Group #32916643 — subgroups

Subgroups — C₇₃×#7239 #32916643

Order 18688.

148 conjugacy classes of subgroups; only the 70 normal subgroups are shown.

normal subgroups only

This group has 148 conjugacy classes of subgroups — too many to draw. Only its 70 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×#2166	C₂	1	2	yes	maximal
2	C₇₃×#2151	C₂	1	1	yes	maximal, characteristic
3	C₂×M₃₂×C₇₃	C₄	1	1	yes	characteristic
4	C₂²×C₁₆×C₇₃	C₄	1	1	yes	characteristic
5	#405×C₇₃	C₂²	1	1	yes	characteristic
6	#364×C₇₃	—	2	2	no	—
7	C₄⋊C₁₆×C₇₃	—	2	2	no	—
8	#349×C₇₃	—	2	2	no	—
9	#347×C₇₃	C₄	1	4	yes	—
10	C₂×M₁₆×C₇₃	D₄	1	1	yes	characteristic
11	C₂²×C₈×C₇₃	C₂×C₄	1	1	yes	characteristic
12	#117×C₇₃	Q₈	1	1	yes	characteristic
13	M₃₂×C₇₃	—	2	4	no	—
14	M₃₂×C₇₃	—	4	1	no	—
15	C₂×C₁₆×C₇₃	—	2	1	no	—
16	C₂×C₁₆×C₇₃	—	2	1	no	—
17	C₂×C₁₆×C₇₃	—	4	1	no	—
18	#97×C₇₃	C₂×C₄	1	2	yes	—
19	C₄×C₈×C₇₃	D₄	1	2	yes	—
20	C₂²×C₄×C₇₃	C₄⋊C₄	1	1	yes	characteristic
21	M₁₆×C₇₃	—	4	1	no	—
22	C₂×C₈×C₇₃	—	2	1	no	—
23	C₂×C₈×C₇₃	—	2	1	no	—
24	C₂×C₈×C₇₃	C₄²	1	1	yes	characteristic
25	C₂×C₈×C₇₃	C₂²⋊C₄	1	2	yes	—
26	C₂×C₈×C₇₃	C₂²⋊C₄	1	1	yes	characteristic
27	C₄⋊C₄×C₇₃	—	2	1	no	—
28	C₂²⋊C₄×C₇₃	—	2	1	no	—
29	C₄²×C₇₃	C₄⋊C₄	1	2	yes	—
30	C₁₆×C₇₃	—	4	4	no	—
31	C₁₆×C₇₃	—	4	1	no	—
32	C₁₆×C₇₃	—	4	1	no	—
33	C₂³×C₇₃	#108	1	1	yes	characteristic
34	C₂×C₄×C₇₃	—	2	1	no	—
35	C₂×C₄×C₇₃	—	2	1	no	—
36	C₂×C₄×C₇₃	—	2	2	no	—
37	C₂×C₄×C₇₃	#95	1	1	yes	characteristic
38	C₂×C₄×C₇₃	#108	1	1	yes	characteristic
39	C₈×C₇₃	—	2	2	no	—
40	C₈×C₇₃	—	2	1	no	—
41	C₈×C₇₃	#100	1	1	yes	characteristic
42	C₈×C₇₃	#101	1	1	yes	characteristic
43	C₂²×C₇₃	—	2	1	no	—
44	C₂²×C₇₃	—	4	1	no	—
45	C₂²×C₇₃	#341	1	1	yes	characteristic
46	C₄×C₇₃	—	2	1	no	—
47	C₄×C₇₃	—	4	2	no	—
48	C₄×C₇₃	H(3,ℤ/4)	1	1	yes	characteristic
