Group #11728217 — subgroups

Subgroups — #11728217

Order 896.

224 conjugacy classes of subgroups; only the 89 normal subgroups are shown.

normal subgroups only

This group has 224 conjugacy classes of subgroups — too many to draw. Only its 89 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×C₇×C₄⋊C₈	C₂	1	1	yes	maximal, characteristic
2	C₂×#6392	C₂	1	1	yes	maximal, characteristic
3	#89700	C₂	1	2	yes	maximal
4	#89346	C₂	1	2	yes	maximal
5	#89346	C₂	1	1	yes	maximal, characteristic
6	C₂²×C₇×C₈	C₂²	1	1	yes	characteristic
7	C₂²×C₇×C₈	C₂²	1	1	yes	characteristic
8	C₂×C₄²×C₇	C₂²	1	1	yes	characteristic
9	C₂×C₄⋊Dic₇	C₂²	1	2	yes	—
10	C₂×C₄⋊Dic₇	C₂²	1	2	yes	—
11	C₂×Dic₇⋊C₄	—	2	2	no	—
12	C₇×C₄⋊C₈	—	2	2	no	—
13	C₇⋊#95	—	2	2	no	—
14	#6392	—	2	2	no	—
15	#2087	—	7	1	no	maximal, Sylow
16	C₂²×C₄×C₇	D₄	1	1	yes	characteristic
17	C₂²×C₄×C₇	D₄	1	1	yes	characteristic
18	C₂²×C₄×C₇	C₂³	1	1	yes	characteristic
19	C₂²×Dic₇	—	2	2	no	—
20	C₂²×Dic₇	—	2	2	no	—
21	C₂×C₇×C₈	—	2	2	no	—
22	C₂×C₇×C₈	—	2	1	no	—
23	C₂×C₇×C₈	—	2	2	no	—
24	C₂×C₇×C₈	D₄	1	2	yes	—
25	C₄²×C₇	—	2	1	no	—
26	C₄²×C₇	—	2	1	no	—
27	C₄⋊Dic₇	—	2	2	no	—
28	C₄⋊Dic₇	—	2	2	no	—
29	C₄⋊Dic₇	—	2	2	no	—
30	C₄⋊Dic₇	Q₈	1	4	yes	—
31	Dic₇⋊C₄	—	4	4	no	—
32	C₂×#107	—	7	1	no	—
33	C₂×C₄⋊C₈	D₇	1	1	yes	characteristic
34	#389	—	7	2	no	—
35	#340	—	7	1	no	—
36	#340	—	7	2	no	—
37	C₂³×C₇	C₂×D₄	1	1	yes	characteristic
38	C₂×C₄×C₇	—	2	1	no	—
39	C₂×C₄×C₇	—	2	1	no	—
40	C₂×C₄×C₇	—	2	1	no	—
41	C₂×C₄×C₇	—	2	2	no	—
42	C₂×C₄×C₇	Q₁₆	1	2	yes	—
43	C₂×C₄×C₇	C₂×D₄	1	1	yes	characteristic
44	C₂×C₄×C₇	C₂×Q₈	1	2	yes	—
45	C₂×C₄×C₇	#41	1	1	yes	characteristic
46	C₂×C₄×C₇	#41	1	2	yes	—
47	C₂×Dic₇	—	2	4	no	—
48	C₂×Dic₇	—	4	2	no	—
49	C₂×Dic₇	—	4	2	no	—
50	C₂×Dic₇	—	4	2	no	—
51	C₂×Dic₇	—	4	2	no	—
52	C₂×Dic₇	—	4	2	no	—
53	C₇×C₈	—	2	2	no	—
54	C₇×C₈	—	4	1	no	—
55	C₂²×C₈	C₂×D₇	1	1	yes	characteristic
56	C₂²×C₈	C₂×D₇	1	1	yes	characteristic
57	C₂×C₄⋊C₄	—	7	2	no	—
58	C₂×C₄⋊C₄	—	7	2	no	—
59	C₂×C₄⋊C₄	—	14	2	no	—
60	C₂×C₄²	C₂×D₇	1	1	yes	characteristic
