Group #24905397 — subgroups

Subgroups — #24905397

Order 10624.

134 conjugacy classes of subgroups; only the 41 normal subgroups are shown.

normal subgroups only

This group has 134 conjugacy classes of subgroups — too many to draw. Only its 41 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	#19127427	C₂	1	2	yes	maximal
2	#391×C₈₃	C₂	1	1	yes	maximal, characteristic
3	C₂²⋊(C₈₃⋊C₈)	—	2	2	no	—
4	C₂²×Q₈×C₈₃	C₄	1	1	yes	characteristic
5	C₂×C₄⋊C₄×C₈₃	C₂²	1	1	yes	characteristic
6	C₂×C₄²×C₈₃	C₄	1	1	yes	characteristic
7	#95×C₈₃	—	2	2	no	—
8	C₂×C₈₃⋊C₈	—	4	2	no	—
9	C₂×Q₈×C₈₃	—	2	1	no	—
10	C₂×Q₈×C₈₃	—	4	1	no	—
11	C₂×Q₈×C₈₃	—	4	1	no	—
12	C₂×Q₈×C₈₃	C₈	1	2	yes	complemented
13	C₂²×C₄×C₈₃	—	2	1	no	—
14	C₂²×C₄×C₈₃	—	2	1	no	—
15	C₂²×C₄×C₈₃	C₂×C₄	1	1	yes	characteristic
16	C₂²×C₄×C₈₃	D₄	1	2	yes	—
17	C₄⋊C₄×C₈₃	—	4	1	no	—
18	C₄²×C₈₃	—	4	1	no	—
19	C₂³×C₈₃	C₂²⋊C₄	1	1	yes	characteristic
20	Q₈×C₈₃	—	4	2	no	—
21	Q₈×C₈₃	—	4	2	no	—
22	C₂×C₄×C₈₃	—	2	1	no	—
23	C₂×C₄×C₈₃	—	2	1	no	—
24	C₂×C₄×C₈₃	—	2	2	no	—
25	C₂×C₄×C₈₃	—	2	2	no	—
26	C₂×C₄×C₈₃	—	2	2	no	—
27	C₂×C₄×C₈₃	—	4	1	no	—
28	C₂×C₄×C₈₃	—	4	1	no	—
29	C₂×C₄×C₈₃	—	4	1	no	—
30	C₂×C₄×C₈₃	—	4	1	no	—
31	C₂×C₄×C₈₃	—	4	2	no	—
32	C₂×C₄×C₈₃	C₂×C₈	1	1	yes	characteristic
33	C₂×C₄×C₈₃	M₁₆	1	1	yes	characteristic
34	C₈₃⋊C₈	—	8	2	no	—
35	C₂²×C₈₃	—	2	1	no	—
36	C₂²×C₈₃	—	2	1	no	—
37	C₂²×C₈₃	C₂²⋊C₈	1	1	yes	characteristic
38	C₂²×C₈₃	#99	1	1	yes	characteristic
39	C₂²×C₈₃	#100	1	1	yes	characteristic
40	C₄×C₈₃	—	2	1	no	—
41	C₄×C₈₃	—	2	1	no	—
42	C₄×C₈₃	—	4	2	no	—
43	C₄×C₈₃	—	4	2	no	—
