Group #24906218 — subgroups

Subgroups — #24906218

Order 10624.

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

normal subgroups only

This group has 184 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	#19127725	C₂	1	1	yes	maximal, characteristic
2	C₂×C₄×C₈₃⋊C₈	C₂	1	1	yes	maximal, characteristic
3	#19127451	C₂	1	4	yes	maximal, complemented
4	#423×C₈₃	C₂	1	1	yes	maximal, characteristic
5	C₂×#13050574	—	2	1	no	—
6	C₂²×C₈₃⋊C₈	C₄	1	2	yes	—
7	C₄⋊(C₈₃⋊C₈)	—	2	1	no	—
8	C₄⋊(C₈₃⋊C₈)	C₂²	1	2	yes	—
9	C₄×C₈₃⋊C₈	C₄	1	1	yes	characteristic
10	C₄×C₈₃⋊C₈	C₄	1	1	yes	characteristic
11	C₄×C₈₃⋊C₈	C₂²	1	2	yes	—
12	C₂×M₁₆×C₈₃	—	2	1	no	—
13	C₂×C₄²×C₈₃	C₂²	1	1	yes	characteristic
14	C₄⋊C₈×C₈₃	—	2	1	no	—
15	C₄⋊C₈×C₈₃	C₂²	1	2	yes	—
16	#13050574	—	4	2	no	—
17	C₂×C₈₃⋊C₈	—	2	4	no	—
18	C₂×C₈₃⋊C₈	—	2	2	no	—
19	C₂×C₈₃⋊C₈	C₂×C₄	1	2	yes	—
20	C₂×C₈₃⋊C₈	C₂×C₄	1	2	yes	—
21	C₂²×C₄×C₈₃	C₂×C₄	1	1	yes	characteristic
22	C₂²×C₄×C₈₃	D₄	1	1	yes	characteristic
23	C₂²×C₄×C₈₃	Q₈	1	1	yes	characteristic
24	M₁₆×C₈₃	—	4	2	no	—
25	C₂×C₈×C₈₃	—	2	2	no	—
26	C₄²×C₈₃	C₂×C₄	1	1	yes	characteristic
27	C₄²×C₈₃	D₄	1	1	yes	characteristic
28	C₄²×C₈₃	Q₈	1	1	yes	characteristic
29	C₄²×C₈₃	C₂³	1	1	yes	characteristic
30	C₂³×C₈₃	C₄⋊C₄	1	1	yes	characteristic
31	C₂×C₄×C₈₃	—	2	2	no	—
32	C₂×C₄×C₈₃	—	2	2	no	—
33	C₂×C₄×C₈₃	C₄⋊C₄	1	1	yes	characteristic
34	C₂×C₄×C₈₃	C₄⋊C₄	1	1	yes	characteristic
35	C₂×C₄×C₈₃	C₄⋊C₄	1	1	yes	characteristic
36	C₂×C₄×C₈₃	SD₁₆	1	2	yes	—
37	C₂×C₄×C₈₃	SD₁₆	1	2	yes	—
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	C₈×C₈₃	—	4	2	no	—
42	C₈₃⋊C₈	—	2	2	no	—
43	C₈₃⋊C₈	—	4	2	no	—
44	C₈₃⋊C₈	M₁₆	1	4	yes	complemented
45	C₂²×C₈₃	—	2	1	no	—
46	C₂²×C₈₃	—	2	1	no	—
47	C₂²×C₈₃	#106	1	2	yes	—
48	C₂²×C₈₃	C₂×C₄⋊C₄	1	1	yes	characteristic
49	C₄×C₈₃	—	2	1	no	—
50	C₄×C₈₃	—	2	1	no	—
51	C₄×C₈₃	#106	1	2	yes	—
52	C₄×C₈₃	#108	1	2	yes	—
53	C₄×C₈₃	C₂×M₁₆	1	2	yes	—
54	C₄×C₈₃	C₂×SD₁₆	1	1	yes	characteristic
55	C₄×C₈₃	C₂×SD₁₆	1	1	yes	characteristic
56	C₂×C₈₃	—	2	2	no	—
