Group #11740643 — subgroups

Subgroups — #11740643

Order 896.

380 conjugacy classes of subgroups; only the 141 normal subgroups are shown.

normal subgroups only

This group has 380 conjugacy classes of subgroups — too many to draw. Only its 141 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⋊Q₈	C₂	1	1	yes	maximal, characteristic
2	#90161	C₂	1	1	yes	maximal, characteristic
3	#89997	C₂	1	4	yes	maximal
4	#89854	C₂	1	4	yes	maximal, complemented
5	#89853	C₂	1	4	yes	maximal, complemented
6	C₄×#1136	C₂	1	1	yes	maximal, characteristic
7	C₂²×C₇×Q₈	C₂²	1	2	yes	—
8	C₇×C₄⋊Q₈	—	2	2	no	—
9	C₇×C₄⋊Q₈	C₂²	1	2	yes	—
10	C₇×C₄⋊Q₈	C₂²	1	2	yes	—
11	C₂×C₇×C₄⋊C₄	—	2	2	no	—
12	C₂×C₄²×C₇	C₂²	1	1	yes	characteristic
13	#6489	C₂²	1	2	yes	—
14	#6487	—	2	2	no	—
15	C₂×#1136	C₂²	1	2	yes	—
16	#6433	C₂²	1	2	yes	—
17	#6432	C₂²	1	2	yes	—
18	C₄×Dic₁₄	—	2	2	no	—
19	#6409	C₂²	1	16	yes	—
20	#6377	C₂²	1	2	yes	—
21	C₄×C₇⋊C₈	C₂²	1	2	yes	—
22	#3139	—	7	1	no	maximal, Sylow
23	C₂×C₇×Q₈	—	2	2	no	—
24	C₂×C₇×Q₈	—	2	8	no	—
25	C₂×C₇×Q₈	C₂³	1	4	yes	—
26	C₂²×C₄×C₇	—	2	2	no	—
27	C₂²×C₄×C₇	D₄	1	2	yes	—
28	C₂²×C₄×C₇	C₂³	1	1	yes	characteristic
29	C₂×Dic₁₄	—	2	2	no	—
30	C₇×C₄⋊C₄	—	2	4	no	—
31	C₇×C₄⋊C₄	—	2	4	no	—
32	C₄²×C₇	D₄	1	2	yes	—
33	C₄²×C₇	C₂³	1	1	yes	characteristic
34	C₄²×C₇	C₂³	1	1	yes	characteristic
35	C₂²⋊Dic₇	—	2	2	no	—
36	C₄⋊Dic₇	—	2	2	no	—
37	C₄⋊Dic₇	C₂³	1	4	yes	—
38	Dic₇⋊C₄	—	2	4	no	—
39	C₄×Dic₇	—	2	2	no	—
40	#1136	—	2	4	no	—
41	C₂×C₇⋊C₈	C₂³	1	4	yes	—
42	#533	—	7	1	no	—
43	C₂×C₄⋊Q₈	D₇	1	1	yes	characteristic
44	#491	—	7	4	no	—
45	#487	—	7	4	no	—
46	#419	—	7	4	no	—
47	C₄×M₁₆	—	7	1	no	—
48	C₂³×C₇	C₂×D₄	1	1	yes	characteristic
49	C₇×Q₈	—	2	8	no	—
50	C₇×Q₈	—	4	4	no	—
51	C₂×C₄×C₇	—	2	4	no	—
52	C₂×C₄×C₇	—	2	4	no	—
53	C₂×C₄×C₇	—	4	4	no	—
54	C₂×C₄×C₇	C₂×D₄	1	2	yes	—
55	C₂×C₄×C₇	C₂×D₄	1	1	yes	characteristic
56	C₂×C₄×C₇	C₂×D₄	1	2	yes	—
57	C₂×C₄×C₇	#41	1	4	yes	—
58	C₂×C₄×C₇	C₂⁴	1	1	yes	characteristic
59	C₂×Dic₇	—	2	4	no	—
