Group #49094978 — subgroups

Subgroups — #49094978

Order 35968.

176 conjugacy classes of subgroups; only the 61 normal subgroups are shown.

normal subgroups only

This group has 176 conjugacy classes of subgroups — too many to draw. Only its 61 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	#31823866	C₂	1	1	yes	maximal, characteristic
2	#31823866	C₂	1	1	yes	maximal, characteristic
3	#31823859	C₂	1	1	yes	maximal, characteristic
4	#31823815	C₂	1	1	yes	maximal, characteristic
5	#31823814	C₂	1	1	yes	maximal, characteristic
6	#31823185	C₂	1	1	yes	maximal, characteristic
7	#494×C₂₈₁	C₂	1	1	yes	maximal, characteristic
8	#22582879	C₂²	1	1	yes	characteristic
9	#22582879	C₂²	1	1	yes	characteristic
10	#22582812	—	2	2	no	—
11	#22582812	—	2	2	no	—
12	#22582810	—	2	2	no	—
13	#22582804	—	2	2	no	—
14	#22582802	—	2	2	no	—
15	C₄×#14555886	C₂²	1	1	yes	characteristic
16	C₄×#14555886	C₂²	1	1	yes	characteristic
17	C₂×Q₁₆×C₂₈₁	—	2	2	no	—
18	C₄⋊Q₈×C₂₈₁	C₂²	1	1	yes	characteristic
19	C₄⋊Q₈×C₂₈₁	C₂²	1	1	yes	characteristic
20	C₄×C₈×C₂₈₁	C₂²	1	1	yes	characteristic
21	C₂×Dic₅₆₂	—	2	1	no	—
22	C₂×Dic₅₆₂	—	2	1	no	—
23	C₂×#14555886	D₄	1	2	yes	—
24	C₂×#14555886	D₄	1	2	yes	—
25	C₄⋊Dic₂₈₁	—	2	1	no	—
26	C₄⋊Dic₂₈₁	—	2	1	no	—
27	C₄⋊Dic₂₈₁	—	2	1	no	—
28	C₄⋊Dic₂₈₁	—	2	1	no	—
29	C₂×Q₈×C₂₈₁	—	2	1	no	—
30	C₂×Q₈×C₂₈₁	—	2	1	no	—
31	Q₁₆×C₂₈₁	—	4	4	no	—
32	C₂×C₈×C₂₈₁	D₄	1	2	yes	—
33	C₄⋊C₄×C₂₈₁	—	2	1	no	—
34	C₄⋊C₄×C₂₈₁	—	2	1	no	—
35	C₄⋊C₄×C₂₈₁	—	2	1	no	—
36	C₄⋊C₄×C₂₈₁	—	2	1	no	—
37	C₄²×C₂₈₁	C₂³	1	1	yes	characteristic
38	Dic₅₆₂	—	4	2	no	—
39	Dic₅₆₂	—	4	2	no	—
40	C₂×Dic₂₈₁	—	4	1	no	—
41	C₂×Dic₂₈₁	—	4	1	no	—
42	#14555886	—	2	2	no	—
43	#14555886	Q₁₆	1	4	yes	complemented
44	Q₈×C₂₈₁	—	4	2	no	—
45	Q₈×C₂₈₁	—	4	2	no	—
46	C₂×C₄×C₂₈₁	—	4	1	no	—
47	C₂×C₄×C₂₈₁	—	4	1	no	—
48	C₂×C₄×C₂₈₁	C₂×D₄	1	1	yes	characteristic
49	C₂×C₄×C₂₈₁	C₂×D₄	1	1	yes	characteristic
50	C₂×C₄×C₂₈₁	C₂×D₄	1	1	yes	characteristic
51	C₈×C₂₈₁	—	2	2	no	—
52	C₂²×C₂₈₁	#127	1	1	yes	characteristic
53	C₄×C₂₈₁	—	8	1	no	—
54	C₄×C₂₈₁	—	8	1	no	—
55	C₄×C₂₈₁	C₂×Q₁₆	1	2	yes	—
56	C₄×C₂₈₁	AGL(1,ℤ/8)	1	2	yes	—
57	C₄×C₂₈₁	#137	1	2	yes	—
58	Dic₂₈₁	—	8	1	no	—
59	Dic₂₈₁	—	8	1	no	—
60	C₂×C₂₈₁	#494	1	1	yes	characteristic
61	C₂×C₂₈₁	#496	1	1	yes	characteristic
62	C₂×C₂₈₁	#497	1	1	yes	characteristic
63	C₂₈₁	#1723	1	1	yes	characteristic, complemented, Sylow
64	#1723	—	281	1	no	maximal, Sylow
65	#500	—	281	1	no	—
66	#500	—	281	1	no	—
67	#498	—	281	1	no	—
68	#494	D₂₈₁	1	1	yes	characteristic
69	#487	—	281	1	no	—
70	#487	—	281	1	no	—
71	#322	—	281	1	no	—
72	C₂×Q₁₆	—	2	2	no	—
73	C₄⋊Q₈	—	281	1	no	—
74	C₄⋊Q₈	—	281	1	no	—
75	C₄⋊Q₈	C₂×D₂₈₁	1	1	yes	characteristic
76	C₄⋊Q₈	C₂×D₂₈₁	1	1	yes	characteristic
77	#107	—	562	2	no	—
78	#107	—	562	2	no	—
79	#106	—	562	2	no	—
80	#103	—	562	2	no	—
81	#103	—	562	2	no	—
82	C₄×C₈	—	281	1	no	—
83	C₄×C₈	—	281	1	no	—
84	C₄×C₈	C₂×D₂₈₁	1	1	yes	characteristic
85	C₂×Q₈	—	2	1	no	—
86	C₂×Q₈	—	2	1	no	—
87	C₂×Q₈	—	562	1	no	—
88	C₂×Q₈	—	562	1	no	—
89	Q₁₆	—	4	4	no	—
90	C₂×C₈	—	281	2	no	—
91	C₂×C₈	—	281	2	no	—
92	C₂×C₈	C₂²⋊D₂₈₁	1	2	yes	—
93	C₄⋊C₄	—	2	1	no	—
94	C₄⋊C₄	—	2	1	no	—
95	C₄⋊C₄	—	2	1	no	—
96	C₄⋊C₄	—	2	1	no	—
97	C₄⋊C₄	—	562	1	no	—
98	C₄⋊C₄	—	562	1	no	—
99	C₄⋊C₄	—	562	1	no	—
100	C₄⋊C₄	—	562	1	no	—
101	C₄²	C₂²×D₂₈₁	1	1	yes	characteristic
102	Q₈	—	4	2	no	—
103	Q₈	—	4	2	no	—
104	Q₈	—	1124	2	no	—
105	Q₈	—	1124	2	no	—
106	C₂×C₄	—	4	1	no	—
107	C₂×C₄	—	4	1	no	—
108	C₂×C₄	—	1124	1	no	—
109	C₂×C₄	—	1124	1	no	—
110	C₂×C₄	D₄×D₂₈₁	1	1	yes	characteristic
111	C₂×C₄	D₄×D₂₈₁	1	1	yes	characteristic
112	C₂×C₄	C₂×C₂²⋊D₂₈₁	1	1	yes	characteristic
113	C₈	—	2	2	no	—
114	C₈	—	281	4	no	—
115	C₈	—	562	2	no	—
116	C₂²	#22582875	1	1	yes	characteristic
117	C₄	—	8	1	no	—
118	C₄	—	8	1	no	—
119	C₄	—	2248	1	no	—
120	C₄	—	2248	1	no	—
121	C₄	Q₁₆×D₂₈₁	1	2	yes	—
122	C₄	#22582905	1	2	yes	—
123	C₄	#22582911	1	2	yes	—
124	C₂	#31823841	1	1	yes	characteristic
125	C₂	#31823850	1	1	yes	characteristic
126	C₂	#31823856	1	1	yes	characteristic

