Group #19136443 — subgroups

Subgroups — C₄×D₆₈₈ #19136443

Order 5504.

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×D₆₈₈	C₂	1	1	yes	maximal, characteristic
2	C₄×D₃₄₄	C₂	1	2	yes	maximal, complemented
3	#14582725	C₂	1	1	yes	maximal, characteristic, complemented
4	#14582701	C₂	1	2	yes	maximal, complemented
5	C₄×C₁₆×C₄₃	C₂	1	1	yes	maximal, characteristic, complemented
6	C₂×D₃₄₄	C₂²	1	2	yes	—
7	C₄×D₁₇₂	—	2	2	no	—
8	D₆₈₈	C₄	1	4	yes	direct product factor
9	#13063259	C₂²	1	2	yes	—
10	#13063247	—	2	2	no	—
11	C₂×C₁₆×C₄₃	C₂²	1	1	yes	characteristic
12	C₂×C₁₆×C₄₃	C₂²	1	1	yes	characteristic
13	C₄×C₈×C₄₃	C₂²	1	1	yes	characteristic
14	C₂×D₁₇₂	—	2	2	no	—
15	C₂×C₄×D₄₃	—	4	2	no	—
16	C₂×C₈×C₄₃	D₄	1	1	yes	characteristic
17	C₂×C₈×C₄₃	C₂³	1	1	yes	characteristic
18	C₄²×C₄₃	D₄	1	1	yes	characteristic
19	(C₂×D₄₃)⋊C₄	—	4	2	no	—
20	C₄⋊Dic₄₃	—	2	2	no	—
21	D₃₄₄	—	2	2	no	—
22	D₃₄₄	C₂×C₄	1	4	yes	complemented
23	C₁₆×C₄₃	—	2	1	no	—
24	C₁₆×C₄₃	C₂×C₄	1	2	yes	complemented
25	C₂²×D₄₃	—	4	2	no	—
26	C₂×C₄×C₄₃	D₈	1	1	yes	characteristic
27	C₂×C₄×C₄₃	D₈	1	1	yes	characteristic
28	C₂×C₄×C₄₃	C₂×D₄	1	1	yes	characteristic
29	C₂×Dic₄₃	—	4	2	no	—
30	D₁₇₂	—	2	4	no	—
31	D₁₇₂	—	4	2	no	—
32	C₄×D₄₃	—	8	4	no	—
33	C₈×C₄₃	—	2	1	no	—
34	C₈×C₄₃	C₂²×C₄	1	1	yes	characteristic
35	C₈×C₄₃	#41	1	1	yes	characteristic
36	C₂²×C₄₃	C₂×D₈	1	1	yes	characteristic
37	C₂×D₄₃	—	4	4	no	—
38	C₂×D₄₃	—	8	2	no	—
39	C₂×D₄₃	—	8	2	no	—
40	C₄×C₄₃	—	2	1	no	—
41	C₄×C₄₃	D₁₆	1	2	yes	complemented
42	C₄×C₄₃	C₄×D₄	1	1	yes	characteristic
43	C₄×C₄₃	#135	1	1	yes	characteristic
44	Dic₄₃	—	8	2	no	—
45	C₄×D₁₆	—	43	1	no	maximal, Sylow
46	C₂×C₄₃	C₄×D₈	1	1	yes	characteristic
47	C₂×C₄₃	C₂×D₁₆	1	1	yes	characteristic
48	C₂×C₄₃	#508	1	1	yes	characteristic
49	D₄₃	—	8	4	no	—
50	C₂×D₁₆	—	43	1	no	—
51	C₄×D₈	—	43	2	no	—
52	#366	—	43	1	no	—
53	#357	—	43	2	no	—
54	C₄×C₁₆	D₄₃	1	1	yes	characteristic, complemented
55	C₄₃	C₄×D₁₆	1	1	yes	characteristic, complemented, Sylow
56	C₂×D₈	—	43	2	no	—
57	C₄×D₄	—	86	2	no	—
58	D₁₆	—	43	4	no	—
59	C₂×C₁₆	C₂×D₄₃	1	1	yes	characteristic
60	C₂×C₁₆	C₂×D₄₃	1	1	yes	characteristic
61	#107	—	43	2	no	—
62	#102	—	86	2	no	—
63	C₄×C₈	C₂×D₄₃	1	1	yes	characteristic
64	C₂×D₄	—	86	2	no	—
65	C₂²×C₄	—	172	2	no	—
66	D₈	—	43	4	no	—
67	D₈	—	86	2	no	—
68	C₂×C₈	D₁₇₂	1	1	yes	characteristic
69	C₂×C₈	C₂²×D₄₃	1	1	yes	characteristic
70	C₄⋊C₄	—	86	2	no	—
71	C₂²⋊C₄	—	172	2	no	—
72	C₄²	D₁₇₂	1	1	yes	characteristic
73	C₁₆	—	2	1	no	—
74	C₁₆	C₄×D₄₃	1	2	yes	complemented
75	C₂³	—	172	2	no	—
76	D₄	—	86	4	no	—
77	D₄	—	172	2	no	—
78	C₂×C₄	—	344	4	no	—
79	C₂×C₄	—	172	2	no	—
80	C₂×C₄	D₃₄₄	1	1	yes	characteristic
81	C₂×C₄	D₃₄₄	1	1	yes	characteristic
82	C₂×C₄	C₂×D₁₇₂	1	1	yes	characteristic
83	C₈	—	2	1	no	—
84	C₈	C₂×C₄×D₄₃	1	1	yes	characteristic
85	C₈	#10621898	1	1	yes	characteristic
86	C₂²	—	344	2	no	—
87	C₂²	—	344	2	no	—
88	C₂²	—	172	4	no	—
89	C₂²	C₂×D₃₄₄	1	1	yes	characteristic
90	C₄	—	2	1	no	—
91	C₄	—	344	2	no	—
92	C₄	D₆₈₈	1	2	yes	direct product factor
93	C₄	C₄×D₁₇₂	1	1	yes	characteristic
94	C₄	#13063347	1	1	yes	characteristic
95	C₂	—	344	4	no	—
96	C₂	C₄×D₃₄₄	1	1	yes	characteristic
97	C₂	C₂×D₆₈₈	1	1	yes	characteristic
98	C₂	#14583331	1	1	yes	characteristic

