Group #24908440 — subgroups

Subgroups — #24908440

Order 10624.

284 conjugacy classes of subgroups; only the 101 normal subgroups are shown.

normal subgroups only

This group has 284 conjugacy classes of subgroups — too many to draw. Only its 101 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	#19127725	C₂	1	1	yes	maximal, characteristic
3	#19127461	C₂	1	4	yes	maximal
4	C₄³×C₈₃	C₂	1	1	yes	maximal, characteristic
5	C₂×#13050574	C₂²	1	2	yes	—
6	#14581792	C₄	1	2	yes	complemented
7	#14581784	C₄	1	2	yes	complemented
8	#14581771	—	2	2	no	—
9	C₄×Dic₁₆₆	—	2	2	no	—
10	#14581751	C₂²	1	2	yes	—
11	C₄⋊(C₈₃⋊C₈)	—	2	2	no	—
12	C₂×C₄²×C₈₃	—	2	2	no	—
13	C₂×C₄²×C₈₃	C₂²	1	2	yes	—
14	C₂×C₄²×C₈₃	C₂²	1	1	yes	characteristic
15	C₂×Dic₁₆₆	—	2	2	no	—
16	#13050574	—	2	4	no	—
17	C₂×C₈₃⋊C₈	—	2	2	no	—
18	C₄⋊Dic₈₃	—	2	2	no	—
19	C₄⋊Dic₈₃	C₂×C₄	1	4	yes	—
20	Dic₈₃⋊C₄	—	2	4	no	—
21	C₂²⋊Dic₈₃	—	2	2	no	—
22	C₄×Dic₈₃	—	2	2	no	—
23	C₂²×C₄×C₈₃	—	2	2	no	—
24	C₂²×C₄×C₈₃	D₄	1	2	yes	—
25	C₂²×C₄×C₈₃	C₂³	1	1	yes	characteristic
26	C₄²×C₈₃	—	2	8	no	—
27	C₄²×C₈₃	—	2	2	no	—
28	C₄²×C₈₃	C₂×C₄	1	1	yes	characteristic
29	C₄²×C₈₃	C₂×C₄	1	1	yes	characteristic
30	C₄²×C₈₃	D₄	1	2	yes	—
31	C₄²×C₈₃	Q₈	1	4	yes	complemented
32	C₂³×C₈₃	C₂×D₄	1	1	yes	characteristic
33	C₂×C₄×C₈₃	—	2	8	no	—
34	C₂×C₄×C₈₃	—	2	4	no	—
35	C₂×C₄×C₈₃	—	2	4	no	—
36	C₂×C₄×C₈₃	C₂²⋊C₄	1	2	yes	—
37	C₂×C₄×C₈₃	C₂²⋊C₄	1	2	yes	—
38	C₂×C₄×C₈₃	C₂²×C₄	1	1	yes	characteristic
39	C₂×C₄×C₈₃	C₂×D₄	1	1	yes	characteristic
40	C₂×C₄×C₈₃	C₂×Q₈	1	2	yes	—
41	C₂×C₄×C₈₃	#41	1	2	yes	—
42	C₂×Dic₈₃	—	2	4	no	—
43	Dic₁₆₆	—	4	4	no	—
44	C₈₃⋊C₈	—	4	2	no	—
45	C₂²×C₈₃	—	2	2	no	—
46	C₂²×C₈₃	C₂×C₂²⋊C₄	1	1	yes	characteristic
47	C₂²×C₈₃	C₂²⋊Q₈	1	2	yes	—
48	C₄×C₈₃	—	2	8	no	—
49	C₄×C₈₃	—	2	2	no	—
50	C₄×C₈₃	#104	1	4	yes	—
51	C₄×C₈₃	C₄×Q₈	1	2	yes	—
