Group #11728331 — subgroups

Subgroups — #11728331

Order 896.

336 conjugacy classes of subgroups; only the 117 normal subgroups are shown.

normal subgroups only

This group has 336 conjugacy classes of subgroups — too many to draw. Only its 117 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×C₈×D₇	C₂	1	1	yes	maximal, characteristic
2	C₂×C₇×#106	C₂	1	1	yes	maximal, characteristic, complemented
3	C₂×#6391	C₂	1	1	yes	maximal, characteristic, complemented
4	#89759	C₂	1	2	yes	maximal
5	#89324	C₂	1	2	yes	maximal, complemented
6	C₂²×C₄×D₇	C₂²	1	1	yes	characteristic
7	C₂²×C₇×C₈	C₂²	1	1	yes	characteristic
8	C₂×C₇×C₄⋊C₄	C₂²	1	2	yes	—
9	C₂×(C₂×D₇)⋊C₄	—	2	2	no	—
10	C₂×C₄⋊Dic₇	C₂²	1	2	yes	—
11	C₂²×C₇⋊C₈	C₂²	1	1	yes	characteristic
12	C₂×C₈×D₇	—	2	2	no	—
13	C₂×C₈×D₇	—	2	2	no	—
14	C₂×C₈×D₇	C₄	1	4	yes	complemented
15	C₇×#106	—	2	2	no	—
16	C₇⋊#95	—	2	2	no	—
17	#6391	—	2	2	no	—
18	#1807	—	7	1	no	maximal, Sylow
19	C₂³×D₇	D₄	1	1	yes	characteristic
20	C₂²×C₄×C₇	—	2	2	no	—
21	C₂²×C₄×C₇	C₂³	1	1	yes	characteristic
22	C₂²×Dic₇	—	2	2	no	—
23	C₂²×Dic₇	D₄	1	1	yes	characteristic
24	C₂×C₄×D₇	—	2	1	no	—
25	C₂×C₄×D₇	—	2	1	no	—
26	C₂×C₄×D₇	—	2	1	no	—
27	C₂×C₄×D₇	—	2	1	no	—
28	C₂×C₄×D₇	C₂×C₄	1	2	yes	—
29	C₂×C₄×D₇	Q₈	1	2	yes	—
30	C₂×C₇×C₈	C₂×C₄	1	2	yes	—
31	C₂×C₇×C₈	D₄	1	2	yes	—
32	C₂×C₇×C₈	D₄	1	2	yes	—
33	C₇×C₄⋊C₄	—	2	2	no	—
34	C₇×C₄⋊C₄	—	2	2	no	—
35	(C₂×D₇)⋊C₄	—	4	4	no	—
36	C₄⋊Dic₇	—	2	2	no	—
37	C₄⋊Dic₇	—	2	2	no	—
38	C₂×C₇⋊C₈	—	2	1	no	—
39	C₂×C₇⋊C₈	—	2	1	no	—
40	C₂×C₇⋊C₈	C₂×C₄	1	2	yes	—
41	C₈×D₇	—	2	8	no	—
42	C₂³×C₈	—	7	1	no	—
43	C₂×#106	—	7	1	no	—
44	C₂×#106	D₇	1	1	yes	characteristic, complemented
45	#380	—	7	2	no	—
46	#340	—	7	2	no	—
47	C₂³×C₇	C₂×D₄	1	1	yes	characteristic
48	C₂²×D₇	—	2	1	no	—
49	C₂²×D₇	—	2	1	no	—
50	C₂²×D₇	—	2	2	no	—
51	C₂²×D₇	C₄⋊C₄	1	2	yes	—
