Group #32916770 — subgroups

Subgroups — C₇₃×#7366 #32916770

Order 18688.

158 conjugacy classes of subgroups; only the 50 normal subgroups are shown.

normal subgroups only

This group has 158 conjugacy classes of subgroups — too many to draw. Only its 50 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×#1835	C₂	1	1	yes	maximal, characteristic
2	C₇₃×#1324	C₂	1	2	yes	maximal
3	C₂×C₄×C₈×C₇₃	C₄	1	1	yes	characteristic
4	#382×C₇₃	C₄	1	1	yes	characteristic
5	#382×C₇₃	C₂²	1	1	yes	characteristic
6	#340×C₇₃	—	2	2	no	—
7	#324×C₇₃	—	2	2	no	—
8	C₂²×C₈×C₇₃	—	2	1	no	—
9	C₂×C₄⋊C₄×C₇₃	—	2	1	no	—
10	C₂×C₄⋊C₄×C₇₃	D₄	1	2	yes	—
11	C₂×C₄²×C₇₃	C₂×C₄	1	1	yes	characteristic
12	C₂²⋊C₈×C₇₃	—	4	2	no	—
13	C₄×C₈×C₇₃	—	2	1	no	—
14	C₄×C₈×C₇₃	C₈	1	2	yes	complemented
15	#95×C₇₃	—	4	1	no	—
16	#95×C₇₃	—	4	1	no	—
17	C₂²×C₄×C₇₃	—	2	1	no	—
18	C₂²×C₄×C₇₃	—	2	2	no	—
19	C₂²×C₄×C₇₃	—	4	1	no	—
20	C₂²×C₄×C₇₃	C₂²⋊C₄	1	1	yes	characteristic
21	C₂×C₈×C₇₃	—	2	2	no	—
22	C₂×C₈×C₇₃	—	4	1	no	—
23	C₂×C₈×C₇₃	—	4	1	no	—
24	C₂×C₈×C₇₃	—	8	2	no	—
25	C₄⋊C₄×C₇₃	—	4	2	no	—
26	C₄⋊C₄×C₇₃	—	4	2	no	—
27	C₄²×C₇₃	—	2	1	no	—
28	C₄²×C₇₃	C₂×C₈	1	1	yes	characteristic
29	C₄²×C₇₃	M₁₆	1	1	yes	characteristic
30	C₂³×C₇₃	#99	1	1	yes	characteristic
31	C₂×C₄×C₇₃	—	2	1	no	—
32	C₂×C₄×C₇₃	—	2	1	no	—
33	C₂×C₄×C₇₃	—	2	1	no	—
34	C₂×C₄×C₇₃	—	2	1	no	—
35	C₂×C₄×C₇₃	—	4	2	no	—
36	C₂×C₄×C₇₃	—	4	2	no	—
37	C₂×C₄×C₇₃	—	8	2	no	—
38	C₂×C₄×C₇₃	—	8	2	no	—
39	C₂×C₄×C₇₃	—	8	1	no	—
40	C₂×C₄×C₇₃	C₂²⋊C₈	1	1	yes	characteristic
41	C₂×C₄×C₇₃	#101	1	1	yes	characteristic
42	C₈×C₇₃	—	4	2	no	—
43	C₈×C₇₃	—	16	2	no	—
44	C₂²×C₇₃	—	2	2	no	—
45	C₂²×C₇₃	#324	1	1	yes	characteristic
46	C₂²×C₇₃	#354	1	1	yes	characteristic
47	C₂²×C₇₃	#355	1	1	yes	characteristic
48	C₄×C₇₃	—	2	1	no	—
49	C₄×C₇₃	—	2	1	no	—
50	C₄×C₇₃	—	4	1	no	—
51	C₄×C₇₃	—	4	1	no	—
