Group #49096276 — subgroups

Subgroups — #49096276

Order 35968.

276 conjugacy classes of subgroups; only the 109 normal subgroups are shown.

normal subgroups only

This group has 276 conjugacy classes of subgroups — too many to draw. Only its 109 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	C₂×#22582879	C₂	1	1	yes	maximal, characteristic
2	#31823329	C₂	1	1	yes	maximal, characteristic
3	#31823226	C₂	1	4	yes	maximal
4	C₂×#97×C₂₈₁	C₂	1	1	yes	maximal, characteristic
5	C₂²×Dic₅₆₂	—	2	1	no	—
6	#22582879	—	2	2	no	—
7	#22582879	C₄	1	4	yes	complemented
8	C₂×C₄⋊Dic₂₈₁	—	2	1	no	—
9	C₂×C₄⋊Dic₂₈₁	C₂²	1	2	yes	—
10	C₂×C₄⋊Dic₂₈₁	C₂²	1	2	yes	—
11	#22582786	—	2	1	no	—
12	#22582786	—	2	1	no	—
13	C₂²×C₈×C₂₈₁	C₂²	1	2	yes	—
14	C₂×C₄²×C₂₈₁	C₂²	1	1	yes	characteristic
15	#97×C₂₈₁	—	2	2	no	—
16	C₂×Dic₅₆₂	—	2	4	no	—
17	C₂×Dic₅₆₂	—	4	4	no	—
18	C₂²×Dic₂₈₁	—	2	2	no	—
19	C₂²×Dic₂₈₁	—	2	2	no	—
20	C₄⋊Dic₂₈₁	—	2	4	no	—
21	C₄⋊Dic₂₈₁	—	2	2	no	—
22	C₄⋊Dic₂₈₁	—	2	4	no	—
23	C₄⋊Dic₂₈₁	C₂×C₄	1	4	yes	—
24	C₂²×C₄×C₂₈₁	D₄	1	1	yes	characteristic
25	C₂²×C₄×C₂₈₁	D₄	1	1	yes	characteristic
26	C₂²×C₄×C₂₈₁	C₂³	1	1	yes	characteristic
27	C₂×C₈×C₂₈₁	—	2	2	no	—
28	C₂×C₈×C₂₈₁	—	2	2	no	—
29	C₂×C₈×C₂₈₁	Q₈	1	4	yes	complemented
30	C₄²×C₂₈₁	C₂×C₄	1	2	yes	—
31	C₄²×C₂₈₁	D₄	1	2	yes	—
32	Dic₅₆₂	—	4	8	no	—
33	C₂×Dic₂₈₁	—	2	4	no	—
34	C₂×Dic₂₈₁	—	4	4	no	—
35	C₂×Dic₂₈₁	—	4	2	no	—
36	C₂×Dic₂₈₁	—	4	2	no	—
37	C₂×Dic₂₈₁	—	4	2	no	—
38	C₂³×C₂₈₁	C₂×D₄	1	1	yes	characteristic
39	C₂×C₄×C₂₈₁	—	2	2	no	—
40	C₂×C₄×C₂₈₁	—	2	2	no	—
41	C₂×C₄×C₂₈₁	C₂²⋊C₄	1	2	yes	—
42	C₂×C₄×C₂₈₁	C₂²⋊C₄	1	2	yes	—
43	C₂×C₄×C₂₈₁	C₂²×C₄	1	1	yes	characteristic
44	C₂×C₄×C₂₈₁	C₂×D₄	1	1	yes	characteristic
45	C₂×C₄×C₂₈₁	C₂×Q₈	1	2	yes	—
46	C₂×C₄×C₂₈₁	#41	1	2	yes	—
47	C₈×C₂₈₁	—	2	4	no	—
48	C₂²×C₂₈₁	C₂×C₂²⋊C₄	1	1	yes	characteristic
49	C₂²×C₂₈₁	#124	1	1	yes	characteristic
50	C₂²×C₂₈₁	C₄⋊Q₈	1	1	yes	characteristic
51	C₂²×C₂₈₁	AGL(1,ℤ/8)	1	2	yes	—
52	C₂²×C₂₈₁	#137	1	2	yes	—
53	C₄×C₂₈₁	—	2	2	no	—
54	C₄×C₂₈₁	—	2	2	no	—
55	C₄×C₂₈₁	C₄×Q₈	1	2	yes	—
56	C₄×C₂₈₁	C₂²⋊Q₈	1	2	yes	—
57	Dic₂₈₁	—	4	4	no	—
58	Dic₂₈₁	—	8	2	no	—