57	C₂×C₈₃	#423	1	1	yes	characteristic
58	C₂×C₈₃	C₂×#106	1	1	yes	characteristic
59	C₂×C₈₃	C₂×#108	1	1	yes	characteristic
60	#1566	—	83	1	no	maximal, Sylow
61	C₈₃	#1566	1	1	yes	characteristic, complemented, Sylow
62	#423	—	83	1	no	—
63	#423	D₈₃	1	1	yes	characteristic
64	C₂×C₄×C₈	—	83	1	no	—
65	#334	—	83	4	no	—
66	C₂×M₁₆	—	2	1	no	—
67	C₂×M₁₆	—	166	1	no	—
68	C₂²×C₈	—	83	2	no	—
69	C₂×C₄²	C₂×D₈₃	1	1	yes	characteristic
70	C₄⋊C₈	—	2	1	no	—
71	C₄⋊C₈	—	83	2	no	—
72	C₄⋊C₈	—	166	1	no	—
73	C₄⋊C₈	C₂×D₈₃	1	2	yes	—
74	C₄×C₈	—	83	1	no	—
75	C₄×C₈	—	83	1	no	—
76	C₄×C₈	—	83	2	no	—
77	C₂²×C₄	Dic₁₆₆	1	1	yes	characteristic
78	C₂²×C₄	C₄×D₈₃	1	1	yes	characteristic
79	C₂²×C₄	C₂²⋊D₈₃	1	1	yes	characteristic
80	M₁₆	—	4	2	no	—
81	M₁₆	—	332	2	no	—
82	C₂×C₈	—	2	2	no	—
83	C₂×C₈	—	83	2	no	—
84	C₂×C₈	—	83	2	no	—
85	C₂×C₈	—	166	2	no	—
86	C₂×C₈	—	166	4	no	—
87	C₄²	Dic₁₆₆	1	1	yes	characteristic
88	C₄²	C₄×D₈₃	1	1	yes	characteristic
89	C₄²	C₂²⋊D₈₃	1	1	yes	characteristic
90	C₄²	C₂²×D₈₃	1	1	yes	characteristic
91	C₂³	Dic₈₃⋊C₄	1	1	yes	characteristic
92	C₂×C₄	—	2	2	no	—
93	C₂×C₄	—	2	2	no	—
94	C₂×C₄	Dic₈₃⋊C₄	1	1	yes	characteristic
95	C₂×C₄	Dic₈₃⋊C₄	1	1	yes	characteristic
96	C₂×C₄	Dic₈₃⋊C₄	1	1	yes	characteristic
97	C₂×C₄	#13050577	1	2	yes	—
98	C₂×C₄	#13050578	1	2	yes	—
99	C₂×C₄	C₂×C₄×D₈₃	1	1	yes	characteristic
100	C₂×C₄	C₂×C₂²⋊D₈₃	1	1	yes	characteristic
101	C₂×C₄	C₂×Dic₁₆₆	1	1	yes	characteristic
102	C₈	—	4	2	no	—
103	C₈	—	332	2	no	—
104	C₈	—	83	4	no	—
105	C₈	—	166	2	no	—
106	C₂²	—	2	1	no	—
107	C₂²	—	2	1	no	—
108	C₂²	#14581723	1	2	yes	—
109	C₂²	C₂×Dic₈₃⋊C₄	1	1	yes	characteristic
110	C₄	—	2	1	no	—
111	C₄	—	2	1	no	—
112	C₄	#14581723	1	2	yes	—
113	C₄	#14581727	1	2	yes	—
114	C₄	M₁₆×D₈₃	1	2	yes	—
115	C₄	C₂×#13050577	1	1	yes	characteristic
116	C₄	C₂×#13050578	1	1	yes	characteristic
117	C₂	—	2	2	no	—
118	C₂	#19127724	1	1	yes	characteristic
119	C₂	C₂×#14581723	1	1	yes	characteristic
120	C₂	C₂×#14581727	1	1	yes	characteristic

subgroup

G/H

class size

multiplicity

normal?

