Group #46813467 — subgroups

Subgroups — #46813467

Order 34432.

1104 conjugacy classes of subgroups; only the 309 normal subgroups are shown.

normal subgroups only

This group has 1104 conjugacy classes of subgroups — too many to draw. Only its 309 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×#22575412	C₂	1	2	yes	maximal, complemented
2	#31770508	C₂	1	8	yes	maximal, complemented
3	#31770461	C₂	1	4	yes	maximal, complemented
4	C₂³×D₄×C₂₆₉	C₂	1	1	yes	maximal, characteristic
5	#22575412	—	2	8	no	—
6	C₂×C₄⋊Dic₂₆₉	C₂²	1	4	yes	complemented
7	C₂×C₂²⋊Dic₂₆₉	C₂²	1	4	yes	—
8	C₂×C₂²⋊Dic₂₆₉	C₂²	1	16	yes	complemented
9	C₂×C₄×Dic₂₆₉	C₂²	1	4	yes	complemented
10	C₂⁵×C₂₆₉	C₂²	1	2	yes	—
11	C₂²×D₄×C₂₆₉	—	2	8	no	—
12	C₂²×D₄×C₂₆₉	C₄	1	8	yes	complemented
13	C₂²×D₄×C₂₆₉	C₂²	1	4	yes	—
14	C₂³×C₄×C₂₆₉	C₂²	1	1	yes	characteristic
15	C₂²×Dic₂₆₉	C₂³	1	8	yes	—
16	C₄⋊Dic₂₆₉	—	2	8	no	—
17	C₂²⋊Dic₂₆₉	—	2	8	no	—
18	C₂²⋊Dic₂₆₉	—	4	16	no	—
19	C₄×Dic₂₆₉	—	2	8	no	—
20	C₂⁴×C₂₆₉	—	2	8	no	—
21	C₂⁴×C₂₆₉	—	4	8	no	—
22	C₂⁴×C₂₆₉	C₂×C₄	1	8	yes	—
23	C₂⁴×C₂₆₉	C₂³	1	4	yes	—
24	C₂⁴×C₂₆₉	C₂³	1	1	yes	characteristic
25	C₂×D₄×C₂₆₉	—	2	32	no	—
26	C₂×D₄×C₂₆₉	—	2	8	no	—
27	C₂×D₄×C₂₆₉	—	2	8	no	—
28	C₂×D₄×C₂₆₉	C₂×C₄	1	16	yes	—
29	C₂²×C₄×C₂₆₉	C₂×C₄	1	4	yes	—
30	C₂²×C₄×C₂₆₉	D₄	1	8	yes	—
31	C₂²×C₄×C₂₆₉	C₂³	1	2	yes	—
32	C₂×Dic₂₆₉	—	2	8	no	—
33	C₂×Dic₂₆₉	—	2	16	no	—
34	C₂³×C₂₆₉	—	2	4	no	—
35	C₂³×C₂₆₉	—	2	16	no	—
36	C₂³×C₂₆₉	—	2	8	no	—
37	C₂³×C₂₆₉	—	2	16	no	—
38	C₂³×C₂₆₉	—	4	16	no	—
39	C₂³×C₂₆₉	—	4	32	no	—
40	C₂³×C₂₆₉	C₂²×C₄	1	8	yes	—
41	C₂³×C₂₆₉	C₂²×C₄	1	2	yes	—
42	C₂³×C₂₆₉	C₂×D₄	1	4	yes	—
43	C₂³×C₂₆₉	C₂⁴	1	1	yes	characteristic
44	D₄×C₂₆₉	—	2	32	no	—
45	C₂×C₄×C₂₆₉	C₂²⋊C₄	1	16	yes	complemented
46	C₂×C₄×C₂₆₉	C₂²×C₄	1	4	yes	—
47	C₂×C₄×C₂₆₉	C₂×D₄	1	4	yes	—
48	C₂×C₄×C₂₆₉	C₂×D₄	1	4	yes	—
49	C₂²×C₂₆₉	—	2	4	no	—
50	C₂²×C₂₆₉	—	2	8	no	—
51	C₂²×C₂₆₉	—	2	8	no	—
52	C₂²×C₂₆₉	—	2	16	no	—
53	C₂²×C₂₆₉	—	4	32	no	—
54	C₂²×C₂₆₉	—	4	16	no	—
55	C₂²×C₂₆₉	C₂×C₂²⋊C₄	1	4	yes	—
56	C₂²×C₂₆₉	C₂³×C₄	1	1	yes	characteristic
57	C₂²×C₂₆₉	C₂²×D₄	1	2	yes	—
