Group #37524441 — subgroups

Subgroups — #37524441

Order 22932.

156 conjugacy classes of subgroups; only the 45 normal subgroups are shown.

normal subgroups only

This group has 156 conjugacy classes of subgroups — too many to draw. Only its 45 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	#26039472	C₂	1	1	yes	maximal, characteristic, complemented
2	#26039457	C₂	1	1	yes	maximal, characteristic, complemented
3	#26039455	C₂	1	1	yes	maximal, characteristic, complemented
4	C₃×C₇×D₇×D₁₃	C₃	1	1	yes	maximal, characteristic
5	#19153199	C₂²	1	1	yes	characteristic, complemented, Hall
6	C₃×C₇×D₉₁	C₂×C₃	1	1	yes	characteristic
7	C₃×C₇×C₁₃×D₇	C₂×C₃	1	1	yes	characteristic
8	C₃×C₇²×D₁₃	C₂×C₃	1	1	yes	characteristic
9	C₂×#13109897	—	7	1	no	maximal
10	#13109897⋊C₂	—	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	C₂×#11725076	—	13	1	no	maximal, Hall
14	C₂×#11717942	—	7	1	no	—
15	#13109903	—	7	1	no	—
16	#13109901	—	7	1	no	—
17	#13109897	—	7	1	no	—
18	#13109897	—	7	1	no	—
19	#13109897	—	7	1	no	—
20	#13109897	—	14	3	no	—
21	C₇×D₉₁	C₂×C₉	1	1	yes	characteristic, complemented
22	C₇×C₁₃×D₇	C₂×C₉	1	1	yes	characteristic, complemented
23	C₇²×D₁₃	C₂×C₉	1	1	yes	characteristic, complemented
24	C₂×C₃×C₇×D₁₃	—	21	1	no	—
25	C₃×D₇×D₁₃	ΓL(1,𝔽₈)	1	1	yes	characteristic
26	C₂×#89204	—	13	1	no	—
27	#11725076	—	13	1	no	—
28	#11725076	—	13	1	no	—
29	#11717942	—	7	1	no	—
30	#11717942	—	7	1	no	—
31	#11717942	—	14	3	no	—
32	C₇²×C₁₃	C₂²×C₉	1	1	yes	characteristic, complemented, Hall
33	C₂×C₃×C₇×D₇	—	39	1	no	—
34	C₂×C₃×C₇×C₁₃	—	21	1	no	—
35	C₃×D₉₁	C₂×ΓL(1,𝔽₈)	1	1	yes	characteristic
36	C₃×C₁₃×D₇	C₂×ΓL(1,𝔽₈)	1	1	yes	characteristic
37	C₃×C₇×D₁₃	—	2	3	no	—
38	C₃×C₇×D₁₃	—	21	1	no	—
39	C₃×C₇×D₁₃	AGL(1,𝔽₇)	1	1	yes	characteristic
40	C₃×C₇×D₁₃	C₂×ΓL(1,𝔽₈)	1	1	yes	characteristic
41	C₂×D₁₃⋊C₉	—	49	1	no	Hall
42	#89204	—	13	1	no	Hall
43	C₂×C₇×D₁₃	—	21	1	no	—
44	D₇×D₁₃	C₇⋊C₉	1	1	yes	characteristic, complemented
45	C₂×C₃×C₇²	—	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×AGL(1,𝔽₇)	1	1	yes	characteristic
50	C₃×C₇×C₁₃	C₂²×ΓL(1,𝔽₈)	1	1	yes	characteristic
51	C₂²×C₇⋊C₉	—	91	1	no	—
52	C₂×D₇⋊C₉	—	91	1	no	—
53	C₂×C₁₃⋊C₉	—	49	1	no	—
54	D₁₃⋊C₉	—	49	1	no	—
55	D₁₃⋊C₉	—	49	1	no	—
