Group #45665139 — subgroups

Subgroups — #45665139

Order 32896.

308 conjugacy classes of subgroups; only the 105 normal subgroups are shown.

normal subgroups only

This group has 308 conjugacy classes of subgroups — too many to draw. Only its 105 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×#22551515	C₂	1	4	yes	maximal
3	C₂×#22551503	C₂	1	1	yes	maximal, characteristic
4	#382×C₂₅₇	C₂	1	1	yes	maximal, characteristic
5	C₂²×Dic₅₁₄	C₂²	1	2	yes	—
6	C₂²×#14523339	C₂²	1	2	yes	—
7	#22551592	—	2	2	no	—
8	#22551592	C₄	1	4	yes	complemented
9	C₂×C₄⋊Dic₂₅₇	—	2	2	no	—
10	#22551515	—	2	8	no	—
11	#22551503	—	2	2	no	—
12	C₂×C₄⋊C₄×C₂₅₇	C₂²	1	2	yes	—
13	C₂×C₄²×C₂₅₇	C₂²	1	1	yes	characteristic
14	#95×C₂₅₇	—	2	2	no	—
15	C₂×Dic₅₁₄	—	2	2	no	—
16	C₂×Dic₅₁₄	—	2	4	no	—
17	C₂×Dic₅₁₄	—	2	4	no	—
18	C₂×Dic₅₁₄	C₂×C₄	1	4	yes	—
19	C₂²×Dic₂₅₇	—	2	2	no	—
20	C₂×#14523339	—	2	2	no	—
21	C₂×#14523339	—	2	2	no	—
22	C₂×#14523339	D₄	1	4	yes	—
23	C₄⋊Dic₂₅₇	—	2	8	no	—
24	C₂²×C₄×C₂₅₇	—	2	2	no	—
25	C₂²×C₄×C₂₅₇	D₄	1	2	yes	—
26	C₂²×C₄×C₂₅₇	C₂³	1	1	yes	characteristic
27	C₄⋊C₄×C₂₅₇	—	2	4	no	—
28	C₄²×C₂₅₇	C₂×C₄	1	2	yes	—
29	C₄²×C₂₅₇	D₄	1	2	yes	—
30	Dic₅₁₄	—	2	8	no	—
31	Dic₅₁₄	—	4	4	no	—
32	C₂×Dic₂₅₇	—	2	4	no	—
33	C₂×Dic₂₅₇	—	4	2	no	—
34	C₂×Dic₂₅₇	—	4	2	no	—
35	#14523339	—	2	4	no	—
36	C₂³×C₂₅₇	C₂×D₄	1	1	yes	characteristic
37	C₂×C₄×C₂₅₇	—	2	4	no	—
38	C₂×C₄×C₂₅₇	—	4	2	no	—
39	C₂×C₄×C₂₅₇	—	4	4	no	—
40	C₂×C₄×C₂₅₇	C₂²⋊C₄	1	4	yes	—
41	C₂×C₄×C₂₅₇	C₂²×C₄	1	1	yes	characteristic
42	C₂×C₄×C₂₅₇	C₂×D₄	1	2	yes	—
43	C₂×C₄×C₂₅₇	C₂×D₄	1	1	yes	characteristic
44	C₂×C₄×C₂₅₇	#41	1	2	yes	—
45	C₂²×C₂₅₇	C₂×C₂²⋊C₄	1	1	yes	characteristic
46	C₂²×C₂₅₇	#124	1	1	yes	characteristic
47	C₂²×C₂₅₇	#127	1	1	yes	characteristic
48	C₂²×C₂₅₇	#137	1	4	yes	—
49	C₄×C₂₅₇	—	2	4	no	—
50	C₄×C₂₅₇	—	8	2	no	—
51	C₄×C₂₅₇	C₄×D₄	1	2	yes	—