49	C₄×C₇₃	#343	1	1	yes	characteristic
50	#7239	C₇₃	1	1	yes	maximal, characteristic, direct product factor, Sylow
51	C₂×C₇₃	—	2	1	no	—
52	C₂×C₇₃	—	4	1	no	—
53	C₂×C₇₃	#1290	1	1	yes	characteristic
54	#2166	C₂×C₇₃	1	2	yes	—
55	#2151	C₂×C₇₃	1	1	yes	characteristic
56	C₇₃	#7239	1	1	yes	characteristic, direct product factor, Sylow
57	C₂×M₃₂	C₄×C₇₃	1	1	yes	characteristic
58	C₂²×C₁₆	C₄×C₇₃	1	1	yes	characteristic
59	#405	C₂²×C₇₃	1	1	yes	characteristic
60	#364	—	2	2	no	—
61	C₄⋊C₁₆	—	2	2	no	—
62	#349	—	2	2	no	—
63	#347	C₄×C₇₃	1	4	yes	—
64	C₂×M₁₆	D₄×C₇₃	1	1	yes	characteristic
65	C₂²×C₈	C₂×C₄×C₇₃	1	1	yes	characteristic
66	#117	Q₈×C₇₃	1	1	yes	characteristic
67	M₃₂	—	2	4	no	—
68	M₃₂	—	4	1	no	—
69	C₂×C₁₆	—	2	1	no	—
70	C₂×C₁₆	—	2	1	no	—
71	C₂×C₁₆	—	4	1	no	—
72	#97	C₂×C₄×C₇₃	1	2	yes	—
73	C₄×C₈	D₄×C₇₃	1	2	yes	—
74	C₂²×C₄	C₄⋊C₄×C₇₃	1	1	yes	characteristic
75	M₁₆	—	4	1	no	—
76	C₂×C₈	—	2	1	no	—
77	C₂×C₈	—	2	1	no	—
78	C₂×C₈	C₄²×C₇₃	1	1	yes	characteristic
79	C₂×C₈	C₂²⋊C₄×C₇₃	1	2	yes	—
80	C₂×C₈	C₂²⋊C₄×C₇₃	1	1	yes	characteristic
81	C₄⋊C₄	—	2	1	no	—
82	C₂²⋊C₄	—	2	1	no	—
83	C₄²	C₄⋊C₄×C₇₃	1	2	yes	—
84	C₁₆	—	4	4	no	—
85	C₁₆	—	4	1	no	—
86	C₁₆	—	4	1	no	—
87	C₂³	#108×C₇₃	1	1	yes	characteristic
88	C₂×C₄	—	2	1	no	—
89	C₂×C₄	—	2	1	no	—
90	C₂×C₄	—	2	2	no	—
91	C₂×C₄	#95×C₇₃	1	1	yes	characteristic
92	C₂×C₄	#108×C₇₃	1	1	yes	characteristic
93	C₈	—	2	2	no	—
94	C₈	—	2	1	no	—
95	C₈	#100×C₇₃	1	1	yes	characteristic
96	C₈	#101×C₇₃	1	1	yes	characteristic
97	C₂²	—	2	1	no	—
98	C₂²	—	4	1	no	—
99	C₂²	#341×C₇₃	1	1	yes	characteristic
100	C₄	—	2	1	no	—
101	C₄	—	4	2	no	—
102	C₄	H(3,ℤ/4)×C₇₃	1	1	yes	characteristic
103	C₄	#343×C₇₃	1	1	yes	characteristic
104	C₂	—	2	1	no	—
105	C₂	—	4	1	no	—
106	C₂	C₇₃×#1290	1	1	yes	characteristic

