Group #24909540 — subgroups

Subgroups — #24909540

Order 10624.

170 conjugacy classes of subgroups; only the 47 normal subgroups are shown.

normal subgroups only

This group has 170 conjugacy classes of subgroups — too many to draw. Only its 47 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	#19128139	C₂	1	2	yes	maximal, complemented
2	#19128078	C₂	1	1	yes	maximal, characteristic, complemented
3	#19127819	C₂	1	1	yes	maximal, characteristic, complemented
4	#19127515	C₂	1	2	yes	maximal, complemented
5	#364×C₈₃	C₂	1	1	yes	maximal, characteristic, complemented
6	#14581814	C₂²	1	1	yes	characteristic
7	C₂×D₆₆₄	—	2	1	no	—
8	C₂×D₆₆₄	C₂²	1	2	yes	—
9	#14581798	—	2	1	no	—
10	#14581789	—	2	1	no	—
11	#14581737	—	2	2	no	—
12	D₁₃₂₈	—	2	2	no	—
13	#14581728	C₂²	1	1	yes	characteristic
14	#14581720	—	2	1	no	—
15	M₃₂×C₈₃	C₂²	1	2	yes	—
16	C₄×C₈×C₈₃	C₂²	1	1	yes	characteristic
17	#13050590	—	2	1	no	—
18	C₂×D₃₃₂	—	2	2	no	—
19	C₂×D₃₃₂	—	4	2	no	—
20	Dic₃₃₂	D₄	1	1	yes	characteristic, complemented
21	#13050579	—	2	1	no	—
22	D₆₆₄	—	2	2	no	—
23	D₆₆₄	—	2	2	no	—
24	D₆₆₄	—	4	1	no	—
25	D₆₆₄	—	4	1	no	—
26	D₆₆₄	D₄	1	1	yes	characteristic, complemented
27	#13050574	—	2	1	no	—
28	#13050573	—	4	1	no	—
29	C₈×D₈₃	—	4	1	no	—
30	C₂×C₈×C₈₃	D₄	1	1	yes	characteristic
31	C₂×C₈×C₈₃	C₂³	1	1	yes	characteristic
32	C₄²×C₈₃	D₄	1	1	yes	characteristic
33	C₁₆×C₈₃	—	2	2	no	—
34	C₂²×D₈₃	—	4	2	no	—
35	C₂×C₄×C₈₃	—	2	1	no	—
36	C₂×C₄×C₈₃	C₂×D₄	1	1	yes	characteristic
37	C₂²⋊D₈₃	—	4	1	no	—
38	D₃₃₂	—	2	1	no	—
39	D₃₃₂	—	4	2	no	—
40	D₃₃₂	—	4	2	no	—
41	D₃₃₂	—	8	4	no	—
42	C₄×D₈₃	—	4	1	no	—
43	Dic₁₆₆	—	2	1	no	—
44	C₈×C₈₃	D₈	1	1	yes	characteristic
45	C₈×C₈₃	D₈	1	1	yes	characteristic
46	C₈×C₈₃	C₂×D₄	1	1	yes	characteristic
47	C₈×C₈₃	#41	1	1	yes	characteristic
48	C₈₃⋊C₈	—	4	1	no	—
49	C₂²×C₈₃	AGL(1,ℤ/8)	1	1	yes	characteristic
50	C₂×D₈₃	—	4	1	no	—
51	C₂×D₈₃	—	8	4	no	—
52	C₂×D₈₃	—	8	2	no	—
53	C₄×C₈₃	—	2	2	no	—
54	C₄×C₈₃	#121	1	1	yes	characteristic
55	C₄×C₈₃	C₂×D₈	1	1	yes	characteristic
56	Dic₈₃	—	4	1	no	—
57	C₂×C₈₃	—	2	1	no	—
58	C₂×C₈₃	#459	1	1	yes	characteristic
59	D₈₃	—	8	1	no	—
60	D₈₃	—	16	2	no	—
61	#2213	—	83	1	no	maximal, Sylow
62	C₈₃	#2213	1	1	yes	characteristic, complemented, Sylow
63	#509	—	83	2	no	—
64	#493	—	83	1	no	—
65	#443	—	83	1	no	—
66	#364	D₈₃	1	1	yes	characteristic, complemented
67	#361	—	83	2	no	—
68	#135	—	83	1	no	—
69	C₂×D₈	—	83	2	no	—
70	C₂×D₈	—	166	1	no	—
71	#131	—	166	1	no	—
72	#127	—	166	1	no	—
73	SD₃₂	—	166	2	no	—
74	D₁₆	—	166	2	no	—
75	M₃₂	C₂×D₈₃	1	2	yes	—
76	#108	—	83	1	no	—
77	#104	—	166	1	no	—
78	C₄×C₈	C₂×D₈₃	1	1	yes	characteristic
79	#41	—	166	1	no	—
80	C₂×D₄	—	332	2	no	—
81	C₂×D₄	—	166	2	no	—
82	Q₁₆	—	83	1	no	—
83	SD₁₆	—	166	1	no	—
84	D₈	—	332	1	no	—
85	D₈	—	332	1	no	—
86	D₈	—	83	1	no	—
87	D₈	—	166	2	no	—
88	D₈	—	166	2	no	—
89	M₁₆	—	332	1	no	—
90	M₁₆	—	166	1	no	—
91	C₂×C₈	—	332	1	no	—
92	C₂×C₈	D₃₃₂	1	1	yes	characteristic
93	C₂×C₈	C₂²×D₈₃	1	1	yes	characteristic
94	C₄²	D₃₃₂	1	1	yes	characteristic
95	C₁₆	—	2	2	no	—
96	C₂³	—	332	2	no	—
97	Q₈	—	166	1	no	—
98	D₄	—	332	1	no	—
99	D₄	—	332	2	no	—
100	D₄	—	332	2	no	—
101	D₄	—	664	4	no	—
102	D₄	—	166	1	no	—
103	C₂×C₄	—	2	1	no	—
104	C₂×C₄	—	332	1	no	—
105	C₂×C₄	C₂×D₃₃₂	1	1	yes	characteristic
106	C₈	—	332	1	no	—
107	C₈	D₆₆₄	1	1	yes	characteristic
108	C₈	D₆₆₄	1	1	yes	characteristic
109	C₈	D₄×D₈₃	1	1	yes	characteristic
110	C₈	#13050591	1	1	yes	characteristic
111	C₂²	—	332	1	no	—
112	C₂²	—	664	4	no	—
113	C₂²	—	664	2	no	—
114	C₂²	#14581820	1	1	yes	characteristic
115	C₄	—	2	2	no	—
116	C₄	—	332	1	no	—
117	C₄	#14581767	1	1	yes	characteristic
118	C₄	C₂×D₆₆₄	1	1	yes	characteristic
119	C₂	—	2	1	no	—
120	C₂	—	1328	2	no	—
121	C₂	—	664	1	no	—
122	C₂	#19127897	1	1	yes	characteristic

