Group #11740684 — subgroups

Subgroups — #11740684

Order 896.

540 conjugacy classes of subgroups; only the 189 normal subgroups are shown.

normal subgroups only

This group has 540 conjugacy classes of subgroups — too many to draw. Only its 189 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×C₇×C₄⋊Q₈	C₂	1	1	yes	maximal, characteristic
2	C₂×#6507	C₂	1	2	yes	maximal
3	C₂×C₄×Dic₁₄	C₂	1	1	yes	maximal, characteristic
4	#90002	C₂	1	2	yes	maximal
5	#89752	C₂	1	4	yes	maximal
6	#89745	C₂	1	4	yes	maximal
7	C₄×C₄⋊Dic₇	C₂	1	1	yes	maximal, characteristic
8	C₂²×C₇×Q₈	C₂²	1	2	yes	—
9	C₂²×Dic₁₄	C₂²	1	1	yes	characteristic
10	C₇×C₄⋊Q₈	—	2	4	no	—
11	C₂×C₇×C₄⋊C₄	C₂²	1	2	yes	—
12	C₂×C₇×C₄⋊C₄	C₂²	1	2	yes	—
13	C₂×C₄²×C₇	C₂²	1	1	yes	characteristic
14	#6507	—	2	8	no	—
15	C₂×C₄⋊Dic₇	C₂²	1	1	yes	characteristic
16	C₂×C₄⋊Dic₇	C₂²	1	2	yes	—
17	C₂×Dic₇⋊C₄	C₂²	1	2	yes	—
18	C₂×Dic₇⋊C₄	C₂²	1	8	yes	—
19	C₂×C₄×Dic₇	C₂²	1	2	yes	—
20	C₂×C₄×Dic₇	C₂²	1	2	yes	—
21	C₄×Dic₁₄	—	2	4	no	—
22	C₇⋊#95	C₂²	1	2	yes	—
23	C₇⋊#95	C₂²	1	8	yes	—
24	#2546	—	7	1	no	maximal, Sylow
25	C₂×C₇×Q₈	—	2	4	no	—
26	C₂×C₇×Q₈	—	2	4	no	—
27	C₂×C₇×Q₈	—	2	4	no	—
28	C₂²×C₄×C₇	C₂³	1	4	yes	—
29	C₂²×C₄×C₇	C₂³	1	1	yes	characteristic
30	C₂²×C₄×C₇	C₂³	1	1	yes	characteristic
31	C₂²×C₄×C₇	C₂³	1	1	yes	characteristic
32	C₂²×Dic₇	C₂³	1	2	yes	—
33	C₂²×Dic₇	C₂³	1	2	yes	—
34	C₂²×Dic₇	C₂³	1	4	yes	—
35	C₂×Dic₁₄	—	2	2	no	—
36	C₂×Dic₁₄	—	2	2	no	—
37	C₂×Dic₁₄	D₄	1	4	yes	—
38	C₇×C₄⋊C₄	—	2	4	no	—
39	C₇×C₄⋊C₄	—	2	4	no	—
40	C₄²×C₇	D₄	1	4	yes	—
41	C₄⋊Dic₇	—	2	2	no	—
42	C₄⋊Dic₇	Q₈	1	8	yes	complemented
43	Dic₇⋊C₄	—	2	4	no	—
44	Dic₇⋊C₄	—	2	16	no	—
45	C₄×Dic₇	—	2	4	no	—
46	C₄×Dic₇	—	2	4	no	—
47	C₂×C₄⋊Q₈	—	7	2	no	—
48	C₂×C₄⋊Q₈	D₇	1	1	yes	characteristic
49	C₂×C₄×Q₈	—	7	1	no	—
50	#398	—	7	4	no	—
51	#391	—	7	2	no	—
52	#389	—	7	4	no	—
53	C₄×C₄⋊C₄	—	7	1	no	—
54	C₂³×C₇	C₂⁴	1	1	yes	characteristic
55	C₇×Q₈	—	4	8	no	—
56	C₂×C₄×C₇	—	2	2	no	—
57	C₂×C₄×C₇	—	2	4	no	—
58	C₂×C₄×C₇	—	2	4	no	—
59	C₂×C₄×C₇	—	2	4	no	—
60	C₂×C₄×C₇	C₂×D₄	1	2	yes	—
61	C₂×C₄×C₇	C₂×D₄	1	2	yes	—
62	C₂×C₄×C₇	C₂×D₄	1	2	yes	—
63	C₂×C₄×C₇	C₂×D₄	1	2	yes	—
64	C₂×C₄×C₇	C₂×Q₈	1	4	yes	—
65	C₂×C₄×C₇	#41	1	2	yes	—
66	C₂×Dic₇	—	2	4	no	—
67	C₂×Dic₇	—	2	2	no	—
68	C₂×Dic₇	—	2	2	no	—
69	C₂×Dic₇	—	2	4	no	—
70	C₂×Dic₇	—	2	2	no	—
71	C₂×Dic₇	—	2	4	no	—
72	C₂×Dic₇	C₂×D₄	1	4	yes	—
73	C₂×Dic₇	C₂×Q₈	1	8	yes	—
74	Dic₁₄	—	2	8	no	—
75	C₂²×Q₈	—	7	1	no	—
76	C₂²×Q₈	C₂×D₇	1	2	yes	—
77	C₄⋊Q₈	—	2	4	no	—
