Group #46813471 — subgroups

Subgroups — #46813471

Order 34432.

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

normal subgroups only

This group has 784 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
2	#31770512	C₂	1	8	yes	maximal, complemented
3	#31770468	C₂	1	4	yes	maximal
4	C₂³×Q₈×C₂₆₉	C₂	1	1	yes	maximal, characteristic
5	#22575412	—	2	8	no	—
6	C₂×C₄⋊Dic₂₆₉	C₂²	1	4	yes	—
7	C₂×C₂²⋊Dic₂₆₉	C₂²	1	4	yes	—
8	C₂×C₄×Dic₂₆₉	C₂²	1	4	yes	—
9	#22575360	C₂²	1	16	yes	—
10	C₂²×Q₈×C₂₆₉	—	2	8	no	—
11	C₂²×Q₈×C₂₆₉	C₄	1	8	yes	complemented
12	C₂²×Q₈×C₂₆₉	C₂²	1	4	yes	—
13	C₂³×C₄×C₂₆₉	C₂²	1	2	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	C₄×Dic₂₆₉	—	2	8	no	—
19	C₂⁴×C₂₆₉	C₂³	1	1	yes	characteristic
20	C₂×Q₈×C₂₆₉	—	2	32	no	—
21	C₂×Q₈×C₂₆₉	—	2	8	no	—
22	C₂×Q₈×C₂₆₉	—	2	8	no	—
23	C₂×Q₈×C₂₆₉	C₂×C₄	1	16	yes	—
24	C₂²×C₄×C₂₆₉	—	2	8	no	—
25	C₂²×C₄×C₂₆₉	C₂×C₄	1	8	yes	—
26	C₂²×C₄×C₂₆₉	C₂×C₄	1	4	yes	—
27	C₂²×C₄×C₂₆₉	D₄	1	8	yes	—
28	C₂²×C₄×C₂₆₉	C₂³	1	4	yes	—
29	C₂²×C₄×C₂₆₉	C₂³	1	2	yes	—
30	C₂×Dic₂₆₉	—	2	8	no	—
31	C₂×Dic₂₆₉	—	2	16	no	—
32	C₂³×C₂₆₉	—	2	4	no	—
33	C₂³×C₂₆₉	C₂²×C₄	1	2	yes	—
34	C₂³×C₂₆₉	C₂×D₄	1	4	yes	—
35	C₂³×C₂₆₉	C₂⁴	1	1	yes	characteristic
36	Q₈×C₂₆₉	—	2	32	no	—
37	C₂×C₄×C₂₆₉	—	2	16	no	—
38	C₂×C₄×C₂₆₉	—	2	8	no	—
39	C₂×C₄×C₂₆₉	C₂²⋊C₄	1	16	yes	—
40	C₂×C₄×C₂₆₉	C₂²×C₄	1	8	yes	—
41	C₂×C₄×C₂₆₉	C₂²×C₄	1	4	yes	—
42	C₂×C₄×C₂₆₉	C₂×D₄	1	4	yes	—
43	C₂×C₄×C₂₆₉	C₂×D₄	1	4	yes	—
44	C₂²×C₂₆₉	—	2	4	no	—
45	C₂²×C₂₆₉	—	2	8	no	—
46	C₂²×C₂₆₉	C₂×C₂²⋊C₄	1	4	yes	—
47	C₂²×C₂₆₉	C₂³×C₄	1	1	yes	characteristic
48	C₂²×C₂₆₉	C₂²×D₄	1	2	yes	—
49	C₂²×C₂₆₉	2₋¹⁺⁴	1	4	yes	—
50	C₄×C₂₆₉	—	2	8	no	—
51	C₄×C₂₆₉	C₂×C₂²⋊C₄	1	8	yes	—
52	Dic₂₆₉	—	4	8	no	—
53	C₂×C₂₆₉	—	2	4	no	—
