Group #13253562 — subgroups

Subgroups — #13253562

Order 1792.

558 conjugacy classes of subgroups; only the 199 normal subgroups are shown.

normal subgroups only

This group has 558 conjugacy classes of subgroups — too many to draw. Only its 199 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×#89471	C₂	1	1	yes	maximal, characteristic
2	C₇×#1845	C₂	1	1	yes	maximal, characteristic
3	C₂×C₈×C₇⋊C₈	C₂	1	1	yes	maximal, characteristic
4	#11727336	C₂	1	4	yes	maximal
5	C₂²×Dic₂₈	—	2	2	no	—
6	C₂×#6432	C₂²	1	2	yes	—
7	C₂×C₇×#107	—	2	2	no	—
8	C₂×C₄×C₇×C₈	C₂²	1	1	yes	characteristic
9	C₇×#384	C₂²	1	2	yes	—
10	C₂×C₄×C₇⋊C₈	C₂²	1	2	yes	—
11	#89471	—	2	2	no	—
12	#89471	C₄	1	4	yes	complemented
13	#89324	—	2	8	no	—
14	C₈×C₇⋊C₈	—	2	2	no	—
15	#9825	—	7	1	no	maximal, Sylow
16	C₂²×Dic₁₄	—	2	2	no	—
17	C₂²×C₇×C₈	D₄	1	2	yes	—
18	C₂×C₇×C₄⋊C₄	—	2	2	no	—
19	C₂×C₇×C₄⋊C₄	—	2	4	no	—
20	C₂×C₄²×C₇	C₂³	1	1	yes	characteristic
21	C₂×C₄⋊Dic₇	—	2	4	no	—
22	C₂²×C₇⋊C₈	D₄	1	4	yes	—
23	C₂×Dic₂₈	—	2	8	no	—
24	C₂×Dic₂₈	—	4	4	no	—
25	C₂×Dic₂₈	—	4	4	no	—
26	#6432	—	2	4	no	—
27	#6432	—	2	2	no	—
28	#6432	C₂×C₄	1	4	yes	—
29	C₇×#107	—	4	4	no	—
30	C₄×C₇×C₈	C₂×C₄	1	2	yes	—
31	C₄×C₇×C₈	D₄	1	2	yes	—
32	C₄×C₇⋊C₈	—	2	2	no	—
33	C₄×C₇⋊C₈	—	2	2	no	—
34	C₂×#494	—	7	1	no	—
35	#1990	—	7	4	no	—
36	#1845	D₇	1	1	yes	characteristic
37	C₂×C₈²	—	7	1	no	—
38	C₂²×C₄×C₇	—	4	2	no	—
39	C₂²×C₄×C₇	C₂×D₄	1	1	yes	characteristic
40	C₂²×C₄×C₇	C₂×D₄	1	2	yes	—
41	C₂²×Dic₇	—	4	2	no	—
42	C₂×Dic₁₄	—	2	4	no	—
43	C₂×Dic₁₄	—	4	4	no	—
44	C₂×Dic₁₄	—	4	4	no	—
45	C₂×Dic₁₄	—	4	2	no	—
46	C₂×C₇×C₈	C₂²⋊C₄	1	4	yes	—
47	C₂×C₇×C₈	D₈	1	4	yes	—
48	C₂×C₇×C₈	SD₁₆	1	4	yes	—
49	C₇×C₄⋊C₄	—	4	4	no	—
50	C₇×C₄⋊C₄	—	4	8	no	—
51	C₄²×C₇	C₂²×C₄	1	1	yes	characteristic