subgroup

G/H

class size

multiplicity

normal?

properties

#19128139

C₂

yes

maximal, complemented

#19128078

C₂

yes

maximal, characteristic, complemented

#19127819

C₂

yes

maximal, characteristic, complemented

#19127515

C₂

yes

maximal, complemented

#364×C₈₃

C₂

yes

maximal, characteristic, complemented

#14581814

C₂²

yes

characteristic

C₂×D₆₆₄

—

C₂×D₆₆₄

C₂²

yes

—

#14581798

—

#14581789

—

#14581737

—

D₁₃₂₈

—

#14581728

C₂²

yes

characteristic

#14581720

—

M₃₂×C₈₃

C₂²

yes

—

C₄×C₈×C₈₃

C₂²

yes

characteristic

#13050590

—

C₂×D₃₃₂

—

C₂×D₃₃₂

—

Dic₃₃₂

D₄

yes

characteristic, complemented

#13050579

—

D₆₆₄

—

D₆₆₄

—

D₆₆₄

—

D₆₆₄

—

D₆₆₄

D₄

yes

characteristic, complemented

#13050574

—

#13050573

—

C₈×D₈₃

—

C₂×C₈×C₈₃

D₄

yes

characteristic

C₂×C₈×C₈₃

C₂³

yes

characteristic

C₄²×C₈₃

D₄

yes

characteristic

C₁₆×C₈₃

—

C₂²×D₈₃

—

C₂×C₄×C₈₃

—

C₂×C₄×C₈₃

C₂×D₄

yes

characteristic

C₂²⋊D₈₃

—

D₃₃₂

—

D₃₃₂

—

D₃₃₂

—

D₃₃₂

—

C₄×D₈₃

—

Dic₁₆₆

—

C₈×C₈₃

D₈

yes

characteristic

C₈×C₈₃

D₈

yes

characteristic

C₈×C₈₃

C₂×D₄

yes

characteristic

C₈×C₈₃

#41

yes

characteristic

C₈₃⋊C₈

—

C₂²×C₈₃

AGL(1,ℤ/8)

