Group #49094784 — subgroups

Subgroups — #49094784

Order 35968.

184 conjugacy classes of subgroups; only the 55 normal subgroups are shown.

normal subgroups only

This group has 184 conjugacy classes of subgroups — too many to draw. Only its 55 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	#31823660	C₂	1	2	yes	maximal, complemented
2	#31823593	C₂	1	1	yes	maximal, characteristic, complemented
3	#31823214	C₂	1	1	yes	maximal, characteristic, complemented
4	#31823207	C₂	1	2	yes	maximal, complemented
5	#493×C₂₈₁	C₂	1	1	yes	maximal, characteristic
6	C₂×#17985126	—	2	2	no	—
7	C₄×Dic₅₆₂	C₂²	1	1	yes	characteristic
8	C₄⋊#14555886	C₂²	1	2	yes	—
9	C₄⋊#14555886	C₂²	1	1	yes	characteristic
10	#22582808	—	2	1	no	—
11	#22582800	—	2	2	no	—
12	#22582789	—	2	1	no	—
13	C₂×D₈×C₂₈₁	—	2	1	no	—
14	C₂×D₈×C₂₈₁	—	2	1	no	—
15	#127×C₂₈₁	C₂²	1	2	yes	—
16	C₄×C₈×C₂₈₁	C₂²	1	1	yes	characteristic
17	C₂×Dic₅₆₂	D₄	1	1	yes	characteristic
18	#17985126	—	4	4	no	—
19	C₂×#14555886	—	2	1	no	—
20	C₂×#14555886	—	2	2	no	—
21	C₄⋊Dic₂₈₁	D₄	1	1	yes	characteristic
22	#17985117	—	2	1	no	—
23	C₄×Dic₂₈₁	—	2	1	no	—
24	C₂×D₄×C₂₈₁	—	2	2	no	—
25	C₂×D₄×C₂₈₁	—	2	2	no	—
26	C₂×D₄×C₂₈₁	—	2	2	no	—
27	D₈×C₂₈₁	—	4	2	no	—
28	D₈×C₂₈₁	—	4	2	no	—
29	C₂×C₈×C₂₈₁	D₄	1	1	yes	characteristic
30	C₂×C₈×C₂₈₁	D₄	1	1	yes	characteristic
31	C₄²×C₂₈₁	C₂³	1	1	yes	characteristic
32	Dic₅₆₂	—	2	2	no	—
33	C₂×Dic₂₈₁	—	2	1	no	—
34	C₂×Dic₂₈₁	—	2	1	no	—
35	#14555886	—	4	1	no	—
36	#14555886	—	4	2	no	—
37	C₂³×C₂₈₁	—	4	2	no	—
38	D₄×C₂₈₁	—	4	2	no	—
39	D₄×C₂₈₁	—	4	2	no	—
40	D₄×C₂₈₁	—	4	2	no	—
41	D₄×C₂₈₁	—	4	2	no	—
42	D₄×C₂₈₁	—	8	2	no	—
43	C₂×C₄×C₂₈₁	C₂×D₄	1	1	yes	characteristic
44	C₂×C₄×C₂₈₁	C₂×D₄	1	1	yes	characteristic
45	C₂×C₄×C₂₈₁	#41	1	1	yes	characteristic
46	C₈×C₂₈₁	—	2	1	no	—
47	C₈×C₂₈₁	SD₁₆	1	2	yes	complemented
48	C₂²×C₂₈₁	—	8	2	no	—
49	C₂²×C₂₈₁	—	8	2	no	—
50	C₂²×C₂₈₁	—	8	2	no	—
51	C₂²×C₂₈₁	#121	1	1	yes	characteristic
52	C₄×C₂₈₁	—	2	1	no	—
53	C₄×C₂₈₁	C₂×SD₁₆	1	1	yes	characteristic
54	C₄×C₂₈₁	AGL(1,ℤ/8)	1	2	yes	—
55	C₄×C₂₈₁	AGL(1,ℤ/8)	1	1	yes	characteristic
56	Dic₂₈₁	—	4	1	no	—
57	Dic₂₈₁	—	4	1	no	—
58	C₂×C₂₈₁	—	16	2	no	—
59	C₂×C₂₈₁	#460	1	1	yes	characteristic
60	C₂×C₂₈₁	#469	1	1	yes	characteristic
61	C₂×C₂₈₁	#472	1	1	yes	characteristic
62	C₂₈₁	#1688	1	1	yes	characteristic, complemented, Sylow
63	#1688	—	281	1	no	maximal, Sylow
64	#493	D₂₈₁	1	1	yes	characteristic
65	#460	—	281	2	no	—
66	#446	—	281	1	no	—
67	#334	—	281	1	no	—
68	#331	—	281	2	no	—
69	C₂×SD₁₆	—	562	2	no	—
70	C₂×D₈	—	2	1	no	—
71	C₂×D₈	—	2	1	no	—
72	#127	C₂×D₂₈₁	1	2	yes	—
73	C₄×Q₈	—	281	1	no	—
74	C₄⋊C₈	—	281	1	no	—
75	C₄⋊C₈	—	281	2	no	—
76	C₄⋊C₈	—	562	1	no	—
77	#102	—	562	2	no	—
78	#97	—	562	1	no	—
79	C₄×C₈	C₂×D₂₈₁	1	1	yes	characteristic
80	C₂×Q₈	—	281	1	no	—
81	C₂×D₄	—	2	2	no	—
82	C₂×D₄	—	2	2	no	—
83	C₂×D₄	—	2	2	no	—
84	SD₁₆	—	1124	4	no	—
85	D₈	—	4	2	no	—
86	D₈	—	4	2	no	—
87	C₂×C₈	—	562	2	no	—
88	C₂×C₈	—	562	1	no	—
89	C₂×C₈	C₂²⋊D₂₈₁	1	1	yes	characteristic
90	C₂×C₈	C₂²⋊D₂₈₁	1	1	yes	characteristic
91	C₄⋊C₄	—	281	1	no	—
92	C₄⋊C₄	—	562	1	no	—
93	C₄²	—	562	1	no	—
94	C₄²	C₂²×D₂₈₁	1	1	yes	characteristic
95	C₂³	—	4	2	no	—
96	Q₈	—	562	2	no	—
97	D₄	—	4	2	no	—
98	D₄	—	4	2	no	—
99	D₄	—	4	2	no	—
100	D₄	—	4	2	no	—
101	D₄	—	8	2	no	—
102	C₂×C₄	—	562	1	no	—
103	C₂×C₄	—	562	1	no	—
104	C₂×C₄	D₄×D₂₈₁	1	1	yes	characteristic
105	C₂×C₄	C₂×C₂²⋊D₂₈₁	1	1	yes	characteristic
106	C₂×C₄	#17985137	1	1	yes	characteristic
107	C₈	—	2	1	no	—
108	C₈	—	1124	2	no	—
109	C₈	—	1124	1	no	—
110	C₈	#17985126	1	2	yes	complemented
111	C₂²	—	8	2	no	—
112	C₂²	—	8	2	no	—
113	C₂²	—	8	2	no	—
114	C₂²	#22582850	1	1	yes	characteristic
115	C₄	—	2	1	no	—
116	C₄	—	1124	1	no	—
117	C₄	—	1124	1	no	—
118	C₄	C₂×#17985126	1	1	yes	characteristic
119	C₄	#22582903	1	2	yes	—
120	C₄	#22582905	1	1	yes	characteristic
121	C₂	—	16	2	no	—
122	C₂	#31823660	1	1	yes	characteristic
123	C₂	#31823713	1	1	yes	characteristic
124	C₂	#31823730	1	1	yes	characteristic

