Group #44547463 — subgroups

Subgroups — C₂²×C₂²⋊C₈×C₂₅₁ #44547463

Order 32128.

1072 conjugacy classes of subgroups; only the 520 normal subgroups are shown.

normal subgroups only

This group has 1072 conjugacy classes of subgroups — too many to draw. Only its 520 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	C₂⁴×C₄×C₂₅₁	C₂	1	1	yes	maximal, characteristic
2	C₂³×C₈×C₂₅₁	C₂	1	2	yes	maximal, complemented
3	C₂×C₂²⋊C₈×C₂₅₁	C₂	1	12	yes	maximal, direct product factor
4	C₂⁵×C₂₅₁	C₄	1	1	yes	characteristic
5	C₂³×C₄×C₂₅₁	—	2	8	no	—
6	C₂³×C₄×C₂₅₁	C₄	1	1	yes	characteristic
7	C₂³×C₄×C₂₅₁	C₄	1	6	yes	—
8	C₂³×C₄×C₂₅₁	C₂²	1	6	yes	—
9	C₂³×C₄×C₂₅₁	C₂²	1	1	yes	characteristic
10	C₂²×C₈×C₂₅₁	—	2	8	no	—
11	C₂²×C₈×C₂₅₁	C₂²	1	12	yes	complemented
12	C₂²⋊C₈×C₂₅₁	C₂²	1	16	yes	direct product factor
13	C₂⁴×C₂₅₁	—	2	4	no	—
14	C₂⁴×C₂₅₁	—	2	4	no	—
15	C₂⁴×C₂₅₁	C₈	1	8	yes	complemented
16	C₂⁴×C₂₅₁	C₂×C₄	1	6	yes	—
17	C₂⁴×C₂₅₁	C₂×C₄	1	1	yes	characteristic
18	C₂²×C₄×C₂₅₁	—	2	4	no	—
19	C₂²×C₄×C₂₅₁	—	2	24	no	—
20	C₂²×C₄×C₂₅₁	—	2	24	no	—
21	C₂²×C₄×C₂₅₁	C₂×C₄	1	6	yes	—
22	C₂²×C₄×C₂₅₁	C₂×C₄	1	12	yes	—
23	C₂²×C₄×C₂₅₁	C₂×C₄	1	3	yes	—
24	C₂²×C₄×C₂₅₁	D₄	1	8	yes	—
25	C₂²×C₄×C₂₅₁	C₂³	1	4	yes	—
26	C₂²×C₄×C₂₅₁	C₂³	1	3	yes	—
27	C₂×C₈×C₂₅₁	—	2	24	no	—
28	C₂×C₈×C₂₅₁	C₂³	1	8	yes	complemented
29	C₂³×C₂₅₁	—	2	32	no	—
30	C₂³×C₂₅₁	—	2	12	no	—
31	C₂³×C₂₅₁	—	2	12	no	—
32	C₂³×C₂₅₁	C₂²⋊C₄	1	4	yes	—
33	C₂³×C₂₅₁	C₂×C₈	1	4	yes	—
34	C₂³×C₂₅₁	C₂×C₈	1	24	yes	complemented
35	C₂³×C₂₅₁	M₁₆	1	4	yes	—
36	C₂³×C₂₅₁	C₂²×C₄	1	4	yes	—
