Group #44545865 — subgroups

Subgroups — #1278×C₂₅₁ #44545865

Order 32128.

172 conjugacy classes of subgroups; only the 96 normal subgroups are shown.

normal subgroups only

This group has 172 conjugacy classes of subgroups — too many to draw. Only its 96 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	#431×C₂₅₁	C₂	1	1	yes	maximal, characteristic
2	C₄×M₁₆×C₂₅₁	C₂	1	1	yes	maximal, characteristic
3	C₂×C₄×C₈×C₂₅₁	C₂	1	1	yes	maximal, characteristic
4	C₂×M₁₆×C₂₅₁	C₄	1	2	yes	—
5	C₂²×C₈×C₂₅₁	—	2	1	no	—
6	C₂×C₄²×C₂₅₁	C₂²	1	1	yes	characteristic
7	C₄⋊C₈×C₂₅₁	C₄	1	2	yes	—
8	C₂²⋊C₈×C₂₅₁	—	2	1	no	—
9	#97×C₂₅₁	—	2	1	no	—
10	C₄×C₈×C₂₅₁	—	2	1	no	—
11	C₄×C₈×C₂₅₁	—	2	1	no	—
12	C₄×C₈×C₂₅₁	—	2	1	no	—
13	C₄×C₈×C₂₅₁	C₄	1	2	yes	—
14	C₂²×C₄×C₂₅₁	C₂×C₄	1	1	yes	characteristic
15	C₂²×C₄×C₂₅₁	D₄	1	1	yes	characteristic
16	C₂²×C₄×C₂₅₁	Q₈	1	1	yes	characteristic
17	M₁₆×C₂₅₁	—	2	2	no	—
18	M₁₆×C₂₅₁	C₈	1	4	yes	complemented
19	C₂×C₈×C₂₅₁	—	2	4	no	—
20	C₂×C₈×C₂₅₁	—	2	2	no	—
21	C₂×C₈×C₂₅₁	—	2	2	no	—
22	C₂×C₈×C₂₅₁	—	2	2	no	—
23	C₄²×C₂₅₁	C₂×C₄	1	1	yes	characteristic
24	C₄²×C₂₅₁	C₂×C₄	1	1	yes	characteristic
25	C₄²×C₂₅₁	D₄	1	1	yes	characteristic
26	C₄²×C₂₅₁	D₄	1	1	yes	characteristic
27	C₂³×C₂₅₁	C₄⋊C₄	1	1	yes	characteristic
28	C₂×C₄×C₂₅₁	—	2	2	no	—
29	C₂×C₄×C₂₅₁	—	2	2	no	—
30	C₂×C₄×C₂₅₁	C₄²	1	1	yes	characteristic
31	C₂×C₄×C₂₅₁	C₂²⋊C₄	1	1	yes	characteristic
32	C₂×C₄×C₂₅₁	C₂²⋊C₄	1	1	yes	characteristic
33	C₂×C₄×C₂₅₁	C₂²⋊C₄	1	1	yes	characteristic
34	C₂×C₄×C₂₅₁	C₄⋊C₄	1	1	yes	characteristic
35	C₂×C₄×C₂₅₁	C₄⋊C₄	1	1	yes	characteristic
36	C₂×C₄×C₂₅₁	C₂×C₈	1	2	yes	—
37	C₂×C₄×C₂₅₁	M₁₆	1	2	yes	—
38	C₈×C₂₅₁	—	2	4	no	—
39	C₈×C₂₅₁	—	2	4	no	—
40	C₈×C₂₅₁	—	4	2	no	—
41	C₂²×C₂₅₁	—	2	1	no	—
42	C₂²×C₂₅₁	—	2	1	no	—
43	C₂²×C₂₅₁	#95	1	1	yes	characteristic
44	C₂²×C₂₅₁	C₄⋊C₈	1	2	yes	—
45	C₄×C₂₅₁	—	2	1	no	—
46	C₄×C₂₅₁	—	2	1	no	—
47	C₄×C₂₅₁	C₄×C₈	1	1	yes	characteristic
48	C₄×C₂₅₁	#97	1	1	yes	characteristic
49	C₄×C₂₅₁	C₂²⋊C₈	1	2	yes	—
50	C₄×C₂₅₁	#104	1	2	yes	—
51	C₄×C₂₅₁	#108	1	2	yes	—
52	C₂×C₂₅₁	—	2	2	no	—
