Subgroups — C₆₁×#29887 #29533320

Order 15616.

620 conjugacy classes of subgroups.

The subgroup lattice has 258 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×C₆₁×#498	C₂	1	1	yes	maximal, characteristic, complemented
2	C₆₁×#3079	C₂	1	2	yes	maximal
3	C₆₁×#2922	C₂	1	1	yes	maximal, characteristic
4	C₆₁×#2090	C₂	1	4	yes	maximal, complemented
5	C₆₁×#2079	C₂	1	4	yes	maximal
6	C₆₁×#1997	C₂	1	2	yes	maximal, complemented
7	C₆₁×#1837	C₂	1	1	yes	maximal, characteristic, complemented
8	C₂×C₄⋊Q₈×C₆₁	C₂²	1	2	yes	—
9	C₄×#41×C₆₁	C₂²	1	1	yes	characteristic
10	C₆₁×#498	—	2	2	no	—
11	C₆₁×#498	—	2	2	no	—
12	C₆₁×#477	—	2	2	no	—
13	C₆₁×#476	—	2	2	no	—
14	C₆₁×#475	—	2	2	no	—
15	C₆₁×#474	—	2	2	no	—
16	C₆₁×#433	—	2	2	no	—
17	C₆₁×#431	—	2	2	no	—
18	C₆₁×#428	C₂²	1	4	yes	—
19	C₂×#106×C₆₁	C₂²	1	4	yes	—
20	C₆₁×#424	C₂²	1	2	yes	—
21	C₆₁×#423	C₂²	1	2	yes	—
22	C₆₁×#417	C₂²	1	4	yes	—
23	C₆₁×#408	C₂²	1	2	yes	—
24	C₄×M₁₆×C₆₁	C₂²	1	1	yes	characteristic
25	C₂×C₄×C₈×C₆₁	C₂²	1	1	yes	characteristic
26	C₆₁×#343	C₂²	1	8	yes	—
27	C₆₁×#341	C₂²	1	4	yes	—
28	C₂×#41×C₆₁	C₂³	1	1	yes	characteristic
29	C₂²×Q₈×C₆₁	—	2	2	no	—
30	C₂×M₁₆×C₆₁	C₂³	1	4	yes	—
31	C₂×M₁₆×C₆₁	C₂³	1	2	yes	—
32	C₂²×C₈×C₆₁	C₂³	1	2	yes	—
33	C₄⋊Q₈×C₆₁	—	2	2	no	—
34	C₄⋊Q₈×C₆₁	—	2	2	no	—
35	C₄⋊Q₈×C₆₁	—	2	2	no	—
36	C₄⋊Q₈×C₆₁	—	2	2	no	—
37	C₄×Q₈×C₆₁	D₄	1	2	yes	—
38	C₄×D₄×C₆₁	—	2	2	no	—
39	C₄×D₄×C₆₁	D₄	1	2	yes	—
40	#117×C₆₁	—	2	1	no	—
41	#117×C₆₁	C₂³	1	1	yes	characteristic
42	C₂×C₄⋊C₄×C₆₁	—	2	2	no	—
43	C₂×C₄⋊C₄×C₆₁	C₂³	1	4	yes	—
44	C₂×C₄²×C₆₁	—	2	1	no	—
45	C₂×C₄²×C₆₁	C₂³	1	1	yes	characteristic
46	#107×C₆₁	—	2	4	no	—
47	#106×C₆₁	—	2	8	no	—
48	#106×C₆₁	—	2	4	no	—
49	C₄⋊C₈×C₆₁	—	2	4	no	—
50	C₄⋊C₈×C₆₁	—	2	2	no	—
51	C₄⋊C₈×C₆₁	—	2	2	no	—
52	#103×C₆₁	—	2	4	no	—
53	#102×C₆₁	—	2	4	no	—
54	C₂²⋊C₈×C₆₁	—	2	4	no	—
55	#97×C₆₁	—	2	1	no	—
56	C₄×C₈×C₆₁	—	2	1	no	—
57	C₄×C₈×C₆₁	D₄	1	2	yes	—
58	C₄×C₈×C₆₁	D₄	1	2	yes	—
59	#41×C₆₁	—	2	4	no	—
60	C₂×Q₈×C₆₁	—	4	2	no	—
61	C₂×Q₈×C₆₁	—	4	2	no	—
62	C₂×Q₈×C₆₁	—	8	4	no	—
63	C₂×Q₈×C₆₁	C₂×D₄	1	1	yes	characteristic
