Group #46822339 — subgroups

Subgroups — #46822339

Order 34432.

322 conjugacy classes of subgroups; only the 117 normal subgroups are shown.

normal subgroups only

This group has 322 conjugacy classes of subgroups — too many to draw. Only its 117 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	#31771217	C₂	1	2	yes	maximal
2	#31770913	C₂	1	4	yes	maximal, complemented
3	#31770863	C₂	1	2	yes	maximal
4	#31770846	C₂	1	2	yes	maximal
5	#31770816	C₂	1	4	yes	maximal, complemented
6	#424×C₂₆₉	C₂	1	1	yes	maximal, characteristic
7	C₂×Dic₁₀₇₆	C₂²	1	2	yes	—
8	C₂×Dic₁₀₇₆	C₂²	1	2	yes	—
9	#22575453	C₂²	1	2	yes	—
10	#22575446	—	2	4	no	—
11	#22575445	C₂²	1	2	yes	—
12	#22575443	—	2	4	no	—
13	#22575432	C₂²	1	4	yes	—
14	C₄×Dic₅₃₈	C₂²	1	4	yes	—
15	#22575387	C₂²	1	2	yes	—
16	#22575385	C₂²	1	2	yes	—
17	#22575378	C₂²	1	4	yes	—
18	#22575378	C₂²	1	4	yes	—
19	C₂×M₁₆×C₂₆₉	C₂²	1	1	yes	characteristic
20	C₂²×C₈×C₂₆₉	C₂²	1	1	yes	characteristic
21	#117×C₂₆₉	C₂²	1	1	yes	characteristic
22	C₄⋊C₈×C₂₆₉	C₂²	1	4	yes	—
23	C₂×Dic₅₃₈	C₂³	1	4	yes	—
24	Dic₁₀₇₆	—	4	2	no	—
25	Dic₁₀₇₆	—	4	2	no	—
26	C₄⋊Dic₂₆₉	—	2	4	no	—
27	C₄⋊Dic₂₆₉	C₂³	1	4	yes	—
28	#16864381	—	2	4	no	—
29	#16864381	—	2	4	no	—
30	#16864381	—	2	4	no	—
31	C₂²⋊Dic₂₆₉	—	2	4	no	—
32	C₄×Dic₂₆₉	—	2	4	no	—
33	C₂²×C₄×C₂₆₉	C₂³	1	1	yes	characteristic
34	M₁₆×C₂₆₉	—	4	1	no	—
35	C₂×C₈×C₂₆₉	—	4	1	no	—
36	C₂×C₈×C₂₆₉	C₂³	1	2	yes	—
37	C₂×C₈×C₂₆₉	C₂³	1	2	yes	—
38	C₄⋊C₄×C₂₆₉	D₄	1	2	yes	—
39	C₂²⋊C₄×C₂₆₉	D₄	1	2	yes	—
40	C₄²×C₂₆₉	C₂³	1	2	yes	—
41	Dic₅₃₈	—	2	4	no	—
42	Dic₅₃₈	—	2	4	no	—
43	C₂×Dic₂₆₉	—	2	4	no	—
44	C₂×Dic₂₆₉	—	2	4	no	—
45	C₂³×C₂₆₉	C₂×D₄	1	1	yes	characteristic
46	C₂×C₄×C₂₆₉	—	2	1	no	—
47	C₂×C₄×C₂₆₉	—	2	1	no	—
48	C₂×C₄×C₂₆₉	C₂×D₄	1	4	yes	—
49	C₂×C₄×C₂₆₉	C₂×D₄	1	1	yes	characteristic
50	C₂×C₄×C₂₆₉	C₂⁴	1	1	yes	characteristic
51	C₈×C₂₆₉	—	2	2	no	—
52	C₈×C₂₆₉	—	2	2	no	—
53	C₂²×C₂₆₉	—	2	1	no	—
54	C₂²×C₂₆₉	—	2	2	no	—
55	C₂²×C₂₆₉	C₂²×D₄	1	1	yes	characteristic
56	C₄×C₂₆₉	—	2	1	no	—
57	C₄×C₂₆₉	—	2	4	no	—
