Group #46810046 — subgroups

Subgroups — #46810046

Order 34432.

188 conjugacy classes of subgroups; only the 57 normal subgroups are shown.

normal subgroups only

This group has 188 conjugacy classes of subgroups — too many to draw. Only its 57 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	#31771334	C₂	1	1	yes	maximal, characteristic
2	#31770956	C₂	1	1	yes	maximal, characteristic
3	#31770901	C₂	1	1	yes	maximal, characteristic
4	#31770900	C₂	1	1	yes	maximal, characteristic
5	#31770352	C₂	1	1	yes	maximal, characteristic
6	#31770337	C₂	1	1	yes	maximal, characteristic
7	Q₈⋊C₈×C₂₆₉	C₂	1	1	yes	maximal, characteristic
8	#22575453	—	2	1	no	—
9	#22575453	C₂²	1	1	yes	characteristic
10	#22575453	C₂²	1	1	yes	characteristic
11	#22575444	—	2	2	no	—
12	#22575444	—	2	1	no	—
13	Q₈×Dic₂₆₉	—	2	1	no	—
14	C₄×Dic₅₃₈	—	2	1	no	—
15	C₄×Dic₅₃₈	C₂²	1	1	yes	characteristic
16	#22575387	—	2	1	no	—
17	#22575386	—	2	1	no	—
18	C₄⋊#14524130	C₂²	1	1	yes	characteristic
19	#22575378	—	2	1	no	—
20	#22575378	—	2	2	no	—
21	#22575377	—	2	1	no	—
22	C₄×Q₈×C₂₆₉	C₂²	1	1	yes	characteristic
23	C₄⋊C₈×C₂₆₉	C₂²	1	1	yes	characteristic
24	C₄×C₈×C₂₆₉	C₂²	1	1	yes	characteristic
25	C₂×Dic₅₃₈	—	2	1	no	—
26	C₂×Dic₅₃₈	—	2	1	no	—
27	C₂×Dic₅₃₈	D₄	1	1	yes	characteristic
28	C₂×#14524130	—	2	1	no	—
29	C₄⋊Dic₂₆₉	—	2	2	no	—
30	C₄⋊Dic₂₆₉	—	2	2	no	—
31	C₄⋊Dic₂₆₉	—	2	1	no	—
32	C₄⋊Dic₂₆₉	—	2	1	no	—
33	C₄⋊Dic₂₆₉	—	2	1	no	—
34	C₄⋊Dic₂₆₉	D₄	1	2	yes	—
35	C₄⋊Dic₂₆₉	D₄	1	1	yes	characteristic
36	#16864381	—	2	2	no	—
37	#16864381	—	2	1	no	—
38	C₄×Dic₂₆₉	—	2	2	no	—
39	C₄×Dic₂₆₉	—	2	1	no	—
40	C₂×Q₈×C₂₆₉	D₄	1	1	yes	characteristic
41	C₂×C₈×C₂₆₉	—	2	1	no	—
42	C₂×C₈×C₂₆₉	—	2	1	no	—
43	C₄⋊C₄×C₂₆₉	—	2	1	no	—
44	C₄⋊C₄×C₂₆₉	D₄	1	1	yes	characteristic
45	C₄²×C₂₆₉	—	2	1	no	—
46	C₄²×C₂₆₉	C₂³	1	1	yes	characteristic
47	Dic₅₃₈	—	2	1	no	—
48	Dic₅₃₈	—	2	1	no	—
49	Dic₅₃₈	—	4	1	no	—
50	Dic₅₃₈	—	4	1	no	—
51	Dic₅₃₈	—	4	2	no	—
52	C₂×Dic₂₆₉	—	2	1	no	—
53	C₂×Dic₂₆₉	—	2	1	no	—
54	C₂×Dic₂₆₉	—	2	2	no	—
55	C₂×Dic₂₆₉	—	4	1	no	—
56	#14524130	—	4	1	no	—
57	Q₈×C₂₆₉	—	2	1	no	—
58	Q₈×C₂₆₉	Q₁₆	1	2	yes	—
59	C₂×C₄×C₂₆₉	—	2	1	no	—
60	C₂×C₄×C₂₆₉	—	2	1	no	—
61	C₂×C₄×C₂₆₉	C₂×D₄	1	1	yes	characteristic
62	C₂×C₄×C₂₆₉	C₂×D₄	1	1	yes	characteristic
63	C₂×C₄×C₂₆₉	C₂×D₄	1	1	yes	characteristic
64	C₈×C₂₆₉	—	4	1	no	—
65	C₈×C₂₆₉	—	4	1	no	—
66	C₂²×C₂₆₉	#120	1	1	yes	characteristic
67	C₄×C₂₆₉	—	2	2	no	—
68	C₄×C₂₆₉	—	2	1	no	—
69	C₄×C₂₆₉	—	4	1	no	—
70	C₄×C₂₆₉	C₂×Q₁₆	1	1	yes	characteristic
71	C₄×C₂₆₉	#135	1	1	yes	characteristic
