Group #46822266 — subgroups

Subgroups — #46822266

Order 34432.

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

normal subgroups only

This group has 430 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	#31771155	C₂	1	2	yes	maximal, complemented
2	#31770896	C₂	1	4	yes	maximal, complemented
3	#31770858	C₂	1	4	yes	maximal, complemented
4	#31770801	C₂	1	4	yes	maximal, complemented
5	#423×C₂₆₉	C₂	1	1	yes	maximal, characteristic, complemented
6	C₂×#16864402	—	2	2	no	—
7	C₂×D₂₁₅₂	C₂²	1	4	yes	—
8	#22575453	C₂²	1	2	yes	—
9	#22575450	C₂²	1	2	yes	—
10	#22575441	—	2	2	no	—
11	#22575427	—	2	2	no	—
12	#22575427	C₂²	1	4	yes	—
13	C₄×D₁₀₇₆	—	2	2	no	—
14	C₄×D₁₀₇₆	C₂²	1	4	yes	—
15	#22575385	C₂²	1	4	yes	—
16	#22575375	C₂²	1	8	yes	—
17	C₂×M₁₆×C₂₆₉	C₂²	1	2	yes	—
18	C₂×C₄²×C₂₆₉	C₂²	1	1	yes	characteristic
19	C₄⋊C₈×C₂₆₉	C₂²	1	4	yes	—
20	#16864402	—	4	8	no	—
21	C₂×Dic₅₃₈	—	2	2	no	—
22	C₂×D₁₀₇₆	—	2	2	no	—
23	C₂×D₁₀₇₆	—	2	2	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	4	no	—
28	C₄⋊Dic₂₆₉	—	2	2	no	—
29	C₄⋊Dic₂₆₉	C₂³	1	4	yes	—
30	#16864379	—	2	4	no	—
31	#16864379	—	2	4	no	—
32	C₂²×C₄×C₂₆₉	D₄	1	2	yes	—
33	C₂²×C₄×C₂₆₉	C₂³	1	1	yes	characteristic
34	M₁₆×C₂₆₉	—	4	2	no	—
35	C₂×C₈×C₂₆₉	C₂³	1	4	yes	—
36	C₄²×C₂₆₉	D₄	1	1	yes	characteristic
37	C₄²×C₂₆₉	D₄	1	1	yes	characteristic
38	C₄²×C₂₆₉	C₂³	1	2	yes	—
39	C₂²×D₂₆₉	—	2	4	no	—
40	Dic₅₃₈	—	4	4	no	—
41	D₁₀₇₆	—	2	4	no	—
42	D₁₀₇₆	—	2	4	no	—
43	D₁₀₇₆	—	4	4	no	—
44	D₁₀₇₆	—	4	4	no	—
45	C₂²⋊D₂₆₉	—	4	8	no	—
46	C₂×Dic₂₆₉	—	2	4	no	—
47	C₄×D₂₆₉	—	4	8	no	—
48	C₂³×C₂₆₉	C₂×D₄	1	1	yes	characteristic
49	C₂×C₄×C₂₆₉	—	2	4	no	—
50	C₂×C₄×C₂₆₉	—	2	1	no	—
51	C₂×C₄×C₂₆₉	—	2	1	no	—
52	C₂×C₄×C₂₆₉	C₂×D₄	1	1	yes	characteristic
53	C₂×C₄×C₂₆₉	C₂×D₄	1	2	yes	—
54	C₂×C₄×C₂₆₉	C₂×D₄	1	2	yes	—
55	C₂×C₄×C₂₆₉	C₂⁴	1	1	yes	characteristic
56	C₈×C₂₆₉	—	2	4	no	—
57	C₂²×C₂₆₉	—	2	1	no	—
58	C₂²×C₂₆₉	—	2	2	no	—
59	C₂²×C₂₆₉	C₂²×D₄	1	1	yes	characteristic
60	C₂×D₂₆₉	—	4	4	no	—
61	C₂×D₂₆₉	—	4	4	no	—
62	C₂×D₂₆₉	—	4	4	no	—
63	C₄×C₂₆₉	—	2	1	no	—
64	C₄×C₂₆₉	—	2	2	no	—
65	C₄×C₂₆₉	AGL(1,ℤ/8)	1	4	yes	—
66	C₄×C₂₆₉	H(5,𝔽₂)	1	1	yes	characteristic
67	C₄×C₂₆₉	2₋¹⁺⁴	1	1	yes	characteristic
