Group #37524455 — subgroups

Subgroups — #13109902⋊D₇ #37524455

Order 22932.

128 conjugacy classes of subgroups; only the 37 normal subgroups are shown.

normal subgroups only

This group has 128 conjugacy classes of subgroups — too many to draw. Only its 37 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	#26039477	C₂	1	1	yes	maximal, characteristic, complemented
2	#26039468	C₂	1	1	yes	maximal, characteristic, complemented
3	C₇×#13109902	C₂	1	1	yes	maximal, characteristic, complemented
4	C₃×(C₇⋊D₇)⋊D₁₃	C₃	1	1	yes	maximal, characteristic
5	C₇×#11717941	C₂²	1	1	yes	characteristic, complemented, Hall
6	C₃×C₁₃×C₇⋊D₇	C₂×C₃	1	1	yes	characteristic
7	C₃×C₇×D₉₁	C₂×C₃	1	1	yes	characteristic
8	C₃×C₇×D₉₁	C₂×C₃	1	1	yes	characteristic
9	#13109896⋊C₂	—	7	1	no	maximal
10	D₇×D₁₃⋊C₉	—	7	1	no	maximal
11	(C₇⋊D₇)⋊D₁₃	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	#13109902	D₇	1	1	yes	characteristic, complemented
15	#13109900	—	7	1	no	—
16	#13109896	—	7	1	no	—
17	(C₁₃⋊C₉)⋊D₇	—	7	1	no	—
18	D₇×C₁₃⋊C₉	—	7	1	no	—
19	C₇×D₁₃⋊C₉	—	7	1	no	—
20	C₁₃×C₇⋊D₇	C₂×C₉	1	1	yes	characteristic, complemented
21	C₇×D₉₁	C₂×C₉	1	1	yes	characteristic, complemented
22	C₇×D₉₁	C₂×C₉	1	1	yes	characteristic, complemented
23	C₃×D₇×D₁₃	—	7	1	no	—
24	C₃×D₇×D₁₃	—	21	1	no	—
25	(C₇⋊C₉)⋊D₇	—	13	1	no	—
26	D₇×C₇⋊C₉	—	13	1	no	—
27	C₇×D₇⋊C₉	—	13	1	no	—
28	#11717941	C₂×D₇	1	1	yes	characteristic, complemented
29	C₇×C₁₃⋊C₉	—	7	1	no	—
30	C₇²×C₁₃	C₂²×C₉	1	1	yes	characteristic, complemented, Hall
31	C₃×D₇²	—	39	1	no	—
32	C₃×D₉₁	AGL(1,𝔽₇)	1	1	yes	characteristic
33	C₃×D₉₁	C₃×D₇	1	1	yes	characteristic
34	C₃×C₁₃×D₇	—	7	1	no	—
35	C₃×C₁₃×D₇	—	7	1	no	—
36	C₃×C₁₃×D₇	—	42	1	no	—
37	C₃×C₇×D₁₃	—	7	1	no	—
38	C₃×C₇×D₁₃	—	21	1	no	—
39	C₂×D₁₃⋊C₉	—	49	1	no	Hall
40	C₇×C₇⋊C₉	—	13	1	no	Hall
41	D₇×D₁₃	—	7	1	no	—
42	D₇×D₁₃	—	21	1	no	—
43	C₃×C₇⋊D₇	D₁₃⋊C₃	1	1	yes	characteristic
44	C₃×C₇×D₇	—	13	1	no	—
45	C₃×C₇×D₇	—	39	1	no	—
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×C₁₃⋊C₉	—	49	1	no	—
