Group #22612247 — subgroups

Subgroups — #22612247

Order 9344.

104 conjugacy classes of subgroups; only the 35 normal subgroups are shown.

normal subgroups only

This group has 104 conjugacy classes of subgroups — too many to draw. Only its 35 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	#17989016	C₂	1	1	yes	maximal, characteristic
2	Q₃₂×D₇₃	C₂	1	1	yes	maximal, characteristic, complemented
3	Dic₁₁₆₈	C₂	1	1	yes	maximal, characteristic, complemented
4	#17988427	C₂	1	1	yes	maximal, characteristic, complemented
5	#17988425	C₂	1	1	yes	maximal, characteristic, complemented
6	#17988420	C₂	1	1	yes	maximal, characteristic, complemented
7	SD₆₄×C₇₃	C₂	1	1	yes	maximal, characteristic, complemented
8	#14556610	—	2	1	no	—
9	Q₁₆×D₇₃	—	2	1	no	—
10	Dic₅₈₄	C₂²	1	1	yes	characteristic
11	Dic₅₈₄	C₂²	1	1	yes	characteristic
12	#14556537	—	2	1	no	—
13	#14556535	—	2	1	no	—
14	C₁₆×D₇₃	C₂²	1	1	yes	characteristic
15	#14556497	C₂²	1	1	yes	characteristic
16	Q₃₂×C₇₃	C₂²	1	1	yes	characteristic
17	D₁₆×C₇₃	C₂²	1	1	yes	characteristic
18	C₃₂×C₇₃	C₂²	1	1	yes	characteristic
19	#11924027	—	4	1	no	—
20	Q₈×D₇₃	—	4	1	no	—
21	Dic₂₉₂	—	2	1	no	—
22	Dic₂₉₂	—	2	1	no	—
23	#11924018	—	4	1	no	—
24	#11924016	—	4	1	no	—
25	C₈×D₇₃	D₄	1	1	yes	characteristic
26	#11924003	D₄	1	1	yes	characteristic
27	Q₁₆×C₇₃	—	2	1	no	—
28	D₈×C₇₃	—	2	1	no	—
29	C₁₆×C₇₃	C₂³	1	1	yes	characteristic
30	Q₈×C₇₃	—	4	1	no	—
31	D₄×C₇₃	—	4	1	no	—
32	C₂²⋊D₇₃	—	8	1	no	—
33	C₂×Dic₇₃	—	8	1	no	—
34	C₄×D₇₃	—	8	1	no	—
35	C₄×D₇₃	D₈	1	1	yes	characteristic
36	Dic₁₄₆	—	4	1	no	—
37	Dic₁₄₆	—	4	1	no	—
38	Dic₁₄₆	—	8	1	no	—
39	C₈×C₇₃	C₂×D₄	1	1	yes	characteristic
40	#10596831	D₈	1	1	yes	characteristic
41	C₂²×C₇₃	—	8	1	no	—
42	C₂×D₇₃	D₁₆	1	1	yes	characteristic
43	C₄×C₇₃	—	8	1	no	—
44	C₄×C₇₃	C₂×D₈	1	1	yes	characteristic
45	Dic₇₃	—	8	1	no	—
46	Dic₇₃	—	8	1	no	—
47	Dic₇₃	D₁₆	1	1	yes	characteristic
48	C₂×C₇₃	—	16	1	no	—
49	C₂×C₇₃	C₂×D₁₆	1	1	yes	characteristic
50	D₇₃	—	2	1	no	—
51	#2264	—	73	1	no	maximal, Sylow
52	C₇₃	#2264	1	1	yes	characteristic, complemented, Sylow
53	#508	—	73	1	no	—
54	C₂×Q₃₂	—	73	1	no	—
55	Q₆₄	—	73	1	no	—
56	Q₆₄	—	73	1	no	—
57	SD₆₄	—	73	1	no	—
58	SD₆₄	D₇₃	1	1	yes	characteristic, complemented
59	M₆₄	—	73	1	no	—
60	#135	—	146	1	no	—
61	C₂×Q₁₆	—	146	1	no	—
62	Q₃₂	—	73	1	no	—
63	Q₃₂	—	73	1	no	—
64	Q₃₂	—	146	1	no	—
65	Q₃₂	C₂×D₇₃	1	1	yes	characteristic
66	SD₃₂	—	146	1	no	—
67	D₁₆	C₂×D₇₃	1	1	yes	characteristic
68	C₂×C₁₆	—	73	1	no	—
69	C₃₂	—	73	1	no	—
70	C₃₂	C₂×D₇₃	1	1	yes	characteristic
71	#41	—	292	1	no	—
72	C₂×Q₈	—	292	1	no	—
73	Q₁₆	—	2	1	no	—
74	Q₁₆	—	292	1	no	—
75	Q₁₆	—	146	1	no	—
76	Q₁₆	—	146	1	no	—
77	SD₁₆	—	292	1	no	—
78	D₈	—	2	1	no	—
79	C₂×C₈	—	73	1	no	—
80	C₁₆	—	73	1	no	—
81	C₁₆	C₂²×D₇₃	1	1	yes	characteristic
82	Q₈	—	4	1	no	—
83	Q₈	—	292	1	no	—
84	Q₈	—	292	1	no	—
85	Q₈	—	584	1	no	—
86	D₄	—	4	1	no	—
87	D₄	—	584	1	no	—
88	C₂×C₄	—	584	1	no	—
89	C₂×C₄	—	584	1	no	—
90	C₂×C₄	—	73	1	no	—
91	C₈	—	73	1	no	—
92	C₈	D₄×D₇₃	1	1	yes	characteristic
93	C₂²	—	8	1	no	—
94	C₂²	—	73	1	no	—
95	C₄	—	8	1	no	—
96	C₄	—	584	1	no	—
97	C₄	—	584	1	no	—
98	C₄	—	73	1	no	—
99	C₄	D₈×D₇₃	1	1	yes	characteristic
100	C₂	—	16	1	no	—
101	C₂	—	146	1	no	—
102	C₂	D₁₆×D₇₃	1	1	yes	characteristic

