Group #24916552 — subgroups

Subgroups — #24916552

Order 10624.

364 conjugacy classes of subgroups; only the 221 normal subgroups are shown.

normal subgroups only

This group has 364 conjugacy classes of subgroups — too many to draw. Only its 221 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	#19127832	C₂	1	6	yes	maximal, complemented
2	Q₈×C₈₃⋊C₈	C₂	1	2	yes	maximal, complemented
3	#19127758	C₂	1	3	yes	maximal
4	#19127754	C₂	1	3	yes	maximal
5	C₂×C₄×Q₈×C₈₃	C₂	1	1	yes	maximal, characteristic
6	C₄⋊(C₈₃⋊C₈)	C₂²	1	6	yes	—
7	C₄⋊(C₈₃⋊C₈)	C₂²	1	6	yes	—
8	C₂²⋊(C₈₃⋊C₈)	C₂²	1	1	yes	characteristic
9	C₂²⋊(C₈₃⋊C₈)	C₂²	1	3	yes	—
10	#14581703	C₂²	1	6	yes	—
11	C₄×C₈₃⋊C₈	C₂²	1	6	yes	—
12	C₂²×Q₈×C₈₃	C₄	1	1	yes	characteristic
13	C₄×Q₈×C₈₃	C₄	1	3	yes	—
14	C₄×Q₈×C₈₃	C₄	1	1	yes	characteristic
15	C₄×Q₈×C₈₃	C₂²	1	1	yes	characteristic
16	C₄×Q₈×C₈₃	C₂²	1	3	yes	—
17	C₂×C₄⋊C₄×C₈₃	C₄	1	3	yes	—
18	C₂×C₄²×C₈₃	C₂²	1	3	yes	—
19	C₂×C₈₃⋊C₈	C₂³	1	2	yes	—
20	C₂×C₈₃⋊C₈	C₂³	1	6	yes	—
21	C₂×Q₈×C₈₃	—	2	4	no	—
22	C₂×Q₈×C₈₃	C₂×C₄	1	3	yes	—
23	C₂×Q₈×C₈₃	C₂×C₄	1	1	yes	characteristic
24	C₂²×C₄×C₈₃	C₂×C₄	1	3	yes	—
25	C₂²×C₄×C₈₃	C₂×C₄	1	3	yes	—
26	C₂²×C₄×C₈₃	C₂³	1	1	yes	characteristic
27	C₄⋊C₄×C₈₃	C₂×C₄	1	6	yes	—
28	C₄⋊C₄×C₈₃	C₂×C₄	1	3	yes	—
29	C₄⋊C₄×C₈₃	C₂×C₄	1	3	yes	—
30	C₄²×C₈₃	C₂×C₄	1	3	yes	—
31	C₄²×C₈₃	C₂×C₄	1	3	yes	—
32	C₄²×C₈₃	C₂³	1	3	yes	—
33	C₄²×C₈₃	C₂³	1	3	yes	—
34	C₂³×C₈₃	C₂²×C₄	1	1	yes	characteristic
35	Q₈×C₈₃	—	2	6	no	—
36	Q₈×C₈₃	M₁₆	1	4	yes	—
37	C₂×C₄×C₈₃	—	2	6	no	—
38	C₂×C₄×C₈₃	—	2	6	no	—
39	C₂×C₄×C₈₃	—	2	2	no	—
40	C₂×C₄×C₈₃	C₂²×C₄	1	1	yes	characteristic
41	C₂×C₄×C₈₃	C₂²×C₄	1	3	yes	—
42	C₂×C₄×C₈₃	C₂²×C₄	1	3	yes	—
43	C₂×C₄×C₈₃	C₂²×C₄	1	3	yes	—
44	C₂×C₄×C₈₃	C₂²×C₄	1	3	yes	—
45	C₂×C₄×C₈₃	C₂⁴	1	1	yes	characteristic
46	C₈₃⋊C₈	—	2	2	no	—
47	C₈₃⋊C₈	—	2	6	no	—
48	C₂²×C₈₃	—	2	1	no	—
49	C₂²×C₈₃	—	2	1	no	—
50	C₂²×C₈₃	#131	1	2	yes	—
51	C₂²×C₈₃	C₂³×C₄	1	1	yes	characteristic
52	C₄×C₈₃	—	2	1	no	—
53	C₄×C₈₃	—	2	3	no	—
54	C₄×C₈₃	—	2	3	no	—
55	C₄×C₈₃	—	2	3	no	—
56	C₄×C₈₃	C₂×M₁₆	1	6	yes	—
