Group #33006553 — subgroups

Subgroups — #33006553

Order 18688.

184 conjugacy classes of subgroups; only the 105 normal subgroups are shown.

normal subgroups only

This group has 184 conjugacy classes of subgroups — too many to draw. Only its 105 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	#22612225	C₂	1	2	yes	maximal, complemented
2	#22608129	C₂	1	1	yes	maximal, characteristic, complemented
3	#22608073	C₂	1	2	yes	maximal, complemented
4	#22608065	C₂	1	1	yes	maximal, characteristic, complemented
5	C₇₃×#2170	C₂	1	1	yes	maximal, characteristic
6	#17988421	—	2	2	no	—
7	#17988421	C₂²	1	2	yes	—
8	C₂×#14556497	—	2	2	no	—
9	C₂×#14556497	C₂²	1	1	yes	characteristic
10	C₂×#14556497	C₂²	1	1	yes	characteristic
11	#504×C₇₃	C₂²	1	2	yes	—
12	#443×C₇₃	C₄	1	1	yes	characteristic
13	#364×C₇₃	C₄	1	1	yes	characteristic
14	#350×C₇₃	C₄	1	2	yes	—
15	C₄×C₁₆×C₇₃	C₂²	1	1	yes	characteristic
16	#14556497	—	2	2	no	—
17	#14556497	—	2	1	no	—
18	#14556497	D₄	1	2	yes	—
19	#135×C₇₃	C₈	1	1	yes	characteristic
20	#131×C₇₃	C₂×C₄	1	2	yes	—
21	M₃₂×C₇₃	—	2	2	no	—
22	M₃₂×C₇₃	C₂×C₄	1	2	yes	—
23	C₂×C₁₆×C₇₃	—	2	2	no	—
24	C₂×C₁₆×C₇₃	C₂×C₄	1	1	yes	characteristic
25	C₂×C₁₆×C₇₃	C₂³	1	1	yes	characteristic
26	#108×C₇₃	C₈	1	1	yes	characteristic
27	#104×C₇₃	C₈	1	2	yes	—
28	C₄×C₈×C₇₃	C₂×C₄	1	1	yes	characteristic
29	#41×C₇₃	C₂×C₈	1	2	yes	—
30	Q₁₆×C₇₃	C₁₆	1	1	yes	characteristic
31	SD₁₆×C₇₃	C₁₆	1	2	yes	—
32	D₈×C₇₃	C₁₆	1	1	yes	characteristic
33	M₁₆×C₇₃	—	2	2	no	—
34	M₁₆×C₇₃	C₂×C₈	1	2	yes	—
35	C₂×C₈×C₇₃	—	2	2	no	—
36	C₂×C₈×C₇₃	C₂×C₈	1	1	yes	characteristic
37	C₂×C₈×C₇₃	C₂²×C₄	1	1	yes	characteristic
38	C₄²×C₇₃	C₂×C₈	1	1	yes	characteristic
39	C₁₆×C₇₃	—	2	2	no	—
40	C₁₆×C₇₃	—	2	1	no	—
41	C₁₆×C₇₃	C₂×D₄	1	1	yes	characteristic
42	C₁₆×C₇₃	#41	1	1	yes	characteristic
43	Q₈×C₇₃	C₂×C₁₆	1	2	yes	—
44	D₄×C₇₃	—	2	2	no	—
45	D₄×C₇₃	C₂×C₁₆	1	2	yes	—
46	C₂×C₄×C₇₃	—	2	2	no	—
47	C₂×C₄×C₇₃	—	2	1	no	—
48	C₂×C₄×C₇₃	C₂²×C₈	1	1	yes	characteristic
49	C₈×C₇₃	—	2	2	no	—
50	C₈×C₇₃	C₂×C₁₆	1	1	yes	characteristic
51	C₈×C₇₃	C₂×C₁₆	1	1	yes	characteristic
52	C₈×C₇₃	C₄×D₄	1	1	yes	characteristic
53	C₈×C₇₃	#131	1	1	yes	characteristic
54	C₂²×C₇₃	—	2	2	no	—
55	C₂²×C₇₃	#504	1	1	yes	characteristic
56	C₄×C₇₃	—	2	2	no	—
