Group #46809613 — subgroups

Subgroups — #46809613

Order 34432.

202 conjugacy classes of subgroups; only the 73 normal subgroups are shown.

normal subgroups only

This group has 202 conjugacy classes of subgroups — too many to draw. Only its 73 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	#31771162	C₂	1	1	yes	maximal, characteristic
2	#31770662	C₂	1	1	yes	maximal, characteristic
3	#31770353	C₂	1	2	yes	maximal, complemented
4	#31770352	C₂	1	2	yes	maximal, complemented
5	#423×C₂₆₉	C₂	1	1	yes	maximal, characteristic
6	#22575453	C₂²	1	1	yes	characteristic
7	#22575453	C₂²	1	1	yes	characteristic
8	#22575445	—	2	1	no	—
9	#22575432	—	2	1	no	—
10	#22575432	C₄	1	2	yes	—
11	C₄×Dic₅₃₈	—	2	1	no	—
12	C₄×Dic₅₃₈	C₄	1	2	yes	—
13	#22575412	—	2	1	no	—
14	C₄⋊#14524130	C₂²	1	2	yes	—
15	C₂²⋊#14524130	—	2	1	no	—
16	#22575364	—	2	1	no	—
17	C₂×M₁₆×C₂₆₉	—	2	1	no	—
18	C₂×C₄²×C₂₆₉	C₂²	1	1	yes	characteristic
19	C₄⋊C₈×C₂₆₉	—	2	1	no	—
20	C₄⋊C₈×C₂₆₉	C₂²	1	2	yes	—
21	C₂×Dic₅₃₈	—	2	1	no	—
22	C₂×Dic₅₃₈	C₂×C₄	1	2	yes	—
23	C₂×#14524130	—	2	2	no	—
24	C₄⋊Dic₂₆₉	—	2	1	no	—
25	C₄⋊Dic₂₆₉	—	2	1	no	—
26	C₄⋊Dic₂₆₉	—	2	1	no	—
27	C₄⋊Dic₂₆₉	C₂×C₄	1	2	yes	—
28	#16864381	—	2	2	no	—
29	#16864381	—	2	2	no	—
30	C₂²⋊Dic₂₆₉	—	2	2	no	—
31	C₄×Dic₂₆₉	—	2	2	no	—
32	C₂²×C₄×C₂₆₉	C₂×C₄	1	1	yes	characteristic
33	C₂²×C₄×C₂₆₉	D₄	1	1	yes	characteristic
34	C₂²×C₄×C₂₆₉	D₄	1	1	yes	characteristic
35	M₁₆×C₂₆₉	—	4	2	no	—
36	C₂×C₈×C₂₆₉	—	2	2	no	—
37	C₄²×C₂₆₉	C₂×C₄	1	1	yes	characteristic
38	C₄²×C₂₆₉	D₄	1	1	yes	characteristic
39	C₄²×C₂₆₉	D₄	1	1	yes	characteristic
40	C₄²×C₂₆₉	C₂³	1	1	yes	characteristic
41	Dic₅₃₈	—	4	1	no	—
42	Dic₅₃₈	—	4	1	no	—
43	Dic₅₃₈	—	4	2	no	—
44	C₂×Dic₂₆₉	—	2	2	no	—
45	C₂×Dic₂₆₉	—	2	2	no	—
46	#14524130	—	4	2	no	—
47	C₂³×C₂₆₉	C₂²⋊C₄	1	1	yes	characteristic
48	C₂×C₄×C₂₆₉	—	2	1	no	—
49	C₂×C₄×C₂₆₉	—	2	1	no	—
50	C₂×C₄×C₂₆₉	—	2	2	no	—
51	C₂×C₄×C₂₆₉	—	4	1	no	—
52	C₂×C₄×C₂₆₉	C₂²⋊C₄	1	1	yes	characteristic
53	C₂×C₄×C₂₆₉	C₂²⋊C₄	1	1	yes	characteristic
54	C₂×C₄×C₂₆₉	C₂²⋊C₄	1	1	yes	characteristic
55	C₂×C₄×C₂₆₉	C₂²×C₄	1	1	yes	characteristic
56	C₂×C₄×C₂₆₉	C₂×D₄	1	1	yes	characteristic
57	C₂×C₄×C₂₆₉	C₂×D₄	1	1	yes	characteristic
58	C₈×C₂₆₉	—	4	2	no	—
59	C₂²×C₂₆₉	—	2	1	no	—
60	C₂²×C₂₆₉	—	2	1	no	—
61	C₂²×C₂₆₉	—	2	1	no	—
62	C₂²×C₂₆₉	C₂×C₂²⋊C₄	1	1	yes	characteristic
63	C₄×C₂₆₉	—	2	1	no	—
64	C₄×C₂₆₉	—	2	1	no	—
65	C₄×C₂₆₉	—	2	1	no	—
66	C₄×C₂₆₉	—	2	1	no	—
67	C₄×C₂₆₉	AGL(1,ℤ/8)	1	1	yes	characteristic
68	C₄×C₂₆₉	#137	1	1	yes	characteristic
