Group #24912705 — subgroups

Subgroups — #24912705

Order 10624.

380 conjugacy classes of subgroups; only the 165 normal subgroups are shown.

normal subgroups only

This group has 380 conjugacy classes of subgroups — too many to draw. Only its 165 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	#19127640	C₂	1	4	yes	maximal
2	#19127600	C₂	1	4	yes	maximal
3	#19127566	C₂	1	4	yes	maximal
4	C₄×C₄⋊Dic₈₃	C₂	1	2	yes	maximal
5	#384×C₈₃	C₂	1	1	yes	maximal, characteristic
6	C₂×C₄⋊Dic₈₃	C₂²	1	2	yes	—
7	C₂×C₄⋊Dic₈₃	C₂²	1	2	yes	—
8	C₂×C₄⋊Dic₈₃	C₂²	1	4	yes	—
9	C₂×C₄×Dic₈₃	C₂²	1	4	yes	—
10	C₈₃⋊#95	C₂²	1	4	yes	—
11	C₈₃⋊#95	C₂²	1	4	yes	—
12	C₈₃⋊#95	C₂²	1	8	yes	—
13	C₂×C₄⋊C₄×C₈₃	C₂²	1	4	yes	—
14	C₂×C₄⋊C₄×C₈₃	C₂²	1	2	yes	—
15	C₂×C₄²×C₈₃	C₂²	1	1	yes	characteristic
16	C₂²×Dic₈₃	C₂³	1	4	yes	—
17	C₂²×Dic₈₃	C₂³	1	4	yes	—
18	C₄⋊Dic₈₃	—	2	4	no	—
19	C₄⋊Dic₈₃	—	2	4	no	—
20	C₄⋊Dic₈₃	—	4	4	no	—
21	C₄×Dic₈₃	—	2	8	no	—
22	C₂²×C₄×C₈₃	C₂³	1	4	yes	—
23	C₂²×C₄×C₈₃	C₂³	1	1	yes	characteristic
24	C₂²×C₄×C₈₃	C₂³	1	2	yes	—
25	C₄⋊C₄×C₈₃	—	2	8	no	—
26	C₄⋊C₄×C₈₃	—	4	2	no	—
27	C₄²×C₈₃	Q₈	1	4	yes	—
28	C₂³×C₈₃	C₂⁴	1	1	yes	characteristic
29	C₂×C₄×C₈₃	—	2	1	no	—
30	C₂×C₄×C₈₃	—	2	1	no	—
31	C₂×C₄×C₈₃	—	2	8	no	—
32	C₂×C₄×C₈₃	—	2	4	no	—
33	C₂×C₄×C₈₃	C₂×Q₈	1	2	yes	—
34	C₂×C₄×C₈₃	C₂×Q₈	1	4	yes	—
35	C₂×C₄×C₈₃	#41	1	4	yes	—
36	C₂×C₄×C₈₃	#41	1	4	yes	—
37	C₂×Dic₈₃	—	2	4	no	—
38	C₂×Dic₈₃	—	2	4	no	—
39	C₂×Dic₈₃	—	2	4	no	—
40	C₂×Dic₈₃	—	2	4	no	—
41	C₂×Dic₈₃	—	2	4	no	—
42	C₂×Dic₈₃	—	2	4	no	—
43	C₂²×C₈₃	C₂²×Q₈	1	1	yes	characteristic
44	C₂²×C₈₃	C₂×#41	1	2	yes	—
45	C₂²×C₈₃	C₂×#41	1	2	yes	—
46	C₂²×C₈₃	H(5,𝔽₂)	1	1	yes	characteristic
47	C₂²×C₈₃	2₋¹⁺⁴	1	1	yes	characteristic
48	C₄×C₈₃	—	2	2	no	—
49	C₄×C₈₃	—	4	4	no	—
50	C₄×C₈₃	#125	1	8	yes	—
51	Dic₈₃	—	4	4	no	—
52	Dic₈₃	—	4	4	no	—
53	C₂×C₈₃	C₂×#125	1	2	yes	—
54	C₂×C₈₃	#544	1	1	yes	characteristic
55	C₂×C₈₃	#555	1	1	yes	characteristic
56	C₂×C₈₃	#556	1	2	yes	—
57	C₂×C₈₃	#559	1	1	yes	characteristic
58	#2571	—	83	1	no	maximal, Sylow
59	C₈₃	#2571	1	1	yes	characteristic, complemented, Sylow
60	#400	—	83	4	no	—
61	#389	—	83	4	no	—
62	#384	D₈₃	1	1	yes	characteristic
63	#382	—	83	4	no	—
64	C₄×C₄⋊C₄	—	83	2	no	—
65	C₂×C₄⋊C₄	—	83	4	no	—
66	C₂×C₄⋊C₄	—	83	2	no	—
67	C₂×C₄⋊C₄	—	83	2	no	—
68	C₂×C₄⋊C₄	C₂×D₈₃	1	4	yes	—
69	C₂×C₄⋊C₄	C₂×D₈₃	1	2	yes	—
70	C₂×C₄²	—	83	4	no	—
71	C₂×C₄²	C₂×D₈₃	1	1	yes	characteristic
72	#95	—	83	4	no	—
73	#95	—	83	4	no	—
74	#95	—	83	8	no	—
75	C₂²×C₄	—	83	4	no	—
76	C₂²×C₄	—	83	4	no	—
77	C₂²×C₄	C₂²×D₈₃	1	4	yes	—
78	C₂²×C₄	C₂²×D₈₃	1	1	yes	characteristic
79	C₂²×C₄	C₂²×D₈₃	1	2	yes	—
80	C₄⋊C₄	—	2	8	no	—
81	C₄⋊C₄	—	4	2	no	—
82	C₄⋊C₄	—	332	4	no	—
83	C₄⋊C₄	—	166	4	no	—
84	C₄⋊C₄	—	166	4	no	—
85	C₄²	—	166	8	no	—
86	C₄²	Dic₁₆₆	1	4	yes	—
87	C₂³	C₂³×D₈₃	1	1	yes	characteristic
88	C₂×C₄	—	2	1	no	—
89	C₂×C₄	—	2	1	no	—
90	C₂×C₄	—	2	8	no	—
91	C₂×C₄	—	2	4	no	—
92	C₂×C₄	—	166	4	no	—
93	C₂×C₄	—	166	4	no	—
94	C₂×C₄	—	166	4	no	—
95	C₂×C₄	—	166	4	no	—
96	C₂×C₄	—	166	4	no	—
97	C₂×C₄	—	166	4	no	—
98	C₂×C₄	C₂×Dic₁₆₆	1	2	yes	—
99	C₂×C₄	C₂×Dic₁₆₆	1	4	yes	—
100	C₂×C₄	#13050589	1	4	yes	—
101	C₂×C₄	#13050591	1	4	yes	—
102	C₂²	C₂²×Dic₁₆₆	1	1	yes	characteristic
103	C₂²	C₂×#13050589	1	2	yes	—
104	C₂²	C₂×#13050591	1	2	yes	—
105	C₂²	#14581837	1	1	yes	characteristic
106	C₂²	#14581840	1	1	yes	characteristic
107	C₄	—	2	2	no	—
108	C₄	—	4	4	no	—
109	C₄	—	332	4	no	—
110	C₄	—	332	4	no	—
111	C₄	#14581783	1	8	yes	—
112	C₂	C₂×#14581783	1	2	yes	—
113	C₂	#19128332	1	1	yes	characteristic
114	C₂	#19128417	1	1	yes	characteristic
115	C₂	#19128425	1	2	yes	—
116	C₂	#19128437	1	1	yes	characteristic

