Group #19147705 — subgroups

Subgroups — C₂²×#13063270 #19147705

Order 5504.

400 conjugacy classes of subgroups; only the 149 normal subgroups are shown.

normal subgroups only

This group has 400 conjugacy classes of subgroups — too many to draw. Only its 149 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	C₂²×Dic₁₇₂	C₂	1	1	yes	maximal, characteristic, complemented
2	C₂²×D₃₄₄	C₂	1	1	yes	maximal, characteristic
3	C₂×#13063270	C₂	1	12	yes	maximal, direct product factor
4	C₂²×C₁₆×C₄₃	C₂	1	1	yes	maximal, characteristic, complemented
5	C₂²×Dic₈₆	—	2	1	no	—
6	C₂²×D₁₇₂	—	2	1	no	—
7	C₂×Dic₁₇₂	—	2	3	no	—
8	C₂×Dic₁₇₂	C₂²	1	6	yes	complemented
9	C₂×D₃₄₄	—	2	3	no	—
10	C₂×D₃₄₄	C₂²	1	6	yes	—
11	#13063270	C₂²	1	16	yes	direct product factor
12	C₂²×C₈×C₄₃	C₂²	1	1	yes	characteristic
13	C₂×C₁₆×C₄₃	C₂²	1	6	yes	complemented
14	C₂³×D₄₃	—	4	1	no	—
15	C₂²×C₄×C₄₃	D₄	1	1	yes	characteristic
16	C₂²×Dic₄₃	—	4	1	no	—
17	C₂×D₁₇₂	—	2	6	no	—
18	C₂×D₁₇₂	—	4	3	no	—
19	C₂×Dic₈₆	—	2	6	no	—
20	C₂×Dic₈₆	—	4	3	no	—
21	C₂×C₈×C₄₃	D₄	1	3	yes	—
22	C₂×C₈×C₄₃	C₂³	1	3	yes	—
23	Dic₁₇₂	—	2	6	no	—
24	Dic₁₇₂	C₂³	1	4	yes	complemented
25	D₃₄₄	—	2	6	no	—
26	D₃₄₄	C₂³	1	4	yes	—
27	C₁₆×C₄₃	C₂³	1	4	yes	complemented
28	C₂³×C₄₃	D₈	1	1	yes	characteristic
29	C₂²×D₄₃	—	4	6	no	—
30	C₂²×D₄₃	—	8	4	no	—
31	C₂×C₄×C₄₃	D₈	1	3	yes	—
32	C₂×C₄×C₄₃	C₂×D₄	1	3	yes	—
33	C₂×Dic₄₃	—	4	6	no	—
34	D₁₇₂	—	2	4	no	—
35	D₁₇₂	—	4	6	no	—
36	Dic₈₆	—	2	4	no	—
37	Dic₈₆	—	4	6	no	—
38	C₈×C₄₃	C₂×D₄	1	3	yes	—
39	C₈×C₄₃	C₂⁴	1	1	yes	characteristic
40	C₂²×C₄₃	SD₃₂	1	4	yes	complemented
41	C₂²×C₄₃	C₂×D₈	1	3	yes	—
42	C₂×D₄₃	—	4	4	no	—
43	C₂×D₄₃	—	8	12	no	—
44	C₄×C₄₃	C₂×D₈	1	3	yes	—
45	C₄×C₄₃	C₂²×D₄	1	1	yes	characteristic
46	Dic₄₃	—	4	4	no	—
47	C₂²×SD₃₂	—	43	1	no	maximal, Sylow
48	C₂×C₄₃	C₂×SD₃₂	1	6	yes	complemented
49	C₂×C₄₃	C₂²×D₈	1	1	yes	characteristic
50	D₄₃	—	8	4	no	—
51	C₂²×Q₁₆	—	43	1	no	—
52	C₂²×D₈	—	43	1	no	—
53	C₂×SD₃₂	—	43	12	no	—
54	C₂²×C₁₆	D₄₃	1	1	yes	characteristic, complemented
55	C₄₃	C₂²×SD₃₂	1	1	yes	characteristic, complemented, Sylow
56	C₂²×Q₈	—	86	1	no	—
57	C₂²×D₄	—	86	1	no	—
58	C₂×Q₁₆	—	43	6	no	—
59	C₂×Q₁₆	—	86	3	no	—
60	C₂×D₈	—	43	6	no	—
61	C₂×D₈	—	86	3	no	—
62	C₂²×C₈	C₂×D₄₃	1	1	yes	characteristic
63	SD₃₂	—	43	16	no	—
64	C₂×C₁₆	C₂×D₄₃	1	6	yes	complemented
65	C₂⁴	—	172	1	no	—
66	C₂×Q₈	—	86	6	no	—
67	C₂×Q₈	—	172	3	no	—
68	C₂×D₄	—	86	6	no	—
69	C₂×D₄	—	172	3	no	—
70	C₂²×C₄	—	172	1	no	—
71	C₂²×C₄	D₁₇₂	1	1	yes	characteristic
72	Q₁₆	—	43	4	no	—
73	Q₁₆	—	86	6	no	—
74	D₈	—	43	4	no	—
75	D₈	—	86	6	no	—
76	C₂×C₈	D₁₇₂	1	3	yes	—
77	C₂×C₈	C₂²×D₄₃	1	3	yes	—
78	C₁₆	C₂²×D₄₃	1	4	yes	complemented
79	C₂³	—	344	4	no	—
80	C₂³	—	172	6	no	—
81	C₂³	D₃₄₄	1	1	yes	characteristic
82	Q₈	—	86	4	no	—
83	Q₈	—	172	6	no	—
84	D₄	—	86	4	no	—
85	D₄	—	172	6	no	—
86	C₂×C₄	—	172	6	no	—
87	C₂×C₄	D₃₄₄	1	3	yes	—
88	C₂×C₄	C₂×D₁₇₂	1	3	yes	—
89	C₈	C₂×D₁₇₂	1	3	yes	—
90	C₈	C₂³×D₄₃	1	1	yes	characteristic
91	C₂²	—	344	12	no	—
92	C₂²	—	172	4	no	—
93	C₂²	#13063270	1	4	yes	direct product factor
94	C₂²	C₂×D₃₄₄	1	3	yes	—
95	C₄	—	172	4	no	—
96	C₄	C₂×D₃₄₄	1	3	yes	—
97	C₄	C₂²×D₁₇₂	1	1	yes	characteristic
98	C₂	—	344	4	no	—
99	C₂	C₂×#13063270	1	6	yes	direct product factor
100	C₂	C₂²×D₃₄₄	1	1	yes	characteristic

