Subgroups — #30624295

Order 15616.

646 conjugacy classes of subgroups.

The subgroup lattice has 284 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#21462580	C₂	1	2	yes	maximal
2	#21462216	C₂	1	4	yes	maximal, complemented
3	Q₃₂×#6923	C₂	1	4	yes	maximal, complemented
4	#21462170	C₂	1	1	yes	maximal, characteristic
5	#21462129	C₂	1	2	yes	maximal
6	#21462027	C₂	1	1	yes	maximal, characteristic
7	C₂×Q₃₂×D₆₁	C₂	1	1	yes	maximal, characteristic
8	#16840830	—	2	2	no	—
9	D₆₁⋊M₃₂	C₂²	1	1	yes	characteristic
10	#16840802	—	2	4	no	—
11	Q₁₆×#6923	C₂²	1	8	yes	—
12	#16840772	C₂²	1	2	yes	—
13	#16840755	—	2	2	no	—
14	#16840725	C₂²	1	1	yes	characteristic
15	#16840689	C₂²	1	2	yes	—
16	#16840688	C₂²	1	2	yes	—
17	#16840675	C₂²	1	8	yes	—
18	#16840649	C₂²	1	2	yes	—
19	C₁₆×#6923	C₂²	1	2	yes	—
20	C₂×Q₁₆×D₆₁	C₂²	1	2	yes	—
21	C₂×Dic₄₈₈	C₄	1	1	yes	characteristic, complemented
22	C₂×#14519723	C₄	1	2	yes	complemented
23	Q₃₂×D₆₁	C₄	1	2	yes	complemented
24	Q₃₂×D₆₁	C₄	1	2	yes	complemented
25	Q₃₂×D₆₁	C₂²	1	2	yes	—
26	Q₃₂×D₆₁	C₂²	1	2	yes	—
27	C₂×C₁₆×D₆₁	C₂²	1	1	yes	characteristic
28	C₂×Q₃₂×C₆₁	C₄	1	1	yes	characteristic, complemented
29	D₆₁⋊M₁₆	D₄	1	1	yes	characteristic
30	Q₈×#6923	—	2	8	no	—
31	Q₈×#6923	—	4	4	no	—
32	#14519856	—	2	2	no	—
33	#14519856	—	4	4	no	—
34	#14519856	D₄	1	1	yes	characteristic
35	#14519852	—	2	1	no	—
36	#14519851	—	2	1	no	—
37	D₆₁⋊C₁₆	C₂³	1	2	yes	—
38	#14519846	C₂³	1	4	yes	—
39	#14519845	—	2	2	no	—
40	#14519840	—	2	8	no	—
41	#14519831	D₄	1	2	yes	—
42	C₈×#6923	C₂³	1	2	yes	—
43	C₂×Q₈×D₆₁	—	2	2	no	—
44	C₂×Dic₂₄₄	C₂×C₄	1	2	yes	—
45	C₂×#11758202	—	2	2	no	—
46	Q₁₆×D₆₁	—	2	2	no	—
47	Q₁₆×D₆₁	—	2	2	no	—
48	Q₁₆×D₆₁	C₂×C₄	1	4	yes	—
49	Q₁₆×D₆₁	C₂³	1	4	yes	—
50	C₂×C₈×D₆₁	C₂³	1	1	yes	characteristic
51	Dic₄₈₈	C₂×C₄	1	2	yes	—
52	Dic₄₈₈	C₂×C₄	1	2	yes	—
53	#14519723	C₂×C₄	1	4	yes	—
54	#14519723	C₂×C₄	1	4	yes	—
55	C₂×#11758185	C₂×C₄	1	1	yes	characteristic
56	C₁₆×D₆₁	C₂×C₄	1	1	yes	characteristic
57	C₁₆×D₆₁	C₂×C₄	1	1	yes	characteristic
58	C₁₆×D₆₁	C₂³	1	1	yes	characteristic
59	C₁₆×D₆₁	C₂³	1	1	yes	characteristic
60	C₂×Q₁₆×C₆₁	C₂×C₄	1	2	yes	—
61	Q₃₂×C₆₁	C₂×C₄	1	2	yes	—
62	Q₃₂×C₆₁	C₂×C₄	1	2	yes	—
63	C₂×C₁₆×C₆₁	C₂×C₄	1	1	yes	characteristic
64	C₂×Q₈×C₆₁	—	2	2	no	—
65	Q₈×D₆₁	—	2	4	no	—
66	Q₈×D₆₁	—	2	4	no	—
67	Q₈×D₆₁	—	4	2	no	—
68	Q₈×D₆₁	—	4	2	no	—
69	C₂×C₄×D₆₁	—	4	2	no	—
70	C₂×C₄×D₆₁	C₂×D₄	1	1	yes	characteristic
71	C₂×Dic₁₂₂	—	2	2	no	—
72	C₂×Dic₁₂₂	—	4	2	no	—
73	C₂²⋊#6923	—	4	2	no	—
74	C₂²⋊#6923	D₈	1	2	yes	—
75	#11758217	—	2	1	no	—
76	C₄⋊#6923	—	2	4	no	—
77	C₄⋊#6923	—	4	8	no	—
78	C₄⋊#6923	D₈	1	2	yes	—
79	C₄×#6923	—	4	8	no	—
80	C₄×#6923	—	4	2	no	—
81	C₄×#6923	C₂×D₄	1	2	yes	—
82	#11758213	—	2	1	no	—
