Group #24909016 — subgroups

Subgroups — #24909016

Order 10624.

210 conjugacy classes of subgroups; only the 65 normal subgroups are shown.

normal subgroups only

This group has 210 conjugacy classes of subgroups — too many to draw. Only its 65 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	#19127750	C₂	1	1	yes	maximal, characteristic, complemented
2	#19127727	C₂	1	2	yes	maximal, complemented
3	#19127718	C₂	1	2	yes	maximal
4	#19127458	C₂	1	1	yes	maximal, characteristic, complemented
5	#408×C₈₃	C₂	1	1	yes	maximal, characteristic
6	C₂×#13050574	C₂²	1	2	yes	—
7	#14581751	C₂²	1	2	yes	—
8	#14581728	—	2	2	no	—
9	Dic₈₃⋊C₈	—	2	4	no	—
10	#14581719	—	2	4	no	—
11	C₂×#41×C₈₃	C₂²	1	1	yes	characteristic
12	C₂×M₁₆×C₈₃	C₂²	1	1	yes	characteristic
13	C₂²×C₈×C₈₃	C₂²	1	1	yes	characteristic
14	C₂²⋊C₈×C₈₃	—	2	2	no	—
15	#13050574	—	2	2	no	—
16	#13050574	—	2	2	no	—
17	C₂×C₈₃⋊C₈	—	2	2	no	—
18	C₄⋊Dic₈₃	—	2	2	no	—
19	C₂²⋊Dic₈₃	—	2	2	no	—
20	C₄×Dic₈₃	D₄	1	4	yes	—
21	#41×C₈₃	—	2	2	no	—
22	#41×C₈₃	—	2	2	no	—
23	C₂×Q₈×C₈₃	Q₈	1	1	yes	characteristic
24	C₂×D₄×C₈₃	—	2	1	no	—
25	C₂×D₄×C₈₃	Q₈	1	1	yes	characteristic
26	C₂²×C₄×C₈₃	—	2	1	no	—
27	C₂²×C₄×C₈₃	C₂³	1	1	yes	characteristic
28	M₁₆×C₈₃	—	2	2	no	—
29	C₂×C₈×C₈₃	—	2	1	no	—
30	C₂×C₈×C₈₃	—	2	2	no	—
31	C₂×C₈×C₈₃	D₄	1	2	yes	—
32	C₂³×C₈₃	—	2	1	no	—
33	C₂³×C₈₃	#41	1	1	yes	characteristic
34	Q₈×C₈₃	—	2	2	no	—
35	D₄×C₈₃	—	2	2	no	—
36	D₄×C₈₃	—	4	2	no	—
37	C₂×C₄×C₈₃	—	2	1	no	—
38	C₂×C₄×C₈₃	—	4	1	no	—
39	C₂×C₄×C₈₃	—	4	1	no	—
40	C₂×C₄×C₈₃	C₂×D₄	1	2	yes	—
41	C₂×C₄×C₈₃	C₂×D₄	1	1	yes	characteristic
42	C₂×C₄×C₈₃	C₂×Q₈	1	1	yes	characteristic
43	C₂×C₄×C₈₃	#41	1	2	yes	—
44	C₂×Dic₈₃	—	2	4	no	—
45	C₈×C₈₃	—	2	2	no	—
46	C₈×C₈₃	—	4	1	no	—
47	C₈₃⋊C₈	—	4	2	no	—
48	C₂²×C₈₃	—	4	1	no	—
49	C₂²×C₈₃	—	4	2	no	—
50	C₂²×C₈₃	—	4	1	no	—
51	C₂²×C₈₃	C₂²⋊Q₈	1	1	yes	characteristic
52	C₂²×C₈₃	#123	1	2	yes	—
53	C₄×C₈₃	—	4	1	no	—
