Subgroups — C₆₁×#33947 #29537380

Order 15616.

476 conjugacy classes of subgroups.

The subgroup lattice has 220 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₆₁×#3380	C₂	1	2	yes	maximal, complemented
2	C₆₁×#2253	C₂	1	2	yes	maximal, complemented
3	C₆₁×#2252	C₂	1	1	yes	maximal, characteristic, complemented
4	C₆₁×#2228	C₂	1	4	yes	maximal, complemented
5	C₆₁×#2226	C₂	1	2	yes	maximal, complemented
6	Q₈×C₁₆×C₆₁	C₂	1	1	yes	maximal, characteristic, complemented
7	C₄×D₁₆×C₆₁	C₂	1	1	yes	maximal, characteristic
8	C₄×D₁₆×C₆₁	C₂	1	2	yes	maximal
9	C₆₁×#554	—	2	2	no	—
10	C₂×D₁₆×C₆₁	C₂²	1	1	yes	characteristic
11	C₆₁×#500	C₂²	1	2	yes	—
12	C₆₁×#499	C₂²	1	4	yes	—
13	C₆₁×#478	—	2	4	no	—
14	C₆₁×#476	—	2	2	no	—
15	C₈×Q₈×C₆₁	C₂²	1	1	yes	characteristic
16	C₄×D₈×C₆₁	C₂²	1	2	yes	—
17	C₄×D₈×C₆₁	C₂²	1	4	yes	—
18	C₆₁×#367	C₂²	1	4	yes	—
19	C₆₁×#366	C₂²	1	2	yes	—
20	C₆₁×#366	C₂²	1	1	yes	characteristic
21	C₆₁×#366	C₂²	1	2	yes	—
22	C₆₁×C₄⋊C₁₆	C₂²	1	1	yes	characteristic
23	C₆₁×C₄⋊C₁₆	C₂²	1	2	yes	—
24	C₆₁×#357	C₂²	1	4	yes	—
25	C₆₁×#357	C₂²	1	2	yes	—
26	C₄×C₁₆×C₆₁	C₂²	1	1	yes	characteristic
27	C₄×C₁₆×C₆₁	C₂²	1	2	yes	—
28	C₂×D₈×C₆₁	C₂³	1	2	yes	—
29	C₄⋊Q₈×C₆₁	—	2	2	no	—
30	#125×C₆₁	—	2	4	no	—
31	C₂²⋊Q₈×C₆₁	—	4	2	no	—
32	C₂²⋊Q₈×C₆₁	—	4	4	no	—
33	C₄×Q₈×C₆₁	D₄	1	1	yes	characteristic
34	C₄×D₄×C₆₁	—	2	2	no	—
35	C₄×D₄×C₆₁	—	2	4	no	—
36	C₂×C₄⋊C₄×C₆₁	—	4	2	no	—
37	D₁₆×C₆₁	Q₈	1	4	yes	complemented
38	C₂×C₁₆×C₆₁	C₂³	1	1	yes	characteristic
39	C₂×C₁₆×C₆₁	C₂³	1	2	yes	—
40	C₂×C₁₆×C₆₁	C₂³	1	1	yes	characteristic
41	#107×C₆₁	—	2	2	no	—
42	#107×C₆₁	C₂³	1	4	yes	—
43	#107×C₆₁	C₂³	1	2	yes	—
44	#106×C₆₁	—	2	4	no	—
45	C₄⋊C₈×C₆₁	D₄	1	1	yes	characteristic
46	C₄⋊C₈×C₆₁	D₄	1	2	yes	—
47	#102×C₆₁	—	2	4	no	—
48	#102×C₆₁	—	2	2	no	—
49	C₄×C₈×C₆₁	C₂³	1	2	yes	—
50	C₄×C₈×C₆₁	C₂³	1	1	yes	characteristic
51	C₂×Q₈×C₆₁	—	4	2	no	—
52	C₂×Q₈×C₆₁	D₈	1	1	yes	characteristic
53	C₂×D₄×C₆₁	—	2	2	no	—
54	C₂²×C₄×C₆₁	—	4	2	no	—
55	C₂²×C₄×C₆₁	—	4	4	no	—
56	D₈×C₆₁	—	2	2	no	—
57	D₈×C₆₁	C₂×Q₈	1	4	yes	—
58	C₂×C₈×C₆₁	C₂×D₄	1	2	yes	—
59	C₂×C₈×C₆₁	C₂×D₄	1	1	yes	characteristic
60	C₂×C₈×C₆₁	C₂⁴	1	1	yes	characteristic
61	C₄⋊C₄×C₆₁	—	2	2	no	—
62	C₄⋊C₄×C₆₁	—	2	4	no	—
63	C₄⋊C₄×C₆₁	—	4	4	no	—
64	C₄⋊C₄×C₆₁	—	4	2	no	—
65	C₄⋊C₄×C₆₁	—	4	4	no	—
66	C₄⋊C₄×C₆₁	D₈	1	1	yes	characteristic
