Group #13268548 — subgroups

Subgroups — #13268548

Order 1792.

340 conjugacy classes of subgroups; only the 91 normal subgroups are shown.

normal subgroups only

This group has 340 conjugacy classes of subgroups — too many to draw. Only its 91 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	#11726814	C₂	1	2	yes	maximal, complemented
2	#11726812	C₂	1	2	yes	maximal, complemented
3	#11725816	C₂	1	1	yes	maximal, characteristic, complemented
4	#11725812	C₂	1	1	yes	maximal, characteristic, complemented
5	C₇×#1312	C₂	1	1	yes	maximal, characteristic, complemented
6	C₂×D₁₁₂	—	2	4	no	—
7	#89467	—	2	1	no	—
8	#89467	—	2	1	no	—
9	#89467	C₂²	1	2	yes	—
10	#89467	C₂²	1	1	yes	characteristic
11	#89466	—	2	1	no	—
12	#89465	—	2	1	no	—
13	#89454	C₂²	1	1	yes	characteristic
14	C₄×C₇×C₁₆	C₂²	1	2	yes	—
15	C₇×C₈²	C₂²	1	1	yes	characteristic
16	#89304	—	2	4	no	—
17	#12525	—	7	1	no	maximal, Sylow
18	C₂×D₅₆	—	2	2	no	—
19	C₂×D₅₆	—	2	2	no	—
20	C₂×D₅₆	—	2	2	no	—
21	C₂×D₅₆	—	4	2	no	—
22	C₂×D₅₆	—	4	2	no	—
23	C₂×#1132	—	4	2	no	—
24	#6437	—	2	1	no	—
25	#6437	—	2	2	no	—
26	#6432	—	2	1	no	—
27	C₂×C₇×C₁₆	D₄	1	4	yes	—
28	C₄×C₇×C₈	D₄	1	1	yes	characteristic
29	C₄×C₇×C₈	D₄	1	1	yes	characteristic
30	C₄×C₇×C₈	C₂³	1	1	yes	characteristic
31	#6395	—	4	2	no	—
32	#6392	—	2	2	no	—
33	D₁₁₂	—	4	8	no	—
34	#2246	—	7	2	no	—
35	#2240	—	7	2	no	—
36	#1712	—	7	1	no	—
37	#1706	—	7	1	no	—
38	#1312	D₇	1	1	yes	characteristic, complemented
39	C₂×D₂₈	—	4	1	no	—
40	C₂×D₂₈	—	4	2	no	—
41	C₂×D₂₈	—	4	1	no	—
42	C₂×D₂₈	—	4	2	no	—
43	C₂×D₂₈	—	4	2	no	—
44	C₂×D₂₈	—	4	1	no	—
45	C₂×Dic₁₄	—	4	1	no	—
46	C₂×C₇×C₈	D₈	1	2	yes	—
47	C₂×C₇×C₈	D₈	1	2	yes	—
48	C₂×C₇×C₈	C₂×D₄	1	2	yes	—
49	C₄²×C₇	C₂×D₄	1	1	yes	characteristic
50	C₄⋊Dic₇	—	4	1	no	—
51	C₄⋊Dic₇	—	4	1	no	—
52	D₅₆	—	4	4	no	—
53	D₅₆	—	4	4	no	—
54	D₅₆	—	4	4	no	—
55	D₅₆	—	8	4	no	—
56	D₅₆	—	8	4	no	—
57	#1132	—	8	4	no	—
58	C₇×C₁₆	—	2	4	no	—
59	C₂×D₁₆	—	14	4	no	—
60	#500	—	7	1	no	—
61	#493	—	7	1	no	—
62	#493	—	7	2	no	—
63	#493	—	14	1	no	—
64	#493	—	14	1	no	—
65	#492	—	14	1	no	—
66	#486	—	14	1	no	—
67	#357	—	14	4	no	—
68	C₄×C₁₆	C₂×D₇	1	2	yes	—
69	C₈²	C₂×D₇	1	1	yes	characteristic
70	C₂²×D₇	—	8	1	no	—
71	C₂²×D₇	—	8	2	no	—
72	C₂×C₄×C₇	#127	1	1	yes	characteristic
73	C₂×C₄×C₇	C₂×D₈	1	1	yes	characteristic
74	C₂×C₄×C₇	C₂×D₈	1	1	yes	characteristic
75	C₂×Dic₇	—	8	1	no	—
76	D₂₈	—	8	2	no	—
77	D₂₈	—	8	2	no	—
78	D₂₈	—	8	2	no	—
