Group #11727445 — subgroups

Subgroups — #11727445

Order 896.

370 conjugacy classes of subgroups; only the 97 normal subgroups are shown.

normal subgroups only

This group has 370 conjugacy classes of subgroups — too many to draw. Only its 97 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	C₂²×#1144	C₂	1	2	yes	maximal
2	C₂×#6432	C₂	1	1	yes	maximal, characteristic
3	C₇×#391	C₂	1	1	yes	maximal, characteristic
4	C₂×#6384	C₂	1	2	yes	maximal
5	#89319	C₂	1	1	yes	maximal, characteristic
6	C₂²×C₇×Q₈	C₂²	1	1	yes	characteristic
7	C₂²×Dic₁₄	C₂²	1	2	yes	—
8	C₂×C₇×C₄⋊C₄	C₂²	1	1	yes	characteristic
9	C₂×C₄²×C₇	C₂²	1	1	yes	characteristic
10	C₂×#1144	—	2	4	no	—
11	C₂×#1144	—	2	4	no	—
12	C₂×#1144	—	2	4	no	—
13	C₂×C₄⋊Dic₇	—	2	2	no	—
14	C₂²×C₇⋊C₈	C₂²	1	2	yes	—
15	#6432	—	2	4	no	—
16	C₇×#95	—	2	2	no	—
17	#6384	—	2	4	no	—
18	#2016	—	7	1	no	maximal, Sylow
19	C₂×C₇×Q₈	—	2	1	no	—
20	C₂×C₇×Q₈	—	2	2	no	—
21	C₂×C₇×Q₈	—	2	2	no	—
22	C₂×C₇×Q₈	D₄	1	2	yes	—
23	C₂²×C₄×C₇	—	2	1	no	—
24	C₂²×C₄×C₇	—	2	1	no	—
25	C₂²×C₄×C₇	D₄	1	2	yes	—
26	C₂²×C₄×C₇	C₂³	1	1	yes	characteristic
27	C₂²×Dic₇	—	2	2	no	—
28	C₂×Dic₁₄	—	2	2	no	—
29	C₂×Dic₁₄	—	2	2	no	—
30	C₂×Dic₁₄	—	2	2	no	—
31	C₂×Dic₁₄	—	2	2	no	—
32	C₂×Dic₁₄	—	2	2	no	—
33	C₂×Dic₁₄	D₄	1	4	yes	—
34	C₇×C₄⋊C₄	—	2	2	no	—
35	C₄²×C₇	—	2	2	no	—
36	#1144	—	2	16	no	—
37	C₄⋊Dic₇	—	2	8	no	—
38	C₂×C₇⋊C₈	—	2	2	no	—
39	C₂×C₇⋊C₈	—	2	2	no	—
40	C₂×C₇⋊C₈	D₄	1	4	yes	—
41	C₂²×Q₁₆	—	7	2	no	—
42	C₂×C₄⋊Q₈	—	7	1	no	—
43	C₂×#103	—	7	2	no	—
44	#391	D₇	1	1	yes	characteristic
45	#336	—	7	1	no	—
46	C₂³×C₇	C₂×D₄	1	1	yes	characteristic
47	C₇×Q₈	—	2	4	no	—
48	C₇×Q₈	—	4	2	no	—
49	C₂×C₄×C₇	—	2	2	no	—
50	C₂×C₄×C₇	—	2	2	no	—
51	C₂×C₄×C₇	—	2	2	no	—
52	C₂×C₄×C₇	—	4	1	no	—
53	C₂×C₄×C₇	—	4	1	no	—
54	C₂×C₄×C₇	—	4	1	no	—
55	C₂×C₄×C₇	—	4	2	no	—
56	C₂×C₄×C₇	Q₁₆	1	4	yes	complemented
57	C₂×C₄×C₇	C₂×D₄	1	1	yes	characteristic
58	C₂×C₄×C₇	C₂×D₄	1	2	yes	—
59	C₂×C₄×C₇	C₂×D₄	1	2	yes	—
60	C₂×C₄×C₇	#41	1	1	yes	characteristic
61	C₂×Dic₇	—	2	4	no	—
62	C₂×Dic₇	—	4	2	no	—
63	C₂×Dic₇	—	4	2	no	—
64	Dic₁₄	—	2	4	no	—
65	Dic₁₄	—	2	4	no	—
66	Dic₁₄	—	4	2	no	—
67	Dic₁₄	—	4	2	no	—
68	C₇⋊C₈	—	2	4	no	—
69	C₂²×Q₈	—	7	2	no	—
70	C₂²×Q₈	C₂×D₇	1	1	yes	characteristic
71	C₂×Q₁₆	—	14	4	no	—
72	C₂×Q₁₆	—	14	4	no	—
