Group #45665296 — subgroups

Subgroups — #45665296

Order 32896.

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

normal subgroups only

This group has 284 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	#30657294	C₂	1	1	yes	maximal, characteristic
2	C₄²×Dic₂₅₇	C₂	1	1	yes	maximal, characteristic
3	#30656498	C₂	1	4	yes	maximal
4	#423×C₂₅₇	C₂	1	1	yes	maximal, characteristic
5	#22551592	C₄	1	2	yes	complemented
6	#22551584	C₄	1	2	yes	complemented
7	#22551571	—	2	2	no	—
8	C₄×Dic₅₁₄	—	2	2	no	—
9	#22551551	C₂²	1	2	yes	—
10	C₂×C₄×Dic₂₅₇	—	2	2	no	—
11	C₂×C₄×Dic₂₅₇	C₂²	1	2	yes	—
12	C₂×M₁₆×C₂₅₇	C₂²	1	2	yes	—
13	C₂×C₄²×C₂₅₇	C₂²	1	1	yes	characteristic
14	C₄⋊C₈×C₂₅₇	—	2	2	no	—
15	C₂×Dic₅₁₄	—	2	2	no	—
16	C₂²×Dic₂₅₇	—	2	2	no	—
17	C₄⋊Dic₂₅₇	—	2	2	no	—
18	C₄⋊Dic₂₅₇	C₂×C₄	1	4	yes	—
19	#16861597	—	2	4	no	—
20	C₂²⋊Dic₂₅₇	—	2	2	no	—
21	C₄×Dic₂₅₇	—	2	8	no	—
22	C₄×Dic₂₅₇	—	2	2	no	—
23	C₄×Dic₂₅₇	—	2	2	no	—
24	C₄×Dic₂₅₇	Q₈	1	4	yes	complemented
25	C₂²×C₄×C₂₅₇	D₄	1	2	yes	—
26	C₂²×C₄×C₂₅₇	C₂³	1	1	yes	characteristic
27	M₁₆×C₂₅₇	—	2	4	no	—
28	C₂×C₈×C₂₅₇	—	2	2	no	—
29	C₄²×C₂₅₇	C₂×C₄	1	1	yes	characteristic
30	C₄²×C₂₅₇	C₂×C₄	1	1	yes	characteristic
31	C₄²×C₂₅₇	D₄	1	2	yes	—
32	Dic₅₁₄	—	4	4	no	—
33	C₂×Dic₂₅₇	—	2	8	no	—
34	C₂×Dic₂₅₇	—	2	4	no	—
35	C₂×Dic₂₅₇	—	2	4	no	—
36	C₂³×C₂₅₇	C₂×D₄	1	1	yes	characteristic
37	C₂×C₄×C₂₅₇	—	2	4	no	—
38	C₂×C₄×C₂₅₇	C₂²⋊C₄	1	2	yes	—
39	C₂×C₄×C₂₅₇	C₂²⋊C₄	1	2	yes	—
40	C₂×C₄×C₂₅₇	C₂²×C₄	1	1	yes	characteristic
41	C₂×C₄×C₂₅₇	C₂×D₄	1	1	yes	characteristic
42	C₂×C₄×C₂₅₇	C₂×Q₈	1	2	yes	—
43	C₂×C₄×C₂₅₇	#41	1	2	yes	—
44	C₈×C₂₅₇	—	4	2	no	—
45	C₂²×C₂₅₇	—	2	2	no	—
46	C₂²×C₂₅₇	C₂×C₂²⋊C₄	1	1	yes	characteristic
47	C₂²×C₂₅₇	C₂²⋊Q₈	1	2	yes	—
48	C₄×C₂₅₇	—	2	2	no	—
49	C₄×C₂₅₇	#104	1	4	yes	—
50	C₄×C₂₅₇	C₄×Q₈	1	2	yes	—
51	C₄×C₂₅₇	#124	1	1	yes	characteristic
