Group #24908217 — subgroups

Subgroups — #24908217

Order 10624.

200 conjugacy classes of subgroups; only the 97 normal subgroups are shown.

normal subgroups only

This group has 200 conjugacy classes of subgroups — too many to draw. Only its 97 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×#14581728	C₂	1	1	yes	maximal, characteristic
2	#19127728	C₂	1	2	yes	maximal
3	C₂×#14581702	C₂	1	1	yes	maximal, characteristic
4	#19127460	C₂	1	2	yes	maximal
5	#427×C₈₃	C₂	1	1	yes	maximal, characteristic
6	C₂×#13050574	C₂²	1	2	yes	—
7	C₂²×C₈₃⋊C₈	C₂²	1	1	yes	characteristic
8	C₂×C₄×Dic₈₃	C₂²	1	1	yes	characteristic
9	#14581728	—	2	2	no	—
10	C₄⋊(C₈₃⋊C₈)	—	2	4	no	—
11	#14581702	—	2	2	no	—
12	C₂²×C₈×C₈₃	C₂²	1	1	yes	characteristic
13	#117×C₈₃	C₂²	1	2	yes	—
14	#107×C₈₃	C₄	1	2	yes	complemented
15	#106×C₈₃	C₄	1	2	yes	complemented
16	C₂²×Dic₈₃	—	2	1	no	—
17	#13050574	—	2	2	no	—
18	#13050574	—	2	2	no	—
19	C₂×C₈₃⋊C₈	—	2	1	no	—
20	C₂×C₈₃⋊C₈	—	2	1	no	—
21	C₂×C₈₃⋊C₈	—	2	2	no	—
22	C₂×C₈₃⋊C₈	D₄	1	2	yes	—
23	C₄×Dic₈₃	—	2	1	no	—
24	C₄×Dic₈₃	Q₈	1	2	yes	—
25	C₂²×C₄×C₈₃	C₂³	1	1	yes	characteristic
26	C₂×C₈×C₈₃	C₂×C₄	1	1	yes	characteristic
27	C₂×C₈×C₈₃	C₂×C₄	1	1	yes	characteristic
28	C₂×C₈×C₈₃	D₄	1	2	yes	—
29	C₂×C₈×C₈₃	Q₈	1	2	yes	—
30	C₄⋊C₄×C₈₃	C₂×C₄	1	4	yes	—
31	C₂²⋊C₄×C₈₃	—	2	2	no	—
32	C₄²×C₈₃	—	2	2	no	—
33	C₂³×C₈₃	#41	1	1	yes	characteristic
34	C₂×C₄×C₈₃	—	2	4	no	—
35	C₂×C₄×C₈₃	C₂²×C₄	1	1	yes	characteristic
36	C₂×C₄×C₈₃	C₂×D₄	1	1	yes	characteristic
37	C₂×C₄×C₈₃	C₂×D₄	1	1	yes	characteristic
38	C₂×C₄×C₈₃	C₂×Q₈	1	1	yes	characteristic
39	C₂×C₄×C₈₃	C₂×Q₈	1	1	yes	characteristic
40	C₂×C₄×C₈₃	#41	1	1	yes	characteristic
41	C₂×Dic₈₃	—	2	2	no	—
42	C₂×Dic₈₃	—	4	2	no	—
43	C₈×C₈₃	C₄⋊C₄	1	2	yes	—
44	C₈×C₈₃	C₄⋊C₄	1	2	yes	—
45	C₈₃⋊C₈	—	2	2	no	—
46	C₈₃⋊C₈	—	4	2	no	—
47	C₂²×C₈₃	—	2	2	no	—
48	C₂²×C₈₃	C₄×D₄	1	1	yes	characteristic
49	C₂²×C₈₃	C₂²⋊Q₈	1	1	yes	characteristic
50	C₂²×C₈₃	#125	1	1	yes	characteristic
51	C₄×C₈₃	—	4	4	no	—
52	C₄×C₈₃	C₂×C₄⋊C₄	1	1	yes	characteristic
53	C₄×C₈₃	C₄×Q₈	1	1	yes	characteristic
54	C₄×C₈₃	#121	1	1	yes	characteristic