subgroup

G/H

class size

multiplicity

normal?

properties

#31823660

C₂

yes

maximal, complemented

#31823593

C₂

yes

maximal, characteristic, complemented

#31823214

C₂

yes

maximal, characteristic, complemented

#31823207

C₂

yes

maximal, complemented

#493×C₂₈₁

C₂

yes

maximal, characteristic

C₂×#17985126

—

C₄×Dic₅₆₂

C₂²

yes

characteristic

C₄⋊#14555886

C₂²

yes

—

C₄⋊#14555886

C₂²

yes

—

—

—

C₂×D₈×C₂₈₁

—

C₂×D₈×C₂₈₁

—

#127×C₂₈₁

C₂²

yes

—

C₄×C₈×C₂₈₁

C₂²

yes

characteristic

C₂×Dic₅₆₂

D₄

yes

—

—

—

C₄⋊Dic₂₈₁

D₄

yes

characteristic

#17985117

—

C₄×Dic₂₈₁

—

C₂×D₄×C₂₈₁

—

C₂×D₄×C₂₈₁

—

C₂×D₄×C₂₈₁

—

D₈×C₂₈₁

—

D₈×C₂₈₁

—

C₂×C₈×C₂₈₁

D₄

yes

characteristic

C₂×C₈×C₂₈₁

D₄

yes

characteristic

C₄²×C₂₈₁

C₂³

yes

characteristic

Dic₅₆₂

—

C₂×Dic₂₈₁

—

C₂×Dic₂₈₁

—

#14555886

—

#14555886

—

C₂³×C₂₈₁

—

D₄×C₂₈₁

—

D₄×C₂₈₁

—

D₄×C₂₈₁

—

D₄×C₂₈₁

—

D₄×C₂₈₁

—

C₂×C₄×C₂₈₁

C₂×D₄

yes

characteristic

C₂×C₄×C₂₈₁

C₂×D₄

yes

characteristic

C₂×C₄×C₂₈₁

#41

yes

characteristic

C₈×C₂₈₁

—

C₈×C₂₈₁

SD₁₆

yes

complemented

C₂²×C₂₈₁

—

C₂²×C₂₈₁

—

C₂²×C₂₈₁

—

C₂²×C₂₈₁

#121

yes

characteristic

C₄×C₂₈₁

—

C₄×C₂₈₁

C₂×SD₁₆

yes

characteristic

C₄×C₂₈₁

AGL(1,ℤ/8)

