Group #24906233 — subgroups

Subgroups — #24906233

Order 10624.

168 conjugacy classes of subgroups; only the 89 normal subgroups are shown.

normal subgroups only

This group has 168 conjugacy classes of subgroups — too many to draw. Only its 89 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	#19127754	C₂	1	1	yes	maximal, characteristic
2	#19127725	C₂	1	1	yes	maximal, characteristic
3	#19127451	C₂	1	4	yes	maximal, complemented
4	C₂×C₄⋊C₈×C₈₃	C₂	1	1	yes	maximal, characteristic
5	C₂×#13050574	—	2	1	no	—
6	C₄⋊(C₈₃⋊C₈)	—	2	1	no	—
7	C₄⋊(C₈₃⋊C₈)	C₄	1	2	yes	—
8	C₄⋊(C₈₃⋊C₈)	C₂²	1	2	yes	—
9	C₂²⋊(C₈₃⋊C₈)	C₄	1	2	yes	—
10	C₄×C₈₃⋊C₈	C₂²	1	2	yes	—
11	C₂²×C₈×C₈₃	—	2	1	no	—
12	C₂×C₄²×C₈₃	C₂²	1	1	yes	characteristic
13	C₄⋊C₈×C₈₃	—	2	1	no	—
14	C₄⋊C₈×C₈₃	C₂²	1	2	yes	—
15	#13050574	—	4	2	no	—
16	C₂×C₈₃⋊C₈	—	2	2	no	—
17	C₂×C₈₃⋊C₈	C₂×C₄	1	4	yes	—
18	C₂²×C₄×C₈₃	C₂×C₄	1	1	yes	characteristic
19	C₂²×C₄×C₈₃	D₄	1	1	yes	characteristic
20	C₂²×C₄×C₈₃	Q₈	1	1	yes	characteristic
21	C₂×C₈×C₈₃	—	2	2	no	—
22	C₂×C₈×C₈₃	—	4	2	no	—
23	C₄²×C₈₃	C₂×C₄	1	1	yes	characteristic
24	C₄²×C₈₃	D₄	1	1	yes	characteristic
25	C₄²×C₈₃	Q₈	1	1	yes	characteristic
26	C₄²×C₈₃	C₂³	1	1	yes	characteristic
27	C₂³×C₈₃	C₄⋊C₄	1	1	yes	characteristic
28	C₂×C₄×C₈₃	—	2	2	no	—
29	C₂×C₄×C₈₃	—	4	1	no	—
30	C₂×C₄×C₈₃	C₄⋊C₄	1	1	yes	characteristic
31	C₂×C₄×C₈₃	C₄⋊C₄	1	1	yes	characteristic
32	C₂×C₄×C₈₃	C₄⋊C₄	1	1	yes	characteristic
33	C₂×C₄×C₈₃	SD₁₆	1	2	yes	—
34	C₂×C₄×C₈₃	SD₁₆	1	2	yes	—
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×Q₈	1	1	yes	characteristic
38	C₈×C₈₃	—	4	2	no	—
39	C₈₃⋊C₈	—	2	4	no	—
40	C₈₃⋊C₈	—	4	2	no	—
41	C₂²×C₈₃	—	2	1	no	—
42	C₂²×C₈₃	—	2	1	no	—
43	C₂²×C₈₃	#106	1	2	yes	—
44	C₂²×C₈₃	C₂×C₄⋊C₄	1	1	yes	characteristic
45	C₄×C₈₃	—	2	1	no	—
46	C₄×C₈₃	—	2	1	no	—
47	C₄×C₈₃	—	2	1	no	—
48	C₄×C₈₃	#106	1	2	yes	—
49	C₄×C₈₃	#131	1	2	yes	—
50	C₄×C₈₃	C₂×SD₁₆	1	1	yes	characteristic
