Group #44566891 — subgroups

Subgroups — #44566891

Order 32192.

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

normal subgroups only

This group has 192 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	#30632862	C₂	1	1	yes	maximal, characteristic, complemented
2	#30632857	C₂	1	2	yes	maximal
3	#30632857	C₂	1	2	yes	maximal
4	#30632855	C₂	1	1	yes	maximal, characteristic, complemented
5	#30632855	C₂	1	2	yes	maximal, complemented
6	#30632854	C₂	1	2	yes	maximal, complemented
7	#30632843	C₂	1	1	yes	maximal, characteristic
8	Q₈×Dic₅₀₃	C₂	1	1	yes	maximal, characteristic, complemented
9	#30632829	C₂	1	1	yes	maximal, characteristic, complemented
10	#30632823	C₂	1	1	yes	maximal, characteristic
11	C₂²⋊Q₈×C₅₀₃	C₂	1	1	yes	maximal, characteristic
12	C₂×Dic₁₀₀₆	C₂²	1	1	yes	characteristic
13	C₄⋊Dic₅₀₃	C₂²	1	1	yes	characteristic
14	C₄⋊Dic₅₀₃	C₂²	1	1	yes	characteristic
15	C₄⋊Dic₅₀₃	C₂²	1	1	yes	characteristic
16	C₄⋊Dic₅₀₃	C₂²	1	2	yes	—
17	C₄⋊Dic₅₀₃	C₂²	1	2	yes	—
18	Dic₅₀₃⋊C₄	C₂²	1	2	yes	—
19	Dic₅₀₃⋊C₄	C₂²	1	2	yes	—
20	Dic₅₀₃⋊C₄	C₂²	1	2	yes	—
21	Dic₅₀₃⋊C₄	C₂²	1	2	yes	—
22	Dic₅₀₃⋊C₄	C₂²	1	2	yes	—
23	C₂²⋊Dic₅₀₃	C₂²	1	1	yes	characteristic
24	C₂²⋊Dic₅₀₃	C₂²	1	1	yes	characteristic
25	C₂²⋊Dic₅₀₃	C₂²	1	2	yes	—
26	C₄×Dic₅₀₃	C₂²	1	1	yes	characteristic
27	C₄×Dic₅₀₃	C₂²	1	1	yes	characteristic
28	C₄×Dic₅₀₃	C₂²	1	2	yes	—
29	C₄×Dic₅₀₃	C₂²	1	2	yes	—
30	C₂×Q₈×C₅₀₃	C₂²	1	1	yes	characteristic
31	C₂²×C₄×C₅₀₃	C₂²	1	1	yes	characteristic
32	C₄⋊C₄×C₅₀₃	C₂²	1	1	yes	characteristic
33	C₄⋊C₄×C₅₀₃	C₂²	1	2	yes	—
34	C₂²⋊C₄×C₅₀₃	C₂²	1	2	yes	—
35	Dic₁₀₀₆	—	2	2	no	—
36	C₂×Dic₅₀₃	C₂³	1	1	yes	characteristic
37	C₂×Dic₅₀₃	C₂³	1	1	yes	characteristic
38	C₂×Dic₅₀₃	C₂³	1	1	yes	characteristic
39	C₂×Dic₅₀₃	C₂³	1	2	yes	—
40	C₂×Dic₅₀₃	C₂³	1	1	yes	characteristic
41	C₂×Dic₅₀₃	C₂³	1	2	yes	—
42	C₂³×C₅₀₃	C₂³	1	1	yes	characteristic
43	Q₈×C₅₀₃	—	2	2	no	—
44	C₂×C₄×C₅₀₃	—	2	2	no	—
45	C₂×C₄×C₅₀₃	C₂³	1	2	yes	—
46	C₂×C₄×C₅₀₃	C₂³	1	2	yes	—
47	C₂×C₄×C₅₀₃	C₂³	1	1	yes	characteristic
48	C₂×C₄×C₅₀₃	C₂³	1	1	yes	characteristic
49	C₂²×C₅₀₃	—	2	1	no	—
50	C₂²×C₅₀₃	—	2	1	no	—
51	C₂²×C₅₀₃	—	2	1	no	—
52	C₂²×C₅₀₃	C₂⁴	1	1	yes	characteristic
53	C₄×C₅₀₃	—	2	1	no	—
