Group #46812448 — subgroups

Subgroups — #46812448

Order 34432.

214 conjugacy classes of subgroups; only the 89 normal subgroups are shown.

normal subgroups only

This group has 214 conjugacy classes of subgroups — too many to draw. Only its 89 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	#31770475	C₂	1	1	yes	maximal, characteristic
2	#31770367	C₂	1	2	yes	maximal
3	#31770367	C₂	1	2	yes	maximal
4	#31770360	C₂	1	1	yes	maximal, characteristic
5	#389×C₂₆₉	C₂	1	1	yes	maximal, characteristic
6	C₂²×#14524130	C₂²	1	2	yes	—
7	C₂×C₄⋊Dic₂₆₉	—	2	1	no	—
8	C₂×C₄⋊Dic₂₆₉	—	2	1	no	—
9	C₂×C₄⋊Dic₂₆₉	C₂²	1	2	yes	—
10	C₂×C₄⋊C₄×C₂₆₉	—	2	1	no	—
11	C₂×C₄⋊C₄×C₂₆₉	C₂²	1	1	yes	characteristic
12	C₂×C₄⋊C₄×C₂₆₉	C₂²	1	1	yes	characteristic
13	C₂×C₄²×C₂₆₉	C₂²	1	1	yes	characteristic
14	#95×C₂₆₉	—	2	1	no	—
15	C₂²×Dic₂₆₉	—	2	2	no	—
16	C₂×#14524130	—	2	2	no	—
17	C₂×#14524130	—	2	2	no	—
18	C₂×#14524130	—	2	2	no	—
19	C₄⋊Dic₂₆₉	—	2	4	no	—
20	C₄⋊Dic₂₆₉	—	4	2	no	—
21	C₄⋊Dic₂₆₉	—	4	2	no	—
22	C₂²×C₄×C₂₆₉	—	2	1	no	—
23	C₂²×C₄×C₂₆₉	—	2	1	no	—
24	C₂²×C₄×C₂₆₉	D₄	1	1	yes	characteristic
25	C₂²×C₄×C₂₆₉	D₄	1	1	yes	characteristic
26	C₂²×C₄×C₂₆₉	C₂³	1	1	yes	characteristic
27	C₄⋊C₄×C₂₆₉	—	2	1	no	—
28	C₄⋊C₄×C₂₆₉	—	2	2	no	—
29	C₄⋊C₄×C₂₆₉	—	4	2	no	—
30	C₄⋊C₄×C₂₆₉	Q₈	1	2	yes	—
31	C₄²×C₂₆₉	—	2	2	no	—
32	C₂×Dic₂₆₉	—	4	2	no	—
33	C₂×Dic₂₆₉	—	4	2	no	—
34	C₂×Dic₂₆₉	—	4	2	no	—
35	#14524130	—	4	2	no	—
36	C₂³×C₂₆₉	C₂×D₄	1	1	yes	characteristic
37	C₂×C₄×C₂₆₉	—	2	1	no	—
38	C₂×C₄×C₂₆₉	—	2	1	no	—
39	C₂×C₄×C₂₆₉	—	2	1	no	—
40	C₂×C₄×C₂₆₉	—	2	1	no	—
41	C₂×C₄×C₂₆₉	—	2	2	no	—
42	C₂×C₄×C₂₆₉	—	4	1	no	—
43	C₂×C₄×C₂₆₉	—	4	1	no	—
44	C₂×C₄×C₂₆₉	—	4	1	no	—
45	C₂×C₄×C₂₆₉	—	4	1	no	—
46	C₂×C₄×C₂₆₉	—	4	1	no	—
47	C₂×C₄×C₂₆₉	SD₁₆	1	2	yes	—
48	C₂×C₄×C₂₆₉	Q₁₆	1	2	yes	—
49	C₂×C₄×C₂₆₉	C₂×Q₈	1	1	yes	characteristic
50	C₂×C₄×C₂₆₉	#41	1	2	yes	—
51	C₂×C₄×C₂₆₉	#41	1	1	yes	characteristic
52	C₂×C₄×C₂₆₉	#41	1	2	yes	—
53	C₂²×C₂₆₉	C₂²⋊Q₈	1	1	yes	characteristic
54	C₂²×C₂₆₉	#123	1	1	yes	characteristic
55	C₂²×C₂₆₉	#124	1	1	yes	characteristic
56	C₂²×C₂₆₉	C₂×SD₁₆	1	1	yes	characteristic
57	C₂²×C₂₆₉	C₂×Q₁₆	1	1	yes	characteristic
58	C₂²×C₂₆₉	AGL(1,ℤ/8)	1	1	yes	characteristic
59	C₂²×C₂₆₉	AGL(1,ℤ/8)	1	1	yes	characteristic
60	C₄×C₂₆₉	—	2	2	no	—
61	C₄×C₂₆₉	—	2	2	no	—
62	C₄×C₂₆₉	—	4	2	no	—
63	C₄×C₂₆₉	—	8	1	no	—
64	C₄×C₂₆₉	#125	1	2	yes	—
65	C₄×C₂₆₉	#126	1	2	yes	—
66	Dic₂₆₉	—	8	2	no	—
67	C₂×C₂₆₉	#400	1	1	yes	characteristic
68	C₂×C₂₆₉	#476	1	1	yes	characteristic