subgroup

G/H

class size

multiplicity

normal?

properties

C₂×D₆₈₈

C₂

yes

maximal, characteristic

C₄×D₃₄₄

C₂

yes

maximal, complemented

#14582725

C₂

yes

maximal, characteristic, complemented

#14582701

C₂

yes

maximal, complemented

C₄×C₁₆×C₄₃

C₂

yes

maximal, characteristic, complemented

C₂×D₃₄₄

C₂²

yes

—

C₄×D₁₇₂

—

D₆₈₈

C₄

yes

direct product factor

#13063259

C₂²

yes

—

#13063247

—

C₂×C₁₆×C₄₃

C₂²

yes

characteristic

C₂×C₁₆×C₄₃

C₂²

yes

characteristic

C₄×C₈×C₄₃

C₂²

yes

characteristic

C₂×D₁₇₂

—

C₂×C₄×D₄₃

—

C₂×C₈×C₄₃

D₄

yes

characteristic

C₂×C₈×C₄₃

C₂³

yes

characteristic

C₄²×C₄₃

D₄

yes

characteristic

(C₂×D₄₃)⋊C₄

—

C₄⋊Dic₄₃

—

D₃₄₄

—

D₃₄₄

C₂×C₄

yes

complemented

C₁₆×C₄₃

—

C₁₆×C₄₃

C₂×C₄

yes

complemented

C₂²×D₄₃

—

C₂×C₄×C₄₃

D₈

yes

characteristic

C₂×C₄×C₄₃

D₈

yes

characteristic

C₂×C₄×C₄₃

C₂×D₄

yes

characteristic

C₂×Dic₄₃

—

D₁₇₂

—

D₁₇₂

—

C₄×D₄₃

—

C₈×C₄₃

—

C₈×C₄₃

C₂²×C₄

yes

characteristic

C₈×C₄₃

#41

yes

characteristic

C₂²×C₄₃

C₂×D₈

yes

characteristic

C₂×D₄₃

—

C₂×D₄₃

—

C₂×D₄₃

—

C₄×C₄₃

—

C₄×C₄₃

D₁₆

yes

complemented

C₄×C₄₃

C₄×D₄

yes

characteristic

C₄×C₄₃

#135

yes

characteristic

Dic₄₃

—

C₄×D₁₆

—

maximal, Sylow

C₂×C₄₃

C₄×D₈

yes

characteristic

C₂×C₄₃

C₂×D₁₆

yes

characteristic

C₂×C₄₃

#508

yes

characteristic

D₄₃

—

C₂×D₁₆

—

C₄×D₈

—

#366

—

#357

—

C₄×C₁₆

D₄₃

yes

characteristic, complemented

C₄₃

C₄×D₁₆

yes

characteristic, complemented, Sylow

C₂×D₈

—

C₄×D₄

—

D₁₆

—

C₂×C₁₆

C₂×D₄₃

yes

characteristic

C₂×C₁₆

C₂×D₄₃

yes

characteristic

#107

—

#102

—

C₄×C₈

C₂×D₄₃

yes

characteristic

C₂×D₄

—

C₂²×C₄

—

172

—

D₈

—

D₈

—

C₂×C₈

D₁₇₂

yes

characteristic

C₂×C₈

C₂²×D₄₃

yes

characteristic

C₄⋊C₄

—

C₂²⋊C₄

—

172

—

C₄²

D₁₇₂

yes

characteristic

C₁₆

—

C₁₆

C₄×D₄₃

yes

complemented

C₂³

—

172

—

D₄

—

D₄

—

172

—

C₂×C₄

—

344

—

