Group #46822684 — subgroups

Subgroups — #46822684

Order 34432.

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

normal subgroups only

This group has 476 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	#31771406	C₂	1	4	yes	maximal, complemented
2	#31770872	C₂	1	4	yes	maximal, complemented
3	#31770772	C₂	1	4	yes	maximal, complemented
4	#31770723	C₂	1	2	yes	maximal, complemented
5	#493×C₂₆₉	C₂	1	1	yes	maximal, characteristic, complemented
6	#22575498	C₂²	1	4	yes	complemented
7	#22575479	C₂²	1	8	yes	complemented
8	#22575477	—	2	8	no	—
9	#22575475	C₂²	1	2	yes	—
10	#22575472	—	2	4	no	—
11	D₈×D₂₆₉	—	2	4	no	—
12	#22575459	C₂²	1	4	yes	complemented
13	#22575389	C₂²	1	2	yes	complemented
14	#22575381	C₂²	1	4	yes	complemented
15	#22575370	C₂²	1	4	yes	complemented
16	C₂×D₈×C₂₆₉	C₂²	1	4	yes	complemented
17	#127×C₂₆₉	C₂²	1	2	yes	complemented
18	C₄×C₈×C₂₆₉	C₂²	1	1	yes	characteristic, complemented
19	#16864402	C₂³	1	4	yes	—
20	#16864401	—	2	8	no	—
21	C₂×C₂²⋊D₂₆₉	—	4	4	no	—
22	D₄×D₂₆₉	—	2	8	no	—
23	Dic₁₀₇₆	D₄	1	2	yes	complemented
24	#16864391	D₄	1	4	yes	complemented
25	#16864390	—	2	8	no	—
26	D₂₁₅₂	D₄	1	2	yes	complemented
27	#16864387	—	2	8	no	—
28	#16864386	C₂³	1	4	yes	—
29	#16864385	—	2	4	no	—
30	C₈×D₂₆₉	—	2	4	no	—
31	C₂×D₄×C₂₆₉	—	4	2	no	—
32	C₂×D₄×C₂₆₉	C₂³	1	4	yes	—
33	D₈×C₂₆₉	—	2	4	no	—
34	D₈×C₂₆₉	—	2	4	no	—
35	C₂×C₈×C₂₆₉	C₂³	1	2	yes	—
36	C₄²×C₂₆₉	C₂³	1	1	yes	characteristic
37	C₂²×D₂₆₉	—	4	4	no	—
38	Dic₅₃₈	C₂×D₄	1	4	yes	—
39	D₁₀₇₆	C₂×D₄	1	4	yes	—
40	C₂²⋊D₂₆₉	—	2	4	no	—
41	C₂²⋊D₂₆₉	—	4	8	no	—
42	C₂×Dic₂₆₉	—	4	4	no	—
43	C₄×D₂₆₉	—	2	4	no	—
44	#14524130	—	2	4	no	—
45	C₂³×C₂₆₉	—	2	4	no	—
46	D₄×C₂₆₉	—	2	8	no	—
47	D₄×C₂₆₉	—	8	4	no	—
48	C₂×C₄×C₂₆₉	—	2	1	no	—
49	C₂×C₄×C₂₆₉	C₂⁴	1	1	yes	characteristic
50	C₈×C₂₆₉	C₂×D₄	1	2	yes	—
51	C₈×C₂₆₉	C₂×D₄	1	2	yes	—
52	C₂²×C₂₆₉	—	4	4	no	—
53	C₂²×C₂₆₉	—	8	4	no	—