69	C₂×C₂₆₉	#477	1	1	yes	characteristic
70	C₂×C₂₆₉	#481	1	1	yes	characteristic
71	C₂×C₂₆₉	#484	1	1	yes	characteristic
72	C₂×C₂₆₉	#487	1	1	yes	characteristic
73	C₂×C₂₆₉	#490	1	1	yes	characteristic
74	C₂₆₉	#2100	1	1	yes	characteristic, complemented, Sylow
75	#2100	—	269	1	no	maximal, Sylow
76	#389	D₂₆₉	1	1	yes	characteristic
77	#384	—	269	1	no	—
78	#340	—	269	2	no	—
79	#340	—	269	2	no	—
80	#336	—	269	1	no	—
81	C₂²×C₈	—	269	2	no	—
82	C₂×C₄⋊C₄	—	2	1	no	—
83	C₂×C₄⋊C₄	—	269	2	no	—
84	C₂×C₄⋊C₄	—	538	1	no	—
85	C₂×C₄⋊C₄	—	538	1	no	—
86	C₂×C₄⋊C₄	C₂×D₂₆₉	1	1	yes	characteristic
87	C₂×C₄⋊C₄	C₂×D₂₆₉	1	1	yes	characteristic
88	C₂×C₄²	C₂×D₂₆₉	1	1	yes	characteristic
89	#95	—	2	1	no	—
90	C₂²×C₄	—	2	1	no	—
91	C₂²×C₄	—	2	1	no	—
92	C₂²×C₄	—	538	2	no	—
93	C₂²×C₄	C₂²⋊D₂₆₉	1	1	yes	characteristic
94	C₂²×C₄	C₂²⋊D₂₆₉	1	1	yes	characteristic
95	C₂²×C₄	C₂²×D₂₆₉	1	1	yes	characteristic
96	C₂×C₈	—	538	2	no	—
97	C₂×C₈	—	538	2	no	—
98	C₂×C₈	—	538	2	no	—
99	C₄⋊C₄	—	2	1	no	—
100	C₄⋊C₄	—	2	2	no	—
101	C₄⋊C₄	—	4	2	no	—
102	C₄⋊C₄	—	538	4	no	—
103	C₄⋊C₄	—	1076	2	no	—
104	C₄⋊C₄	—	1076	2	no	—
105	C₄⋊C₄	Dic₅₃₈	1	2	yes	—
106	C₄²	—	2	2	no	—
107	C₂³	C₂×C₂²⋊D₂₆₉	1	1	yes	characteristic
108	C₂×C₄	—	2	1	no	—
109	C₂×C₄	—	2	1	no	—
110	C₂×C₄	—	2	1	no	—
111	C₂×C₄	—	2	1	no	—
112	C₂×C₄	—	2	2	no	—
113	C₂×C₄	—	4	1	no	—
114	C₂×C₄	—	4	1	no	—
115	C₂×C₄	—	4	1	no	—
116	C₂×C₄	—	4	1	no	—
117	C₂×C₄	—	4	1	no	—
118	C₂×C₄	—	1076	2	no	—
119	C₂×C₄	—	1076	2	no	—
120	C₂×C₄	—	1076	2	no	—
121	C₂×C₄	#16864390	1	2	yes	—
122	C₂×C₄	#16864392	1	2	yes	—
123	C₂×C₄	C₂×Dic₅₃₈	1	1	yes	characteristic
124	C₂×C₄	#16864401	1	2	yes	—
125	C₂×C₄	#16864402	1	1	yes	characteristic
126	C₂×C₄	#16864403	1	2	yes	—
127	C₈	—	1076	2	no	—
128	C₂²	#22575432	1	1	yes	characteristic
129	C₂²	#22575434	1	1	yes	characteristic
130	C₂²	#22575442	1	1	yes	characteristic
131	C₂²	C₂×#16864390	1	1	yes	characteristic
132	C₂²	C₂×#16864392	1	1	yes	characteristic
133	C₂²	#22575478	1	1	yes	characteristic
134	C₂²	#22575478	1	1	yes	characteristic
135	C₄	—	2	2	no	—
136	C₄	—	2	2	no	—
137	C₄	—	4	2	no	—
138	C₄	—	8	1	no	—
139	C₄	—	2152	2	no	—
140	C₄	#22575444	1	2	yes	—
141	C₄	#22575448	1	2	yes	—
142	C₂	#31770544	1	1	yes	characteristic
143	C₂	#31770895	1	1	yes	characteristic
144	C₂	#31770900	1	1	yes	characteristic
145	C₂	#31770921	1	1	yes	characteristic
146	C₂	#31770938	1	1	yes	characteristic
147	C₂	#31770957	1	1	yes	characteristic
148	C₂	#31770971	1	1	yes	characteristic

