Subgroups — #37524364

Order 22932.

180 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×#19153200	C₂	1	1	yes	maximal, characteristic
2	C₃×C₇²×Dic₁₃	C₃	1	1	yes	maximal, characteristic
3	#19153200	C₄	1	1	yes	characteristic, complemented, Hall
4	C₂×C₃×C₇²×C₁₃	C₂×C₃	1	1	yes	characteristic
5	#15692565	—	7	8	no	maximal
6	C₇²×Dic₁₃	C₉	1	1	yes	characteristic, complemented, Hall
7	C₃×C₇²×C₁₃	C₃×C₄	1	1	yes	characteristic
8	C₄×#89204	—	13	1	no	maximal, Hall
9	C₂×#11717941	—	7	8	no	—
10	C₂×C₇²×C₁₃	C₂×C₉	1	1	yes	characteristic
11	C₃×C₇×Dic₁₃	ΓL(1,𝔽₈)	1	8	yes	—
12	C₂×#89204	—	13	1	no	—
13	#11717941	—	7	8	no	—
14	C₇²×C₁₃	C₄×C₉	1	1	yes	characteristic, complemented, Hall
15	C₃×C₄×C₇²	—	39	1	no	—
16	C₂×C₃×C₇×C₁₃	C₂×ΓL(1,𝔽₈)	1	8	yes	—
17	Dic₁₃⋊C₉	—	49	1	no	Hall
18	#89204	—	13	1	no	Hall
19	C₇×Dic₁₃	C₇⋊C₉	1	8	yes	complemented
20	C₂×C₃×C₇²	D₁₃⋊C₃	1	1	yes	characteristic
21	C₃×C₇×C₁₃	C₄×ΓL(1,𝔽₈)	1	8	yes	—
22	C₄×C₇⋊C₉	—	91	8	no	—
23	C₂×C₁₃⋊C₉	—	49	1	no	—
24	C₄×C₇²	—	39	1	no	Hall
25	C₂×C₇×C₁₃	C₂×C₇⋊C₉	1	8	yes	—
26	C₃×Dic₁₃	#3861	1	1	yes	characteristic
27	C₃×C₇²	Dic₁₃⋊C₃	1	1	yes	characteristic
28	C₂×C₇⋊C₉	—	91	8	no	—
29	C₁₃⋊C₉	—	49	1	no	Hall
30	C₂×C₇²	D₁₃⋊C₉	1	1	yes	characteristic
31	C₇×C₁₃	C₄×C₇⋊C₉	1	8	yes	complemented
32	C₃×C₄×C₇	—	39	8	no	—
33	C₂×C₃×C₁₃	C₂×#3861	1	1	yes	characteristic
34	C₇⋊C₉	—	91	8	no	—
35	Dic₁₃	#89204	1	1	yes	characteristic, complemented, Hall
36	C₇²	Dic₁₃⋊C₉	1	1	yes	characteristic, complemented, Sylow
37	C₂×C₃×C₇	#10587716	1	8	yes	—
38	C₃×C₁₃	C₄×#3861	1	1	yes	characteristic
39	C₄×C₉	—	637	1	no	Hall
40	C₄×C₇	—	39	8	no	—
41	C₂×C₁₃	C₂×#89204	1	1	yes	characteristic
42	C₃×C₇	#11763957	1	8	yes	—
43	C₂×C₉	—	637	1	no	—
44	C₂×C₇	#13109896	1	8	yes	—
45	C₁₃	C₄×#89204	1	1	yes	characteristic, complemented, Sylow
46	C₃×C₄	—	39	1	no	—
47	C₉	—	637	1	no	Sylow
48	C₇	#15692565	1	8	yes	complemented
49	C₂×C₃	#16838473	1	1	yes	characteristic
50	C₄	—	39	1	no	Sylow
51	C₃	#21439549	1	1	yes	characteristic
52	C₂	#26039471	1	1	yes	characteristic

	subgroup	G/H
center, Frattini, hypercenter	C₂×C₃	#16838473
commutator, hypocenter	C₇²×C₁₃	C₄×C₉
Fitting, socle, prime radical, cosocle	C₂×C₃×C₇²×C₁₃	C₂×C₃
LCS	G ▷ C₇²×C₁₃
UCS	C₁ ◁ C₂×C₃

Complements

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

N	H ≅ G/N	type
3. #19153200	C₄	semidirect
6. C₇²×Dic₁₃	C₉	semidirect
14. C₇²×C₁₃	C₄×C₉	semidirect
19. C₇×Dic₁₃	C₇⋊C₉	semidirect
31. C₇×C₁₃	C₄×C₇⋊C₉	semidirect
35. Dic₁₃	#89204	semidirect
36. C₇²	Dic₁₃⋊C₉	semidirect
45. C₁₃	C₄×#89204	semidirect
48. C₇	#15692565	semidirect

Maximal subgroups

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

subgroup	index
C₂×#19153200	2
C₃×C₇²×Dic₁₃	3
#15692565	7
C₄×#89204	13