Group #44546221 — subgroups

Subgroups — #1634×C₂₅₁ #44546221

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	#555×C₂₅₁	C₂	1	1	yes	maximal, characteristic
2	#494×C₂₅₁	C₂	1	1	yes	maximal, characteristic, complemented
3	#476×C₂₅₁	C₂	1	2	yes	maximal, complemented
4	#333×C₂₅₁	C₂	1	1	yes	maximal, characteristic, complemented
5	D₄⋊C₈×C₂₅₁	C₂	1	2	yes	maximal, complemented
6	C₂×Q₁₆×C₂₅₁	—	2	2	no	—
7	C₄⋊Q₈×C₂₅₁	C₂²	1	1	yes	characteristic
8	C₄⋊Q₈×C₂₅₁	C₂²	1	1	yes	characteristic
9	#124×C₂₅₁	—	2	2	no	—
10	#121×C₂₅₁	—	2	2	no	—
11	C₄×D₄×C₂₅₁	C₂²	1	2	yes	—
12	#117×C₂₅₁	—	2	2	no	—
13	#106×C₂₅₁	—	2	2	no	—
14	C₄⋊C₈×C₂₅₁	C₂²	1	2	yes	—
15	#102×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	Q₁₆×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	—	4	2	no	—
34	Q₈×C₂₅₁	—	4	2	no	—
35	D₄×C₂₅₁	—	2	2	no	—
36	D₄×C₂₅₁	—	2	2	no	—
37	D₄×C₂₅₁	—	8	2	no	—
38	C₂×C₄×C₂₅₁	—	2	2	no	—
39	C₂×C₄×C₂₅₁	—	2	2	no	—
40	C₂×C₄×C₂₅₁	—	4	2	no	—
41	C₂×C₄×C₂₅₁	—	4	2	no	—
42	C₂×C₄×C₂₅₁	—	4	1	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	2	no	—
48	C₂²×C₂₅₁	—	4	2	no	—
49	C₂²×C₂₅₁	—	4	2	no	—
50	C₂²×C₂₅₁	#120	1	1	yes	characteristic
51	C₄×C₂₅₁	—	2	1	no	—
52	C₄×C₂₅₁	—	4	2	no	—
53	C₄×C₂₅₁	—	4	2	no	—
54	C₄×C₂₅₁	—	8	1	no	—
55	C₄×C₂₅₁	#135	1	2	yes	—
56	C₄×C₂₅₁	#137	1	2	yes	—
57	C₂×C₂₅₁	—	8	2	no	—
58	C₂×C₂₅₁	#452	1	2	yes	—
59	C₂×C₂₅₁	#453	1	1	yes	characteristic
60	C₂₅₁	#1634	1	1	yes	characteristic, direct product factor, Sylow
61	#1634	C₂₅₁	1	1	yes	maximal, characteristic, direct product factor, Sylow
