Group #37524481 — subgroups

Subgroups — D₇×(C₁₃⋊C₉)⋊D₇ #37524481

Order 22932.

156 conjugacy classes of subgroups; only the 42 normal subgroups are shown.

normal subgroups only

This group has 156 conjugacy classes of subgroups — too many to draw. Only its 42 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	(C₁₃⋊C₉)⋊(C₇⋊D₇)	C₂	1	1	yes	maximal, characteristic, complemented
2	C₇×D₇×C₁₃⋊C₉	C₂	1	1	yes	maximal, characteristic, complemented
3	C₇×(C₁₃⋊C₉)⋊D₇	C₂	1	1	yes	maximal, characteristic, complemented
4	C₃×D₇×D₉₁	C₃	1	1	yes	maximal, characteristic
5	C₇²×C₁₃⋊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×(C₁₃⋊C₉)⋊D₇	—	7	1	no	maximal
10	D₇×D₁₃⋊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	C₉×D₇²	—	13	1	no	maximal, Hall
14	C₂×C₇×C₁₃⋊C₉	—	7	1	no	—
15	(C₁₃⋊C₉)⋊D₇	—	7	1	no	—
16	(C₁₃⋊C₉)⋊D₇	—	7	1	no	—
17	(C₁₃⋊C₉)⋊D₇	—	14	3	no	—
18	(C₁₃⋊C₉)⋊D₇	D₇	1	1	yes	characteristic, direct product factor
19	D₇×C₁₃⋊C₉	D₇	1	1	yes	characteristic, complemented
20	C₇×D₁₃⋊C₉	—	7	1	no	—
21	C₇⋊D₉₁	C₂×C₉	1	1	yes	characteristic, complemented
22	C₇×D₉₁	C₂×C₉	1	1	yes	characteristic, complemented
23	C₇×C₁₃×D₇	C₂×C₉	1	1	yes	characteristic, complemented
24	C₂×C₃×D₉₁	—	7	1	no	—
25	C₃×D₇×D₁₃	—	7	1	no	—
26	C₉×C₇⋊D₇	—	13	1	no	—
27	C₇×C₉×D₇	—	13	1	no	—
28	C₇×C₉×D₇	—	13	1	no	—
29	C₇×C₁₃⋊C₉	—	2	3	no	—
30	C₇×C₁₃⋊C₉	C₂×D₇	1	1	yes	characteristic, complemented
31	C₇×C₁₃⋊C₉	C₂×D₇	1	1	yes	characteristic, complemented
32	C₇²×C₁₃	C₂²×C₉	1	1	yes	characteristic, complemented, Hall
33	C₃×D₇²	—	39	1	no	—
34	C₂×C₃×C₇×C₁₃	—	7	1	no	—
35	C₃×D₉₁	—	7	1	no	—
36	C₃×D₉₁	—	7	1	no	—
37	C₃×D₉₁	—	14	3	no	—
38	C₃×D₉₁	C₃×D₇	1	1	yes	characteristic
39	C₃×C₁₃×D₇	C₃×D₇	1	1	yes	characteristic
40	C₃×C₇×D₁₃	—	7	1	no	—
41	C₂×D₁₃⋊C₉	—	49	1	no	Hall
42	C₇²×C₉	—	13	1	no	Hall
43	C₂×D₉₁	—	7	1	no	—
44	D₇×D₁₃	—	7	1	no	—
45	C₃×C₇⋊D₇	—	39	1	no	—
46	C₃×C₇×D₇	—	39	1	no	—
47	C₃×C₇×D₇	D₁₃⋊C₃	1	1	yes	characteristic
48	C₃×C₇×C₁₃	—	2	3	no	—
49	C₃×C₇×C₁₃	C₂×C₃×D₇	1	1	yes	characteristic
50	C₃×C₇×C₁₃	C₂×C₃×D₇	1	1	yes	characteristic
51	C₂×C₉×D₇	—	91	1	no	—
52	C₂×C₉×D₇	—	91	1	no	—
53	C₂×C₁₃⋊C₉	—	7	1	no	—
54	D₁₃⋊C₉	—	7	1	no	—
55	D₁₃⋊C₉	—	49	1	no	—
56	D₇²	—	39	1	no	Hall
