Subgroups — C₆₁×#8806 #29512239

Order 15616.

462 conjugacy classes of subgroups.

The subgroup lattice has 174 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₆₁×#1842	C₂	1	2	yes	maximal
2	C₄²×C₈×C₆₁	C₂	1	1	yes	maximal, characteristic
3	C₂×C₆₁×#335	C₂	1	4	yes	maximal
4	C₂×C₄⋊C₈×C₆₁	—	2	4	no	—
5	C₂×C₄⋊C₈×C₆₁	C₂²	1	4	yes	—
6	C₂×C₄×C₈×C₆₁	C₄	1	2	yes	—
7	C₂×C₄×C₈×C₆₁	C₄	1	1	yes	characteristic
8	C₂×C₄×C₈×C₆₁	C₄	1	1	yes	characteristic
9	C₂×C₄×C₈×C₆₁	C₂²	1	2	yes	—
10	C₄³×C₆₁	C₂²	1	1	yes	characteristic
11	C₆₁×#335	—	2	8	no	—
12	C₂²×C₈×C₆₁	—	2	4	no	—
13	C₂²×C₈×C₆₁	C₂×C₄	1	2	yes	—
14	C₂²×C₈×C₆₁	C₂×C₄	1	2	yes	—
15	C₂×C₄²×C₆₁	C₂×C₄	1	1	yes	characteristic
16	C₂×C₄²×C₆₁	C₂×C₄	1	1	yes	characteristic
17	C₂×C₄²×C₆₁	D₄	1	2	yes	—
18	C₂×C₄²×C₆₁	Q₈	1	2	yes	—
19	C₂×C₄²×C₆₁	C₂³	1	1	yes	characteristic
20	C₄⋊C₈×C₆₁	—	2	8	no	—
21	C₄⋊C₈×C₆₁	—	4	8	no	—
22	C₄×C₈×C₆₁	—	2	4	no	—
23	C₄×C₈×C₆₁	—	2	2	no	—
24	C₄×C₈×C₆₁	C₈	1	4	yes	complemented
25	C₄×C₈×C₆₁	D₄	1	4	yes	—
26	C₄×C₈×C₆₁	Q₈	1	4	yes	—
27	C₂²×C₄×C₆₁	C₄⋊C₄	1	2	yes	—
28	C₂²×C₄×C₆₁	C₄⋊C₄	1	2	yes	—
29	C₂²×C₄×C₆₁	C₂²×C₄	1	1	yes	characteristic
30	C₂²×C₄×C₆₁	C₂×D₄	1	1	yes	characteristic
31	C₂²×C₄×C₆₁	C₂×Q₈	1	1	yes	characteristic
32	C₂×C₈×C₆₁	—	2	2	no	—
33	C₂×C₈×C₆₁	—	2	2	no	—
34	C₂×C₈×C₆₁	—	4	4	no	—
35	C₂×C₈×C₆₁	—	4	8	no	—
36	C₂×C₈×C₆₁	C₄⋊C₄	1	4	yes	—
37	C₂×C₈×C₆₁	C₄⋊C₄	1	4	yes	—
38	C₂×C₈×C₆₁	C₂×C₈	1	4	yes	—
39	C₂×C₈×C₆₁	M₁₆	1	4	yes	—
40	C₄²×C₆₁	—	2	4	no	—
41	C₄²×C₆₁	—	2	2	no	—
42	C₄²×C₆₁	C₂×C₈	1	2	yes	—
43	C₄²×C₆₁	M₁₆	1	2	yes	—
44	C₄²×C₆₁	D₈	1	4	yes	—
45	C₄²×C₆₁	Q₁₆	1	4	yes	—
46	C₄²×C₆₁	C₂×D₄	1	2	yes	—
47	C₄²×C₆₁	C₂×Q₈	1	2	yes	—
48	C₂³×C₆₁	C₂×C₄⋊C₄	1	1	yes	characteristic
49	C₂×C₄×C₆₁	—	2	2	no	—
50	C₂×C₄×C₆₁	—	2	2	no	—
51	C₂×C₄×C₆₁	—	2	2	no	—
52	C₂×C₄×C₆₁	C₄⋊C₈	1	4	yes	—
53	C₂×C₄×C₆₁	#107	1	4	yes	—
54	C₂×C₄×C₆₁	#107	1	4	yes	—
55	C₂×C₄×C₆₁	#108	1	4	yes	—
56	C₂×C₄×C₆₁	C₂×C₄⋊C₄	1	2	yes	—
57	C₂×C₄×C₆₁	#127	1	1	yes	characteristic
58	C₂×C₄×C₆₁	C₄⋊Q₈	1	2	yes	—
59	C₂×C₄×C₆₁	C₄⋊Q₈	1	1	yes	characteristic
60	C₂×C₄×C₆₁	C₂²×C₈	1	1	yes	characteristic
61	C₂×C₄×C₆₁	C₂×M₁₆	1	2	yes	—
62	C₂×C₄×C₆₁	C₂×M₁₆	1	1	yes	characteristic
63	C₂×C₄×C₆₁	C₂×D₈	1	2	yes	—
