Group #45665163 — subgroups

Subgroups — #45665163

Order 32896.

292 conjugacy classes of subgroups; only the 97 normal subgroups are shown.

normal subgroups only

This group has 292 conjugacy classes of subgroups — too many to draw. Only its 97 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	C₂×#22551510	C₂	1	4	yes	maximal
3	C₄×#16861602	C₂	1	1	yes	maximal, characteristic
4	#423×C₂₅₇	C₂	1	1	yes	maximal, characteristic
5	C₂²×Dic₅₁₄	C₂²	1	2	yes	—
6	C₂×#16861602	C₂²	1	2	yes	—
7	#22551592	—	2	2	no	—
8	#22551592	—	2	1	no	—
9	#22551592	—	2	1	no	—
10	C₂×C₄⋊Dic₂₅₇	C₄	1	4	yes	—
11	#22551510	—	2	8	no	—
12	#22551503	—	2	1	no	—
13	C₄×#14523339	—	2	1	no	—
14	C₂×M₁₆×C₂₅₇	C₂²	1	2	yes	—
15	C₂×C₄²×C₂₅₇	C₂²	1	1	yes	characteristic
16	C₄⋊C₈×C₂₅₇	—	2	2	no	—
17	C₂×Dic₅₁₄	—	2	2	no	—
18	C₂×Dic₅₁₄	—	2	2	no	—
19	C₂×Dic₅₁₄	—	2	4	no	—
20	C₂×Dic₅₁₄	—	2	4	no	—
21	C₂²×Dic₂₅₇	C₂×C₄	1	4	yes	—
22	#16861602	—	2	2	no	—
23	#16861602	D₄	1	4	yes	—
24	C₂×#14523339	—	2	2	no	—
25	C₄⋊Dic₂₅₇	—	2	4	no	—
26	C₄⋊Dic₂₅₇	—	2	4	no	—
27	C₂²×C₄×C₂₅₇	C₂×C₄	1	2	yes	—
28	C₂²×C₄×C₂₅₇	C₂³	1	1	yes	characteristic
29	M₁₆×C₂₅₇	—	2	4	no	—
30	C₂×C₈×C₂₅₇	—	2	2	no	—
31	C₄²×C₂₅₇	D₄	1	2	yes	—
32	C₄²×C₂₅₇	D₄	1	1	yes	characteristic
33	C₄²×C₂₅₇	D₄	1	1	yes	characteristic
34	Dic₅₁₄	—	4	4	no	—
35	Dic₅₁₄	—	4	2	no	—
36	Dic₅₁₄	—	4	2	no	—
37	C₂×Dic₂₅₇	—	2	4	no	—
38	C₂×Dic₂₅₇	—	2	8	no	—
39	#14523339	—	2	4	no	—
40	C₂³×C₂₅₇	C₂²×C₄	1	1	yes	characteristic
41	C₂×C₄×C₂₅₇	—	2	4	no	—
42	C₂×C₄×C₂₅₇	C₂²⋊C₄	1	2	yes	—
43	C₂×C₄×C₂₅₇	C₂²⋊C₄	1	2	yes	—
44	C₂×C₄×C₂₅₇	C₂×D₄	1	2	yes	—
45	C₂×C₄×C₂₅₇	C₂×D₄	1	1	yes	characteristic
46	C₂×C₄×C₂₅₇	C₂×D₄	1	1	yes	characteristic
47	C₂×C₄×C₂₅₇	#41	1	2	yes	—
48	C₈×C₂₅₇	—	4	2	no	—
49	C₂²×C₂₅₇	—	2	2	no	—
50	C₂²×C₂₅₇	C₂×C₂²⋊C₄	1	1	yes	characteristic
51	C₂²×C₂₅₇	C₄×D₄	1	2	yes	—
52	C₄×C₂₅₇	—	2	2	no	—
53	C₄×C₂₅₇	#101	1	4	yes	complemented
54	C₄×C₂₅₇	#121	1	2	yes	—
55	C₄×C₂₅₇	#124	1	1	yes	characteristic