54	C₂²×C₂₆₉	H(5,𝔽₂)	1	1	yes	characteristic
55	C₂×D₂₆₉	—	2	4	no	—
56	C₂×D₂₆₉	—	16	4	no	—
57	C₄×C₂₆₉	—	2	2	no	—
58	C₄×C₂₆₉	C₂²×D₄	1	2	yes	—
59	Dic₂₆₉	—	2	4	no	—
60	C₂×C₂₆₉	—	2	1	no	—
61	C₂×C₂₆₉	—	8	4	no	—
62	C₂×C₂₆₉	D₄²	1	1	yes	characteristic
63	D₂₆₉	—	4	4	no	—
64	C₂₆₉	#3292	1	1	yes	characteristic, complemented, Sylow
65	#3292	—	269	1	no	maximal, Sylow
66	#576	—	269	4	no	—
67	#493	D₂₆₉	1	1	yes	characteristic, complemented
68	#472	—	269	4	no	—
69	#453	—	269	4	no	—
70	#443	—	269	2	no	—
71	H(5,𝔽₂)	—	269	4	no	—
72	AGL(1,ℤ/8)	—	269	8	no	—
73	AGL(1,ℤ/8)	—	538	8	no	—
74	#135	—	269	2	no	—
75	#135	—	538	4	no	—
76	C₂×D₈	—	538	4	no	—
77	C₂×D₈	C₂×D₂₆₉	1	4	yes	complemented
78	#131	—	269	4	no	—
79	#127	C₂×D₂₆₉	1	2	yes	complemented
80	#108	—	269	2	no	—
81	#104	—	269	4	no	—
82	#100	—	269	4	no	—
83	C₄×C₈	C₂×D₂₆₉	1	1	yes	characteristic, complemented
84	#41	—	269	4	no	—
85	#41	—	538	8	no	—
86	C₂×D₄	—	4	2	no	—
87	C₂×D₄	—	538	8	no	—
88	C₂×D₄	—	1076	4	no	—
89	C₂×D₄	C₂²×D₂₆₉	1	4	yes	—
90	Q₁₆	—	269	2	no	—
91	SD₁₆	—	269	4	no	—
92	SD₁₆	—	538	8	no	—
93	D₈	—	2	4	no	—
94	D₈	—	2	4	no	—
95	D₈	—	269	2	no	—
96	D₈	—	538	8	no	—
97	M₁₆	—	269	4	no	—
98	M₁₆	—	538	4	no	—
99	C₂×C₈	—	538	4	no	—
100	C₂×C₈	C₂²×D₂₆₉	1	2	yes	—
101	C₄²	C₂²×D₂₆₉	1	1	yes	characteristic
102	C₂³	—	2	4	no	—
103	C₂³	—	1076	4	no	—
104	Q₈	—	269	4	no	—
105	D₄	—	2	8	no	—
106	D₄	—	8	4	no	—
107	D₄	—	269	4	no	—
108	D₄	—	538	4	no	—
109	D₄	—	1076	8	no	—
110	C₂×C₄	—	2	1	no	—
111	C₂×C₄	—	538	4	no	—
112	C₂×C₄	—	1076	4	no	—
113	C₂×C₄	C₂³×D₂₆₉	1	1	yes	characteristic
114	C₈	—	538	4	no	—
115	C₈	D₄×D₂₆₉	1	2	yes	—
116	C₈	D₄×D₂₆₉	1	2	yes	—
117	C₂²	—	4	4	no	—
118	C₂²	—	8	4	no	—
119	C₂²	—	538	4	no	—
120	C₂²	—	4304	4	no	—
121	C₂²	#22575498	1	1	yes	characteristic
122	C₄	—	2	2	no	—
123	C₄	—	538	4	no	—
124	C₄	C₂×D₄×D₂₆₉	1	2	yes	—
125	C₂	—	2	1	no	—
126	C₂	—	8	4	no	—
127	C₂	—	1076	4	no	—
128	C₂	#31771240	1	1	yes	characteristic