55	C₄×C₈₃	C₄⋊Q₈	1	1	yes	characteristic
56	Dic₈₃	—	4	2	no	—
57	C₂×C₈₃	—	2	1	no	—
58	C₂×C₈₃	—	2	1	no	—
59	C₂×C₈₃	#389	1	1	yes	characteristic
60	C₂×C₈₃	#444	1	2	yes	—
61	#1946	—	83	1	no	maximal, Sylow
62	C₈₃	#1946	1	1	yes	characteristic, complemented, Sylow
63	C₂×#108	—	83	1	no	—
64	#427	D₈₃	1	1	yes	characteristic
65	#424	—	83	2	no	—
66	C₂×#97	—	83	1	no	—
67	#339	—	83	2	no	—
68	C₂×M₁₆	—	83	2	no	—
69	C₂²×C₈	—	83	1	no	—
70	C₂²×C₈	C₂×D₈₃	1	1	yes	characteristic
71	#117	C₂×D₈₃	1	2	yes	—
72	C₂×C₄²	—	83	1	no	—
73	#108	—	166	2	no	—
74	#107	Dic₈₃	1	2	yes	complemented
75	#106	Dic₈₃	1	2	yes	complemented
76	C₄⋊C₈	—	166	4	no	—
77	#97	—	166	2	no	—
78	C₂²×C₄	—	166	1	no	—
79	C₂²×C₄	C₂²×D₈₃	1	1	yes	characteristic
80	M₁₆	—	166	2	no	—
81	M₁₆	—	166	2	no	—
82	C₂×C₈	—	83	2	no	—
83	C₂×C₈	—	166	2	no	—
84	C₂×C₈	—	166	1	no	—
85	C₂×C₈	—	166	1	no	—
86	C₂×C₈	Dic₁₆₆	1	2	yes	—
87	C₂×C₈	D₃₃₂	1	2	yes	—
88	C₂×C₈	C₂×Dic₈₃	1	1	yes	characteristic
89	C₂×C₈	C₂×Dic₈₃	1	1	yes	characteristic
90	C₄⋊C₄	C₂×Dic₈₃	1	4	yes	—
91	C₂²⋊C₄	—	2	2	no	—
92	C₄²	—	2	2	no	—
93	C₄²	—	83	2	no	—
94	C₄²	—	166	1	no	—
95	C₂³	#13050589	1	1	yes	characteristic
96	C₂×C₄	—	2	4	no	—
97	C₂×C₄	—	332	2	no	—
98	C₂×C₄	—	166	2	no	—
99	C₂×C₄	C₂²×Dic₈₃	1	1	yes	characteristic
100	C₂×C₄	D₄×D₈₃	1	1	yes	characteristic
101	C₂×C₄	C₂×D₃₃₂	1	1	yes	characteristic
102	C₂×C₄	Q₈×D₈₃	1	1	yes	characteristic
103	C₂×C₄	C₂×Dic₁₆₆	1	1	yes	characteristic
104	C₂×C₄	#13050591	1	1	yes	characteristic
105	C₈	—	332	2	no	—
106	C₈	—	166	2	no	—
107	C₈	C₄⋊Dic₈₃	1	2	yes	—
108	C₈	C₄⋊Dic₈₃	1	2	yes	—
109	C₂²	—	2	2	no	—
110	C₂²	D₄×Dic₈₃	1	1	yes	characteristic
111	C₂²	#14581772	1	1	yes	characteristic
112	C₂²	#14581783	1	1	yes	characteristic
113	C₄	—	4	4	no	—
114	C₄	—	332	2	no	—
115	C₄	C₂×C₄⋊Dic₈₃	1	1	yes	characteristic
116	C₄	Q₈×Dic₈₃	1	1	yes	characteristic
117	C₄	#14581767	1	1	yes	characteristic
118	C₄	#14581791	1	1	yes	characteristic
119	C₂	—	2	1	no	—
120	C₂	—	2	1	no	—
121	C₂	#19127600	1	1	yes	characteristic
122	C₂	#19127823	1	2	yes	—

subgroup

G/H

class size

multiplicity

normal?