73	C₂×Q₁₆	—	14	4	no	—
74	C₂²×C₈	—	7	2	no	—
75	C₄⋊Q₈	—	14	4	no	—
76	C₂×C₄⋊C₄	—	14	2	no	—
77	C₂×C₄⋊C₄	C₂×D₇	1	1	yes	characteristic
78	C₂×C₄²	C₂×D₇	1	1	yes	characteristic
79	#103	—	14	4	no	—
80	#95	—	2	2	no	—
81	C₂²×C₇	#120	1	1	yes	characteristic
82	C₂²×C₇	#121	1	1	yes	characteristic
83	C₂²×C₇	#127	1	1	yes	characteristic
84	C₂²×C₇	C₂×Q₁₆	1	2	yes	—
85	C₂²×C₇	#137	1	2	yes	—
86	C₄×C₇	—	2	4	no	—
87	C₄×C₇	—	4	2	no	—
88	C₄×C₇	—	8	1	no	—
89	C₄×C₇	#120	1	2	yes	—
90	C₄×C₇	#121	1	2	yes	—
91	Dic₇	—	4	4	no	—
92	C₂×Q₈	—	2	1	no	—
93	C₂×Q₈	—	2	2	no	—
94	C₂×Q₈	—	2	2	no	—
95	C₂×Q₈	—	7	4	no	—
96	C₂×Q₈	—	14	2	no	—
97	C₂×Q₈	—	14	2	no	—
98	C₂×Q₈	—	14	2	no	—
99	C₂×Q₈	—	14	2	no	—
100	C₂×Q₈	—	14	2	no	—
101	C₂×Q₈	D₂₈	1	2	yes	—
102	C₂²×C₄	—	2	1	no	—
103	C₂²×C₄	—	2	1	no	—
104	C₂²×C₄	—	14	2	no	—
105	C₂²×C₄	C₂²⋊D₇	1	2	yes	—
106	C₂²×C₄	C₂²×D₇	1	1	yes	characteristic
107	Q₁₆	—	14	16	no	—
108	C₂×C₈	—	7	4	no	—
109	C₂×C₈	—	14	2	no	—
110	C₂×C₈	—	14	2	no	—
111	C₄⋊C₄	—	2	2	no	—
112	C₄⋊C₄	—	14	8	no	—
113	C₄²	—	2	2	no	—
114	C₂×C₇	#392	1	1	yes	characteristic
115	C₂×C₇	#451	1	2	yes	—
116	C₂×C₇	#462	1	2	yes	—
117	C₂×C₇	#494	1	1	yes	characteristic
118	C₂×C₇	#497	1	1	yes	characteristic
119	C₂³	C₂×C₂²⋊D₇	1	1	yes	characteristic
120	Q₈	—	2	4	no	—
121	Q₈	—	4	2	no	—
122	Q₈	—	14	4	no	—
123	Q₈	—	14	4	no	—
124	Q₈	—	28	2	no	—
125	Q₈	—	28	2	no	—
126	C₂×C₄	—	2	2	no	—
127	C₂×C₄	—	2	2	no	—
128	C₂×C₄	—	2	2	no	—
129	C₂×C₄	—	4	1	no	—
130	C₂×C₄	—	4	1	no	—
131	C₂×C₄	—	4	1	no	—
132	C₂×C₄	—	4	2	no	—
133	C₂×C₄	—	14	4	no	—
134	C₂×C₄	—	28	2	no	—
135	C₂×C₄	—	28	2	no	—
136	C₂×C₄	#1144	1	4	yes	complemented
137	C₂×C₄	C₂×D₂₈	1	1	yes	characteristic
138	C₂×C₄	#1157	1	1	yes	characteristic
139	C₂×C₄	D₄×D₇	1	2	yes	—
140	C₂×C₄	D₄×D₇	1	2	yes	—
141	C₈	—	14	4	no	—
142	C₇	#2016	1	1	yes	characteristic, complemented, Sylow
143	C₂²	#6493	1	1	yes	characteristic
144	C₂²	#6500	1	1	yes	characteristic
145	C₂²	#6503	1	1	yes	characteristic
146	C₂²	C₂×#1144	1	2	yes	—
147	C₂²	#6514	1	2	yes	—
148	C₄	—	2	4	no	—
149	C₄	—	4	2	no	—
150	C₄	—	8	1	no	—
151	C₄	—	28	4	no	—
152	C₄	#6445	1	2	yes	—
153	C₄	#6447	1	2	yes	—
154	C₂	#89732	1	1	yes	characteristic
155	C₂	#89803	1	2	yes	—
156	C₂	#89822	1	2	yes	—
157	C₂	#89832	1	1	yes	characteristic
158	C₂	#89864	1	1	yes	characteristic

