Group #24908910 — subgroups

Subgroups — #24908910

Order 10624.

198 conjugacy classes of subgroups; only the 63 normal subgroups are shown.

normal subgroups only

This group has 198 conjugacy classes of subgroups — too many to draw. Only its 63 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	C₂×#14581724	C₂	1	1	yes	maximal, characteristic, complemented
2	#19127703	C₂	1	2	yes	maximal
3	#19127667	C₂	1	1	yes	maximal, characteristic
4	#19127469	C₂	1	2	yes	maximal, complemented
5	#341×C₈₃	C₂	1	1	yes	maximal, characteristic, complemented
6	C₂×#13050590	C₂²	1	1	yes	characteristic
7	C₂×#13050574	C₂²	1	1	yes	characteristic
8	C₂×C₄⋊Dic₈₃	C₂²	1	2	yes	—
9	#14581724	—	2	2	no	—
10	#14581717	—	2	2	no	—
11	#14581714	—	2	2	no	—
12	(C₂×D₈₃)⋊C₈	—	2	2	no	—
13	C₂×M₁₆×C₈₃	C₂²	1	2	yes	—
14	C₂²×C₈×C₈₃	C₂²	1	1	yes	characteristic
15	#13050590	—	2	2	no	—
16	#13050590	—	2	2	no	—
17	C₂×Dic₁₆₆	D₄	1	1	yes	characteristic
18	C₂×D₃₃₂	D₄	1	1	yes	characteristic
19	C₂×C₂²⋊D₈₃	—	2	1	no	—
20	C₂²×Dic₈₃	—	2	2	no	—
21	C₂×C₄×D₈₃	—	2	1	no	—
22	#13050574	—	2	2	no	—
23	C₂×C₈₃⋊C₈	—	2	1	no	—
24	C₄⋊Dic₈₃	—	2	2	no	—
25	C₄⋊Dic₈₃	—	2	2	no	—
26	C₂²×C₄×C₈₃	C₂³	1	1	yes	characteristic
27	M₁₆×C₈₃	—	2	2	no	—
28	M₁₆×C₈₃	—	2	2	no	—
29	C₂×C₈×C₈₃	—	2	2	no	—
30	C₂×C₈×C₈₃	—	2	2	no	—
31	C₂×C₈×C₈₃	D₄	1	2	yes	—
32	C₂³×C₈₃	#41	1	1	yes	characteristic
33	C₂²×D₈₃	—	2	1	no	—
34	C₂×C₄×C₈₃	C₂×D₄	1	1	yes	characteristic
35	C₂×C₄×C₈₃	C₂×D₄	1	1	yes	characteristic
36	C₂×C₄×C₈₃	#41	1	2	yes	—
37	C₂×C₄×C₈₃	#41	1	2	yes	—
38	C₂²⋊D₈₃	—	4	2	no	—
39	C₂×Dic₈₃	—	2	1	no	—
40	C₂×Dic₈₃	—	4	2	no	—
41	C₂×Dic₈₃	—	8	2	no	—
42	D₃₃₂	—	2	2	no	—
43	C₄×D₈₃	—	4	1	no	—
44	C₄×D₈₃	—	4	1	no	—
45	Dic₁₆₆	—	2	2	no	—
46	C₈×C₈₃	—	2	2	no	—
47	C₈×C₈₃	—	4	2	no	—
48	C₈₃⋊C₈	—	4	1	no	—
49	C₂²×C₈₃	—	2	2	no	—
50	C₂²×C₈₃	#121	1	1	yes	characteristic
51	C₂²×C₈₃	#126	1	2	yes	—
52	C₂×D₈₃	—	4	1	no	—
53	C₂×D₈₃	—	4	2	no	—
54	C₄×C₈₃	#123	1	2	yes	—
