Group #24907345 — subgroups

Subgroups — #24907345

Order 10624.

232 conjugacy classes of subgroups; only the 117 normal subgroups are shown.

normal subgroups only

This group has 232 conjugacy classes of subgroups — too many to draw. Only its 117 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	#19127739	C₂	1	1	yes	maximal, characteristic
2	#19127659	C₂	1	2	yes	maximal
3	C₂×C₈×Dic₈₃	C₂	1	1	yes	maximal, characteristic
4	#19127462	C₂	1	2	yes	maximal
5	C₂×#108×C₈₃	C₂	1	1	yes	maximal, characteristic
6	C₂²×C₈₃⋊C₈	C₂²	1	1	yes	characteristic
7	#14581751	C₂²	1	2	yes	—
8	C₂×C₄×Dic₈₃	C₂²	1	1	yes	characteristic
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	C₈×Dic₈₃	C₄	1	4	yes	complemented
14	C₂×M₁₆×C₈₃	C₂²	1	2	yes	—
15	C₂²×C₈×C₈₃	C₂²	1	1	yes	characteristic
16	#108×C₈₃	C₄	1	4	yes	complemented
17	C₂²×Dic₈₃	—	2	1	no	—
18	C₂×C₈₃⋊C₈	C₂×C₄	1	2	yes	—
19	C₂×C₈₃⋊C₈	D₄	1	1	yes	characteristic
20	C₂×C₈₃⋊C₈	D₄	1	1	yes	characteristic
21	C₂×C₈₃⋊C₈	Q₈	1	2	yes	—
22	C₄⋊Dic₈₃	C₂×C₄	1	4	yes	—
23	C₂²⋊Dic₈₃	—	2	2	no	—
24	C₄×Dic₈₃	—	2	2	no	—
25	C₄×Dic₈₃	—	2	1	no	—
26	C₄×Dic₈₃	C₂×C₄	1	2	yes	—
27	C₂²×C₄×C₈₃	C₂³	1	1	yes	characteristic
28	M₁₆×C₈₃	—	2	2	no	—
29	M₁₆×C₈₃	C₂×C₄	1	4	yes	—
30	C₂×C₈×C₈₃	—	2	2	no	—
31	C₂×C₈×C₈₃	C₂×C₄	1	2	yes	—
32	C₂×C₈×C₈₃	C₂×C₄	1	1	yes	characteristic
33	C₂×C₈×C₈₃	C₂×C₄	1	1	yes	characteristic
34	C₂×C₈×C₈₃	C₂×C₄	1	2	yes	—
35	C₂³×C₈₃	#41	1	1	yes	characteristic
36	C₂×C₄×C₈₃	C₂²×C₄	1	1	yes	characteristic
37	C₂×C₄×C₈₃	C₂²×C₄	1	1	yes	characteristic
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	1	yes	characteristic
41	C₂×C₄×C₈₃	#41	1	1	yes	characteristic
42	C₂×Dic₈₃	—	2	4	no	—
43	C₂×Dic₈₃	—	2	2	no	—
44	C₂×Dic₈₃	—	2	4	no	—
45	C₈×C₈₃	—	2	4	no	—
46	C₈×C₈₃	C₄²	1	2	yes	—
47	C₈×C₈₃	C₄²	1	2	yes	—
48	C₈₃⋊C₈	C₄⋊C₄	1	2	yes	—
49	C₈₃⋊C₈	C₄⋊C₄	1	2	yes	—
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	C₂×C₄²	1	1	yes	characteristic
55	C₄×C₈₃	C₂×C₄⋊C₄	1	1	yes	characteristic
56	C₄×C₈₃	C₄×D₄	1	1	yes	characteristic
57	C₄×C₈₃	C₄×Q₈	1	1	yes	characteristic
58	Dic₈₃	—	2	4	no	—
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	D₈₃	1	1	yes	characteristic
67	#427	—	83	1	no	—
68	#405	—	83	2	no	—
69	C₂×C₄×C₈	—	83	1	no	—
70	#339	—	83	2	no	—
71	C₂×M₁₆	C₂×D₈₃	1	2	yes	—
72	C₂²×C₈	—	83	1	no	—
73	C₂²×C₈	C₂×D₈₃	1	1	yes	characteristic
74	#117	—	83	2	no	—
75	C₂×C₄²	—	83	1	no	—
76	#108	Dic₈₃	1	4	yes	complemented
77	#107	—	83	2	no	—
78	#106	—	83	2	no	—
79	#97	—	166	2	no	—
80	C₄×C₈	—	83	4	no	—
81	C₄×C₈	—	166	2	no	—
82	C₂²×C₄	—	166	1	no	—
83	C₂²×C₄	C₂²×D₈₃	1	1	yes	characteristic
84	M₁₆	—	2	2	no	—
85	M₁₆	C₂×Dic₈₃	1	4	yes	—
86	C₂×C₈	—	2	2	no	—
87	C₂×C₈	—	83	2	no	—
88	C₂×C₈	—	83	1	no	—
89	C₂×C₈	—	83	1	no	—
90	C₂×C₈	—	83	2	no	—
91	C₂×C₈	C₄×D₈₃	1	2	yes	—
92	C₂×C₈	C₄×D₈₃	1	1	yes	characteristic
93	C₂×C₈	C₄×D₈₃	1	1	yes	characteristic
94	C₂×C₈	C₂×Dic₈₃	1	2	yes	—
95	C₄⋊C₄	—	83	4	no	—
96	C₂²⋊C₄	—	166	2	no	—
97	C₄²	—	83	2	no	—
98	C₄²	—	166	2	no	—
99	C₄²	—	166	1	no	—
100	C₂³	#13050591	1	1	yes	characteristic
101	C₂×C₄	—	166	4	no	—
102	C₂×C₄	—	166	4	no	—
103	C₂×C₄	—	166	2	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	D₄×D₈₃	1	1	yes	characteristic
108	C₂×C₄	Q₈×D₈₃	1	1	yes	characteristic
109	C₂×C₄	#13050589	1	1	yes	characteristic
110	C₈	—	2	4	no	—
111	C₈	—	83	2	no	—
112	C₈	—	83	2	no	—
113	C₈	C₄×Dic₈₃	1	2	yes	—
114	C₈	C₄×Dic₈₃	1	2	yes	—
115	C₂²	—	2	2	no	—
116	C₂²	#14581750	1	1	yes	characteristic
117	C₂²	#14581754	1	1	yes	characteristic
118	C₂²	Q₈×Dic₈₃	1	1	yes	characteristic
119	C₄	—	332	4	no	—
120	C₄	—	166	4	no	—
121	C₄	C₂×C₄×Dic₈₃	1	1	yes	characteristic
122	C₄	C₄⋊C₄×D₈₃	1	1	yes	characteristic
123	C₄	D₄×Dic₈₃	1	1	yes	characteristic
124	C₄	#14581757	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	#19127817	1	2	yes	—