78	C₄⋊Q₈	—	14	8	no	—
79	C₄×Q₈	—	14	4	no	—
80	C₂×C₄⋊C₄	—	7	2	no	—
81	C₂×C₄⋊C₄	—	7	2	no	—
82	C₂×C₄⋊C₄	—	7	1	no	—
83	C₂×C₄⋊C₄	—	7	8	no	—
84	C₂×C₄⋊C₄	C₂×D₇	1	2	yes	—
85	C₂×C₄⋊C₄	C₂×D₇	1	2	yes	—
86	C₂×C₄²	—	7	2	no	—
87	C₂×C₄²	—	7	2	no	—
88	C₂×C₄²	C₂×D₇	1	1	yes	characteristic
89	#95	—	7	2	no	—
90	#95	—	7	8	no	—
91	C₂²×C₇	C₂²×D₄	1	1	yes	characteristic
92	C₂²×C₇	C₂²×D₄	1	1	yes	characteristic
93	C₂²×C₇	C₂²×Q₈	1	2	yes	—
94	C₂²×C₇	C₂×#41	1	1	yes	characteristic
95	C₂²×C₇	2₋¹⁺⁴	1	2	yes	—
96	C₄×C₇	—	2	2	no	—
97	C₄×C₇	—	4	4	no	—
98	C₄×C₇	#121	1	4	yes	—
99	C₄×C₇	C₄⋊Q₈	1	4	yes	complemented
100	Dic₇	—	2	4	no	—
101	Dic₇	—	2	8	no	—
102	Dic₇	—	4	2	no	—
103	C₂×Q₈	—	2	4	no	—
104	C₂×Q₈	—	2	4	no	—
105	C₂×Q₈	—	2	4	no	—
106	C₂×Q₈	—	7	4	no	—
107	C₂×Q₈	—	14	2	no	—
108	C₂×Q₈	—	14	2	no	—
109	C₂²×C₄	—	7	4	no	—
110	C₂²×C₄	—	7	2	no	—
111	C₂²×C₄	—	7	2	no	—
112	C₂²×C₄	C₂²×D₇	1	4	yes	—
113	C₂²×C₄	C₂²×D₇	1	1	yes	characteristic
114	C₂²×C₄	C₂²×D₇	1	1	yes	characteristic
115	C₂²×C₄	C₂²×D₇	1	1	yes	characteristic
116	C₄⋊C₄	—	2	4	no	—
117	C₄⋊C₄	—	2	4	no	—
118	C₄⋊C₄	—	7	8	no	—
119	C₄⋊C₄	—	14	16	no	—
120	C₄⋊C₄	—	14	4	no	—
121	C₄⋊C₄	—	14	2	no	—
122	C₄²	—	14	4	no	—
123	C₄²	—	14	4	no	—
124	C₄²	C₂²⋊D₇	1	4	yes	—
125	C₂×C₇	C₂×#121	1	1	yes	characteristic
126	C₂×C₇	C₂×C₄⋊Q₈	1	1	yes	characteristic
127	C₂×C₇	#536	1	1	yes	characteristic
128	C₂×C₇	D₄×Q₈	1	2	yes	—
129	C₂×C₇	#557	1	2	yes	—
130	C₂³	C₂³×D₇	1	1	yes	characteristic
131	Q₈	—	4	8	no	—
132	Q₈	—	14	8	no	—
133	C₂×C₄	—	2	2	no	—
134	C₂×C₄	—	2	4	no	—
135	C₂×C₄	—	2	4	no	—
136	C₂×C₄	—	2	4	no	—
137	C₂×C₄	—	7	4	no	—
138	C₂×C₄	—	7	8	no	—
139	C₂×C₄	—	14	4	no	—
140	C₂×C₄	—	14	2	no	—
141	C₂×C₄	—	14	4	no	—
142	C₂×C₄	—	14	2	no	—
143	C₂×C₄	—	14	4	no	—
144	C₂×C₄	—	14	2	no	—
145	C₂×C₄	D₄×D₇	1	2	yes	—
146	C₂×C₄	#1159	1	2	yes	—
147	C₂×C₄	Q₈×D₇	1	4	yes	—
148	C₂×C₄	C₂×C₂²⋊D₇	1	2	yes	—
149	C₂×C₄	C₂×C₂²⋊D₇	1	2	yes	—
150	C₂×C₄	C₂×C₂²⋊D₇	1	2	yes	—
151	C₇	#2546	1	1	yes	characteristic, complemented, Sylow
152	C₂²	C₂×D₄×D₇	1	1	yes	characteristic
153	C₂²	C₂×#1159	1	1	yes	characteristic
154	C₂²	C₂×Q₈×D₇	1	2	yes	—
155	C₂²	#6551	1	2	yes	—
156	C₂²	C₂²×C₂²⋊D₇	1	1	yes	characteristic
157	C₄	—	2	2	no	—
158	C₄	—	4	4	no	—
159	C₄	—	14	8	no	—
160	C₄	—	14	4	no	—
161	C₄	—	28	2	no	—
162	C₄	#6501	1	4	yes	—
163	C₄	#6507	1	4	yes	complemented
164	C₂	#90412	1	2	yes	—
165	C₂	#90418	1	2	yes	—
166	C₂	C₂×#6501	1	1	yes	characteristic
167	C₂	C₂×#6507	1	1	yes	characteristic
168	C₂	#90507	1	1	yes	characteristic