68	Dic₂₆₉	—	4	4	no	—
69	C₂×C₂₆₉	—	4	1	no	—
70	C₂×C₂₆₉	#537	1	1	yes	characteristic
71	C₂×C₂₆₉	C₂×AGL(1,ℤ/8)	1	2	yes	—
72	D₂₆₉	—	8	4	no	—
73	C₂₆₉	#3253	1	1	yes	characteristic, complemented, Sylow
74	#3253	—	269	1	no	maximal, Sylow
75	#532	—	269	2	no	—
76	#476	—	269	4	no	—
77	#469	—	269	4	no	—
78	#459	—	269	4	no	—
79	#423	D₂₆₉	1	1	yes	characteristic, complemented
80	C₂×#41	—	538	2	no	—
81	C₂×D₈	—	269	4	no	—
82	C₂×M₁₆	C₂×D₂₆₉	1	2	yes	—
83	C₄⋊Q₈	—	269	2	no	—
84	#127	—	269	2	no	—
85	#124	—	538	2	no	—
86	#121	—	269	4	no	—
87	#121	—	538	2	no	—
88	C₄×D₄	—	269	4	no	—
89	C₄×D₄	—	538	2	no	—
90	C₂×C₄²	C₂×D₂₆₉	1	1	yes	characteristic
91	#106	—	269	4	no	—
92	C₄⋊C₈	C₂×D₂₆₉	1	4	yes	—
93	#102	—	269	8	no	—
94	#41	—	1076	8	no	—
95	C₂×Q₈	—	538	2	no	—
96	C₂×D₄	—	269	4	no	—
97	C₂×D₄	—	538	4	no	—
98	C₂×D₄	—	538	2	no	—
99	C₂×D₄	—	538	2	no	—
100	C₂²×C₄	—	538	4	no	—
101	C₂²×C₄	D₁₀₇₆	1	2	yes	—
102	C₂²×C₄	C₂²×D₂₆₉	1	1	yes	characteristic
103	D₈	—	1076	4	no	—
104	M₁₆	—	4	2	no	—
105	C₂×C₈	C₂²×D₂₆₉	1	4	yes	—
106	C₄⋊C₄	—	269	4	no	—
107	C₄⋊C₄	—	538	2	no	—
108	C₂²⋊C₄	—	538	4	no	—
109	C₂²⋊C₄	—	538	4	no	—
110	C₄²	D₁₀₇₆	1	1	yes	characteristic
111	C₄²	D₁₀₇₆	1	1	yes	characteristic
112	C₄²	C₂²×D₂₆₉	1	2	yes	—
113	C₂³	—	538	4	no	—
114	C₂³	C₂×D₁₀₇₆	1	1	yes	characteristic
115	Q₈	—	1076	4	no	—
116	D₄	—	538	4	no	—
117	D₄	—	538	4	no	—
118	D₄	—	1076	4	no	—
119	D₄	—	1076	8	no	—
120	D₄	—	1076	4	no	—
121	C₂×C₄	—	2	4	no	—
122	C₂×C₄	—	2	1	no	—
123	C₂×C₄	—	2	1	no	—
124	C₂×C₄	—	538	4	no	—
125	C₂×C₄	—	1076	8	no	—
126	C₂×C₄	C₂×D₁₀₇₆	1	1	yes	characteristic
127	C₂×C₄	C₂×D₁₀₇₆	1	2	yes	—
128	C₂×C₄	C₂×D₁₀₇₆	1	2	yes	—
129	C₂×C₄	C₂³×D₂₆₉	1	1	yes	characteristic
130	C₈	—	2	4	no	—
131	C₂²	—	2	2	no	—
132	C₂²	—	2	1	no	—
133	C₂²	—	1076	4	no	—
134	C₂²	—	1076	4	no	—
135	C₂²	—	1076	4	no	—
136	C₂²	C₂²×D₁₀₇₆	1	1	yes	characteristic
137	C₄	—	2	2	no	—
138	C₄	—	2	1	no	—
139	C₄	—	1076	4	no	—
140	C₄	#22575481	1	4	yes	—
141	C₄	#22575498	1	1	yes	characteristic
142	C₄	#22575501	1	1	yes	characteristic
143	C₂	—	4	1	no	—
144	C₂	—	2152	4	no	—
145	C₂	#31771186	1	1	yes	characteristic
146	C₂	C₂×#22575481	1	2	yes	—

subgroup

G/H

class size

multiplicity

normal?

