Group #11726751 — subgroups

Subgroups — #11726751

Order 896.

272 conjugacy classes of subgroups; only the 87 normal subgroups are shown.

normal subgroups only

This group has 272 conjugacy classes of subgroups — too many to draw. Only its 87 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×#6450	C₂	1	1	yes	maximal, characteristic
2	C₂×#6398	C₂	1	2	yes	maximal
3	#89890	C₂	1	2	yes	maximal
4	C₇×#343	C₂	1	1	yes	maximal, characteristic
5	#89325	C₂	1	1	yes	maximal, characteristic
6	C₂²×Dic₁₄	C₂²	1	1	yes	characteristic
7	C₂×C₇×M₁₆	C₂²	1	2	yes	—
8	C₂×C₇×C₄⋊C₄	C₂²	1	1	yes	characteristic
9	C₂×Dic₇⋊C₄	C₄	1	2	yes	—
10	C₂×C₄×Dic₇	C₄	1	2	yes	—
11	C₂×C₄×Dic₇	C₂²	1	1	yes	characteristic
12	C₂×#1136	C₂²	1	2	yes	—
13	#6450	—	2	2	no	—
14	#6450	—	2	2	no	—
15	#6398	—	2	4	no	—
16	Dic₇⋊C₈	—	2	2	no	—
17	Dic₇⋊C₈	—	2	2	no	—
18	#1801	—	7	1	no	maximal, Sylow
19	C₂²×C₄×C₇	C₂×C₄	1	2	yes	—
20	C₂²×C₄×C₇	C₂³	1	1	yes	characteristic
21	C₂²×Dic₇	C₂×C₄	1	1	yes	characteristic
22	C₂²×Dic₇	C₂×C₄	1	2	yes	—
23	C₂²×Dic₇	C₂×C₄	1	1	yes	characteristic
24	C₂×Dic₁₄	—	2	2	no	—
25	C₂×Dic₁₄	—	2	2	no	—
26	C₂×Dic₁₄	D₄	1	2	yes	—
27	C₂×Dic₁₄	Q₈	1	2	yes	—
28	C₇×M₁₆	—	2	2	no	—
29	C₇×M₁₆	—	2	2	no	—
30	C₂×C₇×C₈	—	2	2	no	—
31	C₇×C₄⋊C₄	—	2	2	no	—
32	Dic₇⋊C₄	—	2	4	no	—
33	C₄×Dic₇	—	2	2	no	—
34	C₄×Dic₇	—	2	2	no	—
35	C₄×Dic₇	D₄	1	2	yes	—
36	C₄×Dic₇	D₄	1	2	yes	—
37	#1136	—	2	2	no	—
38	#1136	—	2	2	no	—
39	C₂×C₇⋊C₈	—	2	2	no	—
40	C₂×C₄×Q₈	—	7	1	no	—
41	#423	—	7	2	no	—
42	C₂×#101	—	7	2	no	—
43	#343	—	7	1	no	—
44	#343	D₇	1	1	yes	characteristic
45	C₂³×C₇	C₂²×C₄	1	1	yes	characteristic
46	C₂×C₄×C₇	—	2	2	no	—
47	C₂×C₄×C₇	—	4	2	no	—
48	C₂×C₄×C₇	C₂×D₄	1	1	yes	characteristic
49	C₂×C₄×C₇	C₂×D₄	1	2	yes	—
50	C₂×C₄×C₇	C₂×Q₈	1	1	yes	characteristic
51	C₂×C₄×C₇	#41	1	2	yes	—
52	C₂×Dic₇	—	2	2	no	—
53	C₂×Dic₇	—	2	2	no	—
54	C₂×Dic₇	—	2	2	no	—
55	C₂×Dic₇	—	4	1	no	—
56	C₂×Dic₇	C₂²⋊C₄	1	1	yes	characteristic
57	C₂×Dic₇	C₂²⋊C₄	1	2	yes	—
58	C₂×Dic₇	C₂²⋊C₄	1	1	yes	characteristic
59	C₂×Dic₇	C₄⋊C₄	1	4	yes	—
60	Dic₁₄	—	2	4	no	—
61	Dic₁₄	—	2	4	no	—
62	C₇×C₈	—	4	2	no	—
63	C₇⋊C₈	—	4	2	no	—
64	C₂²×Q₈	—	7	1	no	—
65	C₂×M₁₆	—	7	2	no	—
