Group #44548665 — subgroups

Subgroups — #44548665

Order 32128.

178 conjugacy classes of subgroups; only the 101 normal subgroups are shown.

normal subgroups only

This group has 178 conjugacy classes of subgroups — too many to draw. Only its 101 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	#30631740	C₂	1	1	yes	maximal, characteristic
2	#30631740	C₂	1	1	yes	maximal, characteristic
3	#30631409	C₂	1	1	yes	maximal, characteristic, complemented
4	#30631409	C₂	1	1	yes	maximal, characteristic, complemented
5	#30631408	C₂	1	1	yes	maximal, characteristic, complemented
6	C₈×C₂₅₁⋊C₈	C₂	1	1	yes	maximal, characteristic, complemented
7	#431×C₂₅₁	C₂	1	1	yes	maximal, characteristic
8	C₄⋊(C₂₅₁⋊C₈)	C₄	1	2	yes	—
9	C₄⋊(C₂₅₁⋊C₈)	C₄	1	2	yes	—
10	C₂²⋊(C₂₅₁⋊C₈)	C₄	1	2	yes	—
11	C₂²⋊(C₂₅₁⋊C₈)	C₄	1	2	yes	—
12	C₄×C₂₅₁⋊C₈	C₂²	1	1	yes	characteristic
13	C₄×C₂₅₁⋊C₈	C₂²	1	1	yes	characteristic
14	C₄×C₂₅₁⋊C₈	C₂²	1	1	yes	characteristic
15	C₄×C₂₅₁⋊C₈	C₂²	1	1	yes	characteristic
16	C₂×C₄²×C₂₅₁	C₂²	1	1	yes	characteristic
17	C₄⋊C₈×C₂₅₁	C₄	1	2	yes	—
18	C₂²⋊C₈×C₂₅₁	C₄	1	2	yes	—
19	C₄×C₈×C₂₅₁	C₂²	1	1	yes	characteristic
20	C₄×C₈×C₂₅₁	C₂²	1	1	yes	characteristic
21	C₂×C₂₅₁⋊C₈	C₂×C₄	1	2	yes	—
22	C₂×C₂₅₁⋊C₈	C₂×C₄	1	2	yes	—
23	C₂×C₂₅₁⋊C₈	C₂×C₄	1	2	yes	—
24	C₂×C₂₅₁⋊C₈	C₂×C₄	1	2	yes	—
25	C₂²×C₄×C₂₅₁	C₂×C₄	1	1	yes	characteristic
26	C₂²×C₄×C₂₅₁	C₂×C₄	1	1	yes	characteristic
27	C₂²×C₄×C₂₅₁	C₂×C₄	1	1	yes	characteristic
28	C₂×C₈×C₂₅₁	C₂×C₄	1	2	yes	—
29	C₂×C₈×C₂₅₁	C₂×C₄	1	2	yes	—
30	C₄²×C₂₅₁	C₂×C₄	1	1	yes	characteristic
31	C₄²×C₂₅₁	C₂×C₄	1	1	yes	characteristic
32	C₄²×C₂₅₁	C₂×C₄	1	1	yes	characteristic
33	C₄²×C₂₅₁	C₂³	1	1	yes	characteristic
34	C₂₅₁⋊C₈	—	2	2	no	—
35	C₂₅₁⋊C₈	—	2	2	no	—
36	C₂₅₁⋊C₈	—	2	2	no	—
37	C₂₅₁⋊C₈	—	2	2	no	—
38	C₂³×C₂₅₁	C₄²	1	1	yes	characteristic
39	C₂×C₄×C₂₅₁	—	2	2	no	—
40	C₂×C₄×C₂₅₁	—	2	2	no	—
41	C₂×C₄×C₂₅₁	—	2	2	no	—
42	C₂×C₄×C₂₅₁	C₄²	1	1	yes	characteristic
43	C₂×C₄×C₂₅₁	C₄²	1	1	yes	characteristic
44	C₂×C₄×C₂₅₁	C₄²	1	1	yes	characteristic
45	C₂×C₄×C₂₅₁	C₂²×C₄	1	1	yes	characteristic
46	C₂×C₄×C₂₅₁	C₂²×C₄	1	1	yes	characteristic
47	C₂×C₄×C₂₅₁	C₂²×C₄	1	1	yes	characteristic
48	C₈×C₂₅₁	—	2	2	no	—
49	C₈×C₂₅₁	—	2	2	no	—
50	C₂²×C₂₅₁	—	2	1	no	—
51	C₂²×C₂₅₁	—	2	1	no	—
52	C₂²×C₂₅₁	—	2	1	no	—
53	C₂²×C₂₅₁	C₂×C₄²	1	1	yes	characteristic
