Group #49096280 — subgroups

Subgroups — #49096280

Order 35968.

284 conjugacy classes of subgroups; only the 101 normal subgroups are shown.

normal subgroups only

This group has 284 conjugacy classes of subgroups — too many to draw. Only its 101 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	#31824020	C₂	1	1	yes	maximal, characteristic
2	#31823487	C₂	1	1	yes	maximal, characteristic
3	#31823223	C₂	1	4	yes	maximal
4	C₄³×C₂₈₁	C₂	1	1	yes	maximal, characteristic
5	C₂×#17985122	C₂²	1	2	yes	—
6	#22582879	C₄	1	2	yes	complemented
7	#22582871	C₄	1	2	yes	complemented
8	#22582858	—	2	2	no	—
9	C₄×Dic₅₆₂	—	2	2	no	—
10	#22582838	C₂²	1	2	yes	—
11	C₄⋊#14555886	—	2	2	no	—
12	C₂×C₄²×C₂₈₁	—	2	2	no	—
13	C₂×C₄²×C₂₈₁	C₂²	1	2	yes	—
14	C₂×C₄²×C₂₈₁	C₂²	1	1	yes	characteristic
15	C₂×Dic₅₆₂	—	2	2	no	—
16	#17985122	—	2	4	no	—
17	C₂×#14555886	—	2	2	no	—
18	C₄⋊Dic₂₈₁	—	2	2	no	—
19	C₄⋊Dic₂₈₁	C₂×C₄	1	4	yes	—
20	#17985117	—	2	4	no	—
21	C₂²⋊Dic₂₈₁	—	2	2	no	—
22	C₄×Dic₂₈₁	—	2	2	no	—
23	C₂²×C₄×C₂₈₁	—	2	2	no	—
24	C₂²×C₄×C₂₈₁	D₄	1	2	yes	—
25	C₂²×C₄×C₂₈₁	C₂³	1	1	yes	characteristic
26	C₄²×C₂₈₁	—	2	8	no	—
27	C₄²×C₂₈₁	—	2	2	no	—
28	C₄²×C₂₈₁	C₂×C₄	1	1	yes	characteristic
29	C₄²×C₂₈₁	C₂×C₄	1	1	yes	characteristic
30	C₄²×C₂₈₁	D₄	1	2	yes	—
31	C₄²×C₂₈₁	Q₈	1	4	yes	complemented
32	Dic₅₆₂	—	4	4	no	—
33	C₂×Dic₂₈₁	—	2	4	no	—
34	#14555886	—	4	2	no	—
35	C₂³×C₂₈₁	C₂×D₄	1	1	yes	characteristic
36	C₂×C₄×C₂₈₁	—	2	8	no	—
37	C₂×C₄×C₂₈₁	—	2	4	no	—
38	C₂×C₄×C₂₈₁	—	2	4	no	—
39	C₂×C₄×C₂₈₁	C₂²⋊C₄	1	2	yes	—
40	C₂×C₄×C₂₈₁	C₂²⋊C₄	1	2	yes	—
41	C₂×C₄×C₂₈₁	C₂²×C₄	1	1	yes	characteristic
42	C₂×C₄×C₂₈₁	C₂×D₄	1	1	yes	characteristic
43	C₂×C₄×C₂₈₁	C₂×Q₈	1	2	yes	—
44	C₂×C₄×C₂₈₁	#41	1	2	yes	—
45	C₂²×C₂₈₁	—	2	2	no	—
46	C₂²×C₂₈₁	C₂×C₂²⋊C₄	1	1	yes	characteristic
47	C₂²×C₂₈₁	C₂²⋊Q₈	1	2	yes	—
48	C₄×C₂₈₁	—	2	8	no	—
49	C₄×C₂₈₁	—	2	2	no	—
50	C₄×C₂₈₁	#104	1	4	yes	—
51	C₄×C₂₈₁	C₄×Q₈	1	2	yes	—
