Group #44560887 — subgroups

Subgroups — #44560887

Order 32128.

348 conjugacy classes of subgroups; only the 149 normal subgroups are shown.

normal subgroups only

This group has 348 conjugacy classes of subgroups — too many to draw. Only its 149 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	#30632244	C₂	1	1	yes	maximal, characteristic
2	#30632097	C₂	1	4	yes	maximal, complemented
3	#30632093	C₂	1	4	yes	maximal, complemented
4	#30631723	C₂	1	4	yes	maximal
5	C₂×#22548626	C₂	1	1	yes	maximal, characteristic
6	#533×C₂₅₁	C₂	1	1	yes	maximal, characteristic
7	C₂²×C₂₅₁⋊C₈	C₂²	1	2	yes	—
8	#22548715	C₂²	1	2	yes	—
9	#22548707	C₂²	1	2	yes	—
10	#22548694	—	2	2	no	—
11	C₄×Dic₅₀₂	—	2	2	no	—
12	#22548674	C₂²	1	2	yes	—
13	#22548648	C₂²	1	8	yes	—
14	#22548646	C₂²	1	8	yes	—
15	#22548626	C₂²	1	2	yes	—
16	#22548626	C₂²	1	2	yes	—
17	C₄⋊Q₈×C₂₅₁	C₂²	1	2	yes	—
18	#125×C₂₅₁	C₂²	1	2	yes	—
19	C₂²⋊Q₈×C₂₅₁	—	2	2	no	—
20	C₄×Q₈×C₂₅₁	—	2	2	no	—
21	#117×C₂₅₁	C₂²	1	2	yes	—
22	C₂×C₄²×C₂₅₁	C₂²	1	1	yes	characteristic
23	C₂×Dic₅₀₂	—	2	2	no	—
24	C₂×C₂₅₁⋊C₈	Q₈	1	8	yes	complemented
25	C₂×C₂₅₁⋊C₈	C₂³	1	4	yes	—
26	C₄⋊Dic₂₅₁	—	2	2	no	—
27	C₄⋊Dic₂₅₁	C₂³	1	4	yes	—
28	Dic₂₅₁⋊C₄	—	2	4	no	—
29	C₂²⋊Dic₂₅₁	—	2	2	no	—
30	C₄×Dic₂₅₁	—	2	2	no	—
31	C₂×Q₈×C₂₅₁	—	2	2	no	—
32	C₂²×C₄×C₂₅₁	D₄	1	2	yes	—
33	C₂²×C₄×C₂₅₁	C₂³	1	1	yes	characteristic
34	C₄⋊C₄×C₂₅₁	—	2	4	no	—
35	C₄⋊C₄×C₂₅₁	—	2	2	no	—
36	C₄⋊C₄×C₂₅₁	C₂³	1	4	yes	—
37	C₂²⋊C₄×C₂₅₁	—	2	2	no	—
38	C₄²×C₂₅₁	—	2	2	no	—
39	C₄²×C₂₅₁	D₄	1	2	yes	—
40	C₄²×C₂₅₁	C₂³	1	1	yes	characteristic
41	C₄²×C₂₅₁	C₂³	1	1	yes	characteristic
42	Dic₅₀₂	—	4	4	no	—
43	C₂×Dic₂₅₁	—	2	4	no	—
44	C₂₅₁⋊C₈	C₂×Q₈	1	8	yes	—
45	C₂³×C₂₅₁	C₂×D₄	1	1	yes	characteristic
46	Q₈×C₂₅₁	—	4	4	no	—
47	C₂×C₄×C₂₅₁	—	2	4	no	—
48	C₂×C₄×C₂₅₁	—	4	2	no	—
49	C₂×C₄×C₂₅₁	C₂×D₄	1	2	yes	—
50	C₂×C₄×C₂₅₁	C₂×D₄	1	2	yes	—
51	C₂×C₄×C₂₅₁	C₂×D₄	1	1	yes	characteristic
52	C₂×C₄×C₂₅₁	C₂×Q₈	1	4	yes	—
53	C₂×C₄×C₂₅₁	C₂⁴	1	1	yes	characteristic
54	C₂²×C₂₅₁	—	2	2	no	—
55	C₂²×C₂₅₁	C₄⋊Q₈	1	2	yes	—
56	C₂²×C₂₅₁	C₂²×D₄	1	1	yes	characteristic
57	C₄×C₂₅₁	—	2	2	no	—