64	C₂×C₄×C₆₁	C₂×Q₁₆	1	2	yes	—
65	C₈×C₆₁	—	2	4	no	—
66	C₈×C₆₁	—	8	4	no	—
67	C₈×C₆₁	C₄⋊C₈	1	8	yes	complemented
68	#8806	C₆₁	1	1	yes	maximal, characteristic, direct product factor, Sylow
69	C₂²×C₆₁	#384	1	1	yes	characteristic
70	C₂²×C₆₁	C₂×C₄⋊C₈	1	1	yes	characteristic
71	C₂²×C₆₁	#423	1	1	yes	characteristic
72	C₂²×C₆₁	C₂×#107	1	2	yes	—
73	C₂²×C₆₁	C₂×#108	1	2	yes	—
74	C₄×C₆₁	—	2	4	no	—
75	C₄×C₆₁	—	2	2	no	—
76	C₄×C₆₁	#335	1	8	yes	complemented
77	C₄×C₆₁	C₂×C₄⋊C₈	1	2	yes	—
78	C₄×C₆₁	#423	1	2	yes	—
79	C₄×C₆₁	#493	1	1	yes	characteristic
80	C₄×C₆₁	#494	1	1	yes	characteristic
81	C₄×C₆₁	#500	1	2	yes	—
82	#1842	C₂×C₆₁	1	2	yes	—
83	C₄²×C₈	C₂×C₆₁	1	1	yes	characteristic
84	C₂×#335	C₂×C₆₁	1	4	yes	—
85	C₂×C₆₁	C₂×#335	1	2	yes	—
86	C₂×C₆₁	#1567	1	2	yes	—
87	C₂×C₆₁	#1842	1	1	yes	characteristic
88	C₂×C₆₁	#1845	1	1	yes	characteristic
89	C₂×C₆₁	#1848	1	1	yes	characteristic
90	C₂×C₄⋊C₈	—	2	4	no	—
91	C₂×C₄⋊C₈	C₂²×C₆₁	1	4	yes	—
92	C₂×C₄×C₈	C₄×C₆₁	1	2	yes	—
93	C₂×C₄×C₈	C₄×C₆₁	1	1	yes	characteristic
94	C₂×C₄×C₈	C₄×C₆₁	1	1	yes	characteristic
95	C₂×C₄×C₈	C₂²×C₆₁	1	2	yes	—
96	C₄³	C₂²×C₆₁	1	1	yes	characteristic
97	#335	—	2	8	no	—
98	C₆₁	#8806	1	1	yes	characteristic, direct product factor, Sylow
99	C₂²×C₈	—	2	4	no	—
100	C₂²×C₈	C₂×C₄×C₆₁	1	2	yes	—
101	C₂²×C₈	C₂×C₄×C₆₁	1	2	yes	—
102	C₂×C₄²	C₂×C₄×C₆₁	1	1	yes	characteristic
103	C₂×C₄²	C₂×C₄×C₆₁	1	1	yes	characteristic
104	C₂×C₄²	D₄×C₆₁	1	2	yes	—
105	C₂×C₄²	Q₈×C₆₁	1	2	yes	—
106	C₂×C₄²	C₂³×C₆₁	1	1	yes	characteristic
107	C₄⋊C₈	—	2	8	no	—
108	C₄⋊C₈	—	4	8	no	—
109	C₄×C₈	—	2	4	no	—
110	C₄×C₈	—	2	2	no	—
111	C₄×C₈	C₈×C₆₁	1	4	yes	complemented
112	C₄×C₈	D₄×C₆₁	1	4	yes	—
113	C₄×C₈	Q₈×C₆₁	1	4	yes	—
114	C₂²×C₄	C₄⋊C₄×C₆₁	1	2	yes	—
115	C₂²×C₄	C₄⋊C₄×C₆₁	1	2	yes	—
116	C₂²×C₄	C₂²×C₄×C₆₁	1	1	yes	characteristic
117	C₂²×C₄	C₂×D₄×C₆₁	1	1	yes	characteristic
118	C₂²×C₄	C₂×Q₈×C₆₁	1	1	yes	characteristic
119	C₂×C₈	—	2	2	no	—
120	C₂×C₈	—	2	2	no	—
121	C₂×C₈	—	4	4	no	—
122	C₂×C₈	—	4	8	no	—
123	C₂×C₈	C₄⋊C₄×C₆₁	1	4	yes	—
124	C₂×C₈	C₄⋊C₄×C₆₁	1	4	yes	—
125	C₂×C₈	C₂×C₈×C₆₁	1	4	yes	—
126	C₂×C₈	M₁₆×C₆₁	1	4	yes	—
127	C₄²	—	2	4	no	—
128	C₄²	—	2	2	no	—
129	C₄²	C₂×C₈×C₆₁	1	2	yes	—
130	C₄²	M₁₆×C₆₁	1	2	yes	—
