Group #46812041 — subgroups

Subgroups — #46812041

Order 34432.

282 conjugacy classes of subgroups; only the 69 normal subgroups are shown.

normal subgroups only

This group has 282 conjugacy classes of subgroups — too many to draw. Only its 69 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	#31771236	C₂	1	1	yes	maximal, characteristic, complemented
2	#31770610	C₂	1	2	yes	maximal
3	#31770374	C₂	1	1	yes	maximal, characteristic, complemented
4	#31770365	C₂	1	2	yes	maximal, complemented
5	#390×C₂₆₉	C₂	1	1	yes	maximal, characteristic
6	C₂×#16864386	C₂²	1	2	yes	—
7	#22575452	—	2	2	no	—
8	#22575444	—	2	2	no	—
9	#22575432	—	2	2	no	—
10	#22575412	C₂²	1	2	yes	—
11	#22575374	—	2	2	no	—
12	#22575374	—	2	2	no	—
13	C₂²×D₄×C₂₆₉	C₂²	1	1	yes	characteristic
14	C₂×C₄⋊C₄×C₂₆₉	C₂²	1	1	yes	characteristic
15	C₂×C₂²⋊C₄×C₂₆₉	—	2	2	no	—
16	C₂×C₄²×C₂₆₉	C₂²	1	1	yes	characteristic
17	C₂×Dic₅₃₈	—	2	2	no	—
18	#16864386	—	2	2	no	—
19	#16864386	—	2	2	no	—
20	C₂×#14524130	—	2	2	no	—
21	C₄⋊Dic₂₆₉	—	2	2	no	—
22	C₄⋊Dic₂₆₉	D₄	1	2	yes	—
23	C₄⋊Dic₂₆₉	D₄	1	2	yes	—
24	#16864381	—	2	4	no	—
25	C₂²⋊Dic₂₆₉	—	2	2	no	—
26	C₄×Dic₂₆₉	—	2	2	no	—
27	C₂⁴×C₂₆₉	—	2	1	no	—
28	C₂×D₄×C₂₆₉	—	2	2	no	—
29	C₂×D₄×C₂₆₉	—	2	2	no	—
30	C₂×D₄×C₂₆₉	—	2	1	no	—
31	C₂×D₄×C₂₆₉	Q₈	1	2	yes	complemented
32	C₂²×C₄×C₂₆₉	—	2	1	no	—
33	C₂²×C₄×C₂₆₉	D₄	1	2	yes	—
34	C₂²×C₄×C₂₆₉	C₂³	1	1	yes	characteristic
35	C₄⋊C₄×C₂₆₉	—	2	2	no	—
36	C₂²⋊C₄×C₂₆₉	—	4	4	no	—
37	C₄²×C₂₆₉	—	2	2	no	—
38	Dic₅₃₈	—	8	2	no	—
39	C₂×Dic₂₆₉	—	2	2	no	—
40	C₂×Dic₂₆₉	—	2	2	no	—
41	#14524130	—	4	2	no	—
42	C₂³×C₂₆₉	—	2	2	no	—
43	C₂³×C₂₆₉	—	4	2	no	—
44	C₂³×C₂₆₉	—	4	2	no	—
45	C₂³×C₂₆₉	—	4	1	no	—
46	C₂³×C₂₆₉	—	4	1	no	—
47	C₂³×C₂₆₉	C₂×D₄	1	1	yes	characteristic
48	D₄×C₂₆₉	—	2	4	no	—
49	D₄×C₂₆₉	—	4	2	no	—
50	C₂×C₄×C₂₆₉	—	2	2	no	—
51	C₂×C₄×C₂₆₉	—	2	2	no	—
52	C₂×C₄×C₂₆₉	—	4	1	no	—
53	C₂×C₄×C₂₆₉	—	4	1	no	—
54	C₂×C₄×C₂₆₉	—	4	2	no	—
55	C₂×C₄×C₂₆₉	C₂×D₄	1	2	yes	—
56	C₂×C₄×C₂₆₉	C₂×Q₈	1	1	yes	characteristic
57	C₂×C₄×C₂₆₉	#41	1	2	yes	—
58	C₂×C₄×C₂₆₉	#41	1	1	yes	characteristic
59	C₂²×C₂₆₉	—	2	1	no	—
60	C₂²×C₂₆₉	—	2	1	no	—
61	C₂²×C₂₆₉	—	4	4	no	—
62	C₂²×C₂₆₉	—	4	2	no	—
63	C₂²×C₂₆₉	—	8	2	no	—
64	C₂²×C₂₆₉	—	8	1	no	—
65	C₂²×C₂₆₉	—	8	1	no	—
66	C₂²×C₂₆₉	#120	1	1	yes	characteristic
67	C₂²×C₂₆₉	C₂²⋊Q₈	1	1	yes	characteristic
68	C₂²×C₂₆₉	#124	1	1	yes	characteristic
69	C₄×C₂₆₉	—	4	2	no	—
70	C₄×C₂₆₉	—	4	2	no	—
71	C₄×C₂₆₉	C₂²⋊Q₈	1	2	yes	—
72	C₄×C₂₆₉	#124	1	2	yes	—
