Group #11731790 — subgroups

Subgroups — #11731790

Order 896.

360 conjugacy classes of subgroups; only the 149 normal subgroups are shown.

normal subgroups only

This group has 360 conjugacy classes of subgroups — too many to draw. Only its 149 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×#384	C₂	1	1	yes	maximal, characteristic
2	#89758	C₂	1	4	yes	maximal
3	#89757	C₂	1	2	yes	maximal
4	#89756	C₂	1	4	yes	maximal
5	#89745	C₂	1	2	yes	maximal
6	#89708	C₂	1	1	yes	maximal, characteristic
7	#89704	C₂	1	1	yes	maximal, characteristic
8	C₂×C₇×C₄⋊C₄	C₂²	1	2	yes	—
9	C₂×C₇×C₄⋊C₄	C₂²	1	4	yes	—
10	C₂×C₄²×C₇	C₂²	1	1	yes	characteristic
11	C₂×C₄⋊Dic₇	C₂²	1	2	yes	—
12	C₂×C₄⋊Dic₇	C₂²	1	4	yes	—
13	C₂×Dic₇⋊C₄	C₂²	1	4	yes	—
14	C₂×C₄×Dic₇	C₂²	1	1	yes	characteristic
15	C₂×C₄×Dic₇	C₂²	1	1	yes	characteristic
16	C₇⋊#95	C₂²	1	2	yes	—
17	C₇⋊#95	C₂²	1	2	yes	—
18	C₇⋊#95	C₂²	1	4	yes	—
19	C₇⋊#95	C₂²	1	4	yes	—
20	C₇⋊#95	C₂²	1	4	yes	—
21	#2647	—	7	1	no	maximal, Sylow
22	C₂²×C₄×C₇	C₂³	1	4	yes	—
23	C₂²×C₄×C₇	C₂³	1	2	yes	—
24	C₂²×C₄×C₇	C₂³	1	1	yes	characteristic
25	C₂²×Dic₇	C₂³	1	1	yes	characteristic
26	C₂²×Dic₇	C₂³	1	2	yes	—
27	C₂²×Dic₇	C₂³	1	4	yes	—
28	C₂²×Dic₇	C₂³	1	1	yes	characteristic
29	C₇×C₄⋊C₄	—	2	4	no	—
30	C₇×C₄⋊C₄	—	4	4	no	—
31	C₄²×C₇	Q₈	1	4	yes	—
32	C₄⋊Dic₇	—	2	4	no	—
33	C₄⋊Dic₇	—	4	4	no	—
34	Dic₇⋊C₄	—	4	4	no	—
35	C₄×Dic₇	—	2	2	no	—
36	C₄×Dic₇	—	2	2	no	—
37	#400	—	7	4	no	—
38	#398	—	7	4	no	—
39	#389	—	7	2	no	—
40	#389	—	7	2	no	—
41	#384	D₇	1	1	yes	characteristic
42	#382	—	7	1	no	—
43	#376	—	7	1	no	—
44	C₂³×C₇	C₂⁴	1	1	yes	characteristic
45	C₂×C₄×C₇	—	2	2	no	—
46	C₂×C₄×C₇	—	2	2	no	—
47	C₂×C₄×C₇	—	2	4	no	—
48	C₂×C₄×C₇	—	2	4	no	—
49	C₂×C₄×C₇	—	2	4	no	—
50	C₂×C₄×C₇	C₂×Q₈	1	4	yes	—
51	C₂×C₄×C₇	C₂×Q₈	1	2	yes	—
52	C₂×C₄×C₇	#41	1	2	yes	—
53	C₂×C₄×C₇	#41	1	2	yes	—
54	C₂×Dic₇	—	2	1	no	—
55	C₂×Dic₇	—	2	1	no	—
56	C₂×Dic₇	—	2	2	no	—
57	C₂×Dic₇	—	2	4	no	—
58	C₂×Dic₇	—	2	2	no	—
59	C₂×Dic₇	—	2	4	no	—
60	C₂×Dic₇	—	2	2	no	—
61	C₂×Dic₇	—	2	2	no	—
62	C₂×Dic₇	—	2	2	no	—
63	C₂×Dic₇	—	2	4	no	—
64	C₂×C₄⋊C₄	—	7	2	no	—
65	C₂×C₄⋊C₄	—	7	4	no	—
66	C₂×C₄⋊C₄	—	7	4	no	—
