Group #31824092 — subgroups

Subgroups — #31824092

Order 17984.

206 conjugacy classes of subgroups; only the 103 normal subgroups are shown.

normal subgroups only

This group has 206 conjugacy classes of subgroups — too many to draw. Only its 103 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	#22582878	C₂	1	2	yes	maximal, complemented
2	#22582877	C₂	1	2	yes	maximal, complemented
3	#22582869	C₂	1	2	yes	maximal, complemented
4	#22582869	C₂	1	2	yes	maximal, complemented
5	#22582858	C₂	1	1	yes	maximal, characteristic
6	#22582856	C₂	1	2	yes	maximal
7	#22582835	C₂	1	2	yes	maximal
8	C₂×#17985117	C₂	1	1	yes	maximal, characteristic
9	C₂²⋊Q₈×C₂₈₁	C₂	1	1	yes	maximal, characteristic
10	C₂×Dic₅₆₂	C₂²	1	2	yes	—
11	C₂×Dic₅₆₂	C₂²	1	1	yes	characteristic
12	C₂²×Dic₂₈₁	C₂²	1	1	yes	characteristic
13	C₂²×Dic₂₈₁	C₂²	1	1	yes	characteristic
14	C₄⋊Dic₂₈₁	C₂²	1	2	yes	—
15	C₄⋊Dic₂₈₁	C₂²	1	1	yes	characteristic
16	#17985117	C₂²	1	2	yes	—
17	#17985117	C₂²	1	2	yes	—
18	#17985117	C₂²	1	2	yes	—
19	#17985117	C₂²	1	2	yes	—
20	#17985117	C₂²	1	2	yes	—
21	#17985117	C₂²	1	2	yes	—
22	#17985117	C₂²	1	2	yes	—
23	C₂²⋊Dic₂₈₁	C₂²	1	2	yes	—
24	C₄×Dic₂₈₁	C₂²	1	2	yes	—
25	C₄×Dic₂₈₁	C₂²	1	2	yes	—
26	C₂×Q₈×C₂₈₁	C₂²	1	1	yes	characteristic
27	C₂²×C₄×C₂₈₁	C₂²	1	1	yes	characteristic
28	C₄⋊C₄×C₂₈₁	C₂²	1	1	yes	characteristic
29	C₄⋊C₄×C₂₈₁	C₂²	1	2	yes	—
30	C₂²⋊C₄×C₂₈₁	C₂²	1	2	yes	—
31	Dic₅₆₂	—	4	2	no	—
32	Dic₅₆₂	—	4	1	no	—
33	C₂×Dic₂₈₁	—	4	1	no	—
34	C₂×Dic₂₈₁	Q₈	1	4	yes	complemented
35	C₂×Dic₂₈₁	C₂³	1	2	yes	—
36	C₂×Dic₂₈₁	C₂³	1	2	yes	—
37	C₂×Dic₂₈₁	C₂³	1	2	yes	—
38	C₂×Dic₂₈₁	C₂³	1	2	yes	—
39	C₂³×C₂₈₁	C₂³	1	1	yes	characteristic
40	Q₈×C₂₈₁	—	4	1	no	—
41	C₂×C₄×C₂₈₁	—	4	1	no	—
42	C₂×C₄×C₂₈₁	C₂³	1	2	yes	—
43	C₂×C₄×C₂₈₁	C₂³	1	2	yes	—
44	C₂×C₄×C₂₈₁	C₂³	1	1	yes	characteristic
45	C₂×C₄×C₂₈₁	C₂³	1	1	yes	characteristic
46	C₂²×C₂₈₁	—	2	1	no	—
47	C₂²×C₂₈₁	—	2	1	no	—
48	C₂²×C₂₈₁	C₂×Q₈	1	2	yes	—
49	C₂²×C₂₈₁	C₂⁴	1	1	yes	characteristic
50	C₄×C₂₈₁	—	2	1	no	—
51	C₄×C₂₈₁	—	2	1	no	—
