Group #45664223 — subgroups

Subgroups — C₄×#22551527 #45664223

Order 32896.

228 conjugacy classes of subgroups; only the 121 normal subgroups are shown.

normal subgroups only

This group has 228 conjugacy classes of subgroups — too many to draw. Only its 121 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×#22551527	C₂	1	2	yes	maximal
2	C₄×#16861602	C₂	1	1	yes	maximal, characteristic
3	C₂×C₄×#14523339	C₂	1	1	yes	maximal, characteristic
4	#30656501	C₂	1	2	yes	maximal
5	C₄×M₁₆×C₂₅₇	C₂	1	1	yes	maximal, characteristic
6	C₂×#16861602	C₂²	1	2	yes	—
7	C₂²×#14523339	C₂²	1	1	yes	characteristic
8	C₂²×#14523339	C₂²	1	1	yes	characteristic
9	#22551527	C₄	1	8	yes	direct product factor
10	#22551503	—	2	1	no	—
11	C₄×#14523339	—	2	1	no	—
12	C₄×#14523339	C₄	1	4	yes	—
13	C₂×M₁₆×C₂₅₇	C₂²	1	2	yes	—
14	C₂×C₄²×C₂₅₇	C₂²	1	1	yes	characteristic
15	#97×C₂₅₇	—	2	1	no	—
16	C₄×C₈×C₂₅₇	—	2	1	no	—
17	#16861602	—	2	2	no	—
18	#16861602	C₂×C₄	1	4	yes	—
19	C₂×#14523339	—	2	2	no	—
20	C₂×#14523339	—	2	2	no	—
21	C₂×#14523339	C₂×C₄	1	2	yes	—
22	C₂×#14523339	C₂×C₄	1	2	yes	—
23	C₂×#14523339	C₂×C₄	1	4	yes	—
24	C₂²×C₄×C₂₅₇	Q₈	1	2	yes	—
25	C₂²×C₄×C₂₅₇	C₂³	1	1	yes	characteristic
26	M₁₆×C₂₅₇	—	2	2	no	—
27	M₁₆×C₂₅₇	C₂×C₄	1	4	yes	—
28	C₂×C₈×C₂₅₇	—	2	2	no	—
29	C₄²×C₂₅₇	C₂×C₄	1	2	yes	—
30	C₄²×C₂₅₇	D₄	1	1	yes	characteristic
31	C₄²×C₂₅₇	D₄	1	1	yes	characteristic
32	#14523339	—	2	2	no	—
33	#14523339	—	2	4	no	—
34	#14523339	C₄²	1	4	yes	—
35	C₂³×C₂₅₇	C₂×Q₈	1	1	yes	characteristic
36	C₂×C₄×C₂₅₇	—	2	4	no	—
37	C₂×C₄×C₂₅₇	C₄⋊C₄	1	2	yes	—
38	C₂×C₄×C₂₅₇	C₄⋊C₄	1	2	yes	—
39	C₂×C₄×C₂₅₇	C₂²×C₄	1	1	yes	characteristic
40	C₂×C₄×C₂₅₇	C₂²×C₄	1	2	yes	—
41	C₂×C₄×C₂₅₇	C₂×D₄	1	1	yes	characteristic
42	C₂×C₄×C₂₅₇	#41	1	2	yes	—
43	C₈×C₂₅₇	—	2	4	no	—
44	C₂²×C₂₅₇	—	2	2	no	—
45	C₂²×C₂₅₇	C₂×C₄⋊C₄	1	1	yes	characteristic
46	C₂²×C₂₅₇	C₄×Q₈	1	2	yes	—
47	C₄×C₂₅₇	—	2	2	no	—