44	C₄×C₈₃	—	4	2	no	—
45	C₂×C₈₃	—	2	1	no	—
46	C₂×C₈₃	—	2	1	no	—
47	C₂×C₈₃	#323	1	1	yes	characteristic
48	C₂×C₈₃	#354	1	1	yes	characteristic
49	C₂×C₈₃	#356	1	1	yes	characteristic
50	#1319	—	83	1	no	maximal, Sylow
51	C₈₃	#1319	1	1	yes	characteristic, complemented, Sylow
52	#391	D₈₃	1	1	yes	characteristic
53	#324	—	83	2	no	—
54	C₂²×Q₈	Dic₈₃	1	1	yes	characteristic
55	C₂×C₄⋊C₄	C₂×D₈₃	1	1	yes	characteristic
56	C₂×C₄²	Dic₈₃	1	1	yes	characteristic
57	C₂²⋊C₈	—	166	2	no	—
58	#95	—	2	2	no	—
59	C₂×Q₈	—	2	1	no	—
60	C₂×Q₈	—	4	1	no	—
61	C₂×Q₈	—	4	1	no	—
62	C₂×Q₈	C₈₃⋊C₈	1	2	yes	complemented
63	C₂²×C₄	—	2	1	no	—
64	C₂²×C₄	—	2	1	no	—
65	C₂²×C₄	C₂×Dic₈₃	1	1	yes	characteristic
66	C₂²×C₄	C₂²⋊D₈₃	1	2	yes	—
67	C₂×C₈	—	332	2	no	—
68	C₄⋊C₄	—	4	1	no	—
69	C₄²	—	4	1	no	—
70	C₂³	C₂²⋊Dic₈₃	1	1	yes	characteristic
71	Q₈	—	4	2	no	—
72	Q₈	—	4	2	no	—
73	C₂×C₄	—	2	1	no	—
74	C₂×C₄	—	2	1	no	—
75	C₂×C₄	—	2	2	no	—
76	C₂×C₄	—	2	2	no	—
77	C₂×C₄	—	2	2	no	—
78	C₂×C₄	—	4	1	no	—
79	C₂×C₄	—	4	1	no	—
80	C₂×C₄	—	4	1	no	—
81	C₂×C₄	—	4	1	no	—
82	C₂×C₄	—	4	2	no	—
83	C₂×C₄	C₂×C₈₃⋊C₈	1	1	yes	characteristic
84	C₂×C₄	#13050574	1	1	yes	characteristic
85	C₈	—	664	2	no	—
86	C₂²	—	2	1	no	—
87	C₂²	—	2	1	no	—
88	C₂²	C₂²⋊(C₈₃⋊C₈)	1	1	yes	characteristic
89	C₂²	#14581707	1	1	yes	characteristic
90	C₂²	#14581709	1	1	yes	characteristic
91	C₄	—	2	1	no	—
92	C₄	—	2	1	no	—
93	C₄	—	4	2	no	—
94	C₄	—	4	2	no	—
95	C₄	—	4	2	no	—
96	C₂	—	2	1	no	—
97	C₂	—	2	1	no	—
98	C₂	#19127425	1	1	yes	characteristic
99	C₂	#19127496	1	1	yes	characteristic
100	C₂	#19127500	1	1	yes	characteristic