58	C₄×C₂₅₁	—	2	2	no	—
59	C₄×C₂₅₁	—	4	4	no	—
60	C₄×C₂₅₁	C₄⋊Q₈	1	2	yes	—
61	C₄×C₂₅₁	C₂²×Q₈	1	2	yes	—
62	Dic₂₅₁	—	4	4	no	—
63	C₂×C₂₅₁	—	2	2	no	—
64	C₂×C₂₅₁	C₂×C₄⋊Q₈	1	1	yes	characteristic
65	C₂×C₂₅₁	#575	1	2	yes	—
66	C₂₅₁	#3162	1	1	yes	characteristic, complemented, Sylow
67	#3162	—	251	1	no	maximal, Sylow
68	#533	—	251	1	no	—
69	#533	D₂₅₁	1	1	yes	characteristic
70	#501	—	251	4	no	—
71	#501	—	251	4	no	—
72	#427	—	251	4	no	—
73	C₂×#97	—	251	1	no	—
74	C₂²×C₈	—	251	2	no	—
75	C₄⋊Q₈	—	251	2	no	—
76	C₄⋊Q₈	C₂×D₂₅₁	1	2	yes	—
77	#125	—	251	2	no	—
78	#125	C₂×D₂₅₁	1	2	yes	—
79	C₂²⋊Q₈	—	2	2	no	—
80	C₂²⋊Q₈	—	502	2	no	—
81	C₄×Q₈	—	2	2	no	—
82	C₄×Q₈	—	502	2	no	—
83	#117	—	251	2	no	—
84	#117	C₂×D₂₅₁	1	2	yes	—
85	C₂×C₄²	C₂×D₂₅₁	1	1	yes	characteristic
86	#107	—	251	8	no	—
87	#106	—	251	8	no	—
88	#97	—	251	2	no	—
89	#97	—	251	2	no	—
90	C₂×Q₈	—	2	2	no	—
91	C₂×Q₈	—	502	2	no	—
92	C₂²×C₄	C₂²⋊D₂₅₁	1	2	yes	—
93	C₂²×C₄	C₂²×D₂₅₁	1	1	yes	characteristic
94	C₂×C₈	—	251	8	no	—
95	C₂×C₈	—	251	4	no	—
96	C₄⋊C₄	—	2	4	no	—
97	C₄⋊C₄	—	2	2	no	—
98	C₄⋊C₄	—	502	4	no	—
99	C₄⋊C₄	—	502	2	no	—
100	C₄⋊C₄	—	251	4	no	—
101	C₄⋊C₄	C₂²×D₂₅₁	1	4	yes	—
102	C₂²⋊C₄	—	2	2	no	—
103	C₂²⋊C₄	—	502	2	no	—
104	C₄²	—	2	2	no	—
105	C₄²	—	502	2	no	—
106	C₄²	C₂²⋊D₂₅₁	1	2	yes	—
107	C₄²	C₂²×D₂₅₁	1	1	yes	characteristic
108	C₄²	C₂²×D₂₅₁	1	1	yes	characteristic
109	C₂³	C₂×C₂²⋊D₂₅₁	1	1	yes	characteristic
110	Q₈	—	4	4	no	—
111	Q₈	—	1004	4	no	—
112	C₂×C₄	—	2	4	no	—
113	C₂×C₄	—	4	2	no	—
114	C₂×C₄	—	502	4	no	—
115	C₂×C₄	C₂×C₂²⋊D₂₅₁	1	2	yes	—
116	C₂×C₄	C₂×C₂²⋊D₂₅₁	1	2	yes	—
117	C₂×C₄	C₂×C₂²⋊D₂₅₁	1	1	yes	characteristic
118	C₂×C₄	Q₈×D₂₅₁	1	4	yes	—
119	C₂×C₄	C₂³×D₂₅₁	1	1	yes	characteristic
120	C₈	—	251	8	no	—
121	C₂²	—	2	2	no	—
122	C₂²	#22548713	1	2	yes	—
123	C₂²	C₂²×C₂²⋊D₂₅₁	1	1	yes	characteristic
124	C₄	—	2	2	no	—
125	C₄	—	2	2	no	—
126	C₄	—	4	4	no	—
127	C₄	—	1004	4	no	—
128	C₄	#22548713	1	2	yes	—
129	C₄	C₂×Q₈×D₂₅₁	1	2	yes	—
130	C₂	—	2	2	no	—
131	C₂	C₂×#22548713	1	1	yes	characteristic
132	C₂	#30632485	1	2	yes	—

subgroup

G/H

class size

multiplicity

normal?

