Group #45667500 — subgroups

Subgroups — #45667500

Order 32896.

424 conjugacy classes of subgroups; only the 173 normal subgroups are shown.

normal subgroups only

This group has 424 conjugacy classes of subgroups — too many to draw. Only its 173 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×#22551592	C₂	1	1	yes	maximal, characteristic
2	#30656641	C₂	1	4	yes	maximal
3	#30656622	C₂	1	4	yes	maximal
4	#30656620	C₂	1	4	yes	maximal
5	#30656605	C₂	1	1	yes	maximal, characteristic
6	#383×C₂₅₇	C₂	1	1	yes	maximal, characteristic
7	C₂²×Dic₅₁₄	C₂²	1	2	yes	—
8	#22551592	C₄	1	8	yes	complemented
9	C₂×C₄⋊Dic₂₅₇	C₂²	1	2	yes	—
10	C₂×C₄⋊Dic₂₅₇	C₂²	1	2	yes	—
11	C₂×#16861597	C₂²	1	4	yes	—
12	C₂×C₄×Dic₂₅₇	C₂²	1	4	yes	—
13	#22551499	C₂²	1	2	yes	—
14	#22551499	C₂²	1	2	yes	—
15	#22551499	C₂²	1	2	yes	—
16	#22551499	C₂²	1	8	yes	—
17	C₂×C₄²×C₂₅₇	C₂²	1	1	yes	characteristic
18	#95×C₂₅₇	C₂²	1	2	yes	—
19	#95×C₂₅₇	C₂²	1	2	yes	—
20	#95×C₂₅₇	C₂²	1	2	yes	—
21	C₂×Dic₅₁₄	—	4	2	no	—
22	C₂×Dic₅₁₄	—	4	2	no	—
23	C₂×Dic₅₁₄	C₂×C₄	1	8	yes	—
24	C₂²×Dic₂₅₇	C₂³	1	4	yes	—
25	C₂²×Dic₂₅₇	C₂³	1	4	yes	—
26	C₄⋊Dic₂₅₇	C₂×C₄	1	8	yes	—
27	C₄⋊Dic₂₅₇	C₂×C₄	1	8	yes	—
28	#16861597	—	4	4	no	—
29	C₄×Dic₂₅₇	—	4	4	no	—
30	C₂²×C₄×C₂₅₇	C₂³	1	4	yes	—
31	C₂²×C₄×C₂₅₇	C₂³	1	1	yes	characteristic
32	C₂²×C₄×C₂₅₇	C₂³	1	1	yes	characteristic
33	C₂²×C₄×C₂₅₇	C₂³	1	1	yes	characteristic
34	C₄²×C₂₅₇	C₂×C₄	1	4	yes	—
35	Dic₅₁₄	—	4	8	no	—
36	C₂×Dic₂₅₇	—	2	4	no	—
37	C₂×Dic₂₅₇	—	2	4	no	—
38	C₂×Dic₂₅₇	—	2	4	no	—
39	C₂×Dic₂₅₇	—	2	4	no	—
40	C₂×Dic₂₅₇	—	2	4	no	—
41	C₂×Dic₂₅₇	C₂²×C₄	1	8	yes	—
42	C₂³×C₂₅₇	C₂⁴	1	1	yes	characteristic
43	C₂×C₄×C₂₅₇	—	2	2	no	—
44	C₂×C₄×C₂₅₇	—	2	2	no	—
45	C₂×C₄×C₂₅₇	—	2	2	no	—
46	C₂×C₄×C₂₅₇	—	2	4	no	—
47	C₂×C₄×C₂₅₇	—	2	4	no	—
48	C₂×C₄×C₂₅₇	—	2	4	no	—
49	C₂×C₄×C₂₅₇	C₂²×C₄	1	2	yes	—
50	C₂×C₄×C₂₅₇	C₂²×C₄	1	2	yes	—
51	C₂×C₄×C₂₅₇	C₂²×C₄	1	2	yes	—
52	C₂²×C₂₅₇	C₂³×C₄	1	1	yes	characteristic
53	C₂²×C₂₅₇	H(5,𝔽₂)	1	2	yes	—
54	C₂²×C₂₅₇	2₋¹⁺⁴	1	2	yes	—
55	C₂²×C₂₅₇	2₋¹⁺⁴	1	2	yes	—
56	C₄×C₂₅₇	—	2	2	no	—
57	C₄×C₂₅₇	—	2	2	no	—
58	C₄×C₂₅₇	—	2	2	no	—