37	C₂³×C₂₅₁	C₂²×C₄	1	3	yes	—
38	C₂×C₄×C₂₅₁	—	2	12	no	—
39	C₂×C₄×C₂₅₁	—	2	24	no	—
40	C₂×C₄×C₂₅₁	—	2	8	no	—
41	C₂×C₄×C₂₅₁	C₂²⋊C₄	1	12	yes	—
42	C₂×C₄×C₂₅₁	C₂²×C₄	1	4	yes	—
43	C₂×C₄×C₂₅₁	C₂²×C₄	1	3	yes	—
44	C₂×C₄×C₂₅₁	C₂×D₄	1	12	yes	—
45	C₂×C₄×C₂₅₁	C₂⁴	1	1	yes	characteristic
46	C₈×C₂₅₁	—	2	8	no	—
47	C₂²×C₂₅₁	—	2	4	no	—
48	C₂²×C₂₅₁	—	2	4	no	—
49	C₂²×C₂₅₁	—	2	48	no	—
50	C₂²×C₂₅₁	C₂²⋊C₈	1	16	yes	direct product factor
51	C₂²×C₂₅₁	C₂×C₂²⋊C₄	1	6	yes	—
52	C₂²×C₂₅₁	C₂²×C₈	1	6	yes	—
53	C₂²×C₂₅₁	C₂²×C₈	1	8	yes	complemented
54	C₂²×C₂₅₁	C₂×M₁₆	1	6	yes	—
55	C₂²×C₂₅₁	C₂³×C₄	1	1	yes	characteristic
56	C₄×C₂₅₁	—	2	4	no	—
57	C₄×C₂₅₁	C₂×C₂²⋊C₄	1	6	yes	—
58	C₄×C₂₅₁	C₂²×D₄	1	2	yes	—
59	C₂×C₂₅₁	—	2	8	no	—
60	C₂×C₂₅₁	C₂×C₂²⋊C₈	1	12	yes	direct product factor
61	C₂×C₂₅₁	C₂²×C₂²⋊C₄	1	1	yes	characteristic
62	C₂×C₂₅₁	C₂³×C₈	1	1	yes	characteristic
63	C₂×C₂₅₁	C₂²×M₁₆	1	1	yes	characteristic
64	C₂₅₁	C₂²×C₂²⋊C₈	1	1	yes	characteristic, direct product factor, Sylow
65	C₂²×C₂²⋊C₈	C₂₅₁	1	1	yes	maximal, characteristic, direct product factor, Sylow
66	C₂⁴×C₄	C₂×C₂₅₁	1	1	yes	characteristic
67	C₂³×C₈	C₂×C₂₅₁	1	2	yes	complemented
68	C₂×C₂²⋊C₈	C₂×C₂₅₁	1	12	yes	direct product factor
69	C₂⁵	C₄×C₂₅₁	1	1	yes	characteristic
70	C₂³×C₄	—	2	8	no	—
71	C₂³×C₄	C₄×C₂₅₁	1	1	yes	characteristic
72	C₂³×C₄	C₄×C₂₅₁	1	6	yes	—
73	C₂³×C₄	C₂²×C₂₅₁	1	6	yes	—
74	C₂³×C₄	C₂²×C₂₅₁	1	1	yes	characteristic
75	C₂²×C₈	—	2	8	no	—
76	C₂²×C₈	C₂²×C₂₅₁	1	12	yes	complemented
77	C₂²⋊C₈	C₂²×C₂₅₁	1	16	yes	direct product factor
78	C₂⁴	—	2	4	no	—
