Group #11737378 — subgroups

Subgroups — #11737378

Order 896.

336 conjugacy classes of subgroups; only the 139 normal subgroups are shown.

normal subgroups only

This group has 336 conjugacy classes of subgroups — too many to draw. Only its 139 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	#90261	C₂	1	1	yes	maximal, characteristic
2	Q₈×Dic₁₄	C₂	1	1	yes	maximal, characteristic
3	C₇×C₈×Q₈	C₂	1	1	yes	maximal, characteristic
4	#89622	C₂	1	3	yes	maximal
5	#89621	C₂	1	3	yes	maximal
6	C₄×Dic₂₈	C₂	1	3	yes	maximal
7	#89453	C₂	1	3	yes	maximal
8	C₄×C₇×Q₈	C₂²	1	1	yes	characteristic
9	Q₈×Dic₇	—	2	1	no	—
10	Q₈×Dic₇	—	2	1	no	—
11	C₂×Dic₂₈	C₂²	1	3	yes	—
12	#6453	—	2	3	no	—
13	#6451	—	2	3	no	—
14	#6432	C₂²	1	3	yes	—
15	#6432	C₂²	1	3	yes	—
16	C₄×Dic₁₄	C₂²	1	3	yes	—
17	C₄×Dic₁₄	C₂²	1	3	yes	—
18	C₇×C₄⋊C₈	C₂²	1	3	yes	—
19	C₄×C₇×C₈	C₂²	1	3	yes	—
20	#6392	C₂²	1	3	yes	—
21	#6390	C₂²	1	6	yes	—
22	#6390	C₂²	1	3	yes	—
23	#6390	C₂²	1	1	yes	characteristic
24	#3363	—	7	1	no	maximal, Sylow
25	C₂×C₇×Q₈	D₄	1	1	yes	characteristic
26	C₂×Dic₁₄	—	2	3	no	—
27	C₂×Dic₁₄	C₂³	1	3	yes	—
28	C₂×Dic₁₄	C₂³	1	1	yes	characteristic
29	C₂×C₇×C₈	C₂³	1	1	yes	characteristic
30	C₂×C₇×C₈	C₂³	1	3	yes	—
31	C₇×C₄⋊C₄	D₄	1	3	yes	—
32	C₄²×C₇	C₂³	1	3	yes	—
33	C₄⋊Dic₇	—	2	3	no	—
34	C₄⋊Dic₇	—	2	3	no	—
35	C₄⋊Dic₇	—	2	3	no	—
36	C₄⋊Dic₇	C₂³	1	1	yes	characteristic
37	C₄⋊Dic₇	C₂³	1	3	yes	—
38	Dic₇⋊C₄	—	2	3	no	—
39	Dic₇⋊C₄	—	2	3	no	—
40	C₄×Dic₇	—	2	3	no	—
41	C₄×Dic₇	—	2	3	no	—
42	Dic₂₈	—	2	6	no	—
43	Q₈²	—	7	1	no	—
44	#557	—	7	1	no	—
45	#487	—	7	3	no	—
46	#477	—	7	3	no	—
47	#462	—	7	3	no	—
48	C₈×Q₈	D₇	1	1	yes	characteristic
49	C₄×Q₁₆	—	7	3	no	—
50	C₇×Q₈	Q₁₆	1	4	yes	complemented
51	C₂×C₄×C₇	C₂×D₄	1	3	yes	—
52	C₂×C₄×C₇	C₂×D₄	1	3	yes	—
53	C₂×C₄×C₇	C₂⁴	1	1	yes	characteristic
54	C₂×Dic₇	—	2	1	no	—
55	C₂×Dic₇	—	2	1	no	—
56	C₂×Dic₇	—	2	3	no	—
57	C₂×Dic₇	—	2	3	no	—
58	Dic₁₄	—	2	1	no	—
59	Dic₁₄	—	2	3	no	—
60	Dic₁₄	—	2	3	no	—
61	Dic₁₄	—	8	3	no	—
62	Dic₁₄	#41	1	2	yes	—
63	C₇×C₈	—	2	3	no	—
