Group #11731667 — subgroups

Subgroups — #11731667

Order 896.

400 conjugacy classes of subgroups; only the 157 normal subgroups are shown.

normal subgroups only

This group has 400 conjugacy classes of subgroups — too many to draw. Only its 157 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×#382	C₂	1	1	yes	maximal, characteristic
2	#89757	C₂	1	2	yes	maximal
3	#89745	C₂	1	2	yes	maximal
4	#89708	C₂	1	1	yes	maximal, characteristic
5	C₄²×Dic₇	C₂	1	1	yes	maximal, characteristic
6	#89427	C₂	1	4	yes	maximal
7	#89426	C₂	1	4	yes	maximal
8	C₂×C₇×C₄⋊C₄	C₂²	1	2	yes	—
9	C₂×C₄²×C₇	C₂²	1	1	yes	characteristic
10	C₂×C₄⋊Dic₇	C₂²	1	4	yes	—
11	C₂×C₄⋊Dic₇	C₂²	1	2	yes	—
12	C₂×Dic₇⋊C₄	C₂²	1	4	yes	—
13	C₂×C₄×Dic₇	C₂²	1	2	yes	—
14	C₂×C₄×Dic₇	C₂²	1	2	yes	—
15	C₂×C₄×Dic₇	C₂²	1	1	yes	characteristic
16	C₂×C₄×Dic₇	C₂²	1	1	yes	characteristic
17	C₇×#95	C₂²	1	2	yes	—
18	C₇×#95	C₂²	1	2	yes	—
19	C₇⋊#95	C₂²	1	2	yes	—
20	C₇⋊#95	C₂²	1	4	yes	—
21	C₇⋊#95	C₂²	1	4	yes	—
22	C₇⋊#95	C₂²	1	2	yes	—
23	#2864	—	7	1	no	maximal, Sylow
24	C₂²×C₄×C₇	C₂³	1	4	yes	—
25	C₂²×C₄×C₇	C₂³	1	1	yes	characteristic
26	C₂²×C₄×C₇	C₂³	1	1	yes	characteristic
27	C₂²×C₄×C₇	C₂³	1	1	yes	characteristic
28	C₂²×Dic₇	C₂³	1	4	yes	—
29	C₂²×Dic₇	C₂³	1	2	yes	—
30	C₂²×Dic₇	C₂³	1	2	yes	—
31	C₇×C₄⋊C₄	—	2	4	no	—
32	C₄²×C₇	Q₈	1	4	yes	—
33	C₄⋊Dic₇	—	2	4	no	—
34	C₄⋊Dic₇	—	2	8	no	—
35	Dic₇⋊C₄	—	2	8	no	—
36	C₄×Dic₇	—	2	2	no	—
37	C₄×Dic₇	—	2	2	no	—
38	C₄×Dic₇	—	2	4	no	—
39	C₄×Dic₇	—	2	4	no	—
40	#389	—	7	2	no	—
41	#389	—	7	2	no	—
42	#387	—	7	4	no	—
43	#387	—	7	4	no	—
44	#382	—	7	1	no	—
45	#382	D₇	1	1	yes	characteristic
46	C₄³	—	7	1	no	—
47	C₂³×C₇	C₂⁴	1	1	yes	characteristic
48	C₂×C₄×C₇	—	2	2	no	—
49	C₂×C₄×C₇	—	2	2	no	—
50	C₂×C₄×C₇	—	2	4	no	—
51	C₂×C₄×C₇	—	2	4	no	—
52	C₂×C₄×C₇	—	2	4	no	—
53	C₂×C₄×C₇	C₂×Q₈	1	2	yes	—
54	C₂×C₄×C₇	C₂×Q₈	1	2	yes	—
55	C₂×C₄×C₇	C₂×Q₈	1	2	yes	—
56	C₂×C₄×C₇	#41	1	2	yes	—
57	C₂×C₄×C₇	#41	1	2	yes	—
58	C₂×Dic₇	—	2	2	no	—
59	C₂×Dic₇	—	2	2	no	—
60	C₂×Dic₇	—	2	2	no	—
61	C₂×Dic₇	—	2	2	no	—
62	C₂×Dic₇	—	2	4	no	—
63	C₂×Dic₇	—	2	4	no	—
64	C₂×Dic₇	—	2	4	no	—
65	C₂×Dic₇	#41	1	4	yes	—
66	C₂×Dic₇	#41	1	4	yes	—
67	C₂×C₄⋊C₄	—	7	4	no	—
68	C₂×C₄⋊C₄	—	7	2	no	—
69	C₂×C₄⋊C₄	—	7	4	no	—