subgroup

G/H

class size

multiplicity

normal?

properties

#31770475

C₂

yes

maximal, characteristic

#31770367

C₂

yes

maximal

#31770367

C₂

yes

maximal

#31770360

C₂

yes

maximal, characteristic

#389×C₂₆₉

C₂

yes

maximal, characteristic

C₂²×#14524130

C₂²

yes

—

C₂×C₄⋊Dic₂₆₉

—

C₂×C₄⋊Dic₂₆₉

—

C₂×C₄⋊Dic₂₆₉

C₂²

yes

—

C₂×C₄⋊C₄×C₂₆₉

—

C₂×C₄⋊C₄×C₂₆₉

C₂²

yes

characteristic

C₂×C₄⋊C₄×C₂₆₉

C₂²

yes

characteristic

C₂×C₄²×C₂₆₉

C₂²

yes

characteristic

#95×C₂₆₉

—

C₂²×Dic₂₆₉

—

C₂×#14524130

—

C₂×#14524130

—

C₂×#14524130

—

C₄⋊Dic₂₆₉

—

C₄⋊Dic₂₆₉

—

C₄⋊Dic₂₆₉

—

C₂²×C₄×C₂₆₉

—

C₂²×C₄×C₂₆₉

—

C₂²×C₄×C₂₆₉

D₄

yes

characteristic

C₂²×C₄×C₂₆₉

D₄

yes

characteristic

C₂²×C₄×C₂₆₉

C₂³

yes

characteristic

C₄⋊C₄×C₂₆₉

—

C₄⋊C₄×C₂₆₉

—

C₄⋊C₄×C₂₆₉

—

C₄⋊C₄×C₂₆₉

Q₈

yes

—

C₄²×C₂₆₉

—

C₂×Dic₂₆₉

—

C₂×Dic₂₆₉

—

C₂×Dic₂₆₉

—

#14524130

—

C₂³×C₂₆₉

C₂×D₄

yes

characteristic

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×C₄×C₂₆₉

—

C₂×C₄×C₂₆₉

SD₁₆

yes

—

C₂×C₄×C₂₆₉

Q₁₆

yes

—

C₂×C₄×C₂₆₉

C₂×Q₈

yes

characteristic

C₂×C₄×C₂₆₉

#41

yes

—

C₂×C₄×C₂₆₉

#41

yes

characteristic

C₂×C₄×C₂₆₉

#41

yes

—

C₂²×C₂₆₉

C₂²⋊Q₈

yes

C₂²×C₂₆₉

yes

C₂²×C₂₆₉

yes

C₂²×C₂₆₉

C₂×SD₁₆

yes

characteristic

C₂²×C₂₆₉

C₂×Q₁₆

yes

C₂²×C₂₆₉

yes

C₂²×C₂₆₉

yes

C₄×C₂₆₉

—

C₄×C₂₆₉

—

C₄×C₂₆₉

—

C₄×C₂₆₉

—

C₄×C₂₆₉

#125

yes

—

C₄×C₂₆₉

#126

yes

—

Dic₂₆₉

—

C₂×C₂₆₉

yes

C₂×C₂₆₉

yes

C₂×C₂₆₉

yes

C₂×C₂₆₉

yes

C₂×C₂₆₉

yes

C₂×C₂₆₉

yes

C₂×C₂₆₉

yes

C₂₆₉

yes

characteristic, complemented, Sylow

#2100

—

269

maximal, Sylow

#389

D₂₆₉

yes

characteristic

#384

—

269

—

#340

—

269

—

#340

—

269

—

#336

—

269

—

C₂²×C₈

—

269

—

C₂×C₄⋊C₄

—

C₂×C₄⋊C₄

—

269

—

C₂×C₄⋊C₄

—

538

—

C₂×C₄⋊C₄

—

538

—

C₂×C₄⋊C₄

C₂×D₂₆₉

yes

characteristic

C₂×C₄⋊C₄

C₂×D₂₆₉

yes

characteristic

C₂×C₄²

C₂×D₂₆₉

yes

characteristic

#95

—

C₂²×C₄

—

C₂²×C₄

—

C₂²×C₄

—

538

—

C₂²×C₄

C₂²⋊D₂₆₉

yes

characteristic

C₂²×C₄

C₂²⋊D₂₆₉

yes

characteristic

C₂²×C₄

C₂²×D₂₆₉

yes

characteristic

C₂×C₈

—

538

—

C₂×C₈

—

538

—

C₂×C₈

—

538

—

C₄⋊C₄

—

100

C₄⋊C₄

—

101

C₄⋊C₄

