Group #24908282 — subgroups

Subgroups — #24908282

Order 10624.

308 conjugacy classes of subgroups; only the 105 normal subgroups are shown.

normal subgroups only

This group has 308 conjugacy classes of subgroups — too many to draw. Only its 105 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	C₂×#14581792	C₂	1	1	yes	maximal, characteristic
2	C₂×#14581715	C₂	1	4	yes	maximal
3	C₂×#14581703	C₂	1	1	yes	maximal, characteristic
4	#382×C₈₃	C₂	1	1	yes	maximal, characteristic
5	C₂²×Dic₁₆₆	C₂²	1	2	yes	—
6	C₂²×C₈₃⋊C₈	C₂²	1	2	yes	—
7	#14581792	—	2	2	no	—
8	#14581792	C₄	1	4	yes	complemented
9	C₂×C₄⋊Dic₈₃	—	2	2	no	—
10	#14581715	—	2	8	no	—
11	#14581703	—	2	2	no	—
12	C₂×C₄⋊C₄×C₈₃	C₂²	1	2	yes	—
13	C₂×C₄²×C₈₃	C₂²	1	1	yes	characteristic
14	#95×C₈₃	—	2	2	no	—
15	C₂×Dic₁₆₆	—	2	4	no	—
16	C₂×Dic₁₆₆	—	2	4	no	—
17	C₂×Dic₁₆₆	—	2	2	no	—
18	C₂×Dic₁₆₆	C₂×C₄	1	4	yes	—
19	C₂²×Dic₈₃	—	2	2	no	—
20	C₂×C₈₃⋊C₈	—	2	2	no	—
21	C₂×C₈₃⋊C₈	—	2	2	no	—
22	C₂×C₈₃⋊C₈	D₄	1	4	yes	—
23	C₄⋊Dic₈₃	—	2	8	no	—
24	C₂²×C₄×C₈₃	—	2	2	no	—
25	C₂²×C₄×C₈₃	D₄	1	2	yes	—
26	C₂²×C₄×C₈₃	C₂³	1	1	yes	characteristic
27	C₄⋊C₄×C₈₃	—	2	4	no	—
28	C₄²×C₈₃	C₂×C₄	1	2	yes	—
29	C₄²×C₈₃	D₄	1	2	yes	—
30	C₂³×C₈₃	C₂×D₄	1	1	yes	characteristic
31	C₂×C₄×C₈₃	—	2	4	no	—
32	C₂×C₄×C₈₃	—	4	2	no	—
33	C₂×C₄×C₈₃	—	4	4	no	—
34	C₂×C₄×C₈₃	C₂²⋊C₄	1	4	yes	—
35	C₂×C₄×C₈₃	C₂²×C₄	1	1	yes	characteristic
36	C₂×C₄×C₈₃	C₂×D₄	1	2	yes	—
37	C₂×C₄×C₈₃	C₂×D₄	1	1	yes	characteristic
38	C₂×C₄×C₈₃	#41	1	2	yes	—
39	C₂×Dic₈₃	—	2	4	no	—
40	C₂×Dic₈₃	—	4	2	no	—
41	C₂×Dic₈₃	—	4	2	no	—
42	Dic₁₆₆	—	2	8	no	—
43	Dic₁₆₆	—	4	4	no	—
44	C₈₃⋊C₈	—	2	4	no	—
45	C₂²×C₈₃	C₂×C₂²⋊C₄	1	1	yes	characteristic
46	C₂²×C₈₃	#124	1	1	yes	characteristic
47	C₂²×C₈₃	#127	1	1	yes	characteristic
48	C₂²×C₈₃	#137	1	4	yes	—
49	C₄×C₈₃	—	2	4	no	—
50	C₄×C₈₃	—	8	2	no	—
