Group #24907342 — subgroups

Subgroups — #24907342

Order 10624.

232 conjugacy classes of subgroups; only the 125 normal subgroups are shown.

normal subgroups only

This group has 232 conjugacy classes of subgroups — too many to draw. Only its 125 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	#19127739	C₂	1	1	yes	maximal, characteristic
3	#19127656	C₂	1	2	yes	maximal
4	#19127461	C₂	1	2	yes	maximal
5	C₂×C₄×C₈×C₈₃	C₂	1	1	yes	maximal, characteristic
6	C₂×#13050574	C₂²	1	2	yes	—
7	#14581751	C₂²	1	2	yes	—
8	#14581728	C₄	1	4	yes	complemented
9	#14581726	C₄	1	2	yes	complemented
10	#14581724	C₄	1	2	yes	complemented
11	#14581702	—	2	2	no	—
12	C₈×Dic₈₃	—	2	2	no	—
13	C₂²×C₈×C₈₃	C₂²	1	1	yes	characteristic
14	C₂²×C₈×C₈₃	C₂²	1	1	yes	characteristic
15	C₂×C₄²×C₈₃	C₂²	1	1	yes	characteristic
16	C₄×C₈×C₈₃	C₄	1	4	yes	complemented
17	#13050574	—	2	2	no	—
18	#13050574	C₂×C₄	1	4	yes	—
19	C₂×C₈₃⋊C₈	—	2	2	no	—
20	C₄⋊Dic₈₃	C₂×C₄	1	4	yes	—
21	C₂²⋊Dic₈₃	—	2	2	no	—
22	C₄×Dic₈₃	—	2	2	no	—
23	C₂²×C₄×C₈₃	—	2	1	no	—
24	C₂²×C₄×C₈₃	C₂³	1	1	yes	characteristic
25	C₂×C₈×C₈₃	C₂×C₄	1	1	yes	characteristic
26	C₂×C₈×C₈₃	C₂×C₄	1	1	yes	characteristic
27	C₂×C₈×C₈₃	C₂×C₄	1	2	yes	—
28	C₂×C₈×C₈₃	C₂×C₄	1	2	yes	—
29	C₂×C₈×C₈₃	C₂×C₄	1	2	yes	—
30	C₂×C₈×C₈₃	D₄	1	1	yes	characteristic
31	C₂×C₈×C₈₃	D₄	1	1	yes	characteristic
32	C₂×C₈×C₈₃	Q₈	1	2	yes	—
33	C₄²×C₈₃	—	2	1	no	—
34	C₄²×C₈₃	C₂×C₄	1	2	yes	—
35	C₂³×C₈₃	#41	1	1	yes	characteristic
36	C₂×C₄×C₈₃	—	2	4	no	—
37	C₂×C₄×C₈₃	—	2	2	no	—
38	C₂×C₄×C₈₃	C₂²×C₄	1	1	yes	characteristic
39	C₂×C₄×C₈₃	C₂²×C₄	1	1	yes	characteristic
40	C₂×C₄×C₈₃	C₂²×C₄	1	1	yes	characteristic
41	C₂×C₄×C₈₃	C₂×D₄	1	1	yes	characteristic
42	C₂×C₄×C₈₃	C₂×Q₈	1	1	yes	characteristic
43	C₂×C₄×C₈₃	#41	1	1	yes	characteristic
44	C₂×Dic₈₃	—	2	4	no	—
45	C₈×C₈₃	C₄²	1	2	yes	—
46	C₈×C₈₃	C₄²	1	2	yes	—
47	C₈×C₈₃	C₄⋊C₄	1	2	yes	—
48	C₈×C₈₃	C₄⋊C₄	1	2	yes	—
49	C₈₃⋊C₈	—	2	4	no	—
50	C₂²×C₈₃	—	2	2	no	—
51	C₂²×C₈₃	#117	1	1	yes	characteristic
52	C₂²×C₈₃	C₄×D₄	1	1	yes	characteristic
53	C₂²×C₈₃	C₄×Q₈	1	1	yes	characteristic
54	C₄×C₈₃	—	2	4	no	—
55	C₄×C₈₃	C₂×C₄²	1	1	yes	characteristic
56	C₄×C₈₃	C₂×C₄⋊C₄	1	1	yes	characteristic
57	C₄×C₈₃	C₄×D₄	1	1	yes	characteristic
