Subgroups — #37524337

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×C₇⋊Dic₉₁	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	#15692571	—	7	1	no	maximal
6	#15692571	—	7	7	no	maximal
7	C₇⋊Dic₉₁	C₉	1	1	yes	characteristic, complemented, Hall
8	C₃×C₇²×C₁₃	C₃×C₄	1	1	yes	characteristic
9	#13183533	—	13	1	no	maximal, Hall
10	C₂×#11717941	—	7	1	no	—
11	C₂×#11717941	—	7	7	no	—
12	C₂×C₇²×C₁₃	C₂×C₉	1	1	yes	characteristic
13	C₃×Dic₉₁	—	7	1	no	—
14	C₃×Dic₉₁	—	21	7	no	—
15	C₂×#89204	—	13	1	no	—
16	#11717941	—	7	1	no	—
17	#11717941	—	7	7	no	—
18	C₇²×C₁₃	C₄×C₉	1	1	yes	characteristic, complemented, Hall
19	C₃×C₇⋊Dic₇	—	39	1	no	—
20	C₂×C₃×C₇×C₁₃	AGL(1,𝔽₇)	1	1	yes	characteristic
21	C₂×C₃×C₇×C₁₃	AGL(1,𝔽₇)	1	7	yes	—
22	Dic₁₃⋊C₉	—	49	1	no	Hall
23	#89204	—	13	1	no	Hall
24	Dic₉₁	—	7	1	no	—
25	Dic₉₁	—	21	7	no	—
26	C₂×C₃×C₇²	D₁₃⋊C₃	1	1	yes	characteristic
27	C₃×C₇×C₁₃	ΓL(1,𝔽₈)⋊C₄	1	1	yes	characteristic
28	C₃×C₇×C₁₃	ΓL(1,𝔽₈)⋊C₄	1	7	yes	—
29	Dic₇⋊C₉	—	91	1	no	—
30	Dic₇⋊C₉	—	91	7	no	—
31	C₂×C₁₃⋊C₉	—	49	1	no	—
32	C₇⋊Dic₇	—	39	1	no	Hall
33	C₂×C₇×C₁₃	D₇⋊C₉	1	1	yes	characteristic
34	C₂×C₇×C₁₃	D₇⋊C₉	1	7	yes	—
35	C₃×Dic₁₃	—	147	1	no	—
36	C₃×C₇²	Dic₁₃⋊C₃	1	1	yes	characteristic
37	C₂×C₇⋊C₉	—	91	1	no	—
38	C₂×C₇⋊C₉	—	91	7	no	—
39	C₁₃⋊C₉	—	49	1	no	Hall
40	C₂×C₇²	D₁₃⋊C₉	1	1	yes	characteristic
41	C₇×C₁₃	Dic₇⋊C₉	1	1	yes	characteristic, complemented
42	C₇×C₁₃	Dic₇⋊C₉	1	7	yes	complemented
43	C₃×Dic₇	—	273	1	no	—
44	C₃×Dic₇	—	273	7	no	—
45	C₂×C₃×C₁₃	#64424	1	1	yes	characteristic
46	C₇⋊C₉	—	91	1	no	—
47	C₇⋊C₉	—	91	7	no	—
48	Dic₁₃	—	147	1	no	Hall
49	C₇²	Dic₁₃⋊C₉	1	1	yes	characteristic, complemented, Sylow
50	C₂×C₃×C₇	#10587722	1	1	yes	characteristic
51	C₂×C₃×C₇	#10587722	1	7	yes	—
52	C₃×C₁₃	#10596874	1	1	yes	characteristic
53	C₄×C₉	—	637	1	no	Hall
54	Dic₇	—	273	1	no	—
55	Dic₇	—	273	7	no	—
56	C₂×C₁₃	#11725077	1	1	yes	characteristic
57	C₃×C₇	#11763963	1	1	yes	characteristic
58	C₃×C₇	#11763963	1	7	yes	—
59	C₂×C₉	—	637	1	no	—
60	C₂×C₇	#13109902	1	1	yes	characteristic
61	C₂×C₇	#13109902	1	7	yes	—
62	C₁₃	#13183533	1	1	yes	characteristic, complemented, Sylow
63	C₃×C₄	—	1911	1	no	—
64	C₉	—	637	1	no	Sylow
65	C₇	#15692571	1	1	yes	characteristic, complemented
66	C₇	#15692571	1	7	yes	complemented
67	C₂×C₃	#16838446	1	1	yes	characteristic
68	C₄	—	1911	1	no	Sylow
69	C₃	#21439522	1	1	yes	characteristic
70	C₂	#26039444	1	1	yes	characteristic

	subgroup	G/H
center, Frattini, hypercenter	C₂×C₃	#16838446
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
7. C₇⋊Dic₉₁	C₉	semidirect
18. C₇²×C₁₃	C₄×C₉	semidirect
41. C₇×C₁₃	Dic₇⋊C₉	semidirect
42. C₇×C₁₃	Dic₇⋊C₉	semidirect
49. C₇²	Dic₁₃⋊C₉	semidirect
62. C₁₃	#13183533	semidirect
65. C₇	#15692571	semidirect
66. C₇	#15692571	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×C₇⋊Dic₉₁	3
#15692571	7
#13183533	13