Subgroups — #29711002

Order 15616.

604 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₆₁⋊#2104	C₂	1	7	yes	maximal
2	#21445550	C₂	1	7	yes	maximal
3	C₆₁×#2104	C₂	1	1	yes	maximal, characteristic
4	C₂×#14519733	C₂²	1	7	yes	—
5	#16839529	C₂²	1	7	yes	—
6	#16839528	C₂²	1	14	yes	—
7	C₆₁×#342	C₂²	1	7	yes	—
8	C₂³×Dic₆₁	C₂³	1	1	yes	characteristic
9	#14519733	—	2	14	no	—
10	#14519733	—	2	14	no	—
11	C₂×#11758196	—	2	7	no	—
12	C₂×C₂²⋊Dic₆₁	C₂³	1	7	yes	—
13	C₂×C₄×Dic₆₁	—	2	7	no	—
14	C₂×C₂²⋊C₄×C₆₁	C₂³	1	7	yes	—
15	C₂⁴×C₆₁	C₂⁴	1	1	yes	characteristic
16	C₂²×C₄×C₆₁	—	2	7	no	—
17	C₂²×Dic₆₁	—	2	7	no	—
18	C₂²×Dic₆₁	#41	1	7	yes	—
19	C₂²×Dic₆₁	#41	1	7	yes	—
20	C₂²⋊C₄×C₆₁	—	2	14	no	—
21	C₂²⋊Dic₆₁	—	2	14	no	—
22	#11758196	—	4	14	no	—
23	C₄×Dic₆₁	—	4	7	no	—
24	C₄×Dic₆₁	—	4	7	no	—
25	C₂³×C₆₁	—	8	1	no	—
26	C₂³×C₆₁	C₂×#41	1	7	yes	—
27	C₂×C₄×C₆₁	—	4	7	no	—
28	C₂×C₄×C₆₁	—	8	7	no	—
29	C₂×Dic₆₁	—	4	7	no	—
30	C₂×Dic₆₁	—	8	7	no	—
31	C₂×Dic₆₁	#126	1	7	yes	—
32	C₂×Dic₆₁	#126	1	7	yes	—
33	C₂×Dic₆₁	#126	1	14	yes	—
34	#23847	—	61	1	no	maximal, Sylow
35	C₂²×C₆₁	—	4	7	no	—
36	C₂²×C₆₁	C₂×#126	1	7	yes	—
37	C₄×C₆₁	—	8	7	no	—
38	Dic₆₁	—	8	7	no	—
39	Dic₆₁	#401	1	1	yes	characteristic
40	Dic₆₁	#401	1	7	yes	—
41	#2104	—	61	7	no	—
42	#2104	D₆₁	1	1	yes	characteristic
43	#1740	—	61	7	no	—
44	C₂×C₆₁	—	2	7	no	—
45	C₂×C₆₁	C₂×#401	1	1	yes	characteristic
46	C₂×#117	—	61	7	no	—
47	#342	—	61	14	no	—
48	#342	—	61	7	no	—
49	#342	C₂×D₆₁	1	7	yes	—
50	C₆₁	#23847	1	1	yes	characteristic, complemented, Sylow
51	C₂³×C₄	—	61	1	no	—
52	#117	—	122	14	no	—
53	#117	—	122	14	no	—
54	C₂×C₄⋊C₄	—	122	7	no	—
55	C₂×C₂²⋊C₄	—	61	7	no	—
56	C₂×C₂²⋊C₄	C₂²×D₆₁	1	7	yes	—
57	C₂×C₄²	—	122	7	no	—
58	C₂⁴	C₂³×D₆₁	1	1	yes	characteristic
59	C₂²×C₄	—	2	7	no	—
60	C₂²×C₄	—	61	7	no	—
61	C₂²×C₄	—	61	7	no	—
62	C₂²×C₄	—	122	7	no	—
63	C₄⋊C₄	—	244	14	no	—
64	C₂²⋊C₄	—	2	14	no	—
65	C₂²⋊C₄	—	122	14	no	—
66	C₄²	—	244	7	no	—
67	C₄²	—	244	7	no	—
68	C₂³	—	8	1	no	—
69	C₂³	#41×D₆₁	1	7	yes	—
70	C₂×C₄	—	4	7	no	—
71	C₂×C₄	—	8	7	no	—
72	C₂×C₄	—	61	7	no	—
73	C₂×C₄	—	61	7	no	—
74	C₂×C₄	—	61	14	no	—
75	C₂×C₄	—	488	7	no	—
76	C₂×C₄	—	244	7	no	—
77	C₂²	—	4	7	no	—
78	C₂²	#126×D₆₁	1	7	yes	—
79	C₄	—	8	7	no	—
80	C₄	—	61	1	no	—
81	C₄	—	61	7	no	—
82	C₄	—	488	7	no	—
83	C₂	—	2	7	no	—
84	C₂	#401×D₆₁	1	1	yes	characteristic

	subgroup	G/H
center	C₂	#401×D₆₁
commutator, prime radical, cosocle	C₂⁴×C₆₁	C₂⁴
Frattini	C₂⁴	C₂³×D₆₁
Fitting	C₆₁×#2104	C₂
socle	C₂×C₆₁	C₂×#401
hypercenter	#2104	D₆₁
hypocenter	C₆₁	#23847
LCS	G ▷ C₂⁴×C₆₁ ▷ C₂×C₆₁ ▷ C₆₁
UCS	C₁ ◁ C₂ ◁ C₂⁴ ◁ #2104

Complements

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

N	H ≅ G/N	type
50. C₆₁	#23847	semidirect

Maximal subgroups

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

subgroup	index
C₆₁⋊#2104	2
#21445550	2
C₆₁×#2104	2
#23847	61