69	C₄×C₂₆₉	#137	1	2	yes	—
70	Dic₂₆₉	—	4	2	no	—
71	Dic₂₆₉	—	4	2	no	—
72	C₂×C₂₆₉	—	4	1	no	—
73	C₂×C₂₆₉	#413	1	1	yes	characteristic
74	C₂×C₂₆₉	#417	1	1	yes	characteristic
75	C₂×C₂₆₉	#419	1	1	yes	characteristic
76	C₂₆₉	#1559	1	1	yes	characteristic, complemented, Sylow
77	#1559	—	269	1	no	maximal, Sylow
78	#533	—	269	1	no	—
79	#432	—	269	1	no	—
80	#423	D₂₆₉	1	1	yes	characteristic
81	#333	—	269	2	no	—
82	#332	—	269	2	no	—
83	C₂×M₁₆	—	2	1	no	—
84	C₄⋊Q₈	—	269	1	no	—
85	C₄⋊Q₈	—	269	1	no	—
86	#125	—	538	1	no	—
87	C₂²⋊Q₈	—	269	2	no	—
88	C₂²⋊Q₈	—	538	1	no	—
89	C₄×Q₈	—	269	2	no	—
90	C₄×Q₈	—	538	1	no	—
91	#117	—	538	1	no	—
92	C₂×C₄²	C₂×D₂₆₉	1	1	yes	characteristic
93	C₄⋊C₈	—	2	1	no	—
94	C₄⋊C₈	—	269	2	no	—
95	C₄⋊C₈	C₂×D₂₆₉	1	2	yes	—
96	C₂²⋊C₈	—	538	1	no	—
97	#97	—	538	1	no	—
98	C₂×Q₈	—	269	2	no	—
99	C₂×Q₈	—	538	1	no	—
100	C₂²×C₄	C₄×D₂₆₉	1	1	yes	characteristic
101	C₂²×C₄	C₂²⋊D₂₆₉	1	1	yes	characteristic
102	C₂²×C₄	D₁₀₇₆	1	1	yes	characteristic
103	M₁₆	—	4	2	no	—
104	C₂×C₈	—	2	2	no	—
105	C₂×C₈	—	538	2	no	—
106	C₄⋊C₄	—	269	2	no	—
107	C₄⋊C₄	—	538	2	no	—
108	C₄⋊C₄	—	538	2	no	—
109	C₄⋊C₄	—	538	1	no	—
110	C₄⋊C₄	—	538	1	no	—
111	C₄⋊C₄	—	538	1	no	—
112	C₂²⋊C₄	—	538	2	no	—
113	C₄²	—	538	2	no	—
114	C₄²	C₄×D₂₆₉	1	1	yes	characteristic
115	C₄²	C₂²⋊D₂₆₉	1	1	yes	characteristic
116	C₄²	D₁₀₇₆	1	1	yes	characteristic
117	C₄²	C₂²×D₂₆₉	1	1	yes	characteristic
118	C₂³	#16864379	1	1	yes	characteristic
119	Q₈	—	1076	2	no	—
120	Q₈	—	1076	1	no	—
121	Q₈	—	1076	1	no	—
122	C₂×C₄	—	2	1	no	—
123	C₂×C₄	—	2	1	no	—
124	C₂×C₄	—	2	2	no	—
125	C₂×C₄	—	4	1	no	—
126	C₂×C₄	—	538	2	no	—
127	C₂×C₄	—	538	2	no	—
128	C₂×C₄	#16864379	1	1	yes	characteristic
129	C₂×C₄	#16864379	1	1	yes	characteristic
130	C₂×C₄	#16864379	1	1	yes	characteristic
131	C₂×C₄	C₂×C₄×D₂₆₉	1	1	yes	characteristic
132	C₂×C₄	C₂×C₂²⋊D₂₆₉	1	1	yes	characteristic
133	C₂×C₄	C₂×D₁₀₇₆	1	1	yes	characteristic
134	C₈	—	4	2	no	—
135	C₈	—	1076	2	no	—
136	C₂²	—	2	1	no	—
137	C₂²	—	2	1	no	—
138	C₂²	—	2	1	no	—
139	C₂²	C₂×#16864379	1	1	yes	characteristic
140	C₄	—	2	1	no	—
141	C₄	—	2	1	no	—
142	C₄	—	2	1	no	—
143	C₄	—	2	1	no	—
144	C₄	—	1076	2	no	—
145	C₄	—	1076	2	no	—
146	C₄	#22575479	1	1	yes	characteristic
147	C₄	#22575484	1	1	yes	characteristic
148	C₄	#22575486	1	2	yes	—
149	C₂	—	4	1	no	—
150	C₂	#31770587	1	1	yes	characteristic
151	C₂	#31770604	1	1	yes	characteristic
152	C₂	#31770615	1	1	yes	characteristic

subgroup

G/H

class size

multiplicity

normal?