subgroup

G/H

class size

multiplicity

normal?

properties

#31823866

C₂

yes

maximal, characteristic

#31823866

C₂

yes

maximal, characteristic

#31823859

C₂

yes

maximal, characteristic

#31823815

C₂

yes

maximal, characteristic

#31823814

C₂

yes

maximal, characteristic

#31823185

C₂

yes

maximal, characteristic

#494×C₂₈₁

C₂

yes

maximal, characteristic

#22582879

C₂²

yes

characteristic

#22582879

C₂²

yes

—

—

—

—

—

C₂²

yes

characteristic

C₄×#14555886

C₂²

yes

characteristic

C₂×Q₁₆×C₂₈₁

—

C₄⋊Q₈×C₂₈₁

C₂²

yes

characteristic

C₄⋊Q₈×C₂₈₁

C₂²

yes

characteristic

C₄×C₈×C₂₈₁

C₂²

yes

characteristic

C₂×Dic₅₆₂

—

C₂×Dic₅₆₂

—

C₂×#14555886

D₄

yes

—

C₂×#14555886

D₄

yes

—

C₄⋊Dic₂₈₁

—

C₄⋊Dic₂₈₁

—

C₄⋊Dic₂₈₁

—

C₄⋊Dic₂₈₁

—

C₂×Q₈×C₂₈₁

—

C₂×Q₈×C₂₈₁

—

Q₁₆×C₂₈₁

—

C₂×C₈×C₂₈₁

D₄

yes

—

C₄⋊C₄×C₂₈₁

—

C₄⋊C₄×C₂₈₁

—

C₄⋊C₄×C₂₈₁

—

C₄⋊C₄×C₂₈₁

—

C₄²×C₂₈₁

C₂³

yes

characteristic

Dic₅₆₂

—

Dic₅₆₂

—

C₂×Dic₂₈₁

—

C₂×Dic₂₈₁

—

#14555886

—

#14555886

Q₁₆

yes

complemented

Q₈×C₂₈₁

—

Q₈×C₂₈₁

—

C₂×C₄×C₂₈₁

—

C₂×C₄×C₂₈₁

—

C₂×C₄×C₂₈₁

C₂×D₄

yes

characteristic

C₂×C₄×C₂₈₁

C₂×D₄

yes

characteristic

C₂×C₄×C₂₈₁

C₂×D₄

yes

characteristic

C₈×C₂₈₁

—

C₂²×C₂₈₁

#127

yes

characteristic

C₄×C₂₈₁

—

C₄×C₂₈₁

—

C₄×C₂₈₁

C₂×Q₁₆

yes

—

C₄×C₂₈₁

AGL(1,ℤ/8)

