Group #49094127 — subgroups

Subgroups — #49094127

Order 35968.

192 conjugacy classes of subgroups; only the 87 normal subgroups are shown.

normal subgroups only

This group has 192 conjugacy classes of subgroups — too many to draw. Only its 87 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	D₄×#14555886	C₂	1	2	yes	maximal, complemented
2	#31823214	C₂	1	1	yes	maximal, characteristic, complemented
3	D₄⋊#14555886	C₂	1	2	yes	maximal, complemented
4	#31823184	C₂	1	1	yes	maximal, characteristic, complemented
5	C₄×D₈×C₂₈₁	C₂	1	1	yes	maximal, characteristic
6	C₂²×#14555886	—	2	2	no	—
7	C₄⋊#14555886	C₂²	1	2	yes	—
8	C₂²⋊#14555886	—	2	2	no	—
9	C₄×#14555886	C₂²	1	1	yes	characteristic
10	C₄×#14555886	C₂²	1	1	yes	characteristic
11	C₂×D₈×C₂₈₁	C₄	1	1	yes	characteristic
12	C₄×D₄×C₂₈₁	C₂²	1	2	yes	—
13	#107×C₂₈₁	C₄	1	1	yes	characteristic
14	#102×C₂₈₁	C₄	1	2	yes	—
15	C₄×C₈×C₂₈₁	C₂²	1	1	yes	characteristic
16	C₂×#14555886	—	2	2	no	—
17	C₂×#14555886	—	2	1	no	—
18	C₂×#14555886	—	4	4	no	—
19	C₂×#14555886	D₄	1	2	yes	—
20	C₂×D₄×C₂₈₁	C₂×C₄	1	2	yes	—
21	C₂²×C₄×C₂₈₁	—	2	2	no	—
22	D₈×C₂₈₁	C₈	1	4	yes	complemented
23	C₂×C₈×C₂₈₁	C₂×C₄	1	1	yes	characteristic
24	C₂×C₈×C₂₈₁	C₂×C₄	1	1	yes	characteristic
25	C₄⋊C₄×C₂₈₁	C₂×C₄	1	2	yes	—
26	C₂²⋊C₄×C₂₈₁	—	2	2	no	—
27	C₄²×C₂₈₁	C₂³	1	1	yes	characteristic
28	#14555886	—	2	2	no	—
29	#14555886	—	4	1	no	—
30	#14555886	—	4	2	no	—
31	C₂³×C₂₈₁	—	2	2	no	—
32	D₄×C₂₈₁	—	2	2	no	—
33	D₄×C₂₈₁	C₂×C₈	1	4	yes	complemented
34	C₂×C₄×C₂₈₁	—	2	2	no	—
35	C₂×C₄×C₂₈₁	—	4	2	no	—
36	C₂×C₄×C₂₈₁	—	4	2	no	—
37	C₂×C₄×C₂₈₁	C₂²×C₄	1	1	yes	characteristic
38	C₂×C₄×C₂₈₁	C₂×D₄	1	1	yes	characteristic
39	C₂×C₄×C₂₈₁	#41	1	1	yes	characteristic
40	C₈×C₂₈₁	—	2	1	no	—
41	C₈×C₂₈₁	C₂×C₈	1	2	yes	complemented
42	C₂²×C₂₈₁	—	2	4	no	—
43	C₂²×C₂₈₁	—	4	2	no	—
44	C₂²×C₂₈₁	—	4	2	no	—
45	C₂²×C₂₈₁	C₄×D₄	1	1	yes	characteristic
46	C₄×C₂₈₁	—	2	1	no	—
47	C₄×C₂₈₁	—	4	2	no	—
