Group #44546220 — subgroups

Subgroups — #1633×C₂₅₁ #44546220

Order 32128.

206 conjugacy classes of subgroups; only the 76 normal subgroups are shown.

normal subgroups only

This group has 206 conjugacy classes of subgroups — too many to draw. Only its 76 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	Q₈²×C₂₅₁	C₂	1	1	yes	maximal, characteristic
2	#494×C₂₅₁	C₂	1	1	yes	maximal, characteristic
3	#477×C₂₅₁	C₂	1	2	yes	maximal
4	#333×C₂₅₁	C₂	1	1	yes	maximal, characteristic
5	Q₈⋊C₈×C₂₅₁	C₂	1	2	yes	maximal
6	C₂×Q₁₆×C₂₅₁	—	2	2	no	—
7	C₄⋊Q₈×C₂₅₁	—	2	2	no	—
8	C₄⋊Q₈×C₂₅₁	—	2	2	no	—
9	C₄⋊Q₈×C₂₅₁	C₂²	1	1	yes	characteristic
10	C₄⋊Q₈×C₂₅₁	C₂²	1	1	yes	characteristic
11	C₄×Q₈×C₂₅₁	—	2	2	no	—
12	C₄×Q₈×C₂₅₁	C₂²	1	2	yes	—
13	#107×C₂₅₁	—	2	2	no	—
14	C₄⋊C₈×C₂₅₁	C₂²	1	2	yes	—
15	#103×C₂₅₁	—	2	2	no	—
16	C₄×C₈×C₂₅₁	C₂²	1	1	yes	characteristic
17	C₂×Q₈×C₂₅₁	—	2	2	no	—
18	C₂×Q₈×C₂₅₁	—	2	1	no	—
19	C₂×Q₈×C₂₅₁	D₄	1	2	yes	—
20	C₂×Q₈×C₂₅₁	D₄	1	2	yes	—
21	Q₁₆×C₂₅₁	—	4	4	no	—
22	C₂×C₈×C₂₅₁	—	2	1	no	—
23	C₂×C₈×C₂₅₁	—	2	2	no	—
24	C₄⋊C₄×C₂₅₁	—	2	4	no	—
25	C₄⋊C₄×C₂₅₁	—	2	2	no	—
26	C₄⋊C₄×C₂₅₁	—	2	2	no	—
27	C₄⋊C₄×C₂₅₁	—	2	2	no	—
28	C₄⋊C₄×C₂₅₁	D₄	1	2	yes	—
29	C₄²×C₂₅₁	—	2	2	no	—
30	C₄²×C₂₅₁	—	2	2	no	—
31	C₄²×C₂₅₁	C₂³	1	1	yes	characteristic
32	Q₈×C₂₅₁	—	2	2	no	—
33	Q₈×C₂₅₁	—	4	2	no	—
34	Q₈×C₂₅₁	—	4	2	no	—
35	Q₈×C₂₅₁	—	8	2	no	—
36	Q₈×C₂₅₁	Q₁₆	1	4	yes	complemented
37	C₂×C₄×C₂₅₁	—	2	2	no	—
38	C₂×C₄×C₂₅₁	—	2	2	no	—
39	C₂×C₄×C₂₅₁	—	2	2	no	—
40	C₂×C₄×C₂₅₁	—	4	1	no	—
41	C₂×C₄×C₂₅₁	C₂×D₄	1	1	yes	characteristic
42	C₂×C₄×C₂₅₁	C₂×D₄	1	2	yes	—
43	C₈×C₂₅₁	—	2	2	no	—
44	C₈×C₂₅₁	—	4	2	no	—
45	C₂²×C₂₅₁	#120	1	1	yes	characteristic
46	C₄×C₂₅₁	—	2	4	no	—
47	C₄×C₂₅₁	—	2	1	no	—
48	C₄×C₂₅₁	—	4	2	no	—
