Group #24908253 — subgroups

Subgroups — #24908253

Order 10624.

184 conjugacy classes of subgroups; only the 89 normal subgroups are shown.

normal subgroups only

This group has 184 conjugacy classes of subgroups — too many to draw. Only its 89 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	#19127749	C₂	1	2	yes	maximal
2	C₂×#14581728	C₂	1	1	yes	maximal, characteristic
3	C₂×#14581728	C₂	1	1	yes	maximal, characteristic
4	#19127471	C₂	1	2	yes	maximal
5	#405×C₈₃	C₂	1	1	yes	maximal, characteristic
6	C₂×#13050574	C₂²	1	2	yes	—
7	C₂×#13050574	C₂²	1	2	yes	—
8	#14581728	—	2	1	no	—
9	#14581728	—	2	1	no	—
10	#14581728	C₄	1	4	yes	complemented
11	#14581727	—	2	4	no	—
12	C₂×M₁₆×C₈₃	C₂²	1	1	yes	characteristic
13	C₂²×C₈×C₈₃	C₂²	1	1	yes	characteristic
14	#117×C₈₃	C₂²	1	1	yes	characteristic
15	#97×C₈₃	—	2	1	no	—
16	C₄×C₈×C₈₃	—	2	1	no	—
17	#13050574	—	2	2	no	—
18	#13050574	—	2	2	no	—
19	#13050574	—	2	2	no	—
20	#13050574	C₂×C₄	1	4	yes	—
21	C₂×C₈₃⋊C₈	—	2	2	no	—
22	C₂×C₈₃⋊C₈	—	2	2	no	—
23	C₂²×C₄×C₈₃	C₂³	1	1	yes	characteristic
24	M₁₆×C₈₃	—	2	1	no	—
25	M₁₆×C₈₃	D₄	1	2	yes	—
26	C₂×C₈×C₈₃	—	2	1	no	—
27	C₂×C₈×C₈₃	C₂×C₄	1	2	yes	—
28	C₂×C₈×C₈₃	D₄	1	1	yes	characteristic
29	C₂×C₈×C₈₃	D₄	1	1	yes	characteristic
30	C₂×C₈×C₈₃	Q₈	1	2	yes	—
31	C₄⋊C₄×C₈₃	—	2	1	no	—
32	C₂²⋊C₄×C₈₃	Q₈	1	2	yes	—
33	C₄²×C₈₃	—	2	1	no	—
34	C₂³×C₈₃	C₂×Q₈	1	1	yes	characteristic
35	C₂×C₄×C₈₃	—	2	2	no	—
36	C₂×C₄×C₈₃	C₂²×C₄	1	1	yes	characteristic
37	C₂×C₄×C₈₃	C₂×D₄	1	1	yes	characteristic
38	C₂×C₄×C₈₃	C₂×D₄	1	1	yes	characteristic
39	C₂×C₄×C₈₃	C₂×Q₈	1	1	yes	characteristic
40	C₂×C₄×C₈₃	#41	1	1	yes	characteristic
41	C₂×C₄×C₈₃	#41	1	1	yes	characteristic
42	C₈×C₈₃	—	2	2	no	—
43	C₈×C₈₃	C₄⋊C₄	1	2	yes	—
44	C₈×C₈₃	C₄⋊C₄	1	2	yes	—
45	C₈₃⋊C₈	—	2	4	no	—
46	C₈₃⋊C₈	—	4	2	no	—
47	C₂²×C₈₃	—	4	1	no	—
48	C₂²×C₈₃	C₄×Q₈	1	1	yes	characteristic
49	C₂²×C₈₃	C₂²⋊Q₈	1	1	yes	characteristic
50	C₂²×C₈₃	C₄⋊Q₈	1	1	yes	characteristic
51	C₄×C₈₃	—	4	2	no	—
52	C₄×C₈₃	C₂×C₄⋊C₄	1	1	yes	characteristic
