Subgroups — #16840374

Order 3904.

192 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#14519777	C₂	1	1	yes	maximal, characteristic, complemented
2	#14519770	C₂	1	4	yes	maximal
3	#14519769	C₂	1	1	yes	maximal, characteristic, complemented
4	#14519767	C₂	1	4	yes	maximal, complemented
5	#14519756	C₂	1	2	yes	maximal
6	C₄×Dic₁₂₂	C₂	1	2	yes	maximal, complemented
7	#117×C₆₁	C₂	1	1	yes	maximal, characteristic
8	C₂²×C₄×C₆₁	C₂²	1	1	yes	characteristic
9	C₂×Dic₁₂₂	C₂²	1	2	yes	—
10	C₄⋊C₄×C₆₁	C₂²	1	2	yes	—
11	C₂²⋊C₄×C₆₁	C₂²	1	2	yes	—
12	C₄²×C₆₁	C₂²	1	1	yes	characteristic
13	C₄²×C₆₁	C₂²	1	1	yes	characteristic
14	C₂²⋊Dic₆₁	C₂²	1	4	yes	—
15	C₄⋊Dic₆₁	C₂²	1	2	yes	—
16	C₄⋊Dic₆₁	C₂²	1	4	yes	—
17	#11758196	C₂²	1	4	yes	—
18	#11758196	C₂²	1	4	yes	—
19	#11758196	C₂²	1	4	yes	—
20	C₄×Dic₆₁	C₂²	1	4	yes	—
21	C₂³×C₆₁	C₂³	1	1	yes	characteristic
22	C₂×C₄×C₆₁	—	2	2	no	—
23	C₂×C₄×C₆₁	C₂³	1	2	yes	—
24	C₂×C₄×C₆₁	C₂³	1	1	yes	characteristic
25	C₂×C₄×C₆₁	C₂³	1	2	yes	—
26	C₂×C₄×C₆₁	C₂³	1	1	yes	characteristic
27	C₂×Dic₆₁	C₂³	1	4	yes	—
28	C₂×Dic₆₁	C₂³	1	4	yes	—
29	Dic₁₂₂	—	2	4	no	—
30	C₂²×C₆₁	—	2	1	no	—
31	C₂²×C₆₁	—	2	2	no	—
32	C₂²×C₆₁	C₂⁴	1	1	yes	characteristic
33	C₄×C₆₁	—	2	1	no	—
34	C₄×C₆₁	—	2	2	no	—
35	C₄×C₆₁	—	2	2	no	—
36	C₄×C₆₁	#41	1	2	yes	—
37	Dic₆₁	—	2	4	no	—
38	Dic₆₁	—	2	4	no	—
39	C₂×C₆₁	—	4	1	no	—
40	C₂×C₆₁	C₂×#41	1	1	yes	characteristic
41	C₂×C₆₁	2₋¹⁺⁴	1	2	yes	—
42	#541	—	61	1	no	maximal, Sylow
43	C₆₁	#541	1	1	yes	characteristic, complemented, Sylow
44	C₄⋊Q₈	—	61	1	no	—
45	#126	—	61	4	no	—
46	#125	—	61	4	no	—
47	#125	—	61	1	no	—
48	C₂²⋊Q₈	—	61	2	no	—
49	C₄×Q₈	—	61	2	no	—
50	#117	D₆₁	1	1	yes	characteristic
51	C₂×Q₈	—	61	2	no	—
52	C₂²×C₄	C₂×D₆₁	1	1	yes	characteristic
53	C₄⋊C₄	—	61	4	no	—
54	C₄⋊C₄	—	61	4	no	—
55	C₄⋊C₄	—	61	4	no	—
56	C₄⋊C₄	—	61	2	no	—
57	C₄⋊C₄	—	61	4	no	—
58	C₄⋊C₄	C₂×D₆₁	1	2	yes	—
59	C₂²⋊C₄	—	61	4	no	—
60	C₂²⋊C₄	C₂×D₆₁	1	2	yes	—
61	C₄²	—	61	4	no	—
62	C₄²	C₂×D₆₁	1	1	yes	characteristic
63	C₄²	C₂×D₆₁	1	1	yes	characteristic
64	C₂³	C₂²×D₆₁	1	1	yes	characteristic
65	Q₈	—	122	4	no	—
66	C₂×C₄	—	2	2	no	—
67	C₂×C₄	—	61	4	no	—
68	C₂×C₄	—	61	4	no	—
69	C₂×C₄	C₂²×D₆₁	1	2	yes	—
70	C₂×C₄	C₂²×D₆₁	1	2	yes	—
71	C₂×C₄	C₂²×D₆₁	1	1	yes	characteristic
72	C₂×C₄	C₂²×D₆₁	1	1	yes	characteristic
73	C₂²	—	2	1	no	—
74	C₂²	—	2	2	no	—
75	C₂²	C₂³×D₆₁	1	1	yes	characteristic
76	C₄	—	2	1	no	—
77	C₄	—	2	2	no	—
78	C₄	—	2	2	no	—
79	C₄	—	122	4	no	—
80	C₄	—	122	4	no	—
81	C₄	#11758222	1	2	yes	—
82	C₂	—	4	1	no	—
83	C₂	C₂×#11758222	1	1	yes	characteristic
84	C₂	#14519824	1	2	yes	—

	subgroup	G/H
center, Frattini	C₂²	C₂³×D₆₁
commutator, socle, cosocle	C₂²×C₆₁	C₂⁴
Fitting	#117×C₆₁	C₂
prime radical	C₂³×C₆₁	C₂³
hypercenter	#117	D₆₁
hypocenter	C₆₁	#541
LCS	G ▷ C₂²×C₆₁ ▷ C₆₁
UCS	C₁ ◁ C₂² ◁ #117

Complements

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

N	H ≅ G/N	type
1. #14519777	C₂	semidirect
3. #14519769	C₂	semidirect
4. #14519767	C₂	semidirect
6. C₄×Dic₁₂₂	C₂	semidirect
43. C₆₁	#541	semidirect

Maximal subgroups

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

subgroup	index
#14519777	2
#14519770	2
#14519769	2
#14519767	2
#14519756	2
C₄×Dic₁₂₂	2
#117×C₆₁	2
#541	61