Subgroups — #29596494

Order 15616.

370 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	#21458921	C₂	1	1	yes	maximal, characteristic
2	#21447585	C₂	1	1	yes	maximal, characteristic, complemented
3	#21447584	C₂	1	1	yes	maximal, characteristic, complemented
4	#21447432	C₂	1	1	yes	maximal, characteristic
5	#21445620	C₂	1	1	yes	maximal, characteristic
6	#21443907	C₂	1	1	yes	maximal, characteristic, complemented
7	C₆₁×#1379	C₂	1	1	yes	maximal, characteristic, complemented
8	C₂×#14519783	C₂²	1	1	yes	characteristic
9	#16840180	—	2	2	no	—
10	#16840178	—	2	2	no	—
11	C₂×#14519717	C₂²	1	1	yes	characteristic
12	C₂×#14519717	C₂²	1	1	yes	characteristic
13	M₃₂×D₆₁	—	2	2	no	—
14	#16839744	C₄	1	1	yes	characteristic
15	#16839735	C₄	1	1	yes	characteristic
16	#16839717	C₂²	1	1	yes	characteristic
17	#16839583	—	2	2	no	—
18	#16839582	—	2	2	no	—
19	#16839541	—	2	2	no	—
20	#16839534	C₄	1	1	yes	characteristic
21	#16839532	C₄	1	1	yes	characteristic
22	C₂×M₃₂×C₆₁	C₂²	1	1	yes	characteristic
23	C₂²×C₁₆×C₆₁	C₂²	1	1	yes	characteristic
24	C₆₁×#405	C₂²	1	1	yes	characteristic
25	C₂×#11758222	C₂×C₄	1	1	yes	characteristic
26	#14519783	—	2	1	no	—
27	#14519783	—	2	1	no	—
28	#14519783	—	2	1	no	—
29	#14519783	—	2	1	no	—
30	C₂×#11758194	C₂×C₄	1	1	yes	characteristic
31	C₂×#11758194	C₂×C₄	1	1	yes	characteristic
32	C₂×#11758189	Q₈	1	2	yes	—
33	C₂×C₈×D₆₁	D₄	1	2	yes	—
34	#14519736	C₂×C₄	1	1	yes	characteristic
35	#14519717	—	2	1	no	—
36	#14519717	—	2	1	no	—
37	#14519717	—	2	1	no	—
38	#14519717	D₄	1	2	yes	—
39	#14519716	—	2	4	no	—
40	C₂×#11758185	—	2	1	no	—
41	C₂×#11758185	—	2	1	no	—
42	C₁₆×D₆₁	—	2	4	no	—
43	#14519695	C₈	1	2	yes	—
44	#14519693	C₈	1	2	yes	—
45	#14519691	—	2	2	no	—
46	#14519687	—	2	1	no	—
47	C₈×Dic₆₁	—	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	D₄	1	2	yes	—
53	C₂×C₁₆×C₆₁	—	2	1	no	—
54	C₂×C₁₆×C₆₁	—	2	1	no	—
55	C₂×C₁₆×C₆₁	—	2	1	no	—
56	C₂×C₁₆×C₆₁	—	2	1	no	—
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	M₁₆	1	2	yes	—
61	#11758222	—	2	1	no	—
62	#11758222	—	2	1	no	—
63	#11758222	—	2	1	no	—
64	#11758222	—	2	1	no	—
65	C₂×D₂₄₄	C₂×C₈	1	1	yes	characteristic
66	C₂×C₄×D₆₁	C₄⋊C₄	1	2	yes	—
67	C₂×Dic₁₂₂	C₂×C₈	1	1	yes	characteristic
