Subgroups — #16840366

Order 3904.

240 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#14519769	C₂	1	1	yes	maximal, characteristic, complemented
2	#14519758	C₂	1	4	yes	maximal, complemented
3	#14519749	C₂	1	4	yes	maximal, complemented
4	#14519748	C₂	1	2	yes	maximal, complemented
5	D₄×Dic₆₁	C₂	1	2	yes	maximal, complemented
6	#14519734	C₂	1	1	yes	maximal, characteristic, complemented
7	#127×C₆₁	C₂	1	1	yes	maximal, characteristic, complemented
8	C₂×D₄×C₆₁	C₂²	1	2	yes	—
9	C₂×D₄×C₆₁	C₂²	1	4	yes	—
10	C₂×C₂²⋊D₆₁	C₂²	1	4	yes	—
11	C₂²×Dic₆₁	C₂²	1	4	yes	—
12	C₂×C₄×D₆₁	C₂²	1	1	yes	characteristic
13	C₄²×C₆₁	C₂²	1	1	yes	characteristic
14	C₂²⋊Dic₆₁	C₂²	1	4	yes	—
15	C₂²⋊Dic₆₁	C₂²	1	4	yes	—
16	#11758198	C₂²	1	2	yes	—
17	C₄⋊Dic₆₁	C₂²	1	2	yes	—
18	#11758196	C₂²	1	2	yes	—
19	#11758196	C₂²	1	4	yes	—
20	C₄×Dic₆₁	C₂²	1	1	yes	characteristic
21	C₂³×C₆₁	C₂³	1	4	yes	—
22	C₂²×D₆₁	C₂³	1	1	yes	characteristic
23	D₄×C₆₁	—	2	4	no	—
24	D₄×C₆₁	—	4	4	no	—
25	C₂×C₄×C₆₁	C₂³	1	2	yes	—
26	C₂×C₄×C₆₁	C₂³	1	1	yes	characteristic
27	C₂²⋊D₆₁	—	4	4	no	—
28	C₂×Dic₆₁	—	4	4	no	—
29	C₂×Dic₆₁	C₂³	1	2	yes	—
30	C₂×Dic₆₁	C₂³	1	4	yes	—
31	C₂×Dic₆₁	C₂³	1	1	yes	characteristic
32	C₄×D₆₁	—	2	2	no	—
33	C₂²×C₆₁	—	2	4	no	—
34	C₂²×C₆₁	—	2	4	no	—
35	C₂²×C₆₁	—	2	4	no	—
36	C₂²×C₆₁	C₂⁴	1	1	yes	characteristic
37	C₂×D₆₁	—	2	1	no	—
38	C₂×D₆₁	—	2	2	no	—
39	C₄×C₆₁	—	2	2	no	—
40	C₄×C₆₁	#41	1	2	yes	—
41	Dic₆₁	—	2	1	no	—
42	Dic₆₁	—	2	2	no	—
43	Dic₆₁	—	2	4	no	—
44	C₂×C₆₁	—	4	4	no	—
45	C₂×C₆₁	C₂×#41	1	1	yes	characteristic
46	C₂×C₆₁	H(5,𝔽₂)	1	2	yes	—
47	D₆₁	—	4	1	no	—
48	#540	—	61	1	no	maximal, Sylow
49	C₆₁	#540	1	1	yes	characteristic, complemented, Sylow
50	#127	D₆₁	1	1	yes	characteristic, complemented
51	#125	—	61	1	no	—
52	#123	—	61	4	no	—
53	#121	—	61	2	no	—
54	#121	—	61	4	no	—
55	C₄×D₄	—	61	2	no	—
56	#117	—	61	1	no	—
57	C₂×D₄	—	61	4	no	—
58	C₂×D₄	C₂×D₆₁	1	2	yes	—
59	C₂×D₄	C₂×D₆₁	1	4	yes	—
60	C₂²×C₄	—	61	4	no	—
61	C₂²×C₄	—	61	1	no	—
62	C₄⋊C₄	—	61	2	no	—
63	C₄⋊C₄	—	61	4	no	—
64	C₄⋊C₄	—	61	2	no	—
65	C₂²⋊C₄	—	61	4	no	—
66	C₂²⋊C₄	—	61	4	no	—
67	C₂²⋊C₄	—	61	2	no	—
68	C₄²	—	61	1	no	—
69	C₄²	C₂×D₆₁	1	1	yes	characteristic
70	C₂³	—	61	1	no	—
71	C₂³	C₂²×D₆₁	1	4	yes	—
72	D₄	—	2	4	no	—
73	D₄	—	4	4	no	—
74	D₄	—	244	4	no	—
75	C₂×C₄	—	61	2	no	—
76	C₂×C₄	—	61	1	no	—
77	C₂×C₄	—	61	4	no	—
78	C₂×C₄	—	122	2	no	—
79	C₂×C₄	—	244	4	no	—
80	C₂×C₄	C₂²×D₆₁	1	2	yes	—
81	C₂×C₄	C₂²×D₆₁	1	1	yes	characteristic
82	C₂²	—	2	4	no	—
83	C₂²	—	2	4	no	—
84	C₂²	—	2	4	no	—
85	C₂²	—	122	1	no	—
86	C₂²	—	122	2	no	—
87	C₂²	C₂³×D₆₁	1	1	yes	characteristic
88	C₄	—	2	2	no	—
89	C₄	—	122	1	no	—
90	C₄	—	122	2	no	—
91	C₄	—	122	4	no	—
92	C₄	#11758224	1	2	yes	—
93	C₂	—	4	4	no	—
94	C₂	—	244	1	no	—
95	C₂	C₂×#11758224	1	1	yes	characteristic
96	C₂	#14519822	1	2	yes	—

	subgroup	G/H
center, Frattini	C₂²	C₂³×D₆₁
commutator, socle, cosocle	C₂²×C₆₁	C₂⁴
Fitting	#127×C₆₁	C₂
hypercenter	#127	D₆₁
hypocenter	C₆₁	#540
LCS	G ▷ C₂²×C₆₁ ▷ C₆₁
UCS	C₁ ◁ C₂² ◁ #127

Complements

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

N	H ≅ G/N	type
1. #14519769	C₂	semidirect
2. #14519758	C₂	semidirect
3. #14519749	C₂	semidirect
4. #14519748	C₂	semidirect
5. D₄×Dic₆₁	C₂	semidirect
6. #14519734	C₂	semidirect
7. #127×C₆₁	C₂	semidirect
49. C₆₁	#540	semidirect
50. #127	D₆₁	semidirect

Maximal subgroups

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

subgroup	index
#14519769	2
#14519758	2
#14519749	2
#14519748	2
D₄×Dic₆₁	2
#14519734	2
#127×C₆₁	2
#540	61