Group #37524459 — subgroups

Subgroups — D₇⋊C₉×D₉₁ #37524459

Order 22932.

128 conjugacy classes of subgroups; only the 43 normal subgroups are shown.

normal subgroups only

This group has 128 conjugacy classes of subgroups — too many to draw. Only its 43 normal subgroups are shown.

Dotted edges: containments with no intermediate normal subgroup — omitted non-normal subgroups may lie between.

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₇⋊C₉×D₉₁	C₂	1	1	yes	maximal, characteristic, complemented
2	(C₇⋊C₉)⋊D₉₁	C₂	1	1	yes	maximal, characteristic, complemented
3	C₇×C₁₃×D₇⋊C₉	C₂	1	1	yes	maximal, characteristic, complemented
4	C₃×D₇×D₉₁	C₃	1	1	yes	maximal, characteristic
5	C₇×C₁₃×C₇⋊C₉	C₂²	1	1	yes	characteristic, complemented, Hall
6	C₃×C₇⋊D₉₁	C₂×C₃	1	1	yes	characteristic
7	C₃×C₇×D₉₁	C₂×C₃	1	1	yes	characteristic
8	C₃×C₇×C₁₃×D₇	C₂×C₃	1	1	yes	characteristic
9	C₂×C₉×D₉₁	—	7	1	no	maximal
10	D₁₃×D₇⋊C₉	—	7	1	no	maximal
11	D₇×D₉₁	C₉	1	1	yes	characteristic, complemented, Hall
12	C₃×C₇²×C₁₃	C₂²×C₃	1	1	yes	characteristic
13	D₇×D₇⋊C₉	—	13	1	no	maximal, Hall
14	C₂×C₇×C₉×C₁₃	—	7	1	no	—
15	C₉×D₉₁	—	7	1	no	—
16	C₉×D₉₁	—	7	1	no	—
17	(C₇⋊C₉)⋊D₁₃	—	7	1	no	—
18	C₁₃×D₇⋊C₉	D₇	1	1	yes	characteristic, complemented
19	D₁₃×C₇⋊C₉	—	7	1	no	—
20	C₇⋊D₉₁	C₂×C₉	1	1	yes	characteristic, complemented
21	C₇×D₉₁	C₂×C₉	1	1	yes	characteristic, complemented
22	C₇×C₁₃×D₇	C₂×C₉	1	1	yes	characteristic, complemented
23	C₂×C₃×D₉₁	—	21	1	no	—
24	C₃×D₇×D₁₃	—	7	1	no	—
25	(C₇⋊C₉)⋊D₇	—	13	1	no	—
26	D₇×C₇⋊C₉	—	13	1	no	—
27	C₇×D₇⋊C₉	D₁₃	1	1	yes	characteristic, complemented
28	C₇×C₉×C₁₃	—	7	1	no	—
29	C₁₃×C₇⋊C₉	C₂×D₇	1	1	yes	characteristic, complemented
30	C₇²×C₁₃	C₂²×C₉	1	1	yes	characteristic, complemented, Hall
31	C₃×D₇²	—	13	1	no	—
32	C₂×C₃×C₇×C₁₃	—	7	1	no	—
33	C₃×D₉₁	—	7	1	no	—
34	C₃×D₉₁	—	21	1	no	—
35	C₃×D₉₁	—	42	1	no	—
36	C₃×D₉₁	AGL(1,𝔽₇)	1	1	yes	characteristic
37	C₃×C₁₃×D₇	C₃×D₇	1	1	yes	characteristic
38	C₃×C₇×D₁₃	—	7	1	no	—
39	C₂×C₉×D₁₃	—	49	1	no	Hall
40	C₇×C₇⋊C₉	C₂×D₁₃	1	1	yes	characteristic, complemented, Hall
41	C₂×D₉₁	—	21	1	no	—
42	D₇×D₁₃	—	7	1	no	—
43	C₃×C₇⋊D₇	—	13	1	no	—
44	C₃×C₇×D₇	—	13	1	no	—
45	C₃×C₇×D₇	C₃×D₁₃	1	1	yes	characteristic
46	C₃×C₇×C₁₃	—	6	1	no	—
47	C₃×C₇×C₁₃	C₂×AGL(1,𝔽₇)	1	1	yes	characteristic
48	C₃×C₇×C₁₃	C₂×C₃×D₇	1	1	yes	characteristic