C₂×C₄

—

172

—

C₂×C₄

D₃₄₄

yes

characteristic

C₂×C₄

D₃₄₄

yes

characteristic

C₂×C₄

C₂×D₁₇₂

yes

characteristic

C₈

—

C₈

C₂×C₄×D₄₃

yes

characteristic

C₈

#10621898

yes

characteristic

C₂²

—

344

—

C₂²

—

344

—

C₂²

—

172

—

C₂²

C₂×D₃₄₄

yes

characteristic

C₄

—

C₄

—

344

—

C₄

D₆₈₈

yes

direct product factor

C₄

C₄×D₁₇₂

yes

characteristic

C₄

#13063347

yes

characteristic

C₂

—

344

—

C₂

C₄×D₃₄₄

yes

characteristic

C₂

C₂×D₆₈₈

yes

characteristic

C₂

#14583331

yes

characteristic

	subgroup	G/H
center	C₂×C₄	D₃₄₄
commutator	C₈×C₄₃	C₂²×C₄
Frattini	C₂×C₈	C₂²×D₄₃
Fitting	C₄×C₁₆×C₄₃	C₂
socle	C₂²×C₄₃	C₂×D₈
hypercenter	C₄×C₁₆	D₄₃
hypocenter	C₄₃	C₄×D₁₆
LCS	G ▷ C₈×C₄₃ ▷ C₄×C₄₃ ▷ C₂×C₄₃ ▷ C₄₃
UCS	C₁ ◁ C₂×C₄ ◁ C₄² ◁ C₄×C₈ ◁ C₄×C₁₆

subgroup

G/H

center

C₂×C₄

D₃₄₄

commutator

C₈×C₄₃

C₂²×C₄

Frattini

C₂×C₈

C₂²×D₄₃

Fitting

C₄×C₁₆×C₄₃

C₂

socle

C₂²×C₄₃

C₂×D₈

hypercenter

C₄×C₁₆

D₄₃

hypocenter

C₄₃

C₄×D₁₆

LCS

G ▷ C₈×C₄₃ ▷ C₄×C₄₃ ▷ C₂×C₄₃ ▷ C₄₃

UCS

C₁ ◁ C₂×C₄ ◁ C₄² ◁ C₄×C₈ ◁ C₄×C₁₆

N	H ≅ G/N	type
2. C₄×D₃₄₄	C₂	semidirect
3. #14582725	C₂	semidirect
4. #14582701	C₂	semidirect
5. C₄×C₁₆×C₄₃	C₂	semidirect
8. D₆₈₈	C₄	direct (N × H)
22. D₃₄₄	C₂×C₄	semidirect
24. C₁₆×C₄₃	C₂×C₄	semidirect
41. C₄×C₄₃	D₁₆	semidirect
54. C₄×C₁₆	D₄₃	semidirect
55. C₄₃	C₄×D₁₆	semidirect
74. C₁₆	C₄×D₄₃	semidirect
92. C₄	D₆₈₈	direct (N × H)

H ≅ G/N

type

2. C₄×D₃₄₄

C₂

semidirect

3. #14582725

C₂

semidirect

4. #14582701

C₂

semidirect

5. C₄×C₁₆×C₄₃

C₂

semidirect

8. D₆₈₈

C₄

direct (N × H)

22. D₃₄₄

C₂×C₄

semidirect

24. C₁₆×C₄₃

C₂×C₄

semidirect

41. C₄×C₄₃

D₁₆

semidirect

54. C₄×C₁₆

D₄₃

semidirect

55. C₄₃

C₄×D₁₆

semidirect

74. C₁₆

C₄×D₄₃

semidirect

92. C₄

D₆₈₈

direct (N × H)

Subgroups — C₄×D₆₈₈ #19136443

Conjugacy classes of subgroups

Complements

Subgroups — C4×D688 #19136443

Conjugacy classes of subgroups

Complements

Subgroups — C₄×D₆₈₈ #19136443