54	C₂×C₂₆₉	C₂²×C₂²⋊C₄	1	1	yes	characteristic
55	C₂×C₂₆₉	#519	1	2	yes	—
56	C₂×C₂₆₉	#536	1	4	yes	—
57	C₂₆₉	#2310	1	1	yes	characteristic, complemented, Sylow
58	#2310	—	269	1	no	maximal, Sylow
59	C₂³×Q₈	D₂₆₉	1	1	yes	characteristic
60	C₂×#117	—	269	2	no	—
61	#391	—	269	8	no	—
62	#381	—	269	4	no	—
63	C₂²×Q₈	—	2	8	no	—
64	C₂²×Q₈	Dic₂₆₉	1	8	yes	complemented
65	C₂²×Q₈	C₂×D₂₆₉	1	4	yes	—
66	C₂³×C₄	C₂×D₂₆₉	1	2	yes	—
67	C₂³×C₄	C₂×D₂₆₉	1	1	yes	characteristic
68	#117	—	538	8	no	—
69	C₂×C₄⋊C₄	—	269	4	no	—
70	C₂×C₂²⋊C₄	—	269	4	no	—
71	C₂×C₄²	—	269	4	no	—
72	#95	—	269	16	no	—
73	C₂⁴	C₂²×D₂₆₉	1	1	yes	characteristic
74	C₂×Q₈	—	2	32	no	—
75	C₂×Q₈	—	2	8	no	—
76	C₂×Q₈	—	2	8	no	—
77	C₂×Q₈	C₂×Dic₂₆₉	1	16	yes	—
78	C₂²×C₄	—	2	8	no	—
79	C₂²×C₄	—	269	8	no	—
80	C₂²×C₄	C₂×Dic₂₆₉	1	8	yes	—
81	C₂²×C₄	C₂×Dic₂₆₉	1	4	yes	—
82	C₂²×C₄	C₂²⋊D₂₆₉	1	8	yes	—
83	C₂²×C₄	C₂²×D₂₆₉	1	4	yes	—
84	C₂²×C₄	C₂²×D₂₆₉	1	2	yes	—
85	C₄⋊C₄	—	538	8	no	—
86	C₂²⋊C₄	—	538	8	no	—
87	C₄²	—	538	8	no	—
88	C₂³	—	2	4	no	—
89	C₂³	C₂²×Dic₂₆₉	1	2	yes	—
90	C₂³	C₂×C₂²⋊D₂₆₉	1	4	yes	—
91	C₂³	C₂³×D₂₆₉	1	1	yes	characteristic
92	Q₈	—	2	32	no	—
93	C₂×C₄	—	2	16	no	—
94	C₂×C₄	—	2	8	no	—
95	C₂×C₄	—	538	8	no	—
96	C₂×C₄	—	538	16	no	—
97	C₂×C₄	C₂²⋊Dic₂₆₉	1	16	yes	—
98	C₂×C₄	C₂²×Dic₂₆₉	1	8	yes	—
99	C₂×C₄	C₂²×Dic₂₆₉	1	4	yes	—
100	C₂×C₄	C₂×C₂²⋊D₂₆₉	1	4	yes	—
101	C₂×C₄	C₂×C₂²⋊D₂₆₉	1	4	yes	—
102	C₂²	—	2	8	no	—
103	C₂²	—	2	4	no	—
104	C₂²	C₂×C₂²⋊Dic₂₆₉	1	4	yes	—
105	C₂²	C₂³×Dic₂₆₉	1	1	yes	characteristic
106	C₂²	C₂²×C₂²⋊D₂₆₉	1	2	yes	—
107	C₂²	#22575501	1	4	yes	—
108	C₄	—	2	8	no	—
109	C₄	—	1076	8	no	—
110	C₄	C₂×C₂²⋊Dic₂₆₉	1	8	yes	—
111	C₂	—	2	4	no	—
112	C₂	C₂²×C₂²⋊Dic₂₆₉	1	1	yes	characteristic
113	C₂	#31771083	1	2	yes	—
114	C₂	#31771179	1	4	yes	—