56	C₄×C₂₅₇	#127	1	1	yes	characteristic
57	Dic₂₅₇	—	4	4	no	—
58	C₂×C₂₅₇	—	2	2	no	—
59	C₂×C₂₅₇	#390	1	1	yes	characteristic
60	C₂×C₂₅₇	C₂×#101	1	2	yes	—
61	C₂₅₇	#1966	1	1	yes	characteristic, complemented, Sylow
62	#1966	—	257	1	no	maximal, Sylow
63	C₂×C₄⋊Q₈	—	257	1	no	—
64	#423	D₂₅₇	1	1	yes	characteristic
65	C₂×#101	—	257	4	no	—
66	C₄×M₁₆	—	257	1	no	—
67	C₂²×Q₈	—	257	2	no	—
68	C₂×M₁₆	—	257	2	no	—
69	C₂×M₁₆	C₂×D₂₅₇	1	2	yes	—
70	C₄⋊Q₈	—	514	2	no	—
71	C₄⋊Q₈	—	514	1	no	—
72	C₄⋊Q₈	—	514	1	no	—
73	C₂×C₄⋊C₄	—	257	4	no	—
74	C₂×C₄²	C₂×D₂₅₇	1	1	yes	characteristic
75	C₄⋊C₈	—	2	2	no	—
76	#101	—	514	8	no	—
77	#97	—	514	1	no	—
78	C₄×C₈	—	514	1	no	—
79	C₂×Q₈	—	514	4	no	—
80	C₂×Q₈	—	514	2	no	—
81	C₂×Q₈	—	514	2	no	—
82	C₂×Q₈	—	514	4	no	—
83	C₂²×C₄	—	257	4	no	—
84	C₂²×C₄	C₄×D₂₅₇	1	2	yes	—
85	C₂²×C₄	C₂²×D₂₅₇	1	1	yes	characteristic
86	M₁₆	—	257	4	no	—
87	M₁₆	—	2	4	no	—
88	M₁₆	—	514	2	no	—
89	C₂×C₈	—	2	2	no	—
90	C₂×C₈	—	514	2	no	—
91	C₄⋊C₄	—	514	4	no	—
92	C₄⋊C₄	—	514	4	no	—
93	C₄²	C₂²⋊D₂₅₇	1	2	yes	—
94	C₄²	D₁₀₂₈	1	1	yes	characteristic
95	C₄²	D₁₀₂₈	1	1	yes	characteristic
96	C₂³	C₂×C₄×D₂₅₇	1	1	yes	characteristic
97	Q₈	—	1028	4	no	—
98	Q₈	—	1028	2	no	—
99	Q₈	—	1028	2	no	—
100	C₂×C₄	—	2	4	no	—
101	C₂×C₄	—	514	8	no	—
102	C₂×C₄	—	514	4	no	—
103	C₂×C₄	#16861595	1	2	yes	—
104	C₂×C₄	#16861595	1	2	yes	—
105	C₂×C₄	D₄×D₂₅₇	1	2	yes	—
106	C₂×C₄	C₂×C₂²⋊D₂₅₇	1	1	yes	characteristic
107	C₂×C₄	C₂×D₁₀₂₈	1	1	yes	characteristic
108	C₂×C₄	#16861619	1	2	yes	—
109	C₈	—	514	4	no	—
110	C₈	—	4	2	no	—
111	C₂²	—	2	2	no	—
112	C₂²	C₂×#16861595	1	1	yes	characteristic
113	C₂²	#22551554	1	2	yes	—
114	C₄	—	2	2	no	—
115	C₄	—	1028	4	no	—
116	C₄	#22551510	1	4	yes	complemented
117	C₄	#22551567	1	2	yes	—
118	C₄	#22551581	1	1	yes	characteristic
119	C₄	#22551588	1	1	yes	characteristic
120	C₂	—	2	2	no	—
121	C₂	#30656638	1	1	yes	characteristic
122	C₂	C₂×#22551510	1	2	yes	—

subgroup

G/H

class size

multiplicity

normal?

