Group #24908731 — subgroups

Subgroups — #24908731

Order 10624.

256 conjugacy classes of subgroups; only the 69 normal subgroups are shown.

normal subgroups only

This group has 256 conjugacy classes of subgroups — too many to draw. Only its 69 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	C₂²×Dic₃₃₂	C₂	1	1	yes	maximal, characteristic
2	#19127709	C₂	1	2	yes	maximal
3	C₂×#14581710	C₂	1	2	yes	maximal
4	#19127466	C₂	1	1	yes	maximal, characteristic
5	#424×C₈₃	C₂	1	1	yes	maximal, characteristic
6	C₂²×Dic₁₆₆	C₂²	1	2	yes	—
7	C₂×Dic₃₃₂	—	2	4	no	—
8	C₂×Dic₃₃₂	—	2	2	no	—
9	C₂×#13050574	C₂²	1	2	yes	—
10	#14581715	—	2	4	no	—
11	#14581710	—	2	4	no	—
12	C₂×M₁₆×C₈₃	C₂²	1	1	yes	characteristic
13	C₂²×C₈×C₈₃	C₂²	1	1	yes	characteristic
14	#117×C₈₃	C₂²	1	1	yes	characteristic
15	C₄⋊C₈×C₈₃	—	2	2	no	—
16	C₂×Dic₁₆₆	—	2	2	no	—
17	C₂×Dic₁₆₆	—	2	2	no	—
18	C₂×Dic₁₆₆	—	2	2	no	—
19	C₂×Dic₁₆₆	—	2	2	no	—
20	C₂×Dic₁₆₆	—	2	2	no	—
21	C₂×Dic₁₆₆	D₄	1	4	yes	—
22	C₂²×Dic₈₃	—	2	2	no	—
23	Dic₃₃₂	—	2	8	no	—
24	#13050574	—	2	2	no	—
25	#13050574	—	2	2	no	—
26	C₂×C₈₃⋊C₈	—	2	2	no	—
27	C₂²×C₄×C₈₃	C₂³	1	1	yes	characteristic
28	M₁₆×C₈₃	—	2	2	no	—
29	C₂×C₈×C₈₃	—	2	2	no	—
30	C₂×C₈×C₈₃	—	2	1	no	—
31	C₂×C₈×C₈₃	D₄	1	2	yes	—
32	C₄⋊C₄×C₈₃	D₄	1	2	yes	—
33	C₂²⋊C₄×C₈₃	—	2	1	no	—
34	C₄²×C₈₃	—	2	1	no	—
35	C₂³×C₈₃	#41	1	1	yes	characteristic
36	C₂×C₄×C₈₃	—	2	2	no	—
37	C₂×C₄×C₈₃	C₂×D₄	1	2	yes	—
38	C₂×C₄×C₈₃	C₂×D₄	1	1	yes	characteristic
39	C₂×C₄×C₈₃	C₂×D₄	1	1	yes	characteristic
40	C₂×C₄×C₈₃	#41	1	2	yes	—
41	C₂×Dic₈₃	—	2	4	no	—
42	C₂×Dic₈₃	—	4	2	no	—
43	C₂×Dic₈₃	—	4	2	no	—
44	Dic₁₆₆	—	2	4	no	—
45	Dic₁₆₆	—	2	4	no	—
46	Dic₁₆₆	—	4	2	no	—
47	Dic₁₆₆	—	4	2	no	—
48	C₈×C₈₃	—	2	2	no	—
49	C₈×C₈₃	—	4	1	no	—
50	C₈₃⋊C₈	—	4	2	no	—
51	C₂²×C₈₃	—	2	2	no	—
52	C₂²×C₈₃	#121	1	1	yes	characteristic
53	C₂²×C₈₃	#123	1	2	yes	—
54	C₄×C₈₃	—	4	2	no	—
