Group #24918857 — subgroups

Subgroups — #24918857

Order 10624.

342 conjugacy classes of subgroups; only the 125 normal subgroups are shown.

normal subgroups only

This group has 342 conjugacy classes of subgroups — too many to draw. Only its 125 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	#19128258	C₂	1	2	yes	maximal
2	#19127987	C₂	1	4	yes	maximal, complemented
3	#19127959	C₂	1	4	yes	maximal
4	#19127912	C₂	1	4	yes	maximal, complemented
5	#423×C₈₃	C₂	1	1	yes	maximal, characteristic
6	C₂×Dic₃₃₂	C₂²	1	4	yes	—
7	#14581792	C₂²	1	2	yes	—
8	#14581792	C₂²	1	2	yes	—
9	#14581784	—	2	2	no	—
10	#14581771	—	2	2	no	—
11	#14581771	C₂²	1	4	yes	—
12	C₄×Dic₁₆₆	—	2	2	no	—
13	C₄×Dic₁₆₆	C₂²	1	4	yes	—
14	#14581751	—	2	2	no	—
15	#14581724	C₂²	1	4	yes	—
16	#14581717	C₂²	1	8	yes	—
17	C₂×M₁₆×C₈₃	C₂²	1	2	yes	—
18	C₂×C₄²×C₈₃	C₂²	1	1	yes	characteristic
19	C₄⋊C₈×C₈₃	C₂²	1	4	yes	—
20	C₂×Dic₁₆₆	—	2	2	no	—
21	C₂×Dic₁₆₆	C₂³	1	4	yes	—
22	Dic₃₃₂	—	4	4	no	—
23	C₄⋊Dic₈₃	—	2	2	no	—
24	C₄⋊Dic₈₃	—	2	2	no	—
25	C₄⋊Dic₈₃	—	2	2	no	—
26	C₄⋊Dic₈₃	C₂³	1	4	yes	—
27	Dic₈₃⋊C₄	—	2	4	no	—
28	Dic₈₃⋊C₄	—	2	4	no	—
29	C₂²⋊Dic₈₃	—	2	4	no	—
30	C₄×Dic₈₃	—	2	4	no	—
31	C₂²×C₄×C₈₃	D₄	1	2	yes	—
32	C₂²×C₄×C₈₃	C₂³	1	1	yes	characteristic
33	M₁₆×C₈₃	—	4	2	no	—
34	C₂×C₈×C₈₃	C₂³	1	4	yes	—
35	C₄²×C₈₃	D₄	1	1	yes	characteristic
36	C₄²×C₈₃	D₄	1	1	yes	characteristic
37	C₄²×C₈₃	C₂³	1	2	yes	—
38	C₂³×C₈₃	C₂×D₄	1	1	yes	characteristic
39	C₂×C₄×C₈₃	—	2	4	no	—
40	C₂×C₄×C₈₃	—	2	1	no	—
41	C₂×C₄×C₈₃	—	2	1	no	—
42	C₂×C₄×C₈₃	C₂×D₄	1	1	yes	characteristic
43	C₂×C₄×C₈₃	C₂×D₄	1	2	yes	—
44	C₂×C₄×C₈₃	C₂×D₄	1	2	yes	—
45	C₂×C₄×C₈₃	C₂⁴	1	1	yes	characteristic
46	C₂×Dic₈₃	—	2	4	no	—
47	C₂×Dic₈₃	—	2	4	no	—
48	Dic₁₆₆	—	2	4	no	—
49	Dic₁₆₆	—	2	4	no	—
50	Dic₁₆₆	—	4	4	no	—
51	C₈×C₈₃	—	2	4	no	—
52	C₂²×C₈₃	—	2	1	no	—
53	C₂²×C₈₃	—	2	2	no	—
54	C₂²×C₈₃	C₂²×D₄	1	1	yes	characteristic
55	C₄×C₈₃	—	2	1	no	—
56	C₄×C₈₃	—	2	2	no	—
57	C₄×C₈₃	#137	1	4	yes	—
58	C₄×C₈₃	H(5,𝔽₂)	1	1	yes	characteristic
59	C₄×C₈₃	2₋¹⁺⁴	1	1	yes	characteristic
60	Dic₈₃	—	4	4	no	—
61	Dic₈₃	—	4	4	no	—
62	C₂×C₈₃	—	4	1	no	—
63	C₂×C₈₃	#537	1	1	yes	characteristic
64	C₂×C₈₃	C₂×#137	1	2	yes	—
65	#3255	—	83	1	no	maximal, Sylow
66	C₈₃	#3255	1	1	yes	characteristic, complemented, Sylow
67	#533	—	83	2	no	—
68	#475	—	83	4	no	—
69	#470	—	83	4	no	—
70	#462	—	83	4	no	—
71	#423	D₈₃	1	1	yes	characteristic
72	C₂×Q₁₆	—	83	4	no	—
73	C₂×M₁₆	C₂×D₈₃	1	2	yes	—
74	C₄⋊Q₈	—	83	2	no	—
75	C₄⋊Q₈	—	83	2	no	—
76	#125	—	166	2	no	—
77	C₂²⋊Q₈	—	83	4	no	—
78	C₂²⋊Q₈	—	166	2	no	—
79	C₄×Q₈	—	83	4	no	—
80	C₄×Q₈	—	166	2	no	—
81	#117	—	166	2	no	—
82	C₂×C₄²	C₂×D₈₃	1	1	yes	characteristic
83	#106	—	83	4	no	—
84	C₄⋊C₈	C₂×D₈₃	1	4	yes	—
85	#103	—	83	8	no	—
86	C₂×Q₈	—	83	4	no	—
87	C₂×Q₈	—	166	2	no	—
88	C₂²×C₄	D₃₃₂	1	2	yes	—
89	C₂²×C₄	C₂²×D₈₃	1	1	yes	characteristic
90	Q₁₆	—	332	4	no	—
91	M₁₆	—	4	2	no	—
92	C₂×C₈	C₂²×D₈₃	1	4	yes	—
93	C₄⋊C₄	—	83	4	no	—
94	C₄⋊C₄	—	166	2	no	—
95	C₄⋊C₄	—	166	2	no	—
96	C₄⋊C₄	—	166	4	no	—
97	C₄⋊C₄	—	166	4	no	—
98	C₄⋊C₄	—	166	2	no	—
99	C₂²⋊C₄	—	166	4	no	—
100	C₄²	—	166	4	no	—
101	C₄²	D₃₃₂	1	1	yes	characteristic
102	C₄²	D₃₃₂	1	1	yes	characteristic
103	C₄²	C₂²×D₈₃	1	2	yes	—
104	C₂³	C₂×D₃₃₂	1	1	yes	characteristic
105	Q₈	—	332	4	no	—
106	Q₈	—	166	4	no	—
107	Q₈	—	166	4	no	—
108	C₂×C₄	—	2	4	no	—
109	C₂×C₄	—	2	1	no	—
110	C₂×C₄	—	2	1	no	—
111	C₂×C₄	—	166	4	no	—
112	C₂×C₄	—	166	4	no	—
113	C₂×C₄	C₂×D₃₃₂	1	1	yes	characteristic
114	C₂×C₄	C₂×D₃₃₂	1	2	yes	—
115	C₂×C₄	C₂×D₃₃₂	1	2	yes	—
116	C₂×C₄	C₂³×D₈₃	1	1	yes	characteristic
117	C₈	—	2	4	no	—
118	C₂²	—	2	2	no	—
119	C₂²	—	2	1	no	—
120	C₂²	C₂²×D₃₃₂	1	1	yes	characteristic
121	C₄	—	2	2	no	—
122	C₄	—	2	1	no	—
123	C₄	—	332	4	no	—
124	C₄	—	332	4	no	—
125	C₄	#14581825	1	4	yes	—
126	C₄	#14581837	1	1	yes	characteristic
127	C₄	#14581840	1	1	yes	characteristic
128	C₂	—	4	1	no	—
129	C₂	#19128282	1	1	yes	characteristic
130	C₂	C₂×#14581825	1	2	yes	—

