Group #49097380 — subgroups

Subgroups — #49097380

Order 35968.

170 conjugacy classes of subgroups; only the 47 normal subgroups are shown.

normal subgroups only

This group has 170 conjugacy classes of subgroups — too many to draw. Only its 47 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	#31823901	C₂	1	2	yes	maximal, complemented
2	#31823840	C₂	1	1	yes	maximal, characteristic, complemented
3	#31823581	C₂	1	1	yes	maximal, characteristic, complemented
4	#31823277	C₂	1	2	yes	maximal, complemented
5	#364×C₂₈₁	C₂	1	1	yes	maximal, characteristic, complemented
6	#22582901	C₂²	1	1	yes	characteristic
7	C₂×D₂₂₄₈	—	2	1	no	—
8	C₂×D₂₂₄₈	C₂²	1	2	yes	—
9	#22582885	—	2	1	no	—
10	#22582876	—	2	1	no	—
11	#22582824	—	2	2	no	—
12	D₄₄₉₆	—	2	2	no	—
13	#22582815	C₂²	1	1	yes	characteristic
14	#22582807	—	2	1	no	—
15	M₃₂×C₂₈₁	C₂²	1	2	yes	—
16	C₄×C₈×C₂₈₁	C₂²	1	1	yes	characteristic
17	#17985138	—	2	1	no	—
18	C₂×D₁₁₂₄	—	2	2	no	—
19	C₂×D₁₁₂₄	—	4	2	no	—
20	Dic₁₁₂₄	D₄	1	1	yes	characteristic, complemented
21	#17985127	—	2	1	no	—
22	D₂₂₄₈	—	2	2	no	—
23	D₂₂₄₈	—	2	2	no	—
24	D₂₂₄₈	—	4	1	no	—
25	D₂₂₄₈	—	4	1	no	—
26	D₂₂₄₈	D₄	1	1	yes	characteristic, complemented
27	#17985122	—	2	1	no	—
28	#17985121	—	4	1	no	—
29	C₈×D₂₈₁	—	4	1	no	—
30	C₂×C₈×C₂₈₁	D₄	1	1	yes	characteristic
31	C₂×C₈×C₂₈₁	C₂³	1	1	yes	characteristic
32	C₄²×C₂₈₁	D₄	1	1	yes	characteristic
33	C₁₆×C₂₈₁	—	2	2	no	—
34	C₂²×D₂₈₁	—	4	2	no	—
35	Dic₅₆₂	—	2	1	no	—
36	D₁₁₂₄	—	2	1	no	—
37	D₁₁₂₄	—	4	2	no	—
38	D₁₁₂₄	—	4	2	no	—
39	D₁₁₂₄	—	8	4	no	—
40	C₂²⋊D₂₈₁	—	4	1	no	—
41	C₄×D₂₈₁	—	4	1	no	—
42	#14555886	—	4	1	no	—
43	C₂×C₄×C₂₈₁	—	2	1	no	—
44	C₂×C₄×C₂₈₁	C₂×D₄	1	1	yes	characteristic
45	C₈×C₂₈₁	D₈	1	1	yes	characteristic
46	C₈×C₂₈₁	D₈	1	1	yes	characteristic
47	C₈×C₂₈₁	C₂×D₄	1	1	yes	characteristic
48	C₈×C₂₈₁	#41	1	1	yes	characteristic
49	C₂²×C₂₈₁	AGL(1,ℤ/8)	1	1	yes	characteristic
50	C₂×D₂₈₁	—	4	1	no	—
51	C₂×D₂₈₁	—	8	4	no	—
52	C₂×D₂₈₁	—	8	2	no	—
53	C₄×C₂₈₁	—	2	2	no	—
54	C₄×C₂₈₁	#121	1	1	yes	characteristic
55	C₄×C₂₈₁	C₂×D₈	1	1	yes	characteristic
56	Dic₂₈₁	—	4	1	no	—
57	C₂×C₂₈₁	—	2	1	no	—
58	C₂×C₂₈₁	#459	1	1	yes	characteristic
59	D₂₈₁	—	8	1	no	—
60	D₂₈₁	—	16	2	no	—
61	C₂₈₁	#2213	1	1	yes	characteristic, complemented, Sylow
62	#2213	—	281	1	no	maximal, Sylow
63	#509	—	281	2	no	—
64	#493	—	281	1	no	—
65	#443	—	281	1	no	—
66	#364	D₂₈₁	1	1	yes	characteristic, complemented
67	#361	—	281	2	no	—
68	#135	—	281	1	no	—
69	C₂×D₈	—	281	2	no	—
70	C₂×D₈	—	562	1	no	—
71	#131	—	562	1	no	—
72	#127	—	562	1	no	—
73	SD₃₂	—	562	2	no	—
74	D₁₆	—	562	2	no	—
75	M₃₂	C₂×D₂₈₁	1	2	yes	—
76	#108	—	281	1	no	—
77	#104	—	562	1	no	—
78	C₄×C₈	C₂×D₂₈₁	1	1	yes	characteristic
79	#41	—	562	1	no	—
80	C₂×D₄	—	562	2	no	—
81	C₂×D₄	—	1124	2	no	—
82	Q₁₆	—	281	1	no	—
83	SD₁₆	—	562	1	no	—
84	D₈	—	281	1	no	—
85	D₈	—	562	2	no	—
86	D₈	—	562	2	no	—
87	D₈	—	1124	1	no	—
88	D₈	—	1124	1	no	—
89	M₁₆	—	562	1	no	—
90	M₁₆	—	1124	1	no	—
91	C₂×C₈	—	1124	1	no	—
92	C₂×C₈	D₁₁₂₄	1	1	yes	characteristic
93	C₂×C₈	C₂²×D₂₈₁	1	1	yes	characteristic
94	C₄²	D₁₁₂₄	1	1	yes	characteristic
95	C₁₆	—	2	2	no	—
96	C₂³	—	1124	2	no	—
97	Q₈	—	562	1	no	—
98	D₄	—	562	1	no	—
99	D₄	—	1124	1	no	—
100	D₄	—	1124	2	no	—
101	D₄	—	1124	2	no	—
102	D₄	—	2248	4	no	—
103	C₂×C₄	—	2	1	no	—
104	C₂×C₄	—	1124	1	no	—
105	C₂×C₄	C₂×D₁₁₂₄	1	1	yes	characteristic
106	C₈	—	1124	1	no	—
107	C₈	D₂₂₄₈	1	1	yes	characteristic
108	C₈	D₂₂₄₈	1	1	yes	characteristic
109	C₈	D₄×D₂₈₁	1	1	yes	characteristic
110	C₈	#17985139	1	1	yes	characteristic
111	C₂²	—	1124	1	no	—
112	C₂²	—	2248	4	no	—
113	C₂²	—	2248	2	no	—
114	C₂²	#22582907	1	1	yes	characteristic
115	C₄	—	2	2	no	—
116	C₄	—	1124	1	no	—
117	C₄	#22582854	1	1	yes	characteristic
118	C₄	C₂×D₂₂₄₈	1	1	yes	characteristic
119	C₂	—	2	1	no	—
120	C₂	—	4496	2	no	—
121	C₂	—	2248	1	no	—
122	C₂	#31823659	1	1	yes	characteristic

