Subgroups — C₇⋊D₇×D₁₁₇ #37524581

Order 22932.

300 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₇⋊D₈₁₉	C₂	1	1	yes	maximal, characteristic, complemented
2	C₇²×D₁₁₇	C₂	1	1	yes	maximal, characteristic, complemented
3	C₉×C₁₃×C₇⋊D₇	C₂	1	1	yes	maximal, characteristic, complemented
4	D₃₉×C₇⋊D₇	—	3	1	no	maximal
5	C₇²×C₉×C₁₃	C₂²	1	1	yes	characteristic, complemented, Hall
6	C₃⋊(C₇⋊D₉₁)	—	3	1	no	—
7	C₇²×D₃₉	—	3	1	no	—
8	C₃×C₁₃×C₇⋊D₇	D₃	1	1	yes	characteristic
9	D₇×D₁₁₇	—	7	8	no	maximal
10	D₁₃×C₇⋊D₇	—	9	1	no	Hall
11	C₃×C₇²×C₁₃	C₂×D₃	1	1	yes	characteristic
12	D₉×C₇⋊D₇	—	13	1	no	maximal, Hall
13	D₈₁₉	—	7	8	no	—
14	C₇×D₁₁₇	D₇	1	8	yes	complemented
15	C₉×C₁₃×D₇	—	7	8	no	—
16	C₇⋊D₉₁	—	9	1	no	—
17	C₁₃×C₇⋊D₇	D₉	1	1	yes	characteristic, complemented
18	C₇²×D₁₃	—	9	1	no	—
19	D₇×D₃₉	—	21	8	no	—
20	C₇⋊D₆₃	—	13	1	no	—
21	C₇²×D₉	—	13	1	no	—
22	C₉×C₇⋊D₇	D₁₃	1	1	yes	characteristic, complemented
23	C₇×C₉×C₁₃	C₂×D₇	1	8	yes	complemented
24	C₇²×C₁₃	C₂×D₉	1	1	yes	characteristic, complemented, Hall
25	D₃×C₇⋊D₇	—	39	1	no	—
26	D₂₇₃	—	21	8	no	—
27	C₇×D₃₉	—	3	8	no	—
28	C₃×C₁₃×D₇	—	7	8	no	—
29	C₂×D₁₁₇	—	49	1	no	Hall
30	C₇²×C₉	C₂×D₁₃	1	1	yes	characteristic, complemented, Hall
31	D₇×D₁₃	—	63	8	no	—
32	C₃⋊(C₇⋊D₇)	—	39	1	no	—
33	D₃×C₇²	—	39	1	no	—
34	C₃×C₇⋊D₇	D₃₉	1	1	yes	characteristic
35	C₃×C₇×C₁₃	D₃×D₇	1	8	yes	—
36	D₇×D₉	—	91	8	no	—
37	C₂×C₉×C₁₃	—	49	1	no	—
38	D₁₁₇	—	49	1	no	—
39	D₁₁₇	C₇⋊D₇	1	1	yes	characteristic, direct product factor
40	C₂×C₇⋊D₇	—	117	1	no	Hall
41	D₉₁	—	63	8	no	—
42	C₇×D₁₃	—	9	8	no	—
43	C₁₃×D₇	—	7	8	no	—
44	C₂×D₃₉	—	147	1	no	—
45	C₃×C₇²	C₂×D₃₉	1	1	yes	characteristic
46	D₆₃	—	91	8	no	—
47	C₉×D₇	—	7	8	no	—
48	C₇×D₉	—	13	8	no	—
49	C₉×C₁₃	C₂×C₇⋊D₇	1	1	yes	characteristic, complemented, Hall
50	C₂×C₇²	—	117	1	no	—
51	C₇⋊D₇	—	117	1	no	—
52	C₇⋊D₇	D₁₁₇	1	1	yes	characteristic, direct product factor
53	C₇×C₁₃	D₇×D₉	1	8	yes	complemented
54	D₃×D₇	—	273	8	no	—
55	C₂×C₃×C₁₃	—	49	1	no	—
56	D₃₉	—	3	1	no	—
57	D₃₉	—	147	1	no	—
58	C₇×C₉	D₇×D₁₃	1	8	yes	complemented
59	C₂×D₁₃	—	441	1	no	Hall
60	C₇²	C₂×D₁₁₇	1	1	yes	characteristic, complemented, Sylow
61	D₂₁	—	273	8	no	—
62	C₃×D₇	—	7	8	no	—
63	D₃×C₇	—	39	8	no	—
64	C₃×C₁₃	D₃×C₇⋊D₇	1	1	yes	characteristic
65	C₂×D₉	—	637	1	no	Hall
66	C₂×D₇	—	819	8	no	—
67	C₂×C₁₃	—	49	1	no	—
68	D₁₃	—	9	1	no	—
69	D₁₃	—	441	1	no	—
70	C₃×C₇	D₇×D₃₉	1	8	yes	—
71	C₂×C₉	—	49	1	no	—
72	D₉	—	13	1	no	—
73	D₉	—	637	1	no	—
74	C₂×C₇	—	117	8	no	—
75	D₇	—	7	8	no	—
76	D₇	—	819	8	no	—
77	C₁₃	D₉×C₇⋊D₇	1	1	yes	characteristic, complemented, Sylow
78	C₂×D₃	—	1911	1	no	—
79	C₉	D₁₃×C₇⋊D₇	1	1	yes	characteristic, complemented, Sylow
80	C₇	D₇×D₁₁₇	1	8	yes	complemented
81	C₂×C₃	—	49	1	no	—
82	D₃	—	39	1	no	—
83	D₃	—	1911	1	no	—
84	C₂²	—	5733	1	no	Sylow
85	C₃	D₃₉×C₇⋊D₇	1	1	yes	characteristic
86	C₂	—	49	1	no	—
87	C₂	—	5733	1	no	—
88	C₂	—	117	1	no	—

