Group #13268563 — subgroups

Subgroups — #13268563

Order 1792.

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

normal subgroups only

This group has 274 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×#1707	C₂	1	1	yes	maximal, characteristic
2	#11729136	C₂	1	2	yes	maximal
3	#11729066	C₂	1	2	yes	maximal
4	#11725819	C₂	1	1	yes	maximal, characteristic
5	#11725244	C₂	1	1	yes	maximal, characteristic
6	C₇×#500	C₂²	1	1	yes	characteristic
7	C₇×#498	—	2	1	no	—
8	C₇×#494	C₂²	1	1	yes	characteristic
9	C₇×#487	—	2	1	no	—
10	C₂×#6403	—	2	4	no	—
11	#89471	C₂²	1	2	yes	—
12	#89454	—	2	1	no	—
13	#89454	—	2	1	no	—
14	C₇×C₈²	C₂²	1	1	yes	characteristic
15	#89287	—	2	4	no	—
16	C₄×C₇⋊C₁₆	C₂²	1	2	yes	—
17	#12528	—	7	1	no	maximal, Sylow
18	C₂×C₇×Q₁₆	—	2	2	no	—
19	C₇×C₄⋊Q₈	—	2	1	no	—
20	C₇×C₄⋊Q₈	—	2	1	no	—
21	C₂×Dic₂₈	—	2	2	no	—
22	C₂×Dic₂₈	—	2	2	no	—
23	#6432	—	2	2	no	—
24	C₇×#107	—	2	2	no	—
25	C₇×#106	—	4	2	no	—
26	C₇×#103	—	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	#6403	—	4	8	no	—
31	#6392	—	4	2	no	—
32	#6392	—	4	2	no	—
33	C₂×C₇⋊C₁₆	D₄	1	4	yes	—
34	#2247	—	7	2	no	—
35	#2241	—	7	2	no	—
36	#1713	—	7	1	no	—
37	#1707	D₇	1	1	yes	characteristic
38	#1312	—	7	1	no	—
39	C₂×C₇×Q₈	—	4	1	no	—
40	C₂×C₇×Q₈	—	4	1	no	—
41	C₂×Dic₁₄	—	4	2	no	—
42	C₇×Q₁₆	—	4	4	no	—
43	C₂×C₇×C₈	D₈	1	2	yes	—
44	C₂×C₇×C₈	D₈	1	2	yes	—
45	C₂×C₇×C₈	C₂×D₄	1	2	yes	—
46	C₇×C₄⋊C₄	—	4	1	no	—
47	C₇×C₄⋊C₄	—	4	1	no	—
48	C₇×C₄⋊C₄	—	4	1	no	—
49	C₇×C₄⋊C₄	—	4	1	no	—
50	C₄²×C₇	C₂×D₄	1	1	yes	characteristic
51	C₄⋊Dic₇	—	4	2	no	—
52	C₄⋊Dic₇	—	4	2	no	—
53	Dic₂₈	—	4	4	no	—
54	Dic₂₈	—	4	4	no	—
55	C₇⋊C₁₆	—	2	4	no	—
56	C₂×Q₃₂	—	14	4	no	—
57	#500	—	14	1	no	—
58	#500	—	14	1	no	—
59	#500	C₂×D₇	1	1	yes	characteristic
60	#498	—	2	1	no	—
61	#494	—	7	2	no	—
62	#494	C₂×D₇	1	1	yes	characteristic
63	#487	—	2	1	no	—
64	#358	—	14	4	no	—
65	C₄×C₁₆	—	7	2	no	—
66	C₈²	C₂×D₇	1	1	yes	characteristic
67	C₇×Q₈	—	8	2	no	—
68	C₇×Q₈	—	8	2	no	—
69	C₂×C₄×C₇	—	8	1	no	—
70	C₂×C₄×C₇	—	8	1	no	—
71	C₂×C₄×C₇	#127	1	1	yes	characteristic
