Group #24908728 — subgroups

Subgroups — #24908728

Order 10624.

256 conjugacy classes of subgroups; only the 71 normal subgroups are shown.

normal subgroups only

This group has 256 conjugacy classes of subgroups — too many to draw. Only its 71 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	#19127727	C₂	1	1	yes	maximal, characteristic
2	#19127710	C₂	1	2	yes	maximal
3	C₂×#14581711	C₂	1	2	yes	maximal
4	#19127466	C₂	1	1	yes	maximal, characteristic
5	C₂²×Q₁₆×C₈₃	C₂	1	1	yes	maximal, characteristic
6	C₂×#13050574	C₂²	1	1	yes	characteristic
7	C₂×#13050574	C₂²	1	2	yes	—
8	#14581751	C₂²	1	1	yes	characteristic
9	Dic₈₃⋊C₈	—	2	2	no	—
10	#14581716	—	2	4	no	—
11	#14581711	—	2	4	no	—
12	C₂²×Q₈×C₈₃	C₂²	1	2	yes	—
13	C₂×Q₁₆×C₈₃	—	2	4	no	—
14	C₂×Q₁₆×C₈₃	—	2	2	no	—
15	C₂²×C₈×C₈₃	C₂²	1	1	yes	characteristic
16	#13050574	—	2	2	no	—
17	#13050574	—	2	2	no	—
18	#13050574	—	2	2	no	—
19	C₂×C₈₃⋊C₈	—	2	2	no	—
20	C₂×C₈₃⋊C₈	—	2	1	no	—
21	C₄⋊Dic₈₃	D₄	1	2	yes	—
22	C₂²⋊Dic₈₃	—	2	1	no	—
23	C₄×Dic₈₃	—	2	1	no	—
24	C₂×Q₈×C₈₃	—	2	2	no	—
25	C₂×Q₈×C₈₃	—	2	2	no	—
26	C₂×Q₈×C₈₃	—	2	2	no	—
27	C₂×Q₈×C₈₃	—	2	2	no	—
28	C₂×Q₈×C₈₃	—	2	2	no	—
29	C₂×Q₈×C₈₃	D₄	1	4	yes	—
30	C₂²×C₄×C₈₃	—	2	2	no	—
31	C₂²×C₄×C₈₃	C₂³	1	1	yes	characteristic
32	Q₁₆×C₈₃	—	2	8	no	—
33	C₂×C₈×C₈₃	—	2	2	no	—
34	C₂×C₈×C₈₃	D₄	1	2	yes	—
35	C₂³×C₈₃	#41	1	1	yes	characteristic
36	Q₈×C₈₃	—	2	4	no	—
37	Q₈×C₈₃	—	2	4	no	—
38	Q₈×C₈₃	—	4	2	no	—
39	Q₈×C₈₃	—	4	2	no	—
40	C₂×C₄×C₈₃	—	2	4	no	—
41	C₂×C₄×C₈₃	—	4	2	no	—
42	C₂×C₄×C₈₃	—	4	2	no	—
43	C₂×C₄×C₈₃	C₂×D₄	1	1	yes	characteristic
44	C₂×C₄×C₈₃	C₂×D₄	1	2	yes	—
45	C₂×C₄×C₈₃	C₂×D₄	1	1	yes	characteristic
46	C₂×C₄×C₈₃	#41	1	2	yes	—
47	C₂×Dic₈₃	—	2	2	no	—
48	C₈×C₈₃	—	2	2	no	—
49	C₈₃⋊C₈	—	4	1	no	—
50	C₈₃⋊C₈	—	4	2	no	—
51	C₂²×C₈₃	—	2	2	no	—
52	C₂²×C₈₃	#121	1	1	yes	characteristic
53	C₂²×C₈₃	#123	1	2	yes	—
54	C₄×C₈₃	—	4	4	no	—
