Group #13268588 — subgroups

Subgroups — #13268588

Order 1792.

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

normal subgroups only

This group has 272 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	#11729139	C₂	1	2	yes	maximal, complemented
3	#11729135	C₂	1	2	yes	maximal
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	C₂²	1	2	yes	—
8	C₇×#494	—	2	1	no	—
9	#89467	C₂²	1	1	yes	characteristic
10	#89466	—	2	1	no	—
11	#89465	—	2	1	no	—
12	#89454	C₂²	1	1	yes	characteristic
13	C₇×C₈²	C₂²	1	1	yes	characteristic
14	#89286	—	2	4	no	—
15	#89285	—	2	4	no	—
16	C₄×C₇⋊C₁₆	C₂²	1	2	yes	—
17	#12532	—	7	1	no	maximal, Sylow
18	C₂×C₇×Q₁₆	—	4	2	no	—
19	C₇×C₄⋊Q₈	—	2	2	no	—
20	C₂×D₅₆	—	2	2	no	—
21	C₂×#1132	—	4	2	no	—
22	#6437	—	2	1	no	—
23	#6432	—	2	1	no	—
24	C₇×#107	—	2	2	no	—
25	C₇×#107	—	2	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	#6395	—	4	2	no	—
31	#6392	—	2	2	no	—
32	C₂×C₇⋊C₁₆	D₄	1	4	yes	—
33	#2254	—	7	2	no	—
34	#2240	—	7	2	no	—
35	#1713	D₇	1	1	yes	characteristic, complemented
36	#1706	—	7	1	no	—
37	#1312	—	7	1	no	—
38	C₂×C₇×Q₈	—	4	2	no	—
39	C₂×D₂₈	—	4	1	no	—
40	C₂×D₂₈	—	4	1	no	—
41	C₂×D₂₈	—	4	1	no	—
42	C₂×Dic₁₄	—	4	1	no	—
43	C₇×Q₁₆	—	8	4	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	C₇×C₄⋊C₄	—	4	2	no	—
48	C₇×C₄⋊C₄	—	4	2	no	—
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	#1132	—	8	4	no	—
54	C₇⋊C₁₆	—	2	4	no	—
55	#500	—	2	1	no	—
56	#500	—	7	1	no	—
57	#500	C₂×D₇	1	2	yes	—
58	#494	—	2	1	no	—
59	#493	—	7	1	no	—
60	#492	—	14	1	no	—
61	#486	—	14	1	no	—
62	#367	—	14	4	no	—
63	#357	—	14	4	no	—
64	C₄×C₁₆	—	7	2	no	—
65	C₈²	C₂×D₇	1	1	yes	characteristic
66	C₂²×D₇	—	8	1	no	—
67	C₇×Q₈	—	8	4	no	—
68	C₂×C₄×C₇	—	8	2	no	—
69	C₂×C₄×C₇	#127	1	1	yes	characteristic
70	C₂×C₄×C₇	C₂×D₈	1	1	yes	characteristic
71	C₂×C₄×C₇	C₂×D₈	1	1	yes	characteristic
72	C₂×Dic₇	—	8	1	no	—
73	D₂₈	—	8	2	no	—
74	D₂₈	—	8	2	no	—
75	D₂₈	—	8	2	no	—
76	Dic₁₄	—	8	2	no	—
77	C₇×C₈	—	2	2	no	—