subgroup

G/H

class size

multiplicity

normal?

properties

C₇₃×#2166

C₂

yes

maximal

C₇₃×#2151

C₂

yes

maximal, characteristic

C₂×M₃₂×C₇₃

C₄

yes

characteristic

C₂²×C₁₆×C₇₃

C₄

yes

characteristic

#405×C₇₃

C₂²

yes

characteristic

#364×C₇₃

—

C₄⋊C₁₆×C₇₃

—

#349×C₇₃

—

#347×C₇₃

C₄

yes

—

C₂×M₁₆×C₇₃

D₄

yes

characteristic

C₂²×C₈×C₇₃

C₂×C₄

yes

characteristic

#117×C₇₃

Q₈

yes

characteristic

M₃₂×C₇₃

—

M₃₂×C₇₃

—

C₂×C₁₆×C₇₃

—

C₂×C₁₆×C₇₃

—

C₂×C₁₆×C₇₃

—

#97×C₇₃

C₂×C₄

yes

—

C₄×C₈×C₇₃

D₄

yes

—

C₂²×C₄×C₇₃

C₄⋊C₄

yes

characteristic

M₁₆×C₇₃

—

C₂×C₈×C₇₃

—

C₂×C₈×C₇₃

—

C₂×C₈×C₇₃

C₄²

yes

characteristic

C₂×C₈×C₇₃

C₂²⋊C₄

yes

—

C₂×C₈×C₇₃

C₂²⋊C₄

yes

characteristic

C₄⋊C₄×C₇₃

—

C₂²⋊C₄×C₇₃

—

C₄²×C₇₃

C₄⋊C₄

yes

—

C₁₆×C₇₃

—

C₁₆×C₇₃

—

C₁₆×C₇₃

—

C₂³×C₇₃

#108

yes

characteristic

C₂×C₄×C₇₃

—

C₂×C₄×C₇₃

—

C₂×C₄×C₇₃

—

C₂×C₄×C₇₃

yes

C₂×C₄×C₇₃

yes

C₈×C₇₃

—

C₈×C₇₃

—

C₈×C₇₃

yes

C₈×C₇₃

yes

C₂²×C₇₃

—

C₂²×C₇₃

—

C₂²×C₇₃

#341

yes

characteristic

C₄×C₇₃

—

C₄×C₇₃

—

C₄×C₇₃

yes

C₄×C₇₃

yes

C₇₃

yes

maximal, characteristic, direct product factor, Sylow

C₂×C₇₃

—

C₂×C₇₃

—

C₂×C₇₃

#1290

yes

characteristic

#2166

C₂×C₇₃

yes

—

#2151

C₂×C₇₃

yes

characteristic

C₇₃

#7239

yes

characteristic, direct product factor, Sylow

C₂×M₃₂

C₄×C₇₃

yes

characteristic

C₂²×C₁₆

C₄×C₇₃

yes

characteristic

#405

C₂²×C₇₃

yes

characteristic

#364

—

C₄⋊C₁₆

—

#349

—

#347

C₄×C₇₃

yes

—

C₂×M₁₆

D₄×C₇₃

yes

characteristic

C₂²×C₈

C₂×C₄×C₇₃

yes

characteristic

#117

Q₈×C₇₃

yes

characteristic

M₃₂

—

M₃₂

—

C₂×C₁₆

—

C₂×C₁₆

—

C₂×C₁₆

—

#97

C₂×C₄×C₇₃

yes

—

C₄×C₈

D₄×C₇₃

yes

—

C₂²×C₄

C₄⋊C₄×C₇₃

yes

characteristic

M₁₆

—

C₂×C₈

—

C₂×C₈

—

C₂×C₈

C₄²×C₇₃

yes

characteristic

C₂×C₈

C₂²⋊C₄×C₇₃

yes

—

C₂×C₈

C₂²⋊C₄×C₇₃

yes

characteristic

C₄⋊C₄

—

C₂²⋊C₄

—

C₄²

C₄⋊C₄×C₇₃

yes

—

C₁₆

—

C₁₆

—

C₁₆

—

C₂³

#108×C₇₃

yes

characteristic

C₂×C₄

—

C₂×C₄

—

C₂×C₄

—

C₂×C₄

#95×C₇₃

yes

characteristic

C₂×C₄

#108×C₇₃

yes

characteristic

C₈

—

C₈

—

C₈

#100×C₇₃

yes

characteristic

C₈

#101×C₇₃

yes

characteristic

C₂²

—

C₂²

—

C₂²

#341×C₇₃

yes

characteristic

100

C₄

—

101

C₄

—

102

C₄

H(3,ℤ/4)×C₇₃

yes

characteristic

103

C₄

#343×C₇₃

yes

characteristic

104

C₂

—

105

C₂

—

106

C₂

C₇₃×#1290

yes

characteristic

	subgroup	G/H
center	C₄×C₇₃	H(3,ℤ/4)
commutator	C₂×C₈	C₄²×C₇₃
Frattini, cosocle	#405	C₂²×C₇₃
socle	C₂×C₇₃	#1290
prime radical	C₂³×C₇₃	#108
LCS	G ▷ C₂×C₈ ▷ C₄ ▷ C₁
UCS	C₁ ◁ C₄×C₇₃ ◁ C₂×C₈×C₇₃ ◁ G

subgroup

G/H

center

C₄×C₇₃

H(3,ℤ/4)

commutator

C₂×C₈

C₄²×C₇₃

C₂²×C₇₃

C₂×C₇₃

C₂³×C₇₃

G ▷ C₂×C₈ ▷ C₄ ▷ C₁

C₁ ◁ C₄×C₇₃ ◁ C₂×C₈×C₇₃ ◁ G

N	H ≅ G/N	type
50. #7239	C₇₃	direct (N × H)
56. C₇₃	#7239	direct (N × H)

H ≅ G/N

type

50. #7239

C₇₃

direct (N × H)

56. C₇₃

#7239

direct (N × H)

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

subgroup	index
C₇₃×#2166	2
C₇₃×#2151	2
#7239	73

subgroup

index

C₇₃×#2166

C₇₃×#2151

#7239

Subgroups — C₇₃×#7239 #32916643

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C73×#7239 #32916643

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₇₃×#7239 #32916643