79	C₂⁴	—	2	4	no	—
80	C₂⁴	C₈×C₂₅₁	1	8	yes	complemented
81	C₂⁴	C₂×C₄×C₂₅₁	1	6	yes	—
82	C₂⁴	C₂×C₄×C₂₅₁	1	1	yes	characteristic
83	C₂²×C₄	—	2	4	no	—
84	C₂²×C₄	—	2	24	no	—
85	C₂²×C₄	—	2	24	no	—
86	C₂²×C₄	C₂×C₄×C₂₅₁	1	6	yes	—
87	C₂²×C₄	C₂×C₄×C₂₅₁	1	12	yes	—
88	C₂²×C₄	C₂×C₄×C₂₅₁	1	3	yes	—
89	C₂²×C₄	D₄×C₂₅₁	1	8	yes	—
90	C₂²×C₄	C₂³×C₂₅₁	1	4	yes	—
91	C₂²×C₄	C₂³×C₂₅₁	1	3	yes	—
92	C₂×C₈	—	2	24	no	—
93	C₂×C₈	C₂³×C₂₅₁	1	8	yes	complemented
94	C₂³	—	2	32	no	—
95	C₂³	—	2	12	no	—
96	C₂³	—	2	12	no	—
97	C₂³	C₂²⋊C₄×C₂₅₁	1	4	yes	—
98	C₂³	C₂×C₈×C₂₅₁	1	4	yes	—
99	C₂³	C₂×C₈×C₂₅₁	1	24	yes	complemented
100	C₂³	M₁₆×C₂₅₁	1	4	yes	—
101	C₂³	C₂²×C₄×C₂₅₁	1	4	yes	—
102	C₂³	C₂²×C₄×C₂₅₁	1	3	yes	—
103	C₂×C₄	—	2	12	no	—
104	C₂×C₄	—	2	24	no	—
105	C₂×C₄	—	2	8	no	—
106	C₂×C₄	C₂²⋊C₄×C₂₅₁	1	12	yes	—
107	C₂×C₄	C₂²×C₄×C₂₅₁	1	4	yes	—
108	C₂×C₄	C₂²×C₄×C₂₅₁	1	3	yes	—
109	C₂×C₄	C₂×D₄×C₂₅₁	1	12	yes	—
110	C₂×C₄	C₂⁴×C₂₅₁	1	1	yes	characteristic
111	C₈	—	2	8	no	—
112	C₂²	—	2	48	no	—
113	C₂²	—	2	4	no	—
114	C₂²	—	2	4	no	—
115	C₂²	C₂²⋊C₈×C₂₅₁	1	16	yes	direct product factor
116	C₂²	C₂×C₂²⋊C₄×C₂₅₁	1	6	yes	—
117	C₂²	C₂²×C₈×C₂₅₁	1	6	yes	—
118	C₂²	C₂²×C₈×C₂₅₁	1	8	yes	complemented
119	C₂²	C₂×M₁₆×C₂₅₁	1	6	yes	—
120	C₂²	C₂³×C₄×C₂₅₁	1	1	yes	characteristic
121	C₄	—	2	4	no	—
122	C₄	C₂×C₂²⋊C₄×C₂₅₁	1	6	yes	—
123	C₄	C₂²×D₄×C₂₅₁	1	2	yes	—
124	C₂	—	2	8	no	—
125	C₂	C₂×C₂²⋊C₈×C₂₅₁	1	12	yes	direct product factor
126	C₂	C₂²×C₂²⋊C₄×C₂₅₁	1	1	yes	characteristic
127	C₂	C₂³×C₈×C₂₅₁	1	1	yes	characteristic
128	C₂	C₂²×M₁₆×C₂₅₁	1	1	yes	characteristic

