Group #45688166 — subgroups

Subgroups — #45688166

Order 33344.

236 conjugacy classes of subgroups; only the 95 normal subgroups are shown.

normal subgroups only

This group has 236 conjugacy classes of subgroups — too many to draw. Only its 95 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	#30659704	C₂	1	1	yes	maximal, characteristic
2	#30659694	C₂	1	2	yes	maximal, complemented
3	#30659694	C₂	1	2	yes	maximal, complemented
4	#30659693	C₂	1	2	yes	maximal, complemented
5	#30659689	C₂	1	1	yes	maximal, characteristic, complemented
6	#30659673	C₂	1	2	yes	maximal, complemented
7	C₄×C₂²⋊D₅₂₁	C₂	1	2	yes	maximal, complemented
8	#30659668	C₂	1	2	yes	maximal, complemented
9	#123×C₅₂₁	C₂	1	1	yes	maximal, characteristic, complemented
10	C₂×Dic₁₀₄₂	C₂²	1	2	yes	—
11	C₂×D₂₀₈₄	C₂²	1	1	yes	characteristic
12	C₂×C₂²⋊D₅₂₁	C₂²	1	2	yes	—
13	C₂×C₂²⋊D₅₂₁	C₂²	1	2	yes	—
14	C₂²×Dic₅₂₁	C₂²	1	1	yes	characteristic
15	C₂×C₄×D₅₂₁	C₂²	1	2	yes	—
16	#22552343	C₂²	1	2	yes	—
17	#22552343	C₂²	1	2	yes	—
18	C₂²⋊Dic₅₂₁	C₂²	1	2	yes	—
19	C₂²⋊Dic₅₂₁	C₂²	1	1	yes	characteristic
20	#22552341	C₂²	1	2	yes	—
21	#22552341	C₂²	1	2	yes	—
22	#22552341	C₂²	1	2	yes	—
23	C₄×Dic₅₂₁	C₂²	1	2	yes	—
24	C₄×Dic₅₂₁	C₂²	1	2	yes	—
25	C₄×Dic₅₂₁	C₂²	1	1	yes	characteristic
26	C₂×D₄×C₅₂₁	C₂²	1	1	yes	characteristic
27	C₂²×C₄×C₅₂₁	C₂²	1	1	yes	characteristic
28	C₄⋊C₄×C₅₂₁	C₂²	1	2	yes	—
29	C₂²⋊C₄×C₅₂₁	C₂²	1	2	yes	—
30	C₂²⋊C₄×C₅₂₁	C₂²	1	1	yes	characteristic
31	C₂²×D₅₂₁	C₂³	1	2	yes	—
32	Dic₁₀₄₂	—	2	4	no	—
33	D₂₀₈₄	—	4	1	no	—
34	C₂²⋊D₅₂₁	—	2	4	no	—
35	C₂²⋊D₅₂₁	—	2	4	no	—
36	C₂×Dic₅₂₁	—	4	1	no	—
37	C₂×Dic₅₂₁	C₂³	1	2	yes	—
38	C₂×Dic₅₂₁	C₂³	1	2	yes	—
39	C₂×Dic₅₂₁	C₂³	1	2	yes	—
40	C₄×D₅₂₁	—	4	2	no	—
41	C₂³×C₅₂₁	C₂³	1	1	yes	characteristic
42	C₂³×C₅₂₁	C₂³	1	1	yes	characteristic
43	D₄×C₅₂₁	—	4	1	no	—
44	C₂×C₄×C₅₂₁	—	4	1	no	—
45	C₂×C₄×C₅₂₁	C₂³	1	2	yes	—
46	C₂×C₄×C₅₂₁	C₂³	1	2	yes	—
47	C₂×C₄×C₅₂₁	C₂³	1	1	yes	characteristic
48	C₂²×C₅₂₁	—	2	2	no	—
49	C₂²×C₅₂₁	—	2	2	no	—
50	C₂²×C₅₂₁	—	2	1	no	—
51	C₂²×C₅₂₁	—	2	1	no	—
52	C₂²×C₅₂₁	C₂⁴	1	1	yes	characteristic
53	C₂×D₅₂₁	—	2	2	no	—
54	C₂×D₅₂₁	—	2	2	no	—
