Group #11727393 — subgroups

Subgroups — #11727393

Order 896.

294 conjugacy classes of subgroups; only the 85 normal subgroups are shown.

normal subgroups only

This group has 294 conjugacy classes of subgroups — too many to draw. Only its 85 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	C₂×#6433	C₂	1	1	yes	maximal, characteristic, complemented
2	C₇×#390	C₂	1	1	yes	maximal, characteristic
3	C₂×#6406	C₂	1	2	yes	maximal
4	#89324	C₂	1	2	yes	maximal, complemented
5	#89319	C₂	1	1	yes	maximal, characteristic, complemented
6	C₂²×C₇×D₄	C₂²	1	1	yes	characteristic
7	C₂×C₇×C₄⋊C₄	C₂²	1	1	yes	characteristic
8	C₂×C₇×C₂²⋊C₄	—	2	2	no	—
9	C₂×C₄²×C₇	C₂²	1	1	yes	characteristic
10	C₂×C₄⋊Dic₇	C₂²	1	2	yes	—
11	C₂×Dic₇⋊C₄	—	2	2	no	—
12	C₂²×C₇⋊C₈	C₂²	1	2	yes	—
13	#6433	—	2	4	no	—
14	#6406	—	2	4	no	—
15	#2025	—	7	1	no	maximal, Sylow
16	C₂⁴×C₇	—	2	1	no	—
17	C₂×C₇×D₄	—	2	2	no	—
18	C₂×C₇×D₄	—	2	2	no	—
19	C₂×C₇×D₄	—	2	1	no	—
20	C₂×C₇×D₄	Q₈	1	2	yes	—
21	C₂²×C₄×C₇	—	2	1	no	—
22	C₂²×C₄×C₇	D₄	1	2	yes	—
23	C₂²×C₄×C₇	C₂³	1	1	yes	characteristic
24	C₂²×Dic₇	—	2	2	no	—
25	C₇×C₄⋊C₄	—	2	2	no	—
26	C₇×C₂²⋊C₄	—	4	4	no	—
27	C₄²×C₇	—	2	2	no	—
28	C₄⋊Dic₇	—	2	2	no	—
29	C₄⋊Dic₇	D₄	1	4	yes	—
30	Dic₇⋊C₄	—	2	8	no	—
31	C₂×C₇⋊C₈	—	2	2	no	—
32	C₂×C₇⋊C₈	—	2	2	no	—
33	C₂×C₇⋊C₈	—	2	2	no	—
34	C₂×#125	—	7	1	no	—
35	C₂×#102	—	7	2	no	—
36	#390	D₇	1	1	yes	characteristic
37	#340	—	7	2	no	—
38	#336	—	7	1	no	—
39	C₂³×C₇	—	2	2	no	—
40	C₂³×C₇	—	4	1	no	—
41	C₂³×C₇	—	4	1	no	—
42	C₂³×C₇	—	4	2	no	—
43	C₂³×C₇	—	4	2	no	—
44	C₂³×C₇	C₂×D₄	1	1	yes	characteristic
45	C₇×D₄	—	2	4	no	—
46	C₇×D₄	—	4	2	no	—
47	C₂×C₄×C₇	—	2	2	no	—
48	C₂×C₄×C₇	—	2	2	no	—
49	C₂×C₄×C₇	—	2	2	no	—
50	C₂×C₄×C₇	—	4	1	no	—
51	C₂×C₄×C₇	—	4	2	no	—
52	C₂×C₄×C₇	C₂×D₄	1	2	yes	—
53	C₂×C₄×C₇	C₂×Q₈	1	1	yes	characteristic
54	C₂×C₄×C₇	#41	1	1	yes	characteristic
55	C₂×C₄×C₇	#41	1	2	yes	—
56	C₂×Dic₇	—	2	4	no	—
57	C₂×Dic₇	—	4	2	no	—
58	C₂×Dic₇	—	4	2	no	—
59	C₇⋊C₈	—	4	2	no	—
60	C₂²×D₄	C₂×D₇	1	1	yes	characteristic
61	C₂²×C₈	—	7	2	no	—
62	#125	—	14	4	no	—
63	C₂×C₄⋊C₄	—	7	2	no	—
64	C₂×C₄⋊C₄	—	14	2	no	—
65	C₂×C₄⋊C₄	C₂×D₇	1	1	yes	characteristic
66	C₂×C₂²⋊C₄	—	2	2	no	—
