Group #11736514 — subgroups

Subgroups — #11736514

Order 896.

342 conjugacy classes of subgroups; only the 125 normal subgroups are shown.

normal subgroups only

This group has 342 conjugacy classes of subgroups — too many to draw. Only its 125 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	#90161	C₂	1	2	yes	maximal
2	C₇×#423	C₂	1	1	yes	maximal, characteristic
3	#89909	C₂	1	4	yes	maximal
4	#89623	C₂	1	4	yes	maximal, complemented
5	#89621	C₂	1	4	yes	maximal, complemented
6	C₂×C₇×M₁₆	C₂²	1	2	yes	—
7	C₂×C₄²×C₇	C₂²	1	1	yes	characteristic
8	#6489	—	2	2	no	—
9	#6487	—	2	2	no	—
10	#6487	C₂²	1	4	yes	—
11	C₂×Dic₂₈	C₂²	1	4	yes	—
12	#6433	—	2	2	no	—
13	#6432	C₂²	1	2	yes	—
14	#6432	C₂²	1	2	yes	—
15	C₄×Dic₁₄	—	2	2	no	—
16	C₄×Dic₁₄	C₂²	1	4	yes	—
17	C₇×C₄⋊C₈	C₂²	1	4	yes	—
18	#6391	C₂²	1	4	yes	—
19	#6390	C₂²	1	8	yes	—
20	#3255	—	7	1	no	maximal, Sylow
21	C₂²×C₄×C₇	D₄	1	2	yes	—
22	C₂²×C₄×C₇	C₂³	1	1	yes	characteristic
23	C₂×Dic₁₄	—	2	2	no	—
24	C₂×Dic₁₄	C₂³	1	4	yes	—
25	C₇×M₁₆	—	4	2	no	—
26	C₂×C₇×C₈	C₂³	1	4	yes	—
27	C₄²×C₇	D₄	1	1	yes	characteristic
28	C₄²×C₇	D₄	1	1	yes	characteristic
29	C₄²×C₇	C₂³	1	2	yes	—
30	C₂²⋊Dic₇	—	2	4	no	—
31	C₄⋊Dic₇	—	2	2	no	—
32	C₄⋊Dic₇	—	2	2	no	—
33	C₄⋊Dic₇	—	2	2	no	—
34	C₄⋊Dic₇	C₂³	1	4	yes	—
35	Dic₇⋊C₄	—	2	4	no	—
36	Dic₇⋊C₄	—	2	4	no	—
37	C₄×Dic₇	—	2	4	no	—
38	Dic₂₈	—	4	4	no	—
39	#533	—	7	2	no	—
40	#475	—	7	4	no	—
41	#470	—	7	4	no	—
42	#462	—	7	4	no	—
43	#423	D₇	1	1	yes	characteristic
44	C₂³×C₇	C₂×D₄	1	1	yes	characteristic
45	C₂×C₄×C₇	—	2	4	no	—
46	C₂×C₄×C₇	—	2	1	no	—
47	C₂×C₄×C₇	—	2	1	no	—
48	C₂×C₄×C₇	C₂×D₄	1	1	yes	characteristic
49	C₂×C₄×C₇	C₂×D₄	1	2	yes	—
50	C₂×C₄×C₇	C₂×D₄	1	2	yes	—
51	C₂×C₄×C₇	C₂⁴	1	1	yes	characteristic
52	C₂×Dic₇	—	2	4	no	—
53	C₂×Dic₇	—	2	4	no	—
54	Dic₁₄	—	2	4	no	—
55	Dic₁₄	—	2	4	no	—
56	Dic₁₄	—	4	4	no	—
57	C₇×C₈	—	2	4	no	—
58	C₂×Q₁₆	—	7	4	no	—
