₂×M₁₆	—	2	1	no	—
179	C₂×M₁₆	—	2	2	no	—
180	C₂×M₁₆	—	2	2	no	—
181	C₂×M₁₆	—	61	4	no	—
182	C₂×M₁₆	—	122	2	no	—
183	C₂×M₁₆	—	122	2	no	—
184	C₂×M₁₆	—	122	2	no	—
185	C₂×M₁₆	—	122	2	no	—
186	C₂×M₁₆	—	122	2	no	—
187	C₂×M₁₆	Dic₁₂₂	1	2	yes	complemented
188	C₂²×C₈	—	2	1	no	—
189	C₂²×C₈	—	2	2	no	—
190	C₂²×C₈	—	2	2	no	—
191	C₂²×C₈	—	2	1	no	—
192	C₂²×C₈	—	2	1	no	—
193	C₂²×C₈	—	122	2	no	—
194	C₂²×C₈	Dic₁₂₂	1	2	yes	complemented
195	C₄⋊Q₈	—	122	4	no	—
196	C₄⋊Q₈	—	122	8	no	—
197	#125	—	122	4	no	—
198	#125	—	122	8	no	—
199	C₂²⋊Q₈	—	122	16	no	—
200	C₂²⋊Q₈	—	122	8	no	—
201	C₄×Q₈	—	122	8	no	—
202	#117	—	2	2	no	—
203	#117	—	2	2	no	—
204	#117	—	122	2	no	—
205	#117	—	122	2	no	—
206	#117	—	122	2	no	—
207	#117	—	122	2	no	—
208	C₂×C₄⋊C₄	—	61	4	no	—
209	C₂×C₄⋊C₄	D₂₄₄	1	2	yes	—
210	C₂×C₂²⋊C₄	—	61	4	no	—
211	C₂×C₂²⋊C₄	C₄×D₆₁	1	2	yes	—
212	C₂×C₄²	—	61	4	no	—
213	C₂×C₄²	C₂²⋊D₆₁	1	2	yes	—
214	C₂⁴	C₂×C₄×D₆₁	1	1	yes	characteristic
215	C₂×Q₈	—	122	8	no	—
216	C₂²×C₄	—	61	8	no	—
217	C₂²×C₄	#11758198	1	4	yes	—
218	C₂²×C₄	C₂×Dic₁₂₂	1	1	yes	characteristic
219	C₂²×C₄	C₂×Dic₁₂₂	1	1	yes	characteristic
220	C₂²×C₄	C₂×D₂₄₄	1	1	yes	characteristic
221	C₂²×C₄	#11758222	1	1	yes	characteristic
222	C₂²×C₄	#11758222	1	1	yes	characteristic
223	C₂²×C₄	D₄×D₆₁	1	2	yes	—
224	C₂²×C₄	D₄×D₆₁	1	2	yes	—
225	C₂²×C₄	#11758224	1	2	yes	—
226	C₂²×C₄	Q₈×D₆₁	1	2	yes	—
227	C₂²×C₄	C₂×C₂²⋊D₆₁	1	1	yes	characteristic
228	M₁₆	—	2	4	no	—
229	M₁₆	—	4	2	no	—
230	M₁₆	—	122	4	no	—
231	M₁₆	—	122	4	no	—
232	M₁₆	—	244	2	no	—
233	M₁₆	—	244	2	no	—
234	C₂×C₈	—	2	4	no	—
235	C₂×C₈	—	2	2	no	—
236	C₂×C₈	—	2	2	no	—
237	C₂×C₈	—	4	1	no	—
238	C₂×C₈	—	4	1	no	—
239	C₂×C₈	—	4	2	no	—
240	C₂×C₈	—	4	1	no	—
241	C₂×C₈	—	4	1	no	—
242	C₂×C₈	—	122	4	no	—
243	C₂×C₈	—	244	2	no	—
244	C₂×C₈	—	244	2	no	—
245	C₄⋊C₄	—	2	4	no	—
246	C₄⋊C₄	—	122	16	no	—
247	C₄⋊C₄	—	122	8	no	—
248	C₄⋊C₄	—	122	4	no	—
249	C₄⋊C₄	—	122	4	no	—
250	C₂²⋊C₄	—	2	4	no	—
251	C₂²⋊C₄	—	122	4	no	—
252	C₂²⋊C₄	—	122	4	no	—
253	C₄²	—	2	2	no	—
254	C₄²	—	2	2	no	—
255	C₄²	—	122	4	no	—
256	C₄²	—	122	4	no	—
257	C₂³	—	4	2	no	—
258	C₂³	C₂²⋊C₄×D₆₁	1	2	yes	—
259	C₂³	C₂×#11758198	1	1	yes	characteristic
260	C₂³	#14519742	1	2	yes	—
261	C₂³	C₄×Dic₁₂₂	1	1	yes	characteristic
262	C₂³	C₄×Dic₁₂₂	1	1	yes	characteristic