subgroup

G/H

class size

multiplicity

normal?

properties

C₂⁴×C₄×C₂₅₁

C₂

yes

maximal, characteristic

C₂³×C₈×C₂₅₁

C₂

yes

maximal, complemented

C₂×C₂²⋊C₈×C₂₅₁

C₂

yes

maximal, direct product factor

C₂⁵×C₂₅₁

C₄

yes

characteristic

C₂³×C₄×C₂₅₁

—

C₂³×C₄×C₂₅₁

C₄

yes

characteristic

C₂³×C₄×C₂₅₁

C₄

yes

—

C₂³×C₄×C₂₅₁

C₂²

yes

—

C₂³×C₄×C₂₅₁

C₂²

yes

characteristic

C₂²×C₈×C₂₅₁

—

C₂²×C₈×C₂₅₁

C₂²

yes

complemented

C₂²⋊C₈×C₂₅₁

C₂²

yes

direct product factor

C₂⁴×C₂₅₁

—

C₂⁴×C₂₅₁

—

C₂⁴×C₂₅₁

C₈

yes

complemented

C₂⁴×C₂₅₁

C₂×C₄

yes

—

C₂⁴×C₂₅₁

C₂×C₄

yes

characteristic

C₂²×C₄×C₂₅₁

—

C₂²×C₄×C₂₅₁

—

C₂²×C₄×C₂₅₁

—

C₂²×C₄×C₂₅₁

C₂×C₄

yes

—

C₂²×C₄×C₂₅₁

C₂×C₄

yes

—

C₂²×C₄×C₂₅₁

C₂×C₄

yes

—

C₂²×C₄×C₂₅₁

D₄

yes

—

C₂²×C₄×C₂₅₁

C₂³

yes

—

C₂²×C₄×C₂₅₁

C₂³

yes

—

C₂×C₈×C₂₅₁

—

C₂×C₈×C₂₅₁

C₂³

yes

complemented

C₂³×C₂₅₁

—

C₂³×C₂₅₁

—

C₂³×C₂₅₁

—

C₂³×C₂₅₁

C₂²⋊C₄

yes

—

C₂³×C₂₅₁

C₂×C₈

yes

—

C₂³×C₂₅₁

C₂×C₈

yes

complemented

C₂³×C₂₅₁

M₁₆

yes

—

C₂³×C₂₅₁

C₂²×C₄

yes

—

C₂³×C₂₅₁

C₂²×C₄

yes

—

C₂×C₄×C₂₅₁

—

C₂×C₄×C₂₅₁

—

C₂×C₄×C₂₅₁

—

C₂×C₄×C₂₅₁

C₂²⋊C₄

yes

—

C₂×C₄×C₂₅₁

C₂²×C₄

yes

—

C₂×C₄×C₂₅₁

C₂²×C₄

yes

—

C₂×C₄×C₂₅₁

C₂×D₄

yes

—

C₂×C₄×C₂₅₁

C₂⁴

yes

characteristic

C₈×C₂₅₁

—

C₂²×C₂₅₁

—

C₂²×C₂₅₁

—

C₂²×C₂₅₁

—

C₂²×C₂₅₁

C₂²⋊C₈

yes

direct product factor

C₂²×C₂₅₁

C₂×C₂²⋊C₄

yes

—

C₂²×C₂₅₁

C₂²×C₈

yes

—

C₂²×C₂₅₁

C₂²×C₈

yes

complemented

C₂²×C₂₅₁

C₂×M₁₆

yes

—

C₂²×C₂₅₁

C₂³×C₄

yes

characteristic

C₄×C₂₅₁

—

C₄×C₂₅₁

C₂×C₂²⋊C₄

yes

—

C₄×C₂₅₁

C₂²×D₄

yes

—

C₂×C₂₅₁

—

C₂×C₂₅₁

C₂×C₂²⋊C₈

yes

direct product factor

C₂×C₂₅₁

C₂²×C₂²⋊C₄

yes

characteristic

C₂×C₂₅₁

C₂³×C₈

yes

characteristic

C₂×C₂₅₁

C₂²×M₁₆

yes

characteristic

C₂₅₁

C₂²×C₂²⋊C₈

yes

characteristic, direct product factor, Sylow

C₂²×C₂²⋊C₈

C₂₅₁

yes

maximal, characteristic, direct product factor, Sylow

C₂⁴×C₄

C₂×C₂₅₁