properties

C₂×#14581728

C₂

yes

maximal, characteristic

#19127728

C₂

yes

maximal

C₂×#14581702

C₂

yes

maximal, characteristic

#19127460

C₂

yes

maximal

#427×C₈₃

C₂

yes

maximal, characteristic

C₂×#13050574

C₂²

yes

—

C₂²×C₈₃⋊C₈

C₂²

yes

characteristic

C₂×C₄×Dic₈₃

C₂²

yes

characteristic

#14581728

—

C₄⋊(C₈₃⋊C₈)

—

#14581702

—

C₂²×C₈×C₈₃

C₂²

yes

characteristic

#117×C₈₃

C₂²

yes

—

#107×C₈₃

C₄

yes

complemented

#106×C₈₃

C₄

yes

complemented

C₂²×Dic₈₃

—

#13050574

—

#13050574

—

C₂×C₈₃⋊C₈

—

C₂×C₈₃⋊C₈

—

C₂×C₈₃⋊C₈

—

C₂×C₈₃⋊C₈

D₄

yes

—

C₄×Dic₈₃

—

C₄×Dic₈₃

Q₈

yes

—

C₂²×C₄×C₈₃

C₂³

yes

characteristic

C₂×C₈×C₈₃

C₂×C₄

yes

characteristic

C₂×C₈×C₈₃

C₂×C₄

yes

characteristic

C₂×C₈×C₈₃

D₄

yes

—

C₂×C₈×C₈₃

Q₈

yes

—

C₄⋊C₄×C₈₃

C₂×C₄

yes

—

C₂²⋊C₄×C₈₃

—

C₄²×C₈₃

—

C₂³×C₈₃

#41

yes

characteristic

C₂×C₄×C₈₃

—

C₂×C₄×C₈₃

C₂²×C₄

yes

characteristic

C₂×C₄×C₈₃

C₂×D₄

yes

characteristic

C₂×C₄×C₈₃

C₂×D₄

yes

characteristic

C₂×C₄×C₈₃

C₂×Q₈

yes

characteristic

C₂×C₄×C₈₃

C₂×Q₈

yes

characteristic

C₂×C₄×C₈₃

#41

yes

characteristic

C₂×Dic₈₃

—

C₂×Dic₈₃

—

C₈×C₈₃

C₄⋊C₄

yes

—

C₈×C₈₃

C₄⋊C₄

yes

—

C₈₃⋊C₈

—

C₈₃⋊C₈

—

C₂²×C₈₃

—

C₂²×C₈₃

C₄×D₄

yes

characteristic

C₂²×C₈₃

C₂²⋊Q₈

yes

characteristic

C₂²×C₈₃

#125

yes

characteristic

C₄×C₈₃

—

C₄×C₈₃

C₂×C₄⋊C₄

yes

characteristic

C₄×C₈₃

C₄×Q₈

yes

characteristic

C₄×C₈₃

#121

yes

characteristic

C₄×C₈₃

C₄⋊Q₈

yes

characteristic

Dic₈₃

—

C₂×C₈₃

—

C₂×C₈₃

—

C₂×C₈₃

#389

yes

characteristic

C₂×C₈₃

#444

yes

—

#1946

—

maximal, Sylow

C₈₃

#1946

yes

characteristic, complemented, Sylow

C₂×#108

—

#427

D₈₃

yes

characteristic

#424

—

C₂×#97

—

#339

—

C₂×M₁₆

—

C₂²×C₈

—

C₂²×C₈

C₂×D₈₃

yes

characteristic

#117

C₂×D₈₃

yes

—

C₂×C₄²

—

#108

—

166

—

#107

Dic₈₃

yes

complemented

#106

Dic₈₃

yes

complemented

C₄⋊C₈

—

166

—

#97

—

166

—

C₂²×C₄

—

166

—

C₂²×C₄

C₂²×D₈₃

yes

characteristic

M₁₆

—

166

—

M₁₆

—

166

—

C₂×C₈

—

C₂×C₈

—

166

—

C₂×C₈

—

166

—

C₂×C₈

—

166

—

C₂×C₈

Dic₁₆₆

yes

—

C₂×C₈