53	C₂×C₂₅₁	#336	1	1	yes	characteristic
54	C₂×C₂₅₁	#339	1	1	yes	characteristic
55	C₂×C₂₅₁	#341	1	1	yes	characteristic
56	C₂₅₁	#1278	1	1	yes	characteristic, direct product factor, Sylow
57	#1278	C₂₅₁	1	1	yes	maximal, characteristic, direct product factor, Sylow
58	#431	C₂×C₂₅₁	1	1	yes	characteristic
59	C₄×M₁₆	C₂×C₂₅₁	1	1	yes	characteristic
60	C₂×C₄×C₈	C₂×C₂₅₁	1	1	yes	characteristic
61	C₂×M₁₆	C₄×C₂₅₁	1	2	yes	—
62	C₂²×C₈	—	2	1	no	—
63	C₂×C₄²	C₂²×C₂₅₁	1	1	yes	characteristic
64	C₄⋊C₈	C₄×C₂₅₁	1	2	yes	—
65	C₂²⋊C₈	—	2	1	no	—
66	#97	—	2	1	no	—
67	C₄×C₈	—	2	1	no	—
68	C₄×C₈	—	2	1	no	—
69	C₄×C₈	—	2	1	no	—
70	C₄×C₈	C₄×C₂₅₁	1	2	yes	—
71	C₂²×C₄	C₂×C₄×C₂₅₁	1	1	yes	characteristic
72	C₂²×C₄	D₄×C₂₅₁	1	1	yes	characteristic
73	C₂²×C₄	Q₈×C₂₅₁	1	1	yes	characteristic
74	M₁₆	—	2	2	no	—
75	M₁₆	C₈×C₂₅₁	1	4	yes	complemented
76	C₂×C₈	—	2	4	no	—
77	C₂×C₈	—	2	2	no	—
78	C₂×C₈	—	2	2	no	—
79	C₂×C₈	—	2	2	no	—
80	C₄²	C₂×C₄×C₂₅₁	1	1	yes	characteristic
81	C₄²	C₂×C₄×C₂₅₁	1	1	yes	characteristic
82	C₄²	D₄×C₂₅₁	1	1	yes	characteristic
83	C₄²	D₄×C₂₅₁	1	1	yes	characteristic
84	C₂³	C₄⋊C₄×C₂₅₁	1	1	yes	characteristic
85	C₂×C₄	—	2	2	no	—
86	C₂×C₄	—	2	2	no	—
87	C₂×C₄	C₄²×C₂₅₁	1	1	yes	characteristic
88	C₂×C₄	C₂²⋊C₄×C₂₅₁	1	1	yes	characteristic
89	C₂×C₄	C₂²⋊C₄×C₂₅₁	1	1	yes	characteristic
90	C₂×C₄	C₂²⋊C₄×C₂₅₁	1	1	yes	characteristic
91	C₂×C₄	C₄⋊C₄×C₂₅₁	1	1	yes	characteristic
92	C₂×C₄	C₄⋊C₄×C₂₅₁	1	1	yes	characteristic
93	C₂×C₄	C₂×C₈×C₂₅₁	1	2	yes	—
94	C₂×C₄	M₁₆×C₂₅₁	1	2	yes	—
95	C₈	—	2	4	no	—
96	C₈	—	2	4	no	—
97	C₈	—	4	2	no	—
98	C₂²	—	2	1	no	—
99	C₂²	—	2	1	no	—
100	C₂²	#95×C₂₅₁	1	1	yes	characteristic
101	C₂²	C₄⋊C₈×C₂₅₁	1	2	yes	—
102	C₄	—	2	1	no	—
103	C₄	—	2	1	no	—
104	C₄	C₄×C₈×C₂₅₁	1	1	yes	characteristic
105	C₄	#97×C₂₅₁	1	1	yes	characteristic
106	C₄	C₂²⋊C₈×C₂₅₁	1	2	yes	—
107	C₄	#104×C₂₅₁	1	2	yes	—
108	C₄	#108×C₂₅₁	1	2	yes	—
109	C₂	—	2	2	no	—
110	C₂	#336×C₂₅₁	1	1	yes	characteristic
111	C₂	#339×C₂₅₁	1	1	yes	characteristic
112	C₂	#341×C₂₅₁	1	1	yes	characteristic

subgroup

G/H

class size

multiplicity

normal?