subgroup

G/H

class size

multiplicity

normal?

properties

#19127739

C₂

yes

maximal, characteristic

#19127659

C₂

yes

maximal

C₂×C₈×Dic₈₃

C₂

yes

maximal, characteristic

#19127462

C₂

yes

maximal

C₂×#108×C₈₃

C₂

yes

maximal, characteristic

C₂²×C₈₃⋊C₈

C₂²

yes

characteristic

#14581751

C₂²

yes

—

C₂×C₄×Dic₈₃

C₂²

yes

characteristic

#14581726

C₄

yes

complemented

#14581724

C₄

yes

complemented

#14581702

—

C₈×Dic₈₃

—

C₈×Dic₈₃

C₄

yes

complemented

C₂×M₁₆×C₈₃

C₂²

yes

—

C₂²×C₈×C₈₃

C₂²

yes

characteristic

#108×C₈₃

C₄

yes

complemented

C₂²×Dic₈₃

—

C₂×C₈₃⋊C₈

C₂×C₄

yes

—

C₂×C₈₃⋊C₈

D₄

yes

characteristic

C₂×C₈₃⋊C₈

D₄

yes

characteristic

C₂×C₈₃⋊C₈

Q₈

yes

—

C₄⋊Dic₈₃

C₂×C₄

yes

—

C₂²⋊Dic₈₃

—

C₄×Dic₈₃

—

C₄×Dic₈₃

—

C₄×Dic₈₃

C₂×C₄

yes

—

C₂²×C₄×C₈₃

C₂³

yes

characteristic

M₁₆×C₈₃

—

M₁₆×C₈₃

C₂×C₄

yes

—

C₂×C₈×C₈₃

—

C₂×C₈×C₈₃

C₂×C₄

yes

—

C₂×C₈×C₈₃

C₂×C₄

yes

characteristic

C₂×C₈×C₈₃

C₂×C₄

yes

characteristic

C₂×C₈×C₈₃

C₂×C₄

yes

—

C₂³×C₈₃

#41

yes

characteristic

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

—

C₂×Dic₈₃

—

C₂×Dic₈₃

—

C₈×C₈₃

—

C₈×C₈₃

C₄²

yes

—

C₈×C₈₃

C₄²

yes

—

C₈₃⋊C₈

C₄⋊C₄

yes

—

C₈₃⋊C₈

C₄⋊C₄

yes

—

C₂²×C₈₃

—

C₂²×C₈₃

#117

yes

characteristic

C₂²×C₈₃

C₄×D₄

yes

characteristic

C₂²×C₈₃

C₄×Q₈

yes

characteristic

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

—

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

D₈₃

yes

characteristic

#427

—

#405

—

C₂×C₄×C₈

—

#339

—

C₂×M₁₆

C₂×D₈₃

yes

—

C₂²×C₈

—

C₂²×C₈

C₂×D₈₃

yes

characteristic

#117

—

C₂×C₄²

—

#108

Dic₈₃

yes

—

—

—

166

—

C₄×C₈

—

C₄×C₈

—

166

—

C₂²×C₄

—

166

—

C₂²×C₄

C₂²×D₈₃

yes

characteristic

M₁₆

—

M₁₆

C₂×Dic₈₃

yes

—

C₂×C₈

—

C₂×C₈

—

C₂×C₈

—

C₂×C₈

—

C₂×C₈

—

C₂×C₈

C₄×D₈₃

yes

—

C₂×C₈

C₄×D₈₃

yes

characteristic

C₂×C₈

C₄×D₈₃

yes

characteristic

C₂×C₈

C₂×Dic₈₃

yes

—

C₄⋊C₄

—

C₂²⋊C₄

—

166

—

C₄²

—

C₄²

—

