Group #46811796 — subgroups

Subgroups — #46811796

Order 34432.

316 conjugacy classes of subgroups; only the 97 normal subgroups are shown.

normal subgroups only

This group has 316 conjugacy classes of subgroups — too many to draw. Only its 97 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×#22575389	C₂	1	1	yes	maximal, characteristic, complemented
2	#31770610	C₂	1	2	yes	maximal, complemented
3	C₂×#22575370	C₂	1	2	yes	maximal, complemented
4	#31770560	C₂	1	1	yes	maximal, characteristic, complemented
5	C₂²×D₈×C₂₆₉	C₂	1	1	yes	maximal, characteristic
6	C₂×#16864386	C₂²	1	2	yes	—
7	C₂×#16864386	C₂²	1	1	yes	characteristic
8	#22575412	C₂²	1	1	yes	characteristic
9	#22575389	—	2	2	no	—
10	#22575374	—	2	4	no	—
11	#22575370	—	2	4	no	—
12	#22575363	—	2	1	no	—
13	C₈×Dic₂₆₉	—	2	1	no	—
14	C₂²×D₄×C₂₆₉	C₂²	1	2	yes	—
15	C₂×D₈×C₂₆₉	—	2	1	no	—
16	C₂×D₈×C₂₆₉	—	2	1	no	—
17	C₂×D₈×C₂₆₉	—	2	2	no	—
18	C₂×D₈×C₂₆₉	C₄	1	4	yes	complemented
19	C₂²×C₈×C₂₆₉	C₂²	1	1	yes	characteristic
20	#16864386	—	2	2	no	—
21	#16864386	—	2	2	no	—
22	#16864386	—	2	1	no	—
23	#16864386	D₄	1	2	yes	—
24	C₂×#14524130	—	2	1	no	—
25	C₂×#14524130	—	2	2	no	—
26	C₄⋊Dic₂₆₉	D₄	1	2	yes	—
27	C₂²⋊Dic₂₆₉	—	2	1	no	—
28	C₄×Dic₂₆₉	—	2	1	no	—
29	C₂⁴×C₂₆₉	—	2	2	no	—
30	C₂×D₄×C₂₆₉	—	2	2	no	—
31	C₂×D₄×C₂₆₉	—	2	2	no	—
32	C₂×D₄×C₂₆₉	—	2	2	no	—
33	C₂×D₄×C₂₆₉	—	2	2	no	—
34	C₂×D₄×C₂₆₉	—	2	2	no	—
35	C₂×D₄×C₂₆₉	C₂×C₄	1	4	yes	—
36	C₂²×C₄×C₂₆₉	C₂³	1	1	yes	characteristic
37	D₈×C₂₆₉	—	2	4	no	—
38	D₈×C₂₆₉	—	2	4	no	—
39	C₂×C₈×C₂₆₉	C₂×C₄	1	2	yes	—
40	C₂×C₈×C₂₆₉	D₄	1	1	yes	characteristic
41	C₂×C₈×C₂₆₉	D₄	1	1	yes	characteristic
42	C₂×C₈×C₂₆₉	D₄	1	2	yes	—
43	C₂×Dic₂₆₉	—	2	2	no	—