52	C₄×C₂₅₇	C₄⋊Q₈	1	1	yes	characteristic
53	Dic₂₅₇	—	2	8	no	—
54	Dic₂₅₇	—	4	4	no	—
55	C₂×C₂₅₇	—	2	2	no	—
56	C₂×C₂₅₇	#391	1	1	yes	characteristic
57	C₂×C₂₅₇	C₂×#104	1	2	yes	—
58	C₂₅₇	#1994	1	1	yes	characteristic, complemented, Sylow
59	#1994	—	257	1	no	maximal, Sylow
60	#533	—	257	1	no	—
61	#423	D₂₅₇	1	1	yes	characteristic
62	C₄³	—	257	1	no	—
63	#339	—	257	4	no	—
64	C₂×M₁₆	C₂×D₂₅₇	1	2	yes	—
65	C₄⋊Q₈	—	257	2	no	—
66	#125	—	257	2	no	—
67	C₂²⋊Q₈	—	514	2	no	—
68	C₄×Q₈	—	514	2	no	—
69	#117	—	257	2	no	—
70	C₂×C₄²	—	257	2	no	—
71	C₂×C₄²	—	514	2	no	—
72	C₂×C₄²	C₂×D₂₅₇	1	1	yes	characteristic
73	C₄⋊C₈	—	2	2	no	—
74	C₂×Q₈	—	514	2	no	—
75	C₂²×C₄	—	514	2	no	—
76	C₂²×C₄	D₁₀₂₈	1	2	yes	—
77	C₂²×C₄	C₂²×D₂₅₇	1	1	yes	characteristic
78	M₁₆	—	2	4	no	—
79	C₂×C₈	—	2	2	no	—
80	C₄⋊C₄	—	257	4	no	—
81	C₄⋊C₄	—	514	4	no	—
82	C₄⋊C₄	—	514	2	no	—
83	C₂²⋊C₄	—	514	2	no	—
84	C₄²	—	257	4	no	—
85	C₄²	—	514	8	no	—
86	C₄²	—	514	2	no	—
87	C₄²	—	514	2	no	—
88	C₄²	C₄×D₂₅₇	1	1	yes	characteristic
89	C₄²	C₄×D₂₅₇	1	1	yes	characteristic
90	C₄²	C₂²⋊D₂₅₇	1	2	yes	—
91	C₂³	C₂×D₁₀₂₈	1	1	yes	characteristic
92	Q₈	—	1028	4	no	—
93	C₂×C₄	—	2	4	no	—
94	C₂×C₄	—	514	8	no	—
95	C₂×C₄	—	514	4	no	—
96	C₂×C₄	—	514	4	no	—
97	C₂×C₄	#16861595	1	2	yes	—
98	C₂×C₄	#16861595	1	2	yes	—
99	C₂×C₄	C₂×C₄×D₂₅₇	1	1	yes	characteristic
100	C₂×C₄	C₂×C₂²⋊D₂₅₇	1	1	yes	characteristic
101	C₂×C₄	Q₈×D₂₅₇	1	2	yes	—
102	C₂×C₄	#16861617	1	2	yes	—
103	C₈	—	4	2	no	—
104	C₂²	—	2	2	no	—
105	C₂²	C₂×#16861595	1	1	yes	characteristic
106	C₂²	#22551572	1	2	yes	—
107	C₄	—	2	2	no	—
108	C₄	—	514	8	no	—
109	C₄	—	1028	4	no	—
110	C₄	#22551518	1	4	yes	—
111	C₄	#22551557	1	2	yes	—
112	C₄	#22551579	1	1	yes	characteristic
113	C₄	#22551590	1	1	yes	characteristic
114	C₂	—	2	2	no	—
115	C₂	#30656642	1	1	yes	characteristic
116	C₂	C₂×#22551518	1	2	yes	—