properties

#19127725

C₂

yes

maximal, characteristic

C₂×C₄×C₈₃⋊C₈

C₂

yes

maximal, characteristic

#19127451

C₂

yes

maximal, complemented

#423×C₈₃

C₂

yes

maximal, characteristic

C₂×#13050574

—

C₂²×C₈₃⋊C₈

C₄

yes

—

C₄⋊(C₈₃⋊C₈)

—

C₄⋊(C₈₃⋊C₈)

C₂²

yes

—

C₄×C₈₃⋊C₈

C₄

yes

characteristic

C₄×C₈₃⋊C₈

C₄

yes

characteristic

C₄×C₈₃⋊C₈

C₂²

yes

—

C₂×M₁₆×C₈₃

—

C₂×C₄²×C₈₃

C₂²

yes

characteristic

C₄⋊C₈×C₈₃

—

C₄⋊C₈×C₈₃

C₂²

yes

—

#13050574

—

C₂×C₈₃⋊C₈

—

C₂×C₈₃⋊C₈

—

C₂×C₈₃⋊C₈

C₂×C₄

yes

—

C₂×C₈₃⋊C₈

C₂×C₄

yes

—

C₂²×C₄×C₈₃

C₂×C₄

yes

characteristic

C₂²×C₄×C₈₃

D₄

yes

characteristic

C₂²×C₄×C₈₃

Q₈

yes

characteristic

M₁₆×C₈₃

—

C₂×C₈×C₈₃

—

C₄²×C₈₃

C₂×C₄

yes

characteristic

C₄²×C₈₃

D₄

yes

characteristic

C₄²×C₈₃

Q₈

yes

characteristic

C₄²×C₈₃

C₂³

yes

characteristic

C₂³×C₈₃

C₄⋊C₄

yes

characteristic

C₂×C₄×C₈₃

—

C₂×C₄×C₈₃

—

C₂×C₄×C₈₃

C₄⋊C₄

yes

characteristic

C₂×C₄×C₈₃

C₄⋊C₄

yes

characteristic

C₂×C₄×C₈₃

C₄⋊C₄

yes

characteristic

C₂×C₄×C₈₃

SD₁₆

yes

—

C₂×C₄×C₈₃

SD₁₆

yes

—

C₂×C₄×C₈₃

C₂²×C₄

yes

characteristic

C₂×C₄×C₈₃

C₂×D₄

yes

characteristic

C₂×C₄×C₈₃

C₂×Q₈

yes

characteristic

C₈×C₈₃

—

C₈₃⋊C₈

—

C₈₃⋊C₈

—

C₈₃⋊C₈

M₁₆

yes

complemented

C₂²×C₈₃

—

C₂²×C₈₃

—

C₂²×C₈₃

#106

yes

—

C₂²×C₈₃

C₂×C₄⋊C₄

yes

characteristic

C₄×C₈₃

—

C₄×C₈₃

—

C₄×C₈₃

#106

yes

—

C₄×C₈₃

#108

yes

—

C₄×C₈₃

C₂×M₁₆

yes

—

C₄×C₈₃

C₂×SD₁₆

yes

characteristic

C₄×C₈₃

C₂×SD₁₆

yes

characteristic

C₂×C₈₃

—

C₂×C₈₃

yes

C₂×C₈₃

yes

C₂×C₈₃

yes

—

C₈₃

yes

characteristic, complemented, Sylow

#423

—

#423

D₈₃

yes

characteristic

C₂×C₄×C₈

—

#334

—

C₂×M₁₆

—

C₂×M₁₆

—

166

—

C₂²×C₈

—

C₂×C₄²

C₂×D₈₃

yes

characteristic

C₄⋊C₈

—

C₄⋊C₈

—

C₄⋊C₈

—

166

—

C₄⋊C₈

C₂×D₈₃

yes

—

C₄×C₈

—

C₄×C₈

—

C₄×C₈

—

C₂²×C₄

Dic₁₆₆

yes

characteristic

C₂²×C₄

C₄×D₈₃

yes

characteristic

C₂²×C₄

