Group #46819974 — subgroups

Subgroups — #46819974

Order 34432.

364 conjugacy classes of subgroups; only the 221 normal subgroups are shown.

normal subgroups only

This group has 364 conjugacy classes of subgroups — too many to draw. Only its 221 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	#31770736	C₂	1	6	yes	maximal, complemented
2	Q₈×#14524130	C₂	1	2	yes	maximal, complemented
3	#31770662	C₂	1	3	yes	maximal
4	#31770658	C₂	1	3	yes	maximal
5	C₂×C₄×Q₈×C₂₆₉	C₂	1	1	yes	maximal, characteristic
6	C₄⋊#14524130	C₂²	1	6	yes	—
7	C₄⋊#14524130	C₂²	1	6	yes	—
8	C₂²⋊#14524130	C₂²	1	1	yes	characteristic
9	C₂²⋊#14524130	C₂²	1	3	yes	—
10	#22575364	C₂²	1	6	yes	—
11	C₄×#14524130	C₂²	1	6	yes	—
12	C₂²×Q₈×C₂₆₉	C₄	1	1	yes	characteristic
13	C₄×Q₈×C₂₆₉	C₄	1	3	yes	—
14	C₄×Q₈×C₂₆₉	C₄	1	1	yes	characteristic
15	C₄×Q₈×C₂₆₉	C₂²	1	1	yes	characteristic
16	C₄×Q₈×C₂₆₉	C₂²	1	3	yes	—
17	C₂×C₄⋊C₄×C₂₆₉	C₄	1	3	yes	—
18	C₂×C₄²×C₂₆₉	C₂²	1	3	yes	—
19	C₂×#14524130	C₂³	1	2	yes	—
20	C₂×#14524130	C₂³	1	6	yes	—
21	C₂×Q₈×C₂₆₉	—	2	4	no	—
22	C₂×Q₈×C₂₆₉	C₂×C₄	1	3	yes	—
23	C₂×Q₈×C₂₆₉	C₂×C₄	1	1	yes	characteristic
24	C₂²×C₄×C₂₆₉	C₂×C₄	1	3	yes	—
25	C₂²×C₄×C₂₆₉	C₂×C₄	1	3	yes	—
26	C₂²×C₄×C₂₆₉	C₂³	1	1	yes	characteristic
27	C₄⋊C₄×C₂₆₉	C₂×C₄	1	6	yes	—
28	C₄⋊C₄×C₂₆₉	C₂×C₄	1	3	yes	—
29	C₄⋊C₄×C₂₆₉	C₂×C₄	1	3	yes	—
30	C₄²×C₂₆₉	C₂×C₄	1	3	yes	—
31	C₄²×C₂₆₉	C₂×C₄	1	3	yes	—
32	C₄²×C₂₆₉	C₂³	1	3	yes	—
33	C₄²×C₂₆₉	C₂³	1	3	yes	—
34	#14524130	—	2	2	no	—
35	#14524130	—	2	6	no	—
36	C₂³×C₂₆₉	C₂²×C₄	1	1	yes	characteristic
37	Q₈×C₂₆₉	—	2	6	no	—
38	Q₈×C₂₆₉	M₁₆	1	4	yes	—
39	C₂×C₄×C₂₆₉	—	2	6	no	—
40	C₂×C₄×C₂₆₉	—	2	6	no	—
41	C₂×C₄×C₂₆₉	—	2	2	no	—
42	C₂×C₄×C₂₆₉	C₂²×C₄	1	1	yes	characteristic
43	C₂×C₄×C₂₆₉	C₂²×C₄	1	3	yes	—
44	C₂×C₄×C₂₆₉	C₂²×C₄	1	3	yes	—
45	C₂×C₄×C₂₆₉	C₂²×C₄	1	3	yes	—
46	C₂×C₄×C₂₆₉	C₂²×C₄	1	3	yes	—
47	C₂×C₄×C₂₆₉	C₂⁴	1	1	yes	characteristic
48	C₂²×C₂₆₉	—	2	1	no	—
49	C₂²×C₂₆₉	—	2	1	no	—
50	C₂²×C₂₆₉	#131	1	2	yes	—
51	C₂²×C₂₆₉	C₂³×C₄	1	1	yes	characteristic
52	C₄×C₂₆₉	—	2	1	no	—
53	C₄×C₂₆₉	—	2	3	no	—
54	C₄×C₂₆₉	—	2	3	no	—
55	C₄×C₂₆₉	—	2	3	no	—