53	C₄×C₈₃	C₄×D₄	1	1	yes	characteristic
54	C₄×C₈₃	#121	1	1	yes	characteristic
55	C₄×C₈₃	#125	1	1	yes	characteristic
56	C₂×C₈₃	—	2	1	no	—
57	C₂×C₈₃	—	2	1	no	—
58	C₂×C₈₃	—	2	1	no	—
59	C₂×C₈₃	#389	1	1	yes	characteristic
60	#1953	—	83	1	no	maximal, Sylow
61	C₈₃	#1953	1	1	yes	characteristic, complemented, Sylow
62	#430	—	83	2	no	—
63	C₂×#108	—	83	1	no	—
64	C₂×#108	—	83	1	no	—
65	#405	D₈₃	1	1	yes	characteristic
66	#344	—	83	2	no	—
67	C₂×M₁₆	—	83	2	no	—
68	C₂×M₁₆	—	83	2	no	—
69	C₂×M₁₆	C₂×D₈₃	1	1	yes	characteristic
70	C₂²×C₈	C₂×D₈₃	1	1	yes	characteristic
71	#117	C₂×D₈₃	1	1	yes	characteristic
72	#108	—	83	4	no	—
73	#108	—	166	4	no	—
74	#108	—	166	1	no	—
75	#108	—	166	1	no	—
76	#97	—	2	1	no	—
77	C₄×C₈	—	2	1	no	—
78	C₂²×C₄	C₂²×D₈₃	1	1	yes	characteristic
79	M₁₆	—	2	1	no	—
80	M₁₆	—	83	4	no	—
81	M₁₆	—	166	2	no	—
82	M₁₆	—	166	2	no	—
83	M₁₆	—	166	2	no	—
84	M₁₆	D₃₃₂	1	2	yes	—
85	C₂×C₈	—	2	1	no	—
86	C₂×C₈	—	166	2	no	—
87	C₂×C₈	—	166	2	no	—
88	C₂×C₈	Dic₁₆₆	1	2	yes	—
89	C₂×C₈	C₄×D₈₃	1	2	yes	—
90	C₂×C₈	C₂²⋊D₈₃	1	1	yes	characteristic
91	C₂×C₈	C₂²⋊D₈₃	1	1	yes	characteristic
92	C₄⋊C₄	—	2	1	no	—
93	C₂²⋊C₄	Dic₁₆₆	1	2	yes	—
94	C₄²	—	2	1	no	—
95	C₂³	C₂×Dic₁₆₆	1	1	yes	characteristic
96	C₂×C₄	—	2	2	no	—
97	C₂×C₄	C₂×C₄×D₈₃	1	1	yes	characteristic
98	C₂×C₄	C₂×C₂²⋊D₈₃	1	1	yes	characteristic
99	C₂×C₄	C₂×D₃₃₂	1	1	yes	characteristic
100	C₂×C₄	C₂×Dic₁₆₆	1	1	yes	characteristic
101	C₂×C₄	#13050590	1	1	yes	characteristic
102	C₂×C₄	#13050590	1	1	yes	characteristic
103	C₈	—	2	2	no	—
104	C₈	—	332	2	no	—
105	C₈	—	166	4	no	—
106	C₈	Dic₈₃⋊C₄	1	2	yes	—
107	C₈	Dic₈₃⋊C₄	1	2	yes	—
108	C₂²	—	4	1	no	—
109	C₂²	C₄×Dic₁₆₆	1	1	yes	characteristic
110	C₂²	#14581771	1	1	yes	characteristic
111	C₂²	#14581792	1	1	yes	characteristic
112	C₄	—	4	2	no	—
113	C₄	C₂×Dic₈₃⋊C₄	1	1	yes	characteristic
114	C₄	C₄×D₃₃₂	1	1	yes	characteristic
115	C₄	#14581766	1	1	yes	characteristic
116	C₄	#14581784	1	1	yes	characteristic
117	C₂	—	2	1	no	—
118	C₂	—	2	1	no	—
119	C₂	—	2	1	no	—
120	C₂	#19127599	1	1	yes	characteristic