52	C₄×C₂₈₁	—	2	2	no	—
53	C₄×C₂₈₁	—	2	2	no	—
54	Dic₂₈₁	—	2	2	no	—
55	Dic₂₈₁	—	2	2	no	—
56	Dic₂₈₁	—	2	2	no	—
57	Dic₂₈₁	C₂×Q₈	1	4	yes	—
58	C₂×C₂₈₁	—	2	2	no	—
59	C₂×C₂₈₁	C₂²×Q₈	1	1	yes	characteristic
60	C₂×C₂₈₁	H(5,𝔽₂)	1	1	yes	characteristic
61	C₂×C₂₈₁	2₋¹⁺⁴	1	1	yes	characteristic
62	C₂₈₁	#544	1	1	yes	characteristic, complemented, Sylow
63	#544	—	281	1	no	maximal, Sylow
64	C₄⋊Q₈	—	281	2	no	—
65	C₄⋊Q₈	—	281	2	no	—
66	#125	—	281	2	no	—
67	#125	—	281	2	no	—
68	C₂²⋊Q₈	—	281	1	no	—
69	C₂²⋊Q₈	—	281	2	no	—
70	C₂²⋊Q₈	D₂₈₁	1	1	yes	characteristic
71	#117	—	281	2	no	—
72	C₂×C₄⋊C₄	—	281	1	no	—
73	C₂×Q₈	—	281	1	no	—
74	C₂×Q₈	—	281	2	no	—
75	C₂×Q₈	C₂×D₂₈₁	1	1	yes	characteristic
76	C₂²×C₄	—	281	1	no	—
77	C₂²×C₄	—	281	1	no	—
78	C₂²×C₄	C₂×D₂₈₁	1	1	yes	characteristic
79	C₄⋊C₄	—	281	2	no	—
80	C₄⋊C₄	—	281	2	no	—
81	C₄⋊C₄	—	281	2	no	—
82	C₄⋊C₄	—	281	2	no	—
83	C₄⋊C₄	—	281	2	no	—
84	C₄⋊C₄	—	281	2	no	—
85	C₄⋊C₄	—	281	2	no	—
86	C₄⋊C₄	—	281	1	no	—
87	C₄⋊C₄	—	281	2	no	—
88	C₄⋊C₄	C₂×D₂₈₁	1	1	yes	characteristic
89	C₄⋊C₄	C₂×D₂₈₁	1	2	yes	—
90	C₂²⋊C₄	—	281	2	no	—
91	C₂²⋊C₄	C₂×D₂₈₁	1	2	yes	—
92	C₄²	—	281	2	no	—
93	C₄²	—	281	2	no	—
94	C₂³	C₂²×D₂₈₁	1	1	yes	characteristic
95	Q₈	—	4	1	no	—
96	Q₈	—	1124	1	no	—
97	Q₈	—	1124	2	no	—
98	C₂×C₄	—	4	1	no	—
99	C₂×C₄	—	281	4	no	—
100	C₂×C₄	—	281	2	no	—
101	C₂×C₄	—	281	2	no	—
102	C₂×C₄	—	281	2	no	—
103	C₂×C₄	—	281	2	no	—
104	C₂×C₄	—	1124	1	no	—
105	C₂×C₄	C₂²×D₂₈₁	1	2	yes	—
106	C₂×C₄	C₂²×D₂₈₁	1	2	yes	—
107	C₂×C₄	C₂²×D₂₈₁	1	1	yes	characteristic
108	C₂×C₄	C₂²×D₂₈₁	1	1	yes	characteristic
109	C₂²	—	2	1	no	—
110	C₂²	—	2	1	no	—
111	C₂²	Q₈×D₂₈₁	1	2	yes	—
112	C₂²	C₂³×D₂₈₁	1	1	yes	characteristic
113	C₄	—	2	1	no	—
114	C₄	—	2	1	no	—
115	C₄	—	2	2	no	—
116	C₄	—	2	2	no	—
117	C₄	—	281	4	no	—
118	C₄	—	562	2	no	—
119	C₄	—	562	2	no	—
120	C₄	—	562	2	no	—
121	C₂	—	2	2	no	—
122	C₂	C₂×Q₈×D₂₈₁	1	1	yes	characteristic
123	C₂	#22582924	1	1	yes	characteristic
124	C₂	#22582926	1	1	yes	characteristic

subgroup

G/H

class size

multiplicity

normal?

