Subgroups — #16840368

Order 3904.

240 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#14519774	C₂	1	1	yes	maximal, characteristic, complemented
2	#14519762	C₂	1	4	yes	maximal, complemented
3	#14519752	C₂	1	4	yes	maximal, complemented
4	#14519752	C₂	1	2	yes	maximal, complemented
5	#14519739	C₂	1	2	yes	maximal, complemented
6	#14519734	C₂	1	1	yes	maximal, characteristic, complemented
7	#125×C₆₁	C₂	1	1	yes	maximal, characteristic, complemented
8	C₂×D₂₄₄	C₂²	1	2	yes	—
9	C₂×D₂₄₄	C₂²	1	4	yes	—
10	C₂×D₂₄₄	C₂²	1	4	yes	—
11	C₂×C₄×D₆₁	C₂²	1	1	yes	characteristic
12	C₂×C₄×D₆₁	C₂²	1	4	yes	—
13	C₄⋊C₄×C₆₁	C₂²	1	2	yes	—
14	C₄⋊C₄×C₆₁	C₂²	1	4	yes	—
15	C₄²×C₆₁	C₂²	1	1	yes	characteristic
16	#11758198	C₂²	1	2	yes	—
17	#11758198	C₂²	1	4	yes	—
18	#11758198	C₂²	1	4	yes	—
19	#11758196	C₂²	1	2	yes	—
20	C₄×Dic₆₁	C₂²	1	1	yes	characteristic
21	C₂²×D₆₁	C₂³	1	1	yes	characteristic
22	C₂²×D₆₁	C₂³	1	4	yes	—
23	C₂×C₄×C₆₁	C₂³	1	2	yes	—
24	C₂×C₄×C₆₁	C₂³	1	4	yes	—
25	C₂×C₄×C₆₁	C₂³	1	1	yes	characteristic
26	C₂×Dic₆₁	C₂³	1	2	yes	—
27	C₂×Dic₆₁	C₂³	1	1	yes	characteristic
28	D₂₄₄	—	2	4	no	—
29	D₂₄₄	—	4	4	no	—
30	D₂₄₄	—	4	4	no	—
31	C₄×D₆₁	—	2	2	no	—
32	C₄×D₆₁	—	4	4	no	—
33	C₂²×C₆₁	C₂⁴	1	1	yes	characteristic
34	C₂×D₆₁	—	2	1	no	—
35	C₂×D₆₁	—	2	4	no	—
36	C₂×D₆₁	—	2	4	no	—
37	C₂×D₆₁	—	2	2	no	—
38	C₂×D₆₁	—	2	4	no	—
39	C₄×C₆₁	—	2	4	no	—
40	C₄×C₆₁	—	2	2	no	—
41	C₄×C₆₁	#41	1	2	yes	—
42	Dic₆₁	—	2	1	no	—
43	Dic₆₁	—	2	2	no	—
44	C₂×C₆₁	C₂×#41	1	1	yes	characteristic
45	C₂×C₆₁	H(5,𝔽₂)	1	2	yes	—
46	D₆₁	—	4	1	no	—
47	D₆₁	—	4	4	no	—
48	#540	—	61	1	no	maximal, Sylow
49	C₆₁	#540	1	1	yes	characteristic, complemented, Sylow
50	#127	—	61	1	no	—
51	#125	D₆₁	1	1	yes	characteristic, complemented
52	#123	—	61	4	no	—
53	#121	—	61	4	no	—
54	#121	—	61	2	no	—
55	C₄×D₄	—	61	2	no	—
56	#117	—	61	1	no	—
57	C₂×D₄	—	61	2	no	—
58	C₂×D₄	—	61	4	no	—
59	C₂×D₄	—	61	4	no	—
60	C₂²×C₄	—	61	1	no	—
61	C₂²×C₄	—	61	4	no	—
62	C₄⋊C₄	—	61	2	no	—
63	C₄⋊C₄	C₂×D₆₁	1	2	yes	—
64	C₄⋊C₄	C₂×D₆₁	1	4	yes	—
65	C₂²⋊C₄	—	61	2	no	—
66	C₂²⋊C₄	—	61	4	no	—
67	C₂²⋊C₄	—	61	4	no	—
68	C₄²	—	61	1	no	—
69	C₄²	C₂×D₆₁	1	1	yes	characteristic
70	C₂³	—	61	1	no	—
71	C₂³	—	61	4	no	—
72	D₄	—	122	4	no	—
73	D₄	—	244	4	no	—
74	D₄	—	244	4	no	—
75	C₂×C₄	—	61	2	no	—
76	C₂×C₄	—	61	1	no	—
77	C₂×C₄	—	122	2	no	—
78	C₂×C₄	—	244	4	no	—
79	C₂×C₄	C₂²×D₆₁	1	2	yes	—
80	C₂×C₄	C₂²×D₆₁	1	4	yes	—
81	C₂×C₄	C₂²×D₆₁	1	1	yes	characteristic
82	C₂²	—	122	1	no	—
83	C₂²	—	122	4	no	—
84	C₂²	—	122	4	no	—
85	C₂²	—	122	2	no	—
86	C₂²	—	122	4	no	—
87	C₂²	C₂³×D₆₁	1	1	yes	characteristic
88	C₄	—	2	4	no	—
89	C₄	—	2	2	no	—
90	C₄	—	122	1	no	—
91	C₄	—	122	2	no	—
92	C₄	#11758226	1	2	yes	—
93	C₂	—	244	1	no	—
94	C₂	—	244	4	no	—
95	C₂	C₂×#11758226	1	1	yes	characteristic
96	C₂	#14519823	1	2	yes	—

	subgroup	G/H
center, Frattini	C₂²	C₂³×D₆₁
commutator, socle, cosocle	C₂²×C₆₁	C₂⁴
Fitting	#125×C₆₁	C₂
hypercenter	#125	D₆₁
hypocenter	C₆₁	#540
LCS	G ▷ C₂²×C₆₁ ▷ C₆₁
UCS	C₁ ◁ C₂² ◁ #125

Complements

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

N	H ≅ G/N	type
1. #14519774	C₂	semidirect
2. #14519762	C₂	semidirect
3. #14519752	C₂	semidirect
4. #14519752	C₂	semidirect
5. #14519739	C₂	semidirect
6. #14519734	C₂	semidirect
7. #125×C₆₁	C₂	semidirect
49. C₆₁	#540	semidirect
51. #125	D₆₁	semidirect

Maximal subgroups

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

subgroup	index
#14519774	2
#14519762	2
#14519752	2
#14519739	2
#14519734	2
#125×C₆₁	2
#540	61