44	#14524130	—	2	2	no	—
45	#14524130	—	4	2	no	—
46	C₂³×C₂₆₉	—	2	4	no	—
47	C₂³×C₂₆₉	—	4	2	no	—
48	C₂³×C₂₆₉	—	4	2	no	—
49	C₂³×C₂₆₉	—	8	4	no	—
50	C₂³×C₂₆₉	#41	1	1	yes	characteristic
51	D₄×C₂₆₉	—	2	4	no	—
52	D₄×C₂₆₉	—	2	4	no	—
53	D₄×C₂₆₉	—	4	2	no	—
54	D₄×C₂₆₉	—	4	2	no	—
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	C₂×C₄×C₂₆₉	C₂×D₄	1	1	yes	characteristic
59	C₂×C₄×C₂₆₉	C₂×D₄	1	1	yes	characteristic
60	C₂×C₄×C₂₆₉	#41	1	1	yes	characteristic
61	C₈×C₂₆₉	C₂²⋊C₄	1	2	yes	—
62	C₈×C₂₆₉	C₂²⋊C₄	1	2	yes	—
63	C₂²×C₂₆₉	—	2	2	no	—
64	C₂²×C₂₆₉	—	4	4	no	—
65	C₂²×C₂₆₉	—	8	4	no	—
66	C₂²×C₂₆₉	—	8	4	no	—
67	C₂²×C₂₆₉	—	8	4	no	—
68	C₂²×C₂₆₉	C₄×D₄	1	1	yes	characteristic
69	C₂²×C₂₆₉	#121	1	1	yes	characteristic
70	C₂²×C₂₆₉	#124	1	1	yes	characteristic
71	C₄×C₂₆₉	C₂×C₂²⋊C₄	1	1	yes	characteristic
72	C₄×C₂₆₉	C₄×D₄	1	1	yes	characteristic
73	C₄×C₂₆₉	#121	1	1	yes	characteristic
74	C₄×C₂₆₉	#127	1	1	yes	characteristic
75	Dic₂₆₉	—	4	2	no	—
76	C₂×C₂₆₉	—	2	1	no	—
77	C₂×C₂₆₉	—	2	1	no	—
78	C₂×C₂₆₉	—	8	4	no	—
79	C₂×C₂₆₉	#390	1	1	yes	characteristic
80	C₂×C₂₆₉	#472	1	2	yes	—
81	C₂₆₉	#1978	1	1	yes	characteristic, complemented, Sylow
82	#1978	—	269	1	no	maximal, Sylow
83	C₂²×D₈	D₂₆₉	1	1	yes	characteristic
84	C₂×#108	—	269	1	no	—
85	#418	—	269	2	no	—
86	C₂×#100	—	269	2	no	—
87	#405	—	269	1	no	—
88	C₂²×D₄	C₂×D₂₆₉	1	2	yes	—
89	C₂×D₈	—	2	1	no	—
90	C₂×D₈	—	2	1	no	—
91	C₂×D₈	—	2	2	no	—
92	C₂×D₈	Dic₂₆₉	1	4	yes	complemented
93	C₂×M₁₆	—	269	1	no	—
94	C₂×M₁₆	—	269	2	no	—
95	C₂²×C₈	C₂×D₂₆₉	1	1	yes	characteristic
96	#117	—	269	1	no	—
97	#108	—	538	2	no	—
98	#102	—	538	4	no	—
99	#100	—	538	4	no	—
100	#97	—	538	1	no	—
101	C₄×C₈	—	538	1	no	—
102	C₂⁴	—	2	2	no	—
103	C₂×D₄	—	2	2	no	—