55	C₄×C₈₃	#120	1	1	yes	characteristic
56	C₄×C₈₃	#121	1	2	yes	—
57	C₄×C₈₃	#124	1	1	yes	characteristic
58	Dic₈₃	—	4	2	no	—
59	C₂×C₈₃	—	2	2	no	—
60	C₂×C₈₃	#394	1	1	yes	characteristic
61	C₂×C₈₃	#473	1	2	yes	—
62	#2047	—	83	1	no	maximal, Sylow
63	C₈₃	#2047	1	1	yes	characteristic, complemented, Sylow
64	C₂²×Q₁₆	D₈₃	1	1	yes	characteristic
65	#424	—	83	1	no	—
66	#419	—	83	2	no	—
67	C₂×#101	—	83	2	no	—
68	#341	—	83	1	no	—
69	C₂²×Q₈	C₂×D₈₃	1	2	yes	—
70	C₂×Q₁₆	—	2	4	no	—
71	C₂×Q₁₆	—	2	2	no	—
72	C₂×M₁₆	—	83	2	no	—
73	C₂×M₁₆	—	83	1	no	—
74	C₂²×C₈	C₂×D₈₃	1	1	yes	characteristic
75	#117	—	83	1	no	—
76	C₄⋊C₈	—	166	2	no	—
77	#103	—	166	4	no	—
78	#101	—	166	4	no	—
79	C₂×Q₈	—	2	2	no	—
80	C₂×Q₈	—	2	2	no	—
81	C₂×Q₈	—	2	2	no	—
82	C₂×Q₈	—	2	2	no	—
83	C₂×Q₈	—	2	2	no	—
84	C₂×Q₈	C₂²⋊D₈₃	1	4	yes	—
85	C₂²×C₄	—	2	2	no	—
86	C₂²×C₄	C₂²×D₈₃	1	1	yes	characteristic
87	Q₁₆	—	2	8	no	—
88	M₁₆	—	166	2	no	—
89	M₁₆	—	166	2	no	—
90	M₁₆	—	166	2	no	—
91	C₂×C₈	—	2	2	no	—
92	C₂×C₈	—	166	2	no	—
93	C₂×C₈	—	166	1	no	—
94	C₂×C₈	C₂²⋊D₈₃	1	2	yes	—
95	C₄⋊C₄	—	83	2	no	—
96	C₂²⋊C₄	—	166	1	no	—
97	C₄²	—	166	1	no	—
98	C₂³	#13050589	1	1	yes	characteristic
99	Q₈	—	2	4	no	—
100	Q₈	—	2	4	no	—
101	Q₈	—	4	2	no	—
102	Q₈	—	4	2	no	—
103	C₂×C₄	—	2	4	no	—
104	C₂×C₄	—	4	2	no	—
105	C₂×C₄	—	4	2	no	—
106	C₂×C₄	—	166	2	no	—
107	C₂×C₄	D₄×D₈₃	1	1	yes	characteristic
108	C₂×C₄	C₂×C₂²⋊D₈₃	1	2	yes	—
109	C₂×C₄	C₂×C₂²⋊D₈₃	1	1	yes	characteristic
110	C₂×C₄	#13050589	1	2	yes	—
111	C₈	—	2	2	no	—
112	C₈	—	332	1	no	—
113	C₈	—	332	2	no	—
114	C₂²	—	2	2	no	—
115	C₂²	#14581763	1	1	yes	characteristic
116	C₂²	#14581773	1	2	yes	—
117	C₄	—	4	4	no	—
118	C₄	—	332	2	no	—
119	C₄	#14581761	1	1	yes	characteristic
120	C₄	#14581764	1	2	yes	—
121	C₄	#14581779	1	1	yes	characteristic
122	C₂	—	2	2	no	—
123	C₂	#19127615	1	1	yes	characteristic
124	C₂	#19127973	1	2	yes	—