52	D₁₃⋊C₉	—	7	1	no	—
53	D₁₃⋊C₉	—	49	1	no	—
54	D₇²	—	39	1	no	Hall
55	D₉₁	D₇⋊C₉	1	1	yes	characteristic, complemented
56	D₉₁	C₉×D₇	1	1	yes	characteristic, complemented
57	C₇×D₁₃	—	7	1	no	—
58	C₇×D₁₃	—	21	1	no	—
59	C₁₃×D₇	—	7	1	no	—
60	C₁₃×D₇	—	7	1	no	—
61	C₁₃×D₇	—	42	1	no	—
62	C₂×C₃×D₁₃	—	147	1	no	—
63	C₃×C₇²	C₂×D₁₃⋊C₃	1	1	yes	characteristic
64	C₂×C₇×C₉	—	91	1	no	—
65	C₉×D₇	—	91	1	no	—
66	C₉×D₇	—	91	1	no	—
67	C₂×C₇⋊C₉	—	91	1	no	—
68	D₇⋊C₉	—	13	1	no	—
69	D₇⋊C₉	—	91	1	no	—
70	C₁₃⋊C₉	—	7	1	no	Hall
71	C₇⋊D₇	D₁₃⋊C₉	1	1	yes	characteristic, complemented
72	C₇×D₇	—	13	1	no	—
73	C₇×D₇	—	39	1	no	—
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	—	273	1	no	—
78	C₂×C₃×D₇	—	273	1	no	—
79	C₂×C₃×C₁₃	—	147	1	no	—
80	C₃×D₁₃	—	7	1	no	—
81	C₃×D₁₃	—	7	1	no	—
82	C₇×C₉	—	91	1	no	—
83	C₇⋊C₉	—	13	1	no	—
84	C₂×D₁₃	—	147	1	no	Hall
85	C₇²	C₂×D₁₃⋊C₉	1	1	yes	characteristic, complemented, Sylow
86	C₂×C₃×C₇	—	273	1	no	—
87	C₂×C₃×C₇	—	91	1	no	—
88	C₃×D₇	—	7	1	no	—
89	C₃×D₇	—	13	1	no	—
90	C₃×D₇	—	21	1	no	—
91	C₃×D₇	—	39	1	no	—
92	C₃×D₇	—	42	1	no	—
93	C₃×C₁₃	D₇×AGL(1,𝔽₇)	1	1	yes	characteristic
94	C₂²×C₉	—	637	1	no	Hall
95	C₂×D₇	—	273	1	no	—
96	C₂×D₇	—	273	1	no	—
97	C₂×C₁₃	—	147	1	no	—
98	D₁₃	—	7	1	no	—
99	D₁₃	—	7	1	no	—
100	C₃×C₇	—	6	1	no	—
101	C₃×C₇	D₇×D₁₃⋊C₃	1	1	yes	characteristic
102	C₃×C₇	#10587716⋊C₂	1	1	yes	characteristic
103	C₂×C₉	—	91	1	no	—
104	C₂×C₉	—	637	1	no	—
105	C₂×C₉	—	637	1	no	—
106	C₂×C₇	—	273	1	no	—
107	C₂×C₇	—	91	1	no	—
108	D₇	—	7	1	no	—
109	D₇	—	13	1	no	—
110	D₇	—	21	1	no	—
111	D₇	—	39	1	no	—
112	D₇	—	42	1	no	—
113	C₁₃	D₇×D₇⋊C₉	1	1	yes	characteristic, complemented, Sylow
114	C₂²×C₃	—	1911	1	no	—
115	C₉	—	91	1	no	Sylow
116	C₇	—	6	1	no	—
117	C₇	D₇×D₁₃⋊C₉	1	1	yes	characteristic, complemented
118	C₇	#13109896⋊C₂	1	1	yes	characteristic, complemented
119	C₂×C₃	—	273	1	no	—
120	C₂×C₃	—	273	1	no	—
121	C₂×C₃	—	147	1	no	—
122	C₂²	—	1911	1	no	Sylow
123	C₃	#10587722⋊D₇	1	1	yes	characteristic
124	C₂	—	273	1	no	—
125	C₂	—	273	1	no	—
126	C₂	—	147	1	no	—