166

—

C₄²

—

166

—

100

C₂³

#13050591

yes

characteristic

101

C₂×C₄

—

166

—

102

C₂×C₄

—

166

—

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

D₄×D₈₃

yes

characteristic

108

C₂×C₄

Q₈×D₈₃

yes

characteristic

109

C₂×C₄

#13050589

yes

characteristic

110

C₈

—

111

C₈

—

112

C₈

—

113

C₈

C₄×Dic₈₃

yes

—

114

C₈

C₄×Dic₈₃

yes

—

115

C₂²

—

116

C₂²

#14581750

yes

characteristic

117

C₂²

#14581754

yes

characteristic

118

C₂²

Q₈×Dic₈₃

yes

characteristic

119

C₄

—

332

—

120

C₄

—

166

—

121

C₄

C₂×C₄×Dic₈₃

yes

characteristic

122

C₄

C₄⋊C₄×D₈₃

yes

characteristic

123

C₄

D₄×Dic₈₃

yes

characteristic

124

C₄

#14581757

yes

characteristic

125

C₂

—

126

C₂

—

127

C₂

C₄⋊C₄×Dic₈₃

yes

characteristic

128

C₂

#19127817

yes

—

	subgroup	G/H
center	C₂×C₄	D₄×D₈₃
commutator	C₄×C₈₃	C₂×C₄²
Frattini	C₂²×C₄	C₂²×D₈₃
Fitting	C₂×#108×C₈₃	C₂
socle	C₂²×C₈₃	C₄×D₄
prime radical	C₂³×C₈₃	#41
cosocle	C₂²×C₄×C₈₃	C₂³
hypercenter	C₂×#108	D₈₃
hypocenter	C₈₃	#1772
LCS	G ▷ C₄×C₈₃ ▷ C₂×C₈₃ ▷ C₈₃
UCS	C₁ ◁ C₂×C₄ ◁ C₂²×C₄ ◁ C₂×#108

subgroup

G/H

center

C₂×C₄

D₄×D₈₃

commutator

C₄×C₈₃

C₂×C₄²

Frattini

C₂²×C₄

C₂²×D₈₃

Fitting

C₂×#108×C₈₃

C₂

socle

C₂²×C₈₃

C₄×D₄

prime radical

C₂³×C₈₃

#41

cosocle

C₂²×C₄×C₈₃

C₂³

D₈₃

C₈₃

G ▷ C₄×C₈₃ ▷ C₂×C₈₃ ▷ C₈₃

UCS

C₁ ◁ C₂×C₄ ◁ C₂²×C₄ ◁ C₂×#108

N	H ≅ G/N	type
9. #14581726	C₄	semidirect
10. #14581724	C₄	semidirect
13. C₈×Dic₈₃	C₄	semidirect
16. #108×C₈₃	C₄	semidirect
65. C₈₃	#1772	semidirect
76. #108	Dic₈₃	semidirect

H ≅ G/N

type

9. #14581726

C₄

semidirect

10. #14581724

C₄

semidirect

13. C₈×Dic₈₃

C₄

semidirect

16. #108×C₈₃

C₄

semidirect

65. C₈₃

#1772

semidirect

76. #108

Dic₈₃

semidirect

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

subgroup	index
#19127739	2
#19127659	2
C₂×C₈×Dic₈₃	2
#19127462	2
C₂×#108×C₈₃	2
#1772	83

subgroup

index

C₂×C₈×Dic₈₃

C₂×#108×C₈₃

Subgroups — #24907345

Conjugacy classes of subgroups

Complements

Maximal subgroups