57	C₂×C₇×C₁₃	—	7	1	no	—
58	D₉₁	—	7	1	no	—
59	D₉₁	—	7	1	no	—
60	D₉₁	—	14	3	no	—
61	D₉₁	C₉×D₇	1	1	yes	characteristic, complemented
62	C₇×D₁₃	—	7	1	no	—
63	C₁₃×D₇	C₉×D₇	1	1	yes	characteristic, complemented
64	C₂×C₃×D₁₃	—	49	1	no	—
65	C₃×C₇²	C₂×D₁₃⋊C₃	1	1	yes	characteristic
66	C₂×C₇×C₉	—	91	1	no	—
67	C₂×C₇×C₉	—	91	1	no	—
68	C₉×D₇	—	13	1	no	—
69	C₉×D₇	—	13	1	no	—
70	C₉×D₇	—	91	1	no	—
71	C₉×D₇	—	91	1	no	—
72	C₉×D₇	—	182	3	no	—
73	C₁₃⋊C₉	D₇²	1	1	yes	characteristic, complemented, Hall
74	C₇⋊D₇	—	39	1	no	—
75	C₇×D₇	—	39	1	no	—
76	C₇×D₇	D₁₃⋊C₉	1	1	yes	characteristic, complemented
77	C₇×C₁₃	—	2	3	no	—
78	C₇×C₁₃	C₂×C₉×D₇	1	1	yes	characteristic, complemented
79	C₇×C₁₃	C₂×C₉×D₇	1	1	yes	characteristic, complemented
80	C₂×C₃×D₇	—	273	1	no	—
81	C₂×C₃×D₇	—	273	1	no	—
82	C₂×C₃×C₁₃	—	7	1	no	—
83	C₃×D₁₃	—	7	1	no	—
84	C₃×D₁₃	—	49	1	no	—
85	C₇×C₉	—	13	1	no	—
86	C₇×C₉	—	13	1	no	—
87	C₇×C₉	—	26	3	no	—
88	C₂×D₁₃	—	49	1	no	Hall
89	C₇²	C₂×D₁₃⋊C₉	1	1	yes	characteristic, complemented, Sylow
90	C₂×C₃×C₇	—	7	1	no	—
91	C₂×C₃×C₇	—	273	1	no	—
92	C₃×D₇	—	273	1	no	—
93	C₃×D₇	—	546	3	no	—
94	C₃×D₇	—	39	1	no	—
95	C₃×D₇	—	91	1	no	—
96	C₃×D₇	(C₁₃⋊C₃)⋊D₇	1	1	yes	characteristic
97	C₃×C₁₃	C₃×D₇²	1	1	yes	characteristic
98	C₂²×C₉	—	637	1	no	Hall
99	C₂×D₇	—	273	1	no	—
100	C₂×D₇	—	273	1	no	—
101	C₂×C₁₃	—	7	1	no	—
102	D₁₃	—	7	1	no	—
103	D₁₃	—	49	1	no	—
104	C₃×C₇	—	2	3	no	—
105	C₃×C₇	D₇×D₁₃⋊C₃	1	1	yes	characteristic
106	C₃×C₇	C₂×(C₁₃⋊C₃)⋊D₇	1	1	yes	characteristic
107	C₂×C₉	—	91	1	no	—
108	C₂×C₉	—	91	1	no	—
109	C₂×C₉	—	637	1	no	—
110	C₂×C₇	—	7	1	no	—
111	C₂×C₇	—	273	1	no	—
112	D₇	—	273	1	no	—
113	D₇	—	546	3	no	—
114	D₇	—	39	1	no	—
115	D₇	—	91	1	no	—
116	D₇	(C₁₃⋊C₉)⋊D₇	1	1	yes	characteristic, direct product factor
117	C₁₃	C₉×D₇²	1	1	yes	characteristic, complemented, Sylow
118	C₂²×C₃	—	1911	1	no	—
119	C₉	—	13	1	no	Sylow
120	C₇	—	2	3	no	—
121	C₇	D₇×D₁₃⋊C₉	1	1	yes	characteristic, complemented
122	C₇	C₂×(C₁₃⋊C₉)⋊D₇	1	1	yes	characteristic, complemented
123	C₂×C₃	—	7	1	no	—
124	C₂×C₃	—	273	1	no	—
125	C₂×C₃	—	1911	1	no	—
126	C₂²	—	1911	1	no	Sylow
127	C₃	D₇×(C₁₃⋊C₃)⋊D₇	1	1	yes	characteristic
128	C₂	—	7	1	no	—
129	C₂	—	273	1	no	—
130	C₂	—	1911	1	no	—

