Group #13268554 — subgroups

Subgroups — #13268554

Order 1792.

296 conjugacy classes of subgroups; only the 87 normal subgroups are shown.

normal subgroups only

This group has 296 conjugacy classes of subgroups — too many to draw. Only its 87 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	C₇×#1713	C₂	1	1	yes	maximal, characteristic, complemented
2	#11729067	C₂	1	2	yes	maximal
3	#11729064	C₂	1	2	yes	maximal, complemented
4	#11725812	C₂	1	1	yes	maximal, characteristic
5	#11725244	C₂	1	1	yes	maximal, characteristic, complemented
6	C₇×#500	—	2	1	no	—
7	C₇×#500	—	2	1	no	—
8	C₇×#494	C₂²	1	2	yes	—
9	C₂×#6402	—	2	4	no	—
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	C₇×C₈²	C₂²	1	1	yes	characteristic
15	#89359	—	2	4	no	—
16	C₄×C₇⋊C₁₆	C₂²	1	2	yes	—
17	#12526	—	7	1	no	maximal, Sylow
18	C₂×C₇×Q₁₆	—	2	2	no	—
19	C₂×C₇×Q₁₆	—	2	2	no	—
20	C₇×C₄⋊Q₈	—	2	2	no	—
21	C₂×D₅₆	—	2	2	no	—
22	C₂×#1132	—	4	2	no	—
23	#6437	—	2	1	no	—
24	#6432	—	2	1	no	—
25	C₇×#107	—	4	2	no	—
26	C₇×#107	—	4	2	no	—
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	#6402	—	4	8	no	—
31	#6395	—	4	2	no	—
32	#6392	—	2	2	no	—
33	C₂×C₇⋊C₁₆	D₄	1	4	yes	—
34	#2248	—	7	2	no	—
35	#2241	—	7	2	no	—
36	#1713	D₇	1	1	yes	characteristic, complemented
37	#1706	—	7	1	no	—
38	#1312	—	7	1	no	—
39	C₂×C₇×Q₈	—	4	2	no	—
40	C₂×D₂₈	—	4	1	no	—
41	C₂×D₂₈	—	4	1	no	—
42	C₂×D₂₈	—	4	1	no	—
43	C₂×Dic₁₄	—	4	1	no	—
44	C₇×Q₁₆	—	4	4	no	—
45	C₇×Q₁₆	—	4	4	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	C₇×C₄⋊C₄	—	4	2	no	—
50	C₇×C₄⋊C₄	—	4	2	no	—
51	C₄²×C₇	C₂×D₄	1	1	yes	characteristic
52	C₄⋊Dic₇	—	4	1	no	—
53	C₄⋊Dic₇	—	4	1	no	—
54	D₅₆	—	4	4	no	—
55	#1132	—	8	4	no	—
56	C₇⋊C₁₆	—	2	4	no	—
57	C₂×SD₃₂	—	14	4	no	—
58	#500	—	2	1	no	—
59	#500	—	2	1	no	—
60	#500	—	7	1	no	—
61	#494	C₂×D₇	1	2	yes	—
62	#493	—	7	1	no	—
63	#492	—	14	1	no	—
64	#486	—	14	1	no	—
65	#358	—	14	4	no	—
66	C₄×C₁₆	—	7	2	no	—
67	C₈²	C₂×D₇	1	1	yes	characteristic
68	C₂²×D₇	—	8	1	no	—
69	C₇×Q₈	—	8	2	no	—
70	C₇×Q₈	—	8	2	no	—
71	C₂×C₄×C₇	—	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	Dic₁₄	—	8	2	no	—
80	C₇×C₈	—	2	2	no	—
81	C₇×C₈	SD₃₂	1	4	yes	complemented
82	C₇×C₈	AGL(1,ℤ/8)	1	4	yes	—
83	C₂×Q₁₆	—	2	2	no	—
84	C₂×Q₁₆	—	2	2	no	—
85	C₂×SD₁₆	—	28	2	no	—
86	C₂×D₈	—	14	2	no	—
87	C₄⋊Q₈	—	2	2	no	—
88	C₄⋊Q₈	—	14	1	no	—
89	#127	—	14	1	no	—
90	SD₃₂	—	28	8	no	—
91	C₂×C₁₆	—	7	4	no	—
92	#107	—	4	2	no	—
93	#107	—	4	2	no	—
94	#107	—	14	2	no	—
95	#102	—	28	2	no	—
96	C₄×C₈	C₂²⋊D₇	1	1	yes	characteristic
97	C₄×C₈	C₂²⋊D₇	1	1	yes	characteristic
98	C₄×C₈	C₂²×D₇	1	1	yes	characteristic
99	C₂²×C₇	#493	1	1	yes	characteristic
100	C₂×D₇	—	16	1	no	—
101	C₂×D₇	—	16	1	no	—
102	C₂×D₇	—	16	1	no	—
103	C₄×C₇	—	16	2	no	—
104	C₄×C₇	#496	1	2	yes	—
105	C₄×C₇	C₂×SD₃₂	1	2	yes	—
106	C₄×C₇	#510	1	2	yes	—
107	Dic₇	—	16	1	no	—
108	C₂×Q₈	—	4	2	no	—
109	C₂×Q₈	—	28	1	no	—
110	C₂×D₄	—	28	1	no	—
111	C₂×D₄	—	28	1	no	—
112	C₂×D₄	—	28	1	no	—
113	Q₁₆	—	4	4	no	—
114	Q₁₆	—	4	4	no	—
115	SD₁₆	—	56	4	no	—
116	D₈	—	28	4	no	—
117	C₂×C₈	#1141	1	2	yes	—
118	C₂×C₈	#1141	1	2	yes	—
119	C₂×C₈	D₄×D₇	1	2	yes	—
120	C₄⋊C₄	—	4	2	no	—
121	C₄⋊C₄	—	4	2	no	—
122	C₄⋊C₄	—	28	1	no	—
123	C₄⋊C₄	—	28	1	no	—
124	C₄²	C₂×C₂²⋊D₇	1	1	yes	characteristic
125	C₁₆	—	14	4	no	—
126	C₂×C₇	#1721	1	1	yes	characteristic
127	C₂×C₇	#2248	1	1	yes	characteristic
128	C₂×C₇	#2251	1	1	yes	characteristic
129	D₇	—	32	1	no	—
130	C₂³	—	56	1	no	—
131	Q₈	—	8	2	no	—
132	Q₈	—	8	2	no	—
133	Q₈	—	56	2	no	—
134	D₄	—	56	2	no	—
135	D₄	—	56	2	no	—
136	D₄	—	56	2	no	—
137	C₂×C₄	—	8	2	no	—
138	C₂×C₄	—	56	1	no	—
139	C₂×C₄	C₂×#1141	1	1	yes	characteristic
140	C₂×C₄	C₂×#1141	1	1	yes	characteristic
141	C₂×C₄	#6503	1	1	yes	characteristic
142	C₈	—	2	2	no	—
143	C₈	#6402	1	4	yes	complemented
144	C₈	#6474	1	4	yes	—
145	C₇	#12526	1	1	yes	characteristic, complemented, Sylow
146	C₂²	—	112	1	no	—
147	C₂²	—	112	1	no	—
148	C₂²	—	112	1	no	—
149	C₂²	#89845	1	1	yes	characteristic
150	C₄	—	16	2	no	—
151	C₄	—	112	1	no	—
152	C₄	#89926	1	2	yes	—
153	C₄	C₂×#6402	1	2	yes	—
154	C₄	#89951	1	2	yes	—
155	C₂	—	224	1	no	—
156	C₂	#11729034	1	1	yes	characteristic
157	C₂	#11729067	1	1	yes	characteristic
158	C₂	#11729068	1	1	yes	characteristic