52	C₄×C₂₅₇	#121	1	2	yes	—
53	Dic₂₅₇	—	4	4	no	—
54	C₂×C₂₅₇	#390	1	1	yes	characteristic
55	C₂×C₂₅₇	#419	1	2	yes	—
56	C₂×C₂₅₇	#491	1	2	yes	—
57	C₂×C₂₅₇	#497	1	2	yes	—
58	C₂₅₇	#1960	1	1	yes	characteristic, complemented, Sylow
59	#1960	—	257	1	no	maximal, Sylow
60	C₂×C₄⋊Q₈	—	257	1	no	—
61	C₂×#103	—	257	4	no	—
62	C₂×#97	—	257	1	no	—
63	#382	D₂₅₇	1	1	yes	characteristic
64	C₂²×Q₈	—	257	2	no	—
65	C₂²×C₈	—	257	2	no	—
66	C₄⋊Q₈	—	257	4	no	—
67	C₄⋊Q₈	—	514	2	no	—
68	C₂×C₄⋊C₄	—	514	2	no	—
69	C₂×C₄⋊C₄	C₂×D₂₅₇	1	2	yes	—
70	C₂×C₄²	C₂×D₂₅₇	1	1	yes	characteristic
71	#103	—	514	8	no	—
72	#97	—	514	2	no	—
73	#95	—	2	2	no	—
74	C₂×Q₈	—	257	4	no	—
75	C₂×Q₈	—	514	2	no	—
76	C₂×Q₈	—	514	4	no	—
77	C₂×Q₈	—	514	4	no	—
78	C₂²×C₄	—	2	2	no	—
79	C₂²×C₄	—	514	2	no	—
80	C₂²×C₄	C₂²⋊D₂₅₇	1	2	yes	—
81	C₂²×C₄	C₂²×D₂₅₇	1	1	yes	characteristic
82	C₂×C₈	—	257	4	no	—
83	C₂×C₈	—	514	2	no	—
84	C₂×C₈	—	514	2	no	—
85	C₄⋊C₄	—	2	4	no	—
86	C₄⋊C₄	—	514	8	no	—
87	C₄²	C₄×D₂₅₇	1	2	yes	—
88	C₄²	D₁₀₂₈	1	2	yes	—
89	C₂³	C₂×C₂²⋊D₂₅₇	1	1	yes	characteristic
90	Q₈	—	514	8	no	—
91	Q₈	—	1028	4	no	—
92	C₂×C₄	—	2	4	no	—
93	C₂×C₄	—	514	4	no	—
94	C₂×C₄	—	4	2	no	—
95	C₂×C₄	—	4	4	no	—
96	C₂×C₄	—	1028	2	no	—
97	C₂×C₄	—	1028	2	no	—
98	C₂×C₄	#16861595	1	4	yes	—
99	C₂×C₄	C₂×C₄×D₂₅₇	1	1	yes	characteristic
100	C₂×C₄	D₄×D₂₅₇	1	2	yes	—
101	C₂×C₄	C₂×D₁₀₂₈	1	1	yes	characteristic
102	C₂×C₄	#16861619	1	2	yes	—
103	C₈	—	514	4	no	—
104	C₂²	C₂×#16861595	1	1	yes	characteristic
105	C₂²	#22551581	1	1	yes	characteristic
106	C₂²	#22551588	1	1	yes	characteristic
107	C₂²	#22551622	1	4	yes	—
108	C₄	—	2	4	no	—
109	C₄	—	1028	4	no	—
110	C₄	—	8	2	no	—
111	C₄	#22551554	1	2	yes	—
112	C₄	#22551567	1	2	yes	—
113	C₂	#30656638	1	1	yes	characteristic
114	C₂	#30656745	1	2	yes	—
115	C₂	#30657107	1	2	yes	—
116	C₂	#30657132	1	2	yes	—

