Group #24905549 — subgroups

Subgroups — #24905549

Order 10624.

132 conjugacy classes of subgroups; only the 77 normal subgroups are shown.

normal subgroups only

This group has 132 conjugacy classes of subgroups — too many to draw. Only its 77 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×#14581732	C₂	1	1	yes	maximal, characteristic
2	C₂×#14581732	C₂	1	1	yes	maximal, characteristic
3	#405×C₈₃	C₂	1	1	yes	maximal, characteristic
4	#14581732	—	2	1	no	—
5	#14581732	—	2	1	no	—
6	#14581732	—	2	1	no	—
7	#14581732	C₄	1	2	yes	complemented
8	C₂×C₈₃⋊C₁₆	—	2	1	no	—
9	C₂×C₈₃⋊C₁₆	C₄	1	2	yes	complemented
10	C₂×M₁₆×C₈₃	C₄	1	1	yes	characteristic
11	C₂²×C₈×C₈₃	C₂²	1	1	yes	characteristic
12	#117×C₈₃	C₄	1	1	yes	characteristic
13	#97×C₈₃	—	2	1	no	—
14	C₄×C₈×C₈₃	—	2	1	no	—
15	C₈₃⋊C₁₆	—	2	2	no	—
16	C₈₃⋊C₁₆	—	2	2	no	—
17	C₂²×C₄×C₈₃	C₂×C₄	1	1	yes	characteristic
18	M₁₆×C₈₃	—	2	1	no	—
19	M₁₆×C₈₃	C₈	1	2	yes	—
20	C₂×C₈×C₈₃	—	2	1	no	—
21	C₂×C₈×C₈₃	C₂×C₄	1	1	yes	characteristic
22	C₂×C₈×C₈₃	C₂×C₄	1	1	yes	characteristic
23	C₂×C₈×C₈₃	D₄	1	1	yes	characteristic
24	C₂×C₈×C₈₃	D₄	1	1	yes	characteristic
25	C₂×C₈×C₈₃	D₄	1	1	yes	characteristic
26	C₂×C₈×C₈₃	Q₈	1	1	yes	characteristic
27	C₄⋊C₄×C₈₃	—	2	1	no	—
28	C₂²⋊C₄×C₈₃	C₈	1	2	yes	—
29	C₄²×C₈₃	—	2	1	no	—
30	C₂³×C₈₃	C₂×C₈	1	1	yes	characteristic
31	C₂×C₄×C₈₃	—	2	2	no	—
32	C₂×C₄×C₈₃	C₄²	1	1	yes	characteristic
33	C₂×C₄×C₈₃	C₂²⋊C₄	1	1	yes	characteristic
34	C₂×C₄×C₈₃	C₄⋊C₄	1	1	yes	characteristic
35	C₂×C₄×C₈₃	C₂×C₈	1	1	yes	characteristic
36	C₂×C₄×C₈₃	M₁₆	1	1	yes	characteristic
37	C₂×C₄×C₈₃	M₁₆	1	1	yes	characteristic
38	C₈×C₈₃	—	2	2	no	—
39	C₈×C₈₃	C₂²⋊C₄	1	1	yes	characteristic
40	C₈×C₈₃	C₂²⋊C₄	1	1	yes	characteristic
41	C₈×C₈₃	C₄⋊C₄	1	1	yes	characteristic
42	C₈×C₈₃	C₄⋊C₄	1	1	yes	characteristic
43	C₂²×C₈₃	—	4	1	no	—
44	C₂²×C₈₃	C₄×C₈	1	1	yes	characteristic
45	C₂²×C₈₃	C₂²⋊C₈	1	1	yes	characteristic
46	C₂²×C₈₃	C₄⋊C₈	1	1	yes	characteristic
47	C₄×C₈₃	—	4	2	no	—
48	C₄×C₈₃	#95	1	1	yes	characteristic
49	C₄×C₈₃	#97	1	1	yes	characteristic
50	C₄×C₈₃	C₂²⋊C₈	1	1	yes	characteristic