58	C₄×C₈₃	C₄×Q₈	1	1	yes	characteristic
59	Dic₈₃	—	4	4	no	—
60	C₂×C₈₃	—	2	1	no	—
61	C₂×C₈₃	—	2	1	no	—
62	C₂×C₈₃	C₄×C₄⋊C₄	1	1	yes	characteristic
63	C₂×C₈₃	#443	1	2	yes	—
64	#1772	—	83	1	no	maximal, Sylow
65	C₈₃	#1772	1	1	yes	characteristic, complemented, Sylow
66	C₂×#108	—	83	1	no	—
67	#427	—	83	1	no	—
68	#405	—	83	2	no	—
69	C₂×C₄×C₈	D₈₃	1	1	yes	characteristic
70	#339	—	83	2	no	—
71	C₂×M₁₆	—	83	2	no	—
72	C₂²×C₈	C₂×D₈₃	1	1	yes	characteristic
73	C₂²×C₈	C₂×D₈₃	1	1	yes	characteristic
74	#117	—	83	2	no	—
75	C₂×C₄²	C₂×D₈₃	1	1	yes	characteristic
76	#108	—	83	4	no	—
77	#107	—	83	2	no	—
78	#106	—	83	2	no	—
79	#97	—	166	2	no	—
80	C₄×C₈	—	166	2	no	—
81	C₄×C₈	Dic₈₃	1	4	yes	complemented
82	C₂²×C₄	—	2	1	no	—
83	C₂²×C₄	C₂²×D₈₃	1	1	yes	characteristic
84	M₁₆	—	83	4	no	—
85	M₁₆	—	166	2	no	—
86	C₂×C₈	—	166	2	no	—
87	C₂×C₈	Dic₁₆₆	1	2	yes	—
88	C₂×C₈	C₄×D₈₃	1	1	yes	characteristic
89	C₂×C₈	C₄×D₈₃	1	1	yes	characteristic
90	C₂×C₈	C₄×D₈₃	1	2	yes	—
91	C₂×C₈	D₃₃₂	1	1	yes	characteristic
92	C₂×C₈	D₃₃₂	1	1	yes	characteristic
93	C₂×C₈	C₂×Dic₈₃	1	2	yes	—
94	C₂×C₈	C₂×Dic₈₃	1	2	yes	—
95	C₄⋊C₄	—	83	4	no	—
96	C₂²⋊C₄	—	166	2	no	—
97	C₄²	—	2	1	no	—
98	C₄²	—	166	2	no	—
99	C₄²	C₂×Dic₈₃	1	2	yes	—
100	C₂³	#13050590	1	1	yes	characteristic
101	C₂×C₄	—	2	4	no	—
102	C₂×C₄	—	2	2	no	—
103	C₂×C₄	—	166	4	no	—
104	C₂×C₄	C₂×C₄×D₈₃	1	1	yes	characteristic
105	C₂×C₄	C₂×C₄×D₈₃	1	1	yes	characteristic
106	C₂×C₄	C₂²×Dic₈₃	1	1	yes	characteristic
107	C₂×C₄	C₂×D₃₃₂	1	1	yes	characteristic
108	C₂×C₄	C₂×Dic₁₆₆	1	1	yes	characteristic
109	C₂×C₄	#13050590	1	1	yes	characteristic
110	C₈	—	166	4	no	—
111	C₈	C₄×Dic₈₃	1	2	yes	—
112	C₈	C₄×Dic₈₃	1	2	yes	—
113	C₈	C₄⋊Dic₈₃	1	2	yes	—
114	C₈	C₄⋊Dic₈₃	1	2	yes	—
115	C₂²	—	2	2	no	—
116	C₂²	#14581751	1	1	yes	characteristic
117	C₂²	C₄×D₃₃₂	1	1	yes	characteristic
118	C₂²	C₄×Dic₁₆₆	1	1	yes	characteristic
119	C₄	—	2	4	no	—
120	C₄	—	332	4	no	—
121	C₄	C₂×C₄×Dic₈₃	1	1	yes	characteristic
122	C₄	C₂×C₄⋊Dic₈₃	1	1	yes	characteristic
123	C₄	C₄×D₃₃₂	1	1	yes	characteristic
124	C₄	C₄×Dic₁₆₆	1	1	yes	characteristic
125	C₂	—	2	1	no	—
126	C₂	—	2	1	no	—
127	C₂	C₄×C₄⋊Dic₈₃	1	1	yes	characteristic
128	C₂	#19127819	1	2	yes	—

subgroup

G/H

class size

multiplicity

normal?