58	C₂²×C₂₆₉	H(5,𝔽₂)	1	4	yes	—
59	C₄×C₂₆₉	C₂×C₂²⋊C₄	1	8	yes	—
60	Dic₂₆₉	—	4	8	no	—
61	C₂×C₂₆₉	—	2	4	no	—
62	C₂×C₂₆₉	—	4	8	no	—
63	C₂×C₂₆₉	C₂²×C₂²⋊C₄	1	1	yes	characteristic
64	C₂×C₂₆₉	#518	1	2	yes	—
65	C₂×C₂₆₉	#535	1	4	yes	—
66	C₂₆₉	#2309	1	1	yes	characteristic, complemented, Sylow
67	#2309	—	269	1	no	maximal, Sylow
68	C₂³×D₄	D₂₆₉	1	1	yes	characteristic
69	C₂×#117	—	269	2	no	—
70	#390	—	269	8	no	—
71	#379	—	269	4	no	—
72	C₂⁵	C₂×D₂₆₉	1	2	yes	—
73	C₂²×D₄	—	2	8	no	—
74	C₂²×D₄	Dic₂₆₉	1	8	yes	complemented
75	C₂²×D₄	C₂×D₂₆₉	1	4	yes	—
76	C₂³×C₄	C₂×D₂₆₉	1	1	yes	characteristic
77	#117	—	538	8	no	—
78	C₂×C₄⋊C₄	—	269	4	no	—
79	C₂×C₂²⋊C₄	—	269	4	no	—
80	C₂×C₂²⋊C₄	—	269	16	no	—
81	C₂×C₄²	—	269	4	no	—
82	C₂⁴	—	2	8	no	—
83	C₂⁴	—	4	8	no	—
84	C₂⁴	C₂×Dic₂₆₉	1	8	yes	—
85	C₂⁴	C₂²×D₂₆₉	1	4	yes	—
86	C₂⁴	C₂²×D₂₆₉	1	1	yes	characteristic
87	C₂×D₄	—	2	32	no	—
88	C₂×D₄	—	2	8	no	—
89	C₂×D₄	—	2	8	no	—
90	C₂×D₄	C₂×Dic₂₆₉	1	16	yes	—
91	C₂²×C₄	—	269	8	no	—
92	C₂²×C₄	C₂×Dic₂₆₉	1	4	yes	—
93	C₂²×C₄	C₂²⋊D₂₆₉	1	8	yes	—
94	C₂²×C₄	C₂²×D₂₆₉	1	2	yes	—
95	C₄⋊C₄	—	538	8	no	—
96	C₂²⋊C₄	—	538	8	no	—
97	C₂²⋊C₄	—	1076	16	no	—
98	C₄²	—	538	8	no	—
99	C₂³	—	2	4	no	—
100	C₂³	—	2	16	no	—
101	C₂³	—	2	8	no	—
102	C₂³	—	2	16	no	—
103	C₂³	—	4	16	no	—
104	C₂³	—	4	32	no	—
105	C₂³	C₂²×Dic₂₆₉	1	8	yes	—
106	C₂³	C₂²×Dic₂₆₉	1	2	yes	—
107	C₂³	C₂×C₂²⋊D₂₆₉	1	4	yes	—
108	C₂³	C₂³×D₂₆₉	1	1	yes	characteristic
109	D₄	—	2	32	no	—
110	C₂×C₄	—	538	8	no	—
111	C₂×C₄	—	538	16	no	—
112	C₂×C₄	C₂²⋊Dic₂₆₉	1	16	yes	complemented
113	C₂×C₄	C₂²×Dic₂₆₉	1	4	yes	—
114	C₂×C₄	C₂×C₂²⋊D₂₆₉	1	4	yes	—
115	C₂×C₄	C₂×C₂²⋊D₂₆₉	1	4	yes	—
116	C₂²	—	2	8	no	—
117	C₂²	—	2	4	no	—
118	C₂²	—	2	8	no	—
119	C₂²	—	2	16	no	—
120	C₂²	—	4	32	no	—
121	C₂²	—	4	16	no	—
122	C₂²	C₂×C₂²⋊Dic₂₆₉	1	4	yes	—
123	C₂²	C₂³×Dic₂₆₉	1	1	yes	characteristic
124	C₂²	C₂²×C₂²⋊D₂₆₉	1	2	yes	—
125	C₂²	#22575498	1	4	yes	—
126	C₄	—	1076	8	no	—
127	C₄	C₂×C₂²⋊Dic₂₆₉	1	8	yes	—
128	C₂	—	2	4	no	—
129	C₂	—	4	8	no	—
130	C₂	C₂²×C₂²⋊Dic₂₆₉	1	1	yes	characteristic
131	C₂	#31771080	1	2	yes	—
132	C₂	#31771172	1	4	yes	—

