Group #44546217 — subgroups

Subgroups — #1630×C₂₅₁ #44546217

Order 32128.

218 conjugacy classes of subgroups; only the 68 normal subgroups are shown.

normal subgroups only

This group has 218 conjugacy classes of subgroups — too many to draw. Only its 68 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	#559×C₂₅₁	C₂	1	1	yes	maximal, characteristic
2	#492×C₂₅₁	C₂	1	1	yes	maximal, characteristic
3	#477×C₂₅₁	C₂	1	2	yes	maximal, complemented
4	#331×C₂₅₁	C₂	1	1	yes	maximal, characteristic
5	Q₈⋊C₈×C₂₅₁	C₂	1	2	yes	maximal, complemented
6	C₂×SD₁₆×C₂₅₁	—	2	2	no	—
7	C₄⋊Q₈×C₂₅₁	C₂²	1	1	yes	characteristic
8	#127×C₂₅₁	C₂²	1	1	yes	characteristic
9	#124×C₂₅₁	—	2	2	no	—
10	#123×C₂₅₁	—	2	2	no	—
11	C₄×Q₈×C₂₅₁	C₂²	1	2	yes	—
12	C₄×D₄×C₂₅₁	—	2	2	no	—
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	1	no	—
18	C₂×Q₈×C₂₅₁	D₄	1	2	yes	—
19	C₂×D₄×C₂₅₁	—	2	2	no	—
20	C₂×D₄×C₂₅₁	D₄	1	2	yes	—
21	C₂²×C₄×C₂₅₁	—	2	2	no	—
22	SD₁₆×C₂₅₁	—	4	4	no	—
23	C₂×C₈×C₂₅₁	—	2	1	no	—
24	C₂×C₈×C₂₅₁	—	2	2	no	—
25	C₄⋊C₄×C₂₅₁	—	2	2	no	—
26	C₄⋊C₄×C₂₅₁	—	2	2	no	—
27	C₄⋊C₄×C₂₅₁	D₄	1	2	yes	—
28	C₂²⋊C₄×C₂₅₁	—	2	4	no	—
29	C₂²⋊C₄×C₂₅₁	—	2	2	no	—
30	C₄²×C₂₅₁	—	2	2	no	—
31	C₄²×C₂₅₁	C₂³	1	1	yes	characteristic
32	C₂³×C₂₅₁	—	2	2	no	—
33	Q₈×C₂₅₁	—	2	2	no	—
34	Q₈×C₂₅₁	—	2	2	no	—
35	Q₈×C₂₅₁	—	4	2	no	—
36	D₄×C₂₅₁	—	4	2	no	—
37	D₄×C₂₅₁	—	4	2	no	—
38	D₄×C₂₅₁	—	4	2	no	—
39	C₂×C₄×C₂₅₁	—	2	2	no	—
40	C₂×C₄×C₂₅₁	—	2	2	no	—
41	C₂×C₄×C₂₅₁	—	4	1	no	—
42	C₂×C₄×C₂₅₁	—	8	2	no	—
43	C₂×C₄×C₂₅₁	C₂×D₄	1	1	yes	characteristic
44	C₂×C₄×C₂₅₁	C₂×D₄	1	2	yes	—
45	C₈×C₂₅₁	—	2	2	no	—
46	C₈×C₂₅₁	—	4	2	no	—
47	C₂²×C₂₅₁	—	4	4	no	—
48	C₂²×C₂₅₁	—	4	2	no	—
49	C₂²×C₂₅₁	#120	1	1	yes	characteristic
50	C₄×C₂₅₁	—	2	1	no	—
51	C₄×C₂₅₁	—	4	2	no	—
52	C₄×C₂₅₁	—	4	2	no	—
53	C₄×C₂₅₁	—	8	1	no	—