subgroup

G/H

class size

multiplicity

normal?

properties

#30657294

C₂

yes

maximal, characteristic

C₄²×Dic₂₅₇

C₂

yes

maximal, characteristic

#30656498

C₂

yes

maximal

#423×C₂₅₇

C₂

yes

maximal, characteristic

#22551592

C₄

yes

complemented

#22551584

C₄

yes

complemented

#22551571

—

C₄×Dic₅₁₄

—

#22551551

C₂²

yes

—

C₂×C₄×Dic₂₅₇

—

C₂×C₄×Dic₂₅₇

C₂²

yes

—

C₂×M₁₆×C₂₅₇

C₂²

yes

—

C₂×C₄²×C₂₅₇

C₂²

yes

characteristic

C₄⋊C₈×C₂₅₇

—

C₂×Dic₅₁₄

—

C₂²×Dic₂₅₇

—

C₄⋊Dic₂₅₇

—

C₄⋊Dic₂₅₇

C₂×C₄

yes

—

#16861597

—

C₂²⋊Dic₂₅₇

—

C₄×Dic₂₅₇

—

C₄×Dic₂₅₇

—

C₄×Dic₂₅₇

—

C₄×Dic₂₅₇

Q₈

yes

complemented

C₂²×C₄×C₂₅₇

D₄

yes

—

C₂²×C₄×C₂₅₇

C₂³

yes

characteristic

M₁₆×C₂₅₇

—

C₂×C₈×C₂₅₇

—

C₄²×C₂₅₇

C₂×C₄

yes

characteristic

C₄²×C₂₅₇

C₂×C₄

yes

characteristic

C₄²×C₂₅₇

D₄

yes

—

Dic₅₁₄

—

C₂×Dic₂₅₇

—

C₂×Dic₂₅₇

—

C₂×Dic₂₅₇

—

C₂³×C₂₅₇

C₂×D₄

yes

characteristic

C₂×C₄×C₂₅₇

—

C₂×C₄×C₂₅₇

C₂²⋊C₄

yes

—

C₂×C₄×C₂₅₇

C₂²⋊C₄

yes

—

C₂×C₄×C₂₅₇

C₂²×C₄

yes

characteristic

C₂×C₄×C₂₅₇

C₂×D₄

yes

characteristic

C₂×C₄×C₂₅₇

C₂×Q₈

yes

—

C₂×C₄×C₂₅₇

#41

yes

—

C₈×C₂₅₇

—

C₂²×C₂₅₇

—

C₂²×C₂₅₇

C₂×C₂²⋊C₄

yes

characteristic

C₂²×C₂₅₇

C₂²⋊Q₈

yes

—

C₄×C₂₅₇

—

C₄×C₂₅₇

#104

yes

—

C₄×C₂₅₇

C₄×Q₈

yes

—

C₄×C₂₅₇

#124

yes

characteristic

C₄×C₂₅₇

C₄⋊Q₈

yes

characteristic

Dic₂₅₇

—

Dic₂₅₇

—

C₂×C₂₅₇

—

C₂×C₂₅₇

#391

yes

characteristic

C₂×C₂₅₇

C₂×#104

yes

—

C₂₅₇

#1994

yes

characteristic, complemented, Sylow

#1994

—

257

maximal, Sylow

#533

—

257

—

#423

D₂₅₇

yes

characteristic

C₄³

—

257

—

#339

—

257

—

C₂×M₁₆

C₂×D₂₅₇

yes

—

C₄⋊Q₈

—

257

—

#125

—

257

—

C₂²⋊Q₈

—

514

—

C₄×Q₈

—

514

—

#117

—

257

—

C₂×C₄²

—

257

—

C₂×C₄²

—

514

—

C₂×C₄²

C₂×D₂₅₇

yes

characteristic

C₄⋊C₈

—

C₂×Q₈

—

514

—

C₂²×C₄

—

514

—

C₂²×C₄

D₁₀₂₈

yes

—

C₂²×C₄

C₂²×D₂₅₇

yes

characteristic

M₁₆

—

C₂×C₈

—

C₄⋊C₄

—

257

—

C₄⋊C₄

—

514

—

C₄⋊C₄

—

514

—

C₂²⋊C₄