54	C₄×C₈₃	C₂²⋊Q₈	1	2	yes	—
55	C₄×C₈₃	#123	1	1	yes	characteristic
56	C₄×C₈₃	#127	1	1	yes	characteristic
57	Dic₈₃	—	4	4	no	—
58	C₂×C₈₃	—	2	2	no	—
59	C₂×C₈₃	—	2	1	no	—
60	C₂×C₈₃	—	8	1	no	—
61	C₂×C₈₃	#399	1	1	yes	characteristic
62	#2098	—	83	1	no	maximal, Sylow
63	C₈₃	#2098	1	1	yes	characteristic, complemented, Sylow
64	#430	—	83	1	no	—
65	#424	—	83	2	no	—
66	#421	—	83	2	no	—
67	#408	D₈₃	1	1	yes	characteristic
68	H(3,ℤ/4)	—	83	1	no	—
69	C₂×#41	C₂×D₈₃	1	1	yes	characteristic
70	C₂×M₁₆	—	83	2	no	—
71	C₂×M₁₆	C₂×D₈₃	1	1	yes	characteristic
72	C₂²×C₈	C₂×D₈₃	1	1	yes	characteristic
73	#117	—	83	2	no	—
74	#108	—	166	2	no	—
75	C₄⋊C₈	—	166	4	no	—
76	#104	—	166	4	no	—
77	C₂²⋊C₈	—	2	2	no	—
78	#41	—	2	2	no	—
79	#41	—	2	2	no	—
80	C₂×Q₈	Dic₁₆₆	1	1	yes	characteristic
81	C₂×D₄	—	2	1	no	—
82	C₂×D₄	Dic₁₆₆	1	1	yes	characteristic
83	C₂²×C₄	—	2	1	no	—
84	C₂²×C₄	C₂²×D₈₃	1	1	yes	characteristic
85	M₁₆	—	2	2	no	—
86	M₁₆	—	166	2	no	—
87	M₁₆	—	166	2	no	—
88	C₂×C₈	—	2	2	no	—
89	C₂×C₈	—	2	1	no	—
90	C₂×C₈	—	166	2	no	—
91	C₂×C₈	C₂²⋊D₈₃	1	2	yes	—
92	C₄⋊C₄	—	166	2	no	—
93	C₂²⋊C₄	—	166	2	no	—
94	C₄²	—	83	4	no	—
95	C₂³	—	2	1	no	—
96	C₂³	#13050590	1	1	yes	characteristic
97	Q₈	—	2	2	no	—
98	D₄	—	2	2	no	—
99	D₄	—	4	2	no	—
100	C₂×C₄	—	2	1	no	—
101	C₂×C₄	—	4	1	no	—
102	C₂×C₄	—	4	1	no	—
103	C₂×C₄	—	166	4	no	—
104	C₂×C₄	D₄×D₈₃	1	2	yes	—
105	C₂×C₄	C₂×C₂²⋊D₈₃	1	1	yes	characteristic
106	C₂×C₄	C₂×Dic₁₆₆	1	1	yes	characteristic
107	C₂×C₄	#13050589	1	2	yes	—
108	C₈	—	2	2	no	—
109	C₈	—	4	1	no	—
110	C₈	—	332	2	no	—
111	C₂²	—	4	1	no	—
112	C₂²	—	4	2	no	—
113	C₂²	—	4	1	no	—
114	C₂²	#14581771	1	1	yes	characteristic
115	C₂²	#14581774	1	2	yes	—
116	C₄	—	4	1	no	—
117	C₄	—	332	4	no	—
118	C₄	#14581769	1	2	yes	—
119	C₄	#14581773	1	1	yes	characteristic
120	C₄	#14581788	1	1	yes	characteristic
121	C₂	—	2	2	no	—
122	C₂	—	2	1	no	—
123	C₂	—	8	1	no	—
124	C₂	#19127637	1	1	yes	characteristic