subgroup

G/H

class size

multiplicity

normal?

properties

#31771406

C₂

yes

maximal, complemented

#31770872

C₂

yes

maximal, complemented

#31770772

C₂

yes

maximal, complemented

#31770723

C₂

yes

maximal, complemented

#493×C₂₆₉

C₂

yes

maximal, characteristic, complemented

#22575498

C₂²

yes

complemented

#22575479

C₂²

yes

complemented

#22575477

—

#22575475

C₂²

yes

—

#22575472

—

D₈×D₂₆₉

—

#22575459

C₂²

yes

complemented

#22575389

C₂²

yes

complemented

#22575381

C₂²

yes

complemented

#22575370

C₂²

yes

complemented

C₂×D₈×C₂₆₉

C₂²

yes

complemented

#127×C₂₆₉

C₂²

yes

complemented

C₄×C₈×C₂₆₉

C₂²

yes

characteristic, complemented

#16864402

C₂³

yes

—

#16864401

—

C₂×C₂²⋊D₂₆₉

—

D₄×D₂₆₉

—

Dic₁₀₇₆

D₄

yes

complemented

#16864391

D₄

yes

complemented

#16864390

—

D₂₁₅₂

D₄

yes

complemented

#16864387

—

#16864386

C₂³

yes

—

#16864385

—

C₈×D₂₆₉

—

C₂×D₄×C₂₆₉

—

C₂×D₄×C₂₆₉

C₂³

yes

—

D₈×C₂₆₉

—

D₈×C₂₆₉

—

C₂×C₈×C₂₆₉

C₂³

yes

—

C₄²×C₂₆₉

C₂³

yes

characteristic

C₂²×D₂₆₉

—

Dic₅₃₈

C₂×D₄

yes

—

D₁₀₇₆

C₂×D₄

yes

—

C₂²⋊D₂₆₉

—

C₂²⋊D₂₆₉

—

C₂×Dic₂₆₉

—

C₄×D₂₆₉

—

#14524130

—

C₂³×C₂₆₉

—

D₄×C₂₆₉

—

D₄×C₂₆₉

—

C₂×C₄×C₂₆₉

—

C₂×C₄×C₂₆₉

C₂⁴

yes

characteristic

C₈×C₂₆₉

C₂×D₄

yes

—

C₈×C₂₆₉

C₂×D₄

yes

—

C₂²×C₂₆₉

—

C₂²×C₂₆₉

—

C₂²×C₂₆₉

H(5,𝔽₂)

yes

characteristic

C₂×D₂₆₉

—

C₂×D₂₆₉

—

C₄×C₂₆₉

—

C₄×C₂₆₉

C₂²×D₄

yes

—

Dic₂₆₉

—

C₂×C₂₆₉

—

C₂×C₂₆₉

—

C₂×C₂₆₉

D₄²

yes

characteristic

D₂₆₉

—

C₂₆₉

#3292

yes

characteristic, complemented, Sylow

#3292

—

269

maximal, Sylow

#576

—

269

—

#493

D₂₆₉

yes

characteristic, complemented

#472

—

269

—

#453

—

269

—

#443

—

269

—

H(5,𝔽₂)

—

269

—

AGL(1,ℤ/8)

—

269

—

AGL(1,ℤ/8)