72	C₂×C₄×C₇	C₂×D₈	1	1	yes	characteristic
73	C₂×C₄×C₇	C₂×D₈	1	1	yes	characteristic
74	C₂×Dic₇	—	8	2	no	—
75	Dic₁₄	—	8	2	no	—
76	Dic₁₄	—	8	2	no	—
77	C₇×C₈	—	2	2	no	—
78	C₇×C₈	Q₃₂	1	4	yes	complemented
79	C₇×C₈	AGL(1,ℤ/8)	1	4	yes	—
80	C₂×Q₁₆	—	2	2	no	—
81	C₂×Q₁₆	—	14	2	no	—
82	C₂×Q₁₆	—	14	2	no	—
83	C₄⋊Q₈	—	2	1	no	—
84	C₄⋊Q₈	—	2	1	no	—
85	C₄⋊Q₈	—	14	2	no	—
86	Q₃₂	—	28	8	no	—
87	C₂×C₁₆	—	7	4	no	—
88	#107	—	2	2	no	—
89	#107	—	28	2	no	—
90	#107	—	28	2	no	—
91	#106	—	4	2	no	—
92	#103	—	4	2	no	—
93	C₄×C₈	C₂²⋊D₇	1	1	yes	characteristic
94	C₄×C₈	C₂²⋊D₇	1	1	yes	characteristic
95	C₄×C₈	C₂²×D₇	1	1	yes	characteristic
96	C₂²×C₇	#493	1	1	yes	characteristic
97	C₄×C₇	—	16	1	no	—
98	C₄×C₇	—	16	1	no	—
99	C₄×C₇	#496	1	2	yes	—
100	C₄×C₇	C₂×Q₃₂	1	2	yes	—
101	C₄×C₇	#510	1	2	yes	—
102	Dic₇	—	16	2	no	—
103	C₂×Q₈	—	4	1	no	—
104	C₂×Q₈	—	4	1	no	—
105	C₂×Q₈	—	28	2	no	—
106	Q₁₆	—	4	4	no	—
107	Q₁₆	—	28	4	no	—
108	Q₁₆	—	28	4	no	—
109	C₂×C₈	#1141	1	2	yes	—
110	C₂×C₈	#1141	1	2	yes	—
111	C₂×C₈	D₄×D₇	1	2	yes	—
112	C₄⋊C₄	—	4	1	no	—
113	C₄⋊C₄	—	4	1	no	—
114	C₄⋊C₄	—	4	1	no	—
115	C₄⋊C₄	—	4	1	no	—
116	C₄⋊C₄	—	28	2	no	—
117	C₄⋊C₄	—	28	2	no	—
118	C₄²	C₂×C₂²⋊D₇	1	1	yes	characteristic
119	C₁₆	—	14	4	no	—
120	C₂×C₇	#1721	1	1	yes	characteristic
121	C₂×C₇	#2247	1	1	yes	characteristic
122	C₂×C₇	#2251	1	1	yes	characteristic
123	Q₈	—	8	2	no	—
124	Q₈	—	8	2	no	—
125	Q₈	—	56	2	no	—
126	Q₈	—	56	2	no	—
127	C₂×C₄	—	8	1	no	—
128	C₂×C₄	—	8	1	no	—
129	C₂×C₄	—	56	2	no	—
130	C₂×C₄	C₂×#1141	1	1	yes	characteristic
131	C₂×C₄	C₂×#1141	1	1	yes	characteristic
132	C₂×C₄	#6503	1	1	yes	characteristic
133	C₈	—	2	2	no	—
134	C₈	#6403	1	4	yes	complemented
135	C₈	#6477	1	4	yes	—
136	C₇	#12528	1	1	yes	characteristic, complemented, Sylow
137	C₂²	#89845	1	1	yes	characteristic
138	C₄	—	16	1	no	—
139	C₄	—	16	1	no	—
140	C₄	—	112	2	no	—
141	C₄	#89948	1	2	yes	—
142	C₄	C₂×#6403	1	2	yes	—
143	C₄	#89967	1	2	yes	—
144	C₂	#11729016	1	1	yes	characteristic
145	C₂	#11729066	1	1	yes	characteristic
146	C₂	#11729076	1	1	yes	characteristic

