Group #24906241 — subgroups

Subgroups — #24906241

Order 10624.

150 conjugacy classes of subgroups; only the 73 normal subgroups are shown.

normal subgroups only

This group has 150 conjugacy classes of subgroups — too many to draw. Only its 73 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	#19127758	C₂	1	1	yes	maximal, characteristic
2	#19127754	C₂	1	1	yes	maximal, characteristic
3	#19127453	C₂	1	2	yes	maximal, complemented
4	#19127451	C₂	1	2	yes	maximal, complemented
5	#431×C₈₃	C₂	1	1	yes	maximal, characteristic
6	C₄⋊(C₈₃⋊C₈)	C₄	1	2	yes	—
7	C₄⋊(C₈₃⋊C₈)	C₂²	1	2	yes	—
8	C₂²⋊(C₈₃⋊C₈)	—	2	1	no	—
9	C₂²⋊(C₈₃⋊C₈)	C₄	1	2	yes	—
10	#14581703	—	2	1	no	—
11	C₄×C₈₃⋊C₈	C₂²	1	1	yes	characteristic
12	C₄×C₈₃⋊C₈	C₂²	1	1	yes	characteristic
13	C₂×C₄²×C₈₃	C₂²	1	1	yes	characteristic
14	C₄⋊C₈×C₈₃	C₂²	1	2	yes	—
15	C₂²⋊C₈×C₈₃	—	2	1	no	—
16	C₄×C₈×C₈₃	—	2	1	no	—
17	C₂×C₈₃⋊C₈	—	2	2	no	—
18	C₂×C₈₃⋊C₈	C₂×C₄	1	2	yes	—
19	C₂×C₈₃⋊C₈	C₂×C₄	1	2	yes	—
20	C₂²×C₄×C₈₃	C₂×C₄	1	1	yes	characteristic
21	C₂²×C₄×C₈₃	D₄	1	1	yes	characteristic
22	C₂²×C₄×C₈₃	Q₈	1	1	yes	characteristic
23	C₂×C₈×C₈₃	—	2	2	no	—
24	C₄²×C₈₃	C₂×C₄	1	1	yes	characteristic
25	C₄²×C₈₃	D₄	1	1	yes	characteristic
26	C₄²×C₈₃	Q₈	1	1	yes	characteristic
27	C₄²×C₈₃	C₂³	1	1	yes	characteristic
28	C₂³×C₈₃	C₄⋊C₄	1	1	yes	characteristic
29	C₂×C₄×C₈₃	—	2	2	no	—
30	C₂×C₄×C₈₃	—	2	2	no	—
31	C₂×C₄×C₈₃	—	4	1	no	—
32	C₂×C₄×C₈₃	C₄⋊C₄	1	1	yes	characteristic
33	C₂×C₄×C₈₃	C₄⋊C₄	1	1	yes	characteristic
34	C₂×C₄×C₈₃	C₄⋊C₄	1	1	yes	characteristic
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	2	no	—
40	C₈₃⋊C₈	—	2	2	no	—
41	C₈₃⋊C₈	—	4	2	no	—
42	C₂²×C₈₃	—	2	1	no	—
43	C₂²×C₈₃	—	2	1	no	—
44	C₂²×C₈₃	—	2	1	no	—
45	C₂²×C₈₃	C₂×C₄⋊C₄	1	1	yes	characteristic
46	C₄×C₈₃	—	2	1	no	—
47	C₄×C₈₃	—	2	1	no	—
48	C₄×C₈₃	—	2	1	no	—
49	C₄×C₈₃	—	2	1	no	—
50	C₄×C₈₃	#131	1	2	yes	—
51	C₄×C₈₃	#135	1	2	yes	—
52	C₂×C₈₃	—	4	1	no	—
53	C₂×C₈₃	#424	1	1	yes	characteristic
54	C₂×C₈₃	#427	1	1	yes	characteristic
55	C₂×C₈₃	#430	1	1	yes	characteristic
56	#1572	—	83	1	no	maximal, Sylow
57	C₈₃	#1572	1	1	yes	characteristic, complemented, Sylow
58	#432	—	83	1	no	—
59	#431	—	83	1	no	—
60	#431	D₈₃	1	1	yes	characteristic
61	#335	—	83	2	no	—
62	#334	—	83	2	no	—
63	C₂×C₄²	C₂×D₈₃	1	1	yes	characteristic
64	C₄⋊C₈	—	83	2	no	—
65	C₄⋊C₈	—	83	2	no	—
66	C₄⋊C₈	C₂×D₈₃	1	2	yes	—
67	C₂²⋊C₈	—	2	1	no	—
68	C₂²⋊C₈	—	83	2	no	—
69	C₂²⋊C₈	—	166	1	no	—
70	#97	—	166	1	no	—
71	C₄×C₈	—	2	1	no	—
72	C₄×C₈	—	83	1	no	—
73	C₄×C₈	—	83	1	no	—
74	C₂²×C₄	Dic₁₆₆	1	1	yes	characteristic
75	C₂²×C₄	C₄×D₈₃	1	1	yes	characteristic
76	C₂²×C₄	C₂²⋊D₈₃	1	1	yes	characteristic
77	C₂×C₈	—	2	2	no	—
78	C₂×C₈	—	83	2	no	—
79	C₂×C₈	—	83	2	no	—
80	C₂×C₈	—	166	2	no	—
81	C₄²	Dic₁₆₆	1	1	yes	characteristic
82	C₄²	C₄×D₈₃	1	1	yes	characteristic
83	C₄²	C₂²⋊D₈₃	1	1	yes	characteristic
84	C₄²	C₂²×D₈₃	1	1	yes	characteristic
85	C₂³	Dic₈₃⋊C₄	1	1	yes	characteristic
86	C₂×C₄	—	2	2	no	—
87	C₂×C₄	—	2	2	no	—
88	C₂×C₄	—	4	1	no	—
89	C₂×C₄	Dic₈₃⋊C₄	1	1	yes	characteristic
90	C₂×C₄	Dic₈₃⋊C₄	1	1	yes	characteristic
91	C₂×C₄	Dic₈₃⋊C₄	1	1	yes	characteristic
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	2	no	—
98	C₈	—	166	2	no	—
99	C₂²	—	2	1	no	—
100	C₂²	—	2	1	no	—
101	C₂²	—	2	1	no	—
102	C₂²	C₂×Dic₈₃⋊C₄	1	1	yes	characteristic
103	C₄	—	2	1	no	—
104	C₄	—	2	1	no	—
105	C₄	—	2	1	no	—
106	C₄	—	2	1	no	—
107	C₄	#14581798	1	2	yes	—
108	C₄	#14581812	1	2	yes	—
109	C₂	—	4	1	no	—
110	C₂	#19127727	1	1	yes	characteristic
111	C₂	#19127737	1	1	yes	characteristic
112	C₂	#19127749	1	1	yes	characteristic