59	C₄×C₂₅₇	—	4	4	no	—
60	Dic₂₅₇	—	2	8	no	—
61	Dic₂₅₇	—	4	4	no	—
62	C₂×C₂₅₇	#518	1	1	yes	characteristic
63	C₂×C₂₅₇	#519	1	1	yes	characteristic
64	C₂×C₂₅₇	#519	1	1	yes	characteristic
65	C₂×C₂₅₇	#563	1	4	yes	—
66	C₂₅₇	#2381	1	1	yes	characteristic, complemented, Sylow
67	#2381	—	257	1	no	maximal, Sylow
68	C₂×C₄⋊Q₈	—	257	1	no	—
69	#391	—	257	4	no	—
70	#387	—	257	4	no	—
71	#387	—	257	4	no	—
72	#383	—	257	1	no	—
73	#383	D₂₅₇	1	1	yes	characteristic
74	C₂²×Q₈	—	257	2	no	—
75	C₄⋊Q₈	—	257	8	no	—
76	C₂×C₄⋊C₄	—	257	2	no	—
77	C₂×C₄⋊C₄	—	257	2	no	—
78	C₂×C₄⋊C₄	—	257	4	no	—
79	C₂×C₄²	—	257	4	no	—
80	C₂×C₄²	C₂×D₂₅₇	1	1	yes	characteristic
81	#95	—	257	2	no	—
82	#95	—	257	2	no	—
83	#95	—	257	2	no	—
84	#95	—	257	8	no	—
85	#95	C₂×D₂₅₇	1	2	yes	—
86	#95	C₂×D₂₅₇	1	2	yes	—
87	#95	C₂×D₂₅₇	1	2	yes	—
88	C₂×Q₈	—	257	8	no	—
89	C₂×Q₈	—	1028	2	no	—
90	C₂×Q₈	—	1028	2	no	—
91	C₂²×C₄	—	257	4	no	—
92	C₂²×C₄	—	257	4	no	—
93	C₂²×C₄	C₂²×D₂₅₇	1	4	yes	—
94	C₂²×C₄	C₂²×D₂₅₇	1	1	yes	characteristic
95	C₂²×C₄	C₂²×D₂₅₇	1	1	yes	characteristic
96	C₂²×C₄	C₂²×D₂₅₇	1	1	yes	characteristic
97	C₄⋊C₄	—	257	8	no	—
98	C₄⋊C₄	—	257	8	no	—
99	C₄⋊C₄	—	1028	4	no	—
100	C₄²	—	1028	4	no	—
101	C₄²	C₄×D₂₅₇	1	4	yes	—
102	C₂³	C₂³×D₂₅₇	1	1	yes	characteristic
103	Q₈	—	1028	8	no	—
104	C₂×C₄	—	257	8	no	—
105	C₂×C₄	—	2	2	no	—
106	C₂×C₄	—	2	2	no	—
107	C₂×C₄	—	2	2	no	—
108	C₂×C₄	—	2	4	no	—
109	C₂×C₄	—	2	4	no	—
110	C₂×C₄	—	2	4	no	—
111	C₂×C₄	—	514	4	no	—
112	C₂×C₄	—	514	4	no	—
113	C₂×C₄	—	514	4	no	—
114	C₂×C₄	—	514	4	no	—
115	C₂×C₄	—	514	4	no	—
116	C₂×C₄	C₂×C₄×D₂₅₇	1	2	yes	—
117	C₂×C₄	C₂×C₄×D₂₅₇	1	2	yes	—
118	C₂×C₄	C₂×C₄×D₂₅₇	1	2	yes	—
119	C₂²	C₂²×C₄×D₂₅₇	1	1	yes	characteristic
120	C₂²	#22551638	1	2	yes	—
121	C₂²	#22551639	1	2	yes	—
122	C₂²	#22551639	1	2	yes	—
123	C₄	—	2	2	no	—
124	C₄	—	2	2	no	—
125	C₄	—	2	2	no	—
126	C₄	—	514	8	no	—
127	C₄	—	4	4	no	—
128	C₄	—	1028	4	no	—
129	C₂	#30657211	1	1	yes	characteristic
130	C₂	#30657214	1	1	yes	characteristic
131	C₂	#30657214	1	1	yes	characteristic
132	C₂	#30657489	1	4	yes	—

subgroup

G/H

class size

multiplicity

normal?

