Group #24909017 — subgroups

Subgroups — #24909017

Order 10624.

210 conjugacy classes of subgroups; only the 67 normal subgroups are shown.

normal subgroups only

This group has 210 conjugacy classes of subgroups — too many to draw. Only its 67 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	#19127728	C₂	1	2	yes	maximal, complemented
3	#19127719	C₂	1	2	yes	maximal
4	#19127669	C₂	1	1	yes	maximal, characteristic
5	H(3,ℤ/4)×C₈₃	C₂	1	1	yes	maximal, characteristic, complemented
6	C₂×#13050590	C₂²	1	1	yes	characteristic
7	C₂×#13050574	C₂²	1	2	yes	—
8	C₂²×C₈₃⋊C₈	C₂²	1	1	yes	characteristic
9	#14581728	—	2	2	no	—
10	C₄⋊(C₈₃⋊C₈)	—	2	4	no	—
11	#14581720	—	2	4	no	—
12	(C₂×D₈₃)⋊C₈	—	2	2	no	—
13	C₂×M₁₆×C₈₃	C₂²	1	1	yes	characteristic
14	#117×C₈₃	C₂²	1	2	yes	—
15	#13050590	—	2	2	no	—
16	#13050590	—	2	2	no	—
17	C₂×Dic₁₆₆	Q₈	1	1	yes	characteristic
18	C₂×D₃₃₂	Q₈	1	1	yes	characteristic
19	C₂×C₂²⋊D₈₃	—	2	1	no	—
20	C₂×C₄×D₈₃	—	2	1	no	—
21	#13050574	—	2	2	no	—
22	#13050574	—	2	2	no	—
23	C₂×C₈₃⋊C₈	—	2	1	no	—
24	C₂×C₈₃⋊C₈	—	2	2	no	—
25	C₂×C₈₃⋊C₈	—	2	2	no	—
26	C₂²×C₄×C₈₃	C₂³	1	1	yes	characteristic
27	M₁₆×C₈₃	—	2	2	no	—
28	C₂×C₈×C₈₃	D₄	1	2	yes	—
29	C₄⋊C₄×C₈₃	—	2	2	no	—
30	C₂²⋊C₄×C₈₃	—	2	2	no	—
31	C₄²×C₈₃	D₄	1	4	yes	—
32	C₂³×C₈₃	#41	1	1	yes	characteristic
33	C₂²×D₈₃	—	2	1	no	—
34	C₂×C₄×C₈₃	—	2	4	no	—
35	C₂×C₄×C₈₃	C₂×D₄	1	2	yes	—
36	C₂×C₄×C₈₃	C₂×D₄	1	1	yes	characteristic
37	C₂×C₄×C₈₃	C₂×Q₈	1	1	yes	characteristic
38	C₂×C₄×C₈₃	#41	1	2	yes	—
39	C₂²⋊D₈₃	—	4	2	no	—
40	C₂×Dic₈₃	—	2	1	no	—
41	D₃₃₂	—	2	2	no	—
42	C₄×D₈₃	—	4	1	no	—
43	C₄×D₈₃	—	4	1	no	—
44	Dic₁₆₆	—	2	2	no	—
45	C₈×C₈₃	—	2	2	no	—
46	C₈₃⋊C₈	—	4	1	no	—
47	C₈₃⋊C₈	—	4	2	no	—
48	C₂²×C₈₃	—	4	1	no	—
49	C₂²×C₈₃	C₂²⋊Q₈	1	1	yes	characteristic
50	C₂²×C₈₃	#123	1	2	yes	—
51	C₂×D₈₃	—	4	2	no	—
52	C₂×D₈₃	—	4	1	no	—
53	C₄×C₈₃	—	4	4	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	1	no	—
58	C₂×C₈₃	—	2	2	no	—
59	C₂×C₈₃	—	2	1	no	—
60	C₂×C₈₃	#399	1	1	yes	characteristic
61	D₈₃	—	8	1	no	—
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	—	83	1	no	—
68	H(3,ℤ/4)	D₈₃	1	1	yes	characteristic, complemented
69	C₂×#41	—	83	1	no	—
70	C₂×M₁₆	—	83	2	no	—
71	C₂×M₁₆	C₂×D₈₃	1	1	yes	characteristic
72	C₂²×C₈	—	83	1	no	—
73	#117	C₂×D₈₃	1	2	yes	—
74	#108	—	166	2	no	—
75	C₄⋊C₈	—	166	4	no	—
76	#104	—	166	4	no	—
77	C₂²⋊C₈	—	166	2	no	—
78	#41	—	166	2	no	—
79	#41	—	166	2	no	—
80	C₂×Q₈	—	83	1	no	—
81	C₂×D₄	—	83	1	no	—
82	C₂×D₄	—	166	1	no	—
83	C₂²×C₄	—	166	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	—	166	1	no	—
89	C₂×C₈	—	166	2	no	—
90	C₂×C₈	—	166	2	no	—
91	C₂×C₈	D₃₃₂	1	2	yes	—
92	C₄⋊C₄	—	2	2	no	—
93	C₂²⋊C₄	—	2	2	no	—
94	C₄²	D₃₃₂	1	4	yes	—
95	C₂³	—	166	1	no	—
96	C₂³	#13050589	1	1	yes	characteristic
97	Q₈	—	166	2	no	—
98	D₄	—	332	2	no	—
99	D₄	—	166	2	no	—
100	C₂×C₄	—	2	4	no	—
101	C₂×C₄	—	332	1	no	—
102	C₂×C₄	—	332	1	no	—
103	C₂×C₄	—	166	1	no	—
104	C₂×C₄	C₂×D₃₃₂	1	2	yes	—
105	C₂×C₄	C₂×D₃₃₂	1	1	yes	characteristic
106	C₂×C₄	Q₈×D₈₃	1	1	yes	characteristic
107	C₂×C₄	#13050589	1	2	yes	—
108	C₈	—	2	2	no	—
109	C₈	—	332	1	no	—
110	C₈	—	332	2	no	—
111	C₂²	—	4	1	no	—
112	C₂²	—	332	2	no	—
113	C₂²	—	332	1	no	—
114	C₂²	#14581772	1	1	yes	characteristic
115	C₂²	#14581776	1	2	yes	—
116	C₄	—	4	4	no	—
117	C₄	—	332	1	no	—
118	C₄	#14581772	1	2	yes	—
119	C₄	#14581776	1	1	yes	characteristic
120	C₄	#14581789	1	1	yes	characteristic
121	C₂	—	2	2	no	—
122	C₂	—	2	1	no	—
123	C₂	—	664	1	no	—
124	C₂	#19127638	1	1	yes	characteristic

