Subgroups — #29595142

Order 15616.

382 conjugacy classes of subgroups.

The subgroup lattice has 286 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#21458921	C₂	1	1	yes	maximal, characteristic
2	#21447448	C₂	1	1	yes	maximal, characteristic
3	C₂×#16839539	C₂	1	1	yes	maximal, characteristic, complemented
4	#21445665	C₂	1	1	yes	maximal, characteristic
5	#21443905	C₂	1	1	yes	maximal, characteristic, complemented
6	#21443902	C₂	1	1	yes	maximal, characteristic, complemented
7	C₆₁×#2110	C₂	1	1	yes	maximal, characteristic, complemented
8	C₂×#14519783	C₂²	1	1	yes	characteristic
9	#16840180	—	2	2	no	—
10	#16840178	C₄	1	4	yes	complemented
11	C₂×#14519717	C₂²	1	1	yes	characteristic
12	C₂×#14519717	C₂²	1	1	yes	characteristic
13	M₃₂×D₆₁	—	2	2	no	—
14	C₂×#14519703	C₄	1	1	yes	characteristic
15	#16839756	C₄	1	1	yes	characteristic
16	C₂×#14519687	C₂²	1	1	yes	characteristic
17	#16839547	C₄	1	4	yes	complemented
18	#16839539	—	2	2	no	—
19	#16839526	C₄	1	1	yes	characteristic
20	#16839521	C₄	1	1	yes	characteristic
21	C₂×M₃₂×C₆₁	C₂²	1	1	yes	characteristic
22	C₂×M₃₂×C₆₁	C₂²	1	1	yes	characteristic
23	C₆₁×#405	C₂²	1	1	yes	characteristic
24	C₆₁×#347	—	2	2	no	—
25	C₂×#11758222	C₂×C₄	1	1	yes	characteristic
26	#14519783	—	2	1	no	—
27	#14519783	—	2	1	no	—
28	#14519783	C₂×C₄	1	2	yes	—
29	#14519783	C₂×C₄	1	2	yes	—
30	C₂×#11758194	C₂×C₄	1	1	yes	characteristic
31	C₂×#11758189	—	2	1	no	—
32	C₂²×#92418	C₂×C₄	1	1	yes	characteristic
33	C₂×C₈×D₆₁	—	2	1	no	—
34	C₂×C₄×Dic₆₁	C₂×C₄	1	1	yes	characteristic
35	#14519717	—	2	1	no	—
36	#14519717	—	2	1	no	—
37	#14519717	—	2	1	no	—
38	#14519717	C₂×C₄	1	2	yes	—
39	#14519716	—	2	4	no	—
40	C₂×#11758185	—	2	1	no	—
41	C₂×#11758185	C₂×C₄	1	2	yes	—
42	C₁₆×D₆₁	—	2	4	no	—
43	#14519703	—	2	2	no	—
44	#14519700	—	2	1	no	—
45	#14519697	—	2	1	no	—
46	#14519687	—	2	1	no	—
47	#14519687	—	2	1	no	—
48	C₂×M₁₆×C₆₁	C₂×C₄	1	1	yes	characteristic
49	C₂²×C₈×C₆₁	C₂³	1	1	yes	characteristic
50	#117×C₆₁	C₂×C₄	1	1	yes	characteristic
51	M₃₂×C₆₁	—	2	1	no	—
52	M₃₂×C₆₁	C₂×C₄	1	2	yes	—
53	M₃₂×C₆₁	D₄	1	2	yes	—
54	M₃₂×C₆₁	D₄	1	2	yes	—
55	C₂×C₁₆×C₆₁	—	2	1	no	—
56	C₂×C₁₆×C₆₁	C₂×C₄	1	2	yes	—
57	#97×C₆₁	—	2	1	no	—
58	C₄×C₈×C₆₁	—	2	1	no	—
59	C₂²×C₄×C₆₁	C₂²×C₄	1	1	yes	characteristic
60	C₂×C₂²⋊D₆₁	—	2	1	no	—
61	C₂²×Dic₆₁	—	2	1	no	—
62	#11758222	—	2	1	no	—
63	#11758222	—	2	1	no	—
64	#11758222	C₄²	1	2	yes	—
65	#11758222	M₁₆	1	2	yes	—
66	C₂×D₂₄₄	M₁₆	1	1	yes	characteristic
67	C₂×C₄×D₆₁	—	2	1	no	—
68	C₂×Dic₁₂₂	M₁₆	1	1	yes	characteristic
69	M₁₆×C₆₁	—	2	1	no	—
70	M₁₆×C₆₁	—	2	1	no	—
71	C₂×C₈×C₆₁	—	2	1	no	—
72	C₂×C₈×C₆₁	C₂²×C₄	1	1	yes	characteristic
73	C₂×C₈×C₆₁	C₂²×C₄	1	1	yes	characteristic
74	C₂×C₈×C₆₁	C₂×D₄	1	1	yes	characteristic
75	C₂×C₈×C₆₁	C₂×D₄	1	1	yes	characteristic
76	C₂×C₈×C₆₁	#41	1	1	yes	characteristic
77	C₂×C₈×C₆₁	#41	1	1	yes	characteristic
78	C₄⋊C₄×C₆₁	—	2	1	no	—