64	C₂×D₄×C₆₁	—	2	1	no	—
65	C₂×D₄×C₆₁	C₂×D₄	1	1	yes	characteristic
66	C₂²×C₄×C₆₁	—	2	2	no	—
67	C₂²×C₄×C₆₁	—	2	1	no	—
68	C₂²×C₄×C₆₁	—	2	4	no	—
69	C₂²×C₄×C₆₁	#41	1	2	yes	—
70	C₂²×C₄×C₆₁	C₂⁴	1	1	yes	characteristic
71	M₁₆×C₆₁	—	2	8	no	—
72	M₁₆×C₆₁	—	2	4	no	—
73	C₂×C₈×C₆₁	—	2	4	no	—
74	C₂×C₈×C₆₁	—	2	2	no	—
75	C₂×C₈×C₆₁	—	2	4	no	—
76	C₂×C₈×C₆₁	C₂×D₄	1	4	yes	—
77	C₄⋊C₄×C₆₁	—	2	2	no	—
78	C₄⋊C₄×C₆₁	—	2	4	no	—
79	C₄⋊C₄×C₆₁	—	2	4	no	—
80	C₄⋊C₄×C₆₁	—	4	2	no	—
81	C₄⋊C₄×C₆₁	—	4	2	no	—
82	C₄⋊C₄×C₆₁	C₂×D₄	1	2	yes	—
83	C₂²⋊C₄×C₆₁	—	2	1	no	—
84	C₂²⋊C₄×C₆₁	—	2	2	no	—
85	C₄²×C₆₁	—	2	2	no	—
86	C₄²×C₆₁	—	2	1	no	—
87	C₄²×C₆₁	C₂×D₄	1	1	yes	characteristic
88	C₄²×C₆₁	C₂×D₄	1	1	yes	characteristic
89	C₄²×C₆₁	C₂×D₄	1	2	yes	—
90	C₂³×C₆₁	—	2	1	no	—
91	C₂³×C₆₁	C₂×#41	1	1	yes	characteristic
92	Q₈×C₆₁	—	2	2	no	—
93	Q₈×C₆₁	—	8	4	no	—
94	Q₈×C₆₁	—	8	4	no	—
95	D₄×C₆₁	—	2	2	no	—
96	D₄×C₆₁	—	4	2	no	—
97	C₂×C₄×C₆₁	—	2	4	no	—
98	C₂×C₄×C₆₁	—	2	2	no	—
99	C₂×C₄×C₆₁	—	2	1	no	—
100	C₂×C₄×C₆₁	—	4	2	no	—
101	C₂×C₄×C₆₁	—	4	4	no	—
102	C₂×C₄×C₆₁	—	4	4	no	—
103	C₂×C₄×C₆₁	—	8	4	no	—
104	C₂×C₄×C₆₁	#121	1	2	yes	—
105	C₂×C₄×C₆₁	#121	1	2	yes	—
106	C₂×C₄×C₆₁	C₂²×D₄	1	1	yes	characteristic
107	C₂×C₄×C₆₁	C₂²×D₄	1	1	yes	characteristic
108	C₂×C₄×C₆₁	2₋¹⁺⁴	1	4	yes	—
109	C₈×C₆₁	—	2	4	no	—
110	C₈×C₆₁	—	4	2	no	—
111	C₈×C₆₁	—	4	4	no	—
112	#29887	C₆₁	1	1	yes	maximal, characteristic, direct product factor, Sylow
113	C₂²×C₆₁	—	2	2	no	—
114	C₂²×C₆₁	—	4	1	no	—
115	C₂²×C₆₁	—	4	2	no	—
116	C₂²×C₆₁	C₂×#121	1	1	yes	characteristic
117	C₂²×C₆₁	#541	1	2	yes	—
118	C₄×C₆₁	—	2	2	no	—
119	C₄×C₆₁	—	4	1	no	—
120	C₄×C₆₁	—	4	2	no	—
121	C₄×C₆₁	—	8	4	no	—
122	C₄×C₆₁	#471	1	4	yes	—
123	C₄×C₆₁	#536	1	2	yes	—
124	C₄×C₆₁	#536	1	2	yes	—
125	C₂×#498	C₂×C₆₁	1	1	yes	characteristic, complemented
126	#3079	C₂×C₆₁	1	2	yes	—
127	#2922	C₂×C₆₁	1	1	yes	characteristic
128	#2090	C₂×C₆₁	1	4	yes	complemented
129	#2079	C₂×C₆₁	1	4	yes	—
130	#1997	C₂×C₆₁	1	2	yes	complemented
131	#1837	C₂×C₆₁	1	1	yes	characteristic, complemented
132	C₂×C₆₁	—	2	2	no	—
133	C₂×C₆₁	—	8	1	no	—