yes

—

C₄×C₂₈₁

#137

yes

—

Dic₂₈₁

—

Dic₂₈₁

—

C₂×C₂₈₁

yes

C₂×C₂₈₁

yes

C₂×C₂₈₁

yes

C₂₈₁

yes

characteristic, complemented, Sylow

#1723

—

281

maximal, Sylow

#500

—

281

—

#500

—

281

—

#498

—

281

—

#494

D₂₈₁

yes

characteristic

#487

—

281

—

#487

—

281

—

#322

—

281

—

C₂×Q₁₆

—

C₄⋊Q₈

—

281

—

C₄⋊Q₈

—

281

—

C₄⋊Q₈

C₂×D₂₈₁

yes

characteristic

C₄⋊Q₈

C₂×D₂₈₁

yes

characteristic

#107

—

562

—

#107

—

562

—

#106

—

562

—

#103

—

562

—

#103

—

562

—

C₄×C₈

—

281

—

C₄×C₈

—

281

—

C₄×C₈

C₂×D₂₈₁

yes

characteristic

C₂×Q₈

—

C₂×Q₈

—

C₂×Q₈

—

562

—

C₂×Q₈

—

562

—

Q₁₆

—

C₂×C₈

—

281

—

C₂×C₈

—

281

—

C₂×C₈

C₂²⋊D₂₈₁

yes

—

C₄⋊C₄

—

C₄⋊C₄

—

C₄⋊C₄

—

C₄⋊C₄

—

C₄⋊C₄

—

562

—

C₄⋊C₄

—

562

—

C₄⋊C₄

—

562

—

100

C₄⋊C₄

—

562

—

101

C₄²

C₂²×D₂₈₁

yes

characteristic

102

Q₈

—

103

Q₈

—

104

Q₈

—

1124

—

105

Q₈

—

1124

—

106

C₂×C₄

—

107

C₂×C₄

—

108

C₂×C₄

—

1124

—

109

C₂×C₄

—

1124

—

110

C₂×C₄

D₄×D₂₈₁

yes

characteristic

111

C₂×C₄

D₄×D₂₈₁

yes

characteristic

112

C₂×C₄

C₂×C₂²⋊D₂₈₁

yes

characteristic

113

C₈

—

114

C₈

—

281

—

115

C₈

—

562

—

116

C₂²

#22582875

yes

characteristic

117

C₄

—

118

C₄

—

119

C₄

—

2248

—

120

C₄

—

2248

—

121

C₄

Q₁₆×D₂₈₁

yes

—

122

C₄

#22582905

yes

—

123

C₄

#22582911

yes

—

124

C₂

#31823841

yes

characteristic

125

C₂

#31823850

yes

characteristic

126

C₂

#31823856

yes

characteristic

	subgroup	G/H
center	C₂²	#22582875
commutator, cosocle	C₄²×C₂₈₁	C₂³
Frattini	C₄²	C₂²×D₂₈₁
Fitting	#494×C₂₈₁	C₂
socle, prime radical	C₂²×C₂₈₁	#127
hypercenter	#494	D₂₈₁
hypocenter	C₂₈₁	#1723
LCS	G ▷ C₄²×C₂₈₁ ▷ C₂²×C₂₈₁ ▷ C₂₈₁
UCS	C₁ ◁ C₂² ◁ C₄² ◁ #494

subgroup

G/H

center

C₂²

#22582875

commutator, cosocle

C₄²×C₂₈₁

C₂³

Frattini

C₄²

C₂²×D₂₈₁

Fitting

#494×C₂₈₁

C₂

C₂²×C₂₈₁

D₂₈₁

C₂₈₁

G ▷ C₄²×C₂₈₁ ▷ C₂²×C₂₈₁ ▷ C₂₈₁

UCS

C₁ ◁ C₂² ◁ C₄² ◁ #494

N	H ≅ G/N	type
43. #14555886	Q₁₆	semidirect
63. C₂₈₁	#1723	semidirect

H ≅ G/N

type

43. #14555886

Q₁₆

semidirect

63. C₂₈₁

#1723

semidirect

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

subgroup	index
#31823866	2
#31823859	2
#31823815	2
#31823814	2
#31823185	2
#494×C₂₈₁	2
#1723	281

subgroup

index

#494×C₂₈₁

281

Subgroups — #49094978

Conjugacy classes of subgroups

Complements

Maximal subgroups