51	C₄×C₈₃	C₄⋊C₈	1	1	yes	characteristic
52	C₂×C₈₃	—	2	1	no	—
53	C₂×C₈₃	—	2	1	no	—
54	C₂×C₈₃	—	2	1	no	—
55	C₂×C₈₃	#336	1	1	yes	characteristic
56	#1378	—	83	1	no	maximal, Sylow
57	C₈₃	#1378	1	1	yes	characteristic, complemented, Sylow
58	C₂×M₃₂	—	83	1	no	—
59	C₂×M₃₂	—	83	1	no	—
60	#405	D₈₃	1	1	yes	characteristic
61	C₂×M₁₆	Dic₈₃	1	1	yes	characteristic
62	C₂²×C₈	C₂×D₈₃	1	1	yes	characteristic
63	#117	Dic₈₃	1	1	yes	characteristic
64	M₃₂	—	83	2	no	—
65	M₃₂	—	166	1	no	—
66	M₃₂	—	166	1	no	—
67	M₃₂	—	166	1	no	—
68	C₂×C₁₆	—	83	2	no	—
69	C₂×C₁₆	—	166	1	no	—
70	#97	—	2	1	no	—
71	C₄×C₈	—	2	1	no	—
72	C₂²×C₄	C₂×Dic₈₃	1	1	yes	characteristic
73	M₁₆	—	2	1	no	—
74	M₁₆	C₈₃⋊C₈	1	2	yes	—
75	C₂×C₈	—	2	1	no	—
76	C₂×C₈	Dic₁₆₆	1	1	yes	characteristic
77	C₂×C₈	C₄×D₈₃	1	1	yes	characteristic
78	C₂×C₈	C₄×D₈₃	1	1	yes	characteristic
79	C₂×C₈	D₃₃₂	1	1	yes	characteristic
80	C₂×C₈	C₂²⋊D₈₃	1	1	yes	characteristic
81	C₂×C₈	C₂²⋊D₈₃	1	1	yes	characteristic
82	C₄⋊C₄	—	2	1	no	—
83	C₂²⋊C₄	C₈₃⋊C₈	1	2	yes	—
84	C₄²	—	2	1	no	—
85	C₁₆	—	166	2	no	—
86	C₁₆	—	166	2	no	—
87	C₂³	C₂×C₈₃⋊C₈	1	1	yes	characteristic
88	C₂×C₄	—	2	2	no	—
89	C₂×C₄	C₄×Dic₈₃	1	1	yes	characteristic
90	C₂×C₄	C₂²⋊Dic₈₃	1	1	yes	characteristic
91	C₂×C₄	C₄⋊Dic₈₃	1	1	yes	characteristic
92	C₂×C₄	C₂×C₈₃⋊C₈	1	1	yes	characteristic
93	C₂×C₄	#13050574	1	1	yes	characteristic
94	C₂×C₄	#13050574	1	1	yes	characteristic
95	C₈	—	2	2	no	—
96	C₈	(C₂×D₈₃)⋊C₄	1	1	yes	characteristic
97	C₈	(C₂×D₈₃)⋊C₄	1	1	yes	characteristic
98	C₈	Dic₈₃⋊C₄	1	1	yes	characteristic
99	C₈	Dic₈₃⋊C₄	1	1	yes	characteristic
100	C₂²	—	4	1	no	—
101	C₂²	C₄×C₈₃⋊C₈	1	1	yes	characteristic
102	C₂²	C₂²⋊(C₈₃⋊C₈)	1	1	yes	characteristic
103	C₂²	C₄⋊(C₈₃⋊C₈)	1	1	yes	characteristic
104	C₄	—	4	2	no	—
105	C₄	C₈₃⋊#95	1	1	yes	characteristic
106	C₄	#14581703	1	1	yes	characteristic
107	C₄	C₂²⋊(C₈₃⋊C₈)	1	1	yes	characteristic
108	C₄	C₄⋊(C₈₃⋊C₈)	1	1	yes	characteristic
109	C₂	—	2	1	no	—
110	C₂	—	2	1	no	—
111	C₂	—	2	1	no	—
112	C₂	#19127456	1	1	yes	characteristic

subgroup

G/H

class size

multiplicity

normal?

