Group #24905320 — subgroups

Subgroups — #24905320

Order 10624.

164 conjugacy classes of subgroups; only the 61 normal subgroups are shown.

normal subgroups only

This group has 164 conjugacy classes of subgroups — too many to draw. Only its 61 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	C₄×C₄⋊Dic₈₃	C₂	1	1	yes	maximal, characteristic
3	#432×C₈₃	C₂	1	1	yes	maximal, characteristic
4	C₂×C₄⋊Dic₈₃	—	2	1	no	—
5	C₂×C₄×Dic₈₃	C₄	1	2	yes	—
6	C₄⋊(C₈₃⋊C₈)	—	2	1	no	—
7	C₂²⋊(C₈₃⋊C₈)	—	2	1	no	—
8	#14581703	C₄	1	2	yes	complemented
9	C₈₃⋊#95	—	2	1	no	—
10	C₂×C₄²×C₈₃	C₂²	1	1	yes	characteristic
11	C₄⋊C₈×C₈₃	C₄	1	2	yes	complemented
12	C₂²⋊C₈×C₈₃	—	2	1	no	—
13	#97×C₈₃	—	2	1	no	—
14	C₂²×Dic₈₃	—	2	2	no	—
15	C₂×C₈₃⋊C₈	—	2	2	no	—
16	C₄⋊Dic₈₃	—	2	4	no	—
17	C₄×Dic₈₃	—	4	2	no	—
18	C₂²×C₄×C₈₃	C₂×C₄	1	1	yes	characteristic
19	C₂²×C₄×C₈₃	D₄	1	1	yes	characteristic
20	C₂²×C₄×C₈₃	D₄	1	1	yes	characteristic
21	C₂×C₈×C₈₃	—	2	2	no	—
22	C₄²×C₈₃	C₂×C₄	1	1	yes	characteristic
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	C₂³×C₈₃	C₂²⋊C₄	1	1	yes	characteristic
27	C₂×C₄×C₈₃	—	2	1	no	—
28	C₂×C₄×C₈₃	—	2	1	no	—
29	C₂×C₄×C₈₃	—	2	2	no	—
30	C₂×C₄×C₈₃	—	4	1	no	—
31	C₂×C₄×C₈₃	C₄²	1	1	yes	characteristic
32	C₂×C₄×C₈₃	C₂²⋊C₄	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	C₄⋊C₄	1	1	yes	characteristic
36	C₂×C₄×C₈₃	C₄⋊C₄	1	1	yes	characteristic
37	C₂×Dic₈₃	—	2	4	no	—
38	C₂×Dic₈₃	—	4	2	no	—
39	C₂×Dic₈₃	—	4	2	no	—
40	C₈×C₈₃	—	4	2	no	—
41	C₈₃⋊C₈	—	4	2	no	—
42	C₂²×C₈₃	—	2	1	no	—
43	C₂²×C₈₃	—	2	1	no	—
44	C₂²×C₈₃	#95	1	1	yes	characteristic
45	C₂²×C₈₃	#100	1	1	yes	characteristic
46	C₂²×C₈₃	#101	1	1	yes	characteristic
47	C₄×C₈₃	—	2	1	no	—
48	C₄×C₈₃	—	2	1	no	—
49	C₄×C₈₃	—	2	1	no	—
50	C₄×C₈₃	—	2	1	no	—
51	C₄×C₈₃	—	2	1	no	—
52	C₄×C₈₃	#104	1	2	yes	—
53	Dic₈₃	—	4	4	no	—
54	C₂×C₈₃	—	2	1	no	—
55	C₂×C₈₃	—	2	1	no	—
56	C₂×C₈₃	H(3,ℤ/4)	1	1	yes	characteristic