67	C₄⋊C₄×C₆₁	D₈	1	2	yes	—
68	C₂²⋊C₄×C₆₁	—	4	2	no	—
69	C₂²⋊C₄×C₆₁	—	4	4	no	—
70	C₄²×C₆₁	C₂×D₄	1	2	yes	—
71	C₄²×C₆₁	C₂×D₄	1	1	yes	characteristic
72	C₁₆×C₆₁	—	2	1	no	—
73	C₁₆×C₆₁	—	2	2	no	—
74	C₁₆×C₆₁	C₂×Q₈	1	2	yes	—
75	C₂³×C₆₁	—	4	2	no	—
76	Q₈×C₆₁	—	2	2	no	—
77	Q₈×C₆₁	—	16	2	no	—
78	D₄×C₆₁	—	2	4	no	—
79	D₄×C₆₁	—	4	2	no	—
80	C₂×C₄×C₆₁	—	4	2	no	—
81	C₂×C₄×C₆₁	—	4	4	no	—
82	C₂×C₄×C₆₁	—	8	4	no	—
83	C₂×C₄×C₆₁	—	16	4	no	—
84	C₂×C₄×C₆₁	C₂×D₈	1	2	yes	—
85	C₂×C₄×C₆₁	C₂×D₈	1	1	yes	characteristic
86	C₂×C₄×C₆₁	C₂×D₈	1	2	yes	—
87	C₂×C₄×C₆₁	C₂×D₈	1	1	yes	characteristic
88	C₂×C₄×C₆₁	C₂²×D₄	1	1	yes	characteristic
89	C₈×C₆₁	—	2	2	no	—
90	C₈×C₆₁	—	2	1	no	—
91	C₈×C₆₁	C₂²×Q₈	1	1	yes	characteristic
92	C₈×C₆₁	2₋¹⁺⁴	1	1	yes	characteristic
93	#33947	C₆₁	1	1	yes	maximal, characteristic, direct product factor, Sylow
94	C₂²×C₆₁	—	4	4	no	—
95	C₂²×C₆₁	—	8	2	no	—
96	C₂²×C₆₁	—	8	2	no	—
97	C₂²×C₆₁	C₂²×D₈	1	1	yes	characteristic
98	C₄×C₆₁	—	2	2	no	—
99	C₄×C₆₁	—	2	2	no	—
100	C₄×C₆₁	—	2	1	no	—
101	C₄×C₆₁	—	8	2	no	—
102	C₄×C₆₁	—	8	4	no	—
103	C₄×C₆₁	#508	1	2	yes	—
104	C₄×C₆₁	D₄×Q₈	1	1	yes	characteristic
105	C₄×C₆₁	#578	1	1	yes	characteristic
106	#3380	C₂×C₆₁	1	2	yes	complemented
107	#2253	C₂×C₆₁	1	2	yes	complemented
108	#2252	C₂×C₆₁	1	1	yes	characteristic, complemented
109	#2228	C₂×C₆₁	1	4	yes	complemented
110	#2226	C₂×C₆₁	1	2	yes	complemented
111	Q₈×C₁₆	C₂×C₆₁	1	1	yes	characteristic, complemented
112	C₄×D₁₆	C₂×C₆₁	1	1	yes	characteristic
113	C₄×D₁₆	C₂×C₆₁	1	2	yes	—
114	C₂×C₆₁	—	8	4	no	—
115	C₂×C₆₁	Q₈×D₈	1	1	yes	characteristic
116	C₂×C₆₁	C₂×#508	1	1	yes	characteristic
117	C₂×C₆₁	#3415	1	1	yes	characteristic
118	#554	—	2	2	no	—
119	C₂×D₁₆	C₂²×C₆₁	1	1	yes	characteristic
120	#500	C₂²×C₆₁	1	2	yes	—
121	#499	C₂²×C₆₁	1	4	yes	—
122	#478	—	2	4	no	—
123	#476	—	2	2	no	—
124	C₈×Q₈	C₂²×C₆₁	1	1	yes	characteristic
125	C₄×D₈	C₂²×C₆₁	1	2	yes	—
126	C₄×D₈	C₂²×C₆₁	1	4	yes	—
127	#367	C₂²×C₆₁	1	4	yes	—
128	#366	C₂²×C₆₁	1	2	yes	—
129	#366	C₂²×C₆₁	1	1	yes	characteristic
130	#366	C₂²×C₆₁	1	2	yes	—
131	C₄⋊C₁₆	C₂²×C₆₁	1	1	yes	characteristic
132	C₄⋊C₁₆	C₂²×C₆₁	1	2	yes	—
133	#357	C₂²×C₆₁	1	4	yes	—
134	#357	C₂²×C₆₁	1	2	yes	—
135	C₄×C₁₆	C₂²×C₆₁	1	1	yes	characteristic
136	C₄×C₁₆	C₂²×C₆₁	1	2	yes	—
137	C₆₁	#33947	1	1	yes	characteristic, direct product factor, Sylow