59	C₂×C₂₈₁	#391	1	1	yes	characteristic
60	C₂×C₂₈₁	#418	1	1	yes	characteristic
61	C₂×C₂₈₁	#419	1	1	yes	characteristic
62	C₂×C₂₈₁	#489	1	2	yes	—
63	C₂×C₂₈₁	#501	1	2	yes	—
64	C₂₈₁	#1993	1	1	yes	characteristic, complemented, Sylow
65	#1993	—	281	1	no	maximal, Sylow
66	C₂×C₄⋊Q₈	—	281	1	no	—
67	C₂×#97	D₂₈₁	1	1	yes	characteristic
68	#382	—	281	1	no	—
69	#340	—	281	4	no	—
70	C₂²×Q₈	—	562	1	no	—
71	C₂²×C₈	C₂×D₂₈₁	1	2	yes	—
72	C₄⋊Q₈	—	281	4	no	—
73	C₄⋊Q₈	—	562	2	no	—
74	C₂×C₄⋊C₄	—	281	2	no	—
75	C₂×C₄⋊C₄	—	281	2	no	—
76	C₂×C₄⋊C₄	—	562	1	no	—
77	C₂×C₄²	C₂×D₂₈₁	1	1	yes	characteristic
78	#97	—	2	2	no	—
79	#95	—	562	1	no	—
80	#95	—	562	1	no	—
81	C₂×Q₈	—	562	4	no	—
82	C₂×Q₈	—	1124	4	no	—
83	C₂²×C₄	—	562	2	no	—
84	C₂²×C₄	—	562	2	no	—
85	C₂²×C₄	D₁₁₂₄	1	1	yes	characteristic
86	C₂²×C₄	D₁₁₂₄	1	1	yes	characteristic
87	C₂²×C₄	C₂²×D₂₈₁	1	1	yes	characteristic
88	C₂×C₈	—	2	2	no	—
89	C₂×C₈	—	2	2	no	—
90	C₂×C₈	Dic₅₆₂	1	4	yes	complemented
91	C₄⋊C₄	—	281	4	no	—
92	C₄⋊C₄	—	562	4	no	—
93	C₄⋊C₄	—	562	2	no	—
94	C₄⋊C₄	—	562	4	no	—
95	C₄²	C₄×D₂₈₁	1	2	yes	—
96	C₄²	C₂²⋊D₂₈₁	1	2	yes	—
97	C₂³	C₂×D₁₁₂₄	1	1	yes	characteristic
98	Q₈	—	1124	8	no	—
99	C₂×C₄	—	2	2	no	—
100	C₂×C₄	—	2	2	no	—
101	C₂×C₄	—	562	4	no	—
102	C₂×C₄	—	1124	4	no	—
103	C₂×C₄	—	1124	2	no	—
104	C₂×C₄	—	1124	2	no	—
105	C₂×C₄	—	1124	2	no	—
106	C₂×C₄	#17985115	1	2	yes	—
107	C₂×C₄	#17985115	1	2	yes	—
108	C₂×C₄	C₂×C₄×D₂₈₁	1	1	yes	characteristic
109	C₂×C₄	C₂×C₂²⋊D₂₈₁	1	1	yes	characteristic
110	C₂×C₄	C₂×Dic₅₆₂	1	2	yes	—
111	C₂×C₄	#17985138	1	2	yes	—
112	C₈	—	2	4	no	—
113	C₂²	C₂×#17985115	1	1	yes	characteristic
114	C₂²	#22582867	1	1	yes	characteristic
115	C₂²	#22582879	1	1	yes	characteristic
116	C₂²	#22582907	1	2	yes	—
117	C₂²	#22582912	1	2	yes	—
118	C₄	—	2	2	no	—
119	C₄	—	2	2	no	—
120	C₄	—	1124	4	no	—
121	C₄	—	2248	2	no	—
122	C₄	C₄×Dic₅₆₂	1	2	yes	—
123	C₄	#22582858	1	2	yes	—
124	C₂	#31823369	1	1	yes	characteristic
125	C₂	#31823469	1	1	yes	characteristic
126	C₂	#31823473	1	1	yes	characteristic
127	C₂	#31823823	1	2	yes	—
128	C₂	#31823872	1	2	yes	—

subgroup

G/H

class size

multiplicity

normal?