subgroup

G/H

class size

multiplicity

normal?

properties

(C₁₃⋊C₉)⋊(C₇⋊D₇)

C₂

yes

maximal, characteristic, complemented

C₇×D₇×C₁₃⋊C₉

C₂

yes

maximal, characteristic, complemented

C₇×(C₁₃⋊C₉)⋊D₇

C₂

yes

maximal, characteristic, complemented

C₃×D₇×D₉₁

C₃

yes

maximal, characteristic

C₇²×C₁₃⋊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×(C₁₃⋊C₉)⋊D₇

—

maximal

D₇×D₁₃⋊C₉

—

maximal

D₇×D₉₁

C₉

yes

characteristic, complemented, Hall

C₃×C₇²×C₁₃

C₂²×C₃

yes

characteristic

C₉×D₇²

—

maximal, Hall

C₂×C₇×C₁₃⋊C₉

—

(C₁₃⋊C₉)⋊D₇

—

(C₁₃⋊C₉)⋊D₇

—

(C₁₃⋊C₉)⋊D₇

—

(C₁₃⋊C₉)⋊D₇

D₇

yes

characteristic, direct product factor

D₇×C₁₃⋊C₉

D₇

yes

characteristic, complemented

C₇×D₁₃⋊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

—

C₉×C₇⋊D₇

—

C₇×C₉×D₇

—

C₇×C₉×D₇

—

C₇×C₁₃⋊C₉

—

C₇×C₁₃⋊C₉

C₂×D₇

yes

characteristic, complemented

C₇×C₁₃⋊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

C₃×D₇

yes

characteristic

C₃×C₁₃×D₇

C₃×D₇

yes

characteristic

C₃×C₇×D₁₃

—

C₂×D₁₃⋊C₉

—

Hall

C₇²×C₉

—

Hall

C₂×D₉₁

—

D₇×D₁₃

—

C₃×C₇⋊D₇

—

C₃×C₇×D₇

—

C₃×C₇×D₇

D₁₃⋊C₃

yes

characteristic

C₃×C₇×C₁₃

—

C₃×C₇×C₁₃

C₂×C₃×D₇

yes

characteristic

C₃×C₇×C₁₃

C₂×C₃×D₇

yes

characteristic

C₂×C₉×D₇

—

C₂×C₉×D₇

—

C₂×C₁₃⋊C₉

—

D₁₃⋊C₉

—

D₁₃⋊C₉

—

D₇²

—

Hall

C₂×C₇×C₁₃

—

D₉₁

—

D₉₁

—

D₉₁

—

D₉₁

C₉×D₇

yes

characteristic, complemented

C₇×D₁₃

—

C₁₃×D₇

C₉×D₇

yes

characteristic, complemented

C₂×C₃×D₁₃

—

C₃×C₇²

C₂×D₁₃⋊C₃

yes

characteristic

C₂×C₇×C₉

—

C₂×C₇×C₉

—

C₉×D₇

—

C₉×D₇

—

C₉×D₇

—

C₉×D₇

—

C₉×D₇

—

182

—

C₁₃⋊C₉

D₇²

yes

characteristic, complemented, Hall

C₇⋊D₇

—

C₇×D₇

—

C₇×D₇

D₁₃⋊C₉

yes

characteristic, complemented

C₇×C₁₃

—

C₇×C₁₃

C₂×C₉×D₇

yes

characteristic, complemented

C₇×C₁₃

C₂×C₉×D₇

yes

characteristic, complemented

C₂×C₃×D₇

—

273

—

C₂×C₃×D₇

—

273

—

C₂×C₃×C₁₃

—

C₃×D₁₃

—

C₃×D₁₃

—

C₇×C₉

—

C₇×C₉

—

C₇×C₉

—

C₂×D₁₃

—

Hall

C₇²

C₂×D₁₃⋊C₉

yes

characteristic, complemented, Sylow

C₂×C₃×C₇

—

C₂×C₃×C₇

—

273

—

C₃×D₇

—

273

—

C₃×D₇

—

546

—

C₃×D₇

—

C₃×D₇

—

C₃×D₇

(C₁₃⋊C₃)⋊D₇

yes

characteristic

C₃×C₁₃

C₃×D₇²

yes

characteristic

C₂²×C₉

—

637

Hall

C₂×D₇

—

273

—

100

C₂×D₇

—

273

—

101

C₂×C₁₃

—

102

D₁₃

—

103

D₁₃

—

104

C₃×C₇

—

105

C₃×C₇

D₇×D₁₃⋊C₃

yes

characteristic

106

C₃×C₇

C₂×(C₁₃⋊C₃)⋊D₇

yes

characteristic

107

C₂×C₉

—

108

C₂×C₉

—

109

C₂×C₉

—

637

—

110

C₂×C₇

—

111

C₂×C₇

—

273

—

112

D₇

—

273

—

113

D₇

—

546

—

114

D₇

—

115

D₇

—

116

D₇

(C₁₃⋊C₉)⋊D₇

yes

characteristic, direct product factor

117

C₁₃

C₉×D₇²

yes

characteristic, complemented, Sylow

118

C₂²×C₃

—

1911

—

119

C₉

—

Sylow

120

C₇

—

121

C₇

D₇×D₁₃⋊C₉

yes

characteristic, complemented

122

C₇

C₂×(C₁₃⋊C₉)⋊D₇

yes

characteristic, complemented

123

C₂×C₃

—

124

C₂×C₃

—

273

—

125

C₂×C₃

—

1911

—

126

C₂²

—

1911

Sylow

127

C₃

D₇×(C₁₃⋊C₃)⋊D₇

yes

characteristic

128

C₂

—

129

C₂

—

273

—

130

C₂

—

1911

—

	subgroup	G/H
center, Frattini, hypercenter	C₃	D₇×(C₁₃⋊C₃)⋊D₇
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

