Group #24916403 — subgroups

Subgroups — #24916403

Order 10624.

348 conjugacy classes of subgroups; only the 189 normal subgroups are shown.

normal subgroups only

This group has 348 conjugacy classes of subgroups — too many to draw. Only its 189 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	1	yes	maximal, characteristic
2	#19127829	C₂	1	8	yes	maximal, complemented
3	#19127727	C₂	1	2	yes	maximal
4	#19127725	C₂	1	1	yes	maximal, characteristic
5	#19127656	C₂	1	2	yes	maximal
6	C₂×#97×C₈₃	C₂	1	1	yes	maximal, characteristic
7	C₂×#13050574	C₂²	1	2	yes	—
8	#14581792	C₄	1	2	yes	—
9	#14581784	C₄	1	2	yes	—
10	#14581771	C₄	1	4	yes	—
11	C₄×Dic₁₆₆	C₂²	1	4	yes	—
12	#14581751	C₂²	1	2	yes	—
13	C₄⋊(C₈₃⋊C₈)	C₂²	1	4	yes	—
14	Dic₈₃⋊C₈	C₂²	1	8	yes	—
15	#14581702	C₂²	1	4	yes	—
16	C₈×Dic₈₃	C₂²	1	4	yes	—
17	C₂²×C₈×C₈₃	C₂²	1	1	yes	characteristic
18	C₂²×C₈×C₈₃	C₂²	1	1	yes	characteristic
19	C₂×C₄²×C₈₃	C₂²	1	1	yes	characteristic
20	#97×C₈₃	C₂²	1	4	yes	—
21	C₂×Dic₁₆₆	C₂×C₄	1	4	yes	—
22	#13050574	—	2	4	no	—
23	C₂×C₈₃⋊C₈	C₂³	1	4	yes	—
24	C₄⋊Dic₈₃	C₂×C₄	1	4	yes	—
25	C₄⋊Dic₈₃	C₂×C₄	1	4	yes	—
26	Dic₈₃⋊C₄	C₂×C₄	1	8	yes	—
27	C₂²⋊Dic₈₃	C₂×C₄	1	4	yes	—
28	C₄×Dic₈₃	C₂³	1	4	yes	—
29	C₂²×C₄×C₈₃	C₂×C₄	1	2	yes	—
30	C₂²×C₄×C₈₃	C₂³	1	1	yes	characteristic
31	C₂×C₈×C₈₃	—	2	2	no	—
32	C₂×C₈×C₈₃	Q₈	1	4	yes	complemented
33	C₂×C₈×C₈₃	C₂³	1	2	yes	—
34	C₂×C₈×C₈₃	C₂³	1	2	yes	—
35	C₄²×C₈₃	C₂×C₄	1	1	yes	characteristic
36	C₄²×C₈₃	C₂×C₄	1	1	yes	characteristic
37	C₄²×C₈₃	C₂³	1	2	yes	—
38	C₂³×C₈₃	C₂²×C₄	1	1	yes	characteristic
39	C₂×C₄×C₈₃	—	4	2	no	—
40	C₂×C₄×C₈₃	C₂²×C₄	1	2	yes	—
41	C₂×C₄×C₈₃	C₂²×C₄	1	2	yes	—
42	C₂×C₄×C₈₃	C₂²×C₄	1	1	yes	characteristic
43	C₂×C₄×C₈₃	C₂×Q₈	1	2	yes	—
44	C₂×C₄×C₈₃	#41	1	2	yes	—
45	C₂×C₄×C₈₃	C₂⁴	1	1	yes	characteristic
46	C₂×Dic₈₃	C₂²×C₄	1	8	yes	—
47	Dic₁₆₆	—	4	4	no	—
48	C₈×C₈₃	—	2	2	no	—
49	C₈×C₈₃	C₂×Q₈	1	4	yes	—
50	C₈₃⋊C₈	—	2	4	no	—
51	C₂²×C₈₃	—	2	2	no	—
52	C₂²×C₈₃	C₄×Q₈	1	2	yes	—
53	C₂²×C₈₃	C₂³×C₄	1	1	yes	characteristic
54	C₄×C₈₃	—	2	2	no	—
55	C₄×C₈₃	—	2	2	no	—
56	C₄×C₈₃	C₄×Q₈	1	2	yes	—
