Group #32918219 — subgroups

Subgroups — C₇₃×#8815 #32918219

Order 18688.

422 conjugacy classes of subgroups; only the 238 normal subgroups are shown.

normal subgroups only

This group has 422 conjugacy classes of subgroups — too many to draw. Only its 238 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	C₇₃×#1844	C₂	1	2	yes	maximal
2	C₇₃×#1834	C₂	1	4	yes	maximal
3	C₂×C₈²×C₇₃	C₂	1	1	yes	maximal, characteristic
4	C₂×#106×C₇₃	—	2	4	no	—
5	C₂×C₄×C₈×C₇₃	C₂²	1	2	yes	—
6	C₂×C₄×C₈×C₇₃	C₂²	1	1	yes	characteristic
7	#384×C₇₃	C₂²	1	4	yes	—
8	#340×C₇₃	—	2	8	no	—
9	C₈²×C₇₃	C₄	1	4	yes	complemented
10	C₂²×C₈×C₇₃	D₄	1	4	yes	—
11	C₂²×C₈×C₇₃	D₄	1	2	yes	—
12	C₂×C₄⋊C₄×C₇₃	—	2	4	no	—
13	C₂×C₄⋊C₄×C₇₃	—	2	8	no	—
14	C₂×C₄²×C₇₃	C₂³	1	1	yes	characteristic
15	#106×C₇₃	—	4	8	no	—
16	C₄×C₈×C₇₃	C₂×C₄	1	4	yes	—
17	C₄×C₈×C₇₃	C₂×C₄	1	2	yes	—
18	C₄×C₈×C₇₃	Q₈	1	4	yes	—
19	C₄×C₈×C₇₃	Q₈	1	2	yes	—
20	C₂²×C₄×C₇₃	—	4	4	no	—
21	C₂²×C₄×C₇₃	C₂×D₄	1	1	yes	characteristic
22	C₂²×C₄×C₇₃	C₂×D₄	1	2	yes	—
23	C₂×C₈×C₇₃	—	2	8	no	—
24	C₂×C₈×C₇₃	C₄⋊C₄	1	8	yes	—
25	C₂×C₈×C₇₃	C₄⋊C₄	1	4	yes	—
26	C₂×C₈×C₇₃	SD₁₆	1	8	yes	—
27	C₄⋊C₄×C₇₃	—	4	8	no	—
28	C₄⋊C₄×C₇₃	—	4	16	no	—
29	C₄²×C₇₃	C₂²×C₄	1	1	yes	characteristic
30	C₄²×C₇₃	C₂×Q₈	1	1	yes	characteristic
31	C₄²×C₇₃	C₂×Q₈	1	2	yes	—
32	C₂³×C₇₃	#127	1	1	yes	characteristic
33	C₂×C₄×C₇₃	—	8	4	no	—
34	C₂×C₄×C₇₃	—	8	8	no	—
35	C₂×C₄×C₇₃	C₂×C₄⋊C₄	1	1	yes	characteristic
36	C₂×C₄×C₇₃	C₂×C₄⋊C₄	1	2	yes	—
37	C₂×C₄×C₇₃	C₄⋊Q₈	1	1	yes	characteristic
38	C₂×C₄×C₇₃	C₄⋊Q₈	1	2	yes	—
39	C₂×C₄×C₇₃	C₂×SD₁₆	1	4	yes	—
40	C₂×C₄×C₇₃	#135	1	8	yes	—
41	C₈×C₇₃	—	2	8	no	—
42	C₈×C₇₃	#106	1	8	yes	—
43	C₂²×C₇₃	#384	1	1	yes	characteristic
44	C₂²×C₇₃	#492	1	2	yes	—
45	C₂²×C₇₃	#495	1	4	yes	—
46	C₄×C₇₃	—	16	4	no	—
47	C₄×C₇₃	C₂×#106	1	2	yes	—
48	C₄×C₇₃	#427	1	4	yes	—
49	C₄×C₇₃	#498	1	2	yes	—
50	C₄×C₇₃	#499	1	4	yes	—
