Group #34001073 — subgroups

Subgroups — #34001073

Order 18688.

166 conjugacy classes of subgroups; only the 66 normal subgroups are shown.

normal subgroups only

This group has 166 conjugacy classes of subgroups — too many to draw. Only its 66 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×#17989515	C₂	1	2	yes	maximal
2	C₂×C₃₂×D₇₃	C₂	1	1	yes	maximal, characteristic
3	C₂×#14556662	—	2	2	no	—
4	#17989515	—	2	2	no	—
5	#17989515	C₄	1	4	yes	—
6	C₂×C₁₆×D₇₃	C₂²	1	1	yes	characteristic
7	C₂×#14556497	C₄	1	1	yes	characteristic
8	C₃₂×D₇₃	—	2	2	no	—
9	C₂×C₃₂×C₇₃	C₄	1	1	yes	characteristic
10	C₂×#11924038	—	4	2	no	—
11	#14556662	—	2	4	no	—
12	#14556662	—	4	2	no	—
13	C₂×C₈×D₇₃	D₄	1	1	yes	characteristic
14	C₂×#11924003	Q₈	1	1	yes	characteristic
15	C₁₆×D₇₃	C₂×C₄	1	2	yes	—
16	C₁₆×D₇₃	D₄	1	2	yes	—
17	#14556497	—	2	1	no	—
18	C₂×C₁₆×C₇₃	C₂×C₄	1	1	yes	characteristic
19	C₃₂×C₇₃	—	2	1	no	—
20	#11924038	—	4	4	no	—
21	#11924038	—	8	2	no	—
22	C₂×#10596833	—	8	2	no	—
23	C₂×C₄×D₇₃	D₈	1	1	yes	characteristic
24	C₂×#10596831	Q₁₆	1	1	yes	characteristic
25	C₈×D₇₃	C₂²⋊C₄	1	2	yes	—
26	C₈×D₇₃	SD₁₆	1	2	yes	—
27	#11924003	C₄⋊C₄	1	2	yes	—
28	C₂×C₈×C₇₃	C₄⋊C₄	1	1	yes	characteristic
29	C₁₆×C₇₃	C₄²	1	1	yes	characteristic
30	C₁₆×C₇₃	C₂²⋊C₄	1	1	yes	characteristic
31	C₂²×D₇₃	Q₃₂	1	1	yes	characteristic
32	C₂×C₄×C₇₃	#107	1	1	yes	characteristic
33	C₂×Dic₇₃	D₁₆	1	1	yes	characteristic
34	C₄×D₇₃	#102	1	2	yes	—
35	C₄×D₇₃	SD₃₂	1	2	yes	—
36	#10596833	—	8	4	no	—
37	C₈×C₇₃	#95	1	1	yes	characteristic
38	C₈×C₇₃	#106	1	1	yes	characteristic
39	#10596831	#103	1	2	yes	—
40	C₂²×C₇₃	#366	1	1	yes	characteristic
41	C₂×D₇₃	—	2	1	no	—
42	C₂×D₇₃	—	2	1	no	—
43	C₂×D₇₃	#358	1	2	yes	—