properties

C₂×#22551592

C₂

yes

maximal, characteristic

C₂×#22551510

C₂

yes

maximal

C₄×#16861602

C₂

yes

maximal, characteristic

#423×C₂₅₇

C₂

yes

maximal, characteristic

C₂²×Dic₅₁₄

C₂²

yes

—

C₂×#16861602

C₂²

yes

—

#22551592

—

#22551592

—

#22551592

—

C₂×C₄⋊Dic₂₅₇

C₄

yes

—

#22551510

—

#22551503

—

C₄×#14523339

—

C₂×M₁₆×C₂₅₇

C₂²

yes

—

C₂×C₄²×C₂₅₇

C₂²

yes

characteristic

C₄⋊C₈×C₂₅₇

—

C₂×Dic₅₁₄

—

C₂×Dic₅₁₄

—

C₂×Dic₅₁₄

—

C₂×Dic₅₁₄

—

C₂²×Dic₂₅₇

C₂×C₄

yes

—

#16861602

—

#16861602

D₄

yes

—

C₂×#14523339

—

C₄⋊Dic₂₅₇

—

C₄⋊Dic₂₅₇

—

C₂²×C₄×C₂₅₇

C₂×C₄

yes

—

C₂²×C₄×C₂₅₇

C₂³

yes

characteristic

M₁₆×C₂₅₇

—

C₂×C₈×C₂₅₇

—

C₄²×C₂₅₇

D₄

yes

—

C₄²×C₂₅₇

D₄

yes

characteristic

C₄²×C₂₅₇

D₄

yes

characteristic

Dic₅₁₄

—

Dic₅₁₄

—

Dic₅₁₄

—

C₂×Dic₂₅₇

—

C₂×Dic₂₅₇

—

#14523339

—

C₂³×C₂₅₇

C₂²×C₄

yes

characteristic

C₂×C₄×C₂₅₇

—

C₂×C₄×C₂₅₇

C₂²⋊C₄

yes

—

C₂×C₄×C₂₅₇

C₂²⋊C₄

yes

—

C₂×C₄×C₂₅₇

C₂×D₄

yes

—

C₂×C₄×C₂₅₇

C₂×D₄

yes

characteristic

C₂×C₄×C₂₅₇

C₂×D₄

yes

characteristic

C₂×C₄×C₂₅₇

#41

yes

—

C₈×C₂₅₇

—

C₂²×C₂₅₇

—

C₂²×C₂₅₇

C₂×C₂²⋊C₄

yes

characteristic

C₂²×C₂₅₇

C₄×D₄

yes

—

C₄×C₂₅₇

—

C₄×C₂₅₇

#101

yes

complemented

C₄×C₂₅₇

#121

yes

—

C₄×C₂₅₇

yes

C₄×C₂₅₇

yes

Dic₂₅₇

—

C₂×C₂₅₇

—

C₂×C₂₅₇

#390

yes

characteristic

C₂×C₂₅₇

C₂×#101

yes

—

C₂₅₇

#1966

yes

characteristic, complemented, Sylow

#1966

—

257

maximal, Sylow

C₂×C₄⋊Q₈

—

257

—

#423

D₂₅₇

yes

characteristic

C₂×#101

—

257

—

C₄×M₁₆

—

257

—

C₂²×Q₈

—

257

—

C₂×M₁₆

—

257

—

C₂×M₁₆

C₂×D₂₅₇

yes

—

C₄⋊Q₈

—

514

—

C₄⋊Q₈

—

514

—

C₄⋊Q₈

—

514

—

C₂×C₄⋊C₄

—

257

—

C₂×C₄²

C₂×D₂₅₇

yes

characteristic

C₄⋊C₈

—

#101

—

514

—

#97

—

514

—

C₄×C₈

—

514

—

C₂×Q₈

—

514

—

C₂×Q₈

—

514

—

C₂×Q₈

—

514

—

C₂×Q₈

—

514

—

C₂²×C₄

—

257

—

C₂²×C₄

C₄×D₂₅₇

yes

—

C₂²×C₄

C₂²×D₂₅₇

yes

characteristic