48	C₄×C₂₈₁	C₂²×C₈	1	1	yes	characteristic
49	C₄×C₂₈₁	#131	1	1	yes	characteristic
50	C₄×C₂₈₁	AGL(1,ℤ/8)	1	2	yes	—
51	C₂×C₂₈₁	—	4	4	no	—
52	C₂×C₂₈₁	C₈×D₄	1	1	yes	characteristic
53	C₂×C₂₈₁	#442	1	1	yes	characteristic
54	C₂×C₂₈₁	#444	1	1	yes	characteristic
55	C₂₈₁	#1580	1	1	yes	characteristic, complemented, Sylow
56	#1580	—	281	1	no	maximal, Sylow
57	C₄×D₈	D₂₈₁	1	1	yes	characteristic
58	C₈×D₄	—	281	2	no	—
59	#334	—	281	1	no	—
60	D₄⋊C₈	—	281	2	no	—
61	#322	—	281	1	no	—
62	C₂×D₈	Dic₂₈₁	1	1	yes	characteristic
63	C₂²×C₈	—	562	2	no	—
64	C₄×D₄	C₂×D₂₈₁	1	2	yes	—
65	#107	Dic₂₈₁	1	1	yes	characteristic
66	C₄⋊C₈	—	281	2	no	—
67	#102	Dic₂₈₁	1	2	yes	—
68	C₂²⋊C₈	—	562	2	no	—
69	C₄×C₈	—	281	1	no	—
70	C₄×C₈	—	281	1	no	—
71	C₄×C₈	C₂×D₂₈₁	1	1	yes	characteristic
72	C₂×D₄	C₂×Dic₂₈₁	1	2	yes	—
73	C₂²×C₄	—	2	2	no	—
74	D₈	#14555886	1	4	yes	complemented
75	C₂×C₈	—	281	2	no	—
76	C₂×C₈	—	562	2	no	—
77	C₂×C₈	—	562	1	no	—
78	C₂×C₈	—	1124	4	no	—
79	C₂×C₈	C₂×Dic₂₈₁	1	1	yes	characteristic
80	C₂×C₈	C₂×Dic₂₈₁	1	1	yes	characteristic
81	C₄⋊C₄	C₂×Dic₂₈₁	1	2	yes	—
82	C₂²⋊C₄	—	2	2	no	—
83	C₄²	C₂²×D₂₈₁	1	1	yes	characteristic
84	C₂³	—	2	2	no	—
85	D₄	—	2	2	no	—
86	D₄	C₂×#14555886	1	4	yes	complemented
87	C₂×C₄	—	2	2	no	—
88	C₂×C₄	—	4	2	no	—
89	C₂×C₄	—	4	2	no	—
90	C₂×C₄	C₂²×Dic₂₈₁	1	1	yes	characteristic
91	C₂×C₄	D₄×D₂₈₁	1	1	yes	characteristic
92	C₂×C₄	#17985137	1	1	yes	characteristic
93	C₈	—	2	1	no	—
94	C₈	—	562	2	no	—
95	C₈	—	1124	1	no	—
96	C₈	—	1124	2	no	—
97	C₈	C₂×#14555886	1	2	yes	complemented
98	C₂²	—	2	4	no	—
99	C₂²	—	4	2	no	—
100	C₂²	—	4	2	no	—
101	C₂²	D₄×Dic₂₈₁	1	1	yes	characteristic
102	C₄	—	2	1	no	—
103	C₄	—	4	2	no	—
104	C₄	C₂²×#14555886	1	1	yes	characteristic
105	C₄	#22582886	1	1	yes	characteristic
106	C₄	#22582903	1	2	yes	—
107	C₂	—	4	4	no	—
108	C₂	D₄×#14555886	1	1	yes	characteristic
109	C₂	#31823575	1	1	yes	characteristic
110	C₂	#31823585	1	1	yes	characteristic