49	C₂×C₉×D₇	—	91	1	no	—
50	C₂×D₇⋊C₉	—	91	1	no	—
51	C₂×C₉×C₁₃	—	7	1	no	—
52	C₉×D₁₃	—	49	1	no	—
53	C₉×D₁₃	—	49	1	no	—
54	D₇²	—	13	1	no	Hall
55	C₂×C₇×C₁₃	—	7	1	no	—
56	D₉₁	—	7	1	no	—
57	D₉₁	—	21	1	no	—
58	D₉₁	—	42	1	no	—
59	D₉₁	D₇⋊C₉	1	1	yes	characteristic, direct product factor
60	C₇×D₁₃	—	7	1	no	—
61	C₁₃×D₇	C₉×D₇	1	1	yes	characteristic, complemented
62	C₂×C₃×D₁₃	—	147	1	no	—
63	C₃×C₇²	C₂×C₃×D₁₃	1	1	yes	characteristic
64	C₂×C₇×C₉	—	7	1	no	—
65	C₉×D₇	—	91	1	no	—
66	C₉×D₇	—	91	1	no	—
67	C₂×C₇⋊C₉	—	91	1	no	—
68	D₇⋊C₉	—	91	1	no	—
69	D₇⋊C₉	D₉₁	1	1	yes	characteristic, direct product factor
70	C₉×C₁₃	—	7	1	no	Hall
71	C₇⋊D₇	—	13	1	no	—
72	C₇×D₇	—	13	1	no	—
73	C₇×D₇	C₉×D₁₃	1	1	yes	characteristic, complemented
74	C₇×C₁₃	—	6	1	no	—
75	C₇×C₁₃	C₂×D₇⋊C₉	1	1	yes	characteristic, complemented
76	C₇×C₁₃	C₂×C₉×D₇	1	1	yes	characteristic, complemented
77	C₂×C₃×D₇	—	91	1	no	—
78	C₂×C₃×D₇	—	91	1	no	—
79	C₂×C₃×C₁₃	—	21	1	no	—
80	C₃×D₁₃	—	7	1	no	—
81	C₃×D₁₃	—	147	1	no	—
82	C₇×C₉	—	7	1	no	—
83	C₇⋊C₉	C₂×D₉₁	1	1	yes	characteristic, complemented
84	C₂×D₁₃	—	147	1	no	Hall
85	C₇²	C₂×C₉×D₁₃	1	1	yes	characteristic, complemented, Sylow
86	C₂×C₃×C₇	—	21	1	no	—
87	C₂×C₃×C₇	—	91	1	no	—
88	C₃×D₇	—	13	1	no	—
89	C₃×D₇	—	273	1	no	—
90	C₃×D₇	—	546	1	no	—
91	C₃×D₇	—	91	1	no	—
92	C₃×D₇	C₃×D₉₁	1	1	yes	characteristic
93	C₃×C₁₃	D₇×AGL(1,𝔽₇)	1	1	yes	characteristic
94	C₂²×C₉	—	637	1	no	Hall
95	C₂×D₇	—	91	1	no	—
96	C₂×D₇	—	91	1	no	—
97	C₂×C₁₃	—	21	1	no	—
98	D₁₃	—	7	1	no	—
99	D₁₃	—	147	1	no	—
100	C₃×C₇	—	6	1	no	—
101	C₃×C₇	D₁₃×AGL(1,𝔽₇)	1	1	yes	characteristic
102	C₃×C₇	C₂×C₃×D₉₁	1	1	yes	characteristic
103	C₂×C₉	—	7	1	no	—
104	C₂×C₉	—	637	1	no	—
105	C₂×C₉	—	637	1	no	—
106	C₂×C₇	—	21	1	no	—
107	C₂×C₇	—	91	1	no	—
108	D₇	—	13	1	no	—
109	D₇	—	273	1	no	—
110	D₇	—	546	1	no	—
111	D₇	—	91	1	no	—
112	D₇	C₉×D₉₁	1	1	yes	characteristic, complemented
113	C₁₃	D₇×D₇⋊C₉	1	1	yes	characteristic, complemented, Sylow
114	C₂²×C₃	—	1911	1	no	—
115	C₉	—	7	1	no	Sylow
116	C₇	—	6	1	no	—
117	C₇	D₁₃×D₇⋊C₉	1	1	yes	characteristic, complemented
118	C₇	C₂×C₉×D₉₁	1	1	yes	characteristic, complemented
119	C₂×C₃	—	21	1	no	—
120	C₂×C₃	—	91	1	no	—
121	C₂×C₃	—	637	1	no	—
122	C₂²	—	1911	1	no	Sylow
123	C₃	AGL(1,𝔽₇)×D₉₁	1	1	yes	characteristic
124	C₂	—	21	1	no	—
125	C₂	—	91	1	no	—
126	C₂	—	637	1	no	—

