Group #46812378 — subgroups

Subgroups — #46812378

Order 34432.

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

normal subgroups only

This group has 224 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	C₂×#22575387	C₂	1	1	yes	maximal, characteristic
2	#31770503	C₂	1	2	yes	maximal
3	#31770368	C₂	1	2	yes	maximal
4	#31770368	C₂	1	1	yes	maximal, characteristic
5	C₂×C₄⋊C₈×C₂₆₉	C₂	1	1	yes	maximal, characteristic
6	C₂×C₄⋊Dic₂₆₉	C₂²	1	2	yes	—
7	C₂×C₄⋊Dic₂₆₉	C₂²	1	2	yes	—
8	C₂×#16864381	—	2	2	no	—
9	#22575387	—	2	2	no	—
10	#22575360	—	2	2	no	—
11	C₂²×C₈×C₂₆₉	C₂²	1	1	yes	characteristic
12	C₂²×C₈×C₂₆₉	C₂²	1	1	yes	characteristic
13	C₂×C₄²×C₂₆₉	C₂²	1	1	yes	characteristic
14	C₄⋊C₈×C₂₆₉	—	2	2	no	—
15	C₂²×Dic₂₆₉	—	2	2	no	—
16	C₂²×Dic₂₆₉	—	2	2	no	—
17	C₄⋊Dic₂₆₉	—	2	2	no	—
18	C₄⋊Dic₂₆₉	—	2	2	no	—
19	C₄⋊Dic₂₆₉	—	2	2	no	—
20	C₄⋊Dic₂₆₉	Q₈	1	4	yes	—
21	#16864381	—	4	4	no	—
22	C₂²×C₄×C₂₆₉	D₄	1	1	yes	characteristic
23	C₂²×C₄×C₂₆₉	D₄	1	1	yes	characteristic
24	C₂²×C₄×C₂₆₉	C₂³	1	1	yes	characteristic
25	C₂×C₈×C₂₆₉	—	2	2	no	—
26	C₂×C₈×C₂₆₉	—	2	1	no	—
27	C₂×C₈×C₂₆₉	—	2	2	no	—
28	C₂×C₈×C₂₆₉	D₄	1	2	yes	—
29	C₄²×C₂₆₉	—	2	1	no	—
30	C₄²×C₂₆₉	—	2	1	no	—
31	C₂×Dic₂₆₉	—	2	4	no	—
32	C₂×Dic₂₆₉	—	4	2	no	—
33	C₂×Dic₂₆₉	—	4	2	no	—
34	C₂×Dic₂₆₉	—	4	2	no	—
35	C₂×Dic₂₆₉	—	4	2	no	—
36	C₂×Dic₂₆₉	—	4	2	no	—
37	C₂³×C₂₆₉	C₂×D₄	1	1	yes	characteristic
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	Q₁₆	1	2	yes	—
43	C₂×C₄×C₂₆₉	C₂×D₄	1	1	yes	characteristic
44	C₂×C₄×C₂₆₉	C₂×Q₈	1	2	yes	—
45	C₂×C₄×C₂₆₉	#41	1	2	yes	—
46	C₂×C₄×C₂₆₉	#41	1	1	yes	characteristic
47	C₈×C₂₆₉	—	2	2	no	—
48	C₈×C₂₆₉	—	4	1	no	—
49	C₂²×C₂₆₉	#121	1	1	yes	characteristic
50	C₂²×C₂₆₉	C₂²⋊Q₈	1	2	yes	—
51	C₂²×C₂₆₉	C₂×Q₁₆	1	1	yes	characteristic
52	C₂²×C₂₆₉	#135	1	1	yes	characteristic
53	C₂²×C₂₆₉	AGL(1,ℤ/8)	1	2	yes	—
54	C₄×C₂₆₉	—	2	2	no	—
55	C₄×C₂₆₉	—	4	1	no	—
56	C₄×C₂₆₉	#123	1	1	yes	characteristic
57	C₄×C₂₆₉	#125	1	2	yes	—
58	C₄×C₂₆₉	C₄⋊Q₈	1	1	yes	characteristic
59	Dic₂₆₉	—	4	4	no	—
60	Dic₂₆₉	—	8	2	no	—
61	C₂×C₂₆₉	#398	1	1	yes	characteristic
62	C₂×C₂₆₉	#467	1	1	yes	characteristic