54	C₄×C₂₅₁	—	2	1	no	—
55	C₄×C₂₅₁	—	2	1	no	—
56	C₄×C₂₅₁	—	2	1	no	—
57	C₄×C₂₅₁	#131	1	2	yes	—
58	C₄×C₂₅₁	#131	1	2	yes	—
59	C₄×C₂₅₁	#131	1	2	yes	—
60	C₂×C₂₅₁	—	4	1	no	—
61	C₂×C₂₅₁	#405	1	1	yes	characteristic
62	C₂×C₂₅₁	#405	1	1	yes	characteristic
63	C₂×C₂₅₁	#405	1	1	yes	characteristic
64	C₂₅₁	#1454	1	1	yes	characteristic, complemented, Sylow
65	#1454	—	251	1	no	maximal, Sylow
66	#431	—	251	1	no	—
67	#431	—	251	1	no	—
68	#431	D₂₅₁	1	1	yes	characteristic
69	#322	—	251	1	no	—
70	#322	—	251	1	no	—
71	#322	—	251	1	no	—
72	C₈²	—	251	1	no	—
73	C₂×C₄²	C₂×D₂₅₁	1	1	yes	characteristic
74	C₄⋊C₈	—	251	2	no	—
75	C₄⋊C₈	—	251	2	no	—
76	C₄⋊C₈	Dic₂₅₁	1	2	yes	—
77	C₂²⋊C₈	—	251	2	no	—
78	C₂²⋊C₈	—	251	2	no	—
79	C₂²⋊C₈	Dic₂₅₁	1	2	yes	—
80	C₄×C₈	—	251	1	no	—
81	C₄×C₈	—	251	1	no	—
82	C₄×C₈	—	251	1	no	—
83	C₄×C₈	—	251	1	no	—
84	C₄×C₈	C₂×D₂₅₁	1	1	yes	characteristic
85	C₄×C₈	C₂×D₂₅₁	1	1	yes	characteristic
86	C₂²×C₄	C₄×D₂₅₁	1	1	yes	characteristic
87	C₂²×C₄	C₄×D₂₅₁	1	1	yes	characteristic
88	C₂²×C₄	C₂×Dic₂₅₁	1	1	yes	characteristic
89	C₂×C₈	—	251	2	no	—
90	C₂×C₈	—	251	2	no	—
91	C₂×C₈	—	251	2	no	—
92	C₂×C₈	—	251	2	no	—
93	C₂×C₈	C₂×Dic₂₅₁	1	2	yes	—
94	C₂×C₈	C₂×Dic₂₅₁	1	2	yes	—
95	C₄²	C₄×D₂₅₁	1	1	yes	characteristic
96	C₄²	C₄×D₂₅₁	1	1	yes	characteristic
97	C₄²	C₂×Dic₂₅₁	1	1	yes	characteristic
98	C₄²	C₂²×D₂₅₁	1	1	yes	characteristic
99	C₂³	C₄×Dic₂₅₁	1	1	yes	characteristic
100	C₂×C₄	—	2	2	no	—
101	C₂×C₄	—	2	2	no	—
102	C₂×C₄	—	2	2	no	—
103	C₂×C₄	C₄×Dic₂₅₁	1	1	yes	characteristic
104	C₂×C₄	C₄×Dic₂₅₁	1	1	yes	characteristic
105	C₂×C₄	C₄×Dic₂₅₁	1	1	yes	characteristic
106	C₂×C₄	C₂×C₄×D₂₅₁	1	1	yes	characteristic
107	C₂×C₄	C₂×C₄×D₂₅₁	1	1	yes	characteristic
108	C₂×C₄	C₂²×Dic₂₅₁	1	1	yes	characteristic
109	C₈	—	2	2	no	—
110	C₈	—	2	2	no	—
111	C₈	—	502	2	no	—
112	C₈	—	502	2	no	—
113	C₈	—	502	2	no	—
114	C₈	—	502	2	no	—
115	C₂²	—	2	1	no	—
116	C₂²	—	2	1	no	—
117	C₂²	—	2	1	no	—
118	C₂²	C₂×C₄×Dic₂₅₁	1	1	yes	characteristic
119	C₄	—	2	1	no	—
120	C₄	—	2	1	no	—
121	C₄	—	2	1	no	—
122	C₄	#22548721	1	2	yes	—
123	C₄	#22548722	1	2	yes	—
124	C₄	#22548723	1	2	yes	—
125	C₂	—	4	1	no	—
126	C₂	#30631642	1	1	yes	characteristic
127	C₂	#30631643	1	1	yes	characteristic
128	C₂	#30631645	1	1	yes	characteristic

subgroup

G/H

class size

multiplicity

normal?

