Group #44545883 — subgroups

Subgroups — #1296×C₂₅₁ #44545883

Order 32128.

188 conjugacy classes of subgroups; only the 112 normal subgroups are shown.

normal subgroups only

This group has 188 conjugacy classes of subgroups — too many to draw. Only its 112 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×C₄⋊C₈×C₂₅₁	C₂	1	2	yes	maximal
2	#384×C₂₅₁	C₂	1	1	yes	maximal, characteristic
3	C₂²×C₈×C₂₅₁	—	2	2	no	—
4	C₂×C₄⋊C₄×C₂₅₁	—	2	2	no	—
5	C₂×C₄⋊C₄×C₂₅₁	C₄	1	2	yes	—
6	C₂×C₄²×C₂₅₁	C₂²	1	1	yes	characteristic
7	C₄⋊C₈×C₂₅₁	—	2	2	no	—
8	C₄⋊C₈×C₂₅₁	C₄	1	4	yes	complemented
9	C₂²×C₄×C₂₅₁	—	2	2	no	—
10	C₂²×C₄×C₂₅₁	C₂×C₄	1	1	yes	characteristic
11	C₂²×C₄×C₂₅₁	D₄	1	2	yes	—
12	C₂×C₈×C₂₅₁	—	2	4	no	—
13	C₂×C₈×C₂₅₁	—	4	2	no	—
14	C₂×C₈×C₂₅₁	—	4	2	no	—
15	C₄⋊C₄×C₂₅₁	—	4	2	no	—
16	C₄⋊C₄×C₂₅₁	—	4	2	no	—
17	C₄⋊C₄×C₂₅₁	—	4	2	no	—
18	C₄²×C₂₅₁	C₂×C₄	1	2	yes	—
19	C₄²×C₂₅₁	D₄	1	1	yes	characteristic
20	C₄²×C₂₅₁	Q₈	1	1	yes	characteristic
21	C₂³×C₂₅₁	C₂²⋊C₄	1	1	yes	characteristic
22	C₂×C₄×C₂₅₁	—	2	1	no	—
23	C₂×C₄×C₂₅₁	—	2	1	no	—
24	C₂×C₄×C₂₅₁	—	4	2	no	—
25	C₂×C₄×C₂₅₁	—	4	4	no	—
26	C₂×C₄×C₂₅₁	C₄²	1	1	yes	characteristic
27	C₂×C₄×C₂₅₁	C₂²⋊C₄	1	2	yes	—
28	C₂×C₄×C₂₅₁	C₄⋊C₄	1	1	yes	characteristic
29	C₂×C₄×C₂₅₁	C₄⋊C₄	1	2	yes	—
30	C₂×C₄×C₂₅₁	D₈	1	2	yes	—
31	C₂×C₄×C₂₅₁	SD₁₆	1	2	yes	—
32	C₂×C₄×C₂₅₁	SD₁₆	1	2	yes	—
33	C₂×C₄×C₂₅₁	Q₁₆	1	2	yes	—
34	C₈×C₂₅₁	—	4	4	no	—
35	C₂²×C₂₅₁	#95	1	1	yes	characteristic
36	C₂²×C₂₅₁	#100	1	1	yes	characteristic
37	C₂²×C₂₅₁	#101	1	1	yes	characteristic
38	C₂²×C₂₅₁	#102	1	2	yes	—
39	C₂²×C₂₅₁	#103	1	2	yes	—
40	C₄×C₂₅₁	—	2	1	no	—
41	C₄×C₂₅₁	—	2	1	no	—
42	C₄×C₂₅₁	—	8	2	no	—
43	C₄×C₂₅₁	#102	1	2	yes	—
44	C₄×C₂₅₁	#103	1	2	yes	—
45	C₄×C₂₅₁	#106	1	2	yes	—
46	C₄×C₂₅₁	#107	1	2	yes	—
47	C₂×C₂₅₁	#331	1	1	yes	characteristic