—

538

—

#135

—

269

—

#135

—

538

—

C₂×D₈

—

538

—

C₂×D₈

C₂×D₂₆₉

yes

complemented

#131

—

269

—

#127

C₂×D₂₆₉

yes

complemented

#108

—

269

—

#104

—

269

—

#100

—

269

—

C₄×C₈

C₂×D₂₆₉

yes

characteristic, complemented

#41

—

269

—

#41

—

538

—

C₂×D₄

—

C₂×D₄

—

538

—

C₂×D₄

—

1076

—

C₂×D₄

C₂²×D₂₆₉

yes

—

Q₁₆

—

269

—

SD₁₆

—

269

—

SD₁₆

—

538

—

D₈

—

D₈

—

D₈

—

269

—

D₈

—

538

—

M₁₆

—

269

—

M₁₆

—

538

—

C₂×C₈

—

538

—

100

C₂×C₈

C₂²×D₂₆₉

yes

—

101

C₄²

C₂²×D₂₆₉

yes

characteristic

102

C₂³

—

103

C₂³

—

1076

—

104

Q₈

—

269

—

105

D₄

—

106

D₄

—

107

D₄

—

269

—

108

D₄

—

538

—

109

D₄

—

1076

—

110

C₂×C₄

—

111

C₂×C₄

—

538

—

112

C₂×C₄

—

1076

—

113

C₂×C₄

C₂³×D₂₆₉

yes

characteristic

114

C₈

—

538

—

115

C₈

D₄×D₂₆₉

yes

—

116

C₈

D₄×D₂₆₉

yes

—

117

C₂²

—

118

C₂²

—

119

C₂²

—

538

—

120

C₂²

—

4304

—

121

C₂²

#22575498

yes

characteristic

122

C₄

—

123

C₄

—

538

—

124

C₄

C₂×D₄×D₂₆₉

yes

—

125

C₂

—

126

C₂

—

127

C₂

—

1076

—

128

C₂

#31771240

yes

characteristic

	subgroup	G/H
center	C₂	#31771240
commutator, cosocle	C₂×C₄×C₂₆₉	C₂⁴
Frattini	C₂×C₄	C₂³×D₂₆₉
Fitting	#493×C₂₆₉	C₂
socle	C₂×C₂₆₉	D₄²
hypercenter	#493	D₂₆₉
hypocenter	C₂₆₉	#3292
LCS	G ▷ C₂×C₄×C₂₆₉ ▷ C₂×C₂₆₉ ▷ C₂₆₉
UCS	C₁ ◁ C₂ ◁ C₂×C₄ ◁ #493

subgroup

G/H

center

C₂

#31771240

commutator, cosocle

C₂×C₄×C₂₆₉

C₂⁴

Frattini

C₂×C₄

C₂³×D₂₆₉

Fitting

#493×C₂₆₉

C₂

socle

C₂×C₂₆₉

D₄²

D₂₆₉

C₂₆₉

G ▷ C₂×C₄×C₂₆₉ ▷ C₂×C₂₆₉ ▷ C₂₆₉

UCS

C₁ ◁ C₂ ◁ C₂×C₄ ◁ #493

N	H ≅ G/N	type
1. #31771406	C₂	semidirect
2. #31770872	C₂	semidirect
3. #31770772	C₂	semidirect
4. #31770723	C₂	semidirect
5. #493×C₂₆₉	C₂	semidirect
6. #22575498	C₂²	semidirect
7. #22575479	C₂²	semidirect
12. #22575459	C₂²	semidirect
13. #22575389	C₂²	semidirect
14. #22575381	C₂²	semidirect
15. #22575370	C₂²	semidirect
16. C₂×D₈×C₂₆₉	C₂²	semidirect
17. #127×C₂₆₉	C₂²	semidirect
18. C₄×C₈×C₂₆₉	C₂²	semidirect
23. Dic₁₀₇₆	D₄	semidirect
24. #16864391	D₄	semidirect
26. D₂₁₅₂	D₄	semidirect
64. C₂₆₉	#3292	semidirect
67. #493	D₂₆₉	semidirect
77. C₂×D₈	C₂×D₂₆₉	semidirect
79. #127	C₂×D₂₆₉	semidirect
83. C₄×C₈	C₂×D₂₆₉	semidirect

H ≅ G/N

type

1. #31771406

C₂

semidirect

2. #31770872

C₂

semidirect

3. #31770772

C₂

semidirect

4. #31770723

C₂

semidirect

5. #493×C₂₆₉

C₂

semidirect

6. #22575498

C₂²

semidirect

7. #22575479

C₂²

semidirect

12. #22575459

C₂²

semidirect

13. #22575389

C₂²

semidirect

14. #22575381

C₂²

semidirect

15. #22575370

C₂²

semidirect

16. C₂×D₈×C₂₆₉

C₂²

semidirect

17. #127×C₂₆₉

C₂²

semidirect

18. C₄×C₈×C₂₆₉

C₂²

semidirect

23. Dic₁₀₇₆

D₄

semidirect

24. #16864391

D₄

semidirect

26. D₂₁₅₂

D₄

semidirect

64. C₂₆₉

#3292

semidirect

67. #493

D₂₆₉

semidirect

77. C₂×D₈

C₂×D₂₆₉

semidirect

79. #127

C₂×D₂₆₉

semidirect

83. C₄×C₈

C₂×D₂₆₉

semidirect

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

subgroup	index
#31771406	2
#31770872	2
#31770772	2
#31770723	2
#493×C₂₆₉	2
#3292	269

subgroup

index

#493×C₂₆₉

269

Subgroups — #46822684

Conjugacy classes of subgroups

Complements

Maximal subgroups