52	C₄×C₈₃	#124	1	1	yes	characteristic
53	C₄×C₈₃	C₄⋊Q₈	1	1	yes	characteristic
54	Dic₈₃	—	4	4	no	—
55	C₂×C₈₃	—	2	2	no	—
56	C₂×C₈₃	#391	1	1	yes	characteristic
57	C₂×C₈₃	C₂×#104	1	2	yes	—
58	#1994	—	83	1	no	maximal, Sylow
59	C₈₃	#1994	1	1	yes	characteristic, complemented, Sylow
60	#533	—	83	1	no	—
61	#423	—	83	1	no	—
62	C₄³	D₈₃	1	1	yes	characteristic
63	#339	—	83	4	no	—
64	C₂×M₁₆	—	83	2	no	—
65	C₄⋊Q₈	—	83	2	no	—
66	#125	—	83	2	no	—
67	C₂²⋊Q₈	—	166	2	no	—
68	C₄×Q₈	—	166	2	no	—
69	#117	—	83	2	no	—
70	C₂×C₄²	—	2	2	no	—
71	C₂×C₄²	C₂×D₈₃	1	2	yes	—
72	C₂×C₄²	C₂×D₈₃	1	1	yes	characteristic
73	C₄⋊C₈	—	166	2	no	—
74	C₂×Q₈	—	166	2	no	—
75	C₂²×C₄	—	2	2	no	—
76	C₂²×C₄	C₂²⋊D₈₃	1	2	yes	—
77	C₂²×C₄	C₂²×D₈₃	1	1	yes	characteristic
78	M₁₆	—	166	4	no	—
79	C₂×C₈	—	166	2	no	—
80	C₄⋊C₄	—	83	4	no	—
81	C₄⋊C₄	—	166	4	no	—
82	C₄⋊C₄	—	166	2	no	—
83	C₂²⋊C₄	—	166	2	no	—
84	C₄²	—	2	8	no	—
85	C₄²	—	2	2	no	—
86	C₄²	—	166	2	no	—
87	C₄²	Dic₁₆₆	1	4	yes	complemented
88	C₄²	C₄×D₈₃	1	1	yes	characteristic
89	C₄²	C₄×D₈₃	1	1	yes	characteristic
90	C₄²	D₃₃₂	1	2	yes	—
91	C₂³	C₂×C₂²⋊D₈₃	1	1	yes	characteristic
92	Q₈	—	332	4	no	—
93	C₂×C₄	—	2	8	no	—
94	C₂×C₄	—	2	4	no	—
95	C₂×C₄	—	2	4	no	—
96	C₂×C₄	—	166	4	no	—
97	C₂×C₄	(C₂×D₈₃)⋊C₄	1	2	yes	—
98	C₂×C₄	(C₂×D₈₃)⋊C₄	1	2	yes	—
99	C₂×C₄	C₂×C₄×D₈₃	1	1	yes	characteristic
100	C₂×C₄	C₂×D₃₃₂	1	1	yes	characteristic
101	C₂×C₄	C₂×Dic₁₆₆	1	2	yes	—
102	C₂×C₄	#13050590	1	2	yes	—
103	C₈	—	332	2	no	—
104	C₂²	—	2	2	no	—
105	C₂²	C₂×(C₂×D₈₃)⋊C₄	1	1	yes	characteristic
106	C₂²	#14581771	1	2	yes	—
107	C₄	—	2	8	no	—
108	C₄	—	2	2	no	—
109	C₄	—	332	4	no	—
110	C₄	#14581720	1	4	yes	—
111	C₄	C₄×Dic₁₆₆	1	2	yes	—
112	C₄	#14581780	1	1	yes	characteristic
113	C₄	#14581792	1	1	yes	characteristic
114	C₂	—	2	2	no	—
115	C₂	#19127607	1	1	yes	characteristic
116	C₂	C₂×#14581720	1	2	yes	—