yes

—

C₄×C₂₈₁

AGL(1,ℤ/8)

yes

characteristic

Dic₂₈₁

—

Dic₂₈₁

—

C₂×C₂₈₁

—

C₂×C₂₈₁

yes

C₂×C₂₈₁

yes

C₂×C₂₈₁

yes

C₂₈₁

yes

characteristic, complemented, Sylow

#1688

—

281

maximal, Sylow

#493

D₂₈₁

yes

characteristic

#460

—

281

—

#446

—

281

—

#334

—

281

—

#331

—

281

—

C₂×SD₁₆

—

562

—

C₂×D₈

—

C₂×D₈

—

#127

C₂×D₂₈₁

yes

—

C₄×Q₈

—

281

—

C₄⋊C₈

—

281

—

C₄⋊C₈

—

281

—

C₄⋊C₈

—

562

—

#102

—

562

—

#97

—

562

—

C₄×C₈

C₂×D₂₈₁

yes

characteristic

C₂×Q₈

—

281

—

C₂×D₄

—

C₂×D₄

—

C₂×D₄

—

SD₁₆

—

1124

—

D₈

—

D₈

—

C₂×C₈

—

562

—

C₂×C₈

—

562

—

C₂×C₈

C₂²⋊D₂₈₁

yes

characteristic

C₂×C₈

C₂²⋊D₂₈₁

yes

characteristic

C₄⋊C₄

—

281

—

C₄⋊C₄

—

562

—

C₄²

—

562

—

C₄²

C₂²×D₂₈₁

yes

characteristic

C₂³

—

Q₈

—

562

—

D₄

—

D₄

—

D₄

—

100

D₄

—

101

D₄

—

102

C₂×C₄

—

562

—

103

C₂×C₄

—

562

—

104

C₂×C₄

D₄×D₂₈₁

yes

characteristic

105

C₂×C₄

C₂×C₂²⋊D₂₈₁

yes

characteristic

106

C₂×C₄

#17985137

yes

characteristic

107

C₈

—

108

C₈

—

1124

—

109

C₈

—

1124

—

110

C₈

#17985126

yes

complemented

111

C₂²

—

112

C₂²

—

113

C₂²

—

114

C₂²

#22582850

yes

characteristic

115

C₄

—

116

C₄

—

1124

—

117

C₄

—

1124

—

118

C₄

C₂×#17985126

yes

characteristic

119

C₄

#22582903

yes

—

120

C₄

#22582905

yes

characteristic

121

C₂

—

122

C₂

#31823660

yes

characteristic

123

C₂

#31823713

yes

characteristic

124

C₂

#31823730

yes

characteristic

	subgroup	G/H
center	C₂²	#22582850
commutator, cosocle	C₄²×C₂₈₁	C₂³
Frattini	C₄²	C₂²×D₂₈₁
Fitting, prime radical	#493×C₂₈₁	C₂
socle	C₂²×C₂₈₁	#121
hypercenter	#493	D₂₈₁
hypocenter	C₂₈₁	#1688
LCS	G ▷ C₄²×C₂₈₁ ▷ C₂²×C₂₈₁ ▷ C₂₈₁
UCS	C₁ ◁ C₂² ◁ C₄² ◁ #493

subgroup

G/H

center

C₂²

#22582850

commutator, cosocle

C₄²×C₂₈₁

C₂³

Frattini

C₄²

C₂²×D₂₈₁

Fitting, prime radical

#493×C₂₈₁

C₂

C₂²×C₂₈₁

D₂₈₁

C₂₈₁

G ▷ C₄²×C₂₈₁ ▷ C₂²×C₂₈₁ ▷ C₂₈₁

UCS

C₁ ◁ C₂² ◁ C₄² ◁ #493

N	H ≅ G/N	type
1. #31823660	C₂	semidirect
2. #31823593	C₂	semidirect
3. #31823214	C₂	semidirect
4. #31823207	C₂	semidirect
47. C₈×C₂₈₁	SD₁₆	semidirect
62. C₂₈₁	#1688	semidirect
110. C₈	#17985126	semidirect

H ≅ G/N

type

1. #31823660

C₂

semidirect

2. #31823593

C₂

semidirect

3. #31823214

C₂

semidirect

4. #31823207

C₂

semidirect

47. C₈×C₂₈₁

SD₁₆

semidirect

62. C₂₈₁

#1688

semidirect

110. C₈

#17985126

semidirect

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

subgroup	index
#31823660	2
#31823593	2
#31823214	2
#31823207	2
#493×C₂₈₁	2
#1688	281

subgroup

index

#493×C₂₈₁

281

Subgroups — #49094784

Conjugacy classes of subgroups

Complements

Maximal subgroups