56	C₂×C₇×D₇	—	39	1	no	Hall
57	C₂×C₇×C₁₃	—	21	1	no	—
58	D₉₁	C₂×C₇⋊C₉	1	1	yes	characteristic, complemented
59	C₇×D₁₃	—	2	3	no	—
60	C₇×D₁₃	—	21	1	no	—
61	C₇×D₁₃	D₇⋊C₉	1	1	yes	characteristic, complemented
62	C₇×D₁₃	C₂×C₇⋊C₉	1	1	yes	characteristic, complemented
63	C₁₃×D₇	C₂×C₇⋊C₉	1	1	yes	characteristic, complemented
64	C₂×C₃×D₁₃	—	21	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	C₂×C₇⋊C₉	—	91	1	no	—
69	C₂×C₇⋊C₉	—	91	1	no	—
70	C₂×C₇⋊C₉	—	182	3	no	—
71	D₇⋊C₉	—	91	1	no	—
72	D₇⋊C₉	—	91	1	no	—
73	C₁₃⋊C₉	—	49	1	no	Hall
74	C₂×C₇²	—	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×D₇⋊C₉	1	1	yes	characteristic, complemented
79	C₇×C₁₃	C₂²×C₇⋊C₉	1	1	yes	characteristic, complemented
80	C₂²×C₃×C₇	—	273	1	no	—
81	C₂×C₃×D₇	—	39	1	no	—
82	C₂×C₃×C₁₃	—	21	1	no	—
83	C₃×D₁₃	—	21	1	no	—
84	C₃×D₁₃	#64423	1	1	yes	characteristic
85	C₇⋊C₉	—	91	1	no	—
86	C₇⋊C₉	—	91	1	no	—
87	C₇⋊C₉	—	182	3	no	—
88	C₂×D₁₃	—	21	1	no	Hall
89	C₇²	C₂×D₁₃⋊C₉	1	1	yes	characteristic, complemented, Sylow
90	C₂×C₃×C₇	—	273	1	no	—
91	C₂×C₃×C₇	—	21	1	no	—
92	C₂×C₃×C₇	—	39	1	no	—
93	C₂×C₃×C₇	—	39	1	no	—
94	C₂×C₃×C₇	—	78	3	no	—
95	C₃×D₇	—	39	1	no	—
96	C₃×D₇	#10587717	1	1	yes	characteristic
97	C₃×C₁₃	C₂×#64423	1	1	yes	characteristic
98	C₂²×C₉	—	637	1	no	Hall
99	C₂²×C₇	—	273	1	no	—
100	C₂×D₇	—	39	1	no	—
101	C₂×C₁₃	—	21	1	no	—
102	D₁₃	—	21	1	no	—
103	D₁₃	#11725076	1	1	yes	characteristic, complemented
104	C₃×C₇	—	2	3	no	—
105	C₃×C₇	#10587717⋊C₂	1	1	yes	characteristic
106	C₃×C₇	C₂×#10587717	1	1	yes	characteristic
107	C₂×C₉	—	637	1	no	—
108	C₂×C₉	—	637	1	no	—
109	C₂×C₉	—	637	1	no	—
110	C₂×C₇	—	273	1	no	—
111	C₂×C₇	—	21	1	no	—
112	C₂×C₇	—	39	1	no	—
113	C₂×C₇	—	39	1	no	—
114	C₂×C₇	—	78	3	no	—
115	D₇	—	39	1	no	—
116	D₇	#13109897	1	1	yes	characteristic, complemented
117	C₁₃	C₂×#11725076	1	1	yes	characteristic, complemented, Sylow
118	C₂²×C₃	—	273	1	no	—
119	C₉	—	637	1	no	Sylow
120	C₇	—	2	3	no	—
121	C₇	#13109897⋊C₂	1	1	yes	characteristic, complemented
122	C₇	C₂×#13109897	1	1	yes	characteristic, complemented
123	C₂×C₃	—	273	1	no	—
124	C₂×C₃	—	21	1	no	—
125	C₂×C₃	—	39	1	no	—
126	C₂²	—	273	1	no	Sylow
127	C₃	#21439626	1	1	yes	characteristic
128	C₂	—	273	1	no	—
129	C₂	—	21	1	no	—
130	C₂	—	39	1	no	—

