Group #46822689 — subgroups

Subgroups — #46822689

Order 34432.

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

normal subgroups only

This group has 444 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	#31771418	C₂	1	4	yes	maximal, complemented
2	#31770877	C₂	1	4	yes	maximal, complemented
3	#31770787	C₂	1	4	yes	maximal, complemented
4	#31770723	C₂	1	2	yes	maximal, complemented
5	#494×C₂₆₉	C₂	1	1	yes	maximal, characteristic, complemented
6	#22575501	C₂²	1	4	yes	—
7	#22575485	—	2	8	no	—
8	#22575484	C₂²	1	8	yes	—
9	#22575476	—	2	4	no	—
10	#22575475	C₂²	1	2	yes	—
11	Q₁₆×D₂₆₉	—	2	4	no	—
12	#22575459	C₂²	1	4	yes	—
13	#22575389	C₂²	1	2	yes	—
14	#22575381	C₂²	1	4	yes	—
15	#22575372	C₂²	1	4	yes	—
16	C₂×Q₁₆×C₂₆₉	C₂²	1	4	yes	—
17	C₄⋊Q₈×C₂₆₉	C₂²	1	2	yes	—
18	C₄×C₈×C₂₆₉	C₂²	1	1	yes	characteristic
19	#16864403	—	2	8	no	—
20	#16864402	—	4	4	no	—
21	#16864402	C₂³	1	4	yes	—
22	Q₈×D₂₆₉	—	2	8	no	—
23	Dic₁₀₇₆	D₄	1	2	yes	—
24	#16864392	—	2	8	no	—
25	#16864391	D₄	1	4	yes	—
26	#16864389	—	2	8	no	—
27	D₂₁₅₂	D₄	1	2	yes	—
28	#16864386	C₂³	1	4	yes	—
29	#16864385	—	2	4	no	—
30	C₈×D₂₆₉	—	2	4	no	—
31	C₂×Q₈×C₂₆₉	C₂³	1	4	yes	—
32	Q₁₆×C₂₆₉	—	2	4	no	—
33	Q₁₆×C₂₆₉	—	2	4	no	—
34	C₂×C₈×C₂₆₉	C₂³	1	2	yes	—
35	C₄⋊C₄×C₂₆₉	—	4	2	no	—
36	C₄²×C₂₆₉	C₂³	1	1	yes	characteristic
37	Dic₅₃₈	—	4	4	no	—
38	Dic₅₃₈	C₂×D₄	1	4	yes	—
39	D₁₀₇₆	—	4	4	no	—
40	D₁₀₇₆	C₂×D₄	1	4	yes	—
41	C₂²⋊D₂₆₉	—	2	4	no	—
42	C₄×D₂₆₉	—	2	4	no	—
43	C₄×D₂₆₉	—	4	8	no	—
44	#14524130	—	2	4	no	—
45	Q₈×C₂₆₉	—	2	8	no	—
46	C₂×C₄×C₂₆₉	—	2	1	no	—
47	C₂×C₄×C₂₆₉	—	2	4	no	—
48	C₂×C₄×C₂₆₉	C₂⁴	1	1	yes	characteristic
49	C₈×C₂₆₉	C₂×D₄	1	2	yes	—
50	C₈×C₂₆₉	C₂×D₄	1	2	yes	—
51	C₂²×C₂₆₉	H(5,𝔽₂)	1	1	yes	characteristic
52	C₂×D₂₆₉	—	2	4	no	—
53	C₄×C₂₆₉	—	2	2	no	—
54	C₄×C₂₆₉	—	4	4	no	—
55	C₄×C₂₆₉	C₂²×D₄	1	2	yes	—
56	Dic₂₆₉	—	2	4	no	—
57	C₂×C₂₆₉	—	2	1	no	—
58	C₂×C₂₆₉	D₄²	1	1	yes	characteristic
59	D₂₆₉	—	4	4	no	—
60	C₂₆₉	#3293	1	1	yes	characteristic, complemented, Sylow
61	#3293	—	269	1	no	maximal, Sylow
62	#578	—	269	4	no	—
63	#494	D₂₆₉	1	1	yes	characteristic, complemented
64	#473	—	269	4	no	—
65	#456	—	269	4	no	—
66	#443	—	269	2	no	—
67	2₋¹⁺⁴	—	269	4	no	—
68	#137	—	269	8	no	—
69	#137	—	538	8	no	—
70	#135	—	269	2	no	—
71	#135	—	538	4	no	—
72	C₂×Q₁₆	—	538	4	no	—
73	C₂×Q₁₆	C₂×D₂₆₉	1	4	yes	—
74	#131	—	269	4	no	—
75	C₄⋊Q₈	C₂×D₂₆₉	1	2	yes	—
76	#108	—	269	2	no	—
77	#104	—	269	4	no	—
78	#101	—	269	4	no	—
79	C₄×C₈	C₂×D₂₆₉	1	1	yes	characteristic
80	#41	—	269	4	no	—
81	#41	—	538	8	no	—
82	#41	—	1076	4	no	—
83	C₂×Q₈	—	538	8	no	—
84	C₂×Q₈	C₂²×D₂₆₉	1	4	yes	—
85	Q₁₆	—	2	4	no	—
86	Q₁₆	—	2	4	no	—
87	Q₁₆	—	269	2	no	—
88	Q₁₆	—	538	8	no	—
89	SD₁₆	—	269	4	no	—
90	SD₁₆	—	538	8	no	—
91	D₈	—	269	2	no	—
92	M₁₆	—	269	4	no	—
93	M₁₆	—	538	4	no	—
94	C₂×C₈	—	538	4	no	—
95	C₂×C₈	C₂²×D₂₆₉	1	2	yes	—
96	C₄⋊C₄	—	4	2	no	—
97	C₄²	C₂²×D₂₆₉	1	1	yes	characteristic
98	Q₈	—	2	8	no	—
99	Q₈	—	269	4	no	—
100	Q₈	—	1076	4	no	—
101	D₄	—	269	4	no	—
102	D₄	—	538	4	no	—
103	D₄	—	1076	4	no	—
104	C₂×C₄	—	2	1	no	—
105	C₂×C₄	—	2	4	no	—
106	C₂×C₄	—	538	4	no	—
107	C₂×C₄	—	1076	8	no	—
108	C₂×C₄	C₂³×D₂₆₉	1	1	yes	characteristic
109	C₈	—	538	4	no	—
110	C₈	D₄×D₂₆₉	1	2	yes	—
111	C₈	D₄×D₂₆₉	1	2	yes	—
112	C₂²	—	538	4	no	—
113	C₂²	#22575498	1	1	yes	characteristic
114	C₄	—	2	2	no	—
115	C₄	—	4	4	no	—
116	C₄	—	538	4	no	—
117	C₄	C₂×D₄×D₂₆₉	1	2	yes	—
118	C₂	—	2	1	no	—
119	C₂	—	1076	4	no	—
120	C₂	#31771240	1	1	yes	characteristic