48	C₂×C₂₅₁	#332	1	2	yes	—
49	C₂×C₂₅₁	#333	1	1	yes	characteristic
50	C₂×C₂₅₁	#340	1	2	yes	—
51	C₂×C₂₅₁	#343	1	1	yes	characteristic
52	C₂₅₁	#1296	1	1	yes	characteristic, direct product factor, Sylow
53	#1296	C₂₅₁	1	1	yes	maximal, characteristic, direct product factor, Sylow
54	C₂×C₄⋊C₈	C₂×C₂₅₁	1	2	yes	—
55	#384	C₂×C₂₅₁	1	1	yes	characteristic
56	C₂²×C₈	—	2	2	no	—
57	C₂×C₄⋊C₄	—	2	2	no	—
58	C₂×C₄⋊C₄	C₄×C₂₅₁	1	2	yes	—
59	C₂×C₄²	C₂²×C₂₅₁	1	1	yes	characteristic
60	C₄⋊C₈	—	2	2	no	—
61	C₄⋊C₈	C₄×C₂₅₁	1	4	yes	complemented
62	C₂²×C₄	—	2	2	no	—
63	C₂²×C₄	C₂×C₄×C₂₅₁	1	1	yes	characteristic
64	C₂²×C₄	D₄×C₂₅₁	1	2	yes	—
65	C₂×C₈	—	2	4	no	—
66	C₂×C₈	—	4	2	no	—
67	C₂×C₈	—	4	2	no	—
68	C₄⋊C₄	—	4	2	no	—
69	C₄⋊C₄	—	4	2	no	—
70	C₄⋊C₄	—	4	2	no	—
71	C₄²	C₂×C₄×C₂₅₁	1	2	yes	—
72	C₄²	D₄×C₂₅₁	1	1	yes	characteristic
73	C₄²	Q₈×C₂₅₁	1	1	yes	characteristic
74	C₂³	C₂²⋊C₄×C₂₅₁	1	1	yes	characteristic
75	C₂×C₄	—	2	1	no	—
76	C₂×C₄	—	2	1	no	—
77	C₂×C₄	—	4	2	no	—
78	C₂×C₄	—	4	4	no	—
79	C₂×C₄	C₄²×C₂₅₁	1	1	yes	characteristic
80	C₂×C₄	C₂²⋊C₄×C₂₅₁	1	2	yes	—
81	C₂×C₄	C₄⋊C₄×C₂₅₁	1	1	yes	characteristic
82	C₂×C₄	C₄⋊C₄×C₂₅₁	1	2	yes	—
83	C₂×C₄	D₈×C₂₅₁	1	2	yes	—
84	C₂×C₄	SD₁₆×C₂₅₁	1	2	yes	—
85	C₂×C₄	SD₁₆×C₂₅₁	1	2	yes	—
86	C₂×C₄	Q₁₆×C₂₅₁	1	2	yes	—
87	C₈	—	4	4	no	—
88	C₂²	#95×C₂₅₁	1	1	yes	characteristic
89	C₂²	#100×C₂₅₁	1	1	yes	characteristic
90	C₂²	#101×C₂₅₁	1	1	yes	characteristic
91	C₂²	#102×C₂₅₁	1	2	yes	—
92	C₂²	#103×C₂₅₁	1	2	yes	—
93	C₄	—	2	1	no	—
94	C₄	—	2	1	no	—
95	C₄	—	8	2	no	—
96	C₄	#102×C₂₅₁	1	2	yes	—
97	C₄	#103×C₂₅₁	1	2	yes	—
98	C₄	#106×C₂₅₁	1	2	yes	—
99	C₄	#107×C₂₅₁	1	2	yes	—
100	C₂	#331×C₂₅₁	1	1	yes	characteristic
101	C₂	#332×C₂₅₁	1	2	yes	—
102	C₂	#333×C₂₅₁	1	1	yes	characteristic
103	C₂	#340×C₂₅₁	1	2	yes	—
104	C₂	#343×C₂₅₁	1	1	yes	characteristic