79	C₂²⋊C₄×C₆₁	—	2	1	no	—
80	C₄²×C₆₁	—	2	1	no	—
81	C₄×Dic₆₁	—	2	1	no	—
82	C₄×Dic₆₁	—	2	1	no	—
83	#11758194	—	2	1	no	—
84	#11758194	C₄²	1	2	yes	—
85	C₂×#92418	—	2	1	no	—
86	C₂×#92418	—	2	1	no	—
87	C₂×#92418	—	2	1	no	—
88	C₂×#92418	—	2	1	no	—
89	#11758189	—	2	2	no	—
90	#11758189	—	2	2	no	—
91	C₈×D₆₁	—	2	2	no	—
92	C₈×D₆₁	—	2	2	no	—
93	C₁₆×C₆₁	—	2	2	no	—
94	C₁₆×C₆₁	C₂²⋊C₄	1	4	yes	—
95	#11758185	—	2	2	no	—
96	#11758185	—	2	2	no	—
97	C₂³×C₆₁	#131	1	1	yes	characteristic
98	C₂²×D₆₁	—	2	1	no	—
99	C₂×C₄×C₆₁	—	4	1	no	—
100	C₂×C₄×C₆₁	C₂×C₄²	1	1	yes	characteristic
101	C₂×C₄×C₆₁	C₄×D₄	1	1	yes	characteristic
102	C₂×C₄×C₆₁	C₄×D₄	1	1	yes	characteristic
103	C₂×C₄×C₆₁	C₂×M₁₆	1	1	yes	characteristic
104	C₂×C₄×C₆₁	C₂×M₁₆	1	1	yes	characteristic
105	C₂×C₄×C₆₁	#131	1	1	yes	characteristic
106	C₂²⋊D₆₁	—	2	2	no	—
107	C₂²⋊D₆₁	—	2	2	no	—
108	C₂×Dic₆₁	—	2	1	no	—
109	C₂×Dic₆₁	—	4	1	no	—
110	C₂×Dic₆₁	—	8	1	no	—
111	D₂₄₄	—	2	1	no	—
112	D₂₄₄	#97	1	2	yes	—
113	C₄×D₆₁	—	2	2	no	—
114	C₄×D₆₁	—	2	2	no	—
115	Dic₁₂₂	—	2	1	no	—
116	Dic₁₂₂	#97	1	2	yes	—
117	C₈×C₆₁	—	4	1	no	—
118	C₈×C₆₁	C₂×C₂²⋊C₄	1	1	yes	characteristic
119	C₈×C₆₁	#117	1	1	yes	characteristic
120	C₈×C₆₁	C₄×D₄	1	1	yes	characteristic
121	C₈×C₆₁	C₄×D₄	1	1	yes	characteristic
122	#92418	—	2	2	no	—
123	#92418	—	4	1	no	—
124	#12618	—	61	1	no	maximal, Sylow
125	C₂²×C₆₁	—	4	1	no	—
126	C₂²×C₆₁	#405	1	1	yes	characteristic
127	C₂²×C₆₁	#435	1	1	yes	characteristic
128	C₂²×C₆₁	#435	1	1	yes	characteristic
129	C₂×D₆₁	—	2	2	no	—
130	C₂×D₆₁	—	8	1	no	—
131	C₄×C₆₁	—	8	1	no	—
132	C₄×C₆₁	C₄×C₂²⋊C₄	1	1	yes	characteristic
133	C₄×C₆₁	C₂×#97	1	1	yes	characteristic
134	C₄×C₆₁	#435	1	1	yes	characteristic
135	C₄×C₆₁	#435	1	1	yes	characteristic
136	Dic₆₁	—	2	2	no	—
137	Dic₆₁	—	8	1	no	—
138	#3407	—	61	1	no	—
139	#2117	—	61	1	no	—
140	#2110	D₆₁	1	1	yes	characteristic, complemented
141	C₂×#347	—	61	1	no	—
142	#1765	—	61	1	no	—
143	#1378	—	61	1	no	—
144	#1377	—	61	1	no	—
145	C₂×C₆₁	—	2	1	no	—
146	C₂×C₆₁	—	2	1	no	—
147	C₂×C₆₁	—	2	1	no	—
148	C₂×C₆₁	#1752	1	1	yes	characteristic
149	D₆₁	—	4	2	no	—
150	C₂×#131	—	61	1	no	—
151	#504	—	61	4	no	—
152	#504	—	122	2	no	—
153	C₂×M₃₂	—	61	1	no	—
154	C₂×M₃₂	—	61	1	no	—
155	C₂×M₃₂	—	122	2	no	—
156	C₂×M₃₂	C₂×D₆₁	1	1	yes	characteristic
157	C₂×M₃₂	C₂×D₆₁	1	1	yes	characteristic
158	C₂×#104	—	61	1	no	—
159	#416	—	61	1	no	—
160	#405	C₂×D₆₁	1	1	yes	characteristic
161	C₂×#97	—	61	1	no	—
162	#350	—	61	4	no	—
163	#347	—	2	2	no	—
164	#347	—	122	2	no	—
165	#341	—	61	1	no	—
166	#339	—	61	1	no	—
167	C₆₁	#12618	1	1	yes	characteristic, complemented, Sylow
168	C₂×#41	—	61	1	no	—
169	#131	—	61	2	no	—
170	#131	—	61	2	no	—
171	#131	—	122	1	no	—
172	#131	—	122	1	no	—
173	C₂×M₁₆	—	61	1	no	—
174	C₂×M₁₆	—	122	1	no	—
175	C₂×M₁₆	C₄×D₆₁	1	1	yes	characteristic
176	C₂²×C₈	—	61	1	no	—
177	C₂²×C₈	—	122	1	no	—
178	C₂²×C₈	C₂²×D₆₁	1	1	yes	characteristic