subgroup

G/H

class size

multiplicity

normal?

properties

#19127727

C₂

yes

maximal, characteristic

#19127710

C₂

yes

maximal

C₂×#14581711

C₂

yes

maximal

#19127466

C₂

yes

maximal, characteristic

C₂²×Q₁₆×C₈₃

C₂

yes

maximal, characteristic

C₂×#13050574

C₂²

yes

characteristic

C₂×#13050574

C₂²

yes

—

#14581751

C₂²

yes

characteristic

Dic₈₃⋊C₈

—

#14581716

—

#14581711

—

C₂²×Q₈×C₈₃

C₂²

yes

—

C₂×Q₁₆×C₈₃

—

C₂×Q₁₆×C₈₃

—

C₂²×C₈×C₈₃

C₂²

yes

—

—

—

C₂×C₈₃⋊C₈

—

C₂×C₈₃⋊C₈

—

C₄⋊Dic₈₃

D₄

yes

—

C₂²⋊Dic₈₃

—

C₄×Dic₈₃

—

C₂×Q₈×C₈₃

—

C₂×Q₈×C₈₃

—

C₂×Q₈×C₈₃

—

C₂×Q₈×C₈₃

—

C₂×Q₈×C₈₃

—

C₂×Q₈×C₈₃

D₄

yes

—

C₂²×C₄×C₈₃

—

C₂²×C₄×C₈₃

C₂³

yes

characteristic

Q₁₆×C₈₃

—

C₂×C₈×C₈₃

—

C₂×C₈×C₈₃

D₄

yes

—

C₂³×C₈₃

#41

yes

characteristic

Q₈×C₈₃

—

Q₈×C₈₃

—

Q₈×C₈₃

—

Q₈×C₈₃

—

C₂×C₄×C₈₃

—

C₂×C₄×C₈₃

—

C₂×C₄×C₈₃

—

C₂×C₄×C₈₃

C₂×D₄

yes

characteristic

C₂×C₄×C₈₃

C₂×D₄

yes

—

C₂×C₄×C₈₃

C₂×D₄

yes

characteristic

C₂×C₄×C₈₃

#41

yes

—

C₂×Dic₈₃

—

C₈×C₈₃

—

C₈₃⋊C₈

—

C₈₃⋊C₈

—

C₂²×C₈₃

—

C₂²×C₈₃

#121

yes

characteristic

C₂²×C₈₃

#123

yes

—

C₄×C₈₃

—

C₄×C₈₃

#120

yes

characteristic

C₄×C₈₃

#121

yes

—

C₄×C₈₃

#124

yes

characteristic

Dic₈₃

—

C₂×C₈₃

—

C₂×C₈₃

#394

yes

characteristic

C₂×C₈₃

#473

yes

—

#2047

—

maximal, Sylow

C₈₃

#2047

yes

characteristic, complemented, Sylow

C₂²×Q₁₆

D₈₃

yes

—

—

—

—

C₂²×Q₈

C₂×D₈₃

yes

—

C₂×Q₁₆

—

C₂×Q₁₆

—

C₂×M₁₆

—

C₂×M₁₆

—

C₂²×C₈

C₂×D₈₃

yes

characteristic

#117

—

C₄⋊C₈

—

166

—

#103

—

166

—

#101

—

166

—

C₂×Q₈

—

C₂×Q₈

—

C₂×Q₈

—

C₂×Q₈

—

C₂×Q₈

—

C₂×Q₈

C₂²⋊D₈₃

yes

—

C₂²×C₄

—

C₂²×C₄

C₂²×D₈₃

yes

characteristic

Q₁₆

—

M₁₆

—

166

—

M₁₆

—

166

—

M₁₆

—

166

—

C₂×C₈

—

C₂×C₈

—

166

—

C₂×C₈

—

166

—

C₂×C₈

C₂²⋊D₈₃

yes

—

C₄⋊C₄

—

C₂²⋊C₄

—

166

—

C₄²

—

166

—

C₂³

#13050589

yes

characteristic

Q₈

—

100

Q₈

—

101

Q₈

—

102

Q₈

—

103

C₂×C₄

—

104

C₂×C₄

—

105

C₂×C₄

—

106

C₂×C₄

—

166

—

107

C₂×C₄

D₄×D₈₃

yes

characteristic

108

C₂×C₄

C₂×C₂²⋊D₈₃

yes

—

109

C₂×C₄

C₂×C₂²⋊D₈₃

yes

characteristic

110

C₂×C₄

#13050589

yes

—

111

C₈

—

112

C₈

—

332

—

113

C₈

—

332

—

114

C₂²

—

115

C₂²

#14581763

yes

characteristic

116

C₂²

#14581773

yes

—

117

C₄

—

118

C₄

—

332

—

119

C₄

#14581761

yes

characteristic

120

C₄

#14581764

yes

—

121

C₄

#14581779

yes

characteristic

122

C₂

—

123

C₂

#19127615

yes

characteristic

124

C₂

#19127973

yes

—

	subgroup	G/H
center	C₂²	#14581763
commutator, cosocle	C₂²×C₄×C₈₃	C₂³
Frattini	C₂²×C₄	C₂²×D₈₃
Fitting	C₂²×Q₁₆×C₈₃	C₂
socle	C₂²×C₈₃	#121
prime radical	C₂³×C₈₃	#41
hypercenter	C₂²×Q₁₆	D₈₃
hypocenter	C₈₃	#2047
LCS	G ▷ C₂²×C₄×C₈₃ ▷ C₂×C₈₃ ▷ C₈₃
UCS	C₁ ◁ C₂² ◁ C₂²×C₄ ◁ C₂²×Q₁₆

subgroup

G/H

center

C₂²

#14581763

commutator, cosocle

C₂²×C₄×C₈₃

C₂³

Frattini

C₂²×C₄

C₂²×D₈₃

Fitting

C₂²×Q₁₆×C₈₃

C₂

C₂²×C₈₃

C₂³×C₈₃

C₂²×Q₁₆

D₈₃

hypocenter

C₈₃

#2047

LCS

G ▷ C₂²×C₄×C₈₃ ▷ C₂×C₈₃ ▷ C₈₃

UCS

C₁ ◁ C₂² ◁ C₂²×C₄ ◁ C₂²×Q₁₆

N	H ≅ G/N	type
63. C₈₃	#2047	semidirect

H ≅ G/N

type

63. C₈₃

#2047

semidirect

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

subgroup	index
#19127727	2
#19127710	2
C₂×#14581711	2
#19127466	2
C₂²×Q₁₆×C₈₃	2
#2047	83

subgroup

index

C₂²×Q₁₆×C₈₃

Subgroups — #24908728

Conjugacy classes of subgroups

Complements

Maximal subgroups