Group #19134188 — subgroups

Subgroups — #19134188

Order 5504.

224 conjugacy classes of subgroups; only the 69 normal subgroups are shown.

normal subgroups only

This group has 224 conjugacy classes of subgroups — too many to draw. Only its 69 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	#14583276	C₂	1	1	yes	maximal, characteristic
2	#14583273	C₂	1	2	yes	maximal, complemented
3	#14583247	C₂	1	2	yes	maximal, complemented
4	C₈×C₄₃⋊C₈	C₂	1	1	yes	maximal, characteristic, complemented
5	C₄₃×#500	C₂	1	1	yes	maximal, characteristic, complemented
6	C₂×#10621884	—	2	4	no	—
7	C₂×D₃₄₄	—	2	2	no	—
8	#13063322	C₂²	1	2	yes	—
9	#13063245	—	2	4	no	—
10	C₄×C₄₃⋊C₈	C₂²	1	1	yes	characteristic
11	C₄×C₄₃⋊C₈	C₂²	1	1	yes	characteristic
12	C₄⋊Q₈×C₄₃	C₂²	1	2	yes	—
13	#107×C₄₃	—	2	2	no	—
14	C₄×C₈×C₄₃	C₂²	1	1	yes	characteristic
15	C₂×Q₈×C₄₃	—	2	2	no	—
16	C₂×D₁₇₂	—	2	2	no	—
17	C₂×D₁₇₂	—	2	2	no	—
18	C₂×D₁₇₂	—	2	2	no	—
19	C₂×C₈×C₄₃	D₄	1	2	yes	—
20	C₄⋊C₄×C₄₃	—	2	2	no	—
21	C₄⋊C₄×C₄₃	—	2	2	no	—
22	C₄²×C₄₃	C₂³	1	1	yes	characteristic
23	#10621884	—	4	8	no	—
24	C₂×C₄₃⋊C₈	D₄	1	2	yes	—
25	C₂×C₄₃⋊C₈	D₄	1	2	yes	—
26	D₃₄₄	—	4	4	no	—
27	C₂²×D₄₃	—	4	2	no	—
28	Q₈×C₄₃	—	4	4	no	—
29	C₂×C₄×C₄₃	—	4	2	no	—
30	C₂×C₄×C₄₃	C₂×D₄	1	1	yes	characteristic
31	C₂×C₄×C₄₃	C₂×D₄	1	1	yes	characteristic
32	C₂×C₄×C₄₃	C₂×D₄	1	1	yes	characteristic
33	D₁₇₂	—	4	4	no	—
34	D₁₇₂	—	4	4	no	—
35	D₁₇₂	—	4	4	no	—
36	C₈×C₄₃	SD₁₆	1	4	yes	complemented
37	C₄₃⋊C₈	—	2	2	no	—
38	C₄₃⋊C₈	D₈	1	4	yes	complemented
39	C₂²×C₄₃	#127	1	1	yes	characteristic
40	C₂×D₄₃	—	8	2	no	—
41	C₂×D₄₃	—	8	2	no	—
42	C₂×D₄₃	—	8	2	no	—