subgroup

G/H

class size

multiplicity

normal?

properties

C₇⋊C₉×D₉₁

C₂

yes

maximal, characteristic, complemented

(C₇⋊C₉)⋊D₉₁

C₂

yes

maximal, characteristic, complemented

C₇×C₁₃×D₇⋊C₉

C₂

yes

maximal, characteristic, complemented

C₃×D₇×D₉₁

C₃

yes

maximal, characteristic

C₇×C₁₃×C₇⋊C₉

C₂²

yes

characteristic, complemented, Hall

C₃×C₇⋊D₉₁

C₂×C₃

yes

characteristic

C₃×C₇×D₉₁

C₂×C₃

yes

characteristic

C₃×C₇×C₁₃×D₇

C₂×C₃

yes

characteristic

C₂×C₉×D₉₁

—

maximal

D₁₃×D₇⋊C₉

—

maximal

D₇×D₉₁

C₉

yes

characteristic, complemented, Hall

C₃×C₇²×C₁₃

C₂²×C₃

yes

characteristic

D₇×D₇⋊C₉

—

maximal, Hall

C₂×C₇×C₉×C₁₃

—

C₉×D₉₁

—

C₉×D₉₁

—

(C₇⋊C₉)⋊D₁₃

—

C₁₃×D₇⋊C₉

D₇

yes

characteristic, complemented

D₁₃×C₇⋊C₉

—

C₇⋊D₉₁

C₂×C₉

yes

characteristic, complemented

C₇×D₉₁

C₂×C₉

yes

characteristic, complemented

C₇×C₁₃×D₇

C₂×C₉

yes

characteristic, complemented

C₂×C₃×D₉₁

—

C₃×D₇×D₁₃

—

(C₇⋊C₉)⋊D₇

—

D₇×C₇⋊C₉

—

C₇×D₇⋊C₉

D₁₃

yes

characteristic, complemented

C₇×C₉×C₁₃

—

C₁₃×C₇⋊C₉

C₂×D₇

yes

characteristic, complemented

C₇²×C₁₃

C₂²×C₉

yes

characteristic, complemented, Hall

C₃×D₇²

—

C₂×C₃×C₇×C₁₃

—

C₃×D₉₁

—

C₃×D₉₁

—

C₃×D₉₁

—

C₃×D₉₁

AGL(1,𝔽₇)

yes

characteristic

C₃×C₁₃×D₇

C₃×D₇

yes

characteristic

C₃×C₇×D₁₃

—

C₂×C₉×D₁₃

—

Hall

C₇×C₇⋊C₉

C₂×D₁₃

yes

characteristic, complemented, Hall

C₂×D₉₁

—

D₇×D₁₃

—

C₃×C₇⋊D₇

—

C₃×C₇×D₇

—

C₃×C₇×D₇

C₃×D₁₃

yes

characteristic

C₃×C₇×C₁₃

—

C₃×C₇×C₁₃

C₂×AGL(1,𝔽₇)