51	#8815	C₇₃	1	1	yes	maximal, characteristic, direct product factor, Sylow
52	C₂×C₇₃	#1708	1	4	yes	—
53	C₂×C₇₃	#1844	1	1	yes	characteristic
54	C₂×C₇₃	#1846	1	2	yes	—
55	#1844	C₂×C₇₃	1	2	yes	—
56	#1834	C₂×C₇₃	1	4	yes	—
57	C₂×C₈²	C₂×C₇₃	1	1	yes	characteristic
58	C₇₃	#8815	1	1	yes	characteristic, direct product factor, Sylow
59	C₂×#106	—	2	4	no	—
60	C₂×C₄×C₈	C₂²×C₇₃	1	2	yes	—
61	C₂×C₄×C₈	C₂²×C₇₃	1	1	yes	characteristic
62	#384	C₂²×C₇₃	1	4	yes	—
63	#340	—	2	8	no	—
64	C₈²	C₄×C₇₃	1	4	yes	complemented
65	C₂²×C₈	D₄×C₇₃	1	4	yes	—
66	C₂²×C₈	D₄×C₇₃	1	2	yes	—
67	C₂×C₄⋊C₄	—	2	8	no	—
68	C₂×C₄⋊C₄	—	2	4	no	—
69	C₂×C₄²	C₂³×C₇₃	1	1	yes	characteristic
70	#106	—	4	8	no	—
71	C₄×C₈	C₂×C₄×C₇₃	1	4	yes	—
72	C₄×C₈	C₂×C₄×C₇₃	1	2	yes	—
73	C₄×C₈	Q₈×C₇₃	1	4	yes	—
74	C₄×C₈	Q₈×C₇₃	1	2	yes	—
75	C₂²×C₄	—	4	4	no	—
76	C₂²×C₄	C₂×D₄×C₇₃	1	1	yes	characteristic
77	C₂²×C₄	C₂×D₄×C₇₃	1	2	yes	—
78	C₂×C₈	—	2	8	no	—
79	C₂×C₈	C₄⋊C₄×C₇₃	1	8	yes	—
80	C₂×C₈	C₄⋊C₄×C₇₃	1	4	yes	—
81	C₂×C₈	SD₁₆×C₇₃	1	8	yes	—
82	C₄⋊C₄	—	4	8	no	—
83	C₄⋊C₄	—	4	16	no	—
84	C₄²	C₂²×C₄×C₇₃	1	1	yes	characteristic
85	C₄²	C₂×Q₈×C₇₃	1	1	yes	characteristic
86	C₄²	C₂×Q₈×C₇₃	1	2	yes	—
87	C₂³	#127×C₇₃	1	1	yes	characteristic
88	C₂×C₄	—	8	4	no	—
89	C₂×C₄	—	8	8	no	—
90	C₂×C₄	C₂×C₄⋊C₄×C₇₃	1	1	yes	characteristic
91	C₂×C₄	C₂×C₄⋊C₄×C₇₃	1	2	yes	—
92	C₂×C₄	C₄⋊Q₈×C₇₃	1	1	yes	characteristic
93	C₂×C₄	C₄⋊Q₈×C₇₃	1	2	yes	—
94	C₂×C₄	C₂×SD₁₆×C₇₃	1	4	yes	—
95	C₂×C₄	#135×C₇₃	1	8	yes	—
96	C₈	—	2	8	no	—
97	C₈	#106×C₇₃	1	8	yes	—
98	C₂²	#384×C₇₃	1	1	yes	characteristic
99	C₂²	#492×C₇₃	1	2	yes	—
100	C₂²	#495×C₇₃	1	4	yes	—
101	C₄	—	16	4	no	—
102	C₄	C₂×#106×C₇₃	1	2	yes	—
103	C₄	#427×C₇₃	1	4	yes	—
104	C₄	#498×C₇₃	1	2	yes	—
105	C₄	#499×C₇₃	1	4	yes	—
106	C₂	C₇₃×#1708	1	4	yes	—
107	C₂	C₇₃×#1844	1	1	yes	characteristic
108	C₂	C₇₃×#1846	1	2	yes	—

subgroup

G/H

class size

multiplicity

normal?