52	C₂²×D₇	SD₁₆	1	4	yes	—
53	C₂×C₄×C₇	—	4	2	no	—
54	C₂×C₄×C₇	—	4	2	no	—
55	C₂×C₄×C₇	—	4	2	no	—
56	C₂×C₄×C₇	C₂²×C₄	1	1	yes	characteristic
57	C₂×C₄×C₇	C₂×D₄	1	1	yes	characteristic
58	C₂×C₄×C₇	C₂×D₄	1	1	yes	characteristic
59	C₂×C₄×C₇	C₂×Q₈	1	1	yes	characteristic
60	C₂×C₄×C₇	#41	1	1	yes	characteristic
61	C₂×C₄×C₇	#41	1	1	yes	characteristic
62	C₂×Dic₇	—	2	1	no	—
63	C₂×Dic₇	—	2	1	no	—
64	C₂×Dic₇	—	4	2	no	—
65	C₂×Dic₇	—	4	2	no	—
66	C₂×Dic₇	—	4	2	no	—
67	C₂×Dic₇	C₄⋊C₄	1	2	yes	—
68	C₄×D₇	—	2	2	no	—
69	C₄×D₇	—	2	2	no	—
70	C₄×D₇	—	2	2	no	—
71	C₄×D₇	—	2	2	no	—
72	C₇×C₈	C₂²⋊C₄	1	4	yes	complemented
73	C₇⋊C₈	—	2	2	no	—
74	C₂³×C₄	—	7	1	no	—
75	C₂²×C₈	—	7	1	no	—
76	C₂²×C₈	—	7	4	no	—
77	C₂²×C₈	—	14	2	no	—
78	C₂²×C₈	—	14	2	no	—
79	C₂²×C₈	C₂×D₇	1	1	yes	characteristic
80	C₂×C₄⋊C₄	—	7	2	no	—
81	C₂×C₄⋊C₄	C₂×D₇	1	2	yes	—
82	C₂×C₂²⋊C₄	—	14	2	no	—
83	#106	—	2	2	no	—
84	#106	—	14	2	no	—
85	#95	—	14	2	no	—
86	C₂²×C₇	C₂×C₄⋊C₄	1	1	yes	characteristic
87	C₂²×C₇	#121	1	1	yes	characteristic
88	C₂²×C₇	#121	1	1	yes	characteristic
89	C₂²×C₇	C₂×SD₁₆	1	2	yes	—
90	C₂²×C₇	#135	1	2	yes	—
91	C₂×D₇	—	2	2	no	—
92	C₂×D₇	—	2	2	no	—
93	C₂×D₇	—	2	4	no	—
94	C₂×D₇	—	2	4	no	—
95	C₂×D₇	#106	1	4	yes	complemented
96	C₄×C₇	—	8	2	no	—
97	C₄×C₇	C₂×C₂²⋊C₄	1	1	yes	characteristic
98	C₄×C₇	#117	1	1	yes	characteristic
99	C₄×C₇	C₂²⋊Q₈	1	1	yes	characteristic
100	C₄×C₇	C₂²⋊Q₈	1	1	yes	characteristic
101	Dic₇	—	2	2	no	—
102	Dic₇	—	8	2	no	—
103	C₂⁴	—	7	1	no	—
104	C₂²×C₄	—	2	2	no	—
105	C₂²×C₄	—	7	1	no	—
106	C₂²×C₄	—	7	2	no	—
107	C₂²×C₄	—	7	2	no	—
108	C₂²×C₄	—	14	1	no	—
109	C₂²×C₄	—	14	1	no	—
110	C₂²×C₄	—	14	1	no	—
111	C₂²×C₄	—	14	1	no	—
112	C₂²×C₄	—	14	2	no	—
113	C₂²×C₄	C₂²×D₇	1	1	yes	characteristic
114	C₂×C₈	—	7	2	no	—
115	C₂×C₈	—	14	1	no	—
116	C₂×C₈	—	14	1	no	—
117	C₂×C₈	—	14	8	no	—
118	C₂×C₈	C₄×D₇	1	2	yes	—