58	C₄×C₂₆₉	H(5,𝔽₂)	1	1	yes	characteristic
59	C₄×C₂₆₉	2₋¹⁺⁴	1	1	yes	characteristic
60	Dic₂₆₉	—	4	4	no	—
61	Dic₂₆₉	—	4	4	no	—
62	C₂×C₂₆₉	—	4	1	no	—
63	C₂×C₂₆₉	#537	1	1	yes	characteristic
64	C₂×C₂₆₉	#578	1	2	yes	—
65	C₂₆₉	#3261	1	1	yes	characteristic, complemented, Sylow
66	#3261	—	269	1	no	maximal, Sylow
67	#541	—	269	2	no	—
68	#479	—	269	4	no	—
69	#470	—	269	2	no	—
70	#467	—	269	2	no	—
71	#462	—	269	4	no	—
72	#424	D₂₆₉	1	1	yes	characteristic
73	C₂×Q₁₆	—	269	2	no	—
74	C₂×Q₁₆	—	269	2	no	—
75	C₂×M₁₆	C₂×D₂₆₉	1	1	yes	characteristic
76	C₂²×C₈	C₂×D₂₆₉	1	1	yes	characteristic
77	C₄⋊Q₈	—	269	2	no	—
78	#126	—	538	4	no	—
79	#125	—	269	2	no	—
80	#125	—	538	4	no	—
81	C₂²⋊Q₈	—	269	4	no	—
82	C₄×Q₈	—	269	4	no	—
83	#117	C₂×D₂₆₉	1	1	yes	characteristic
84	#107	—	269	2	no	—
85	#106	—	269	2	no	—
86	C₄⋊C₈	C₂×D₂₆₉	1	4	yes	—
87	#103	—	269	4	no	—
88	#103	—	269	4	no	—
89	C₂×Q₈	—	269	4	no	—
90	C₂²×C₄	C₂²×D₂₆₉	1	1	yes	characteristic
91	Q₁₆	—	1076	2	no	—
92	Q₁₆	—	1076	2	no	—
93	M₁₆	—	4	1	no	—
94	C₂×C₈	—	4	1	no	—
95	C₂×C₈	C₂²×D₂₆₉	1	2	yes	—
96	C₂×C₈	C₂²×D₂₆₉	1	2	yes	—
97	C₄⋊C₄	—	269	4	no	—
98	C₄⋊C₄	—	538	4	no	—
99	C₄⋊C₄	—	538	4	no	—
100	C₄⋊C₄	—	538	4	no	—
101	C₄⋊C₄	—	538	4	no	—
102	C₄⋊C₄	D₁₀₇₆	1	2	yes	—
103	C₂²⋊C₄	—	538	4	no	—
104	C₂²⋊C₄	D₁₀₇₆	1	2	yes	—
105	C₄²	—	538	4	no	—
106	C₄²	C₂²×D₂₆₉	1	2	yes	—
107	C₂³	C₂×D₁₀₇₆	1	1	yes	characteristic
108	Q₈	—	538	4	no	—
109	Q₈	—	538	4	no	—
110	C₂×C₄	—	2	1	no	—
111	C₂×C₄	—	2	1	no	—
112	C₂×C₄	—	538	4	no	—
113	C₂×C₄	—	538	4	no	—
114	C₂×C₄	C₂×D₁₀₇₆	1	4	yes	—
115	C₂×C₄	C₂×D₁₀₇₆	1	1	yes	characteristic
116	C₂×C₄	C₂³×D₂₆₉	1	1	yes	characteristic
117	C₈	—	2	2	no	—
118	C₈	—	2	2	no	—
119	C₂²	—	2	2	no	—
120	C₂²	—	2	1	no	—
121	C₂²	C₂²×D₁₀₇₆	1	1	yes	characteristic
122	C₄	—	2	4	no	—
123	C₄	—	2	1	no	—
124	C₄	—	1076	4	no	—
125	C₄	—	1076	4	no	—
126	C₄	#22575498	1	1	yes	characteristic
127	C₄	#22575501	1	1	yes	characteristic
128	C₂	—	4	1	no	—
129	C₂	#31771186	1	1	yes	characteristic
130	C₂	#31771422	1	2	yes	—

subgroup

G/H

class size

multiplicity

normal?