66	C₂×M₁₆	C₂×D₇	1	2	yes	—
67	C₄×Q₈	—	14	2	no	—
68	C₄×Q₈	—	14	2	no	—
69	C₂×C₄⋊C₄	—	7	2	no	—
70	C₂×C₄⋊C₄	C₂×D₇	1	1	yes	characteristic
71	C₂×C₄²	—	7	1	no	—
72	C₂×C₄²	—	7	2	no	—
73	C₄⋊C₈	—	14	2	no	—
74	C₄⋊C₈	—	14	2	no	—
75	#101	—	14	4	no	—
76	C₂²×C₇	—	2	1	no	—
77	C₂²×C₇	—	2	1	no	—
78	C₂²×C₇	C₂×C₂²⋊C₄	1	1	yes	characteristic
79	C₂²×C₇	C₂×C₄⋊C₄	1	1	yes	characteristic
80	C₂²×C₇	#117	1	1	yes	characteristic
81	C₄×C₇	—	4	2	no	—
82	C₄×C₇	#121	1	2	yes	—
83	C₄×C₇	C₂²⋊Q₈	1	2	yes	—
84	Dic₇	—	2	4	no	—
85	Dic₇	—	2	1	no	—
86	Dic₇	—	2	2	no	—
87	Dic₇	#101	1	2	yes	complemented
88	C₂×Q₈	—	7	2	no	—
89	C₂×Q₈	—	7	2	no	—
90	C₂×Q₈	—	14	2	no	—
91	C₂×Q₈	—	14	2	no	—
92	C₂²×C₄	—	7	1	no	—
93	C₂²×C₄	—	7	2	no	—
94	C₂²×C₄	—	7	1	no	—
95	C₂²×C₄	C₄×D₇	1	2	yes	—
96	C₂²×C₄	C₂²×D₇	1	1	yes	characteristic
97	M₁₆	—	2	2	no	—
98	M₁₆	—	2	2	no	—
99	M₁₆	—	14	2	no	—
100	M₁₆	—	14	2	no	—
101	C₂×C₈	—	2	2	no	—
102	C₂×C₈	—	14	2	no	—
103	C₄⋊C₄	—	2	2	no	—
104	C₄⋊C₄	—	14	4	no	—
105	C₄²	—	7	2	no	—
106	C₄²	—	7	2	no	—
107	C₄²	—	14	2	no	—
108	C₄²	—	14	2	no	—
109	C₂×C₇	—	2	1	no	—
110	C₂×C₇	—	2	1	no	—
111	C₂×C₇	#380	1	1	yes	characteristic
112	C₂×C₇	C₂×#101	1	1	yes	characteristic
113	C₂×C₇	#413	1	1	yes	characteristic
114	C₂³	C₂×C₄×D₇	1	1	yes	characteristic
115	Q₈	—	14	4	no	—
116	Q₈	—	14	4	no	—
117	C₂×C₄	—	2	2	no	—
118	C₂×C₄	—	4	2	no	—
119	C₂×C₄	—	7	1	no	—
120	C₂×C₄	—	7	1	no	—
121	C₂×C₄	—	7	2	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	—	28	1	no	—
127	C₂×C₄	#1157	1	2	yes	—
128	C₂×C₄	D₄×D₇	1	1	yes	characteristic
129	C₂×C₄	D₄×D₇	1	2	yes	—
130	C₂×C₄	Q₈×D₇	1	1	yes	characteristic
131	C₈	—	4	2	no	—
132	C₈	—	28	2	no	—
133	C₇	#1801	1	1	yes	characteristic, complemented, Sylow
134	C₂²	—	2	1	no	—
135	C₂²	—	2	1	no	—
136	C₂²	#6435	1	1	yes	characteristic
137	C₂²	D₇×C₂²⋊C₄	1	1	yes	characteristic
138	C₂²	D₇×C₄⋊C₄	1	1	yes	characteristic
139	C₄	—	4	2	no	—
140	C₄	—	7	2	no	—
141	C₄	—	14	4	no	—
142	C₄	—	14	2	no	—
143	C₄	—	14	1	no	—
144	C₄	#6447	1	2	yes	—
145	C₄	#6459	1	2	yes	—
146	C₂	—	2	1	no	—
147	C₂	—	2	1	no	—
148	C₂	#89439	1	1	yes	characteristic
149	C₂	#89517	1	1	yes	characteristic
150	C₂	D₇×#101	1	1	yes	characteristic