properties

#31771162

C₂

yes

maximal, characteristic

#31770662

C₂

yes

maximal, characteristic

#31770353

C₂

yes

maximal, complemented

#31770352

C₂

yes

maximal, complemented

#423×C₂₆₉

C₂

yes

maximal, characteristic

#22575453

C₂²

yes

characteristic

#22575453

C₂²

yes

—

—

C₄

yes

—

C₄×Dic₅₃₈

—

C₄×Dic₅₃₈

C₄

yes

—

#22575412

—

C₄⋊#14524130

C₂²

yes

—

C₂²⋊#14524130

—

#22575364

—

C₂×M₁₆×C₂₆₉

—

C₂×C₄²×C₂₆₉

C₂²

yes

characteristic

C₄⋊C₈×C₂₆₉

—

C₄⋊C₈×C₂₆₉

C₂²

yes

—

C₂×Dic₅₃₈

—

C₂×Dic₅₃₈

C₂×C₄

yes

—

C₂×#14524130

—

C₄⋊Dic₂₆₉

—

C₄⋊Dic₂₆₉

—

C₄⋊Dic₂₆₉

—

C₄⋊Dic₂₆₉

C₂×C₄

yes

—

#16864381

—

#16864381

—

C₂²⋊Dic₂₆₉

—

C₄×Dic₂₆₉

—

C₂²×C₄×C₂₆₉

C₂×C₄

yes

characteristic

C₂²×C₄×C₂₆₉

D₄

yes

characteristic

C₂²×C₄×C₂₆₉

D₄

yes

characteristic

M₁₆×C₂₆₉

—

C₂×C₈×C₂₆₉

—

C₄²×C₂₆₉

C₂×C₄

yes

characteristic

C₄²×C₂₆₉

D₄

yes

characteristic

C₄²×C₂₆₉

D₄

yes

characteristic

C₄²×C₂₆₉

C₂³

yes

characteristic

Dic₅₃₈

—

Dic₅₃₈

—

Dic₅₃₈

—

C₂×Dic₂₆₉

—

C₂×Dic₂₆₉

—

#14524130

—

C₂³×C₂₆₉

C₂²⋊C₄

yes

characteristic

C₂×C₄×C₂₆₉

—

C₂×C₄×C₂₆₉

—

C₂×C₄×C₂₆₉

—

C₂×C₄×C₂₆₉

—

C₂×C₄×C₂₆₉

C₂²⋊C₄

yes

characteristic

C₂×C₄×C₂₆₉

C₂²⋊C₄

yes

characteristic

C₂×C₄×C₂₆₉

C₂²⋊C₄

yes

characteristic

C₂×C₄×C₂₆₉

C₂²×C₄

yes

characteristic

C₂×C₄×C₂₆₉

C₂×D₄

yes

characteristic

C₂×C₄×C₂₆₉

C₂×D₄

yes

characteristic

C₈×C₂₆₉

—

C₂²×C₂₆₉

—

C₂²×C₂₆₉

—

C₂²×C₂₆₉

—

C₂²×C₂₆₉

C₂×C₂²⋊C₄

yes

characteristic

C₄×C₂₆₉

—

C₄×C₂₆₉

—

C₄×C₂₆₉

—

C₄×C₂₆₉

—

C₄×C₂₆₉

yes

C₄×C₂₆₉

yes

C₄×C₂₆₉

yes

—

Dic₂₆₉

—

Dic₂₆₉

—

C₂×C₂₆₉

—

C₂×C₂₆₉

yes

C₂×C₂₆₉

yes

C₂×C₂₆₉

yes

C₂₆₉

yes

characteristic, complemented, Sylow

#1559

—

269

maximal, Sylow

#533

—

269

—

#432

—

269

—

#423

D₂₆₉

yes

characteristic

#333

—

269

—

#332

—

269

—

C₂×M₁₆

—

C₄⋊Q₈

—

269

—

C₄⋊Q₈

—

269

—

#125

—

538

—

C₂²⋊Q₈

—

269

—

C₂²⋊Q₈

—

538

—

C₄×Q₈

—

269

—

C₄×Q₈

—

538

—

#117

—

538

—

C₂×C₄²

C₂×D₂₆₉

yes

characteristic

C₄⋊C₈

—

C₄⋊C₈

—

269

—

C₄⋊C₈

C₂×D₂₆₉

yes

—

C₂²⋊C₈

—

538

—

#97

—

538

—

C₂×Q₈

—

269

—

C₂×Q₈

—

538

—

100

C₂²×C₄

C₄×D₂₆₉

yes

characteristic

101

C₂²×C₄

C₂²⋊D₂₆₉

yes

characteristic

102

C₂²×C₄

D₁₀₇₆

yes

characteristic

103

M₁₆

—

104

C₂×C₈

—

105

C₂×C₈

