Group #11731800 — subgroups

Subgroups — #11731800

Order 896.

472 conjugacy classes of subgroups; only the 197 normal subgroups are shown.

normal subgroups only

This group has 472 conjugacy classes of subgroups — too many to draw. Only its 197 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	C₂×#6453	C₂	1	2	yes	maximal
2	C₇×#384	C₂	1	1	yes	maximal, characteristic
3	#89757	C₂	1	4	yes	maximal
4	#89749	C₂	1	2	yes	maximal
5	#89745	C₂	1	4	yes	maximal
6	#89707	C₂	1	1	yes	maximal, characteristic
7	C₄²×Dic₇	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×C₄⋊Dic₇	C₂²	1	4	yes	—
14	C₂×Dic₇⋊C₄	C₂²	1	4	yes	—
15	C₂×C₄×Dic₇	C₂²	1	2	yes	—
16	C₂×C₄×Dic₇	C₂²	1	2	yes	—
17	C₂×C₄×Dic₇	C₂²	1	1	yes	characteristic
18	C₂×C₄×Dic₇	C₂²	1	1	yes	characteristic
19	#6453	—	2	8	no	—
20	C₇⋊#95	C₂²	1	8	yes	—
21	#2859	—	7	1	no	maximal, Sylow
22	C₂²×C₄×C₇	C₂³	1	4	yes	—
23	C₂²×C₄×C₇	C₂³	1	1	yes	characteristic
24	C₂²×C₄×C₇	C₂³	1	2	yes	—
25	C₂²×Dic₇	C₂³	1	4	yes	—
26	C₂²×Dic₇	C₂³	1	1	yes	characteristic
27	C₂²×Dic₇	C₂³	1	1	yes	characteristic
28	C₂²×Dic₇	C₂³	1	2	yes	—
29	C₇×C₄⋊C₄	—	2	4	no	—
30	C₇×C₄⋊C₄	—	2	8	no	—
31	C₄²×C₇	Q₈	1	4	yes	—
32	C₄⋊Dic₇	—	2	4	no	—
33	C₄⋊Dic₇	—	2	8	no	—
34	C₄⋊Dic₇	—	2	8	no	—
35	Dic₇⋊C₄	—	2	8	no	—
36	C₄×Dic₇	—	2	4	no	—
37	C₄×Dic₇	—	2	4	no	—
38	C₄×Dic₇	D₄	1	4	yes	—
39	C₄×Dic₇	Q₈	1	4	yes	—
40	C₂×#125	—	7	2	no	—
41	#389	—	7	4	no	—
42	#389	—	7	4	no	—
43	#384	—	7	1	no	—
44	#384	D₇	1	1	yes	characteristic
45	#382	—	7	2	no	—
46	C₄³	—	7	1	no	—
47	C₂³×C₇	C₂⁴	1	1	yes	characteristic
48	C₂×C₄×C₇	—	2	4	no	—
49	C₂×C₄×C₇	—	2	8	no	—
50	C₂×C₄×C₇	C₂×D₄	1	2	yes	—
51	C₂×C₄×C₇	C₂×Q₈	1	4	yes	—
52	C₂×C₄×C₇	C₂×Q₈	1	2	yes	—
53	C₂×C₄×C₇	C₂×Q₈	1	2	yes	—
54	C₂×C₄×C₇	#41	1	4	yes	—
55	C₂×C₄×C₇	#41	1	4	yes	—
56	C₂×Dic₇	—	2	2	no	—
57	C₂×Dic₇	—	2	2	no	—
58	C₂×Dic₇	—	2	2	no	—
59	C₂×Dic₇	—	2	2	no	—
60	C₂×Dic₇	—	2	4	no	—
61	C₂×Dic₇	—	2	8	no	—
62	C₂×Dic₇	C₂×D₄	1	2	yes	—
63	C₂×Dic₇	C₂×D₄	1	2	yes	—
64	C₂×Dic₇	C₂×Q₈	1	4	yes	—
65	#125	—	14	8	no	—
66	C₂×C₄⋊C₄	—	7	4	no	—
67	C₂×C₄⋊C₄	—	7	2	no	—
68	C₂×C₄⋊C₄	—	7	4	no	—
69	C₂×C₄⋊C₄	—	7	4	no	—
70	C₂×C₄⋊C₄	C₂×D₇	1	2	yes	—
