Group #11726445 — subgroups

Subgroups — #11726445

Order 896.

254 conjugacy classes of subgroups; only the 95 normal subgroups are shown.

normal subgroups only

This group has 254 conjugacy classes of subgroups — too many to draw. Only its 95 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×#6392	C₂	1	2	yes	maximal
2	C₂×#6390	C₂	1	2	yes	maximal
3	#89707	C₂	1	1	yes	maximal, characteristic
4	#89701	C₂	1	1	yes	maximal, characteristic
5	C₇×#336	C₂	1	1	yes	maximal, characteristic
6	C₂²×Dic₁₄	C₂²	1	1	yes	characteristic
7	C₂²×C₇×C₈	C₂²	1	2	yes	—
8	C₂×C₄²×C₇	C₂²	1	1	yes	characteristic
9	C₂×C₄⋊Dic₇	—	2	2	no	—
10	C₂×C₄⋊Dic₇	C₂²	1	2	yes	—
11	C₂×C₄⋊Dic₇	C₂²	1	1	yes	characteristic
12	C₇⋊#95	—	2	2	no	—
13	#6392	—	2	4	no	—
14	#6390	—	2	4	no	—
15	#2095	—	7	1	no	maximal, Sylow
16	C₂²×C₄×C₇	D₄	1	2	yes	—
17	C₂²×C₄×C₇	C₂³	1	1	yes	characteristic
18	C₂²×Dic₇	—	2	1	no	—
19	C₂²×Dic₇	—	2	1	no	—
20	C₂²×Dic₇	—	2	2	no	—
21	C₂×Dic₁₄	—	2	2	no	—
22	C₂×Dic₁₄	—	2	2	no	—
23	C₂×Dic₁₄	—	2	1	no	—
24	C₂×Dic₁₄	Q₈	1	2	yes	—
25	C₂×C₇×C₈	—	2	2	no	—
26	C₂×C₇×C₈	—	2	2	no	—
27	C₂×C₇×C₈	D₄	1	4	yes	—
28	C₄²×C₇	—	2	2	no	—
29	C₄⋊Dic₇	—	2	2	no	—
30	C₄⋊Dic₇	—	2	2	no	—
31	C₄⋊Dic₇	—	2	2	no	—
32	C₄⋊Dic₇	—	4	4	no	—
33	C₂×#107	—	7	2	no	—
34	C₂×#103	—	7	2	no	—
35	#391	—	7	1	no	—
36	#384	—	7	1	no	—
37	#336	D₇	1	1	yes	characteristic
38	C₂³×C₇	C₂×D₄	1	1	yes	characteristic
39	C₂×C₄×C₇	—	2	2	no	—
40	C₂×C₄×C₇	—	2	2	no	—
41	C₂×C₄×C₇	Q₁₆	1	4	yes	—
42	C₂×C₄×C₇	C₂×D₄	1	2	yes	—
43	C₂×C₄×C₇	C₂×Q₈	1	1	yes	characteristic
44	C₂×C₄×C₇	#41	1	2	yes	—
45	C₂×C₄×C₇	#41	1	1	yes	characteristic
46	C₂×Dic₇	—	2	2	no	—
47	C₂×Dic₇	—	4	1	no	—
48	C₂×Dic₇	—	4	1	no	—
49	C₂×Dic₇	—	4	2	no	—
50	C₂×Dic₇	—	4	1	no	—
51	C₂×Dic₇	—	4	4	no	—
52	C₂×Dic₇	—	4	2	no	—
53	Dic₁₄	—	2	4	no	—
54	Dic₁₄	—	4	2	no	—
55	C₇×C₈	—	2	4	no	—
56	C₂²×Q₈	—	7	1	no	—
57	C₂²×C₈	C₂×D₇	1	2	yes	—
58	C₂×C₄⋊C₄	—	7	2	no	—
59	C₂×C₄⋊C₄	—	7	1	no	—
60	C₂×C₄⋊C₄	—	14	2	no	—