subgroup

G/H

class size

multiplicity

normal?

properties

#26039472

C₂

yes

maximal, characteristic, complemented

#26039457

C₂

yes

maximal, characteristic, complemented

#26039455

C₂

yes

maximal, characteristic, complemented

C₃×C₇×D₇×D₁₃

C₃

yes

maximal, characteristic

#19153199

C₂²

yes

characteristic, complemented, Hall

C₃×C₇×D₉₁

C₂×C₃

yes

characteristic

C₃×C₇×C₁₃×D₇

C₂×C₃

yes

characteristic

C₃×C₇²×D₁₃

C₂×C₃

yes

—

—

C₇×D₇×D₁₃

C₉

yes

characteristic, complemented, Hall

C₃×C₇²×C₁₃

C₂²×C₃

yes

—

—

—

—

—

—

—

—

C₇×D₉₁

C₂×C₉

yes

characteristic, complemented

C₇×C₁₃×D₇

C₂×C₉

yes

characteristic, complemented

C₇²×D₁₃

C₂×C₉

yes

characteristic, complemented

C₂×C₃×C₇×D₁₃

—

C₃×D₇×D₁₃

yes

—

—

—

—

—

—

C₇²×C₁₃

C₂²×C₉

yes

characteristic, complemented, Hall

C₂×C₃×C₇×D₇

—

C₂×C₃×C₇×C₁₃

—

C₃×D₉₁

yes

C₃×C₁₃×D₇

yes

C₃×C₇×D₁₃

—

C₃×C₇×D₁₃

—

C₃×C₇×D₁₃

yes

C₃×C₇×D₁₃

yes

C₂×D₁₃⋊C₉

—

Hall

#89204

—

Hall

C₂×C₇×D₁₃

—

D₇×D₁₃

C₇⋊C₉

yes

characteristic, complemented

C₂×C₃×C₇²

—

C₃×C₇×D₇

—

C₃×C₇×D₇

D₁₃⋊C₃

yes

characteristic

C₃×C₇×C₁₃

—

C₃×C₇×C₁₃

C₂×AGL(1,𝔽₇)

yes

characteristic

C₃×C₇×C₁₃

C₂²×ΓL(1,𝔽₈)