subgroup

G/H

class size

multiplicity

normal?

properties

C₇×#1713

C₂

yes

maximal, characteristic, complemented

#11729067

C₂

yes

maximal

#11729064

C₂

yes

maximal, complemented

#11725812

C₂

yes

maximal, characteristic

#11725244

C₂

yes

maximal, characteristic, complemented

C₇×#500

—

C₇×#500

—

C₇×#494

C₂²

yes

—

C₂×#6402

—

#89467

C₂²

yes

—

—

C₂²

yes

characteristic

C₇×C₈²

C₂²

yes

characteristic

#89359

—

C₄×C₇⋊C₁₆

C₂²

yes

—

#12526

—

maximal, Sylow

C₂×C₇×Q₁₆

—

C₂×C₇×Q₁₆

—

C₇×C₄⋊Q₈

—

C₂×D₅₆

—

—

—

—

—

—

C₄×C₇×C₈

D₄

yes

characteristic

C₄×C₇×C₈

D₄

yes

characteristic

C₄×C₇×C₈

C₂³

yes

—

—

—

C₂×C₇⋊C₁₆

D₄

yes

—

#2248

—

#2241

—

#1713

D₇

yes

characteristic, complemented

#1706

—

#1312

—

C₂×C₇×Q₈

—

C₂×D₂₈

—

C₂×D₂₈

—

C₂×D₂₈

—

C₂×Dic₁₄

—

C₇×Q₁₆

—

C₇×Q₁₆

—

C₂×C₇×C₈

D₈

yes

—

C₂×C₇×C₈

D₈

yes

—

C₂×C₇×C₈

C₂×D₄

yes

—

C₇×C₄⋊C₄

—

C₇×C₄⋊C₄

—

C₄²×C₇

C₂×D₄

yes

characteristic

C₄⋊Dic₇

—

C₄⋊Dic₇

—

D₅₆

—

#1132

—

C₇⋊C₁₆

—

C₂×SD₃₂

—

—

—

—

C₂×D₇

yes

—

—

—

—

—

C₄×C₁₆

—

C₈²

C₂×D₇

yes

characteristic

C₂²×D₇

—

C₇×Q₈

—

C₇×Q₈

—

C₂×C₄×C₇

—

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

—

Dic₁₄

—

C₇×C₈

—

C₇×C₈

SD₃₂

yes

complemented

C₇×C₈

AGL(1,ℤ/8)