subgroup

G/H

class size

multiplicity

normal?

properties

#19127640

C₂

yes

maximal

#19127600

C₂

yes

maximal

#19127566

C₂

yes

maximal

C₄×C₄⋊Dic₈₃

C₂

yes

maximal

#384×C₈₃

C₂

yes

maximal, characteristic

C₂×C₄⋊Dic₈₃

C₂²

yes

—

C₂×C₄⋊Dic₈₃

C₂²

yes

—

C₂×C₄⋊Dic₈₃

C₂²

yes

—

C₂×C₄×Dic₈₃

C₂²

yes

—

C₈₃⋊#95

C₂²

yes

—

C₈₃⋊#95

C₂²

yes

—

C₈₃⋊#95

C₂²

yes

—

C₂×C₄⋊C₄×C₈₃

C₂²

yes

—

C₂×C₄⋊C₄×C₈₃

C₂²

yes

—

C₂×C₄²×C₈₃

C₂²

yes

characteristic

C₂²×Dic₈₃

C₂³

yes

—

C₂²×Dic₈₃

C₂³

yes

—

C₄⋊Dic₈₃

—

C₄⋊Dic₈₃

—

C₄⋊Dic₈₃

—

C₄×Dic₈₃

—

C₂²×C₄×C₈₃

C₂³

yes

—

C₂²×C₄×C₈₃

C₂³

yes

characteristic

C₂²×C₄×C₈₃

C₂³

yes

—

C₄⋊C₄×C₈₃

—

C₄⋊C₄×C₈₃

—

C₄²×C₈₃

Q₈

yes

—

C₂³×C₈₃

C₂⁴

yes

characteristic

C₂×C₄×C₈₃

—

C₂×C₄×C₈₃

—

C₂×C₄×C₈₃

—

C₂×C₄×C₈₃

—

C₂×C₄×C₈₃

C₂×Q₈

yes

—

C₂×C₄×C₈₃

C₂×Q₈

yes

—

C₂×C₄×C₈₃

#41

yes

—

C₂×C₄×C₈₃

#41

yes

—

C₂×Dic₈₃

—

C₂×Dic₈₃

—

C₂×Dic₈₃

—

C₂×Dic₈₃

—

C₂×Dic₈₃

—

C₂×Dic₈₃

—

C₂²×C₈₃

C₂²×Q₈

yes

characteristic

C₂²×C₈₃

C₂×#41

yes

—

C₂²×C₈₃

C₂×#41

yes

—

C₂²×C₈₃

H(5,𝔽₂)

