Group #24913408 — subgroups

Subgroups — #24913408

Order 10624.

584 conjugacy classes of subgroups; only the 149 normal subgroups are shown.

normal subgroups only

This group has 584 conjugacy classes of subgroups — too many to draw. Only its 149 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×#14581784	C₂	1	1	yes	maximal, characteristic, complemented
2	#19127615	C₂	1	8	yes	maximal, complemented
3	#19127604	C₂	1	4	yes	maximal, complemented
4	#19127547	C₂	1	1	yes	maximal, characteristic, complemented
5	C₂×#127×C₈₃	C₂	1	1	yes	maximal, characteristic
6	#14581784	—	2	4	no	—
7	C₂×C₄⋊Dic₈₃	C₂²	1	2	yes	—
8	C₂×Dic₈₃⋊C₄	C₂²	1	4	yes	—
9	C₂×C₂²⋊Dic₈₃	C₂²	1	16	yes	—
10	C₂×C₄×Dic₈₃	C₂²	1	2	yes	—
11	C₈₃⋊#95	C₂²	1	4	yes	—
12	C₂²×D₄×C₈₃	C₂²	1	2	yes	—
13	C₂²×D₄×C₈₃	C₂²	1	4	yes	—
14	#127×C₈₃	—	2	4	no	—
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	Dic₈₃⋊C₄	—	2	8	no	—
20	C₂²⋊Dic₈₃	—	4	16	no	—
21	C₄×Dic₈₃	—	2	4	no	—
22	C₂⁴×C₈₃	C₂³	1	4	yes	—
23	C₂×D₄×C₈₃	—	2	8	no	—
24	C₂×D₄×C₈₃	—	2	4	no	—
25	C₂×D₄×C₈₃	—	2	8	no	—
26	C₂×D₄×C₈₃	—	4	8	no	—
27	C₂²×C₄×C₈₃	C₂³	1	2	yes	—
28	C₂²×C₄×C₈₃	C₂³	1	1	yes	characteristic
29	C₄²×C₈₃	D₄	1	4	yes	—
30	C₂³×C₈₃	—	2	4	no	—
31	C₂³×C₈₃	—	2	8	no	—
32	C₂³×C₈₃	—	2	16	no	—
33	C₂³×C₈₃	C₂⁴	1	1	yes	characteristic
34	D₄×C₈₃	—	4	16	no	—
35	D₄×C₈₃	—	4	8	no	—
36	C₂×C₄×C₈₃	—	2	4	no	—
37	C₂×C₄×C₈₃	C₂×D₄	1	4	yes	—
38	C₂×C₄×C₈₃	C₂×D₄	1	2	yes	—
39	C₂×C₄×C₈₃	#41	1	4	yes	—
40	C₂×Dic₈₃	—	2	4	no	—
41	C₂×Dic₈₃	—	2	4	no	—
42	C₂×Dic₈₃	—	2	8	no	—
43	C₂×Dic₈₃	—	2	8	no	—
44	C₂²×C₈₃	—	4	4	no	—
45	C₂²×C₈₃	—	4	8	no	—
46	C₂²×C₈₃	—	4	16	no	—
47	C₂²×C₈₃	C₂²×D₄	1	1	yes	characteristic
48	C₂²×C₈₃	C₂×#41	1	2	yes	—
49	C₂²×C₈₃	H(5,𝔽₂)	1	4	yes	—
50	C₄×C₈₃	—	2	4	no	—
51	C₄×C₈₃	#124	1	4	yes	—
52	Dic₈₃	—	4	4	no	—
53	Dic₈₃	—	4	4	no	—
54	C₂×C₈₃	—	8	4	no	—
55	C₂×C₈₃	C₂×#124	1	1	yes	characteristic
56	C₂×C₈₃	#535	1	2	yes	—
57	C₂×C₈₃	#540	1	4	yes	—
58	#2641	—	83	1	no	maximal, Sylow
59	C₈₃	#2641	1	1	yes	characteristic, complemented, Sylow
60	C₂×#127	D₈₃	1	1	yes	characteristic
61	C₂×#125	—	83	1	no	—
62	#394	—	83	8	no	—
63	#390	—	83	4	no	—
64	#376	—	83	1	no	—
65	C₂²×D₄	C₂×D₈₃	1	2	yes	—
66	C₂²×D₄	C₂×D₈₃	1	4	yes	—
67	#127	—	2	4	no	—
68	#125	—	166	4	no	—
69	C₂×C₄⋊C₄	—	83	2	no	—
70	C₂×C₄⋊C₄	—	83	4	no	—
71	C₂×C₂²⋊C₄	—	83	16	no	—
72	C₂×C₄²	—	83	2	no	—
73	C₂×C₄²	C₂×D₈₃	1	1	yes	characteristic
74	#95	—	83	4	no	—
75	C₂⁴	C₂²×D₈₃	1	4	yes	—
76	C₂×D₄	—	2	8	no	—
77	C₂×D₄	—	2	4	no	—
78	C₂×D₄	—	2	8	no	—
79	C₂×D₄	—	4	8	no	—
80	C₂²×C₄	—	83	4	no	—
81	C₂²×C₄	—	83	4	no	—
82	C₂²×C₄	C₂²×D₈₃	1	2	yes	—
83	C₂²×C₄	C₂²×D₈₃	1	1	yes	characteristic
84	C₄⋊C₄	—	166	8	no	—
85	C₄⋊C₄	—	166	4	no	—
86	C₂²⋊C₄	—	332	16	no	—
87	C₄²	—	166	4	no	—
88	C₄²	C₂²⋊D₈₃	1	4	yes	—
89	C₂³	—	2	4	no	—
90	C₂³	—	2	8	no	—
91	C₂³	—	2	16	no	—
92	C₂³	C₂³×D₈₃	1	1	yes	characteristic
93	D₄	—	4	16	no	—
94	D₄	—	4	8	no	—
95	C₂×C₄	—	2	4	no	—
96	C₂×C₄	—	166	4	no	—
97	C₂×C₄	—	166	4	no	—
98	C₂×C₄	—	166	8	no	—
99	C₂×C₄	—	166	8	no	—
100	C₂×C₄	C₂×C₂²⋊D₈₃	1	4	yes	—
101	C₂×C₄	C₂×C₂²⋊D₈₃	1	2	yes	—
102	C₂×C₄	#13050589	1	4	yes	—
103	C₂²	—	4	4	no	—
104	C₂²	—	4	8	no	—
105	C₂²	—	4	16	no	—
106	C₂²	C₂²×C₂²⋊D₈₃	1	1	yes	characteristic
107	C₂²	C₂×#13050589	1	2	yes	—
108	C₂²	#14581837	1	4	yes	—
109	C₄	—	2	4	no	—
110	C₄	—	332	4	no	—
111	C₄	—	332	4	no	—
112	C₄	#14581779	1	4	yes	—
113	C₂	—	8	4	no	—
114	C₂	C₂×#14581779	1	1	yes	characteristic
115	C₂	#19128268	1	2	yes	—
116	C₂	#19128305	1	4	yes	—

