Group #24906258 — subgroups

Subgroups — D₈×C₈₃⋊C₈ #24906258

Order 10624.

196 conjugacy classes of subgroups; only the 89 normal subgroups are shown.

normal subgroups only

This group has 196 conjugacy classes of subgroups — too many to draw. Only its 89 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	D₄×C₈₃⋊C₈	C₂	1	2	yes	maximal, complemented
2	#19127454	C₂	1	1	yes	maximal, characteristic, complemented
3	D₄⋊(C₈₃⋊C₈)	C₂	1	2	yes	maximal, complemented
4	C₈×C₈₃⋊C₈	C₂	1	1	yes	maximal, characteristic, complemented
5	C₄×D₈×C₈₃	C₂	1	1	yes	maximal, characteristic
6	C₂²×C₈₃⋊C₈	—	2	2	no	—
7	C₄⋊(C₈₃⋊C₈)	C₂²	1	2	yes	—
8	C₂²⋊(C₈₃⋊C₈)	—	2	2	no	—
9	C₄×C₈₃⋊C₈	C₂²	1	1	yes	characteristic
10	C₄×C₈₃⋊C₈	C₂²	1	1	yes	characteristic
11	C₂×D₈×C₈₃	C₄	1	1	yes	characteristic
12	C₄×D₄×C₈₃	C₂²	1	2	yes	—
13	#107×C₈₃	C₄	1	1	yes	characteristic
14	#102×C₈₃	C₄	1	2	yes	—
15	C₄×C₈×C₈₃	C₂²	1	1	yes	characteristic
16	C₂×C₈₃⋊C₈	—	2	1	no	—
17	C₂×C₈₃⋊C₈	—	2	2	no	—
18	C₂×C₈₃⋊C₈	—	4	4	no	—
19	C₂×C₈₃⋊C₈	D₄	1	1	yes	characteristic
20	C₂×C₈₃⋊C₈	D₄	1	1	yes	characteristic
21	C₂×D₄×C₈₃	C₂×C₄	1	2	yes	—
22	C₂²×C₄×C₈₃	—	2	2	no	—
23	D₈×C₈₃	C₈	1	4	yes	complemented
24	C₂×C₈×C₈₃	C₂×C₄	1	1	yes	characteristic
25	C₂×C₈×C₈₃	C₂×C₄	1	1	yes	characteristic
26	C₄⋊C₄×C₈₃	C₂×C₄	1	2	yes	—
27	C₂²⋊C₄×C₈₃	—	2	2	no	—
28	C₄²×C₈₃	C₂³	1	1	yes	characteristic
29	C₂³×C₈₃	—	2	2	no	—
30	D₄×C₈₃	—	2	2	no	—
31	D₄×C₈₃	C₂×C₈	1	4	yes	complemented
32	C₂×C₄×C₈₃	—	2	2	no	—
33	C₂×C₄×C₈₃	—	4	2	no	—
34	C₂×C₄×C₈₃	—	4	2	no	—
35	C₂×C₄×C₈₃	C₂²×C₄	1	1	yes	characteristic
36	C₂×C₄×C₈₃	C₂×D₄	1	1	yes	characteristic
37	C₂×C₄×C₈₃	#41	1	1	yes	characteristic
38	C₈×C₈₃	—	2	1	no	—
39	C₈×C₈₃	C₂×C₈	1	2	yes	complemented
40	C₈₃⋊C₈	—	2	2	no	—
41	C₈₃⋊C₈	—	2	1	no	—
42	C₈₃⋊C₈	—	4	2	no	—
43	C₈₃⋊C₈	D₈	1	2	yes	direct product factor
44	C₂²×C₈₃	—	2	4	no	—
45	C₂²×C₈₃	—	4	2	no	—
46	C₂²×C₈₃	—	4	2	no	—
47	C₂²×C₈₃	C₄×D₄	1	1	yes	characteristic
48	C₄×C₈₃	—	2	1	no	—
49	C₄×C₈₃	—	4	2	no	—
50	C₄×C₈₃	C₂²×C₈	1	1	yes	characteristic