63	C₂×C₂₆₉	#469	1	1	yes	characteristic
64	C₂×C₂₆₉	#477	1	2	yes	—
65	C₂×C₂₆₉	#478	1	2	yes	—
66	C₂₆₉	#2087	1	1	yes	characteristic, complemented, Sylow
67	#2087	—	269	1	no	maximal, Sylow
68	C₂×#107	—	269	1	no	—
69	C₂×C₄⋊C₈	D₂₆₉	1	1	yes	characteristic
70	#389	—	269	2	no	—
71	#340	—	269	1	no	—
72	#340	—	269	2	no	—
73	C₂²×C₈	C₂×D₂₆₉	1	1	yes	characteristic
74	C₂²×C₈	C₂×D₂₆₉	1	1	yes	characteristic
75	C₂×C₄⋊C₄	—	269	2	no	—
76	C₂×C₄⋊C₄	—	269	2	no	—
77	C₂×C₄⋊C₄	—	538	2	no	—
78	C₂×C₄²	C₂×D₂₆₉	1	1	yes	characteristic
79	#107	—	538	2	no	—
80	C₄⋊C₈	—	2	2	no	—
81	#95	—	538	2	no	—
82	C₂²×C₄	—	538	2	no	—
83	C₂²×C₄	—	538	2	no	—
84	C₂²×C₄	D₁₀₇₆	1	1	yes	characteristic
85	C₂²×C₄	D₁₀₇₆	1	1	yes	characteristic
86	C₂²×C₄	C₂²×D₂₆₉	1	1	yes	characteristic
87	C₂×C₈	—	2	2	no	—
88	C₂×C₈	—	2	1	no	—
89	C₂×C₈	—	2	2	no	—
90	C₂×C₈	C₂²⋊D₂₆₉	1	2	yes	—
91	C₄⋊C₄	—	269	4	no	—
92	C₄⋊C₄	—	538	2	no	—
93	C₄⋊C₄	—	538	2	no	—
94	C₄⋊C₄	—	538	2	no	—
95	C₄⋊C₄	—	1076	4	no	—
96	C₄²	—	2	1	no	—
97	C₄²	—	2	1	no	—
98	C₂³	C₂×D₁₀₇₆	1	1	yes	characteristic
99	C₂×C₄	—	2	1	no	—
100	C₂×C₄	—	2	1	no	—
101	C₂×C₄	—	2	1	no	—
102	C₂×C₄	—	2	2	no	—
103	C₂×C₄	—	538	4	no	—
104	C₂×C₄	—	1076	2	no	—
105	C₂×C₄	—	1076	2	no	—
106	C₂×C₄	—	1076	2	no	—
107	C₂×C₄	—	1076	2	no	—
108	C₂×C₄	—	1076	2	no	—
109	C₂×C₄	Dic₁₀₇₆	1	2	yes	—
110	C₂×C₄	C₂×C₂²⋊D₂₆₉	1	1	yes	characteristic
111	C₂×C₄	Q₈×D₂₆₉	1	2	yes	—
112	C₂×C₄	#16864401	1	2	yes	—
113	C₂×C₄	#16864402	1	1	yes	characteristic
114	C₈	—	2	2	no	—
115	C₈	—	4	1	no	—
116	C₂²	#22575427	1	1	yes	characteristic
117	C₂²	#22575433	1	2	yes	—
118	C₂²	C₂×Dic₁₀₇₆	1	1	yes	characteristic
119	C₂²	#22575475	1	1	yes	characteristic
120	C₂²	#22575481	1	2	yes	—
121	C₄	—	2	2	no	—
122	C₄	—	4	1	no	—
123	C₄	—	1076	4	no	—
124	C₄	—	2152	2	no	—
125	C₄	#22575434	1	1	yes	characteristic
126	C₄	#22575443	1	2	yes	—
127	C₄	#22575451	1	1	yes	characteristic
128	C₂	#31770535	1	1	yes	characteristic
129	C₂	#31770846	1	1	yes	characteristic
130	C₂	#31770858	1	1	yes	characteristic
131	C₂	#31770901	1	2	yes	—
132	C₂	#31770906	1	2	yes	—

subgroup

G/H

class size

multiplicity

normal?

