Group #32973632 — subgroups

Subgroups — #32973632

Order 18688.

126 conjugacy classes of subgroups; only the 47 normal subgroups are shown.

normal subgroups only

This group has 126 conjugacy classes of subgroups — too many to draw. Only its 47 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	#22608129	C₂	1	1	yes	maximal, characteristic
2	#22608007	C₂	1	1	yes	maximal, characteristic
3	C₇₃×#2247	C₂	1	1	yes	maximal, characteristic
4	#17988421	—	2	1	no	—
5	#17988408	—	2	1	no	—
6	C₂×Q₃₂×C₇₃	—	2	1	no	—
7	C₂×Q₃₂×C₇₃	C₄	1	2	yes	—
8	#494×C₇₃	—	2	1	no	—
9	C₄×C₁₆×C₇₃	C₂²	1	1	yes	characteristic
10	#14556531	—	4	1	no	—
11	#14556497	—	4	1	no	—
12	C₂×Q₁₆×C₇₃	—	2	2	no	—
13	C₂×Q₁₆×C₇₃	—	4	1	no	—
14	C₄⋊Q₈×C₇₃	—	4	1	no	—
15	Q₃₂×C₇₃	—	4	2	no	—
16	Q₃₂×C₇₃	—	4	1	no	—
17	Q₃₂×C₇₃	—	4	1	no	—
18	C₂×C₁₆×C₇₃	C₂×C₄	1	1	yes	characteristic
19	C₂×C₁₆×C₇₃	D₄	1	1	yes	characteristic
20	C₄×C₈×C₇₃	D₄	1	1	yes	characteristic
21	#11924003	—	8	1	no	—
22	C₂×Q₈×C₇₃	—	4	2	no	—
23	Q₁₆×C₇₃	—	4	2	no	—
24	Q₁₆×C₇₃	—	4	2	no	—
25	Q₁₆×C₇₃	—	8	1	no	—
26	Q₁₆×C₇₃	—	8	1	no	—
27	C₂×C₈×C₇₃	C₂²⋊C₄	1	1	yes	characteristic
28	C₂×C₈×C₇₃	D₈	1	1	yes	characteristic
29	C₄⋊C₄×C₇₃	—	8	2	no	—
30	C₄²×C₇₃	SD₁₆	1	1	yes	characteristic
31	C₁₆×C₇₃	—	2	1	no	—
32	C₁₆×C₇₃	D₈	1	1	yes	characteristic
33	C₁₆×C₇₃	SD₁₆	1	1	yes	characteristic
34	Q₈×C₇₃	—	8	2	no	—
35	Q₈×C₇₃	—	8	2	no	—
36	C₂×C₄×C₇₃	—	2	1	no	—
37	C₂×C₄×C₇₃	—	8	2	no	—
38	C₂×C₄×C₇₃	#102	1	1	yes	characteristic
39	C₈×C₇₃	#100	1	1	yes	characteristic
40	C₈×C₇₃	#102	1	1	yes	characteristic
41	C₈×C₇₃	D₁₆	1	1	yes	characteristic
42	C₈×C₇₃	SD₃₂	1	1	yes	characteristic
43	C₂²×C₇₃	#361	1	1	yes	characteristic