subgroup

G/H

class size

multiplicity

normal?

properties

#17989016

C₂

yes

maximal, characteristic

Q₃₂×D₇₃

C₂

yes

maximal, characteristic, complemented

Dic₁₁₆₈

C₂

yes

maximal, characteristic, complemented

#17988427

C₂

yes

maximal, characteristic, complemented

#17988425

C₂

yes

maximal, characteristic, complemented

#17988420

C₂

yes

maximal, characteristic, complemented

SD₆₄×C₇₃

C₂

yes

maximal, characteristic, complemented

#14556610

—

Q₁₆×D₇₃

—

Dic₅₈₄

C₂²

yes

characteristic

Dic₅₈₄

C₂²

yes

characteristic

#14556537

—

#14556535

—

C₁₆×D₇₃

C₂²

yes

characteristic

#14556497

C₂²

yes

characteristic

Q₃₂×C₇₃

C₂²

yes

characteristic

D₁₆×C₇₃

C₂²

yes

characteristic

C₃₂×C₇₃

C₂²

yes

characteristic

#11924027

—

Q₈×D₇₃

—

Dic₂₉₂

—

Dic₂₉₂

—

#11924018

—

#11924016

—

C₈×D₇₃

D₄

yes

characteristic

#11924003

D₄

yes

characteristic

Q₁₆×C₇₃

—

D₈×C₇₃

—

C₁₆×C₇₃

C₂³

yes

characteristic

Q₈×C₇₃

—

D₄×C₇₃

—

C₂²⋊D₇₃

—

C₂×Dic₇₃

—

C₄×D₇₃

—

C₄×D₇₃

D₈

yes

characteristic

Dic₁₄₆

—

Dic₁₄₆

—

Dic₁₄₆

—

C₈×C₇₃

C₂×D₄

yes

characteristic

#10596831

D₈

yes

characteristic

C₂²×C₇₃

—

C₂×D₇₃

D₁₆

yes

characteristic

C₄×C₇₃

—

C₄×C₇₃

C₂×D₈

yes

characteristic

Dic₇₃

—

Dic₇₃

—

Dic₇₃

D₁₆

yes

characteristic

C₂×C₇₃

—

C₂×C₇₃

C₂×D₁₆

yes

characteristic

D₇₃

—

#2264

—

maximal, Sylow

C₇₃

#2264

yes

characteristic, complemented, Sylow

#508

—

C₂×Q₃₂

—

Q₆₄

—

Q₆₄

—

SD₆₄

—

SD₆₄

D₇₃

yes

characteristic, complemented

M₆₄

—

#135

—

146

—

C₂×Q₁₆

—

146

—

Q₃₂

—

Q₃₂

—

Q₃₂

—

146

—

Q₃₂

C₂×D₇₃

yes

characteristic