subgroup

G/H

class size

multiplicity

normal?

properties

C₂×C₄⋊C₈×C₂₅₁

C₂

yes

maximal

#384×C₂₅₁

C₂

yes

maximal, characteristic

C₂²×C₈×C₂₅₁

—

C₂×C₄⋊C₄×C₂₅₁

—

C₂×C₄⋊C₄×C₂₅₁

C₄

yes

—

C₂×C₄²×C₂₅₁

C₂²

yes

characteristic

C₄⋊C₈×C₂₅₁

—

C₄⋊C₈×C₂₅₁

C₄

yes

complemented

C₂²×C₄×C₂₅₁

—

C₂²×C₄×C₂₅₁

C₂×C₄

yes

characteristic

C₂²×C₄×C₂₅₁

D₄

yes

—

C₂×C₈×C₂₅₁

—

C₂×C₈×C₂₅₁

—

C₂×C₈×C₂₅₁

—

C₄⋊C₄×C₂₅₁

—

C₄⋊C₄×C₂₅₁

—

C₄⋊C₄×C₂₅₁

—

C₄²×C₂₅₁

C₂×C₄

yes

—

C₄²×C₂₅₁

D₄

yes

characteristic

C₄²×C₂₅₁

Q₈

yes

characteristic

C₂³×C₂₅₁

C₂²⋊C₄

yes

characteristic

C₂×C₄×C₂₅₁

—

C₂×C₄×C₂₅₁

—

C₂×C₄×C₂₅₁

—

C₂×C₄×C₂₅₁

—

C₂×C₄×C₂₅₁

C₄²

yes

characteristic

C₂×C₄×C₂₅₁

C₂²⋊C₄

yes

—

C₂×C₄×C₂₅₁

C₄⋊C₄

yes

characteristic

C₂×C₄×C₂₅₁

C₄⋊C₄

yes

—

C₂×C₄×C₂₅₁

D₈

yes

—

C₂×C₄×C₂₅₁

SD₁₆

yes

—

C₂×C₄×C₂₅₁

SD₁₆

yes

—

C₂×C₄×C₂₅₁

Q₁₆

yes

—

C₈×C₂₅₁

—

C₂²×C₂₅₁

yes

C₂²×C₂₅₁

yes

C₂²×C₂₅₁

yes

C₂²×C₂₅₁

yes

—

C₂²×C₂₅₁

#103

yes

—

C₄×C₂₅₁

—

C₄×C₂₅₁

—

C₄×C₂₅₁

—

C₄×C₂₅₁

#102

yes

—

C₄×C₂₅₁

#103

yes

—

C₄×C₂₅₁

#106

yes

—

C₄×C₂₅₁

#107

yes

—

C₂×C₂₅₁

#331

yes

characteristic

C₂×C₂₅₁

#332

yes

—

C₂×C₂₅₁

#333

yes

characteristic

C₂×C₂₅₁

#340

yes

—

C₂×C₂₅₁

#343

yes

characteristic

C₂₅₁

#1296

yes

characteristic, direct product factor, Sylow

#1296

C₂₅₁

yes

maximal, characteristic, direct product factor, Sylow

C₂×C₄⋊C₈

C₂×C₂₅₁

yes

—

#384

C₂×C₂₅₁

yes

characteristic

C₂²×C₈

—

C₂×C₄⋊C₄

—

C₂×C₄⋊C₄

C₄×C₂₅₁

yes

—

C₂×C₄²

C₂²×C₂₅₁

yes

characteristic

C₄⋊C₈

—

C₄⋊C₈

C₄×C₂₅₁

yes

complemented

C₂²×C₄

—

C₂²×C₄

C₂×C₄×C₂₅₁

yes

characteristic

C₂²×C₄

D₄×C₂₅₁

yes

—

C₂×C₈

—

C₂×C₈

—

C₂×C₈

—

C₄⋊C₄

—

C₄⋊C₄

—

C₄⋊C₄

—

C₄²

C₂×C₄×C₂₅₁

yes

—

C₄²

D₄×C₂₅₁

yes

characteristic

C₄²

Q₈×C₂₅₁

yes

characteristic

C₂³

C₂²⋊C₄×C₂₅₁

yes

characteristic

C₂×C₄

—

C₂×C₄

—

C₂×C₄

—

C₂×C₄

—

C₂×C₄

C₄²×C₂₅₁

yes

characteristic

C₂×C₄

C₂²⋊C₄×C₂₅₁

yes

—

C₂×C₄

C₄⋊C₄×C₂₅₁

yes

characteristic

C₂×C₄

C₄⋊C₄×C₂₅₁

yes

—

C₂×C₄

D₈×C₂₅₁

yes

—

C₂×C₄

SD₁₆×C₂₅₁

yes

—

C₂×C₄

SD₁₆×C₂₅₁

yes

—

C₂×C₄

Q₁₆×C₂₅₁

yes

—

C₈

—

C₂²

#95×C₂₅₁

yes

characteristic

C₂²

#100×C₂₅₁

yes

characteristic

C₂²

#101×C₂₅₁

yes

characteristic

C₂²

#102×C₂₅₁

yes

—

C₂²

#103×C₂₅₁

yes

—

C₄

—

C₄

—

C₄

—

C₄

#102×C₂₅₁

yes

—

C₄

#103×C₂₅₁

yes

—

C₄

#106×C₂₅₁

yes

—

C₄

#107×C₂₅₁

yes

—

100

C₂

#331×C₂₅₁

yes

characteristic

101

C₂

#332×C₂₅₁

yes

—

102

C₂

#333×C₂₅₁

yes

characteristic

103

C₂

#340×C₂₅₁

yes

—

104

C₂

#343×C₂₅₁

yes

characteristic

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

subgroup

G/H

center, socle, prime radical

C₂³×C₂₅₁

C₂²⋊C₄

commutator

C₂×C₄

C₄²×C₂₅₁

Frattini, cosocle

C₂×C₄²

C₂²×C₂₅₁

LCS

G ▷ C₂×C₄ ▷ C₂² ▷ C₁

UCS

C₁ ◁ C₂³×C₂₅₁ ◁ C₂×C₄²×C₂₅₁ ◁ G

N	H ≅ G/N	type
8. C₄⋊C₈×C₂₅₁	C₄	semidirect
52. C₂₅₁	#1296	direct (N × H)
53. #1296	C₂₅₁	direct (N × H)
61. C₄⋊C₈	C₄×C₂₅₁	semidirect

H ≅ G/N

type

8. C₄⋊C₈×C₂₅₁

C₄

semidirect

52. C₂₅₁

#1296

direct (N × H)

53. #1296

C₂₅₁

direct (N × H)

61. C₄⋊C₈

C₄×C₂₅₁

semidirect

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

subgroup	index
C₂×C₄⋊C₈×C₂₅₁	2
#384×C₂₅₁	2
#1296	251

subgroup

index

C₂×C₄⋊C₈×C₂₅₁

#384×C₂₅₁

#1296

251

Subgroups — #1296×C₂₅₁ #44545883

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — #1296×C251 #44545883

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — #1296×C₂₅₁ #44545883