subgroup

G/H

class size

multiplicity

normal?

properties

#19127758

C₂

yes

maximal, characteristic

#19127754

C₂

yes

maximal, characteristic

#19127453

C₂

yes

maximal, complemented

#19127451

C₂

yes

maximal, complemented

#431×C₈₃

C₂

yes

maximal, characteristic

C₄⋊(C₈₃⋊C₈)

C₄

yes

—

C₄⋊(C₈₃⋊C₈)

C₂²

yes

—

C₂²⋊(C₈₃⋊C₈)

—

C₂²⋊(C₈₃⋊C₈)

C₄

yes

—

#14581703

—

C₄×C₈₃⋊C₈

C₂²

yes

characteristic

C₄×C₈₃⋊C₈

C₂²

yes

characteristic

C₂×C₄²×C₈₃

C₂²

yes

characteristic

C₄⋊C₈×C₈₃

C₂²

yes

—

C₂²⋊C₈×C₈₃

—

C₄×C₈×C₈₃

—

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

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

—

C₂×C₄×C₈₃

C₄⋊C₄

yes

characteristic

C₂×C₄×C₈₃

C₄⋊C₄

yes

characteristic

C₂×C₄×C₈₃

C₄⋊C₄

yes

characteristic

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

—

C₂²×C₈₃

—

C₂²×C₈₃

—

C₂²×C₈₃

—

C₂²×C₈₃

C₂×C₄⋊C₄

yes

characteristic

C₄×C₈₃

—

C₄×C₈₃

—

C₄×C₈₃

—

C₄×C₈₃

—

C₄×C₈₃

#131

yes

—

C₄×C₈₃

#135

yes

—

C₂×C₈₃

—

C₂×C₈₃

yes

C₂×C₈₃

yes

C₂×C₈₃

yes

—

C₈₃

yes

characteristic, complemented, Sylow

#432

—

#431

—

#431

D₈₃

yes

characteristic

#335

—

#334

—

C₂×C₄²

C₂×D₈₃

yes

characteristic

C₄⋊C₈

—

C₄⋊C₈

—

C₄⋊C₈

C₂×D₈₃

yes

—

C₂²⋊C₈

—

C₂²⋊C₈

—

C₂²⋊C₈

—

166

—

#97

—

166

—

C₄×C₈

—

C₄×C₈

—

C₄×C₈

—

C₂²×C₄

Dic₁₆₆

yes

characteristic

C₂²×C₄

C₄×D₈₃

yes

characteristic

C₂²×C₄

C₂²⋊D₈₃

yes

characteristic

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

—

C₂×C₄

Dic₈₃⋊C₄

yes

characteristic

C₂×C₄

Dic₈₃⋊C₄

yes

characteristic

C₂×C₄

Dic₈₃⋊C₄

yes

characteristic

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

—

C₈

—

166

—

C₂²

—

100

C₂²

—

101

C₂²

—

102

C₂²

C₂×Dic₈₃⋊C₄

yes

characteristic

103

C₄

—

104

C₄

—

105

C₄

—

106

C₄

—

107

C₄

#14581798

yes

—

108

C₄

#14581812

yes

—

109

C₂

—

110

C₂

#19127727

yes

characteristic

111

C₂

#19127737

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	#431×C₈₃	C₂
socle	C₂²×C₈₃	C₂×C₄⋊C₄
prime radical	C₂³×C₈₃	C₄⋊C₄
cosocle	C₄²×C₈₃	C₂³
hypercenter	#431	D₈₃
hypocenter	C₈₃	#1572
LCS	G ▷ C₂×C₄×C₈₃ ▷ C₂×C₈₃ ▷ C₈₃
UCS	C₁ ◁ C₂×C₄ ◁ C₂×C₄² ◁ #431

subgroup

G/H

center

C₂×C₄

C₂×C₂²⋊D₈₃

commutator

C₂×C₄×C₈₃

C₂²×C₄

Frattini

C₄²

C₂²×D₈₃

Fitting

#431×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 ◁ #431

N	H ≅ G/N	type
3. #19127453	C₂	semidirect
4. #19127451	C₂	semidirect
57. C₈₃	#1572	semidirect

H ≅ G/N

type

3. #19127453

C₂

semidirect

4. #19127451

C₂

semidirect

57. C₈₃

#1572

semidirect

Subgroups — #24906241

Conjugacy classes of subgroups

Complements