54	C₄×C₂₅₁	#135	1	2	yes	—
55	C₄×C₂₅₁	#137	1	2	yes	—
56	C₂×C₂₅₁	—	8	2	no	—
57	C₂×C₂₅₁	#452	1	2	yes	—
58	C₂×C₂₅₁	#453	1	1	yes	characteristic
59	C₂₅₁	#1630	1	1	yes	characteristic, direct product factor, Sylow
60	#1630	C₂₅₁	1	1	yes	maximal, characteristic, direct product factor, Sylow
61	#559	C₂×C₂₅₁	1	1	yes	characteristic
62	#492	C₂×C₂₅₁	1	1	yes	characteristic
63	#477	C₂×C₂₅₁	1	2	yes	complemented
64	#331	C₂×C₂₅₁	1	1	yes	characteristic
65	Q₈⋊C₈	C₂×C₂₅₁	1	2	yes	complemented
66	C₂×SD₁₆	—	2	2	no	—
67	C₄⋊Q₈	C₂²×C₂₅₁	1	1	yes	characteristic
68	#127	C₂²×C₂₅₁	1	1	yes	characteristic
69	#124	—	2	2	no	—
70	#123	—	2	2	no	—
71	C₄×Q₈	C₂²×C₂₅₁	1	2	yes	—
72	C₄×D₄	—	2	2	no	—
73	#107	—	2	2	no	—
74	C₄⋊C₈	C₂²×C₂₅₁	1	2	yes	—
75	#103	—	2	2	no	—
76	C₄×C₈	C₂²×C₂₅₁	1	1	yes	characteristic
77	C₂×Q₈	—	2	1	no	—
78	C₂×Q₈	D₄×C₂₅₁	1	2	yes	—
79	C₂×D₄	—	2	2	no	—
80	C₂×D₄	D₄×C₂₅₁	1	2	yes	—
81	C₂²×C₄	—	2	2	no	—
82	SD₁₆	—	4	4	no	—
83	C₂×C₈	—	2	1	no	—
84	C₂×C₈	—	2	2	no	—
85	C₄⋊C₄	—	2	2	no	—
86	C₄⋊C₄	—	2	2	no	—
87	C₄⋊C₄	D₄×C₂₅₁	1	2	yes	—
88	C₂²⋊C₄	—	2	4	no	—
89	C₂²⋊C₄	—	2	2	no	—
90	C₄²	—	2	2	no	—
91	C₄²	C₂³×C₂₅₁	1	1	yes	characteristic
92	C₂³	—	2	2	no	—
93	Q₈	—	2	2	no	—
94	Q₈	—	2	2	no	—
95	Q₈	—	4	2	no	—
96	D₄	—	4	2	no	—
97	D₄	—	4	2	no	—
98	D₄	—	4	2	no	—
99	C₂×C₄	—	2	2	no	—
100	C₂×C₄	—	2	2	no	—
101	C₂×C₄	—	4	1	no	—
102	C₂×C₄	—	8	2	no	—
103	C₂×C₄	C₂×D₄×C₂₅₁	1	1	yes	characteristic
104	C₂×C₄	C₂×D₄×C₂₅₁	1	2	yes	—
105	C₈	—	2	2	no	—
106	C₈	—	4	2	no	—
107	C₂²	—	4	4	no	—
108	C₂²	—	4	2	no	—
109	C₂²	#120×C₂₅₁	1	1	yes	characteristic
110	C₄	—	2	1	no	—
111	C₄	—	4	2	no	—
112	C₄	—	4	2	no	—
113	C₄	—	8	1	no	—
114	C₄	#135×C₂₅₁	1	2	yes	—
115	C₄	#137×C₂₅₁	1	2	yes	—
116	C₂	—	8	2	no	—
117	C₂	#452×C₂₅₁	1	2	yes	—
118	C₂	#453×C₂₅₁	1	1	yes	characteristic