79	D₂₈	—	8	2	no	—
80	D₂₈	—	8	4	no	—
81	D₂₈	—	8	4	no	—
82	D₂₈	—	8	2	no	—
83	Dic₁₄	—	8	2	no	—
84	C₇×C₈	—	2	2	no	—
85	C₇×C₈	D₁₆	1	4	yes	complemented
86	C₇×C₈	AGL(1,ℤ/8)	1	4	yes	—
87	C₂×SD₁₆	—	28	2	no	—
88	C₂×D₈	—	14	2	no	—
89	C₂×D₈	—	14	2	no	—
90	C₂×D₈	—	14	2	no	—
91	C₂×D₈	—	28	2	no	—
92	C₂×D₈	—	28	2	no	—
93	C₄⋊Q₈	—	14	1	no	—
94	#127	—	14	1	no	—
95	#127	—	14	2	no	—
96	D₁₆	—	28	8	no	—
97	C₂×C₁₆	D₂₈	1	4	yes	—
98	#107	—	14	2	no	—
99	#102	—	28	2	no	—
100	C₄×C₈	D₂₈	1	1	yes	characteristic
101	C₄×C₈	D₂₈	1	1	yes	characteristic
102	C₄×C₈	C₂²×D₇	1	1	yes	characteristic
103	C₂²×C₇	#493	1	1	yes	characteristic
104	C₂×D₇	—	16	2	no	—
105	C₂×D₇	—	16	1	no	—
106	C₂×D₇	—	16	1	no	—
107	C₂×D₇	—	16	2	no	—
108	C₂×D₇	—	16	2	no	—
109	C₂×D₇	—	16	1	no	—
110	C₄×C₇	#496	1	2	yes	—
111	C₄×C₇	C₂×D₁₆	1	2	yes	—
112	C₄×C₇	#509	1	2	yes	—
113	Dic₇	—	16	1	no	—
114	C₂×Q₈	—	28	1	no	—
115	C₂×D₄	—	28	1	no	—
116	C₂×D₄	—	28	2	no	—
117	C₂×D₄	—	28	1	no	—
118	C₂×D₄	—	28	2	no	—
119	C₂×D₄	—	28	2	no	—
120	C₂×D₄	—	28	1	no	—
121	SD₁₆	—	56	4	no	—
122	D₈	—	28	4	no	—
123	D₈	—	28	4	no	—
124	D₈	—	28	4	no	—
125	D₈	—	56	4	no	—
126	D₈	—	56	4	no	—
127	C₂×C₈	D₅₆	1	2	yes	—
128	C₂×C₈	D₅₆	1	2	yes	—
129	C₂×C₈	C₂×D₂₈	1	2	yes	—
130	C₄⋊C₄	—	28	1	no	—
131	C₄⋊C₄	—	28	1	no	—
132	C₄²	C₂×D₂₈	1	1	yes	characteristic
133	C₁₆	—	2	4	no	—
134	C₂×C₇	#1721	1	1	yes	characteristic
135	C₂×C₇	#2246	1	1	yes	characteristic
136	C₂×C₇	#2250	1	1	yes	characteristic
137	D₇	—	32	1	no	—
138	D₇	—	32	2	no	—
139	C₂³	—	56	1	no	—
140	C₂³	—	56	2	no	—
141	Q₈	—	56	2	no	—
142	D₄	—	56	4	no	—
143	D₄	—	56	2	no	—
144	D₄	—	56	4	no	—
145	D₄	—	56	2	no	—
146	D₄	—	56	2	no	—
147	D₄	—	56	2	no	—
148	D₄	—	56	2	no	—
149	C₂×C₄	—	56	1	no	—
150	C₂×C₄	#6437	1	1	yes	characteristic
151	C₂×C₄	C₂×D₅₆	1	1	yes	characteristic
152	C₂×C₄	C₂×D₅₆	1	1	yes	characteristic
153	C₈	—	2	2	no	—
154	C₈	D₁₁₂	1	4	yes	complemented
155	C₈	#6471	1	4	yes	—
156	C₇	#12525	1	1	yes	characteristic, complemented, Sylow
157	C₂²	—	112	1	no	—
158	C₂²	—	112	2	no	—
159	C₂²	—	112	2	no	—
160	C₂²	—	112	1	no	—
161	C₂²	—	112	2	no	—
162	C₂²	—	112	1	no	—
163	C₂²	#89467	1	1	yes	characteristic
164	C₄	—	112	1	no	—
165	C₄	#89484	1	2	yes	—
166	C₄	C₂×D₁₁₂	1	2	yes	—
167	C₄	#89680	1	2	yes	—
168	C₂	—	224	2	no	—
169	C₂	—	224	1	no	—
170	C₂	#11725848	1	1	yes	characteristic
171	C₂	#11726814	1	1	yes	characteristic
172	C₂	#11726835	1	1	yes	characteristic

