Group #11736616 — subgroups

Subgroups — #11736616

Order 896.

348 conjugacy classes of subgroups; only the 149 normal subgroups are shown.

normal subgroups only

This group has 348 conjugacy classes of subgroups — too many to draw. Only its 149 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	#90208	C₂	1	2	yes	maximal
2	C₇×#427	C₂	1	1	yes	maximal, characteristic
3	#89878	C₂	1	1	yes	maximal, characteristic
4	C₂×#6389	C₂	1	1	yes	maximal, characteristic
5	#89770	C₂	1	2	yes	maximal
6	#89648	C₂	1	4	yes	maximal, complemented
7	#89628	C₂	1	4	yes	maximal, complemented
8	C₂²×C₇×C₈	C₂²	1	1	yes	characteristic
9	C₇×#117	C₂²	1	2	yes	—
10	#6489	C₂²	1	2	yes	—
11	C₂×C₄×Dic₇	C₂²	1	1	yes	characteristic
12	C₂²×C₇⋊C₈	C₂²	1	1	yes	characteristic
13	#6453	C₂²	1	4	yes	—
14	#6451	C₂²	1	4	yes	—
15	#6441	—	2	4	no	—
16	C₄×Dic₁₄	—	2	4	no	—
17	C₇×#107	C₂²	1	2	yes	—
18	C₇×#106	C₂²	1	2	yes	—
19	#6392	C₂²	1	2	yes	—
20	#6391	C₂²	1	2	yes	—
21	#6389	C₂²	1	2	yes	—
22	#6389	C₂²	1	2	yes	—
23	#6382	C₂²	1	4	yes	—
24	#6381	C₂²	1	4	yes	—
25	#3162	—	7	1	no	maximal, Sylow
26	C₂²×C₄×C₇	C₂³	1	1	yes	characteristic
27	C₂²×Dic₇	D₄	1	2	yes	—
28	C₂×Dic₁₄	—	2	4	no	—
29	C₂×C₇×C₈	Q₈	1	4	yes	complemented
30	C₂×C₇×C₈	C₂³	1	1	yes	characteristic
31	C₂×C₇×C₈	C₂³	1	1	yes	characteristic
32	C₇×C₄⋊C₄	C₂³	1	4	yes	—
33	C₇×C₂²⋊C₄	—	2	2	no	—
34	C₄²×C₇	—	2	2	no	—
35	C₂²⋊Dic₇	—	2	2	no	—
36	C₄⋊Dic₇	—	2	4	no	—
37	C₄⋊Dic₇	C₂³	1	4	yes	—
38	Dic₇⋊C₄	—	2	4	no	—
39	Dic₇⋊C₄	—	2	4	no	—
40	C₄×Dic₇	—	2	2	no	—
41	C₄×Dic₇	D₄	1	2	yes	—
42	C₄×Dic₇	C₂³	1	2	yes	—
43	C₂×C₇⋊C₈	Q₈	1	4	yes	complemented
44	C₂×C₇⋊C₈	C₂³	1	2	yes	—
45	#533	—	7	2	no	—
46	#501	—	7	4	no	—
47	#501	—	7	4	no	—
48	#427	—	7	1	no	—
49	#427	—	7	2	no	—
50	#427	D₇	1	1	yes	characteristic
51	C₂×#97	—	7	1	no	—
52	C₂³×C₇	C₂×D₄	1	1	yes	characteristic
53	C₂×C₄×C₇	—	2	4	no	—
54	C₂×C₄×C₇	C₂×D₄	1	1	yes	characteristic
55	C₂×C₄×C₇	C₂×Q₈	1	2	yes	—
56	C₂×C₄×C₇	C₂×Q₈	1	2	yes	—
57	C₂×C₄×C₇	C₂⁴	1	1	yes	characteristic
58	C₂×Dic₇	—	2	4	no	—
59	C₂×Dic₇	—	4	2	no	—
60	C₂×Dic₇	C₂×D₄	1	4	yes	—
61	Dic₁₄	—	4	8	no	—