52	C₄²×C₇	C₂×D₄	1	1	yes	characteristic
53	C₄²×C₇	#41	1	2	yes	—
54	C₄⋊Dic₇	—	2	8	no	—
55	C₄⋊Dic₇	—	4	4	no	—
56	C₂×C₇⋊C₈	—	2	4	no	—
57	C₂×C₇⋊C₈	—	2	4	no	—
58	C₂×C₇⋊C₈	Q₁₆	1	8	yes	complemented
59	Dic₂₈	—	4	16	no	—
60	C₂²×Q₁₆	—	14	2	no	—
61	C₂×C₄⋊Q₈	—	7	2	no	—
62	#494	—	7	4	no	—
63	#494	—	14	2	no	—
64	C₂×#107	—	2	2	no	—
65	C₂×C₄×C₈	—	7	2	no	—
66	C₂×C₄×C₈	C₂×D₇	1	1	yes	characteristic
67	#384	C₂×D₇	1	2	yes	—
68	#340	—	14	8	no	—
69	C₈²	—	14	2	no	—
70	C₂³×C₇	#127	1	1	yes	characteristic
71	C₂×C₄×C₇	—	8	2	no	—
72	C₂×C₄×C₇	—	8	4	no	—
73	C₂×C₄×C₇	C₂×C₂²⋊C₄	1	1	yes	characteristic
74	C₂×C₄×C₇	C₄×D₄	1	2	yes	—
75	C₂×C₄×C₇	#121	1	2	yes	—
76	C₂×C₄×C₇	#124	1	1	yes	characteristic
77	C₂×C₄×C₇	C₂×D₈	1	2	yes	—
78	C₂×C₄×C₇	C₂×SD₁₆	1	2	yes	—
79	C₂×C₄×C₇	C₂×Q₁₆	1	4	yes	—
80	C₂×C₄×C₇	#135	1	4	yes	—
81	C₂×Dic₇	—	4	4	no	—
82	C₂×Dic₇	—	8	2	no	—
83	C₂×Dic₇	—	8	2	no	—
84	Dic₁₄	—	4	8	no	—
85	Dic₁₄	—	8	4	no	—
86	C₇×C₈	#102	1	8	yes	—
87	C₇⋊C₈	—	2	8	no	—
88	C₂²×Q₈	—	14	2	no	—
89	C₂×Q₁₆	—	14	8	no	—
90	C₂×Q₁₆	—	28	4	no	—
91	C₂×Q₁₆	—	28	4	no	—
92	C₂²×C₈	—	7	4	no	—
93	C₂²×C₈	C₂²⋊D₇	1	2	yes	—
94	C₄⋊Q₈	—	7	4	no	—
95	C₄⋊Q₈	—	14	2	no	—
96	C₄⋊Q₈	—	14	4	no	—
97	C₂×C₄⋊C₄	—	2	4	no	—
98	C₂×C₄⋊C₄	—	2	2	no	—
99	C₂×C₄⋊C₄	—	14	4	no	—
100	C₂×C₄²	C₂²×D₇	1	1	yes	characteristic
101	#107	—	4	4	no	—
102	C₄×C₈	—	14	2	no	—
103	C₄×C₈	—	14	2	no	—
104	C₄×C₈	C₄×D₇	1	2	yes	—
105	C₄×C₈	D₂₈	1	2	yes	—
106	C₂²×C₇	#390	1	1	yes	characteristic
107	C₂²×C₇	#492	1	1	yes	characteristic
108	C₂²×C₇	#493	1	1	yes	characteristic
109	C₂²×C₇	#494	1	2	yes	—
110	C₂²×C₇	#495	1	2	yes	—
111	C₄×C₇	—	16	2	no	—
112	C₄×C₇	C₂×#102	1	2	yes	—
113	C₄×C₇	C₄×Q₁₆	1	4	yes	—
114	C₄×C₇	#467	1	4	yes	—
115	C₄×C₇	#486	1	2	yes	—
116	Dic₇	—	8	4	no	—
117	C₂×Q₈	—	14	4	no	—
118	C₂×Q₈	—	28	2	no	—