properties

C₂×#14581728

C₂

yes

maximal, characteristic

#19127739

C₂

yes

maximal, characteristic

#19127656

C₂

yes

maximal

#19127461

C₂

yes

maximal

C₂×C₄×C₈×C₈₃

C₂

yes

maximal, characteristic

C₂×#13050574

C₂²

yes

—

#14581751

C₂²

yes

—

#14581728

C₄

yes

complemented

#14581726

C₄

yes

complemented

#14581724

C₄

yes

complemented

#14581702

—

C₈×Dic₈₃

—

C₂²×C₈×C₈₃

C₂²

yes

characteristic

C₂²×C₈×C₈₃

C₂²

yes

characteristic

C₂×C₄²×C₈₃

C₂²

yes

characteristic

C₄×C₈×C₈₃

C₄

yes

complemented

#13050574

—

#13050574

C₂×C₄

yes

—

C₂×C₈₃⋊C₈

—

C₄⋊Dic₈₃

C₂×C₄

yes

—

C₂²⋊Dic₈₃

—

C₄×Dic₈₃

—

C₂²×C₄×C₈₃

—

C₂²×C₄×C₈₃

C₂³

yes

characteristic

C₂×C₈×C₈₃

C₂×C₄

yes

characteristic

C₂×C₈×C₈₃

C₂×C₄

yes

characteristic

C₂×C₈×C₈₃

C₂×C₄

yes

—

C₂×C₈×C₈₃

C₂×C₄

yes

—

C₂×C₈×C₈₃

C₂×C₄

yes

—

C₂×C₈×C₈₃

D₄

yes

characteristic

C₂×C₈×C₈₃

D₄

yes

characteristic

C₂×C₈×C₈₃

Q₈

yes

—

C₄²×C₈₃

—

C₄²×C₈₃

C₂×C₄

yes

—

C₂³×C₈₃

#41

yes

characteristic

C₂×C₄×C₈₃

—

C₂×C₄×C₈₃

—

C₂×C₄×C₈₃

C₂²×C₄

yes

characteristic

C₂×C₄×C₈₃

C₂²×C₄

yes

characteristic

C₂×C₄×C₈₃

C₂²×C₄

yes

characteristic

C₂×C₄×C₈₃

C₂×D₄

yes

characteristic

C₂×C₄×C₈₃

C₂×Q₈

yes

characteristic

C₂×C₄×C₈₃

#41

yes

characteristic

C₂×Dic₈₃

—

C₈×C₈₃

C₄²

yes

—

C₈×C₈₃

C₄²

yes

—

C₈×C₈₃

C₄⋊C₄

yes

—

C₈×C₈₃

C₄⋊C₄

yes

—

C₈₃⋊C₈

—

C₂²×C₈₃

—

C₂²×C₈₃

#117

yes

characteristic

C₂²×C₈₃

C₄×D₄

yes

characteristic

C₂²×C₈₃

C₄×Q₈

yes

characteristic

C₄×C₈₃

—

C₄×C₈₃

C₂×C₄²

yes

characteristic

C₄×C₈₃

C₂×C₄⋊C₄

yes

characteristic

C₄×C₈₃

C₄×D₄

yes

characteristic

C₄×C₈₃

C₄×Q₈

yes

characteristic

Dic₈₃

—

C₂×C₈₃

—

C₂×C₈₃

—

C₂×C₈₃

C₄×C₄⋊C₄

yes

characteristic

C₂×C₈₃

#443

yes

—

#1772

—

maximal, Sylow

C₈₃

#1772

yes

characteristic, complemented, Sylow

C₂×#108

—

#427

—

#405

—

C₂×C₄×C₈

D₈₃

yes

characteristic

#339

—

C₂×M₁₆

—

C₂²×C₈

C₂×D₈₃

yes

characteristic

C₂²×C₈

C₂×D₈₃

yes

characteristic

#117

—

C₂×C₄²

C₂×D₈₃

yes

—

—

—

—

166

—

C₄×C₈

—

166

—

C₄×C₈

Dic₈₃

yes

complemented

C₂²×C₄

—

C₂²×C₄

C₂²×D₈₃

yes

characteristic

M₁₆

—

M₁₆

—

166

—

C₂×C₈

—

166

—

C₂×C₈

Dic₁₆₆

yes

—

C₂×C₈

C₄×D₈₃

yes

characteristic

C₂×C₈

C₄×D₈₃

yes

characteristic

C₂×C₈

C₄×D₈₃

yes

—

C₂×C₈

D₃₃₂

