Group #24915859 — subgroups

Subgroups — #24915859

Order 10624.

648 conjugacy classes of subgroups; only the 245 normal subgroups are shown.

normal subgroups only

This group has 648 conjugacy classes of subgroups — too many to draw. Only its 245 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×#14581792	C₂	1	1	yes	maximal, characteristic
2	C₂×#14581790	C₂	1	4	yes	maximal
3	#19127570	C₂	1	8	yes	maximal
4	C₄²×Dic₈₃	C₂	1	1	yes	maximal, characteristic
5	C₂×C₄⋊Q₈×C₈₃	C₂	1	1	yes	maximal, characteristic
6	C₂²×Dic₁₆₆	C₂²	1	2	yes	—
7	#14581792	—	2	4	no	—
8	#14581790	—	2	16	no	—
9	C₂×C₄⋊Dic₈₃	C₂²	1	4	yes	—
10	C₂×Dic₈₃⋊C₄	C₂²	1	16	yes	—
11	C₂×C₄×Dic₈₃	C₂²	1	2	yes	—
12	C₂×C₄×Dic₈₃	C₂²	1	4	yes	—
13	C₂²×Q₈×C₈₃	C₂²	1	2	yes	—
14	C₄⋊Q₈×C₈₃	—	2	4	no	—
15	C₂×C₄⋊C₄×C₈₃	C₂²	1	4	yes	—
16	C₂×C₄²×C₈₃	C₂²	1	1	yes	characteristic
17	C₂×Dic₁₆₆	—	2	8	no	—
18	C₂×Dic₁₆₆	—	2	4	no	—
19	C₂²×Dic₈₃	C₂³	1	4	yes	—
20	C₂²×Dic₈₃	C₂³	1	4	yes	—
21	C₄⋊Dic₈₃	—	2	8	no	—
22	Dic₈₃⋊C₄	—	2	32	no	—
23	C₄×Dic₈₃	D₄	1	8	yes	—
24	C₄×Dic₈₃	Q₈	1	16	yes	complemented
25	C₂×Q₈×C₈₃	—	2	8	no	—
26	C₂×Q₈×C₈₃	—	2	4	no	—
27	C₂²×C₄×C₈₃	C₂³	1	4	yes	—
28	C₂²×C₄×C₈₃	C₂³	1	1	yes	characteristic
29	C₂²×C₄×C₈₃	C₂³	1	2	yes	—
30	C₄⋊C₄×C₈₃	—	2	8	no	—
31	C₄²×C₈₃	D₄	1	4	yes	—
32	C₂³×C₈₃	C₂⁴	1	1	yes	characteristic
33	Q₈×C₈₃	—	4	8	no	—
34	C₂×C₄×C₈₃	—	2	4	no	—
35	C₂×C₄×C₈₃	—	2	8	no	—
36	C₂×C₄×C₈₃	C₂×D₄	1	4	yes	—
37	C₂×C₄×C₈₃	C₂×D₄	1	2	yes	—
38	C₂×C₄×C₈₃	C₂×D₄	1	4	yes	—
39	C₂×C₄×C₈₃	C₂×Q₈	1	8	yes	—
40	C₂×Dic₈₃	—	2	8	no	—
41	C₂×Dic₈₃	—	2	4	no	—
42	C₂×Dic₈₃	C₂×D₄	1	8	yes	—
43	C₂×Dic₈₃	C₂×Q₈	1	16	yes	—
44	Dic₁₆₆	—	4	8	no	—
45	C₂²×C₈₃	C₂²×D₄	1	2	yes	—
46	C₂²×C₈₃	C₂²×D₄	1	1	yes	characteristic
47	C₂²×C₈₃	C₂²×Q₈	1	4	yes	—
48	C₄×C₈₃	—	4	4	no	—
49	C₄×C₈₃	#127	1	4	yes	—