properties

C₂×#14581732

C₂

yes

maximal, characteristic

C₂×#14581732

C₂

yes

maximal, characteristic

#405×C₈₃

C₂

yes

maximal, characteristic

—

—

—

C₄

yes

complemented

C₂×C₈₃⋊C₁₆

—

C₂×C₈₃⋊C₁₆

C₄

yes

complemented

C₂×M₁₆×C₈₃

C₄

yes

characteristic

C₂²×C₈×C₈₃

C₂²

yes

characteristic

#117×C₈₃

C₄

yes

characteristic

#97×C₈₃

—

C₄×C₈×C₈₃

—

C₈₃⋊C₁₆

—

C₈₃⋊C₁₆

—

C₂²×C₄×C₈₃

C₂×C₄

yes

characteristic

M₁₆×C₈₃

—

M₁₆×C₈₃

C₈

yes

—

C₂×C₈×C₈₃

—

C₂×C₈×C₈₃

C₂×C₄

yes

characteristic

C₂×C₈×C₈₃

C₂×C₄

yes

characteristic

C₂×C₈×C₈₃

D₄

yes

characteristic

C₂×C₈×C₈₃

D₄

yes

characteristic

C₂×C₈×C₈₃

D₄

yes

characteristic

C₂×C₈×C₈₃

Q₈

yes

characteristic

C₄⋊C₄×C₈₃

—

C₂²⋊C₄×C₈₃

C₈

yes

—

C₄²×C₈₃

—

C₂³×C₈₃

C₂×C₈

yes

characteristic

C₂×C₄×C₈₃

—

C₂×C₄×C₈₃

C₄²

yes

characteristic

C₂×C₄×C₈₃

C₂²⋊C₄

yes

characteristic

C₂×C₄×C₈₃

C₄⋊C₄

yes

characteristic

C₂×C₄×C₈₃

C₂×C₈

yes

characteristic

C₂×C₄×C₈₃

M₁₆

yes

characteristic

C₂×C₄×C₈₃

M₁₆

yes

characteristic

C₈×C₈₃

—

C₈×C₈₃

C₂²⋊C₄

yes

characteristic

C₈×C₈₃

C₂²⋊C₄

yes

characteristic

C₈×C₈₃

C₄⋊C₄

yes

characteristic

C₈×C₈₃

C₄⋊C₄

yes

characteristic

C₂²×C₈₃

—

C₂²×C₈₃

C₄×C₈

yes

characteristic

C₂²×C₈₃

C₂²⋊C₈

yes

characteristic

C₂²×C₈₃

C₄⋊C₈

yes

characteristic

C₄×C₈₃

—

C₄×C₈₃

yes

C₄×C₈₃

yes

C₄×C₈₃

C₂²⋊C₈

yes

characteristic

C₄×C₈₃

C₄⋊C₈

yes

characteristic

C₂×C₈₃

—

C₂×C₈₃

—

C₂×C₈₃

—

C₂×C₈₃

yes

—

C₈₃

yes

characteristic, complemented, Sylow

C₂×M₃₂

—

C₂×M₃₂

—

#405

D₈₃

yes

characteristic

C₂×M₁₆

Dic₈₃

yes

characteristic

C₂²×C₈

C₂×D₈₃

yes

characteristic

#117

Dic₈₃

yes

characteristic

M₃₂

—

M₃₂

—

166

—

M₃₂

—

166

—

M₃₂

—

166

—

C₂×C₁₆

—

C₂×C₁₆

—

166

—

#97

—

C₄×C₈

—

C₂²×C₄

C₂×Dic₈₃

yes

characteristic

M₁₆

—

M₁₆

C₈₃⋊C₈

yes

—

C₂×C₈