55	C₂×D₅₂₁	—	2	2	no	—
56	C₄×C₅₂₁	—	2	2	no	—
57	C₄×C₅₂₁	—	2	2	no	—
58	C₄×C₅₂₁	—	2	1	no	—
59	Dic₅₂₁	—	2	2	no	—
60	Dic₅₂₁	—	2	2	no	—
61	Dic₅₂₁	#41	1	4	yes	—
62	C₂×C₅₂₁	—	4	1	no	—
63	C₂×C₅₂₁	—	4	1	no	—
64	C₂×C₅₂₁	C₂×#41	1	2	yes	—
65	C₂×C₅₂₁	H(5,𝔽₂)	1	1	yes	characteristic
66	D₅₂₁	—	4	2	no	—
67	C₅₂₁	#559	1	1	yes	characteristic, complemented, Sylow
68	#559	—	521	1	no	maximal, Sylow
69	#127	—	521	1	no	—
70	#124	—	521	2	no	—
71	#124	—	521	2	no	—
72	#123	—	521	1	no	—
73	#123	—	521	2	no	—
74	#123	D₅₂₁	1	1	yes	characteristic, complemented
75	C₄×Q₈	—	521	2	no	—
76	C₄×D₄	—	521	2	no	—
77	C₄×D₄	—	521	2	no	—
78	C₂×Q₈	—	521	2	no	—
79	C₂×D₄	—	521	2	no	—
80	C₂×D₄	—	521	2	no	—
81	C₂×D₄	—	521	1	no	—
82	C₂×D₄	C₂×D₅₂₁	1	1	yes	characteristic
83	C₂²×C₄	—	521	1	no	—
84	C₂²×C₄	—	521	2	no	—
85	C₂²×C₄	C₂×D₅₂₁	1	1	yes	characteristic
86	C₄⋊C₄	—	521	2	no	—
87	C₄⋊C₄	—	521	2	no	—
88	C₄⋊C₄	C₂×D₅₂₁	1	2	yes	—
89	C₂²⋊C₄	—	521	2	no	—
90	C₂²⋊C₄	—	521	1	no	—
91	C₂²⋊C₄	—	521	2	no	—
92	C₂²⋊C₄	—	521	2	no	—
93	C₂²⋊C₄	—	521	2	no	—
94	C₂²⋊C₄	C₂×D₅₂₁	1	2	yes	—
95	C₂²⋊C₄	C₂×D₅₂₁	1	1	yes	characteristic
96	C₄²	—	521	2	no	—
97	C₄²	—	521	2	no	—
98	C₄²	—	521	1	no	—
99	C₂³	—	521	2	no	—
100	C₂³	C₂²×D₅₂₁	1	1	yes	characteristic
101	C₂³	C₂²×D₅₂₁	1	1	yes	characteristic
102	Q₈	—	1042	4	no	—
103	D₄	—	4	1	no	—
104	D₄	—	1042	4	no	—
105	D₄	—	1042	4	no	—
106	D₄	—	2084	1	no	—
107	C₂×C₄	—	4	1	no	—
108	C₂×C₄	—	521	2	no	—
109	C₂×C₄	—	521	2	no	—
110	C₂×C₄	—	521	2	no	—
111	C₂×C₄	—	2084	2	no	—
112	C₂×C₄	—	2084	1	no	—
113	C₂×C₄	C₂²×D₅₂₁	1	2	yes	—
114	C₂×C₄	C₂²×D₅₂₁	1	2	yes	—
115	C₂×C₄	C₂²×D₅₂₁	1	1	yes	characteristic
116	C₂²	—	2	2	no	—
117	C₂²	—	2	2	no	—
118	C₂²	—	2	1	no	—
119	C₂²	—	2	1	no	—
120	C₂²	—	1042	2	no	—
121	C₂²	—	1042	2	no	—
122	C₂²	—	1042	2	no	—
123	C₂²	C₂³×D₅₂₁	1	1	yes	characteristic
124	C₄	—	2	2	no	—
125	C₄	—	2	2	no	—
126	C₄	—	2	1	no	—
127	C₄	—	521	4	no	—
128	C₄	—	1042	2	no	—
129	C₄	—	1042	2	no	—
130	C₂	—	4	1	no	—
131	C₂	—	4	1	no	—
132	C₂	—	2084	2	no	—
133	C₂	#41×D₅₂₁	1	2	yes	—
134	C₂	#30659753	1	1	yes	characteristic

subgroup

G/H

class size

multiplicity

normal?