properties

C₇₃×#1844

C₂

yes

maximal

C₇₃×#1834

C₂

yes

maximal

C₂×C₈²×C₇₃

C₂

yes

maximal, characteristic

C₂×#106×C₇₃

—

C₂×C₄×C₈×C₇₃

C₂²

yes

—

C₂×C₄×C₈×C₇₃

C₂²

yes

characteristic

#384×C₇₃

C₂²

yes

—

#340×C₇₃

—

C₈²×C₇₃

C₄

yes

complemented

C₂²×C₈×C₇₃

D₄

yes

—

C₂²×C₈×C₇₃

D₄

yes

—

C₂×C₄⋊C₄×C₇₃

—

C₂×C₄⋊C₄×C₇₃

—

C₂×C₄²×C₇₃

C₂³

yes

characteristic

#106×C₇₃

—

C₄×C₈×C₇₃

C₂×C₄

yes

—

C₄×C₈×C₇₃

C₂×C₄

yes

—

C₄×C₈×C₇₃

Q₈

yes

—

C₄×C₈×C₇₃

Q₈

yes

—

C₂²×C₄×C₇₃

—

C₂²×C₄×C₇₃

C₂×D₄

yes

characteristic

C₂²×C₄×C₇₃

C₂×D₄

yes

—

C₂×C₈×C₇₃

—

C₂×C₈×C₇₃

C₄⋊C₄

yes

—

C₂×C₈×C₇₃

C₄⋊C₄

yes

—

C₂×C₈×C₇₃

SD₁₆

yes

—

C₄⋊C₄×C₇₃

—

C₄⋊C₄×C₇₃

—

C₄²×C₇₃

C₂²×C₄

yes

characteristic

C₄²×C₇₃

C₂×Q₈

yes

characteristic

C₄²×C₇₃

C₂×Q₈

yes

—

C₂³×C₇₃

#127

yes

characteristic

C₂×C₄×C₇₃

—

C₂×C₄×C₇₃

—

C₂×C₄×C₇₃

C₂×C₄⋊C₄

yes

characteristic

C₂×C₄×C₇₃

C₂×C₄⋊C₄

yes

—

C₂×C₄×C₇₃

C₄⋊Q₈

yes

characteristic

C₂×C₄×C₇₃

C₄⋊Q₈

yes

—

C₂×C₄×C₇₃

C₂×SD₁₆

yes

—

C₂×C₄×C₇₃

#135

yes

—

C₈×C₇₃

—

C₈×C₇₃

#106

yes

—

C₂²×C₇₃

#384

yes

characteristic

C₂²×C₇₃

#492

yes

—

C₂²×C₇₃

#495

yes

—

C₄×C₇₃

—

C₄×C₇₃

C₂×#106

yes

—

C₄×C₇₃

#427

yes

—

C₄×C₇₃

#498

yes

—

C₄×C₇₃

#499

yes

—

#8815

C₇₃

yes

maximal, characteristic, direct product factor, Sylow

C₂×C₇₃

#1708

yes

—

C₂×C₇₃

#1844

yes

characteristic

C₂×C₇₃

#1846

yes

—

#1844

C₂×C₇₃

yes

—

#1834

C₂×C₇₃

yes

—

C₂×C₈²

C₂×C₇₃

yes

characteristic

C₇₃

#8815

yes

characteristic, direct product factor, Sylow

C₂×#106

—

C₂×C₄×C₈

C₂²×C₇₃

yes

—

C₂×C₄×C₈

C₂²×C₇₃

yes

characteristic

#384

C₂²×C₇₃

yes

—

#340

—

C₈²

C₄×C₇₃

yes

complemented

C₂²×C₈

D₄×C₇₃

yes

—

C₂²×C₈

D₄×C₇₃

yes

—

C₂×C₄⋊C₄

—

C₂×C₄⋊C₄

—

C₂×C₄²

C₂³×C₇₃

yes

characteristic

#106

—

C₄×C₈

C₂×C₄×C₇₃

yes

—

C₄×C₈

C₂×C₄×C₇₃

yes

—

C₄×C₈

Q₈×C₇₃

