Group #11734724 — subgroups

Subgroups — #11734724

Order 896.

380 conjugacy classes of subgroups; only the 115 normal subgroups are shown.

normal subgroups only

This group has 380 conjugacy classes of subgroups — too many to draw. Only its 115 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×#395	C₂	1	1	yes	maximal, characteristic, complemented
2	#90003	C₂	1	1	yes	maximal, characteristic
3	#90001	C₂	1	1	yes	maximal, characteristic, complemented
4	#89761	C₂	1	3	yes	maximal
5	#89754	C₂	1	3	yes	maximal, complemented
6	#89447	C₂	1	3	yes	maximal
7	#89430	C₂	1	3	yes	maximal, complemented
8	C₂²×C₇×Q₈	C₂²	1	1	yes	characteristic
9	C₂²×C₄×D₇	C₂²	1	1	yes	characteristic
10	C₂×C₇×C₄⋊C₄	C₂²	1	3	yes	—
11	C₂×(C₂×D₇)⋊C₄	C₂²	1	3	yes	—
12	C₂×(C₂×D₇)⋊C₄	C₂²	1	3	yes	—
13	C₂×Dic₇⋊C₄	C₂²	1	3	yes	—
14	C₂×Dic₇⋊C₄	C₂²	1	3	yes	—
15	C₇×#95	C₂²	1	3	yes	—
16	C₇⋊#95	C₂²	1	3	yes	—
17	C₇⋊#95	C₂²	1	3	yes	—
18	C₇⋊#95	C₂²	1	3	yes	—
19	C₇⋊#95	C₂²	1	3	yes	—
20	C₇⋊#95	C₂²	1	3	yes	—
21	#2835	—	7	1	no	maximal, Sylow
22	C₂³×D₇	C₂³	1	1	yes	characteristic
23	C₂×C₇×Q₈	—	4	3	no	—
24	C₂²×C₄×C₇	C₂³	1	1	yes	characteristic
25	C₂²×C₄×C₇	C₂³	1	3	yes	—
26	C₂²×C₄×C₇	C₂³	1	3	yes	—
27	C₂²×Dic₇	C₂³	1	3	yes	—
28	C₂²×Dic₇	C₂³	1	3	yes	—
29	C₂²×Dic₇	C₂³	1	1	yes	characteristic
30	C₂×C₄×D₇	—	4	3	no	—
31	C₇×C₄⋊C₄	—	4	3	no	—
32	(C₂×D₇)⋊C₄	—	2	6	no	—
33	(C₂×D₇)⋊C₄	—	4	3	no	—
34	Dic₇⋊C₄	—	4	3	no	—
35	Dic₇⋊C₄	—	4	3	no	—
36	#400	—	7	3	no	—
37	#400	—	7	3	no	—
38	#397	—	7	3	no	—
39	#395	—	7	1	no	—
40	#395	D₇	1	1	yes	characteristic, complemented
41	#393	—	7	1	no	—
42	#385	—	7	3	no	—
43	C₂³×C₇	C₂⁴	1	1	yes	characteristic
44	C₂²×D₇	—	2	3	no	—
45	C₂²×D₇	—	2	3	no	—
46	C₂²×D₇	#41	1	2	yes	—
47	C₇×Q₈	—	4	4	no	—
48	C₂×C₄×C₇	—	2	3	no	—
49	C₂×C₄×C₇	—	2	3	no	—
50	C₂×C₄×C₇	—	2	3	no	—
51	C₂×C₄×C₇	—	2	3	no	—
52	C₂×C₄×C₇	—	2	3	no	—
53	C₂×C₄×C₇	—	2	3	no	—
54	C₂×C₄×C₇	—	2	3	no	—
55	C₂×Dic₇	—	2	3	no	—
56	C₂×Dic₇	—	2	3	no	—
57	C₂×Dic₇	—	2	3	no	—
58	C₂×Dic₇	—	2	3	no	—