57	C₄×C₈₃	C₂²×Q₈	1	1	yes	characteristic
58	C₄×C₈₃	C₂×#41	1	1	yes	characteristic
59	Dic₈₃	—	2	8	no	—
60	C₂×C₈₃	—	2	2	no	—
61	C₂×C₈₃	C₂×C₄×Q₈	1	1	yes	characteristic
62	C₂×C₈₃	#568	1	2	yes	—
63	#2961	—	83	1	no	maximal, Sylow
64	C₈₃	#2961	1	1	yes	characteristic, complemented, Sylow
65	#533	—	83	1	no	—
66	#446	—	83	8	no	—
67	#424	—	83	2	no	—
68	#423	—	83	1	no	—
69	#405	—	83	2	no	—
70	C₂×#97	D₈₃	1	1	yes	characteristic
71	C₂×M₁₆	—	83	2	no	—
72	C₂²×C₈	C₂×D₈₃	1	1	yes	characteristic
73	C₂²×C₈	C₂×D₈₃	1	1	yes	characteristic
74	C₄⋊Q₈	—	83	2	no	—
75	#125	—	83	2	no	—
76	C₂²⋊Q₈	—	83	4	no	—
77	C₄×Q₈	—	83	4	no	—
78	#117	—	83	2	no	—
79	C₂×C₄²	C₂×D₈₃	1	1	yes	characteristic
80	C₄⋊C₈	—	83	8	no	—
81	C₄⋊C₈	—	83	4	no	—
82	#97	—	83	4	no	—
83	#97	C₂×D₈₃	1	4	yes	—
84	C₄×C₈	—	83	4	no	—
85	C₂×Q₈	—	83	4	no	—
86	C₂²×C₄	C₄×D₈₃	1	2	yes	—
87	C₂²×C₄	C₂²×D₈₃	1	1	yes	characteristic
88	M₁₆	—	166	4	no	—
89	C₂×C₈	—	2	2	no	—
90	C₂×C₈	—	83	4	no	—
91	C₂×C₈	Dic₁₆₆	1	4	yes	complemented
92	C₂×C₈	C₂²×D₈₃	1	2	yes	—
93	C₂×C₈	C₂²×D₈₃	1	2	yes	—
94	C₄⋊C₄	—	83	4	no	—
95	C₄⋊C₄	—	83	8	no	—
96	C₄⋊C₄	—	83	4	no	—
97	C₂²⋊C₄	—	83	4	no	—
98	C₄²	—	83	4	no	—
99	C₄²	C₄×D₈₃	1	1	yes	characteristic
100	C₄²	C₄×D₈₃	1	1	yes	characteristic
101	C₄²	C₂²×D₈₃	1	2	yes	—
102	C₂³	C₂×C₄×D₈₃	1	1	yes	characteristic
103	Q₈	—	332	4	no	—
104	C₂×C₄	—	4	2	no	—
105	C₂×C₄	—	83	8	no	—
106	C₂×C₄	C₂×C₄×D₈₃	1	2	yes	—
107	C₂×C₄	C₂×C₄×D₈₃	1	2	yes	—
108	C₂×C₄	C₂×C₄×D₈₃	1	1	yes	characteristic
109	C₂×C₄	C₂×Dic₁₆₆	1	2	yes	—
110	C₂×C₄	#13050590	1	2	yes	—
111	C₂×C₄	C₂³×D₈₃	1	1	yes	characteristic
112	C₈	—	2	2	no	—
113	C₈	—	166	4	no	—
114	C₈	C₂×Dic₁₆₆	1	4	yes	—
115	C₂²	—	2	2	no	—
116	C₂²	C₄×Dic₁₆₆	1	2	yes	—
117	C₂²	C₂²×C₄×D₈₃	1	1	yes	characteristic
118	C₄	—	2	2	no	—
119	C₄	—	2	2	no	—
120	C₄	—	166	8	no	—
121	C₄	C₄×Dic₁₆₆	1	2	yes	—
122	C₄	C₂²×Dic₁₆₆	1	1	yes	characteristic
123	C₄	C₂×#13050590	1	1	yes	characteristic
124	C₂	—	2	2	no	—
125	C₂	C₂×C₄×Dic₁₆₆	1	1	yes	characteristic
126	C₂	#19128467	1	2	yes	—

subgroup

G/H

class size

multiplicity

normal?