subgroup

G/H

class size

multiplicity

normal?

properties

#19127749

C₂

yes

maximal

C₂×#14581728

C₂

yes

maximal, characteristic

C₂×#14581728

C₂

yes

maximal, characteristic

#19127471

C₂

yes

maximal

#405×C₈₃

C₂

yes

maximal, characteristic

C₂×#13050574

C₂²

yes

—

C₂×#13050574

C₂²

yes

—

#14581728

—

#14581728

—

#14581728

C₄

yes

complemented

#14581727

—

C₂×M₁₆×C₈₃

C₂²

yes

characteristic

C₂²×C₈×C₈₃

C₂²

yes

characteristic

#117×C₈₃

C₂²

yes

characteristic

#97×C₈₃

—

C₄×C₈×C₈₃

—

—

—

—

C₂×C₄

yes

—

C₂×C₈₃⋊C₈

—

C₂×C₈₃⋊C₈

—

C₂²×C₄×C₈₃

C₂³

yes

characteristic

M₁₆×C₈₃

—

M₁₆×C₈₃

D₄

yes

—

C₂×C₈×C₈₃

—

C₂×C₈×C₈₃

C₂×C₄

yes

—

C₂×C₈×C₈₃

D₄

yes

characteristic

C₂×C₈×C₈₃

D₄

yes

characteristic

C₂×C₈×C₈₃

Q₈

yes

—

C₄⋊C₄×C₈₃

—

C₂²⋊C₄×C₈₃

Q₈

yes

—

C₄²×C₈₃

—

C₂³×C₈₃

C₂×Q₈

yes

characteristic

C₂×C₄×C₈₃

—

C₂×C₄×C₈₃

C₂²×C₄

yes

characteristic

C₂×C₄×C₈₃

C₂×D₄

yes

characteristic

C₂×C₄×C₈₃

C₂×D₄

yes

characteristic

C₂×C₄×C₈₃

C₂×Q₈

yes

C₂×C₄×C₈₃

yes

C₂×C₄×C₈₃

yes

C₈×C₈₃

—

C₈×C₈₃

C₄⋊C₄

yes

—

C₈×C₈₃

C₄⋊C₄

yes

—

C₈₃⋊C₈

—

C₈₃⋊C₈

—

C₂²×C₈₃

—

C₂²×C₈₃

C₄×Q₈

yes

characteristic

C₂²×C₈₃

C₂²⋊Q₈

yes

characteristic

C₂²×C₈₃

C₄⋊Q₈

yes

characteristic

C₄×C₈₃

—

C₄×C₈₃

C₂×C₄⋊C₄

yes

characteristic

C₄×C₈₃

C₄×D₄

yes

C₄×C₈₃

yes

C₄×C₈₃

yes

C₂×C₈₃

—

C₂×C₈₃

—

C₂×C₈₃

—

C₂×C₈₃

yes

—

C₈₃

yes

characteristic, complemented, Sylow

—

—

—

D₈₃

yes

characteristic

#344

—

C₂×M₁₆

—

C₂×M₁₆

—

C₂×M₁₆

C₂×D₈₃

yes

characteristic

C₂²×C₈

C₂×D₈₃

yes

characteristic

#117

C₂×D₈₃

yes

characteristic

#108

—

#108

—

166

—

#108

—

166

—

#108

—

166

—

#97

—

C₄×C₈

—

C₂²×C₄

C₂²×D₈₃

yes

characteristic

M₁₆

—

M₁₆

—

M₁₆

—

166

—

M₁₆

—

166

—

M₁₆

—

166

—

M₁₆

D₃₃₂

yes

—

C₂×C₈

—

C₂×C₈

—

166

—

C₂×C₈

—

166

—

C₂×C₈

Dic₁₆₆

yes

—

C₂×C₈

C₄×D₈₃

yes

—

C₂×C₈

C₂²⋊D₈₃

yes

characteristic

C₂×C₈

C₂²⋊D₈₃

yes

characteristic

C₄⋊C₄

—

C₂²⋊C₄

Dic₁₆₆

yes

—

C₄²

—

C₂³

C₂×Dic₁₆₆

yes

characteristic

C₂×C₄

—

C₂×C₄

C₂×C₄×D₈₃

yes

characteristic

C₂×C₄

C₂×C₂²⋊D₈₃

yes

characteristic

C₂×C₄

C₂×D₃₃₂

yes

characteristic

100

C₂×C₄

C₂×Dic₁₆₆

yes

characteristic

101

C₂×C₄

#13050590

yes

characteristic

102

C₂×C₄

#13050590

yes

characteristic

103

C₈

—

104

C₈

—

332

—

105

C₈

—

166

—

106

C₈

Dic₈₃⋊C₄

yes

—

107

C₈

Dic₈₃⋊C₄

yes

—

108

C₂²

—

109

C₂²

C₄×Dic₁₆₆

yes

characteristic

110

C₂²

#14581771

yes

characteristic

111

C₂²

#14581792

yes

characteristic

112

C₄

—

113

C₄

C₂×Dic₈₃⋊C₄

yes

characteristic

114

C₄

C₄×D₃₃₂

yes

characteristic

115

C₄

#14581766

yes

characteristic

116

C₄

#14581784

yes

characteristic

117

C₂

—

118

C₂

—

119

C₂

—

120

C₂

#19127599

yes

characteristic

	subgroup	G/H
center	C₄	#14581766
commutator	C₂×C₄×C₈₃	C₂²×C₄
Frattini	C₂²×C₄	C₂²×D₈₃
Fitting	#405×C₈₃	C₂
socle	C₂×C₈₃	#389
prime radical	C₂³×C₈₃	C₂×Q₈
cosocle	C₂²×C₄×C₈₃	C₂³
hypercenter	#405	D₈₃
hypocenter	C₈₃	#1953
LCS	G ▷ C₂×C₄×C₈₃ ▷ C₂×C₈₃ ▷ C₈₃
UCS	C₁ ◁ C₄ ◁ C₂²×C₄ ◁ #405

subgroup

G/H

center

C₄

#14581766

commutator

C₂×C₄×C₈₃

C₂²×C₄

Frattini

C₂²×C₄

C₂²×D₈₃

Fitting

#405×C₈₃

C₂

socle

C₂×C₈₃

#389

prime radical

C₂³×C₈₃

C₂×Q₈

cosocle

C₂²×C₄×C₈₃

C₂³

D₈₃

C₈₃

G ▷ C₂×C₄×C₈₃ ▷ C₂×C₈₃ ▷ C₈₃

UCS

C₁ ◁ C₄ ◁ C₂²×C₄ ◁ #405

N	H ≅ G/N	type
10. #14581728	C₄	semidirect
61. C₈₃	#1953	semidirect

H ≅ G/N

type

10. #14581728

C₄

semidirect

61. C₈₃

#1953

semidirect

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

subgroup	index
#19127749	2
C₂×#14581728	2
#19127471	2
#405×C₈₃	2
#1953	83

subgroup

index

#405×C₈₃

Subgroups — #24908253

Conjugacy classes of subgroups

Complements

Maximal subgroups