subgroup

G/H

class size

multiplicity

normal?

properties

C₂×#6450

C₂

yes

maximal, characteristic

C₂×#6398

C₂

yes

maximal

#89890

C₂

yes

maximal

C₇×#343

C₂

yes

maximal, characteristic

#89325

C₂

yes

maximal, characteristic

C₂²×Dic₁₄

C₂²

yes

characteristic

C₂×C₇×M₁₆

C₂²

yes

—

C₂×C₇×C₄⋊C₄

C₂²

yes

characteristic

C₂×Dic₇⋊C₄

C₄

yes

—

C₂×C₄×Dic₇

C₄

yes

—

C₂×C₄×Dic₇

C₂²

yes

characteristic

C₂×#1136

C₂²

yes

—

#6450

—

#6450

—

#6398

—

Dic₇⋊C₈

—

Dic₇⋊C₈

—

#1801

—

maximal, Sylow

C₂²×C₄×C₇

C₂×C₄

yes

—

C₂²×C₄×C₇

C₂³

yes

characteristic

C₂²×Dic₇

C₂×C₄

yes

characteristic

C₂²×Dic₇

C₂×C₄

yes

—

C₂²×Dic₇

C₂×C₄

yes

characteristic

C₂×Dic₁₄

—

C₂×Dic₁₄

—

C₂×Dic₁₄

D₄

yes

—

C₂×Dic₁₄

Q₈

yes

—

C₇×M₁₆

—

C₇×M₁₆

—

C₂×C₇×C₈

—

C₇×C₄⋊C₄

—

Dic₇⋊C₄

—

C₄×Dic₇

—

C₄×Dic₇

—

C₄×Dic₇

D₄

yes

—

C₄×Dic₇

D₄

yes

—

#1136

—

#1136

—

C₂×C₇⋊C₈

—

C₂×C₄×Q₈

—

—

—

—

D₇

yes

characteristic

C₂³×C₇

C₂²×C₄

yes

characteristic

C₂×C₄×C₇

—

C₂×C₄×C₇

—

C₂×C₄×C₇

C₂×D₄

yes

characteristic

C₂×C₄×C₇

C₂×D₄

yes

—

C₂×C₄×C₇

C₂×Q₈

yes

characteristic

C₂×C₄×C₇

#41

yes

—

C₂×Dic₇

—

C₂×Dic₇

—

C₂×Dic₇

—

C₂×Dic₇

—

C₂×Dic₇

C₂²⋊C₄

yes

characteristic

C₂×Dic₇

C₂²⋊C₄

yes

—

C₂×Dic₇

C₂²⋊C₄

yes

characteristic

C₂×Dic₇

C₄⋊C₄

yes

—

Dic₁₄

—

Dic₁₄

—

C₇×C₈

—

C₇⋊C₈

—

C₂²×Q₈

—

C₂×M₁₆

—

C₂×M₁₆

C₂×D₇

yes

—

C₄×Q₈

—

C₄×Q₈

—

C₂×C₄⋊C₄

—

C₂×C₄⋊C₄

C₂×D₇

yes

characteristic

C₂×C₄²

—

C₂×C₄²

—

C₄⋊C₈

—

C₄⋊C₈

—

#101

—

C₂²×C₇

—

C₂²×C₇

—

C₂²×C₇

C₂×C₂²⋊C₄

yes

characteristic

C₂²×C₇

C₂×C₄⋊C₄

yes

characteristic

C₂²×C₇

#117

yes

characteristic

C₄×C₇

—

C₄×C₇

#121

yes

—

C₄×C₇

C₂²⋊Q₈

yes

—

Dic₇

—

Dic₇

—

Dic₇

—

Dic₇

#101

yes

complemented

C₂×Q₈

—

C₂×Q₈

—

C₂×Q₈

—

C₂×Q₈

—

C₂²×C₄

—

C₂²×C₄

—

C₂²×C₄

—

C₂²×C₄

C₄×D₇

yes

—

C₂²×C₄

C₂²×D₇

yes