—

538

—

106

C₄⋊C₄

—

269

—

107

C₄⋊C₄

—

538

—

108

C₄⋊C₄

—

538

—

109

C₄⋊C₄

—

538

—

110

C₄⋊C₄

—

538

—

111

C₄⋊C₄

—

538

—

112

C₂²⋊C₄

—

538

—

113

C₄²

—

538

—

114

C₄²

C₄×D₂₆₉

yes

characteristic

115

C₄²

C₂²⋊D₂₆₉

yes

characteristic

116

C₄²

D₁₀₇₆

yes

characteristic

117

C₄²

C₂²×D₂₆₉

yes

characteristic

118

C₂³

#16864379

yes

characteristic

119

Q₈

—

1076

—

120

Q₈

—

1076

—

121

Q₈

—

1076

—

122

C₂×C₄

—

123

C₂×C₄

—

124

C₂×C₄

—

125

C₂×C₄

—

126

C₂×C₄

—

538

—

127

C₂×C₄

—

538

—

128

C₂×C₄

#16864379

yes

characteristic

129

C₂×C₄

#16864379

yes

characteristic

130

C₂×C₄

#16864379

yes

characteristic

131

C₂×C₄

C₂×C₄×D₂₆₉

yes

characteristic

132

C₂×C₄

C₂×C₂²⋊D₂₆₉

yes

characteristic

133

C₂×C₄

C₂×D₁₀₇₆

yes

characteristic

134

C₈

—

135

C₈

—

1076

—

136

C₂²

—

137

C₂²

—

138

C₂²

—

139

C₂²

C₂×#16864379

yes

characteristic

140

C₄

—

141

C₄

—

142

C₄

—

143

C₄

—

144

C₄

—

1076

—

145

C₄

—

1076

—

146

C₄

#22575479

yes

characteristic

147

C₄

#22575484

yes

characteristic

148

C₄

#22575486

yes

—

149

C₂

—

150

C₂

#31770587

yes

characteristic

151

C₂

#31770604

yes

characteristic

152

C₂

#31770615

yes

characteristic

	subgroup	G/H
center	C₂²	C₂×#16864379
commutator	C₂×C₄×C₂₆₉	C₂²×C₄
Frattini	C₄²	C₂²×D₂₆₉
Fitting	#423×C₂₆₉	C₂
socle	C₂²×C₂₆₉	C₂×C₂²⋊C₄
prime radical	C₂³×C₂₆₉	C₂²⋊C₄
cosocle	C₄²×C₂₆₉	C₂³
hypercenter	#423	D₂₆₉
hypocenter	C₂₆₉	#1559
LCS	G ▷ C₂×C₄×C₂₆₉ ▷ C₂²×C₂₆₉ ▷ C₂₆₉
UCS	C₁ ◁ C₂² ◁ C₂×C₄² ◁ #423

subgroup

G/H

center

C₂²

C₂×#16864379

commutator

C₂×C₄×C₂₆₉

C₂²×C₄

Frattini

C₄²

C₂²×D₂₆₉

Fitting

#423×C₂₆₉

C₂

socle

C₂²×C₂₆₉

C₂×C₂²⋊C₄

prime radical

C₂³×C₂₆₉

C₂²⋊C₄

cosocle

C₄²×C₂₆₉

C₂³

D₂₆₉

C₂₆₉

G ▷ C₂×C₄×C₂₆₉ ▷ C₂²×C₂₆₉ ▷ C₂₆₉

UCS

C₁ ◁ C₂² ◁ C₂×C₄² ◁ #423

N	H ≅ G/N	type
3. #31770353	C₂	semidirect
4. #31770352	C₂	semidirect
76. C₂₆₉	#1559	semidirect

H ≅ G/N

type

3. #31770353

C₂

semidirect

4. #31770352

C₂

semidirect

76. C₂₆₉

#1559

semidirect

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

subgroup	index
#31771162	2
#31770662	2
#31770353	2
#31770352	2
#423×C₂₆₉	2
#1559	269

subgroup

index

#423×C₂₆₉

269

Subgroups — #46809613

Conjugacy classes of subgroups

Complements

Maximal subgroups