48	C₄×C₂₅₇	#108	1	4	yes	complemented
49	C₄×C₂₅₇	C₂×C₄²	1	1	yes	characteristic
50	C₄×C₂₅₇	#117	1	1	yes	characteristic
51	C₄×C₂₅₇	C₄×D₄	1	2	yes	—
52	C₂×C₂₅₇	—	2	2	no	—
53	C₂×C₂₅₇	C₄×C₄⋊C₄	1	1	yes	characteristic
54	C₂×C₂₅₇	C₂×#108	1	2	yes	—
55	C₂₅₇	C₄×#108	1	1	yes	characteristic, complemented, Sylow
56	C₄×#108	—	257	1	no	maximal, Sylow
57	C₂×#108	—	257	2	no	—
58	C₄×M₁₆	—	257	1	no	—
59	C₄×M₁₆	D₂₅₇	1	1	yes	characteristic
60	C₂×C₄×C₈	—	257	1	no	—
61	#341	—	257	2	no	—
62	C₂×M₁₆	—	257	2	no	—
63	C₂×M₁₆	C₂×D₂₅₇	1	2	yes	—
64	C₂²×C₈	—	257	1	no	—
65	C₂²×C₈	—	257	1	no	—
66	C₂×C₄²	C₂×D₂₅₇	1	1	yes	characteristic
67	#108	—	257	8	no	—
68	#97	—	2	1	no	—
69	#97	—	514	1	no	—
70	C₄×C₈	—	257	4	no	—
71	C₄×C₈	—	2	1	no	—
72	C₄×C₈	—	514	1	no	—
73	C₂²×C₄	Dic₅₁₄	1	2	yes	—
74	C₂²×C₄	C₂²×D₂₅₇	1	1	yes	characteristic
75	M₁₆	—	257	4	no	—
76	M₁₆	—	2	2	no	—
77	M₁₆	—	514	2	no	—
78	M₁₆	C₄×D₂₅₇	1	4	yes	—
79	C₂×C₈	—	257	2	no	—
80	C₂×C₈	—	257	4	no	—
81	C₂×C₈	—	257	2	no	—
82	C₂×C₈	—	2	2	no	—
83	C₂×C₈	—	514	2	no	—
84	C₂×C₈	—	514	2	no	—
85	C₄²	C₄×D₂₅₇	1	2	yes	—
86	C₄²	C₂²⋊D₂₅₇	1	1	yes	characteristic
87	C₄²	C₂²⋊D₂₅₇	1	1	yes	characteristic
88	C₂³	C₂×Dic₅₁₄	1	1	yes	characteristic
89	C₂×C₄	—	2	4	no	—
90	C₂×C₄	#16861597	1	2	yes	—
91	C₂×C₄	#16861597	1	2	yes	—
92	C₂×C₄	C₂×C₄×D₂₅₇	1	2	yes	—
93	C₂×C₄	C₂×C₄×D₂₅₇	1	1	yes	characteristic
94	C₂×C₄	C₂×C₂²⋊D₂₅₇	1	1	yes	characteristic
95	C₂×C₄	#16861618	1	2	yes	—
96	C₈	—	257	4	no	—
97	C₈	—	2	4	no	—
98	C₈	—	514	4	no	—
99	C₈	—	514	2	no	—
100	C₂²	—	2	2	no	—
101	C₂²	C₂×#16861597	1	1	yes	characteristic
102	C₂²	C₄×Dic₅₁₄	1	2	yes	—
103	C₄	—	2	2	no	—
104	C₄	#22551527	1	4	yes	direct product factor
105	C₄	C₄²×D₂₅₇	1	1	yes	characteristic
106	C₄	#22551549	1	1	yes	characteristic
107	C₄	C₄×C₂²⋊D₂₅₇	1	2	yes	—
108	C₂	—	2	2	no	—
109	C₂	C₄×#16861597	1	1	yes	characteristic
110	C₂	C₂×#22551527	1	2	yes	—