62	C₇×C₈	C₂×Q₈	1	2	yes	—
63	C₇×C₈	C₂×Q₈	1	2	yes	—
64	C₇⋊C₈	C₂×Q₈	1	4	yes	—
65	C₂²×C₈	—	7	1	no	—
66	C₂²×C₈	C₂×D₇	1	1	yes	characteristic
67	C₄⋊Q₈	—	7	4	no	—
68	#125	—	7	4	no	—
69	C₂²⋊Q₈	—	14	4	no	—
70	C₄×Q₈	—	14	4	no	—
71	#117	—	7	2	no	—
72	#117	C₂×D₇	1	2	yes	—
73	C₂×C₄²	—	7	1	no	—
74	#107	—	7	2	no	—
75	#107	—	7	4	no	—
76	#107	C₂×D₇	1	2	yes	—
77	#106	—	7	2	no	—
78	#106	—	7	4	no	—
79	#106	C₂×D₇	1	2	yes	—
80	#97	—	7	2	no	—
81	#97	—	7	2	no	—
82	C₂²×C₇	—	2	2	no	—
83	C₂²×C₇	C₄⋊Q₈	1	2	yes	—
84	C₂²×C₇	C₂²×D₄	1	1	yes	characteristic
85	C₄×C₇	—	4	4	no	—
86	C₄×C₇	C₄⋊Q₈	1	2	yes	—
87	C₄×C₇	C₂²×Q₈	1	1	yes	characteristic
88	C₄×C₇	C₂²×Q₈	1	1	yes	characteristic
89	Dic₇	—	2	4	no	—
90	Dic₇	—	4	4	no	—
91	C₂×Q₈	—	14	4	no	—
92	C₂²×C₄	—	7	2	no	—
93	C₂²×C₄	C₂²×D₇	1	1	yes	characteristic
94	C₂×C₈	—	7	4	no	—
95	C₂×C₈	—	7	2	no	—
96	C₂×C₈	Dic₁₄	1	4	yes	complemented
97	C₂×C₈	C₂²×D₇	1	1	yes	characteristic
98	C₂×C₈	C₂²×D₇	1	1	yes	characteristic
99	C₄⋊C₄	—	7	4	no	—
100	C₄⋊C₄	—	14	4	no	—
101	C₄⋊C₄	—	14	4	no	—
102	C₄⋊C₄	—	14	4	no	—
103	C₄⋊C₄	C₂²×D₇	1	4	yes	—
104	C₂²⋊C₄	—	2	2	no	—
105	C₂²⋊C₄	—	14	2	no	—
106	C₄²	—	2	2	no	—
107	C₄²	—	7	2	no	—
108	C₄²	—	7	2	no	—
109	C₄²	—	14	2	no	—
110	C₂×C₇	—	2	2	no	—
111	C₂×C₇	C₂×C₄⋊Q₈	1	1	yes	characteristic
112	C₂×C₇	#575	1	2	yes	—
113	C₂³	D₄×D₇	1	1	yes	characteristic
114	Q₈	—	28	8	no	—
115	C₂×C₄	—	2	4	no	—
116	C₂×C₄	—	7	4	no	—
117	C₂×C₄	—	14	4	no	—
118	C₂×C₄	—	28	2	no	—
119	C₂×C₄	C₂×Dic₁₄	1	2	yes	—
120	C₂×C₄	D₄×D₇	1	1	yes	characteristic
121	C₂×C₄	Q₈×D₇	1	2	yes	—
122	C₂×C₄	C₂³×D₇	1	1	yes	characteristic
123	C₈	—	7	4	no	—
124	C₈	C₂×Dic₁₄	1	2	yes	—
125	C₈	C₂×Dic₁₄	1	2	yes	—
126	C₇	#3162	1	1	yes	characteristic, complemented, Sylow
127	C₂²	—	2	2	no	—
128	C₂²	#6451	1	2	yes	—
129	C₂²	C₂×D₄×D₇	1	1	yes	characteristic
130	C₄	—	4	4	no	—
131	C₄	—	14	4	no	—
132	C₄	—	28	4	no	—
133	C₄	#6451	1	2	yes	—
134	C₄	C₂²×Dic₁₄	1	1	yes	characteristic
135	C₄	C₂×Q₈×D₇	1	1	yes	characteristic
136	C₂	—	2	2	no	—
137	C₂	C₂×#6451	1	1	yes	characteristic
138	C₂	#90459	1	2	yes	—

subgroup

G/H

class size

multiplicity

normal?