51	C₄×C₈₃	#131	1	1	yes	characteristic
52	C₄×C₈₃	C₂×D₈	1	1	yes	characteristic
53	C₄×C₈₃	#135	1	1	yes	characteristic
54	C₂×C₈₃	—	4	4	no	—
55	C₂×C₈₃	C₈×D₄	1	1	yes	characteristic
56	C₂×C₈₃	C₄×D₈	1	1	yes	characteristic
57	C₂×C₈₃	#443	1	1	yes	characteristic
58	C₈×D₈	—	83	1	no	maximal, Sylow
59	C₈₃	C₈×D₈	1	1	yes	characteristic, complemented, Sylow
60	C₄×D₈	D₈₃	1	1	yes	characteristic
61	C₈×D₄	—	83	2	no	—
62	#335	—	83	1	no	—
63	D₄⋊C₈	—	83	2	no	—
64	C₈²	—	83	1	no	—
65	C₂×D₈	Dic₈₃	1	1	yes	characteristic
66	C₂²×C₈	—	166	2	no	—
67	C₄×D₄	C₂×D₈₃	1	2	yes	—
68	#107	Dic₈₃	1	1	yes	characteristic
69	C₄⋊C₈	—	83	2	no	—
70	#102	Dic₈₃	1	2	yes	—
71	C₂²⋊C₈	—	166	2	no	—
72	C₄×C₈	—	83	1	no	—
73	C₄×C₈	—	83	1	no	—
74	C₄×C₈	C₂×D₈₃	1	1	yes	characteristic
75	C₂×D₄	C₂×Dic₈₃	1	2	yes	—
76	C₂²×C₄	—	2	2	no	—
77	D₈	C₈₃⋊C₈	1	4	yes	direct product factor
78	C₂×C₈	—	332	4	no	—
79	C₂×C₈	—	83	1	no	—
80	C₂×C₈	—	83	1	no	—
81	C₂×C₈	—	166	2	no	—
82	C₂×C₈	—	166	1	no	—
83	C₂×C₈	C₂×Dic₈₃	1	1	yes	characteristic
84	C₂×C₈	C₂×Dic₈₃	1	1	yes	characteristic
85	C₄⋊C₄	C₂×Dic₈₃	1	2	yes	—
86	C₂²⋊C₄	—	2	2	no	—
87	C₄²	C₂²×D₈₃	1	1	yes	characteristic
88	C₂³	—	2	2	no	—
89	D₄	—	2	2	no	—
90	D₄	C₂×C₈₃⋊C₈	1	4	yes	complemented
91	C₂×C₄	—	2	2	no	—
92	C₂×C₄	—	4	2	no	—
93	C₂×C₄	—	4	2	no	—
94	C₂×C₄	C₂²×Dic₈₃	1	1	yes	characteristic
95	C₂×C₄	D₄×D₈₃	1	1	yes	characteristic
96	C₂×C₄	#13050589	1	1	yes	characteristic
97	C₈	—	2	1	no	—
98	C₈	—	332	2	no	—
99	C₈	—	83	2	no	—
100	C₈	—	166	2	no	—
101	C₈	—	166	1	no	—
102	C₈	C₂×C₈₃⋊C₈	1	2	yes	complemented
103	C₂²	—	2	4	no	—
104	C₂²	—	4	2	no	—
105	C₂²	—	4	2	no	—
106	C₂²	D₄×Dic₈₃	1	1	yes	characteristic
107	C₄	—	2	1	no	—
108	C₄	—	4	2	no	—
109	C₄	C₂²×C₈₃⋊C₈	1	1	yes	characteristic
110	C₄	#14581799	1	1	yes	characteristic
111	C₄	D₈×D₈₃	1	1	yes	characteristic
112	C₄	#14581811	1	1	yes	characteristic
113	C₂	—	4	4	no	—
114	C₂	D₄×C₈₃⋊C₈	1	1	yes	characteristic
115	C₂	D₈×Dic₈₃	1	1	yes	characteristic
116	C₂	#19127818	1	1	yes	characteristic