52	C₄×C₇₃	—	8	2	no	—
53	C₄×C₇₃	—	16	1	no	—
54	#7366	C₇₃	1	1	yes	maximal, characteristic, direct product factor, Sylow
55	C₂×C₇₃	—	2	1	no	—
56	C₂×C₇₃	—	2	1	no	—
57	C₂×C₇₃	#1324	1	1	yes	characteristic
58	C₂×C₇₃	#1414	1	2	yes	—
59	#1835	C₂×C₇₃	1	1	yes	characteristic
60	#1324	C₂×C₇₃	1	2	yes	—
61	C₇₃	#7366	1	1	yes	characteristic, direct product factor, Sylow
62	C₂×C₄×C₈	C₄×C₇₃	1	1	yes	characteristic
63	#382	C₄×C₇₃	1	1	yes	characteristic
64	#382	C₂²×C₇₃	1	1	yes	characteristic
65	#340	—	2	2	no	—
66	#324	—	2	2	no	—
67	C₂²×C₈	—	2	1	no	—
68	C₂×C₄⋊C₄	—	2	1	no	—
69	C₂×C₄⋊C₄	D₄×C₇₃	1	2	yes	—
70	C₂×C₄²	C₂×C₄×C₇₃	1	1	yes	characteristic
71	C₂²⋊C₈	—	4	2	no	—
72	C₄×C₈	—	2	1	no	—
73	C₄×C₈	C₈×C₇₃	1	2	yes	complemented
74	#95	—	4	1	no	—
75	#95	—	4	1	no	—
76	C₂²×C₄	—	2	1	no	—
77	C₂²×C₄	—	2	2	no	—
78	C₂²×C₄	—	4	1	no	—
79	C₂²×C₄	C₂²⋊C₄×C₇₃	1	1	yes	characteristic
80	C₂×C₈	—	2	2	no	—
81	C₂×C₈	—	4	1	no	—
82	C₂×C₈	—	4	1	no	—
83	C₂×C₈	—	8	2	no	—
84	C₄⋊C₄	—	4	2	no	—
85	C₄⋊C₄	—	4	2	no	—
86	C₄²	—	2	1	no	—
87	C₄²	C₂×C₈×C₇₃	1	1	yes	characteristic
88	C₄²	M₁₆×C₇₃	1	1	yes	characteristic
89	C₂³	#99×C₇₃	1	1	yes	characteristic
90	C₂×C₄	—	2	1	no	—
91	C₂×C₄	—	2	1	no	—
92	C₂×C₄	—	2	1	no	—
93	C₂×C₄	—	2	1	no	—
94	C₂×C₄	—	4	2	no	—
95	C₂×C₄	—	4	2	no	—
96	C₂×C₄	—	8	2	no	—
97	C₂×C₄	—	8	2	no	—
98	C₂×C₄	—	8	1	no	—
99	C₂×C₄	C₂²⋊C₈×C₇₃	1	1	yes	characteristic
100	C₂×C₄	#101×C₇₃	1	1	yes	characteristic
101	C₈	—	4	2	no	—
102	C₈	—	16	2	no	—
103	C₂²	—	2	2	no	—
104	C₂²	#324×C₇₃	1	1	yes	characteristic
105	C₂²	#354×C₇₃	1	1	yes	characteristic
106	C₂²	#355×C₇₃	1	1	yes	characteristic
107	C₄	—	2	1	no	—
108	C₄	—	2	1	no	—
109	C₄	—	4	1	no	—
110	C₄	—	4	1	no	—
111	C₄	—	8	2	no	—
112	C₄	—	16	1	no	—
113	C₂	—	2	1	no	—
114	C₂	—	2	1	no	—
115	C₂	C₇₃×#1324	1	1	yes	characteristic
116	C₂	C₇₃×#1414	1	2	yes	—