56	C₄×C₂₆₉	C₂×M₁₆	1	6	yes	—
57	C₄×C₂₆₉	2₋¹⁺⁴	1	2	yes	—
58	C₂×C₂₆₉	—	2	2	no	—
59	C₂×C₂₆₉	#519	1	1	yes	characteristic
60	C₂×C₂₆₉	C₂²×M₁₆	1	1	yes	characteristic
61	C₂×C₂₆₉	C₂×#131	1	1	yes	characteristic
62	C₂₆₉	#2984	1	1	yes	characteristic, complemented, Sylow
63	#2984	—	269	1	no	maximal, Sylow
64	C₂×C₄×Q₈	D₂₆₉	1	1	yes	characteristic
65	#446	—	269	6	no	—
66	C₈×Q₈	—	269	2	no	—
67	#432	—	269	3	no	—
68	#431	—	269	3	no	—
69	C₂²×Q₈	Dic₂₆₉	1	1	yes	characteristic
70	C₄×Q₈	Dic₂₆₉	1	3	yes	—
71	C₄×Q₈	Dic₂₆₉	1	1	yes	characteristic
72	C₄×Q₈	C₂×D₂₆₉	1	1	yes	characteristic
73	C₄×Q₈	C₂×D₂₆₉	1	3	yes	—
74	C₂×C₄⋊C₄	Dic₂₆₉	1	3	yes	—
75	C₂×C₄²	C₂×D₂₆₉	1	3	yes	—
76	C₄⋊C₈	—	269	6	no	—
77	C₄⋊C₈	—	269	6	no	—
78	C₂²⋊C₈	—	269	1	no	—
79	C₂²⋊C₈	—	269	3	no	—
80	#97	—	269	6	no	—
81	C₄×C₈	—	269	6	no	—
82	C₂×Q₈	—	2	4	no	—
83	C₂×Q₈	C₂×Dic₂₆₉	1	3	yes	—
84	C₂×Q₈	C₂×Dic₂₆₉	1	1	yes	characteristic
85	C₂²×C₄	C₂×Dic₂₆₉	1	3	yes	—
86	C₂²×C₄	C₂×Dic₂₆₉	1	3	yes	—
87	C₂²×C₄	C₂²×D₂₆₉	1	1	yes	characteristic
88	C₂×C₈	—	269	2	no	—
89	C₂×C₈	—	269	6	no	—
90	C₄⋊C₄	C₂×Dic₂₆₉	1	6	yes	—
91	C₄⋊C₄	C₂×Dic₂₆₉	1	3	yes	—
92	C₄⋊C₄	C₂×Dic₂₆₉	1	3	yes	—
93	C₄²	C₂×Dic₂₆₉	1	3	yes	—
94	C₄²	C₂×Dic₂₆₉	1	3	yes	—
95	C₄²	C₂²×D₂₆₉	1	3	yes	—
96	C₄²	C₂²×D₂₆₉	1	3	yes	—
97	C₂³	C₂²×Dic₂₆₉	1	1	yes	characteristic
98	Q₈	—	2	6	no	—
99	Q₈	#16864386	1	4	yes	—
100	C₂×C₄	—	2	6	no	—
101	C₂×C₄	—	2	6	no	—
102	C₂×C₄	—	2	2	no	—
103	C₂×C₄	C₂²×Dic₂₆₉	1	1	yes	characteristic
104	C₂×C₄	C₂²×Dic₂₆₉	1	3	yes	—
105	C₂×C₄	C₂²×Dic₂₆₉	1	3	yes	—
106	C₂×C₄	C₂²×Dic₂₆₉	1	3	yes	—
107	C₂×C₄	C₂²×Dic₂₆₉	1	3	yes	—
108	C₂×C₄	C₂³×D₂₆₉	1	1	yes	characteristic
109	C₈	—	538	2	no	—
110	C₈	—	538	6	no	—
111	C₂²	—	2	1	no	—
112	C₂²	—	2	1	no	—
113	C₂²	#22575460	1	2	yes	—
114	C₂²	C₂³×Dic₂₆₉	1	1	yes	characteristic
115	C₄	—	2	1	no	—
116	C₄	—	2	3	no	—
117	C₄	—	2	3	no	—
118	C₄	—	2	3	no	—
119	C₄	C₂×#16864386	1	6	yes	—
120	C₄	#22575501	1	2	yes	—
121	C₂	—	2	2	no	—
122	C₂	#31771083	1	1	yes	characteristic
123	C₂	C₂²×#16864386	1	1	yes	characteristic
124	C₂	C₂×#22575460	1	1	yes	characteristic

subgroup

G/H

class size

multiplicity

normal?