59	C₂×Dic₇	—	2	3	no	—
60	C₂×Dic₇	—	2	6	no	—
61	C₄×D₇	—	4	4	no	—
62	C₂²×Q₈	C₂×D₇	1	1	yes	characteristic
63	C₂³×C₄	—	7	1	no	—
64	C₂×C₄⋊C₄	—	7	3	no	—
65	C₂×C₄⋊C₄	—	7	3	no	—
66	C₂×C₄⋊C₄	C₂×D₇	1	3	yes	—
67	C₂×C₂²⋊C₄	—	7	3	no	—
68	C₂×C₂²⋊C₄	—	7	3	no	—
69	#95	—	7	3	no	—
70	#95	—	7	3	no	—
71	#95	—	7	3	no	—
72	#95	—	7	3	no	—
73	#95	—	7	3	no	—
74	#95	C₂×D₇	1	3	yes	—
75	C₂²×C₇	C₂×#41	1	1	yes	characteristic
76	C₂²×C₇	H(5,𝔽₂)	1	3	yes	—
77	C₂²×C₇	2₋¹⁺⁴	1	3	yes	—
78	C₂×D₇	—	2	6	no	—
79	C₂×D₇	—	4	1	no	—
80	C₂×D₇	—	4	3	no	—
81	C₄×C₇	—	4	3	no	—
82	C₄×C₇	—	4	1	no	—
83	C₄×C₇	—	4	3	no	—
84	Dic₇	—	4	3	no	—
85	Dic₇	—	4	1	no	—
86	Dic₇	—	4	3	no	—
87	C₂⁴	—	7	1	no	—
88	C₂×Q₈	—	4	3	no	—
89	C₂²×C₄	—	7	3	no	—
90	C₂²×C₄	—	7	3	no	—
91	C₂²×C₄	—	7	1	no	—
92	C₂²×C₄	—	28	3	no	—
93	C₂²×C₄	C₂²×D₇	1	1	yes	characteristic
94	C₂²×C₄	C₂²×D₇	1	3	yes	—
95	C₂²×C₄	C₂²×D₇	1	3	yes	—
96	C₄⋊C₄	—	4	3	no	—
97	C₄⋊C₄	—	28	3	no	—
98	C₄⋊C₄	—	28	3	no	—
99	C₂²⋊C₄	—	14	6	no	—
100	C₂²⋊C₄	—	28	3	no	—
101	C₂×C₇	#539	1	3	yes	—
102	C₂×C₇	H(3,𝔽₄)	1	1	yes	characteristic
103	C₂×C₇	#563	1	3	yes	—
104	D₇	—	4	2	no	—
105	C₂³	—	7	2	no	—
106	C₂³	—	14	3	no	—
107	C₂³	—	14	3	no	—
108	C₂³	C₂³×D₇	1	1	yes	characteristic
109	Q₈	—	4	4	no	—
110	C₂×C₄	—	2	3	no	—
111	C₂×C₄	—	2	3	no	—
112	C₂×C₄	—	2	3	no	—
113	C₂×C₄	—	2	3	no	—
114	C₂×C₄	—	2	3	no	—
115	C₂×C₄	—	2	3	no	—
116	C₂×C₄	—	2	3	no	—
117	C₂×C₄	—	14	3	no	—
118	C₂×C₄	—	14	3	no	—
119	C₂×C₄	—	14	3	no	—
120	C₂×C₄	—	14	6	no	—
121	C₂×C₄	—	14	3	no	—
122	C₂×C₄	—	14	3	no	—
123	C₂×C₄	—	28	4	no	—
124	C₇	#2835	1	1	yes	characteristic, complemented, Sylow
125	C₂²	—	14	6	no	—
126	C₂²	—	28	3	no	—
127	C₂²	—	28	1	no	—
128	C₂²	#6548	1	3	yes	—
129	C₂²	#6551	1	3	yes	—
130	C₂²	D₇×#41	1	1	yes	characteristic
131	C₄	—	4	1	no	—
132	C₄	—	4	3	no	—
133	C₄	—	4	3	no	—
134	C₄	—	28	3	no	—
135	C₄	—	28	1	no	—
136	C₄	—	28	3	no	—
137	C₂	—	28	2	no	—
138	C₂	#90285	1	1	yes	characteristic
139	C₂	#90329	1	3	yes	—
140	C₂	#90426	1	3	yes	—