138	C₂×D₈	C₂³×C₆₁	1	2	yes	—
139	C₄⋊Q₈	—	2	2	no	—
140	#125	—	2	4	no	—
141	C₂²⋊Q₈	—	4	2	no	—
142	C₂²⋊Q₈	—	4	4	no	—
143	C₄×Q₈	D₄×C₆₁	1	1	yes	characteristic
144	C₄×D₄	—	2	2	no	—
145	C₄×D₄	—	2	4	no	—
146	C₂×C₄⋊C₄	—	4	2	no	—
147	D₁₆	Q₈×C₆₁	1	4	yes	complemented
148	C₂×C₁₆	C₂³×C₆₁	1	1	yes	characteristic
149	C₂×C₁₆	C₂³×C₆₁	1	2	yes	—
150	C₂×C₁₆	C₂³×C₆₁	1	1	yes	characteristic
151	#107	—	2	2	no	—
152	#107	C₂³×C₆₁	1	4	yes	—
153	#107	C₂³×C₆₁	1	2	yes	—
154	#106	—	2	4	no	—
155	C₄⋊C₈	D₄×C₆₁	1	1	yes	characteristic
156	C₄⋊C₈	D₄×C₆₁	1	2	yes	—
157	#102	—	2	4	no	—
158	#102	—	2	2	no	—
159	C₄×C₈	C₂³×C₆₁	1	2	yes	—
160	C₄×C₈	C₂³×C₆₁	1	1	yes	characteristic
161	C₂×Q₈	—	4	2	no	—
162	C₂×Q₈	D₈×C₆₁	1	1	yes	characteristic
163	C₂×D₄	—	2	2	no	—
164	C₂²×C₄	—	4	2	no	—
165	C₂²×C₄	—	4	4	no	—
166	D₈	—	2	2	no	—
167	D₈	C₂×Q₈×C₆₁	1	4	yes	—
168	C₂×C₈	C₂×D₄×C₆₁	1	2	yes	—
169	C₂×C₈	C₂×D₄×C₆₁	1	1	yes	characteristic
170	C₂×C₈	C₂⁴×C₆₁	1	1	yes	characteristic
171	C₄⋊C₄	—	2	2	no	—
172	C₄⋊C₄	—	2	4	no	—
173	C₄⋊C₄	—	4	4	no	—
174	C₄⋊C₄	—	4	2	no	—
175	C₄⋊C₄	—	4	4	no	—
176	C₄⋊C₄	D₈×C₆₁	1	1	yes	characteristic
177	C₄⋊C₄	D₈×C₆₁	1	2	yes	—
178	C₂²⋊C₄	—	4	2	no	—
179	C₂²⋊C₄	—	4	4	no	—
180	C₄²	C₂×D₄×C₆₁	1	2	yes	—
181	C₄²	C₂×D₄×C₆₁	1	1	yes	characteristic
182	C₁₆	—	2	1	no	—
183	C₁₆	—	2	2	no	—
184	C₁₆	C₂×Q₈×C₆₁	1	2	yes	—
185	C₂³	—	4	2	no	—
186	Q₈	—	2	2	no	—
187	Q₈	—	16	2	no	—
188	D₄	—	2	4	no	—
189	D₄	—	4	2	no	—
190	C₂×C₄	—	4	2	no	—
191	C₂×C₄	—	4	4	no	—
192	C₂×C₄	—	8	4	no	—
193	C₂×C₄	—	16	4	no	—
194	C₂×C₄	C₂×D₈×C₆₁	1	2	yes	—
195	C₂×C₄	C₂×D₈×C₆₁	1	1	yes	characteristic
196	C₂×C₄	C₂×D₈×C₆₁	1	2	yes	—
197	C₂×C₄	C₂×D₈×C₆₁	1	1	yes	characteristic
198	C₂×C₄	C₂²×D₄×C₆₁	1	1	yes	characteristic
199	C₈	—	2	2	no	—
200	C₈	—	2	1	no	—
201	C₈	C₂²×Q₈×C₆₁	1	1	yes	characteristic
202	C₈	2₋¹⁺⁴×C₆₁	1	1	yes	characteristic
203	C₂²	—	4	4	no	—
204	C₂²	—	8	2	no	—
205	C₂²	—	8	2	no	—
206	C₂²	C₂²×D₈×C₆₁	1	1	yes	characteristic
207	C₄	—	2	2	no	—
208	C₄	—	2	2	no	—
209	C₄	—	2	1	no	—
210	C₄	—	8	2	no	—
211	C₄	—	8	4	no	—
212	C₄	C₆₁×#508	1	2	yes	—
213	C₄	D₄×Q₈×C₆₁	1	1	yes	characteristic
214	C₄	C₆₁×#578	1	1	yes	characteristic
215	C₂	—	8	4	no	—
216	C₂	Q₈×D₈×C₆₁	1	1	yes	characteristic
217	C₂	C₂×C₆₁×#508	1	1	yes	characteristic
218	C₂	C₆₁×#3415	1	1	yes	characteristic