—

514

—

C₄²

—

257

—

C₄²

—

514

—

C₄²

—

514

—

C₄²

—

514

—

C₄²

C₄×D₂₅₇

yes

characteristic

C₄²

C₄×D₂₅₇

yes

characteristic

C₄²

C₂²⋊D₂₅₇

yes

—

C₂³

C₂×D₁₀₂₈

yes

characteristic

Q₈

—

1028

—

C₂×C₄

—

C₂×C₄

—

514

—

C₂×C₄

—

514

—

C₂×C₄

—

514

—

C₂×C₄

#16861595

yes

—

C₂×C₄

#16861595

yes

—

C₂×C₄

C₂×C₄×D₂₅₇

yes

characteristic

100

C₂×C₄

C₂×C₂²⋊D₂₅₇

yes

characteristic

101

C₂×C₄

Q₈×D₂₅₇

yes

—

102

C₂×C₄

#16861617

yes

—

103

C₈

—

104

C₂²

—

105

C₂²

C₂×#16861595

yes

characteristic

106

C₂²

#22551572

yes

—

107

C₄

—

108

C₄

—

514

—

109

C₄

—

1028

—

110

C₄

#22551518

yes

—

111

C₄

#22551557

yes

—

112

C₄

#22551579

yes

characteristic

113

C₄

#22551590

yes

characteristic

114

C₂

—

115

C₂

#30656642

yes

characteristic

116

C₂

C₂×#22551518

yes

—

	subgroup	G/H
center	C₂×C₄	C₂×C₂²⋊D₂₅₇
commutator	C₂×C₄×C₂₅₇	C₂²×C₄
Frattini	C₂²×C₄	C₂²×D₂₅₇
Fitting	#423×C₂₅₇	C₂
socle	C₂²×C₂₅₇	C₂×C₂²⋊C₄
prime radical	C₂³×C₂₅₇	C₂×D₄
cosocle	C₂²×C₄×C₂₅₇	C₂³
hypercenter	#423	D₂₅₇
hypocenter	C₂₅₇	#1994
LCS	G ▷ C₂×C₄×C₂₅₇ ▷ C₂×C₂₅₇ ▷ C₂₅₇
UCS	C₁ ◁ C₂×C₄ ◁ C₂×C₄² ◁ #423

subgroup

G/H

center

C₂×C₄

C₂×C₂²⋊D₂₅₇

commutator

C₂×C₄×C₂₅₇

C₂²×C₄

Frattini

C₂²×C₄

C₂²×D₂₅₇

Fitting

#423×C₂₅₇

C₂

socle

C₂²×C₂₅₇

C₂×C₂²⋊C₄

prime radical

C₂³×C₂₅₇

C₂×D₄

cosocle

C₂²×C₄×C₂₅₇

C₂³

D₂₅₇

C₂₅₇

G ▷ C₂×C₄×C₂₅₇ ▷ C₂×C₂₅₇ ▷ C₂₅₇

UCS

C₁ ◁ C₂×C₄ ◁ C₂×C₄² ◁ #423

N	H ≅ G/N	type
5. #22551592	C₄	semidirect
6. #22551584	C₄	semidirect
24. C₄×Dic₂₅₇	Q₈	semidirect
58. C₂₅₇	#1994	semidirect

H ≅ G/N

type

5. #22551592

C₄

semidirect

6. #22551584

C₄

semidirect

24. C₄×Dic₂₅₇

Q₈

semidirect

58. C₂₅₇

#1994

semidirect

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

subgroup	index
#30657294	2
C₄²×Dic₂₅₇	2
#30656498	2
#423×C₂₅₇	2
#1994	257

subgroup

index

#30657294

C₄²×Dic₂₅₇

#30656498

#423×C₂₅₇

#1994

257

Subgroups — #45665296

Conjugacy classes of subgroups

Complements

Maximal subgroups