70	C₂×C₄⋊C₄	C₂×D₇	1	2	yes	—
71	C₂×C₄²	—	7	2	no	—
72	C₂×C₄²	—	7	2	no	—
73	C₂×C₄²	—	7	1	no	—
74	C₂×C₄²	—	7	1	no	—
75	C₂×C₄²	C₂×D₇	1	1	yes	characteristic
76	#95	—	7	2	no	—
77	#95	—	7	2	no	—
78	#95	—	7	4	no	—
79	#95	—	7	4	no	—
80	#95	C₂×D₇	1	2	yes	—
81	#95	C₂×D₇	1	2	yes	—
82	C₂²×C₇	C₂²×Q₈	1	1	yes	characteristic
83	C₂²×C₇	C₂×#41	1	1	yes	characteristic
84	C₂²×C₇	C₂×#41	1	1	yes	characteristic
85	C₂²×C₇	C₂×#41	1	2	yes	—
86	C₂²×C₇	C₂×#41	1	2	yes	—
87	C₄×C₇	—	2	2	no	—
88	C₄×C₇	—	2	2	no	—
89	C₄×C₇	—	4	4	no	—
90	C₄×C₇	#125	1	4	yes	—
91	Dic₇	—	2	4	no	—
92	Dic₇	—	2	4	no	—
93	Dic₇	—	4	4	no	—
94	C₂²×C₄	—	7	2	no	—
95	C₂²×C₄	—	7	2	no	—
96	C₂²×C₄	—	7	4	no	—
97	C₂²×C₄	C₂²×D₇	1	4	yes	—
98	C₂²×C₄	C₂²×D₇	1	1	yes	characteristic
99	C₂²×C₄	C₂²×D₇	1	1	yes	characteristic
100	C₂²×C₄	C₂²×D₇	1	1	yes	characteristic
101	C₄⋊C₄	—	2	4	no	—
102	C₄⋊C₄	—	14	8	no	—
103	C₄⋊C₄	—	14	8	no	—
104	C₄⋊C₄	—	14	4	no	—
105	C₄²	—	14	4	no	—
106	C₄²	—	14	4	no	—
107	C₄²	—	14	2	no	—
108	C₄²	—	14	2	no	—
109	C₄²	Dic₁₄	1	4	yes	—
110	C₂×C₇	C₂×#125	1	1	yes	characteristic
111	C₂×C₇	#529	1	2	yes	—
112	C₂×C₇	#529	1	2	yes	—
113	C₂×C₇	#533	1	1	yes	characteristic
114	C₂×C₇	#533	1	1	yes	characteristic
115	C₂³	C₂³×D₇	1	1	yes	characteristic
116	C₂×C₄	—	2	2	no	—
117	C₂×C₄	—	2	2	no	—
118	C₂×C₄	—	2	4	no	—
119	C₂×C₄	—	2	4	no	—
120	C₂×C₄	—	2	4	no	—
121	C₂×C₄	—	7	4	no	—
122	C₂×C₄	—	7	4	no	—
123	C₂×C₄	—	14	2	no	—
124	C₂×C₄	—	14	2	no	—
125	C₂×C₄	—	14	2	no	—
126	C₂×C₄	—	14	2	no	—
127	C₂×C₄	—	14	4	no	—
128	C₂×C₄	—	14	4	no	—
129	C₂×C₄	—	14	4	no	—
130	C₂×C₄	C₂×Dic₁₄	1	2	yes	—
131	C₂×C₄	C₂×Dic₁₄	1	2	yes	—
132	C₂×C₄	C₂×Dic₁₄	1	2	yes	—
133	C₂×C₄	#1159	1	2	yes	—
134	C₂×C₄	#1161	1	2	yes	—
135	C₇	#2864	1	1	yes	characteristic, complemented, Sylow
136	C₂²	C₂²×Dic₁₄	1	1	yes	characteristic
137	C₂²	C₂×#1159	1	1	yes	characteristic
138	C₂²	C₂×#1161	1	1	yes	characteristic
139	C₂²	D₇×#41	1	2	yes	—
140	C₂²	D₇×#41	1	2	yes	—
141	C₄	—	2	2	no	—
142	C₄	—	2	2	no	—
143	C₄	—	4	4	no	—
144	C₄	—	14	4	no	—
145	C₄	—	14	4	no	—
146	C₄	—	28	4	no	—
147	C₄	#6453	1	4	yes	—
148	C₂	C₂×#6453	1	1	yes	characteristic
149	C₂	#90208	1	1	yes	characteristic
150	C₂	#90208	1	1	yes	characteristic
151	C₂	#90371	1	2	yes	—
152	C₂	#90382	1	2	yes	—