50	C₄×C₈₃	C₄⋊Q₈	1	8	yes	complemented
51	Dic₈₃	—	4	4	no	—
52	Dic₈₃	C₄⋊Q₈	1	16	yes	complemented
53	C₂×C₈₃	C₂×#127	1	1	yes	characteristic
54	C₂×C₈₃	C₂×C₄⋊Q₈	1	4	yes	—
55	C₂×C₈₃	C₂×C₄⋊Q₈	1	2	yes	—
56	#2868	—	83	1	no	maximal, Sylow
57	C₈₃	#2868	1	1	yes	characteristic, complemented, Sylow
58	C₂×C₄⋊Q₈	—	83	4	no	—
59	C₂×C₄⋊Q₈	—	83	1	no	—
60	C₂×C₄⋊Q₈	D₈₃	1	1	yes	characteristic
61	#384	—	83	8	no	—
62	C₄³	—	83	1	no	—
63	C₂²×Q₈	—	83	2	no	—
64	C₂²×Q₈	C₂×D₈₃	1	2	yes	—
65	C₄⋊Q₈	—	2	4	no	—
66	C₄⋊Q₈	—	166	16	no	—
67	C₄⋊Q₈	—	166	4	no	—
68	C₂×C₄⋊C₄	—	83	16	no	—
69	C₂×C₄⋊C₄	—	83	4	no	—
70	C₂×C₄⋊C₄	C₂×D₈₃	1	4	yes	—
71	C₂×C₄²	—	83	2	no	—
72	C₂×C₄²	—	83	4	no	—
73	C₂×C₄²	C₂×D₈₃	1	1	yes	characteristic
74	C₂×Q₈	—	2	8	no	—
75	C₂×Q₈	—	2	4	no	—
76	C₂×Q₈	—	166	8	no	—
77	C₂×Q₈	—	166	4	no	—
78	C₂²×C₄	—	83	4	no	—
79	C₂²×C₄	—	83	4	no	—
80	C₂²×C₄	C₂²×D₈₃	1	4	yes	—
81	C₂²×C₄	C₂²×D₈₃	1	1	yes	characteristic
82	C₂²×C₄	C₂²×D₈₃	1	2	yes	—
83	C₄⋊C₄	—	2	8	no	—
84	C₄⋊C₄	—	166	32	no	—
85	C₄⋊C₄	—	166	8	no	—
86	C₄²	—	83	8	no	—
87	C₄²	—	83	16	no	—
88	C₄²	C₂²⋊D₈₃	1	4	yes	—
89	C₂³	C₂³×D₈₃	1	1	yes	characteristic
90	Q₈	—	4	8	no	—
91	Q₈	—	332	8	no	—
92	C₂×C₄	—	2	8	no	—
93	C₂×C₄	—	2	4	no	—
94	C₂×C₄	—	83	8	no	—
95	C₂×C₄	—	83	16	no	—
96	C₂×C₄	—	166	8	no	—
97	C₂×C₄	—	166	4	no	—
98	C₂×C₄	D₄×D₈₃	1	4	yes	—
99	C₂×C₄	C₂×C₂²⋊D₈₃	1	2	yes	—
100	C₂×C₄	C₂×C₂²⋊D₈₃	1	4	yes	—
101	C₂×C₄	Q₈×D₈₃	1	8	yes	—
102	C₂²	C₂×D₄×D₈₃	1	2	yes	—
103	C₂²	C₂²×C₂²⋊D₈₃	1	1	yes	characteristic
104	C₂²	C₂×Q₈×D₈₃	1	4	yes	—
105	C₄	—	4	4	no	—
106	C₄	—	332	4	no	—
107	C₄	—	83	16	no	—
108	C₄	#14581788	1	4	yes	—
109	C₄	#14581790	1	8	yes	complemented
110	C₂	C₂×#14581788	1	1	yes	characteristic
111	C₂	C₄⋊Q₈×D₈₃	1	4	yes	—
112	C₂	C₂×#14581790	1	2	yes	—

subgroup

G/H

class size

multiplicity

normal?