subgroup

G/H

class size

multiplicity

normal?

properties

C₂×C₇×C₄⋊Q₈

C₂

yes

maximal, characteristic

C₂×#6507

C₂

yes

maximal

C₂×C₄×Dic₁₄

C₂

yes

maximal, characteristic

#90002

C₂

yes

maximal

#89752

C₂

yes

maximal

#89745

C₂

yes

maximal

C₄×C₄⋊Dic₇

C₂

yes

maximal, characteristic

C₂²×C₇×Q₈

C₂²

yes

—

C₂²×Dic₁₄

C₂²

yes

characteristic

C₇×C₄⋊Q₈

—

C₂×C₇×C₄⋊C₄

C₂²

yes

—

C₂×C₇×C₄⋊C₄

C₂²

yes

—

C₂×C₄²×C₇

C₂²

yes

characteristic

#6507

—

C₂×C₄⋊Dic₇

C₂²

yes

characteristic

C₂×C₄⋊Dic₇

C₂²

yes

—

C₂×Dic₇⋊C₄

C₂²

yes

—

C₂×Dic₇⋊C₄

C₂²

yes

—

C₂×C₄×Dic₇

C₂²

yes

—

C₂×C₄×Dic₇

C₂²

yes

—

C₄×Dic₁₄

—

C₇⋊#95

C₂²

yes

—

C₇⋊#95

C₂²

yes

—

#2546

—

maximal, Sylow

C₂×C₇×Q₈

—

C₂×C₇×Q₈

—

C₂×C₇×Q₈

—

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

—

C₂×Dic₁₄

—

C₂×Dic₁₄

—

C₂×Dic₁₄

D₄

yes

—

C₇×C₄⋊C₄

—

C₇×C₄⋊C₄

—

C₄²×C₇

D₄

yes

—

C₄⋊Dic₇

—

C₄⋊Dic₇

Q₈

yes

complemented

Dic₇⋊C₄

—

Dic₇⋊C₄

—

C₄×Dic₇

—

C₄×Dic₇

—

C₂×C₄⋊Q₈

—

C₂×C₄⋊Q₈

D₇

yes

characteristic

C₂×C₄×Q₈

—

#398

—

#391

—

#389

—

C₄×C₄⋊C₄

—

C₂³×C₇

C₂⁴

yes

characteristic

C₇×Q₈

—

C₂×C₄×C₇

—

C₂×C₄×C₇

—

C₂×C₄×C₇

—

C₂×C₄×C₇

—

C₂×C₄×C₇

C₂×D₄

yes

—

C₂×C₄×C₇

C₂×D₄

yes

—

C₂×C₄×C₇

C₂×D₄

yes

—

C₂×C₄×C₇

C₂×D₄

yes

—

C₂×C₄×C₇

C₂×Q₈

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×Q₈

yes

—

Dic₁₄

—

C₂²×Q₈

—

C₂²×Q₈

C₂×D₇

yes

—

C₄⋊Q₈

—

C₄⋊Q₈

—

C₄×Q₈

—

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×D₇

yes

characteristic

#95

—

#95

—

C₂²×C₇

C₂²×D₄

yes

characteristic

C₂²×C₇

C₂²×D₄

yes

characteristic

C₂²×C₇

C₂²×Q₈

yes

—

C₂²×C₇

C₂×#41

yes

characteristic

C₂²×C₇

2₋¹⁺⁴

yes

—

C₄×C₇

—

C₄×C₇

—

C₄×C₇

#121

yes

—

C₄×C₇

C₄⋊Q₈

yes

complemented

100

Dic₇

—

101

Dic₇

—

102

Dic₇

—

103

C₂×Q₈

—

104

C₂×Q₈

—

105

C₂×Q₈

—

106

C₂×Q₈

—

107

C₂×Q₈

—

108

C₂×Q₈

—

109

C₂²×C₄

—

110

C₂²×C₄

—

111

C₂²×C₄

—

112

C₂²×C₄

C₂²×D₇

yes

—

113

C₂²×C₄

C₂²×D₇

yes

characteristic

114

C₂²×C₄

C₂²×D₇

yes

characteristic

115

C₂²×C₄

C₂²×D₇

yes

characteristic

116

C₄⋊C₄

—

117

C₄⋊C₄

—

118

C₄⋊C₄

