Group #23765248 — subgroups

Subgroups — C₄×D₁₂₆₄ #23765248

Order 10112.

174 conjugacy classes of subgroups; only the 65 normal subgroups are shown.

normal subgroups only

This group has 174 conjugacy classes of subgroups — too many to draw. Only its 65 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×D₁₂₆₄	C₂	1	1	yes	maximal, characteristic
2	C₄×D₆₃₂	C₂	1	2	yes	maximal, complemented
3	#19098389	C₂	1	1	yes	maximal, characteristic, complemented
4	#19098365	C₂	1	2	yes	maximal, complemented
5	C₄×C₁₆×C₇₉	C₂	1	1	yes	maximal, characteristic, complemented
6	C₂×D₆₃₂	C₂²	1	2	yes	—
7	C₄×D₃₁₆	—	2	2	no	—
8	D₁₂₆₄	C₄	1	4	yes	direct product factor
9	#14579095	C₂²	1	2	yes	—
10	#14579083	—	2	2	no	—
11	C₂×C₁₆×C₇₉	C₂²	1	1	yes	characteristic
12	C₂×C₁₆×C₇₉	C₂²	1	1	yes	characteristic
13	C₄×C₈×C₇₉	C₂²	1	1	yes	characteristic
14	C₂×D₃₁₆	—	2	2	no	—
15	C₂×C₄×D₇₉	—	4	2	no	—
16	D₆₃₂	—	2	2	no	—
17	D₆₃₂	C₂×C₄	1	4	yes	complemented
18	C₄⋊Dic₇₉	—	2	2	no	—
19	(C₂×D₇₉)⋊C₄	—	4	2	no	—
20	C₂×C₈×C₇₉	D₄	1	1	yes	characteristic
21	C₂×C₈×C₇₉	C₂³	1	1	yes	characteristic
22	C₄²×C₇₉	D₄	1	1	yes	characteristic
23	C₁₆×C₇₉	—	2	1	no	—
24	C₁₆×C₇₉	C₂×C₄	1	2	yes	complemented
25	C₂²×D₇₉	—	4	2	no	—
26	C₂×C₄×C₇₉	D₈	1	1	yes	characteristic
27	C₂×C₄×C₇₉	D₈	1	1	yes	characteristic
28	C₂×C₄×C₇₉	C₂×D₄	1	1	yes	characteristic
29	C₂×Dic₇₉	—	4	2	no	—
30	D₃₁₆	—	2	4	no	—
31	D₃₁₆	—	4	2	no	—
32	C₄×D₇₉	—	8	4	no	—
33	C₈×C₇₉	—	2	1	no	—
34	C₈×C₇₉	C₂²×C₄	1	1	yes	characteristic
35	C₈×C₇₉	#41	1	1	yes	characteristic
36	C₂²×C₇₉	C₂×D₈	1	1	yes	characteristic
37	C₂×D₇₉	—	4	4	no	—
38	C₂×D₇₉	—	8	2	no	—
39	C₂×D₇₉	—	8	2	no	—
40	C₄×C₇₉	—	2	1	no	—
41	C₄×C₇₉	D₁₆	1	2	yes	complemented
42	C₄×C₇₉	C₄×D₄	1	1	yes	characteristic
43	C₄×C₇₉	#135	1	1	yes	characteristic
44	Dic₇₉	—	8	2	no	—
45	C₂×C₇₉	C₄×D₈	1	1	yes	characteristic
46	C₂×C₇₉	C₂×D₁₆	1	1	yes	characteristic
47	C₂×C₇₉	#508	1	1	yes	characteristic
48	D₇₉	—	8	4	no	—
49	C₄×D₁₆	—	79	1	no	maximal, Sylow
50	C₇₉	C₄×D₁₆	1	1	yes	characteristic, complemented, Sylow
51	C₂×D₁₆	—	79	1	no	—
52	C₄×D₈	—	79	2	no	—
53	#366	—	79	1	no	—
54	#357	—	79	2	no	—
55	C₄×C₁₆	D₇₉	1	1	yes	characteristic, complemented
56	C₂×D₈	—	79	2	no	—
57	C₄×D₄	—	158	2	no	—
58	D₁₆	—	79	4	no	—
59	C₂×C₁₆	C₂×D₇₉	1	1	yes	characteristic
60	C₂×C₁₆	C₂×D₇₉	1	1	yes	characteristic
61	#107	—	79	2	no	—
62	#102	—	158	2	no	—
63	C₄×C₈	C₂×D₇₉	1	1	yes	characteristic
64	C₂×D₄	—	158	2	no	—
65	C₂²×C₄	—	316	2	no	—
66	D₈	—	79	4	no	—
67	D₈	—	158	2	no	—
68	C₂×C₈	D₃₁₆	1	1	yes	characteristic
69	C₂×C₈	C₂²×D₇₉	1	1	yes	characteristic
70	C₄⋊C₄	—	158	2	no	—
71	C₂²⋊C₄	—	316	2	no	—
72	C₄²	D₃₁₆	1	1	yes	characteristic
73	C₁₆	—	2	1	no	—
74	C₁₆	C₄×D₇₉	1	2	yes	complemented
75	C₂³	—	316	2	no	—
76	D₄	—	316	2	no	—
77	D₄	—	158	4	no	—
78	C₂×C₄	—	316	2	no	—
79	C₂×C₄	—	632	4	no	—
80	C₂×C₄	D₆₃₂	1	1	yes	characteristic
81	C₂×C₄	D₆₃₂	1	1	yes	characteristic
82	C₂×C₄	C₂×D₃₁₆	1	1	yes	characteristic
83	C₈	—	2	1	no	—
84	C₈	C₂×C₄×D₇₉	1	1	yes	characteristic
85	C₈	#11929624	1	1	yes	characteristic
86	C₂²	—	316	4	no	—
87	C₂²	—	632	2	no	—
88	C₂²	—	632	2	no	—
89	C₂²	C₂×D₆₃₂	1	1	yes	characteristic
90	C₄	—	2	1	no	—
91	C₄	—	632	2	no	—
92	C₄	D₁₂₆₄	1	2	yes	direct product factor
93	C₄	C₄×D₃₁₆	1	1	yes	characteristic
94	C₄	#14579183	1	1	yes	characteristic
95	C₂	—	632	4	no	—
96	C₂	C₄×D₆₃₂	1	1	yes	characteristic
97	C₂	C₂×D₁₂₆₄	1	1	yes	characteristic
98	C₂	#19098995	1	1	yes	characteristic