yes

characteristic

C₂³×C₈

C₂×C₂₅₁

yes

complemented

C₂×C₂²⋊C₈

C₂×C₂₅₁

yes

direct product factor

C₂⁵

C₄×C₂₅₁

yes

characteristic

C₂³×C₄

—

C₂³×C₄

C₄×C₂₅₁

yes

characteristic

C₂³×C₄

C₄×C₂₅₁

yes

—

C₂³×C₄

C₂²×C₂₅₁

yes

—

C₂³×C₄

C₂²×C₂₅₁

yes

characteristic

C₂²×C₈

—

C₂²×C₈

C₂²×C₂₅₁

yes

complemented

C₂²⋊C₈

C₂²×C₂₅₁

yes

direct product factor

C₂⁴

—

C₂⁴

—

C₂⁴

C₈×C₂₅₁

yes

complemented

C₂⁴

C₂×C₄×C₂₅₁

yes

—

C₂⁴

C₂×C₄×C₂₅₁

yes

characteristic

C₂²×C₄

—

C₂²×C₄

—

C₂²×C₄

—

C₂²×C₄

C₂×C₄×C₂₅₁

yes

—

C₂²×C₄

C₂×C₄×C₂₅₁

yes

—

C₂²×C₄

C₂×C₄×C₂₅₁

yes

—

C₂²×C₄

D₄×C₂₅₁

yes

—

C₂²×C₄

C₂³×C₂₅₁

yes

—

C₂²×C₄

C₂³×C₂₅₁

yes

—

C₂×C₈

—

C₂×C₈

C₂³×C₂₅₁

yes

complemented

C₂³

—

C₂³

—

C₂³

—

C₂³

C₂²⋊C₄×C₂₅₁

yes

—

C₂³

C₂×C₈×C₂₅₁

yes

—

C₂³

C₂×C₈×C₂₅₁

yes

complemented

100

C₂³

M₁₆×C₂₅₁

yes

—

101

C₂³

C₂²×C₄×C₂₅₁

yes

—

102

C₂³

C₂²×C₄×C₂₅₁

yes

—

103

C₂×C₄

—

104

C₂×C₄

—

105

C₂×C₄

—

106

C₂×C₄

C₂²⋊C₄×C₂₅₁

yes

—

107

C₂×C₄

C₂²×C₄×C₂₅₁

yes

—

108

C₂×C₄

C₂²×C₄×C₂₅₁

yes

—

109

C₂×C₄

C₂×D₄×C₂₅₁

yes

—

110

C₂×C₄

C₂⁴×C₂₅₁

yes

characteristic

111

C₈

—

112

C₂²

—

113

C₂²

—

114

C₂²

—

115

C₂²

C₂²⋊C₈×C₂₅₁

yes

direct product factor

116

C₂²

C₂×C₂²⋊C₄×C₂₅₁

yes

—

117

C₂²

C₂²×C₈×C₂₅₁

yes

—

118

C₂²

C₂²×C₈×C₂₅₁

yes

complemented

119

C₂²

C₂×M₁₆×C₂₅₁

yes

—

120

C₂²

C₂³×C₄×C₂₅₁

yes

characteristic

121

C₄

—

122

C₄

C₂×C₂²⋊C₄×C₂₅₁

yes

—

123

C₄

C₂²×D₄×C₂₅₁

yes

—

124

C₂

—

125

C₂

C₂×C₂²⋊C₈×C₂₅₁

yes

direct product factor

126

C₂

C₂²×C₂²⋊C₄×C₂₅₁

yes

characteristic

127

C₂

C₂³×C₈×C₂₅₁

yes

characteristic

128

C₂

C₂²×M₁₆×C₂₅₁

yes

characteristic

	subgroup	G/H
center	C₂³×C₄×C₂₅₁	C₂²
commutator	C₂	C₂³×C₈×C₂₅₁
Frattini, cosocle	C₂×C₄	C₂⁴×C₂₅₁
socle	C₂⁴×C₂₅₁	C₂×C₄
prime radical	C₂⁵×C₂₅₁	C₄
LCS	G ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³×C₄×C₂₅₁ ◁ G