subgroup

G/H

class size

multiplicity

normal?

properties

#19127750

C₂

yes

maximal, characteristic, complemented

#19127728

C₂

yes

maximal, complemented

#19127719

C₂

yes

maximal

#19127669

C₂

yes

maximal, characteristic

H(3,ℤ/4)×C₈₃

C₂

yes

maximal, characteristic, complemented

C₂×#13050590

C₂²

yes

characteristic

C₂×#13050574

C₂²

yes

—

C₂²×C₈₃⋊C₈

C₂²

yes

characteristic

#14581728

—

C₄⋊(C₈₃⋊C₈)

—

#14581720

—

(C₂×D₈₃)⋊C₈

—

C₂×M₁₆×C₈₃

C₂²

yes

characteristic

#117×C₈₃

C₂²

yes

—

#13050590

—

#13050590

—

C₂×Dic₁₆₆

Q₈

yes

characteristic

C₂×D₃₃₂

Q₈

yes

characteristic

C₂×C₂²⋊D₈₃

—

C₂×C₄×D₈₃

—

#13050574

—

#13050574

—

C₂×C₈₃⋊C₈

—

C₂×C₈₃⋊C₈

—

C₂×C₈₃⋊C₈

—

C₂²×C₄×C₈₃

C₂³

yes

characteristic

M₁₆×C₈₃

—

C₂×C₈×C₈₃

D₄

yes

—

C₄⋊C₄×C₈₃

—

C₂²⋊C₄×C₈₃

—

C₄²×C₈₃

D₄

yes

—

C₂³×C₈₃

#41

yes

characteristic

C₂²×D₈₃

—

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

—

C₂²⋊D₈₃

—

C₂×Dic₈₃

—

D₃₃₂

—

C₄×D₈₃

—

C₄×D₈₃

—

Dic₁₆₆

—

C₈×C₈₃

—

C₈₃⋊C₈

—

C₈₃⋊C₈

—

C₂²×C₈₃

—

C₂²×C₈₃

C₂²⋊Q₈

yes

characteristic

C₂²×C₈₃

#123

yes

—

C₂×D₈₃

—

C₂×D₈₃

—

C₄×C₈₃

—

C₄×C₈₃

C₂²⋊Q₈

yes

—

C₄×C₈₃

yes

C₄×C₈₃

yes

Dic₈₃

—

C₂×C₈₃

—

C₂×C₈₃

—

C₂×C₈₃

#399

yes

characteristic

D₈₃

—

#2098

—

maximal, Sylow

C₈₃

#2098

yes

characteristic, complemented, Sylow

—

—

—

—

D₈₃

yes

characteristic, complemented

C₂×#41

—

C₂×M₁₆

—

C₂×M₁₆

C₂×D₈₃

yes

characteristic

C₂²×C₈

—

#117

C₂×D₈₃

yes

—

#108

—

166

—

C₄⋊C₈

—

166

—

#104

—

166

—

C₂²⋊C₈

—

166

—

#41

—

166

—

#41

—

166

—

C₂×Q₈

—

C₂×D₄

—

C₂×D₄

—

166

—

C₂²×C₄

—

166

—

C₂²×C₄

C₂²×D₈₃

yes

characteristic

M₁₆

—

M₁₆

—

166

—

M₁₆

—

166

—

C₂×C₈

—

166

—

C₂×C₈

—

166

—

C₂×C₈

—

166

—

C₂×C₈

D₃₃₂

yes

—

C₄⋊C₄

—