119	C₂×Q₈	—	28	4	no	—
120	C₂×Q₈	—	28	4	no	—
121	C₂²×C₄	—	4	2	no	—
122	C₂²×C₄	—	28	2	no	—
123	C₂²×C₄	D₄×D₇	1	2	yes	—
124	C₂²×C₄	C₂×C₂²⋊D₇	1	1	yes	characteristic
125	Q₁₆	—	28	16	no	—
126	C₂×C₈	—	7	8	no	—
127	C₂×C₈	—	14	4	no	—
128	C₂×C₈	—	14	4	no	—
129	C₂×C₈	(C₂×D₇)⋊C₄	1	4	yes	—
130	C₂×C₈	#1141	1	4	yes	—
131	C₂×C₈	#1143	1	4	yes	—
132	C₄⋊C₄	—	4	8	no	—
133	C₄⋊C₄	—	4	4	no	—
134	C₄⋊C₄	—	14	8	no	—
135	C₄⋊C₄	—	28	4	no	—
136	C₄²	C₂×C₄×D₇	1	1	yes	characteristic
137	C₄²	C₂×D₂₈	1	1	yes	characteristic
138	C₄²	#1161	1	2	yes	—
139	C₂×C₇	#1707	1	2	yes	—
140	C₂×C₇	#1709	1	1	yes	characteristic
141	C₂×C₇	#1713	1	1	yes	characteristic
142	C₂×C₇	#1957	1	1	yes	characteristic
143	C₂×C₇	#1969	1	2	yes	—
144	C₂³	#6503	1	1	yes	characteristic
145	Q₈	—	28	8	no	—
146	Q₈	—	56	4	no	—
147	C₂×C₄	—	8	4	no	—
148	C₂×C₄	—	8	2	no	—
149	C₂×C₄	—	28	4	no	—
150	C₂×C₄	—	56	2	no	—
151	C₂×C₄	—	56	2	no	—
152	C₂×C₄	#6456	1	2	yes	—
153	C₂×C₄	#6458	1	2	yes	—
154	C₂×C₄	#6475	1	4	yes	—
155	C₂×C₄	D₇×Q₁₆	1	4	yes	—
156	C₂×C₄	C₂×(C₂×D₇)⋊C₄	1	1	yes	characteristic
157	C₂×C₄	C₂×#1141	1	2	yes	—
158	C₂×C₄	C₂×#1143	1	2	yes	—
159	C₂×C₄	#6510	1	1	yes	characteristic
160	C₈	—	14	8	no	—
161	C₈	#6383	1	8	yes	—
162	C₇	#9825	1	1	yes	characteristic, complemented, Sylow
163	C₂²	#89760	1	1	yes	characteristic
164	C₂²	#89845	1	1	yes	characteristic
165	C₂²	#89857	1	1	yes	characteristic
166	C₂²	#89927	1	2	yes	—
167	C₂²	#89953	1	2	yes	—
168	C₄	—	16	2	no	—
169	C₄	—	56	4	no	—
170	C₄	#89645	1	4	yes	—
171	C₄	#89659	1	4	yes	—
172	C₄	C₂×#6383	1	2	yes	—
173	C₄	#89860	1	2	yes	—
174	C₂	#11727334	1	1	yes	characteristic
175	C₂	#11728381	1	2	yes	—
176	C₂	#11729029	1	1	yes	characteristic
177	C₂	#11729052	1	1	yes	characteristic
178	C₂	#11729114	1	2	yes	—