52	C₄×C₂₈₁	#124	1	1	yes	characteristic
53	C₄×C₂₈₁	C₄⋊Q₈	1	1	yes	characteristic
54	Dic₂₈₁	—	4	4	no	—
55	C₂×C₂₈₁	—	2	2	no	—
56	C₂×C₂₈₁	#391	1	1	yes	characteristic
57	C₂×C₂₈₁	C₂×#104	1	2	yes	—
58	C₂₈₁	#1994	1	1	yes	characteristic, complemented, Sylow
59	#1994	—	281	1	no	maximal, Sylow
60	#533	—	281	1	no	—
61	#423	—	281	1	no	—
62	C₄³	D₂₈₁	1	1	yes	characteristic
63	#339	—	281	4	no	—
64	C₂×M₁₆	—	281	2	no	—
65	C₄⋊Q₈	—	281	2	no	—
66	#125	—	281	2	no	—
67	C₂²⋊Q₈	—	562	2	no	—
68	C₄×Q₈	—	562	2	no	—
69	#117	—	281	2	no	—
70	C₂×C₄²	—	2	2	no	—
71	C₂×C₄²	C₂×D₂₈₁	1	2	yes	—
72	C₂×C₄²	C₂×D₂₈₁	1	1	yes	characteristic
73	C₄⋊C₈	—	562	2	no	—
74	C₂×Q₈	—	562	2	no	—
75	C₂²×C₄	—	2	2	no	—
76	C₂²×C₄	C₂²⋊D₂₈₁	1	2	yes	—
77	C₂²×C₄	C₂²×D₂₈₁	1	1	yes	characteristic
78	M₁₆	—	562	4	no	—
79	C₂×C₈	—	562	2	no	—
80	C₄⋊C₄	—	281	4	no	—
81	C₄⋊C₄	—	562	4	no	—
82	C₄⋊C₄	—	562	2	no	—
83	C₂²⋊C₄	—	562	2	no	—
84	C₄²	—	2	8	no	—
85	C₄²	—	2	2	no	—
86	C₄²	—	562	2	no	—
87	C₄²	C₄×D₂₈₁	1	1	yes	characteristic
88	C₄²	C₄×D₂₈₁	1	1	yes	characteristic
89	C₄²	D₁₁₂₄	1	2	yes	—
90	C₄²	Dic₅₆₂	1	4	yes	complemented
91	C₂³	C₂×C₂²⋊D₂₈₁	1	1	yes	characteristic
92	Q₈	—	1124	4	no	—
93	C₂×C₄	—	2	8	no	—
94	C₂×C₄	—	2	4	no	—
95	C₂×C₄	—	2	4	no	—
96	C₂×C₄	—	562	4	no	—
97	C₂×C₄	#17985115	1	2	yes	—
98	C₂×C₄	#17985115	1	2	yes	—
99	C₂×C₄	C₂×C₄×D₂₈₁	1	1	yes	characteristic
100	C₂×C₄	C₂×D₁₁₂₄	1	1	yes	characteristic
101	C₂×C₄	C₂×Dic₅₆₂	1	2	yes	—
102	C₂×C₄	#17985138	1	2	yes	—
103	C₈	—	1124	2	no	—
104	C₂²	—	2	2	no	—
105	C₂²	C₂×#17985115	1	1	yes	characteristic
106	C₂²	#22582858	1	2	yes	—
107	C₄	—	2	8	no	—
108	C₄	—	2	2	no	—
109	C₄	—	1124	4	no	—
110	C₄	#22582807	1	4	yes	—
111	C₄	C₄×Dic₅₆₂	1	2	yes	—
112	C₄	#22582867	1	1	yes	characteristic
113	C₄	#22582879	1	1	yes	characteristic
114	C₂	—	2	2	no	—
115	C₂	#31823369	1	1	yes	characteristic
116	C₂	C₂×#22582807	1	2	yes	—

subgroup

G/H

class size

multiplicity

normal?