properties

C₂×#22575387

C₂

yes

maximal, characteristic

#31770503

C₂

yes

maximal

#31770368

C₂

yes

maximal

#31770368

C₂

yes

maximal, characteristic

C₂×C₄⋊C₈×C₂₆₉

C₂

yes

maximal, characteristic

C₂×C₄⋊Dic₂₆₉

C₂²

yes

—

C₂×C₄⋊Dic₂₆₉

C₂²

yes

—

C₂×#16864381

—

#22575387

—

#22575360

—

C₂²×C₈×C₂₆₉

C₂²

yes

characteristic

C₂²×C₈×C₂₆₉

C₂²

yes

characteristic

C₂×C₄²×C₂₆₉

C₂²

yes

characteristic

C₄⋊C₈×C₂₆₉

—

C₂²×Dic₂₆₉

—

C₂²×Dic₂₆₉

—

C₄⋊Dic₂₆₉

—

C₄⋊Dic₂₆₉

—

C₄⋊Dic₂₆₉

—

C₄⋊Dic₂₆₉

Q₈

yes

—

#16864381

—

C₂²×C₄×C₂₆₉

D₄

yes

characteristic

C₂²×C₄×C₂₆₉

D₄

yes

characteristic

C₂²×C₄×C₂₆₉

C₂³

yes

characteristic

C₂×C₈×C₂₆₉

—

C₂×C₈×C₂₆₉

—

C₂×C₈×C₂₆₉

—

C₂×C₈×C₂₆₉

D₄

yes

—

C₄²×C₂₆₉

—

C₄²×C₂₆₉

—

C₂×Dic₂₆₉

—

C₂×Dic₂₆₉

—

C₂×Dic₂₆₉

—

C₂×Dic₂₆₉

—

C₂×Dic₂₆₉

—

C₂×Dic₂₆₉

—

C₂³×C₂₆₉

C₂×D₄

yes

characteristic

C₂×C₄×C₂₆₉

—

C₂×C₄×C₂₆₉

—

C₂×C₄×C₂₆₉

—

C₂×C₄×C₂₆₉

—

C₂×C₄×C₂₆₉

Q₁₆

yes

—

C₂×C₄×C₂₆₉

C₂×D₄

yes

characteristic

C₂×C₄×C₂₆₉

C₂×Q₈

yes

—

C₂×C₄×C₂₆₉

#41

yes

—

C₂×C₄×C₂₆₉

#41

yes

characteristic

C₈×C₂₆₉

—

C₈×C₂₆₉

—

C₂²×C₂₆₉

#121

yes

characteristic

C₂²×C₂₆₉

C₂²⋊Q₈

yes

—

C₂²×C₂₆₉

C₂×Q₁₆

yes

C₂²×C₂₆₉

yes

C₂²×C₂₆₉

yes

—

C₄×C₂₆₉

—

C₄×C₂₆₉

—

C₄×C₂₆₉

#123

yes

characteristic

C₄×C₂₆₉

#125

yes

—

C₄×C₂₆₉

C₄⋊Q₈

yes

characteristic

Dic₂₆₉

—

Dic₂₆₉

—

C₂×C₂₆₉

yes

C₂×C₂₆₉

yes

C₂×C₂₆₉

yes

C₂×C₂₆₉

yes

—

C₂×C₂₆₉

#478

yes

—

C₂₆₉

#2087

yes

characteristic, complemented, Sylow

#2087

—

269

maximal, Sylow

C₂×#107

—

269

—

C₂×C₄⋊C₈

D₂₆₉

yes

characteristic

#389

—

269

—

#340

—

269

—

#340

—

269

—

C₂²×C₈

C₂×D₂₆₉

yes

characteristic

C₂²×C₈

C₂×D₂₆₉

yes

characteristic

C₂×C₄⋊C₄

—

269

—

C₂×C₄⋊C₄

—

269

—

C₂×C₄⋊C₄

—

538

—

C₂×C₄²

C₂×D₂₆₉

yes

characteristic

#107

—

538

—

C₄⋊C₈

—

#95

—

538

—

C₂²×C₄

—

538

—

C₂²×C₄

—

538

—

C₂²×C₄

D₁₀₇₆

yes

characteristic

C₂²×C₄

D₁₀₇₆

yes

characteristic

C₂²×C₄

C₂²×D₂₆₉

yes

characteristic

C₂×C₈

—

C₂×C₈