subgroup

G/H

class size

multiplicity

normal?

properties

C₂×#22575412

C₂

yes

maximal

#31770512

C₂

yes

maximal, complemented

#31770468

C₂

yes

maximal

C₂³×Q₈×C₂₆₉

C₂

yes

maximal, characteristic

#22575412

—

C₂×C₄⋊Dic₂₆₉

C₂²

yes

—

C₂×C₂²⋊Dic₂₆₉

C₂²

yes

—

C₂×C₄×Dic₂₆₉

C₂²

yes

—

#22575360

C₂²

yes

—

C₂²×Q₈×C₂₆₉

—

C₂²×Q₈×C₂₆₉

C₄

yes

complemented

C₂²×Q₈×C₂₆₉

C₂²

yes

—

C₂³×C₄×C₂₆₉

C₂²

yes

—

C₂³×C₄×C₂₆₉

C₂²

yes

characteristic

C₂²×Dic₂₆₉

C₂³

yes

—

C₄⋊Dic₂₆₉

—

C₂²⋊Dic₂₆₉

—

C₄×Dic₂₆₉

—

C₂⁴×C₂₆₉

C₂³

yes

characteristic

C₂×Q₈×C₂₆₉

—

C₂×Q₈×C₂₆₉

—

C₂×Q₈×C₂₆₉

—

C₂×Q₈×C₂₆₉

C₂×C₄

yes

—

C₂²×C₄×C₂₆₉

—

C₂²×C₄×C₂₆₉

C₂×C₄

yes

—

C₂²×C₄×C₂₆₉

C₂×C₄

yes

—

C₂²×C₄×C₂₆₉

D₄

yes

—

C₂²×C₄×C₂₆₉

C₂³

yes

—

C₂²×C₄×C₂₆₉

C₂³

yes

—

C₂×Dic₂₆₉

—

C₂×Dic₂₆₉

—

C₂³×C₂₆₉

—

C₂³×C₂₆₉

C₂²×C₄

yes

—

C₂³×C₂₆₉

C₂×D₄

yes

—

C₂³×C₂₆₉

C₂⁴

yes

characteristic

Q₈×C₂₆₉

—

C₂×C₄×C₂₆₉

—

C₂×C₄×C₂₆₉

—

C₂×C₄×C₂₆₉

C₂²⋊C₄

yes

—

C₂×C₄×C₂₆₉

C₂²×C₄

yes

—

C₂×C₄×C₂₆₉

C₂²×C₄

yes

—

C₂×C₄×C₂₆₉

C₂×D₄

yes

—

C₂×C₄×C₂₆₉

C₂×D₄

yes

—

C₂²×C₂₆₉

—

C₂²×C₂₆₉

—

C₂²×C₂₆₉

C₂×C₂²⋊C₄

yes

—

C₂²×C₂₆₉

C₂³×C₄

yes

characteristic

C₂²×C₂₆₉

C₂²×D₄

yes

—

C₂²×C₂₆₉

2₋¹⁺⁴

yes

—

C₄×C₂₆₉

—

C₄×C₂₆₉

C₂×C₂²⋊C₄

yes

—

Dic₂₆₉

—

C₂×C₂₆₉

—

C₂×C₂₆₉

C₂²×C₂²⋊C₄

yes

characteristic

C₂×C₂₆₉

#519

yes

—

C₂×C₂₆₉

#536

yes

—

C₂₆₉

#2310

yes

characteristic, complemented, Sylow

#2310

—

269

maximal, Sylow

C₂³×Q₈

D₂₆₉

yes

characteristic

C₂×#117

—

269

—

#391

—

269

—

#381

—

269

—

C₂²×Q₈

—

C₂²×Q₈

Dic₂₆₉

yes

complemented

C₂²×Q₈

C₂×D₂₆₉

yes

—

C₂³×C₄

C₂×D₂₆₉

yes

—

C₂³×C₄

C₂×D₂₆₉

yes

characteristic

#117

—

538

—

C₂×C₄⋊C₄

—

269

—

C₂×C₂²⋊C₄

—

269

—

C₂×C₄²

—

269

—

#95

—

269

—

C₂⁴

C₂²×D₂₆₉

yes

characteristic

C₂×Q₈

—

C₂×Q₈

—

C₂×Q₈

—

C₂×Q₈