51	C₄×C₈₃	C₂×SD₁₆	1	1	yes	characteristic
52	C₂×C₈₃	—	2	2	no	—
53	C₂×C₈₃	#424	1	1	yes	characteristic
54	C₂×C₈₃	C₂×#106	1	1	yes	characteristic
55	C₂×C₈₃	#430	1	1	yes	characteristic
56	#1570	—	83	1	no	maximal, Sylow
57	C₈₃	#1570	1	1	yes	characteristic, complemented, Sylow
58	#431	—	83	1	no	—
59	#423	—	83	1	no	—
60	C₂×C₄⋊C₈	D₈₃	1	1	yes	characteristic
61	#334	—	83	4	no	—
62	C₂×M₁₆	—	166	1	no	—
63	C₂²×C₈	—	2	1	no	—
64	C₂×C₄²	C₂×D₈₃	1	1	yes	characteristic
65	C₄⋊C₈	—	2	1	no	—
66	C₄⋊C₈	—	83	2	no	—
67	C₄⋊C₈	—	83	2	no	—
68	C₄⋊C₈	—	166	1	no	—
69	C₄⋊C₈	C₂×D₈₃	1	2	yes	—
70	C₂²⋊C₈	—	83	2	no	—
71	C₄×C₈	—	83	2	no	—
72	C₂²×C₄	Dic₁₆₆	1	1	yes	characteristic
73	C₂²×C₄	C₄×D₈₃	1	1	yes	characteristic
74	C₂²×C₄	C₂²⋊D₈₃	1	1	yes	characteristic
75	M₁₆	—	332	2	no	—
76	C₂×C₈	—	2	2	no	—
77	C₂×C₈	—	4	2	no	—
78	C₂×C₈	—	83	4	no	—
79	C₂×C₈	—	166	2	no	—
80	C₄²	Dic₁₆₆	1	1	yes	characteristic
81	C₄²	C₄×D₈₃	1	1	yes	characteristic
82	C₄²	C₂²⋊D₈₃	1	1	yes	characteristic
83	C₄²	C₂²×D₈₃	1	1	yes	characteristic
84	C₂³	Dic₈₃⋊C₄	1	1	yes	characteristic
85	C₂×C₄	—	2	2	no	—
86	C₂×C₄	—	4	1	no	—
87	C₂×C₄	Dic₈₃⋊C₄	1	1	yes	characteristic
88	C₂×C₄	Dic₈₃⋊C₄	1	1	yes	characteristic
89	C₂×C₄	Dic₈₃⋊C₄	1	1	yes	characteristic
90	C₂×C₄	#13050577	1	2	yes	—
91	C₂×C₄	#13050578	1	2	yes	—
92	C₂×C₄	C₂×C₄×D₈₃	1	1	yes	characteristic
93	C₂×C₄	C₂×C₂²⋊D₈₃	1	1	yes	characteristic
94	C₂×C₄	C₂×Dic₁₆₆	1	1	yes	characteristic
95	C₈	—	4	2	no	—
96	C₈	—	332	2	no	—
97	C₈	—	166	4	no	—
98	C₂²	—	2	1	no	—
99	C₂²	—	2	1	no	—
100	C₂²	#14581723	1	2	yes	—
101	C₂²	C₂×Dic₈₃⋊C₄	1	1	yes	characteristic
102	C₄	—	2	1	no	—
103	C₄	—	2	1	no	—
104	C₄	—	2	1	no	—
105	C₄	#14581723	1	2	yes	—
106	C₄	#14581798	1	2	yes	—
107	C₄	C₂×#13050577	1	1	yes	characteristic
108	C₄	C₂×#13050578	1	1	yes	characteristic
109	C₂	—	2	2	no	—
110	C₂	#19127727	1	1	yes	characteristic
111	C₂	C₂×#14581723	1	1	yes	characteristic
112	C₂	#19127749	1	1	yes	characteristic