179	#117	C₄×D₆₁	1	1	yes	characteristic
180	C₂×C₄²	—	61	1	no	—
181	M₃₂	—	2	1	no	—
182	M₃₂	—	61	2	no	—
183	M₃₂	—	122	1	no	—
184	M₃₂	—	122	1	no	—
185	M₃₂	—	122	1	no	—
186	M₃₂	—	122	4	no	—
187	M₃₂	C₄×D₆₁	1	2	yes	—
188	M₃₂	D₂₄₄	1	2	yes	—
189	M₃₂	C₂²⋊D₆₁	1	2	yes	—
190	C₂×C₁₆	—	2	1	no	—
191	C₂×C₁₆	—	61	2	no	—
192	C₂×C₁₆	—	122	1	no	—
193	C₂×C₁₆	—	122	4	no	—
194	C₂×C₁₆	C₄×D₆₁	1	2	yes	—
195	#104	—	122	2	no	—
196	#103	—	122	1	no	—
197	#102	—	122	1	no	—
198	#97	—	2	1	no	—
199	#97	—	122	1	no	—
200	#97	—	122	1	no	—
201	C₄×C₈	—	2	1	no	—
202	#41	—	61	2	no	—
203	#41	—	61	2	no	—
204	#41	—	122	1	no	—
205	#41	—	122	1	no	—
206	C₂×Q₈	—	61	1	no	—
207	C₂×D₄	—	61	1	no	—
208	C₂×D₄	—	122	1	no	—
209	C₂²×C₄	—	122	1	no	—
210	C₂²×C₄	—	122	1	no	—
211	C₂²×C₄	C₂×C₄×D₆₁	1	1	yes	characteristic
212	M₁₆	—	2	1	no	—
213	M₁₆	—	2	1	no	—
214	M₁₆	—	61	2	no	—
215	M₁₆	—	122	1	no	—
216	M₁₆	—	122	2	no	—
217	M₁₆	—	122	2	no	—
218	C₂×C₈	—	2	1	no	—
219	C₂×C₈	—	122	1	no	—
220	C₂×C₈	—	122	2	no	—
221	C₂×C₈	—	122	2	no	—
222	C₂×C₈	—	122	1	no	—
223	C₂×C₈	—	122	1	no	—
224	C₂×C₈	—	122	1	no	—
225	C₂×C₈	C₂×C₄×D₆₁	1	1	yes	characteristic
226	C₂×C₈	C₂×C₄×D₆₁	1	1	yes	characteristic
227	C₂×C₈	C₂×D₂₄₄	1	1	yes	characteristic
228	C₂×C₈	#11758222	1	1	yes	characteristic
229	C₂×C₈	#11758222	1	1	yes	characteristic
230	C₂×C₈	C₂×C₂²⋊D₆₁	1	1	yes	characteristic
231	C₄⋊C₄	—	2	1	no	—
232	C₂²⋊C₄	—	2	1	no	—
233	C₄²	—	2	1	no	—
234	C₄²	—	122	1	no	—
235	C₄²	—	122	1	no	—
236	C₁₆	—	2	2	no	—
237	C₁₆	—	122	2	no	—
238	C₁₆	—	122	2	no	—
239	C₁₆	#11758198	1	4	yes	—
240	C₂³	—	122	1	no	—
241	C₂³	#14519785	1	1	yes	characteristic
242	Q₈	—	61	2	no	—
243	Q₈	—	122	1	no	—
244	D₄	—	61	2	no	—
245	D₄	—	122	2	no	—
246	D₄	—	122	2	no	—
247	D₄	—	122	1	no	—
248	C₂×C₄	—	4	1	no	—
249	C₂×C₄	—	122	1	no	—
250	C₂×C₄	—	122	2	no	—
251	C₂×C₄	—	122	2	no	—
252	C₂×C₄	—	488	1	no	—
253	C₂×C₄	—	244	1	no	—
254	C₂×C₄	C₄²×D₆₁	1	1	yes	characteristic
255	C₂×C₄	C₄×C₂²⋊D₆₁	1	1	yes	characteristic
256	C₂×C₄	C₄×D₂₄₄	1	1	yes	characteristic
257	C₂×C₄	M₁₆×D₆₁	1	1	yes	characteristic
258	C₂×C₄	M₁₆×D₆₁	1	1	yes	characteristic
259	C₂×C₄	#14519785	1	1	yes	characteristic
260	C₈	—	4	1	no	—
261	C₈	—	122	2	no	—
262	C₈	—	244	1	no	—
263	C₈	C₂×#11758198	1	1	yes	characteristic
264	C₈	#14519734	1	1	yes	characteristic
265	C₈	C₄×C₂²⋊D₆₁	1	1	yes	characteristic
266	C₈	C₄×D₂₄₄	1	1	yes	characteristic
267	C₂²	—	4	1	no	—
268	C₂²	—	122	2	no	—
269	C₂²	—	488	1	no	—
270	C₂²	#16839719	1	1	yes	characteristic
271	C₂²	#16839831	1	1	yes	characteristic
272	C₂²	#16839835	1	1	yes	characteristic
273	C₄	—	8	1	no	—
274	C₄	—	122	2	no	—
275	C₄	—	488	1	no	—
276	C₄	C₄×#11758198	1	1	yes	characteristic
277	C₄	#97×D₆₁	1	1	yes	characteristic
278	C₄	#16839831	1	1	yes	characteristic
279	C₄	#16839835	1	1	yes	characteristic
280	C₂	—	2	1	no	—
281	C₂	—	2	1	no	—
282	C₂	—	2	1	no	—
283	C₂	—	244	2	no	—
284	C₂	#21445596	1	1	yes	characteristic