51	C₄×C₈₃	C₄×D₄	1	2	yes	—
52	C₄×C₈₃	#121	1	2	yes	—
53	Dic₈₃	—	4	4	no	—
54	C₂×C₈₃	#390	1	1	yes	characteristic
55	C₂×C₈₃	#419	1	2	yes	—
56	C₂×C₈₃	#491	1	2	yes	—
57	C₂×C₈₃	#497	1	2	yes	—
58	#1960	—	83	1	no	maximal, Sylow
59	C₈₃	#1960	1	1	yes	characteristic, complemented, Sylow
60	C₂×C₄⋊Q₈	—	83	1	no	—
61	C₂×#103	—	83	4	no	—
62	C₂×#97	—	83	1	no	—
63	#382	D₈₃	1	1	yes	characteristic
64	C₂²×Q₈	—	83	2	no	—
65	C₂²×C₈	—	83	2	no	—
66	C₄⋊Q₈	—	83	4	no	—
67	C₄⋊Q₈	—	166	2	no	—
68	C₂×C₄⋊C₄	—	166	2	no	—
69	C₂×C₄⋊C₄	C₂×D₈₃	1	2	yes	—
70	C₂×C₄²	C₂×D₈₃	1	1	yes	characteristic
71	#103	—	166	8	no	—
72	#97	—	166	2	no	—
73	#95	—	2	2	no	—
74	C₂×Q₈	—	83	4	no	—
75	C₂×Q₈	—	166	4	no	—
76	C₂×Q₈	—	166	4	no	—
77	C₂×Q₈	—	166	2	no	—
78	C₂²×C₄	—	2	2	no	—
79	C₂²×C₄	—	166	2	no	—
80	C₂²×C₄	C₂²⋊D₈₃	1	2	yes	—
81	C₂²×C₄	C₂²×D₈₃	1	1	yes	characteristic
82	C₂×C₈	—	83	4	no	—
83	C₂×C₈	—	166	2	no	—
84	C₂×C₈	—	166	2	no	—
85	C₄⋊C₄	—	2	4	no	—
86	C₄⋊C₄	—	166	8	no	—
87	C₄²	C₄×D₈₃	1	2	yes	—
88	C₄²	D₃₃₂	1	2	yes	—
89	C₂³	C₂×C₂²⋊D₈₃	1	1	yes	characteristic
90	Q₈	—	332	4	no	—
91	Q₈	—	166	8	no	—
92	C₂×C₄	—	2	4	no	—
93	C₂×C₄	—	4	2	no	—
94	C₂×C₄	—	4	4	no	—
95	C₂×C₄	—	332	2	no	—
96	C₂×C₄	—	332	2	no	—
97	C₂×C₄	—	166	4	no	—
98	C₂×C₄	(C₂×D₈₃)⋊C₄	1	4	yes	—
99	C₂×C₄	C₂×C₄×D₈₃	1	1	yes	characteristic
100	C₂×C₄	D₄×D₈₃	1	2	yes	—
101	C₂×C₄	C₂×D₃₃₂	1	1	yes	characteristic
102	C₂×C₄	#13050591	1	2	yes	—
103	C₈	—	166	4	no	—
104	C₂²	C₂×(C₂×D₈₃)⋊C₄	1	1	yes	characteristic
105	C₂²	#14581781	1	1	yes	characteristic
106	C₂²	#14581788	1	1	yes	characteristic
107	C₂²	#14581822	1	4	yes	—
108	C₄	—	2	4	no	—
109	C₄	—	8	2	no	—
110	C₄	—	332	4	no	—
111	C₄	#14581754	1	2	yes	—
112	C₄	#14581767	1	2	yes	—
113	C₂	#19127602	1	1	yes	characteristic
114	C₂	#19127709	1	2	yes	—
115	C₂	#19128071	1	2	yes	—
116	C₂	#19128096	1	2	yes	—