yes

characteristic

C₂²×C₈₃

2₋¹⁺⁴

yes

characteristic

C₄×C₈₃

—

C₄×C₈₃

—

C₄×C₈₃

#125

yes

—

Dic₈₃

—

Dic₈₃

—

C₂×C₈₃

C₂×#125

yes

—

C₂×C₈₃

yes

C₂×C₈₃

yes

C₂×C₈₃

yes

—

C₂×C₈₃

yes

—

C₈₃

yes

characteristic, complemented, Sylow

#400

—

#389

—

#384

D₈₃

yes

characteristic

#382

—

C₄×C₄⋊C₄

—

C₂×C₄⋊C₄

—

C₂×C₄⋊C₄

—

C₂×C₄⋊C₄

—

C₂×C₄⋊C₄

C₂×D₈₃

yes

—

C₂×C₄⋊C₄

C₂×D₈₃

yes

—

C₂×C₄²

—

C₂×C₄²

C₂×D₈₃

yes

—

—

—

C₂²×C₄

—

C₂²×C₄

—

C₂²×C₄

C₂²×D₈₃

yes

—

C₂²×C₄

C₂²×D₈₃

yes

characteristic

C₂²×C₄

C₂²×D₈₃

yes

—

C₄⋊C₄

—

C₄⋊C₄

—

C₄⋊C₄

—

332

—

C₄⋊C₄

—

166

—

C₄⋊C₄

—

166

—

C₄²

—

166

—

C₄²

Dic₁₆₆

yes

—

C₂³

C₂³×D₈₃

yes

characteristic

C₂×C₄

—

C₂×C₄

—

C₂×C₄

—

C₂×C₄

—

C₂×C₄

—

166

—

C₂×C₄

—

166

—

C₂×C₄

—

166

—

C₂×C₄

—

166

—

C₂×C₄

—

166

—

C₂×C₄

—

166

—

C₂×C₄

C₂×Dic₁₆₆

yes

—

C₂×C₄

C₂×Dic₁₆₆

yes

—

100

C₂×C₄

#13050589

yes

—

101

C₂×C₄

#13050591

yes

—

102

C₂²

C₂²×Dic₁₆₆

yes

characteristic

103

C₂²

C₂×#13050589

yes

—

104

C₂²

C₂×#13050591

yes

—

105

C₂²

#14581837

yes

characteristic

106

C₂²

#14581840

yes

characteristic

107

C₄

—

108

C₄

—

109

C₄

—

332

—

110

C₄

—

332

—

111

C₄

#14581783

yes

—

112

C₂

C₂×#14581783

yes

—

113

C₂

#19128332

yes

characteristic

114

C₂

#19128417

yes

characteristic

115

C₂

#19128425

yes

—

116

C₂

#19128437

yes

characteristic

	subgroup	G/H
center, Frattini	C₂³	C₂³×D₈₃
commutator, socle, prime radical, cosocle	C₂³×C₈₃	C₂⁴
Fitting	#384×C₈₃	C₂
hypercenter	#384	D₈₃
hypocenter	C₈₃	#2571
LCS	G ▷ C₂³×C₈₃ ▷ C₈₃
UCS	C₁ ◁ C₂³ ◁ #384

subgroup

G/H

center, Frattini

C₂³

C₂³×D₈₃

commutator, socle, prime radical, cosocle

C₂³×C₈₃

C₂⁴

Fitting

#384×C₈₃

C₂

D₈₃

C₈₃

G ▷ C₂³×C₈₃ ▷ C₈₃

C₁ ◁ C₂³ ◁ #384

N	H ≅ G/N	type
59. C₈₃	#2571	semidirect

H ≅ G/N

type

59. C₈₃

#2571

semidirect

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

subgroup	index
#19127640	2
#19127600	2
#19127566	2
C₄×C₄⋊Dic₈₃	2
#384×C₈₃	2
#2571	83

subgroup

index

#19127640

#19127600

#19127566

C₄×C₄⋊Dic₈₃

#384×C₈₃

#2571

Subgroups — #24912705

Conjugacy classes of subgroups

Complements

Maximal subgroups