57	C₄×C₈₃	2₋¹⁺⁴	1	2	yes	—
58	C₂×C₈₃	—	2	2	no	—
59	C₂×C₈₃	#519	1	1	yes	characteristic
60	C₂×C₈₃	C₂²×M₁₆	1	1	yes	characteristic
61	C₂×C₈₃	C₂×#131	1	1	yes	characteristic
62	#2984	—	83	1	no	maximal, Sylow
63	C₈₃	#2984	1	1	yes	characteristic, complemented, Sylow
64	C₂×C₄×Q₈	D₈₃	1	1	yes	characteristic
65	#446	—	83	6	no	—
66	C₈×Q₈	—	83	2	no	—
67	#432	—	83	3	no	—
68	#431	—	83	3	no	—
69	C₂²×Q₈	Dic₈₃	1	1	yes	characteristic
70	C₄×Q₈	Dic₈₃	1	3	yes	—
71	C₄×Q₈	Dic₈₃	1	1	yes	characteristic
72	C₄×Q₈	C₂×D₈₃	1	1	yes	characteristic
73	C₄×Q₈	C₂×D₈₃	1	3	yes	—
74	C₂×C₄⋊C₄	Dic₈₃	1	3	yes	—
75	C₂×C₄²	C₂×D₈₃	1	3	yes	—
76	C₄⋊C₈	—	83	6	no	—
77	C₄⋊C₈	—	83	6	no	—
78	C₂²⋊C₈	—	83	1	no	—
79	C₂²⋊C₈	—	83	3	no	—
80	#97	—	83	6	no	—
81	C₄×C₈	—	83	6	no	—
82	C₂×Q₈	—	2	4	no	—
83	C₂×Q₈	C₂×Dic₈₃	1	3	yes	—
84	C₂×Q₈	C₂×Dic₈₃	1	1	yes	characteristic
85	C₂²×C₄	C₂×Dic₈₃	1	3	yes	—
86	C₂²×C₄	C₂×Dic₈₃	1	3	yes	—
87	C₂²×C₄	C₂²×D₈₃	1	1	yes	characteristic
88	C₂×C₈	—	83	2	no	—
89	C₂×C₈	—	83	6	no	—
90	C₄⋊C₄	C₂×Dic₈₃	1	6	yes	—
91	C₄⋊C₄	C₂×Dic₈₃	1	3	yes	—
92	C₄⋊C₄	C₂×Dic₈₃	1	3	yes	—
93	C₄²	C₂×Dic₈₃	1	3	yes	—
94	C₄²	C₂×Dic₈₃	1	3	yes	—
95	C₄²	C₂²×D₈₃	1	3	yes	—
96	C₄²	C₂²×D₈₃	1	3	yes	—
97	C₂³	C₂²×Dic₈₃	1	1	yes	characteristic
98	Q₈	—	2	6	no	—
99	Q₈	#13050574	1	4	yes	—
100	C₂×C₄	—	2	6	no	—
101	C₂×C₄	—	2	6	no	—
102	C₂×C₄	—	2	2	no	—
103	C₂×C₄	C₂²×Dic₈₃	1	1	yes	characteristic
104	C₂×C₄	C₂²×Dic₈₃	1	3	yes	—
105	C₂×C₄	C₂²×Dic₈₃	1	3	yes	—
106	C₂×C₄	C₂²×Dic₈₃	1	3	yes	—
107	C₂×C₄	C₂²×Dic₈₃	1	3	yes	—
108	C₂×C₄	C₂³×D₈₃	1	1	yes	characteristic
109	C₈	—	166	2	no	—
110	C₈	—	166	6	no	—
111	C₂²	—	2	1	no	—
112	C₂²	—	2	1	no	—
113	C₂²	#14581799	1	2	yes	—
114	C₂²	C₂³×Dic₈₃	1	1	yes	characteristic
115	C₄	—	2	1	no	—
116	C₄	—	2	3	no	—
117	C₄	—	2	3	no	—
118	C₄	—	2	3	no	—
119	C₄	C₂×#13050574	1	6	yes	—
120	C₄	#14581840	1	2	yes	—
121	C₂	—	2	2	no	—
122	C₂	#19128179	1	1	yes	characteristic
123	C₂	C₂²×#13050574	1	1	yes	characteristic
124	C₂	C₂×#14581799	1	1	yes	characteristic

subgroup

G/H

class size

multiplicity

normal?