subgroup

G/H

class size

multiplicity

normal?

properties

C₂×D₁₂₆₄

C₂

yes

maximal, characteristic

C₄×D₆₃₂

C₂

yes

maximal, complemented

#19098389

C₂

yes

maximal, characteristic, complemented

#19098365

C₂

yes

maximal, complemented

C₄×C₁₆×C₇₉

C₂

yes

maximal, characteristic, complemented

C₂×D₆₃₂

C₂²

yes

—

C₄×D₃₁₆

—

D₁₂₆₄

C₄

yes

direct product factor

#14579095

C₂²

yes

—

#14579083

—

C₂×C₁₆×C₇₉

C₂²

yes

characteristic

C₂×C₁₆×C₇₉

C₂²

yes

characteristic

C₄×C₈×C₇₉

C₂²

yes

characteristic

C₂×D₃₁₆

—

C₂×C₄×D₇₉

—

D₆₃₂

—

D₆₃₂

C₂×C₄

yes

complemented

C₄⋊Dic₇₉

—

(C₂×D₇₉)⋊C₄

—

C₂×C₈×C₇₉

D₄

yes

characteristic

C₂×C₈×C₇₉

C₂³

yes

characteristic

C₄²×C₇₉

D₄

yes

characteristic

C₁₆×C₇₉

—

C₁₆×C₇₉

C₂×C₄

yes

complemented

C₂²×D₇₉

—

C₂×C₄×C₇₉

D₈

yes

characteristic

C₂×C₄×C₇₉

D₈

yes

characteristic

C₂×C₄×C₇₉

C₂×D₄

yes

characteristic

C₂×Dic₇₉

—

D₃₁₆

—

D₃₁₆

—

C₄×D₇₉

—

C₈×C₇₉

—

C₈×C₇₉

C₂²×C₄

yes

characteristic

C₈×C₇₉

#41

yes

characteristic

C₂²×C₇₉

C₂×D₈

yes

characteristic

C₂×D₇₉

—

C₂×D₇₉

—

C₂×D₇₉

—

C₄×C₇₉

—

C₄×C₇₉

D₁₆

yes

complemented

C₄×C₇₉

C₄×D₄

yes

characteristic

C₄×C₇₉

#135

yes

characteristic

Dic₇₉

—

C₂×C₇₉

C₄×D₈

yes

characteristic

C₂×C₇₉

C₂×D₁₆

yes

characteristic

C₂×C₇₉

#508

yes

characteristic

D₇₉

—

C₄×D₁₆

—

maximal, Sylow

C₇₉

C₄×D₁₆

yes

characteristic, complemented, Sylow

C₂×D₁₆

—

C₄×D₈

—

#366

—

#357

—

C₄×C₁₆

D₇₉

yes

characteristic, complemented

C₂×D₈

—

C₄×D₄

—

158

—

D₁₆

—

C₂×C₁₆

C₂×D₇₉

yes

characteristic

C₂×C₁₆

C₂×D₇₉

yes

characteristic

#107

—

#102

—

158

—

C₄×C₈

C₂×D₇₉

yes

characteristic

C₂×D₄

—

158

—

C₂²×C₄

—

316

—

D₈

—

D₈

—

158

—

C₂×C₈

D₃₁₆

yes

characteristic

C₂×C₈

C₂²×D₇₉

yes

characteristic

C₄⋊C₄

—

158

—

C₂²⋊C₄

—

316

—

C₄²

D₃₁₆

yes

characteristic

C₁₆

—

C₁₆

C₄×D₇₉

yes

complemented

C₂³

—

316

—

D₄

—

316

—

D₄

—

158

—

C₂×C₄

—

316

—

C₂×C₄

—

632

—

C₂×C₄

D₆₃₂