subgroup

G/H

class size

multiplicity

normal?

properties

C₂×#89471

C₂

yes

maximal, characteristic

C₇×#1845

C₂

yes

maximal, characteristic

C₂×C₈×C₇⋊C₈

C₂

yes

maximal, characteristic

#11727336

C₂

yes

maximal

C₂²×Dic₂₈

—

C₂×#6432

C₂²

yes

—

C₂×C₇×#107

—

C₂×C₄×C₇×C₈

C₂²

yes

characteristic

C₇×#384

C₂²

yes

—

C₂×C₄×C₇⋊C₈

C₂²

yes

—

#89471

—

#89471

C₄

yes

complemented

#89324

—

C₈×C₇⋊C₈

—

#9825

—

maximal, Sylow

C₂²×Dic₁₄

—

C₂²×C₇×C₈

D₄

yes

—

C₂×C₇×C₄⋊C₄

—

C₂×C₇×C₄⋊C₄

—

C₂×C₄²×C₇

C₂³

yes

characteristic

C₂×C₄⋊Dic₇

—

C₂²×C₇⋊C₈

D₄

yes

—

C₂×Dic₂₈

—

C₂×Dic₂₈

—

C₂×Dic₂₈

—

#6432

—

#6432

—

#6432

C₂×C₄

yes

—

C₇×#107

—

C₄×C₇×C₈

C₂×C₄

yes

—

C₄×C₇×C₈

D₄

yes

—

C₄×C₇⋊C₈

—

C₄×C₇⋊C₈

—

C₂×#494

—

#1990

—

#1845

D₇

yes

characteristic

C₂×C₈²

—

C₂²×C₄×C₇

—

C₂²×C₄×C₇

C₂×D₄

yes

characteristic

C₂²×C₄×C₇

C₂×D₄

yes

—

C₂²×Dic₇

—

C₂×Dic₁₄

—

C₂×Dic₁₄

—

C₂×Dic₁₄

—

C₂×Dic₁₄

—

C₂×C₇×C₈

C₂²⋊C₄

yes

—

C₂×C₇×C₈

D₈

yes

—

C₂×C₇×C₈

SD₁₆

yes

—

C₇×C₄⋊C₄

—

C₇×C₄⋊C₄

—

C₄²×C₇

C₂²×C₄

yes

characteristic

C₄²×C₇

C₂×D₄

yes

characteristic

C₄²×C₇

#41

yes

—

C₄⋊Dic₇

—

C₄⋊Dic₇

—

C₂×C₇⋊C₈

—

C₂×C₇⋊C₈

—

C₂×C₇⋊C₈

Q₁₆

yes

complemented

Dic₂₈

—

C₂²×Q₁₆

—

C₂×C₄⋊Q₈

—

#494

—

#494

—

C₂×#107

—

C₂×C₄×C₈

—

C₂×C₄×C₈

C₂×D₇

yes

characteristic

#384

C₂×D₇

yes

—

#340

—

C₈²

—

C₂³×C₇

#127

yes

characteristic

C₂×C₄×C₇

—

C₂×C₄×C₇

—

C₂×C₄×C₇

C₂×C₂²⋊C₄

yes

characteristic

C₂×C₄×C₇

C₄×D₄

yes

—

C₂×C₄×C₇

#121

yes

—

C₂×C₄×C₇

#124

yes

characteristic

C₂×C₄×C₇

C₂×D₈

yes

—

C₂×C₄×C₇

C₂×SD₁₆

yes

—

C₂×C₄×C₇

C₂×Q₁₆

yes

—

C₂×C₄×C₇

#135

yes

—

C₂×Dic₇

—

C₂×Dic₇

—

C₂×Dic₇

—

Dic₁₄

—

Dic₁₄

—

C₇×C₈

#102

yes

—

C₇⋊C₈

—

C₂²×Q₈

—

C₂×Q₁₆

—

C₂×Q₁₆

—

C₂×Q₁₆

—

C₂²×C₈

—

C₂²×C₈

C₂²⋊D₇

yes

—

C₄⋊Q₈

—

C₄⋊Q₈

—

C₄⋊Q₈

—

C₂×C₄⋊C₄

—

C₂×C₄⋊C₄

—

C₂×C₄⋊C₄

—

100

C₂×C₄²

C₂²×D₇

yes

characteristic

101

#107

—

102

C₄×C₈

—

103

C₄×C₈

—

104

C₄×C₈

C₄×D₇

yes

—

105

C₄×C₈

D₂₈

yes

—

106

C₂²×C₇

#390

yes

characteristic

107

C₂²×C₇

#492

yes

characteristic

108

C₂²×C₇

#493

yes

characteristic

109

C₂²×C₇

#494

yes

—

110

C₂²×C₇

#495

yes

—

111

C₄×C₇

—

112

C₄×C₇

C₂×#102

yes

—

113

C₄×C₇

C₄×Q₁₆

yes

—

114

C₄×C₇

#467

yes

—

115

C₄×C₇

#486

yes

—

116

Dic₇

—

117

C₂×Q₈

—

118

C₂×Q₈

—

119

C₂×Q₈

—

120