subgroup

G/H

class size

multiplicity

normal?

properties

C₂²×#1144

C₂

yes

maximal

C₂×#6432

C₂

yes

maximal, characteristic

C₇×#391

C₂

yes

maximal, characteristic

C₂×#6384

C₂

yes

maximal

#89319

C₂

yes

maximal, characteristic

C₂²×C₇×Q₈

C₂²

yes

characteristic

C₂²×Dic₁₄

C₂²

yes

—

C₂×C₇×C₄⋊C₄

C₂²

yes

characteristic

C₂×C₄²×C₇

C₂²

yes

—

—

—

C₂×C₄⋊Dic₇

—

C₂²×C₇⋊C₈

C₂²

yes

—

#6432

—

C₇×#95

—

#6384

—

#2016

—

maximal, Sylow

C₂×C₇×Q₈

—

C₂×C₇×Q₈

—

C₂×C₇×Q₈

—

C₂×C₇×Q₈

D₄

yes

—

C₂²×C₄×C₇

—

C₂²×C₄×C₇

—

C₂²×C₄×C₇

D₄

yes

—

C₂²×C₄×C₇

C₂³

yes

characteristic

C₂²×Dic₇

—

C₂×Dic₁₄

—

C₂×Dic₁₄

—

C₂×Dic₁₄

—

C₂×Dic₁₄

—

C₂×Dic₁₄

—

C₂×Dic₁₄

D₄

yes

—

C₇×C₄⋊C₄

—

C₄²×C₇

—

#1144

—

C₄⋊Dic₇

—

C₂×C₇⋊C₈

—

C₂×C₇⋊C₈

—

C₂×C₇⋊C₈

D₄

yes

—

C₂²×Q₁₆

—

C₂×C₄⋊Q₈

—

C₂×#103

—

#391

D₇

yes

characteristic

#336

—

C₂³×C₇

C₂×D₄

yes

characteristic

C₇×Q₈

—

C₇×Q₈

—

C₂×C₄×C₇

—

C₂×C₄×C₇

—

C₂×C₄×C₇

—

C₂×C₄×C₇

—

C₂×C₄×C₇

—

C₂×C₄×C₇

—

C₂×C₄×C₇

—

C₂×C₄×C₇

Q₁₆

yes

complemented

C₂×C₄×C₇

C₂×D₄

yes

characteristic

C₂×C₄×C₇

C₂×D₄

yes

—

C₂×C₄×C₇

C₂×D₄

yes

—

C₂×C₄×C₇

#41

yes

characteristic

C₂×Dic₇

—

C₂×Dic₇

—

C₂×Dic₇

—

Dic₁₄

—

Dic₁₄

—

Dic₁₄

—

Dic₁₄

—

C₇⋊C₈

—

C₂²×Q₈

—

C₂²×Q₈

C₂×D₇

yes

characteristic

C₂×Q₁₆

—

C₂×Q₁₆

—

C₂×Q₁₆

—

C₂²×C₈

—

C₄⋊Q₈

—

C₂×C₄⋊C₄

—

C₂×C₄⋊C₄

C₂×D₇

yes

characteristic

C₂×C₄²

C₂×D₇

yes

characteristic

#103

—

#95

—

C₂²×C₇

yes

C₂²×C₇

yes

C₂²×C₇

yes

C₂²×C₇

C₂×Q₁₆

yes

—

C₂²×C₇

#137

yes

—

C₄×C₇

—

C₄×C₇

—

C₄×C₇

—

C₄×C₇

#120

yes

—

C₄×C₇

#121

yes

—

Dic₇

—

C₂×Q₈

—

C₂×Q₈

—

C₂×Q₈

—

C₂×Q₈

—

C₂×Q₈

—

C₂×Q₈

—

C₂×Q₈

—

C₂×Q₈

—

100

C₂×Q₈

—

101

C₂×Q₈

D₂₈

yes

—

102

C₂²×C₄

—

103

C₂²×C₄

—

104

C₂²×C₄

—

105

C₂²×C₄

C₂²⋊D₇

yes

—

106

C₂²×C₄

C₂²×D₇

yes

characteristic

107

Q₁₆

—

108

C₂×C₈

—

109

C₂×C₈

—

110

C₂×C₈

—

111

C₄⋊C₄

—

112

C₄⋊C₄

—

113

C₄²

—

114

C₂×C₇