subgroup

G/H

class size

multiplicity

normal?

properties

D₄×#14555886

C₂

yes

maximal, complemented

#31823214

C₂

yes

maximal, characteristic, complemented

D₄⋊#14555886

C₂

yes

maximal, complemented

#31823184

C₂

yes

maximal, characteristic, complemented

C₄×D₈×C₂₈₁

C₂

yes

maximal, characteristic

C₂²×#14555886

—

C₄⋊#14555886

C₂²

yes

—

C₂²⋊#14555886

—

C₄×#14555886

C₂²

yes

characteristic

C₄×#14555886

C₂²

yes

characteristic

C₂×D₈×C₂₈₁

C₄

yes

characteristic

C₄×D₄×C₂₈₁

C₂²

yes

—

#107×C₂₈₁

C₄

yes

characteristic

#102×C₂₈₁

C₄

yes

—

C₄×C₈×C₂₈₁

C₂²

yes

—

—

—

D₄

yes

—

C₂×D₄×C₂₈₁

C₂×C₄

yes

—

C₂²×C₄×C₂₈₁

—

D₈×C₂₈₁

C₈

yes

complemented

C₂×C₈×C₂₈₁

C₂×C₄

yes

characteristic

C₂×C₈×C₂₈₁

C₂×C₄

yes

characteristic

C₄⋊C₄×C₂₈₁

C₂×C₄

yes

—

C₂²⋊C₄×C₂₈₁

—

C₄²×C₂₈₁

C₂³

yes

—

—

—

C₂³×C₂₈₁

—

D₄×C₂₈₁

—

D₄×C₂₈₁

C₂×C₈

yes

complemented

C₂×C₄×C₂₈₁

—

C₂×C₄×C₂₈₁

—

C₂×C₄×C₂₈₁

—

C₂×C₄×C₂₈₁

C₂²×C₄

yes

characteristic

C₂×C₄×C₂₈₁

C₂×D₄

yes

characteristic

C₂×C₄×C₂₈₁

#41

yes

characteristic

C₈×C₂₈₁

—

C₈×C₂₈₁

C₂×C₈

yes

complemented

C₂²×C₂₈₁

—

C₂²×C₂₈₁

—

C₂²×C₂₈₁

—

C₂²×C₂₈₁

C₄×D₄

yes

characteristic

C₄×C₂₈₁

—

C₄×C₂₈₁

—

C₄×C₂₈₁

C₂²×C₈

yes

C₄×C₂₈₁

yes

C₄×C₂₈₁

yes

—

C₂×C₂₈₁

—

C₂×C₂₈₁

C₈×D₄

yes

C₂×C₂₈₁

yes

C₂×C₂₈₁

yes

C₂₈₁

yes

characteristic, complemented, Sylow

#1580

—

281

maximal, Sylow

C₄×D₈

D₂₈₁

yes

characteristic

C₈×D₄

—

281

—

#334

—

281

—

D₄⋊C₈

—

281

—

#322

—

281

—

C₂×D₈

Dic₂₈₁

yes

characteristic

C₂²×C₈

—

562

—

C₄×D₄

C₂×D₂₈₁

yes

—

#107

Dic₂₈₁

yes

characteristic

C₄⋊C₈

—

281

—

#102

Dic₂₈₁

yes

—

C₂²⋊C₈

—

562

—

C₄×C₈

—

281

—

C₄×C₈

—

281

—

C₄×C₈

C₂×D₂₈₁

yes

characteristic

C₂×D₄

C₂×Dic₂₈₁

yes

—

C₂²×C₄

—

D₈

#14555886

yes

complemented

C₂×C₈

—

281

—

C₂×C₈

—

562

—

C₂×C₈

—

562

—

C₂×C₈

—

1124

—

C₂×C₈

C₂×Dic₂₈₁

yes

characteristic

C₂×C₈

C₂×Dic₂₈₁

yes

characteristic

C₄⋊C₄

C₂×Dic₂₈₁

yes

—

C₂²⋊C₄

—

C₄²

C₂²×D₂₈₁

yes

characteristic

C₂³

—

D₄

—

D₄

C₂×#14555886

yes

complemented

C₂×C₄

—