yes

—

C₄×C₈

Q₈×C₇₃

yes

—

C₂²×C₄

—

C₂²×C₄

C₂×D₄×C₇₃

yes

characteristic

C₂²×C₄

C₂×D₄×C₇₃

yes

—

C₂×C₈

—

C₂×C₈

C₄⋊C₄×C₇₃

yes

—

C₂×C₈

C₄⋊C₄×C₇₃

yes

—

C₂×C₈

SD₁₆×C₇₃

yes

—

C₄⋊C₄

—

C₄⋊C₄

—

C₄²

C₂²×C₄×C₇₃

yes

characteristic

C₄²

C₂×Q₈×C₇₃

yes

characteristic

C₄²

C₂×Q₈×C₇₃

yes

—

C₂³

#127×C₇₃

yes

characteristic

C₂×C₄

—

C₂×C₄

—

C₂×C₄

C₂×C₄⋊C₄×C₇₃

yes

characteristic

C₂×C₄

C₂×C₄⋊C₄×C₇₃

yes

—

C₂×C₄

C₄⋊Q₈×C₇₃

yes

characteristic

C₂×C₄

C₄⋊Q₈×C₇₃

yes

—

C₂×C₄

C₂×SD₁₆×C₇₃

yes

—

C₂×C₄

#135×C₇₃

yes

—

C₈

—

C₈

#106×C₇₃

yes

—

C₂²

#384×C₇₃

yes

characteristic

C₂²

#492×C₇₃

yes

—

100

C₂²

#495×C₇₃

yes

—

101

C₄

—

102

C₄

C₂×#106×C₇₃

yes

—

103

C₄

#427×C₇₃

yes

—

104

C₄

#498×C₇₃

yes

—

105

C₄

#499×C₇₃

yes

—

106

C₂

C₇₃×#1708

yes

—

107

C₂

C₇₃×#1844

yes

characteristic

108

C₂

C₇₃×#1846

yes

—

	subgroup	G/H
center, socle, prime radical	C₂³×C₇₃	#127
commutator	C₄²	C₂²×C₄×C₇₃
Frattini, cosocle	C₂×C₄²	C₂³×C₇₃
LCS	G ▷ C₄² ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂³×C₇₃ ◁ C₂×C₄²×C₇₃ ◁ G

subgroup

G/H

center, socle, prime radical

C₂³×C₇₃

#127

commutator

C₄²

C₂²×C₄×C₇₃

Frattini, cosocle

C₂×C₄²

C₂³×C₇₃

LCS

G ▷ C₄² ▷ C₂² ▷ C₁

UCS

C₁ ◁ C₂³×C₇₃ ◁ C₂×C₄²×C₇₃ ◁ G

N	H ≅ G/N	type
9. C₈²×C₇₃	C₄	semidirect
51. #8815	C₇₃	direct (N × H)
58. C₇₃	#8815	direct (N × H)
64. C₈²	C₄×C₇₃	semidirect

H ≅ G/N

type

9. C₈²×C₇₃

C₄

semidirect

51. #8815

C₇₃

direct (N × H)

58. C₇₃

#8815

direct (N × H)

64. C₈²

C₄×C₇₃

semidirect

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

subgroup	index
C₇₃×#1844	2
C₇₃×#1834	2
C₂×C₈²×C₇₃	2
#8815	73

subgroup

index

C₇₃×#1844

C₇₃×#1834

C₂×C₈²×C₇₃

#8815

Subgroups — C₇₃×#8815 #32918219

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C73×#8815 #32918219

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₇₃×#8815 #32918219