61	#107	—	14	2	no	—
62	C₄⋊C₈	—	2	2	no	—
63	#95	—	14	2	no	—
64	C₂²×C₇	#121	1	1	yes	characteristic
65	C₂²×C₇	C₂²⋊Q₈	1	2	yes	—
66	C₂²×C₇	C₂×Q₁₆	1	1	yes	characteristic
67	C₂²×C₇	#135	1	1	yes	characteristic
68	C₂²×C₇	AGL(1,ℤ/8)	1	2	yes	—
69	C₄×C₇	—	2	2	no	—
70	C₄×C₇	—	4	1	no	—
71	C₄×C₇	#123	1	1	yes	characteristic
72	C₄×C₇	#125	1	2	yes	—
73	C₄×C₇	C₄⋊Q₈	1	1	yes	characteristic
74	Dic₇	—	4	4	no	—
75	Dic₇	—	8	2	no	—
76	C₂²×C₄	—	14	2	no	—
77	C₂²×C₄	—	14	2	no	—
78	C₂²×C₄	D₂₈	1	1	yes	characteristic
79	C₂²×C₄	D₂₈	1	1	yes	characteristic
80	C₂²×C₄	C₂²×D₇	1	1	yes	characteristic
81	C₂×C₈	—	2	2	no	—
82	C₂×C₈	—	2	2	no	—
83	C₂×C₈	—	2	1	no	—
84	C₂×C₈	C₂²⋊D₇	1	2	yes	—
85	C₄⋊C₄	—	7	4	no	—
86	C₄⋊C₄	—	14	2	no	—
87	C₄⋊C₄	—	14	2	no	—
88	C₄⋊C₄	—	14	2	no	—
89	C₄⋊C₄	—	28	4	no	—
90	C₄²	—	2	1	no	—
91	C₄²	—	2	1	no	—
92	C₂×C₇	#398	1	1	yes	characteristic
93	C₂×C₇	#467	1	1	yes	characteristic
94	C₂×C₇	#469	1	1	yes	characteristic
95	C₂×C₇	#477	1	2	yes	—
96	C₂×C₇	#478	1	2	yes	—
97	C₂³	C₂×D₂₈	1	1	yes	characteristic
98	C₂×C₄	—	2	1	no	—
99	C₂×C₄	—	2	1	no	—
100	C₂×C₄	—	2	2	no	—
101	C₂×C₄	—	2	1	no	—
102	C₂×C₄	—	14	4	no	—
103	C₂×C₄	—	28	2	no	—
104	C₂×C₄	—	28	2	no	—
105	C₂×C₄	—	28	2	no	—
106	C₂×C₄	—	28	2	no	—
107	C₂×C₄	—	28	2	no	—
108	C₂×C₄	Dic₂₈	1	2	yes	—
109	C₂×C₄	#1157	1	1	yes	characteristic
110	C₂×C₄	#1159	1	2	yes	—
111	C₂×C₄	Q₈×D₇	1	2	yes	—
112	C₂×C₄	C₂×C₂²⋊D₇	1	1	yes	characteristic
113	C₈	—	2	2	no	—
114	C₈	—	4	1	no	—
115	C₇	#2087	1	1	yes	characteristic, complemented, Sylow
116	C₂²	#6460	1	2	yes	—
117	C₂²	#6467	1	1	yes	characteristic
118	C₂²	C₂×Dic₂₈	1	1	yes	characteristic
119	C₂²	#6471	1	2	yes	—
120	C₂²	#6493	1	1	yes	characteristic
121	C₄	—	2	2	no	—
122	C₄	—	4	1	no	—
123	C₄	—	28	4	no	—
124	C₄	—	56	2	no	—
125	C₄	#6452	1	2	yes	—
126	C₄	#6498	1	1	yes	characteristic
127	C₄	#6507	1	1	yes	characteristic
128	C₂	#89619	1	2	yes	—
129	C₂	#89622	1	2	yes	—
130	C₂	#89755	1	1	yes	characteristic
131	C₂	#89888	1	1	yes	characteristic
132	C₂	#89907	1	1	yes	characteristic