properties

#90208

C₂

yes

maximal

C₇×#427

C₂

yes

maximal, characteristic

#89878

C₂

yes

maximal, characteristic

C₂×#6389

C₂

yes

maximal, characteristic

#89770

C₂

yes

maximal

#89648

C₂

yes

maximal, complemented

#89628

C₂

yes

maximal, complemented

C₂²×C₇×C₈

C₂²

yes

characteristic

C₇×#117

C₂²

yes

—

#6489

C₂²

yes

—

C₂×C₄×Dic₇

C₂²

yes

characteristic

C₂²×C₇⋊C₈

C₂²

yes

characteristic

#6453

C₂²

yes

—

#6451

C₂²

yes

—

#6441

—

C₄×Dic₁₄

—

C₇×#107

C₂²

yes

—

C₇×#106

C₂²

yes

—

#6392

C₂²

yes

—

#6391

C₂²

yes

—

#6389

C₂²

yes

—

#6389

C₂²

yes

—

#6382

C₂²

yes

—

#6381

C₂²

yes

—

#3162

—

maximal, Sylow

C₂²×C₄×C₇

C₂³

yes

characteristic

C₂²×Dic₇

D₄

yes

—

C₂×Dic₁₄

—

C₂×C₇×C₈

Q₈

yes

complemented

C₂×C₇×C₈

C₂³

yes

characteristic

C₂×C₇×C₈

C₂³

yes

characteristic

C₇×C₄⋊C₄

C₂³

yes

—

C₇×C₂²⋊C₄

—

C₄²×C₇

—

C₂²⋊Dic₇

—

C₄⋊Dic₇

—

C₄⋊Dic₇

C₂³

yes

—

Dic₇⋊C₄

—

Dic₇⋊C₄

—

C₄×Dic₇

—

C₄×Dic₇

D₄

yes

—

C₄×Dic₇

C₂³

yes

—

C₂×C₇⋊C₈

Q₈

yes

complemented

C₂×C₇⋊C₈

C₂³

yes

—

—

—

—

—

—

D₇

yes

characteristic

C₂×#97

—

C₂³×C₇

C₂×D₄

yes

characteristic

C₂×C₄×C₇

—

C₂×C₄×C₇

C₂×D₄

yes

characteristic

C₂×C₄×C₇

C₂×Q₈

yes

—

C₂×C₄×C₇

C₂×Q₈

yes

—

C₂×C₄×C₇

C₂⁴

yes

characteristic

C₂×Dic₇

—

C₂×Dic₇

—

C₂×Dic₇

C₂×D₄

yes

—

Dic₁₄

—

C₇×C₈

C₂×Q₈

yes

—

C₇×C₈

C₂×Q₈

yes

—

C₇⋊C₈

C₂×Q₈

yes

—

C₂²×C₈

—

C₂²×C₈

C₂×D₇

yes

characteristic

C₄⋊Q₈

—

#125

—

C₂²⋊Q₈

—

C₄×Q₈

—

#117

—

#117

C₂×D₇

yes

—

C₂×C₄²

—

#107

—

#107

—

#107

C₂×D₇

yes

—

#106

—

#106

—

#106

C₂×D₇

yes

—

#97

—

#97

—

C₂²×C₇

—

C₂²×C₇

C₄⋊Q₈

yes

—

C₂²×C₇

C₂²×D₄

yes

characteristic

C₄×C₇

—

C₄×C₇

C₄⋊Q₈

yes

—

C₄×C₇

C₂²×Q₈

yes

characteristic

C₄×C₇

C₂²×Q₈

yes

characteristic

Dic₇

—

Dic₇

—

C₂×Q₈

—

C₂²×C₄

—

C₂²×C₄

C₂²×D₇

yes

characteristic

C₂×C₈

—

C₂×C₈

—

C₂×C₈

Dic₁₄

yes

complemented

C₂×C₈

C₂²×D₇

yes

characteristic

C₂×C₈

C₂²×D₇

yes

characteristic