73	Dic₂₆₉	—	4	2	no	—
74	Dic₂₆₉	—	4	2	no	—
75	C₂×C₂₆₉	—	2	1	no	—
76	C₂×C₂₆₉	—	2	1	no	—
77	C₂×C₂₆₉	—	8	2	no	—
78	C₂×C₂₆₉	#393	1	1	yes	characteristic
79	C₂×C₂₆₉	#453	1	1	yes	characteristic
80	C₂×C₂₆₉	#454	1	1	yes	characteristic
81	C₂₆₉	#2027	1	1	yes	characteristic, complemented, Sylow
82	#2027	—	269	1	no	maximal, Sylow
83	#544	—	269	1	no	—
84	#418	—	269	2	no	—
85	#390	D₂₆₉	1	1	yes	characteristic
86	#343	—	269	1	no	—
87	#339	—	269	2	no	—
88	C₂²×D₄	C₂×D₂₆₉	1	1	yes	characteristic
89	C₂×M₁₆	—	269	2	no	—
90	C₄⋊Q₈	—	538	2	no	—
91	#125	—	538	2	no	—
92	C₂²⋊Q₈	—	538	2	no	—
93	#117	—	269	2	no	—
94	C₂×C₄⋊C₄	C₂×D₂₆₉	1	1	yes	characteristic
95	C₂×C₂²⋊C₄	—	2	2	no	—
96	C₂×C₄²	C₂×D₂₆₉	1	1	yes	characteristic
97	#102	—	538	2	no	—
98	#102	—	538	2	no	—
99	C₂⁴	—	2	1	no	—
100	C₂×Q₈	—	538	2	no	—
101	C₂×D₄	—	2	2	no	—
102	C₂×D₄	—	2	2	no	—
103	C₂×D₄	—	2	1	no	—
104	C₂×D₄	Dic₅₃₈	1	2	yes	complemented
105	C₂²×C₄	—	2	1	no	—
106	C₂²×C₄	C₂²⋊D₂₆₉	1	2	yes	—
107	C₂²×C₄	C₂²×D₂₆₉	1	1	yes	characteristic
108	M₁₆	—	538	2	no	—
109	M₁₆	—	538	2	no	—
110	C₂×C₈	—	538	2	no	—
111	C₄⋊C₄	—	2	2	no	—
112	C₄⋊C₄	—	269	2	no	—
113	C₄⋊C₄	—	269	2	no	—
114	C₄⋊C₄	—	538	2	no	—
115	C₄⋊C₄	—	538	4	no	—
116	C₂²⋊C₄	—	4	4	no	—
117	C₂²⋊C₄	—	538	2	no	—
118	C₄²	—	2	2	no	—
119	C₄²	—	538	2	no	—
120	C₂³	—	2	2	no	—
121	C₂³	—	4	2	no	—
122	C₂³	—	4	2	no	—
123	C₂³	—	4	1	no	—
124	C₂³	—	4	1	no	—
125	C₂³	C₂×C₂²⋊D₂₆₉	1	1	yes	characteristic
126	Q₈	—	2152	2	no	—
127	D₄	—	2	4	no	—
128	D₄	—	4	2	no	—
129	C₂×C₄	—	2	2	no	—
130	C₂×C₄	—	2	2	no	—
131	C₂×C₄	—	4	1	no	—
132	C₂×C₄	—	4	1	no	—
133	C₂×C₄	—	4	2	no	—
134	C₂×C₄	—	538	2	no	—
135	C₂×C₄	—	538	2	no	—
136	C₂×C₄	D₄×D₂₆₉	1	2	yes	—
137	C₂×C₄	C₂×Dic₅₃₈	1	1	yes	characteristic
138	C₂×C₄	#16864401	1	2	yes	—
139	C₂×C₄	#16864402	1	1	yes	characteristic
140	C₈	—	1076	2	no	—
141	C₂²	—	2	1	no	—
142	C₂²	—	2	1	no	—
143	C₂²	—	4	4	no	—
144	C₂²	—	4	2	no	—
145	C₂²	—	8	2	no	—
146	C₂²	—	8	1	no	—
147	C₂²	—	8	1	no	—
148	C₂²	#22575421	1	1	yes	characteristic
149	C₂²	#22575432	1	1	yes	characteristic
150	C₂²	#22575440	1	1	yes	characteristic
151	C₄	—	4	2	no	—
152	C₄	—	4	2	no	—
153	C₄	—	1076	2	no	—
154	C₄	—	1076	2	no	—
155	C₄	#22575430	1	2	yes	—
156	C₄	#22575439	1	2	yes	—
157	C₂	—	2	1	no	—
158	C₂	—	2	1	no	—
159	C₂	—	8	2	no	—
160	C₂	#31770517	1	1	yes	characteristic
161	C₂	#31770772	1	1	yes	characteristic
162	C₂	#31770777	1	1	yes	characteristic

subgroup

G/H

class size

multiplicity

normal?