subgroup

G/H

class size

multiplicity

normal?

properties

#19127427

C₂

yes

maximal

#391×C₈₃

C₂

yes

maximal, characteristic

C₂²⋊(C₈₃⋊C₈)

—

C₂²×Q₈×C₈₃

C₄

yes

characteristic

C₂×C₄⋊C₄×C₈₃

C₂²

yes

characteristic

C₂×C₄²×C₈₃

C₄

yes

characteristic

#95×C₈₃

—

C₂×C₈₃⋊C₈

—

C₂×Q₈×C₈₃

—

C₂×Q₈×C₈₃

—

C₂×Q₈×C₈₃

—

C₂×Q₈×C₈₃

C₈

yes

complemented

C₂²×C₄×C₈₃

—

C₂²×C₄×C₈₃

—

C₂²×C₄×C₈₃

C₂×C₄

yes

characteristic

C₂²×C₄×C₈₃

D₄

yes

—

C₄⋊C₄×C₈₃

—

C₄²×C₈₃

—

C₂³×C₈₃

C₂²⋊C₄

yes

characteristic

Q₈×C₈₃

—

Q₈×C₈₃

—

C₂×C₄×C₈₃

—

C₂×C₄×C₈₃

—

C₂×C₄×C₈₃

—

C₂×C₄×C₈₃

—

C₂×C₄×C₈₃

—

C₂×C₄×C₈₃

—

C₂×C₄×C₈₃

—

C₂×C₄×C₈₃

—

C₂×C₄×C₈₃

—

C₂×C₄×C₈₃

—

C₂×C₄×C₈₃

C₂×C₈

yes

characteristic

C₂×C₄×C₈₃

M₁₆

yes

characteristic

C₈₃⋊C₈

—

C₂²×C₈₃

—

C₂²×C₈₃

—

C₂²×C₈₃

C₂²⋊C₈

yes

C₂²×C₈₃

yes

C₂²×C₈₃

yes

C₄×C₈₃

—

C₄×C₈₃

—

C₄×C₈₃

—

C₄×C₈₃

—

C₄×C₈₃

—

C₂×C₈₃

—

C₂×C₈₃

—

C₂×C₈₃

yes

C₂×C₈₃

yes

C₂×C₈₃

yes

—

C₈₃

yes

characteristic, complemented, Sylow

#391

D₈₃

yes

characteristic

#324

—

C₂²×Q₈

Dic₈₃

yes

characteristic

C₂×C₄⋊C₄

C₂×D₈₃

yes

characteristic

C₂×C₄²

Dic₈₃

yes

characteristic

C₂²⋊C₈

—

166

—

#95

—

C₂×Q₈

—

C₂×Q₈

—

C₂×Q₈

—

C₂×Q₈

C₈₃⋊C₈

yes

complemented

C₂²×C₄

—

C₂²×C₄

—

C₂²×C₄

C₂×Dic₈₃

yes

characteristic

C₂²×C₄

C₂²⋊D₈₃

yes

—

C₂×C₈

—

332

—

C₄⋊C₄

—

C₄²

—

C₂³

C₂²⋊Dic₈₃

yes

characteristic

Q₈

—

Q₈

—

C₂×C₄

—

C₂×C₄

—

C₂×C₄

—

C₂×C₄

—

C₂×C₄

—

C₂×C₄

—

C₂×C₄

—

C₂×C₄

—

C₂×C₄

—

C₂×C₄

—

C₂×C₄

C₂×C₈₃⋊C₈

yes

characteristic

C₂×C₄

#13050574

yes

characteristic

C₈

—

664

—

C₂²

—

C₂²

—

C₂²

C₂²⋊(C₈₃⋊C₈)

yes

C₂²

yes

C₂²

yes

C₄

—

C₄

—

C₄

—

C₄

—

C₄

—

C₂

—

C₂

—

C₂

yes

C₂

yes

100

C₂

#19127500

yes

characteristic

	subgroup	G/H
center	C₂²	#14581707
commutator	C₂×C₄×C₈₃	C₂×C₈
Frattini	C₂×C₄⋊C₄	C₂×D₈₃
Fitting	#391×C₈₃	C₂
socle	C₂²×C₈₃	#99
prime radical	C₂³×C₈₃	C₂²⋊C₄
cosocle	C₂×C₄⋊C₄×C₈₃	C₂²
hypercenter	#391	D₈₃
hypocenter	C₈₃	#1319
LCS	G ▷ C₂×C₄×C₈₃ ▷ C₂²×C₈₃ ▷ C₂×C₈₃ ▷ C₈₃
UCS	C₁ ◁ C₂² ◁ C₂³ ◁ C₂×C₄⋊C₄ ◁ #391

subgroup

G/H

center

C₂²

#14581707

commutator

C₂×C₄×C₈₃

C₂×C₈

Frattini

C₂×C₄⋊C₄

C₂×D₈₃

Fitting

#391×C₈₃

C₂

socle

C₂²×C₈₃

#99

prime radical

C₂³×C₈₃

C₂²⋊C₄

cosocle

C₂×C₄⋊C₄×C₈₃

C₂²

D₈₃

C₈₃

G ▷ C₂×C₄×C₈₃ ▷ C₂²×C₈₃ ▷ C₂×C₈₃ ▷ C₈₃

UCS

C₁ ◁ C₂² ◁ C₂³ ◁ C₂×C₄⋊C₄ ◁ #391

N	H ≅ G/N	type
12. C₂×Q₈×C₈₃	C₈	semidirect
51. C₈₃	#1319	semidirect
62. C₂×Q₈	C₈₃⋊C₈	semidirect

H ≅ G/N

type

12. C₂×Q₈×C₈₃

C₈

semidirect

51. C₈₃

#1319

semidirect

62. C₂×Q₈

C₈₃⋊C₈

semidirect

Subgroups — #24905397

Conjugacy classes of subgroups

Complements