D₇×(C₁₃⋊C₃)⋊D₇

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. (C₁₃⋊C₉)⋊(C₇⋊D₇)	C₂	semidirect
2. C₇×D₇×C₁₃⋊C₉	C₂	semidirect
3. C₇×(C₁₃⋊C₉)⋊D₇	C₂	semidirect
5. C₇²×C₁₃⋊C₉	C₂²	semidirect
11. D₇×D₉₁	C₉	semidirect
18. (C₁₃⋊C₉)⋊D₇	D₇	direct (N × H)
19. D₇×C₁₃⋊C₉	D₇	semidirect
21. C₇⋊D₉₁	C₂×C₉	semidirect
22. C₇×D₉₁	C₂×C₉	semidirect
23. C₇×C₁₃×D₇	C₂×C₉	semidirect
30. C₇×C₁₃⋊C₉	C₂×D₇	semidirect
31. C₇×C₁₃⋊C₉	C₂×D₇	semidirect
32. C₇²×C₁₃	C₂²×C₉	semidirect
61. D₉₁	C₉×D₇	semidirect
63. C₁₃×D₇	C₉×D₇	semidirect
73. C₁₃⋊C₉	D₇²	semidirect
76. C₇×D₇	D₁₃⋊C₉	semidirect
78. C₇×C₁₃	C₂×C₉×D₇	semidirect
79. C₇×C₁₃	C₂×C₉×D₇	semidirect
89. C₇²	C₂×D₁₃⋊C₉	semidirect
116. D₇	(C₁₃⋊C₉)⋊D₇	direct (N × H)
117. C₁₃	C₉×D₇²	semidirect
121. C₇	D₇×D₁₃⋊C₉	semidirect
122. C₇	C₂×(C₁₃⋊C₉)⋊D₇	semidirect

H ≅ G/N

type

1. (C₁₃⋊C₉)⋊(C₇⋊D₇)

C₂

semidirect

2. C₇×D₇×C₁₃⋊C₉

C₂

semidirect

3. C₇×(C₁₃⋊C₉)⋊D₇

C₂

semidirect

5. C₇²×C₁₃⋊C₉

C₂²

semidirect

11. D₇×D₉₁

C₉

semidirect

18. (C₁₃⋊C₉)⋊D₇

D₇

direct (N × H)

19. D₇×C₁₃⋊C₉

D₇

semidirect

21. C₇⋊D₉₁

C₂×C₉

semidirect

22. C₇×D₉₁

C₂×C₉

semidirect

23. C₇×C₁₃×D₇

C₂×C₉

semidirect

30. C₇×C₁₃⋊C₉

C₂×D₇

semidirect

31. C₇×C₁₃⋊C₉

C₂×D₇

semidirect

32. C₇²×C₁₃

C₂²×C₉

semidirect

61. D₉₁

C₉×D₇

semidirect

63. C₁₃×D₇

C₉×D₇

semidirect

73. C₁₃⋊C₉

D₇²

semidirect

76. C₇×D₇

D₁₃⋊C₉

semidirect

78. C₇×C₁₃

C₂×C₉×D₇

semidirect

79. C₇×C₁₃

C₂×C₉×D₇

semidirect

89. C₇²

C₂×D₁₃⋊C₉

semidirect

116. D₇

(C₁₃⋊C₉)⋊D₇

direct (N × H)

117. C₁₃

C₉×D₇²

semidirect

121. C₇

D₇×D₁₃⋊C₉

semidirect

122. C₇

C₂×(C₁₃⋊C₉)⋊D₇

semidirect

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

subgroup	index
(C₁₃⋊C₉)⋊(C₇⋊D₇)	2
C₇×D₇×C₁₃⋊C₉	2
C₇×(C₁₃⋊C₉)⋊D₇	2
C₃×D₇×D₉₁	3
C₂×(C₁₃⋊C₉)⋊D₇	7
D₇×D₁₃⋊C₉	7
C₉×D₇²	13

subgroup

index

(C₁₃⋊C₉)⋊(C₇⋊D₇)

C₇×D₇×C₁₃⋊C₉

C₇×(C₁₃⋊C₉)⋊D₇

C₃×D₇×D₉₁

C₂×(C₁₃⋊C₉)⋊D₇

D₇×D₁₃⋊C₉

C₉×D₇²

Subgroups — D₇×(C₁₃⋊C₉)⋊D₇ #37524481

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — D7×(C13⋊C9)⋊D7 #37524481

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — D₇×(C₁₃⋊C₉)⋊D₇ #37524481