71	C₂×C₄⋊C₄	C₂×D₇	1	4	yes	—
72	C₂×C₄²	—	7	1	no	—
73	C₂×C₄²	—	7	1	no	—
74	C₂×C₄²	—	7	2	no	—
75	C₂×C₄²	—	7	2	no	—
76	C₂×C₄²	C₂×D₇	1	1	yes	characteristic
77	#95	—	7	8	no	—
78	C₂²×C₇	C₂²×D₄	1	1	yes	characteristic
79	C₂²×C₇	C₂²×Q₈	1	1	yes	characteristic
80	C₂²×C₇	C₂²×Q₈	1	1	yes	characteristic
81	C₂²×C₇	C₂×#41	1	2	yes	—
82	C₂²×C₇	C₂×#41	1	2	yes	—
83	C₄×C₇	—	4	4	no	—
84	C₄×C₇	#125	1	8	yes	complemented
85	C₄×C₇	C₄⋊Q₈	1	4	yes	—
86	Dic₇	—	2	2	no	—
87	Dic₇	—	2	2	no	—
88	Dic₇	—	2	4	no	—
89	Dic₇	—	4	4	no	—
90	C₂²×C₄	—	7	2	no	—
91	C₂²×C₄	—	7	1	no	—
92	C₂²×C₄	—	7	1	no	—
93	C₂²×C₄	—	7	4	no	—
94	C₂²×C₄	C₂²×D₇	1	4	yes	—
95	C₂²×C₄	C₂²×D₇	1	1	yes	characteristic
96	C₂²×C₄	C₂²×D₇	1	2	yes	—
97	C₄⋊C₄	—	2	4	no	—
98	C₄⋊C₄	—	2	8	no	—
99	C₄⋊C₄	—	14	8	no	—
100	C₄⋊C₄	—	14	8	no	—
101	C₄⋊C₄	—	14	4	no	—
102	C₄⋊C₄	—	14	8	no	—
103	C₄²	—	7	4	no	—
104	C₄²	—	7	4	no	—
105	C₄²	—	14	4	no	—
106	C₄²	—	14	4	no	—
107	C₄²	Dic₁₄	1	4	yes	—
108	C₂×C₇	C₂×#125	1	2	yes	—
109	C₂×C₇	C₂×C₄⋊Q₈	1	1	yes	characteristic
110	C₂×C₇	#532	1	1	yes	characteristic
111	C₂×C₇	#532	1	1	yes	characteristic
112	C₂×C₇	#533	1	2	yes	—
113	C₂³	C₂³×D₇	1	1	yes	characteristic
114	C₂×C₄	—	2	4	no	—
115	C₂×C₄	—	2	8	no	—
116	C₂×C₄	—	7	2	no	—
117	C₂×C₄	—	7	2	no	—
118	C₂×C₄	—	7	4	no	—
119	C₂×C₄	—	14	2	no	—
120	C₂×C₄	—	14	2	no	—
121	C₂×C₄	—	14	2	no	—
122	C₂×C₄	—	14	2	no	—
123	C₂×C₄	—	14	4	no	—
124	C₂×C₄	—	14	8	no	—
125	C₂×C₄	C₂×Dic₁₄	1	2	yes	—
126	C₂×C₄	C₂×Dic₁₄	1	4	yes	—
127	C₂×C₄	D₄×D₇	1	2	yes	—
128	C₂×C₄	#1159	1	4	yes	—
129	C₂×C₄	Q₈×D₇	1	2	yes	—
130	C₂×C₄	#1161	1	4	yes	—
131	C₇	#2859	1	1	yes	characteristic, complemented, Sylow
132	C₂²	C₂²×Dic₁₄	1	1	yes	characteristic
133	C₂²	C₂×D₄×D₇	1	1	yes	characteristic
134	C₂²	C₂×#1159	1	2	yes	—
135	C₂²	C₂×Q₈×D₇	1	1	yes	characteristic
136	C₂²	C₂×#1161	1	2	yes	—
137	C₄	—	4	4	no	—
138	C₄	—	14	2	no	—
139	C₄	—	14	2	no	—
140	C₄	—	14	4	no	—
141	C₄	—	28	4	no	—
142	C₄	#6451	1	4	yes	—
143	C₄	#6453	1	8	yes	complemented
144	C₂	C₂×#6451	1	1	yes	characteristic
145	C₂	C₂×#6453	1	2	yes	—
146	C₂	#90407	1	1	yes	characteristic
147	C₂	#90416	1	1	yes	characteristic
148	C₂	#90419	1	2	yes	—