properties

#22582878

C₂

yes

maximal, complemented

#22582877

C₂

yes

maximal, complemented

#22582869

C₂

yes

maximal, complemented

#22582869

C₂

yes

maximal, complemented

#22582858

C₂

yes

maximal, characteristic

#22582856

C₂

yes

maximal

#22582835

C₂

yes

maximal

C₂×#17985117

C₂

yes

maximal, characteristic

C₂²⋊Q₈×C₂₈₁

C₂

yes

maximal, characteristic

C₂×Dic₅₆₂

C₂²

yes

—

C₂×Dic₅₆₂

C₂²

yes

characteristic

C₂²×Dic₂₈₁

C₂²

yes

characteristic

C₂²×Dic₂₈₁

C₂²

yes

characteristic

C₄⋊Dic₂₈₁

C₂²

yes

—

C₄⋊Dic₂₈₁

C₂²

yes

characteristic

#17985117

C₂²

yes

—

#17985117

C₂²

yes

—

#17985117

C₂²

yes

—

#17985117

C₂²

yes

—

#17985117

C₂²

yes

—

#17985117

C₂²

yes

—

#17985117

C₂²

yes

—

C₂²⋊Dic₂₈₁

C₂²

yes

—

C₄×Dic₂₈₁

C₂²

yes

—

C₄×Dic₂₈₁

C₂²

yes

—

C₂×Q₈×C₂₈₁

C₂²

yes

characteristic

C₂²×C₄×C₂₈₁

C₂²

yes

characteristic

C₄⋊C₄×C₂₈₁

C₂²

yes

characteristic

C₄⋊C₄×C₂₈₁

C₂²

yes

—

C₂²⋊C₄×C₂₈₁

C₂²

yes

—

Dic₅₆₂

—

Dic₅₆₂

—

C₂×Dic₂₈₁

—

C₂×Dic₂₈₁

Q₈

yes

complemented

C₂×Dic₂₈₁

C₂³

yes

—

C₂×Dic₂₈₁

C₂³

yes

—

C₂×Dic₂₈₁

C₂³

yes

—

C₂×Dic₂₈₁

C₂³

yes

—

C₂³×C₂₈₁

C₂³

yes

characteristic

Q₈×C₂₈₁

—

C₂×C₄×C₂₈₁

—

C₂×C₄×C₂₈₁

C₂³

yes

—

C₂×C₄×C₂₈₁

C₂³

yes

—

C₂×C₄×C₂₈₁

C₂³

yes

characteristic

C₂×C₄×C₂₈₁

C₂³

yes

characteristic

C₂²×C₂₈₁

—

C₂²×C₂₈₁

—

C₂²×C₂₈₁

C₂×Q₈

yes

—

C₂²×C₂₈₁

C₂⁴

yes

characteristic

C₄×C₂₈₁

—

C₄×C₂₈₁

—

C₄×C₂₈₁

—

C₄×C₂₈₁

—

Dic₂₈₁

—

Dic₂₈₁

—

Dic₂₈₁

—

Dic₂₈₁

C₂×Q₈

yes

—

C₂×C₂₈₁

—

C₂×C₂₈₁

C₂²×Q₈

yes

characteristic

C₂×C₂₈₁

H(5,𝔽₂)