properties

C₂×#22582879

C₂

yes

maximal, characteristic

#31823329

C₂

yes

maximal, characteristic

#31823226

C₂

yes

maximal

C₂×#97×C₂₈₁

C₂

yes

maximal, characteristic

C₂²×Dic₅₆₂

—

#22582879

—

#22582879

C₄

yes

complemented

C₂×C₄⋊Dic₂₈₁

—

C₂×C₄⋊Dic₂₈₁

C₂²

yes

—

C₂×C₄⋊Dic₂₈₁

C₂²

yes

—

#22582786

—

#22582786

—

C₂²×C₈×C₂₈₁

C₂²

yes

—

C₂×C₄²×C₂₈₁

C₂²

yes

characteristic

#97×C₂₈₁

—

C₂×Dic₅₆₂

—

C₂×Dic₅₆₂

—

C₂²×Dic₂₈₁

—

C₂²×Dic₂₈₁

—

C₄⋊Dic₂₈₁

—

C₄⋊Dic₂₈₁

—

C₄⋊Dic₂₈₁

—

C₄⋊Dic₂₈₁

C₂×C₄

yes

—

C₂²×C₄×C₂₈₁

D₄

yes

characteristic

C₂²×C₄×C₂₈₁

D₄

yes

characteristic

C₂²×C₄×C₂₈₁

C₂³

yes

characteristic

C₂×C₈×C₂₈₁

—

C₂×C₈×C₂₈₁

—

C₂×C₈×C₂₈₁

Q₈

yes

complemented

C₄²×C₂₈₁

C₂×C₄

yes

—

C₄²×C₂₈₁

D₄

yes

—

Dic₅₆₂

—

C₂×Dic₂₈₁

—

C₂×Dic₂₈₁

—

C₂×Dic₂₈₁

—

C₂×Dic₂₈₁

—

C₂×Dic₂₈₁

—

C₂³×C₂₈₁

C₂×D₄

yes

characteristic

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

—

C₈×C₂₈₁

—

C₂²×C₂₈₁

C₂×C₂²⋊C₄

yes

characteristic

C₂²×C₂₈₁

#124

yes

characteristic

C₂²×C₂₈₁

C₄⋊Q₈

yes

characteristic

C₂²×C₂₈₁

AGL(1,ℤ/8)