57	C₂×C₈₃	#339	1	1	yes	characteristic
58	C₂×C₈₃	#343	1	1	yes	characteristic
59	#1285	—	83	1	no	maximal, Sylow
60	C₈₃	#1285	1	1	yes	characteristic, complemented, Sylow
61	#432	—	83	1	no	—
62	#432	D₈₃	1	1	yes	characteristic
63	C₄×C₄⋊C₄	—	83	1	no	—
64	C₂×C₄⋊C₄	—	166	1	no	—
65	C₂×C₄²	—	83	2	no	—
66	C₂×C₄²	C₂×D₈₃	1	1	yes	characteristic
67	C₄⋊C₈	—	166	1	no	—
68	C₄⋊C₈	Dic₈₃	1	2	yes	complemented
69	C₂²⋊C₈	—	2	1	no	—
70	C₂²⋊C₈	—	166	1	no	—
71	#97	—	2	1	no	—
72	#97	—	83	2	no	—
73	#95	—	166	1	no	—
74	C₂²×C₄	—	166	2	no	—
75	C₂²×C₄	C₄×D₈₃	1	1	yes	characteristic
76	C₂²×C₄	D₃₃₂	1	1	yes	characteristic
77	C₂²×C₄	C₂²⋊D₈₃	1	1	yes	characteristic
78	C₂×C₈	—	2	2	no	—
79	C₂×C₈	—	166	2	no	—
80	C₄⋊C₄	—	166	4	no	—
81	C₄²	—	332	2	no	—
82	C₄²	Dic₁₆₆	1	1	yes	characteristic
83	C₄²	C₄×D₈₃	1	1	yes	characteristic
84	C₄²	C₂×Dic₈₃	1	1	yes	characteristic
85	C₄²	C₂²⋊D₈₃	1	1	yes	characteristic
86	C₂³	(C₂×D₈₃)⋊C₄	1	1	yes	characteristic
87	C₂×C₄	—	2	1	no	—
88	C₂×C₄	—	2	1	no	—
89	C₂×C₄	—	2	2	no	—
90	C₂×C₄	—	4	1	no	—
91	C₂×C₄	—	332	2	no	—
92	C₂×C₄	—	332	2	no	—
93	C₂×C₄	—	166	4	no	—
94	C₂×C₄	C₄×Dic₈₃	1	1	yes	characteristic
95	C₂×C₄	(C₂×D₈₃)⋊C₄	1	1	yes	characteristic
96	C₂×C₄	C₂²⋊Dic₈₃	1	1	yes	characteristic
97	C₂×C₄	Dic₈₃⋊C₄	1	1	yes	characteristic
98	C₂×C₄	Dic₈₃⋊C₄	1	1	yes	characteristic
99	C₂×C₄	C₄⋊Dic₈₃	1	1	yes	characteristic
100	C₈	—	4	2	no	—
101	C₈	—	332	2	no	—
102	C₂²	—	2	1	no	—
103	C₂²	—	2	1	no	—
104	C₂²	C₈₃⋊#95	1	1	yes	characteristic
105	C₂²	#14581708	1	1	yes	characteristic
106	C₂²	#14581710	1	1	yes	characteristic
107	C₄	—	2	1	no	—
108	C₄	—	2	1	no	—
109	C₄	—	2	1	no	—
110	C₄	—	2	1	no	—
111	C₄	—	2	1	no	—
112	C₄	—	332	4	no	—
113	C₄	#14581719	1	2	yes	—
114	C₂	—	2	1	no	—
115	C₂	—	2	1	no	—
116	C₂	#19127458	1	1	yes	characteristic
117	C₂	#19127460	1	1	yes	characteristic
118	C₂	#19127469	1	1	yes	characteristic