59	C₂×M₁₆	C₂×D₇	1	2	yes	—
60	C₄⋊Q₈	—	7	2	no	—
61	C₄⋊Q₈	—	7	2	no	—
62	#125	—	14	2	no	—
63	C₂²⋊Q₈	—	7	4	no	—
64	C₂²⋊Q₈	—	14	2	no	—
65	C₄×Q₈	—	7	4	no	—
66	C₄×Q₈	—	14	2	no	—
67	#117	—	14	2	no	—
68	C₂×C₄²	C₂×D₇	1	1	yes	characteristic
69	#106	—	7	4	no	—
70	C₄⋊C₈	C₂×D₇	1	4	yes	—
71	#103	—	7	8	no	—
72	C₂²×C₇	—	2	1	no	—
73	C₂²×C₇	—	2	2	no	—
74	C₂²×C₇	C₂²×D₄	1	1	yes	characteristic
75	C₄×C₇	—	2	1	no	—
76	C₄×C₇	—	2	2	no	—
77	C₄×C₇	#137	1	4	yes	—
78	C₄×C₇	H(5,𝔽₂)	1	1	yes	characteristic
79	C₄×C₇	2₋¹⁺⁴	1	1	yes	characteristic
80	Dic₇	—	4	4	no	—
81	Dic₇	—	4	4	no	—
82	C₂×Q₈	—	7	4	no	—
83	C₂×Q₈	—	14	2	no	—
84	C₂²×C₄	D₂₈	1	2	yes	—
85	C₂²×C₄	C₂²×D₇	1	1	yes	characteristic
86	Q₁₆	—	28	4	no	—
87	M₁₆	—	4	2	no	—
88	C₂×C₈	C₂²×D₇	1	4	yes	—
89	C₄⋊C₄	—	7	4	no	—
90	C₄⋊C₄	—	14	4	no	—
91	C₄⋊C₄	—	14	4	no	—
92	C₄⋊C₄	—	14	2	no	—
93	C₄⋊C₄	—	14	2	no	—
94	C₄⋊C₄	—	14	2	no	—
95	C₂²⋊C₄	—	14	4	no	—
96	C₄²	—	14	4	no	—
97	C₄²	D₂₈	1	1	yes	characteristic
98	C₄²	D₂₈	1	1	yes	characteristic
99	C₄²	C₂²×D₇	1	2	yes	—
100	C₂×C₇	—	4	1	no	—
101	C₂×C₇	#537	1	1	yes	characteristic
102	C₂×C₇	C₂×#137	1	2	yes	—
103	C₂³	C₂×D₂₈	1	1	yes	characteristic
104	Q₈	—	14	4	no	—
105	Q₈	—	14	4	no	—
106	Q₈	—	28	4	no	—
107	C₂×C₄	—	2	4	no	—
108	C₂×C₄	—	2	1	no	—
109	C₂×C₄	—	2	1	no	—
110	C₂×C₄	—	14	4	no	—
111	C₂×C₄	—	14	4	no	—
112	C₂×C₄	C₂×D₂₈	1	1	yes	characteristic
113	C₂×C₄	C₂×D₂₈	1	2	yes	—
114	C₂×C₄	C₂×D₂₈	1	2	yes	—
115	C₂×C₄	C₂³×D₇	1	1	yes	characteristic
116	C₈	—	2	4	no	—
117	C₇	#3255	1	1	yes	characteristic, complemented, Sylow
118	C₂²	—	2	2	no	—
119	C₂²	—	2	1	no	—
120	C₂²	C₂²×D₂₈	1	1	yes	characteristic
121	C₄	—	2	2	no	—
122	C₄	—	2	1	no	—
123	C₄	—	28	4	no	—
124	C₄	—	28	4	no	—
125	C₄	#6472	1	4	yes	—
126	C₄	#6548	1	1	yes	characteristic
127	C₄	#6551	1	1	yes	characteristic
128	C₂	—	4	1	no	—
129	C₂	#90201	1	1	yes	characteristic
130	C₂	C₂×#6472	1	2	yes	—