54	C₄×C₅₀₃	—	2	2	no	—
55	C₄×C₅₀₃	—	2	2	no	—
56	C₄×C₅₀₃	#41	1	2	yes	—
57	Dic₅₀₃	—	2	2	no	—
58	Dic₅₀₃	—	2	1	no	—
59	Dic₅₀₃	—	2	1	no	—
60	Dic₅₀₃	—	2	1	no	—
61	Dic₅₀₃	—	2	1	no	—
62	Dic₅₀₃	—	2	2	no	—
63	C₂×C₅₀₃	—	4	1	no	—
64	C₂×C₅₀₃	C₂×#41	1	1	yes	characteristic
65	C₂×C₅₀₃	2₋¹⁺⁴	1	1	yes	characteristic
66	C₂×C₅₀₃	2₋¹⁺⁴	1	1	yes	characteristic
67	C₅₀₃	#541	1	1	yes	characteristic, complemented, Sylow
68	#541	—	503	1	no	maximal, Sylow
69	C₄⋊Q₈	—	503	1	no	—
70	#126	—	503	2	no	—
71	#126	—	503	2	no	—
72	#125	—	503	2	no	—
73	#125	—	503	1	no	—
74	#125	—	503	2	no	—
75	C₂²⋊Q₈	—	503	1	no	—
76	C₂²⋊Q₈	D₅₀₃	1	1	yes	characteristic
77	C₄×Q₈	—	503	1	no	—
78	C₄×Q₈	—	503	1	no	—
79	#117	—	503	1	no	—
80	C₂×Q₈	—	503	1	no	—
81	C₂×Q₈	C₂×D₅₀₃	1	1	yes	characteristic
82	C₂²×C₄	C₂×D₅₀₃	1	1	yes	characteristic
83	C₄⋊C₄	—	503	1	no	—
84	C₄⋊C₄	—	503	1	no	—
85	C₄⋊C₄	—	503	1	no	—
86	C₄⋊C₄	—	503	2	no	—
87	C₄⋊C₄	—	503	2	no	—
88	C₄⋊C₄	—	503	2	no	—
89	C₄⋊C₄	—	503	2	no	—
90	C₄⋊C₄	—	503	2	no	—
91	C₄⋊C₄	—	503	2	no	—
92	C₄⋊C₄	—	503	2	no	—
93	C₄⋊C₄	C₂×D₅₀₃	1	1	yes	characteristic
94	C₄⋊C₄	C₂×D₅₀₃	1	2	yes	—
95	C₂²⋊C₄	—	503	1	no	—
96	C₂²⋊C₄	—	503	1	no	—
97	C₂²⋊C₄	—	503	2	no	—
98	C₂²⋊C₄	C₂×D₅₀₃	1	2	yes	—
99	C₄²	—	503	1	no	—
100	C₄²	—	503	1	no	—
101	C₄²	—	503	2	no	—
102	C₄²	—	503	2	no	—
103	C₂³	C₂²×D₅₀₃	1	1	yes	characteristic
104	Q₈	—	2	2	no	—
105	Q₈	—	1006	2	no	—
106	C₂×C₄	—	2	2	no	—
107	C₂×C₄	—	503	1	no	—
108	C₂×C₄	—	503	1	no	—
109	C₂×C₄	—	503	1	no	—
110	C₂×C₄	—	503	1	no	—
111	C₂×C₄	—	503	2	no	—
112	C₂×C₄	—	503	2	no	—
113	C₂×C₄	C₂²×D₅₀₃	1	2	yes	—
114	C₂×C₄	C₂²×D₅₀₃	1	2	yes	—
115	C₂×C₄	C₂²×D₅₀₃	1	1	yes	characteristic
116	C₂×C₄	C₂²×D₅₀₃	1	1	yes	characteristic
117	C₂²	—	2	1	no	—
118	C₂²	—	2	1	no	—
119	C₂²	—	2	1	no	—
120	C₂²	C₂³×D₅₀₃	1	1	yes	characteristic
121	C₄	—	2	1	no	—
122	C₄	—	2	2	no	—
123	C₄	—	2	2	no	—
124	C₄	—	1006	2	no	—
125	C₄	—	1006	1	no	—
126	C₄	—	1006	1	no	—
127	C₄	—	1006	1	no	—
128	C₄	—	1006	1	no	—
129	C₄	—	1006	2	no	—
130	C₄	#22548882	1	2	yes	—
131	C₂	—	4	1	no	—
132	C₂	C₂×#22548882	1	1	yes	characteristic
133	C₂	#30632911	1	1	yes	characteristic
134	C₂	#30632912	1	1	yes	characteristic

subgroup

G/H

class size

multiplicity

normal?