properties

C₂×#22551592

C₂

yes

maximal, characteristic

#30656641

C₂

yes

maximal

#30656622

C₂

yes

maximal

#30656620

C₂

yes

maximal

#30656605

C₂

yes

maximal, characteristic

#383×C₂₅₇

C₂

yes

maximal, characteristic

C₂²×Dic₅₁₄

C₂²

yes

—

#22551592

C₄

yes

complemented

C₂×C₄⋊Dic₂₅₇

C₂²

yes

—

C₂×C₄⋊Dic₂₅₇

C₂²

yes

—

C₂×#16861597

C₂²

yes

—

C₂×C₄×Dic₂₅₇

C₂²

yes

—

#22551499

C₂²

yes

—

#22551499

C₂²

yes

—

#22551499

C₂²

yes

—

#22551499

C₂²

yes

—

C₂×C₄²×C₂₅₇

C₂²

yes

characteristic

#95×C₂₅₇

C₂²

yes

—

#95×C₂₅₇

C₂²

yes

—

#95×C₂₅₇

C₂²

yes

—

C₂×Dic₅₁₄

—

C₂×Dic₅₁₄

—

C₂×Dic₅₁₄

C₂×C₄

yes

—

C₂²×Dic₂₅₇

C₂³

yes

—

C₂²×Dic₂₅₇

C₂³

yes

—

C₄⋊Dic₂₅₇

C₂×C₄

yes

—

C₄⋊Dic₂₅₇

C₂×C₄

yes

—

#16861597

—

C₄×Dic₂₅₇

—

C₂²×C₄×C₂₅₇

C₂³

yes

—

C₂²×C₄×C₂₅₇

C₂³

yes

characteristic

C₂²×C₄×C₂₅₇

C₂³

yes

characteristic

C₂²×C₄×C₂₅₇

C₂³

yes

characteristic

C₄²×C₂₅₇

C₂×C₄

yes

—

Dic₅₁₄

—

C₂×Dic₂₅₇

—

C₂×Dic₂₅₇

—

C₂×Dic₂₅₇

—

C₂×Dic₂₅₇

—

C₂×Dic₂₅₇

—

C₂×Dic₂₅₇

C₂²×C₄

yes

—

C₂³×C₂₅₇

C₂⁴

yes

characteristic

C₂×C₄×C₂₅₇

—

C₂×C₄×C₂₅₇

—

C₂×C₄×C₂₅₇

—

C₂×C₄×C₂₅₇

—

C₂×C₄×C₂₅₇

—

C₂×C₄×C₂₅₇

—

C₂×C₄×C₂₅₇

C₂²×C₄

yes

—

C₂×C₄×C₂₅₇

C₂²×C₄

yes

—

C₂×C₄×C₂₅₇

C₂²×C₄

yes

—

C₂²×C₂₅₇

C₂³×C₄

yes

characteristic

C₂²×C₂₅₇

H(5,𝔽₂)

