Subgroups — C₇⋊D₇×D₁₅₃ #44472603

Order 29988.

300 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₇⋊D₁₀₇₁	C₂	1	1	yes	maximal, characteristic, complemented
2	C₇²×D₁₅₃	C₂	1	1	yes	maximal, characteristic, complemented
3	C₉×C₁₇×C₇⋊D₇	C₂	1	1	yes	maximal, characteristic, complemented
4	C₇⋊D₇×D₅₁	—	3	1	no	maximal
5	C₇²×C₉×C₁₇	C₂²	1	1	yes	characteristic, complemented, Hall
6	C₃⋊(C₇⋊D₁₁₉)	—	3	1	no	—
7	C₇²×D₅₁	—	3	1	no	—
8	C₃×C₁₇×C₇⋊D₇	D₃	1	1	yes	characteristic
9	D₇×D₁₅₃	—	7	8	no	maximal
10	D₁₇×C₇⋊D₇	—	9	1	no	Hall
11	C₃×C₇²×C₁₇	C₂×D₃	1	1	yes	characteristic
12	D₁₀₇₁	—	7	8	no	—
13	C₇×D₁₅₃	D₇	1	8	yes	complemented
14	C₉×D₇×C₁₇	—	7	8	no	—
15	D₉×C₇⋊D₇	—	17	1	no	maximal, Hall
16	C₇⋊D₁₁₉	—	9	1	no	—
17	C₁₇×C₇⋊D₇	D₉	1	1	yes	characteristic, complemented
18	C₇²×D₁₇	—	9	1	no	—
19	D₇×D₅₁	—	21	8	no	—
20	C₇×C₉×C₁₇	C₂×D₇	1	8	yes	complemented
21	C₇⋊D₆₃	—	17	1	no	—
22	C₇²×D₉	—	17	1	no	—
23	C₉×C₇⋊D₇	D₁₇	1	1	yes	characteristic, complemented
24	C₇²×C₁₇	C₂×D₉	1	1	yes	characteristic, complemented, Hall
25	D₃₅₇	—	21	8	no	—
26	C₇×D₅₁	—	3	8	no	—
27	C₃×D₇×C₁₇	—	7	8	no	—
28	C₂×D₁₅₃	—	49	1	no	Hall
29	D₃×C₇⋊D₇	—	51	1	no	—
30	D₇×D₁₇	—	63	8	no	—
31	C₇²×C₉	C₂×D₁₇	1	1	yes	characteristic, complemented, Hall
32	C₃×C₇×C₁₇	D₃×D₇	1	8	yes	—
33	C₂×C₉×C₁₇	—	49	1	no	—
34	D₁₅₃	—	49	1	no	—
35	D₁₅₃	C₇⋊D₇	1	1	yes	characteristic, direct product factor
36	C₃⋊(C₇⋊D₇)	—	51	1	no	—
37	D₃×C₇²	—	51	1	no	—
38	C₃×C₇⋊D₇	D₅₁	1	1	yes	characteristic
39	D₇×D₉	—	119	8	no	—
40	D₁₁₉	—	63	8	no	—
41	C₇×D₁₇	—	9	8	no	—
42	D₇×C₁₇	—	7	8	no	—
43	C₂×D₅₁	—	147	1	no	—
44	C₂×C₇⋊D₇	—	153	1	no	Hall
45	C₉×C₁₇	C₂×C₇⋊D₇	1	1	yes	characteristic, complemented, Hall
46	C₃×C₇²	C₂×D₅₁	1	1	yes	characteristic
47	D₆₃	—	119	8	no	—
48	C₉×D₇	—	7	8	no	—
49	C₇×D₉	—	17	8	no	—
50	C₇×C₁₇	D₇×D₉	1	8	yes	complemented
51	C₂×C₃×C₁₇	—	49	1	no	—
52	D₅₁	—	3	1	no	—
53	D₅₁	—	147	1	no	—
54	C₂×C₇²	—	153	1	no	—
55	C₇⋊D₇	—	153	1	no	—
56	C₇⋊D₇	D₁₅₃	1	1	yes	characteristic, direct product factor
57	D₃×D₇	—	357	8	no	—
58	C₂×D₁₇	—	441	1	no	Hall
59	C₇×C₉	D₇×D₁₇	1	8	yes	complemented
60	C₃×C₁₇	D₃×C₇⋊D₇	1	1	yes	characteristic
61	C₇²	C₂×D₁₅₃	1	1	yes	characteristic, complemented, Sylow
62	D₂₁	—	357	8	no	—
63	C₃×D₇	—	7	8	no	—
64	D₃×C₇	—	51	8	no	—
65	C₂×D₉	—	833	1	no	Hall
66	C₂×C₁₇	—	49	1	no	—
67	D₁₇	—	9	1	no	—
68	D₁₇	—	441	1	no	—
69	C₂×D₇	—	1071	8	no	—
70	C₃×C₇	D₇×D₅₁	1	8	yes	—
71	C₂×C₉	—	49	1	no	—
72	D₉	—	17	1	no	—
73	D₉	—	833	1	no	—
74	C₁₇	D₉×C₇⋊D₇	1	1	yes	characteristic, complemented, Sylow
75	C₂×C₇	—	153	8	no	—
76	D₇	—	7	8	no	—
77	D₇	—	1071	8	no	—
78	C₂×D₃	—	2499	1	no	—
79	C₉	D₁₇×C₇⋊D₇	1	1	yes	characteristic, complemented, Sylow
80	C₇	D₇×D₁₅₃	1	8	yes	complemented
81	C₂×C₃	—	49	1	no	—
82	D₃	—	51	1	no	—
83	D₃	—	2499	1	no	—
84	C₂²	—	7497	1	no	Sylow
85	C₃	C₇⋊D₇×D₅₁	1	1	yes	characteristic
86	C₂	—	49	1	no	—
87	C₂	—	7497	1	no	—
88	C₂	—	153	1	no	—