properties

#30659704

C₂

yes

maximal, characteristic

#30659694

C₂

yes

maximal, complemented

#30659694

C₂

yes

maximal, complemented

#30659693

C₂

yes

maximal, complemented

#30659689

C₂

yes

maximal, characteristic, complemented

#30659673

C₂

yes

maximal, complemented

C₄×C₂²⋊D₅₂₁

C₂

yes

maximal, complemented

#30659668

C₂

yes

maximal, complemented

#123×C₅₂₁

C₂

yes

maximal, characteristic, complemented

C₂×Dic₁₀₄₂

C₂²

yes

—

C₂×D₂₀₈₄

C₂²

yes

characteristic

C₂×C₂²⋊D₅₂₁

C₂²

yes

—

C₂×C₂²⋊D₅₂₁

C₂²

yes

—

C₂²×Dic₅₂₁

C₂²

yes

characteristic

C₂×C₄×D₅₂₁

C₂²

yes

—

#22552343

C₂²

yes

—

#22552343

C₂²

yes

—

C₂²⋊Dic₅₂₁

C₂²

yes

—

C₂²⋊Dic₅₂₁

C₂²

yes

characteristic

#22552341

C₂²

yes

—

#22552341

C₂²

yes

—

#22552341

C₂²

yes

—

C₄×Dic₅₂₁

C₂²

yes

—

C₄×Dic₅₂₁

C₂²

yes

—

C₄×Dic₅₂₁

C₂²

yes

characteristic

C₂×D₄×C₅₂₁

C₂²

yes

characteristic

C₂²×C₄×C₅₂₁

C₂²

yes

characteristic

C₄⋊C₄×C₅₂₁

C₂²

yes

—

C₂²⋊C₄×C₅₂₁

C₂²

yes

—

C₂²⋊C₄×C₅₂₁

C₂²

yes

characteristic

C₂²×D₅₂₁

C₂³

yes

—

Dic₁₀₄₂

—

D₂₀₈₄

—

C₂²⋊D₅₂₁

—

C₂²⋊D₅₂₁

—

C₂×Dic₅₂₁

—

C₂×Dic₅₂₁

C₂³

yes

—

C₂×Dic₅₂₁

C₂³

yes

—

C₂×Dic₅₂₁

C₂³

yes

—

C₄×D₅₂₁

—

C₂³×C₅₂₁

C₂³

yes

characteristic

C₂³×C₅₂₁

C₂³

yes

characteristic

D₄×C₅₂₁

—

C₂×C₄×C₅₂₁

—

C₂×C₄×C₅₂₁

C₂³

yes

—

C₂×C₄×C₅₂₁

C₂³

yes

—

C₂×C₄×C₅₂₁

C₂³

yes

characteristic

C₂²×C₅₂₁

—

C₂²×C₅₂₁

—

C₂²×C₅₂₁

—

C₂²×C₅₂₁

—

C₂²×C₅₂₁

C₂⁴

yes

characteristic

C₂×D₅₂₁

—

C₂×D₅₂₁

—

C₂×D₅₂₁

—

C₄×C₅₂₁

—

C₄×C₅₂₁

—

C₄×C₅₂₁

—

Dic₅₂₁

—

Dic₅₂₁

—

Dic₅₂₁

#41

yes

—

C₂×C₅₂₁

—

C₂×C₅₂₁

—

C₂×C₅₂₁

C₂×#41

yes

—

C₂×C₅₂₁

H(5,𝔽₂)