properties

#31771217

C₂

yes

maximal

#31770913

C₂

yes

maximal, complemented

#31770863

C₂

yes

maximal

#31770846

C₂

yes

maximal

#31770816

C₂

yes

maximal, complemented

#424×C₂₆₉

C₂

yes

maximal, characteristic

C₂×Dic₁₀₇₆

C₂²

yes

—

C₂×Dic₁₀₇₆

C₂²

yes

—

#22575453

C₂²

yes

—

#22575446

—

#22575445

C₂²

yes

—

#22575443

—

#22575432

C₂²

yes

—

C₄×Dic₅₃₈

C₂²

yes

—

#22575387

C₂²

yes

—

#22575385

C₂²

yes

—

#22575378

C₂²

yes

—

#22575378

C₂²

yes

—

C₂×M₁₆×C₂₆₉

C₂²

yes

characteristic

C₂²×C₈×C₂₆₉

C₂²

yes

characteristic

#117×C₂₆₉

C₂²

yes

characteristic

C₄⋊C₈×C₂₆₉

C₂²

yes

—

C₂×Dic₅₃₈

C₂³

yes

—

Dic₁₀₇₆

—

Dic₁₀₇₆

—

C₄⋊Dic₂₆₉

—

C₄⋊Dic₂₆₉

C₂³

yes

—

#16864381

—

#16864381

—

#16864381

—

C₂²⋊Dic₂₆₉

—

C₄×Dic₂₆₉

—

C₂²×C₄×C₂₆₉

C₂³

yes

characteristic

M₁₆×C₂₆₉

—

C₂×C₈×C₂₆₉

—

C₂×C₈×C₂₆₉

C₂³

yes

—

C₂×C₈×C₂₆₉

C₂³

yes

—

C₄⋊C₄×C₂₆₉

D₄

yes

—

C₂²⋊C₄×C₂₆₉

D₄

yes

—

C₄²×C₂₆₉

C₂³

yes

—

Dic₅₃₈

—

Dic₅₃₈

—

C₂×Dic₂₆₉

—

C₂×Dic₂₆₉

—

C₂³×C₂₆₉

C₂×D₄

yes

characteristic

C₂×C₄×C₂₆₉

—

C₂×C₄×C₂₆₉

—

C₂×C₄×C₂₆₉

C₂×D₄

yes

—

C₂×C₄×C₂₆₉

C₂×D₄

yes

characteristic

C₂×C₄×C₂₆₉

C₂⁴

yes

characteristic

C₈×C₂₆₉

—

C₈×C₂₆₉

—

C₂²×C₂₆₉

—

C₂²×C₂₆₉

—

C₂²×C₂₆₉

C₂²×D₄

yes

characteristic

C₄×C₂₆₉

—

C₄×C₂₆₉

—

C₄×C₂₆₉

H(5,𝔽₂)