134	C₂×C₆₁	#2621	1	1	yes	characteristic
135	C₂×C₆₁	C₂×#471	1	2	yes	—
136	C₂×C₄⋊Q₈	C₂²×C₆₁	1	2	yes	—
137	C₄×#41	C₂²×C₆₁	1	1	yes	characteristic
138	#498	—	2	2	no	—
139	#498	—	2	2	no	—
140	#477	—	2	2	no	—
141	#476	—	2	2	no	—
142	#475	—	2	2	no	—
143	#474	—	2	2	no	—
144	#433	—	2	2	no	—
145	#431	—	2	2	no	—
146	#428	C₂²×C₆₁	1	4	yes	—
147	C₂×#106	C₂²×C₆₁	1	4	yes	—
148	#424	C₂²×C₆₁	1	2	yes	—
149	#423	C₂²×C₆₁	1	2	yes	—
150	#417	C₂²×C₆₁	1	4	yes	—
151	#408	C₂²×C₆₁	1	2	yes	—
152	C₄×M₁₆	C₂²×C₆₁	1	1	yes	characteristic
153	C₂×C₄×C₈	C₂²×C₆₁	1	1	yes	characteristic
154	#343	C₂²×C₆₁	1	8	yes	—
155	#341	C₂²×C₆₁	1	4	yes	—
156	C₆₁	#29887	1	1	yes	characteristic, direct product factor, Sylow
157	C₂×#41	C₂³×C₆₁	1	1	yes	characteristic
158	C₂²×Q₈	—	2	2	no	—
159	C₂×M₁₆	C₂³×C₆₁	1	4	yes	—
160	C₂×M₁₆	C₂³×C₆₁	1	2	yes	—
161	C₂²×C₈	C₂³×C₆₁	1	2	yes	—
162	C₄⋊Q₈	—	2	2	no	—
163	C₄⋊Q₈	—	2	2	no	—
164	C₄⋊Q₈	—	2	2	no	—
165	C₄⋊Q₈	—	2	2	no	—
166	C₄×Q₈	D₄×C₆₁	1	2	yes	—
167	C₄×D₄	—	2	2	no	—
168	C₄×D₄	D₄×C₆₁	1	2	yes	—
169	#117	—	2	1	no	—
170	#117	C₂³×C₆₁	1	1	yes	characteristic
171	C₂×C₄⋊C₄	—	2	2	no	—
172	C₂×C₄⋊C₄	C₂³×C₆₁	1	4	yes	—
173	C₂×C₄²	—	2	1	no	—
174	C₂×C₄²	C₂³×C₆₁	1	1	yes	characteristic
175	#107	—	2	4	no	—
176	#106	—	2	8	no	—
177	#106	—	2	4	no	—
178	C₄⋊C₈	—	2	4	no	—
179	C₄⋊C₈	—	2	2	no	—
180	C₄⋊C₈	—	2	2	no	—
181	#103	—	2	4	no	—
182	#102	—	2	4	no	—
183	C₂²⋊C₈	—	2	4	no	—
184	#97	—	2	1	no	—
185	C₄×C₈	—	2	1	no	—
186	C₄×C₈	D₄×C₆₁	1	2	yes	—
187	C₄×C₈	D₄×C₆₁	1	2	yes	—
188	#41	—	2	4	no	—
189	C₂×Q₈	—	4	2	no	—
190	C₂×Q₈	—	4	2	no	—
191	C₂×Q₈	—	8	4	no	—
192	C₂×Q₈	C₂×D₄×C₆₁	1	1	yes	characteristic
193	C₂×D₄	—	2	1	no	—
194	C₂×D₄	C₂×D₄×C₆₁	1	1	yes	characteristic
195	C₂²×C₄	—	2	2	no	—
196	C₂²×C₄	—	2	1	no	—
197	C₂²×C₄	—	2	4	no	—
198	C₂²×C₄	#41×C₆₁	1	2	yes	—
199	C₂²×C₄	C₂⁴×C₆₁	1	1	yes	characteristic
200	M₁₆	—	2	8	no	—
201	M₁₆	—	2	4	no	—
202	C₂×C₈	—	2	4	no	—
203	C₂×C₈	—	2	2	no	—
204	C₂×C₈	—	2	4	no	—
205	C₂×C₈	C₂×D₄×C₆₁	1	4	yes	—
206	C₄⋊C₄	—	2	2	no	—
207	C₄⋊C₄	—	2	4	no	—
208	C₄⋊C₄	—	2	4	no	—
209	C₄⋊C₄	—	4	2	no	—
210	C₄⋊C₄	—	4	2	no	—