subgroup

G/H

class size

multiplicity

normal?

properties

#26039477

C₂

yes

maximal, characteristic, complemented

#26039468

C₂

yes

maximal, characteristic, complemented

C₇×#13109902

C₂

yes

maximal, characteristic, complemented

C₃×(C₇⋊D₇)⋊D₁₃

C₃

yes

maximal, characteristic

C₇×#11717941

C₂²

yes

characteristic, complemented, Hall

C₃×C₁₃×C₇⋊D₇

C₂×C₃

yes

characteristic

C₃×C₇×D₉₁

C₂×C₃

yes

characteristic

C₃×C₇×D₉₁

C₂×C₃

yes

characteristic

#13109896⋊C₂

—

maximal

D₇×D₁₃⋊C₉

—

maximal

(C₇⋊D₇)⋊D₁₃

C₉

yes

characteristic, complemented, Hall

C₃×C₇²×C₁₃

C₂²×C₃

yes

characteristic

D₇×D₇⋊C₉

—

maximal, Hall

#13109902

D₇

yes

characteristic, complemented

#13109900

—

#13109896

—

(C₁₃⋊C₉)⋊D₇

—

D₇×C₁₃⋊C₉

—

C₇×D₁₃⋊C₉

—

C₁₃×C₇⋊D₇

C₂×C₉

yes

characteristic, complemented

C₇×D₉₁

C₂×C₉

yes

characteristic, complemented

C₇×D₉₁

C₂×C₉

yes

characteristic, complemented

C₃×D₇×D₁₃

—

C₃×D₇×D₁₃

—

(C₇⋊C₉)⋊D₇

—

D₇×C₇⋊C₉

—

C₇×D₇⋊C₉

—

#11717941

C₂×D₇

yes

characteristic, complemented

C₇×C₁₃⋊C₉

—

C₇²×C₁₃

C₂²×C₉

yes

characteristic, complemented, Hall

C₃×D₇²

—

C₃×D₉₁

AGL(1,𝔽₇)

yes

characteristic

C₃×D₉₁

C₃×D₇

yes

characteristic

C₃×C₁₃×D₇

—

C₃×C₁₃×D₇

—

C₃×C₁₃×D₇

—

C₃×C₇×D₁₃

—

C₃×C₇×D₁₃

—

C₂×D₁₃⋊C₉

—

Hall

C₇×C₇⋊C₉

—

Hall

D₇×D₁₃

—

D₇×D₁₃

—

C₃×C₇⋊D₇

D₁₃⋊C₃

yes

characteristic

C₃×C₇×D₇

—

C₃×C₇×D₇

—

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×C₁₃⋊C₉

—

D₁₃⋊C₉

—

D₁₃⋊C₉

—

D₇²

—

Hall

D₉₁

D₇⋊C₉

yes

characteristic, complemented

D₉₁

C₉×D₇

yes

characteristic, complemented

C₇×D₁₃

—

C₇×D₁₃

—

C₁₃×D₇

—

C₁₃×D₇

—

C₁₃×D₇

—

C₂×C₃×D₁₃

—

147

—

C₃×C₇²

C₂×D₁₃⋊C₃

yes

characteristic

C₂×C₇×C₉

—

C₉×D₇

—

C₉×D₇

—

C₂×C₇⋊C₉

—

D₇⋊C₉

—

D₇⋊C₉

—

C₁₃⋊C₉

—

Hall

C₇⋊D₇

D₁₃⋊C₉

yes

characteristic, complemented

C₇×D₇

—

C₇×D₇

—

C₇×C₁₃

—

C₇×C₁₃

C₂×D₇⋊C₉

yes

characteristic, complemented

C₇×C₁₃

C₂×C₉×D₇

yes

characteristic, complemented

C₂×C₃×D₇

—

273

—

C₂×C₃×D₇

—

273

—

C₂×C₃×C₁₃

—

147

—

C₃×D₁₃

—

C₃×D₁₃

—

C₇×C₉

—

C₇⋊C₉

—

C₂×D₁₃

—

147

Hall

C₇²

C₂×D₁₃⋊C₉

yes

characteristic, complemented, Sylow

C₂×C₃×C₇

—

273

—

C₂×C₃×C₇

—

C₃×D₇

—

C₃×D₇

—

C₃×D₇

—

C₃×D₇

—

C₃×D₇

—

C₃×C₁₃

D₇×AGL(1,𝔽₇)

