Group #13268570 — subgroups

Subgroups — #13268570

Order 1792.

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

normal subgroups only

This group has 318 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	#11726815	C₂	1	2	yes	maximal, complemented
2	#11726812	C₂	1	2	yes	maximal, complemented
3	#11725817	C₂	1	1	yes	maximal, characteristic, complemented
4	#11725816	C₂	1	1	yes	maximal, characteristic
5	C₇×#1312	C₂	1	1	yes	maximal, characteristic, complemented
6	C₂×#6374	—	2	4	no	—
7	#89471	C₂²	1	1	yes	characteristic
8	#89467	—	2	1	no	—
9	#89467	—	2	1	no	—
10	#89467	C₂²	1	2	yes	—
11	#89454	C₂²	1	1	yes	characteristic
12	#89453	—	2	1	no	—
13	#89452	—	2	1	no	—
14	C₄×C₇×C₁₆	C₂²	1	2	yes	—
15	C₇×C₈²	C₂²	1	1	yes	characteristic
16	#89304	—	2	4	no	—
17	#12529	—	7	1	no	maximal, Sylow
18	C₂×Dic₂₈	—	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	#6437	—	2	2	no	—
24	#6432	—	2	1	no	—
25	#6432	—	2	1	no	—
26	C₂×C₇×C₁₆	D₄	1	4	yes	—
27	C₄×C₇×C₈	D₄	1	1	yes	characteristic
28	C₄×C₇×C₈	D₄	1	1	yes	characteristic
29	C₄×C₇×C₈	C₂³	1	1	yes	characteristic
30	#6392	—	2	2	no	—
31	#6391	—	4	2	no	—
32	#6390	—	4	2	no	—
33	#6374	—	4	8	no	—
34	#2248	—	7	2	no	—
35	#2240	—	7	2	no	—
36	#1712	—	7	1	no	—
37	#1707	—	7	1	no	—
38	#1312	D₇	1	1	yes	characteristic, complemented
39	C₂×D₂₈	—	4	2	no	—
40	C₂×D₂₈	—	4	2	no	—
41	C₂×D₂₈	—	4	2	no	—
42	C₂×Dic₁₄	—	4	1	no	—
43	C₂×Dic₁₄	—	4	1	no	—
44	C₂×C₇×C₈	D₈	1	2	yes	—
45	C₂×C₇×C₈	D₈	1	2	yes	—
46	C₂×C₇×C₈	C₂×D₄	1	2	yes	—
47	C₄²×C₇	C₂×D₄	1	1	yes	characteristic
48	C₄⋊Dic₇	—	4	1	no	—
49	C₄⋊Dic₇	—	4	1	no	—
50	C₄⋊Dic₇	—	4	1	no	—
51	C₄⋊Dic₇	—	4	1	no	—
52	Dic₂₈	—	4	4	no	—
53	D₅₆	—	4	4	no	—
54	D₅₆	—	4	4	no	—
55	D₅₆	—	8	4	no	—
56	D₅₆	—	8	4	no	—
57	C₇×C₁₆	—	2	4	no	—
58	C₂×SD₃₂	—	14	4	no	—
59	#500	—	7	1	no	—
60	#498	—	14	1	no	—
61	#494	—	7	1	no	—
62	#493	—	7	2	no	—
63	#493	—	14	1	no	—
64	#493	—	14	1	no	—
65	#487	—	14	1	no	—
66	#357	—	14	4	no	—
67	C₄×C₁₆	C₂×D₇	1	2	yes	—
68	C₈²	C₂×D₇	1	1	yes	characteristic
69	C₂²×D₇	—	8	2	no	—
70	C₂×C₄×C₇	#127	1	1	yes	characteristic
71	C₂×C₄×C₇	C₂×D₈	1	1	yes	characteristic
72	C₂×C₄×C₇	C₂×D₈	1	1	yes	characteristic
73	C₂×Dic₇	—	8	1	no	—
74	C₂×Dic₇	—	8	1	no	—
75	D₂₈	—	8	2	no	—