yes

—

C₂²×C₂₅₇

2₋¹⁺⁴

yes

—

C₂²×C₂₅₇

2₋¹⁺⁴

yes

—

C₄×C₂₅₇

—

C₄×C₂₅₇

—

C₄×C₂₅₇

—

C₄×C₂₅₇

—

Dic₂₅₇

—

Dic₂₅₇

—

C₂×C₂₅₇

yes

C₂×C₂₅₇

yes

C₂×C₂₅₇

yes

C₂×C₂₅₇

yes

—

C₂₅₇

#2381

yes

characteristic, complemented, Sylow

#2381

—

257

maximal, Sylow

C₂×C₄⋊Q₈

—

257

—

#391

—

257

—

#387

—

257

—

#387

—

257

—

#383

—

257

—

#383

D₂₅₇

yes

characteristic

C₂²×Q₈

—

257

—

C₄⋊Q₈

—

257

—

C₂×C₄⋊C₄

—

257

—

C₂×C₄⋊C₄

—

257

—

C₂×C₄⋊C₄

—

257

—

C₂×C₄²

—

257

—

C₂×C₄²

C₂×D₂₅₇

yes

characteristic

#95

—

257

—

#95

—

257

—

#95

—

257

—

#95

—

257

—

#95

C₂×D₂₅₇

yes

—

#95

C₂×D₂₅₇

yes

—

#95

C₂×D₂₅₇

yes

—

C₂×Q₈

—

257

—

C₂×Q₈

—

1028

—

C₂×Q₈

—

1028

—

C₂²×C₄

—

257

—

C₂²×C₄

—

257

—

C₂²×C₄

C₂²×D₂₅₇

yes

—

C₂²×C₄

C₂²×D₂₅₇

yes

characteristic

C₂²×C₄

C₂²×D₂₅₇

yes

characteristic

C₂²×C₄

C₂²×D₂₅₇

yes

characteristic

C₄⋊C₄

—

257

—

C₄⋊C₄

—

257

—

C₄⋊C₄

—

1028

—

100

C₄²

—

1028

—

101

C₄²

C₄×D₂₅₇

yes

—

102

C₂³

C₂³×D₂₅₇

yes

characteristic

103

Q₈

—

1028

—

104

C₂×C₄

—

257

—

105

C₂×C₄

—

106

C₂×C₄

—

107

C₂×C₄

—

108

C₂×C₄

—

109

C₂×C₄

—

110

C₂×C₄

—

111

C₂×C₄

—

514

—

112

C₂×C₄

—

514

—

113

C₂×C₄

—

514

—

114

C₂×C₄

—

514

—

115

C₂×C₄

—

514

—

116

C₂×C₄

C₂×C₄×D₂₅₇

yes

—

117

C₂×C₄

C₂×C₄×D₂₅₇

yes

—

118

C₂×C₄

C₂×C₄×D₂₅₇

yes

—

119

C₂²

C₂²×C₄×D₂₅₇

yes

characteristic

120

C₂²

#22551638

yes

—

121

C₂²

#22551639

yes

—

122

C₂²

#22551639

yes

—

123

C₄

—

124

C₄

—

125

C₄

—

126

C₄

—

514

—

127

C₄

—

128

C₄

—

1028

—

129

C₂

#30657211

yes

characteristic

130

C₂

#30657214

yes

characteristic

131

C₂

#30657214

yes

characteristic

132

C₂

#30657489

yes

—

	subgroup	G/H
center, Frattini	C₂³	C₂³×D₂₅₇
commutator	C₂²×C₂₅₇	C₂³×C₄
Fitting	#383×C₂₅₇	C₂
socle, prime radical, cosocle	C₂³×C₂₅₇	C₂⁴
hypercenter	#383	D₂₅₇
hypocenter	C₂₅₇	#2381
LCS	G ▷ C₂²×C₂₅₇ ▷ C₂₅₇
UCS	C₁ ◁ C₂³ ◁ #383

subgroup

G/H

center, Frattini

C₂³

C₂³×D₂₅₇

commutator

C₂²×C₂₅₇

C₂³×C₄

Fitting

#383×C₂₅₇

C₂

socle, prime radical, cosocle

C₂³×C₂₅₇

C₂⁴

D₂₅₇

C₂₅₇

G ▷ C₂²×C₂₅₇ ▷ C₂₅₇

C₁ ◁ C₂³ ◁ #383

N	H ≅ G/N	type
8. #22551592	C₄	semidirect
66. C₂₅₇	#2381	semidirect

H ≅ G/N

type

8. #22551592

C₄

semidirect

66. C₂₅₇

#2381

semidirect

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

subgroup	index
C₂×#22551592	2
#30656641	2
#30656622	2
#30656620	2
#30656605	2
#383×C₂₅₇	2
#2381	257

subgroup

index

#383×C₂₅₇

257

Subgroups — #45667500

Conjugacy classes of subgroups

Complements

Maximal subgroups