properties

#30632862

C₂

yes

maximal, characteristic, complemented

#30632857

C₂

yes

maximal

#30632857

C₂

yes

maximal

#30632855

C₂

yes

maximal, characteristic, complemented

#30632855

C₂

yes

maximal, complemented

#30632854

C₂

yes

maximal, complemented

#30632843

C₂

yes

maximal, characteristic

Q₈×Dic₅₀₃

C₂

yes

maximal, characteristic, complemented

#30632829

C₂

yes

maximal, characteristic, complemented

#30632823

C₂

yes

maximal, characteristic

C₂²⋊Q₈×C₅₀₃

C₂

yes

maximal, characteristic

C₂×Dic₁₀₀₆

C₂²

yes

characteristic

C₄⋊Dic₅₀₃

C₂²

yes

characteristic

C₄⋊Dic₅₀₃

C₂²

yes

characteristic

C₄⋊Dic₅₀₃

C₂²

yes

characteristic

C₄⋊Dic₅₀₃

C₂²

yes

—

C₄⋊Dic₅₀₃

C₂²

yes

—

Dic₅₀₃⋊C₄

C₂²

yes

—

Dic₅₀₃⋊C₄

C₂²

yes

—

Dic₅₀₃⋊C₄

C₂²

yes

—

Dic₅₀₃⋊C₄

C₂²

yes

—

Dic₅₀₃⋊C₄

C₂²

yes

—

C₂²⋊Dic₅₀₃

C₂²

yes

characteristic

C₂²⋊Dic₅₀₃

C₂²

yes

characteristic

C₂²⋊Dic₅₀₃

C₂²

yes

—

C₄×Dic₅₀₃

C₂²

yes

characteristic

C₄×Dic₅₀₃

C₂²

yes

characteristic

C₄×Dic₅₀₃

C₂²

yes

—

C₄×Dic₅₀₃

C₂²

yes

—

C₂×Q₈×C₅₀₃

C₂²

yes

characteristic

C₂²×C₄×C₅₀₃

C₂²

yes

characteristic

C₄⋊C₄×C₅₀₃

C₂²

yes

characteristic

C₄⋊C₄×C₅₀₃

C₂²

yes

—

C₂²⋊C₄×C₅₀₃

C₂²

yes

—

Dic₁₀₀₆

—

C₂×Dic₅₀₃

C₂³

yes

characteristic

C₂×Dic₅₀₃

C₂³

yes

characteristic

C₂×Dic₅₀₃

C₂³

yes

characteristic

C₂×Dic₅₀₃

C₂³

yes

—

C₂×Dic₅₀₃

C₂³

yes

characteristic

C₂×Dic₅₀₃

C₂³

yes

—

C₂³×C₅₀₃

C₂³

yes

characteristic

Q₈×C₅₀₃

—

C₂×C₄×C₅₀₃

—

C₂×C₄×C₅₀₃

C₂³

yes

—

C₂×C₄×C₅₀₃

C₂³

yes

—

C₂×C₄×C₅₀₃

C₂³

yes

characteristic

C₂×C₄×C₅₀₃

C₂³

yes

characteristic

C₂²×C₅₀₃

—

C₂²×C₅₀₃

—

C₂²×C₅₀₃

—

C₂²×C₅₀₃

C₂⁴

yes

characteristic

C₄×C₅₀₃

—

C₄×C₅₀₃

—

C₄×C₅₀₃

—

C₄×C₅₀₃

#41

yes

—

Dic₅₀₃

—

Dic₅₀₃

—

Dic₅₀₃

—

Dic₅₀₃

—

Dic₅₀₃

—

Dic₅₀₃

—

C₂×C₅₀₃

—

C₂×C₅₀₃

C₂×#41

yes

characteristic

C₂×C₅₀₃

2₋¹⁺⁴

yes

characteristic

C₂×C₅₀₃

2₋¹⁺⁴

yes

characteristic

C₅₀₃

#541

yes

characteristic, complemented, Sylow

#541

—

503

maximal, Sylow

C₄⋊Q₈

—

503

—

#126

—

503

—

#126

—

503

—

#125

—

503

—

#125

—

503

—

#125

—

503

—

C₂²⋊Q₈

—

503

—

C₂²⋊Q₈

D₅₀₃

yes

characteristic

C₄×Q₈

—

503

—

C₄×Q₈

—

503

—

#117

—

503

—

C₂×Q₈

—

503

—

C₂×Q₈

C₂×D₅₀₃

yes

characteristic

C₂²×C₄

C₂×D₅₀₃

yes

characteristic

C₄⋊C₄

—

503

—