	subgroup	G/H
center, socle	C₂²×C₆₁	C₂²×D₈
commutator, Frattini, cosocle	C₂×C₈	C₂⁴×C₆₁
prime radical	C₂×D₁₆×C₆₁	C₂²
LCS	G ▷ C₂×C₈ ▷ C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂²×C₆₁ ◁ C₄×Q₈×C₆₁ ◁ C₈×Q₈×C₆₁ ◁ G

Complements

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

N	H ≅ G/N	type
1. C₆₁×#3380	C₂	semidirect
2. C₆₁×#2253	C₂	semidirect
3. C₆₁×#2252	C₂	semidirect
4. C₆₁×#2228	C₂	semidirect
5. C₆₁×#2226	C₂	semidirect
6. Q₈×C₁₆×C₆₁	C₂	semidirect
37. D₁₆×C₆₁	Q₈	semidirect
93. #33947	C₆₁	direct (N × H)
106. #3380	C₂×C₆₁	semidirect
107. #2253	C₂×C₆₁	semidirect
108. #2252	C₂×C₆₁	semidirect
109. #2228	C₂×C₆₁	semidirect
110. #2226	C₂×C₆₁	semidirect
111. Q₈×C₁₆	C₂×C₆₁	semidirect
137. C₆₁	#33947	direct (N × H)
147. D₁₆	Q₈×C₆₁	semidirect

Maximal subgroups

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

subgroup	index
C₆₁×#3380	2
C₆₁×#2253	2
C₆₁×#2252	2
C₆₁×#2228	2
C₆₁×#2226	2
Q₈×C₁₆×C₆₁	2
C₄×D₁₆×C₆₁	2
#33947	61

Subgroups — C61×#33947 #29537380

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₆₁×#33947 #29537380