properties

#31771236

C₂

yes

maximal, characteristic, complemented

#31770610

C₂

yes

maximal

#31770374

C₂

yes

maximal, characteristic, complemented

#31770365

C₂

yes

maximal, complemented

#390×C₂₆₉

C₂

yes

maximal, characteristic

C₂×#16864386

C₂²

yes

—

—

—

—

C₂²

yes

—

#22575374

—

#22575374

—

C₂²×D₄×C₂₆₉

C₂²

yes

characteristic

C₂×C₄⋊C₄×C₂₆₉

C₂²

yes

characteristic

C₂×C₂²⋊C₄×C₂₆₉

—

C₂×C₄²×C₂₆₉

C₂²

yes

characteristic

C₂×Dic₅₃₈

—

#16864386

—

#16864386

—

C₂×#14524130

—

C₄⋊Dic₂₆₉

—

C₄⋊Dic₂₆₉

D₄

yes

—

C₄⋊Dic₂₆₉

D₄

yes

—

#16864381

—

C₂²⋊Dic₂₆₉

—

C₄×Dic₂₆₉

—

C₂⁴×C₂₆₉

—

C₂×D₄×C₂₆₉

—

C₂×D₄×C₂₆₉

—

C₂×D₄×C₂₆₉

—

C₂×D₄×C₂₆₉

Q₈

yes

complemented

C₂²×C₄×C₂₆₉

—

C₂²×C₄×C₂₆₉

D₄

yes

—

C₂²×C₄×C₂₆₉

C₂³

yes

characteristic

C₄⋊C₄×C₂₆₉

—

C₂²⋊C₄×C₂₆₉

—

C₄²×C₂₆₉

—

Dic₅₃₈

—

C₂×Dic₂₆₉

—

C₂×Dic₂₆₉

—

#14524130

—

C₂³×C₂₆₉

—

C₂³×C₂₆₉

—

C₂³×C₂₆₉

—

C₂³×C₂₆₉

—

C₂³×C₂₆₉

—

C₂³×C₂₆₉

C₂×D₄

yes

characteristic

D₄×C₂₆₉

—

D₄×C₂₆₉

—

C₂×C₄×C₂₆₉

—

C₂×C₄×C₂₆₉

—

C₂×C₄×C₂₆₉

—

C₂×C₄×C₂₆₉

—

C₂×C₄×C₂₆₉

—

C₂×C₄×C₂₆₉

C₂×D₄

yes

—

C₂×C₄×C₂₆₉

C₂×Q₈

yes

characteristic

C₂×C₄×C₂₆₉

#41

yes

—

C₂×C₄×C₂₆₉

#41

yes

characteristic

C₂²×C₂₆₉

—

C₂²×C₂₆₉

—

C₂²×C₂₆₉

—

C₂²×C₂₆₉

—

C₂²×C₂₆₉

—

C₂²×C₂₆₉

—

C₂²×C₂₆₉

—

C₂²×C₂₆₉

#120

yes

characteristic

C₂²×C₂₆₉

C₂²⋊Q₈

yes

characteristic

C₂²×C₂₆₉

#124

yes

characteristic

C₄×C₂₆₉

—

C₄×C₂₆₉

—

C₄×C₂₆₉

C₂²⋊Q₈

yes

—

C₄×C₂₆₉

#124

yes

—

Dic₂₆₉

—

Dic₂₆₉

—

C₂×C₂₆₉

—

C₂×C₂₆₉

—

C₂×C₂₆₉

—

C₂×C₂₆₉

yes

C₂×C₂₆₉

yes

C₂×C₂₆₉

yes

C₂₆₉

yes

characteristic, complemented, Sylow

#2027

—

269

maximal, Sylow

#544

—

269

—

#418

—

269

—

#390

D₂₆₉

yes

characteristic

#343

—

269

—

#339

—

269

—

C₂²×D₄

C₂×D₂₆₉

yes

characteristic

C₂×M₁₆

—

269

—

C₄⋊Q₈

—

538

—

#125

—

538

—

C₂²⋊Q₈

—

538

—

#117

—

269

—

C₂×C₄⋊C₄

C₂×D₂₆₉

yes

characteristic

C₂×C₂²⋊C₄

—

C₂×C₄²

C₂×D₂₆₉

yes

characteristic

#102

—

538

—

#102

—

538

—

C₂⁴

—

100

C₂×Q₈

—

538

—

101

C₂×D₄

—

102

C₂×D₄

—

103

C₂×D₄

—

104

C₂×D₄

Dic₅₃₈

yes

complemented

105

C₂²×C₄

—

106

C₂²×C₄

C₂²⋊D₂₆₉

yes

—

107

C₂²×C₄

C₂²×D₂₆₉

yes

characteristic

108

M₁₆

—

538

—

109

M₁₆

—

538

—

110

C₂×C₈

—

538