subgroup

G/H

class size

multiplicity

normal?

properties

#19128258

C₂

yes

maximal, characteristic

#19127725

C₂

yes

maximal, characteristic

#19127461

C₂

yes

maximal

C₄³×C₈₃

C₂

yes

maximal, characteristic

C₂×#13050574

C₂²

yes

—

#14581792

C₄

yes

complemented

#14581784

C₄

yes

complemented

#14581771

—

C₄×Dic₁₆₆

—

#14581751

C₂²

yes

—

C₄⋊(C₈₃⋊C₈)

—

C₂×C₄²×C₈₃

—

C₂×C₄²×C₈₃

C₂²

yes

—

C₂×C₄²×C₈₃

C₂²

yes

characteristic

C₂×Dic₁₆₆

—

#13050574

—

C₂×C₈₃⋊C₈

—

C₄⋊Dic₈₃

—

C₄⋊Dic₈₃

C₂×C₄

yes

—

Dic₈₃⋊C₄

—

C₂²⋊Dic₈₃

—

C₄×Dic₈₃

—

C₂²×C₄×C₈₃

—

C₂²×C₄×C₈₃

D₄

yes

—

C₂²×C₄×C₈₃

C₂³

yes

characteristic

C₄²×C₈₃

—

C₄²×C₈₃

—

C₄²×C₈₃

C₂×C₄

yes

characteristic

C₄²×C₈₃

C₂×C₄

yes

characteristic

C₄²×C₈₃

D₄

yes

—

C₄²×C₈₃

Q₈

yes

complemented

C₂³×C₈₃

C₂×D₄

yes

characteristic

C₂×C₄×C₈₃

—

C₂×C₄×C₈₃

—

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×D₄

yes

characteristic

C₂×C₄×C₈₃

C₂×Q₈

yes

—

C₂×C₄×C₈₃

#41

yes

—

C₂×Dic₈₃

—

Dic₁₆₆

—

C₈₃⋊C₈

—

C₂²×C₈₃

—

C₂²×C₈₃

C₂×C₂²⋊C₄

yes

characteristic

C₂²×C₈₃

C₂²⋊Q₈

yes

—

C₄×C₈₃

—

C₄×C₈₃

—

C₄×C₈₃

#104

yes

—

C₄×C₈₃

C₄×Q₈

yes

—

C₄×C₈₃

#124

yes

characteristic

C₄×C₈₃

C₄⋊Q₈

yes

characteristic

Dic₈₃

—

C₂×C₈₃

—

C₂×C₈₃

#391

yes

characteristic

C₂×C₈₃

C₂×#104

yes

—

#1994

—

maximal, Sylow

C₈₃

#1994

yes

characteristic, complemented, Sylow

#533

—

#423

—

C₄³

D₈₃

yes

characteristic

#339

—

C₂×M₁₆

—

C₄⋊Q₈

—

#125

—

C₂²⋊Q₈

—

166

—

C₄×Q₈

—

166

—

#117

—

C₂×C₄²

—

C₂×C₄²

C₂×D₈₃

yes

—

C₂×C₄²

C₂×D₈₃

yes

characteristic

C₄⋊C₈

—

166

—

C₂×Q₈

—

166

—

C₂²×C₄

—

C₂²×C₄

C₂²⋊D₈₃

yes

—

C₂²×C₄

C₂²×D₈₃

yes

characteristic

M₁₆

—

166

—

C₂×C₈

—

166

—

C₄⋊C₄

—

C₄⋊C₄

—

166

—

C₄⋊C₄

—

166

—

C₂²⋊C₄

—

166

—

C₄²

—

C₄²

—

C₄²

—

166

—

C₄²

Dic₁₆₆

yes

complemented

C₄²

C₄×D₈₃

yes

characteristic

C₄²

C₄×D₈₃

yes

characteristic

C₄²

D₃₃₂

yes

—

C₂³

C₂×C₂²⋊D₈₃

yes

characteristic

Q₈

—

332

—

C₂×C₄

—

C₂×C₄

—

C₂×C₄

—

C₂×C₄

—

166

—

C₂×C₄

(C₂×D₈₃)⋊C₄

yes

—

C₂×C₄

(C₂×D₈₃)⋊C₄

yes

—

C₂×C₄

C₂×C₄×D₈₃

yes

characteristic

100

C₂×C₄

C₂×D₃₃₂

yes

characteristic

101

C₂×C₄

C₂×Dic₁₆₆

yes

—

102

C₂×C₄

#13050590

yes

—

103

C₈

—

332

—

104

C₂²

—

105

C₂²

C₂×(C₂×D₈₃)⋊C₄

yes

characteristic

106

C₂²

#14581771

yes

—

107

C₄

—

108

C₄

—

109

C₄

—

332

—

110

C₄

#14581720

yes

—

111

C₄

C₄×Dic₁₆₆

yes

—

112

C₄

#14581780

yes

characteristic

113

C₄

#14581792

yes

characteristic

114

C₂

—

115

C₂

#19127607

yes

characteristic

116

C₂

C₂×#14581720

yes

—

	subgroup	G/H
center	C₂×C₄	C₂×D₃₃₂
commutator	C₂×C₄×C₈₃	C₂²×C₄
Frattini	C₂²×C₄	C₂²×D₈₃
Fitting	C₄³×C₈₃	C₂
socle	C₂²×C₈₃	C₂×C₂²⋊C₄
prime radical	C₂³×C₈₃	C₂×D₄
cosocle	C₂²×C₄×C₈₃	C₂³
hypercenter	C₄³	D₈₃
hypocenter	C₈₃	#1994
LCS	G ▷ C₂×C₄×C₈₃ ▷ C₂×C₈₃ ▷ C₈₃
UCS	C₁ ◁ C₂×C₄ ◁ C₂×C₄² ◁ C₄³

subgroup

G/H

center

C₂×C₄

C₂×D₃₃₂

commutator

C₂×C₄×C₈₃

C₂²×C₄

Frattini

C₂²×C₄

C₂²×D₈₃

Fitting

C₄³×C₈₃

C₂

socle

C₂²×C₈₃

C₂×C₂²⋊C₄

prime radical

C₂³×C₈₃

C₂×D₄

cosocle

C₂²×C₄×C₈₃

C₂³

hypercenter

C₄³

D₈₃

hypocenter

C₈₃

#1994

LCS

G ▷ C₂×C₄×C₈₃ ▷ C₂×C₈₃ ▷ C₈₃

UCS

C₁ ◁ C₂×C₄ ◁ C₂×C₄² ◁ C₄³

N	H ≅ G/N	type
6. #14581792	C₄	semidirect
7. #14581784	C₄	semidirect
31. C₄²×C₈₃	Q₈	semidirect
59. C₈₃	#1994	semidirect
87. C₄²	Dic₁₆₆	semidirect

H ≅ G/N

type

6. #14581792

C₄

semidirect

7. #14581784

C₄

semidirect

31. C₄²×C₈₃

Q₈

semidirect

59. C₈₃

#1994

semidirect

87. C₄²

Dic₁₆₆

semidirect

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

subgroup	index
#19128258	2
#19127725	2
#19127461	2
C₄³×C₈₃	2
#1994	83

subgroup

index

C₄³×C₈₃

Subgroups — #24908440

Conjugacy classes of subgroups

Complements

Maximal subgroups