64	C₇×C₈	—	2	1	no	—
65	C₂×Q₁₆	—	7	3	no	—
66	C₄⋊Q₈	—	7	3	no	—
67	C₄⋊Q₈	—	7	3	no	—
68	C₄⋊Q₈	—	14	3	no	—
69	#125	—	14	3	no	—
70	C₄×Q₈	—	7	3	no	—
71	C₄×Q₈	—	7	3	no	—
72	C₄×Q₈	—	14	1	no	—
73	C₄×Q₈	—	14	1	no	—
74	C₄×Q₈	C₂×D₇	1	1	yes	characteristic
75	#107	—	7	3	no	—
76	C₄⋊C₈	C₂×D₇	1	3	yes	—
77	#103	—	7	1	no	—
78	#103	—	7	3	no	—
79	#103	—	7	6	no	—
80	C₄×C₈	C₂×D₇	1	3	yes	—
81	C₂²×C₇	C₂²×D₄	1	1	yes	characteristic
82	C₄×C₇	—	2	3	no	—
83	C₄×C₇	C₂×Q₁₆	1	6	yes	—
84	C₄×C₇	C₂×#41	1	1	yes	characteristic
85	C₄×C₇	2₋¹⁺⁴	1	1	yes	characteristic
86	Dic₇	—	2	2	no	—
87	Dic₇	—	4	3	no	—
88	Dic₇	—	4	3	no	—
89	Dic₇	—	4	1	no	—
90	C₂×Q₈	—	7	1	no	—
91	C₂×Q₈	—	7	3	no	—
92	C₂×Q₈	—	14	3	no	—
93	C₂×Q₈	D₂₈	1	1	yes	characteristic
94	Q₁₆	—	14	6	no	—
95	C₂×C₈	C₂²×D₇	1	1	yes	characteristic
96	C₂×C₈	C₂²×D₇	1	3	yes	—
97	C₄⋊C₄	—	7	3	no	—
98	C₄⋊C₄	—	7	1	no	—
99	C₄⋊C₄	—	14	3	no	—
100	C₄⋊C₄	—	14	3	no	—
101	C₄⋊C₄	—	14	3	no	—
102	C₄⋊C₄	—	14	3	no	—
103	C₄⋊C₄	—	14	3	no	—
104	C₄⋊C₄	D₂₈	1	3	yes	—
105	C₄²	—	14	3	no	—
106	C₄²	—	14	3	no	—
107	C₄²	C₂²×D₇	1	3	yes	—
108	C₂×C₇	#548	1	1	yes	characteristic
109	C₂×C₇	C₂²×Q₁₆	1	1	yes	characteristic
110	C₂×C₇	#577	1	1	yes	characteristic
111	Q₈	—	7	2	no	—
112	Q₈	—	14	1	no	—
113	Q₈	—	14	3	no	—
114	Q₈	—	14	3	no	—
115	Q₈	—	56	3	no	—
116	Q₈	Dic₂₈	1	4	yes	complemented
117	C₂×C₄	—	14	1	no	—
118	C₂×C₄	—	14	1	no	—
119	C₂×C₄	—	14	3	no	—
120	C₂×C₄	—	14	3	no	—
121	C₂×C₄	C₂×D₂₈	1	3	yes	—
122	C₂×C₄	C₂×D₂₈	1	3	yes	—
123	C₂×C₄	C₂³×D₇	1	1	yes	characteristic
124	C₈	—	2	1	no	—
125	C₈	—	2	3	no	—
126	C₇	#3363	1	1	yes	characteristic, complemented, Sylow
127	C₂²	C₂²×D₂₈	1	1	yes	characteristic
128	C₄	—	2	3	no	—
129	C₄	—	14	2	no	—
130	C₄	—	28	3	no	—
131	C₄	—	28	1	no	—
132	C₄	—	28	3	no	—
133	C₄	C₂×Dic₂₈	1	6	yes	—
134	C₄	#6551	1	1	yes	characteristic
135	C₄	D₇×#41	1	1	yes	characteristic
136	C₂	#90267	1	1	yes	characteristic
137	C₂	C₂²×Dic₂₈	1	1	yes	characteristic
138	C₂	#90442	1	1	yes	characteristic

subgroup

G/H

class size

multiplicity

normal?