properties

#31770736

C₂

yes

maximal, complemented

Q₈×#14524130

C₂

yes

maximal, complemented

#31770662

C₂

yes

maximal

#31770658

C₂

yes

maximal

C₂×C₄×Q₈×C₂₆₉

C₂

yes

maximal, characteristic

C₄⋊#14524130

C₂²

yes

—

C₄⋊#14524130

C₂²

yes

—

C₂²⋊#14524130

C₂²

yes

characteristic

C₂²⋊#14524130

C₂²

yes

—

#22575364

C₂²

yes

—

C₄×#14524130

C₂²

yes

—

C₂²×Q₈×C₂₆₉

C₄

yes

characteristic

C₄×Q₈×C₂₆₉

C₄

yes

—

C₄×Q₈×C₂₆₉

C₄

yes

characteristic

C₄×Q₈×C₂₆₉

C₂²

yes

characteristic

C₄×Q₈×C₂₆₉

C₂²

yes

—

C₂×C₄⋊C₄×C₂₆₉

C₄

yes

—

C₂×C₄²×C₂₆₉

C₂²

yes

—

C₂×#14524130

C₂³

yes

—

C₂×#14524130

C₂³

yes

—

C₂×Q₈×C₂₆₉

—

C₂×Q₈×C₂₆₉

C₂×C₄

yes

—

C₂×Q₈×C₂₆₉

C₂×C₄

yes

characteristic

C₂²×C₄×C₂₆₉

C₂×C₄

yes

—

C₂²×C₄×C₂₆₉

C₂×C₄

yes

—

C₂²×C₄×C₂₆₉

C₂³

yes

characteristic

C₄⋊C₄×C₂₆₉

C₂×C₄

yes

—

C₄⋊C₄×C₂₆₉

C₂×C₄

yes

—

C₄⋊C₄×C₂₆₉

C₂×C₄

yes

—

C₄²×C₂₆₉

C₂×C₄

yes

—

C₄²×C₂₆₉

C₂×C₄

yes

—

C₄²×C₂₆₉

C₂³

yes

—

C₄²×C₂₆₉

C₂³

yes

—

#14524130

—

#14524130

—

C₂³×C₂₆₉

C₂²×C₄

yes

characteristic

Q₈×C₂₆₉

—

Q₈×C₂₆₉

M₁₆

yes

—

C₂×C₄×C₂₆₉

—

C₂×C₄×C₂₆₉

—

C₂×C₄×C₂₆₉

—

C₂×C₄×C₂₆₉

C₂²×C₄

yes

characteristic

C₂×C₄×C₂₆₉

C₂²×C₄

yes

—

C₂×C₄×C₂₆₉

C₂²×C₄

yes

—

C₂×C₄×C₂₆₉

C₂²×C₄

yes

—

C₂×C₄×C₂₆₉

C₂²×C₄

yes

—

C₂×C₄×C₂₆₉

C₂⁴

yes

characteristic

C₂²×C₂₆₉

—

C₂²×C₂₆₉

—

C₂²×C₂₆₉

#131

yes

—

C₂²×C₂₆₉

C₂³×C₄

yes

characteristic

C₄×C₂₆₉

—

C₄×C₂₆₉

—

C₄×C₂₆₉

—

C₄×C₂₆₉

—

C₄×C₂₆₉

C₂×M₁₆

yes

—

C₄×C₂₆₉

2₋¹⁺⁴

yes

—

C₂×C₂₆₉

—

C₂×C₂₆₉

#519

yes

characteristic

C₂×C₂₆₉

C₂²×M₁₆

yes

C₂×C₂₆₉

yes

C₂₆₉

yes

characteristic, complemented, Sylow

#2984

—

269

maximal, Sylow

C₂×C₄×Q₈

D₂₆₉

yes

characteristic

#446

—

269

—

C₈×Q₈

—

269

—

#432

—

269

—

#431

—

269

—

C₂²×Q₈

Dic₂₆₉

yes

characteristic

C₄×Q₈

Dic₂₆₉

yes

—

C₄×Q₈

Dic₂₆₉

yes

characteristic

C₄×Q₈

C₂×D₂₆₉

yes

characteristic

C₄×Q₈

C₂×D₂₆₉

yes

—

C₂×C₄⋊C₄

Dic₂₆₉

yes

—

C₂×C₄²

C₂×D₂₆₉

yes

—

C₄⋊C₈

—

269

—

C₄⋊C₈

—

269

—

C₂²⋊C₈

—

269

—

C₂²⋊C₈

—

269

—

#97

—

269

—