subgroup

G/H

class size

multiplicity

normal?

properties

#19128258

C₂

yes

maximal

#19127987

C₂

yes

maximal, complemented

#19127959

C₂

yes

maximal

#19127912

C₂

yes

maximal, complemented

#423×C₈₃

C₂

yes

maximal, characteristic

C₂×Dic₃₃₂

C₂²

yes

—

#14581792

C₂²

yes

—

#14581792

C₂²

yes

—

#14581784

—

#14581771

—

#14581771

C₂²

yes

—

C₄×Dic₁₆₆

—

C₄×Dic₁₆₆

C₂²

yes

—

#14581751

—

#14581724

C₂²

yes

—

#14581717

C₂²

yes

—

C₂×M₁₆×C₈₃

C₂²

yes

—

C₂×C₄²×C₈₃

C₂²

yes

characteristic

C₄⋊C₈×C₈₃

C₂²

yes

—

C₂×Dic₁₆₆

—

C₂×Dic₁₆₆

C₂³

yes

—

Dic₃₃₂

—

C₄⋊Dic₈₃

—

C₄⋊Dic₈₃

—

C₄⋊Dic₈₃

—

C₄⋊Dic₈₃

C₂³

yes

—

Dic₈₃⋊C₄

—

Dic₈₃⋊C₄

—

C₂²⋊Dic₈₃

—

C₄×Dic₈₃

—

C₂²×C₄×C₈₃

D₄

yes

—

C₂²×C₄×C₈₃

C₂³

yes

characteristic

M₁₆×C₈₃

—

C₂×C₈×C₈₃

C₂³

yes

—

C₄²×C₈₃

D₄

yes

characteristic

C₄²×C₈₃

D₄

yes

characteristic

C₄²×C₈₃

C₂³

yes

—

C₂³×C₈₃

C₂×D₄

yes

characteristic

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

—

C₂×C₄×C₈₃

C₂⁴

yes

characteristic

C₂×Dic₈₃

—

C₂×Dic₈₃

—

Dic₁₆₆

—

Dic₁₆₆

—

Dic₁₆₆

—

C₈×C₈₃

—

C₂²×C₈₃

—

C₂²×C₈₃

—

C₂²×C₈₃

C₂²×D₄

yes

characteristic

C₄×C₈₃

—

C₄×C₈₃

—

C₄×C₈₃

#137

yes

—

C₄×C₈₃

H(5,𝔽₂)