—

111

C₄⋊C₄

—

112

C₄⋊C₄

—

269

—

113

C₄⋊C₄

—

269

—

114

C₄⋊C₄

—

538

—

115

C₄⋊C₄

—

538

—

116

C₂²⋊C₄

—

117

C₂²⋊C₄

—

538

—

118

C₄²

—

119

C₄²

—

538

—

120

C₂³

—

121

C₂³

—

122

C₂³

—

123

C₂³

—

124

C₂³

—

125

C₂³

C₂×C₂²⋊D₂₆₉

yes

characteristic

126

Q₈

—

2152

—

127

D₄

—

128

D₄

—

129

C₂×C₄

—

130

C₂×C₄

—

131

C₂×C₄

—

132

C₂×C₄

—

133

C₂×C₄

—

134

C₂×C₄

—

538

—

135

C₂×C₄

—

538

—

136

C₂×C₄

D₄×D₂₆₉

yes

—

137

C₂×C₄

C₂×Dic₅₃₈

yes

characteristic

138

C₂×C₄

#16864401

yes

—

139

C₂×C₄

#16864402

yes

characteristic

140

C₈

—

1076

—

141

C₂²

—

142

C₂²

—

143

C₂²

—

144

C₂²

—

145

C₂²

—

146

C₂²

—

147

C₂²

—

148

C₂²

#22575421

yes

characteristic

149

C₂²

#22575432

yes

characteristic

150

C₂²

#22575440

yes

characteristic

151

C₄

—

152

C₄

—

153

C₄

—

1076

—

154

C₄

—

1076

—

155

C₄

#22575430

yes

—

156

C₄

#22575439

yes

—

157

C₂

—

158

C₂

—

159

C₂

—

160

C₂

#31770517

yes

characteristic

161

C₂

#31770772

yes

characteristic

162

C₂

#31770777

yes

characteristic

	subgroup	G/H
center	C₂²	#22575421
commutator, cosocle	C₂²×C₄×C₂₆₉	C₂³
Frattini	C₂²×C₄	C₂²×D₂₆₉
Fitting	#390×C₂₆₉	C₂
socle	C₂²×C₂₆₉	#120
prime radical	C₂²×D₄×C₂₆₉	C₂²
hypercenter	#390	D₂₆₉
hypocenter	C₂₆₉	#2027
LCS	G ▷ C₂²×C₄×C₂₆₉ ▷ C₂×C₂₆₉ ▷ C₂₆₉
UCS	C₁ ◁ C₂² ◁ C₂²×C₄ ◁ #390

subgroup

G/H

center

C₂²

#22575421

commutator, cosocle

C₂²×C₄×C₂₆₉

C₂³

Frattini

C₂²×C₄

C₂²×D₂₆₉

Fitting

#390×C₂₆₉

C₂

socle

C₂²×C₂₆₉

#120

prime radical

C₂²×D₄×C₂₆₉

C₂²

D₂₆₉

C₂₆₉

G ▷ C₂²×C₄×C₂₆₉ ▷ C₂×C₂₆₉ ▷ C₂₆₉

UCS

C₁ ◁ C₂² ◁ C₂²×C₄ ◁ #390

N	H ≅ G/N	type
1. #31771236	C₂	semidirect
3. #31770374	C₂	semidirect
4. #31770365	C₂	semidirect
31. C₂×D₄×C₂₆₉	Q₈	semidirect
81. C₂₆₉	#2027	semidirect
104. C₂×D₄	Dic₅₃₈	semidirect

H ≅ G/N

type

1. #31771236

C₂

semidirect

3. #31770374

C₂

semidirect

4. #31770365

C₂

semidirect

31. C₂×D₄×C₂₆₉

Q₈

semidirect

81. C₂₆₉

#2027

semidirect

104. C₂×D₄

Dic₅₃₈

semidirect

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

subgroup	index
#31771236	2
#31770610	2
#31770374	2
#31770365	2
#390×C₂₆₉	2
#2027	269

subgroup

index

#390×C₂₆₉

269

Subgroups — #46812041

Conjugacy classes of subgroups

Complements

Maximal subgroups