44	C₄×C₇₃	—	2	2	no	—
45	C₄×C₇₃	—	16	2	no	—
46	C₄×C₇₃	#331	1	1	yes	characteristic
47	C₄×C₇₃	#357	1	1	yes	characteristic
48	#7449	—	73	1	no	maximal, Sylow
49	C₂×C₇₃	—	2	1	no	—
50	C₂×C₇₃	#1361	1	1	yes	characteristic
51	#2247	D₇₃	1	1	yes	characteristic
52	#1422	—	73	1	no	—
53	#1373	—	73	1	no	—
54	C₇₃	#7449	1	1	yes	characteristic, complemented, Sylow
55	C₂×Q₃₂	—	2	1	no	—
56	C₂×Q₃₂	Dic₇₃	1	2	yes	—
57	#494	—	2	1	no	—
58	M₆₄	—	146	1	no	—
59	#364	—	146	1	no	—
60	C₄×C₁₆	C₂×D₇₃	1	1	yes	characteristic
61	C₂×Q₁₆	—	2	2	no	—
62	C₂×Q₁₆	—	4	1	no	—
63	C₄⋊Q₈	—	4	1	no	—
64	Q₃₂	—	4	2	no	—
65	Q₃₂	—	4	1	no	—
66	Q₃₂	—	4	1	no	—
67	M₃₂	—	292	1	no	—
68	C₂×C₁₆	C₂×Dic₇₃	1	1	yes	characteristic
69	C₂×C₁₆	C₂²⋊D₇₃	1	1	yes	characteristic
70	C₄×C₈	C₂²⋊D₇₃	1	1	yes	characteristic
71	C₃₂	—	292	1	no	—
72	C₂×Q₈	—	4	2	no	—
73	Q₁₆	—	4	2	no	—
74	Q₁₆	—	4	2	no	—
75	Q₁₆	—	8	1	no	—
76	Q₁₆	—	8	1	no	—
77	C₂×C₈	C₂²⋊Dic₇₃	1	1	yes	characteristic
78	C₂×C₈	#11924013	1	1	yes	characteristic
79	C₄⋊C₄	—	8	2	no	—
80	C₄²	#11924016	1	1	yes	characteristic
81	C₁₆	—	2	1	no	—
82	C₁₆	—	584	1	no	—
83	C₁₆	#11924013	1	1	yes	characteristic
84	C₁₆	#11924016	1	1	yes	characteristic
85	Q₈	—	8	2	no	—
86	Q₈	—	8	2	no	—
87	C₂×C₄	—	2	1	no	—
88	C₂×C₄	—	8	2	no	—
89	C₂×C₄	#14556512	1	1	yes	characteristic
90	C₈	#14556508	1	1	yes	characteristic
91	C₈	#14556512	1	1	yes	characteristic
92	C₈	#14556532	1	1	yes	characteristic
93	C₈	#14556535	1	1	yes	characteristic
94	C₂²	#17988400	1	1	yes	characteristic
95	C₄	—	2	2	no	—
96	C₄	—	16	2	no	—
97	C₄	#17988331	1	1	yes	characteristic
98	C₄	#17988388	1	1	yes	characteristic
99	C₂	—	2	1	no	—
100	C₂	#22607977	1	1	yes	characteristic