68	M₁₆×C₆₁	—	2	1	no	—
69	M₁₆×C₆₁	C₂×C₈	1	2	yes	—
70	C₂×C₈×C₆₁	—	2	1	no	—
71	C₂×C₈×C₆₁	C₂×D₄	1	1	yes	characteristic
72	C₂×C₈×C₆₁	C₂×D₄	1	1	yes	characteristic
73	C₂×C₈×C₆₁	C₂×D₄	1	1	yes	characteristic
74	C₂×C₈×C₆₁	C₂×Q₈	1	1	yes	characteristic
75	C₂×C₈×C₆₁	#41	1	1	yes	characteristic
76	C₂×C₈×C₆₁	#41	1	1	yes	characteristic
77	C₄⋊C₄×C₆₁	—	2	1	no	—
78	C₂²⋊C₄×C₆₁	—	2	1	no	—
79	C₄²×C₆₁	—	2	1	no	—
80	C₂²⋊Dic₆₁	—	2	1	no	—
81	C₄⋊Dic₆₁	—	2	1	no	—
82	C₄×Dic₆₁	—	2	1	no	—
83	#11758194	—	2	1	no	—
84	#11758194	—	2	1	no	—
85	#11758194	—	2	1	no	—
86	#11758194	C₂×C₈	1	2	yes	—
87	C₂×#92418	—	2	1	no	—
88	C₂×#92418	C₄⋊C₄	1	2	yes	—
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	—	4	1	no	—
95	#11758185	—	2	2	no	—
96	#11758185	—	4	1	no	—
97	C₂³×C₆₁	C₂×M₁₆	1	1	yes	characteristic
98	C₂²×D₆₁	C₄⋊C₈	1	2	yes	—
99	C₂×C₄×C₆₁	—	4	1	no	—
100	C₂×C₄×C₆₁	C₂×C₄⋊C₄	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	C₂²×C₈	1	1	yes	characteristic
104	C₂×C₄×C₆₁	#131	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	—	4	1	no	—
109	C₂×Dic₆₁	C₄⋊C₈	1	2	yes	—
110	D₂₄₄	—	2	1	no	—
111	D₂₄₄	—	2	1	no	—
112	C₄×D₆₁	—	2	2	no	—
113	C₄×D₆₁	—	2	2	no	—
114	Dic₁₂₂	—	2	1	no	—
115	Dic₁₂₂	—	2	1	no	—
116	C₈×C₆₁	—	2	2	no	—
117	C₈×C₆₁	#120	1	1	yes	characteristic
118	C₈×C₆₁	C₂²⋊Q₈	1	1	yes	characteristic
119	C₈×C₆₁	C₂²⋊Q₈	1	1	yes	characteristic
120	C₈×C₆₁	#123	1	1	yes	characteristic
121	#92418	—	2	2	no	—
122	#92418	—	2	2	no	—
123	#12916	—	61	1	no	maximal, Sylow
124	C₂²×C₆₁	—	4	1	no	—
125	C₂²×C₆₁	C₂×C₄⋊C₈	1	1	yes	characteristic
126	C₂²×C₆₁	#435	1	1	yes	characteristic
127	C₂²×C₆₁	#435	1	1	yes	characteristic
128	C₂×D₆₁	—	2	2	no	—
129	C₂×D₆₁	—	4	2	no	—
130	C₄×C₆₁	—	8	1	no	—
131	C₄×C₆₁	#385	1	1	yes	characteristic
132	C₄×C₆₁	#424	1	1	yes	characteristic
133	C₄×C₆₁	C₈×D₄	1	1	yes	characteristic
134	C₄×C₆₁	C₈×D₄	1	1	yes	characteristic
135	Dic₆₁	—	2	2	no	—
136	Dic₆₁	—	8	1	no	—
137	#3407	—	61	1	no	—
138	#2153	—	61	1	no	—
139	#2153	—	61	1	no	—
140	#2113	—	61	1	no	—
141	#1757	—	61	1	no	—
142	#1379	—	61	1	no	—
143	#1379	D₆₁	1	1	yes	characteristic, complemented
144	C₂×C₆₁	—	2	1	no	—
145	C₂×C₆₁	—	2	1	no	—
146	C₂×C₆₁	—	2	1	no	—
147	C₂×C₆₁	#1850	1	1	yes	characteristic