C₄⋊C₄

—

503

—

C₄⋊C₄

—

503

—

C₄⋊C₄

—

503

—

C₄⋊C₄

—

503

—

C₄⋊C₄

—

503

—

C₄⋊C₄

—

503

—

C₄⋊C₄

—

503

—

C₄⋊C₄

—

503

—

C₄⋊C₄

—

503

—

C₄⋊C₄

C₂×D₅₀₃

yes

characteristic

C₄⋊C₄

C₂×D₅₀₃

yes

—

C₂²⋊C₄

—

503

—

C₂²⋊C₄

—

503

—

C₂²⋊C₄

—

503

—

C₂²⋊C₄

C₂×D₅₀₃

yes

—

C₄²

—

503

—

100

C₄²

—

503

—

101

C₄²

—

503

—

102

C₄²

—

503

—

103

C₂³

C₂²×D₅₀₃

yes

characteristic

104

Q₈

—

105

Q₈

—

1006

—

106

C₂×C₄

—

107

C₂×C₄

—

503

—

108

C₂×C₄

—

503

—

109

C₂×C₄

—

503

—

110

C₂×C₄

—

503

—

111

C₂×C₄

—

503

—

112

C₂×C₄

—

503

—

113

C₂×C₄

C₂²×D₅₀₃

yes

—

114

C₂×C₄

C₂²×D₅₀₃

yes

—

115

C₂×C₄

C₂²×D₅₀₃

yes

characteristic

116

C₂×C₄

C₂²×D₅₀₃

yes

characteristic

117

C₂²

—

118

C₂²

—

119

C₂²

—

120

C₂²

C₂³×D₅₀₃

yes

characteristic

121

C₄

—

122

C₄

—

123

C₄

—

124

C₄

—

1006

—

125

C₄

—

1006

—

126

C₄

—

1006

—

127

C₄

—

1006

—

128

C₄

—

1006

—

129

C₄

—

1006

—

130

C₄

#22548882

yes

—

131

C₂

—

132

C₂

C₂×#22548882

yes

characteristic

133

C₂

#30632911

yes

characteristic

134

C₂

#30632912

yes

characteristic

	subgroup	G/H
center, Frattini	C₂²	C₂³×D₅₀₃
commutator, socle, cosocle	C₂²×C₅₀₃	C₂⁴
Fitting	C₂²⋊Q₈×C₅₀₃	C₂
prime radical	C₂³×C₅₀₃	C₂³
hypercenter	C₂²⋊Q₈	D₅₀₃
hypocenter	C₅₀₃	#541
LCS	G ▷ C₂²×C₅₀₃ ▷ C₅₀₃
UCS	C₁ ◁ C₂² ◁ C₂²⋊Q₈

subgroup

G/H

center, Frattini

C₂²

C₂³×D₅₀₃

commutator, socle, cosocle

C₂²×C₅₀₃

C₂⁴

Fitting

C₂²⋊Q₈×C₅₀₃

C₂

prime radical

C₂³×C₅₀₃

C₂³

hypercenter

C₂²⋊Q₈

D₅₀₃

C₅₀₃

G ▷ C₂²×C₅₀₃ ▷ C₅₀₃

C₁ ◁ C₂² ◁ C₂²⋊Q₈

N	H ≅ G/N	type
1. #30632862	C₂	semidirect
4. #30632855	C₂	semidirect
5. #30632855	C₂	semidirect
6. #30632854	C₂	semidirect
8. Q₈×Dic₅₀₃	C₂	semidirect
9. #30632829	C₂	semidirect
67. C₅₀₃	#541	semidirect

H ≅ G/N

type

1. #30632862

C₂

semidirect

4. #30632855

C₂

semidirect

5. #30632855

C₂

semidirect

6. #30632854

C₂

semidirect

8. Q₈×Dic₅₀₃

C₂

semidirect

9. #30632829

C₂

semidirect

67. C₅₀₃

#541

semidirect

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

subgroup	index
#30632862	2
#30632857	2
#30632855	2
#30632854	2
#30632843	2
Q₈×Dic₅₀₃	2
#30632829	2
#30632823	2
C₂²⋊Q₈×C₅₀₃	2
#541	503

subgroup

index

Q₈×Dic₅₀₃

C₂²⋊Q₈×C₅₀₃

503

Subgroups — #44566891

Conjugacy classes of subgroups

Complements

Maximal subgroups