subgroup

G/H

class size

multiplicity

normal?

properties

C₇×#395

C₂

yes

maximal, characteristic, complemented

#90003

C₂

yes

maximal, characteristic

#90001

C₂

yes

maximal, characteristic, complemented

#89761

C₂

yes

maximal

#89754

C₂

yes

maximal, complemented

#89447

C₂

yes

maximal

#89430

C₂

yes

maximal, complemented

C₂²×C₇×Q₈

C₂²

yes

characteristic

C₂²×C₄×D₇

C₂²

yes

characteristic

C₂×C₇×C₄⋊C₄

C₂²

yes

—

C₂×(C₂×D₇)⋊C₄

C₂²

yes

—

C₂×(C₂×D₇)⋊C₄

C₂²

yes

—

C₂×Dic₇⋊C₄

C₂²

yes

—

C₂×Dic₇⋊C₄

C₂²

yes

—

C₇×#95

C₂²

yes

—

C₇⋊#95

C₂²

yes

—

C₇⋊#95

C₂²

yes

—

C₇⋊#95

C₂²

yes

—

C₇⋊#95

C₂²

yes

—

C₇⋊#95

C₂²

yes

—

#2835

—

maximal, Sylow

C₂³×D₇

C₂³

yes

characteristic

C₂×C₇×Q₈

—

C₂²×C₄×C₇

C₂³

yes

characteristic

C₂²×C₄×C₇

C₂³

yes

—

C₂²×C₄×C₇

C₂³

yes

—

C₂²×Dic₇

C₂³

yes

—

C₂²×Dic₇

C₂³

yes

—

C₂²×Dic₇

C₂³

yes

characteristic

C₂×C₄×D₇

—

C₇×C₄⋊C₄

—

(C₂×D₇)⋊C₄

—

(C₂×D₇)⋊C₄

—

Dic₇⋊C₄

—

Dic₇⋊C₄

—

—

—

—

—

D₇

yes

characteristic, complemented

#393

—

#385

—

C₂³×C₇

C₂⁴

yes

characteristic

C₂²×D₇

—

C₂²×D₇

—

C₂²×D₇

#41

yes

—

C₇×Q₈

—

C₂×C₄×C₇

—

C₂×C₄×C₇

—

C₂×C₄×C₇

—

C₂×C₄×C₇

—

C₂×C₄×C₇

—

C₂×C₄×C₇

—

C₂×C₄×C₇

—

C₂×Dic₇

—

C₂×Dic₇

—

C₂×Dic₇

—

C₂×Dic₇

—

C₂×Dic₇

—

C₂×Dic₇

—

C₄×D₇

—

C₂²×Q₈

C₂×D₇

yes

characteristic

C₂³×C₄

—

C₂×C₄⋊C₄

—

C₂×C₄⋊C₄

—

C₂×C₄⋊C₄

C₂×D₇

yes

—

C₂×C₂²⋊C₄

—

C₂×C₂²⋊C₄

—

—

—

—

—

—

C₂×D₇

yes

—

C₂²×C₇

C₂×#41

yes

characteristic

C₂²×C₇

H(5,𝔽₂)

yes

—

C₂²×C₇

2₋¹⁺⁴

yes

—

C₂×D₇

—

C₂×D₇

—

C₂×D₇

—

C₄×C₇

—

C₄×C₇

—

C₄×C₇

—

Dic₇

—

Dic₇

—

Dic₇

—

C₂⁴

—

C₂×Q₈

—

C₂²×C₄

—

C₂²×C₄

—

C₂²×C₄

—

C₂²×C₄

—

C₂²×C₄

C₂²×D₇

yes

characteristic

C₂²×C₄

C₂²×D₇

yes

—

C₂²×C₄

C₂²×D₇

yes

—

C₄⋊C₄

—

C₄⋊C₄

—

C₄⋊C₄

—

C₂²⋊C₄

—

100

C₂²⋊C₄

—

101

C₂×C₇

#539

yes

—

102

C₂×C₇

H(3,𝔽₄)

yes

characteristic

103

C₂×C₇

#563

yes

—

104

D₇

—

105

C₂³

—

106

C₂³

—

107

C₂³

—

108

C₂³

C₂³×D₇

yes

characteristic

109

Q₈

—

110

C₂×C₄

—

111

C₂×C₄

—

112

C₂×C₄

—

113

C₂×C₄

—

114

C₂×C₄

—

115

C₂×C₄

—

116

C₂×C₄

—

117

C₂×C₄

—

118

C₂×C₄

—

119

C₂×C₄

—

120

C₂×C₄

—

121

C₂×C₄

—

122

C₂×C₄

—

123

C₂×C₄

—

124

C₇

#2835

yes

characteristic, complemented, Sylow

125

C₂²

—

126

C₂²

—

127

C₂²

—

128

C₂²

#6548

yes

—

129

C₂²

#6551

yes

—

130

C₂²

D₇×#41

yes

characteristic

131

C₄

—

132

C₄

—

133

C₄

—

134

C₄

—

135

C₄

—

136

C₄

—

137

C₂

—

138

C₂

#90285

yes

characteristic

139

C₂

#90329

yes

—

140

C₂

#90426

yes

—

	subgroup	G/H
center, Frattini	C₂³	C₂³×D₇
commutator, socle, cosocle	C₂³×C₇	C₂⁴
Fitting	C₇×#395	C₂
prime radical	C₂³×D₇	C₂³
hypercenter	#395	D₇
hypocenter	C₇	#2835
LCS	G ▷ C₂³×C₇ ▷ C₇
UCS	C₁ ◁ C₂³ ◁ #395

subgroup

G/H

center, Frattini

C₂³

C₂³×D₇

commutator, socle, cosocle

C₂³×C₇

C₂⁴

Fitting

C₇×#395

C₂

prime radical

C₂³×D₇

C₂³

D₇

C₇

G ▷ C₂³×C₇ ▷ C₇

C₁ ◁ C₂³ ◁ #395

N	H ≅ G/N	type
1. C₇×#395	C₂	semidirect
3. #90001	C₂	semidirect
5. #89754	C₂	semidirect
7. #89430	C₂	semidirect
40. #395	D₇	semidirect
124. C₇	#2835	semidirect

H ≅ G/N

type

1. C₇×#395

C₂

semidirect

3. #90001

C₂

semidirect

5. #89754

C₂

semidirect

7. #89430

C₂

semidirect

40. #395

D₇

semidirect

124. C₇

#2835

semidirect

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

subgroup	index
C₇×#395	2
#90003	2
#90001	2
#89761	2
#89754	2
#89447	2
#89430	2
#2835	7

subgroup

index

Subgroups — #11734724

Conjugacy classes of subgroups

Complements

Maximal subgroups