57	C₄×C₇₃	—	4	1	no	—
58	C₄×C₇₃	C₈×D₄	1	1	yes	characteristic
59	C₄×C₇₃	C₂²×C₁₆	1	1	yes	characteristic
60	#13648	—	73	1	no	maximal, Sylow
61	C₂×C₇₃	—	2	1	no	—
62	C₂×C₇₃	—	4	2	no	—
63	C₂×C₇₃	D₄×C₁₆	1	1	yes	characteristic
64	#2258	—	73	2	no	—
65	#2170	D₇₃	1	1	yes	characteristic
66	#1422	—	73	1	no	—
67	#1401	—	73	2	no	—
68	#1397	—	73	1	no	—
69	C₇₃	#13648	1	1	yes	characteristic, complemented, Sylow
70	#504	C₂×D₇₃	1	2	yes	—
71	#443	Dic₇₃	1	1	yes	characteristic
72	M₆₄	—	73	2	no	—
73	M₆₄	—	146	2	no	—
74	C₂×C₃₂	—	73	1	no	—
75	C₂×C₃₂	—	73	1	no	—
76	C₂×C₃₂	—	146	2	no	—
77	#364	Dic₇₃	1	1	yes	characteristic
78	#350	Dic₇₃	1	2	yes	—
79	C₄×C₁₆	C₂×D₇₃	1	1	yes	characteristic
80	#135	#10596831	1	1	yes	characteristic
81	#131	C₂×Dic₇₃	1	2	yes	—
82	M₃₂	—	2	2	no	—
83	M₃₂	C₂×Dic₇₃	1	2	yes	—
84	C₂×C₁₆	—	2	2	no	—
85	C₂×C₁₆	C₂×Dic₇₃	1	1	yes	characteristic
86	C₂×C₁₆	C₂²×D₇₃	1	1	yes	characteristic
87	#108	#10596831	1	1	yes	characteristic
88	#104	#10596831	1	2	yes	—
89	C₄×C₈	C₂×Dic₇₃	1	1	yes	characteristic
90	C₃₂	—	73	2	no	—
91	C₃₂	—	146	2	no	—
92	C₃₂	—	146	1	no	—
93	#41	C₂×#10596831	1	2	yes	—
94	Q₁₆	#11924003	1	1	yes	characteristic
95	SD₁₆	#11924003	1	2	yes	—
96	D₈	#11924003	1	1	yes	characteristic
97	M₁₆	—	2	2	no	—
98	M₁₆	C₂×#10596831	1	2	yes	—
99	C₂×C₈	—	2	2	no	—
100	C₂×C₈	C₂×#10596831	1	1	yes	characteristic
101	C₂×C₈	C₂²×Dic₇₃	1	1	yes	characteristic
102	C₄²	C₂×#10596831	1	1	yes	characteristic
103	C₁₆	—	2	2	no	—
104	C₁₆	—	2	1	no	—
105	C₁₆	D₄×D₇₃	1	1	yes	characteristic
106	C₁₆	#11924027	1	1	yes	characteristic
107	Q₈	C₂×#11924003	1	2	yes	—
108	D₄	—	2	2	no	—
109	D₄	C₂×#11924003	1	2	yes	—
110	C₂×C₄	—	2	2	no	—
111	C₂×C₄	—	2	1	no	—
112	C₂×C₄	C₂²×#10596831	1	1	yes	characteristic
113	C₈	—	2	2	no	—
114	C₈	C₂×#11924003	1	1	yes	characteristic
115	C₈	C₂×#11924003	1	1	yes	characteristic
116	C₈	D₄×Dic₇₃	1	1	yes	characteristic
117	C₈	#14556598	1	1	yes	characteristic
118	C₂²	—	2	2	no	—
119	C₂²	#17989004	1	1	yes	characteristic
120	C₄	—	2	2	no	—
121	C₄	—	4	1	no	—
122	C₄	D₄×#10596831	1	1	yes	characteristic
123	C₄	C₂²×#11924003	1	1	yes	characteristic
124	C₂	—	2	1	no	—
125	C₂	—	4	2	no	—
126	C₂	D₄×#11924003	1	1	yes	characteristic

subgroup

G/H

class size

multiplicity

normal?