subgroup

G/H

class size

multiplicity

normal?

properties

#19127754

C₂

yes

maximal, characteristic

#19127725

C₂

yes

maximal, characteristic

#19127451

C₂

yes

maximal, complemented

C₂×C₄⋊C₈×C₈₃

C₂

yes

maximal, characteristic

C₂×#13050574

—

C₄⋊(C₈₃⋊C₈)

—

C₄⋊(C₈₃⋊C₈)

C₄

yes

—

C₄⋊(C₈₃⋊C₈)

C₂²

yes

—

C₂²⋊(C₈₃⋊C₈)

C₄

yes

—

C₄×C₈₃⋊C₈

C₂²

yes

—

C₂²×C₈×C₈₃

—

C₂×C₄²×C₈₃

C₂²

yes

characteristic

C₄⋊C₈×C₈₃

—

C₄⋊C₈×C₈₃

C₂²

yes

—

#13050574

—

C₂×C₈₃⋊C₈

—

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

C₂×C₈×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

—

C₂²×C₈₃

—

C₂²×C₈₃

—

C₂²×C₈₃

#106

yes

—

C₂²×C₈₃

C₂×C₄⋊C₄

yes

characteristic

C₄×C₈₃

—

C₄×C₈₃

—

C₄×C₈₃

—

C₄×C₈₃

#106

yes

—

C₄×C₈₃

#131

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

#431

—

#423

—

C₂×C₄⋊C₈

D₈₃

yes

characteristic

#334

—

C₂×M₁₆

—

166

—

C₂²×C₈

—

C₂×C₄²

C₂×D₈₃

yes

characteristic

C₄⋊C₈

—

C₄⋊C₈

—

C₄⋊C₈

—

C₄⋊C₈

—

166

—

C₄⋊C₈

C₂×D₈₃

yes

—

C₂²⋊C₈

—

C₄×C₈

—

C₂²×C₄

Dic₁₆₆

yes

characteristic

C₂²×C₄

C₄×D₈₃

yes

characteristic

C₂²×C₄

C₂²⋊D₈₃

yes

characteristic

M₁₆

—

332

—

C₂×C₈

—

C₂×C₈

—

C₂×C₈

—

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

C₂×C₄

C₂×C₂²⋊D₈₃

yes

characteristic

C₂×C₄

C₂×Dic₁₆₆

yes

characteristic

C₈

—

C₈

—

332

—

C₈

—

166

—

C₂²

—

C₂²

—

100

C₂²

#14581723

yes

—

101

C₂²

C₂×Dic₈₃⋊C₄

yes

characteristic

102

C₄

—

103

C₄

—

104

C₄

—

105

C₄

#14581723

yes

—

106

C₄

#14581798

yes

—

107

C₄

C₂×#13050577

yes

characteristic

108

C₄

C₂×#13050578

yes

characteristic

109

C₂

—

110

C₂

#19127727

yes

characteristic

111

C₂

C₂×#14581723

yes

characteristic

112

C₂

#19127749

yes

characteristic

	subgroup	G/H
center	C₂×C₄	C₂×C₂²⋊D₈₃
commutator	C₂×C₄×C₈₃	C₂²×C₄
Frattini	C₄²	C₂²×D₈₃
Fitting	C₂×C₄⋊C₈×C₈₃	C₂
socle	C₂²×C₈₃	C₂×C₄⋊C₄
prime radical	C₂³×C₈₃	C₄⋊C₄
cosocle	C₄²×C₈₃	C₂³
hypercenter	C₂×C₄⋊C₈	D₈₃
hypocenter	C₈₃	#1570
LCS	G ▷ C₂×C₄×C₈₃ ▷ C₂×C₈₃ ▷ C₈₃
UCS	C₁ ◁ C₂×C₄ ◁ C₂×C₄² ◁ C₂×C₄⋊C₈

subgroup

G/H

center

C₂×C₄

C₂×C₂²⋊D₈₃

commutator

C₂×C₄×C₈₃

C₂²×C₄

Frattini

C₄²

C₂²×D₈₃

Fitting

C₂×C₄⋊C₈×C₈₃

C₂

socle

C₂²×C₈₃

C₂×C₄⋊C₄

prime radical

C₂³×C₈₃

C₄⋊C₄

cosocle

C₄²×C₈₃

C₂³

hypercenter

C₂×C₄⋊C₈

D₈₃

hypocenter

C₈₃

#1570

LCS

G ▷ C₂×C₄×C₈₃ ▷ C₂×C₈₃ ▷ C₈₃

UCS

C₁ ◁ C₂×C₄ ◁ C₂×C₄² ◁ C₂×C₄⋊C₈

N	H ≅ G/N	type
3. #19127451	C₂	semidirect
57. C₈₃	#1570	semidirect

H ≅ G/N

type

3. #19127451

C₂

semidirect

57. C₈₃

#1570

semidirect

Subgroups — #24906233

Conjugacy classes of subgroups

Complements