subgroup

G/H

class size

multiplicity

normal?

properties

C₂×#22575412

C₂

yes

maximal, complemented

#31770508

C₂

yes

maximal, complemented

#31770461

C₂

yes

maximal, complemented

C₂³×D₄×C₂₆₉

C₂

yes

maximal, characteristic

#22575412

—

C₂×C₄⋊Dic₂₆₉

C₂²

yes

complemented

C₂×C₂²⋊Dic₂₆₉

C₂²

yes

—

C₂×C₂²⋊Dic₂₆₉

C₂²

yes

complemented

C₂×C₄×Dic₂₆₉

C₂²

yes

complemented

C₂⁵×C₂₆₉

C₂²

yes

—

C₂²×D₄×C₂₆₉

—

C₂²×D₄×C₂₆₉

C₄

yes

complemented

C₂²×D₄×C₂₆₉

C₂²

yes

—

C₂³×C₄×C₂₆₉

C₂²

yes

characteristic

C₂²×Dic₂₆₉

C₂³

yes

—

C₄⋊Dic₂₆₉

—

C₂²⋊Dic₂₆₉

—

C₂²⋊Dic₂₆₉

—

C₄×Dic₂₆₉

—

C₂⁴×C₂₆₉

—

C₂⁴×C₂₆₉

—

C₂⁴×C₂₆₉

C₂×C₄

yes

—

C₂⁴×C₂₆₉

C₂³

yes

—

C₂⁴×C₂₆₉

C₂³

yes

characteristic

C₂×D₄×C₂₆₉

—

C₂×D₄×C₂₆₉

—

C₂×D₄×C₂₆₉

—

C₂×D₄×C₂₆₉

C₂×C₄

yes

—

C₂²×C₄×C₂₆₉

C₂×C₄

yes

—

C₂²×C₄×C₂₆₉

D₄

yes

—

C₂²×C₄×C₂₆₉

C₂³

yes

—

C₂×Dic₂₆₉

—

C₂×Dic₂₆₉

—

C₂³×C₂₆₉

—

C₂³×C₂₆₉

—

C₂³×C₂₆₉

—

C₂³×C₂₆₉

—

C₂³×C₂₆₉

—

C₂³×C₂₆₉

—

C₂³×C₂₆₉

C₂²×C₄

yes

—

C₂³×C₂₆₉

C₂²×C₄

yes

—

C₂³×C₂₆₉

C₂×D₄

yes

—

C₂³×C₂₆₉

C₂⁴

yes

characteristic

D₄×C₂₆₉

—

C₂×C₄×C₂₆₉

C₂²⋊C₄

yes

complemented

C₂×C₄×C₂₆₉

C₂²×C₄

yes

—

C₂×C₄×C₂₆₉

C₂×D₄

yes

—

C₂×C₄×C₂₆₉

C₂×D₄

yes

—

C₂²×C₂₆₉

—

C₂²×C₂₆₉

—

C₂²×C₂₆₉

—

C₂²×C₂₆₉

—

C₂²×C₂₆₉

—

C₂²×C₂₆₉

—

C₂²×C₂₆₉

C₂×C₂²⋊C₄

yes

—

C₂²×C₂₆₉

C₂³×C₄

yes

characteristic

C₂²×C₂₆₉

C₂²×D₄

yes

—

C₂²×C₂₆₉

H(5,𝔽₂)