yes

characteristic

D₅₂₁

—

C₅₂₁

#559

yes

characteristic, complemented, Sylow

#559

—

521

maximal, Sylow

#127

—

521

—

#124

—

521

—

#124

—

521

—

#123

—

521

—

#123

—

521

—

#123

D₅₂₁

yes

characteristic, complemented

C₄×Q₈

—

521

—

C₄×D₄

—

521

—

C₄×D₄

—

521

—

C₂×Q₈

—

521

—

C₂×D₄

—

521

—

C₂×D₄

—

521

—

C₂×D₄

—

521

—

C₂×D₄

C₂×D₅₂₁

yes

characteristic

C₂²×C₄

—

521

—

C₂²×C₄

—

521

—

C₂²×C₄

C₂×D₅₂₁

yes

characteristic

C₄⋊C₄

—

521

—

C₄⋊C₄

—

521

—

C₄⋊C₄

C₂×D₅₂₁

yes

—

C₂²⋊C₄

—

521

—

C₂²⋊C₄

—

521

—

C₂²⋊C₄

—

521

—

C₂²⋊C₄

—

521

—

C₂²⋊C₄

—

521

—

C₂²⋊C₄

C₂×D₅₂₁

yes

—

C₂²⋊C₄

C₂×D₅₂₁

yes

characteristic

C₄²

—

521

—

C₄²

—

521

—

C₄²

—

521

—

C₂³

—

521

—

100

C₂³

C₂²×D₅₂₁

yes

characteristic

101

C₂³

C₂²×D₅₂₁

yes

characteristic

102

Q₈

—

1042

—

103

D₄

—

104

D₄

—

1042

—

105

D₄

—

1042

—

106

D₄

—

2084

—

107

C₂×C₄

—

108

C₂×C₄

—

521

—

109

C₂×C₄

—

521

—

110

C₂×C₄

—

521

—

111

C₂×C₄

—

2084

—

112

C₂×C₄

—

2084

—

113

C₂×C₄

C₂²×D₅₂₁

yes

—

114

C₂×C₄

C₂²×D₅₂₁

yes

—

115

C₂×C₄

C₂²×D₅₂₁

yes

characteristic

116

C₂²

—

117

C₂²

—

118

C₂²

—

119

C₂²

—

120

C₂²

—

1042

—

121

C₂²

—

1042

—

122

C₂²

—

1042

—

123

C₂²

C₂³×D₅₂₁

yes

characteristic

124

C₄

—

125

C₄

—

126

C₄

—

127

C₄

—

521

—

128

C₄

—

1042

—

129

C₄

—

1042

—

130

C₂

—

131

C₂

—

132

C₂

—

2084

—

133

C₂

#41×D₅₂₁

yes

—

134

C₂

#30659753

yes

characteristic

	subgroup	G/H
center, Frattini	C₂²	C₂³×D₅₂₁
commutator, socle, cosocle	C₂²×C₅₂₁	C₂⁴
Fitting	#123×C₅₂₁	C₂
prime radical	#30659704	C₂
hypercenter	#123	D₅₂₁
hypocenter	C₅₂₁	#559
LCS	G ▷ C₂²×C₅₂₁ ▷ C₅₂₁
UCS	C₁ ◁ C₂² ◁ #123

subgroup

G/H

center, Frattini

C₂²

C₂³×D₅₂₁

commutator, socle, cosocle

C₂²×C₅₂₁

C₂⁴

Fitting

#123×C₅₂₁

C₂

C₂

D₅₂₁

C₅₂₁

G ▷ C₂²×C₅₂₁ ▷ C₅₂₁

C₁ ◁ C₂² ◁ #123

N	H ≅ G/N	type
2. #30659694	C₂	semidirect
3. #30659694	C₂	semidirect
4. #30659693	C₂	semidirect
5. #30659689	C₂	semidirect
6. #30659673	C₂	semidirect
7. C₄×C₂²⋊D₅₂₁	C₂	semidirect
8. #30659668	C₂	semidirect
9. #123×C₅₂₁	C₂	semidirect
67. C₅₂₁	#559	semidirect
74. #123	D₅₂₁	semidirect

H ≅ G/N

type

2. #30659694

C₂

semidirect

3. #30659694

C₂

semidirect

4. #30659693

C₂

semidirect

5. #30659689

C₂

semidirect

6. #30659673

C₂

semidirect

7. C₄×C₂²⋊D₅₂₁

C₂

semidirect

8. #30659668

C₂

semidirect

9. #123×C₅₂₁

C₂

semidirect

67. C₅₂₁

#559

semidirect

74. #123

D₅₂₁

semidirect

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

subgroup	index
#30659704	2
#30659694	2
#30659693	2
#30659689	2
#30659673	2
C₄×C₂²⋊D₅₂₁	2
#30659668	2
#123×C₅₂₁	2
#559	521

subgroup

index

C₄×C₂²⋊D₅₂₁

#123×C₅₂₁

521

Subgroups — #45688166

Conjugacy classes of subgroups

Complements

Maximal subgroups