83	D₆₁⋊C₈	C₂×D₄	1	2	yes	—
84	Q₁₆×C₆₁	—	2	2	no	—
85	Q₁₆×C₆₁	C₂²×C₄	1	4	yes	—
86	C₂×C₈×C₆₁	C₂²×C₄	1	1	yes	characteristic
87	#11758202	—	2	4	no	—
88	#11758202	—	2	4	no	—
89	C₂×#92418	#41	1	1	yes	characteristic
90	Dic₂₄₄	—	2	2	no	—
91	Dic₂₄₄	C₂²×C₄	1	4	yes	—
92	C₈×D₆₁	C₂²×C₄	1	1	yes	characteristic
93	C₈×D₆₁	C₂×D₄	1	1	yes	characteristic
94	C₈×D₆₁	#41	1	1	yes	characteristic
95	C₈×D₆₁	C₂⁴	1	1	yes	characteristic
96	#11758187	—	2	2	no	—
97	C₁₆×C₆₁	C₂²×C₄	1	1	yes	characteristic
98	C₁₆×C₆₁	C₂²×C₄	1	1	yes	characteristic
99	#11758185	C₂²×C₄	1	1	yes	characteristic
100	#11758185	C₂²×C₄	1	1	yes	characteristic
101	C₂²×D₆₁	C₂×D₈	1	1	yes	characteristic
102	C₂×#6923	—	4	4	no	—
103	C₂×#6923	C₂×D₈	1	4	yes	—
104	Q₈×C₆₁	—	2	4	no	—
105	Q₈×C₆₁	—	4	2	no	—
106	C₂×C₄×C₆₁	—	4	2	no	—
107	C₂×C₄×C₆₁	C₄×D₄	1	1	yes	characteristic
108	C₂×Dic₆₁	—	4	2	no	—
109	C₂×Dic₆₁	#135	1	1	yes	characteristic
110	C₄×D₆₁	—	4	4	no	—
111	C₄×D₆₁	—	4	4	no	—
112	C₄×D₆₁	C₄×D₄	1	1	yes	characteristic
113	C₄×D₆₁	C₂×D₈	1	1	yes	characteristic
114	C₄×D₆₁	#135	1	1	yes	characteristic
115	C₄×D₆₁	C₂²×D₄	1	1	yes	characteristic
116	Dic₁₂₂	—	2	4	no	—
117	Dic₁₂₂	—	4	2	no	—
118	Dic₁₂₂	—	4	4	no	—
119	Dic₁₂₂	—	4	4	no	—
120	#92420	—	2	2	no	—
121	C₈×C₆₁	C₄×D₄	1	1	yes	characteristic
122	C₈×C₆₁	C₂³×C₄	1	1	yes	characteristic
123	#92418	C₄×D₄	1	1	yes	characteristic
124	#92418	C₂×#41	1	1	yes	characteristic
125	#33458	—	61	1	no	maximal, Sylow
126	C₂²×C₆₁	C₄×D₈	1	1	yes	characteristic
127	C₂×D₆₁	—	2	2	no	—
128	C₂×D₆₁	C₄×D₈	1	1	yes	characteristic
129	C₂×D₆₁	C₂²×D₈	1	1	yes	characteristic
130	#6923	—	2	4	no	—
131	#6923	—	8	4	no	—
132	C₄×C₆₁	—	4	4	no	—
133	C₄×C₆₁	C₄×D₈	1	1	yes	characteristic
134	C₄×C₆₁	C₂×C₄×D₄	1	1	yes	characteristic
135	Dic₆₁	—	4	4	no	—
136	Dic₆₁	C₄×D₈	1	1	yes	characteristic
137	Dic₆₁	C₂×#135	1	1	yes	characteristic
138	C₂²×Q₃₂	—	61	1	no	—
139	#2950	—	61	2	no	—
140	#2176	—	61	4	no	—
141	C₄×Q₃₂	—	61	4	no	—
142	#2158	—	61	1	no	—
143	#2141	—	61	2	no	—
144	#2108	—	61	1	no	—
145	C₂×C₆₁	—	2	1	no	—
146	C₂×C₆₁	C₂×C₄×D₈	1	1	yes	characteristic
147	D₆₁	—	2	1	no	—
148	D₆₁	#3417	1	2	yes	—
149	C₂²×Q₁₆	—	61	2	no	—
150	#519	—	122	2	no	—
151	C₂×Q₃₂	—	61	2	no	—
152	C₂×Q₃₂	—	61	2	no	—
153	C₂×Q₃₂	—	61	2	no	—
154	C₂×Q₃₂	—	61	2	no	—
155	C₂×Q₃₂	—	61	2	no	—
156	C₂×Q₃₂	—	61	1	no	—
157	C₂×Q₃₂	#6923	1	1	yes	characteristic, complemented
158	C₂×M₃₂	—	61	1	no	—
159	C₂²×C₁₆	—	61	1	no	—
160	#441	—	122	4	no	—
161	C₄×Q₁₆	—	61	8	no	—
162	#427	—	61	2	no	—
163	#419	—	122	2	no	—
164	#405	—	61	1	no	—
165	#367	—	61	2	no	—
166	#366	—	61	2	no	—
167	#358	—	61	8	no	—
168	#346	—	61	2	no	—
169	C₄×C₁₆	—	61	2	no	—
170	C₆₁	#33458	1	1	yes	characteristic, complemented, Sylow
171	C₂²×Q₈	—	122	2	no	—