62	#555	C₂×C₂₅₁	1	1	yes	characteristic
63	#494	C₂×C₂₅₁	1	1	yes	characteristic, complemented
64	#476	C₂×C₂₅₁	1	2	yes	complemented
65	#333	C₂×C₂₅₁	1	1	yes	characteristic, complemented
66	D₄⋊C₈	C₂×C₂₅₁	1	2	yes	complemented
67	C₂×Q₁₆	—	2	2	no	—
68	C₄⋊Q₈	C₂²×C₂₅₁	1	1	yes	characteristic
69	C₄⋊Q₈	C₂²×C₂₅₁	1	1	yes	characteristic
70	#124	—	2	2	no	—
71	#121	—	2	2	no	—
72	C₄×D₄	C₂²×C₂₅₁	1	2	yes	—
73	#117	—	2	2	no	—
74	#106	—	2	2	no	—
75	C₄⋊C₈	C₂²×C₂₅₁	1	2	yes	—
76	#102	—	2	2	no	—
77	C₄×C₈	C₂²×C₂₅₁	1	1	yes	characteristic
78	C₂×Q₈	—	2	1	no	—
79	C₂×Q₈	D₄×C₂₅₁	1	2	yes	—
80	C₂×D₄	—	2	2	no	—
81	C₂×D₄	D₄×C₂₅₁	1	2	yes	—
82	C₂²×C₄	—	2	2	no	—
83	Q₁₆	—	4	4	no	—
84	C₂×C₈	—	2	1	no	—
85	C₂×C₈	—	2	2	no	—
86	C₄⋊C₄	—	2	2	no	—
87	C₄⋊C₄	—	2	2	no	—
88	C₄⋊C₄	D₄×C₂₅₁	1	2	yes	—
89	C₂²⋊C₄	—	2	4	no	—
90	C₂²⋊C₄	—	2	2	no	—
91	C₄²	—	2	2	no	—
92	C₄²	C₂³×C₂₅₁	1	1	yes	characteristic
93	C₂³	—	2	2	no	—
94	Q₈	—	4	2	no	—
95	Q₈	—	4	2	no	—
96	D₄	—	2	2	no	—
97	D₄	—	2	2	no	—
98	D₄	—	8	2	no	—
99	C₂×C₄	—	2	2	no	—
100	C₂×C₄	—	2	2	no	—
101	C₂×C₄	—	4	2	no	—
102	C₂×C₄	—	4	2	no	—
103	C₂×C₄	—	4	1	no	—
104	C₂×C₄	C₂×D₄×C₂₅₁	1	1	yes	characteristic
105	C₂×C₄	C₂×D₄×C₂₅₁	1	2	yes	—
106	C₈	—	2	2	no	—
107	C₈	—	4	2	no	—
108	C₂²	—	4	2	no	—
109	C₂²	—	4	2	no	—
110	C₂²	—	4	2	no	—
111	C₂²	#120×C₂₅₁	1	1	yes	characteristic
112	C₄	—	2	1	no	—
113	C₄	—	4	2	no	—
114	C₄	—	4	2	no	—
115	C₄	—	8	1	no	—
116	C₄	#135×C₂₅₁	1	2	yes	—
117	C₄	#137×C₂₅₁	1	2	yes	—
118	C₂	—	8	2	no	—
119	C₂	#452×C₂₅₁	1	2	yes	—
120	C₂	#453×C₂₅₁	1	1	yes	characteristic