subgroup

G/H

class size

multiplicity

normal?

properties

C₂×#14581792

C₂

yes

maximal, characteristic

C₂×#14581715

C₂

yes

maximal

C₂×#14581703

C₂

yes

maximal, characteristic

#382×C₈₃

C₂

yes

maximal, characteristic

C₂²×Dic₁₆₆

C₂²

yes

—

C₂²×C₈₃⋊C₈

C₂²

yes

—

#14581792

—

#14581792

C₄

yes

complemented

C₂×C₄⋊Dic₈₃

—

#14581715

—

#14581703

—

C₂×C₄⋊C₄×C₈₃

C₂²

yes

—

C₂×C₄²×C₈₃

C₂²

yes

characteristic

#95×C₈₃

—

C₂×Dic₁₆₆

—

C₂×Dic₁₆₆

—

C₂×Dic₁₆₆

—

C₂×Dic₁₆₆

C₂×C₄

yes

—

C₂²×Dic₈₃

—

C₂×C₈₃⋊C₈

—

C₂×C₈₃⋊C₈

—

C₂×C₈₃⋊C₈

D₄

yes

—

C₄⋊Dic₈₃

—

C₂²×C₄×C₈₃

—

C₂²×C₄×C₈₃

D₄

yes

—

C₂²×C₄×C₈₃

C₂³

yes

characteristic

C₄⋊C₄×C₈₃

—

C₄²×C₈₃

C₂×C₄

yes

—

C₄²×C₈₃

D₄

yes

—

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

characteristic

C₂×C₄×C₈₃

C₂×D₄

yes

—

C₂×C₄×C₈₃

C₂×D₄

yes

characteristic

C₂×C₄×C₈₃

#41

yes

—

C₂×Dic₈₃

—

C₂×Dic₈₃

—

C₂×Dic₈₃

—

Dic₁₆₆

—

Dic₁₆₆

—

C₈₃⋊C₈

—

C₂²×C₈₃

C₂×C₂²⋊C₄

yes

C₂²×C₈₃

yes

C₂²×C₈₃

yes

C₂²×C₈₃

yes

—

C₄×C₈₃

—

C₄×C₈₃

—

C₄×C₈₃

C₄×D₄

yes

—

C₄×C₈₃

#121

yes

—

Dic₈₃

—

C₂×C₈₃

#390

yes

characteristic

C₂×C₈₃

#419

yes

—

C₂×C₈₃

#491

yes

—

C₂×C₈₃

#497

yes

—

#1960

—

maximal, Sylow

C₈₃

#1960

yes

characteristic, complemented, Sylow

C₂×C₄⋊Q₈

—

C₂×#103

—

C₂×#97

—

#382

D₈₃

yes

characteristic

C₂²×Q₈

—

C₂²×C₈

—

C₄⋊Q₈

—

C₄⋊Q₈

—

166

—

C₂×C₄⋊C₄

—

166

—

C₂×C₄⋊C₄

C₂×D₈₃

yes

—

C₂×C₄²

C₂×D₈₃

yes

characteristic

#103

—

166

—

#97

—

166

—

#95

—

C₂×Q₈

—

C₂×Q₈

—

166

—

C₂×Q₈

—

166

—

C₂×Q₈

—

166

—

C₂²×C₄

—

C₂²×C₄

—

166

—

C₂²×C₄

C₂²⋊D₈₃

yes

—

C₂²×C₄

C₂²×D₈₃

yes

characteristic

C₂×C₈

—

C₂×C₈

—

166

—

C₂×C₈

—

166

—

C₄⋊C₄

—

C₄⋊C₄

—

166

—

C₄²

C₄×D₈₃

yes

—

C₄²

D₃₃₂

yes

—

C₂³

C₂×C₂²⋊D₈₃

yes

characteristic

Q₈

—

332

—

Q₈

—

166

—

C₂×C₄

—

C₂×C₄

—

C₂×C₄

—

C₂×C₄

—

332

—

C₂×C₄

—

332

—

C₂×C₄

—

166

—

C₂×C₄

(C₂×D₈₃)⋊C₄

yes

—

C₂×C₄

C₂×C₄×D₈₃

yes

characteristic

100

C₂×C₄

D₄×D₈₃

yes

—

101

C₂×C₄

C₂×D₃₃₂

yes

characteristic

102

C₂×C₄

#13050591

yes

—

103

C₈

—

166

—

104

C₂²

C₂×(C₂×D₈₃)⋊C₄

yes

characteristic

105

C₂²

#14581781

yes

characteristic

106

C₂²

#14581788

yes

characteristic

107

C₂²

#14581822

yes

—

108

C₄

—

109

C₄

—

110

C₄

—

332

—

111

C₄

#14581754

yes

—

112

C₄

#14581767

yes

—

113

C₂

#19127602

yes

characteristic

114

C₂

#19127709

yes

—

115

C₂

#19128071

yes

—

116

C₂

#19128096

yes

—

	subgroup	G/H
center	C₂³	C₂×C₂²⋊D₈₃
commutator	C₂×C₄×C₈₃	C₂²×C₄
Frattini	C₂²×C₄	C₂²×D₈₃
Fitting	#382×C₈₃	C₂
socle, prime radical	C₂³×C₈₃	C₂×D₄
cosocle	C₂²×C₄×C₈₃	C₂³
hypercenter	#382	D₈₃
hypocenter	C₈₃	#1960
LCS	G ▷ C₂×C₄×C₈₃ ▷ C₂×C₈₃ ▷ C₈₃
UCS	C₁ ◁ C₂³ ◁ C₂×C₄² ◁ #382

subgroup

G/H

center

C₂³

C₂×C₂²⋊D₈₃

commutator

C₂×C₄×C₈₃

C₂²×C₄

Frattini

C₂²×C₄

C₂²×D₈₃

Fitting

#382×C₈₃

C₂

socle, prime radical

C₂³×C₈₃

C₂×D₄

cosocle

C₂²×C₄×C₈₃

C₂³

D₈₃

C₈₃

G ▷ C₂×C₄×C₈₃ ▷ C₂×C₈₃ ▷ C₈₃

UCS

C₁ ◁ C₂³ ◁ C₂×C₄² ◁ #382

N	H ≅ G/N	type
8. #14581792	C₄	semidirect
59. C₈₃	#1960	semidirect

H ≅ G/N

type

8. #14581792

C₄

semidirect

59. C₈₃

#1960

semidirect

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

subgroup	index
C₂×#14581792	2
C₂×#14581715	2
C₂×#14581703	2
#382×C₈₃	2
#1960	83

subgroup

index

#382×C₈₃

Subgroups — #24908282

Conjugacy classes of subgroups

Complements

Maximal subgroups