	subgroup	G/H
center, hypercenter	C₁	G
commutator, Fitting, cosocle, hypocenter	C₇²×C₉×C₁₃	C₂²
Frattini	C₃	D₃₉×C₇⋊D₇
socle	C₃×C₇²×C₁₃	C₂×D₃
LCS	G ▷ C₇²×C₉×C₁₃
UCS	C₁

Complements

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

N	H ≅ G/N	type
1. C₇⋊D₈₁₉	C₂	semidirect
2. C₇²×D₁₁₇	C₂	semidirect
3. C₉×C₁₃×C₇⋊D₇	C₂	semidirect
5. C₇²×C₉×C₁₃	C₂²	semidirect
14. C₇×D₁₁₇	D₇	semidirect
17. C₁₃×C₇⋊D₇	D₉	semidirect
22. C₉×C₇⋊D₇	D₁₃	semidirect
23. C₇×C₉×C₁₃	C₂×D₇	semidirect
24. C₇²×C₁₃	C₂×D₉	semidirect
30. C₇²×C₉	C₂×D₁₃	semidirect
39. D₁₁₇	C₇⋊D₇	direct (N × H)
49. C₉×C₁₃	C₂×C₇⋊D₇	semidirect
52. C₇⋊D₇	D₁₁₇	direct (N × H)
53. C₇×C₁₃	D₇×D₉	semidirect
58. C₇×C₉	D₇×D₁₃	semidirect
60. C₇²	C₂×D₁₁₇	semidirect
77. C₁₃	D₉×C₇⋊D₇	semidirect
79. C₉	D₁₃×C₇⋊D₇	semidirect
80. C₇	D₇×D₁₁₇	semidirect

Maximal subgroups

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

subgroup	index
C₇⋊D₈₁₉	2
C₇²×D₁₁₇	2
C₉×C₁₃×C₇⋊D₇	2
D₃₉×C₇⋊D₇	3
D₇×D₁₁₇	7
D₉×C₇⋊D₇	13

Subgroups — C7⋊D7×D117 #37524581

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₇⋊D₇×D₁₁₇ #37524581