263	Q₈	—	488	8	no	—
264	C₂×C₄	—	2	4	no	—
265	C₂×C₄	—	4	4	no	—
266	C₂×C₄	—	122	8	no	—
267	C₂×C₄	—	244	8	no	—
268	C₂×C₄	#14519745	1	1	yes	characteristic
269	C₂×C₄	#14519745	1	1	yes	characteristic
270	C₂×C₄	#14519747	1	2	yes	—
271	C₂×C₄	#14519754	1	2	yes	—
272	C₂×C₄	#14519754	1	2	yes	—
273	C₂×C₄	#14519755	1	2	yes	—
274	C₂×C₄	#14519755	1	2	yes	—
275	C₂×C₄	#14519756	1	1	yes	characteristic
276	C₂×C₄	#14519756	1	1	yes	characteristic
277	C₂×C₄	#14519757	1	2	yes	—
278	C₂×C₄	#14519763	1	2	yes	—
279	C₂×C₄	#14519763	1	2	yes	—
280	C₂×C₄	#14519764	1	1	yes	characteristic
281	C₂×C₄	#14519765	1	1	yes	characteristic
282	C₂×C₄	#14519775	1	1	yes	characteristic
283	C₂×C₄	#14519776	1	2	yes	—
284	C₂×C₄	#14519776	1	2	yes	—
285	C₂×C₄	#14519777	1	1	yes	characteristic
286	C₈	—	4	2	no	—
287	C₈	—	4	2	no	—
288	C₈	—	244	4	no	—
289	C₂²	—	2	2	no	—
290	C₂²	—	2	4	no	—
291	C₂²	—	4	2	no	—
292	C₂²	—	4	1	no	—
293	C₂²	—	4	1	no	—
294	C₂²	#16839617	1	1	yes	characteristic
295	C₂²	#16839666	1	2	yes	—
296	C₂²	#16839666	1	2	yes	—
297	C₂²	#16839667	1	1	yes	characteristic
298	C₂²	#16839668	1	1	yes	characteristic
299	C₄	—	4	4	no	—
300	C₄	—	244	8	no	—
301	C₄	#16839674	1	1	yes	characteristic
302	C₄	#16839674	1	1	yes	characteristic
303	C₄	#16839675	1	2	yes	—
304	C₄	#16839682	1	1	yes	characteristic
305	C₄	#16839682	1	1	yes	characteristic
306	C₄	#16839683	1	2	yes	—
307	C₂	—	2	2	no	—
308	C₂	—	2	2	no	—
309	C₂	—	2	1	no	—
310	C₂	—	2	1	no	—
311	C₂	#21444089	1	1	yes	characteristic
312	C₂	#21445769	1	2	yes	—

	subgroup	G/H
center	C₂×C₄	#14519745
commutator, prime radical	C₂⁴×C₆₁	C₂²×C₄
Frattini	C₂³×C₄	C₂²×D₆₁
Fitting	C₆₁×#1729	C₂
socle	C₂²×C₆₁	#379
cosocle	C₂³×C₄×C₆₁	C₂³
hypercenter	#1729	D₆₁
hypocenter	C₆₁	#11391
LCS	G ▷ C₂⁴×C₆₁ ▷ C₂×C₆₁ ▷ C₆₁
UCS	C₁ ◁ C₂×C₄ ◁ C₂³×C₄ ◁ #1729

Complements

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

N	H ≅ G/N	type
25. C₂×#11758194	Q₈	semidirect
45. C₂×M₁₆×C₆₁	Q₈	semidirect
51. C₂²×C₈×C₆₁	Q₈	semidirect
176. C₆₁	#11391	semidirect
187. C₂×M₁₆	Dic₁₂₂	semidirect
194. C₂²×C₈	Dic₁₂₂	semidirect

Maximal subgroups

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

subgroup	index
#21459224	2
#21445558	2
#21445557	2
#21445515	2
C₆₁×#1729	2
#11391	61