subgroup

G/H

class size

multiplicity

normal?

properties

#31823901

C₂

yes

maximal, complemented

#31823840

C₂

yes

maximal, characteristic, complemented

#31823581

C₂

yes

maximal, characteristic, complemented

#31823277

C₂

yes

maximal, complemented

#364×C₂₈₁

C₂

yes

maximal, characteristic, complemented

#22582901

C₂²

yes

characteristic

C₂×D₂₂₄₈

—

C₂×D₂₂₄₈

C₂²

yes

—

#22582885

—

#22582876

—

#22582824

—

D₄₄₉₆

—

#22582815

C₂²

yes

characteristic

#22582807

—

M₃₂×C₂₈₁

C₂²

yes

—

C₄×C₈×C₂₈₁

C₂²

yes

characteristic

#17985138

—

C₂×D₁₁₂₄

—

C₂×D₁₁₂₄

—

Dic₁₁₂₄

D₄

yes

characteristic, complemented

#17985127

—

D₂₂₄₈

—

D₂₂₄₈

—

D₂₂₄₈

—

D₂₂₄₈

—

D₂₂₄₈

D₄

yes

characteristic, complemented

#17985122

—

#17985121

—

C₈×D₂₈₁

—

C₂×C₈×C₂₈₁

D₄

yes

characteristic

C₂×C₈×C₂₈₁

C₂³

yes

characteristic

C₄²×C₂₈₁

D₄

yes

characteristic

C₁₆×C₂₈₁

—

C₂²×D₂₈₁

—

Dic₅₆₂

—

D₁₁₂₄

—

D₁₁₂₄

—

D₁₁₂₄

—

D₁₁₂₄

—

C₂²⋊D₂₈₁

—

C₄×D₂₈₁

—

#14555886

—

C₂×C₄×C₂₈₁

—

C₂×C₄×C₂₈₁

C₂×D₄

yes

characteristic

C₈×C₂₈₁

D₈

yes

characteristic

C₈×C₂₈₁

D₈

yes

characteristic

C₈×C₂₈₁

C₂×D₄

yes

C₈×C₂₈₁

yes

C₂²×C₂₈₁

yes

C₂×D₂₈₁

—

C₂×D₂₈₁

—

C₂×D₂₈₁

—

C₄×C₂₈₁

—

C₄×C₂₈₁

#121

yes

characteristic

C₄×C₂₈₁

C₂×D₈

yes

characteristic

Dic₂₈₁

—

C₂×C₂₈₁

—

C₂×C₂₈₁

#459

yes

characteristic

D₂₈₁

—

D₂₈₁

—

C₂₈₁

#2213

yes

characteristic, complemented, Sylow

#2213

—

281

maximal, Sylow

#509

—

281

—

#493

—

281

—

#443

—

281

—

#364

D₂₈₁

yes

characteristic, complemented

#361

—

281

—

#135

—

281

—

C₂×D₈

—

281

—

C₂×D₈

—

562

—

#131

—

562

—

#127

—

562

—

SD₃₂

—

562

—

D₁₆

—

562

—

M₃₂

C₂×D₂₈₁

yes

—

#108

—

281

—

#104

—

562

—

C₄×C₈

C₂×D₂₈₁

yes

characteristic

#41

—

562

—

C₂×D₄

—

562

—

C₂×D₄

—

1124

—

Q₁₆

—

281

—

SD₁₆

—

562

—

D₈

—

281

—

D₈

—

562

—

D₈

—

562

—

D₈

—

1124

—

D₈

—

1124

—

M₁₆

—

562

—

M₁₆

—

1124

—

C₂×C₈

—

1124

—

C₂×C₈

D₁₁₂₄

yes

characteristic