subgroup

G/H

class size

multiplicity

normal?

properties

#555×C₂₅₁

C₂

yes

maximal, characteristic

#494×C₂₅₁

C₂

yes

maximal, characteristic, complemented

#476×C₂₅₁

C₂

yes

maximal, complemented

#333×C₂₅₁

C₂

yes

maximal, characteristic, complemented

D₄⋊C₈×C₂₅₁

C₂

yes

maximal, complemented

C₂×Q₁₆×C₂₅₁

—

C₄⋊Q₈×C₂₅₁

C₂²

yes

characteristic

C₄⋊Q₈×C₂₅₁

C₂²

yes

characteristic

#124×C₂₅₁

—

#121×C₂₅₁

—

C₄×D₄×C₂₅₁

C₂²

yes

—

#117×C₂₅₁

—

#106×C₂₅₁

—

C₄⋊C₈×C₂₅₁

C₂²

yes

—

#102×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

—

Q₁₆×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

—

D₄×C₂₅₁

—

D₄×C₂₅₁

—

D₄×C₂₅₁

—

C₂×C₄×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

—

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

#1634

yes

characteristic, direct product factor, Sylow

#1634

C₂₅₁

yes

maximal, characteristic, direct product factor, Sylow

#555

C₂×C₂₅₁

yes

characteristic

#494

C₂×C₂₅₁

yes

characteristic, complemented

#476

C₂×C₂₅₁

yes

complemented

#333

C₂×C₂₅₁

yes

characteristic, complemented

D₄⋊C₈

C₂×C₂₅₁

yes

complemented

C₂×Q₁₆

—

C₄⋊Q₈

C₂²×C₂₅₁

yes

characteristic

C₄⋊Q₈

C₂²×C₂₅₁

yes

characteristic

#124

—

#121

—

C₄×D₄

C₂²×C₂₅₁

yes

—

#117

—

#106

—

C₄⋊C₈

C₂²×C₂₅₁

yes

—

#102

—

C₄×C₈

C₂²×C₂₅₁

yes

characteristic

C₂×Q₈

—

C₂×Q₈

D₄×C₂₅₁

yes

—

C₂×D₄

—

C₂×D₄

D₄×C₂₅₁

yes

—

C₂²×C₄

—

Q₁₆

—

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

—

D₄

—

D₄

—

D₄

—

C₂×C₄

—

100

C₂×C₄

—

101

C₂×C₄

—

102

C₂×C₄

—

103

C₂×C₄

—

104

C₂×C₄

C₂×D₄×C₂₅₁

yes

characteristic

105

C₂×C₄

C₂×D₄×C₂₅₁

yes

—

106

C₈

—

107

C₈

—

108

C₂²

—

109

C₂²

—

110

C₂²

—

111

C₂²

#120×C₂₅₁

yes

characteristic

112

C₄

—

113

C₄

—

114

C₄

—

115

C₄

—

116

C₄

#135×C₂₅₁

yes

—

117

C₄

#137×C₂₅₁

yes

—

118

C₂

—

119

C₂

#452×C₂₅₁

yes

—

120

C₂

#453×C₂₅₁

yes

characteristic

	subgroup	G/H
center, socle	C₂²×C₂₅₁	#120
commutator, Frattini, cosocle	C₄²	C₂³×C₂₅₁
prime radical	#555×C₂₅₁	C₂
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

#555×C₂₅₁

C₂

LCS

G ▷ C₄² ▷ C₂² ▷ C₁

UCS

C₁ ◁ C₂²×C₂₅₁ ◁ C₄²×C₂₅₁ ◁ G

N	H ≅ G/N	type
2. #494×C₂₅₁	C₂	semidirect
3. #476×C₂₅₁	C₂	semidirect
4. #333×C₂₅₁	C₂	semidirect
5. D₄⋊C₈×C₂₅₁	C₂	semidirect
60. C₂₅₁	#1634	direct (N × H)
61. #1634	C₂₅₁	direct (N × H)
63. #494	C₂×C₂₅₁	semidirect
64. #476	C₂×C₂₅₁	semidirect
65. #333	C₂×C₂₅₁	semidirect
66. D₄⋊C₈	C₂×C₂₅₁	semidirect

H ≅ G/N

type

2. #494×C₂₅₁

C₂

semidirect

3. #476×C₂₅₁

C₂

semidirect

4. #333×C₂₅₁

C₂

semidirect

5. D₄⋊C₈×C₂₅₁

C₂

semidirect

60. C₂₅₁

#1634

direct (N × H)

61. #1634

C₂₅₁

direct (N × H)

63. #494

C₂×C₂₅₁

semidirect

64. #476

C₂×C₂₅₁

semidirect

65. #333

C₂×C₂₅₁

semidirect

66. D₄⋊C₈

C₂×C₂₅₁

semidirect

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

subgroup	index
#555×C₂₅₁	2
#494×C₂₅₁	2
#476×C₂₅₁	2
#333×C₂₅₁	2
D₄⋊C₈×C₂₅₁	2
#1634	251

subgroup

index

#555×C₂₅₁

#494×C₂₅₁

#476×C₂₅₁

#333×C₂₅₁

D₄⋊C₈×C₂₅₁

#1634

251

Subgroups — #1634×C₂₅₁ #44546221

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — #1634×C251 #44546221

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — #1634×C₂₅₁ #44546221