subgroup

G/H

class size

multiplicity

normal?

properties

#31771418

C₂

yes

maximal, complemented

#31770877

C₂

yes

maximal, complemented

#31770787

C₂

yes

maximal, complemented

#31770723

C₂

yes

maximal, complemented

#494×C₂₆₉

C₂

yes

maximal, characteristic, complemented

#22575501

C₂²

yes

—

#22575485

—

#22575484

C₂²

yes

—

#22575476

—

#22575475

C₂²

yes

—

Q₁₆×D₂₆₉

—

#22575459

C₂²

yes

—

#22575389

C₂²

yes

—

#22575381

C₂²

yes

—

#22575372

C₂²

yes

—

C₂×Q₁₆×C₂₆₉

C₂²

yes

—

C₄⋊Q₈×C₂₆₉

C₂²

yes

—

C₄×C₈×C₂₆₉

C₂²

yes

—

—

C₂³

yes

—

Q₈×D₂₆₉

—

Dic₁₀₇₆

D₄

yes

—

#16864392

—

#16864391

D₄

yes

—

#16864389

—

D₂₁₅₂

D₄

yes

—

#16864386

C₂³

yes

—

#16864385

—

C₈×D₂₆₉

—

C₂×Q₈×C₂₆₉

C₂³

yes

—

Q₁₆×C₂₆₉

—

Q₁₆×C₂₆₉

—

C₂×C₈×C₂₆₉

C₂³

yes

—

C₄⋊C₄×C₂₆₉

—

C₄²×C₂₆₉

C₂³

yes

characteristic

Dic₅₃₈

—

Dic₅₃₈

C₂×D₄

yes

—

D₁₀₇₆

—

D₁₀₇₆

C₂×D₄

yes

—

C₂²⋊D₂₆₉

—

C₄×D₂₆₉

—

C₄×D₂₆₉

—

#14524130

—

Q₈×C₂₆₉

—

C₂×C₄×C₂₆₉

—

C₂×C₄×C₂₆₉

—

C₂×C₄×C₂₆₉

C₂⁴

yes

characteristic

C₈×C₂₆₉

C₂×D₄

yes

—

C₈×C₂₆₉

C₂×D₄

yes

—

C₂²×C₂₆₉

H(5,𝔽₂)