yes

characteristic

C₃×C₇×C₁₃

C₂×C₃×D₇

yes

characteristic

C₂×C₉×D₇

—

C₂×D₇⋊C₉

—

C₂×C₉×C₁₃

—

C₉×D₁₃

—

C₉×D₁₃

—

D₇²

—

Hall

C₂×C₇×C₁₃

—

D₉₁

—

D₉₁

—

D₉₁

—

D₉₁

D₇⋊C₉

yes

characteristic, direct product factor

C₇×D₁₃

—

C₁₃×D₇

C₉×D₇

yes

characteristic, complemented

C₂×C₃×D₁₃

—

147

—

C₃×C₇²

C₂×C₃×D₁₃

yes

characteristic

C₂×C₇×C₉

—

C₉×D₇

—

C₉×D₇

—

C₂×C₇⋊C₉

—

D₇⋊C₉

—

D₇⋊C₉

D₉₁

yes

characteristic, direct product factor

C₉×C₁₃

—

Hall

C₇⋊D₇

—

C₇×D₇

—

C₇×D₇

C₉×D₁₃

yes

characteristic, complemented

C₇×C₁₃

—

C₇×C₁₃

C₂×D₇⋊C₉

yes

characteristic, complemented

C₇×C₁₃

C₂×C₉×D₇

yes

characteristic, complemented

C₂×C₃×D₇

—

C₂×C₃×D₇

—

C₂×C₃×C₁₃

—

C₃×D₁₃

—

C₃×D₁₃

—

147

—

C₇×C₉

—

C₇⋊C₉

C₂×D₉₁

yes

characteristic, complemented

C₂×D₁₃

—

147

Hall

C₇²

C₂×C₉×D₁₃

yes

characteristic, complemented, Sylow

C₂×C₃×C₇

—

C₂×C₃×C₇

—

C₃×D₇

—

C₃×D₇

—

273

—

C₃×D₇

—

546

—

C₃×D₇

—

C₃×D₇

C₃×D₉₁

yes

characteristic

C₃×C₁₃

D₇×AGL(1,𝔽₇)

yes

characteristic

C₂²×C₉

—

637

Hall

C₂×D₇

—

C₂×D₇

—

C₂×C₁₃

—

D₁₃

—

D₁₃

—

147

—

100

C₃×C₇

—

101

C₃×C₇

D₁₃×AGL(1,𝔽₇)

yes

characteristic

102

C₃×C₇

C₂×C₃×D₉₁

yes

characteristic

103

C₂×C₉

—

104

C₂×C₉

—

637

—

105

C₂×C₉

—

637

—

106

C₂×C₇

—

107

C₂×C₇

—

108

D₇

—

109

D₇

—

273

—

110

D₇

—

546

—

111

D₇

—

112

D₇

C₉×D₉₁

yes

characteristic, complemented

113

C₁₃

D₇×D₇⋊C₉

yes

characteristic, complemented, Sylow

114

C₂²×C₃

—

1911

—

115

C₉

—

Sylow

116

C₇

—

117

C₇

D₁₃×D₇⋊C₉

yes

characteristic, complemented

118

C₇

C₂×C₉×D₉₁

yes

characteristic, complemented

119

C₂×C₃

—

120

C₂×C₃

—

121

C₂×C₃

—

637

—

122

C₂²

—

1911

Sylow

123

C₃

AGL(1,𝔽₇)×D₉₁

yes

characteristic

124

C₂

—

125

C₂

—

126

C₂

—

637

—

	subgroup	G/H
center, Frattini, hypercenter	C₃	AGL(1,𝔽₇)×D₉₁
commutator, hypocenter	C₇²×C₁₃	C₂²×C₉
Fitting, socle, cosocle	C₃×C₇²×C₁₃	C₂²×C₃
prime radical	C₃×D₇×D₉₁	C₃
LCS	G ▷ C₇²×C₁₃
UCS	C₁ ◁ C₃