subgroup

G/H

class size

multiplicity

normal?

properties

C₂×#14581784

C₂

yes

maximal, characteristic, complemented

#19127615

C₂

yes

maximal, complemented

#19127604

C₂

yes

maximal, complemented

#19127547

C₂

yes

maximal, characteristic, complemented

C₂×#127×C₈₃

C₂

yes

maximal, characteristic

#14581784

—

C₂×C₄⋊Dic₈₃

C₂²

yes

—

C₂×Dic₈₃⋊C₄

C₂²

yes

—

C₂×C₂²⋊Dic₈₃

C₂²

yes

—

C₂×C₄×Dic₈₃

C₂²

yes

—

C₈₃⋊#95

C₂²

yes

—

C₂²×D₄×C₈₃

C₂²

yes

—

C₂²×D₄×C₈₃

C₂²

yes

—

#127×C₈₃

—

C₂×C₄²×C₈₃

C₂²

yes

characteristic

C₂²×Dic₈₃

C₂³

yes

—

C₂²×Dic₈₃

C₂³

yes

—

C₄⋊Dic₈₃

—

Dic₈₃⋊C₄

—

C₂²⋊Dic₈₃

—

C₄×Dic₈₃

—

C₂⁴×C₈₃

C₂³

yes

—

C₂×D₄×C₈₃

—

C₂×D₄×C₈₃

—

C₂×D₄×C₈₃

—

C₂×D₄×C₈₃

—

C₂²×C₄×C₈₃

C₂³

yes

—

C₂²×C₄×C₈₃

C₂³

yes

characteristic

C₄²×C₈₃

D₄

yes

—

C₂³×C₈₃

—

C₂³×C₈₃

—

C₂³×C₈₃

—

C₂³×C₈₃

C₂⁴

yes

characteristic

D₄×C₈₃

—

D₄×C₈₃

—

C₂×C₄×C₈₃

—

C₂×C₄×C₈₃

C₂×D₄

yes

—

C₂×C₄×C₈₃

C₂×D₄

yes

—

C₂×C₄×C₈₃

#41

yes

—

C₂×Dic₈₃

—

C₂×Dic₈₃

—

C₂×Dic₈₃

—

C₂×Dic₈₃

—

C₂²×C₈₃

—

C₂²×C₈₃

—

C₂²×C₈₃

—

C₂²×C₈₃

C₂²×D₄

yes

characteristic

C₂²×C₈₃

C₂×#41

yes

—

C₂²×C₈₃

H(5,𝔽₂)