104	C₂×D₄	—	2	2	no	—
105	C₂×D₄	—	2	2	no	—
106	C₂×D₄	—	2	2	no	—
107	C₂×D₄	—	2	2	no	—
108	C₂×D₄	C₂×Dic₂₆₉	1	4	yes	—
109	C₂²×C₄	C₂²×D₂₆₉	1	1	yes	characteristic
110	D₈	—	2	4	no	—
111	D₈	—	2	4	no	—
112	M₁₆	—	269	2	no	—
113	M₁₆	—	538	1	no	—
114	M₁₆	—	538	2	no	—
115	M₁₆	—	538	2	no	—
116	C₂×C₈	—	538	1	no	—
117	C₂×C₈	—	538	2	no	—
118	C₂×C₈	C₂×Dic₂₆₉	1	2	yes	—
119	C₂×C₈	C₂²⋊D₂₆₉	1	1	yes	characteristic
120	C₂×C₈	C₂²⋊D₂₆₉	1	1	yes	characteristic
121	C₂×C₈	C₂²⋊D₂₆₉	1	2	yes	—
122	C₄⋊C₄	—	269	2	no	—
123	C₂²⋊C₄	—	538	1	no	—
124	C₄²	—	538	1	no	—
125	C₂³	—	2	4	no	—
126	C₂³	—	4	2	no	—
127	C₂³	—	4	2	no	—
128	C₂³	—	8	4	no	—
129	C₂³	#16864401	1	1	yes	characteristic
130	D₄	—	2	4	no	—
131	D₄	—	2	4	no	—
132	D₄	—	4	2	no	—
133	D₄	—	4	2	no	—
134	C₂×C₄	—	538	2	no	—
135	C₂×C₄	C₂²×Dic₂₆₉	1	1	yes	characteristic
136	C₂×C₄	D₄×D₂₆₉	1	1	yes	characteristic
137	C₂×C₄	D₄×D₂₆₉	1	1	yes	characteristic
138	C₂×C₄	C₂×C₂²⋊D₂₆₉	1	1	yes	characteristic
139	C₂×C₄	C₂×C₂²⋊D₂₆₉	1	1	yes	characteristic
140	C₂×C₄	#16864401	1	1	yes	characteristic
141	C₈	—	538	2	no	—
142	C₈	—	1076	2	no	—
143	C₈	C₂²⋊Dic₂₆₉	1	2	yes	—
144	C₈	C₂²⋊Dic₂₆₉	1	2	yes	—
145	C₂²	—	2	2	no	—
146	C₂²	—	4	4	no	—
147	C₂²	—	8	4	no	—
148	C₂²	—	8	4	no	—
149	C₂²	—	8	4	no	—
150	C₂²	D₄×Dic₂₆₉	1	1	yes	characteristic
151	C₂²	#22575424	1	1	yes	characteristic
152	C₂²	#22575440	1	1	yes	characteristic
153	C₄	—	1076	2	no	—
154	C₄	C₂×C₂²⋊Dic₂₆₉	1	1	yes	characteristic
155	C₄	D₄×Dic₂₆₉	1	1	yes	characteristic
156	C₄	#22575424	1	1	yes	characteristic
157	C₄	#22575449	1	1	yes	characteristic
158	C₂	—	2	1	no	—
159	C₂	—	2	1	no	—
160	C₂	—	8	4	no	—
161	C₂	#31770508	1	1	yes	characteristic
162	C₂	#31770872	1	2	yes	—