43	C₄×C₄₃	—	8	2	no	—
44	C₄×C₄₃	C₂×D₈	1	2	yes	—
45	C₄×C₄₃	C₂×SD₁₆	1	2	yes	—
46	C₄×C₄₃	#135	1	2	yes	—
47	#1706	—	43	1	no	maximal, Sylow
48	C₂×C₄₃	#492	1	1	yes	characteristic
49	C₂×C₄₃	#493	1	1	yes	characteristic
50	C₂×C₄₃	#495	1	1	yes	characteristic
51	D₄₃	—	16	2	no	—
52	#500	D₄₃	1	1	yes	characteristic, complemented
53	#493	—	43	1	no	—
54	#492	—	43	2	no	—
55	#486	—	43	2	no	—
56	C₈²	—	43	1	no	—
57	C₄₃	#1706	1	1	yes	characteristic, complemented, Sylow
58	C₂×SD₁₆	—	86	4	no	—
59	C₂×D₈	—	86	2	no	—
60	C₄⋊Q₈	C₂×D₄₃	1	2	yes	—
61	#127	—	43	2	no	—
62	#107	—	2	2	no	—
63	#102	—	86	4	no	—
64	C₄×C₈	—	43	1	no	—
65	C₄×C₈	—	43	1	no	—
66	C₄×C₈	C₂×D₄₃	1	1	yes	characteristic
67	C₂×Q₈	—	2	2	no	—
68	C₂×D₄	—	86	2	no	—
69	C₂×D₄	—	86	2	no	—
70	C₂×D₄	—	86	2	no	—
71	SD₁₆	—	172	8	no	—
72	D₈	—	172	4	no	—
73	C₂×C₈	—	43	2	no	—
74	C₂×C₈	—	43	2	no	—
75	C₂×C₈	C₂²⋊D₄₃	1	2	yes	—
76	C₄⋊C₄	—	2	2	no	—
77	C₄⋊C₄	—	2	2	no	—
78	C₄²	C₂²×D₄₃	1	1	yes	characteristic
79	C₂³	—	172	2	no	—
80	Q₈	—	4	4	no	—
81	D₄	—	172	4	no	—
82	D₄	—	172	4	no	—
83	D₄	—	172	4	no	—
84	C₂×C₄	—	4	2	no	—
85	C₂×C₄	D₄×D₄₃	1	1	yes	characteristic
86	C₂×C₄	D₄×D₄₃	1	1	yes	characteristic
87	C₂×C₄	C₂×C₂²⋊D₄₃	1	1	yes	characteristic
88	C₈	—	43	4	no	—
89	C₈	—	86	2	no	—
90	C₈	#10621884	1	4	yes	complemented
91	C₂²	—	344	2	no	—
92	C₂²	—	344	2	no	—
93	C₂²	—	344	2	no	—
94	C₂²	#13063321	1	1	yes	characteristic
95	C₄	—	8	2	no	—
96	C₄	D₈×D₄₃	1	2	yes	—
97	C₄	C₂×#10621884	1	2	yes	—
98	C₄	#13063348	1	2	yes	—
99	C₂	—	688	2	no	—
100	C₂	#14583273	1	1	yes	characteristic
101	C₂	#14583274	1	1	yes	characteristic
102	C₂	#14583284	1	1	yes	characteristic