119	C₂×C₈	D₂₈	1	2	yes	—
120	C₂×C₈	C₂²⋊D₇	1	2	yes	—
121	C₄⋊C₄	—	2	2	no	—
122	C₄⋊C₄	—	2	2	no	—
123	C₄⋊C₄	—	14	2	no	—
124	C₄⋊C₄	—	14	2	no	—
125	C₂²⋊C₄	—	28	4	no	—
126	C₂×C₇	#380	1	1	yes	characteristic
127	C₂×C₇	C₂×#106	1	1	yes	characteristic
128	C₂×C₇	#427	1	1	yes	characteristic
129	C₂×C₇	#465	1	2	yes	—
130	C₂×C₇	#465	1	2	yes	—
131	D₇	—	2	4	no	—
132	C₂³	—	7	2	no	—
133	C₂³	—	7	4	no	—
134	C₂³	—	14	1	no	—
135	C₂³	—	14	2	no	—
136	C₂³	—	14	1	no	—
137	C₂³	D₄×D₇	1	1	yes	characteristic
138	C₂×C₄	—	4	2	no	—
139	C₂×C₄	—	4	2	no	—
140	C₂×C₄	—	4	2	no	—
141	C₂×C₄	—	7	2	no	—
142	C₂×C₄	—	14	1	no	—
143	C₂×C₄	—	14	1	no	—
144	C₂×C₄	—	14	2	no	—
145	C₂×C₄	—	14	2	no	—
146	C₂×C₄	—	14	2	no	—
147	C₂×C₄	—	14	2	no	—
148	C₂×C₄	—	28	2	no	—
149	C₂×C₄	—	28	2	no	—
150	C₂×C₄	—	28	2	no	—
151	C₂×C₄	C₂×C₄×D₇	1	1	yes	characteristic
152	C₂×C₄	C₂×D₂₈	1	1	yes	characteristic
153	C₂×C₄	#1159	1	1	yes	characteristic
154	C₂×C₄	Q₈×D₇	1	1	yes	characteristic
155	C₂×C₄	#1161	1	1	yes	characteristic
156	C₂×C₄	C₂×C₂²⋊D₇	1	1	yes	characteristic
157	C₈	—	14	2	no	—
158	C₈	(C₂×D₇)⋊C₄	1	4	yes	complemented
159	C₇	#1807	1	1	yes	characteristic, complemented, Sylow
160	C₂²	—	7	4	no	—
161	C₂²	—	14	2	no	—
162	C₂²	—	14	4	no	—
163	C₂²	—	14	4	no	—
164	C₂²	—	14	2	no	—
165	C₂²	D₇×C₄⋊C₄	1	1	yes	characteristic
166	C₂²	#6458	1	1	yes	characteristic
167	C₂²	D₇×SD₁₆	1	2	yes	—
168	C₂²	#6479	1	2	yes	—
169	C₂²	#6501	1	1	yes	characteristic
170	C₄	—	8	2	no	—
171	C₄	—	14	2	no	—
172	C₄	—	56	2	no	—
173	C₄	#6455	1	1	yes	characteristic
174	C₄	#6460	1	1	yes	characteristic
175	C₄	C₂×(C₂×D₇)⋊C₄	1	1	yes	characteristic
176	C₄	#6509	1	1	yes	characteristic
177	C₂	—	14	4	no	—
178	C₂	D₇×#106	1	1	yes	characteristic
179	C₂	#89632	1	1	yes	characteristic
180	C₂	#89636	1	2	yes	—
181	C₂	#89759	1	1	yes	characteristic
182	C₂	#89943	1	2	yes	—