subgroup

G/H

class size

multiplicity

normal?

properties

C₂×#22575389

C₂

yes

maximal, characteristic, complemented

#31770610

C₂

yes

maximal, complemented

C₂×#22575370

C₂

yes

maximal, complemented

#31770560

C₂

yes

maximal, characteristic, complemented

C₂²×D₈×C₂₆₉

C₂

yes

maximal, characteristic

C₂×#16864386

C₂²

yes

—

C₂×#16864386

C₂²

yes

characteristic

#22575412

C₂²

yes

—

—

—

—

C₈×Dic₂₆₉

—

C₂²×D₄×C₂₆₉

C₂²

yes

—

C₂×D₈×C₂₆₉

—

C₂×D₈×C₂₆₉

—

C₂×D₈×C₂₆₉

—

C₂×D₈×C₂₆₉

C₄

yes

complemented

C₂²×C₈×C₂₆₉

C₂²

yes

—

—

—

D₄

yes

—

C₂×#14524130

—

C₂×#14524130

—

C₄⋊Dic₂₆₉

D₄

yes

—

C₂²⋊Dic₂₆₉

—

C₄×Dic₂₆₉

—

C₂⁴×C₂₆₉

—

C₂×D₄×C₂₆₉

—

C₂×D₄×C₂₆₉

—

C₂×D₄×C₂₆₉

—

C₂×D₄×C₂₆₉

—

C₂×D₄×C₂₆₉

—

C₂×D₄×C₂₆₉

C₂×C₄

yes

—

C₂²×C₄×C₂₆₉

C₂³

yes

characteristic

D₈×C₂₆₉

—

D₈×C₂₆₉

—

C₂×C₈×C₂₆₉

C₂×C₄

yes

—

C₂×C₈×C₂₆₉

D₄

yes

characteristic

C₂×C₈×C₂₆₉

D₄

yes

characteristic

C₂×C₈×C₂₆₉

D₄

yes

—

C₂×Dic₂₆₉

—

#14524130

—

#14524130

—

C₂³×C₂₆₉

—

C₂³×C₂₆₉

—

C₂³×C₂₆₉

—

C₂³×C₂₆₉

—

C₂³×C₂₆₉

#41

yes

characteristic

D₄×C₂₆₉

—

D₄×C₂₆₉

—

D₄×C₂₆₉

—

D₄×C₂₆₉

—

C₂×C₄×C₂₆₉

C₂²×C₄

yes

characteristic

C₂×C₄×C₂₆₉

C₂×D₄

yes

characteristic

C₂×C₄×C₂₆₉

C₂×D₄

yes

characteristic

C₂×C₄×C₂₆₉

C₂×D₄

yes

characteristic

C₂×C₄×C₂₆₉

C₂×D₄

yes

characteristic

C₂×C₄×C₂₆₉

#41

yes

characteristic

C₈×C₂₆₉

C₂²⋊C₄

yes

—

C₈×C₂₆₉

C₂²⋊C₄

yes

—

C₂²×C₂₆₉

—

C₂²×C₂₆₉

—

C₂²×C₂₆₉

—

C₂²×C₂₆₉

—

C₂²×C₂₆₉

—

C₂²×C₂₆₉

C₄×D₄

yes

C₂²×C₂₆₉

yes

C₂²×C₂₆₉

yes

C₄×C₂₆₉

C₂×C₂²⋊C₄

yes

characteristic

C₄×C₂₆₉

C₄×D₄

yes

C₄×C₂₆₉

yes

C₄×C₂₆₉

yes

Dic₂₆₉

—

C₂×C₂₆₉

—

C₂×C₂₆₉

—

C₂×C₂₆₉

—

C₂×C₂₆₉

#390

yes

characteristic

C₂×C₂₆₉

#472

yes

—

C₂₆₉

#1978

yes

characteristic, complemented, Sylow

#1978

—

269

maximal, Sylow

C₂²×D₈

D₂₆₉

yes

characteristic

C₂×#108

—

269

—

#418

—

269

—

C₂×#100

—

269

—

#405

—

269

—

C₂²×D₄

C₂×D₂₆₉

yes

—

C₂×D₈

—

C₂×D₈

—

C₂×D₈

—

C₂×D₈

Dic₂₆₉

yes

complemented

C₂×M₁₆

—

269

—

C₂×M₁₆

—

269

—

C₂²×C₈

C₂×D₂₆₉

yes

characteristic

#117

—

269

—

#108

—

538

—

#102

—

538

—

#100

—

538

—

100

#97

—

538

—

101

C₄×C₈

—

538

—

102

C₂⁴

—

103

C₂×D₄

—

104

C₂×D₄

—

105

C₂×D₄

—

106

C₂×D₄

—

107

C₂×D₄

—

108

C₂×D₄

C₂×Dic₂₆₉

yes

—

109

C₂²×C₄

C₂²×D₂₆₉

yes

characteristic

110

D₈

—

111

D₈

—

112

M₁₆

—

269

—

113

M₁₆

—

538

—

114

M₁₆

—

538

—

115

M₁₆

—

538

—

116

C₂×C₈

—

538

—

117

C₂×C₈

—

538

—