subgroup

G/H

class size

multiplicity

normal?

properties

C₂×#6453

C₂

yes

maximal

C₇×#384

C₂

yes

maximal, characteristic

#89757

C₂

yes

maximal

#89749

C₂

yes

maximal

#89745

C₂

yes

maximal

#89707

C₂

yes

maximal, characteristic

C₄²×Dic₇

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×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

#6453

—

C₇⋊#95

C₂²

yes

—

#2859

—

maximal, Sylow

C₂²×C₄×C₇

C₂³

yes

—

C₂²×C₄×C₇

C₂³

yes

characteristic

C₂²×C₄×C₇

C₂³

yes

—

C₂²×Dic₇

C₂³

yes

—

C₂²×Dic₇

C₂³

yes

characteristic

C₂²×Dic₇

C₂³

yes

characteristic

C₂²×Dic₇

C₂³

yes

—

C₇×C₄⋊C₄

—

C₇×C₄⋊C₄

—

C₄²×C₇

Q₈

yes

—

C₄⋊Dic₇

—

C₄⋊Dic₇

—

C₄⋊Dic₇

—

Dic₇⋊C₄

—

C₄×Dic₇

—

C₄×Dic₇

—

C₄×Dic₇

D₄

yes

—

C₄×Dic₇

Q₈

yes

—

—

—

—

—

D₇

yes

characteristic

#382

—

C₄³

—

C₂³×C₇

C₂⁴

yes

characteristic

C₂×C₄×C₇

—

C₂×C₄×C₇

—

C₂×C₄×C₇

C₂×D₄

yes

—

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×D₄

yes

—

C₂×Dic₇

C₂×D₄

yes

—

C₂×Dic₇

C₂×Q₈

yes

—

#125

—

C₂×C₄⋊C₄

—

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×C₄²

—

C₂×C₄²

C₂×D₇

yes

characteristic

#95

—

C₂²×C₇

C₂²×D₄

yes

characteristic

C₂²×C₇

C₂²×Q₈

yes

characteristic

C₂²×C₇

C₂²×Q₈

yes

characteristic

C₂²×C₇

C₂×#41

yes

—

C₂²×C₇

C₂×#41

yes

—

C₄×C₇

—

C₄×C₇

#125

yes

complemented

C₄×C₇

C₄⋊Q₈

yes

—

Dic₇

—

Dic₇

—

Dic₇

—

Dic₇

—

C₂²×C₄

—

C₂²×C₄

—

C₂²×C₄

—

C₂²×C₄

—

C₂²×C₄

C₂²×D₇

yes

—

C₂²×C₄

C₂²×D₇

yes

characteristic

C₂²×C₄

C₂²×D₇

yes

—

C₄⋊C₄

—

C₄⋊C₄

—

C₄⋊C₄

—

100

C₄⋊C₄

—

101

C₄⋊C₄

—

102

C₄⋊C₄

—

103

C₄²

—

104

C₄²

—

105

C₄²

—

106

C₄²

—

107

C₄²

Dic₁₄

yes

—

108

C₂×C₇

C₂×#125

yes

—

109

C₂×C₇

C₂×C₄⋊Q₈

yes

characteristic

110

C₂×C₇

#532

yes

characteristic

111

C₂×C₇

#532

yes

characteristic

112

C₂×C₇

#533

yes

—

113

C₂³

C₂³×D₇

yes

characteristic

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

D₄×D₇

yes

—

128

C₂×C₄

#1159

yes

—

129

C₂×C₄

Q₈×D₇

yes

—

130

C₂×C₄

#1161

yes

—

131

C₇

#2859

yes

characteristic, complemented, Sylow

132

C₂²

C₂²×Dic₁₄

yes

characteristic

133

C₂²

C₂×D₄×D₇

yes

characteristic

134

C₂²

C₂×#1159

yes

—

135

C₂²

C₂×Q₈×D₇

yes

characteristic

136

C₂²

C₂×#1161

yes

—

137

C₄

—

138

C₄

—

139

C₄

—

140

C₄

—

141

C₄

—

142

C₄

#6451

yes

—

143

C₄

#6453

yes

complemented

144

C₂

C₂×#6451

yes

characteristic

145

C₂

C₂×#6453

yes

—

146

C₂

#90407

yes

characteristic

147

C₂

#90416

yes

characteristic

148

C₂

#90419

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	#2859
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
84. C₄×C₇	#125	semidirect
131. C₇	#2859	semidirect
143. C₄	#6453	semidirect

H ≅ G/N

type

84. C₄×C₇

#125

semidirect

131. C₇

#2859

semidirect

143. C₄

#6453

semidirect

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

subgroup	index
C₂×#6453	2
C₇×#384	2
#89757	2
#89749	2
#89745	2
#89707	2
C₄²×Dic₇	2
#2859	7

subgroup

index

C₄²×Dic₇

Subgroups — #11731800

Conjugacy classes of subgroups

Complements

Maximal subgroups