Group #46812046 — subgroups

Subgroups — #46812046

Order 34432.

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

normal subgroups only

This group has 246 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
2	#31770614	C₂	1	2	yes	maximal
3	#31770374	C₂	1	1	yes	maximal, characteristic
4	#31770365	C₂	1	2	yes	maximal
5	#391×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	#22575377	—	2	2	no	—
12	#22575377	—	2	2	no	—
13	C₂²×Q₈×C₂₆₉	C₂²	1	1	yes	characteristic
14	C₂×C₄⋊C₄×C₂₆₉	C₂²	1	1	yes	characteristic
15	C₂×C₄²×C₂₆₉	C₂²	1	1	yes	characteristic
16	#95×C₂₆₉	—	2	2	no	—
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	C₂×Q₈×C₂₆₉	—	2	1	no	—
28	C₂×Q₈×C₂₆₉	—	2	2	no	—
29	C₂×Q₈×C₂₆₉	—	2	2	no	—
30	C₂×Q₈×C₂₆₉	Q₈	1	2	yes	complemented
31	C₂²×C₄×C₂₆₉	—	2	1	no	—
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	C₄²×C₂₆₉	—	2	2	no	—
37	Dic₅₃₈	—	8	2	no	—
38	C₂×Dic₂₆₉	—	2	2	no	—
39	C₂×Dic₂₆₉	—	2	2	no	—
40	#14524130	—	4	2	no	—
41	C₂³×C₂₆₉	C₂×D₄	1	1	yes	characteristic
42	Q₈×C₂₆₉	—	2	4	no	—
43	Q₈×C₂₆₉	—	4	2	no	—
44	C₂×C₄×C₂₆₉	—	2	2	no	—
45	C₂×C₄×C₂₆₉	—	2	2	no	—
46	C₂×C₄×C₂₆₉	—	2	2	no	—
47	C₂×C₄×C₂₆₉	—	4	1	no	—
48	C₂×C₄×C₂₆₉	—	4	1	no	—
49	C₂×C₄×C₂₆₉	—	4	1	no	—
50	C₂×C₄×C₂₆₉	—	4	1	no	—
51	C₂×C₄×C₂₆₉	—	4	2	no	—
52	C₂×C₄×C₂₆₉	C₂×D₄	1	2	yes	—
53	C₂×C₄×C₂₆₉	C₂×Q₈	1	1	yes	characteristic
54	C₂×C₄×C₂₆₉	#41	1	2	yes	—
55	C₂×C₄×C₂₆₉	#41	1	1	yes	characteristic
56	C₂²×C₂₆₉	—	2	1	no	—
57	C₂²×C₂₆₉	—	2	1	no	—
58	C₂²×C₂₆₉	#120	1	1	yes	characteristic
59	C₂²×C₂₆₉	C₂²⋊Q₈	1	1	yes	characteristic
60	C₂²×C₂₆₉	#124	1	1	yes	characteristic
61	C₄×C₂₆₉	—	4	2	no	—
62	C₄×C₂₆₉	—	4	2	no	—
63	C₄×C₂₆₉	—	4	2	no	—
64	C₄×C₂₆₉	C₂²⋊Q₈	1	2	yes	—
65	C₄×C₂₆₉	#124	1	2	yes	—
66	Dic₂₆₉	—	4	2	no	—
67	Dic₂₆₉	—	4	2	no	—
68	C₂×C₂₆₉	—	2	1	no	—
69	C₂×C₂₆₉	—	2	1	no	—
70	C₂×C₂₆₉	#393	1	1	yes	characteristic
71	C₂×C₂₆₉	#455	1	1	yes	characteristic
72	C₂×C₂₆₉	#456	1	1	yes	characteristic
73	C₂₆₉	#2028	1	1	yes	characteristic, complemented, Sylow
