Subgroups — #37525421

Order 22932.

280 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂²×#14274262	C₃	1	1	yes	maximal, characteristic, complemented
2	C₇²×C₂²⋊(C₁₃⋊C₃)	C₃	1	1	yes	maximal, characteristic, complemented
3	C₁₃×C₂²⋊#3861	C₃	1	1	yes	maximal, characteristic, complemented
4	C₂²⋊#14274263	C₃	1	1	yes	maximal, characteristic, complemented
5	C₁₃⋊C₃×#3861	—	4	1	no	maximal, Hall
6	C₂×#14274262	—	3	1	no	—
7	#63271⋊A₄	—	7	8	no	maximal
8	C₂²×C₇²×C₁₃	C₃²	1	1	yes	characteristic, complemented, Hall
9	#14274263	—	4	1	no	—
10	#14274262	A₄	1	1	yes	characteristic, complemented
11	C₁₃×#3861	—	4	1	no	—
12	C₇²×C₁₃⋊C₃	—	4	1	no	—
13	A₄×#3861	—	13	1	no	maximal, Hall
14	C₂×C₇²×C₁₃	—	3	1	no	—
15	C₂²⋊#63270	—	7	8	no	—
16	C₁₃×C₂²⋊ΓL(1,𝔽₈)	—	7	8	no	—
17	C₇×C₂²⋊(C₁₃⋊C₃)	ΓL(1,𝔽₈)	1	8	yes	complemented
18	C₂²×#63271	—	7	8	no	—
19	ΓL(1,𝔽₈)×C₁₃⋊C₃	—	28	8	no	—
20	C₇²×C₁₃	C₃×A₄	1	1	yes	characteristic, complemented, Hall
21	C₂²⋊#3861	—	13	1	no	—
22	C₂²⋊#3861	C₁₃⋊C₃	1	1	yes	characteristic, complemented
23	C₇²×A₄	—	13	1	no	—
24	C₂²×#3861	—	13	1	no	—
25	C₂×#63271	—	21	8	no	—
26	A₄×C₁₃⋊C₃	—	49	1	no	Hall
27	C₃×#3861	—	52	1	no	Hall
28	C₂²×C₇×C₁₃	C₃×ΓL(1,𝔽₈)	1	8	yes	complemented
29	C₂×#3861	—	39	1	no	—
30	#63271	—	7	8	no	—
31	#63270	—	28	8	no	—
32	C₇×C₁₃⋊C₃	—	4	8	no	—
33	C₁₃×ΓL(1,𝔽₈)	—	28	8	no	—
34	A₄×ΓL(1,𝔽₈)	—	91	8	no	—
35	C₂²×C₇²	C₃×C₁₃⋊C₃	1	1	yes	characteristic, complemented, Hall
36	C₂×C₇×C₁₃	—	3	8	no	—
37	C₂²⋊(C₁₃⋊C₃)	—	49	1	no	—
38	C₂²⋊(C₁₃⋊C₃)	#3861	1	1	yes	characteristic, complemented
39	A₄×C₁₃	—	49	1	no	—
40	C₂²×C₁₃⋊C₃	—	49	1	no	—
41	C₃×C₇²	—	52	1	no	—
42	#3861	—	4	1	no	—
43	#3861	—	13	1	no	—
44	#3861	—	52	1	no	—
45	C₃×C₁₃⋊C₃	—	196	1	no	Hall
46	C₂×C₇²	—	3	1	no	—
47	C₇×C₁₃	A₄×ΓL(1,𝔽₈)	1	8	yes	complemented
48	C₂²⋊ΓL(1,𝔽₈)	—	7	8	no	—
49	C₂²⋊ΓL(1,𝔽₈)	—	91	8	no	—
50	C₇×A₄	—	13	8	no	—
51	C₂²×ΓL(1,𝔽₈)	—	91	8	no	—
52	C₂×C₁₃⋊C₃	—	147	1	no	—
53	C₃×ΓL(1,𝔽₈)	—	364	8	no	—
54	C₂²×C₁₃	C₃×#3861	1	1	yes	characteristic, complemented, Hall
55	C₇²	A₄×C₁₃⋊C₃	1	1	yes	characteristic, complemented, Sylow
56	C₂×ΓL(1,𝔽₈)	—	273	8	no	—
57	C₃×C₁₃	—	196	1	no	—
58	C₁₃⋊C₃	—	4	1	no	—
59	C₁₃⋊C₃	—	49	1	no	—
60	C₁₃⋊C₃	—	196	1	no	—
61	C₃×A₄	—	637	1	no	Hall
62	C₂²×C₇	ΓL(1,𝔽₈)×C₁₃⋊C₃	1	8	yes	complemented
63	C₂×C₁₃	—	3	1	no	—
64	C₃×C₇	—	52	8	no	—
65	ΓL(1,𝔽₈)	—	28	8	no	—
66	ΓL(1,𝔽₈)	—	91	8	no	—
67	ΓL(1,𝔽₈)	—	364	8	no	—
68	C₂×C₇	—	3	8	no	—
69	C₁₃	A₄×#3861	1	1	yes	characteristic, complemented, Sylow
70	C₂²×C₃	—	637	1	no	—
71	A₄	—	13	1	no	—
72	A₄	—	49	1	no	—
73	A₄	—	637	1	no	—
74	C₃²	—	2548	1	no	Sylow
75	C₇	#63271⋊A₄	1	8	yes	complemented
76	C₂×C₃	—	1911	1	no	—
77	C₂²	C₁₃⋊C₃×#3861	1	1	yes	characteristic, complemented, Sylow
78	C₃	—	52	1	no	—
79	C₃	—	637	1	no	—
80	C₃	—	196	1	no	—
81	C₃	—	2548	1	no	—
82	C₂	—	3	1	no	—