properties

#19128258

C₂

yes

maximal, characteristic

#19127829

C₂

yes

maximal, complemented

#19127727

C₂

yes

maximal

#19127725

C₂

yes

maximal, characteristic

#19127656

C₂

yes

maximal

C₂×#97×C₈₃

C₂

yes

maximal, characteristic

C₂×#13050574

C₂²

yes

—

#14581792

C₄

yes

—

#14581784

C₄

yes

—

#14581771

C₄

yes

—

C₄×Dic₁₆₆

C₂²

yes

—

#14581751

C₂²

yes

—

C₄⋊(C₈₃⋊C₈)

C₂²

yes

—

Dic₈₃⋊C₈

C₂²

yes

—

#14581702

C₂²

yes

—

C₈×Dic₈₃

C₂²

yes

—

C₂²×C₈×C₈₃

C₂²

yes

characteristic

C₂²×C₈×C₈₃

C₂²

yes

characteristic

C₂×C₄²×C₈₃

C₂²

yes

characteristic

#97×C₈₃

C₂²

yes

—

C₂×Dic₁₆₆

C₂×C₄

yes

—

#13050574

—

C₂×C₈₃⋊C₈

C₂³

yes

—

C₄⋊Dic₈₃

C₂×C₄

yes

—

C₄⋊Dic₈₃

C₂×C₄

yes

—

Dic₈₃⋊C₄

C₂×C₄

yes

—

C₂²⋊Dic₈₃

C₂×C₄

yes

—

C₄×Dic₈₃

C₂³

yes

—

C₂²×C₄×C₈₃

C₂×C₄

yes

—

C₂²×C₄×C₈₃

C₂³

yes

characteristic

C₂×C₈×C₈₃

—

C₂×C₈×C₈₃

Q₈

yes

complemented

C₂×C₈×C₈₃

C₂³

yes

—

C₂×C₈×C₈₃

C₂³

yes

—

C₄²×C₈₃

C₂×C₄

yes

characteristic

C₄²×C₈₃

C₂×C₄

yes

characteristic

C₄²×C₈₃

C₂³

yes

—

C₂³×C₈₃

C₂²×C₄

yes

characteristic

C₂×C₄×C₈₃

—

C₂×C₄×C₈₃

C₂²×C₄

yes

—

C₂×C₄×C₈₃

C₂²×C₄

yes

—

C₂×C₄×C₈₃

C₂²×C₄

yes

characteristic

C₂×C₄×C₈₃

C₂×Q₈

yes

—

C₂×C₄×C₈₃

#41

yes

—

C₂×C₄×C₈₃

C₂⁴

yes

characteristic

C₂×Dic₈₃

C₂²×C₄

yes

—

Dic₁₆₆

—

C₈×C₈₃

—

C₈×C₈₃

C₂×Q₈

yes

—

C₈₃⋊C₈

—

C₂²×C₈₃

—

C₂²×C₈₃

C₄×Q₈

yes

—

C₂²×C₈₃

C₂³×C₄

yes

characteristic

C₄×C₈₃

—

C₄×C₈₃

—

C₄×C₈₃

C₄×Q₈

yes

—

C₄×C₈₃

C₂²×Q₈

yes

characteristic

C₄×C₈₃

C₂×#41

yes

characteristic

Dic₈₃

—

C₂×C₈₃

—

C₂×C₈₃

C₂×C₄×Q₈

yes

characteristic

C₂×C₈₃

#568

yes

—

#2961

—

maximal, Sylow

C₈₃

#2961

yes

characteristic, complemented, Sylow

—

—

—

—

—

C₂×#97

D₈₃

yes

characteristic

C₂×M₁₆

—

C₂²×C₈

C₂×D₈₃

yes

characteristic

C₂²×C₈

C₂×D₈₃

yes

characteristic

C₄⋊Q₈

—

#125

—

C₂²⋊Q₈

—

C₄×Q₈

—

#117

—

C₂×C₄²

C₂×D₈₃

yes

characteristic

C₄⋊C₈

—

C₄⋊C₈

—

#97

—

#97

C₂×D₈₃

yes

—

C₄×C₈

—

C₂×Q₈

