Subgroups — #37525407

Order 22932.

152 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂²×#14274263	C₃	1	1	yes	maximal, characteristic, complemented
2	C₇×C₂²⋊#63271	C₃	1	2	yes	maximal, complemented
3	C₁₃×C₂²⋊#3862	C₃	1	1	yes	maximal, characteristic, complemented
4	#19153212	—	4	1	no	maximal, Hall
5	C₂×#14274263	—	3	1	no	—
6	#63270⋊A₄	—	7	2	no	maximal
7	C₂²×C₇²×C₁₃	C₃²	1	1	yes	characteristic, complemented, Hall
8	C₁₃×#3862	—	4	1	no	—
9	#14274263	A₄	1	1	yes	characteristic, complemented
10	C₇×#63271	—	4	2	no	—
11	#3861⋊A₄	—	13	1	no	maximal, Hall
12	C₂×C₇²×C₁₃	—	3	1	no	—
13	C₂²⋊#63271	ΓL(1,𝔽₈)	1	2	yes	complemented
14	C₁₃×C₂²⋊ΓL(1,𝔽₈)	—	7	2	no	—
15	C₇×C₂²⋊(C₁₃⋊C₃)	—	7	2	no	—
16	C₂²×#63270	—	7	2	no	—
17	C₂²×#63270	—	21	2	no	—
18	ΓL(1,𝔽₈)×C₁₃⋊C₃	—	28	2	no	—
19	C₇²×C₁₃	C₃×A₄	1	1	yes	characteristic, complemented, Hall
20	C₂²⋊#3862	C₁₃⋊C₃	1	1	yes	characteristic, complemented
21	C₇×C₂²⋊ΓL(1,𝔽₈)	—	13	2	no	—
22	C₂²×#3861	—	13	1	no	—
23	C₂×#63270	—	21	2	no	—
24	C₂×#63270	—	21	6	no	—
25	A₄×C₁₃⋊C₃	—	49	1	no	Hall
26	ΓL(1,𝔽₈)²	—	52	1	no	Hall
27	C₂²×C₇×C₁₃	—	3	2	no	—
28	C₂²×C₇×C₁₃	C₃×ΓL(1,𝔽₈)	1	2	yes	complemented
29	C₂×#3861	—	39	1	no	—
30	#63271	—	4	2	no	—
31	#63270	—	7	2	no	—
32	#63270	—	21	2	no	—
33	C₇×C₁₃⋊C₃	—	28	2	no	—
34	C₁₃×ΓL(1,𝔽₈)	—	28	2	no	—
35	A₄×ΓL(1,𝔽₈)	—	91	2	no	—
36	C₂²×C₇²	C₃×C₁₃⋊C₃	1	1	yes	characteristic, complemented, Hall
37	C₂×C₇×C₁₃	—	3	6	no	—
38	C₂×C₇×C₁₃	—	3	2	no	—
39	C₂²⋊(C₁₃⋊C₃)	—	7	2	no	—
40	A₄×C₁₃	—	49	1	no	—
41	C₂²×C₁₃⋊C₃	—	49	1	no	—
42	#3862	—	4	1	no	—
43	#3861	—	13	1	no	—
44	C₇×ΓL(1,𝔽₈)	—	52	2	no	—
45	C₃×C₁₃⋊C₃	—	196	1	no	Hall
46	C₂×C₇²	—	3	1	no	—
47	C₇×C₁₃	—	3	2	no	—
48	C₇×C₁₃	A₄×ΓL(1,𝔽₈)	1	2	yes	complemented
49	C₂²⋊ΓL(1,𝔽₈)	—	7	2	no	—
50	C₂²⋊ΓL(1,𝔽₈)	—	13	2	no	—
51	C₇×A₄	—	91	2	no	—
52	C₂²×ΓL(1,𝔽₈)	—	273	2	no	—
53	C₂²×ΓL(1,𝔽₈)	—	91	2	no	—
54	C₂×C₁₃⋊C₃	—	147	1	no	—
55	C₃×ΓL(1,𝔽₈)	—	364	2	no	—
56	C₂²×C₁₃	ΓL(1,𝔽₈)²	1	1	yes	characteristic, complemented, Hall
57	C₇²	A₄×C₁₃⋊C₃	1	1	yes	characteristic, complemented, Sylow
58	C₂×ΓL(1,𝔽₈)	—	273	6	no	—
59	C₂×ΓL(1,𝔽₈)	—	273	2	no	—
60	C₃×C₁₃	—	196	1	no	—
61	C₁₃⋊C₃	—	28	2	no	—
62	C₁₃⋊C₃	—	49	1	no	—
63	C₃×A₄	—	637	1	no	Hall
64	C₂²×C₇	—	3	2	no	—
65	C₂²×C₇	ΓL(1,𝔽₈)×C₁₃⋊C₃	1	2	yes	complemented
66	C₂×C₁₃	—	3	1	no	—
67	C₃×C₇	—	364	2	no	—
68	ΓL(1,𝔽₈)	—	273	2	no	—
69	ΓL(1,𝔽₈)	—	28	2	no	—
70	ΓL(1,𝔽₈)	—	52	2	no	—
71	ΓL(1,𝔽₈)	—	91	2	no	—
72	C₂×C₇	—	3	6	no	—
73	C₂×C₇	—	3	2	no	—
74	C₁₃	#3861⋊A₄	1	1	yes	characteristic, complemented, Sylow
75	C₂²×C₃	—	637	1	no	—
76	A₄	—	49	1	no	—
77	A₄	—	91	2	no	—
78	C₃²	—	2548	1	no	Sylow
79	C₇	—	3	2	no	—
80	C₇	#63270⋊A₄	1	2	yes	complemented
81	C₂×C₃	—	1911	1	no	—
82	C₂²	#19153212	1	1	yes	characteristic, complemented, Sylow
83	C₃	—	364	2	no	—
84	C₃	—	637	1	no	—
85	C₃	—	196	1	no	—
86	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×#14274263	C₃	semidirect
2. C₇×C₂²⋊#63271	C₃	semidirect
3. C₁₃×C₂²⋊#3862	C₃	semidirect
7. C₂²×C₇²×C₁₃	C₃²	semidirect
9. #14274263	A₄	semidirect
13. C₂²⋊#63271	ΓL(1,𝔽₈)	semidirect
19. C₇²×C₁₃	C₃×A₄	semidirect
20. C₂²⋊#3862	C₁₃⋊C₃	semidirect
28. C₂²×C₇×C₁₃	C₃×ΓL(1,𝔽₈)	semidirect
36. C₂²×C₇²	C₃×C₁₃⋊C₃	semidirect
48. C₇×C₁₃	A₄×ΓL(1,𝔽₈)	semidirect
56. C₂²×C₁₃	ΓL(1,𝔽₈)²	semidirect
57. C₇²	A₄×C₁₃⋊C₃	semidirect
65. C₂²×C₇	ΓL(1,𝔽₈)×C₁₃⋊C₃	semidirect
74. C₁₃	#3861⋊A₄	semidirect
80. C₇	#63270⋊A₄	semidirect
82. C₂²	#19153212	semidirect

Maximal subgroups

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

subgroup	index
C₂²×#14274263	3
C₇×C₂²⋊#63271	3
C₁₃×C₂²⋊#3862	3
#19153212	4
#63270⋊A₄	7
#3861⋊A₄	13