properties

#31824020

C₂

yes

maximal, characteristic

#31823487

C₂

yes

maximal, characteristic

#31823223

C₂

yes

maximal

C₄³×C₂₈₁

C₂

yes

maximal, characteristic

C₂×#17985122

C₂²

yes

—

#22582879

C₄

yes

complemented

#22582871

C₄

yes

complemented

#22582858

—

C₄×Dic₅₆₂

—

#22582838

C₂²

yes

—

C₄⋊#14555886

—

C₂×C₄²×C₂₈₁

—

C₂×C₄²×C₂₈₁

C₂²

yes

—

C₂×C₄²×C₂₈₁

C₂²

yes

characteristic

C₂×Dic₅₆₂

—

#17985122

—

C₂×#14555886

—

C₄⋊Dic₂₈₁

—

C₄⋊Dic₂₈₁

C₂×C₄

yes

—

#17985117

—

C₂²⋊Dic₂₈₁

—

C₄×Dic₂₈₁

—

C₂²×C₄×C₂₈₁

—

C₂²×C₄×C₂₈₁

D₄

yes

—

C₂²×C₄×C₂₈₁

C₂³

yes

characteristic

C₄²×C₂₈₁

—

C₄²×C₂₈₁

—

C₄²×C₂₈₁

C₂×C₄

yes

characteristic

C₄²×C₂₈₁

C₂×C₄

yes

characteristic

C₄²×C₂₈₁

D₄

yes

—

C₄²×C₂₈₁

Q₈

yes

complemented

Dic₅₆₂

—

C₂×Dic₂₈₁

—

#14555886

—

C₂³×C₂₈₁

C₂×D₄

yes

characteristic

C₂×C₄×C₂₈₁

—

C₂×C₄×C₂₈₁

—

C₂×C₄×C₂₈₁

—

C₂×C₄×C₂₈₁

C₂²⋊C₄

yes

—

C₂×C₄×C₂₈₁

C₂²⋊C₄

yes

—

C₂×C₄×C₂₈₁

C₂²×C₄

yes

characteristic

C₂×C₄×C₂₈₁

C₂×D₄

yes

characteristic

C₂×C₄×C₂₈₁

C₂×Q₈

yes

—

C₂×C₄×C₂₈₁

#41

yes

—

C₂²×C₂₈₁

—

C₂²×C₂₈₁

C₂×C₂²⋊C₄

yes

characteristic

C₂²×C₂₈₁

C₂²⋊Q₈

yes

—

C₄×C₂₈₁

—

C₄×C₂₈₁

—

C₄×C₂₈₁

#104

yes

—

C₄×C₂₈₁

C₄×Q₈

yes

—

C₄×C₂₈₁

#124

yes

characteristic

C₄×C₂₈₁

C₄⋊Q₈

yes

characteristic

Dic₂₈₁

—

C₂×C₂₈₁

—

C₂×C₂₈₁

#391

yes

characteristic

C₂×C₂₈₁

C₂×#104

yes

—

C₂₈₁

#1994

yes

characteristic, complemented, Sylow

#1994

—

281

maximal, Sylow

#533

—

281

—

#423

—

281

—

C₄³

D₂₈₁

yes

characteristic

#339

—

281

—

C₂×M₁₆

—

281

—

C₄⋊Q₈

—

281

—

#125

—

281

—

C₂²⋊Q₈

—

562

—

C₄×Q₈

—

562

—

#117

—

281

—

C₂×C₄²

—

C₂×C₄²

C₂×D₂₈₁

yes

—

C₂×C₄²

C₂×D₂₈₁

yes

characteristic

C₄⋊C₈

—

562

—

C₂×Q₈

—

562

—

C₂²×C₄

—

C₂²×C₄

C₂²⋊D₂₈₁

yes

—

C₂²×C₄

C₂²×D₂₈₁

yes

characteristic

M₁₆

—

562

—

C₂×C₈

—

562

—

C₄⋊C₄

—

281

—

C₄⋊C₄

—

562

—

C₄⋊C₄

—

562

—

C₂²⋊C₄

—

562

—

C₄²

—

C₄²

—