C₄×C₈

—

269

—

C₂×Q₈

—

C₂×Q₈

C₂×Dic₂₆₉

yes

—

C₂×Q₈

C₂×Dic₂₆₉

yes

characteristic

C₂²×C₄

C₂×Dic₂₆₉

yes

—

C₂²×C₄

C₂×Dic₂₆₉

yes

—

C₂²×C₄

C₂²×D₂₆₉

yes

characteristic

C₂×C₈

—

269

—

C₂×C₈

—

269

—

C₄⋊C₄

C₂×Dic₂₆₉

yes

—

C₄⋊C₄

C₂×Dic₂₆₉

yes

—

C₄⋊C₄

C₂×Dic₂₆₉

yes

—

C₄²

C₂×Dic₂₆₉

yes

—

C₄²

C₂×Dic₂₆₉

yes

—

C₄²

C₂²×D₂₆₉

yes

—

C₄²

C₂²×D₂₆₉

yes

—

C₂³

C₂²×Dic₂₆₉

yes

characteristic

Q₈

—

Q₈

#16864386

yes

—

100

C₂×C₄

—

101

C₂×C₄

—

102

C₂×C₄

—

103

C₂×C₄

C₂²×Dic₂₆₉

yes

characteristic

104

C₂×C₄

C₂²×Dic₂₆₉

yes

—

105

C₂×C₄

C₂²×Dic₂₆₉

yes

—

106

C₂×C₄

C₂²×Dic₂₆₉

yes

—

107

C₂×C₄

C₂²×Dic₂₆₉

yes

—

108

C₂×C₄

C₂³×D₂₆₉

yes

characteristic

109

C₈

—

538

—

110

C₈

—

538

—

111

C₂²

—

112

C₂²

—

113

C₂²

#22575460

yes

—

114

C₂²

C₂³×Dic₂₆₉

yes

characteristic

115

C₄

—

116

C₄

—

117

C₄

—

118

C₄

—

119

C₄

C₂×#16864386

yes

—

120

C₄

#22575501

yes

—

121

C₂

—

122

C₂

#31771083

yes

characteristic

123

C₂

C₂²×#16864386

yes

characteristic

124

C₂

C₂×#22575460

yes

characteristic

	subgroup	G/H
center, Frattini	C₂×C₄	C₂³×D₂₆₉
commutator, socle	C₂²×C₂₆₉	C₂³×C₄
Fitting	C₂×C₄×Q₈×C₂₆₉	C₂
prime radical	C₂³×C₂₆₉	C₂²×C₄
cosocle	C₂×C₄×C₂₆₉	C₂⁴
hypercenter	C₂×C₄×Q₈	D₂₆₉
hypocenter	C₂₆₉	#2984
LCS	G ▷ C₂²×C₂₆₉ ▷ C₂₆₉
UCS	C₁ ◁ C₂×C₄ ◁ C₂×C₄×Q₈

subgroup

G/H

center, Frattini

C₂×C₄

C₂³×D₂₆₉

commutator, socle

C₂²×C₂₆₉

C₂³×C₄

Fitting

C₂×C₄×Q₈×C₂₆₉

C₂

prime radical

C₂³×C₂₆₉

C₂²×C₄

cosocle

C₂×C₄×C₂₆₉

C₂⁴

hypercenter

C₂×C₄×Q₈

D₂₆₉

C₂₆₉

G ▷ C₂²×C₂₆₉ ▷ C₂₆₉

C₁ ◁ C₂×C₄ ◁ C₂×C₄×Q₈

N	H ≅ G/N	type
1. #31770736	C₂	semidirect
2. Q₈×#14524130	C₂	semidirect
62. C₂₆₉	#2984	semidirect

H ≅ G/N

type

1. #31770736

C₂

semidirect

2. Q₈×#14524130

C₂

semidirect

62. C₂₆₉

#2984

semidirect

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

subgroup	index
#31770736	2
Q₈×#14524130	2
#31770662	2
#31770658	2
C₂×C₄×Q₈×C₂₆₉	2
#2984	269

subgroup

index

C₂×C₄×Q₈×C₂₆₉

269

Subgroups — #46819974

Conjugacy classes of subgroups

Complements

Maximal subgroups