148	D₆₁	—	4	2	no	—
149	C₂×#131	—	61	1	no	—
150	#504	—	122	2	no	—
151	#504	—	122	2	no	—
152	C₂×M₃₂	—	61	1	no	—
153	C₂×M₃₂	—	61	1	no	—
154	C₂×M₃₂	—	122	2	no	—
155	C₂×M₃₂	C₂×D₆₁	1	1	yes	characteristic
156	C₂²×C₁₆	C₂×D₆₁	1	1	yes	characteristic
157	#413	—	61	1	no	—
158	#410	—	61	1	no	—
159	#405	—	61	1	no	—
160	#405	C₂×D₆₁	1	1	yes	characteristic
161	#364	—	122	2	no	—
162	#364	—	122	2	no	—
163	C₂²⋊C₁₆	—	122	2	no	—
164	#344	—	61	1	no	—
165	#344	—	61	1	no	—
166	C₆₁	#12916	1	1	yes	characteristic, complemented, Sylow
167	C₂×#41	—	61	1	no	—
168	#131	—	122	1	no	—
169	#131	—	122	1	no	—
170	#131	—	122	1	no	—
171	#131	—	122	1	no	—
172	C₂×M₁₆	—	61	1	no	—
173	C₂×M₁₆	—	61	2	no	—
174	C₂×M₁₆	—	61	1	no	—
175	C₂×M₁₆	C₄×D₆₁	1	1	yes	characteristic
176	C₂²×C₈	—	61	2	no	—
177	C₂²×C₈	C₂²×D₆₁	1	1	yes	characteristic
178	#117	—	61	1	no	—
179	#117	C₄×D₆₁	1	1	yes	characteristic
180	M₃₂	—	2	1	no	—
181	M₃₂	—	61	2	no	—
182	M₃₂	—	122	4	no	—
183	M₃₂	—	122	1	no	—
184	M₃₂	—	122	1	no	—
185	M₃₂	—	122	1	no	—
186	M₃₂	D₂₄₄	1	2	yes	—
187	C₂×C₁₆	—	2	1	no	—
188	C₂×C₁₆	—	2	1	no	—
189	C₂×C₁₆	—	2	1	no	—
190	C₂×C₁₆	—	2	1	no	—
191	C₂×C₁₆	—	122	4	no	—
192	C₂×C₁₆	—	122	1	no	—
193	C₂×C₁₆	—	122	1	no	—
194	#101	—	61	2	no	—
195	#100	—	61	2	no	—
196	#99	—	122	2	no	—
197	#97	—	2	1	no	—
198	#97	—	122	1	no	—
199	C₄×C₈	—	2	1	no	—
200	C₄×C₈	—	122	1	no	—
201	#41	—	122	1	no	—
202	#41	—	122	1	no	—
203	#41	—	122	1	no	—
204	#41	—	122	1	no	—
205	C₂×Q₈	—	61	1	no	—
206	C₂×D₄	—	61	2	no	—
207	C₂×D₄	—	61	1	no	—
208	C₂²×C₄	—	61	2	no	—
209	C₂²×C₄	C₂×C₄×D₆₁	1	1	yes	characteristic
210	M₁₆	—	2	1	no	—
211	M₁₆	—	61	2	no	—
212	M₁₆	—	122	1	no	—
213	M₁₆	—	122	1	no	—
214	M₁₆	—	122	1	no	—
215	M₁₆	—	122	2	no	—
216	M₁₆	—	122	2	no	—
217	M₁₆	C₈×D₆₁	1	2	yes	—
218	C₂×C₈	—	2	1	no	—
219	C₂×C₈	—	61	2	no	—
220	C₂×C₈	—	122	1	no	—
221	C₂×C₈	—	122	2	no	—
222	C₂×C₈	—	122	2	no	—
223	C₂×C₈	C₂×D₂₄₄	1	1	yes	characteristic
224	C₂×C₈	#11758222	1	1	yes	characteristic
225	C₂×C₈	D₄×D₆₁	1	1	yes	characteristic
226	C₂×C₈	D₄×D₆₁	1	1	yes	characteristic
227	C₂×C₈	#11758224	1	1	yes	characteristic
228	C₂×C₈	Q₈×D₆₁	1	1	yes	characteristic
229	C₄⋊C₄	—	2	1	no	—
230	C₄⋊C₄	—	122	1	no	—
231	C₂²⋊C₄	—	2	1	no	—