yes

characteristic

C₂×C₈

D₃₃₂

yes

characteristic

C₂×C₈

C₂×Dic₈₃

yes

—

C₂×C₈

C₂×Dic₈₃

yes

—

C₄⋊C₄

—

C₂²⋊C₄

—

166

—

C₄²

—

C₄²

—

166

—

C₄²

C₂×Dic₈₃

yes

—

100

C₂³

#13050590

yes

characteristic

101

C₂×C₄

—

102

C₂×C₄

—

103

C₂×C₄

—

166

—

104

C₂×C₄

C₂×C₄×D₈₃

yes

characteristic

105

C₂×C₄

C₂×C₄×D₈₃

yes

characteristic

106

C₂×C₄

C₂²×Dic₈₃

yes

characteristic

107

C₂×C₄

C₂×D₃₃₂

yes

characteristic

108

C₂×C₄

C₂×Dic₁₆₆

yes

characteristic

109

C₂×C₄

#13050590

yes

characteristic

110

C₈

—

166

—

111

C₈

C₄×Dic₈₃

yes

—

112

C₈

C₄×Dic₈₃

yes

—

113

C₈

C₄⋊Dic₈₃

yes

—

114

C₈

C₄⋊Dic₈₃

yes

—

115

C₂²

—

116

C₂²

#14581751

yes

characteristic

117

C₂²

C₄×D₃₃₂

yes

characteristic

118

C₂²

C₄×Dic₁₆₆

yes

characteristic

119

C₄

—

120

C₄

—

332

—

121

C₄

C₂×C₄×Dic₈₃

yes

characteristic

122

C₄

C₂×C₄⋊Dic₈₃

yes

characteristic

123

C₄

C₄×D₃₃₂

yes

characteristic

124

C₄

C₄×Dic₁₆₆

yes

characteristic

125

C₂

—

126

C₂

—

127

C₂

C₄×C₄⋊Dic₈₃

yes

characteristic

128

C₂

#19127819

yes

—

	subgroup	G/H
center	C₂×C₈	D₃₃₂
commutator	C₄×C₈₃	C₂×C₄²
Frattini	C₂²×C₄	C₂²×D₈₃
Fitting	C₂×C₄×C₈×C₈₃	C₂
socle	C₂²×C₈₃	C₄×D₄
prime radical	C₂³×C₈₃	#41
cosocle	C₂²×C₄×C₈₃	C₂³
hypercenter	C₂×C₄×C₈	D₈₃
hypocenter	C₈₃	#1772
LCS	G ▷ C₄×C₈₃ ▷ C₂×C₈₃ ▷ C₈₃
UCS	C₁ ◁ C₂×C₈ ◁ C₂²×C₈ ◁ C₂×C₄×C₈

subgroup

G/H

center

C₂×C₈

D₃₃₂

commutator

C₄×C₈₃

C₂×C₄²

Frattini

C₂²×C₄

C₂²×D₈₃

Fitting

C₂×C₄×C₈×C₈₃

C₂

socle

C₂²×C₈₃

C₄×D₄

prime radical

C₂³×C₈₃

#41

cosocle

C₂²×C₄×C₈₃

C₂³

hypercenter

C₂×C₄×C₈

D₈₃

hypocenter

C₈₃

#1772

LCS

G ▷ C₄×C₈₃ ▷ C₂×C₈₃ ▷ C₈₃

UCS

C₁ ◁ C₂×C₈ ◁ C₂²×C₈ ◁ C₂×C₄×C₈

N	H ≅ G/N	type
8. #14581728	C₄	semidirect
9. #14581726	C₄	semidirect
10. #14581724	C₄	semidirect
16. C₄×C₈×C₈₃	C₄	semidirect
65. C₈₃	#1772	semidirect
81. C₄×C₈	Dic₈₃	semidirect

H ≅ G/N

type

8. #14581728

C₄

semidirect

9. #14581726

C₄

semidirect

10. #14581724

C₄

semidirect

16. C₄×C₈×C₈₃

C₄

semidirect

65. C₈₃

#1772

semidirect

81. C₄×C₈

Dic₈₃

semidirect

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

subgroup	index
C₂×#14581728	2
#19127739	2
#19127656	2
#19127461	2
C₂×C₄×C₈×C₈₃	2
#1772	83

subgroup

index

C₂×C₄×C₈×C₈₃

Subgroups — #24907342

Conjugacy classes of subgroups

Complements

Maximal subgroups