C₂²⋊C₄

—

C₄²

D₃₃₂

yes

—

C₂³

—

166

—

C₂³

#13050589

yes

characteristic

Q₈

—

166

—

D₄

—

332

—

D₄

—

166

—

100

C₂×C₄

—

101

C₂×C₄

—

332

—

102

C₂×C₄

—

332

—

103

C₂×C₄

—

166

—

104

C₂×C₄

C₂×D₃₃₂

yes

—

105

C₂×C₄

C₂×D₃₃₂

yes

characteristic

106

C₂×C₄

Q₈×D₈₃

yes

characteristic

107

C₂×C₄

#13050589

yes

—

108

C₈

—

109

C₈

—

332

—

110

C₈

—

332

—

111

C₂²

—

112

C₂²

—

332

—

113

C₂²

—

332

—

114

C₂²

#14581772

yes

characteristic

115

C₂²

#14581776

yes

—

116

C₄

—

117

C₄

—

332

—

118

C₄

#14581772

yes

—

119

C₄

#14581776

yes

characteristic

120

C₄

#14581789

yes

characteristic

121

C₂

—

122

C₂

—

123

C₂

—

664

—

124

C₂

#19127638

yes

characteristic

	subgroup	G/H
center	C₄	#14581789
commutator, cosocle	C₂²×C₄×C₈₃	C₂³
Frattini	C₂²×C₄	C₂²×D₈₃
Fitting	H(3,ℤ/4)×C₈₃	C₂
socle	C₂×C₈₃	#399
prime radical	C₂×#13050590	C₂²
hypercenter	H(3,ℤ/4)	D₈₃
hypocenter	C₈₃	#2098
LCS	G ▷ C₂²×C₄×C₈₃ ▷ C₂×C₈₃ ▷ C₈₃
UCS	C₁ ◁ C₄ ◁ C₂²×C₄ ◁ H(3,ℤ/4)

subgroup

G/H

center

C₄

#14581789

commutator, cosocle

C₂²×C₄×C₈₃

C₂³

Frattini

C₂²×C₄

C₂²×D₈₃

Fitting

H(3,ℤ/4)×C₈₃

C₂

C₂×C₈₃

C₂²

D₈₃

C₈₃

G ▷ C₂²×C₄×C₈₃ ▷ C₂×C₈₃ ▷ C₈₃

UCS

C₁ ◁ C₄ ◁ C₂²×C₄ ◁ H(3,ℤ/4)

N	H ≅ G/N	type
1. #19127750	C₂	semidirect
2. #19127728	C₂	semidirect
5. H(3,ℤ/4)×C₈₃	C₂	semidirect
63. C₈₃	#2098	semidirect
68. H(3,ℤ/4)	D₈₃	semidirect

H ≅ G/N

type

1. #19127750

C₂

semidirect

2. #19127728

C₂

semidirect

5. H(3,ℤ/4)×C₈₃

C₂

semidirect

63. C₈₃

#2098

semidirect

68. H(3,ℤ/4)

D₈₃

semidirect

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

subgroup	index
#19127750	2
#19127728	2
#19127719	2
#19127669	2
H(3,ℤ/4)×C₈₃	2
#2098	83

subgroup

index

H(3,ℤ/4)×C₈₃

Subgroups — #24909017

Conjugacy classes of subgroups

Complements

Maximal subgroups