Group #46822324 — subgroups

Subgroups — #46822324

Order 34432.

370 conjugacy classes of subgroups; only the 117 normal subgroups are shown.

normal subgroups only

This group has 370 conjugacy classes of subgroups — too many to draw. Only its 117 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	#31771210	C₂	1	2	yes	maximal, complemented
2	#31770906	C₂	1	4	yes	maximal, complemented
3	#31770858	C₂	1	2	yes	maximal, complemented
4	#31770841	C₂	1	2	yes	maximal, complemented
5	#31770801	C₂	1	4	yes	maximal, complemented
6	#424×C₂₆₉	C₂	1	1	yes	maximal, characteristic, complemented
7	C₂×D₂₁₅₂	C₂²	1	2	yes	—
8	C₂×D₂₁₅₂	C₂²	1	2	yes	—
9	#22575450	C₂²	1	2	yes	—
10	#22575445	C₂²	1	2	yes	—
11	#22575438	—	2	4	no	—
12	#22575427	C₂²	1	4	yes	—
13	#22575426	—	2	4	no	—
14	C₄×D₁₀₇₆	C₂²	1	4	yes	—
15	#22575387	C₂²	1	2	yes	—
16	#22575385	C₂²	1	2	yes	—
17	#22575375	C₂²	1	4	yes	—
18	#22575375	C₂²	1	4	yes	—
19	C₂×M₁₆×C₂₆₉	C₂²	1	1	yes	characteristic
20	C₂²×C₈×C₂₆₉	C₂²	1	1	yes	characteristic
21	#117×C₂₆₉	C₂²	1	1	yes	characteristic
22	C₄⋊C₈×C₂₆₉	C₂²	1	4	yes	—
23	C₂×D₁₀₇₆	—	2	4	no	—
24	C₂×D₁₀₇₆	C₂³	1	4	yes	—
25	C₂×C₂²⋊D₂₆₉	—	2	4	no	—
26	C₂×C₄×D₂₆₉	—	2	4	no	—
27	D₂₁₅₂	—	4	2	no	—
28	D₂₁₅₂	—	4	2	no	—
29	C₄⋊Dic₂₆₉	C₂³	1	4	yes	—
30	#16864381	—	2	4	no	—
31	#16864379	—	2	4	no	—
32	#16864379	—	2	4	no	—
33	C₂²×C₄×C₂₆₉	C₂³	1	1	yes	characteristic
34	M₁₆×C₂₆₉	—	4	1	no	—
35	C₂×C₈×C₂₆₉	—	4	1	no	—
36	C₂×C₈×C₂₆₉	C₂³	1	2	yes	—
37	C₂×C₈×C₂₆₉	C₂³	1	2	yes	—
38	C₄⋊C₄×C₂₆₉	D₄	1	2	yes	—
39	C₂²⋊C₄×C₂₆₉	D₄	1	2	yes	—
40	C₄²×C₂₆₉	C₂³	1	2	yes	—
41	C₂²×D₂₆₉	—	2	4	no	—
42	D₁₀₇₆	—	2	4	no	—
43	D₁₀₇₆	—	2	4	no	—
44	D₁₀₇₆	—	8	4	no	—
45	C₂²⋊D₂₆₉	—	8	4	no	—
46	C₂×Dic₂₆₉	—	2	4	no	—
47	C₄×D₂₆₉	—	8	4	no	—
48	C₂³×C₂₆₉	C₂×D₄	1	1	yes	characteristic
49	C₂×C₄×C₂₆₉	—	2	1	no	—
50	C₂×C₄×C₂₆₉	—	2	1	no	—
51	C₂×C₄×C₂₆₉	C₂×D₄	1	4	yes	—
52	C₂×C₄×C₂₆₉	C₂×D₄	1	1	yes	characteristic
53	C₂×C₄×C₂₆₉	C₂⁴	1	1	yes	characteristic
54	C₈×C₂₆₉	—	2	2	no	—
55	C₈×C₂₆₉	—	2	2	no	—
56	C₂²×C₂₆₉	—	2	1	no	—
57	C₂²×C₂₆₉	—	2	2	no	—
58	C₂²×C₂₆₉	C₂²×D₄	1	1	yes	characteristic
59	C₂×D₂₆₉	—	4	4	no	—
60	C₂×D₂₆₉	—	4	4	no	—
61	C₂×D₂₆₉	—	4	4	no	—
62	C₄×C₂₆₉	—	2	1	no	—
63	C₄×C₂₆₉	—	2	4	no	—