C₂×Dic₂₆₉

yes

—

C₂²×C₄

—

C₂²×C₄

—

269

—

C₂²×C₄

C₂×Dic₂₆₉

yes

—

C₂²×C₄

C₂×Dic₂₆₉

yes

—

C₂²×C₄

C₂²⋊D₂₆₉

yes

—

C₂²×C₄

C₂²×D₂₆₉

yes

—

C₂²×C₄

C₂²×D₂₆₉

yes

—

C₄⋊C₄

—

538

—

C₂²⋊C₄

—

538

—

C₄²

—

538

—

C₂³

—

C₂³

C₂²×Dic₂₆₉

yes

—

C₂³

C₂×C₂²⋊D₂₆₉

yes

—

C₂³

C₂³×D₂₆₉

yes

characteristic

Q₈

—

C₂×C₄

—

C₂×C₄

—

C₂×C₄

—

538

—

C₂×C₄

—

538

—

C₂×C₄

C₂²⋊Dic₂₆₉

yes

—

C₂×C₄

C₂²×Dic₂₆₉

yes

—

C₂×C₄

C₂²×Dic₂₆₉

yes

—

100

C₂×C₄

C₂×C₂²⋊D₂₆₉

yes

—

101

C₂×C₄

C₂×C₂²⋊D₂₆₉

yes

—

102

C₂²

—

103

C₂²

—

104

C₂²

C₂×C₂²⋊Dic₂₆₉

yes

—

105

C₂²

C₂³×Dic₂₆₉

yes

characteristic

106

C₂²

C₂²×C₂²⋊D₂₆₉

yes

—

107

C₂²

#22575501

yes

—

108

C₄

—

109

C₄

—

1076

—

110

C₄

C₂×C₂²⋊Dic₂₆₉

yes

—

111

C₂

—

112

C₂

C₂²×C₂²⋊Dic₂₆₉

yes

characteristic

113

C₂

#31771083

yes

—

114

C₂

#31771179

yes

—

	subgroup	G/H
center, Frattini	C₂³	C₂³×D₂₆₉
commutator	C₂²×C₂₆₉	C₂³×C₄
Fitting	C₂³×Q₈×C₂₆₉	C₂
socle, cosocle	C₂³×C₂₆₉	C₂⁴
prime radical	C₂⁴×C₂₆₉	C₂³
hypercenter	C₂³×Q₈	D₂₆₉
hypocenter	C₂₆₉	#2310
LCS	G ▷ C₂²×C₂₆₉ ▷ C₂₆₉
UCS	C₁ ◁ C₂³ ◁ C₂³×Q₈

subgroup

G/H

center, Frattini

C₂³

C₂³×D₂₆₉

commutator

C₂²×C₂₆₉

C₂³×C₄

Fitting

C₂³×Q₈×C₂₆₉

C₂

socle, cosocle

C₂³×C₂₆₉

C₂⁴

prime radical

C₂⁴×C₂₆₉

C₂³

hypercenter

C₂³×Q₈

D₂₆₉

C₂₆₉

G ▷ C₂²×C₂₆₉ ▷ C₂₆₉

C₁ ◁ C₂³ ◁ C₂³×Q₈

N	H ≅ G/N	type
2. #31770512	C₂	semidirect
11. C₂²×Q₈×C₂₆₉	C₄	semidirect
57. C₂₆₉	#2310	semidirect
64. C₂²×Q₈	Dic₂₆₉	semidirect

H ≅ G/N

type

2. #31770512

C₂

semidirect

11. C₂²×Q₈×C₂₆₉

C₄

semidirect

57. C₂₆₉

#2310

semidirect

64. C₂²×Q₈

Dic₂₆₉

semidirect

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

subgroup	index
C₂×#22575412	2
#31770512	2
#31770468	2
C₂³×Q₈×C₂₆₉	2
#2310	269

subgroup

index

C₂³×Q₈×C₂₆₉

269

Subgroups — #46813471

Conjugacy classes of subgroups

Complements

Maximal subgroups