49	C₄×C₂₅₁	—	4	2	no	—
50	C₄×C₂₅₁	—	8	1	no	—
51	C₄×C₂₅₁	C₂×Q₁₆	1	2	yes	—
52	C₄×C₂₅₁	#137	1	2	yes	—
53	C₂×C₂₅₁	#451	1	2	yes	—
54	C₂×C₂₅₁	#453	1	1	yes	characteristic
55	C₂₅₁	#1633	1	1	yes	characteristic, direct product factor, Sylow
56	#1633	C₂₅₁	1	1	yes	maximal, characteristic, direct product factor, Sylow
57	Q₈²	C₂×C₂₅₁	1	1	yes	characteristic
58	#494	C₂×C₂₅₁	1	1	yes	characteristic
59	#477	C₂×C₂₅₁	1	2	yes	—
60	#333	C₂×C₂₅₁	1	1	yes	characteristic
61	Q₈⋊C₈	C₂×C₂₅₁	1	2	yes	—
62	C₂×Q₁₆	—	2	2	no	—
63	C₄⋊Q₈	—	2	2	no	—
64	C₄⋊Q₈	—	2	2	no	—
65	C₄⋊Q₈	C₂²×C₂₅₁	1	1	yes	characteristic
66	C₄⋊Q₈	C₂²×C₂₅₁	1	1	yes	characteristic
67	C₄×Q₈	—	2	2	no	—
68	C₄×Q₈	C₂²×C₂₅₁	1	2	yes	—
69	#107	—	2	2	no	—
70	C₄⋊C₈	C₂²×C₂₅₁	1	2	yes	—
71	#103	—	2	2	no	—
72	C₄×C₈	C₂²×C₂₅₁	1	1	yes	characteristic
73	C₂×Q₈	—	2	2	no	—
74	C₂×Q₈	—	2	1	no	—
75	C₂×Q₈	D₄×C₂₅₁	1	2	yes	—
76	C₂×Q₈	D₄×C₂₅₁	1	2	yes	—
77	Q₁₆	—	4	4	no	—
78	C₂×C₈	—	2	1	no	—
79	C₂×C₈	—	2	2	no	—
80	C₄⋊C₄	—	2	4	no	—
81	C₄⋊C₄	—	2	2	no	—
82	C₄⋊C₄	—	2	2	no	—
83	C₄⋊C₄	—	2	2	no	—
84	C₄⋊C₄	D₄×C₂₅₁	1	2	yes	—
85	C₄²	—	2	2	no	—
86	C₄²	—	2	2	no	—
87	C₄²	C₂³×C₂₅₁	1	1	yes	characteristic
88	Q₈	—	2	2	no	—
89	Q₈	—	4	2	no	—
90	Q₈	—	4	2	no	—
91	Q₈	—	8	2	no	—
92	Q₈	Q₁₆×C₂₅₁	1	4	yes	complemented
93	C₂×C₄	—	2	2	no	—
94	C₂×C₄	—	2	2	no	—
95	C₂×C₄	—	2	2	no	—
96	C₂×C₄	—	4	1	no	—
97	C₂×C₄	C₂×D₄×C₂₅₁	1	1	yes	characteristic
98	C₂×C₄	C₂×D₄×C₂₅₁	1	2	yes	—
99	C₈	—	2	2	no	—
100	C₈	—	4	2	no	—
101	C₂²	#120×C₂₅₁	1	1	yes	characteristic
102	C₄	—	2	4	no	—
103	C₄	—	2	1	no	—
104	C₄	—	4	2	no	—
105	C₄	—	4	2	no	—
106	C₄	—	8	1	no	—
107	C₄	C₂×Q₁₆×C₂₅₁	1	2	yes	—
108	C₄	#137×C₂₅₁	1	2	yes	—
109	C₂	#451×C₂₅₁	1	2	yes	—
110	C₂	#453×C₂₅₁	1	1	yes	characteristic