subgroup

G/H

class size

multiplicity

normal?

properties

C₂²×C₈×D₇

C₂

yes

maximal, characteristic

C₂×C₇×#106

C₂

yes

maximal, characteristic, complemented

C₂×#6391

C₂

yes

maximal, characteristic, complemented

#89759

C₂

yes

maximal

#89324

C₂

yes

maximal, complemented

C₂²×C₄×D₇

C₂²

yes

characteristic

C₂²×C₇×C₈

C₂²

yes

characteristic

C₂×C₇×C₄⋊C₄

C₂²

yes

—

C₂×(C₂×D₇)⋊C₄

—

C₂×C₄⋊Dic₇

C₂²

yes

—

C₂²×C₇⋊C₈

C₂²

yes

characteristic

C₂×C₈×D₇

—

C₂×C₈×D₇

—

C₂×C₈×D₇

C₄

yes

—

—

—

—

C₂³×D₇

D₄

yes

characteristic

C₂²×C₄×C₇

—

C₂²×C₄×C₇

C₂³

yes

characteristic

C₂²×Dic₇

—

C₂²×Dic₇

D₄

yes

characteristic

C₂×C₄×D₇

—

C₂×C₄×D₇

—

C₂×C₄×D₇

—

C₂×C₄×D₇

—

C₂×C₄×D₇

C₂×C₄

yes

—

C₂×C₄×D₇

Q₈

yes

—

C₂×C₇×C₈

C₂×C₄

yes

—

C₂×C₇×C₈

D₄

yes

—

C₂×C₇×C₈

D₄

yes

—

C₇×C₄⋊C₄

—

C₇×C₄⋊C₄

—

(C₂×D₇)⋊C₄

—

C₄⋊Dic₇

—

C₄⋊Dic₇

—

C₂×C₇⋊C₈

—

C₂×C₇⋊C₈

—

C₂×C₇⋊C₈

C₂×C₄

yes

—

C₈×D₇

—

C₂³×C₈

—

C₂×#106

—

C₂×#106

D₇

yes

characteristic, complemented

#380

—

#340

—

C₂³×C₇

C₂×D₄

yes

characteristic

C₂²×D₇

—

C₂²×D₇

—

C₂²×D₇

—

C₂²×D₇

C₄⋊C₄

yes

—

C₂²×D₇

SD₁₆

yes

—

C₂×C₄×C₇

—

C₂×C₄×C₇

—

C₂×C₄×C₇

—

C₂×C₄×C₇

C₂²×C₄

yes

characteristic

C₂×C₄×C₇

C₂×D₄

yes

characteristic

C₂×C₄×C₇

C₂×D₄

yes

characteristic

C₂×C₄×C₇

C₂×Q₈

yes

C₂×C₄×C₇

yes

C₂×C₄×C₇

yes

C₂×Dic₇

—

C₂×Dic₇

—

C₂×Dic₇

—

C₂×Dic₇

—

C₂×Dic₇

—

C₂×Dic₇

C₄⋊C₄

yes

—

C₄×D₇

—

C₄×D₇

—

C₄×D₇

—

C₄×D₇

—

C₇×C₈

C₂²⋊C₄

yes

complemented

C₇⋊C₈

—

C₂³×C₄

—

C₂²×C₈

—

C₂²×C₈

—

C₂²×C₈

—

C₂²×C₈

—

C₂²×C₈

C₂×D₇

yes

characteristic

C₂×C₄⋊C₄

—

C₂×C₄⋊C₄

C₂×D₇

yes

—

C₂×C₂²⋊C₄

—

#106

—

#106

—

#95

—

C₂²×C₇

C₂×C₄⋊C₄

yes

C₂²×C₇

yes

C₂²×C₇

yes

C₂²×C₇

C₂×SD₁₆

yes

—

C₂²×C₇

#135

yes

—

C₂×D₇

—

C₂×D₇

—

C₂×D₇

—

C₂×D₇

—

C₂×D₇

#106

yes

complemented

C₄×C₇

—

C₄×C₇

C₂×C₂²⋊C₄

yes

characteristic

C₄×C₇

#117

yes

characteristic

C₄×C₇

C₂²⋊Q₈

yes

characteristic

100

C₄×C₇

C₂²⋊Q₈

yes

characteristic

101

Dic₇

—

102

Dic₇

—

103

C₂⁴

—

104

C₂²×C₄

—

105

C₂²×C₄

—

106

C₂²×C₄

—

107

C₂²×C₄

—

108

C₂²×C₄

—

109

C₂²×C₄

—

110

C₂²×C₄

—

111

C₂²×C₄

—

112

C₂²×C₄

—

113

C₂²×C₄

C₂²×D₇

yes

characteristic

114

C₂×C₈

—

115

C₂×C₈

—

116

C₂×C₈

—

117

C₂×C₈

—

118

C₂×C₈

C₄×D₇

yes

—

119

C₂×C₈

D₂₈

yes

—

120

C₂×C₈

C₂²⋊D₇

yes

—

121

C₄⋊C₄

—

122

C₄⋊C₄

—

123

C₄⋊C₄

—

124

C₄⋊C₄

—

125

C₂²⋊C₄

—

126

C₂×C₇

#380

yes

characteristic

127

C₂×C₇

C₂×#106

yes

characteristic

128

C₂×C₇

#427

yes

characteristic

129

C₂×C₇

#465

yes

—

130