55	C₄×C₈₃	#123	1	1	yes	characteristic
56	C₄×C₈₃	#124	1	1	yes	characteristic
57	Dic₈₃	—	4	1	no	—
58	Dic₈₃	—	8	2	no	—
59	C₂×C₈₃	—	2	2	no	—
60	C₂×C₈₃	#397	1	1	yes	characteristic
61	C₂×C₈₃	#471	1	2	yes	—
62	D₈₃	—	8	1	no	—
63	#2079	—	83	1	no	maximal, Sylow
64	C₈₃	#2079	1	1	yes	characteristic, complemented, Sylow
65	C₂×#106	—	83	1	no	—
66	#417	—	83	2	no	—
67	#408	—	83	1	no	—
68	#343	—	83	2	no	—
69	#341	D₈₃	1	1	yes	characteristic, complemented
70	C₂×#41	—	83	1	no	—
71	C₂×M₁₆	—	83	1	no	—
72	C₂×M₁₆	C₂×D₈₃	1	2	yes	—
73	C₂²×C₈	C₂×D₈₃	1	1	yes	characteristic
74	C₂×C₄⋊C₄	—	83	2	no	—
75	#106	—	166	2	no	—
76	#103	—	166	2	no	—
77	#102	—	166	2	no	—
78	C₂²⋊C₈	—	166	2	no	—
79	#41	—	166	2	no	—
80	#41	—	166	2	no	—
81	C₂×Q₈	—	83	1	no	—
82	C₂×D₄	—	83	1	no	—
83	C₂×D₄	—	166	1	no	—
84	C₂²×C₄	—	166	1	no	—
85	C₂²×C₄	—	166	2	no	—
86	C₂²×C₄	C₂²×D₈₃	1	1	yes	characteristic
87	M₁₆	—	2	2	no	—
88	M₁₆	—	2	2	no	—
89	M₁₆	—	166	2	no	—
90	C₂×C₈	—	2	2	no	—
91	C₂×C₈	—	2	2	no	—
92	C₂×C₈	—	166	1	no	—
93	C₂×C₈	D₃₃₂	1	2	yes	—
94	C₄⋊C₄	—	166	2	no	—
95	C₄⋊C₄	—	166	2	no	—
96	C₂³	—	166	1	no	—
97	C₂³	#13050591	1	1	yes	characteristic
98	Q₈	—	166	2	no	—
99	D₄	—	332	2	no	—
100	D₄	—	166	2	no	—
101	C₂×C₄	—	332	1	no	—
102	C₂×C₄	—	332	1	no	—
103	C₂×C₄	—	332	2	no	—
104	C₂×C₄	—	664	2	no	—
105	C₂×C₄	—	166	1	no	—
106	C₂×C₄	D₄×D₈₃	1	1	yes	characteristic
107	C₂×C₄	C₂×D₃₃₂	1	1	yes	characteristic
108	C₂×C₄	#13050589	1	2	yes	—
109	C₂×C₄	#13050590	1	2	yes	—
110	C₈	—	2	2	no	—
111	C₈	—	4	2	no	—
112	C₈	—	332	1	no	—
113	C₂²	—	2	2	no	—
114	C₂²	—	332	1	no	—
115	C₂²	—	332	2	no	—
116	C₂²	#14581767	1	1	yes	characteristic
117	C₂²	#14581787	1	2	yes	—
118	C₄	—	332	1	no	—
119	C₄	—	664	2	no	—
120	C₄	#14581774	1	2	yes	—
121	C₄	#14581776	1	1	yes	characteristic
122	C₄	#14581780	1	1	yes	characteristic
123	C₂	—	2	2	no	—
124	C₂	—	664	1	no	—
125	C₂	#19127628	1	1	yes	characteristic
126	C₂	#19127965	1	2	yes	—