yes

characteristic

C₂×D₂₆₉

—

C₄×C₂₆₉

—

C₄×C₂₆₉

—

C₄×C₂₆₉

C₂²×D₄

yes

—

Dic₂₆₉

—

C₂×C₂₆₉

—

C₂×C₂₆₉

D₄²

yes

characteristic

D₂₆₉

—

C₂₆₉

#3293

yes

characteristic, complemented, Sylow

#3293

—

269

maximal, Sylow

#578

—

269

—

#494

D₂₆₉

yes

characteristic, complemented

#473

—

269

—

#456

—

269

—

#443

—

269

—

2₋¹⁺⁴

—

269

—

#137

—

269

—

#137

—

538

—

#135

—

269

—

#135

—

538

—

C₂×Q₁₆

—

538

—

C₂×Q₁₆

C₂×D₂₆₉

yes

—

#131

—

269

—

C₄⋊Q₈

C₂×D₂₆₉

yes

—

#108

—

269

—

#104

—

269

—

#101

—

269

—

C₄×C₈

C₂×D₂₆₉

yes

characteristic

#41

—

269

—

#41

—

538

—

#41

—

1076

—

C₂×Q₈

—

538

—

C₂×Q₈

C₂²×D₂₆₉

yes

—

Q₁₆

—

Q₁₆

—

Q₁₆

—

269

—

Q₁₆

—

538

—

SD₁₆

—

269

—

SD₁₆

—

538

—

D₈

—

269

—

M₁₆

—

269

—

M₁₆

—

538

—

C₂×C₈

—

538

—

C₂×C₈

C₂²×D₂₆₉

yes

—

C₄⋊C₄

—

C₄²

C₂²×D₂₆₉

yes

characteristic

Q₈

—

Q₈

—

269

—

100

Q₈

—

1076

—

101

D₄

—

269

—

102

D₄

—

538

—

103

D₄

—

1076

—

104

C₂×C₄

—

105

C₂×C₄

—

106

C₂×C₄

—

538

—

107

C₂×C₄

—

1076

—

108

C₂×C₄

C₂³×D₂₆₉

yes

characteristic

109

C₈

—

538

—

110

C₈

D₄×D₂₆₉

yes

—

111

C₈

D₄×D₂₆₉

yes

—

112

C₂²

—

538

—

113

C₂²

#22575498

yes

characteristic

114

C₄

—

115

C₄

—

116

C₄

—

538

—

117

C₄

C₂×D₄×D₂₆₉

yes

—

118

C₂

—

119

C₂

—

1076

—

120

C₂

#31771240

yes

characteristic

	subgroup	G/H
center	C₂	#31771240
commutator, cosocle	C₂×C₄×C₂₆₉	C₂⁴
Frattini	C₂×C₄	C₂³×D₂₆₉
Fitting	#494×C₂₆₉	C₂
socle	C₂×C₂₆₉	D₄²
hypercenter	#494	D₂₆₉
hypocenter	C₂₆₉	#3293
LCS	G ▷ C₂×C₄×C₂₆₉ ▷ C₂×C₂₆₉ ▷ C₂₆₉
UCS	C₁ ◁ C₂ ◁ C₂×C₄ ◁ #494

subgroup

G/H

center

C₂

#31771240

commutator, cosocle

C₂×C₄×C₂₆₉

C₂⁴

Frattini

C₂×C₄

C₂³×D₂₆₉

Fitting

#494×C₂₆₉

C₂

socle

C₂×C₂₆₉

D₄²

D₂₆₉

C₂₆₉

G ▷ C₂×C₄×C₂₆₉ ▷ C₂×C₂₆₉ ▷ C₂₆₉

UCS

C₁ ◁ C₂ ◁ C₂×C₄ ◁ #494

N	H ≅ G/N	type
1. #31771418	C₂	semidirect
2. #31770877	C₂	semidirect
3. #31770787	C₂	semidirect
4. #31770723	C₂	semidirect
5. #494×C₂₆₉	C₂	semidirect
60. C₂₆₉	#3293	semidirect
63. #494	D₂₆₉	semidirect

H ≅ G/N

type

1. #31771418

C₂

semidirect

2. #31770877

C₂

semidirect

3. #31770787

C₂

semidirect

4. #31770723

C₂

semidirect

5. #494×C₂₆₉

C₂

semidirect

60. C₂₆₉

#3293

semidirect

63. #494

D₂₆₉

semidirect

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

subgroup	index
#31771418	2
#31770877	2
#31770787	2
#31770723	2
#494×C₂₆₉	2
#3293	269

subgroup

index

#494×C₂₆₉

269

Subgroups — #46822689

Conjugacy classes of subgroups

Complements

Maximal subgroups