—

C₂²×C₄

C₄×D₈₃

yes

—

C₂²×C₄

C₂²×D₈₃

yes

characteristic

M₁₆

—

166

—

C₂×C₈

—

C₂×C₈

—

C₂×C₈

Dic₁₆₆

yes

complemented

C₂×C₈

C₂²×D₈₃

yes

—

C₂×C₈

C₂²×D₈₃

yes

—

C₄⋊C₄

—

C₄⋊C₄

—

C₄⋊C₄

—

C₂²⋊C₄

—

C₄²

—

C₄²

C₄×D₈₃

yes

characteristic

100

C₄²

C₄×D₈₃

yes

characteristic

101

C₄²

C₂²×D₈₃

yes

—

102

C₂³

C₂×C₄×D₈₃

yes

characteristic

103

Q₈

—

332

—

104

C₂×C₄

—

105

C₂×C₄

—

106

C₂×C₄

C₂×C₄×D₈₃

yes

—

107

C₂×C₄

C₂×C₄×D₈₃

yes

—

108

C₂×C₄

C₂×C₄×D₈₃

yes

characteristic

109

C₂×C₄

C₂×Dic₁₆₆

yes

—

110

C₂×C₄

#13050590

yes

—

111

C₂×C₄

C₂³×D₈₃

yes

characteristic

112

C₈

—

113

C₈

—

166

—

114

C₈

C₂×Dic₁₆₆

yes

—

115

C₂²

—

116

C₂²

C₄×Dic₁₆₆

yes

—

117

C₂²

C₂²×C₄×D₈₃

yes

characteristic

118

C₄

—

119

C₄

—

120

C₄

—

166

—

121

C₄

C₄×Dic₁₆₆

yes

—

122

C₄

C₂²×Dic₁₆₆

yes

characteristic

123

C₄

C₂×#13050590

yes

characteristic

124

C₂

—

125

C₂

C₂×C₄×Dic₁₆₆

yes

characteristic

126

C₂

#19128467

yes

—

	subgroup	G/H
center, Frattini	C₂×C₄	C₂³×D₈₃
commutator, socle	C₂²×C₈₃	C₂³×C₄
Fitting	C₂×#97×C₈₃	C₂
prime radical	C₂³×C₈₃	C₂²×C₄
cosocle	C₂×C₄×C₈₃	C₂⁴
hypercenter	C₂×#97	D₈₃
hypocenter	C₈₃	#2961
LCS	G ▷ C₂²×C₈₃ ▷ C₈₃
UCS	C₁ ◁ C₂×C₄ ◁ C₂×#97

subgroup

G/H

center, Frattini

C₂×C₄

C₂³×D₈₃

commutator, socle

C₂²×C₈₃

C₂³×C₄

Fitting

C₂×#97×C₈₃

C₂

prime radical

C₂³×C₈₃

C₂²×C₄

cosocle

C₂×C₄×C₈₃

C₂⁴

hypercenter

C₂×#97

D₈₃

C₈₃

G ▷ C₂²×C₈₃ ▷ C₈₃

C₁ ◁ C₂×C₄ ◁ C₂×#97

N	H ≅ G/N	type
2. #19127829	C₂	semidirect
32. C₂×C₈×C₈₃	Q₈	semidirect
64. C₈₃	#2961	semidirect
91. C₂×C₈	Dic₁₆₆	semidirect

H ≅ G/N

type

2. #19127829

C₂

semidirect

32. C₂×C₈×C₈₃

Q₈

semidirect

64. C₈₃

#2961

semidirect

91. C₂×C₈

Dic₁₆₆

semidirect

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

subgroup	index
#19128258	2
#19127829	2
#19127727	2
#19127725	2
#19127656	2
C₂×#97×C₈₃	2
#2961	83

subgroup

index

C₂×#97×C₈₃

Subgroups — #24916403

Conjugacy classes of subgroups

Complements

Maximal subgroups