subgroup

G/H

class size

multiplicity

normal?

properties

C₇×#382

C₂

yes

maximal, characteristic

#89757

C₂

yes

maximal

#89745

C₂

yes

maximal

#89708

C₂

yes

maximal, characteristic

C₄²×Dic₇

C₂

yes

maximal, characteristic

#89427

C₂

yes

maximal

#89426

C₂

yes

maximal

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

—

C₂×C₄×Dic₇

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

—

C₇⋊#95

C₂²

yes

—

#2864

—

maximal, Sylow

C₂²×C₄×C₇

C₂³

yes

—

C₂²×C₄×C₇

C₂³

yes

characteristic

C₂²×C₄×C₇

C₂³

yes

characteristic

C₂²×C₄×C₇

C₂³

yes

characteristic

C₂²×Dic₇

C₂³

yes

—

C₂²×Dic₇

C₂³

yes

—

C₂²×Dic₇

C₂³

yes

—

C₇×C₄⋊C₄

—

C₄²×C₇

Q₈

yes

—

C₄⋊Dic₇

—

C₄⋊Dic₇

—

Dic₇⋊C₄

—

C₄×Dic₇

—

C₄×Dic₇

—

C₄×Dic₇

—

C₄×Dic₇

—

—

—

—

—

—

D₇

yes

characteristic

C₄³

—

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

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

#41

yes

—

C₂×Dic₇

#41

yes

—

C₂×C₄⋊C₄

—

C₂×C₄⋊C₄

—

C₂×C₄⋊C₄

—

C₂×C₄⋊C₄

C₂×D₇

yes

—

C₂×C₄²

—

C₂×C₄²

—

C₂×C₄²

—

C₂×C₄²

—

C₂×C₄²

C₂×D₇

yes

—

—

—

—

C₂×D₇

yes

—

#95

C₂×D₇

yes

—

C₂²×C₇

C₂²×Q₈

yes

characteristic

C₂²×C₇

C₂×#41

yes

characteristic

C₂²×C₇

C₂×#41

yes

characteristic

C₂²×C₇

C₂×#41

yes

—

C₂²×C₇

C₂×#41

yes

—

C₄×C₇

—

C₄×C₇

—

C₄×C₇

—

C₄×C₇

#125

yes

—

Dic₇

—

Dic₇

—

Dic₇

—

C₂²×C₄

—

C₂²×C₄

—

C₂²×C₄

—

C₂²×C₄

C₂²×D₇

yes

—

C₂²×C₄

C₂²×D₇

yes

characteristic

C₂²×C₄

C₂²×D₇

yes

characteristic

100

C₂²×C₄

C₂²×D₇

yes

characteristic

101

C₄⋊C₄

—

102

C₄⋊C₄

—

103

C₄⋊C₄

—

104

C₄⋊C₄

—

105

C₄²

—

106

C₄²

—

107

C₄²

—

108

C₄²

—

109

C₄²

Dic₁₄

yes

—

110

C₂×C₇

C₂×#125

yes

characteristic

111

C₂×C₇

#529

yes

—

112

C₂×C₇

#529

yes

—

113

C₂×C₇

#533

yes

characteristic

114

C₂×C₇

#533

yes

characteristic

115

C₂³

C₂³×D₇

yes

characteristic

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

—

126

C₂×C₄

—

127

C₂×C₄

—

128

C₂×C₄

—

129

C₂×C₄

—

130

C₂×C₄

C₂×Dic₁₄

yes

—

131

C₂×C₄

C₂×Dic₁₄

yes

—

132

C₂×C₄

C₂×Dic₁₄

yes

—

133

C₂×C₄

#1159

yes

—

134

C₂×C₄

#1161

yes

—

135

C₇

#2864

yes

characteristic, complemented, Sylow

136

C₂²

C₂²×Dic₁₄

yes

characteristic

137

C₂²

C₂×#1159

yes

characteristic

138

C₂²

C₂×#1161

yes

characteristic

139

C₂²

D₇×#41

yes

—

140

C₂²

D₇×#41

yes

—

141

C₄

—

142

C₄

—

143

C₄

—

144

C₄

—

145

C₄

—

146

C₄

—

147

C₄

#6453

yes

—

148

C₂

C₂×#6453

yes

characteristic

149

C₂

#90208

yes

characteristic

150

C₂

#90208

yes

characteristic

151

C₂

#90371

yes

—

152

C₂

#90382

yes

—

	subgroup	G/H
center, Frattini	C₂³	C₂³×D₇
commutator, socle, prime radical, cosocle	C₂³×C₇	C₂⁴
Fitting	C₇×#382	C₂
hypercenter	#382	D₇
hypocenter	C₇	#2864
LCS	G ▷ C₂³×C₇ ▷ C₇
UCS	C₁ ◁ C₂³ ◁ #382

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 ◁ #382

N	H ≅ G/N	type
135. C₇	#2864	semidirect

H ≅ G/N

type

135. C₇

#2864

semidirect

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

subgroup	index
C₇×#382	2
#89757	2
#89745	2
#89708	2
C₄²×Dic₇	2
#89427	2
#89426	2
#2864	7

subgroup

index

C₄²×Dic₇

Subgroups — #11731667

Conjugacy classes of subgroups

Complements

Maximal subgroups