131	C₄²	D₈×C₆₁	1	4	yes	—
132	C₄²	Q₁₆×C₆₁	1	4	yes	—
133	C₄²	C₂×D₄×C₆₁	1	2	yes	—
134	C₄²	C₂×Q₈×C₆₁	1	2	yes	—
135	C₂³	C₂×C₄⋊C₄×C₆₁	1	1	yes	characteristic
136	C₂×C₄	—	2	2	no	—
137	C₂×C₄	—	2	2	no	—
138	C₂×C₄	—	2	2	no	—
139	C₂×C₄	C₄⋊C₈×C₆₁	1	4	yes	—
140	C₂×C₄	#107×C₆₁	1	4	yes	—
141	C₂×C₄	#107×C₆₁	1	4	yes	—
142	C₂×C₄	#108×C₆₁	1	4	yes	—
143	C₂×C₄	C₂×C₄⋊C₄×C₆₁	1	2	yes	—
144	C₂×C₄	#127×C₆₁	1	1	yes	characteristic
145	C₂×C₄	C₄⋊Q₈×C₆₁	1	2	yes	—
146	C₂×C₄	C₄⋊Q₈×C₆₁	1	1	yes	characteristic
147	C₂×C₄	C₂²×C₈×C₆₁	1	1	yes	characteristic
148	C₂×C₄	C₂×M₁₆×C₆₁	1	2	yes	—
149	C₂×C₄	C₂×M₁₆×C₆₁	1	1	yes	characteristic
150	C₂×C₄	C₂×D₈×C₆₁	1	2	yes	—
151	C₂×C₄	C₂×Q₁₆×C₆₁	1	2	yes	—
152	C₈	—	2	4	no	—
153	C₈	—	8	4	no	—
154	C₈	C₄⋊C₈×C₆₁	1	8	yes	complemented
155	C₂²	C₆₁×#384	1	1	yes	characteristic
156	C₂²	C₂×C₄⋊C₈×C₆₁	1	1	yes	characteristic
157	C₂²	C₆₁×#423	1	1	yes	characteristic
158	C₂²	C₂×#107×C₆₁	1	2	yes	—
159	C₂²	C₂×#108×C₆₁	1	2	yes	—
160	C₄	—	2	4	no	—
161	C₄	—	2	2	no	—
162	C₄	C₆₁×#335	1	8	yes	complemented
163	C₄	C₂×C₄⋊C₈×C₆₁	1	2	yes	—
164	C₄	C₆₁×#423	1	2	yes	—
165	C₄	C₆₁×#493	1	1	yes	characteristic
166	C₄	C₆₁×#494	1	1	yes	characteristic
167	C₄	C₆₁×#500	1	2	yes	—
168	C₂	C₂×C₆₁×#335	1	2	yes	—
169	C₂	C₆₁×#1567	1	2	yes	—
170	C₂	C₆₁×#1842	1	1	yes	characteristic
171	C₂	C₆₁×#1845	1	1	yes	characteristic
172	C₂	C₆₁×#1848	1	1	yes	characteristic

	subgroup	G/H
center	C₂²×C₄×C₆₁	C₂×D₄
commutator	C₂×C₄	C₂²×C₈×C₆₁
Frattini, cosocle	C₂×C₄²	C₂³×C₆₁
socle, prime radical	C₂³×C₆₁	C₂×C₄⋊C₄
LCS	G ▷ C₂×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂²×C₄×C₆₁ ◁ C₄³×C₆₁ ◁ G

Complements

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

N	H ≅ G/N	type
24. C₄×C₈×C₆₁	C₈	semidirect
67. C₈×C₆₁	C₄⋊C₈	semidirect
68. #8806	C₆₁	direct (N × H)
76. C₄×C₆₁	#335	semidirect
98. C₆₁	#8806	direct (N × H)
111. C₄×C₈	C₈×C₆₁	semidirect
154. C₈	C₄⋊C₈×C₆₁	semidirect
162. C₄	C₆₁×#335	semidirect

Maximal subgroups

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

subgroup	index
C₆₁×#1842	2
C₄²×C₈×C₆₁	2
C₂×C₆₁×#335	2
#8806	61

Subgroups — C61×#8806 #29512239

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₆₁×#8806 #29512239