subgroup

G/H

class size

multiplicity

normal?

properties

C₂×#14581724

C₂

yes

maximal, characteristic, complemented

#19127703

C₂

yes

maximal

#19127667

C₂

yes

maximal, characteristic

#19127469

C₂

yes

maximal, complemented

#341×C₈₃

C₂

yes

maximal, characteristic, complemented

C₂×#13050590

C₂²

yes

characteristic

C₂×#13050574

C₂²

yes

characteristic

C₂×C₄⋊Dic₈₃

C₂²

yes

—

#14581724

—

#14581717

—

#14581714

—

(C₂×D₈₃)⋊C₈

—

C₂×M₁₆×C₈₃

C₂²

yes

—

C₂²×C₈×C₈₃

C₂²

yes

characteristic

#13050590

—

#13050590

—

C₂×Dic₁₆₆

D₄

yes

characteristic

C₂×D₃₃₂

D₄

yes

characteristic

C₂×C₂²⋊D₈₃

—

C₂²×Dic₈₃

—

C₂×C₄×D₈₃

—

#13050574

—

C₂×C₈₃⋊C₈

—

C₄⋊Dic₈₃

—

C₄⋊Dic₈₃

—

C₂²×C₄×C₈₃

C₂³

yes

characteristic

M₁₆×C₈₃

—

M₁₆×C₈₃

—

C₂×C₈×C₈₃

—

C₂×C₈×C₈₃

—

C₂×C₈×C₈₃

D₄

yes

—

C₂³×C₈₃

#41

yes

characteristic

C₂²×D₈₃

—

C₂×C₄×C₈₃

C₂×D₄

yes

characteristic

C₂×C₄×C₈₃

C₂×D₄

yes

characteristic

C₂×C₄×C₈₃

#41

yes

—

C₂×C₄×C₈₃

#41

yes

—

C₂²⋊D₈₃

—

C₂×Dic₈₃

—

C₂×Dic₈₃

—

C₂×Dic₈₃

—

D₃₃₂

—

C₄×D₈₃

—

C₄×D₈₃

—

Dic₁₆₆

—

C₈×C₈₃

—

C₈×C₈₃

—

C₈₃⋊C₈

—

C₂²×C₈₃

—

C₂²×C₈₃

#121

yes

characteristic

C₂²×C₈₃

#126

yes

—

C₂×D₈₃

—

C₂×D₈₃

—

C₄×C₈₃

#123

yes

—

C₄×C₈₃

yes

C₄×C₈₃

yes

Dic₈₃

—

Dic₈₃

—

C₂×C₈₃

—

C₂×C₈₃

#397

yes

characteristic

C₂×C₈₃

#471

yes

—

D₈₃

—

#2079

—

maximal, Sylow

C₈₃

#2079

yes

characteristic, complemented, Sylow

—

—

—

—

D₈₃

yes

characteristic, complemented

C₂×#41

—

C₂×M₁₆

—

C₂×M₁₆

C₂×D₈₃

yes

—

C₂²×C₈

C₂×D₈₃

yes

characteristic

C₂×C₄⋊C₄

—

#106

—

166

—

#103

—

166

—

#102

—

166

—

C₂²⋊C₈

—

166

—

#41

—

166

—

#41

—

166

—

C₂×Q₈

—

C₂×D₄

—

C₂×D₄

—

166

—

C₂²×C₄

—

166

—

C₂²×C₄

—

166

—

C₂²×C₄

C₂²×D₈₃

yes

characteristic

M₁₆

—

M₁₆

—

M₁₆

—

166

—

C₂×C₈

—

C₂×C₈

—

C₂×C₈

—

166

—

C₂×C₈

D₃₃₂

yes

—

C₄⋊C₄

—

166

—

C₄⋊C₄

—

166

—

C₂³

—

166

—

C₂³

#13050591

yes

characteristic

Q₈

—

166

—

D₄

—

332

—

100

D₄

—

166

—

101

C₂×C₄

—

332

—

102

C₂×C₄

—

332

—

103

C₂×C₄

—

332

—

104

C₂×C₄

—

664

—

105

C₂×C₄

—

166

—

106

C₂×C₄

D₄×D₈₃

yes

characteristic

107

C₂×C₄

C₂×D₃₃₂

yes

characteristic

108

C₂×C₄

#13050589

yes

—

109

C₂×C₄

#13050590

yes

—

110

C₈

—

111

C₈

—

112

C₈

—

332

—

113

C₂²

—

114

C₂²

—

332

—

115

C₂²

—

332

—

116

C₂²

#14581767

yes

characteristic

117

C₂²

#14581787

yes

—

118

C₄

—

332

—

119

C₄

—

664

—

120

C₄

#14581774

yes

—

121

C₄

#14581776

yes

characteristic

122

C₄

#14581780

yes

characteristic

123

C₂

—

124

C₂

—

664

—

125

C₂

#19127628

yes

characteristic

126

C₂

#19127965

yes

—

	subgroup	G/H
center	C₂²	#14581767
commutator, cosocle	C₂²×C₄×C₈₃	C₂³
Frattini	C₂²×C₄	C₂²×D₈₃
Fitting	#341×C₈₃	C₂
socle	C₂²×C₈₃	#121
prime radical	C₂×#13050590	C₂²
hypercenter	#341	D₈₃
hypocenter	C₈₃	#2079
LCS	G ▷ C₂²×C₄×C₈₃ ▷ C₂×C₈₃ ▷ C₈₃
UCS	C₁ ◁ C₂² ◁ C₂²×C₄ ◁ #341

subgroup

G/H

center

C₂²

#14581767

commutator, cosocle

C₂²×C₄×C₈₃

C₂³

Frattini

C₂²×C₄

C₂²×D₈₃

Fitting

#341×C₈₃

C₂

C₂²×C₈₃

C₂²

D₈₃

C₈₃

G ▷ C₂²×C₄×C₈₃ ▷ C₂×C₈₃ ▷ C₈₃

UCS

C₁ ◁ C₂² ◁ C₂²×C₄ ◁ #341

N	H ≅ G/N	type
1. C₂×#14581724	C₂	semidirect
4. #19127469	C₂	semidirect
5. #341×C₈₃	C₂	semidirect
64. C₈₃	#2079	semidirect
69. #341	D₈₃	semidirect

H ≅ G/N

type

1. C₂×#14581724

C₂

semidirect

4. #19127469

C₂

semidirect

5. #341×C₈₃

C₂

semidirect

64. C₈₃

#2079

semidirect

69. #341

D₈₃

semidirect

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

subgroup	index
C₂×#14581724	2
#19127703	2
#19127667	2
#19127469	2
#341×C₈₃	2
#2079	83

subgroup

index

#341×C₈₃

Subgroups — #24908910

Conjugacy classes of subgroups

Complements

Maximal subgroups