118

C₂×C₈

C₂×Dic₂₆₉

yes

—

119

C₂×C₈

C₂²⋊D₂₆₉

yes

characteristic

120

C₂×C₈

C₂²⋊D₂₆₉

yes

characteristic

121

C₂×C₈

C₂²⋊D₂₆₉

yes

—

122

C₄⋊C₄

—

269

—

123

C₂²⋊C₄

—

538

—

124

C₄²

—

538

—

125

C₂³

—

126

C₂³

—

127

C₂³

—

128

C₂³

—

129

C₂³

#16864401

yes

characteristic

130

D₄

—

131

D₄

—

132

D₄

—

133

D₄

—

134

C₂×C₄

—

538

—

135

C₂×C₄

C₂²×Dic₂₆₉

yes

characteristic

136

C₂×C₄

D₄×D₂₆₉

yes

characteristic

137

C₂×C₄

D₄×D₂₆₉

yes

characteristic

138

C₂×C₄

C₂×C₂²⋊D₂₆₉

yes

characteristic

139

C₂×C₄

C₂×C₂²⋊D₂₆₉

yes

characteristic

140

C₂×C₄

#16864401

yes

characteristic

141

C₈

—

538

—

142

C₈

—

1076

—

143

C₈

C₂²⋊Dic₂₆₉

yes

—

144

C₈

C₂²⋊Dic₂₆₉

yes

—

145

C₂²

—

146

C₂²

—

147

C₂²

—

148

C₂²

—

149

C₂²

—

150

C₂²

D₄×Dic₂₆₉

yes

characteristic

151

C₂²

#22575424

yes

characteristic

152

C₂²

#22575440

yes

characteristic

153

C₄

—

1076

—

154

C₄

C₂×C₂²⋊Dic₂₆₉

yes

characteristic

155

C₄

D₄×Dic₂₆₉

yes

characteristic

156

C₄

#22575424

yes

characteristic

157

C₄

#22575449

yes

characteristic

158

C₂

—

159

C₂

—

160

C₂

—

161

C₂

#31770508

yes

characteristic

162

C₂

#31770872

yes

—

	subgroup	G/H
center	C₂²	#22575424
commutator	C₂×C₄×C₂₆₉	C₂²×C₄
Frattini	C₂²×C₄	C₂²×D₂₆₉
Fitting, prime radical	C₂²×D₈×C₂₆₉	C₂
socle	C₂²×C₂₆₉	#121
cosocle	C₂²×C₄×C₂₆₉	C₂³
hypercenter	C₂²×D₈	D₂₆₉
hypocenter	C₂₆₉	#1978
LCS	G ▷ C₂×C₄×C₂₆₉ ▷ C₂×C₂₆₉ ▷ C₂₆₉
UCS	C₁ ◁ C₂² ◁ C₂²×C₄ ◁ C₂²×D₈

subgroup

G/H

center

C₂²

#22575424

commutator

C₂×C₄×C₂₆₉

C₂²×C₄

Frattini

C₂²×C₄

C₂²×D₂₆₉

Fitting, prime radical

C₂²×D₈×C₂₆₉

C₂

socle

C₂²×C₂₆₉

#121

cosocle

C₂²×C₄×C₂₆₉

C₂³

hypercenter

C₂²×D₈

D₂₆₉

hypocenter

C₂₆₉

#1978

LCS

G ▷ C₂×C₄×C₂₆₉ ▷ C₂×C₂₆₉ ▷ C₂₆₉

UCS

C₁ ◁ C₂² ◁ C₂²×C₄ ◁ C₂²×D₈

N	H ≅ G/N	type
1. C₂×#22575389	C₂	semidirect
2. #31770610	C₂	semidirect
3. C₂×#22575370	C₂	semidirect
4. #31770560	C₂	semidirect
18. C₂×D₈×C₂₆₉	C₄	semidirect
81. C₂₆₉	#1978	semidirect
92. C₂×D₈	Dic₂₆₉	semidirect

H ≅ G/N

type

1. C₂×#22575389

C₂

semidirect

2. #31770610

C₂

semidirect

3. C₂×#22575370

C₂

semidirect

4. #31770560

C₂

semidirect

18. C₂×D₈×C₂₆₉

C₄

semidirect

81. C₂₆₉

#1978

semidirect

92. C₂×D₈

Dic₂₆₉

semidirect

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

subgroup	index
C₂×#22575389	2
#31770610	2
C₂×#22575370	2
#31770560	2
C₂²×D₈×C₂₆₉	2
#1978	269

subgroup

index

C₂²×D₈×C₂₆₉

269

Subgroups — #46811796

Conjugacy classes of subgroups

Complements

Maximal subgroups