subgroup

G/H

class size

multiplicity

normal?

properties

#22608129

C₂

yes

maximal, characteristic

#22608007

C₂

yes

maximal, characteristic

C₇₃×#2247

C₂

yes

maximal, characteristic

#17988421

—

#17988408

—

C₂×Q₃₂×C₇₃

—

C₂×Q₃₂×C₇₃

C₄

yes

—

#494×C₇₃

—

C₄×C₁₆×C₇₃

C₂²

yes

characteristic

#14556531

—

#14556497

—

C₂×Q₁₆×C₇₃

—

C₂×Q₁₆×C₇₃

—

C₄⋊Q₈×C₇₃

—

Q₃₂×C₇₃

—

Q₃₂×C₇₃

—

Q₃₂×C₇₃

—

C₂×C₁₆×C₇₃

C₂×C₄

yes

characteristic

C₂×C₁₆×C₇₃

D₄

yes

characteristic

C₄×C₈×C₇₃

D₄

yes

characteristic

#11924003

—

C₂×Q₈×C₇₃

—

Q₁₆×C₇₃

—

Q₁₆×C₇₃

—

Q₁₆×C₇₃

—

Q₁₆×C₇₃

—

C₂×C₈×C₇₃

C₂²⋊C₄

yes

characteristic

C₂×C₈×C₇₃

D₈

yes

characteristic

C₄⋊C₄×C₇₃

—

C₄²×C₇₃

SD₁₆

yes

characteristic

C₁₆×C₇₃

—

C₁₆×C₇₃

D₈

yes

characteristic

C₁₆×C₇₃

SD₁₆

yes

characteristic

Q₈×C₇₃

—

Q₈×C₇₃

—

C₂×C₄×C₇₃

—

C₂×C₄×C₇₃

—

C₂×C₄×C₇₃

yes

C₈×C₇₃

yes

C₈×C₇₃

yes

C₈×C₇₃

D₁₆

yes

characteristic

C₈×C₇₃

SD₃₂

yes

characteristic

C₂²×C₇₃

#361

yes

characteristic

C₄×C₇₃

—

C₄×C₇₃

—

C₄×C₇₃

yes

C₄×C₇₃

yes

—

C₂×C₇₃

—

C₂×C₇₃

#1361

yes

characteristic

#2247

D₇₃

yes

characteristic

#1422

—

#1373

—

C₇₃

#7449

yes

characteristic, complemented, Sylow

C₂×Q₃₂

—

C₂×Q₃₂

Dic₇₃

yes

—

#494

—

M₆₄

—

146

—

#364

—

146

—

C₄×C₁₆

C₂×D₇₃

yes

characteristic

C₂×Q₁₆

—

C₂×Q₁₆

—

C₄⋊Q₈

—

Q₃₂

—

Q₃₂

—

Q₃₂

—

M₃₂

—

292

—

C₂×C₁₆

C₂×Dic₇₃

yes

characteristic

C₂×C₁₆

C₂²⋊D₇₃

yes

characteristic

C₄×C₈

C₂²⋊D₇₃

yes

characteristic

C₃₂

—

292

—

C₂×Q₈

—

Q₁₆

—

Q₁₆

—

Q₁₆

—

Q₁₆

—

C₂×C₈

C₂²⋊Dic₇₃

yes

characteristic

C₂×C₈

#11924013

yes

characteristic

C₄⋊C₄

—

C₄²

#11924016

yes

characteristic

C₁₆

—

C₁₆

—

584

—

C₁₆

yes

C₁₆

yes

Q₈

—

Q₈

—

C₂×C₄

—

C₂×C₄

—

C₂×C₄

yes

C₈

yes

C₈

yes

C₈

yes

C₈

yes

C₂²

yes

C₄

—

C₄

—

C₄

yes

C₄

yes

C₂

—

100

C₂

#22607977

yes

characteristic

	subgroup	G/H
center	C₂	#22607977
commutator	C₂×C₁₆×C₇₃	C₂×C₄
Frattini	C₄×C₁₆	C₂×D₇₃
Fitting	C₇₃×#2247	C₂
socle	C₂×C₇₃	#1361
prime radical	C₂²×C₇₃	#361
cosocle	C₄×C₁₆×C₇₃	C₂²
hypercenter	#2247	D₇₃
hypocenter	C₇₃	#7449
LCS	G ▷ C₂×C₁₆×C₇₃ ▷ C₂×C₈×C₇₃ ▷ C₄×C₇₃ ▷ C₂×C₇₃ ▷ C₇₃
UCS	C₁ ◁ C₂ ◁ C₂×C₄ ◁ C₄×C₈ ◁ C₄×C₁₆ ◁ #2247

subgroup

G/H

center

C₂

#22607977

commutator

C₂×C₁₆×C₇₃

C₂×C₄

Frattini

C₄×C₁₆

C₂×D₇₃

Fitting

C₇₃×#2247

C₂

C₂×C₇₃

C₂²×C₇₃

C₄×C₁₆×C₇₃

C₂²

D₇₃

C₇₃

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

UCS

C₁ ◁ C₂ ◁ C₂×C₄ ◁ C₄×C₈ ◁ C₄×C₁₆ ◁ #2247

N	H ≅ G/N	type
54. C₇₃	#7449	semidirect

H ≅ G/N

type

54. C₇₃

#7449

semidirect

By isomorphism type — maximal subgroups of the same type appear once, whether or not they are normal.

subgroup	index
#22608129	2
#22608007	2
C₇₃×#2247	2
#7449	73

subgroup

index

#22608129

#22608007

C₇₃×#2247

#7449

Subgroups — #32973632

Conjugacy classes of subgroups

Complements

Maximal subgroups