	subgroup	G/H
center, Frattini, hypercenter	C₁	G
commutator, Fitting, socle, cosocle, hypocenter	C₂²×C₇²×C₁₃	C₃²
LCS	G ▷ C₂²×C₇²×C₁₃
UCS	C₁

Complements

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

N	H ≅ G/N	type
1. C₂²×#14274262	C₃	semidirect
2. C₇²×C₂²⋊(C₁₃⋊C₃)	C₃	semidirect
3. C₁₃×C₂²⋊#3861	C₃	semidirect
4. C₂²⋊#14274263	C₃	semidirect
8. C₂²×C₇²×C₁₃	C₃²	semidirect
10. #14274262	A₄	semidirect
17. C₇×C₂²⋊(C₁₃⋊C₃)	ΓL(1,𝔽₈)	semidirect
20. C₇²×C₁₃	C₃×A₄	semidirect
22. C₂²⋊#3861	C₁₃⋊C₃	semidirect
28. C₂²×C₇×C₁₃	C₃×ΓL(1,𝔽₈)	semidirect
35. C₂²×C₇²	C₃×C₁₃⋊C₃	semidirect
38. C₂²⋊(C₁₃⋊C₃)	#3861	semidirect
47. C₇×C₁₃	A₄×ΓL(1,𝔽₈)	semidirect
54. C₂²×C₁₃	C₃×#3861	semidirect
55. C₇²	A₄×C₁₃⋊C₃	semidirect
62. C₂²×C₇	ΓL(1,𝔽₈)×C₁₃⋊C₃	semidirect
69. C₁₃	A₄×#3861	semidirect
75. C₇	#63271⋊A₄	semidirect
77. C₂²	C₁₃⋊C₃×#3861	semidirect

Maximal subgroups

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

subgroup	index
C₂²×#14274262	3
C₇²×C₂²⋊(C₁₃⋊C₃)	3
C₁₃×C₂²⋊#3861	3
C₂²⋊#14274263	3
C₁₃⋊C₃×#3861	4
#63271⋊A₄	7
A₄×#3861	13