67	C₂×C₄²	C₂×D₇	1	1	yes	characteristic
68	#102	—	14	4	no	—
69	C₂²×C₇	—	4	2	no	—
70	C₂²×C₇	—	4	4	no	—
71	C₂²×C₇	—	8	2	no	—
72	C₂²×C₇	—	8	1	no	—
73	C₂²×C₇	—	8	1	no	—
74	C₂²×C₇	#120	1	1	yes	characteristic
75	C₂²×C₇	C₂²⋊Q₈	1	1	yes	characteristic
76	C₂²×C₇	#124	1	1	yes	characteristic
77	C₂²×C₇	#135	1	2	yes	—
78	C₂²×C₇	AGL(1,ℤ/8)	1	2	yes	—
79	C₄×C₇	—	4	2	no	—
80	C₄×C₇	—	8	1	no	—
81	C₄×C₇	C₂²⋊Q₈	1	2	yes	—
82	C₄×C₇	#124	1	2	yes	—
83	Dic₇	—	4	4	no	—
84	C₂⁴	—	2	1	no	—
85	C₂×D₄	—	2	1	no	—
86	C₂×D₄	—	2	2	no	—
87	C₂×D₄	—	2	2	no	—
88	C₂×D₄	Dic₁₄	1	2	yes	—
89	C₂²×C₄	—	2	1	no	—
90	C₂²×C₄	—	14	2	no	—
91	C₂²×C₄	C₂²⋊D₇	1	2	yes	—
92	C₂²×C₄	C₂²×D₇	1	1	yes	characteristic
93	C₂×C₈	—	14	2	no	—
94	C₂×C₈	—	14	2	no	—
95	C₂×C₈	—	14	2	no	—
96	C₄⋊C₄	—	2	2	no	—
97	C₄⋊C₄	—	7	4	no	—
98	C₄⋊C₄	—	14	8	no	—
99	C₄⋊C₄	—	14	2	no	—
100	C₂²⋊C₄	—	4	4	no	—
101	C₄²	—	2	2	no	—
102	C₂×C₇	—	8	2	no	—
103	C₂×C₇	#393	1	1	yes	characteristic
104	C₂×C₇	#449	1	2	yes	—
105	C₂×C₇	#478	1	2	yes	—
106	C₂×C₇	#488	1	1	yes	characteristic
107	C₂×C₇	#490	1	1	yes	characteristic
108	C₂³	—	2	2	no	—
109	C₂³	—	4	2	no	—
110	C₂³	—	4	1	no	—
111	C₂³	—	4	1	no	—
112	C₂³	—	4	2	no	—
113	C₂³	C₂×C₂²⋊D₇	1	1	yes	characteristic
114	D₄	—	2	4	no	—
115	D₄	—	4	2	no	—
116	C₂×C₄	—	2	2	no	—
117	C₂×C₄	—	2	2	no	—
118	C₂×C₄	—	2	2	no	—
119	C₂×C₄	—	4	2	no	—
120	C₂×C₄	—	4	1	no	—
121	C₂×C₄	—	14	4	no	—
122	C₂×C₄	—	28	2	no	—
123	C₂×C₄	—	28	2	no	—
124	C₂×C₄	C₂×Dic₁₄	1	1	yes	characteristic
125	C₂×C₄	#1157	1	1	yes	characteristic
126	C₂×C₄	D₄×D₇	1	2	yes	—
127	C₂×C₄	#1159	1	2	yes	—
128	C₈	—	28	2	no	—
129	C₇	#2025	1	1	yes	characteristic, complemented, Sylow
130	C₂²	—	4	2	no	—
131	C₂²	—	4	4	no	—
132	C₂²	—	8	2	no	—
133	C₂²	—	8	1	no	—
134	C₂²	—	8	1	no	—
135	C₂²	#6487	1	1	yes	characteristic
136	C₂²	#6495	1	2	yes	—
137	C₂²	#6499	1	1	yes	characteristic
138	C₂²	#6500	1	1	yes	characteristic
139	C₂²	#6513	1	2	yes	—
140	C₄	—	4	2	no	—
141	C₄	—	8	1	no	—
142	C₄	—	28	4	no	—
143	C₄	#6441	1	2	yes	—
144	C₄	#6448	1	2	yes	—
145	C₂	—	8	2	no	—
146	C₂	#89719	1	1	yes	characteristic
147	C₂	#89781	1	2	yes	—
148	C₂	#89809	1	2	yes	—
149	C₂	#89828	1	1	yes	characteristic
150	C₂	#89843	1	1	yes	characteristic