55	C₄×C₈₃	#120	1	1	yes	characteristic
56	C₄×C₈₃	#121	1	2	yes	—
57	C₄×C₈₃	#124	1	1	yes	characteristic
58	Dic₈₃	—	4	4	no	—
59	C₂×C₈₃	—	2	2	no	—
60	C₂×C₈₃	#394	1	1	yes	characteristic
61	C₂×C₈₃	#473	1	2	yes	—
62	#2047	—	83	1	no	maximal, Sylow
63	C₈₃	#2047	1	1	yes	characteristic, complemented, Sylow
64	C₂²×Q₁₆	—	83	1	no	—
65	#424	D₈₃	1	1	yes	characteristic
66	#419	—	83	2	no	—
67	C₂×#101	—	83	2	no	—
68	#341	—	83	1	no	—
69	C₂²×Q₈	—	83	2	no	—
70	C₂×Q₁₆	—	166	4	no	—
71	C₂×Q₁₆	—	166	2	no	—
72	C₂×M₁₆	—	83	2	no	—
73	C₂×M₁₆	C₂×D₈₃	1	1	yes	characteristic
74	C₂²×C₈	C₂×D₈₃	1	1	yes	characteristic
75	#117	C₂×D₈₃	1	1	yes	characteristic
76	C₄⋊C₈	—	2	2	no	—
77	#103	—	166	4	no	—
78	#101	—	166	4	no	—
79	C₂×Q₈	—	83	4	no	—
80	C₂×Q₈	—	166	2	no	—
81	C₂×Q₈	—	166	2	no	—
82	C₂×Q₈	—	166	2	no	—
83	C₂×Q₈	—	166	2	no	—
84	C₂×Q₈	—	166	2	no	—
85	C₂²×C₄	—	166	2	no	—
86	C₂²×C₄	C₂²×D₈₃	1	1	yes	characteristic
87	Q₁₆	—	166	8	no	—
88	M₁₆	—	2	2	no	—
89	M₁₆	—	166	2	no	—
90	M₁₆	—	166	2	no	—
91	C₂×C₈	—	2	2	no	—
92	C₂×C₈	—	2	1	no	—
93	C₂×C₈	—	166	2	no	—
94	C₂×C₈	C₂²⋊D₈₃	1	2	yes	—
95	C₄⋊C₄	D₃₃₂	1	2	yes	—
96	C₂²⋊C₄	—	2	1	no	—
97	C₄²	—	2	1	no	—
98	C₂³	#13050590	1	1	yes	characteristic
99	Q₈	—	332	2	no	—
100	Q₈	—	332	2	no	—
101	Q₈	—	166	4	no	—
102	Q₈	—	166	4	no	—
103	C₂×C₄	—	2	2	no	—
104	C₂×C₄	—	332	2	no	—
105	C₂×C₄	—	332	2	no	—
106	C₂×C₄	—	166	4	no	—
107	C₂×C₄	D₄×D₈₃	1	2	yes	—
108	C₂×C₄	C₂×C₂²⋊D₈₃	1	1	yes	characteristic
109	C₂×C₄	C₂×D₃₃₂	1	1	yes	characteristic
110	C₂×C₄	#13050591	1	2	yes	—
111	C₈	—	2	2	no	—
112	C₈	—	4	1	no	—
113	C₈	—	332	2	no	—
114	C₂²	—	2	2	no	—
115	C₂²	#14581766	1	1	yes	characteristic
116	C₂²	#14581777	1	2	yes	—
117	C₄	—	4	2	no	—
118	C₄	—	332	4	no	—
119	C₄	#14581760	1	1	yes	characteristic
120	C₄	#14581767	1	2	yes	—
121	C₄	#14581781	1	1	yes	characteristic
122	C₂	—	2	2	no	—
123	C₂	#19127618	1	1	yes	characteristic
124	C₂	#19127976	1	2	yes	—