64	C₄×C₂₆₉	H(5,𝔽₂)	1	1	yes	characteristic
65	C₄×C₂₆₉	2₋¹⁺⁴	1	1	yes	characteristic
66	Dic₂₆₉	—	4	4	no	—
67	C₂×C₂₆₉	—	4	1	no	—
68	C₂×C₂₆₉	#537	1	1	yes	characteristic
69	C₂×C₂₆₉	#576	1	2	yes	—
70	D₂₆₉	—	8	4	no	—
71	C₂₆₉	#3259	1	1	yes	characteristic, complemented, Sylow
72	#3259	—	269	1	no	maximal, Sylow
73	#540	—	269	2	no	—
74	#478	—	269	4	no	—
75	#469	—	269	2	no	—
76	#466	—	269	2	no	—
77	#459	—	269	4	no	—
78	#424	D₂₆₉	1	1	yes	characteristic, complemented
79	C₂×D₈	—	269	2	no	—
80	C₂×D₈	—	269	2	no	—
81	C₂×M₁₆	C₂×D₂₆₉	1	1	yes	characteristic
82	C₂²×C₈	C₂×D₂₆₉	1	1	yes	characteristic
83	#127	—	269	2	no	—
84	#125	—	269	2	no	—
85	#123	—	538	4	no	—
86	#121	—	269	4	no	—
87	#121	—	538	4	no	—
88	C₄×D₄	—	269	4	no	—
89	#117	C₂×D₂₆₉	1	1	yes	characteristic
90	#107	—	269	2	no	—
91	#106	—	269	2	no	—
92	C₄⋊C₈	C₂×D₂₆₉	1	4	yes	—
93	#102	—	269	4	no	—
94	#102	—	269	4	no	—
95	C₂×D₄	—	269	4	no	—
96	C₂×D₄	—	538	4	no	—
97	C₂×D₄	—	538	4	no	—
98	C₂²×C₄	—	538	4	no	—
99	C₂²×C₄	C₂²×D₂₆₉	1	1	yes	characteristic
100	D₈	—	1076	2	no	—
101	D₈	—	1076	2	no	—
102	M₁₆	—	4	1	no	—
103	C₂×C₈	—	4	1	no	—
104	C₂×C₈	C₂²×D₂₆₉	1	2	yes	—
105	C₂×C₈	C₂²×D₂₆₉	1	2	yes	—
106	C₄⋊C₄	—	269	4	no	—
107	C₄⋊C₄	—	538	4	no	—
108	C₄⋊C₄	D₁₀₇₆	1	2	yes	—
109	C₂²⋊C₄	—	538	4	no	—
110	C₂²⋊C₄	—	538	4	no	—
111	C₂²⋊C₄	D₁₀₇₆	1	2	yes	—
112	C₄²	C₂²×D₂₆₉	1	2	yes	—
113	C₂³	—	538	4	no	—
114	C₂³	C₂×D₁₀₇₆	1	1	yes	characteristic
115	D₄	—	538	4	no	—
116	D₄	—	538	4	no	—
117	D₄	—	2152	4	no	—
118	D₄	—	2152	4	no	—
119	C₂×C₄	—	2	1	no	—
120	C₂×C₄	—	2	1	no	—
121	C₂×C₄	—	538	4	no	—
122	C₂×C₄	—	2152	4	no	—
123	C₂×C₄	C₂×D₁₀₇₆	1	4	yes	—
124	C₂×C₄	C₂×D₁₀₇₆	1	1	yes	characteristic
125	C₂×C₄	C₂³×D₂₆₉	1	1	yes	characteristic
126	C₈	—	2	2	no	—
127	C₈	—	2	2	no	—
128	C₂²	—	2	2	no	—
129	C₂²	—	2	1	no	—
130	C₂²	—	1076	4	no	—
131	C₂²	—	1076	4	no	—
132	C₂²	—	1076	4	no	—
133	C₂²	C₂²×D₁₀₇₆	1	1	yes	characteristic
134	C₄	—	2	4	no	—
135	C₄	—	2	1	no	—
136	C₄	—	1076	4	no	—
137	C₄	#22575498	1	1	yes	characteristic
138	C₄	#22575501	1	1	yes	characteristic
139	C₂	—	4	1	no	—
140	C₂	—	2152	4	no	—
141	C₂	#31771186	1	1	yes	characteristic
142	C₂	#31771410	1	2	yes	—

subgroup

G/H

class size

multiplicity

normal?