yes

characteristic

C₂×C₄

D₆₃₂

yes

characteristic

C₂×C₄

C₂×D₃₁₆

yes

characteristic

C₈

—

C₈

C₂×C₄×D₇₉

yes

characteristic

C₈

#11929624

yes

characteristic

C₂²

—

316

—

C₂²

—

632

—

C₂²

—

632

—

C₂²

C₂×D₆₃₂

yes

characteristic

C₄

—

C₄

—

632

—

C₄

D₁₂₆₄

yes

direct product factor

C₄

C₄×D₃₁₆

yes

characteristic

C₄

#14579183

yes

characteristic

C₂

—

632

—

C₂

C₄×D₆₃₂

yes

characteristic

C₂

C₂×D₁₂₆₄

yes

characteristic

C₂

#19098995

yes

characteristic

	subgroup	G/H
center	C₂×C₄	D₆₃₂
commutator	C₈×C₇₉	C₂²×C₄
Frattini	C₂×C₈	C₂²×D₇₉
Fitting	C₄×C₁₆×C₇₉	C₂
socle	C₂²×C₇₉	C₂×D₈
prime radical	C₂×D₁₂₆₄	C₂
cosocle	C₂×C₈×C₇₉	C₂³
hypercenter	C₄×C₁₆	D₇₉
hypocenter	C₇₉	C₄×D₁₆
LCS	G ▷ C₈×C₇₉ ▷ C₄×C₇₉ ▷ C₂×C₇₉ ▷ C₇₉
UCS	C₁ ◁ C₂×C₄ ◁ C₄² ◁ C₄×C₈ ◁ C₄×C₁₆

subgroup

G/H

center

C₂×C₄

D₆₃₂

commutator

C₈×C₇₉

C₂²×C₄

Frattini

C₂×C₈

C₂²×D₇₉

Fitting

C₄×C₁₆×C₇₉

C₂

socle

C₂²×C₇₉

C₂×D₈

prime radical

C₂×D₁₂₆₄

C₂

cosocle

C₂×C₈×C₇₉

C₂³

hypercenter

C₄×C₁₆

D₇₉

hypocenter

C₇₉

C₄×D₁₆

LCS

G ▷ C₈×C₇₉ ▷ C₄×C₇₉ ▷ C₂×C₇₉ ▷ C₇₉

UCS

C₁ ◁ C₂×C₄ ◁ C₄² ◁ C₄×C₈ ◁ C₄×C₁₆

N	H ≅ G/N	type
2. C₄×D₆₃₂	C₂	semidirect
3. #19098389	C₂	semidirect
4. #19098365	C₂	semidirect
5. C₄×C₁₆×C₇₉	C₂	semidirect
8. D₁₂₆₄	C₄	direct (N × H)
17. D₆₃₂	C₂×C₄	semidirect
24. C₁₆×C₇₉	C₂×C₄	semidirect
41. C₄×C₇₉	D₁₆	semidirect
50. C₇₉	C₄×D₁₆	semidirect
55. C₄×C₁₆	D₇₉	semidirect
74. C₁₆	C₄×D₇₉	semidirect
92. C₄	D₁₂₆₄	direct (N × H)

H ≅ G/N

type

2. C₄×D₆₃₂

C₂

semidirect

3. #19098389

C₂

semidirect

4. #19098365

C₂

semidirect

5. C₄×C₁₆×C₇₉

C₂

semidirect

8. D₁₂₆₄

C₄

direct (N × H)

17. D₆₃₂

C₂×C₄

semidirect

24. C₁₆×C₇₉

C₂×C₄

semidirect

41. C₄×C₇₉

D₁₆

semidirect

50. C₇₉

C₄×D₁₆

semidirect

55. C₄×C₁₆

D₇₉

semidirect

74. C₁₆

C₄×D₇₉

semidirect

92. C₄

D₁₂₆₄

direct (N × H)

Subgroups — C₄×D₁₂₆₄ #23765248

Conjugacy classes of subgroups

Complements

Subgroups — C4×D1264 #23765248

Conjugacy classes of subgroups

Complements

Subgroups — C₄×D₁₂₆₄ #23765248