—

102

C₄⋊C₄

—

538

—

103

C₄⋊C₄

—

1076

—

104

C₄⋊C₄

—

1076

—

105

C₄⋊C₄

Dic₅₃₈

yes

—

106

C₄²

—

107

C₂³

C₂×C₂²⋊D₂₆₉

yes

characteristic

108

C₂×C₄

—

109

C₂×C₄

—

110

C₂×C₄

—

111

C₂×C₄

—

112

C₂×C₄

—

113

C₂×C₄

—

114

C₂×C₄

—

115

C₂×C₄

—

116

C₂×C₄

—

117

C₂×C₄

—

118

C₂×C₄

—

1076

—

119

C₂×C₄

—

1076

—

120

C₂×C₄

—

1076

—

121

C₂×C₄

#16864390

yes

—

122

C₂×C₄

#16864392

yes

—

123

C₂×C₄

C₂×Dic₅₃₈

yes

characteristic

124

C₂×C₄

#16864401

yes

—

125

C₂×C₄

#16864402

yes

characteristic

126

C₂×C₄

#16864403

yes

—

127

C₈

—

1076

—

128

C₂²

#22575432

yes

characteristic

129

C₂²

#22575434

yes

characteristic

130

C₂²

#22575442

yes

characteristic

131

C₂²

C₂×#16864390

yes

characteristic

132

C₂²

C₂×#16864392

yes

characteristic

133

C₂²

#22575478

yes

characteristic

134

C₂²

#22575478

yes

characteristic

135

C₄

—

136

C₄

—

137

C₄

—

138

C₄

—

139

C₄

—

2152

—

140

C₄

#22575444

yes

—

141

C₄

#22575448

yes

—

142

C₂

#31770544

yes

characteristic

143

C₂

#31770895

yes

characteristic

144

C₂

#31770900

yes

characteristic

145

C₂

#31770921

yes

characteristic

146

C₂

#31770938

yes

characteristic

147

C₂

#31770957

yes

characteristic

148

C₂

#31770971

yes

characteristic

	subgroup	G/H
center	C₂³	C₂×C₂²⋊D₂₆₉
commutator, cosocle	C₂²×C₄×C₂₆₉	C₂³
Frattini	C₂²×C₄	C₂²×D₂₆₉
Fitting	#389×C₂₆₉	C₂
socle, prime radical	C₂³×C₂₆₉	C₂×D₄
hypercenter	#389	D₂₆₉
hypocenter	C₂₆₉	#2100
LCS	G ▷ C₂²×C₄×C₂₆₉ ▷ C₂×C₂₆₉ ▷ C₂₆₉
UCS	C₁ ◁ C₂³ ◁ C₂×C₄² ◁ #389

subgroup

G/H

center

C₂³

C₂×C₂²⋊D₂₆₉

commutator, cosocle

C₂²×C₄×C₂₆₉

C₂³

Frattini

C₂²×C₄

C₂²×D₂₆₉

Fitting

#389×C₂₆₉

C₂

socle, prime radical

C₂³×C₂₆₉

C₂×D₄

D₂₆₉

C₂₆₉

G ▷ C₂²×C₄×C₂₆₉ ▷ C₂×C₂₆₉ ▷ C₂₆₉

UCS

C₁ ◁ C₂³ ◁ C₂×C₄² ◁ #389

N	H ≅ G/N	type
74. C₂₆₉	#2100	semidirect

H ≅ G/N

type

74. C₂₆₉

#2100

semidirect

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

subgroup	index
#31770475	2
#31770367	2
#31770360	2
#389×C₂₆₉	2
#2100	269

subgroup

index

#389×C₂₆₉

269

Subgroups — #46812448

Conjugacy classes of subgroups

Complements

Maximal subgroups