Group #24907217 — subgroups

Subgroups — #24907217

Order 10624.

200 conjugacy classes of subgroups; only the 93 normal subgroups are shown.

normal subgroups only

This group has 200 conjugacy classes of subgroups — too many to draw. Only its 93 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	#19127659	C₂	1	1	yes	maximal, characteristic
2	#19127657	C₂	1	2	yes	maximal
3	C₈₃⋊#338	C₂	1	1	yes	maximal, characteristic
4	#19127457	C₂	1	2	yes	maximal
5	H(3,ℤ/4)×C₈₃	C₂	1	1	yes	maximal, characteristic
6	C₂×#13050574	C₂²	1	2	yes	—
7	C₂²×C₈₃⋊C₈	C₂²	1	1	yes	characteristic
8	#14581751	C₂²	1	1	yes	characteristic
9	#14581703	C₄	1	4	yes	complemented
10	#14581702	C₄	1	2	yes	complemented
11	C₄×C₈₃⋊C₈	C₄	1	4	yes	complemented
12	C₈×Dic₈₃	C₄	1	2	yes	complemented
13	C₂×M₁₆×C₈₃	C₂²	1	1	yes	characteristic
14	#117×C₈₃	C₂²	1	2	yes	—
15	#13050574	—	2	2	no	—
16	#13050574	—	2	2	no	—
17	C₂×C₈₃⋊C₈	C₂×C₄	1	4	yes	—
18	C₂×C₈₃⋊C₈	C₂×C₄	1	1	yes	characteristic
19	C₂×C₈₃⋊C₈	C₂×C₄	1	2	yes	—
20	C₂×C₈₃⋊C₈	C₂×C₄	1	1	yes	characteristic
21	C₂×C₈₃⋊C₈	C₂×C₄	1	2	yes	—
22	C₄⋊Dic₈₃	—	2	1	no	—
23	C₂²⋊Dic₈₃	—	2	1	no	—
24	C₄×Dic₈₃	C₂×C₄	1	2	yes	—
25	C₂²×C₄×C₈₃	C₂³	1	1	yes	characteristic
26	M₁₆×C₈₃	—	2	2	no	—
27	C₂×C₈×C₈₃	C₂×C₄	1	2	yes	—
28	C₄⋊C₄×C₈₃	—	2	2	no	—
29	C₂²⋊C₄×C₈₃	—	2	2	no	—
30	C₄²×C₈₃	C₂×C₄	1	4	yes	—
31	C₂³×C₈₃	#41	1	1	yes	characteristic
32	C₂×C₄×C₈₃	—	2	4	no	—
33	C₂×C₄×C₈₃	C₂²×C₄	1	2	yes	—
34	C₂×C₄×C₈₃	C₂²×C₄	1	1	yes	characteristic
35	C₂×C₄×C₈₃	#41	1	2	yes	—
36	C₂×C₄×C₈₃	#41	1	1	yes	characteristic
37	C₂×Dic₈₃	—	2	2	no	—
38	C₈×C₈₃	—	2	2	no	—
39	C₈₃⋊C₈	—	2	4	no	—
40	C₈₃⋊C₈	C₄²	1	1	yes	characteristic
41	C₈₃⋊C₈	C₄²	1	1	yes	characteristic
42	C₈₃⋊C₈	C₄²	1	2	yes	—
43	C₂²×C₈₃	—	4	1	no	—