60	Dic₁₄	—	4	4	no	—
61	C₇⋊C₈	—	2	4	no	—
62	C₂²×Q₈	C₂×D₇	1	2	yes	—
63	C₂×M₁₆	—	7	2	no	—
64	C₄⋊Q₈	—	2	2	no	—
65	C₄⋊Q₈	—	7	2	no	—
66	C₄⋊Q₈	C₂×D₇	1	2	yes	—
67	C₄⋊Q₈	C₂×D₇	1	2	yes	—
68	#125	—	7	2	no	—
69	C₂²⋊Q₈	—	14	2	no	—
70	C₄×Q₈	—	14	2	no	—
71	#117	—	7	2	no	—
72	C₂×C₄⋊C₄	—	2	2	no	—
73	C₂×C₄²	C₂×D₇	1	1	yes	characteristic
74	#103	—	7	16	no	—
75	#97	—	7	2	no	—
76	C₄×C₈	—	7	2	no	—
77	C₂²×C₇	—	2	2	no	—
78	C₂²×C₇	#124	1	2	yes	—
79	C₂²×C₇	C₂²×D₄	1	1	yes	characteristic
80	C₄×C₇	—	2	2	no	—
81	C₄×C₇	—	4	4	no	—
82	C₄×C₇	#124	1	2	yes	—
83	C₄×C₇	#137	1	4	yes	—
84	C₄×C₇	C₂×#41	1	2	yes	—
85	Dic₇	—	4	4	no	—
86	C₂×Q₈	—	2	2	no	—
87	C₂×Q₈	—	2	8	no	—
88	C₂×Q₈	—	14	2	no	—
89	C₂×Q₈	C₂²×D₇	1	4	yes	—
90	C₂²×C₄	—	2	2	no	—
91	C₂²×C₄	C₂²⋊D₇	1	2	yes	—
92	C₂²×C₄	C₂²×D₇	1	1	yes	characteristic
93	M₁₆	—	14	4	no	—
94	C₂×C₈	—	7	4	no	—
95	C₄⋊C₄	—	2	4	no	—
96	C₄⋊C₄	—	2	4	no	—
97	C₄⋊C₄	—	7	4	no	—
98	C₄⋊C₄	—	14	4	no	—
99	C₄⋊C₄	—	14	2	no	—
100	C₂²⋊C₄	—	14	2	no	—
101	C₄²	—	14	2	no	—
102	C₄²	C₂²⋊D₇	1	2	yes	—
103	C₄²	C₂²×D₇	1	1	yes	characteristic
104	C₄²	C₂²×D₇	1	1	yes	characteristic
105	C₂×C₇	—	2	2	no	—
106	C₂×C₇	C₂×#124	1	1	yes	characteristic
107	C₂×C₇	C₂×#137	1	2	yes	—
108	C₂³	C₂×C₂²⋊D₇	1	1	yes	characteristic
109	Q₈	—	2	8	no	—
110	Q₈	—	4	4	no	—
111	Q₈	—	28	4	no	—
112	C₂×C₄	—	2	4	no	—
113	C₂×C₄	—	2	4	no	—
114	C₂×C₄	—	4	4	no	—
115	C₂×C₄	—	14	4	no	—
116	C₂×C₄	#1159	1	4	yes	—
117	C₂×C₄	C₂×C₂²⋊D₇	1	2	yes	—
118	C₂×C₄	C₂×C₂²⋊D₇	1	1	yes	characteristic
119	C₂×C₄	C₂×C₂²⋊D₇	1	2	yes	—
120	C₂×C₄	C₂³×D₇	1	1	yes	characteristic
121	C₈	—	14	4	no	—
122	C₇	#3139	1	1	yes	characteristic, complemented, Sylow
123	C₂²	—	2	2	no	—
124	C₂²	#6499	1	2	yes	—
125	C₂²	C₂²×C₂²⋊D₇	1	1	yes	characteristic
126	C₄	—	2	2	no	—
127	C₄	—	4	4	no	—
128	C₄	—	28	4	no	—
129	C₄	#6499	1	2	yes	—
130	C₄	#6505	1	4	yes	—
131	C₄	C₂×#1159	1	2	yes	—
132	C₂	—	2	2	no	—
133	C₂	C₂×#6499	1	1	yes	characteristic
134	C₂	C₂×#6505	1	2	yes	—

subgroup

G/H

class size

multiplicity

normal?