properties

C₂×#14581792

C₂

yes

maximal, characteristic

C₂×#14581790

C₂

yes

maximal

#19127570

C₂

yes

maximal

C₄²×Dic₈₃

C₂

yes

maximal, characteristic

C₂×C₄⋊Q₈×C₈₃

C₂

yes

maximal, characteristic

C₂²×Dic₁₆₆

C₂²

yes

—

#14581792

—

#14581790

—

C₂×C₄⋊Dic₈₃

C₂²

yes

—

C₂×Dic₈₃⋊C₄

C₂²

yes

—

C₂×C₄×Dic₈₃

C₂²

yes

—

C₂×C₄×Dic₈₃

C₂²

yes

—

C₂²×Q₈×C₈₃

C₂²

yes

—

C₄⋊Q₈×C₈₃

—

C₂×C₄⋊C₄×C₈₃

C₂²

yes

—

C₂×C₄²×C₈₃

C₂²

yes

characteristic

C₂×Dic₁₆₆

—

C₂×Dic₁₆₆

—

C₂²×Dic₈₃

C₂³

yes

—

C₂²×Dic₈₃

C₂³

yes

—

C₄⋊Dic₈₃

—

Dic₈₃⋊C₄

—

C₄×Dic₈₃

D₄

yes

—

C₄×Dic₈₃

Q₈

yes

complemented

C₂×Q₈×C₈₃

—

C₂×Q₈×C₈₃

—

C₂²×C₄×C₈₃

C₂³

yes

—

C₂²×C₄×C₈₃

C₂³

yes

characteristic

C₂²×C₄×C₈₃

C₂³

yes

—

C₄⋊C₄×C₈₃

—

C₄²×C₈₃

D₄

yes

—

C₂³×C₈₃

C₂⁴

yes

characteristic

Q₈×C₈₃

—

C₂×C₄×C₈₃

—

C₂×C₄×C₈₃

—

C₂×C₄×C₈₃

C₂×D₄

yes

—

C₂×C₄×C₈₃

C₂×D₄

yes

—

C₂×C₄×C₈₃

C₂×D₄

yes

—

C₂×C₄×C₈₃

C₂×Q₈

yes

—

C₂×Dic₈₃

—

C₂×Dic₈₃

—

C₂×Dic₈₃

C₂×D₄

yes

—

C₂×Dic₈₃

C₂×Q₈

yes

—

Dic₁₆₆

—

C₂²×C₈₃

C₂²×D₄

yes

—

C₂²×C₈₃

C₂²×D₄

yes

characteristic

C₂²×C₈₃

C₂²×Q₈

yes

—

C₄×C₈₃

—

C₄×C₈₃

#127

yes

—

C₄×C₈₃

C₄⋊Q₈

yes

complemented

Dic₈₃

—

Dic₈₃

C₄⋊Q₈

yes

complemented

C₂×C₈₃

C₂×#127

yes

characteristic

C₂×C₈₃

C₂×C₄⋊Q₈

yes

—

C₂×C₈₃

C₂×C₄⋊Q₈

yes

—

#2868

—

maximal, Sylow

C₈₃

#2868

yes

characteristic, complemented, Sylow

C₂×C₄⋊Q₈

—

C₂×C₄⋊Q₈

—

C₂×C₄⋊Q₈

D₈₃

yes

characteristic

#384

—

C₄³

—

C₂²×Q₈

—

C₂²×Q₈

C₂×D₈₃

yes

—

C₄⋊Q₈

—

C₄⋊Q₈

—

166

—

C₄⋊Q₈

—

166

—

C₂×C₄⋊C₄

—

C₂×C₄⋊C₄

—

C₂×C₄⋊C₄

C₂×D₈₃

yes

—

C₂×C₄²

—

C₂×C₄²

—

C₂×C₄²

C₂×D₈₃

yes

characteristic

C₂×Q₈

—

C₂×Q₈

—

C₂×Q₈

—

166

—

C₂×Q₈

—

166

—

C₂²×C₄

—

C₂²×C₄

—

C₂²×C₄