yes

—

C₂²×C₂₈₁

#137

yes

—

C₄×C₂₈₁

—

C₄×C₂₈₁

—

C₄×C₂₈₁

C₄×Q₈

yes

—

C₄×C₂₈₁

C₂²⋊Q₈

yes

—

Dic₂₈₁

—

Dic₂₈₁

—

C₂×C₂₈₁

yes

C₂×C₂₈₁

yes

C₂×C₂₈₁

yes

C₂×C₂₈₁

yes

—

C₂×C₂₈₁

#501

yes

—

C₂₈₁

#1993

yes

characteristic, complemented, Sylow

#1993

—

281

maximal, Sylow

C₂×C₄⋊Q₈

—

281

—

C₂×#97

D₂₈₁

yes

characteristic

#382

—

281

—

#340

—

281

—

C₂²×Q₈

—

562

—

C₂²×C₈

C₂×D₂₈₁

yes

—

C₄⋊Q₈

—

281

—

C₄⋊Q₈

—

562

—

C₂×C₄⋊C₄

—

281

—

C₂×C₄⋊C₄

—

281

—

C₂×C₄⋊C₄

—

562

—

C₂×C₄²

C₂×D₂₈₁

yes

characteristic

#97

—

#95

—

562

—

#95

—

562

—

C₂×Q₈

—

562

—

C₂×Q₈

—

1124

—

C₂²×C₄

—

562

—

C₂²×C₄

—

562

—

C₂²×C₄

D₁₁₂₄

yes

characteristic

C₂²×C₄

D₁₁₂₄

yes

characteristic

C₂²×C₄

C₂²×D₂₈₁

yes

characteristic

C₂×C₈

—

C₂×C₈

—

C₂×C₈

Dic₅₆₂

yes

complemented

C₄⋊C₄

—

281

—

C₄⋊C₄

—

562

—

C₄⋊C₄

—

562

—

C₄⋊C₄

—

562

—

C₄²

C₄×D₂₈₁

yes

—

C₄²

C₂²⋊D₂₈₁

yes

—

C₂³

C₂×D₁₁₂₄

yes

characteristic

Q₈

—

1124

—

C₂×C₄

—

100

C₂×C₄

—

101

C₂×C₄

—

562

—

102

C₂×C₄

—

1124

—

103

C₂×C₄

—

1124

—

104

C₂×C₄

—

1124

—

105

C₂×C₄

—

1124

—

106

C₂×C₄

#17985115

yes

—

107

C₂×C₄

#17985115

yes

—

108

C₂×C₄

C₂×C₄×D₂₈₁

yes

characteristic

109

C₂×C₄

C₂×C₂²⋊D₂₈₁

yes

characteristic

110

C₂×C₄

C₂×Dic₅₆₂

yes

—

111

C₂×C₄

#17985138

yes

—

112

C₈

—

113

C₂²

C₂×#17985115

yes

characteristic

114

C₂²

#22582867

yes

characteristic

115

C₂²

#22582879

yes

characteristic

116

C₂²

#22582907

yes

—

117

C₂²

#22582912

yes

—

118

C₄

—

119

C₄

—

120

C₄

—

1124

—

121

C₄

—

2248

—

122

C₄

C₄×Dic₅₆₂

yes

—

123

C₄

#22582858

yes

—

124

C₂

#31823369

yes

characteristic

125

C₂

#31823469

yes

characteristic

126

C₂

#31823473

yes

characteristic

127

C₂

#31823823

yes

—

128

C₂

#31823872

yes

—

	subgroup	G/H
center	C₂³	C₂×D₁₁₂₄
commutator	C₂×C₄×C₂₈₁	C₂²×C₄
Frattini	C₂²×C₄	C₂²×D₂₈₁
Fitting	C₂×#97×C₂₈₁	C₂
socle, prime radical	C₂³×C₂₈₁	C₂×D₄
cosocle	C₂²×C₄×C₂₈₁	C₂³
hypercenter	C₂×#97	D₂₈₁
hypocenter	C₂₈₁	#1993
LCS	G ▷ C₂×C₄×C₂₈₁ ▷ C₂×C₂₈₁ ▷ C₂₈₁
UCS	C₁ ◁ C₂³ ◁ C₂×C₄² ◁ C₂×#97

subgroup

G/H

center

C₂³

C₂×D₁₁₂₄

commutator

C₂×C₄×C₂₈₁

C₂²×C₄

Frattini

C₂²×C₄

C₂²×D₂₈₁

Fitting

C₂×#97×C₂₈₁

C₂

socle, prime radical

C₂³×C₂₈₁

C₂×D₄

cosocle

C₂²×C₄×C₂₈₁

C₂³

hypercenter

C₂×#97

D₂₈₁

hypocenter

C₂₈₁

#1993

LCS

G ▷ C₂×C₄×C₂₈₁ ▷ C₂×C₂₈₁ ▷ C₂₈₁

UCS

C₁ ◁ C₂³ ◁ C₂×C₄² ◁ C₂×#97

N	H ≅ G/N	type
7. #22582879	C₄	semidirect
29. C₂×C₈×C₂₈₁	Q₈	semidirect
64. C₂₈₁	#1993	semidirect
90. C₂×C₈	Dic₅₆₂	semidirect

H ≅ G/N

type

7. #22582879

C₄

semidirect

29. C₂×C₈×C₂₈₁

Q₈

semidirect

64. C₂₈₁

#1993

semidirect

90. C₂×C₈

Dic₅₆₂

semidirect

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

subgroup	index
C₂×#22582879	2
#31823329	2
#31823226	2
C₂×#97×C₂₈₁	2
#1993	281

subgroup

index

C₂×#97×C₂₈₁

281

Subgroups — #49096276

Conjugacy classes of subgroups

Complements

Maximal subgroups