—

C₂×C₈

—

C₂×C₈

C₂²⋊D₂₆₉

yes

—

C₄⋊C₄

—

269

—

C₄⋊C₄

—

538

—

C₄⋊C₄

—

538

—

C₄⋊C₄

—

538

—

C₄⋊C₄

—

1076

—

C₄²

—

C₄²

—

C₂³

C₂×D₁₀₇₆

yes

characteristic

C₂×C₄

—

100

C₂×C₄

—

101

C₂×C₄

—

102

C₂×C₄

—

103

C₂×C₄

—

538

—

104

C₂×C₄

—

1076

—

105

C₂×C₄

—

1076

—

106

C₂×C₄

—

1076

—

107

C₂×C₄

—

1076

—

108

C₂×C₄

—

1076

—

109

C₂×C₄

Dic₁₀₇₆

yes

—

110

C₂×C₄

C₂×C₂²⋊D₂₆₉

yes

characteristic

111

C₂×C₄

Q₈×D₂₆₉

yes

—

112

C₂×C₄

#16864401

yes

—

113

C₂×C₄

#16864402

yes

characteristic

114

C₈

—

115

C₈

—

116

C₂²

#22575427

yes

characteristic

117

C₂²

#22575433

yes

—

118

C₂²

C₂×Dic₁₀₇₆

yes

characteristic

119

C₂²

#22575475

yes

characteristic

120

C₂²

#22575481

yes

—

121

C₄

—

122

C₄

—

123

C₄

—

1076

—

124

C₄

—

2152

—

125

C₄

#22575434

yes

characteristic

126

C₄

#22575443

yes

—

127

C₄

#22575451

yes

characteristic

128

C₂

#31770535

yes

characteristic

129

C₂

#31770846

yes

characteristic

130

C₂

#31770858

yes

characteristic

131

C₂

#31770901

yes

—

132

C₂

#31770906

yes

—

	subgroup	G/H
center	C₂³	C₂×D₁₀₇₆
commutator, cosocle	C₂²×C₄×C₂₆₉	C₂³
Frattini	C₂²×C₄	C₂²×D₂₆₉
Fitting	C₂×C₄⋊C₈×C₂₆₉	C₂
socle, prime radical	C₂³×C₂₆₉	C₂×D₄
hypercenter	C₂×C₄⋊C₈	D₂₆₉
hypocenter	C₂₆₉	#2087
LCS	G ▷ C₂²×C₄×C₂₆₉ ▷ C₂×C₂₆₉ ▷ C₂₆₉
UCS	C₁ ◁ C₂³ ◁ C₂×C₄² ◁ C₂×C₄⋊C₈

subgroup

G/H

center

C₂³

C₂×D₁₀₇₆

commutator, cosocle

C₂²×C₄×C₂₆₉

C₂³

Frattini

C₂²×C₄

C₂²×D₂₆₉

Fitting

C₂×C₄⋊C₈×C₂₆₉

C₂

socle, prime radical

C₂³×C₂₆₉

C₂×D₄

hypercenter

C₂×C₄⋊C₈

D₂₆₉

hypocenter

C₂₆₉

#2087

LCS

G ▷ C₂²×C₄×C₂₆₉ ▷ C₂×C₂₆₉ ▷ C₂₆₉

UCS

C₁ ◁ C₂³ ◁ C₂×C₄² ◁ C₂×C₄⋊C₈

N	H ≅ G/N	type
66. C₂₆₉	#2087	semidirect

H ≅ G/N

type

66. C₂₆₉

#2087

semidirect

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

subgroup	index
C₂×#22575387	2
#31770503	2
#31770368	2
C₂×C₄⋊C₈×C₂₆₉	2
#2087	269

subgroup

index

C₂×C₄⋊C₈×C₂₆₉

269

Subgroups — #46812378

Conjugacy classes of subgroups

Complements

Maximal subgroups