properties

#19127832

C₂

yes

maximal, complemented

Q₈×C₈₃⋊C₈

C₂

yes

maximal, complemented

#19127758

C₂

yes

maximal

#19127754

C₂

yes

maximal

C₂×C₄×Q₈×C₈₃

C₂

yes

maximal, characteristic

C₄⋊(C₈₃⋊C₈)

C₂²

yes

—

C₄⋊(C₈₃⋊C₈)

C₂²

yes

—

C₂²⋊(C₈₃⋊C₈)

C₂²

yes

characteristic

C₂²⋊(C₈₃⋊C₈)

C₂²

yes

—

#14581703

C₂²

yes

—

C₄×C₈₃⋊C₈

C₂²

yes

—

C₂²×Q₈×C₈₃

C₄

yes

characteristic

C₄×Q₈×C₈₃

C₄

yes

—

C₄×Q₈×C₈₃

C₄

yes

characteristic

C₄×Q₈×C₈₃

C₂²

yes

characteristic

C₄×Q₈×C₈₃

C₂²

yes

—

C₂×C₄⋊C₄×C₈₃

C₄

yes

—

C₂×C₄²×C₈₃

C₂²

yes

—

C₂×C₈₃⋊C₈

C₂³

yes

—

C₂×C₈₃⋊C₈

C₂³

yes

—

C₂×Q₈×C₈₃

—

C₂×Q₈×C₈₃

C₂×C₄

yes

—

C₂×Q₈×C₈₃

C₂×C₄

yes

characteristic

C₂²×C₄×C₈₃

C₂×C₄

yes

—

C₂²×C₄×C₈₃

C₂×C₄

yes

—

C₂²×C₄×C₈₃

C₂³

yes

characteristic

C₄⋊C₄×C₈₃

C₂×C₄

yes

—

C₄⋊C₄×C₈₃

C₂×C₄

yes

—

C₄⋊C₄×C₈₃

C₂×C₄

yes

—

C₄²×C₈₃

C₂×C₄

yes

—

C₄²×C₈₃

C₂×C₄

yes

—

C₄²×C₈₃

C₂³

yes

—

C₄²×C₈₃

C₂³

yes

—

C₂³×C₈₃

C₂²×C₄

yes

characteristic

Q₈×C₈₃

—

Q₈×C₈₃

M₁₆

yes

—

C₂×C₄×C₈₃

—

C₂×C₄×C₈₃

—

C₂×C₄×C₈₃

—

C₂×C₄×C₈₃

C₂²×C₄

yes

characteristic

C₂×C₄×C₈₃

C₂²×C₄

yes

—

C₂×C₄×C₈₃

C₂²×C₄

yes

—

C₂×C₄×C₈₃

C₂²×C₄

yes

—

C₂×C₄×C₈₃

C₂²×C₄

yes

—

C₂×C₄×C₈₃

C₂⁴

yes

characteristic

C₈₃⋊C₈

—

C₈₃⋊C₈

—

C₂²×C₈₃

—

C₂²×C₈₃

—

C₂²×C₈₃

#131

yes

—

C₂²×C₈₃

C₂³×C₄

yes

characteristic

C₄×C₈₃

—

C₄×C₈₃

—

C₄×C₈₃

—

C₄×C₈₃

—

C₄×C₈₃

C₂×M₁₆

yes

—

C₄×C₈₃

2₋¹⁺⁴

yes

—

C₂×C₈₃

—

C₂×C₈₃

#519

yes

characteristic

C₂×C₈₃

C₂²×M₁₆

yes

C₂×C₈₃

yes

—

C₈₃

yes

characteristic, complemented, Sylow

C₂×C₄×Q₈

D₈₃

yes

characteristic

#446

—

C₈×Q₈

—

#432

—

#431

—

C₂²×Q₈

Dic₈₃

yes

characteristic

C₄×Q₈

Dic₈₃

yes

—

C₄×Q₈

Dic₈₃

yes

characteristic

C₄×Q₈

C₂×D₈₃

yes

characteristic

C₄×Q₈

C₂×D₈₃

yes

—

C₂×C₄⋊C₄

Dic₈₃

yes

—

C₂×C₄²

C₂×D₈₃

yes

—

C₄⋊C₈

—

C₄⋊C₈

—

C₂²⋊C₈

—

C₂²⋊C₈

—

#97

—

C₄×C₈

—

C₂×Q₈

—

C₂×Q₈

C₂×Dic₈₃

yes

—

C₂×Q₈