properties

#431×C₂₅₁

C₂

yes

maximal, characteristic

C₄×M₁₆×C₂₅₁

C₂

yes

maximal, characteristic

C₂×C₄×C₈×C₂₅₁

C₂

yes

maximal, characteristic

C₂×M₁₆×C₂₅₁

C₄

yes

—

C₂²×C₈×C₂₅₁

—

C₂×C₄²×C₂₅₁

C₂²

yes

characteristic

C₄⋊C₈×C₂₅₁

C₄

yes

—

C₂²⋊C₈×C₂₅₁

—

#97×C₂₅₁

—

C₄×C₈×C₂₅₁

—

C₄×C₈×C₂₅₁

—

C₄×C₈×C₂₅₁

—

C₄×C₈×C₂₅₁

C₄

yes

—

C₂²×C₄×C₂₅₁

C₂×C₄

yes

characteristic

C₂²×C₄×C₂₅₁

D₄

yes

characteristic

C₂²×C₄×C₂₅₁

Q₈

yes

characteristic

M₁₆×C₂₅₁

—

M₁₆×C₂₅₁

C₈

yes

complemented

C₂×C₈×C₂₅₁

—

C₂×C₈×C₂₅₁

—

C₂×C₈×C₂₅₁

—

C₂×C₈×C₂₅₁

—

C₄²×C₂₅₁

C₂×C₄

yes

characteristic

C₄²×C₂₅₁

C₂×C₄

yes

characteristic

C₄²×C₂₅₁

D₄

yes

characteristic

C₄²×C₂₅₁

D₄

yes

characteristic

C₂³×C₂₅₁

C₄⋊C₄

yes

characteristic

C₂×C₄×C₂₅₁

—

C₂×C₄×C₂₅₁

—

C₂×C₄×C₂₅₁

C₄²

yes

characteristic

C₂×C₄×C₂₅₁

C₂²⋊C₄

yes

characteristic

C₂×C₄×C₂₅₁

C₂²⋊C₄

yes

characteristic

C₂×C₄×C₂₅₁

C₂²⋊C₄

yes

characteristic

C₂×C₄×C₂₅₁

C₄⋊C₄

yes

characteristic

C₂×C₄×C₂₅₁

C₄⋊C₄

yes

characteristic

C₂×C₄×C₂₅₁

C₂×C₈

yes

—

C₂×C₄×C₂₅₁

M₁₆

yes

—

C₈×C₂₅₁

—

C₈×C₂₅₁

—

C₈×C₂₅₁

—

C₂²×C₂₅₁

—

C₂²×C₂₅₁

—

C₂²×C₂₅₁

#95

yes

characteristic

C₂²×C₂₅₁

C₄⋊C₈

yes

—

C₄×C₂₅₁

—

C₄×C₂₅₁

—

C₄×C₂₅₁

C₄×C₈

yes

characteristic

C₄×C₂₅₁

#97

yes

characteristic

C₄×C₂₅₁

C₂²⋊C₈

yes

—

C₄×C₂₅₁

#104

yes

—

C₄×C₂₅₁

#108

yes

—

C₂×C₂₅₁

—

C₂×C₂₅₁

yes

C₂×C₂₅₁

yes

C₂×C₂₅₁

yes

C₂₅₁

yes

characteristic, direct product factor, Sylow

#1278

C₂₅₁

yes

maximal, characteristic, direct product factor, Sylow

#431

C₂×C₂₅₁

yes

characteristic

C₄×M₁₆

C₂×C₂₅₁

yes

characteristic

C₂×C₄×C₈

C₂×C₂₅₁

yes

characteristic

C₂×M₁₆

C₄×C₂₅₁

yes

—

C₂²×C₈

—

C₂×C₄²

C₂²×C₂₅₁

yes

characteristic

C₄⋊C₈

C₄×C₂₅₁

yes

—

C₂²⋊C₈

—

#97

—

C₄×C₈

—

C₄×C₈

—

C₄×C₈

—

C₄×C₈

C₄×C₂₅₁

yes

—

C₂²×C₄

C₂×C₄×C₂₅₁

yes

characteristic

C₂²×C₄

D₄×C₂₅₁

yes

characteristic

C₂²×C₄

Q₈×C₂₅₁

yes

characteristic

M₁₆

—

M₁₆

C₈×C₂₅₁

yes

complemented

C₂×C₈

—

C₂×C₈

—

C₂×C₈

—