properties

C₂×C₇×C₄⋊Q₈

C₂

yes

maximal, characteristic

#90161

C₂

yes

maximal, characteristic

#89997

C₂

yes

maximal

#89854

C₂

yes

maximal, complemented

#89853

C₂

yes

maximal, complemented

C₄×#1136

C₂

yes

maximal, characteristic

C₂²×C₇×Q₈

C₂²

yes

—

C₇×C₄⋊Q₈

—

C₇×C₄⋊Q₈

C₂²

yes

—

C₇×C₄⋊Q₈

C₂²

yes

—

C₂×C₇×C₄⋊C₄

—

C₂×C₄²×C₇

C₂²

yes

characteristic

#6489

C₂²

yes

—

#6487

—

C₂×#1136

C₂²

yes

—

#6433

C₂²

yes

—

#6432

C₂²

yes

—

C₄×Dic₁₄

—

#6409

C₂²

yes

—

#6377

C₂²

yes

—

C₄×C₇⋊C₈

C₂²

yes

—

#3139

—

maximal, Sylow

C₂×C₇×Q₈

—

C₂×C₇×Q₈

—

C₂×C₇×Q₈

C₂³

yes

—

C₂²×C₄×C₇

—

C₂²×C₄×C₇

D₄

yes

—

C₂²×C₄×C₇

C₂³

yes

characteristic

C₂×Dic₁₄

—

C₇×C₄⋊C₄

—

C₇×C₄⋊C₄

—

C₄²×C₇

D₄

yes

—

C₄²×C₇

C₂³

yes

characteristic

C₄²×C₇

C₂³

yes

characteristic

C₂²⋊Dic₇

—

C₄⋊Dic₇

—

C₄⋊Dic₇

C₂³

yes

—

Dic₇⋊C₄

—

C₄×Dic₇

—

#1136

—

C₂×C₇⋊C₈

C₂³

yes

—

#533

—

C₂×C₄⋊Q₈

D₇

yes

—

—

—

C₄×M₁₆

—

C₂³×C₇

C₂×D₄

yes

characteristic

C₇×Q₈

—

C₇×Q₈

—

C₂×C₄×C₇

—

C₂×C₄×C₇

—

C₂×C₄×C₇

—

C₂×C₄×C₇

C₂×D₄

yes

—

C₂×C₄×C₇

C₂×D₄

yes

characteristic

C₂×C₄×C₇

C₂×D₄

yes

—

C₂×C₄×C₇

#41

yes

—

C₂×C₄×C₇

C₂⁴

yes

characteristic

C₂×Dic₇

—

Dic₁₄

—

C₇⋊C₈

—

C₂²×Q₈

C₂×D₇

yes

—

C₂×M₁₆

—

C₄⋊Q₈

—

C₄⋊Q₈

—

C₄⋊Q₈

C₂×D₇

yes

—

C₄⋊Q₈

C₂×D₇

yes

—

#125

—

C₂²⋊Q₈

—

C₄×Q₈

—

#117

—

C₂×C₄⋊C₄

—

C₂×C₄²

C₂×D₇

yes

characteristic

#103

—

#97

—

C₄×C₈

—

C₂²×C₇

—

C₂²×C₇

#124

yes

—

C₂²×C₇

C₂²×D₄

yes

characteristic

C₄×C₇

—

C₄×C₇

—

C₄×C₇

#124

yes

—

C₄×C₇

#137

yes

—

C₄×C₇

C₂×#41

yes

—

Dic₇

—

C₂×Q₈

—

C₂×Q₈

—

C₂×Q₈

—

C₂×Q₈

C₂²×D₇

yes

—

C₂²×C₄

—

C₂²×C₄

C₂²⋊D₇

yes

—

C₂²×C₄

C₂²×D₇

yes

characteristic

M₁₆

—

C₂×C₈

—

C₄⋊C₄

—

C₄⋊C₄

—

C₄⋊C₄

—

C₄⋊C₄

—

C₄⋊C₄

—

100

C₂²⋊C₄

—

101

C₄²