78	C₇×C₈	SD₃₂	1	4	yes	—
79	C₇×C₈	#137	1	4	yes	—
80	C₂×Q₁₆	—	4	2	no	—
81	C₂×SD₁₆	—	28	2	no	—
82	C₂×D₈	—	14	2	no	—
83	C₄⋊Q₈	—	2	2	no	—
84	C₄⋊Q₈	—	14	1	no	—
85	#127	—	14	1	no	—
86	C₂×C₁₆	—	7	4	no	—
87	#107	—	2	2	no	—
88	#107	—	2	2	no	—
89	#107	—	4	2	no	—
90	#107	—	14	2	no	—
91	#102	—	28	2	no	—
92	C₄×C₈	C₂²⋊D₇	1	1	yes	characteristic
93	C₄×C₈	C₂²⋊D₇	1	1	yes	characteristic
94	C₄×C₈	C₂²×D₇	1	1	yes	characteristic
95	C₂²×C₇	#493	1	1	yes	characteristic
96	C₂×D₇	—	16	1	no	—
97	C₂×D₇	—	16	1	no	—
98	C₂×D₇	—	16	1	no	—
99	C₄×C₇	—	16	2	no	—
100	C₄×C₇	#497	1	2	yes	—
101	C₄×C₇	C₂×SD₃₂	1	2	yes	—
102	C₄×C₇	#509	1	2	yes	—
103	Dic₇	—	16	1	no	—
104	C₂×Q₈	—	4	2	no	—
105	C₂×Q₈	—	28	1	no	—
106	C₂×D₄	—	28	1	no	—
107	C₂×D₄	—	28	1	no	—
108	C₂×D₄	—	28	1	no	—
109	Q₁₆	—	8	4	no	—
110	SD₁₆	—	56	4	no	—
111	D₈	—	28	4	no	—
112	C₂×C₈	#1141	1	2	yes	—
113	C₂×C₈	#1141	1	2	yes	—
114	C₂×C₈	D₄×D₇	1	2	yes	—
115	C₄⋊C₄	—	4	2	no	—
116	C₄⋊C₄	—	4	2	no	—
117	C₄⋊C₄	—	28	1	no	—
118	C₄⋊C₄	—	28	1	no	—
119	C₄²	C₂×C₂²⋊D₇	1	1	yes	characteristic
120	C₁₆	—	14	4	no	—
121	C₂×C₇	#1722	1	1	yes	characteristic
122	C₂×C₇	#2248	1	1	yes	characteristic
123	C₂×C₇	#2250	1	1	yes	characteristic
124	D₇	—	32	1	no	—
125	C₂³	—	56	1	no	—
126	Q₈	—	8	4	no	—
127	Q₈	—	56	2	no	—
128	D₄	—	56	2	no	—
129	D₄	—	56	2	no	—
130	D₄	—	56	2	no	—
131	C₂×C₄	—	8	2	no	—
132	C₂×C₄	—	56	1	no	—
133	C₂×C₄	C₂×#1141	1	1	yes	characteristic
134	C₂×C₄	C₂×#1141	1	1	yes	characteristic
135	C₂×C₄	#6503	1	1	yes	characteristic
136	C₈	—	2	2	no	—
137	C₈	#6402	1	4	yes	—
138	C₈	#6481	1	4	yes	—
139	C₇	#12532	1	1	yes	characteristic, complemented, Sylow
140	C₂²	—	112	1	no	—
141	C₂²	—	112	1	no	—
142	C₂²	—	112	1	no	—
143	C₂²	#89845	1	1	yes	characteristic
144	C₄	—	16	2	no	—
145	C₄	—	112	1	no	—
146	C₄	#89919	1	2	yes	—
147	C₄	C₂×#6402	1	2	yes	—
148	C₄	#89962	1	2	yes	—
149	C₂	—	224	1	no	—
150	C₂	#11729046	1	1	yes	characteristic
151	C₂	#11729055	1	1	yes	characteristic
152	C₂	#11729067	1	1	yes	characteristic