properties

#22612225

C₂

yes

maximal, complemented

#22608129

C₂

yes

maximal, characteristic, complemented

#22608073

C₂

yes

maximal, complemented

#22608065

C₂

yes

maximal, characteristic, complemented

C₇₃×#2170

C₂

yes

maximal, characteristic

#17988421

—

#17988421

C₂²

yes

—

C₂×#14556497

—

C₂×#14556497

C₂²

yes

characteristic

C₂×#14556497

C₂²

yes

characteristic

#504×C₇₃

C₂²

yes

—

#443×C₇₃

C₄

yes

characteristic

#364×C₇₃

C₄

yes

characteristic

#350×C₇₃

C₄

yes

—

C₄×C₁₆×C₇₃

C₂²

yes

—

—

D₄

yes

—

#135×C₇₃

C₈

yes

characteristic

#131×C₇₃

C₂×C₄

yes

—

M₃₂×C₇₃

—

M₃₂×C₇₃

C₂×C₄

yes

—

C₂×C₁₆×C₇₃

—

C₂×C₁₆×C₇₃

C₂×C₄

yes

characteristic

C₂×C₁₆×C₇₃

C₂³

yes

characteristic

#108×C₇₃

C₈

yes

characteristic

#104×C₇₃

C₈

yes

—

C₄×C₈×C₇₃

C₂×C₄

yes

characteristic

#41×C₇₃

C₂×C₈

yes

—

Q₁₆×C₇₃

C₁₆

yes

characteristic

SD₁₆×C₇₃

C₁₆

yes

—

D₈×C₇₃

C₁₆

yes

characteristic

M₁₆×C₇₃

—

M₁₆×C₇₃

C₂×C₈

yes

—

C₂×C₈×C₇₃

—

C₂×C₈×C₇₃

C₂×C₈

yes

characteristic

C₂×C₈×C₇₃

C₂²×C₄

yes

characteristic

C₄²×C₇₃

C₂×C₈

yes

characteristic

C₁₆×C₇₃

—

C₁₆×C₇₃

—

C₁₆×C₇₃

C₂×D₄

yes

characteristic

C₁₆×C₇₃

#41

yes

characteristic

Q₈×C₇₃

C₂×C₁₆

yes

—

D₄×C₇₃

—

D₄×C₇₃

C₂×C₁₆

yes

—

C₂×C₄×C₇₃

—

C₂×C₄×C₇₃

—

C₂×C₄×C₇₃

C₂²×C₈

yes

characteristic

C₈×C₇₃

—

C₈×C₇₃

C₂×C₁₆

yes

characteristic

C₈×C₇₃

C₂×C₁₆

yes

characteristic

C₈×C₇₃

C₄×D₄

yes

characteristic

C₈×C₇₃

#131

yes

characteristic

C₂²×C₇₃

—

C₂²×C₇₃

#504

yes

characteristic

C₄×C₇₃

—

C₄×C₇₃

—

C₄×C₇₃

C₈×D₄

yes

characteristic

C₄×C₇₃

C₂²×C₁₆

yes

characteristic

#13648

—

maximal, Sylow

C₂×C₇₃

—

C₂×C₇₃

—

C₂×C₇₃

D₄×C₁₆

yes

characteristic

#2258

—

#2170

D₇₃

yes

—

—

—

C₇₃

#13648

yes

characteristic, complemented, Sylow

#504

C₂×D₇₃

yes

—

#443

Dic₇₃

yes

characteristic

M₆₄

—

M₆₄

—

146

—

C₂×C₃₂

—

C₂×C₃₂

—

C₂×C₃₂

—

146

—