subgroup

G/H

class size

multiplicity

normal?

properties

D₄×C₈₃⋊C₈

C₂

yes

maximal, complemented

#19127454

C₂

yes

maximal, characteristic, complemented

D₄⋊(C₈₃⋊C₈)

C₂

yes

maximal, complemented

C₈×C₈₃⋊C₈

C₂

yes

maximal, characteristic, complemented

C₄×D₈×C₈₃

C₂

yes

maximal, characteristic

C₂²×C₈₃⋊C₈

—

C₄⋊(C₈₃⋊C₈)

C₂²

yes

—

C₂²⋊(C₈₃⋊C₈)

—

C₄×C₈₃⋊C₈

C₂²

yes

characteristic

C₄×C₈₃⋊C₈

C₂²

yes

characteristic

C₂×D₈×C₈₃

C₄

yes

characteristic

C₄×D₄×C₈₃

C₂²

yes

—

#107×C₈₃

C₄

yes

characteristic

#102×C₈₃

C₄

yes

—

C₄×C₈×C₈₃

C₂²

yes

characteristic

C₂×C₈₃⋊C₈

—

C₂×C₈₃⋊C₈

—

C₂×C₈₃⋊C₈

—

C₂×C₈₃⋊C₈

D₄

yes

characteristic

C₂×C₈₃⋊C₈

D₄

yes

characteristic

C₂×D₄×C₈₃

C₂×C₄

yes

—

C₂²×C₄×C₈₃

—

D₈×C₈₃

C₈

yes

complemented

C₂×C₈×C₈₃

C₂×C₄

yes

characteristic

C₂×C₈×C₈₃

C₂×C₄

yes

characteristic

C₄⋊C₄×C₈₃

C₂×C₄

yes

—

C₂²⋊C₄×C₈₃

—

C₄²×C₈₃

C₂³

yes

characteristic

C₂³×C₈₃

—

D₄×C₈₃

—

D₄×C₈₃

C₂×C₈

yes

complemented

C₂×C₄×C₈₃

—

C₂×C₄×C₈₃

—

C₂×C₄×C₈₃

—

C₂×C₄×C₈₃

C₂²×C₄

yes

characteristic

C₂×C₄×C₈₃

C₂×D₄

yes

characteristic

C₂×C₄×C₈₃

#41

yes

characteristic

C₈×C₈₃

—

C₈×C₈₃

C₂×C₈

yes

complemented

C₈₃⋊C₈

—

C₈₃⋊C₈

—

C₈₃⋊C₈

—

C₈₃⋊C₈

D₈

yes

direct product factor

C₂²×C₈₃

—

C₂²×C₈₃

—

C₂²×C₈₃

—

C₂²×C₈₃

C₄×D₄

yes

characteristic

C₄×C₈₃

—

C₄×C₈₃

—

C₄×C₈₃

C₂²×C₈

yes

characteristic

C₄×C₈₃

#131

yes

characteristic

C₄×C₈₃

C₂×D₈

yes

characteristic

C₄×C₈₃

#135

yes

characteristic

C₂×C₈₃

—

C₂×C₈₃

C₈×D₄

yes

characteristic

C₂×C₈₃

C₄×D₈

yes

characteristic

C₂×C₈₃

#443

yes

characteristic

C₈×D₈

—

maximal, Sylow

C₈₃

C₈×D₈

yes

characteristic, complemented, Sylow

C₄×D₈

D₈₃

yes

characteristic

C₈×D₄

—

#335

—

D₄⋊C₈

—

C₈²

—

C₂×D₈

Dic₈₃

yes

characteristic

C₂²×C₈

—

166

—

C₄×D₄

C₂×D₈₃

yes

—

#107

Dic₈₃

yes

characteristic

C₄⋊C₈

—

#102

Dic₈₃

yes

—

C₂²⋊C₈

—

166

—

C₄×C₈

—

C₄×C₈

—

C₄×C₈

C₂×D₈₃

yes

characteristic

C₂×D₄

C₂×Dic₈₃

yes

—

C₂²×C₄

—

D₈

C₈₃⋊C₈

yes

direct product factor

C₂×C₈

—

332

—

C₂×C₈

—

C₂×C₈

—

C₂×C₈

—

166

—

C₂×C₈

—

166

—

C₂×C₈

C₂×Dic₈₃

yes

characteristic

C₂×C₈

C₂×Dic₈₃

yes

characteristic

C₄⋊C₄

C₂×Dic₈₃

yes