D₃₃₂

yes

—

C₂×C₈

C₂×Dic₈₃

yes

characteristic

C₂×C₈

C₂×Dic₈₃

yes

characteristic

C₄⋊C₄

C₂×Dic₈₃

yes

—

C₂²⋊C₄

—

C₄²

—

C₄²

—

C₄²

—

166

—

C₂³

#13050589

yes

characteristic

C₂×C₄

—

C₂×C₄

—

332

—

C₂×C₄

—

166

—

C₂×C₄

C₂²×Dic₈₃

yes

characteristic

100

C₂×C₄

D₄×D₈₃

yes

characteristic

101

C₂×C₄

C₂×D₃₃₂

yes

characteristic

102

C₂×C₄

Q₈×D₈₃

yes

characteristic

103

C₂×C₄

C₂×Dic₁₆₆

yes

characteristic

104

C₂×C₄

#13050591

yes

characteristic

105

C₈

—

332

—

106

C₈

—

166

—

107

C₈

C₄⋊Dic₈₃

yes

—

108

C₈

C₄⋊Dic₈₃

yes

—

109

C₂²

—

110

C₂²

D₄×Dic₈₃

yes

characteristic

111

C₂²

#14581772

yes

characteristic

112

C₂²

#14581783

yes

characteristic

113

C₄

—

114

C₄

—

332

—

115

C₄

C₂×C₄⋊Dic₈₃

yes

characteristic

116

C₄

Q₈×Dic₈₃

yes

characteristic

117

C₄

#14581767

yes

characteristic

118

C₄

#14581791

yes

characteristic

119

C₂

—

120

C₂

—

121

C₂

#19127600

yes

characteristic

122

C₂

#19127823

yes

—

	subgroup	G/H
center	C₂×C₄	D₄×D₈₃
commutator	C₂×C₄×C₈₃	C₂²×C₄
Frattini	C₂²×C₄	C₂²×D₈₃
Fitting	#427×C₈₃	C₂
socle	C₂²×C₈₃	C₄×D₄
prime radical	C₂³×C₈₃	#41
cosocle	C₂²×C₄×C₈₃	C₂³
hypercenter	#427	D₈₃
hypocenter	C₈₃	#1946
LCS	G ▷ C₂×C₄×C₈₃ ▷ C₂×C₈₃ ▷ C₈₃
UCS	C₁ ◁ C₂×C₄ ◁ C₂²×C₄ ◁ #427

subgroup

G/H

center

C₂×C₄

D₄×D₈₃

commutator

C₂×C₄×C₈₃

C₂²×C₄

Frattini

C₂²×C₄

C₂²×D₈₃

Fitting

#427×C₈₃

C₂

socle

C₂²×C₈₃

C₄×D₄

prime radical

C₂³×C₈₃

#41

cosocle

C₂²×C₄×C₈₃

C₂³

D₈₃

C₈₃

G ▷ C₂×C₄×C₈₃ ▷ C₂×C₈₃ ▷ C₈₃

UCS

C₁ ◁ C₂×C₄ ◁ C₂²×C₄ ◁ #427

N	H ≅ G/N	type
14. #107×C₈₃	C₄	semidirect
15. #106×C₈₃	C₄	semidirect
62. C₈₃	#1946	semidirect
74. #107	Dic₈₃	semidirect
75. #106	Dic₈₃	semidirect

H ≅ G/N

type

14. #107×C₈₃

C₄

semidirect

15. #106×C₈₃

C₄

semidirect

62. C₈₃

#1946

semidirect

74. #107

Dic₈₃

semidirect

75. #106

Dic₈₃

semidirect

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

subgroup	index
C₂×#14581728	2
#19127728	2
C₂×#14581702	2
#19127460	2
#427×C₈₃	2
#1946	83

subgroup

index

#427×C₈₃

Subgroups — #24908217

Conjugacy classes of subgroups

Complements

Maximal subgroups