properties

#30632244

C₂

yes

maximal, characteristic

#30632097

C₂

yes

maximal, complemented

#30632093

C₂

yes

maximal, complemented

#30631723

C₂

yes

maximal

C₂×#22548626

C₂

yes

maximal, characteristic

#533×C₂₅₁

C₂

yes

maximal, characteristic

C₂²×C₂₅₁⋊C₈

C₂²

yes

—

#22548715

C₂²

yes

—

#22548707

C₂²

yes

—

#22548694

—

C₄×Dic₅₀₂

—

#22548674

C₂²

yes

—

#22548648

C₂²

yes

—

#22548646

C₂²

yes

—

#22548626

C₂²

yes

—

#22548626

C₂²

yes

—

C₄⋊Q₈×C₂₅₁

C₂²

yes

—

#125×C₂₅₁

C₂²

yes

—

C₂²⋊Q₈×C₂₅₁

—

C₄×Q₈×C₂₅₁

—

#117×C₂₅₁

C₂²

yes

—

C₂×C₄²×C₂₅₁

C₂²

yes

characteristic

C₂×Dic₅₀₂

—

C₂×C₂₅₁⋊C₈

Q₈

yes

complemented

C₂×C₂₅₁⋊C₈

C₂³

yes

—

C₄⋊Dic₂₅₁

—

C₄⋊Dic₂₅₁

C₂³

yes

—

Dic₂₅₁⋊C₄

—

C₂²⋊Dic₂₅₁

—

C₄×Dic₂₅₁

—

C₂×Q₈×C₂₅₁

—

C₂²×C₄×C₂₅₁

D₄

yes

—

C₂²×C₄×C₂₅₁

C₂³

yes

characteristic

C₄⋊C₄×C₂₅₁

—

C₄⋊C₄×C₂₅₁

—

C₄⋊C₄×C₂₅₁

C₂³

yes

—

C₂²⋊C₄×C₂₅₁

—

C₄²×C₂₅₁

—

C₄²×C₂₅₁

D₄

yes

—

C₄²×C₂₅₁

C₂³

yes

characteristic

C₄²×C₂₅₁

C₂³

yes

characteristic

Dic₅₀₂

—

C₂×Dic₂₅₁

—

C₂₅₁⋊C₈

C₂×Q₈

yes

—

C₂³×C₂₅₁

C₂×D₄

yes

characteristic

Q₈×C₂₅₁

—

C₂×C₄×C₂₅₁

—

C₂×C₄×C₂₅₁

—

C₂×C₄×C₂₅₁

C₂×D₄

yes

—

C₂×C₄×C₂₅₁

C₂×D₄

yes

—

C₂×C₄×C₂₅₁

C₂×D₄

yes

characteristic

C₂×C₄×C₂₅₁

C₂×Q₈

yes

—

C₂×C₄×C₂₅₁

C₂⁴

yes

characteristic

C₂²×C₂₅₁

—

C₂²×C₂₅₁

C₄⋊Q₈

yes

—

C₂²×C₂₅₁

C₂²×D₄

yes

characteristic

C₄×C₂₅₁

—

C₄×C₂₅₁

—

C₄×C₂₅₁

—

C₄×C₂₅₁

C₄⋊Q₈

yes

—

C₄×C₂₅₁

C₂²×Q₈

yes

—

Dic₂₅₁

—

C₂×C₂₅₁

—

C₂×C₂₅₁

C₂×C₄⋊Q₈

yes

characteristic

C₂×C₂₅₁

#575

yes

—

C₂₅₁

#3162

yes

characteristic, complemented, Sylow

#3162

—

251

maximal, Sylow

#533

—

251

—

#533

D₂₅₁

yes

characteristic

#501

—

251

—

#501

—

251

—

#427

—

251

—

C₂×#97

—

251

—

C₂²×C₈

—

251

—

C₄⋊Q₈

—

251

—

C₄⋊Q₈

C₂×D₂₅₁

yes

—

#125

—

251

—

#125

C₂×D₂₅₁

yes

—

C₂²⋊Q₈

—

C₂²⋊Q₈

—

502

—

C₄×Q₈

—

C₄×Q₈

—

502

—

#117

—

251

—

#117

C₂×D₂₅₁

yes

—

C₂×C₄²

C₂×D₂₅₁

yes

characteristic

#107

—

251

—

#106

—

251

—

#97

—

251

—

#97

—

251

—

C₂×Q₈

—

C₂×Q₈

—