subgroup

G/H

class size

multiplicity

normal?

properties

C₇×#1713

C₂

yes

maximal, characteristic, complemented

#11729139

C₂

yes

maximal, complemented

#11729135

C₂

yes

maximal

#11725812

C₂

yes

maximal, characteristic

#11725244

C₂

yes

maximal, characteristic, complemented

C₇×#500

—

C₇×#500

C₂²

yes

—

C₇×#494

—

#89467

C₂²

yes

—

—

C₂²

yes

characteristic

C₇×C₈²

C₂²

yes

characteristic

#89286

—

#89285

—

C₄×C₇⋊C₁₆

C₂²

yes

—

#12532

—

maximal, Sylow

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

characteristic

#6395

—

#6392

—

C₂×C₇⋊C₁₆

D₄

yes

—

#2254

—

#2240

—

#1713

D₇

yes

characteristic, complemented

#1706

—

#1312

—

C₂×C₇×Q₈

—

C₂×D₂₈

—

C₂×D₂₈

—

C₂×D₂₈

—

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

—

C₄²×C₇

C₂×D₄

yes

characteristic

C₄⋊Dic₇

—

C₄⋊Dic₇

—

D₅₆

—

#1132

—

C₇⋊C₁₆

—

#500

—

#500

—

#500

C₂×D₇

yes

—

—

—

—

—

—

—

C₄×C₁₆

—

C₈²

C₂×D₇

yes

characteristic

C₂²×D₇

—

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

—

C₇×C₈

#137

yes

—

C₂×Q₁₆

—

C₂×SD₁₆

—

C₂×D₈

—

C₄⋊Q₈

—

C₄⋊Q₈

—

#127

—

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

C₂×D₇

—

C₂×D₇

—

C₂×D₇

—

C₄×C₇

—

100

C₄×C₇

#497

yes

—

101

C₄×C₇

C₂×SD₃₂

yes

—

102

C₄×C₇

#509

yes

—

103

Dic₇

—

104

C₂×Q₈

—

105

C₂×Q₈

—

106

C₂×D₄

—

107

C₂×D₄

—

108

C₂×D₄

—

109

Q₁₆

—

110

SD₁₆

—

111

D₈

—

112

C₂×C₈

#1141

yes

—

113

C₂×C₈

#1141

yes

—

114

C₂×C₈

D₄×D₇

yes

—

115

C₄⋊C₄

—

116

C₄⋊C₄

—

117

C₄⋊C₄

—

118

C₄⋊C₄

—

119

C₄²

C₂×C₂²⋊D₇

yes

characteristic

120

C₁₆

—

121

C₂×C₇

#1722

yes

characteristic

122

C₂×C₇

#2248

yes

characteristic

123

C₂×C₇

#2250

yes

characteristic

124

D₇

—

125

C₂³

—

126

Q₈

—

127

Q₈

—

128

D₄

—

129

D₄

—

130

D₄

—

131

C₂×C₄

—

132

C₂×C₄

—

133

C₂×C₄

C₂×#1141

yes

characteristic

134

C₂×C₄

C₂×#1141

yes

characteristic

135

C₂×C₄

#6503

yes

characteristic

136

C₈

—

137

C₈

#6402

yes

—

138

C₈

#6481

yes

—

139

C₇

#12532

yes

characteristic, complemented, Sylow

140

C₂²

—

112

—

141

C₂²

—

112

—

142

C₂²

—

112

—

143

C₂²

#89845

yes

characteristic

144

C₄

—

145

C₄

—

112

—

146

C₄

#89919

yes

—

147

C₄

C₂×#6402

yes

—

148

C₄

#89962

yes

—

149

C₂

—

224

—

150

C₂

#11729046

yes

characteristic

151

C₂

#11729055

yes

characteristic

152

C₂

#11729067

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	#12532
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
2. #11729139	C₂	semidirect
5. #11725244	C₂	semidirect
35. #1713	D₇	semidirect
139. C₇	#12532	semidirect

H ≅ G/N

type

1. C₇×#1713

C₂

semidirect

2. #11729139

C₂

semidirect

5. #11725244

C₂

semidirect

35. #1713

D₇

semidirect

139. C₇

#12532

semidirect

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

subgroup	index
C₇×#1713	2
#11729139	2
#11729135	2
#11725812	2
#11725244	2
#12532	7

subgroup

index

Subgroups — #13268588

Conjugacy classes of subgroups

Complements

Maximal subgroups