subgroup

G/H

class size

multiplicity

normal?

properties

C₂×#22551527

C₂

yes

maximal

C₄×#16861602

C₂

yes

maximal, characteristic

C₂×C₄×#14523339

C₂

yes

maximal, characteristic

#30656501

C₂

yes

maximal

C₄×M₁₆×C₂₅₇

C₂

yes

maximal, characteristic

C₂×#16861602

C₂²

yes

—

C₂²×#14523339

C₂²

yes

characteristic

C₂²×#14523339

C₂²

yes

characteristic

#22551527

C₄

yes

direct product factor

#22551503

—

C₄×#14523339

—

C₄×#14523339

C₄

yes

—

C₂×M₁₆×C₂₅₇

C₂²

yes

—

C₂×C₄²×C₂₅₇

C₂²

yes

characteristic

#97×C₂₅₇

—

C₄×C₈×C₂₅₇

—

#16861602

—

#16861602

C₂×C₄

yes

—

C₂×#14523339

—

C₂×#14523339

—

C₂×#14523339

C₂×C₄

yes

—

C₂×#14523339

C₂×C₄

yes

—

C₂×#14523339

C₂×C₄

yes

—

C₂²×C₄×C₂₅₇

Q₈

yes

—

C₂²×C₄×C₂₅₇

C₂³

yes

characteristic

M₁₆×C₂₅₇

—

M₁₆×C₂₅₇

C₂×C₄

yes

—

C₂×C₈×C₂₅₇

—

C₄²×C₂₅₇

C₂×C₄

yes

—

C₄²×C₂₅₇

D₄

yes

characteristic

C₄²×C₂₅₇

D₄

yes

—

—

C₄²

yes

—

C₂³×C₂₅₇

C₂×Q₈

yes

characteristic

C₂×C₄×C₂₅₇

—

C₂×C₄×C₂₅₇

C₄⋊C₄

yes

—

C₂×C₄×C₂₅₇

C₄⋊C₄

yes

—

C₂×C₄×C₂₅₇

C₂²×C₄

yes

characteristic

C₂×C₄×C₂₅₇

C₂²×C₄

yes

—

C₂×C₄×C₂₅₇

C₂×D₄

yes

characteristic

C₂×C₄×C₂₅₇

#41

yes

—

C₈×C₂₅₇

—

C₂²×C₂₅₇

—

C₂²×C₂₅₇

C₂×C₄⋊C₄

yes

characteristic

C₂²×C₂₅₇

C₄×Q₈

yes

—

C₄×C₂₅₇

—

C₄×C₂₅₇

#108

yes

complemented

C₄×C₂₅₇

C₂×C₄²

yes

characteristic

C₄×C₂₅₇

#117

yes

characteristic

C₄×C₂₅₇

C₄×D₄

yes

—

C₂×C₂₅₇

—

C₂×C₂₅₇

C₄×C₄⋊C₄

yes

characteristic

C₂×C₂₅₇

C₂×#108

yes

—

C₂₅₇

C₄×#108

yes

characteristic, complemented, Sylow

C₄×#108

—

257

maximal, Sylow

C₂×#108

—

257

—

C₄×M₁₆

—

257

—

C₄×M₁₆

D₂₅₇

yes

characteristic

C₂×C₄×C₈

—

257

—

#341

—

257

—

C₂×M₁₆

—

257

—

C₂×M₁₆

C₂×D₂₅₇

yes

—

C₂²×C₈

—

257

—

C₂²×C₈

—

257

—

C₂×C₄²

C₂×D₂₅₇

yes

characteristic

#108

—

257

—

#97

—

#97

—

514

—

C₄×C₈

—

257

—

C₄×C₈

—

C₄×C₈

—

514

—

C₂²×C₄

Dic₅₁₄

yes

—

C₂²×C₄

C₂²×D₂₅₇

yes

characteristic

M₁₆

—

257

—

M₁₆

—

M₁₆

—

514

—

M₁₆

C₄×D₂₅₇

yes

—

C₂×C₈

—

257

—

C₂×C₈

—

257

—

