Subgroups — #16840239

Order 3904.

266 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂²×Dic₁₂₂	C₂	1	1	yes	maximal, characteristic
2	C₄×Dic₁₂₂	C₂	1	4	yes	maximal, complemented
3	#14519742	C₂	1	4	yes	maximal, complemented
4	#14519736	C₂	1	1	yes	maximal, characteristic
5	#14519733	C₂	1	4	yes	maximal
6	#117×C₆₁	C₂	1	1	yes	maximal, characteristic
7	C₂²×C₄×C₆₁	C₂²	1	1	yes	characteristic
8	C₂²×Dic₆₁	C₂²	1	2	yes	—
9	C₂×Dic₁₂₂	C₄	1	8	yes	complemented
10	C₂×Dic₁₂₂	C₂²	1	2	yes	—
11	C₂×Dic₁₂₂	C₂²	1	2	yes	—
12	C₄⋊C₄×C₆₁	C₂²	1	2	yes	—
13	C₂²⋊C₄×C₆₁	C₂²	1	2	yes	—
14	C₄²×C₆₁	C₂²	1	2	yes	—
15	C₂²⋊Dic₆₁	C₂²	1	2	yes	—
16	C₄⋊Dic₆₁	C₂²	1	2	yes	—
17	#11758196	C₂²	1	8	yes	—
18	C₄×Dic₆₁	C₂²	1	8	yes	—
19	C₄×Dic₆₁	C₂²	1	2	yes	—
20	C₂³×C₆₁	C₂³	1	1	yes	characteristic
21	C₂×C₄×C₆₁	C₂×C₄	1	4	yes	—
22	C₂×C₄×C₆₁	C₂³	1	4	yes	—
23	C₂×C₄×C₆₁	C₂³	1	1	yes	characteristic
24	C₂×C₄×C₆₁	C₂³	1	1	yes	characteristic
25	C₂×Dic₆₁	C₂×C₄	1	8	yes	—
26	C₂×Dic₆₁	C₂³	1	4	yes	—
27	C₂×Dic₆₁	C₂³	1	4	yes	—
28	Dic₁₂₂	C₂×C₄	1	8	yes	—
29	Dic₁₂₂	C₂×C₄	1	8	yes	—
30	C₂²×C₆₁	—	2	2	no	—
31	C₂²×C₆₁	C₂²×C₄	1	2	yes	—
32	C₂²×C₆₁	C₂⁴	1	1	yes	characteristic
33	C₄×C₆₁	—	2	4	no	—
34	C₄×C₆₁	C₂²×C₄	1	2	yes	—
35	C₄×C₆₁	C₂²×C₄	1	2	yes	—
36	Dic₆₁	—	2	4	no	—
37	Dic₆₁	C₂²×C₄	1	8	yes	—
38	C₂×C₆₁	—	2	2	no	—
39	C₂×C₆₁	C₂³×C₄	1	1	yes	characteristic
40	C₂×C₆₁	2₋¹⁺⁴	1	2	yes	—
41	#519	—	61	1	no	maximal, Sylow
42	C₆₁	#519	1	1	yes	characteristic, complemented, Sylow
43	C₂²×Q₈	—	61	1	no	—
44	C₄×Q₈	—	61	4	no	—
45	C₄×Q₈	—	61	4	no	—
46	#117	—	61	1	no	—
47	#117	—	61	4	no	—
48	#117	D₆₁	1	1	yes	characteristic
49	C₂×Q₈	—	61	8	no	—
50	C₂×Q₈	—	61	2	no	—
51	C₂×Q₈	—	61	2	no	—
52	C₂²×C₄	—	61	2	no	—
53	C₂²×C₄	C₂×D₆₁	1	1	yes	characteristic
54	C₄⋊C₄	—	61	8	no	—
55	C₄⋊C₄	—	61	2	no	—
56	C₄⋊C₄	C₂×D₆₁	1	2	yes	—
57	C₂²⋊C₄	—	61	2	no	—
58	C₂²⋊C₄	C₂×D₆₁	1	2	yes	—
59	C₄²	—	61	8	no	—
60	C₄²	—	61	2	no	—
61	C₄²	C₂×D₆₁	1	2	yes	—
62	C₂³	C₂²×D₆₁	1	1	yes	characteristic
63	Q₈	—	61	8	no	—
64	Q₈	—	61	8	no	—
65	C₂×C₄	—	61	8	no	—
66	C₂×C₄	—	61	4	no	—
67	C₂×C₄	—	61	4	no	—
68	C₂×C₄	C₄×D₆₁	1	4	yes	—
69	C₂×C₄	C₂²×D₆₁	1	4	yes	—
70	C₂×C₄	C₂²×D₆₁	1	1	yes	characteristic
71	C₂×C₄	C₂²×D₆₁	1	1	yes	characteristic
72	C₂²	—	2	2	no	—
73	C₂²	C₂×C₄×D₆₁	1	2	yes	—
74	C₂²	C₂³×D₆₁	1	1	yes	characteristic
75	C₄	—	2	4	no	—
76	C₄	—	61	8	no	—
77	C₄	—	122	4	no	—
78	C₄	C₂×C₄×D₆₁	1	2	yes	—
79	C₄	C₂×C₄×D₆₁	1	2	yes	—
80	C₂	—	2	2	no	—
81	C₂	C₂²×C₄×D₆₁	1	1	yes	characteristic
82	C₂	#14519824	1	2	yes	—

	subgroup	G/H
center, Frattini	C₂²	C₂³×D₆₁
commutator	C₂×C₆₁	C₂³×C₄
Fitting	#117×C₆₁	C₂
socle, cosocle	C₂²×C₆₁	C₂⁴
prime radical	C₂³×C₆₁	C₂³
hypercenter	#117	D₆₁
hypocenter	C₆₁	#519
LCS	G ▷ C₂×C₆₁ ▷ C₆₁
UCS	C₁ ◁ C₂² ◁ #117

Complements

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

N	H ≅ G/N	type
2. C₄×Dic₁₂₂	C₂	semidirect
3. #14519742	C₂	semidirect
9. C₂×Dic₁₂₂	C₄	semidirect
42. C₆₁	#519	semidirect

Maximal subgroups

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

subgroup	index
C₂²×Dic₁₂₂	2
C₄×Dic₁₂₂	2
#14519742	2
#14519736	2
#14519733	2
#117×C₆₁	2
#519	61