C₄²

—

562

—

C₄²

C₄×D₂₈₁

yes

characteristic

C₄²

C₄×D₂₈₁

yes

characteristic

C₄²

D₁₁₂₄

yes

—

C₄²

Dic₅₆₂

yes

complemented

C₂³

C₂×C₂²⋊D₂₈₁

yes

characteristic

Q₈

—

1124

—

C₂×C₄

—

C₂×C₄

—

C₂×C₄

—

C₂×C₄

—

562

—

C₂×C₄

#17985115

yes

—

C₂×C₄

#17985115

yes

—

C₂×C₄

C₂×C₄×D₂₈₁

yes

characteristic

100

C₂×C₄

C₂×D₁₁₂₄

yes

characteristic

101

C₂×C₄

C₂×Dic₅₆₂

yes

—

102

C₂×C₄

#17985138

yes

—

103

C₈

—

1124

—

104

C₂²

—

105

C₂²

C₂×#17985115

yes

characteristic

106

C₂²

#22582858

yes

—

107

C₄

—

108

C₄

—

109

C₄

—

1124

—

110

C₄

#22582807

yes

—

111

C₄

C₄×Dic₅₆₂

yes

—

112

C₄

#22582867

yes

characteristic

113

C₄

#22582879

yes

characteristic

114

C₂

—

115

C₂

#31823369

yes

characteristic

116

C₂

C₂×#22582807

yes

—

	subgroup	G/H
center	C₂×C₄	C₂×D₁₁₂₄
commutator	C₂×C₄×C₂₈₁	C₂²×C₄
Frattini	C₂²×C₄	C₂²×D₂₈₁
Fitting	C₄³×C₂₈₁	C₂
socle	C₂²×C₂₈₁	C₂×C₂²⋊C₄
prime radical	C₂³×C₂₈₁	C₂×D₄
cosocle	C₂²×C₄×C₂₈₁	C₂³
hypercenter	C₄³	D₂₈₁
hypocenter	C₂₈₁	#1994
LCS	G ▷ C₂×C₄×C₂₈₁ ▷ C₂×C₂₈₁ ▷ C₂₈₁
UCS	C₁ ◁ C₂×C₄ ◁ C₂×C₄² ◁ C₄³

subgroup

G/H

center

C₂×C₄

C₂×D₁₁₂₄

commutator

C₂×C₄×C₂₈₁

C₂²×C₄

Frattini

C₂²×C₄

C₂²×D₂₈₁

Fitting

C₄³×C₂₈₁

C₂

socle

C₂²×C₂₈₁

C₂×C₂²⋊C₄

prime radical

C₂³×C₂₈₁

C₂×D₄

cosocle

C₂²×C₄×C₂₈₁

C₂³

hypercenter

C₄³

D₂₈₁

hypocenter

C₂₈₁

#1994

LCS

G ▷ C₂×C₄×C₂₈₁ ▷ C₂×C₂₈₁ ▷ C₂₈₁

UCS

C₁ ◁ C₂×C₄ ◁ C₂×C₄² ◁ C₄³

N	H ≅ G/N	type
6. #22582879	C₄	semidirect
7. #22582871	C₄	semidirect
31. C₄²×C₂₈₁	Q₈	semidirect
58. C₂₈₁	#1994	semidirect
90. C₄²	Dic₅₆₂	semidirect

H ≅ G/N

type

6. #22582879

C₄

semidirect

7. #22582871

C₄

semidirect

31. C₄²×C₂₈₁

Q₈

semidirect

58. C₂₈₁

#1994

semidirect

90. C₄²

Dic₅₆₂

semidirect

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

subgroup	index
#31824020	2
#31823487	2
#31823223	2
C₄³×C₂₈₁	2
#1994	281

subgroup

index

C₄³×C₂₈₁

281

Subgroups — #49096280

Conjugacy classes of subgroups

Complements

Maximal subgroups