properties

#30631740

C₂

yes

maximal, characteristic

#30631740

C₂

yes

maximal, characteristic

#30631409

C₂

yes

maximal, characteristic, complemented

#30631409

C₂

yes

maximal, characteristic, complemented

#30631408

C₂

yes

maximal, characteristic, complemented

C₈×C₂₅₁⋊C₈

C₂

yes

maximal, characteristic, complemented

#431×C₂₅₁

C₂

yes

maximal, characteristic

C₄⋊(C₂₅₁⋊C₈)

C₄

yes

—

C₄⋊(C₂₅₁⋊C₈)

C₄

yes

—

C₂²⋊(C₂₅₁⋊C₈)

C₄

yes

—

C₂²⋊(C₂₅₁⋊C₈)

C₄

yes

—

C₄×C₂₅₁⋊C₈

C₂²

yes

characteristic

C₄×C₂₅₁⋊C₈

C₂²

yes

characteristic

C₄×C₂₅₁⋊C₈

C₂²

yes

characteristic

C₄×C₂₅₁⋊C₈

C₂²

yes

characteristic

C₂×C₄²×C₂₅₁

C₂²

yes

characteristic

C₄⋊C₈×C₂₅₁

C₄

yes

—

C₂²⋊C₈×C₂₅₁

C₄

yes

—

C₄×C₈×C₂₅₁

C₂²

yes

characteristic

C₄×C₈×C₂₅₁

C₂²

yes

characteristic

C₂×C₂₅₁⋊C₈

C₂×C₄

yes

—

C₂×C₂₅₁⋊C₈

C₂×C₄

yes

—

C₂×C₂₅₁⋊C₈

C₂×C₄

yes

—

C₂×C₂₅₁⋊C₈

C₂×C₄

yes

—

C₂²×C₄×C₂₅₁

C₂×C₄

yes

characteristic

C₂²×C₄×C₂₅₁

C₂×C₄

yes

characteristic

C₂²×C₄×C₂₅₁

C₂×C₄

yes

characteristic

C₂×C₈×C₂₅₁

C₂×C₄

yes

—

C₂×C₈×C₂₅₁

C₂×C₄

yes

—

C₄²×C₂₅₁

C₂×C₄

yes

characteristic

C₄²×C₂₅₁

C₂×C₄

yes

characteristic

C₄²×C₂₅₁

C₂×C₄

yes

characteristic

C₄²×C₂₅₁

C₂³

yes

characteristic

C₂₅₁⋊C₈

—

C₂₅₁⋊C₈

—

C₂₅₁⋊C₈

—

C₂₅₁⋊C₈

—

C₂³×C₂₅₁

C₄²

yes

characteristic

C₂×C₄×C₂₅₁

—

C₂×C₄×C₂₅₁

—

C₂×C₄×C₂₅₁

—

C₂×C₄×C₂₅₁

C₄²

yes

characteristic

C₂×C₄×C₂₅₁

C₄²

yes

characteristic

C₂×C₄×C₂₅₁

C₄²

yes

characteristic

C₂×C₄×C₂₅₁

C₂²×C₄

yes

characteristic

C₂×C₄×C₂₅₁

C₂²×C₄

yes

characteristic

C₂×C₄×C₂₅₁

C₂²×C₄

yes

characteristic

C₈×C₂₅₁

—

C₈×C₂₅₁

—

C₂²×C₂₅₁

—

C₂²×C₂₅₁

—

C₂²×C₂₅₁

—

C₂²×C₂₅₁

C₂×C₄²

yes

characteristic

C₄×C₂₅₁

—

C₄×C₂₅₁

—

C₄×C₂₅₁

—

C₄×C₂₅₁

#131

yes

—

C₄×C₂₅₁

#131

yes

—

C₄×C₂₅₁

#131

yes

—

C₂×C₂₅₁

—

C₂×C₂₅₁

yes

C₂×C₂₅₁

yes

C₂×C₂₅₁

yes

C₂₅₁

yes

characteristic, complemented, Sylow

#1454

—

251

maximal, Sylow

#431

—

251

—

#431

—

251

—

#431

D₂₅₁

yes

characteristic

#322

—

251

—

#322

—

251

—

#322

—

251

—

C₈²

—

251

—

C₂×C₄²

C₂×D₂₅₁

yes

characteristic

C₄⋊C₈

—

251

—

C₄⋊C₈

—

251

—

C₄⋊C₈

Dic₂₅₁

yes

—

C₂²⋊C₈

—

251

—

C₂²⋊C₈

—

251

—

C₂²⋊C₈

Dic₂₅₁

yes

—

C₄×C₈

—

251

—

C₄×C₈

—

251

—

C₄×C₈

—

251

—

C₄×C₈

—

251

—

C₄×C₈