properties

#31771210

C₂

yes

maximal, complemented

#31770906

C₂

yes

maximal, complemented

#31770858

C₂

yes

maximal, complemented

#31770841

C₂

yes

maximal, complemented

#31770801

C₂

yes

maximal, complemented

#424×C₂₆₉

C₂

yes

maximal, characteristic, complemented

C₂×D₂₁₅₂

C₂²

yes

—

C₂×D₂₁₅₂

C₂²

yes

—

#22575450

C₂²

yes

—

#22575445

C₂²

yes

—

#22575438

—

#22575427

C₂²

yes

—

#22575426

—

C₄×D₁₀₇₆

C₂²

yes

—

#22575387

C₂²

yes

—

#22575385

C₂²

yes

—

#22575375

C₂²

yes

—

#22575375

C₂²

yes

—

C₂×M₁₆×C₂₆₉

C₂²

yes

characteristic

C₂²×C₈×C₂₆₉

C₂²

yes

characteristic

#117×C₂₆₉

C₂²

yes

characteristic

C₄⋊C₈×C₂₆₉

C₂²

yes

—

C₂×D₁₀₇₆

—

C₂×D₁₀₇₆

C₂³

yes

—

C₂×C₂²⋊D₂₆₉

—

C₂×C₄×D₂₆₉

—

D₂₁₅₂

—

D₂₁₅₂

—

C₄⋊Dic₂₆₉

C₂³

yes

—

#16864381

—

#16864379

—

#16864379

—

C₂²×C₄×C₂₆₉

C₂³

yes

characteristic

M₁₆×C₂₆₉

—

C₂×C₈×C₂₆₉

—

C₂×C₈×C₂₆₉

C₂³

yes

—

C₂×C₈×C₂₆₉

C₂³

yes

—

C₄⋊C₄×C₂₆₉

D₄

yes

—

C₂²⋊C₄×C₂₆₉

D₄

yes

—

C₄²×C₂₆₉

C₂³

yes

—

C₂²×D₂₆₉

—

D₁₀₇₆

—

D₁₀₇₆

—

D₁₀₇₆

—

C₂²⋊D₂₆₉

—

C₂×Dic₂₆₉

—

C₄×D₂₆₉

—

C₂³×C₂₆₉

C₂×D₄

yes

characteristic

C₂×C₄×C₂₆₉

—

C₂×C₄×C₂₆₉

—

C₂×C₄×C₂₆₉

C₂×D₄

yes

—

C₂×C₄×C₂₆₉

C₂×D₄

yes

characteristic

C₂×C₄×C₂₆₉

C₂⁴

yes

characteristic

C₈×C₂₆₉

—

C₈×C₂₆₉

—

C₂²×C₂₆₉

—

C₂²×C₂₆₉

—

C₂²×C₂₆₉

C₂²×D₄

yes

characteristic

C₂×D₂₆₉

—

C₂×D₂₆₉

—

C₂×D₂₆₉

—

C₄×C₂₆₉

—

C₄×C₂₆₉

—

C₄×C₂₆₉

H(5,𝔽₂)