74	#2028	—	269	1	no	maximal, Sylow
75	#544	—	269	1	no	—
76	#419	—	269	2	no	—
77	#391	D₂₆₉	1	1	yes	characteristic
78	#343	—	269	1	no	—
79	#339	—	269	2	no	—
80	C₂²×Q₈	C₂×D₂₆₉	1	1	yes	characteristic
81	C₂×M₁₆	—	269	2	no	—
82	C₄⋊Q₈	—	538	2	no	—
83	#125	—	538	2	no	—
84	C₂²⋊Q₈	—	538	2	no	—
85	#117	—	269	2	no	—
86	C₂×C₄⋊C₄	C₂×D₂₆₉	1	1	yes	characteristic
87	C₂×C₄²	C₂×D₂₆₉	1	1	yes	characteristic
88	#103	—	538	2	no	—
89	#103	—	538	2	no	—
90	#95	—	2	2	no	—
91	C₂×Q₈	—	2	1	no	—
92	C₂×Q₈	—	2	2	no	—
93	C₂×Q₈	—	2	2	no	—
94	C₂×Q₈	—	538	2	no	—
95	C₂×Q₈	Dic₅₃₈	1	2	yes	complemented
96	C₂²×C₄	—	2	1	no	—
97	C₂²×C₄	—	2	1	no	—
98	C₂²×C₄	C₂²⋊D₂₆₉	1	2	yes	—
99	C₂²×C₄	C₂²×D₂₆₉	1	1	yes	characteristic
100	M₁₆	—	538	2	no	—
101	M₁₆	—	538	2	no	—
102	C₂×C₈	—	538	2	no	—
103	C₄⋊C₄	—	2	2	no	—
104	C₄⋊C₄	—	269	2	no	—
105	C₄⋊C₄	—	269	2	no	—
106	C₄⋊C₄	—	538	2	no	—
107	C₄⋊C₄	—	538	4	no	—
108	C₂²⋊C₄	—	538	2	no	—
109	C₄²	—	2	2	no	—
110	C₄²	—	538	2	no	—
111	C₂³	C₂×C₂²⋊D₂₆₉	1	1	yes	characteristic
112	Q₈	—	2	4	no	—
113	Q₈	—	4	2	no	—
114	Q₈	—	2152	2	no	—
115	C₂×C₄	—	2	2	no	—
116	C₂×C₄	—	2	2	no	—
117	C₂×C₄	—	2	2	no	—
118	C₂×C₄	—	4	1	no	—
119	C₂×C₄	—	4	1	no	—
120	C₂×C₄	—	4	1	no	—
121	C₂×C₄	—	4	1	no	—
122	C₂×C₄	—	4	2	no	—
123	C₂×C₄	—	538	2	no	—
124	C₂×C₄	—	538	2	no	—
125	C₂×C₄	D₄×D₂₆₉	1	2	yes	—
126	C₂×C₄	C₂×Dic₅₃₈	1	1	yes	characteristic
127	C₂×C₄	#16864401	1	2	yes	—
128	C₂×C₄	#16864402	1	1	yes	characteristic
129	C₈	—	1076	2	no	—
130	C₂²	—	2	1	no	—
131	C₂²	—	2	1	no	—
132	C₂²	#22575421	1	1	yes	characteristic
133	C₂²	#22575432	1	1	yes	characteristic
134	C₂²	#22575440	1	1	yes	characteristic
135	C₄	—	4	2	no	—
136	C₄	—	4	2	no	—
137	C₄	—	4	2	no	—
138	C₄	—	1076	2	no	—
139	C₄	—	1076	2	no	—
140	C₄	#22575430	1	2	yes	—
141	C₄	#22575439	1	2	yes	—
142	C₂	—	2	1	no	—
143	C₂	—	2	1	no	—
144	C₂	#31770517	1	1	yes	characteristic
145	C₂	#31770782	1	1	yes	characteristic
146	C₂	#31770787	1	1	yes	characteristic