yes

—

C₂×Q₁₆

—

C₂×Q₁₆

—

C₂×SD₁₆

—

C₂×D₈

—

C₄⋊Q₈

—

C₄⋊Q₈

—

#127

—

SD₃₂

—

C₂×C₁₆

—

—

—

—

—

C₄×C₈

C₂²⋊D₇

yes

characteristic

C₄×C₈

C₂²⋊D₇

yes

characteristic

C₄×C₈

C₂²×D₇

yes

characteristic

C₂²×C₇

#493

yes

characteristic

100

C₂×D₇

—

101

C₂×D₇

—

102

C₂×D₇

—

103

C₄×C₇

—

104

C₄×C₇

#496

yes

—

105

C₄×C₇

C₂×SD₃₂

yes

—

106

C₄×C₇

#510

yes

—

107

Dic₇

—

108

C₂×Q₈

—

109

C₂×Q₈

—

110

C₂×D₄

—

111

C₂×D₄

—

112

C₂×D₄

—

113

Q₁₆

—

114

Q₁₆

—

115

SD₁₆

—

116

D₈

—

117

C₂×C₈

#1141

yes

—

118

C₂×C₈

#1141

yes

—

119

C₂×C₈

D₄×D₇

yes

—

120

C₄⋊C₄

—

121

C₄⋊C₄

—

122

C₄⋊C₄

—

123

C₄⋊C₄

—

124

C₄²

C₂×C₂²⋊D₇

yes

characteristic

125

C₁₆

—

126

C₂×C₇

#1721

yes

characteristic

127

C₂×C₇

#2248

yes

characteristic

128

C₂×C₇

#2251

yes

characteristic

129

D₇

—

130

C₂³

—

131

Q₈

—

132

Q₈

—

133

Q₈

—

134

D₄

—

135

D₄

—

136

D₄

—

137

C₂×C₄

—

138

C₂×C₄

—

139

C₂×C₄

C₂×#1141

yes

characteristic

140

C₂×C₄

C₂×#1141

yes

characteristic

141

C₂×C₄

#6503

yes

characteristic

142

C₈

—

143

C₈

#6402

yes

complemented

144

C₈

#6474

yes

—

145

C₇

#12526

yes

characteristic, complemented, Sylow

146

C₂²

—

112

—

147

C₂²

—

112

—

148

C₂²

—

112

—

149

C₂²

#89845

yes

characteristic

150

C₄

—

151

C₄

—

112

—

152

C₄

#89926

yes

—

153

C₄

C₂×#6402

yes

—

154

C₄

#89951

yes

—

155

C₂

—

224

—

156

C₂

#11729034

yes

characteristic

157

C₂

#11729067

yes

characteristic

158

C₂

#11729068

yes

characteristic

	subgroup	G/H
center	C₂²	#89845
commutator, cosocle	C₄×C₇×C₈	C₂³
Frattini	C₄×C₈	C₂²×D₇
Fitting	C₇×#1713	C₂
socle	C₂²×C₇	#493
prime radical	#89467	C₂²
hypercenter	#1713	D₇
hypocenter	C₇	#12526
LCS	G ▷ C₄×C₇×C₈ ▷ C₂×C₄×C₇ ▷ C₂×C₇ ▷ C₇
UCS	C₁ ◁ C₂² ◁ C₄² ◁ C₈² ◁ #1713

subgroup

G/H

center

C₂²

#89845

commutator, cosocle

C₄×C₇×C₈

C₂³

Frattini

C₄×C₈

C₂²×D₇

C₂

C₂²×C₇

C₂²

D₇

C₇

G ▷ C₄×C₇×C₈ ▷ C₂×C₄×C₇ ▷ C₂×C₇ ▷ C₇

UCS

C₁ ◁ C₂² ◁ C₄² ◁ C₈² ◁ #1713

N	H ≅ G/N	type
1. C₇×#1713	C₂	semidirect
3. #11729064	C₂	semidirect
5. #11725244	C₂	semidirect
36. #1713	D₇	semidirect
81. C₇×C₈	SD₃₂	semidirect
143. C₈	#6402	semidirect
145. C₇	#12526	semidirect

H ≅ G/N

type

1. C₇×#1713

C₂

semidirect

3. #11729064

C₂

semidirect

5. #11725244

C₂

semidirect

36. #1713

D₇

semidirect

81. C₇×C₈

SD₃₂

semidirect

143. C₈

#6402

semidirect

145. C₇

#12526

semidirect

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

subgroup	index
C₇×#1713	2
#11729067	2
#11729064	2
#11725812	2
#11725244	2
#12526	7

subgroup

index

Subgroups — #13268554

Conjugacy classes of subgroups

Complements

Maximal subgroups