subgroup

G/H

class size

multiplicity

normal?

properties

C₇₃×#1835

C₂

yes

maximal, characteristic

C₇₃×#1324

C₂

yes

maximal

C₂×C₄×C₈×C₇₃

C₄

yes

characteristic

#382×C₇₃

C₄

yes

characteristic

#382×C₇₃

C₂²

yes

characteristic

#340×C₇₃

—

#324×C₇₃

—

C₂²×C₈×C₇₃

—

C₂×C₄⋊C₄×C₇₃

—

C₂×C₄⋊C₄×C₇₃

D₄

yes

—

C₂×C₄²×C₇₃

C₂×C₄

yes

characteristic

C₂²⋊C₈×C₇₃

—

C₄×C₈×C₇₃

—

C₄×C₈×C₇₃

C₈

yes

complemented

#95×C₇₃

—

#95×C₇₃

—

C₂²×C₄×C₇₃

—

C₂²×C₄×C₇₃

—

C₂²×C₄×C₇₃

—

C₂²×C₄×C₇₃

C₂²⋊C₄

yes

characteristic

C₂×C₈×C₇₃

—

C₂×C₈×C₇₃

—

C₂×C₈×C₇₃

—

C₂×C₈×C₇₃

—

C₄⋊C₄×C₇₃

—

C₄⋊C₄×C₇₃

—

C₄²×C₇₃

—

C₄²×C₇₃

C₂×C₈

yes

characteristic

C₄²×C₇₃

M₁₆

yes

characteristic

C₂³×C₇₃

#99

yes

characteristic

C₂×C₄×C₇₃

—

C₂×C₄×C₇₃

—

C₂×C₄×C₇₃

—

C₂×C₄×C₇₃

—

C₂×C₄×C₇₃

—

C₂×C₄×C₇₃

—

C₂×C₄×C₇₃

—

C₂×C₄×C₇₃

—

C₂×C₄×C₇₃

—

C₂×C₄×C₇₃

C₂²⋊C₈

yes

characteristic

C₂×C₄×C₇₃

#101

yes

characteristic

C₈×C₇₃

—

C₈×C₇₃

—

C₂²×C₇₃

—

C₂²×C₇₃

yes

C₂²×C₇₃

yes

C₂²×C₇₃

yes

C₄×C₇₃

—

C₄×C₇₃

—

C₄×C₇₃

—

C₄×C₇₃

—

C₄×C₇₃

—

C₄×C₇₃

—

#7366

C₇₃

yes

maximal, characteristic, direct product factor, Sylow

C₂×C₇₃

—

C₂×C₇₃

—

C₂×C₇₃

#1324

yes

characteristic

C₂×C₇₃

#1414

yes

—

#1835

C₂×C₇₃

yes

characteristic

#1324

C₂×C₇₃

yes

—

C₇₃

#7366

yes

characteristic, direct product factor, Sylow

C₂×C₄×C₈

C₄×C₇₃

yes

characteristic

#382

C₄×C₇₃

yes

characteristic

#382

C₂²×C₇₃

yes

characteristic

#340

—

#324

—

C₂²×C₈

—

C₂×C₄⋊C₄

—

C₂×C₄⋊C₄

D₄×C₇₃

yes

—

C₂×C₄²

C₂×C₄×C₇₃

yes

characteristic

C₂²⋊C₈

—

C₄×C₈

—

C₄×C₈

C₈×C₇₃

yes

complemented

#95

—

#95

—

C₂²×C₄

—

C₂²×C₄

—

C₂²×C₄

—

C₂²×C₄

C₂²⋊C₄×C₇₃

yes

characteristic

C₂×C₈

—

C₂×C₈

—

C₂×C₈

—

C₂×C₈

—

C₄⋊C₄

—

C₄⋊C₄

—

C₄²

—

C₄²

C₂×C₈×C₇₃

yes

characteristic

C₄²

M₁₆×C₇₃

yes

characteristic

C₂³

#99×C₇₃

yes

characteristic

C₂×C₄

—

C₂×C₄

—

C₂×C₄

—

C₂×C₄

—

C₂×C₄

—

C₂×C₄

—

C₂×C₄

—

C₂×C₄

—

C₂×C₄

—

C₂×C₄

C₂²⋊C₈×C₇₃

yes

characteristic

100

C₂×C₄

#101×C₇₃

yes

characteristic

101

C₈

—

102

C₈

—

103

C₂²

—

104

C₂²

#324×C₇₃

yes

characteristic

105

C₂²

#354×C₇₃

yes

characteristic

106

C₂²

#355×C₇₃

yes

characteristic

107

C₄

—

108

C₄

—

109

C₄

—

110

C₄

—

111

C₄

—

112

C₄

—

113

C₂

—

114

C₂

—

115

C₂

C₇₃×#1324

yes

characteristic

116

C₂

C₇₃×#1414

yes

—

	subgroup	G/H
center, socle	C₂²×C₇₃	#354
commutator	C₄²	C₂×C₈×C₇₃
Frattini, cosocle	#382	C₂²×C₇₃
prime radical	C₂³×C₇₃	#99
LCS	G ▷ C₄² ▷ C₂×C₄ ▷ C₂² ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂²×C₇₃ ◁ C₂³×C₇₃ ◁ C₂²×C₄×C₇₃ ◁ #382×C₇₃ ◁ G

subgroup

G/H

center, socle

C₂²×C₇₃

#354

commutator

C₄²

C₂×C₈×C₇₃

C₂²×C₇₃

C₂³×C₇₃

G ▷ C₄² ▷ C₂×C₄ ▷ C₂² ▷ C₂ ▷ C₁

UCS

C₁ ◁ C₂²×C₇₃ ◁ C₂³×C₇₃ ◁ C₂²×C₄×C₇₃ ◁ #382×C₇₃ ◁ G

N	H ≅ G/N	type
14. C₄×C₈×C₇₃	C₈	semidirect
54. #7366	C₇₃	direct (N × H)
61. C₇₃	#7366	direct (N × H)
73. C₄×C₈	C₈×C₇₃	semidirect

H ≅ G/N

type

14. C₄×C₈×C₇₃

C₈

semidirect

54. #7366

C₇₃

direct (N × H)

61. C₇₃

#7366

direct (N × H)

73. C₄×C₈

C₈×C₇₃

semidirect

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

subgroup	index
C₇₃×#1835	2
C₇₃×#1324	2
#7366	73

subgroup

index

C₇₃×#1835

C₇₃×#1324

#7366

Subgroups — C₇₃×#7366 #32916770

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C73×#7366 #32916770

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₇₃×#7366 #32916770