211	C₄⋊C₄	C₂×D₄×C₆₁	1	2	yes	—
212	C₂²⋊C₄	—	2	1	no	—
213	C₂²⋊C₄	—	2	2	no	—
214	C₄²	—	2	2	no	—
215	C₄²	—	2	1	no	—
216	C₄²	C₂×D₄×C₆₁	1	1	yes	characteristic
217	C₄²	C₂×D₄×C₆₁	1	1	yes	characteristic
218	C₄²	C₂×D₄×C₆₁	1	2	yes	—
219	C₂³	—	2	1	no	—
220	C₂³	C₂×#41×C₆₁	1	1	yes	characteristic
221	Q₈	—	2	2	no	—
222	Q₈	—	8	4	no	—
223	Q₈	—	8	4	no	—
224	D₄	—	2	2	no	—
225	D₄	—	4	2	no	—
226	C₂×C₄	—	2	4	no	—
227	C₂×C₄	—	2	2	no	—
228	C₂×C₄	—	2	1	no	—
229	C₂×C₄	—	4	2	no	—
230	C₂×C₄	—	4	4	no	—
231	C₂×C₄	—	4	4	no	—
232	C₂×C₄	—	8	4	no	—
233	C₂×C₄	#121×C₆₁	1	2	yes	—
234	C₂×C₄	#121×C₆₁	1	2	yes	—
235	C₂×C₄	C₂²×D₄×C₆₁	1	1	yes	characteristic
236	C₂×C₄	C₂²×D₄×C₆₁	1	1	yes	characteristic
237	C₂×C₄	2₋¹⁺⁴×C₆₁	1	4	yes	—
238	C₈	—	2	4	no	—
239	C₈	—	4	2	no	—
240	C₈	—	4	4	no	—
241	C₂²	—	2	2	no	—
242	C₂²	—	4	2	no	—
243	C₂²	—	4	1	no	—
244	C₂²	C₂×#121×C₆₁	1	1	yes	characteristic
245	C₂²	C₆₁×#541	1	2	yes	—
246	C₄	—	2	2	no	—
247	C₄	—	4	2	no	—
248	C₄	—	4	1	no	—
249	C₄	—	8	4	no	—
250	C₄	C₆₁×#471	1	4	yes	—
251	C₄	C₆₁×#536	1	2	yes	—
252	C₄	C₆₁×#536	1	2	yes	—
253	C₂	—	2	2	no	—
254	C₂	—	8	1	no	—
255	C₂	C₆₁×#2621	1	1	yes	characteristic
256	C₂	C₂×C₆₁×#471	1	2	yes	—

	subgroup	G/H
center, socle	C₂²×C₆₁	C₂×#121
commutator, Frattini, cosocle	C₂²×C₄	C₂⁴×C₆₁
prime radical	C₂×#41×C₆₁	C₂³
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂²×C₆₁ ◁ C₂×C₄²×C₆₁ ◁ G

Complements

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

N	H ≅ G/N	type
1. C₂×C₆₁×#498	C₂	semidirect
4. C₆₁×#2090	C₂	semidirect
6. C₆₁×#1997	C₂	semidirect
7. C₆₁×#1837	C₂	semidirect
112. #29887	C₆₁	direct (N × H)
125. C₂×#498	C₂×C₆₁	semidirect
128. #2090	C₂×C₆₁	semidirect
130. #1997	C₂×C₆₁	semidirect
131. #1837	C₂×C₆₁	semidirect
156. C₆₁	#29887	direct (N × H)

Maximal subgroups

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

subgroup	index
C₂×C₆₁×#498	2
C₆₁×#3079	2
C₆₁×#2922	2
C₆₁×#2090	2
C₆₁×#2079	2
C₆₁×#1997	2
C₆₁×#1837	2
#29887	61

Subgroups — C61×#29887 #29533320

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₆₁×#29887 #29533320