subgroup

G/H

class size

multiplicity

normal?

properties

#559×C₂₅₁

C₂

yes

maximal, characteristic

#492×C₂₅₁

C₂

yes

maximal, characteristic

#477×C₂₅₁

C₂

yes

maximal, complemented

#331×C₂₅₁

C₂

yes

maximal, characteristic

Q₈⋊C₈×C₂₅₁

C₂

yes

maximal, complemented

C₂×SD₁₆×C₂₅₁

—

C₄⋊Q₈×C₂₅₁

C₂²

yes

characteristic

#127×C₂₅₁

C₂²

yes

characteristic

#124×C₂₅₁

—

#123×C₂₅₁

—

C₄×Q₈×C₂₅₁

C₂²

yes

—

C₄×D₄×C₂₅₁

—

#107×C₂₅₁

—

C₄⋊C₈×C₂₅₁

C₂²

yes

—

#103×C₂₅₁

—

C₄×C₈×C₂₅₁

C₂²

yes

characteristic

C₂×Q₈×C₂₅₁

—

C₂×Q₈×C₂₅₁

D₄

yes

—

C₂×D₄×C₂₅₁

—

C₂×D₄×C₂₅₁

D₄

yes

—

C₂²×C₄×C₂₅₁

—

SD₁₆×C₂₅₁

—

C₂×C₈×C₂₅₁

—

C₂×C₈×C₂₅₁

—

C₄⋊C₄×C₂₅₁

—

C₄⋊C₄×C₂₅₁

—

C₄⋊C₄×C₂₅₁

D₄

yes

—

C₂²⋊C₄×C₂₅₁

—

C₂²⋊C₄×C₂₅₁

—

C₄²×C₂₅₁

—

C₄²×C₂₅₁

C₂³

yes

characteristic

C₂³×C₂₅₁

—

Q₈×C₂₅₁

—

Q₈×C₂₅₁

—

Q₈×C₂₅₁

—

D₄×C₂₅₁

—

D₄×C₂₅₁

—

D₄×C₂₅₁

—

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

—

C₂²×C₂₅₁

—

C₂²×C₂₅₁

#120

yes

characteristic

C₄×C₂₅₁

—

C₄×C₂₅₁

—

C₄×C₂₅₁

—

C₄×C₂₅₁

—

C₄×C₂₅₁

#135

yes

—

C₄×C₂₅₁

#137

yes

—

C₂×C₂₅₁

—

C₂×C₂₅₁

#452

yes

—

C₂×C₂₅₁

#453

yes

characteristic

C₂₅₁

#1630

yes

characteristic, direct product factor, Sylow

#1630

C₂₅₁

yes

maximal, characteristic, direct product factor, Sylow

#559

C₂×C₂₅₁

yes

characteristic

#492

C₂×C₂₅₁

yes

characteristic

#477

C₂×C₂₅₁

yes

complemented

#331

C₂×C₂₅₁

yes

characteristic

Q₈⋊C₈

C₂×C₂₅₁

yes

complemented

C₂×SD₁₆

—

C₄⋊Q₈

C₂²×C₂₅₁

yes

characteristic

#127

C₂²×C₂₅₁

yes

characteristic

#124

—

#123

—

C₄×Q₈

C₂²×C₂₅₁

yes

—

C₄×D₄

—

#107

—

C₄⋊C₈

C₂²×C₂₅₁

yes

—

#103

—

C₄×C₈

C₂²×C₂₅₁

yes

characteristic

C₂×Q₈

—

C₂×Q₈

D₄×C₂₅₁

yes

—

C₂×D₄

—

C₂×D₄

D₄×C₂₅₁

yes

—

C₂²×C₄

—

SD₁₆

—

C₂×C₈

—

C₂×C₈

—

C₄⋊C₄

—

C₄⋊C₄

—

C₄⋊C₄

D₄×C₂₅₁

yes

—

C₂²⋊C₄

—

C₂²⋊C₄

—

C₄²

—

C₄²

C₂³×C₂₅₁

yes

characteristic

C₂³

—

Q₈

—

Q₈

—

Q₈

—

D₄

—

D₄

—

D₄

—

C₂×C₄

—

100

C₂×C₄

—

101

C₂×C₄

—

102

C₂×C₄

—

103

C₂×C₄

C₂×D₄×C₂₅₁

yes

characteristic

104

C₂×C₄

C₂×D₄×C₂₅₁

yes

—

105

C₈

—

106

C₈

—

107

C₂²

—

108

C₂²

—

109

C₂²

#120×C₂₅₁

yes

characteristic

110

C₄

—

111

C₄

—

112

C₄

—

113

C₄

—

114

C₄

#135×C₂₅₁

yes

—

115

C₄

#137×C₂₅₁

yes

—

116

C₂

—

117

C₂

#452×C₂₅₁

yes

—

118

C₂

#453×C₂₅₁

yes

characteristic

	subgroup	G/H
center, socle	C₂²×C₂₅₁	#120
commutator, Frattini, cosocle	C₄²	C₂³×C₂₅₁
prime radical	#127×C₂₅₁	C₂²
LCS	G ▷ C₄² ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂²×C₂₅₁ ◁ C₄²×C₂₅₁ ◁ G

subgroup

G/H

center, socle

C₂²×C₂₅₁

#120

commutator, Frattini, cosocle

C₄²

C₂³×C₂₅₁

prime radical

#127×C₂₅₁

C₂²

LCS

G ▷ C₄² ▷ C₂² ▷ C₁

UCS

C₁ ◁ C₂²×C₂₅₁ ◁ C₄²×C₂₅₁ ◁ G

N	H ≅ G/N	type
3. #477×C₂₅₁	C₂	semidirect
5. Q₈⋊C₈×C₂₅₁	C₂	semidirect
59. C₂₅₁	#1630	direct (N × H)
60. #1630	C₂₅₁	direct (N × H)
63. #477	C₂×C₂₅₁	semidirect
65. Q₈⋊C₈	C₂×C₂₅₁	semidirect

H ≅ G/N

type

3. #477×C₂₅₁

C₂

semidirect

5. Q₈⋊C₈×C₂₅₁

C₂

semidirect

59. C₂₅₁

#1630

direct (N × H)

60. #1630

C₂₅₁

direct (N × H)

63. #477

C₂×C₂₅₁

semidirect

65. Q₈⋊C₈

C₂×C₂₅₁

semidirect

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

subgroup	index
#559×C₂₅₁	2
#492×C₂₅₁	2
#477×C₂₅₁	2
#331×C₂₅₁	2
Q₈⋊C₈×C₂₅₁	2
#1630	251

subgroup

index

#559×C₂₅₁

#492×C₂₅₁

#477×C₂₅₁

#331×C₂₅₁

Q₈⋊C₈×C₂₅₁

#1630

251

Subgroups — #1630×C₂₅₁ #44546217

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — #1630×C251 #44546217

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — #1630×C₂₅₁ #44546217