72	C₄×C₂₆₉	#137	1	1	yes	characteristic
73	C₄×C₂₆₉	#137	1	1	yes	characteristic
74	Dic₂₆₉	—	4	1	no	—
75	Dic₂₆₉	—	4	2	no	—
76	Dic₂₆₉	—	4	1	no	—
77	Dic₂₆₉	—	8	1	no	—
78	C₂×C₂₆₉	#451	1	1	yes	characteristic
79	C₂×C₂₆₉	#452	1	1	yes	characteristic
80	C₂×C₂₆₉	#454	1	1	yes	characteristic
81	C₂₆₉	#1636	1	1	yes	characteristic, complemented, Sylow
82	#1636	—	269	1	no	maximal, Sylow
83	#557	—	269	1	no	—
84	#487	—	269	1	no	—
85	#477	—	269	1	no	—
86	#477	—	269	1	no	—
87	#332	—	269	1	no	—
88	Q₈⋊C₈	—	269	1	no	—
89	Q₈⋊C₈	D₂₆₉	1	1	yes	characteristic
90	C₄⋊Q₈	—	269	1	no	—
91	C₄⋊Q₈	—	269	1	no	—
92	C₄⋊Q₈	—	538	1	no	—
93	#125	—	538	2	no	—
94	#125	—	538	1	no	—
95	C₄×Q₈	—	269	1	no	—
96	C₄×Q₈	—	538	1	no	—
97	C₄×Q₈	—	538	1	no	—
98	C₄×Q₈	C₂×D₂₆₉	1	1	yes	characteristic
99	#107	—	538	1	no	—
100	#107	—	538	1	no	—
101	C₄⋊C₈	—	269	1	no	—
102	C₄⋊C₈	C₂×D₂₆₉	1	1	yes	characteristic
103	#103	—	538	2	no	—
104	#103	—	538	1	no	—
105	#103	—	538	1	no	—
106	C₄×C₈	C₂×D₂₆₉	1	1	yes	characteristic
107	C₂×Q₈	—	269	1	no	—
108	C₂×Q₈	—	538	1	no	—
109	C₂×Q₈	—	538	1	no	—
110	C₂×Q₈	D₁₀₇₆	1	1	yes	characteristic
111	C₂×C₈	—	2	1	no	—
112	C₂×C₈	—	2	1	no	—
113	C₂×C₈	—	538	1	no	—
114	C₄⋊C₄	—	2	1	no	—
115	C₄⋊C₄	—	269	2	no	—
116	C₄⋊C₄	—	269	1	no	—
117	C₄⋊C₄	—	538	2	no	—
118	C₄⋊C₄	—	538	2	no	—
119	C₄⋊C₄	—	538	2	no	—
120	C₄⋊C₄	—	538	1	no	—
121	C₄⋊C₄	—	538	1	no	—
122	C₄⋊C₄	—	538	1	no	—
123	C₄⋊C₄	—	538	1	no	—
124	C₄⋊C₄	D₁₀₇₆	1	1	yes	characteristic
125	C₄²	—	2	1	no	—
126	C₄²	—	538	2	no	—
127	C₄²	—	538	1	no	—
128	C₄²	C₂²×D₂₆₉	1	1	yes	characteristic
129	Q₈	—	2	1	no	—
130	Q₈	—	538	1	no	—
131	Q₈	—	538	1	no	—
132	Q₈	—	1076	2	no	—
133	Q₈	—	1076	1	no	—
134	Q₈	—	1076	1	no	—
135	Q₈	Dic₁₀₇₆	1	2	yes	—
136	C₂×C₄	—	2	1	no	—
137	C₂×C₄	—	2	1	no	—
138	C₂×C₄	—	538	1	no	—
139	C₂×C₄	—	538	2	no	—
140	C₂×C₄	—	538	1	no	—
141	C₂×C₄	—	1076	1	no	—
142	C₂×C₄	D₄×D₂₆₉	1	1	yes	characteristic
143	C₂×C₄	D₄×D₂₆₉	1	1	yes	characteristic
144	C₂×C₄	C₂×D₁₀₇₆	1	1	yes	characteristic
145	C₈	—	4	1	no	—
146	C₈	—	4	1	no	—
147	C₈	—	1076	1	no	—
148	C₂²	#22575423	1	1	yes	characteristic
149	C₄	—	2	2	no	—
150	C₄	—	2	1	no	—
151	C₄	—	4	1	no	—
152	C₄	—	1076	1	no	—
153	C₄	—	1076	2	no	—
154	C₄	—	1076	1	no	—
155	C₄	—	2152	1	no	—
156	C₄	C₂×Dic₁₀₇₆	1	1	yes	characteristic
157	C₄	#22575474	1	1	yes	characteristic
158	C₄	#22575485	1	1	yes	characteristic
159	C₄	#22575486	1	1	yes	characteristic
160	C₂	#31770763	1	1	yes	characteristic
161	C₂	#31770770	1	1	yes	characteristic
162	C₂	#31770779	1	1	yes	characteristic