subgroup

G/H

class size

multiplicity

normal?

properties

C₂×C₇×C₄⋊C₈

C₂

yes

maximal, characteristic

C₂×#6392

C₂

yes

maximal, characteristic

#89700

C₂

yes

maximal

#89346

C₂

yes

maximal

#89346

C₂

yes

maximal, characteristic

C₂²×C₇×C₈

C₂²

yes

characteristic

C₂²×C₇×C₈

C₂²

yes

characteristic

C₂×C₄²×C₇

C₂²

yes

characteristic

C₂×C₄⋊Dic₇

C₂²

yes

—

C₂×C₄⋊Dic₇

C₂²

yes

—

C₂×Dic₇⋊C₄

—

C₇×C₄⋊C₈

—

—

—

—

C₂²×C₄×C₇

D₄

yes

characteristic

C₂²×C₄×C₇

D₄

yes

characteristic

C₂²×C₄×C₇

C₂³

yes

characteristic

C₂²×Dic₇

—

C₂²×Dic₇

—

C₂×C₇×C₈

—

C₂×C₇×C₈

—

C₂×C₇×C₈

—

C₂×C₇×C₈

D₄

yes

—

C₄²×C₇

—

C₄²×C₇

—

C₄⋊Dic₇

—

C₄⋊Dic₇

—

C₄⋊Dic₇

—

C₄⋊Dic₇

Q₈

yes

—

Dic₇⋊C₄

—

C₂×#107

—

C₂×C₄⋊C₈

D₇

yes

—

—

—

C₂³×C₇

C₂×D₄

yes

characteristic

C₂×C₄×C₇

—

C₂×C₄×C₇

—

C₂×C₄×C₇

—

C₂×C₄×C₇

—

C₂×C₄×C₇

Q₁₆

yes

—

C₂×C₄×C₇

C₂×D₄

yes

characteristic

C₂×C₄×C₇

C₂×Q₈

yes

—

C₂×C₄×C₇

#41

yes

characteristic

C₂×C₄×C₇

#41

yes

—

C₂×Dic₇

—

C₂×Dic₇

—

C₂×Dic₇

—

C₂×Dic₇

—

C₂×Dic₇

—

C₂×Dic₇

—

C₇×C₈

—

C₇×C₈

—

C₂²×C₈

C₂×D₇

yes

characteristic

C₂²×C₈

C₂×D₇

yes

characteristic

C₂×C₄⋊C₄

—

C₂×C₄⋊C₄

—

C₂×C₄⋊C₄

—

C₂×C₄²

C₂×D₇

yes

characteristic

#107

—

C₄⋊C₈

—

#95

—

C₂²×C₇

#121

yes

characteristic

C₂²×C₇

C₂²⋊Q₈

yes

—

C₂²×C₇

C₂×Q₁₆

yes

C₂²×C₇

yes

C₂²×C₇

yes

—

C₄×C₇

—

C₄×C₇

—

C₄×C₇

#123

yes

characteristic

C₄×C₇

#125

yes

—

C₄×C₇

C₄⋊Q₈

yes

characteristic

Dic₇

—

Dic₇

—

C₂²×C₄

—

C₂²×C₄

—

C₂²×C₄

D₂₈

yes

characteristic

C₂²×C₄

D₂₈

yes

characteristic

C₂²×C₄

C₂²×D₇

yes

characteristic

C₂×C₈

—

C₂×C₈

—

C₂×C₈

—

C₂×C₈

C₂²⋊D₇

yes

—

C₄⋊C₄

—

C₄⋊C₄

—

C₄⋊C₄

—

C₄⋊C₄

—

C₄⋊C₄

—

C₄²

—

C₄²

—

C₂×C₇

yes

C₂×C₇

yes

C₂×C₇

yes

C₂×C₇

yes

—

C₂×C₇

#478

yes

—

C₂³

C₂×D₂₈

yes

characteristic

C₂×C₄

—

C₂×C₄

—

100

C₂×C₄

—

101

C₂×C₄

—

102

C₂×C₄

—

103

C₂×C₄

—

104

C₂×C₄

—

105

C₂×C₄

—

106

C₂×C₄

—

107

C₂×C₄

—

108

C₂×C₄

Dic₂₈

yes

—

109

C₂×C₄

#1157

yes

characteristic

110

C₂×C₄

#1159

yes

—

111

C₂×C₄

Q₈×D₇

yes

—

112

C₂×C₄

C₂×C₂²⋊D₇

yes

characteristic

113

C₈

—

114

C₈

—

115

C₇

#2087

yes

characteristic, complemented, Sylow

116

C₂²

#6460

yes

—

117

C₂²

#6467

yes

characteristic

118

C₂²

C₂×Dic₂₈

yes

characteristic

119

C₂²

#6471

yes

—

120

C₂²

#6493

yes

characteristic

121

C₄

—

122

C₄

—

123

C₄

—

124

C₄

—

125

C₄

#6452

yes

—

126

C₄

#6498

yes

characteristic

127

C₄

#6507

yes

characteristic

128

C₂

#89619

yes

—

129

C₂

#89622

yes

—

130

C₂

#89755

yes

characteristic

131

C₂

#89888

yes

characteristic

132

C₂

#89907

yes

characteristic

	subgroup	G/H
center	C₂³	C₂×D₂₈
commutator, cosocle	C₂²×C₄×C₇	C₂³
Frattini	C₂²×C₄	C₂²×D₇
Fitting	C₂×C₇×C₄⋊C₈	C₂
socle, prime radical	C₂³×C₇	C₂×D₄
hypercenter	C₂×C₄⋊C₈	D₇
hypocenter	C₇	#2087
LCS	G ▷ C₂²×C₄×C₇ ▷ C₂×C₇ ▷ C₇
UCS	C₁ ◁ C₂³ ◁ C₂×C₄² ◁ C₂×C₄⋊C₈

subgroup

G/H

center

C₂³

C₂×D₂₈

commutator, cosocle

C₂²×C₄×C₇

C₂³

Frattini

C₂²×C₄

C₂²×D₇

Fitting

C₂×C₇×C₄⋊C₈

C₂

socle, prime radical

C₂³×C₇

C₂×D₄

hypercenter

C₂×C₄⋊C₈

D₇

hypocenter

C₇

#2087

LCS

G ▷ C₂²×C₄×C₇ ▷ C₂×C₇ ▷ C₇

UCS

C₁ ◁ C₂³ ◁ C₂×C₄² ◁ C₂×C₄⋊C₈

N	H ≅ G/N	type
115. C₇	#2087	semidirect

H ≅ G/N

type

115. C₇

#2087

semidirect

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

subgroup	index
C₂×C₇×C₄⋊C₈	2
C₂×#6392	2
#89700	2
#89346	2
#2087	7

subgroup

index

C₂×C₇×C₄⋊C₈

Subgroups — #11728217

Conjugacy classes of subgroups

Complements

Maximal subgroups