	subgroup	G/H
center	C₄	C₄×#11758198
commutator	C₂×C₄×C₆₁	C₂×C₄²
Frattini	C₂²×C₈	C₂²×D₆₁
Fitting	C₆₁×#2110	C₂
socle	C₂×C₆₁	#1752
prime radical	C₂×#11758222	C₂×C₄
cosocle	C₂²×C₈×C₆₁	C₂³
hypercenter	#2110	D₆₁
hypocenter	C₆₁	#12618
LCS	G ▷ C₂×C₄×C₆₁ ▷ C₂×C₆₁ ▷ C₆₁
UCS	C₁ ◁ C₄ ◁ C₂×M₃₂ ◁ #2110

Complements

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

N	H ≅ G/N	type
3. C₂×#16839539	C₂	semidirect
5. #21443905	C₂	semidirect
6. #21443902	C₂	semidirect
7. C₆₁×#2110	C₂	semidirect
10. #16840178	C₄	semidirect
17. #16839547	C₄	semidirect
140. #2110	D₆₁	semidirect
167. C₆₁	#12618	semidirect

Maximal subgroups

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

subgroup	index
#21458921	2
#21447448	2
C₂×#16839539	2
#21445665	2
#21443905	2
#21443902	2
C₆₁×#2110	2
#12618	61