	subgroup	G/H
center, hypercenter	C₁	G
commutator, Fitting, cosocle, hypocenter	C₇²×C₉×C₁₇	C₂²
Frattini	C₃	C₇⋊D₇×D₅₁
socle	C₃×C₇²×C₁₇	C₂×D₃
LCS	G ▷ C₇²×C₉×C₁₇
UCS	C₁

Complements

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

N	H ≅ G/N	type
1. C₇⋊D₁₀₇₁	C₂	semidirect
2. C₇²×D₁₅₃	C₂	semidirect
3. C₉×C₁₇×C₇⋊D₇	C₂	semidirect
5. C₇²×C₉×C₁₇	C₂²	semidirect
13. C₇×D₁₅₃	D₇	semidirect
17. C₁₇×C₇⋊D₇	D₉	semidirect
20. C₇×C₉×C₁₇	C₂×D₇	semidirect
23. C₉×C₇⋊D₇	D₁₇	semidirect
24. C₇²×C₁₇	C₂×D₉	semidirect
31. C₇²×C₉	C₂×D₁₇	semidirect
35. D₁₅₃	C₇⋊D₇	direct (N × H)
45. C₉×C₁₇	C₂×C₇⋊D₇	semidirect
50. C₇×C₁₇	D₇×D₉	semidirect
56. C₇⋊D₇	D₁₅₃	direct (N × H)
59. C₇×C₉	D₇×D₁₇	semidirect
61. C₇²	C₂×D₁₅₃	semidirect
74. C₁₇	D₉×C₇⋊D₇	semidirect
79. C₉	D₁₇×C₇⋊D₇	semidirect
80. C₇	D₇×D₁₅₃	semidirect

Maximal subgroups

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

subgroup	index
C₇⋊D₁₀₇₁	2
C₇²×D₁₅₃	2
C₉×C₁₇×C₇⋊D₇	2
C₇⋊D₇×D₅₁	3
D₇×D₁₅₃	7
D₉×C₇⋊D₇	17

Subgroups — C7⋊D7×D153 #44472603

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₇⋊D₇×D₁₅₃ #44472603