C₂×C₈

—

C₄²

C₂×C₄×C₂₅₁

yes

characteristic

C₄²

C₂×C₄×C₂₅₁

yes

characteristic

C₄²

D₄×C₂₅₁

yes

characteristic

C₄²

D₄×C₂₅₁

yes

characteristic

C₂³

C₄⋊C₄×C₂₅₁

yes

characteristic

C₂×C₄

—

C₂×C₄

—

C₂×C₄

C₄²×C₂₅₁

yes

characteristic

C₂×C₄

C₂²⋊C₄×C₂₅₁

yes

characteristic

C₂×C₄

C₂²⋊C₄×C₂₅₁

yes

characteristic

C₂×C₄

C₂²⋊C₄×C₂₅₁

yes

characteristic

C₂×C₄

C₄⋊C₄×C₂₅₁

yes

characteristic

C₂×C₄

C₄⋊C₄×C₂₅₁

yes

characteristic

C₂×C₄

C₂×C₈×C₂₅₁

yes

—

C₂×C₄

M₁₆×C₂₅₁

yes

—

C₈

—

C₈

—

C₈

—

C₂²

—

C₂²

—

100

C₂²

#95×C₂₅₁

yes

characteristic

101

C₂²

C₄⋊C₈×C₂₅₁

yes

—

102

C₄

—

103

C₄

—

104

C₄

C₄×C₈×C₂₅₁

yes

characteristic

105

C₄

#97×C₂₅₁

yes

characteristic

106

C₄

C₂²⋊C₈×C₂₅₁

yes

—

107

C₄

#104×C₂₅₁

yes

—

108

C₄

#108×C₂₅₁

yes

—

109

C₂

—

110

C₂

#336×C₂₅₁

yes

characteristic

111

C₂

#339×C₂₅₁

yes

characteristic

112

C₂

#341×C₂₅₁

yes

characteristic

	subgroup	G/H
center	C₄²×C₂₅₁	D₄
commutator	C₄	C₄×C₈×C₂₅₁
Frattini, cosocle	C₂×C₄²	C₂²×C₂₅₁
socle	C₂²×C₂₅₁	#95
prime radical	C₂³×C₂₅₁	C₄⋊C₄
LCS	G ▷ C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₄²×C₂₅₁ ◁ C₂×C₄²×C₂₅₁ ◁ G

subgroup

G/H

center

C₄²×C₂₅₁

D₄

commutator

C₄

C₄×C₈×C₂₅₁

Frattini, cosocle

C₂×C₄²

C₂²×C₂₅₁

socle

C₂²×C₂₅₁

#95

prime radical

C₂³×C₂₅₁

C₄⋊C₄

LCS

G ▷ C₄ ▷ C₂ ▷ C₁

UCS

C₁ ◁ C₄²×C₂₅₁ ◁ C₂×C₄²×C₂₅₁ ◁ G

N	H ≅ G/N	type
18. M₁₆×C₂₅₁	C₈	semidirect
56. C₂₅₁	#1278	direct (N × H)
57. #1278	C₂₅₁	direct (N × H)
75. M₁₆	C₈×C₂₅₁	semidirect

H ≅ G/N

type

18. M₁₆×C₂₅₁

C₈

semidirect

56. C₂₅₁

#1278

direct (N × H)

57. #1278

C₂₅₁

direct (N × H)

75. M₁₆

C₈×C₂₅₁

semidirect

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

subgroup	index
#431×C₂₅₁	2
C₄×M₁₆×C₂₅₁	2
C₂×C₄×C₈×C₂₅₁	2
#1278	251

subgroup

index

#431×C₂₅₁

C₄×M₁₆×C₂₅₁

C₂×C₄×C₈×C₂₅₁

#1278

251

Subgroups — #1278×C₂₅₁ #44545865

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — #1278×C251 #44545865

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — #1278×C₂₅₁ #44545865