subgroup

G/H

class size

multiplicity

normal?

properties

C₂×#6433

C₂

yes

maximal, characteristic, complemented

C₇×#390

C₂

yes

maximal, characteristic

C₂×#6406

C₂

yes

maximal

#89324

C₂

yes

maximal, complemented

#89319

C₂

yes

maximal, characteristic, complemented

C₂²×C₇×D₄

C₂²

yes

characteristic

C₂×C₇×C₄⋊C₄

C₂²

yes

characteristic

C₂×C₇×C₂²⋊C₄

—

C₂×C₄²×C₇

C₂²

yes

characteristic

C₂×C₄⋊Dic₇

C₂²

yes

—

C₂×Dic₇⋊C₄

—

C₂²×C₇⋊C₈

C₂²

yes

—

—

—

—

C₂⁴×C₇

—

C₂×C₇×D₄

—

C₂×C₇×D₄

—

C₂×C₇×D₄

—

C₂×C₇×D₄

Q₈

yes

—

C₂²×C₄×C₇

—

C₂²×C₄×C₇

D₄

yes

—

C₂²×C₄×C₇

C₂³

yes

characteristic

C₂²×Dic₇

—

C₇×C₄⋊C₄

—

C₇×C₂²⋊C₄

—

C₄²×C₇

—

C₄⋊Dic₇

—

C₄⋊Dic₇

D₄

yes

—

Dic₇⋊C₄

—

C₂×C₇⋊C₈

—

C₂×C₇⋊C₈

—

C₂×C₇⋊C₈

—

C₂×#125

—

C₂×#102

—

#390

D₇

yes

characteristic

#340

—

#336

—

C₂³×C₇

—

C₂³×C₇

—

C₂³×C₇

—

C₂³×C₇

—

C₂³×C₇

—

C₂³×C₇

C₂×D₄

yes

characteristic

C₇×D₄

—

C₇×D₄

—

C₂×C₄×C₇

—

C₂×C₄×C₇

—

C₂×C₄×C₇

—

C₂×C₄×C₇

—

C₂×C₄×C₇

—

C₂×C₄×C₇

C₂×D₄

yes

—

C₂×C₄×C₇

C₂×Q₈

yes

C₂×C₄×C₇

yes

C₂×C₄×C₇

yes

—

C₂×Dic₇

—

C₂×Dic₇

—

C₂×Dic₇

—

C₇⋊C₈

—

C₂²×D₄

C₂×D₇

yes

characteristic

C₂²×C₈

—

#125

—

C₂×C₄⋊C₄

—

C₂×C₄⋊C₄

—

C₂×C₄⋊C₄

C₂×D₇

yes

characteristic

C₂×C₂²⋊C₄

—

C₂×C₄²

C₂×D₇

yes

characteristic

#102

—

C₂²×C₇

—

C₂²×C₇

—

C₂²×C₇

—

C₂²×C₇

—

C₂²×C₇

—

C₂²×C₇

#120

yes

characteristic

C₂²×C₇

C₂²⋊Q₈

yes

C₂²×C₇

yes

C₂²×C₇

yes

—

C₂²×C₇

AGL(1,ℤ/8)