C₄⋊C₄

—

100

C₄⋊C₄

—

101

C₄⋊C₄

—

102

C₄⋊C₄

—

103

C₄⋊C₄

C₂²×D₇

yes

—

104

C₂²⋊C₄

—

105

C₂²⋊C₄

—

106

C₄²

—

107

C₄²

—

108

C₄²

—

109

C₄²

—

110

C₂×C₇

—

111

C₂×C₇

C₂×C₄⋊Q₈

yes

characteristic

112

C₂×C₇

#575

yes

—

113

C₂³

D₄×D₇

yes

characteristic

114

Q₈

—

115

C₂×C₄

—

116

C₂×C₄

—

117

C₂×C₄

—

118

C₂×C₄

—

119

C₂×C₄

C₂×Dic₁₄

yes

—

120

C₂×C₄

D₄×D₇

yes

characteristic

121

C₂×C₄

Q₈×D₇

yes

—

122

C₂×C₄

C₂³×D₇

yes

characteristic

123

C₈

—

124

C₈

C₂×Dic₁₄

yes

—

125

C₈

C₂×Dic₁₄

yes

—

126

C₇

#3162

yes

characteristic, complemented, Sylow

127

C₂²

—

128

C₂²

#6451

yes

—

129

C₂²

C₂×D₄×D₇

yes

characteristic

130

C₄

—

131

C₄

—

132

C₄

—

133

C₄

#6451

yes

—

134

C₄

C₂²×Dic₁₄

yes

characteristic

135

C₄

C₂×Q₈×D₇

yes

characteristic

136

C₂

—

137

C₂

C₂×#6451

yes

characteristic

138

C₂

#90459

yes

—

	subgroup	G/H
center	C₂×C₄	D₄×D₇
commutator, cosocle	C₂×C₄×C₇	C₂⁴
Frattini	C₂×C₄	C₂³×D₇
Fitting	C₇×#427	C₂
socle	C₂²×C₇	C₂²×D₄
prime radical	C₂³×C₇	C₂×D₄
hypercenter	#427	D₇
hypocenter	C₇	#3162
LCS	G ▷ C₂×C₄×C₇ ▷ C₂×C₇ ▷ C₇
UCS	C₁ ◁ C₂×C₄ ◁ C₂²×C₄ ◁ #427

subgroup

G/H

center

C₂×C₄

D₄×D₇

commutator, cosocle

C₂×C₄×C₇

C₂⁴

Frattini

C₂×C₄

C₂³×D₇

Fitting

C₇×#427

C₂

socle

C₂²×C₇

C₂²×D₄

prime radical

C₂³×C₇

C₂×D₄

D₇

C₇

G ▷ C₂×C₄×C₇ ▷ C₂×C₇ ▷ C₇

UCS

C₁ ◁ C₂×C₄ ◁ C₂²×C₄ ◁ #427

N	H ≅ G/N	type
6. #89648	C₂	semidirect
7. #89628	C₂	semidirect
29. C₂×C₇×C₈	Q₈	semidirect
43. C₂×C₇⋊C₈	Q₈	semidirect
96. C₂×C₈	Dic₁₄	semidirect
126. C₇	#3162	semidirect

H ≅ G/N

type

6. #89648

C₂

semidirect

7. #89628

C₂

semidirect

29. C₂×C₇×C₈

Q₈

semidirect

43. C₂×C₇⋊C₈

Q₈

semidirect

96. C₂×C₈

Dic₁₄

semidirect

126. C₇

#3162

semidirect

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

subgroup	index
#90208	2
C₇×#427	2
#89878	2
C₂×#6389	2
#89770	2
#89648	2
#89628	2
#3162	7

subgroup

index

Subgroups — #11736616

Conjugacy classes of subgroups

Complements

Maximal subgroups