172	C₂×Q₁₆	—	61	4	no	—
173	C₂×Q₁₆	—	61	4	no	—
174	C₂×Q₁₆	—	61	2	no	—
175	C₂×Q₁₆	—	122	2	no	—
176	C₂×Q₁₆	—	122	2	no	—
177	C₂×Q₁₆	—	122	2	no	—
178	C₂×Q₁₆	C₂×#6923	1	2	yes	—
179	C₂×M₁₆	—	61	1	no	—
180	C₂²×C₈	—	61	1	no	—
181	C₄×Q₈	—	122	8	no	—
182	C₄×Q₈	—	244	4	no	—
183	#117	—	61	1	no	—
184	#117	—	122	2	no	—
185	#117	—	244	4	no	—
186	Q₃₂	—	61	4	no	—
187	Q₃₂	—	61	4	no	—
188	Q₃₂	—	61	2	no	—
189	Q₃₂	—	61	2	no	—
190	Q₃₂	C₂×#6923	1	2	yes	—
191	Q₃₂	C₂×#6923	1	2	yes	—
192	M₃₂	—	122	1	no	—
193	M₃₂	—	122	1	no	—
194	C₂×C₁₆	—	61	1	no	—
195	C₂×C₁₆	—	61	1	no	—
196	C₂×C₁₆	—	61	1	no	—
197	C₂×C₁₆	—	61	1	no	—
198	C₂×C₁₆	—	61	1	no	—
199	C₂×C₁₆	—	61	2	no	—
200	C₂×C₁₆	C₂×#6923	1	1	yes	characteristic
201	#107	—	61	4	no	—
202	#106	—	122	2	no	—
203	#103	—	122	8	no	—
204	#97	—	61	2	no	—
205	C₄×C₈	—	61	2	no	—
206	C₂×Q₈	—	2	2	no	—
207	C₂×Q₈	—	122	4	no	—
208	C₂×Q₈	—	122	4	no	—
209	C₂×Q₈	—	122	2	no	—
210	C₂×Q₈	—	244	2	no	—
211	C₂×Q₈	—	244	2	no	—
212	C₂×Q₈	—	244	2	no	—
213	C₂²×C₄	—	61	1	no	—
214	C₂²×C₄	—	244	2	no	—
215	Q₁₆	—	2	2	no	—
216	Q₁₆	—	61	4	no	—
217	Q₁₆	—	122	4	no	—
218	Q₁₆	—	122	4	no	—
219	Q₁₆	—	122	2	no	—
220	Q₁₆	C₂²×#6923	1	4	yes	—
221	M₁₆	—	122	1	no	—
222	M₁₆	—	122	1	no	—
223	C₂×C₈	—	61	1	no	—
224	C₂×C₈	—	61	1	no	—
225	C₂×C₈	—	61	1	no	—
226	C₂×C₈	—	61	1	no	—
227	C₂×C₈	—	61	1	no	—
228	C₂×C₈	—	61	2	no	—
229	C₂×C₈	C₂²×#6923	1	1	yes	characteristic
230	C₄⋊C₄	—	61	2	no	—
231	C₄⋊C₄	—	122	4	no	—
232	C₄⋊C₄	—	244	8	no	—
233	C₂²⋊C₄	—	61	2	no	—
234	C₂²⋊C₄	—	244	2	no	—
235	C₄²	—	61	2	no	—
236	C₄²	—	244	2	no	—
237	C₄²	—	244	8	no	—
238	C₁₆	—	61	1	no	—
239	C₁₆	—	61	1	no	—
240	C₁₆	—	122	2	no	—
241	C₁₆	C₂²×#6923	1	1	yes	characteristic
242	C₁₆	C₂²×#6923	1	1	yes	characteristic
243	C₂³	—	61	1	no	—
244	Q₈	—	2	4	no	—
245	Q₈	—	4	2	no	—
246	Q₈	—	122	4	no	—
247	Q₈	—	244	4	no	—
248	Q₈	—	244	4	no	—
249	Q₈	—	244	2	no	—
250	C₂×C₄	—	4	2	no	—
251	C₂×C₄	—	61	1	no	—
252	C₂×C₄	—	61	1	no	—
253	C₂×C₄	—	61	1	no	—
254	C₂×C₄	—	61	1	no	—
255	C₂×C₄	—	61	1	no	—
256	C₂×C₄	—	61	4	no	—
257	C₂×C₄	—	244	4	no	—
258	C₂×C₄	—	244	4	no	—
259	C₂×C₄	—	244	4	no	—
260	C₂×C₄	—	244	2	no	—
261	C₂×C₄	D₄×#6923	1	1	yes	characteristic
262	C₈	—	61	1	no	—
263	C₈	—	61	1	no	—
264	C₈	—	122	2	no	—
265	C₈	D₄×#6923	1	1	yes	characteristic
266	C₈	C₂³×#6923	1	1	yes	characteristic
267	C₂²	—	61	1	no	—
268	C₂²	—	61	1	no	—
269	C₂²	—	122	2	no	—
270	C₂²	D₈×#6923	1	1	yes	characteristic
271	C₄	—	4	4	no	—
272	C₄	—	61	1	no	—
273	C₄	—	61	1	no	—
274	C₄	—	122	4	no	—
275	C₄	—	488	4	no	—
276	C₄	—	244	4	no	—
277	C₄	D₈×#6923	1	1	yes	characteristic
278	C₄	C₂×D₄×#6923	1	1	yes	characteristic
279	C₂	—	2	1	no	—
280	C₂	—	61	2	no	—
281	C₂	—	122	1	no	—
282	C₂	C₂×D₈×#6923	1	1	yes	characteristic