properties

#90261

C₂

yes

maximal, characteristic

Q₈×Dic₁₄

C₂

yes

maximal, characteristic

C₇×C₈×Q₈

C₂

yes

maximal, characteristic

#89622

C₂

yes

maximal

#89621

C₂

yes

maximal

C₄×Dic₂₈

C₂

yes

maximal

#89453

C₂

yes

maximal

C₄×C₇×Q₈

C₂²

yes

characteristic

Q₈×Dic₇

—

Q₈×Dic₇

—

C₂×Dic₂₈

C₂²

yes

—

#6453

—

#6451

—

#6432

C₂²

yes

—

#6432

C₂²

yes

—

C₄×Dic₁₄

C₂²

yes

—

C₄×Dic₁₄

C₂²

yes

—

C₇×C₄⋊C₈

C₂²

yes

—

C₄×C₇×C₈

C₂²

yes

—

#6392

C₂²

yes

—

#6390

C₂²

yes

—

#6390

C₂²

yes

—

#6390

C₂²

yes

characteristic

#3363

—

maximal, Sylow

C₂×C₇×Q₈

D₄

yes

characteristic

C₂×Dic₁₄

—

C₂×Dic₁₄

C₂³

yes

—

C₂×Dic₁₄

C₂³

yes

characteristic

C₂×C₇×C₈

C₂³

yes

characteristic

C₂×C₇×C₈

C₂³

yes

—

C₇×C₄⋊C₄

D₄

yes

—

C₄²×C₇

C₂³

yes

—

C₄⋊Dic₇

—

C₄⋊Dic₇

—

C₄⋊Dic₇

—

C₄⋊Dic₇

C₂³

yes

characteristic

C₄⋊Dic₇

C₂³

yes

—

Dic₇⋊C₄

—

Dic₇⋊C₄

—

C₄×Dic₇

—

C₄×Dic₇

—

Dic₂₈

—

Q₈²

—

—

—

—

—

C₈×Q₈

D₇

yes

characteristic

C₄×Q₁₆

—

C₇×Q₈

Q₁₆

yes

complemented

C₂×C₄×C₇

C₂×D₄

yes

—

C₂×C₄×C₇

C₂×D₄

yes

—

C₂×C₄×C₇

C₂⁴

yes

characteristic

C₂×Dic₇

—

C₂×Dic₇

—

C₂×Dic₇

—

C₂×Dic₇

—

Dic₁₄

—

Dic₁₄

—

Dic₁₄

—

Dic₁₄

—

Dic₁₄

#41

yes

—

C₇×C₈

—

C₇×C₈

—

C₂×Q₁₆

—

C₄⋊Q₈

—

C₄⋊Q₈

—

C₄⋊Q₈

—

#125

—

C₄×Q₈

—

C₄×Q₈

—

C₄×Q₈

—

C₄×Q₈

—

C₄×Q₈

C₂×D₇

yes

characteristic

#107

—

C₄⋊C₈

C₂×D₇

yes

—

#103

—

#103

—

#103

—

C₄×C₈

C₂×D₇

yes

—

C₂²×C₇

C₂²×D₄

yes

characteristic

C₄×C₇

—

C₄×C₇

C₂×Q₁₆

yes

—

C₄×C₇

C₂×#41

yes

characteristic

C₄×C₇

2₋¹⁺⁴

yes

characteristic

Dic₇

—

Dic₇

—

Dic₇

—

Dic₇

—

C₂×Q₈

—

C₂×Q₈

—

C₂×Q₈

—

C₂×Q₈

D₂₈

yes

characteristic

Q₁₆

—

C₂×C₈

C₂²×D₇

yes

characteristic

C₂×C₈

C₂²×D₇

yes

—

C₄⋊C₄

—

C₄⋊C₄

—

C₄⋊C₄

—

100

C₄⋊C₄

—

101

C₄⋊C₄

—

102

C₄⋊C₄

—