subgroup

G/H

class size

multiplicity

normal?

properties

#11726814

C₂

yes

maximal, complemented

#11726812

C₂

yes

maximal, complemented

#11725816

C₂

yes

maximal, characteristic, complemented

#11725812

C₂

yes

maximal, characteristic, complemented

C₇×#1312

C₂

yes

maximal, characteristic, complemented

C₂×D₁₁₂

—

#89467

—

#89467

—

#89467

C₂²

yes

—

#89467

C₂²

yes

—

—

C₂²

yes

characteristic

C₄×C₇×C₁₆

C₂²

yes

—

C₇×C₈²

C₂²

yes

—

—

C₂×D₅₆

—

C₂×D₅₆

—

C₂×D₅₆

—

C₂×D₅₆

—

C₂×D₅₆

—

—

—

—

—

C₂×C₇×C₁₆

D₄

yes

—

C₄×C₇×C₈

D₄

yes

characteristic

C₄×C₇×C₈

D₄

yes

characteristic

C₄×C₇×C₈

C₂³

yes

characteristic

#6395

—

#6392

—

D₁₁₂

—

—

—

—

—

D₇

yes

characteristic, complemented

C₂×D₂₈

—

C₂×D₂₈

—

C₂×D₂₈

—

C₂×D₂₈

—

C₂×D₂₈

—

C₂×D₂₈

—

C₂×Dic₁₄

—

C₂×C₇×C₈

D₈

yes

—

C₂×C₇×C₈

D₈

yes

—

C₂×C₇×C₈

C₂×D₄

yes

—

C₄²×C₇

C₂×D₄

yes

characteristic

C₄⋊Dic₇

—

C₄⋊Dic₇

—

D₅₆

—

D₅₆

—

D₅₆

—

D₅₆

—

D₅₆

—

#1132

—

C₇×C₁₆

—

C₂×D₁₆

—

—

—

—

—

—

—

—

—

C₄×C₁₆

C₂×D₇

yes

—

C₈²

C₂×D₇

yes

characteristic

C₂²×D₇

—

C₂²×D₇

—

C₂×C₄×C₇

#127

yes

characteristic

C₂×C₄×C₇

C₂×D₈

yes

characteristic

C₂×C₄×C₇

C₂×D₈

yes

characteristic

C₂×Dic₇

—

D₂₈

—

D₂₈

—

D₂₈

—

D₂₈

—

D₂₈

—

D₂₈

—

D₂₈

—

Dic₁₄

—

C₇×C₈

—

C₇×C₈

D₁₆

yes

complemented

C₇×C₈

AGL(1,ℤ/8)