67	C₂×C₄⋊C₄	C₂×D₇	1	2	yes	—
68	C₂×C₄⋊C₄	C₂×D₇	1	4	yes	—
69	C₂×C₄²	—	7	1	no	—
70	C₂×C₄²	—	7	1	no	—
71	C₂×C₄²	C₂×D₇	1	1	yes	characteristic
72	#95	—	7	2	no	—
73	#95	—	7	2	no	—
74	#95	—	7	4	no	—
75	#95	—	7	4	no	—
76	#95	—	7	4	no	—
77	C₂²×C₇	C₂²×Q₈	1	1	yes	characteristic
78	C₂²×C₇	C₂×#41	1	1	yes	characteristic
79	C₂²×C₇	C₂×#41	1	1	yes	characteristic
80	C₂²×C₇	H(5,𝔽₂)	1	2	yes	—
81	C₂²×C₇	2₋¹⁺⁴	1	2	yes	—
82	C₄×C₇	—	2	4	no	—
83	C₄×C₇	—	4	4	no	—
84	C₄×C₇	#125	1	4	yes	—
85	Dic₇	—	4	1	no	—
86	Dic₇	—	4	1	no	—
87	Dic₇	—	4	2	no	—
88	Dic₇	—	4	4	no	—
89	C₂²×C₄	—	7	1	no	—
90	C₂²×C₄	—	7	2	no	—
91	C₂²×C₄	—	7	4	no	—
92	C₂²×C₄	—	7	1	no	—
93	C₂²×C₄	C₂²×D₇	1	4	yes	—
94	C₂²×C₄	C₂²×D₇	1	2	yes	—
95	C₂²×C₄	C₂²×D₇	1	1	yes	characteristic
96	C₄⋊C₄	—	2	4	no	—
97	C₄⋊C₄	—	4	4	no	—
98	C₄⋊C₄	—	14	4	no	—
99	C₄⋊C₄	—	28	4	no	—
100	C₄⋊C₄	—	28	4	no	—
101	C₄²	—	14	2	no	—
102	C₄²	—	14	2	no	—
103	C₄²	Dic₁₄	1	4	yes	—
104	C₂×C₇	C₂×#125	1	1	yes	characteristic
105	C₂×C₇	#540	1	1	yes	characteristic
106	C₂×C₇	#541	1	1	yes	characteristic
107	C₂×C₇	#542	1	2	yes	—
108	C₂×C₇	#544	1	2	yes	—
109	C₂³	C₂³×D₇	1	1	yes	characteristic
110	C₂×C₄	—	2	2	no	—
111	C₂×C₄	—	2	2	no	—
112	C₂×C₄	—	2	4	no	—
113	C₂×C₄	—	2	4	no	—
114	C₂×C₄	—	2	4	no	—
115	C₂×C₄	—	14	1	no	—
116	C₂×C₄	—	14	1	no	—
117	C₂×C₄	—	14	2	no	—
118	C₂×C₄	—	14	4	no	—
119	C₂×C₄	—	14	2	no	—
120	C₂×C₄	—	14	2	no	—
121	C₂×C₄	—	14	4	no	—
122	C₂×C₄	—	14	2	no	—
123	C₂×C₄	—	14	2	no	—
124	C₂×C₄	—	14	4	no	—
125	C₂×C₄	C₂×Dic₁₄	1	4	yes	—
126	C₂×C₄	C₂×Dic₁₄	1	2	yes	—
127	C₂×C₄	#1159	1	2	yes	—
128	C₂×C₄	#1161	1	2	yes	—
129	C₇	#2647	1	1	yes	characteristic, complemented, Sylow
130	C₂²	C₂²×Dic₁₄	1	1	yes	characteristic
131	C₂²	C₂×#1159	1	1	yes	characteristic
132	C₂²	#6548	1	2	yes	—
133	C₂²	C₂×#1161	1	1	yes	characteristic
134	C₂²	#6551	1	2	yes	—
135	C₄	—	2	4	no	—
136	C₄	—	4	4	no	—
137	C₄	—	28	1	no	—
138	C₄	—	28	1	no	—
139	C₄	—	28	2	no	—
140	C₄	—	28	4	no	—
141	C₄	#6453	1	4	yes	—
142	C₂	C₂×#6453	1	1	yes	characteristic
143	C₂	#90191	1	2	yes	—
144	C₂	#90410	1	1	yes	characteristic
145	C₂	#90422	1	1	yes	characteristic
146	C₂	#90424	1	2	yes	—