C₂×C₈

—

257

—

C₂×C₈

—

C₂×C₈

—

514

—

C₂×C₈

—

514

—

C₄²

C₄×D₂₅₇

yes

—

C₄²

C₂²⋊D₂₅₇

yes

characteristic

C₄²

C₂²⋊D₂₅₇

yes

characteristic

C₂³

C₂×Dic₅₁₄

yes

characteristic

C₂×C₄

—

C₂×C₄

#16861597

yes

—

C₂×C₄

#16861597

yes

—

C₂×C₄

C₂×C₄×D₂₅₇

yes

—

C₂×C₄

C₂×C₄×D₂₅₇

yes

characteristic

C₂×C₄

C₂×C₂²⋊D₂₅₇

yes

characteristic

C₂×C₄

#16861618

yes

—

C₈

—

257

—

C₈

—

C₈

—

514

—

C₈

—

514

—

100

C₂²

—

101

C₂²

C₂×#16861597

yes

characteristic

102

C₂²

C₄×Dic₅₁₄

yes

—

103

C₄

—

104

C₄

#22551527

yes

direct product factor

105

C₄

C₄²×D₂₅₇

yes

characteristic

106

C₄

#22551549

yes

characteristic

107

C₄

C₄×C₂²⋊D₂₅₇

yes

—

108

C₂

—

109

C₂

C₄×#16861597

yes

characteristic

110

C₂

C₂×#22551527

yes

—

	subgroup	G/H
center	C₄²	C₂²⋊D₂₅₇
commutator	C₄×C₂₅₇	C₂×C₄²
Frattini	C₂²×C₄	C₂²×D₂₅₇
Fitting	C₄×M₁₆×C₂₅₇	C₂
socle	C₂²×C₂₅₇	C₂×C₄⋊C₄
prime radical	C₂³×C₂₅₇	C₂×Q₈
cosocle	C₂²×C₄×C₂₅₇	C₂³
hypercenter	C₄×M₁₆	D₂₅₇
hypocenter	C₂₅₇	C₄×#108
LCS	G ▷ C₄×C₂₅₇ ▷ C₂×C₂₅₇ ▷ C₂₅₇
UCS	C₁ ◁ C₄² ◁ C₂×C₄² ◁ C₄×M₁₆

subgroup

G/H

center

C₄²

C₂²⋊D₂₅₇

commutator

C₄×C₂₅₇

C₂×C₄²

Frattini

C₂²×C₄

C₂²×D₂₅₇

Fitting

C₄×M₁₆×C₂₅₇

C₂

socle

C₂²×C₂₅₇

C₂×C₄⋊C₄

prime radical

C₂³×C₂₅₇

C₂×Q₈

cosocle

C₂²×C₄×C₂₅₇

C₂³

hypercenter

C₄×M₁₆

D₂₅₇

hypocenter

C₂₅₇

C₄×#108

LCS

G ▷ C₄×C₂₅₇ ▷ C₂×C₂₅₇ ▷ C₂₅₇

UCS

C₁ ◁ C₄² ◁ C₂×C₄² ◁ C₄×M₁₆

N	H ≅ G/N	type
9. #22551527	C₄	direct (N × H)
48. C₄×C₂₅₇	#108	semidirect
55. C₂₅₇	C₄×#108	semidirect
104. C₄	#22551527	direct (N × H)

H ≅ G/N

type

9. #22551527

C₄

direct (N × H)

48. C₄×C₂₅₇

#108

semidirect

55. C₂₅₇

C₄×#108

semidirect

104. C₄

#22551527

direct (N × H)

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

subgroup	index
C₂×#22551527	2
C₄×#16861602	2
C₂×C₄×#14523339	2
#30656501	2
C₄×M₁₆×C₂₅₇	2
C₄×#108	257

subgroup

index

C₄×M₁₆×C₂₅₇

257

Subgroups — C₄×#22551527 #45664223

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C4×#22551527 #45664223

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₄×#22551527 #45664223