yes

characteristic

C₂×D₈₃

—

C₂×D₈₃

—

C₂×D₈₃

—

C₄×C₈₃

—

C₄×C₈₃

#121

yes

characteristic

C₄×C₈₃

C₂×D₈

yes

characteristic

Dic₈₃

—

C₂×C₈₃

—

C₂×C₈₃

#459

yes

characteristic

D₈₃

—

D₈₃

—

#2213

—

maximal, Sylow

C₈₃

#2213

yes

characteristic, complemented, Sylow

—

—

—

D₈₃

yes

characteristic, complemented

#361

—

#135

—

C₂×D₈

—

C₂×D₈

—

166

—

#131

—

166

—

#127

—

166

—

SD₃₂

—

166

—

D₁₆

—

166

—

M₃₂

C₂×D₈₃

yes

—

#108

—

#104

—

166

—

C₄×C₈

C₂×D₈₃

yes

characteristic

#41

—

166

—

C₂×D₄

—

332

—

C₂×D₄

—

166

—

Q₁₆

—

SD₁₆

—

166

—

D₈

—

332

—

D₈

—

332

—

D₈

—

D₈

—

166

—

D₈

—

166

—

M₁₆

—

332

—

M₁₆

—

166

—

C₂×C₈

—

332

—

C₂×C₈

D₃₃₂

yes

characteristic

C₂×C₈

C₂²×D₈₃

yes

characteristic

C₄²

D₃₃₂

yes

characteristic

C₁₆

—

C₂³

—

332

—

Q₈

—

166

—

D₄

—

332

—

D₄

—

332

—

100

D₄

—

332

—

101

D₄

—

664

—

102

D₄

—

166

—

103

C₂×C₄

—

104

C₂×C₄

—

332

—

105

C₂×C₄

C₂×D₃₃₂

yes

characteristic

106

C₈

—

332

—

107

C₈

D₆₆₄

yes

characteristic

108

C₈

D₆₆₄

yes

characteristic

109

C₈

D₄×D₈₃

yes

characteristic

110

C₈

#13050591

yes

characteristic

111

C₂²

—

332

—

112

C₂²

—

664

—

113

C₂²

—

664

—

114

C₂²

#14581820

yes

characteristic

115

C₄

—

116

C₄

—

332

—

117

C₄

#14581767

yes

characteristic

118

C₄

C₂×D₆₆₄

yes

characteristic

119

C₂

—

120

C₂

—

1328

—

121

C₂

—

664

—

122

C₂

#19127897

yes

characteristic

	subgroup	G/H
center	C₂	#19127897
commutator, cosocle	C₂×C₈×C₈₃	C₂³
Frattini	C₂×C₈	C₂²×D₈₃
Fitting	#364×C₈₃	C₂
socle	C₂×C₈₃	#459
hypercenter	#364	D₈₃
hypocenter	C₈₃	#2213
LCS	G ▷ C₂×C₈×C₈₃ ▷ C₄×C₈₃ ▷ C₂×C₈₃ ▷ C₈₃
UCS	C₁ ◁ C₂ ◁ C₂×C₄ ◁ C₄×C₈ ◁ #364

subgroup

G/H

center

C₂

#19127897

commutator, cosocle

C₂×C₈×C₈₃

C₂³

Frattini

C₂×C₈

C₂²×D₈₃

Fitting

#364×C₈₃

C₂

C₂×C₈₃

D₈₃

C₈₃

G ▷ C₂×C₈×C₈₃ ▷ C₄×C₈₃ ▷ C₂×C₈₃ ▷ C₈₃

UCS

C₁ ◁ C₂ ◁ C₂×C₄ ◁ C₄×C₈ ◁ #364

N	H ≅ G/N	type
1. #19128139	C₂	semidirect
2. #19128078	C₂	semidirect
3. #19127819	C₂	semidirect
4. #19127515	C₂	semidirect
5. #364×C₈₃	C₂	semidirect
20. Dic₃₃₂	D₄	semidirect
26. D₆₆₄	D₄	semidirect
62. C₈₃	#2213	semidirect
66. #364	D₈₃	semidirect

H ≅ G/N

type

1. #19128139

C₂

semidirect

2. #19128078

C₂

semidirect

3. #19127819

C₂

semidirect

4. #19127515

C₂

semidirect

5. #364×C₈₃

C₂

semidirect

20. Dic₃₃₂

D₄

semidirect

26. D₆₆₄

D₄

semidirect

62. C₈₃

#2213

semidirect

66. #364

D₈₃

semidirect

By isomorphism type — maximal subgroups of the same type appear once, whether or not they are normal.

subgroup	index
#19128139	2
#19128078	2
#19127819	2
#19127515	2
#364×C₈₃	2
#2213	83

subgroup

index

#364×C₈₃

Subgroups — #24909540

Conjugacy classes of subgroups

Complements

Maximal subgroups