subgroup

G/H

class size

multiplicity

normal?

properties

C₇×#384

C₂

yes

maximal, characteristic

#89758

C₂

yes

maximal

#89757

C₂

yes

maximal

#89756

C₂

yes

maximal

#89745

C₂

yes

maximal

#89708

C₂

yes

maximal, characteristic

#89704

C₂

yes

maximal, characteristic

C₂×C₇×C₄⋊C₄

C₂²

yes

—

C₂×C₇×C₄⋊C₄

C₂²

yes

—

C₂×C₄²×C₇

C₂²

yes

characteristic

C₂×C₄⋊Dic₇

C₂²

yes

—

C₂×C₄⋊Dic₇

C₂²

yes

—

C₂×Dic₇⋊C₄

C₂²

yes

—

C₂×C₄×Dic₇

C₂²

yes

characteristic

C₂×C₄×Dic₇

C₂²

yes

characteristic

C₇⋊#95

C₂²

yes

—

C₇⋊#95

C₂²

yes

—

C₇⋊#95

C₂²

yes

—

C₇⋊#95

C₂²

yes

—

C₇⋊#95

C₂²

yes

—

#2647

—

maximal, Sylow

C₂²×C₄×C₇

C₂³

yes

—

C₂²×C₄×C₇

C₂³

yes

—

C₂²×C₄×C₇

C₂³

yes

characteristic

C₂²×Dic₇

C₂³

yes

characteristic

C₂²×Dic₇

C₂³

yes

—

C₂²×Dic₇

C₂³

yes

—

C₂²×Dic₇

C₂³

yes

characteristic

C₇×C₄⋊C₄

—

C₇×C₄⋊C₄

—

C₄²×C₇

Q₈

yes

—

C₄⋊Dic₇

—

C₄⋊Dic₇

—

Dic₇⋊C₄

—

C₄×Dic₇

—

C₄×Dic₇

—

—

—

—

—

D₇

yes

characteristic

#382

—

#376

—

C₂³×C₇

C₂⁴

yes

characteristic

C₂×C₄×C₇

—

C₂×C₄×C₇

—

C₂×C₄×C₇

—

C₂×C₄×C₇

—

C₂×C₄×C₇

—

C₂×C₄×C₇

C₂×Q₈

yes

—

C₂×C₄×C₇

C₂×Q₈

yes

—

C₂×C₄×C₇

#41

yes

—

C₂×C₄×C₇

#41

yes

—

C₂×Dic₇

—

C₂×Dic₇

—

C₂×Dic₇

—

C₂×Dic₇

—

C₂×Dic₇

—

C₂×Dic₇

—

C₂×Dic₇

—

C₂×Dic₇

—

C₂×Dic₇

—

C₂×Dic₇

—

C₂×C₄⋊C₄

—

C₂×C₄⋊C₄

—

C₂×C₄⋊C₄

—

C₂×C₄⋊C₄

C₂×D₇

yes

—

C₂×C₄⋊C₄

C₂×D₇

yes

—

C₂×C₄²

—

C₂×C₄²

—

C₂×C₄²

C₂×D₇

yes

—

—

—

—

—

C₂²×C₇

C₂²×Q₈

yes

characteristic

C₂²×C₇

C₂×#41

yes

characteristic

C₂²×C₇

C₂×#41

yes

characteristic

C₂²×C₇

H(5,𝔽₂)

yes

—

C₂²×C₇

2₋¹⁺⁴

yes

—

C₄×C₇

—

C₄×C₇

—

C₄×C₇

#125

yes

—

Dic₇

—

Dic₇

—

Dic₇

—

Dic₇

—

C₂²×C₄

—

C₂²×C₄

—

C₂²×C₄

—

C₂²×C₄

—

C₂²×C₄

C₂²×D₇

yes

—

C₂²×C₄

C₂²×D₇

yes

—

C₂²×C₄

C₂²×D₇

yes

characteristic

C₄⋊C₄

—

C₄⋊C₄

—

C₄⋊C₄

—

C₄⋊C₄

—

100

C₄⋊C₄

—

101

C₄²

—

102

C₄²

—

103

C₄²

Dic₁₄

yes

—

104

C₂×C₇

C₂×#125

yes

characteristic

105

C₂×C₇

#540

yes

characteristic

106

C₂×C₇

#541

yes

characteristic

107

C₂×C₇

#542

yes

—

108

C₂×C₇

#544

yes

—

109

C₂³

C₂³×D₇

yes

characteristic

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

C₂×C₄

—

125

C₂×C₄

C₂×Dic₁₄

yes

—

126

C₂×C₄

C₂×Dic₁₄

yes

—

127

C₂×C₄

#1159

yes

—

128

C₂×C₄

#1161

yes

—

129

C₇

#2647

yes

characteristic, complemented, Sylow

130

C₂²

C₂²×Dic₁₄

yes

characteristic

131

C₂²

C₂×#1159

yes

characteristic

132

C₂²

#6548

yes

—

133

C₂²

C₂×#1161

yes

characteristic

134

C₂²

#6551

yes

—

135

C₄

—

136

C₄

—

137

C₄

—

138

C₄

—

139

C₄

—

140

C₄

—

141

C₄

#6453

yes

—

142

C₂

C₂×#6453

yes

characteristic

143

C₂

#90191

yes

—

144

C₂

#90410

yes

characteristic

145

C₂

#90422

yes

characteristic

146

C₂

#90424

yes

—

	subgroup	G/H
center, Frattini	C₂³	C₂³×D₇
commutator, socle, prime radical, cosocle	C₂³×C₇	C₂⁴
Fitting	C₇×#384	C₂
hypercenter	#384	D₇
hypocenter	C₇	#2647
LCS	G ▷ C₂³×C₇ ▷ C₇
UCS	C₁ ◁ C₂³ ◁ #384

subgroup

G/H

center, Frattini

C₂³

C₂³×D₇

commutator, socle, prime radical, cosocle

C₂³×C₇

C₂⁴

C₂

D₇

C₇

G ▷ C₂³×C₇ ▷ C₇

C₁ ◁ C₂³ ◁ #384

N	H ≅ G/N	type
129. C₇	#2647	semidirect

H ≅ G/N

type

129. C₇

#2647

semidirect

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

subgroup	index
C₇×#384	2
#89758	2
#89757	2
#89756	2
#89745	2
#89708	2
#89704	2
#2647	7

subgroup

index

Subgroups — #11731790

Conjugacy classes of subgroups

Complements

Maximal subgroups