C₂×Dic₈₃

yes

characteristic

C₂²×C₄

C₂×Dic₈₃

yes

—

C₂²×C₄

C₂×Dic₈₃

yes

—

C₂²×C₄

C₂²×D₈₃

yes

characteristic

C₂×C₈

—

C₂×C₈

—

C₄⋊C₄

C₂×Dic₈₃

yes

—

C₄⋊C₄

C₂×Dic₈₃

yes

—

C₄⋊C₄

C₂×Dic₈₃

yes

—

C₄²

C₂×Dic₈₃

yes

—

C₄²

C₂×Dic₈₃

yes

—

C₄²

C₂²×D₈₃

yes

—

C₄²

C₂²×D₈₃

yes

—

C₂³

C₂²×Dic₈₃

yes

characteristic

Q₈

—

Q₈

#13050574

yes

—

100

C₂×C₄

—

101

C₂×C₄

—

102

C₂×C₄

—

103

C₂×C₄

C₂²×Dic₈₃

yes

characteristic

104

C₂×C₄

C₂²×Dic₈₃

yes

—

105

C₂×C₄

C₂²×Dic₈₃

yes

—

106

C₂×C₄

C₂²×Dic₈₃

yes

—

107

C₂×C₄

C₂²×Dic₈₃

yes

—

108

C₂×C₄

C₂³×D₈₃

yes

characteristic

109

C₈

—

166

—

110

C₈

—

166

—

111

C₂²

—

112

C₂²

—

113

C₂²

#14581799

yes

—

114

C₂²

C₂³×Dic₈₃

yes

characteristic

115

C₄

—

116

C₄

—

117

C₄

—

118

C₄

—

119

C₄

C₂×#13050574

yes

—

120

C₄

#14581840

yes

—

121

C₂

—

122

C₂

#19128179

yes

characteristic

123

C₂

C₂²×#13050574

yes

characteristic

124

C₂

C₂×#14581799

yes

characteristic

	subgroup	G/H
center, Frattini	C₂×C₄	C₂³×D₈₃
commutator, socle	C₂²×C₈₃	C₂³×C₄
Fitting	C₂×C₄×Q₈×C₈₃	C₂
prime radical	C₂³×C₈₃	C₂²×C₄
cosocle	C₂×C₄×C₈₃	C₂⁴
hypercenter	C₂×C₄×Q₈	D₈₃
hypocenter	C₈₃	#2984
LCS	G ▷ C₂²×C₈₃ ▷ C₈₃
UCS	C₁ ◁ C₂×C₄ ◁ C₂×C₄×Q₈

subgroup

G/H

center, Frattini

C₂×C₄

C₂³×D₈₃

commutator, socle

C₂²×C₈₃

C₂³×C₄

Fitting

C₂×C₄×Q₈×C₈₃

C₂

prime radical

C₂³×C₈₃

C₂²×C₄

cosocle

C₂×C₄×C₈₃

C₂⁴

hypercenter

C₂×C₄×Q₈

D₈₃

C₈₃

G ▷ C₂²×C₈₃ ▷ C₈₃

C₁ ◁ C₂×C₄ ◁ C₂×C₄×Q₈

N	H ≅ G/N	type
1. #19127832	C₂	semidirect
2. Q₈×C₈₃⋊C₈	C₂	semidirect
63. C₈₃	#2984	semidirect

H ≅ G/N

type

1. #19127832

C₂

semidirect

2. Q₈×C₈₃⋊C₈

C₂

semidirect

63. C₈₃

#2984

semidirect

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

subgroup	index
#19127832	2
Q₈×C₈₃⋊C₈	2
#19127758	2
#19127754	2
C₂×C₄×Q₈×C₈₃	2
#2984	83

subgroup

index

Q₈×C₈₃⋊C₈

C₂×C₄×Q₈×C₈₃

Subgroups — #24916552

Conjugacy classes of subgroups

Complements

Maximal subgroups