61	C₂×C₄²	C₂×D₇	1	1	yes	characteristic
62	#107	—	14	4	no	—
63	#103	—	14	4	no	—
64	#95	—	14	2	no	—
65	C₂²×C₇	C₂²⋊Q₈	1	1	yes	characteristic
66	C₂²×C₇	#123	1	1	yes	characteristic
67	C₂²×C₇	#127	1	1	yes	characteristic
68	C₂²×C₇	C₂×Q₁₆	1	2	yes	—
69	C₂²×C₇	AGL(1,ℤ/8)	1	2	yes	—
70	C₄×C₇	—	2	4	no	—
71	C₄×C₇	C₂²⋊Q₈	1	2	yes	—
72	C₄×C₇	#123	1	2	yes	—
73	Dic₇	—	4	2	no	—
74	Dic₇	—	8	2	no	—
75	Dic₇	—	8	1	no	—
76	C₂×Q₈	—	7	2	no	—
77	C₂×Q₈	—	14	1	no	—
78	C₂×Q₈	—	14	2	no	—
79	C₂×Q₈	—	14	2	no	—
80	C₂²×C₄	—	14	1	no	—
81	C₂²×C₄	—	14	1	no	—
82	C₂²×C₄	—	14	2	no	—
83	C₂²×C₄	D₂₈	1	2	yes	—
84	C₂²×C₄	C₂²×D₇	1	1	yes	characteristic
85	C₂×C₈	—	2	2	no	—
86	C₂×C₈	—	2	2	no	—
87	C₂×C₈	D₂₈	1	4	yes	—
88	C₄⋊C₄	—	14	2	no	—
89	C₄⋊C₄	—	14	2	no	—
90	C₄⋊C₄	—	14	2	no	—
91	C₄⋊C₄	—	28	4	no	—
92	C₄²	—	2	2	no	—
93	C₂×C₇	#399	1	1	yes	characteristic
94	C₂×C₇	#477	1	2	yes	—
95	C₂×C₇	#484	1	2	yes	—
96	C₂×C₇	#494	1	1	yes	characteristic
97	C₂×C₇	#496	1	1	yes	characteristic
98	C₂³	C₂×D₂₈	1	1	yes	characteristic
99	Q₈	—	14	4	no	—
100	Q₈	—	28	2	no	—
101	C₂×C₄	—	2	2	no	—
102	C₂×C₄	—	2	2	no	—
103	C₂×C₄	—	14	2	no	—
104	C₂×C₄	—	28	1	no	—
105	C₂×C₄	—	28	1	no	—
106	C₂×C₄	—	28	2	no	—
107	C₂×C₄	—	28	1	no	—
108	C₂×C₄	—	28	4	no	—
109	C₂×C₄	—	28	2	no	—
110	C₂×C₄	Dic₂₈	1	4	yes	—
111	C₂×C₄	C₂×D₂₈	1	2	yes	—
112	C₂×C₄	#1159	1	1	yes	characteristic
113	C₂×C₄	#1159	1	2	yes	—
114	C₂×C₄	Q₈×D₇	1	1	yes	characteristic
115	C₈	—	2	4	no	—
116	C₇	#2095	1	1	yes	characteristic, complemented, Sylow
117	C₂²	#6437	1	1	yes	characteristic
118	C₂²	#6449	1	1	yes	characteristic
119	C₂²	#6460	1	1	yes	characteristic
120	C₂²	C₂×Dic₂₈	1	2	yes	—
121	C₂²	#6471	1	2	yes	—
122	C₄	—	2	4	no	—
123	C₄	—	28	2	no	—
124	C₄	—	56	2	no	—
125	C₄	—	56	1	no	—
126	C₄	#6449	1	2	yes	—
127	C₄	#6460	1	2	yes	—
128	C₂	#89449	1	1	yes	characteristic
129	C₂	#89471	1	1	yes	characteristic
130	C₂	#89484	1	1	yes	characteristic
131	C₂	#89494	1	2	yes	—
132	C₂	#89622	1	2	yes	—