subgroup

G/H

center, Frattini, hypercenter

C₃

AGL(1,𝔽₇)×D₉₁

commutator, hypocenter

C₇²×C₁₃

C₂²×C₉

Fitting, socle, cosocle

C₃×C₇²×C₁₃

C₂²×C₃

prime radical

C₃×D₇×D₉₁

C₃

LCS

G ▷ C₇²×C₁₃

UCS

C₁ ◁ C₃

N	H ≅ G/N	type
1. C₇⋊C₉×D₉₁	C₂	semidirect
2. (C₇⋊C₉)⋊D₉₁	C₂	semidirect
3. C₇×C₁₃×D₇⋊C₉	C₂	semidirect
5. C₇×C₁₃×C₇⋊C₉	C₂²	semidirect
11. D₇×D₉₁	C₉	semidirect
18. C₁₃×D₇⋊C₉	D₇	semidirect
20. C₇⋊D₉₁	C₂×C₉	semidirect
21. C₇×D₉₁	C₂×C₉	semidirect
22. C₇×C₁₃×D₇	C₂×C₉	semidirect
27. C₇×D₇⋊C₉	D₁₃	semidirect
29. C₁₃×C₇⋊C₉	C₂×D₇	semidirect
30. C₇²×C₁₃	C₂²×C₉	semidirect
40. C₇×C₇⋊C₉	C₂×D₁₃	semidirect
59. D₉₁	D₇⋊C₉	direct (N × H)
61. C₁₃×D₇	C₉×D₇	semidirect
69. D₇⋊C₉	D₉₁	direct (N × H)
73. C₇×D₇	C₉×D₁₃	semidirect
75. C₇×C₁₃	C₂×D₇⋊C₉	semidirect
76. C₇×C₁₃	C₂×C₉×D₇	semidirect
83. C₇⋊C₉	C₂×D₉₁	semidirect
85. C₇²	C₂×C₉×D₁₃	semidirect
112. D₇	C₉×D₉₁	semidirect
113. C₁₃	D₇×D₇⋊C₉	semidirect
117. C₇	D₁₃×D₇⋊C₉	semidirect
118. C₇	C₂×C₉×D₉₁	semidirect

H ≅ G/N

type

1. C₇⋊C₉×D₉₁

C₂

semidirect

2. (C₇⋊C₉)⋊D₉₁

C₂

semidirect

3. C₇×C₁₃×D₇⋊C₉

C₂

semidirect

5. C₇×C₁₃×C₇⋊C₉

C₂²

semidirect

11. D₇×D₉₁

C₉

semidirect

18. C₁₃×D₇⋊C₉

D₇

semidirect

20. C₇⋊D₉₁

C₂×C₉

semidirect

21. C₇×D₉₁

C₂×C₉

semidirect

22. C₇×C₁₃×D₇

C₂×C₉

semidirect

27. C₇×D₇⋊C₉

D₁₃

semidirect

29. C₁₃×C₇⋊C₉

C₂×D₇

semidirect

30. C₇²×C₁₃

C₂²×C₉

semidirect

40. C₇×C₇⋊C₉

C₂×D₁₃

semidirect

59. D₉₁

D₇⋊C₉

direct (N × H)

61. C₁₃×D₇

C₉×D₇

semidirect

69. D₇⋊C₉

D₉₁

direct (N × H)

73. C₇×D₇

C₉×D₁₃

semidirect

75. C₇×C₁₃

C₂×D₇⋊C₉

semidirect

76. C₇×C₁₃

C₂×C₉×D₇

semidirect

83. C₇⋊C₉

C₂×D₉₁

semidirect

85. C₇²

C₂×C₉×D₁₃

semidirect

112. D₇

C₉×D₉₁

semidirect

113. C₁₃

D₇×D₇⋊C₉

semidirect

117. C₇

D₁₃×D₇⋊C₉

semidirect

118. C₇

C₂×C₉×D₉₁

semidirect

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

subgroup	index
C₇⋊C₉×D₉₁	2
(C₇⋊C₉)⋊D₉₁	2
C₇×C₁₃×D₇⋊C₉	2
C₃×D₇×D₉₁	3
C₂×C₉×D₉₁	7
D₁₃×D₇⋊C₉	7
D₇×D₇⋊C₉	13

subgroup

index

C₇⋊C₉×D₉₁

(C₇⋊C₉)⋊D₉₁

C₇×C₁₃×D₇⋊C₉

C₃×D₇×D₉₁

C₂×C₉×D₉₁

D₁₃×D₇⋊C₉

D₇×D₇⋊C₉

Subgroups — D₇⋊C₉×D₉₁ #37524459

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — D7⋊C9×D91 #37524459

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — D₇⋊C₉×D₉₁ #37524459