subgroup

G/H

class size

multiplicity

normal?

properties

Q₈²×C₂₅₁

C₂

yes

maximal, characteristic

#494×C₂₅₁

C₂

yes

maximal, characteristic

#477×C₂₅₁

C₂

yes

maximal

#333×C₂₅₁

C₂

yes

maximal, characteristic

Q₈⋊C₈×C₂₅₁

C₂

yes

maximal

C₂×Q₁₆×C₂₅₁

—

C₄⋊Q₈×C₂₅₁

—

C₄⋊Q₈×C₂₅₁

—

C₄⋊Q₈×C₂₅₁

C₂²

yes

characteristic

C₄⋊Q₈×C₂₅₁

C₂²

yes

characteristic

C₄×Q₈×C₂₅₁

—

C₄×Q₈×C₂₅₁

C₂²

yes

—

#107×C₂₅₁

—

C₄⋊C₈×C₂₅₁

C₂²

yes

—

#103×C₂₅₁

—

C₄×C₈×C₂₅₁

C₂²

yes

characteristic

C₂×Q₈×C₂₅₁

—

C₂×Q₈×C₂₅₁

—

C₂×Q₈×C₂₅₁

D₄

yes

—

C₂×Q₈×C₂₅₁

D₄

yes

—

Q₁₆×C₂₅₁

—

C₂×C₈×C₂₅₁

—

C₂×C₈×C₂₅₁

—

C₄⋊C₄×C₂₅₁

—

C₄⋊C₄×C₂₅₁

—

C₄⋊C₄×C₂₅₁

—

C₄⋊C₄×C₂₅₁

—

C₄⋊C₄×C₂₅₁

D₄

yes

—

C₄²×C₂₅₁

—

C₄²×C₂₅₁

—

C₄²×C₂₅₁

C₂³

yes

characteristic

Q₈×C₂₅₁

—

Q₈×C₂₅₁

—

Q₈×C₂₅₁

—

Q₈×C₂₅₁

—

Q₈×C₂₅₁

Q₁₆

yes

complemented

C₂×C₄×C₂₅₁

—

C₂×C₄×C₂₅₁

—

C₂×C₄×C₂₅₁

—

C₂×C₄×C₂₅₁

—

C₂×C₄×C₂₅₁

C₂×D₄

yes

characteristic

C₂×C₄×C₂₅₁

C₂×D₄

yes

—

C₈×C₂₅₁

—

C₈×C₂₅₁

—

C₂²×C₂₅₁

#120

yes

characteristic

C₄×C₂₅₁

—

C₄×C₂₅₁

—

C₄×C₂₅₁

—

C₄×C₂₅₁

—

C₄×C₂₅₁

—

C₄×C₂₅₁

C₂×Q₁₆

yes

—

C₄×C₂₅₁

#137

yes

—

C₂×C₂₅₁

#451

yes

—

C₂×C₂₅₁

#453

yes

characteristic

C₂₅₁

#1633

yes

characteristic, direct product factor, Sylow

#1633

C₂₅₁

yes

maximal, characteristic, direct product factor, Sylow

Q₈²

C₂×C₂₅₁

yes

characteristic

#494

C₂×C₂₅₁

yes

characteristic

#477

C₂×C₂₅₁

yes

—

#333

C₂×C₂₅₁

yes

characteristic

Q₈⋊C₈

C₂×C₂₅₁

yes

—

C₂×Q₁₆

—

C₄⋊Q₈

—

C₄⋊Q₈

—

C₄⋊Q₈

C₂²×C₂₅₁

yes

characteristic

C₄⋊Q₈

C₂²×C₂₅₁

yes

characteristic

C₄×Q₈

—

C₄×Q₈

C₂²×C₂₅₁

yes

—

#107

—

C₄⋊C₈

C₂²×C₂₅₁

yes

—

#103

—

C₄×C₈

C₂²×C₂₅₁

yes

characteristic

C₂×Q₈

—

C₂×Q₈

—

C₂×Q₈

D₄×C₂₅₁

yes

—

C₂×Q₈

D₄×C₂₅₁

yes

—

Q₁₆

—

C₂×C₈

—

C₂×C₈

—

C₄⋊C₄

—

C₄⋊C₄

—

C₄⋊C₄

—

C₄⋊C₄

—

C₄⋊C₄

D₄×C₂₅₁

yes

—

C₄²

—

C₄²

—

C₄²

C₂³×C₂₅₁

yes

characteristic

Q₈

—

Q₈

—

Q₈

—

Q₈

—

Q₈

Q₁₆×C₂₅₁

yes

complemented

C₂×C₄

—

C₂×C₄

—

C₂×C₄

—

C₂×C₄

—

C₂×C₄

C₂×D₄×C₂₅₁

yes

characteristic

C₂×C₄

C₂×D₄×C₂₅₁

yes

—

C₈

—

100

C₈

—

101

C₂²

#120×C₂₅₁

yes

characteristic

102

C₄

—

103

C₄

—

104

C₄

—

105

C₄

—

106

C₄

—

107

C₄

C₂×Q₁₆×C₂₅₁

yes

—

108

C₄

#137×C₂₅₁

yes

—

109

C₂

#451×C₂₅₁

yes

—

110

C₂

#453×C₂₅₁

yes

characteristic

	subgroup	G/H
center, socle, prime radical	C₂²×C₂₅₁	#120
commutator, Frattini, cosocle	C₄²	C₂³×C₂₅₁
LCS	G ▷ C₄² ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂²×C₂₅₁ ◁ C₄²×C₂₅₁ ◁ G

subgroup

G/H

center, socle, prime radical

C₂²×C₂₅₁

#120

commutator, Frattini, cosocle

C₄²

C₂³×C₂₅₁

LCS

G ▷ C₄² ▷ C₂² ▷ C₁

UCS

C₁ ◁ C₂²×C₂₅₁ ◁ C₄²×C₂₅₁ ◁ G

N	H ≅ G/N	type
36. Q₈×C₂₅₁	Q₁₆	semidirect
55. C₂₅₁	#1633	direct (N × H)
56. #1633	C₂₅₁	direct (N × H)
92. Q₈	Q₁₆×C₂₅₁	semidirect

H ≅ G/N

type

36. Q₈×C₂₅₁

Q₁₆

semidirect

55. C₂₅₁

#1633

direct (N × H)

56. #1633

C₂₅₁

direct (N × H)

92. Q₈

Q₁₆×C₂₅₁

semidirect

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

subgroup	index
Q₈²×C₂₅₁	2
#494×C₂₅₁	2
#477×C₂₅₁	2
#333×C₂₅₁	2
Q₈⋊C₈×C₂₅₁	2
#1633	251

subgroup

index

Q₈²×C₂₅₁

#494×C₂₅₁

#477×C₂₅₁

#333×C₂₅₁

Q₈⋊C₈×C₂₅₁

#1633

251

Subgroups — #1633×C₂₅₁ #44546220

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — #1633×C251 #44546220

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — #1633×C₂₅₁ #44546220