yes

characteristic

C₂×C₂₈₁

2₋¹⁺⁴

yes

characteristic

C₂₈₁

#544

yes

characteristic, complemented, Sylow

#544

—

281

maximal, Sylow

C₄⋊Q₈

—

281

—

C₄⋊Q₈

—

281

—

#125

—

281

—

#125

—

281

—

C₂²⋊Q₈

—

281

—

C₂²⋊Q₈

—

281

—

C₂²⋊Q₈

D₂₈₁

yes

characteristic

#117

—

281

—

C₂×C₄⋊C₄

—

281

—

C₂×Q₈

—

281

—

C₂×Q₈

—

281

—

C₂×Q₈

C₂×D₂₈₁

yes

characteristic

C₂²×C₄

—

281

—

C₂²×C₄

—

281

—

C₂²×C₄

C₂×D₂₈₁

yes

characteristic

C₄⋊C₄

—

281

—

C₄⋊C₄

—

281

—

C₄⋊C₄

—

281

—

C₄⋊C₄

—

281

—

C₄⋊C₄

—

281

—

C₄⋊C₄

—

281

—

C₄⋊C₄

—

281

—

C₄⋊C₄

—

281

—

C₄⋊C₄

—

281

—

C₄⋊C₄

C₂×D₂₈₁

yes

characteristic

C₄⋊C₄

C₂×D₂₈₁

yes

—

C₂²⋊C₄

—

281

—

C₂²⋊C₄

C₂×D₂₈₁

yes

—

C₄²

—

281

—

C₄²

—

281

—

C₂³

C₂²×D₂₈₁

yes

characteristic

Q₈

—

Q₈

—

1124

—

Q₈

—

1124

—

C₂×C₄

—

C₂×C₄

—

281

—

100

C₂×C₄

—

281

—

101

C₂×C₄

—

281

—

102

C₂×C₄

—

281

—

103

C₂×C₄

—

281

—

104

C₂×C₄

—

1124

—

105

C₂×C₄

C₂²×D₂₈₁

yes

—

106

C₂×C₄

C₂²×D₂₈₁

yes

—

107

C₂×C₄

C₂²×D₂₈₁

yes

characteristic

108

C₂×C₄

C₂²×D₂₈₁

yes

characteristic

109

C₂²

—

110

C₂²

—

111

C₂²

Q₈×D₂₈₁

yes

—

112

C₂²

C₂³×D₂₈₁

yes

characteristic

113

C₄

—

114

C₄

—

115

C₄

—

116

C₄

—

117

C₄

—

281

—

118

C₄

—

562

—

119

C₄

—

562

—

120

C₄

—

562

—

121

C₂

—

122

C₂

C₂×Q₈×D₂₈₁

yes

characteristic

123

C₂

#22582924

yes

characteristic

124

C₂

#22582926

yes

characteristic

	subgroup	G/H
center, Frattini	C₂²	C₂³×D₂₈₁
commutator, socle, cosocle	C₂²×C₂₈₁	C₂⁴
Fitting	C₂²⋊Q₈×C₂₈₁	C₂
prime radical	C₂³×C₂₈₁	C₂³
hypercenter	C₂²⋊Q₈	D₂₈₁
hypocenter	C₂₈₁	#544
LCS	G ▷ C₂²×C₂₈₁ ▷ C₂₈₁
UCS	C₁ ◁ C₂² ◁ C₂²⋊Q₈

subgroup

G/H

center, Frattini

C₂²

C₂³×D₂₈₁

commutator, socle, cosocle

C₂²×C₂₈₁

C₂⁴

Fitting

C₂²⋊Q₈×C₂₈₁

C₂

prime radical

C₂³×C₂₈₁

C₂³

hypercenter

C₂²⋊Q₈

D₂₈₁

C₂₈₁

G ▷ C₂²×C₂₈₁ ▷ C₂₈₁

C₁ ◁ C₂² ◁ C₂²⋊Q₈

N	H ≅ G/N	type
1. #22582878	C₂	semidirect
2. #22582877	C₂	semidirect
3. #22582869	C₂	semidirect
4. #22582869	C₂	semidirect
34. C₂×Dic₂₈₁	Q₈	semidirect
62. C₂₈₁	#544	semidirect

H ≅ G/N

type

1. #22582878

C₂

semidirect

2. #22582877

C₂

semidirect

3. #22582869

C₂

semidirect

4. #22582869

C₂

semidirect

34. C₂×Dic₂₈₁

Q₈

semidirect

62. C₂₈₁

#544

semidirect

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

subgroup	index
#22582878	2
#22582877	2
#22582869	2
#22582858	2
#22582856	2
#22582835	2
C₂×#17985117	2
C₂²⋊Q₈×C₂₈₁	2
#544	281

subgroup

index

C₂²⋊Q₈×C₂₈₁

281

Subgroups — #31824092

Conjugacy classes of subgroups

Complements

Maximal subgroups