—

C₂×C₈

Dic₁₆₆

yes

characteristic

C₂×C₈

C₄×D₈₃

yes

characteristic

C₂×C₈

C₄×D₈₃

yes

characteristic

C₂×C₈

D₃₃₂

yes

characteristic

C₂×C₈

C₂²⋊D₈₃

yes

characteristic

C₂×C₈

C₂²⋊D₈₃

yes

characteristic

C₄⋊C₄

—

C₂²⋊C₄

C₈₃⋊C₈

yes

—

C₄²

—

C₁₆

—

166

—

C₁₆

—

166

—

C₂³

C₂×C₈₃⋊C₈

yes

characteristic

C₂×C₄

—

C₂×C₄

C₄×Dic₈₃

yes

characteristic

C₂×C₄

C₂²⋊Dic₈₃

yes

characteristic

C₂×C₄

C₄⋊Dic₈₃

yes

characteristic

C₂×C₄

C₂×C₈₃⋊C₈

yes

C₂×C₄

yes

C₂×C₄

yes

C₈

—

C₈

(C₂×D₈₃)⋊C₄

yes

characteristic

C₈

(C₂×D₈₃)⋊C₄

yes

characteristic

C₈

Dic₈₃⋊C₄

yes

characteristic

C₈

Dic₈₃⋊C₄

yes

characteristic

100

C₂²

—

101

C₂²

C₄×C₈₃⋊C₈

yes

characteristic

102

C₂²

C₂²⋊(C₈₃⋊C₈)

yes

characteristic

103

C₂²

C₄⋊(C₈₃⋊C₈)

yes

characteristic

104

C₄

—

105

C₄

C₈₃⋊#95

yes

characteristic

106

C₄

#14581703

yes

characteristic

107

C₄

C₂²⋊(C₈₃⋊C₈)

yes

characteristic

108

C₄

C₄⋊(C₈₃⋊C₈)

yes

characteristic

109

C₂

—

110

C₂

—

111

C₂

—

112

C₂

#19127456

yes

characteristic

	subgroup	G/H
center	C₈	(C₂×D₈₃)⋊C₄
commutator	C₂²×C₈₃	C₄×C₈
Frattini	C₂²×C₈	C₂×D₈₃
Fitting	#405×C₈₃	C₂
socle	C₂×C₈₃	#336
prime radical	C₂³×C₈₃	C₂×C₈
cosocle	C₂²×C₈×C₈₃	C₂²
hypercenter	#405	D₈₃
hypocenter	C₈₃	#1378
LCS	G ▷ C₂²×C₈₃ ▷ C₂×C₈₃ ▷ C₈₃
UCS	C₁ ◁ C₈ ◁ C₂²×C₈ ◁ #405

subgroup

G/H

center

C₈

(C₂×D₈₃)⋊C₄

commutator

C₂²×C₈₃

C₄×C₈

Frattini

C₂²×C₈

C₂×D₈₃

Fitting

#405×C₈₃

C₂

socle

C₂×C₈₃

#336

prime radical

C₂³×C₈₃

C₂×C₈

cosocle

C₂²×C₈×C₈₃

C₂²

D₈₃

C₈₃

G ▷ C₂²×C₈₃ ▷ C₂×C₈₃ ▷ C₈₃

UCS

C₁ ◁ C₈ ◁ C₂²×C₈ ◁ #405

N	H ≅ G/N	type
7. #14581732	C₄	semidirect
9. C₂×C₈₃⋊C₁₆	C₄	semidirect
57. C₈₃	#1378	semidirect

H ≅ G/N

type

7. #14581732

C₄

semidirect

9. C₂×C₈₃⋊C₁₆

C₄

semidirect

57. C₈₃

#1378

semidirect

Subgroups — #24905549

Conjugacy classes of subgroups

Complements