C₂²⋊D₈₃

yes

characteristic

M₁₆

—

M₁₆

—

332

—

C₂×C₈

—

C₂×C₈

—

C₂×C₈

—

C₂×C₈

—

166

—

C₂×C₈

—

166

—

C₄²

Dic₁₆₆

yes

characteristic

C₄²

C₄×D₈₃

yes

characteristic

C₄²

C₂²⋊D₈₃

yes

characteristic

C₄²

C₂²×D₈₃

yes

characteristic

C₂³

Dic₈₃⋊C₄

yes

characteristic

C₂×C₄

—

C₂×C₄

—

C₂×C₄

Dic₈₃⋊C₄

yes

characteristic

C₂×C₄

Dic₈₃⋊C₄

yes

characteristic

C₂×C₄

Dic₈₃⋊C₄

yes

characteristic

C₂×C₄

#13050577

yes

—

C₂×C₄

#13050578

yes

—

C₂×C₄

C₂×C₄×D₈₃

yes

characteristic

100

C₂×C₄

C₂×C₂²⋊D₈₃

yes

characteristic

101

C₂×C₄

C₂×Dic₁₆₆

yes

characteristic

102

C₈

—

103

C₈

—

332

—

104

C₈

—

105

C₈

—

166

—

106

C₂²

—

107

C₂²

—

108

C₂²

#14581723

yes

—

109

C₂²

C₂×Dic₈₃⋊C₄

yes

characteristic

110

C₄

—

111

C₄

—

112

C₄

#14581723

yes

—

113

C₄

#14581727

yes

—

114

C₄

M₁₆×D₈₃

yes

—

115

C₄

C₂×#13050577

yes

characteristic

116

C₄

C₂×#13050578

yes

characteristic

117

C₂

—

118

C₂

#19127724

yes

characteristic

119

C₂

C₂×#14581723

yes

characteristic

120

C₂

C₂×#14581727

yes

characteristic

	subgroup	G/H
center	C₂×C₄	C₂×C₂²⋊D₈₃
commutator	C₂×C₄×C₈₃	C₂²×C₄
Frattini	C₄²	C₂²×D₈₃
Fitting	#423×C₈₃	C₂
socle	C₂²×C₈₃	C₂×C₄⋊C₄
prime radical	C₂³×C₈₃	C₄⋊C₄
cosocle	C₄²×C₈₃	C₂³
hypercenter	#423	D₈₃
hypocenter	C₈₃	#1566
LCS	G ▷ C₂×C₄×C₈₃ ▷ C₂×C₈₃ ▷ C₈₃
UCS	C₁ ◁ C₂×C₄ ◁ C₂×C₄² ◁ #423

subgroup

G/H

center

C₂×C₄

C₂×C₂²⋊D₈₃

commutator

C₂×C₄×C₈₃

C₂²×C₄

Frattini

C₄²

C₂²×D₈₃

Fitting

#423×C₈₃

C₂

socle

C₂²×C₈₃

C₂×C₄⋊C₄

prime radical

C₂³×C₈₃

C₄⋊C₄

cosocle

C₄²×C₈₃

C₂³

D₈₃

C₈₃

G ▷ C₂×C₄×C₈₃ ▷ C₂×C₈₃ ▷ C₈₃

UCS

C₁ ◁ C₂×C₄ ◁ C₂×C₄² ◁ #423

N	H ≅ G/N	type
3. #19127451	C₂	semidirect
44. C₈₃⋊C₈	M₁₆	semidirect
61. C₈₃	#1566	semidirect

H ≅ G/N

type

3. #19127451

C₂

semidirect

44. C₈₃⋊C₈

M₁₆

semidirect

61. C₈₃

#1566

semidirect

Subgroups — #24906218

Conjugacy classes of subgroups

Complements