—

C₂²⋊C₄

—

C₄²

C₂²×D₈₃

yes

characteristic

C₂³

—

D₄

—

D₄

C₂×C₈₃⋊C₈

yes

complemented

C₂×C₄

—

C₂×C₄

—

C₂×C₄

—

C₂×C₄

C₂²×Dic₈₃

yes

characteristic

C₂×C₄

D₄×D₈₃

yes

characteristic

C₂×C₄

#13050589

yes

characteristic

C₈

—

C₈

—

332

—

C₈

—

100

C₈

—

166

—

101

C₈

—

166

—

102

C₈

C₂×C₈₃⋊C₈

yes

complemented

103

C₂²

—

104

C₂²

—

105

C₂²

—

106

C₂²

D₄×Dic₈₃

yes

characteristic

107

C₄

—

108

C₄

—

109

C₄

C₂²×C₈₃⋊C₈

yes

characteristic

110

C₄

#14581799

yes

characteristic

111

C₄

D₈×D₈₃

yes

characteristic

112

C₄

#14581811

yes

characteristic

113

C₂

—

114

C₂

D₄×C₈₃⋊C₈

yes

characteristic

115

C₂

D₈×Dic₈₃

yes

characteristic

116

C₂

#19127818

yes

characteristic

	subgroup	G/H
center	C₂×C₄	D₄×D₈₃
commutator	C₄×C₈₃	C₂²×C₈
Frattini	C₄²	C₂²×D₈₃
Fitting	C₄×D₈×C₈₃	C₂
socle	C₂²×C₈₃	C₄×D₄
prime radical	C₂×D₈×C₈₃	C₄
cosocle	C₄²×C₈₃	C₂³
hypercenter	C₄×D₈	D₈₃
hypocenter	C₈₃	C₈×D₈
LCS	G ▷ C₄×C₈₃ ▷ C₂×C₈₃ ▷ C₈₃
UCS	C₁ ◁ C₂×C₄ ◁ C₄² ◁ C₄×D₈

subgroup

G/H

center

C₂×C₄

D₄×D₈₃

commutator

C₄×C₈₃

C₂²×C₈

Frattini

C₄²

C₂²×D₈₃

Fitting

C₄×D₈×C₈₃

C₂

socle

C₂²×C₈₃

C₄×D₄

prime radical

C₂×D₈×C₈₃

C₄

cosocle

C₄²×C₈₃

C₂³

hypercenter

C₄×D₈

D₈₃

hypocenter

C₈₃

C₈×D₈

LCS

G ▷ C₄×C₈₃ ▷ C₂×C₈₃ ▷ C₈₃

UCS

C₁ ◁ C₂×C₄ ◁ C₄² ◁ C₄×D₈

N	H ≅ G/N	type
1. D₄×C₈₃⋊C₈	C₂	semidirect
2. #19127454	C₂	semidirect
3. D₄⋊(C₈₃⋊C₈)	C₂	semidirect
4. C₈×C₈₃⋊C₈	C₂	semidirect
23. D₈×C₈₃	C₈	semidirect
31. D₄×C₈₃	C₂×C₈	semidirect
39. C₈×C₈₃	C₂×C₈	semidirect
43. C₈₃⋊C₈	D₈	direct (N × H)
59. C₈₃	C₈×D₈	semidirect
77. D₈	C₈₃⋊C₈	direct (N × H)
90. D₄	C₂×C₈₃⋊C₈	semidirect
102. C₈	C₂×C₈₃⋊C₈	semidirect

H ≅ G/N

type

1. D₄×C₈₃⋊C₈

C₂

semidirect

2. #19127454

C₂

semidirect

3. D₄⋊(C₈₃⋊C₈)

C₂

semidirect

4. C₈×C₈₃⋊C₈

C₂

semidirect

23. D₈×C₈₃

C₈

semidirect

31. D₄×C₈₃

C₂×C₈

semidirect

39. C₈×C₈₃

C₂×C₈

semidirect

43. C₈₃⋊C₈

D₈

direct (N × H)

59. C₈₃

C₈×D₈

semidirect

77. D₈

C₈₃⋊C₈

direct (N × H)

90. D₄

C₂×C₈₃⋊C₈

semidirect

102. C₈

C₂×C₈₃⋊C₈

semidirect

Subgroups — D₈×C₈₃⋊C₈ #24906258

Conjugacy classes of subgroups

Complements

Subgroups — D8×C83⋊C8 #24906258

Conjugacy classes of subgroups

Complements

Subgroups — D₈×C₈₃⋊C₈ #24906258