44	C₂²×C₈₃	#117	1	2	yes	—
45	C₂²×C₈₃	#117	1	1	yes	characteristic
46	C₄×C₈₃	—	4	4	no	—
47	C₄×C₈₃	C₂×C₄²	1	1	yes	characteristic
48	C₄×C₈₃	#117	1	1	yes	characteristic
49	C₄×C₈₃	#117	1	2	yes	—
50	Dic₈₃	—	4	2	no	—
51	C₂×C₈₃	—	2	1	no	—
52	C₂×C₈₃	—	2	2	no	—
53	C₂×C₈₃	#376	1	1	yes	characteristic
54	#1747	—	83	1	no	maximal, Sylow
55	C₈₃	#1747	1	1	yes	characteristic, complemented, Sylow
56	#405	—	83	1	no	—
57	#405	—	83	2	no	—
58	#338	—	83	1	no	—
59	H(3,ℤ/4)	—	83	2	no	—
60	H(3,ℤ/4)	D₈₃	1	1	yes	characteristic
61	C₂×M₁₆	—	83	2	no	—
62	C₂×M₁₆	C₂×D₈₃	1	1	yes	characteristic
63	C₂²×C₈	—	83	1	no	—
64	#117	—	83	1	no	—
65	#117	C₂×D₈₃	1	2	yes	—
66	#97	—	83	2	no	—
67	#97	—	83	4	no	—
68	C₄×C₈	—	83	2	no	—
69	C₄×C₈	—	83	4	no	—
70	C₂²×C₄	C₂²×D₈₃	1	1	yes	characteristic
71	M₁₆	—	2	2	no	—
72	M₁₆	—	166	2	no	—
73	M₁₆	—	166	2	no	—
74	C₂×C₈	—	83	1	no	—
75	C₂×C₈	—	83	2	no	—
76	C₂×C₈	—	83	1	no	—
77	C₂×C₈	—	83	2	no	—
78	C₂×C₈	—	83	4	no	—
79	C₂×C₈	C₄×D₈₃	1	2	yes	—
80	C₄⋊C₄	—	2	2	no	—
81	C₄⋊C₄	—	166	1	no	—
82	C₂²⋊C₄	—	2	2	no	—
83	C₂²⋊C₄	—	166	1	no	—
84	C₄²	—	83	2	no	—
85	C₄²	C₄×D₈₃	1	4	yes	—
86	C₂³	#13050589	1	1	yes	characteristic
87	C₂×C₄	—	2	4	no	—
88	C₂×C₄	—	166	2	no	—
89	C₂×C₄	C₂×C₄×D₈₃	1	1	yes	characteristic
90	C₂×C₄	C₂×C₄×D₈₃	1	2	yes	—
91	C₂×C₄	#13050589	1	2	yes	—
92	C₂×C₄	#13050591	1	1	yes	characteristic
93	C₈	—	2	2	no	—
94	C₈	—	83	1	no	—
95	C₈	—	83	1	no	—
96	C₈	—	83	2	no	—
97	C₈	—	166	4	no	—
98	C₂²	—	4	1	no	—
99	C₂²	#14581748	1	2	yes	—
100	C₂²	#14581750	1	1	yes	characteristic
101	C₄	—	4	4	no	—
102	C₄	—	332	2	no	—
103	C₄	C₄²×D₈₃	1	1	yes	characteristic
104	C₄	#14581748	1	1	yes	characteristic
105	C₄	#14581750	1	2	yes	—
106	C₂	—	2	2	no	—
107	C₂	—	2	1	no	—
108	C₂	#19127546	1	1	yes	characteristic