76	D₂₈	—	8	2	no	—
77	D₂₈	—	8	4	no	—
78	D₂₈	—	8	4	no	—
79	Dic₁₄	—	8	2	no	—
80	Dic₁₄	—	8	2	no	—
81	C₇×C₈	—	2	2	no	—
82	C₇×C₈	SD₃₂	1	4	yes	complemented
83	C₇×C₈	AGL(1,ℤ/8)	1	4	yes	—
84	C₂×Q₁₆	—	14	2	no	—
85	C₂×D₈	—	14	2	no	—
86	C₂×D₈	—	14	2	no	—
87	C₂×D₈	—	28	2	no	—
88	C₂×D₈	—	28	2	no	—
89	C₄⋊Q₈	—	14	1	no	—
90	C₄⋊Q₈	—	14	1	no	—
91	#127	—	14	2	no	—
92	SD₃₂	—	28	8	no	—
93	C₂×C₁₆	D₂₈	1	4	yes	—
94	#107	—	14	2	no	—
95	#106	—	28	2	no	—
96	#103	—	28	2	no	—
97	C₄×C₈	D₂₈	1	1	yes	characteristic
98	C₄×C₈	D₂₈	1	1	yes	characteristic
99	C₄×C₈	C₂²×D₇	1	1	yes	characteristic
100	C₂²×C₇	#493	1	1	yes	characteristic
101	C₂×D₇	—	16	2	no	—
102	C₂×D₇	—	16	2	no	—
103	C₂×D₇	—	16	2	no	—
104	C₄×C₇	#496	1	2	yes	—
105	C₄×C₇	C₂×SD₃₂	1	2	yes	—
106	C₄×C₇	#509	1	2	yes	—
107	Dic₇	—	16	1	no	—
108	Dic₇	—	16	1	no	—
109	C₂×Q₈	—	28	1	no	—
110	C₂×Q₈	—	28	1	no	—
111	C₂×D₄	—	28	2	no	—
112	C₂×D₄	—	28	2	no	—
113	C₂×D₄	—	28	2	no	—
114	Q₁₆	—	28	4	no	—
115	D₈	—	28	4	no	—
116	D₈	—	28	4	no	—
117	D₈	—	56	4	no	—
118	D₈	—	56	4	no	—
119	C₂×C₈	D₅₆	1	2	yes	—
120	C₂×C₈	D₅₆	1	2	yes	—
121	C₂×C₈	C₂×D₂₈	1	2	yes	—
122	C₄⋊C₄	—	28	1	no	—
123	C₄⋊C₄	—	28	1	no	—
124	C₄⋊C₄	—	28	1	no	—
125	C₄⋊C₄	—	28	1	no	—
126	C₄²	C₂×D₂₈	1	1	yes	characteristic
127	C₁₆	—	2	4	no	—
128	C₂×C₇	#1721	1	1	yes	characteristic
129	C₂×C₇	#2248	1	1	yes	characteristic
130	C₂×C₇	#2250	1	1	yes	characteristic
131	D₇	—	32	2	no	—
132	C₂³	—	56	2	no	—
133	Q₈	—	56	2	no	—
134	Q₈	—	56	2	no	—
135	D₄	—	56	4	no	—
136	D₄	—	56	4	no	—
137	D₄	—	56	2	no	—
138	D₄	—	56	2	no	—
139	C₂×C₄	—	56	1	no	—
140	C₂×C₄	—	56	1	no	—
141	C₂×C₄	#6437	1	1	yes	characteristic
142	C₂×C₄	C₂×D₅₆	1	1	yes	characteristic
143	C₂×C₄	C₂×D₅₆	1	1	yes	characteristic
144	C₈	—	2	2	no	—
145	C₈	#6374	1	4	yes	complemented
146	C₈	#6471	1	4	yes	—
147	C₇	#12529	1	1	yes	characteristic, complemented, Sylow
148	C₂²	—	112	2	no	—
149	C₂²	—	112	2	no	—
150	C₂²	—	112	2	no	—
151	C₂²	#89467	1	1	yes	characteristic
152	C₄	—	112	1	no	—
153	C₄	—	112	1	no	—
154	C₄	#89484	1	2	yes	—
155	C₄	C₂×#6374	1	2	yes	—
156	C₄	#89680	1	2	yes	—
157	C₂	—	224	2	no	—
158	C₂	#11725848	1	1	yes	characteristic
159	C₂	#11726815	1	1	yes	characteristic
160	C₂	#11726835	1	1	yes	characteristic