subgroup

G/H

class size

multiplicity

normal?

properties

C₂²×Dic₁₇₂

C₂

yes

maximal, characteristic, complemented

C₂²×D₃₄₄

C₂

yes

maximal, characteristic

C₂×#13063270

C₂

yes

maximal, direct product factor

C₂²×C₁₆×C₄₃

C₂

yes

maximal, characteristic, complemented

C₂²×Dic₈₆

—

C₂²×D₁₇₂

—

C₂×Dic₁₇₂

—

C₂×Dic₁₇₂

C₂²

yes

complemented

C₂×D₃₄₄

—

C₂×D₃₄₄

C₂²

yes

—

#13063270

C₂²

yes

direct product factor

C₂²×C₈×C₄₃

C₂²

yes

characteristic

C₂×C₁₆×C₄₃

C₂²

yes

complemented

C₂³×D₄₃

—

C₂²×C₄×C₄₃

D₄

yes

characteristic

C₂²×Dic₄₃

—

C₂×D₁₇₂

—

C₂×D₁₇₂

—

C₂×Dic₈₆

—

C₂×Dic₈₆

—

C₂×C₈×C₄₃

D₄

yes

—

C₂×C₈×C₄₃

C₂³

yes

—

Dic₁₇₂

—

Dic₁₇₂

C₂³

yes

complemented

D₃₄₄

—

D₃₄₄

C₂³

yes

—

C₁₆×C₄₃

C₂³

yes

complemented

C₂³×C₄₃

D₈

yes

characteristic

C₂²×D₄₃

—

C₂²×D₄₃

—

C₂×C₄×C₄₃

D₈

yes

—

C₂×C₄×C₄₃

C₂×D₄

yes

—

C₂×Dic₄₃

—

D₁₇₂

—

D₁₇₂

—

Dic₈₆

—

Dic₈₆

—

C₈×C₄₃

C₂×D₄

yes

—

C₈×C₄₃

C₂⁴

yes

characteristic

C₂²×C₄₃

SD₃₂

yes

complemented

C₂²×C₄₃

C₂×D₈

yes

—

C₂×D₄₃

—

C₂×D₄₃

—

C₄×C₄₃

C₂×D₈

yes

—

C₄×C₄₃

C₂²×D₄

yes

characteristic

Dic₄₃

—

C₂²×SD₃₂

—

maximal, Sylow

C₂×C₄₃

C₂×SD₃₂

yes

complemented

C₂×C₄₃

C₂²×D₈

yes

characteristic

D₄₃

—

C₂²×Q₁₆

—

C₂²×D₈

—

C₂×SD₃₂

—

C₂²×C₁₆

D₄₃

yes

characteristic, complemented

C₄₃

C₂²×SD₃₂

yes

characteristic, complemented, Sylow

C₂²×Q₈

—

C₂²×D₄

—

C₂×Q₁₆

—

C₂×Q₁₆

—

C₂×D₈

—

C₂×D₈

—

C₂²×C₈

C₂×D₄₃

yes

characteristic

SD₃₂

—

C₂×C₁₆

C₂×D₄₃

yes

complemented

C₂⁴

—

172

—

C₂×Q₈

—

C₂×Q₈

—

172

—

C₂×D₄

—

C₂×D₄

—

172

—

C₂²×C₄

—

172

—

C₂²×C₄

D₁₇₂

yes

characteristic

Q₁₆

—

Q₁₆

—

D₈

—

D₈

—

C₂×C₈

D₁₇₂

yes

—

C₂×C₈

C₂²×D₄₃

yes

—

C₁₆

C₂²×D₄₃

yes

complemented

C₂³

—

344

—

C₂³

—

172

—

C₂³

D₃₄₄

yes

characteristic

Q₈

—

Q₈

—

172

—

D₄

—

D₄

—

172

—

C₂×C₄

—

172

—

C₂×C₄

D₃₄₄

yes

—

C₂×C₄

C₂×D₁₇₂

yes

—

C₈

C₂×D₁₇₂

yes

—

C₈

C₂³×D₄₃

yes

characteristic