subgroup

G/H

class size

multiplicity

normal?

properties

C₂×#22551592

C₂

yes

maximal, characteristic

C₂×#22551515

C₂

yes

maximal

C₂×#22551503

C₂

yes

maximal, characteristic

#382×C₂₅₇

C₂

yes

maximal, characteristic

C₂²×Dic₅₁₄

C₂²

yes

—

C₂²×#14523339

C₂²

yes

—

#22551592

—

#22551592

C₄

yes

complemented

C₂×C₄⋊Dic₂₅₇

—

#22551515

—

#22551503

—

C₂×C₄⋊C₄×C₂₅₇

C₂²

yes

—

C₂×C₄²×C₂₅₇

C₂²

yes

characteristic

#95×C₂₅₇

—

C₂×Dic₅₁₄

—

C₂×Dic₅₁₄

—

C₂×Dic₅₁₄

—

C₂×Dic₅₁₄

C₂×C₄

yes

—

C₂²×Dic₂₅₇

—

C₂×#14523339

—

C₂×#14523339

—

C₂×#14523339

D₄

yes

—

C₄⋊Dic₂₅₇

—

C₂²×C₄×C₂₅₇

—

C₂²×C₄×C₂₅₇

D₄

yes

—

C₂²×C₄×C₂₅₇

C₂³

yes

characteristic

C₄⋊C₄×C₂₅₇

—

C₄²×C₂₅₇

C₂×C₄

yes

—

C₄²×C₂₅₇

D₄

yes

—

Dic₅₁₄

—

Dic₅₁₄

—

C₂×Dic₂₅₇

—

C₂×Dic₂₅₇

—

C₂×Dic₂₅₇

—

#14523339

—

C₂³×C₂₅₇

C₂×D₄

yes

characteristic

C₂×C₄×C₂₅₇

—

C₂×C₄×C₂₅₇

—

C₂×C₄×C₂₅₇

—

C₂×C₄×C₂₅₇

C₂²⋊C₄

yes

—

C₂×C₄×C₂₅₇

C₂²×C₄

yes

characteristic

C₂×C₄×C₂₅₇

C₂×D₄

yes

—

C₂×C₄×C₂₅₇

C₂×D₄

yes

characteristic

C₂×C₄×C₂₅₇

#41

yes

—

C₂²×C₂₅₇

C₂×C₂²⋊C₄

yes

C₂²×C₂₅₇

yes

C₂²×C₂₅₇

yes

C₂²×C₂₅₇

yes

—

C₄×C₂₅₇

—

C₄×C₂₅₇

—

C₄×C₂₅₇

C₄×D₄

yes

—

C₄×C₂₅₇

#121

yes

—

Dic₂₅₇

—

C₂×C₂₅₇

#390

yes

characteristic

C₂×C₂₅₇

#419

yes

—

C₂×C₂₅₇

#491

yes

—

C₂×C₂₅₇

#497

yes

—

C₂₅₇

#1960

yes

characteristic, complemented, Sylow

#1960

—

257

maximal, Sylow

C₂×C₄⋊Q₈

—

257

—

C₂×#103

—

257

—

C₂×#97

—

257

—

#382

D₂₅₇

yes

characteristic

C₂²×Q₈

—

257

—

C₂²×C₈

—

257

—

C₄⋊Q₈

—

257

—

C₄⋊Q₈

—

514

—

C₂×C₄⋊C₄

—

514

—

C₂×C₄⋊C₄

C₂×D₂₅₇

yes

—

C₂×C₄²

C₂×D₂₅₇

yes

characteristic

#103

—

514

—

#97

—

514

—

#95

—

C₂×Q₈

—

257

—

C₂×Q₈

—

514

—

C₂×Q₈

—

514

—

C₂×Q₈

—

514

—

C₂²×C₄

—

C₂²×C₄

—

514

—

C₂²×C₄

C₂²⋊D₂₅₇

yes

—

C₂²×C₄

C₂²×D₂₅₇

yes

characteristic

C₂×C₈

—

257

—

C₂×C₈

—

514

—

C₂×C₈

—

514

—

C₄⋊C₄

—

C₄⋊C₄

—

514

—

C₄²

C₄×D₂₅₇

yes

—

C₄²

D₁₀₂₈

yes

—

C₂³

C₂×C₂²⋊D₂₅₇

yes

characteristic

Q₈

—

514

—

Q₈

—

1028

—

C₂×C₄

—

C₂×C₄

—

514

—

C₂×C₄

—

C₂×C₄

—

C₂×C₄

—

1028

—

C₂×C₄

—

1028

—

C₂×C₄

#16861595

yes

—

C₂×C₄

C₂×C₄×D₂₅₇

yes

characteristic

100

C₂×C₄

D₄×D₂₅₇

yes

—

101

C₂×C₄

C₂×D₁₀₂₈

yes

characteristic

102

C₂×C₄

#16861619

yes

—

103

C₈

—

514

—

104

C₂²

C₂×#16861595

yes

characteristic

105

C₂²

#22551581

yes

characteristic

106

C₂²

#22551588

yes

characteristic

107

C₂²

#22551622

yes

—

108

C₄

—

109

C₄

—

1028

—

110

C₄

—

111

C₄

#22551554

yes

—

112

C₄

#22551567

yes

—

113

C₂

#30656638

yes

characteristic

114

C₂

#30656745

yes

—

115

C₂

#30657107

yes

—

116

C₂

#30657132

yes

—

	subgroup	G/H
center	C₂³	C₂×C₂²⋊D₂₅₇
commutator	C₂×C₄×C₂₅₇	C₂²×C₄
Frattini	C₂²×C₄	C₂²×D₂₅₇
Fitting	#382×C₂₅₇	C₂
socle, prime radical	C₂³×C₂₅₇	C₂×D₄
cosocle	C₂²×C₄×C₂₅₇	C₂³
hypercenter	#382	D₂₅₇
hypocenter	C₂₅₇	#1960
LCS	G ▷ C₂×C₄×C₂₅₇ ▷ C₂×C₂₅₇ ▷ C₂₅₇
UCS	C₁ ◁ C₂³ ◁ C₂×C₄² ◁ #382

subgroup

G/H

center

C₂³

C₂×C₂²⋊D₂₅₇

commutator

C₂×C₄×C₂₅₇

C₂²×C₄

Frattini

C₂²×C₄

C₂²×D₂₅₇

Fitting

#382×C₂₅₇

C₂

socle, prime radical

C₂³×C₂₅₇

C₂×D₄

cosocle

C₂²×C₄×C₂₅₇

C₂³

D₂₅₇

C₂₅₇

G ▷ C₂×C₄×C₂₅₇ ▷ C₂×C₂₅₇ ▷ C₂₅₇

UCS

C₁ ◁ C₂³ ◁ C₂×C₄² ◁ #382

N	H ≅ G/N	type
8. #22551592	C₄	semidirect
58. C₂₅₇	#1960	semidirect

H ≅ G/N

type

8. #22551592

C₄

semidirect

58. C₂₅₇

#1960

semidirect

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

subgroup	index
C₂×#22551592	2
C₂×#22551515	2
C₂×#22551503	2
#382×C₂₅₇	2
#1960	257

subgroup

index

#382×C₂₅₇

257

Subgroups — #45665139

Conjugacy classes of subgroups

Complements

Maximal subgroups