Subgroups — C₆₁×#7532 #29510965

Order 15616.

136 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₆₁×#1713	C₂	1	1	yes	maximal, characteristic
2	C₆₁×#1411	C₂	1	2	yes	maximal
3	C₆₁×#500	—	2	2	no	—
4	C₆₁×#494	C₄	1	1	yes	characteristic
5	C₆₁×#494	C₂²	1	1	yes	characteristic
6	C₆₁×#356	—	2	2	no	—
7	C₈²×C₆₁	C₄	1	1	yes	characteristic
8	C₂×Q₁₆×C₆₁	—	4	1	no	—
9	C₂×Q₁₆×C₆₁	—	4	1	no	—
10	C₄⋊Q₈×C₆₁	—	2	1	no	—
11	C₄⋊Q₈×C₆₁	D₄	1	2	yes	—
12	#107×C₆₁	—	4	4	no	—
13	#101×C₆₁	—	4	2	no	—
14	C₄×C₈×C₆₁	—	2	1	no	—
15	C₄×C₈×C₆₁	C₂×C₄	1	1	yes	characteristic
16	C₂×Q₈×C₆₁	—	2	2	no	—
17	C₂×Q₈×C₆₁	—	4	1	no	—
18	Q₁₆×C₆₁	—	8	2	no	—
19	Q₁₆×C₆₁	—	8	2	no	—
20	M₁₆×C₆₁	—	8	2	no	—
21	C₂×C₈×C₆₁	—	2	2	no	—
22	C₂×C₈×C₆₁	—	2	1	no	—
23	C₄⋊C₄×C₆₁	—	4	2	no	—
24	C₄⋊C₄×C₆₁	—	4	2	no	—
25	C₄²×C₆₁	C₂²⋊C₄	1	1	yes	characteristic
26	Q₈×C₆₁	—	8	2	no	—
27	Q₈×C₆₁	—	8	2	no	—
28	C₂×C₄×C₆₁	—	2	1	no	—
29	C₂×C₄×C₆₁	—	4	2	no	—
30	C₂×C₄×C₆₁	—	8	1	no	—
31	C₂×C₄×C₆₁	#99	1	1	yes	characteristic
32	C₈×C₆₁	—	2	4	no	—
33	C₈×C₆₁	—	2	2	no	—
34	C₈×C₆₁	—	16	2	no	—
35	#7532	C₆₁	1	1	yes	maximal, characteristic, direct product factor, Sylow
36	C₂²×C₆₁	#353	1	1	yes	characteristic
37	C₄×C₆₁	—	2	2	no	—
38	C₄×C₆₁	—	2	1	no	—
39	C₄×C₆₁	—	8	2	no	—
40	C₄×C₆₁	—	16	1	no	—
41	#1713	C₂×C₆₁	1	1	yes	characteristic
42	#1411	C₂×C₆₁	1	2	yes	—
43	C₂×C₆₁	—	2	1	no	—
44	C₂×C₆₁	#1408	1	1	yes	characteristic
45	#500	—	2	2	no	—
46	#494	C₄×C₆₁	1	1	yes	characteristic
47	#494	C₂²×C₆₁	1	1	yes	characteristic
48	#356	—	2	2	no	—
49	C₈²	C₄×C₆₁	1	1	yes	characteristic
50	C₆₁	#7532	1	1	yes	characteristic, direct product factor, Sylow
51	C₂×Q₁₆	—	4	1	no	—
52	C₂×Q₁₆	—	4	1	no	—
53	C₄⋊Q₈	—	2	1	no	—
54	C₄⋊Q₈	D₄×C₆₁	1	2	yes	—
55	#107	—	4	4	no	—
56	#101	—	4	2	no	—
57	C₄×C₈	—	2	1	no	—
58	C₄×C₈	C₂×C₄×C₆₁	1	1	yes	characteristic
59	C₂×Q₈	—	2	2	no	—
60	C₂×Q₈	—	4	1	no	—
61	Q₁₆	—	8	2	no	—
62	Q₁₆	—	8	2	no	—
63	M₁₆	—	8	2	no	—
64	C₂×C₈	—	2	2	no	—
65	C₂×C₈	—	2	1	no	—
66	C₄⋊C₄	—	4	2	no	—
67	C₄⋊C₄	—	4	2	no	—
68	C₄²	C₂²⋊C₄×C₆₁	1	1	yes	characteristic
69	Q₈	—	8	2	no	—
70	Q₈	—	8	2	no	—
71	C₂×C₄	—	2	1	no	—
72	C₂×C₄	—	4	2	no	—
73	C₂×C₄	—	8	1	no	—
74	C₂×C₄	#99×C₆₁	1	1	yes	characteristic
75	C₈	—	2	4	no	—
76	C₈	—	2	2	no	—
77	C₈	—	16	2	no	—
78	C₂²	C₆₁×#353	1	1	yes	characteristic
79	C₄	—	2	2	no	—
80	C₄	—	2	1	no	—
81	C₄	—	8	2	no	—
82	C₄	—	16	1	no	—
83	C₂	—	2	1	no	—
84	C₂	C₆₁×#1408	1	1	yes	characteristic

	subgroup	G/H
center, socle	C₂×C₆₁	#1408
commutator	C₄×C₈	C₂×C₄×C₆₁
Frattini, cosocle	#494	C₂²×C₆₁
prime radical	C₂²×C₆₁	#353
LCS	G ▷ C₄×C₈ ▷ C₄² ▷ C₂×C₄ ▷ C₂² ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂×C₆₁ ◁ C₂²×C₆₁ ◁ C₂×C₄×C₆₁ ◁ C₄²×C₆₁ ◁ C₆₁×#494 ◁ G

Complements

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

N	H ≅ G/N	type
35. #7532	C₆₁	direct (N × H)
50. C₆₁	#7532	direct (N × H)

Maximal subgroups

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

subgroup	index
C₆₁×#1713	2
C₆₁×#1411	2
#7532	61

Subgroups — C61×#7532 #29510965

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₆₁×#7532 #29510965