44	C₄×C₇₃	#340	1	1	yes	characteristic
45	C₄×C₇₃	#367	1	1	yes	characteristic
46	Dic₇₃	#357	1	2	yes	—
47	#7472	—	73	1	no	maximal, Sylow
48	C₂×C₇₃	#1380	1	1	yes	characteristic
49	C₂×C₇₃	#1425	1	1	yes	characteristic
50	C₂×C₇₃	#1425	1	1	yes	characteristic
51	D₇₃	—	2	2	no	—
52	C₂²×C₃₂	—	73	1	no	—
53	C₂×#368	—	73	2	no	—
54	C₇₃	#7472	1	1	yes	characteristic, complemented, Sylow
55	C₂²×C₁₆	—	73	1	no	—
56	C₂×#108	—	146	2	no	—
57	C₂×C₃₂	—	73	1	no	—
58	C₂×C₃₂	—	146	2	no	—
59	C₂×C₃₂	#64408	1	1	yes	characteristic
60	#368	—	73	4	no	—
61	#368	—	146	2	no	—
62	C₂×M₁₆	—	292	2	no	—
63	C₂²×C₈	—	73	1	no	—
64	C₂×C₁₆	—	73	1	no	—
65	C₂×C₁₆	—	73	2	no	—
66	C₂×C₁₆	—	73	2	no	—
67	C₂×C₁₆	C₂×#64408	1	1	yes	characteristic
68	#108	—	292	2	no	—
69	#108	—	146	4	no	—
70	C₃₂	—	2	1	no	—
71	C₃₂	—	146	1	no	—
72	C₂²×C₄	—	73	1	no	—
73	M₁₆	—	292	4	no	—
74	M₁₆	—	584	2	no	—
75	C₂×C₈	—	584	2	no	—
76	C₂×C₈	—	73	1	no	—
77	C₂×C₈	—	73	2	no	—
78	C₂×C₈	—	73	2	no	—
79	C₂×C₈	C₄⋊#64408	1	1	yes	characteristic
80	C₁₆	—	73	2	no	—
81	C₁₆	C₄×#64408	1	1	yes	characteristic
82	C₁₆	C₂²⋊#64408	1	1	yes	characteristic
83	C₂³	—	73	1	no	—
84	C₂×C₄	—	73	1	no	—
85	C₂×C₄	—	73	2	no	—
86	C₂×C₄	—	73	2	no	—
87	C₂×C₄	#14556660	1	1	yes	characteristic
88	C₈	—	584	4	no	—
89	C₈	—	73	2	no	—
90	C₈	(C₂×#64408)⋊C₄	1	1	yes	characteristic
91	C₈	#14556659	1	1	yes	characteristic
92	C₂²	—	73	2	no	—
93	C₂²	—	146	1	no	—
94	C₂²	—	146	1	no	—
95	C₂²	#17989513	1	1	yes	characteristic
96	C₄	—	73	2	no	—
97	C₄	#17989461	1	1	yes	characteristic
98	C₄	#17989514	1	1	yes	characteristic
99	C₂	—	146	2	no	—
100	C₂	#22624989	1	1	yes	characteristic
101	C₂	#22625094	1	1	yes	characteristic
102	C₂	#22625095	1	1	yes	characteristic