#392

yes

characteristic

115

C₂×C₇

#451

yes

—

116

C₂×C₇

#462

yes

—

117

C₂×C₇

#494

yes

characteristic

118

C₂×C₇

#497

yes

characteristic

119

C₂³

C₂×C₂²⋊D₇

yes

characteristic

120

Q₈

—

121

Q₈

—

122

Q₈

—

123

Q₈

—

124

Q₈

—

125

Q₈

—

126

C₂×C₄

—

127

C₂×C₄

—

128

C₂×C₄

—

129

C₂×C₄

—

130

C₂×C₄

—

131

C₂×C₄

—

132

C₂×C₄

—

133

C₂×C₄

—

134

C₂×C₄

—

135

C₂×C₄

—

136

C₂×C₄

#1144

yes

complemented

137

C₂×C₄

C₂×D₂₈

yes

characteristic

138

C₂×C₄

#1157

yes

characteristic

139

C₂×C₄

D₄×D₇

yes

—

140

C₂×C₄

D₄×D₇

yes

—

141

C₈

—

142

C₇

#2016

yes

characteristic, complemented, Sylow

143

C₂²

#6493

yes

characteristic

144

C₂²

#6500

yes

characteristic

145

C₂²

#6503

yes

characteristic

146

C₂²

C₂×#1144

yes

—

147

C₂²

#6514

yes

—

148

C₄

—

149

C₄

—

150

C₄

—

151

C₄

—

152

C₄

#6445

yes

—

153

C₄

#6447

yes

—

154

C₂

#89732

yes

characteristic

155

C₂

#89803

yes

—

156

C₂

#89822

yes

—

157

C₂

#89832

yes

characteristic

158

C₂

#89864

yes

characteristic

	subgroup	G/H
center	C₂³	C₂×C₂²⋊D₇
commutator, cosocle	C₂²×C₄×C₇	C₂³
Frattini	C₂²×C₄	C₂²×D₇
Fitting	C₇×#391	C₂
socle, prime radical	C₂³×C₇	C₂×D₄
hypercenter	#391	D₇
hypocenter	C₇	#2016
LCS	G ▷ C₂²×C₄×C₇ ▷ C₂×C₇ ▷ C₇
UCS	C₁ ◁ C₂³ ◁ C₂×C₄² ◁ #391

subgroup

G/H

center

C₂³

C₂×C₂²⋊D₇

commutator, cosocle

C₂²×C₄×C₇

C₂³

Frattini

C₂²×C₄

C₂²×D₇

Fitting

C₇×#391

C₂

socle, prime radical

C₂³×C₇

C₂×D₄

D₇

C₇

G ▷ C₂²×C₄×C₇ ▷ C₂×C₇ ▷ C₇

UCS

C₁ ◁ C₂³ ◁ C₂×C₄² ◁ #391

N	H ≅ G/N	type
56. C₂×C₄×C₇	Q₁₆	semidirect
136. C₂×C₄	#1144	semidirect
142. C₇	#2016	semidirect

H ≅ G/N

type

56. C₂×C₄×C₇

Q₁₆

semidirect

136. C₂×C₄

#1144

semidirect

142. C₇

#2016

semidirect

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

subgroup	index
C₂²×#1144	2
C₂×#6432	2
C₇×#391	2
C₂×#6384	2
#89319	2
#2016	7

subgroup

index

C₂²×#1144

Subgroups — #11727445

Conjugacy classes of subgroups

Complements

Maximal subgroups