subgroup

G/H

class size

multiplicity

normal?

properties

#31771236

C₂

yes

maximal, characteristic

#31770614

C₂

yes

maximal

#31770374

C₂

yes

maximal, characteristic

#31770365

C₂

yes

maximal

#391×C₂₆₉

C₂

yes

maximal, characteristic

C₂×#16864386

C₂²

yes

—

—

—

—

C₂²

yes

—

#22575377

—

#22575377

—

C₂²×Q₈×C₂₆₉

C₂²

yes

characteristic

C₂×C₄⋊C₄×C₂₆₉

C₂²

yes

characteristic

C₂×C₄²×C₂₆₉

C₂²

yes

characteristic

#95×C₂₆₉

—

C₂×Dic₅₃₈

—

#16864386

—

#16864386

—

C₂×#14524130

—

C₄⋊Dic₂₆₉

—

C₄⋊Dic₂₆₉

D₄

yes

—

C₄⋊Dic₂₆₉

D₄

yes

—

#16864381

—

C₂²⋊Dic₂₆₉

—

C₄×Dic₂₆₉

—

C₂×Q₈×C₂₆₉

—

C₂×Q₈×C₂₆₉

—

C₂×Q₈×C₂₆₉

—

C₂×Q₈×C₂₆₉

Q₈

yes

complemented

C₂²×C₄×C₂₆₉

—

C₂²×C₄×C₂₆₉

—

C₂²×C₄×C₂₆₉

D₄

yes

—

C₂²×C₄×C₂₆₉

C₂³

yes

characteristic

C₄⋊C₄×C₂₆₉

—

C₄²×C₂₆₉

—

Dic₅₃₈

—

C₂×Dic₂₆₉

—

C₂×Dic₂₆₉

—

#14524130

—

C₂³×C₂₆₉

C₂×D₄

yes

characteristic

Q₈×C₂₆₉

—

Q₈×C₂₆₉

—

C₂×C₄×C₂₆₉

—

C₂×C₄×C₂₆₉

—

C₂×C₄×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

#120

yes

characteristic

C₂²×C₂₆₉

C₂²⋊Q₈

yes

characteristic

C₂²×C₂₆₉

#124

yes

characteristic

C₄×C₂₆₉

—

C₄×C₂₆₉

—

C₄×C₂₆₉

—

C₄×C₂₆₉

C₂²⋊Q₈

yes

—

C₄×C₂₆₉

#124

yes

—

Dic₂₆₉

—

Dic₂₆₉

—

C₂×C₂₆₉

—

C₂×C₂₆₉

—

C₂×C₂₆₉

yes

C₂×C₂₆₉

yes

C₂×C₂₆₉

yes

C₂₆₉

yes

characteristic, complemented, Sylow

#2028

—

269

maximal, Sylow

#544

—

269

—

#419

—

269

—

#391

D₂₆₉

yes

characteristic

#343

—

269

—

#339

—

269

—

C₂²×Q₈

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×C₄²

C₂×D₂₆₉

yes

characteristic

#103

—

538

—

#103

—

538

—

#95

—

C₂×Q₈

—

C₂×Q₈

—

C₂×Q₈

—

C₂×Q₈

—

538

—

C₂×Q₈

Dic₅₃₈

yes

complemented

C₂²×C₄

—

C₂²×C₄

—

C₂²×C₄

C₂²⋊D₂₆₉

yes

—

C₂²×C₄

C₂²×D₂₆₉

yes

characteristic

100

M₁₆

—

538

—

101

M₁₆

—

538

—

102

C₂×C₈

—

538

—

103

C₄⋊C₄

—

104

C₄⋊C₄

—

269

—

105

C₄⋊C₄

—

269

—

106

C₄⋊C₄

—

538

—

107

C₄⋊C₄

—

538

—

108

C₂²⋊C₄

—

538

—

109

C₄²

—

110

C₄²

—

538

—

111

C₂³

C₂×C₂²⋊D₂₆₉

yes

characteristic

112

Q₈

—

113

Q₈

—

114

Q₈

—

2152

—

115

C₂×C₄

—

116

C₂×C₄

—

117

C₂×C₄

—

118

C₂×C₄

—

119

C₂×C₄

—

120

C₂×C₄

—

121

C₂×C₄

—

122

C₂×C₄

—

123

C₂×C₄

—

538

—

124

C₂×C₄

—

538

—

125

C₂×C₄

D₄×D₂₆₉

yes

—

126

C₂×C₄

C₂×Dic₅₃₈

yes

characteristic

127

C₂×C₄

#16864401

yes

—

128

C₂×C₄

#16864402

yes

characteristic

129

C₈

—

1076

—

130

C₂²

—

131

C₂²

—

132

C₂²

#22575421

yes

characteristic

133

C₂²

#22575432

yes

characteristic

134

C₂²

#22575440

yes

characteristic

135

C₄

—

136

C₄

—

137

C₄

—

138

C₄

—

1076

—

139

C₄

—

1076

—

140

C₄

#22575430

yes

—

141

C₄

#22575439

yes

—

142

C₂

—

143

C₂

—

144

C₂

#31770517

yes

characteristic

145

C₂

#31770782

yes

characteristic

146

C₂

#31770787

yes

characteristic

	subgroup	G/H
center	C₂²	#22575421
commutator, cosocle	C₂²×C₄×C₂₆₉	C₂³
Frattini	C₂²×C₄	C₂²×D₂₆₉
Fitting	#391×C₂₆₉	C₂
socle	C₂²×C₂₆₉	#120
prime radical	C₂³×C₂₆₉	C₂×D₄
hypercenter	#391	D₂₆₉
hypocenter	C₂₆₉	#2028
LCS	G ▷ C₂²×C₄×C₂₆₉ ▷ C₂×C₂₆₉ ▷ C₂₆₉
UCS	C₁ ◁ C₂² ◁ C₂²×C₄ ◁ #391

subgroup

G/H

center

C₂²

#22575421

commutator, cosocle

C₂²×C₄×C₂₆₉

C₂³

Frattini

C₂²×C₄

C₂²×D₂₆₉

Fitting

#391×C₂₆₉

C₂

socle

C₂²×C₂₆₉

#120

prime radical

C₂³×C₂₆₉

C₂×D₄

D₂₆₉

C₂₆₉

G ▷ C₂²×C₄×C₂₆₉ ▷ C₂×C₂₆₉ ▷ C₂₆₉

UCS

C₁ ◁ C₂² ◁ C₂²×C₄ ◁ #391

N	H ≅ G/N	type
30. C₂×Q₈×C₂₆₉	Q₈	semidirect
73. C₂₆₉	#2028	semidirect
95. C₂×Q₈	Dic₅₃₈	semidirect

H ≅ G/N

type

30. C₂×Q₈×C₂₆₉

Q₈

semidirect

73. C₂₆₉

#2028

semidirect

95. C₂×Q₈

Dic₅₃₈

semidirect

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

subgroup	index
#31771236	2
#31770614	2
#31770374	2
#31770365	2
#391×C₂₆₉	2
#2028	269

subgroup

index

#391×C₂₆₉

269

Subgroups — #46812046

Conjugacy classes of subgroups

Complements

Maximal subgroups