properties

#31771155

C₂

yes

maximal, complemented

#31770896

C₂

yes

maximal, complemented

#31770858

C₂

yes

maximal, complemented

#31770801

C₂

yes

maximal, complemented

#423×C₂₆₉

C₂

yes

maximal, characteristic, complemented

C₂×#16864402

—

C₂×D₂₁₅₂

C₂²

yes

—

#22575453

C₂²

yes

—

#22575450

C₂²

yes

—

#22575441

—

#22575427

—

#22575427

C₂²

yes

—

C₄×D₁₀₇₆

—

C₄×D₁₀₇₆

C₂²

yes

—

#22575385

C₂²

yes

—

#22575375

C₂²

yes

—

C₂×M₁₆×C₂₆₉

C₂²

yes

—

C₂×C₄²×C₂₆₉

C₂²

yes

characteristic

C₄⋊C₈×C₂₆₉

C₂²

yes

—

#16864402

—

C₂×Dic₅₃₈

—

C₂×D₁₀₇₆

—

C₂×D₁₀₇₆

—

C₂×D₁₀₇₆

C₂³

yes

—

C₂×C₂²⋊D₂₆₉

—

C₂×C₄×D₂₆₉

—

D₂₁₅₂

—

C₄⋊Dic₂₆₉

—

C₄⋊Dic₂₆₉

C₂³

yes

—

#16864379

—

#16864379

—

C₂²×C₄×C₂₆₉

D₄

yes

—

C₂²×C₄×C₂₆₉

C₂³

yes

characteristic

M₁₆×C₂₆₉

—

C₂×C₈×C₂₆₉

C₂³

yes

—

C₄²×C₂₆₉

D₄

yes

characteristic

C₄²×C₂₆₉

D₄

yes

characteristic

C₄²×C₂₆₉

C₂³

yes

—

C₂²×D₂₆₉

—

Dic₅₃₈

—

D₁₀₇₆

—

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×C₄×C₂₆₉

C₂×D₄

yes

characteristic

C₂×C₄×C₂₆₉

C₂×D₄

yes

—

C₂×C₄×C₂₆₉

C₂×D₄

yes

—

C₂×C₄×C₂₆₉

C₂⁴

yes

characteristic

C₈×C₂₆₉

—

C₂²×C₂₆₉

—

C₂²×C₂₆₉

—

C₂²×C₂₆₉

C₂²×D₄

yes

characteristic

C₂×D₂₆₉

—

C₂×D₂₆₉

—

C₂×D₂₆₉

—

C₄×C₂₆₉

—

C₄×C₂₆₉

—

C₄×C₂₆₉

AGL(1,ℤ/8)

yes

—

C₄×C₂₆₉

H(5,𝔽₂)

yes

characteristic

C₄×C₂₆₉

2₋¹⁺⁴

yes

characteristic

Dic₂₆₉

—

C₂×C₂₆₉

—

C₂×C₂₆₉

#537

yes

characteristic

C₂×C₂₆₉

C₂×AGL(1,ℤ/8)