M₁₆

—

257

—

M₁₆

—

M₁₆

—

514

—

C₂×C₈

—

C₂×C₈

—

514

—

C₄⋊C₄

—

514

—

C₄⋊C₄

—

514

—

C₄²

C₂²⋊D₂₅₇

yes

—

C₄²

D₁₀₂₈

yes

characteristic

C₄²

D₁₀₂₈

yes

characteristic

C₂³

C₂×C₄×D₂₅₇

yes

characteristic

Q₈

—

1028

—

Q₈

—

1028

—

Q₈

—

1028

—

100

C₂×C₄

—

101

C₂×C₄

—

514

—

102

C₂×C₄

—

514

—

103

C₂×C₄

#16861595

yes

—

104

C₂×C₄

#16861595

yes

—

105

C₂×C₄

D₄×D₂₅₇

yes

—

106

C₂×C₄

C₂×C₂²⋊D₂₅₇

yes

characteristic

107

C₂×C₄

C₂×D₁₀₂₈

yes

characteristic

108

C₂×C₄

#16861619

yes

—

109

C₈

—

514

—

110

C₈

—

111

C₂²

—

112

C₂²

C₂×#16861595

yes

characteristic

113

C₂²

#22551554

yes

—

114

C₄

—

115

C₄

—

1028

—

116

C₄

#22551510

yes

complemented

117

C₄

#22551567

yes

—

118

C₄

#22551581

yes

characteristic

119

C₄

#22551588

yes

characteristic

120

C₂

—

121

C₂

#30656638

yes

characteristic

122

C₂

C₂×#22551510

yes

—

	subgroup	G/H
center	C₂²	C₂×#16861595
commutator, prime radical	C₂³×C₂₅₇	C₂²×C₄
Frattini	C₂²×C₄	C₂²×D₂₅₇
Fitting	#423×C₂₅₇	C₂
socle	C₂²×C₂₅₇	C₂×C₂²⋊C₄
cosocle	C₂²×C₄×C₂₅₇	C₂³
hypercenter	#423	D₂₅₇
hypocenter	C₂₅₇	#1966
LCS	G ▷ C₂³×C₂₅₇ ▷ C₂×C₂₅₇ ▷ C₂₅₇
UCS	C₁ ◁ C₂² ◁ C₂×C₄² ◁ #423

subgroup

G/H

center

C₂²

C₂×#16861595

commutator, prime radical

C₂³×C₂₅₇

C₂²×C₄

Frattini

C₂²×C₄

C₂²×D₂₅₇

Fitting

#423×C₂₅₇

C₂

socle

C₂²×C₂₅₇

C₂×C₂²⋊C₄

cosocle

C₂²×C₄×C₂₅₇

C₂³

D₂₅₇

C₂₅₇

G ▷ C₂³×C₂₅₇ ▷ C₂×C₂₅₇ ▷ C₂₅₇

UCS

C₁ ◁ C₂² ◁ C₂×C₄² ◁ #423

N	H ≅ G/N	type
53. C₄×C₂₅₇	#101	semidirect
61. C₂₅₇	#1966	semidirect
116. C₄	#22551510	semidirect

H ≅ G/N

type

53. C₄×C₂₅₇

#101

semidirect

61. C₂₅₇

#1966

semidirect

116. C₄

#22551510

semidirect

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

subgroup	index
C₂×#22551592	2
C₂×#22551510	2
C₄×#16861602	2
#423×C₂₅₇	2
#1966	257

subgroup

index

#423×C₂₅₇

257

Subgroups — #45665163

Conjugacy classes of subgroups

Complements

Maximal subgroups