yes

—

C₄×C₈₃

—

C₄×C₈₃

#124

yes

—

Dic₈₃

—

Dic₈₃

—

C₂×C₈₃

—

C₂×C₈₃

C₂×#124

yes

characteristic

C₂×C₈₃

#535

yes

—

C₂×C₈₃

#540

yes

—

#2641

—

maximal, Sylow

C₈₃

#2641

yes

characteristic, complemented, Sylow

C₂×#127

D₈₃

yes

—

—

—

—

C₂²×D₄

C₂×D₈₃

yes

—

C₂²×D₄

C₂×D₈₃

yes

—

#127

—

#125

—

166

—

C₂×C₄⋊C₄

—

C₂×C₄⋊C₄

—

C₂×C₂²⋊C₄

—

C₂×C₄²

—

C₂×C₄²

C₂×D₈₃

yes

characteristic

#95

—

C₂⁴

C₂²×D₈₃

yes

—

C₂×D₄

—

C₂×D₄

—

C₂×D₄

—

C₂×D₄

—

C₂²×C₄

—

C₂²×C₄

—

C₂²×C₄

C₂²×D₈₃

yes

—

C₂²×C₄

C₂²×D₈₃

yes

characteristic

C₄⋊C₄

—

166

—

C₄⋊C₄

—

166

—

C₂²⋊C₄

—

332

—

C₄²

—

166

—

C₄²

C₂²⋊D₈₃

yes

—

C₂³

—

C₂³

—

C₂³

—

C₂³

C₂³×D₈₃

yes

characteristic

D₄

—

D₄

—

C₂×C₄

—

C₂×C₄

—

166

—

C₂×C₄

—

166

—

C₂×C₄

—

166

—

C₂×C₄

—

166

—

100

C₂×C₄

C₂×C₂²⋊D₈₃

yes

—

101

C₂×C₄

C₂×C₂²⋊D₈₃

yes

—

102

C₂×C₄

#13050589

yes

—

103

C₂²

—

104

C₂²

—

105

C₂²

—

106

C₂²

C₂²×C₂²⋊D₈₃

yes

characteristic

107

C₂²

C₂×#13050589

yes

—

108

C₂²

#14581837

yes

—

109

C₄

—

110

C₄

—

332

—

111

C₄

—

332

—

112

C₄

#14581779

yes

—

113

C₂

—

114

C₂

C₂×#14581779

yes

characteristic

115

C₂

#19128268

yes

—

116

C₂

#19128305

yes

—

	subgroup	G/H
center, Frattini	C₂³	C₂³×D₈₃
commutator, socle, cosocle	C₂³×C₈₃	C₂⁴
Fitting, prime radical	C₂×#127×C₈₃	C₂
hypercenter	C₂×#127	D₈₃
hypocenter	C₈₃	#2641
LCS	G ▷ C₂³×C₈₃ ▷ C₈₃
UCS	C₁ ◁ C₂³ ◁ C₂×#127

subgroup

G/H

center, Frattini

C₂³

C₂³×D₈₃

commutator, socle, cosocle

C₂³×C₈₃

C₂⁴

Fitting, prime radical

C₂×#127×C₈₃

C₂

D₈₃

C₈₃

G ▷ C₂³×C₈₃ ▷ C₈₃

C₁ ◁ C₂³ ◁ C₂×#127

N	H ≅ G/N	type
1. C₂×#14581784	C₂	semidirect
2. #19127615	C₂	semidirect
3. #19127604	C₂	semidirect
4. #19127547	C₂	semidirect
59. C₈₃	#2641	semidirect

H ≅ G/N

type

1. C₂×#14581784

C₂

semidirect

2. #19127615

C₂

semidirect

3. #19127604

C₂

semidirect

4. #19127547

C₂

semidirect

59. C₈₃

#2641

semidirect

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

subgroup	index
C₂×#14581784	2
#19127615	2
#19127604	2
#19127547	2
C₂×#127×C₈₃	2
#2641	83

subgroup

index

C₂×#127×C₈₃

Subgroups — #24913408

Conjugacy classes of subgroups

Complements

Maximal subgroups