subgroup

G/H

class size

multiplicity

normal?

properties

C₂×#17989515

C₂

yes

maximal

C₂×C₃₂×D₇₃

C₂

yes

maximal, characteristic

C₂×#14556662

—

#17989515

—

#17989515

C₄

yes

—

C₂×C₁₆×D₇₃

C₂²

yes

characteristic

C₂×#14556497

C₄

yes

characteristic

C₃₂×D₇₃

—

C₂×C₃₂×C₇₃

C₄

yes

—

—

—

C₂×C₈×D₇₃

D₄

yes

characteristic

C₂×#11924003

Q₈

yes

characteristic

C₁₆×D₇₃

C₂×C₄

yes

—

C₁₆×D₇₃

D₄

yes

—

#14556497

—

C₂×C₁₆×C₇₃

C₂×C₄

yes

characteristic

C₃₂×C₇₃

—

#11924038

—

#11924038

—

C₂×#10596833

—

C₂×C₄×D₇₃

D₈

yes

characteristic

C₂×#10596831

Q₁₆

yes

characteristic

C₈×D₇₃

C₂²⋊C₄

yes

—

C₈×D₇₃

SD₁₆

yes

—

#11924003

C₄⋊C₄

yes

—

C₂×C₈×C₇₃

C₄⋊C₄

yes

characteristic

C₁₆×C₇₃

C₄²

yes

characteristic

C₁₆×C₇₃

C₂²⋊C₄

yes

characteristic

C₂²×D₇₃

Q₃₂

yes

characteristic

C₂×C₄×C₇₃

#107

yes

characteristic

C₂×Dic₇₃

D₁₆

yes

characteristic

C₄×D₇₃

#102

yes

—

C₄×D₇₃

SD₃₂

yes

—

#10596833

—

C₈×C₇₃

yes

C₈×C₇₃

yes

yes

—

C₂²×C₇₃

#366

yes

characteristic

C₂×D₇₃

—

C₂×D₇₃

—

C₂×D₇₃

#358

yes

—

C₄×C₇₃

yes

C₄×C₇₃

yes

Dic₇₃

yes

—

—

C₂×C₇₃

yes

C₂×C₇₃

yes

C₂×C₇₃

yes

D₇₃

—

C₂²×C₃₂

—

C₂×#368

—

C₇₃

#7472

yes

characteristic, complemented, Sylow

C₂²×C₁₆

—

C₂×#108

—

146

—

C₂×C₃₂

—

C₂×C₃₂

—

146

—

C₂×C₃₂

yes

—

—

146

—

C₂×M₁₆

—

292

—

C₂²×C₈

—

C₂×C₁₆

—

C₂×C₁₆

—

C₂×C₁₆

—

C₂×C₁₆

C₂×#64408

yes

characteristic

#108

—

292

—

#108

—

146

—

C₃₂

—

C₃₂

—

146

—

C₂²×C₄

—

M₁₆

—

292

—

M₁₆

—

584

—

C₂×C₈

—

584

—

C₂×C₈

—

C₂×C₈

—

C₂×C₈

—

C₂×C₈

C₄⋊#64408

yes

characteristic

C₁₆

—

C₁₆

yes

C₁₆

yes

C₂³

—

C₂×C₄

—

C₂×C₄

—

C₂×C₄

—

C₂×C₄

#14556660

yes

characteristic

C₈

—

584

—

C₈

—

C₈

yes

C₈

yes

C₂²

—

C₂²

—

146

—

C₂²

—

146

—

C₂²

#17989513

yes

characteristic

C₄

—

C₄

yes

C₄

yes

C₂

—

146

—

100

C₂

#22624989

yes

characteristic

101

C₂

#22625094

yes

characteristic

102

C₂

#22625095

yes

characteristic

	subgroup	G/H
center	C₂²	#17989513
commutator	C₁₆×C₇₃	C₄²
Frattini	C₂×C₁₆	C₂×#64408
Fitting	C₂×C₃₂×C₇₃	C₄
socle	C₂²×C₇₃	#366
prime radical	C₂²×D₇₃	Q₃₂
cosocle	C₂×C₁₆×D₇₃	C₂²
hypercenter	C₂×C₃₂	#64408
hypocenter	C₇₃	#7472
LCS	G ▷ C₁₆×C₇₃ ▷ C₈×C₇₃ ▷ C₄×C₇₃ ▷ C₂×C₇₃ ▷ C₇₃
UCS	C₁ ◁ C₂² ◁ C₂×C₄ ◁ C₂×C₈ ◁ C₂×C₁₆ ◁ C₂×C₃₂

subgroup

G/H

center

C₂²

#17989513

commutator

C₁₆×C₇₃

C₄²

Frattini

C₂×C₁₆

C₂×#64408

Fitting

C₂×C₃₂×C₇₃

C₄

socle

C₂²×C₇₃

#366

prime radical

C₂²×D₇₃

Q₃₂

cosocle

C₂×C₁₆×D₇₃

C₂²

C₂×C₃₂

C₇₃

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

UCS

C₁ ◁ C₂² ◁ C₂×C₄ ◁ C₂×C₈ ◁ C₂×C₁₆ ◁ C₂×C₃₂

N	H ≅ G/N	type
54. C₇₃	#7472	semidirect

H ≅ G/N

type

54. C₇₃

#7472

semidirect

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

subgroup	index
C₂×#17989515	2
C₂×C₃₂×D₇₃	2
#7472	73

subgroup

index

C₂×#17989515

C₂×C₃₂×D₇₃

#7472

Subgroups — #34001073

Conjugacy classes of subgroups

Complements

Maximal subgroups