C₂×Q₈

—

121

C₂²×C₄

—

122

C₂²×C₄

—

123

C₂²×C₄

D₄×D₇

yes

—

124

C₂²×C₄

C₂×C₂²⋊D₇

yes

characteristic

125

Q₁₆

—

126

C₂×C₈

—

127

C₂×C₈

—

128

C₂×C₈

—

129

C₂×C₈

(C₂×D₇)⋊C₄

yes

—

130

C₂×C₈

#1141

yes

—

131

C₂×C₈

#1143

yes

—

132

C₄⋊C₄

—

133

C₄⋊C₄

—

134

C₄⋊C₄

—

135

C₄⋊C₄

—

136

C₄²

C₂×C₄×D₇

yes

characteristic

137

C₄²

C₂×D₂₈

yes

characteristic

138

C₄²

#1161

yes

—

139

C₂×C₇

#1707

yes

—

140

C₂×C₇

#1709

yes

characteristic

141

C₂×C₇

#1713

yes

characteristic

142

C₂×C₇

#1957

yes

characteristic

143

C₂×C₇

#1969

yes

—

144

C₂³

#6503

yes

characteristic

145

Q₈

—

146

Q₈

—

147

C₂×C₄

—

148

C₂×C₄

—

149

C₂×C₄

—

150

C₂×C₄

—

151

C₂×C₄

—

152

C₂×C₄

#6456

yes

—

153

C₂×C₄

#6458

yes

—

154

C₂×C₄

#6475

yes

—

155

C₂×C₄

D₇×Q₁₆

yes

—

156

C₂×C₄

C₂×(C₂×D₇)⋊C₄

yes

characteristic

157

C₂×C₄

C₂×#1141

yes

—

158

C₂×C₄

C₂×#1143

yes

—

159

C₂×C₄

#6510

yes

characteristic

160

C₈

—

161

C₈

#6383

yes

—

162

C₇

#9825

yes

characteristic, complemented, Sylow

163

C₂²

#89760

yes

characteristic

164

C₂²

#89845

yes

characteristic

165

C₂²

#89857

yes

characteristic

166

C₂²

#89927

yes

—

167

C₂²

#89953

yes

—

168

C₄

—

169

C₄

—

170

C₄

#89645

yes

—

171

C₄

#89659

yes

—

172

C₄

C₂×#6383

yes

—

173

C₄

#89860

yes

—

174

C₂

#11727334

yes

characteristic

175

C₂

#11728381

yes

—

176

C₂

#11729029

yes

characteristic

177

C₂

#11729052

yes

characteristic

178

C₂

#11729114

yes

—

	subgroup	G/H
center	C₂³	#6503
commutator	C₄²×C₇	C₂²×C₄
Frattini	C₂×C₄²	C₂²×D₇
Fitting	C₇×#1845	C₂
socle, prime radical	C₂³×C₇	#127
cosocle	C₂×C₄²×C₇	C₂³
hypercenter	#1845	D₇
hypocenter	C₇	#9825
LCS	G ▷ C₄²×C₇ ▷ C₂²×C₇ ▷ C₇
UCS	C₁ ◁ C₂³ ◁ C₂×C₄² ◁ #1845

subgroup

G/H

center

C₂³

#6503

commutator

C₄²×C₇

C₂²×C₄

Frattini

C₂×C₄²

C₂²×D₇

C₂

C₂³×C₇

C₂×C₄²×C₇

C₂³

D₇

C₇

G ▷ C₄²×C₇ ▷ C₂²×C₇ ▷ C₇

UCS

C₁ ◁ C₂³ ◁ C₂×C₄² ◁ #1845

N	H ≅ G/N	type
12. #89471	C₄	semidirect
58. C₂×C₇⋊C₈	Q₁₆	semidirect
162. C₇	#9825	semidirect

H ≅ G/N

type

12. #89471

C₄

semidirect

58. C₂×C₇⋊C₈

Q₁₆

semidirect

162. C₇

#9825

semidirect

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

subgroup	index
C₂×#89471	2
C₇×#1845	2
C₂×C₈×C₇⋊C₈	2
#11727336	2
#9825	7

subgroup

index

C₂×C₈×C₇⋊C₈

Subgroups — #13253562

Conjugacy classes of subgroups

Complements

Maximal subgroups