C₂²×D₈₃

yes

—

C₂²×C₄

C₂²×D₈₃

yes

characteristic

C₂²×C₄

C₂²×D₈₃

yes

—

C₄⋊C₄

—

C₄⋊C₄

—

166

—

C₄⋊C₄

—

166

—

C₄²

—

C₄²

—

C₄²

C₂²⋊D₈₃

yes

—

C₂³

C₂³×D₈₃

yes

characteristic

Q₈

—

Q₈

—

332

—

C₂×C₄

—

C₂×C₄

—

C₂×C₄

—

C₂×C₄

—

C₂×C₄

—

166

—

C₂×C₄

—

166

—

C₂×C₄

D₄×D₈₃

yes

—

C₂×C₄

C₂×C₂²⋊D₈₃

yes

—

100

C₂×C₄

C₂×C₂²⋊D₈₃

yes

—

101

C₂×C₄

Q₈×D₈₃

yes

—

102

C₂²

C₂×D₄×D₈₃

yes

—

103

C₂²

C₂²×C₂²⋊D₈₃

yes

characteristic

104

C₂²

C₂×Q₈×D₈₃

yes

—

105

C₄

—

106

C₄

—

332

—

107

C₄

—

108

C₄

#14581788

yes

—

109

C₄

#14581790

yes

complemented

110

C₂

C₂×#14581788

yes

characteristic

111

C₂

C₄⋊Q₈×D₈₃

yes

—

112

C₂

C₂×#14581790

yes

—

	subgroup	G/H
center, Frattini	C₂³	C₂³×D₈₃
commutator, socle, prime radical, cosocle	C₂³×C₈₃	C₂⁴
Fitting	C₂×C₄⋊Q₈×C₈₃	C₂
hypercenter	C₂×C₄⋊Q₈	D₈₃
hypocenter	C₈₃	#2868
LCS	G ▷ C₂³×C₈₃ ▷ C₈₃
UCS	C₁ ◁ C₂³ ◁ C₂×C₄⋊Q₈

subgroup

G/H

center, Frattini

C₂³

C₂³×D₈₃

commutator, socle, prime radical, cosocle

C₂³×C₈₃

C₂⁴

Fitting

C₂×C₄⋊Q₈×C₈₃

C₂

hypercenter

C₂×C₄⋊Q₈

D₈₃

C₈₃

G ▷ C₂³×C₈₃ ▷ C₈₃

C₁ ◁ C₂³ ◁ C₂×C₄⋊Q₈

N	H ≅ G/N	type
24. C₄×Dic₈₃	Q₈	semidirect
50. C₄×C₈₃	C₄⋊Q₈	semidirect
52. Dic₈₃	C₄⋊Q₈	semidirect
57. C₈₃	#2868	semidirect
109. C₄	#14581790	semidirect

H ≅ G/N

type

24. C₄×Dic₈₃

Q₈

semidirect

50. C₄×C₈₃

C₄⋊Q₈

semidirect

52. Dic₈₃

C₄⋊Q₈

semidirect

57. C₈₃

#2868

semidirect

109. C₄

#14581790

semidirect

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

subgroup	index
C₂×#14581792	2
C₂×#14581790	2
#19127570	2
C₄²×Dic₈₃	2
C₂×C₄⋊Q₈×C₈₃	2
#2868	83

subgroup

index

C₂×#14581792

C₂×#14581790

#19127570

C₄²×Dic₈₃

C₂×C₄⋊Q₈×C₈₃

#2868

Subgroups — #24915859

Conjugacy classes of subgroups

Complements

Maximal subgroups