C₂×D₂₅₁

yes

characteristic

C₄×C₈

C₂×D₂₅₁

yes

characteristic

C₂²×C₄

C₄×D₂₅₁

yes

characteristic

C₂²×C₄

C₄×D₂₅₁

yes

characteristic

C₂²×C₄

C₂×Dic₂₅₁

yes

characteristic

C₂×C₈

—

251

—

C₂×C₈

—

251

—

C₂×C₈

—

251

—

C₂×C₈

—

251

—

C₂×C₈

C₂×Dic₂₅₁

yes

—

C₂×C₈

C₂×Dic₂₅₁

yes

—

C₄²

C₄×D₂₅₁

yes

characteristic

C₄²

C₄×D₂₅₁

yes

characteristic

C₄²

C₂×Dic₂₅₁

yes

characteristic

C₄²

C₂²×D₂₅₁

yes

characteristic

C₂³

C₄×Dic₂₅₁

yes

characteristic

100

C₂×C₄

—

101

C₂×C₄

—

102

C₂×C₄

—

103

C₂×C₄

C₄×Dic₂₅₁

yes

characteristic

104

C₂×C₄

C₄×Dic₂₅₁

yes

characteristic

105

C₂×C₄

C₄×Dic₂₅₁

yes

characteristic

106

C₂×C₄

C₂×C₄×D₂₅₁

yes

characteristic

107

C₂×C₄

C₂×C₄×D₂₅₁

yes

characteristic

108

C₂×C₄

C₂²×Dic₂₅₁

yes

characteristic

109

C₈

—

110

C₈

—

111

C₈

—

502

—

112

C₈

—

502

—

113

C₈

—

502

—

114

C₈

—

502

—

115

C₂²

—

116

C₂²

—

117

C₂²

—

118

C₂²

C₂×C₄×Dic₂₅₁

yes

characteristic

119

C₄

—

120

C₄

—

121

C₄

—

122

C₄

#22548721

yes

—

123

C₄

#22548722

yes

—

124

C₄

#22548723

yes

—

125

C₂

—

126

C₂

#30631642

yes

characteristic

127

C₂

#30631643

yes

characteristic

128

C₂

#30631645

yes

characteristic

	subgroup	G/H
center, Frattini	C₄²	C₂²×D₂₅₁
commutator, socle	C₂²×C₂₅₁	C₂×C₄²
Fitting	#431×C₂₅₁	C₂
prime radical	C₂³×C₂₅₁	C₄²
cosocle	C₄²×C₂₅₁	C₂³
hypercenter	#431	D₂₅₁
hypocenter	C₂₅₁	#1454
LCS	G ▷ C₂²×C₂₅₁ ▷ C₂₅₁
UCS	C₁ ◁ C₄² ◁ #431

subgroup

G/H

center, Frattini

C₄²

C₂²×D₂₅₁

commutator, socle

C₂²×C₂₅₁

C₂×C₄²

Fitting

#431×C₂₅₁

C₂

prime radical

C₂³×C₂₅₁

C₄²

cosocle

C₄²×C₂₅₁

C₂³

D₂₅₁

C₂₅₁

G ▷ C₂²×C₂₅₁ ▷ C₂₅₁

C₁ ◁ C₄² ◁ #431

N	H ≅ G/N	type
3. #30631409	C₂	semidirect
4. #30631409	C₂	semidirect
5. #30631408	C₂	semidirect
6. C₈×C₂₅₁⋊C₈	C₂	semidirect
64. C₂₅₁	#1454	semidirect

H ≅ G/N

type

3. #30631409

C₂

semidirect

4. #30631409

C₂

semidirect

5. #30631408

C₂

semidirect

6. C₈×C₂₅₁⋊C₈

C₂

semidirect

64. C₂₅₁

#1454

semidirect

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

subgroup	index
#30631740	2
#30631409	2
#30631408	2
C₈×C₂₅₁⋊C₈	2
#431×C₂₅₁	2
#1454	251

subgroup

index

#30631740

#30631409

#30631408

C₈×C₂₅₁⋊C₈

#431×C₂₅₁

#1454

251

Subgroups — #44548665

Conjugacy classes of subgroups

Complements

Maximal subgroups