yes

characteristic

C₂²×C₉

—

637

Hall

C₂×D₇

—

273

—

C₂×D₇

—

273

—

C₂×C₁₃

—

147

—

D₁₃

—

D₁₃

—

100

C₃×C₇

—

101

C₃×C₇

D₇×D₁₃⋊C₃

yes

characteristic

102

C₃×C₇

#10587716⋊C₂

yes

characteristic

103

C₂×C₉

—

104

C₂×C₉

—

637

—

105

C₂×C₉

—

637

—

106

C₂×C₇

—

273

—

107

C₂×C₇

—

108

D₇

—

109

D₇

—

110

D₇

—

111

D₇

—

112

D₇

—

113

C₁₃

D₇×D₇⋊C₉

yes

characteristic, complemented, Sylow

114

C₂²×C₃

—

1911

—

115

C₉

—

Sylow

116

C₇

—

117

C₇

D₇×D₁₃⋊C₉

yes

characteristic, complemented

118

C₇

#13109896⋊C₂

yes

characteristic, complemented

119

C₂×C₃

—

273

—

120

C₂×C₃

—

273

—

121

C₂×C₃

—

147

—

122

C₂²

—

1911

Sylow

123

C₃

#10587722⋊D₇

yes

characteristic

124

C₂

—

273

—

125

C₂

—

273

—

126

C₂

—

147

—

	subgroup	G/H
center, Frattini, hypercenter	C₃	#10587722⋊D₇
commutator, hypocenter	C₇²×C₁₃	C₂²×C₉
Fitting, socle, cosocle	C₃×C₇²×C₁₃	C₂²×C₃
prime radical	C₃×(C₇⋊D₇)⋊D₁₃	C₃
LCS	G ▷ C₇²×C₁₃
UCS	C₁ ◁ C₃

subgroup

G/H

center, Frattini, hypercenter

C₃

#10587722⋊D₇

commutator, hypocenter

C₇²×C₁₃

C₂²×C₉

Fitting, socle, cosocle

C₃×C₇²×C₁₃

C₂²×C₃

prime radical

C₃×(C₇⋊D₇)⋊D₁₃

C₃

LCS

G ▷ C₇²×C₁₃

UCS

C₁ ◁ C₃

N	H ≅ G/N	type
1. #26039477	C₂	semidirect
2. #26039468	C₂	semidirect
3. C₇×#13109902	C₂	semidirect
5. C₇×#11717941	C₂²	semidirect
11. (C₇⋊D₇)⋊D₁₃	C₉	semidirect
14. #13109902	D₇	semidirect
20. C₁₃×C₇⋊D₇	C₂×C₉	semidirect
21. C₇×D₉₁	C₂×C₉	semidirect
22. C₇×D₉₁	C₂×C₉	semidirect
28. #11717941	C₂×D₇	semidirect
30. C₇²×C₁₃	C₂²×C₉	semidirect
55. D₉₁	D₇⋊C₉	semidirect
56. D₉₁	C₉×D₇	semidirect
71. C₇⋊D₇	D₁₃⋊C₉	semidirect
75. C₇×C₁₃	C₂×D₇⋊C₉	semidirect
76. C₇×C₁₃	C₂×C₉×D₇	semidirect
85. C₇²	C₂×D₁₃⋊C₉	semidirect
113. C₁₃	D₇×D₇⋊C₉	semidirect
117. C₇	D₇×D₁₃⋊C₉	semidirect
118. C₇	#13109896⋊C₂	semidirect

H ≅ G/N

type

1. #26039477

C₂

semidirect

2. #26039468

C₂

semidirect

3. C₇×#13109902

C₂

semidirect

5. C₇×#11717941

C₂²

semidirect

11. (C₇⋊D₇)⋊D₁₃

C₉

semidirect

14. #13109902

D₇

semidirect

20. C₁₃×C₇⋊D₇

C₂×C₉

semidirect

21. C₇×D₉₁

C₂×C₉

semidirect

22. C₇×D₉₁

C₂×C₉

semidirect

28. #11717941

C₂×D₇

semidirect

30. C₇²×C₁₃

C₂²×C₉

semidirect

55. D₉₁

D₇⋊C₉

semidirect

56. D₉₁

C₉×D₇

semidirect

71. C₇⋊D₇

D₁₃⋊C₉

semidirect

75. C₇×C₁₃

C₂×D₇⋊C₉

semidirect

76. C₇×C₁₃

C₂×C₉×D₇

semidirect

85. C₇²

C₂×D₁₃⋊C₉

semidirect

113. C₁₃

D₇×D₇⋊C₉

semidirect

117. C₇

D₇×D₁₃⋊C₉

semidirect

118. C₇

#13109896⋊C₂

semidirect

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

subgroup	index
#26039477	2
#26039468	2
C₇×#13109902	2
C₃×(C₇⋊D₇)⋊D₁₃	3
#13109896⋊C₂	7
D₇×D₁₃⋊C₉	7
D₇×D₇⋊C₉	13

subgroup

index

C₃×(C₇⋊D₇)⋊D₁₃

D₇×D₁₃⋊C₉

D₇×D₇⋊C₉

Subgroups — #13109902⋊D₇ #37524455

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — #13109902⋊D7 #37524455

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — #13109902⋊D₇ #37524455