	subgroup	G/H
center	C₂	C₂×D₈×#6923
commutator	C₈×C₆₁	C₂³×C₄
Frattini	C₈	C₂³×#6923
Fitting	C₂×Q₃₂×C₆₁	C₄
socle	C₂×C₆₁	C₂×C₄×D₈
prime radical	C₂²×D₆₁	C₂×D₈
cosocle	C₈×D₆₁	C₂⁴
hypercenter	C₂×Q₃₂	#6923
hypocenter	C₆₁	#33458
LCS	G ▷ C₈×C₆₁ ▷ C₄×C₆₁ ▷ C₂×C₆₁ ▷ C₆₁
UCS	C₁ ◁ C₂ ◁ C₂×C₄ ◁ C₂×C₈ ◁ C₂×Q₃₂

Complements

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

N	H ≅ G/N	type
2. #21462216	C₂	semidirect
3. Q₃₂×#6923	C₂	semidirect
21. C₂×Dic₄₈₈	C₄	semidirect
22. C₂×#14519723	C₄	semidirect
23. Q₃₂×D₆₁	C₄	semidirect
24. Q₃₂×D₆₁	C₄	semidirect
28. C₂×Q₃₂×C₆₁	C₄	semidirect
157. C₂×Q₃₂	#6923	semidirect
170. C₆₁	#33458	semidirect

Maximal subgroups

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

subgroup	index
#21462580	2
#21462216	2
Q₃₂×#6923	2
#21462170	2
#21462129	2
#21462027	2
C₂×Q₃₂×D₆₁	2
#33458	61