yes

—

C₄×C₂₆₉

C₂×C₂²⋊C₄

yes

—

Dic₂₆₉

—

C₂×C₂₆₉

—

C₂×C₂₆₉

—

C₂×C₂₆₉

C₂²×C₂²⋊C₄

yes

characteristic

C₂×C₂₆₉

#518

yes

—

C₂×C₂₆₉

#535

yes

—

C₂₆₉

#2309

yes

characteristic, complemented, Sylow

#2309

—

269

maximal, Sylow

C₂³×D₄

D₂₆₉

yes

characteristic

C₂×#117

—

269

—

#390

—

269

—

#379

—

269

—

C₂⁵

C₂×D₂₆₉

yes

—

C₂²×D₄

—

C₂²×D₄

Dic₂₆₉

yes

complemented

C₂²×D₄

C₂×D₂₆₉

yes

—

C₂³×C₄

C₂×D₂₆₉

yes

characteristic

#117

—

538

—

C₂×C₄⋊C₄

—

269

—

C₂×C₂²⋊C₄

—

269

—

C₂×C₂²⋊C₄

—

269

—

C₂×C₄²

—

269

—

C₂⁴

—

C₂⁴

—

C₂⁴

C₂×Dic₂₆₉

yes

—

C₂⁴

C₂²×D₂₆₉

yes

—

C₂⁴

C₂²×D₂₆₉

yes

characteristic

C₂×D₄

—

C₂×D₄

—

C₂×D₄

—

C₂×D₄

C₂×Dic₂₆₉

yes

—

C₂²×C₄

—

269

—

C₂²×C₄

C₂×Dic₂₆₉

yes

—

C₂²×C₄

C₂²⋊D₂₆₉

yes

—

C₂²×C₄

C₂²×D₂₆₉

yes

—

C₄⋊C₄

—

538

—

C₂²⋊C₄

—

538

—

C₂²⋊C₄

—

1076

—

C₄²

—

538

—

C₂³

—

100

C₂³

—

101

C₂³

—

102

C₂³

—

103

C₂³

—

104

C₂³

—

105

C₂³

C₂²×Dic₂₆₉

yes

—

106

C₂³

C₂²×Dic₂₆₉

yes

—

107

C₂³

C₂×C₂²⋊D₂₆₉

yes

—

108

C₂³

C₂³×D₂₆₉

yes

characteristic

109

D₄

—

110

C₂×C₄

—

538

—

111

C₂×C₄

—

538

—

112

C₂×C₄

C₂²⋊Dic₂₆₉

yes

complemented

113

C₂×C₄

C₂²×Dic₂₆₉

yes

—

114

C₂×C₄

C₂×C₂²⋊D₂₆₉

yes

—

115

C₂×C₄

C₂×C₂²⋊D₂₆₉

yes

—

116

C₂²

—

117

C₂²

—

118

C₂²

—

119

C₂²

—

120

C₂²

—

121

C₂²

—

122

C₂²

C₂×C₂²⋊Dic₂₆₉

yes

—

123

C₂²

C₂³×Dic₂₆₉

yes

characteristic

124

C₂²

C₂²×C₂²⋊D₂₆₉

yes

—

125

C₂²

#22575498

yes

—

126

C₄

—

1076

—

127

C₄

C₂×C₂²⋊Dic₂₆₉

yes

—

128

C₂

—

129

C₂

—

130

C₂

C₂²×C₂²⋊Dic₂₆₉

yes

characteristic

131

C₂

#31771080

yes

—

132

C₂

#31771172

yes

—

	subgroup	G/H
center, Frattini	C₂³	C₂³×D₂₆₉
commutator	C₂²×C₂₆₉	C₂³×C₄
Fitting, prime radical	C₂³×D₄×C₂₆₉	C₂
socle, cosocle	C₂³×C₂₆₉	C₂⁴
hypercenter	C₂³×D₄	D₂₆₉
hypocenter	C₂₆₉	#2309
LCS	G ▷ C₂²×C₂₆₉ ▷ C₂₆₉
UCS	C₁ ◁ C₂³ ◁ C₂³×D₄

subgroup

G/H

center, Frattini

C₂³

C₂³×D₂₆₉

commutator

C₂²×C₂₆₉

C₂³×C₄

Fitting, prime radical

C₂³×D₄×C₂₆₉

C₂

socle, cosocle

C₂³×C₂₆₉

C₂⁴

hypercenter

C₂³×D₄

D₂₆₉

C₂₆₉

G ▷ C₂²×C₂₆₉ ▷ C₂₆₉

C₁ ◁ C₂³ ◁ C₂³×D₄

N	H ≅ G/N	type
1. C₂×#22575412	C₂	semidirect
2. #31770508	C₂	semidirect
3. #31770461	C₂	semidirect
6. C₂×C₄⋊Dic₂₆₉	C₂²	semidirect
8. C₂×C₂²⋊Dic₂₆₉	C₂²	semidirect
9. C₂×C₄×Dic₂₆₉	C₂²	semidirect
12. C₂²×D₄×C₂₆₉	C₄	semidirect
45. C₂×C₄×C₂₆₉	C₂²⋊C₄	semidirect
66. C₂₆₉	#2309	semidirect
74. C₂²×D₄	Dic₂₆₉	semidirect
112. C₂×C₄	C₂²⋊Dic₂₆₉	semidirect

H ≅ G/N

type

1. C₂×#22575412

C₂

semidirect

2. #31770508

C₂

semidirect

3. #31770461

C₂

semidirect

6. C₂×C₄⋊Dic₂₆₉

C₂²

semidirect

8. C₂×C₂²⋊Dic₂₆₉

C₂²

semidirect

9. C₂×C₄×Dic₂₆₉

C₂²

semidirect

12. C₂²×D₄×C₂₆₉

C₄

semidirect

45. C₂×C₄×C₂₆₉

C₂²⋊C₄

semidirect

66. C₂₆₉

#2309

semidirect

74. C₂²×D₄

Dic₂₆₉

semidirect

112. C₂×C₄

C₂²⋊Dic₂₆₉

semidirect

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

subgroup	index
C₂×#22575412	2
#31770508	2
#31770461	2
C₂³×D₄×C₂₆₉	2
#2309	269

subgroup

index

C₂³×D₄×C₂₆₉

269

Subgroups — #46813467

Conjugacy classes of subgroups

Complements

Maximal subgroups