Subgroups — C₆₁×#7318 #29510751

Order 15616.

484 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₆₁×#1782	C₂	1	3	yes	maximal
2	C₂²×#41×C₆₁	C₂²	1	1	yes	characteristic
3	C₂×#99×C₆₁	—	2	6	no	—
4	C₆₁×#381	C₄	1	6	yes	complemented
5	C₂×#41×C₆₁	—	4	6	no	—
6	C₂²×Q₈×C₆₁	Q₈	1	1	yes	characteristic
7	C₂²×D₄×C₆₁	D₄	1	3	yes	—
8	C₂³×C₄×C₆₁	C₂×C₄	1	3	yes	—
9	C₂×C₂²⋊C₄×C₆₁	—	2	6	no	—
10	#99×C₆₁	—	4	12	no	—
11	#95×C₆₁	—	4	6	no	—
12	C₂⁴×C₆₁	C₂²⋊C₄	1	3	yes	—
13	#41×C₆₁	—	4	16	no	—
14	C₂×Q₈×C₆₁	—	4	3	no	—
15	C₂×D₄×C₆₁	—	2	12	no	—
16	C₂×D₄×C₆₁	—	4	3	no	—
17	C₂²×C₄×C₆₁	—	2	6	no	—
18	C₂²×C₄×C₆₁	—	4	6	no	—
19	C₂²×C₄×C₆₁	—	4	6	no	—
20	C₂²×C₄×C₆₁	C₄²	1	1	yes	characteristic
21	C₂²×C₄×C₆₁	C₄⋊C₄	1	3	yes	—
22	C₂²⋊C₄×C₆₁	—	4	12	no	—
23	C₂³×C₆₁	—	2	6	no	—
24	C₂³×C₆₁	—	8	3	no	—
25	C₂³×C₆₁	#95	1	1	yes	characteristic
26	C₂³×C₆₁	#99	1	6	yes	—
27	Q₈×C₆₁	—	4	4	no	—
28	D₄×C₆₁	—	4	12	no	—
29	C₂×C₄×C₆₁	—	2	3	no	—
30	C₂×C₄×C₆₁	—	2	3	no	—
31	C₂×C₄×C₆₁	—	2	6	no	—
32	C₂×C₄×C₆₁	—	4	12	no	—
33	C₂×C₄×C₆₁	—	8	6	no	—
34	C₂×C₄×C₆₁	—	8	12	no	—
35	#7318	C₆₁	1	1	yes	maximal, characteristic, direct product factor, Sylow
36	C₂²×C₆₁	—	2	6	no	—
37	C₂²×C₆₁	—	4	1	no	—
38	C₂²×C₆₁	—	4	6	no	—
39	C₂²×C₆₁	—	8	6	no	—
40	C₂²×C₆₁	#342	1	3	yes	—
41	C₄×C₆₁	—	4	1	no	—
42	C₄×C₆₁	—	4	3	no	—
43	C₄×C₆₁	—	16	6	no	—
44	#1782	C₂×C₆₁	1	3	yes	—
45	C₂×C₆₁	—	2	3	no	—
46	C₂×C₆₁	—	4	6	no	—
47	C₂×C₆₁	#1304	1	1	yes	characteristic
48	C₂²×#41	C₂²×C₆₁	1	1	yes	characteristic
49	C₂×#99	—	2	6	no	—
50	#381	C₄×C₆₁	1	6	yes	complemented
51	C₆₁	#7318	1	1	yes	characteristic, direct product factor, Sylow
52	C₂×#41	—	4	6	no	—
53	C₂²×Q₈	Q₈×C₆₁	1	1	yes	characteristic
54	C₂²×D₄	D₄×C₆₁	1	3	yes	—
55	C₂³×C₄	C₂×C₄×C₆₁	1	3	yes	—
56	C₂×C₂²⋊C₄	—	2	6	no	—
57	#99	—	4	12	no	—
58	#95	—	4	6	no	—
59	C₂⁴	C₂²⋊C₄×C₆₁	1	3	yes	—
60	#41	—	4	16	no	—
61	C₂×Q₈	—	4	3	no	—
62	C₂×D₄	—	2	12	no	—
63	C₂×D₄	—	4	3	no	—
64	C₂²×C₄	—	2	6	no	—
65	C₂²×C₄	—	4	6	no	—
66	C₂²×C₄	—	4	6	no	—
67	C₂²×C₄	C₄²×C₆₁	1	1	yes	characteristic
68	C₂²×C₄	C₄⋊C₄×C₆₁	1	3	yes	—
69	C₂²⋊C₄	—	4	12	no	—
70	C₂³	—	2	6	no	—
71	C₂³	—	8	3	no	—
72	C₂³	#95×C₆₁	1	1	yes	characteristic
73	C₂³	#99×C₆₁	1	6	yes	—
74	Q₈	—	4	4	no	—
75	D₄	—	4	12	no	—
76	C₂×C₄	—	2	3	no	—
77	C₂×C₄	—	2	3	no	—
78	C₂×C₄	—	2	6	no	—
79	C₂×C₄	—	4	12	no	—
80	C₂×C₄	—	8	6	no	—
81	C₂×C₄	—	8	12	no	—
82	C₂²	—	2	6	no	—
83	C₂²	—	4	1	no	—
84	C₂²	—	4	6	no	—
85	C₂²	—	8	6	no	—
86	C₂²	C₆₁×#342	1	3	yes	—
87	C₄	—	4	1	no	—
88	C₄	—	4	3	no	—
89	C₄	—	16	6	no	—
90	C₂	—	2	3	no	—
91	C₂	—	4	6	no	—
92	C₂	C₆₁×#1304	1	1	yes	characteristic

	subgroup	G/H
center, socle	C₂×C₆₁	#1304
commutator	C₂²×C₄	C₄²×C₆₁
Frattini, cosocle	C₂²×#41	C₂²×C₆₁
prime radical	C₂²×#41×C₆₁	C₂²
LCS	G ▷ C₂²×C₄ ▷ C₂³ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂×C₆₁ ◁ C₂³×C₆₁ ◁ C₂²×#41×C₆₁ ◁ G

Complements

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

N	H ≅ G/N	type
4. C₆₁×#381	C₄	semidirect
35. #7318	C₆₁	direct (N × H)
50. #381	C₄×C₆₁	semidirect
51. C₆₁	#7318	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×#1782	2
#7318	61

Subgroups — C61×#7318 #29510751

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₆₁×#7318 #29510751