yes

characteristic

C₄×C₈₃

2₋¹⁺⁴

yes

characteristic

Dic₈₃

—

Dic₈₃

—

C₂×C₈₃

—

C₂×C₈₃

#537

yes

characteristic

C₂×C₈₃

C₂×#137

yes

—

#3255

—

maximal, Sylow

C₈₃

#3255

yes

characteristic, complemented, Sylow

—

—

—

—

D₈₃

yes

characteristic

C₂×Q₁₆

—

C₂×M₁₆

C₂×D₈₃

yes

—

C₄⋊Q₈

—

C₄⋊Q₈

—

#125

—

166

—

C₂²⋊Q₈

—

C₂²⋊Q₈

—

166

—

C₄×Q₈

—

C₄×Q₈

—

166

—

#117

—

166

—

C₂×C₄²

C₂×D₈₃

yes

characteristic

#106

—

C₄⋊C₈

C₂×D₈₃

yes

—

#103

—

C₂×Q₈

—

C₂×Q₈

—

166

—

C₂²×C₄

D₃₃₂

yes

—

C₂²×C₄

C₂²×D₈₃

yes

characteristic

Q₁₆

—

332

—

M₁₆

—

C₂×C₈

C₂²×D₈₃

yes

—

C₄⋊C₄

—

C₄⋊C₄

—

166

—

C₄⋊C₄

—

166

—

C₄⋊C₄

—

166

—

C₄⋊C₄

—

166

—

C₄⋊C₄

—

166

—

C₂²⋊C₄

—

166

—

100

C₄²

—

166

—

101

C₄²

D₃₃₂

yes

characteristic

102

C₄²

D₃₃₂

yes

characteristic

103

C₄²

C₂²×D₈₃

yes

—

104

C₂³

C₂×D₃₃₂

yes

characteristic

105

Q₈

—

332

—

106

Q₈

—

166

—

107

Q₈

—

166

—

108

C₂×C₄

—

109

C₂×C₄

—

110

C₂×C₄

—

111

C₂×C₄

—

166

—

112

C₂×C₄

—

166

—

113

C₂×C₄

C₂×D₃₃₂

yes

characteristic

114

C₂×C₄

C₂×D₃₃₂

yes

—

115

C₂×C₄

C₂×D₃₃₂

yes

—

116

C₂×C₄

C₂³×D₈₃

yes

characteristic

117

C₈

—

118

C₂²

—

119

C₂²

—

120

C₂²

C₂²×D₃₃₂

yes

characteristic

121

C₄

—

122

C₄

—

123

C₄

—

332

—

124

C₄

—

332

—

125

C₄

#14581825

yes

—

126

C₄

#14581837

yes

characteristic

127

C₄

#14581840

yes

characteristic

128

C₂

—

129

C₂

#19128282

yes

characteristic

130

C₂

C₂×#14581825

yes

—

	subgroup	G/H
center	C₂²	C₂²×D₃₃₂
commutator, cosocle	C₂×C₄×C₈₃	C₂⁴
Frattini	C₂×C₄	C₂³×D₈₃
Fitting	#423×C₈₃	C₂
socle	C₂²×C₈₃	C₂²×D₄
prime radical	C₂³×C₈₃	C₂×D₄
hypercenter	#423	D₈₃
hypocenter	C₈₃	#3255
LCS	G ▷ C₂×C₄×C₈₃ ▷ C₂×C₈₃ ▷ C₈₃
UCS	C₁ ◁ C₂² ◁ C₂×C₄² ◁ #423

subgroup

G/H

center

C₂²

C₂²×D₃₃₂

commutator, cosocle

C₂×C₄×C₈₃

C₂⁴

Frattini

C₂×C₄

C₂³×D₈₃

Fitting

#423×C₈₃

C₂

socle

C₂²×C₈₃

C₂²×D₄

prime radical

C₂³×C₈₃

C₂×D₄

D₈₃

C₈₃

G ▷ C₂×C₄×C₈₃ ▷ C₂×C₈₃ ▷ C₈₃

UCS

C₁ ◁ C₂² ◁ C₂×C₄² ◁ #423

N	H ≅ G/N	type
2. #19127987	C₂	semidirect
4. #19127912	C₂	semidirect
66. C₈₃	#3255	semidirect

H ≅ G/N

type

2. #19127987

C₂

semidirect

4. #19127912

C₂

semidirect

66. C₈₃

#3255

semidirect

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

subgroup	index
#19128258	2
#19127987	2
#19127959	2
#19127912	2
#423×C₈₃	2
#3255	83

subgroup

index

#423×C₈₃

Subgroups — #24918857

Conjugacy classes of subgroups

Complements

Maximal subgroups