Group #24908905 — subgroups

Subgroups — #24908905

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×#14581726	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	#14581726	—	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	1	no	—
50	C₂²×C₈₃	—	2	1	no	—
51	C₂²×C₈₃	#121	1	1	yes	characteristic
52	C₂²×C₈₃	#126	1	2	yes	—
53	C₂×D₈₃	—	4	1	no	—
54	C₂×D₈₃	—	4	1	no	—
55	C₂×D₈₃	—	4	1	no	—
56	C₄×C₈₃	#123	1	2	yes	—
57	C₄×C₈₃	#123	1	1	yes	characteristic
58	C₄×C₈₃	#124	1	1	yes	characteristic
59	Dic₈₃	—	4	1	no	—
60	Dic₈₃	—	8	2	no	—
61	C₂×C₈₃	—	2	2	no	—
62	C₂×C₈₃	#397	1	1	yes	characteristic
63	C₂×C₈₃	#472	1	1	yes	characteristic
64	C₂×C₈₃	#473	1	1	yes	characteristic
65	D₈₃	—	8	1	no	—
66	#2078	—	83	1	no	maximal, Sylow
67	C₈₃	#2078	1	1	yes	characteristic, complemented, Sylow
68	C₂×#107	—	83	1	no	—
69	#417	—	83	2	no	—
70	#408	—	83	1	no	—
71	#343	—	83	2	no	—
72	#341	D₈₃	1	1	yes	characteristic, complemented
73	C₂×#41	—	83	1	no	—
74	C₂×M₁₆	—	83	1	no	—
75	C₂×M₁₆	C₂×D₈₃	1	2	yes	—
76	C₂²×C₈	C₂×D₈₃	1	1	yes	characteristic
77	C₂×C₄⋊C₄	—	83	2	no	—
78	#107	—	166	2	no	—
79	#103	—	166	2	no	—
80	#102	—	166	2	no	—
81	C₂²⋊C₈	—	166	2	no	—
82	#41	—	166	2	no	—
83	#41	—	166	2	no	—
84	C₂×Q₈	—	83	1	no	—
85	C₂×D₄	—	83	1	no	—
86	C₂×D₄	—	166	1	no	—
87	C₂²×C₄	—	166	1	no	—
88	C₂²×C₄	—	166	2	no	—
89	C₂²×C₄	C₂²×D₈₃	1	1	yes	characteristic
90	M₁₆	—	2	2	no	—
91	M₁₆	—	2	2	no	—
92	M₁₆	—	166	2	no	—
93	C₂×C₈	—	2	2	no	—
94	C₂×C₈	—	2	2	no	—
95	C₂×C₈	—	166	1	no	—
96	C₂×C₈	D₃₃₂	1	2	yes	—
97	C₄⋊C₄	—	166	2	no	—
98	C₄⋊C₄	—	166	2	no	—
99	C₂³	—	166	1	no	—
100	C₂³	#13050591	1	1	yes	characteristic
101	Q₈	—	166	2	no	—
102	D₄	—	332	2	no	—
103	D₄	—	166	2	no	—
104	C₂×C₄	—	332	1	no	—
105	C₂×C₄	—	332	1	no	—
106	C₂×C₄	—	332	2	no	—
107	C₂×C₄	—	664	2	no	—
108	C₂×C₄	—	166	1	no	—
109	C₂×C₄	D₄×D₈₃	1	1	yes	characteristic
110	C₂×C₄	C₂×D₃₃₂	1	1	yes	characteristic
111	C₂×C₄	#13050589	1	2	yes	—
112	C₂×C₄	#13050590	1	2	yes	—
113	C₈	—	2	2	no	—
114	C₈	—	4	2	no	—
115	C₈	—	332	1	no	—
116	C₂²	—	2	1	no	—
117	C₂²	—	2	1	no	—
118	C₂²	—	332	1	no	—
119	C₂²	—	332	1	no	—
120	C₂²	—	332	1	no	—
121	C₂²	#14581767	1	1	yes	characteristic
122	C₂²	#14581787	1	2	yes	—
123	C₄	—	332	1	no	—
124	C₄	—	664	2	no	—
125	C₄	#14581774	1	2	yes	—
126	C₄	#14581776	1	1	yes	characteristic
127	C₄	#14581780	1	1	yes	characteristic
128	C₂	—	2	2	no	—
129	C₂	—	664	1	no	—
130	C₂	#19127628	1	1	yes	characteristic
131	C₂	#19127972	1	1	yes	characteristic
132	C₂	#19127977	1	1	yes	characteristic

subgroup

G/H

class size

multiplicity

normal?

properties

C₂×#14581726

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

—

#14581726

—

#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

—

C₂²×C₈₃

#121

yes

characteristic

C₂²×C₈₃

#126

yes

—

C₂×D₈₃

—

C₂×D₈₃

—

C₂×D₈₃

—

C₄×C₈₃

#123

yes

—

C₄×C₈₃

yes

C₄×C₈₃

yes

Dic₈₃

—

Dic₈₃

—

C₂×C₈₃

—

C₂×C₈₃

yes

C₂×C₈₃

yes

C₂×C₈₃

yes

D₈₃

—

#2078

—

maximal, Sylow

C₈₃

#2078

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

—

#107

—

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

—

100

C₂³

#13050591

yes

characteristic

101

Q₈

—

166

—

102

D₄

—

332

—

103

D₄

—

166

—

104

C₂×C₄

—

332

—

105

C₂×C₄

—

332

—

106

C₂×C₄

—

332

—

107

C₂×C₄

—

664

—

108

C₂×C₄

—

166

—

109

C₂×C₄

D₄×D₈₃

yes

characteristic

110

C₂×C₄

C₂×D₃₃₂

yes

characteristic

111

C₂×C₄

#13050589

yes

—

112

C₂×C₄

#13050590

yes

—

113

C₈

—

114

C₈

—

115

C₈

—

332

—

116

C₂²

—

117

C₂²

—

118

C₂²

—

332

—

119

C₂²

—

332

—

120

C₂²

—

332

—

121

C₂²

#14581767

yes

characteristic

122

C₂²

#14581787

yes

—

123

C₄

—

332

—

124

C₄

—

664

—

125

C₄

#14581774

yes

—

126

C₄

#14581776

yes

characteristic

127

C₄

#14581780

yes

characteristic

128

C₂

—

129

C₂

—

664

—

130

C₂

#19127628

yes

characteristic

131

C₂

#19127972

yes

characteristic

132

C₂

#19127977

yes

characteristic

	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	#2078
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×#14581726	C₂	semidirect
4. #19127469	C₂	semidirect
5. #341×C₈₃	C₂	semidirect
67. C₈₃	#2078	semidirect
72. #341	D₈₃	semidirect

H ≅ G/N

type

1. C₂×#14581726

C₂

semidirect

4. #19127469

C₂

semidirect

5. #341×C₈₃

C₂

semidirect

67. C₈₃

#2078

semidirect

72. #341

D₈₃

semidirect

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

subgroup	index
C₂×#14581726	2
#19127703	2
#19127667	2
#19127469	2
#341×C₈₃	2
#2078	83

subgroup

index

#341×C₈₃

Subgroups — #24908905

Conjugacy classes of subgroups

Complements

Maximal subgroups