103

C₄⋊C₄

—

104

C₄⋊C₄

D₂₈

yes

—

105

C₄²

—

106

C₄²

—

107

C₄²

C₂²×D₇

yes

—

108

C₂×C₇

#548

yes

characteristic

109

C₂×C₇

C₂²×Q₁₆

yes

characteristic

110

C₂×C₇

#577

yes

characteristic

111

Q₈

—

112

Q₈

—

113

Q₈

—

114

Q₈

—

115

Q₈

—

116

Q₈

Dic₂₈

yes

complemented

117

C₂×C₄

—

118

C₂×C₄

—

119

C₂×C₄

—

120

C₂×C₄

—

121

C₂×C₄

C₂×D₂₈

yes

—

122

C₂×C₄

C₂×D₂₈

yes

—

123

C₂×C₄

C₂³×D₇

yes

characteristic

124

C₈

—

125

C₈

—

126

C₇

#3363

yes

characteristic, complemented, Sylow

127

C₂²

C₂²×D₂₈

yes

characteristic

128

C₄

—

129

C₄

—

130

C₄

—

131

C₄

—

132

C₄

—

133

C₄

C₂×Dic₂₈

yes

—

134

C₄

#6551

yes

characteristic

135

C₄

D₇×#41

yes

characteristic

136

C₂

#90267

yes

characteristic

137

C₂

C₂²×Dic₂₈

yes

characteristic

138

C₂

#90442

yes

characteristic

	subgroup	G/H
center	C₂²	C₂²×D₂₈
commutator, cosocle	C₂×C₄×C₇	C₂⁴
Frattini	C₂×C₄	C₂³×D₇
Fitting	C₇×C₈×Q₈	C₂
socle, prime radical	C₂²×C₇	C₂²×D₄
hypercenter	C₈×Q₈	D₇
hypocenter	C₇	#3363
LCS	G ▷ C₂×C₄×C₇ ▷ C₂×C₇ ▷ C₇
UCS	C₁ ◁ C₂² ◁ C₄×Q₈ ◁ C₈×Q₈

subgroup

G/H

center

C₂²

C₂²×D₂₈

commutator, cosocle

C₂×C₄×C₇

C₂⁴

Frattini

C₂×C₄

C₂³×D₇

Fitting

C₇×C₈×Q₈

C₂

socle, prime radical

C₂²×C₇

C₂²×D₄

hypercenter

C₈×Q₈

D₇

hypocenter

C₇

#3363

LCS

G ▷ C₂×C₄×C₇ ▷ C₂×C₇ ▷ C₇

UCS

C₁ ◁ C₂² ◁ C₄×Q₈ ◁ C₈×Q₈

N	H ≅ G/N	type
50. C₇×Q₈	Q₁₆	semidirect
116. Q₈	Dic₂₈	semidirect
126. C₇	#3363	semidirect

H ≅ G/N

type

50. C₇×Q₈

Q₁₆

semidirect

116. Q₈

Dic₂₈

semidirect

126. C₇

#3363

semidirect

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

subgroup	index
#90261	2
Q₈×Dic₁₄	2
C₇×C₈×Q₈	2
#89622	2
#89621	2
C₄×Dic₂₈	2
#89453	2
#3363	7

subgroup

index

#90261

Q₈×Dic₁₄

C₇×C₈×Q₈

C₄×Dic₂₈

Subgroups — #11737378

Conjugacy classes of subgroups

Complements

Maximal subgroups