—

119

C₄⋊C₄

—

120

C₄⋊C₄

—

121

C₄⋊C₄

—

122

C₄²

—

123

C₄²

—

124

C₄²

C₂²⋊D₇

yes

—

125

C₂×C₇

C₂×#121

yes

characteristic

126

C₂×C₇

C₂×C₄⋊Q₈

yes

characteristic

127

C₂×C₇

#536

yes

characteristic

128

C₂×C₇

D₄×Q₈

yes

—

129

C₂×C₇

#557

yes

—

130

C₂³

C₂³×D₇

yes

characteristic

131

Q₈

—

132

Q₈

—

133

C₂×C₄

—

134

C₂×C₄

—

135

C₂×C₄

—

136

C₂×C₄

—

137

C₂×C₄

—

138

C₂×C₄

—

139

C₂×C₄

—

140

C₂×C₄

—

141

C₂×C₄

—

142

C₂×C₄

—

143

C₂×C₄

—

144

C₂×C₄

—

145

C₂×C₄

D₄×D₇

yes

—

146

C₂×C₄

#1159

yes

—

147

C₂×C₄

Q₈×D₇

yes

—

148

C₂×C₄

C₂×C₂²⋊D₇

yes

—

149

C₂×C₄

C₂×C₂²⋊D₇

yes

—

150

C₂×C₄

C₂×C₂²⋊D₇

yes

—

151

C₇

#2546

yes

characteristic, complemented, Sylow

152

C₂²

C₂×D₄×D₇

yes

characteristic

153

C₂²

C₂×#1159

yes

characteristic

154

C₂²

C₂×Q₈×D₇

yes

—

155

C₂²

#6551

yes

—

156

C₂²

C₂²×C₂²⋊D₇

yes

characteristic

157

C₄

—

158

C₄

—

159

C₄

—

160

C₄

—

161

C₄

—

162

C₄

#6501

yes

—

163

C₄

#6507

yes

complemented

164

C₂

#90412

yes

—

165

C₂

#90418

yes

—

166

C₂

C₂×#6501

yes

characteristic

167

C₂

C₂×#6507

yes

characteristic

168

C₂

#90507

yes

characteristic

	subgroup	G/H
center, Frattini	C₂³	C₂³×D₇
commutator, socle, prime radical, cosocle	C₂³×C₇	C₂⁴
Fitting	C₂×C₇×C₄⋊Q₈	C₂
hypercenter	C₂×C₄⋊Q₈	D₇
hypocenter	C₇	#2546
LCS	G ▷ C₂³×C₇ ▷ C₇
UCS	C₁ ◁ C₂³ ◁ C₂×C₄⋊Q₈

subgroup

G/H

center, Frattini

C₂³

C₂³×D₇

commutator, socle, prime radical, cosocle

C₂³×C₇

C₂⁴

Fitting

C₂×C₇×C₄⋊Q₈

C₂

hypercenter

C₂×C₄⋊Q₈

D₇

C₇

G ▷ C₂³×C₇ ▷ C₇

C₁ ◁ C₂³ ◁ C₂×C₄⋊Q₈

N	H ≅ G/N	type
42. C₄⋊Dic₇	Q₈	semidirect
99. C₄×C₇	C₄⋊Q₈	semidirect
151. C₇	#2546	semidirect
163. C₄	#6507	semidirect

H ≅ G/N

type

42. C₄⋊Dic₇

Q₈

semidirect

99. C₄×C₇

C₄⋊Q₈

semidirect

151. C₇

#2546

semidirect

163. C₄

#6507

semidirect

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

subgroup	index
C₂×C₇×C₄⋊Q₈	2
C₂×#6507	2
C₂×C₄×Dic₁₄	2
#90002	2
#89752	2
#89745	2
C₄×C₄⋊Dic₇	2
#2546	7

subgroup

index

C₂×C₇×C₄⋊Q₈

C₂×C₄×Dic₁₄

C₄×C₄⋊Dic₇

Subgroups — #11740684

Conjugacy classes of subgroups

Complements

Maximal subgroups