yes

—

C₄×C₇

—

C₄×C₇

—

C₄×C₇

C₂²⋊Q₈

yes

—

C₄×C₇

#124

yes

—

Dic₇

—

C₂⁴

—

C₂×D₄

—

C₂×D₄

—

C₂×D₄

—

C₂×D₄

Dic₁₄

yes

—

C₂²×C₄

—

C₂²×C₄

—

C₂²×C₄

C₂²⋊D₇

yes

—

C₂²×C₄

C₂²×D₇

yes

characteristic

C₂×C₈

—

C₂×C₈

—

C₂×C₈

—

C₄⋊C₄

—

C₄⋊C₄

—

C₄⋊C₄

—

C₄⋊C₄

—

100

C₂²⋊C₄

—

101

C₄²

—

102

C₂×C₇

—

103

C₂×C₇

#393

yes

characteristic

104

C₂×C₇

#449

yes

—

105

C₂×C₇

#478

yes

—

106

C₂×C₇

#488

yes

characteristic

107

C₂×C₇

#490

yes

characteristic

108

C₂³

—

109

C₂³

—

110

C₂³

—

111

C₂³

—

112

C₂³

—

113

C₂³

C₂×C₂²⋊D₇

yes

characteristic

114

D₄

—

115

D₄

—

116

C₂×C₄

—

117

C₂×C₄

—

118

C₂×C₄

—

119

C₂×C₄

—

120

C₂×C₄

—

121

C₂×C₄

—

122

C₂×C₄

—

123

C₂×C₄

—

124

C₂×C₄

C₂×Dic₁₄

yes

characteristic

125

C₂×C₄

#1157

yes

characteristic

126

C₂×C₄

D₄×D₇

yes

—

127

C₂×C₄

#1159

yes

—

128

C₈

—

129

C₇

#2025

yes

characteristic, complemented, Sylow

130

C₂²

—

131

C₂²

—

132

C₂²

—

133

C₂²

—

134

C₂²

—

135

C₂²

#6487

yes

characteristic

136

C₂²

#6495

yes

—

137

C₂²

#6499

yes

characteristic

138

C₂²

#6500

yes

characteristic

139

C₂²

#6513

yes

—

140

C₄

—

141

C₄

—

142

C₄

—

143

C₄

#6441

yes

—

144

C₄

#6448

yes

—

145

C₂

—

146

C₂

#89719

yes

characteristic

147

C₂

#89781

yes

—

148

C₂

#89809

yes

—

149

C₂

#89828

yes

characteristic

150

C₂

#89843

yes

characteristic

	subgroup	G/H
center	C₂³	C₂×C₂²⋊D₇
commutator, cosocle	C₂²×C₄×C₇	C₂³
Frattini	C₂²×C₄	C₂²×D₇
Fitting	C₇×#390	C₂
socle	C₂³×C₇	C₂×D₄
prime radical	C₂²×C₇×D₄	C₂²
hypercenter	#390	D₇
hypocenter	C₇	#2025
LCS	G ▷ C₂²×C₄×C₇ ▷ C₂×C₇ ▷ C₇
UCS	C₁ ◁ C₂³ ◁ C₂×C₄² ◁ #390

subgroup

G/H

center

C₂³

C₂×C₂²⋊D₇

commutator, cosocle

C₂²×C₄×C₇

C₂³

Frattini

C₂²×C₄

C₂²×D₇

Fitting

C₇×#390

C₂

socle

C₂³×C₇

C₂×D₄

prime radical

C₂²×C₇×D₄

C₂²

D₇

C₇

G ▷ C₂²×C₄×C₇ ▷ C₂×C₇ ▷ C₇

UCS

C₁ ◁ C₂³ ◁ C₂×C₄² ◁ #390

N	H ≅ G/N	type
1. C₂×#6433	C₂	semidirect
4. #89324	C₂	semidirect
5. #89319	C₂	semidirect
129. C₇	#2025	semidirect

H ≅ G/N

type

1. C₂×#6433

C₂

semidirect

4. #89324

C₂

semidirect

5. #89319

C₂

semidirect

129. C₇

#2025

semidirect

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

subgroup	index
C₂×#6433	2
C₇×#390	2
C₂×#6406	2
#89324	2
#89319	2
#2025	7

subgroup

index

Subgroups — #11727393

Conjugacy classes of subgroups

Complements

Maximal subgroups