yes

—

C₂×SD₁₆

—

C₂×D₈

—

C₂×D₈

—

C₂×D₈

—

C₂×D₈

—

C₂×D₈

—

C₄⋊Q₈

—

#127

—

#127

—

D₁₆

—

C₂×C₁₆

D₂₈

yes

—

#107

—

#102

—

100

C₄×C₈

D₂₈

yes

characteristic

101

C₄×C₈

D₂₈

yes

characteristic

102

C₄×C₈

C₂²×D₇

yes

characteristic

103

C₂²×C₇

#493

yes

characteristic

104

C₂×D₇

—

105

C₂×D₇

—

106

C₂×D₇

—

107

C₂×D₇

—

108

C₂×D₇

—

109

C₂×D₇

—

110

C₄×C₇

#496

yes

—

111

C₄×C₇

C₂×D₁₆

yes

—

112

C₄×C₇

#509

yes

—

113

Dic₇

—

114

C₂×Q₈

—

115

C₂×D₄

—

116

C₂×D₄

—

117

C₂×D₄

—

118

C₂×D₄

—

119

C₂×D₄

—

120

C₂×D₄

—

121

SD₁₆

—

122

D₈

—

123

D₈

—

124

D₈

—

125

D₈

—

126

D₈

—

127

C₂×C₈

D₅₆

yes

—

128

C₂×C₈

D₅₆

yes

—

129

C₂×C₈

C₂×D₂₈

yes

—

130

C₄⋊C₄

—

131

C₄⋊C₄

—

132

C₄²

C₂×D₂₈

yes

characteristic

133

C₁₆

—

134

C₂×C₇

#1721

yes

characteristic

135

C₂×C₇

#2246

yes

characteristic

136

C₂×C₇

#2250

yes

characteristic

137

D₇

—

138

D₇

—

139

C₂³

—

140

C₂³

—

141

Q₈

—

142

D₄

—

143

D₄

—

144

D₄

—

145

D₄

—

146

D₄

—

147

D₄

—

148

D₄

—

149

C₂×C₄

—

150

C₂×C₄

#6437

yes

characteristic

151

C₂×C₄

C₂×D₅₆

yes

characteristic

152

C₂×C₄

C₂×D₅₆

yes

characteristic

153

C₈

—

154

C₈

D₁₁₂

yes

complemented

155

C₈

#6471

yes

—

156

C₇

#12525

yes

characteristic, complemented, Sylow

157

C₂²

—

112

—

158

C₂²

—

112

—

159

C₂²

—

112

—

160

C₂²

—

112

—

161

C₂²

—

112

—

162

C₂²

—

112

—

163

C₂²

#89467

yes

characteristic

164

C₄

—

112

—

165

C₄

#89484

yes

—

166

C₄

C₂×D₁₁₂

yes

—

167

C₄

#89680

yes

—

168

C₂

—

224

—

169

C₂

—

224

—

170

C₂

#11725848

yes

characteristic

171

C₂

#11726814

yes

characteristic

172

C₂

#11726835

yes

characteristic

	subgroup	G/H
center	C₂²	#89467
commutator, cosocle	C₄×C₇×C₈	C₂³
Frattini	C₄×C₈	C₂²×D₇
Fitting	C₇×#1312	C₂
socle	C₂²×C₇	#493
hypercenter	#1312	D₇
hypocenter	C₇	#12525
LCS	G ▷ C₄×C₇×C₈ ▷ C₂×C₄×C₇ ▷ C₂×C₇ ▷ C₇
UCS	C₁ ◁ C₂² ◁ C₄² ◁ C₈² ◁ #1312

subgroup

G/H

center

C₂²

#89467

commutator, cosocle

C₄×C₇×C₈

C₂³

Frattini

C₄×C₈

C₂²×D₇

C₂

C₂²×C₇

D₇

C₇

G ▷ C₄×C₇×C₈ ▷ C₂×C₄×C₇ ▷ C₂×C₇ ▷ C₇

UCS

C₁ ◁ C₂² ◁ C₄² ◁ C₈² ◁ #1312

N	H ≅ G/N	type
1. #11726814	C₂	semidirect
2. #11726812	C₂	semidirect
3. #11725816	C₂	semidirect
4. #11725812	C₂	semidirect
5. C₇×#1312	C₂	semidirect
38. #1312	D₇	semidirect
85. C₇×C₈	D₁₆	semidirect
154. C₈	D₁₁₂	semidirect
156. C₇	#12525	semidirect

H ≅ G/N

type

1. #11726814

C₂

semidirect

2. #11726812

C₂

semidirect

3. #11725816

C₂

semidirect

4. #11725812

C₂

semidirect

5. C₇×#1312

C₂

semidirect

38. #1312

D₇

semidirect

85. C₇×C₈

D₁₆

semidirect

154. C₈

D₁₁₂

semidirect

156. C₇

#12525

semidirect

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

subgroup	index
#11726814	2
#11726812	2
#11725816	2
#11725812	2
C₇×#1312	2
#12525	7

subgroup

index

Subgroups — #13268548

Conjugacy classes of subgroups

Complements

Maximal subgroups