Subgroups — C₆₁×#1708 #21441749

Order 7808.

190 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₆₁×#498	C₂	1	1	yes	maximal, characteristic, complemented
2	C₆₁×#492	C₂	1	1	yes	maximal, characteristic
3	C₆₁×#487	C₂	1	2	yes	maximal, complemented
4	C₆₁×#486	C₂	1	2	yes	maximal
5	C₈²×C₆₁	C₂	1	1	yes	maximal, characteristic, complemented
6	C₂×SD₁₆×C₆₁	—	2	2	no	—
7	C₄⋊Q₈×C₆₁	C₂²	1	2	yes	—
8	C₄⋊Q₈×C₆₁	C₂²	1	1	yes	characteristic
9	#127×C₆₁	C₂²	1	1	yes	characteristic
10	#106×C₆₁	—	2	2	no	—
11	#103×C₆₁	—	2	4	no	—
12	#102×C₆₁	—	2	4	no	—
13	C₄×C₈×C₆₁	C₂²	1	2	yes	—
14	C₄×C₈×C₆₁	C₂²	1	1	yes	characteristic
15	C₂×Q₈×C₆₁	—	2	2	no	—
16	C₂×Q₈×C₆₁	—	2	1	no	—
17	C₂×D₄×C₆₁	—	2	2	no	—
18	C₂×D₄×C₆₁	—	2	1	no	—
19	SD₁₆×C₆₁	—	4	4	no	—
20	C₂×C₈×C₆₁	D₄	1	4	yes	—
21	C₂×C₈×C₆₁	D₄	1	2	yes	—
22	C₄⋊C₄×C₆₁	—	2	2	no	—
23	C₄⋊C₄×C₆₁	—	2	2	no	—
24	C₄⋊C₄×C₆₁	—	2	2	no	—
25	C₄²×C₆₁	C₂³	1	1	yes	characteristic
26	C₂³×C₆₁	—	4	1	no	—
27	Q₈×C₆₁	—	4	2	no	—
28	Q₈×C₆₁	—	4	4	no	—
29	D₄×C₆₁	—	4	2	no	—
30	D₄×C₆₁	—	4	4	no	—
31	C₂×C₄×C₆₁	—	4	1	no	—
32	C₂×C₄×C₆₁	—	4	2	no	—
33	C₂×C₄×C₆₁	C₂×D₄	1	1	yes	characteristic
34	C₂×C₄×C₆₁	C₂×D₄	1	2	yes	—
35	C₈×C₆₁	—	2	4	no	—
36	C₈×C₆₁	SD₁₆	1	4	yes	—
37	C₂²×C₆₁	—	8	1	no	—
38	C₂²×C₆₁	—	8	2	no	—
39	C₂²×C₆₁	#127	1	1	yes	characteristic
40	C₄×C₆₁	—	8	1	no	—
41	C₄×C₆₁	—	8	2	no	—
42	C₄×C₆₁	C₂×SD₁₆	1	2	yes	—
43	C₄×C₆₁	#135	1	4	yes	—
44	#1708	C₆₁	1	1	yes	maximal, characteristic, direct product factor, Sylow
45	C₂×C₆₁	—	16	1	no	—
46	C₂×C₆₁	#492	1	1	yes	characteristic
47	C₂×C₆₁	#495	1	2	yes	—
48	#498	C₂×C₆₁	1	1	yes	characteristic, complemented
49	#492	C₂×C₆₁	1	1	yes	characteristic
50	#487	C₂×C₆₁	1	2	yes	complemented
51	#486	C₂×C₆₁	1	2	yes	—
52	C₈²	C₂×C₆₁	1	1	yes	characteristic, complemented
53	C₆₁	#1708	1	1	yes	characteristic, direct product factor, Sylow
54	C₂×SD₁₆	—	2	2	no	—
55	C₄⋊Q₈	C₂²×C₆₁	1	2	yes	—
56	C₄⋊Q₈	C₂²×C₆₁	1	1	yes	characteristic
57	#127	C₂²×C₆₁	1	1	yes	characteristic
58	#106	—	2	2	no	—
59	#103	—	2	4	no	—
60	#102	—	2	4	no	—
61	C₄×C₈	C₂²×C₆₁	1	2	yes	—
62	C₄×C₈	C₂²×C₆₁	1	1	yes	characteristic
63	C₂×Q₈	—	2	2	no	—
64	C₂×Q₈	—	2	1	no	—
65	C₂×D₄	—	2	2	no	—
66	C₂×D₄	—	2	1	no	—
67	SD₁₆	—	4	4	no	—
68	C₂×C₈	D₄×C₆₁	1	4	yes	—
69	C₂×C₈	D₄×C₆₁	1	2	yes	—
70	C₄⋊C₄	—	2	2	no	—
71	C₄⋊C₄	—	2	2	no	—
72	C₄⋊C₄	—	2	2	no	—
73	C₄²	C₂³×C₆₁	1	1	yes	characteristic
74	C₂³	—	4	1	no	—
75	Q₈	—	4	2	no	—
76	Q₈	—	4	4	no	—
77	D₄	—	4	2	no	—
78	D₄	—	4	4	no	—
79	C₂×C₄	—	4	1	no	—
80	C₂×C₄	—	4	2	no	—
81	C₂×C₄	C₂×D₄×C₆₁	1	1	yes	characteristic
82	C₂×C₄	C₂×D₄×C₆₁	1	2	yes	—
83	C₈	—	2	4	no	—
84	C₈	SD₁₆×C₆₁	1	4	yes	—
85	C₂²	—	8	1	no	—
86	C₂²	—	8	2	no	—
87	C₂²	#127×C₆₁	1	1	yes	characteristic
88	C₄	—	8	1	no	—
89	C₄	—	8	2	no	—
90	C₄	C₂×SD₁₆×C₆₁	1	2	yes	—
91	C₄	#135×C₆₁	1	4	yes	—
92	C₂	—	16	1	no	—
93	C₂	C₆₁×#492	1	1	yes	characteristic
94	C₂	C₆₁×#495	1	2	yes	—

	subgroup	G/H
center, socle	C₂²×C₆₁	#127
commutator, Frattini, cosocle	C₄²	C₂³×C₆₁
prime radical	#127×C₆₁	C₂²
LCS	G ▷ C₄² ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂²×C₆₁ ◁ C₄²×C₆₁ ◁ G

Complements

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

N	H ≅ G/N	type
1. C₆₁×#498	C₂	semidirect
3. C₆₁×#487	C₂	semidirect
5. C₈²×C₆₁	C₂	semidirect
44. #1708	C₆₁	direct (N × H)
48. #498	C₂×C₆₁	semidirect
50. #487	C₂×C₆₁	semidirect
52. C₈²	C₂×C₆₁	semidirect
53. C₆₁	#1708	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×#498	2
C₆₁×#492	2
C₆₁×#487	2
C₆₁×#486	2
C₈²×C₆₁	2
#1708	61

Subgroups — C61×#1708 #21441749

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₆₁×#1708 #21441749