yes

characteristic

C₄×C₂₆₉

2₋¹⁺⁴

yes

characteristic

Dic₂₆₉

—

C₂×C₂₆₉

—

C₂×C₂₆₉

#537

yes

characteristic

C₂×C₂₆₉

#576

yes

—

D₂₆₉

—

C₂₆₉

#3259

yes

characteristic, complemented, Sylow

#3259

—

269

maximal, Sylow

#540

—

269

—

#478

—

269

—

#469

—

269

—

#466

—

269

—

#459

—

269

—

#424

D₂₆₉

yes

characteristic, complemented

C₂×D₈

—

269

—

C₂×D₈

—

269

—

C₂×M₁₆

C₂×D₂₆₉

yes

characteristic

C₂²×C₈

C₂×D₂₆₉

yes

characteristic

#127

—

269

—

#125

—

269

—

#123

—

538

—

#121

—

269

—

#121

—

538

—

C₄×D₄

—

269

—

#117

C₂×D₂₆₉

yes

characteristic

#107

—

269

—

#106

—

269

—

C₄⋊C₈

C₂×D₂₆₉

yes

—

#102

—

269

—

#102

—

269

—

C₂×D₄

—

269

—

C₂×D₄

—

538

—

C₂×D₄

—

538

—

C₂²×C₄

—

538

—

C₂²×C₄

C₂²×D₂₆₉

yes

characteristic

100

D₈

—

1076

—

101

D₈

—

1076

—

102

M₁₆

—

103

C₂×C₈

—

104

C₂×C₈

C₂²×D₂₆₉

yes

—

105

C₂×C₈

C₂²×D₂₆₉

yes

—

106

C₄⋊C₄

—

269

—

107

C₄⋊C₄

—

538

—

108

C₄⋊C₄

D₁₀₇₆

yes

—

109

C₂²⋊C₄

—

538

—

110

C₂²⋊C₄

—

538

—

111

C₂²⋊C₄

D₁₀₇₆

yes

—

112

C₄²

C₂²×D₂₆₉

yes

—

113

C₂³

—

538

—

114

C₂³

C₂×D₁₀₇₆

yes

characteristic

115

D₄

—

538

—

116

D₄

—

538

—

117

D₄

—

2152

—

118

D₄

—

2152

—

119

C₂×C₄

—

120

C₂×C₄

—

121

C₂×C₄

—

538

—

122

C₂×C₄

—

2152

—

123

C₂×C₄

C₂×D₁₀₇₆

yes

—

124

C₂×C₄

C₂×D₁₀₇₆

yes

characteristic

125

C₂×C₄

C₂³×D₂₆₉

yes

characteristic

126

C₈

—

127

C₈

—

128

C₂²

—

129

C₂²

—

130

C₂²

—

1076

—

131

C₂²

—

1076

—

132

C₂²

—

1076

—

133

C₂²

C₂²×D₁₀₇₆

yes

characteristic

134

C₄

—

135

C₄

—

136

C₄

—

1076

—

137

C₄

#22575498

yes

characteristic

138

C₄

#22575501

yes

characteristic

139

C₂

—

140

C₂

—

2152

—

141

C₂

#31771186

yes

characteristic

142

C₂

#31771410

yes

—

	subgroup	G/H
center	C₂²	C₂²×D₁₀₇₆
commutator, cosocle	C₂×C₄×C₂₆₉	C₂⁴
Frattini	C₂×C₄	C₂³×D₂₆₉
Fitting	#424×C₂₆₉	C₂
socle	C₂²×C₂₆₉	C₂²×D₄
hypercenter	#424	D₂₆₉
hypocenter	C₂₆₉	#3259
LCS	G ▷ C₂×C₄×C₂₆₉ ▷ C₂×C₂₆₉ ▷ C₂₆₉
UCS	C₁ ◁ C₂² ◁ #117 ◁ #424

subgroup

G/H

center

C₂²

C₂²×D₁₀₇₆

commutator, cosocle

C₂×C₄×C₂₆₉

C₂⁴

Frattini

C₂×C₄

C₂³×D₂₆₉

Fitting

#424×C₂₆₉

C₂

socle

C₂²×C₂₆₉

C₂²×D₄

D₂₆₉

C₂₆₉

G ▷ C₂×C₄×C₂₆₉ ▷ C₂×C₂₆₉ ▷ C₂₆₉

UCS

C₁ ◁ C₂² ◁ #117 ◁ #424

N	H ≅ G/N	type
1. #31771210	C₂	semidirect
2. #31770906	C₂	semidirect
3. #31770858	C₂	semidirect
4. #31770841	C₂	semidirect
5. #31770801	C₂	semidirect
6. #424×C₂₆₉	C₂	semidirect
71. C₂₆₉	#3259	semidirect
78. #424	D₂₆₉	semidirect

H ≅ G/N

type

1. #31771210

C₂

semidirect

2. #31770906

C₂

semidirect

3. #31770858

C₂

semidirect

4. #31770841

C₂

semidirect

5. #31770801

C₂

semidirect

6. #424×C₂₆₉

C₂

semidirect

71. C₂₆₉

#3259

semidirect

78. #424

D₂₆₉

semidirect

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

subgroup	index
#31771210	2
#31770906	2
#31770858	2
#31770841	2
#31770801	2
#424×C₂₆₉	2
#3259	269

subgroup

index

#424×C₂₆₉

269

Subgroups — #46822324

Conjugacy classes of subgroups

Complements

Maximal subgroups