C₂×C₈

C₂²×D₂₈₁

yes

characteristic

C₄²

D₁₁₂₄

yes

characteristic

C₁₆

—

C₂³

—

1124

—

Q₈

—

562

—

D₄

—

562

—

D₄

—

1124

—

100

D₄

—

1124

—

101

D₄

—

1124

—

102

D₄

—

2248

—

103

C₂×C₄

—

104

C₂×C₄

—

1124

—

105

C₂×C₄

C₂×D₁₁₂₄

yes

characteristic

106

C₈

—

1124

—

107

C₈

D₂₂₄₈

yes

characteristic

108

C₈

D₂₂₄₈

yes

characteristic

109

C₈

D₄×D₂₈₁

yes

characteristic

110

C₈

#17985139

yes

characteristic

111

C₂²

—

1124

—

112

C₂²

—

2248

—

113

C₂²

—

2248

—

114

C₂²

#22582907

yes

characteristic

115

C₄

—

116

C₄

—

1124

—

117

C₄

#22582854

yes

characteristic

118

C₄

C₂×D₂₂₄₈

yes

characteristic

119

C₂

—

120

C₂

—

4496

—

121

C₂

—

2248

—

122

C₂

#31823659

yes

characteristic

	subgroup	G/H
center	C₂	#31823659
commutator, cosocle	C₂×C₈×C₂₈₁	C₂³
Frattini	C₂×C₈	C₂²×D₂₈₁
Fitting	#364×C₂₈₁	C₂
socle	C₂×C₂₈₁	#459
hypercenter	#364	D₂₈₁
hypocenter	C₂₈₁	#2213
LCS	G ▷ C₂×C₈×C₂₈₁ ▷ C₄×C₂₈₁ ▷ C₂×C₂₈₁ ▷ C₂₈₁
UCS	C₁ ◁ C₂ ◁ C₂×C₄ ◁ C₄×C₈ ◁ #364

subgroup

G/H

center

C₂

#31823659

commutator, cosocle

C₂×C₈×C₂₈₁

C₂³

Frattini

C₂×C₈

C₂²×D₂₈₁

Fitting

#364×C₂₈₁

C₂

C₂×C₂₈₁

D₂₈₁

C₂₈₁

G ▷ C₂×C₈×C₂₈₁ ▷ C₄×C₂₈₁ ▷ C₂×C₂₈₁ ▷ C₂₈₁

UCS

C₁ ◁ C₂ ◁ C₂×C₄ ◁ C₄×C₈ ◁ #364

N	H ≅ G/N	type
1. #31823901	C₂	semidirect
2. #31823840	C₂	semidirect
3. #31823581	C₂	semidirect
4. #31823277	C₂	semidirect
5. #364×C₂₈₁	C₂	semidirect
20. Dic₁₁₂₄	D₄	semidirect
26. D₂₂₄₈	D₄	semidirect
61. C₂₈₁	#2213	semidirect
66. #364	D₂₈₁	semidirect

H ≅ G/N

type

1. #31823901

C₂

semidirect

2. #31823840

C₂

semidirect

3. #31823581

C₂

semidirect

4. #31823277

C₂

semidirect

5. #364×C₂₈₁

C₂

semidirect

20. Dic₁₁₂₄

D₄

semidirect

26. D₂₂₄₈

D₄

semidirect

61. C₂₈₁

#2213

semidirect

66. #364

D₂₈₁

semidirect

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

subgroup	index
#31823901	2
#31823840	2
#31823581	2
#31823277	2
#364×C₂₈₁	2
#2213	281

subgroup

index

#364×C₂₈₁

281

Subgroups — #49097380

Conjugacy classes of subgroups

Complements

Maximal subgroups