subgroup

G/H

center

C₂³×C₄×C₂₅₁

C₂²

commutator

C₂

C₂³×C₈×C₂₅₁

Frattini, cosocle

C₂×C₄

C₂⁴×C₂₅₁

socle

C₂⁴×C₂₅₁

C₂×C₄

prime radical

C₂⁵×C₂₅₁

C₄

LCS

G ▷ C₂ ▷ C₁

UCS

C₁ ◁ C₂³×C₄×C₂₅₁ ◁ G

N	H ≅ G/N	type
2. C₂³×C₈×C₂₅₁	C₂	semidirect
3. C₂×C₂²⋊C₈×C₂₅₁	C₂	direct (N × H)
11. C₂²×C₈×C₂₅₁	C₂²	semidirect
12. C₂²⋊C₈×C₂₅₁	C₂²	direct (N × H)
15. C₂⁴×C₂₅₁	C₈	semidirect
28. C₂×C₈×C₂₅₁	C₂³	semidirect
34. C₂³×C₂₅₁	C₂×C₈	semidirect
50. C₂²×C₂₅₁	C₂²⋊C₈	direct (N × H)
53. C₂²×C₂₅₁	C₂²×C₈	semidirect
60. C₂×C₂₅₁	C₂×C₂²⋊C₈	direct (N × H)
64. C₂₅₁	C₂²×C₂²⋊C₈	direct (N × H)
65. C₂²×C₂²⋊C₈	C₂₅₁	direct (N × H)
67. C₂³×C₈	C₂×C₂₅₁	semidirect
68. C₂×C₂²⋊C₈	C₂×C₂₅₁	direct (N × H)
76. C₂²×C₈	C₂²×C₂₅₁	semidirect
77. C₂²⋊C₈	C₂²×C₂₅₁	direct (N × H)
80. C₂⁴	C₈×C₂₅₁	semidirect
93. C₂×C₈	C₂³×C₂₅₁	semidirect
99. C₂³	C₂×C₈×C₂₅₁	semidirect
115. C₂²	C₂²⋊C₈×C₂₅₁	direct (N × H)
118. C₂²	C₂²×C₈×C₂₅₁	semidirect
125. C₂	C₂×C₂²⋊C₈×C₂₅₁	direct (N × H)

H ≅ G/N

type

2. C₂³×C₈×C₂₅₁

C₂

semidirect

3. C₂×C₂²⋊C₈×C₂₅₁

C₂

direct (N × H)

11. C₂²×C₈×C₂₅₁

C₂²

semidirect

12. C₂²⋊C₈×C₂₅₁

C₂²

direct (N × H)

15. C₂⁴×C₂₅₁

C₈

semidirect

28. C₂×C₈×C₂₅₁

C₂³

semidirect

34. C₂³×C₂₅₁

C₂×C₈

semidirect

50. C₂²×C₂₅₁

C₂²⋊C₈

direct (N × H)

53. C₂²×C₂₅₁

C₂²×C₈

semidirect

60. C₂×C₂₅₁

C₂×C₂²⋊C₈

direct (N × H)

64. C₂₅₁

C₂²×C₂²⋊C₈

direct (N × H)

65. C₂²×C₂²⋊C₈

C₂₅₁

direct (N × H)

67. C₂³×C₈

C₂×C₂₅₁

semidirect

68. C₂×C₂²⋊C₈

C₂×C₂₅₁

direct (N × H)

76. C₂²×C₈

C₂²×C₂₅₁

semidirect

77. C₂²⋊C₈

C₂²×C₂₅₁

direct (N × H)

80. C₂⁴

C₈×C₂₅₁

semidirect

93. C₂×C₈

C₂³×C₂₅₁

semidirect

99. C₂³

C₂×C₈×C₂₅₁

semidirect

115. C₂²

C₂²⋊C₈×C₂₅₁

direct (N × H)

118. C₂²

C₂²×C₈×C₂₅₁

semidirect

125. C₂

C₂×C₂²⋊C₈×C₂₅₁

direct (N × H)

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

subgroup	index
C₂⁴×C₄×C₂₅₁	2
C₂³×C₈×C₂₅₁	2
C₂×C₂²⋊C₈×C₂₅₁	2
C₂²×C₂²⋊C₈	251

subgroup

index

C₂⁴×C₄×C₂₅₁

C₂³×C₈×C₂₅₁

C₂×C₂²⋊C₈×C₂₅₁

C₂²×C₂²⋊C₈

251

Subgroups — C₂²×C₂²⋊C₈×C₂₅₁ #44547463

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C22×C22⋊C8×C251 #44547463

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₂²×C₂²⋊C₈×C₂₅₁ #44547463