232	C₂²⋊C₄	—	122	1	no	—
233	C₄²	—	2	1	no	—
234	C₄²	—	122	1	no	—
235	C₁₆	—	2	2	no	—
236	C₁₆	—	4	1	no	—
237	C₁₆	—	122	2	no	—
238	C₁₆	—	244	1	no	—
239	C₂³	—	61	2	no	—
240	C₂³	M₁₆×D₆₁	1	1	yes	characteristic
241	Q₈	—	122	1	no	—
242	Q₈	—	122	1	no	—
243	D₄	—	122	2	no	—
244	D₄	—	122	2	no	—
245	D₄	—	122	1	no	—
246	D₄	—	122	1	no	—
247	C₂×C₄	—	4	1	no	—
248	C₂×C₄	—	61	2	no	—
249	C₂×C₄	—	122	2	no	—
250	C₂×C₄	—	122	2	no	—
251	C₂×C₄	—	244	1	no	—
252	C₂×C₄	C₄⋊C₄×D₆₁	1	1	yes	characteristic
253	C₂×C₄	#14519737	1	1	yes	characteristic
254	C₂×C₄	C₄×D₂₄₄	1	1	yes	characteristic
255	C₂×C₄	C₂×C₈×D₆₁	1	1	yes	characteristic
256	C₂×C₄	#14519783	1	1	yes	characteristic
257	C₂×C₄	#14519785	1	1	yes	characteristic
258	C₈	—	2	2	no	—
259	C₈	—	122	2	no	—
260	C₈	—	122	2	no	—
261	C₈	#14519747	1	1	yes	characteristic
262	C₈	#14519755	1	1	yes	characteristic
263	C₈	#14519757	1	1	yes	characteristic
264	C₈	#14519759	1	1	yes	characteristic
265	C₂²	—	4	1	no	—
266	C₂²	—	122	2	no	—
267	C₂²	—	244	2	no	—
268	C₂²	C₄⋊C₈×D₆₁	1	1	yes	characteristic
269	C₂²	#16839835	1	1	yes	characteristic
270	C₂²	#16839836	1	1	yes	characteristic
271	C₄	—	8	1	no	—
272	C₄	—	122	2	no	—
273	C₄	—	488	1	no	—
274	C₄	#16839636	1	1	yes	characteristic
275	C₄	#16839787	1	1	yes	characteristic
276	C₄	#16839825	1	1	yes	characteristic
277	C₄	C₈×D₂₄₄	1	1	yes	characteristic
278	C₂	—	2	1	no	—
279	C₂	—	2	1	no	—
280	C₂	—	2	1	no	—
281	C₂	—	244	2	no	—
282	C₂	#21445998	1	1	yes	characteristic

	subgroup	G/H
center	C₈	#14519747
commutator	C₂×C₄×C₆₁	C₂²×C₈
Frattini	C₂²×C₈	C₂²×D₆₁
Fitting	C₆₁×#1379	C₂
socle	C₂×C₆₁	#1850
prime radical	C₂×#11758222	C₂×C₄
cosocle	C₂²×C₈×C₆₁	C₂³
hypercenter	#1379	D₆₁
hypocenter	C₆₁	#12916
LCS	G ▷ C₂×C₄×C₆₁ ▷ C₂×C₆₁ ▷ C₆₁
UCS	C₁ ◁ C₈ ◁ C₂²×C₈ ◁ #1379

Complements

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

N	H ≅ G/N	type
2. #21447585	C₂	semidirect
3. #21447584	C₂	semidirect
6. #21443907	C₂	semidirect
7. C₆₁×#1379	C₂	semidirect
143. #1379	D₆₁	semidirect
166. C₆₁	#12916	semidirect

Maximal subgroups

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

subgroup	index
#21458921	2
#21447585	2
#21447584	2
#21447432	2
#21445620	2
#21443907	2
C₆₁×#1379	2
#12916	61