yes

characteristic

C₂²×C₇⋊C₉

—

C₂×D₇⋊C₉

—

C₂×C₁₃⋊C₉

—

D₁₃⋊C₉

—

D₁₃⋊C₉

—

C₂×C₇×D₇

—

Hall

C₂×C₇×C₁₃

—

D₉₁

C₂×C₇⋊C₉

yes

characteristic, complemented

C₇×D₁₃

—

C₇×D₁₃

—

C₇×D₁₃

D₇⋊C₉

yes

characteristic, complemented

C₇×D₁₃

C₂×C₇⋊C₉

yes

characteristic, complemented

C₁₃×D₇

C₂×C₇⋊C₉

yes

characteristic, complemented

C₂×C₃×D₁₃

—

C₃×C₇²

C₂×D₁₃⋊C₃

yes

characteristic

C₂×C₇⋊C₉

—

C₂×C₇⋊C₉

—

C₂×C₇⋊C₉

—

C₂×C₇⋊C₉

—

C₂×C₇⋊C₉

—

182

—

D₇⋊C₉

—

D₇⋊C₉

—

C₁₃⋊C₉

—

Hall

C₂×C₇²

—

C₇×D₇

—

C₇×D₇

D₁₃⋊C₉

yes

characteristic, complemented

C₇×C₁₃

—

C₇×C₁₃

C₂×D₇⋊C₉

yes

characteristic, complemented

C₇×C₁₃

C₂²×C₇⋊C₉

yes

characteristic, complemented

C₂²×C₃×C₇

—

273

—

C₂×C₃×D₇

—

C₂×C₃×C₁₃

—

C₃×D₁₃

—

C₃×D₁₃

#64423

yes

characteristic

C₇⋊C₉

—

C₇⋊C₉

—

C₇⋊C₉

—

182

—

C₂×D₁₃

—

Hall

C₇²

C₂×D₁₃⋊C₉

yes

characteristic, complemented, Sylow

C₂×C₃×C₇

—

273

—

C₂×C₃×C₇

—

C₂×C₃×C₇

—

C₂×C₃×C₇

—

C₂×C₃×C₇

—

C₃×D₇

—

C₃×D₇

yes

C₃×C₁₃

yes

C₂²×C₉

—

637

Hall

C₂²×C₇

—

273

—

100

C₂×D₇

—

101

C₂×C₁₃

—

102

D₁₃

—

103

D₁₃

#11725076

yes

characteristic, complemented

104

C₃×C₇

—

105

C₃×C₇

#10587717⋊C₂

yes

characteristic

106

C₃×C₇

C₂×#10587717

yes

characteristic

107

C₂×C₉

—

637

—

108

C₂×C₉

—

637

—

109

C₂×C₉

—

637

—

110

C₂×C₇

—

273

—

111

C₂×C₇

—

112

C₂×C₇

—

113

C₂×C₇

—

114

C₂×C₇

—

115

D₇

—

116

D₇

#13109897

yes

characteristic, complemented

117

C₁₃

C₂×#11725076

yes

characteristic, complemented, Sylow

118

C₂²×C₃

—

273

—

119

C₉

—

637

Sylow

120

C₇

—

121

C₇

#13109897⋊C₂

yes

characteristic, complemented

122

C₇

C₂×#13109897

yes

characteristic, complemented

123

C₂×C₃

—

273

—

124

C₂×C₃

—

125

C₂×C₃

—

126

C₂²

—

273

Sylow

127

C₃

#21439626

yes

characteristic

128

C₂

—

273

—

129

C₂

—

130

C₂

—

	subgroup	G/H
center, Frattini, hypercenter	C₃	#21439626
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

#21439626

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. #26039472	C₂	semidirect
2. #26039457	C₂	semidirect
3. #26039455	C₂	semidirect
5. #19153199	C₂²	semidirect
11. C₇×D₇×D₁₃	C₉	semidirect
21. C₇×D₉₁	C₂×C₉	semidirect
22. C₇×C₁₃×D₇	C₂×C₉	semidirect
23. C₇²×D₁₃	C₂×C₉	semidirect
32. C₇²×C₁₃	C₂²×C₉	semidirect
44. D₇×D₁₃	C₇⋊C₉	semidirect
58. D₉₁	C₂×C₇⋊C₉	semidirect
61. C₇×D₁₃	D₇⋊C₉	semidirect
62. C₇×D₁₃	C₂×C₇⋊C₉	semidirect
63. C₁₃×D₇	C₂×C₇⋊C₉	semidirect
76. C₇×D₇	D₁₃⋊C₉	semidirect
78. C₇×C₁₃	C₂×D₇⋊C₉	semidirect
79. C₇×C₁₃	C₂²×C₇⋊C₉	semidirect
89. C₇²	C₂×D₁₃⋊C₉	semidirect
103. D₁₃	#11725076	semidirect
116. D₇	#13109897	semidirect
117. C₁₃	C₂×#11725076	semidirect
121. C₇	#13109897⋊C₂	semidirect
122. C₇	C₂×#13109897	semidirect

H ≅ G/N

type

1. #26039472

C₂

semidirect

2. #26039457

C₂

semidirect

3. #26039455

C₂

semidirect

5. #19153199

C₂²

semidirect

11. C₇×D₇×D₁₃

C₉

semidirect

21. C₇×D₉₁

C₂×C₉

semidirect

22. C₇×C₁₃×D₇

C₂×C₉

semidirect

23. C₇²×D₁₃

C₂×C₉

semidirect

32. C₇²×C₁₃

C₂²×C₉

semidirect

44. D₇×D₁₃

C₇⋊C₉

semidirect

58. D₉₁

C₂×C₇⋊C₉

semidirect

61. C₇×D₁₃

D₇⋊C₉

semidirect

62. C₇×D₁₃

C₂×C₇⋊C₉

semidirect

63. C₁₃×D₇

C₂×C₇⋊C₉

semidirect

76. C₇×D₇

D₁₃⋊C₉

semidirect

78. C₇×C₁₃

C₂×D₇⋊C₉

semidirect

79. C₇×C₁₃

C₂²×C₇⋊C₉

semidirect

89. C₇²

C₂×D₁₃⋊C₉

semidirect

103. D₁₃

#11725076

semidirect

116. D₇

#13109897

semidirect

117. C₁₃

C₂×#11725076

semidirect

121. C₇

#13109897⋊C₂

semidirect

122. C₇

C₂×#13109897

semidirect

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

subgroup	index
#26039472	2
#26039457	2
#26039455	2
C₃×C₇×D₇×D₁₃	3
C₂×#13109897	7
#13109897⋊C₂	7
C₂×#11725076	13

subgroup

index

C₃×C₇×D₇×D₁₃

Subgroups — #37524441

Conjugacy classes of subgroups

Complements

Maximal subgroups