subgroup

G/H

class size

multiplicity

normal?

properties

#11726815

C₂

yes

maximal, complemented

#11726812

C₂

yes

maximal, complemented

#11725817

C₂

yes

maximal, characteristic, complemented

#11725816

C₂

yes

maximal, characteristic

C₇×#1312

C₂

yes

maximal, characteristic, complemented

C₂×#6374

—

#89471

C₂²

yes

—

—

C₂²

yes

—

#89454

C₂²

yes

characteristic

#89453

—

#89452

—

C₄×C₇×C₁₆

C₂²

yes

—

C₇×C₈²

C₂²

yes

—

—

C₂×Dic₂₈

—

C₂×D₅₆

—

C₂×D₅₆

—

C₂×D₅₆

—

C₂×D₅₆

—

#6437

—

#6432

—

#6432

—

C₂×C₇×C₁₆

D₄

yes

—

C₄×C₇×C₈

D₄

yes

characteristic

C₄×C₇×C₈

D₄

yes

characteristic

C₄×C₇×C₈

C₂³

yes

—

—

—

—

—

—

—

—

D₇

yes

characteristic, complemented

C₂×D₂₈

—

C₂×D₂₈

—

C₂×D₂₈

—

C₂×Dic₁₄

—

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

—

C₄⋊Dic₇

—

C₄⋊Dic₇

—

Dic₂₈

—

D₅₆

—

D₅₆

—

D₅₆

—

D₅₆

—

C₇×C₁₆

—

C₂×SD₃₂

—

—

—

—

—

—

—

—

—

C₄×C₁₆

C₂×D₇

yes

—

C₈²

C₂×D₇

yes

characteristic

C₂²×D₇

—

C₂×C₄×C₇

#127

yes

characteristic

C₂×C₄×C₇

C₂×D₈

yes

characteristic

C₂×C₄×C₇

C₂×D₈

yes

characteristic

C₂×Dic₇

—

C₂×Dic₇

—

D₂₈

—

D₂₈

—

D₂₈

—

D₂₈

—

Dic₁₄

—

Dic₁₄

—

C₇×C₈

—

C₇×C₈

SD₃₂

yes

complemented

C₇×C₈

AGL(1,ℤ/8)

yes

—

C₂×Q₁₆

—

C₂×D₈

—

C₂×D₈

—

C₂×D₈

—

C₂×D₈

—

C₄⋊Q₈

—

C₄⋊Q₈

—

#127

—

SD₃₂

—

C₂×C₁₆

D₂₈

yes

—

#107

—

#106

—

#103

—

C₄×C₈

D₂₈

yes

characteristic

C₄×C₈

D₂₈

yes

characteristic

C₄×C₈

C₂²×D₇

yes

characteristic

100

C₂²×C₇

#493

yes

characteristic

101

C₂×D₇

—

102

C₂×D₇

—

103

C₂×D₇

—

104

C₄×C₇

#496

yes

—

105

C₄×C₇

C₂×SD₃₂

yes

—

106

C₄×C₇

#509

yes

—

107

Dic₇

—

108

Dic₇

—

109

C₂×Q₈

—

110

C₂×Q₈

—

111

C₂×D₄

—

112

C₂×D₄

—

113

C₂×D₄

—

114

Q₁₆

—

115

D₈

—

116

D₈

—

117

D₈

—

118

D₈

—

119

C₂×C₈

D₅₆

yes

—

120

C₂×C₈

D₅₆

yes

—

121

C₂×C₈

C₂×D₂₈

yes

—

122

C₄⋊C₄

—

123

C₄⋊C₄

—

124

C₄⋊C₄

—

125

C₄⋊C₄

—

126

C₄²

C₂×D₂₈

yes

characteristic

127

C₁₆

—

128

C₂×C₇

#1721

yes

characteristic

129

C₂×C₇

#2248

yes

characteristic

130

C₂×C₇

#2250

yes

characteristic

131

D₇

—

132

C₂³

—

133

Q₈

—

134

Q₈

—

135

D₄

—

136

D₄

—

137

D₄

—

138

D₄

—

139

C₂×C₄

—

140

C₂×C₄

—

141

C₂×C₄

#6437

yes

characteristic

142

C₂×C₄

C₂×D₅₆

yes

characteristic

143

C₂×C₄

C₂×D₅₆

yes

characteristic

144

C₈

—

145

C₈

#6374

yes

complemented

146

C₈

#6471

yes

—

147

C₇

#12529

yes

characteristic, complemented, Sylow

148

C₂²

—

112

—

149

C₂²

—

112

—

150

C₂²

—

112

—

151

C₂²

#89467

yes

characteristic

152

C₄

—

112

—

153

C₄

—

112

—

154

C₄

#89484

yes

—

155

C₄

C₂×#6374

yes

—

156

C₄

#89680

yes

—

157

C₂

—

224

—

158

C₂

#11725848

yes

characteristic

159

C₂

#11726815

yes

characteristic

160

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
prime radical	#11725816	C₂
hypercenter	#1312	D₇
hypocenter	C₇	#12529
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

C₂

D₇

C₇

G ▷ C₄×C₇×C₈ ▷ C₂×C₄×C₇ ▷ C₂×C₇ ▷ C₇

UCS

C₁ ◁ C₂² ◁ C₄² ◁ C₈² ◁ #1312

N	H ≅ G/N	type
1. #11726815	C₂	semidirect
2. #11726812	C₂	semidirect
3. #11725817	C₂	semidirect
5. C₇×#1312	C₂	semidirect
38. #1312	D₇	semidirect
82. C₇×C₈	SD₃₂	semidirect
145. C₈	#6374	semidirect
147. C₇	#12529	semidirect

H ≅ G/N

type

1. #11726815

C₂

semidirect

2. #11726812

C₂

semidirect

3. #11725817

C₂

semidirect

5. C₇×#1312

C₂

semidirect

38. #1312

D₇

semidirect

82. C₇×C₈

SD₃₂

semidirect

145. C₈

#6374

semidirect

147. C₇

#12529

semidirect

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

subgroup	index
#11726815	2
#11726812	2
#11725817	2
#11725816	2
C₇×#1312	2
#12529	7

subgroup

index

Subgroups — #13268570

Conjugacy classes of subgroups

Complements

Maximal subgroups