subgroup

G/H

class size

multiplicity

normal?

properties

#31771334

C₂

yes

maximal, characteristic

#31770956

C₂

yes

maximal, characteristic

#31770901

C₂

yes

maximal, characteristic

#31770900

C₂

yes

maximal, characteristic

#31770352

C₂

yes

maximal, characteristic

#31770337

C₂

yes

maximal, characteristic

Q₈⋊C₈×C₂₆₉

C₂

yes

maximal, characteristic

#22575453

—

#22575453

C₂²

yes

characteristic

#22575453

C₂²

yes

characteristic

#22575444

—

#22575444

—

Q₈×Dic₂₆₉

—

C₄×Dic₅₃₈

—

C₄×Dic₅₃₈

C₂²

yes

—

—

C₂²

yes

—

—

—

C₄×Q₈×C₂₆₉

C₂²

yes

characteristic

C₄⋊C₈×C₂₆₉

C₂²

yes

characteristic

C₄×C₈×C₂₆₉

C₂²

yes

characteristic

C₂×Dic₅₃₈

—

C₂×Dic₅₃₈

—

C₂×Dic₅₃₈

D₄

yes

characteristic

C₂×#14524130

—

C₄⋊Dic₂₆₉

—

C₄⋊Dic₂₆₉

—

C₄⋊Dic₂₆₉

—

C₄⋊Dic₂₆₉

—

C₄⋊Dic₂₆₉

—

C₄⋊Dic₂₆₉

D₄

yes

—

C₄⋊Dic₂₆₉

D₄

yes

characteristic

#16864381

—

#16864381

—

C₄×Dic₂₆₉

—

C₄×Dic₂₆₉

—

C₂×Q₈×C₂₆₉

D₄

yes

characteristic

C₂×C₈×C₂₆₉

—

C₂×C₈×C₂₆₉

—

C₄⋊C₄×C₂₆₉

—

C₄⋊C₄×C₂₆₉

D₄

yes

characteristic

C₄²×C₂₆₉

—

C₄²×C₂₆₉

C₂³

yes

characteristic

Dic₅₃₈

—

Dic₅₃₈

—

Dic₅₃₈

—

Dic₅₃₈

—

Dic₅₃₈

—

C₂×Dic₂₆₉

—

C₂×Dic₂₆₉

—

C₂×Dic₂₆₉

—

C₂×Dic₂₆₉

—

#14524130

—

Q₈×C₂₆₉

—

Q₈×C₂₆₉

Q₁₆

yes

—

C₂×C₄×C₂₆₉

—

C₂×C₄×C₂₆₉

—

C₂×C₄×C₂₆₉

C₂×D₄

yes

characteristic

C₂×C₄×C₂₆₉

C₂×D₄

yes

characteristic

C₂×C₄×C₂₆₉

C₂×D₄

yes

characteristic

C₈×C₂₆₉

—

C₈×C₂₆₉

—

C₂²×C₂₆₉

#120

yes

characteristic

C₄×C₂₆₉

—

C₄×C₂₆₉

—

C₄×C₂₆₉

—

C₄×C₂₆₉

C₂×Q₁₆

yes

C₄×C₂₆₉

yes

C₄×C₂₆₉

yes

C₄×C₂₆₉

yes

Dic₂₆₉

—

Dic₂₆₉

—

Dic₂₆₉

—

Dic₂₆₉

—

C₂×C₂₆₉

yes

C₂×C₂₆₉

yes

C₂×C₂₆₉

yes

C₂₆₉

yes

characteristic, complemented, Sylow

#1636

—

269

maximal, Sylow

#557

—

269

—

#487

—

269

—

#477

—

269

—

#477

—

269

—

#332

—

269

—

Q₈⋊C₈

—

269

—

Q₈⋊C₈

D₂₆₉

yes

characteristic

C₄⋊Q₈

—

269

—

C₄⋊Q₈

—

269

—

C₄⋊Q₈

—

538

—

#125

—

538

—

#125

—

538

—

C₄×Q₈

—

269

—

C₄×Q₈

—

538

—

C₄×Q₈

—

538

—

C₄×Q₈

C₂×D₂₆₉

yes

characteristic

#107

—

538

—

100

#107

—

538

—

101

C₄⋊C₈

—

269

—

102

C₄⋊C₈

C₂×D₂₆₉

yes

characteristic

103

#103

—

538

—

104

#103

—

538

—

105

#103

—

538

—

106

C₄×C₈

C₂×D₂₆₉

yes

characteristic

107

C₂×Q₈

—