C₂²

—

344

—

C₂²

—

172

—

C₂²

#13063270

yes

direct product factor

C₂²

C₂×D₃₄₄

yes

—

C₄

—

172

—

C₄

C₂×D₃₄₄

yes

—

C₄

C₂²×D₁₇₂

yes

characteristic

C₂

—

344

—

C₂

C₂×#13063270

yes

direct product factor

100

C₂

C₂²×D₃₄₄

yes

characteristic

	subgroup	G/H
center	C₂³	D₃₄₄
commutator	C₈×C₄₃	C₂⁴
Frattini	C₈	C₂³×D₄₃
Fitting	C₂²×C₁₆×C₄₃	C₂
socle	C₂³×C₄₃	D₈
hypercenter	C₂²×C₁₆	D₄₃
hypocenter	C₄₃	C₂²×SD₃₂
LCS	G ▷ C₈×C₄₃ ▷ C₄×C₄₃ ▷ C₂×C₄₃ ▷ C₄₃
UCS	C₁ ◁ C₂³ ◁ C₂²×C₄ ◁ C₂²×C₈ ◁ C₂²×C₁₆

subgroup

G/H

center

C₂³

D₃₄₄

commutator

C₈×C₄₃

C₂⁴

Frattini

C₈

C₂³×D₄₃

Fitting

C₂²×C₁₆×C₄₃

C₂

socle

C₂³×C₄₃

D₈

hypercenter

C₂²×C₁₆

D₄₃

hypocenter

C₄₃

C₂²×SD₃₂

LCS

G ▷ C₈×C₄₃ ▷ C₄×C₄₃ ▷ C₂×C₄₃ ▷ C₄₃

UCS

C₁ ◁ C₂³ ◁ C₂²×C₄ ◁ C₂²×C₈ ◁ C₂²×C₁₆

N	H ≅ G/N	type
1. C₂²×Dic₁₇₂	C₂	semidirect
3. C₂×#13063270	C₂	direct (N × H)
4. C₂²×C₁₆×C₄₃	C₂	semidirect
8. C₂×Dic₁₇₂	C₂²	semidirect
11. #13063270	C₂²	direct (N × H)
13. C₂×C₁₆×C₄₃	C₂²	semidirect
24. Dic₁₇₂	C₂³	semidirect
27. C₁₆×C₄₃	C₂³	semidirect
40. C₂²×C₄₃	SD₃₂	semidirect
48. C₂×C₄₃	C₂×SD₃₂	semidirect
54. C₂²×C₁₆	D₄₃	semidirect
55. C₄₃	C₂²×SD₃₂	semidirect
64. C₂×C₁₆	C₂×D₄₃	semidirect
78. C₁₆	C₂²×D₄₃	semidirect
93. C₂²	#13063270	direct (N × H)
99. C₂	C₂×#13063270	direct (N × H)

H ≅ G/N

type

1. C₂²×Dic₁₇₂

C₂

semidirect

3. C₂×#13063270

C₂

direct (N × H)

4. C₂²×C₁₆×C₄₃

C₂

semidirect

8. C₂×Dic₁₇₂

C₂²

semidirect

11. #13063270

C₂²

direct (N × H)

13. C₂×C₁₆×C₄₃

C₂²

semidirect

24. Dic₁₇₂

C₂³

semidirect

27. C₁₆×C₄₃

C₂³

semidirect

40. C₂²×C₄₃

SD₃₂

semidirect

48. C₂×C₄₃

C₂×SD₃₂

semidirect

54. C₂²×C₁₆

D₄₃

semidirect

55. C₄₃

C₂²×SD₃₂

semidirect

64. C₂×C₁₆

C₂×D₄₃

semidirect

78. C₁₆

C₂²×D₄₃

semidirect

93. C₂²

#13063270

direct (N × H)

99. C₂

C₂×#13063270

direct (N × H)

Subgroups — C₂²×#13063270 #19147705

Conjugacy classes of subgroups

Complements

Subgroups — C22×#13063270 #19147705

Conjugacy classes of subgroups

Complements

Subgroups — C₂²×#13063270 #19147705