subgroup

G/H

class size

multiplicity

normal?

properties

#14583276

C₂

yes

maximal, characteristic

#14583273

C₂

yes

maximal, complemented

#14583247

C₂

yes

maximal, complemented

C₈×C₄₃⋊C₈

C₂

yes

maximal, characteristic, complemented

C₄₃×#500

C₂

yes

maximal, characteristic, complemented

C₂×#10621884

—

C₂×D₃₄₄

—

#13063322

C₂²

yes

—

#13063245

—

C₄×C₄₃⋊C₈

C₂²

yes

characteristic

C₄×C₄₃⋊C₈

C₂²

yes

characteristic

C₄⋊Q₈×C₄₃

C₂²

yes

—

#107×C₄₃

—

C₄×C₈×C₄₃

C₂²

yes

characteristic

C₂×Q₈×C₄₃

—

C₂×D₁₇₂

—

C₂×D₁₇₂

—

C₂×D₁₇₂

—

C₂×C₈×C₄₃

D₄

yes

—

C₄⋊C₄×C₄₃

—

C₄⋊C₄×C₄₃

—

C₄²×C₄₃

C₂³

yes

characteristic

#10621884

—

C₂×C₄₃⋊C₈

D₄

yes

—

C₂×C₄₃⋊C₈

D₄

yes

—

D₃₄₄

—

C₂²×D₄₃

—

Q₈×C₄₃

—

C₂×C₄×C₄₃

—

C₂×C₄×C₄₃

C₂×D₄

yes

characteristic

C₂×C₄×C₄₃

C₂×D₄

yes

characteristic

C₂×C₄×C₄₃

C₂×D₄

yes

characteristic

D₁₇₂

—

D₁₇₂

—

D₁₇₂

—

C₈×C₄₃

SD₁₆

yes

complemented

C₄₃⋊C₈

—

C₄₃⋊C₈

D₈

yes

complemented

C₂²×C₄₃

#127

yes

characteristic

C₂×D₄₃

—

C₂×D₄₃

—

C₂×D₄₃

—

C₄×C₄₃

—

C₄×C₄₃

C₂×D₈

yes

—

C₄×C₄₃

C₂×SD₁₆

yes

—

C₄×C₄₃

#135

yes

—

—

C₂×C₄₃

yes

C₂×C₄₃

yes

C₂×C₄₃

yes

D₄₃

—

#500

D₄₃

yes

characteristic, complemented

#493

—

#492

—

#486

—

C₈²

—

C₄₃

#1706

yes

characteristic, complemented, Sylow

C₂×SD₁₆

—

C₂×D₈

—

C₄⋊Q₈

C₂×D₄₃

yes

—

#127

—

#107

—

#102

—

C₄×C₈

—

C₄×C₈

—

C₄×C₈

C₂×D₄₃

yes

characteristic

C₂×Q₈

—

C₂×D₄

—

C₂×D₄

—

C₂×D₄

—

SD₁₆

—

172

—

D₈

—

172

—

C₂×C₈

—

C₂×C₈

—

C₂×C₈

C₂²⋊D₄₃

yes

—

C₄⋊C₄

—

C₄⋊C₄

—

C₄²

C₂²×D₄₃

yes

characteristic

C₂³

—

172

—

Q₈

—

D₄

—

172

—

D₄

—

172

—

D₄

—

172

—

C₂×C₄

—

C₂×C₄

D₄×D₄₃

yes

characteristic

C₂×C₄

D₄×D₄₃

yes

characteristic

C₂×C₄

C₂×C₂²⋊D₄₃

yes

characteristic

C₈

—

C₈

—

C₈

#10621884

yes

complemented

C₂²

—

344

—

C₂²

—

344

—

C₂²

—

344

—

C₂²

#13063321

yes

characteristic

C₄

—

C₄

D₈×D₄₃

yes

—

C₄

C₂×#10621884

yes

—

C₄

#13063348

yes

—

C₂

—

688

—

100

C₂

#14583273

yes

characteristic

101

C₂

#14583274

yes

characteristic

102

C₂

#14583284

yes

characteristic

	subgroup	G/H
center	C₂²	#13063321
commutator	C₄²×C₄₃	C₂³
Frattini	C₄²	C₂²×D₄₃
Fitting	C₄₃×#500	C₂
socle	C₂²×C₄₃	#127
hypercenter	#500	D₄₃
hypocenter	C₄₃	#1706
LCS	G ▷ C₄²×C₄₃ ▷ C₂²×C₄₃ ▷ C₄₃
UCS	C₁ ◁ C₂² ◁ C₄² ◁ #500

subgroup

G/H

center

C₂²

#13063321

commutator

C₄²×C₄₃

C₂³

Frattini

C₄²

C₂²×D₄₃

Fitting

C₄₃×#500

C₂

C₂²×C₄₃

D₄₃

C₄₃

G ▷ C₄²×C₄₃ ▷ C₂²×C₄₃ ▷ C₄₃

UCS

C₁ ◁ C₂² ◁ C₄² ◁ #500

N	H ≅ G/N	type
2. #14583273	C₂	semidirect
3. #14583247	C₂	semidirect
4. C₈×C₄₃⋊C₈	C₂	semidirect
5. C₄₃×#500	C₂	semidirect
36. C₈×C₄₃	SD₁₆	semidirect
38. C₄₃⋊C₈	D₈	semidirect
52. #500	D₄₃	semidirect
57. C₄₃	#1706	semidirect
90. C₈	#10621884	semidirect

H ≅ G/N

type

2. #14583273

C₂

semidirect

3. #14583247

C₂

semidirect

4. C₈×C₄₃⋊C₈

C₂

semidirect

5. C₄₃×#500

C₂

semidirect

36. C₈×C₄₃

SD₁₆

semidirect

38. C₄₃⋊C₈

D₈

semidirect

52. #500

D₄₃

semidirect

57. C₄₃

#1706

semidirect

90. C₈

#10621884

semidirect

Subgroups — #19134188

Conjugacy classes of subgroups

Complements