502

—

C₂²×C₄

C₂²⋊D₂₅₁

yes

—

C₂²×C₄

C₂²×D₂₅₁

yes

characteristic

C₂×C₈

—

251

—

C₂×C₈

—

251

—

C₄⋊C₄

—

C₄⋊C₄

—

C₄⋊C₄

—

502

—

C₄⋊C₄

—

502

—

100

C₄⋊C₄

—

251

—

101

C₄⋊C₄

C₂²×D₂₅₁

yes

—

102

C₂²⋊C₄

—

103

C₂²⋊C₄

—

502

—

104

C₄²

—

105

C₄²

—

502

—

106

C₄²

C₂²⋊D₂₅₁

yes

—

107

C₄²

C₂²×D₂₅₁

yes

characteristic

108

C₄²

C₂²×D₂₅₁

yes

characteristic

109

C₂³

C₂×C₂²⋊D₂₅₁

yes

characteristic

110

Q₈

—

111

Q₈

—

1004

—

112

C₂×C₄

—

113

C₂×C₄

—

114

C₂×C₄

—

502

—

115

C₂×C₄

C₂×C₂²⋊D₂₅₁

yes

—

116

C₂×C₄

C₂×C₂²⋊D₂₅₁

yes

—

117

C₂×C₄

C₂×C₂²⋊D₂₅₁

yes

characteristic

118

C₂×C₄

Q₈×D₂₅₁

yes

—

119

C₂×C₄

C₂³×D₂₅₁

yes

characteristic

120

C₈

—

251

—

121

C₂²

—

122

C₂²

#22548713

yes

—

123

C₂²

C₂²×C₂²⋊D₂₅₁

yes

characteristic

124

C₄

—

125

C₄

—

126

C₄

—

127

C₄

—

1004

—

128

C₄

#22548713

yes

—

129

C₄

C₂×Q₈×D₂₅₁

yes

—

130

C₂

—

131

C₂

C₂×#22548713

yes

characteristic

132

C₂

#30632485

yes

—

	subgroup	G/H
center	C₂×C₄	C₂×C₂²⋊D₂₅₁
commutator, cosocle	C₂×C₄×C₂₅₁	C₂⁴
Frattini	C₂×C₄	C₂³×D₂₅₁
Fitting	#533×C₂₅₁	C₂
socle	C₂²×C₂₅₁	C₂²×D₄
prime radical	C₂³×C₂₅₁	C₂×D₄
hypercenter	#533	D₂₅₁
hypocenter	C₂₅₁	#3162
LCS	G ▷ C₂×C₄×C₂₅₁ ▷ C₂×C₂₅₁ ▷ C₂₅₁
UCS	C₁ ◁ C₂×C₄ ◁ C₂×C₄² ◁ #533

subgroup

G/H

center

C₂×C₄

C₂×C₂²⋊D₂₅₁

commutator, cosocle

C₂×C₄×C₂₅₁

C₂⁴

Frattini

C₂×C₄

C₂³×D₂₅₁

Fitting

#533×C₂₅₁

C₂

socle

C₂²×C₂₅₁

C₂²×D₄

prime radical

C₂³×C₂₅₁

C₂×D₄

D₂₅₁

C₂₅₁

G ▷ C₂×C₄×C₂₅₁ ▷ C₂×C₂₅₁ ▷ C₂₅₁

UCS

C₁ ◁ C₂×C₄ ◁ C₂×C₄² ◁ #533

N	H ≅ G/N	type
2. #30632097	C₂	semidirect
3. #30632093	C₂	semidirect
24. C₂×C₂₅₁⋊C₈	Q₈	semidirect
66. C₂₅₁	#3162	semidirect

H ≅ G/N

type

2. #30632097

C₂

semidirect

3. #30632093

C₂

semidirect

24. C₂×C₂₅₁⋊C₈

Q₈

semidirect

66. C₂₅₁

#3162

semidirect

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

subgroup	index
#30632244	2
#30632097	2
#30632093	2
#30631723	2
C₂×#22548626	2
#533×C₂₅₁	2
#3162	251

subgroup

index

#533×C₂₅₁

251

Subgroups — #44560887

Conjugacy classes of subgroups

Complements

Maximal subgroups