subgroup

G/H

class size

multiplicity

normal?

properties

C₂²×Dic₃₃₂

C₂

yes

maximal, characteristic

#19127709

C₂

yes

maximal

C₂×#14581710

C₂

yes

maximal

#19127466

C₂

yes

maximal, characteristic

#424×C₈₃

C₂

yes

maximal, characteristic

C₂²×Dic₁₆₆

C₂²

yes

—

C₂×Dic₃₃₂

—

C₂×Dic₃₃₂

—

C₂×#13050574

C₂²

yes

—

#14581715

—

#14581710

—

C₂×M₁₆×C₈₃

C₂²

yes

characteristic

C₂²×C₈×C₈₃

C₂²

yes

characteristic

#117×C₈₃

C₂²

yes

characteristic

C₄⋊C₈×C₈₃

—

C₂×Dic₁₆₆

—

C₂×Dic₁₆₆

—

C₂×Dic₁₆₆

—

C₂×Dic₁₆₆

—

C₂×Dic₁₆₆

—

C₂×Dic₁₆₆

D₄

yes

—

C₂²×Dic₈₃

—

Dic₃₃₂

—

#13050574

—

#13050574

—

C₂×C₈₃⋊C₈

—

C₂²×C₄×C₈₃

C₂³

yes

characteristic

M₁₆×C₈₃

—

C₂×C₈×C₈₃

—

C₂×C₈×C₈₃

—

C₂×C₈×C₈₃

D₄

yes

—

C₄⋊C₄×C₈₃

D₄

yes

—

C₂²⋊C₄×C₈₃

—

C₄²×C₈₃

—

C₂³×C₈₃

#41

yes

characteristic

C₂×C₄×C₈₃

—

C₂×C₄×C₈₃

C₂×D₄

yes

—

C₂×C₄×C₈₃

C₂×D₄

yes

characteristic

C₂×C₄×C₈₃

C₂×D₄

yes

characteristic

C₂×C₄×C₈₃

#41

yes

—

C₂×Dic₈₃

—

C₂×Dic₈₃

—

C₂×Dic₈₃

—

Dic₁₆₆

—

Dic₁₆₆

—

Dic₁₆₆

—

Dic₁₆₆

—

C₈×C₈₃

—

C₈×C₈₃

—

C₈₃⋊C₈

—

C₂²×C₈₃

—

C₂²×C₈₃

#121

yes

characteristic

C₂²×C₈₃

#123

yes

—

C₄×C₈₃

—

C₄×C₈₃

#120

yes

characteristic

C₄×C₈₃

#121

yes

—

C₄×C₈₃

#124

yes

characteristic

Dic₈₃

—

C₂×C₈₃

—

C₂×C₈₃

#394

yes

characteristic

C₂×C₈₃

#473

yes

—

#2047

—

maximal, Sylow

C₈₃

#2047

yes

characteristic, complemented, Sylow

C₂²×Q₁₆

—

#424

D₈₃

yes

—

—

—

C₂²×Q₈

—

C₂×Q₁₆

—

166

—

C₂×Q₁₆

—

166

—

C₂×M₁₆

—

C₂×M₁₆

C₂×D₈₃

yes

characteristic

C₂²×C₈

C₂×D₈₃

yes

characteristic

#117

C₂×D₈₃

yes

characteristic

C₄⋊C₈

—

#103

—

166

—

#101

—

166

—

C₂×Q₈

—

C₂×Q₈

—

166

—

C₂×Q₈

—

166

—

C₂×Q₈

—

166

—

C₂×Q₈

—

166

—

C₂×Q₈

—

166

—

C₂²×C₄

—

166

—

C₂²×C₄

C₂²×D₈₃

yes

characteristic

Q₁₆

—

166

—

M₁₆

—

M₁₆

—

166

—

M₁₆

—

166

—

C₂×C₈

—

C₂×C₈

—

C₂×C₈

—

166

—

C₂×C₈

C₂²⋊D₈₃

yes

—

C₄⋊C₄

D₃₃₂

yes

—

C₂²⋊C₄

—

C₄²

—

C₂³

#13050590

yes

characteristic

Q₈

—

332

—

100

Q₈

—

332

—

101

Q₈

—

166

—

102

Q₈

—

166

—

103

C₂×C₄

—

104

C₂×C₄

—

332

—

105

C₂×C₄

—

332

—

106

C₂×C₄

—

166

—

107

C₂×C₄

D₄×D₈₃

yes

—

108

C₂×C₄

C₂×C₂²⋊D₈₃

yes

characteristic

109

C₂×C₄

C₂×D₃₃₂

yes

characteristic

110

C₂×C₄

#13050591

yes

—

111

C₈

—

112

C₈

—

113

C₈

—

332

—

114

C₂²

—

115

C₂²

#14581766

yes

characteristic

116

C₂²

#14581777

yes

—

117

C₄

—

118

C₄

—

332

—

119

C₄

#14581760

yes

characteristic

120

C₄

#14581767

yes

—

121

C₄

#14581781

yes

characteristic

122

C₂

—

123

C₂

#19127618

yes

characteristic

124

C₂

#19127976

yes

—

	subgroup	G/H
center	C₂²	#14581766
commutator, cosocle	C₂²×C₄×C₈₃	C₂³
Frattini	C₂²×C₄	C₂²×D₈₃
Fitting	#424×C₈₃	C₂
socle	C₂²×C₈₃	#121
prime radical	C₂³×C₈₃	#41
hypercenter	#424	D₈₃
hypocenter	C₈₃	#2047
LCS	G ▷ C₂²×C₄×C₈₃ ▷ C₂×C₈₃ ▷ C₈₃
UCS	C₁ ◁ C₂² ◁ C₂²×C₄ ◁ #424

subgroup

G/H

center

C₂²

#14581766

commutator, cosocle

C₂²×C₄×C₈₃

C₂³

Frattini

C₂²×C₄

C₂²×D₈₃

Fitting

#424×C₈₃

C₂

C₂²×C₈₃

C₂³×C₈₃

D₈₃

C₈₃

G ▷ C₂²×C₄×C₈₃ ▷ C₂×C₈₃ ▷ C₈₃

UCS

C₁ ◁ C₂² ◁ C₂²×C₄ ◁ #424

N	H ≅ G/N	type
63. C₈₃	#2047	semidirect

H ≅ G/N

type

63. C₈₃

#2047

semidirect

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

subgroup	index
C₂²×Dic₃₃₂	2
#19127709	2
C₂×#14581710	2
#19127466	2
#424×C₈₃	2
#2047	83

subgroup

index

C₂²×Dic₃₃₂

#424×C₈₃

Subgroups — #24908731

Conjugacy classes of subgroups

Complements

Maximal subgroups