C₂×C₇

#465

yes

—

131

D₇

—

132

C₂³

—

133

C₂³

—

134

C₂³

—

135

C₂³

—

136

C₂³

—

137

C₂³

D₄×D₇

yes

characteristic

138

C₂×C₄

—

139

C₂×C₄

—

140

C₂×C₄

—

141

C₂×C₄

—

142

C₂×C₄

—

143

C₂×C₄

—

144

C₂×C₄

—

145

C₂×C₄

—

146

C₂×C₄

—

147

C₂×C₄

—

148

C₂×C₄

—

149

C₂×C₄

—

150

C₂×C₄

—

151

C₂×C₄

C₂×C₄×D₇

yes

characteristic

152

C₂×C₄

C₂×D₂₈

yes

characteristic

153

C₂×C₄

#1159

yes

characteristic

154

C₂×C₄

Q₈×D₇

yes

characteristic

155

C₂×C₄

#1161

yes

characteristic

156

C₂×C₄

C₂×C₂²⋊D₇

yes

characteristic

157

C₈

—

158

C₈

(C₂×D₇)⋊C₄

yes

complemented

159

C₇

#1807

yes

characteristic, complemented, Sylow

160

C₂²

—

161

C₂²

—

162

C₂²

—

163

C₂²

—

164

C₂²

—

165

C₂²

D₇×C₄⋊C₄

yes

characteristic

166

C₂²

#6458

yes

characteristic

167

C₂²

D₇×SD₁₆

yes

—

168

C₂²

#6479

yes

—

169

C₂²

#6501

yes

characteristic

170

C₄

—

171

C₄

—

172

C₄

—

173

C₄

#6455

yes

characteristic

174

C₄

#6460

yes

characteristic

175

C₄

C₂×(C₂×D₇)⋊C₄

yes

characteristic

176

C₄

#6509

yes

characteristic

177

C₂

—

178

C₂

D₇×#106

yes

characteristic

179

C₂

#89632

yes

characteristic

180

C₂

#89636

yes

—

181

C₂

#89759

yes

characteristic

182

C₂

#89943

yes

—

	subgroup	G/H
center	C₂³	D₄×D₇
commutator	C₂×C₄×C₇	C₂²×C₄
Frattini	C₂²×C₄	C₂²×D₇
Fitting	C₂×C₇×#106	C₂
socle	C₂³×C₇	C₂×D₄
prime radical	C₂³×D₇	D₄
cosocle	C₂²×C₄×C₇	C₂³
hypercenter	C₂×#106	D₇
hypocenter	C₇	#1807
LCS	G ▷ C₂×C₄×C₇ ▷ C₂×C₇ ▷ C₇
UCS	C₁ ◁ C₂³ ◁ C₂²×C₄ ◁ C₂×#106

subgroup

G/H

center

C₂³

D₄×D₇

commutator

C₂×C₄×C₇

C₂²×C₄

Frattini

C₂²×C₄

C₂²×D₇

Fitting

C₂×C₇×#106

C₂

socle

C₂³×C₇

C₂×D₄

prime radical

C₂³×D₇

D₄

cosocle

C₂²×C₄×C₇

C₂³

D₇

C₇

G ▷ C₂×C₄×C₇ ▷ C₂×C₇ ▷ C₇

UCS

C₁ ◁ C₂³ ◁ C₂²×C₄ ◁ C₂×#106

N	H ≅ G/N	type
2. C₂×C₇×#106	C₂	semidirect
3. C₂×#6391	C₂	semidirect
5. #89324	C₂	semidirect
14. C₂×C₈×D₇	C₄	semidirect
44. C₂×#106	D₇	semidirect
72. C₇×C₈	C₂²⋊C₄	semidirect
95. C₂×D₇	#106	semidirect
158. C₈	(C₂×D₇)⋊C₄	semidirect
159. C₇	#1807	semidirect

H ≅ G/N

type

2. C₂×C₇×#106

C₂

semidirect

3. C₂×#6391

C₂

semidirect

5. #89324

C₂

semidirect

14. C₂×C₈×D₇

C₄

semidirect

44. C₂×#106

D₇

semidirect

72. C₇×C₈

C₂²⋊C₄

semidirect

95. C₂×D₇

#106

semidirect

158. C₈

(C₂×D₇)⋊C₄

semidirect

159. C₇

#1807

semidirect

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

subgroup	index
C₂²×C₈×D₇	2
C₂×C₇×#106	2
C₂×#6391	2
#89759	2
#89324	2
#1807	7

subgroup

index

C₂²×C₈×D₇

C₂×C₇×#106

Subgroups — #11728331

Conjugacy classes of subgroups

Complements

Maximal subgroups