SD₃₂

—

146

—

D₁₆

C₂×D₇₃

yes

characteristic

C₂×C₁₆

—

C₃₂

—

C₃₂

C₂×D₇₃

yes

characteristic

#41

—

292

—

C₂×Q₈

—

292

—

Q₁₆

—

Q₁₆

—

292

—

Q₁₆

—

146

—

Q₁₆

—

146

—

SD₁₆

—

292

—

D₈

—

C₂×C₈

—

C₁₆

—

C₁₆

C₂²×D₇₃

yes

characteristic

Q₈

—

Q₈

—

292

—

Q₈

—

292

—

Q₈

—

584

—

D₄

—

D₄

—

584

—

C₂×C₄

—

584

—

C₂×C₄

—

584

—

C₂×C₄

—

C₈

—

C₈

D₄×D₇₃

yes

characteristic

C₂²

—

C₂²

—

C₄

—

C₄

—

584

—

C₄

—

584

—

C₄

—

C₄

D₈×D₇₃

yes

characteristic

100

C₂

—

101

C₂

—

146

—

102

C₂

D₁₆×D₇₃

yes

characteristic

	subgroup	G/H
center	C₂	D₁₆×D₇₃
commutator	C₁₆×C₇₃	C₂³
Frattini	C₁₆	C₂²×D₇₃
Fitting	SD₆₄×C₇₃	C₂
socle	C₂×C₇₃	C₂×D₁₆
hypercenter	SD₆₄	D₇₃
hypocenter	C₇₃	#2264
LCS	G ▷ C₁₆×C₇₃ ▷ C₈×C₇₃ ▷ C₄×C₇₃ ▷ C₂×C₇₃ ▷ C₇₃
UCS	C₁ ◁ C₂ ◁ C₄ ◁ C₈ ◁ C₁₆ ◁ SD₆₄

subgroup

G/H

center

C₂

D₁₆×D₇₃

commutator

C₁₆×C₇₃

C₂³

Frattini

C₁₆

C₂²×D₇₃

Fitting

SD₆₄×C₇₃

C₂

socle

C₂×C₇₃

C₂×D₁₆

hypercenter

SD₆₄

D₇₃

hypocenter

C₇₃

#2264

LCS

G ▷ C₁₆×C₇₃ ▷ C₈×C₇₃ ▷ C₄×C₇₃ ▷ C₂×C₇₃ ▷ C₇₃

UCS

C₁ ◁ C₂ ◁ C₄ ◁ C₈ ◁ C₁₆ ◁ SD₆₄

N	H ≅ G/N	type
2. Q₃₂×D₇₃	C₂	semidirect
3. Dic₁₁₆₈	C₂	semidirect
4. #17988427	C₂	semidirect
5. #17988425	C₂	semidirect
6. #17988420	C₂	semidirect
7. SD₆₄×C₇₃	C₂	semidirect
52. C₇₃	#2264	semidirect
58. SD₆₄	D₇₃	semidirect

H ≅ G/N

type

2. Q₃₂×D₇₃

C₂

semidirect

3. Dic₁₁₆₈

C₂

semidirect

4. #17988427

C₂

semidirect

5. #17988425

C₂

semidirect

6. #17988420

C₂

semidirect

7. SD₆₄×C₇₃

C₂

semidirect

52. C₇₃

#2264

semidirect

58. SD₆₄

D₇₃

semidirect

Subgroups — #22612247

Conjugacy classes of subgroups

Complements