269

—

108

C₂×Q₈

—

538

—

109

C₂×Q₈

—

538

—

110

C₂×Q₈

D₁₀₇₆

yes

characteristic

111

C₂×C₈

—

112

C₂×C₈

—

113

C₂×C₈

—

538

—

114

C₄⋊C₄

—

115

C₄⋊C₄

—

269

—

116

C₄⋊C₄

—

269

—

117

C₄⋊C₄

—

538

—

118

C₄⋊C₄

—

538

—

119

C₄⋊C₄

—

538

—

120

C₄⋊C₄

—

538

—

121

C₄⋊C₄

—

538

—

122

C₄⋊C₄

—

538

—

123

C₄⋊C₄

—

538

—

124

C₄⋊C₄

D₁₀₇₆

yes

characteristic

125

C₄²

—

126

C₄²

—

538

—

127

C₄²

—

538

—

128

C₄²

C₂²×D₂₆₉

yes

characteristic

129

Q₈

—

130

Q₈

—

538

—

131

Q₈

—

538

—

132

Q₈

—

1076

—

133

Q₈

—

1076

—

134

Q₈

—

1076

—

135

Q₈

Dic₁₀₇₆

yes

—

136

C₂×C₄

—

137

C₂×C₄

—

138

C₂×C₄

—

538

—

139

C₂×C₄

—

538

—

140

C₂×C₄

—

538

—

141

C₂×C₄

—

1076

—

142

C₂×C₄

D₄×D₂₆₉

yes

characteristic

143

C₂×C₄

D₄×D₂₆₉

yes

characteristic

144

C₂×C₄

C₂×D₁₀₇₆

yes

characteristic

145

C₈

—

146

C₈

—

147

C₈

—

1076

—

148

C₂²

#22575423

yes

characteristic

149

C₄

—

150

C₄

—

151

C₄

—

152

C₄

—

1076

—

153

C₄

—

1076

—

154

C₄

—

1076

—

155

C₄

—

2152

—

156

C₄

C₂×Dic₁₀₇₆

yes

characteristic

157

C₄

#22575474

yes

characteristic

158

C₄

#22575485

yes

characteristic

159

C₄

#22575486

yes

characteristic

160

C₂

#31770763

yes

characteristic

161

C₂

#31770770

yes

characteristic

162

C₂

#31770779

yes

characteristic

	subgroup	G/H
center	C₂²	#22575423
commutator, cosocle	C₄²×C₂₆₉	C₂³
Frattini	C₄²	C₂²×D₂₆₉
Fitting	Q₈⋊C₈×C₂₆₉	C₂
socle, prime radical	C₂²×C₂₆₉	#120
hypercenter	Q₈⋊C₈	D₂₆₉
hypocenter	C₂₆₉	#1636
LCS	G ▷ C₄²×C₂₆₉ ▷ C₂²×C₂₆₉ ▷ C₂₆₉
UCS	C₁ ◁ C₂² ◁ C₄² ◁ Q₈⋊C₈

subgroup

G/H

center

C₂²

#22575423

commutator, cosocle

C₄²×C₂₆₉

C₂³

Frattini

C₄²

C₂²×D₂₆₉

Fitting

Q₈⋊C₈×C₂₆₉

C₂

socle, prime radical

C₂²×C₂₆₉

#120

hypercenter

Q₈⋊C₈

D₂₆₉

hypocenter

C₂₆₉

#1636

LCS

G ▷ C₄²×C₂₆₉ ▷ C₂²×C₂₆₉ ▷ C₂₆₉

UCS

C₁ ◁ C₂² ◁ C₄² ◁ Q₈⋊C₈

N	H ≅ G/N	type
81. C₂₆₉	#1636	semidirect

H ≅ G/N

type

81. C₂₆₉

#1636

semidirect

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

subgroup	index
#31771334	2
#31770956	2
#31770901	2
#31770900	2
#31770352	2
#31770337	2
Q₈⋊C₈×C₂₆₉	2
#1636	269

subgroup

index

Q₈⋊C₈×C₂₆₉

269

Subgroups — #46810046

Conjugacy classes of subgroups

Complements

Maximal subgroups