subgroup

G/H

class size

multiplicity

normal?

properties

#19127758

C₂

yes

maximal, characteristic

C₄×C₄⋊Dic₈₃

C₂

yes

maximal, characteristic

#432×C₈₃

C₂

yes

maximal, characteristic

C₂×C₄⋊Dic₈₃

—

C₂×C₄×Dic₈₃

C₄

yes

—

C₄⋊(C₈₃⋊C₈)

—

C₂²⋊(C₈₃⋊C₈)

—

#14581703

C₄

yes

complemented

C₈₃⋊#95

—

C₂×C₄²×C₈₃

C₂²

yes

characteristic

C₄⋊C₈×C₈₃

C₄

yes

complemented

C₂²⋊C₈×C₈₃

—

#97×C₈₃

—

C₂²×Dic₈₃

—

C₂×C₈₃⋊C₈

—

C₄⋊Dic₈₃

—

C₄×Dic₈₃

—

C₂²×C₄×C₈₃

C₂×C₄

yes

characteristic

C₂²×C₄×C₈₃

D₄

yes

characteristic

C₂²×C₄×C₈₃

D₄

yes

characteristic

C₂×C₈×C₈₃

—

C₄²×C₈₃

C₂×C₄

yes

characteristic

C₄²×C₈₃

C₂×C₄

yes

characteristic

C₄²×C₈₃

D₄

yes

characteristic

C₄²×C₈₃

Q₈

yes

characteristic

C₂³×C₈₃

C₂²⋊C₄

yes

characteristic

C₂×C₄×C₈₃

—

C₂×C₄×C₈₃

—

C₂×C₄×C₈₃

—

C₂×C₄×C₈₃

—

C₂×C₄×C₈₃

C₄²

yes

characteristic

C₂×C₄×C₈₃

C₂²⋊C₄

yes

characteristic

C₂×C₄×C₈₃

C₂²⋊C₄

yes

characteristic

C₂×C₄×C₈₃

C₄⋊C₄

yes

characteristic

C₂×C₄×C₈₃

C₄⋊C₄

yes

characteristic

C₂×C₄×C₈₃

C₄⋊C₄

yes

characteristic

C₂×Dic₈₃

—

C₂×Dic₈₃

—

C₂×Dic₈₃

—

C₈×C₈₃

—

C₈₃⋊C₈

—

C₂²×C₈₃

—

C₂²×C₈₃

—

C₂²×C₈₃

yes

C₂²×C₈₃

yes

C₂²×C₈₃

yes

C₄×C₈₃

—

C₄×C₈₃

—

C₄×C₈₃

—

C₄×C₈₃

—

C₄×C₈₃

—

C₄×C₈₃

#104

yes

—

Dic₈₃

—

C₂×C₈₃

—

C₂×C₈₃

—

C₂×C₈₃

yes

C₂×C₈₃

yes

C₂×C₈₃

yes

—

C₈₃

yes

characteristic, complemented, Sylow

#432

—

#432

D₈₃

yes

characteristic

C₄×C₄⋊C₄

—

C₂×C₄⋊C₄

—

166

—

C₂×C₄²

—

C₂×C₄²

C₂×D₈₃

yes

characteristic

C₄⋊C₈

—

166

—

C₄⋊C₈

Dic₈₃

yes

complemented

C₂²⋊C₈

—

C₂²⋊C₈

—

166

—

#97

—

#97

—

#95

—

166

—

C₂²×C₄

—

166

—

C₂²×C₄

C₄×D₈₃

yes

characteristic

C₂²×C₄

D₃₃₂

yes

characteristic

C₂²×C₄

C₂²⋊D₈₃

yes

characteristic

C₂×C₈

—

C₂×C₈

—

166

—

C₄⋊C₄

—

166

—

C₄²

—

332

—

C₄²

Dic₁₆₆

yes

characteristic

C₄²

C₄×D₈₃

yes

characteristic

C₄²

C₂×Dic₈₃

yes

characteristic

C₄²

C₂²⋊D₈₃

yes

characteristic

C₂³

(C₂×D₈₃)⋊C₄

yes

characteristic

C₂×C₄

—

C₂×C₄

—

C₂×C₄

—

C₂×C₄

—

C₂×C₄

—

332

—

C₂×C₄

—

332

—

C₂×C₄

—

166

—

C₂×C₄

C₄×Dic₈₃

yes

characteristic

C₂×C₄

(C₂×D₈₃)⋊C₄

yes

characteristic

C₂×C₄

C₂²⋊Dic₈₃

yes

characteristic

C₂×C₄

Dic₈₃⋊C₄

yes

characteristic

C₂×C₄

Dic₈₃⋊C₄

yes

characteristic

C₂×C₄

C₄⋊Dic₈₃

yes

characteristic

100

C₈

—

101

C₈

—

332

—

102

C₂²

—

103

C₂²

—

104

C₂²

C₈₃⋊#95

yes

characteristic

105

C₂²

#14581708

yes

characteristic

106

C₂²

#14581710

yes

characteristic

107

C₄

—

108

C₄

—

109

C₄

—

110

C₄

—

111

C₄

—

112

C₄

—

332

—

113

C₄

#14581719

yes

—

114

C₂

—

115

C₂

—

116

C₂

#19127458

yes

characteristic

117

C₂

#19127460

yes

characteristic

118

C₂

#19127469

yes

characteristic

	subgroup	G/H
center	C₂²	C₈₃⋊#95
commutator	C₂×C₄×C₈₃	C₄²
Frattini	C₂×C₄²	C₂×D₈₃
Fitting	#432×C₈₃	C₂
socle	C₂²×C₈₃	#95
prime radical	C₂³×C₈₃	C₂²⋊C₄
cosocle	C₂×C₄²×C₈₃	C₂²
hypercenter	#432	D₈₃
hypocenter	C₈₃	#1285
LCS	G ▷ C₂×C₄×C₈₃ ▷ C₂×C₈₃ ▷ C₈₃
UCS	C₁ ◁ C₂² ◁ C₂×C₄² ◁ #432

subgroup

G/H

center

C₂²

C₈₃⋊#95

commutator

C₂×C₄×C₈₃

C₄²

Frattini

C₂×C₄²

C₂×D₈₃

Fitting

#432×C₈₃

C₂

socle

C₂²×C₈₃

#95

prime radical

C₂³×C₈₃

C₂²⋊C₄

cosocle

C₂×C₄²×C₈₃

C₂²

D₈₃

C₈₃

G ▷ C₂×C₄×C₈₃ ▷ C₂×C₈₃ ▷ C₈₃

UCS

C₁ ◁ C₂² ◁ C₂×C₄² ◁ #432

N	H ≅ G/N	type
8. #14581703	C₄	semidirect
11. C₄⋊C₈×C₈₃	C₄	semidirect
60. C₈₃	#1285	semidirect
68. C₄⋊C₈	Dic₈₃	semidirect

H ≅ G/N

type

8. #14581703

C₄

semidirect

11. C₄⋊C₈×C₈₃

C₄

semidirect

60. C₈₃

#1285

semidirect

68. C₄⋊C₈

Dic₈₃

semidirect

Subgroups — #24905320

Conjugacy classes of subgroups

Complements