subgroup

G/H

class size

multiplicity

normal?

properties

#90161

C₂

yes

maximal

C₇×#423

C₂

yes

maximal, characteristic

#89909

C₂

yes

maximal

#89623

C₂

yes

maximal, complemented

#89621

C₂

yes

maximal, complemented

C₂×C₇×M₁₆

C₂²

yes

—

C₂×C₄²×C₇

C₂²

yes

—

—

C₂²

yes

—

C₂×Dic₂₈

C₂²

yes

—

#6433

—

#6432

C₂²

yes

—

#6432

C₂²

yes

—

C₄×Dic₁₄

—

C₄×Dic₁₄

C₂²

yes

—

C₇×C₄⋊C₈

C₂²

yes

—

#6391

C₂²

yes

—

#6390

C₂²

yes

—

#3255

—

maximal, Sylow

C₂²×C₄×C₇

D₄

yes

—

C₂²×C₄×C₇

C₂³

yes

characteristic

C₂×Dic₁₄

—

C₂×Dic₁₄

C₂³

yes

—

C₇×M₁₆

—

C₂×C₇×C₈

C₂³

yes

—

C₄²×C₇

D₄

yes

characteristic

C₄²×C₇

D₄

yes

characteristic

C₄²×C₇

C₂³

yes

—

C₂²⋊Dic₇

—

C₄⋊Dic₇

—

C₄⋊Dic₇

—

C₄⋊Dic₇

—

C₄⋊Dic₇

C₂³

yes

—

Dic₇⋊C₄

—

Dic₇⋊C₄

—

C₄×Dic₇

—

Dic₂₈

—

—

—

—

—

D₇

yes

characteristic

C₂³×C₇

C₂×D₄

yes

characteristic

C₂×C₄×C₇

—

C₂×C₄×C₇

—

C₂×C₄×C₇

—

C₂×C₄×C₇

C₂×D₄

yes

characteristic

C₂×C₄×C₇

C₂×D₄

yes

—

C₂×C₄×C₇

C₂×D₄

yes

—

C₂×C₄×C₇

C₂⁴

yes

characteristic

C₂×Dic₇

—

C₂×Dic₇

—

Dic₁₄

—

Dic₁₄

—

Dic₁₄

—

C₇×C₈

—

C₂×Q₁₆

—

C₂×M₁₆

C₂×D₇

yes

—

C₄⋊Q₈

—

C₄⋊Q₈

—

#125

—

C₂²⋊Q₈

—

C₂²⋊Q₈

—

C₄×Q₈

—

C₄×Q₈

—

#117

—

C₂×C₄²

C₂×D₇

yes

characteristic

#106

—

C₄⋊C₈

C₂×D₇

yes

—

#103

—

C₂²×C₇

—

C₂²×C₇

—

C₂²×C₇

C₂²×D₄

yes

characteristic

C₄×C₇

—

C₄×C₇

—

C₄×C₇

#137

yes

—

C₄×C₇

H(5,𝔽₂)

yes

characteristic

C₄×C₇

2₋¹⁺⁴

yes

characteristic

Dic₇

—

Dic₇

—

C₂×Q₈

—

C₂×Q₈

—

C₂²×C₄

D₂₈

yes

—

C₂²×C₄

C₂²×D₇

yes

characteristic

Q₁₆

—

M₁₆

—

C₂×C₈

C₂²×D₇

yes

—

C₄⋊C₄

—

C₄⋊C₄

—

C₄⋊C₄

—

C₄⋊C₄

—

C₄⋊C₄

—

C₄⋊C₄

—

C₂²⋊C₄

—

C₄²

—

C₄²

D₂₈

yes

characteristic

C₄²

D₂₈

yes

characteristic

C₄²

C₂²×D₇

yes

—

100

C₂×C₇

—

101

C₂×C₇

#537

yes

characteristic

102

C₂×C₇

C₂×#137

yes

—

103

C₂³

C₂×D₂₈

yes

characteristic

104

Q₈

—

105

Q₈

—

106

Q₈

—

107

C₂×C₄

—

108

C₂×C₄

—

109

C₂×C₄

—

110

C₂×C₄

—

111

C₂×C₄

—

112

C₂×C₄

C₂×D₂₈

yes

characteristic

113

C₂×C₄

C₂×D₂₈

yes

—

114

C₂×C₄

C₂×D₂₈

yes

—

115

C₂×C₄

C₂³×D₇

yes

characteristic

116

C₈

—

117

C₇

#3255

yes

characteristic, complemented, Sylow

118

C₂²

—

119

C₂²

—

120

C₂²

C₂²×D₂₈

yes

characteristic

121

C₄

—

122

C₄

—

123

C₄

—

124

C₄

—

125

C₄

#6472

yes

—

126

C₄

#6548

yes

characteristic

127

C₄

#6551

yes

characteristic

128

C₂

—

129

C₂

#90201

yes

characteristic

130

C₂

C₂×#6472

yes

—

	subgroup	G/H
center	C₂²	C₂²×D₂₈
commutator, cosocle	C₂×C₄×C₇	C₂⁴
Frattini	C₂×C₄	C₂³×D₇
Fitting	C₇×#423	C₂
socle	C₂²×C₇	C₂²×D₄
prime radical	C₂³×C₇	C₂×D₄
hypercenter	#423	D₇
hypocenter	C₇	#3255
LCS	G ▷ C₂×C₄×C₇ ▷ C₂×C₇ ▷ C₇
UCS	C₁ ◁ C₂² ◁ C₂×C₄² ◁ #423

subgroup

G/H

center

C₂²

C₂²×D₂₈

commutator, cosocle

C₂×C₄×C₇

C₂⁴

Frattini

C₂×C₄

C₂³×D₇

Fitting

C₇×#423

C₂

socle

C₂²×C₇

C₂²×D₄

prime radical

C₂³×C₇

C₂×D₄

D₇

C₇

G ▷ C₂×C₄×C₇ ▷ C₂×C₇ ▷ C₇

UCS

C₁ ◁ C₂² ◁ C₂×C₄² ◁ #423

N	H ≅ G/N	type
4. #89623	C₂	semidirect
5. #89621	C₂	semidirect
117. C₇	#3255	semidirect

H ≅ G/N

type

4. #89623

C₂

semidirect

5. #89621

C₂

semidirect

117. C₇

#3255

semidirect

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

subgroup	index
#90161	2
C₇×#423	2
#89909	2
#89623	2
#89621	2
#3255	7

subgroup

index

Subgroups — #11736514

Conjugacy classes of subgroups

Complements

Maximal subgroups