subgroup

G/H

class size

multiplicity

normal?

properties

C₂×#6392

C₂

yes

maximal

C₂×#6390

C₂

yes

maximal

#89707

C₂

yes

maximal, characteristic

#89701

C₂

yes

maximal, characteristic

C₇×#336

C₂

yes

maximal, characteristic

C₂²×Dic₁₄

C₂²

yes

characteristic

C₂²×C₇×C₈

C₂²

yes

—

C₂×C₄²×C₇

C₂²

yes

characteristic

C₂×C₄⋊Dic₇

—

C₂×C₄⋊Dic₇

C₂²

yes

—

C₂×C₄⋊Dic₇

C₂²

yes

—

—

—

—

C₂²×C₄×C₇

D₄

yes

—

C₂²×C₄×C₇

C₂³

yes

characteristic

C₂²×Dic₇

—

C₂²×Dic₇

—

C₂²×Dic₇

—

C₂×Dic₁₄

—

C₂×Dic₁₄

—

C₂×Dic₁₄

—

C₂×Dic₁₄

Q₈

yes

—

C₂×C₇×C₈

—

C₂×C₇×C₈

—

C₂×C₇×C₈

D₄

yes

—

C₄²×C₇

—

C₄⋊Dic₇

—

C₄⋊Dic₇

—

C₄⋊Dic₇

—

C₄⋊Dic₇

—

—

—

—

—

D₇

yes

characteristic

C₂³×C₇

C₂×D₄

yes

characteristic

C₂×C₄×C₇

—

C₂×C₄×C₇

—

C₂×C₄×C₇

Q₁₆

yes

—

C₂×C₄×C₇

C₂×D₄

yes

—

C₂×C₄×C₇

C₂×Q₈

yes

characteristic

C₂×C₄×C₇

#41

yes

—

C₂×C₄×C₇

#41

yes

characteristic

C₂×Dic₇

—

C₂×Dic₇

—

C₂×Dic₇

—

C₂×Dic₇

—

C₂×Dic₇

—

C₂×Dic₇

—

C₂×Dic₇

—

Dic₁₄

—

Dic₁₄

—

C₇×C₈

—

C₂²×Q₈

—

C₂²×C₈

C₂×D₇

yes

—

C₂×C₄⋊C₄

—

C₂×C₄⋊C₄

—

C₂×C₄⋊C₄

—

C₂×C₄²

C₂×D₇

yes

—

—

—

C₂²×C₇

C₂²⋊Q₈

yes

C₂²×C₇

yes

C₂²×C₇

yes

C₂²×C₇

C₂×Q₁₆

yes

—

C₂²×C₇

AGL(1,ℤ/8)

yes

—

C₄×C₇

—

C₄×C₇

C₂²⋊Q₈

yes

—

C₄×C₇

#123

yes

—

Dic₇

—

Dic₇

—

Dic₇

—

C₂×Q₈

—

C₂×Q₈

—

C₂×Q₈

—

C₂×Q₈

—

C₂²×C₄

—

C₂²×C₄

—

C₂²×C₄

—

C₂²×C₄

D₂₈

yes

—

C₂²×C₄

C₂²×D₇

yes

characteristic

C₂×C₈

—

C₂×C₈

—

C₂×C₈

D₂₈

yes

—

C₄⋊C₄

—

C₄⋊C₄

—

C₄⋊C₄

—

C₄⋊C₄

—

C₄²

—

C₂×C₇

#399

yes

characteristic

C₂×C₇

#477

yes

—

C₂×C₇

#484

yes

—

C₂×C₇

yes

C₂×C₇

yes

C₂³

C₂×D₂₈

yes

characteristic

Q₈

—

100

Q₈

—

101

C₂×C₄

—

102

C₂×C₄

—

103

C₂×C₄

—

104

C₂×C₄

—

105

C₂×C₄

—

106

C₂×C₄

—

107

C₂×C₄

—

108

C₂×C₄

—

109

C₂×C₄

—

110

C₂×C₄

Dic₂₈

yes

—

111

C₂×C₄

C₂×D₂₈

yes

—

112

C₂×C₄

#1159

yes

characteristic

113

C₂×C₄

#1159

yes

—

114

C₂×C₄

Q₈×D₇

yes

characteristic

115

C₈

—

116

C₇

#2095

yes

characteristic, complemented, Sylow

117

C₂²

#6437

yes

characteristic

118

C₂²

#6449

yes

characteristic

119

C₂²

#6460

yes

characteristic

120

C₂²

C₂×Dic₂₈

yes

—

121

C₂²

#6471

yes

—

122

C₄

—

123

C₄

—

124

C₄

—

125

C₄

—

126

C₄

#6449

yes

—

127

C₄

#6460

yes

—

128

C₂

#89449

yes

characteristic

129

C₂

#89471

yes

characteristic

130

C₂

#89484

yes

characteristic

131

C₂

#89494

yes

—

132

C₂

#89622

yes

—

	subgroup	G/H
center	C₂³	C₂×D₂₈
commutator, cosocle	C₂²×C₄×C₇	C₂³
Frattini	C₂²×C₄	C₂²×D₇
Fitting	C₇×#336	C₂
socle, prime radical	C₂³×C₇	C₂×D₄
hypercenter	#336	D₇
hypocenter	C₇	#2095
LCS	G ▷ C₂²×C₄×C₇ ▷ C₂×C₇ ▷ C₇
UCS	C₁ ◁ C₂³ ◁ C₂×C₄² ◁ #336

subgroup

G/H

center

C₂³

C₂×D₂₈

commutator, cosocle

C₂²×C₄×C₇

C₂³

Frattini

C₂²×C₄

C₂²×D₇

Fitting

C₇×#336

C₂

socle, prime radical

C₂³×C₇

C₂×D₄

D₇

C₇

G ▷ C₂²×C₄×C₇ ▷ C₂×C₇ ▷ C₇

UCS

C₁ ◁ C₂³ ◁ C₂×C₄² ◁ #336

N	H ≅ G/N	type
116. C₇	#2095	semidirect

H ≅ G/N

type

116. C₇

#2095

semidirect

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

subgroup	index
C₂×#6392	2
C₂×#6390	2
#89707	2
#89701	2
C₇×#336	2
#2095	7

subgroup

index

Subgroups — #11726445

Conjugacy classes of subgroups

Complements

Maximal subgroups