C₂×C₄

—

C₂×C₄

—

C₂×C₄

C₂²×Dic₂₈₁

yes

characteristic

C₂×C₄

D₄×D₂₈₁

yes

characteristic

C₂×C₄

#17985137

yes

characteristic

C₈

—

C₈

—

562

—

C₈

—

1124

—

C₈

—

1124

—

C₈

C₂×#14555886

yes

complemented

C₂²

—

C₂²

—

100

C₂²

—

101

C₂²

D₄×Dic₂₈₁

yes

characteristic

102

C₄

—

103

C₄

—

104

C₄

C₂²×#14555886

yes

characteristic

105

C₄

#22582886

yes

characteristic

106

C₄

#22582903

yes

—

107

C₂

—

108

C₂

D₄×#14555886

yes

characteristic

109

C₂

#31823575

yes

characteristic

110

C₂

#31823585

yes

characteristic

	subgroup	G/H
center	C₂×C₄	D₄×D₂₈₁
commutator	C₄×C₂₈₁	C₂²×C₈
Frattini	C₄²	C₂²×D₂₈₁
Fitting	C₄×D₈×C₂₈₁	C₂
socle	C₂²×C₂₈₁	C₄×D₄
prime radical	C₂×D₈×C₂₈₁	C₄
cosocle	C₄²×C₂₈₁	C₂³
hypercenter	C₄×D₈	D₂₈₁
hypocenter	C₂₈₁	#1580
LCS	G ▷ C₄×C₂₈₁ ▷ C₂×C₂₈₁ ▷ C₂₈₁
UCS	C₁ ◁ C₂×C₄ ◁ C₄² ◁ C₄×D₈

subgroup

G/H

center

C₂×C₄

D₄×D₂₈₁

commutator

C₄×C₂₈₁

C₂²×C₈

Frattini

C₄²

C₂²×D₂₈₁

Fitting

C₄×D₈×C₂₈₁

C₂

socle

C₂²×C₂₈₁

C₄×D₄

prime radical

C₂×D₈×C₂₈₁

C₄

cosocle

C₄²×C₂₈₁

C₂³

hypercenter

C₄×D₈

D₂₈₁

hypocenter

C₂₈₁

#1580

LCS

G ▷ C₄×C₂₈₁ ▷ C₂×C₂₈₁ ▷ C₂₈₁

UCS

C₁ ◁ C₂×C₄ ◁ C₄² ◁ C₄×D₈

N	H ≅ G/N	type
1. D₄×#14555886	C₂	semidirect
2. #31823214	C₂	semidirect
3. D₄⋊#14555886	C₂	semidirect
4. #31823184	C₂	semidirect
22. D₈×C₂₈₁	C₈	semidirect
33. D₄×C₂₈₁	C₂×C₈	semidirect
41. C₈×C₂₈₁	C₂×C₈	semidirect
55. C₂₈₁	#1580	semidirect
74. D₈	#14555886	semidirect
86. D₄	C₂×#14555886	semidirect
97. C₈	C₂×#14555886	semidirect

H ≅ G/N

type

1. D₄×#14555886

C₂

semidirect

2. #31823214

C₂

semidirect

3. D₄⋊#14555886

C₂

semidirect

4. #31823184

C₂

semidirect

22. D₈×C₂₈₁

C₈

semidirect

33. D₄×C₂₈₁

C₂×C₈

semidirect

41. C₈×C₂₈₁

C₂×C₈

semidirect

55. C₂₈₁

#1580

semidirect

74. D₈

#14555886

semidirect

86. D₄

C₂×#14555886

semidirect

97. C₈

C₂×#14555886

semidirect

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

subgroup	index
D₄×#14555886	2
#31823214	2
D₄⋊#14555886	2
#31823184	2
C₄×D₈×C₂₈₁	2
#1580	281

subgroup

index

C₄×D₈×C₂₈₁

281

Subgroups — #49094127

Conjugacy classes of subgroups

Complements

Maximal subgroups