—

102

C₄²

C₂²⋊D₇

yes

—

103

C₄²

C₂²×D₇

yes

characteristic

104

C₄²

C₂²×D₇

yes

characteristic

105

C₂×C₇

—

106

C₂×C₇

C₂×#124

yes

characteristic

107

C₂×C₇

C₂×#137

yes

—

108

C₂³

C₂×C₂²⋊D₇

yes

characteristic

109

Q₈

—

110

Q₈

—

111

Q₈

—

112

C₂×C₄

—

113

C₂×C₄

—

114

C₂×C₄

—

115

C₂×C₄

—

116

C₂×C₄

#1159

yes

—

117

C₂×C₄

C₂×C₂²⋊D₇

yes

—

118

C₂×C₄

C₂×C₂²⋊D₇

yes

characteristic

119

C₂×C₄

C₂×C₂²⋊D₇

yes

—

120

C₂×C₄

C₂³×D₇

yes

characteristic

121

C₈

—

122

C₇

#3139

yes

characteristic, complemented, Sylow

123

C₂²

—

124

C₂²

#6499

yes

—

125

C₂²

C₂²×C₂²⋊D₇

yes

characteristic

126

C₄

—

127

C₄

—

128

C₄

—

129

C₄

#6499

yes

—

130

C₄

#6505

yes

—

131

C₄

C₂×#1159

yes

—

132

C₂

—

133

C₂

C₂×#6499

yes

characteristic

134

C₂

C₂×#6505

yes

—

	subgroup	G/H
center	C₂²	C₂²×C₂²⋊D₇
commutator, cosocle	C₂×C₄×C₇	C₂⁴
Frattini	C₂×C₄	C₂³×D₇
Fitting	C₂×C₇×C₄⋊Q₈	C₂
socle	C₂²×C₇	C₂²×D₄
prime radical	C₂³×C₇	C₂×D₄
hypercenter	C₂×C₄⋊Q₈	D₇
hypocenter	C₇	#3139
LCS	G ▷ C₂×C₄×C₇ ▷ C₂×C₇ ▷ C₇
UCS	C₁ ◁ C₂² ◁ C₂×C₄² ◁ C₂×C₄⋊Q₈

subgroup

G/H

center

C₂²

C₂²×C₂²⋊D₇

commutator, cosocle

C₂×C₄×C₇

C₂⁴

Frattini

C₂×C₄

C₂³×D₇

Fitting

C₂×C₇×C₄⋊Q₈

C₂

socle

C₂²×C₇

C₂²×D₄

prime radical

C₂³×C₇

C₂×D₄

hypercenter

C₂×C₄⋊Q₈

D₇

hypocenter

C₇

#3139

LCS

G ▷ C₂×C₄×C₇ ▷ C₂×C₇ ▷ C₇

UCS

C₁ ◁ C₂² ◁ C₂×C₄² ◁ C₂×C₄⋊Q₈

N	H ≅ G/N	type
4. #89854	C₂	semidirect
5. #89853	C₂	semidirect
122. C₇	#3139	semidirect

H ≅ G/N

type

4. #89854

C₂

semidirect

5. #89853

C₂

semidirect

122. C₇

#3139

semidirect

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

subgroup	index
C₂×C₇×C₄⋊Q₈	2
#90161	2
#89997	2
#89854	2
#89853	2
C₄×#1136	2
#3139	7

subgroup

index

C₂×C₇×C₄⋊Q₈

Subgroups — #11740643

Conjugacy classes of subgroups

Complements

Maximal subgroups