subgroup

G/H

class size

multiplicity

normal?

properties

#19127750

C₂

yes

maximal, characteristic, complemented

#19127727

C₂

yes

maximal, complemented

#19127718

C₂

yes

maximal

#19127458

C₂

yes

maximal, characteristic, complemented

#408×C₈₃

C₂

yes

maximal, characteristic

C₂×#13050574

C₂²

yes

—

#14581751

C₂²

yes

—

#14581728

—

Dic₈₃⋊C₈

—

#14581719

—

C₂×#41×C₈₃

C₂²

yes

characteristic

C₂×M₁₆×C₈₃

C₂²

yes

characteristic

C₂²×C₈×C₈₃

C₂²

yes

characteristic

C₂²⋊C₈×C₈₃

—

#13050574

—

#13050574

—

C₂×C₈₃⋊C₈

—

C₄⋊Dic₈₃

—

C₂²⋊Dic₈₃

—

C₄×Dic₈₃

D₄

yes

—

#41×C₈₃

—

#41×C₈₃

—

C₂×Q₈×C₈₃

Q₈

yes

characteristic

C₂×D₄×C₈₃

—

C₂×D₄×C₈₃

Q₈

yes

characteristic

C₂²×C₄×C₈₃

—

C₂²×C₄×C₈₃

C₂³

yes

characteristic

M₁₆×C₈₃

—

C₂×C₈×C₈₃

—

C₂×C₈×C₈₃

—

C₂×C₈×C₈₃

D₄

yes

—

C₂³×C₈₃

—

C₂³×C₈₃

#41

yes

characteristic

Q₈×C₈₃

—

D₄×C₈₃

—

D₄×C₈₃

—

C₂×C₄×C₈₃

—

C₂×C₄×C₈₃

—

C₂×C₄×C₈₃

—

C₂×C₄×C₈₃

C₂×D₄

yes

—

C₂×C₄×C₈₃

C₂×D₄

yes

characteristic

C₂×C₄×C₈₃

C₂×Q₈

yes

characteristic

C₂×C₄×C₈₃

#41

yes

—

C₂×Dic₈₃

—

C₈×C₈₃

—

C₈×C₈₃

—

C₈₃⋊C₈

—

C₂²×C₈₃

—

C₂²×C₈₃

—

C₂²×C₈₃

—

C₂²×C₈₃

C₂²⋊Q₈

yes

characteristic

C₂²×C₈₃

#123

yes

—

C₄×C₈₃

—

C₄×C₈₃

C₂²⋊Q₈

yes

—

C₄×C₈₃

yes

C₄×C₈₃

yes

Dic₈₃

—

C₂×C₈₃

—

C₂×C₈₃

—

C₂×C₈₃

—

C₂×C₈₃

yes

—

C₈₃

yes

characteristic, complemented, Sylow

—

—

—

D₈₃

yes

characteristic

H(3,ℤ/4)

—

C₂×#41

C₂×D₈₃

yes

characteristic

C₂×M₁₆

—

C₂×M₁₆

C₂×D₈₃

yes

characteristic

C₂²×C₈

C₂×D₈₃

yes

characteristic

#117

—

#108

—

166

—

C₄⋊C₈

—

166

—

#104

—

166

—

C₂²⋊C₈

—

#41

—

#41

—

C₂×Q₈

Dic₁₆₆

yes

characteristic

C₂×D₄

—

C₂×D₄

Dic₁₆₆

yes

characteristic

C₂²×C₄

—

C₂²×C₄

C₂²×D₈₃

yes

characteristic

M₁₆

—

M₁₆

—

166

—

M₁₆

—

166

—

C₂×C₈

—

C₂×C₈

—

C₂×C₈

—

166

—

C₂×C₈

C₂²⋊D₈₃

yes

—

C₄⋊C₄

—

166

—

C₂²⋊C₄

—

166

—

C₄²

—

C₂³

—

C₂³

#13050590

yes

characteristic

Q₈

—

D₄

—

D₄

—

100

C₂×C₄

—

101

C₂×C₄

—

102

C₂×C₄

—

103

C₂×C₄

—

166

—

104

C₂×C₄

D₄×D₈₃

yes

—

105

C₂×C₄

C₂×C₂²⋊D₈₃

yes

characteristic

106

C₂×C₄

C₂×Dic₁₆₆

yes

characteristic

107

C₂×C₄

#13050589

yes

—

108

C₈

—

109

C₈

—

110

C₈

—

332

—

111

C₂²

—

112

C₂²

—

113

C₂²

—

114

C₂²

#14581771

yes

characteristic

115

C₂²

#14581774

yes

—

116

C₄

—

117

C₄

—

332

—

118

C₄

#14581769

yes

—

119

C₄

#14581773

yes

characteristic

120

C₄

#14581788

yes

characteristic

121

C₂

—

122

C₂

—

123

C₂

—

124

C₂

#19127637

yes

characteristic

	subgroup	G/H
center	C₄	#14581788
commutator, cosocle	C₂²×C₄×C₈₃	C₂³
Frattini	C₂²×C₄	C₂²×D₈₃
Fitting	#408×C₈₃	C₂
socle	C₂×C₈₃	#399
prime radical	C₂×#41×C₈₃	C₂²
hypercenter	#408	D₈₃
hypocenter	C₈₃	#2098
LCS	G ▷ C₂²×C₄×C₈₃ ▷ C₂×C₈₃ ▷ C₈₃
UCS	C₁ ◁ C₄ ◁ C₂²×C₄ ◁ #408

subgroup

G/H

center

C₄

#14581788

commutator, cosocle

C₂²×C₄×C₈₃

C₂³

Frattini

C₂²×C₄

C₂²×D₈₃

Fitting

#408×C₈₃

C₂

socle

C₂×C₈₃

#399

prime radical

C₂×#41×C₈₃

C₂²

D₈₃

C₈₃

G ▷ C₂²×C₄×C₈₃ ▷ C₂×C₈₃ ▷ C₈₃

UCS

C₁ ◁ C₄ ◁ C₂²×C₄ ◁ #408

N	H ≅ G/N	type
1. #19127750	C₂	semidirect
2. #19127727	C₂	semidirect
4. #19127458	C₂	semidirect
63. C₈₃	#2098	semidirect

H ≅ G/N

type

1. #19127750

C₂

semidirect

2. #19127727

C₂

semidirect

4. #19127458

C₂

semidirect

63. C₈₃

#2098

semidirect

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

subgroup	index
#19127750	2
#19127727	2
#19127718	2
#19127458	2
#408×C₈₃	2
#2098	83

subgroup

index

#408×C₈₃

Subgroups — #24909016

Conjugacy classes of subgroups

Complements

Maximal subgroups