yes

characteristic

C₄×C₂₆₉

2₋¹⁺⁴

yes

characteristic

Dic₂₆₉

—

Dic₂₆₉

—

C₂×C₂₆₉

—

C₂×C₂₆₉

#537

yes

characteristic

C₂×C₂₆₉

#578

yes

—

C₂₆₉

#3261

yes

characteristic, complemented, Sylow

#3261

—

269

maximal, Sylow

#541

—

269

—

#479

—

269

—

#470

—

269

—

#467

—

269

—

#462

—

269

—

#424

D₂₆₉

yes

characteristic

C₂×Q₁₆

—

269

—

C₂×Q₁₆

—

269

—

C₂×M₁₆

C₂×D₂₆₉

yes

characteristic

C₂²×C₈

C₂×D₂₆₉

yes

characteristic

C₄⋊Q₈

—

269

—

#126

—

538

—

#125

—

269

—

#125

—

538

—

C₂²⋊Q₈

—

269

—

C₄×Q₈

—

269

—

#117

C₂×D₂₆₉

yes

characteristic

#107

—

269

—

#106

—

269

—

C₄⋊C₈

C₂×D₂₆₉

yes

—

#103

—

269

—

#103

—

269

—

C₂×Q₈

—

269

—

C₂²×C₄

C₂²×D₂₆₉

yes

characteristic

Q₁₆

—

1076

—

Q₁₆

—

1076

—

M₁₆

—

C₂×C₈

—

C₂×C₈

C₂²×D₂₆₉

yes

—

C₂×C₈

C₂²×D₂₆₉

yes

—

C₄⋊C₄

—

269

—

C₄⋊C₄

—

538

—

C₄⋊C₄

—

538

—

100

C₄⋊C₄

—

538

—

101

C₄⋊C₄

—

538

—

102

C₄⋊C₄

D₁₀₇₆

yes

—

103

C₂²⋊C₄

—

538

—

104

C₂²⋊C₄

D₁₀₇₆

yes

—

105

C₄²

—

538

—

106

C₄²

C₂²×D₂₆₉

yes

—

107

C₂³

C₂×D₁₀₇₆

yes

characteristic

108

Q₈

—

538

—

109

Q₈

—

538

—

110

C₂×C₄

—

111

C₂×C₄

—

112

C₂×C₄

—

538

—

113

C₂×C₄

—

538

—

114

C₂×C₄

C₂×D₁₀₇₆

yes

—

115

C₂×C₄

C₂×D₁₀₇₆

yes

characteristic

116

C₂×C₄

C₂³×D₂₆₉

yes

characteristic

117

C₈

—

118

C₈

—

119

C₂²

—

120

C₂²

—

121

C₂²

C₂²×D₁₀₇₆

yes

characteristic

122

C₄

—

123

C₄

—

124

C₄

—

1076

—

125

C₄

—

1076

—

126

C₄

#22575498

yes

characteristic

127

C₄

#22575501

yes

characteristic

128

C₂

—

129

C₂

#31771186

yes

characteristic

130

C₂

#31771422

yes

—

	subgroup	G/H
center	C₂²	C₂²×D₁₀₇₆
commutator, cosocle	C₂×C₄×C₂₆₉	C₂⁴
Frattini	C₂×C₄	C₂³×D₂₆₉
Fitting	#424×C₂₆₉	C₂
socle	C₂²×C₂₆₉	C₂²×D₄
prime radical	C₂³×C₂₆₉	C₂×D₄
hypercenter	#424	D₂₆₉
hypocenter	C₂₆₉	#3261
LCS	G ▷ C₂×C₄×C₂₆₉ ▷ C₂×C₂₆₉ ▷ C₂₆₉
UCS	C₁ ◁ C₂² ◁ #117 ◁ #424

subgroup

G/H

center

C₂²

C₂²×D₁₀₇₆

commutator, cosocle

C₂×C₄×C₂₆₉

C₂⁴

Frattini

C₂×C₄

C₂³×D₂₆₉

Fitting

#424×C₂₆₉

C₂

socle

C₂²×C₂₆₉

C₂²×D₄

prime radical

C₂³×C₂₆₉

C₂×D₄

D₂₆₉

C₂₆₉

G ▷ C₂×C₄×C₂₆₉ ▷ C₂×C₂₆₉ ▷ C₂₆₉

UCS

C₁ ◁ C₂² ◁ #117 ◁ #424

N	H ≅ G/N	type
2. #31770913	C₂	semidirect
5. #31770816	C₂	semidirect
65. C₂₆₉	#3261	semidirect

H ≅ G/N

type

2. #31770913

C₂

semidirect

5. #31770816

C₂

semidirect

65. C₂₆₉

#3261

semidirect

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

subgroup	index
#31771217	2
#31770913	2
#31770863	2
#31770846	2
#31770816	2
#424×C₂₆₉	2
#3261	269

subgroup

index

#424×C₂₆₉

269

Subgroups — #46822339

Conjugacy classes of subgroups

Complements

Maximal subgroups