subgroup

G/H

class size

multiplicity

normal?

properties

#19127659

C₂

yes

maximal, characteristic

#19127657

C₂

yes

maximal

C₈₃⋊#338

C₂

yes

maximal, characteristic

#19127457

C₂

yes

maximal

H(3,ℤ/4)×C₈₃

C₂

yes

maximal, characteristic

C₂×#13050574

C₂²

yes

—

C₂²×C₈₃⋊C₈

C₂²

yes

characteristic

#14581751

C₂²

yes

characteristic

#14581703

C₄

yes

complemented

#14581702

C₄

yes

complemented

C₄×C₈₃⋊C₈

C₄

yes

complemented

C₈×Dic₈₃

C₄

yes

complemented

C₂×M₁₆×C₈₃

C₂²

yes

characteristic

#117×C₈₃

C₂²

yes

—

#13050574

—

#13050574

—

C₂×C₈₃⋊C₈

C₂×C₄

yes

—

C₂×C₈₃⋊C₈

C₂×C₄

yes

characteristic

C₂×C₈₃⋊C₈

C₂×C₄

yes

—

C₂×C₈₃⋊C₈

C₂×C₄

yes

characteristic

C₂×C₈₃⋊C₈

C₂×C₄

yes

—

C₄⋊Dic₈₃

—

C₂²⋊Dic₈₃

—

C₄×Dic₈₃

C₂×C₄

yes

—

C₂²×C₄×C₈₃

C₂³

yes

characteristic

M₁₆×C₈₃

—

C₂×C₈×C₈₃

C₂×C₄

yes

—

C₄⋊C₄×C₈₃

—

C₂²⋊C₄×C₈₃

—

C₄²×C₈₃

C₂×C₄

yes

—

C₂³×C₈₃

#41

yes

characteristic

C₂×C₄×C₈₃

—

C₂×C₄×C₈₃

C₂²×C₄

yes

—

C₂×C₄×C₈₃

C₂²×C₄

yes

characteristic

C₂×C₄×C₈₃

#41

yes

—

C₂×C₄×C₈₃

#41

yes

characteristic

C₂×Dic₈₃

—

C₈×C₈₃

—

C₈₃⋊C₈

—

C₈₃⋊C₈

C₄²

yes

characteristic

C₈₃⋊C₈

C₄²

yes

characteristic

C₈₃⋊C₈

C₄²

yes

—

C₂²×C₈₃

—

C₂²×C₈₃

#117

yes

—

C₂²×C₈₃

#117

yes

characteristic

C₄×C₈₃

—

C₄×C₈₃

C₂×C₄²

yes

C₄×C₈₃

yes

C₄×C₈₃

yes

—

Dic₈₃

—

C₂×C₈₃

—

C₂×C₈₃

—

C₂×C₈₃

yes

—

C₈₃

yes

characteristic, complemented, Sylow

—

—

—

—

D₈₃

yes

characteristic

C₂×M₁₆

—

C₂×M₁₆

C₂×D₈₃

yes

characteristic

C₂²×C₈

—

#117

—

#117

C₂×D₈₃

yes

—

#97

—

#97

—

C₄×C₈

—

C₄×C₈

—

C₂²×C₄

C₂²×D₈₃

yes

characteristic

M₁₆

—

M₁₆

—

166

—

M₁₆

—

166

—

C₂×C₈

—

C₂×C₈

—

C₂×C₈

—

C₂×C₈

—

C₂×C₈

—

C₂×C₈

C₄×D₈₃

yes

—

C₄⋊C₄

—

C₄⋊C₄

—

166

—

C₂²⋊C₄

—

C₂²⋊C₄

—

166

—

C₄²

—

C₄²

C₄×D₈₃

yes

—

C₂³

#13050589

yes

characteristic

C₂×C₄

—

C₂×C₄

—

166

—

C₂×C₄

C₂×C₄×D₈₃

yes

characteristic

C₂×C₄

C₂×C₄×D₈₃

yes

—

C₂×C₄

#13050589

yes

—

C₂×C₄

#13050591

yes

characteristic

C₈

—

C₈

—

C₈

—

C₈

—

C₈

—

166

—

C₂²

—

C₂²

#14581748

yes

—

100

C₂²

#14581750

yes

characteristic

101

C₄

—

102

C₄

—

332

—

103

C₄

C₄²×D₈₃

yes

characteristic

104

C₄

#14581748

yes

characteristic

105

C₄

#14581750

yes

—

106

C₂

—

107

C₂

—

108

C₂

#19127546

yes

characteristic

	subgroup	G/H
center	C₄	C₄²×D₈₃
commutator	C₄×C₈₃	C₂×C₄²
Frattini	C₂²×C₄	C₂²×D₈₃
Fitting	H(3,ℤ/4)×C₈₃	C₂
socle	C₂×C₈₃	#376
prime radical	C₂³×C₈₃	#41
cosocle	C₂²×C₄×C₈₃	C₂³
hypercenter	H(3,ℤ/4)	D₈₃
hypocenter	C₈₃	#1747
LCS	G ▷ C₄×C₈₃ ▷ C₈₃
UCS	C₁ ◁ C₄ ◁ H(3,ℤ/4)

subgroup

G/H

center

C₄

C₄²×D₈₃

commutator

C₄×C₈₃

C₂×C₄²

Frattini

C₂²×C₄

C₂²×D₈₃

Fitting

H(3,ℤ/4)×C₈₃

C₂

C₂×C₈₃

C₂³×C₈₃

C₂²×C₄×C₈₃

C₂³

D₈₃

C₈₃

G ▷ C₄×C₈₃ ▷ C₈₃

C₁ ◁ C₄ ◁ H(3,ℤ/4)

N	H ≅ G/N	type
9. #14581703	C₄	semidirect
10. #14581702	C₄	semidirect
11. C₄×C₈₃⋊C₈	C₄	semidirect
12. C₈×Dic₈₃	C₄	semidirect
55. C₈₃	#1747	semidirect

H ≅ G/N

type

9. #14581703

C₄

semidirect

10. #14581702

C₄

semidirect

11. C₄×C₈₃⋊C₈

C₄

semidirect

12. C₈×Dic₈₃

C₄

semidirect

55. C₈₃

#1747

semidirect

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

subgroup	index
#19127659	2
#19127657	2
C₈₃⋊#338	2
#19127457	2
H(3,ℤ/4)×C₈₃	2
#1747	83

subgroup

index

C₈₃⋊#338

H(3,ℤ/4)×C₈₃

Subgroups — #24907217

Conjugacy classes of subgroups

Complements

Maximal subgroups