subgroup

G/H

class size

multiplicity

normal?

properties

C₇×#1707

C₂

yes

maximal, characteristic

#11729136

C₂

yes

maximal

#11729066

C₂

yes

maximal

#11725819

C₂

yes

maximal, characteristic

#11725244

C₂

yes

maximal, characteristic

C₇×#500

C₂²

yes

characteristic

C₇×#498

—

C₇×#494

C₂²

yes

—

—

C₂²

yes

—

#89454

—

#89454

—

C₇×C₈²

C₂²

yes

characteristic

#89287

—

C₄×C₇⋊C₁₆

C₂²

yes

—

#12528

—

maximal, Sylow

C₂×C₇×Q₁₆

—

C₇×C₄⋊Q₈

—

C₇×C₄⋊Q₈

—

C₂×Dic₂₈

—

C₂×Dic₂₈

—

—

—

—

—

C₄×C₇×C₈

D₄

yes

characteristic

C₄×C₇×C₈

D₄

yes

characteristic

C₄×C₇×C₈

C₂³

yes

—

—

—

C₂×C₇⋊C₁₆

D₄

yes

—

—

—

—

D₇

yes

characteristic

#1312

—

C₂×C₇×Q₈

—

C₂×C₇×Q₈

—

C₂×Dic₁₄

—

C₇×Q₁₆

—

C₂×C₇×C₈

D₈

yes

—

C₂×C₇×C₈

D₈

yes

—

C₂×C₇×C₈

C₂×D₄

yes

—

C₇×C₄⋊C₄

—

C₇×C₄⋊C₄

—

C₇×C₄⋊C₄

—

C₇×C₄⋊C₄

—

C₄²×C₇

C₂×D₄

yes

characteristic

C₄⋊Dic₇

—

C₄⋊Dic₇

—

Dic₂₈

—

Dic₂₈

—

C₇⋊C₁₆

—

C₂×Q₃₂

—

#500

—

#500

—

#500

C₂×D₇

yes

—

—

C₂×D₇

yes

characteristic

#487

—

#358

—

C₄×C₁₆

—

C₈²

C₂×D₇

yes

characteristic

C₇×Q₈

—

C₇×Q₈

—

C₂×C₄×C₇

—

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

—

Dic₁₄

—

Dic₁₄

—

C₇×C₈

—

C₇×C₈

Q₃₂

yes

complemented

C₇×C₈

AGL(1,ℤ/8)

yes

—

C₂×Q₁₆

—

C₂×Q₁₆

—

C₂×Q₁₆

—

C₄⋊Q₈

—

C₄⋊Q₈

—

C₄⋊Q₈

—

Q₃₂

—

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

C₄×C₇

—

C₄×C₇

—

C₄×C₇

#496

yes

—

100

C₄×C₇

C₂×Q₃₂

yes

—

101

C₄×C₇

#510

yes

—

102

Dic₇

—

103

C₂×Q₈

—

104

C₂×Q₈

—

105

C₂×Q₈

—

106

Q₁₆

—

107

Q₁₆

—

108

Q₁₆

—

109

C₂×C₈

#1141

yes

—

110

C₂×C₈

#1141

yes

—

111

C₂×C₈

D₄×D₇

yes

—

112

C₄⋊C₄

—

113

C₄⋊C₄

—

114

C₄⋊C₄

—

115

C₄⋊C₄

—

116

C₄⋊C₄

—

117

C₄⋊C₄

—

118

C₄²

C₂×C₂²⋊D₇

yes

characteristic

119

C₁₆

—

120

C₂×C₇

#1721

yes

characteristic

121

C₂×C₇

#2247

yes

characteristic

122

C₂×C₇

#2251

yes

characteristic

123

Q₈

—

124

Q₈

—

125

Q₈

—

126

Q₈

—

127

C₂×C₄

—

128

C₂×C₄

—

129

C₂×C₄

—

130

C₂×C₄

C₂×#1141

yes

characteristic

131

C₂×C₄

C₂×#1141

yes

characteristic

132

C₂×C₄

#6503

yes

characteristic

133

C₈

—

134

C₈

#6403

yes

complemented

135

C₈

#6477

yes

—

136

C₇

#12528

yes

characteristic, complemented, Sylow

137

C₂²

#89845

yes

characteristic

138

C₄

—

139

C₄

—

140

C₄

—

112

—

141

C₄

#89948

yes

—

142

C₄

C₂×#6403

yes

—

143

C₄

#89967

yes

—

144

C₂

#11729016

yes

characteristic

145

C₂

#11729066

yes

characteristic

146

C₂

#11729076

yes

characteristic

	subgroup	G/H
center	C₂²	#89845
commutator, cosocle	C₄×C₇×C₈	C₂³
Frattini	C₄×C₈	C₂²×D₇
Fitting	C₇×#1707	C₂
socle, prime radical	C₂²×C₇	#493
hypercenter	#1707	D₇
hypocenter	C₇	#12528
LCS	G ▷ C₄×C₇×C₈ ▷ C₂×C₄×C₇ ▷ C₂×C₇ ▷ C₇
UCS	C₁ ◁ C₂² ◁ C₄² ◁ C₈² ◁ #1707

subgroup

G/H

center

C₂²

#89845

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 ◁ #1707

N	H ≅ G/N	type
78. C₇×C₈	Q₃₂	semidirect
134. C₈	#6403	semidirect
136. C₇	#12528	semidirect

H ≅ G/N

type

78. C₇×C₈

Q₃₂

semidirect

134. C₈

#6403

semidirect

136. C₇

#12528

semidirect

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

subgroup	index
C₇×#1707	2
#11729136	2
#11729066	2
#11725819	2
#11725244	2
#12528	7

subgroup

index

Subgroups — #13268563

Conjugacy classes of subgroups

Complements

Maximal subgroups