yes

—

D₂₆₉

—

C₂₆₉

#3253

yes

characteristic, complemented, Sylow

#3253

—

269

maximal, Sylow

#532

—

269

—

#476

—

269

—

#469

—

269

—

#459

—

269

—

#423

D₂₆₉

yes

characteristic, complemented

C₂×#41

—

538

—

C₂×D₈

—

269

—

C₂×M₁₆

C₂×D₂₆₉

yes

—

C₄⋊Q₈

—

269

—

#127

—

269

—

#124

—

538

—

#121

—

269

—

#121

—

538

—

C₄×D₄

—

269

—

C₄×D₄

—

538

—

C₂×C₄²

C₂×D₂₆₉

yes

characteristic

#106

—

269

—

C₄⋊C₈

C₂×D₂₆₉

yes

—

#102

—

269

—

#41

—

1076

—

C₂×Q₈

—

538

—

C₂×D₄

—

269

—

C₂×D₄

—

538

—

C₂×D₄

—

538

—

C₂×D₄

—

538

—

100

C₂²×C₄

—

538

—

101

C₂²×C₄

D₁₀₇₆

yes

—

102

C₂²×C₄

C₂²×D₂₆₉

yes

characteristic

103

D₈

—

1076

—

104

M₁₆

—

105

C₂×C₈

C₂²×D₂₆₉

yes

—

106

C₄⋊C₄

—

269

—

107

C₄⋊C₄

—

538

—

108

C₂²⋊C₄

—

538

—

109

C₂²⋊C₄

—

538

—

110

C₄²

D₁₀₇₆

yes

characteristic

111

C₄²

D₁₀₇₆

yes

characteristic

112

C₄²

C₂²×D₂₆₉

yes

—

113

C₂³

—

538

—

114

C₂³

C₂×D₁₀₇₆

yes

characteristic

115

Q₈

—

1076

—

116

D₄

—

538

—

117

D₄

—

538

—

118

D₄

—

1076

—

119

D₄

—

1076

—

120

D₄

—

1076

—

121

C₂×C₄

—

122

C₂×C₄

—

123

C₂×C₄

—

124

C₂×C₄

—

538

—

125

C₂×C₄

—

1076

—

126

C₂×C₄

C₂×D₁₀₇₆

yes

characteristic

127

C₂×C₄

C₂×D₁₀₇₆

yes

—

128

C₂×C₄

C₂×D₁₀₇₆

yes

—

129

C₂×C₄

C₂³×D₂₆₉

yes

characteristic

130

C₈

—

131

C₂²

—

132

C₂²

—

133

C₂²

—

1076

—

134

C₂²

—

1076

—

135

C₂²

—

1076

—

136

C₂²

C₂²×D₁₀₇₆

yes

characteristic

137

C₄

—

138

C₄

—

139

C₄

—

1076

—

140

C₄

#22575481

yes

—

141

C₄

#22575498

yes

characteristic

142

C₄

#22575501

yes

characteristic

143

C₂

—

144

C₂

—

2152

—

145

C₂

#31771186

yes

characteristic

146

C₂

C₂×#22575481

yes

—

	subgroup	G/H
center	C₂²	C₂²×D₁₀₇₆
commutator, cosocle	C₂×C₄×C₂₆₉	C₂⁴
Frattini	C₂×C₄	C₂³×D₂₆₉
Fitting	#423×C₂₆₉	C₂
socle	C₂²×C₂₆₉	C₂²×D₄
hypercenter	#423	D₂₆₉
hypocenter	C₂₆₉	#3253
LCS	G ▷ C₂×C₄×C₂₆₉ ▷ C₂×C₂₆₉ ▷ C₂₆₉
UCS	C₁ ◁ C₂² ◁ C₂×C₄² ◁ #423

subgroup

G/H

center

C₂²

C₂²×D₁₀₇₆

commutator, cosocle

C₂×C₄×C₂₆₉

C₂⁴

Frattini

C₂×C₄

C₂³×D₂₆₉

Fitting

#423×C₂₆₉

C₂

socle

C₂²×C₂₆₉

C₂²×D₄

D₂₆₉

C₂₆₉

G ▷ C₂×C₄×C₂₆₉ ▷ C₂×C₂₆₉ ▷ C₂₆₉

UCS

C₁ ◁ C₂² ◁ C₂×C₄² ◁ #423

N	H ≅ G/N	type
1. #31771155	C₂	semidirect
2. #31770896	C₂	semidirect
3. #31770858	C₂	semidirect
4. #31770801	C₂	semidirect
5. #423×C₂₆₉	C₂	semidirect
73. C₂₆₉	#3253	semidirect
79. #423	D₂₆₉	semidirect

H ≅ G/N

type

1. #31771155

C₂

semidirect

2. #31770896

C₂

semidirect

3. #31770858

C₂

semidirect

4. #31770801

C₂

semidirect

5. #423×C₂₆₉

C₂

semidirect

73. C₂₆₉

#3253

semidirect

79. #423

D₂₆₉

semidirect

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

subgroup	index
#31771155	2
#31770896	2
#31770858	2
#31770801	2
#423×C₂₆₉	2
#3253	269

subgroup

index

#423×C₂₆₉

269

Subgroups — #46822266

Conjugacy classes of subgroups

Complements

Maximal subgroups