characteristic

M₁₆

—

M₁₆

—

M₁₆

—

100

M₁₆

—

101

C₂×C₈

—

102

C₂×C₈

—

103

C₄⋊C₄

—

104

C₄⋊C₄

—

105

C₄²

—

106

C₄²

—

107

C₄²

—

108

C₄²

—

109

C₂×C₇

—

110

C₂×C₇

—

111

C₂×C₇

#380

yes

characteristic

112

C₂×C₇

C₂×#101

yes

characteristic

113

C₂×C₇

#413

yes

characteristic

114

C₂³

C₂×C₄×D₇

yes

characteristic

115

Q₈

—

116

Q₈

—

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

#1157

yes

—

128

C₂×C₄

D₄×D₇

yes

characteristic

129

C₂×C₄

D₄×D₇

yes

—

130

C₂×C₄

Q₈×D₇

yes

characteristic

131

C₈

—

132

C₈

—

133

C₇

#1801

yes

characteristic, complemented, Sylow

134

C₂²

—

135

C₂²

—

136

C₂²

#6435

yes

characteristic

137

C₂²

D₇×C₂²⋊C₄

yes

characteristic

138

C₂²

D₇×C₄⋊C₄

yes

characteristic

139

C₄

—

140

C₄

—

141

C₄

—

142

C₄

—

143

C₄

—

144

C₄

#6447

yes

—

145

C₄

#6459

yes

—

146

C₂

—

147

C₂

—

148

C₂

#89439

yes

characteristic

149

C₂

#89517

yes

characteristic

150

C₂

D₇×#101

yes

characteristic

	subgroup	G/H
center	C₂²	D₇×C₂²⋊C₄
commutator, prime radical	C₂³×C₇	C₂²×C₄
Frattini	C₂²×C₄	C₂²×D₇
Fitting	C₇×#343	C₂
socle	C₂²×C₇	C₂×C₂²⋊C₄
cosocle	C₂²×C₄×C₇	C₂³
hypercenter	#343	D₇
hypocenter	C₇	#1801
LCS	G ▷ C₂³×C₇ ▷ C₂×C₇ ▷ C₇
UCS	C₁ ◁ C₂² ◁ C₂²×C₄ ◁ #343

subgroup

G/H

center

C₂²

D₇×C₂²⋊C₄

commutator, prime radical

C₂³×C₇

C₂²×C₄

Frattini

C₂²×C₄

C₂²×D₇

Fitting

C₇×#343

C₂

socle

C₂²×C₇

C₂×C₂²⋊C₄

cosocle

C₂²×C₄×C₇

C₂³

D₇

C₇

G ▷ C₂³×C₇ ▷ C₂×C₇ ▷ C₇

UCS

C₁ ◁ C₂² ◁ C₂²×C₄ ◁ #343

N	H ≅ G/N	type
87. Dic₇	#101	semidirect
133. C₇	#1801	semidirect

H ≅ G/N

type

87. Dic₇

#101

semidirect

133. C₇

#1801

semidirect

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

subgroup	index
C₂×#6450	2
C₂×#6398	2
#89890	2
C₇×#343	2
#89325	2
#1801	7

subgroup

index

Subgroups — #11726751

Conjugacy classes of subgroups

Complements

Maximal subgroups