#364

Dic₇₃

yes

characteristic

#350

Dic₇₃

yes

—

C₄×C₁₆

C₂×D₇₃

yes

yes

C₂×Dic₇₃

yes

—

M₃₂

—

M₃₂

C₂×Dic₇₃

yes

—

C₂×C₁₆

—

C₂×C₁₆

C₂×Dic₇₃

yes

characteristic

C₂×C₁₆

C₂²×D₇₃

yes

yes

yes

—

C₄×C₈

C₂×Dic₇₃

yes

characteristic

C₃₂

—

C₃₂

—

146

—

C₃₂

—

146

—

#41

C₂×#10596831

yes

—

Q₁₆

#11924003

yes

characteristic

SD₁₆

#11924003

yes

—

D₈

#11924003

yes

characteristic

M₁₆

—

M₁₆

C₂×#10596831

yes

—

C₂×C₈

—

100

C₂×C₈

C₂×#10596831

yes

characteristic

101

C₂×C₈

C₂²×Dic₇₃

yes

characteristic

102

C₄²

C₂×#10596831

yes

characteristic

103

C₁₆

—

104

C₁₆

—

105

C₁₆

D₄×D₇₃

yes

characteristic

106

C₁₆

#11924027

yes

characteristic

107

Q₈

C₂×#11924003

yes

—

108

D₄

—

109

D₄

C₂×#11924003

yes

—

110

C₂×C₄

—

111

C₂×C₄

—

112

C₂×C₄

C₂²×#10596831

yes

characteristic

113

C₈

—

114

C₈

C₂×#11924003

yes

characteristic

115

C₈

C₂×#11924003

yes

characteristic

116

C₈

D₄×Dic₇₃

yes

characteristic

117

C₈

#14556598

yes

characteristic

118

C₂²

—

119

C₂²

#17989004

yes

characteristic

120

C₄

—

121

C₄

—

122

C₄

D₄×#10596831

yes

characteristic

123

C₄

C₂²×#11924003

yes

characteristic

124

C₂

—

125

C₂

—

126

C₂

D₄×#11924003

yes

characteristic

	subgroup	G/H
center	C₁₆	D₄×D₇₃
commutator	C₄×C₇₃	C₂²×C₁₆
Frattini	C₂×C₁₆	C₂²×D₇₃
Fitting	C₇₃×#2170	C₂
socle	C₂×C₇₃	D₄×C₁₆
prime radical	#135×C₇₃	C₈
cosocle	C₂×C₁₆×C₇₃	C₂³
hypercenter	#2170	D₇₃
hypocenter	C₇₃	#13648
LCS	G ▷ C₄×C₇₃ ▷ C₂×C₇₃ ▷ C₇₃
UCS	C₁ ◁ C₁₆ ◁ C₂×C₁₆ ◁ #2170

subgroup

G/H

center

C₁₆

D₄×D₇₃

commutator

C₄×C₇₃

C₂²×C₁₆

Frattini

C₂×C₁₆

C₂²×D₇₃

Fitting

C₇₃×#2170

C₂

socle

C₂×C₇₃

D₄×C₁₆

prime radical

#135×C₇₃

C₈

cosocle

C₂×C₁₆×C₇₃

C₂³

D₇₃

C₇₃

G ▷ C₄×C₇₃ ▷ C₂×C₇₃ ▷ C₇₃

UCS

C₁ ◁ C₁₆ ◁ C₂×C₁₆ ◁ #2170

N	H ≅ G/N	type
1. #22612225	C₂	semidirect
2. #22608129	C₂	semidirect
3. #22608073	C₂	semidirect
4. #22608065	C₂	semidirect
69. C₇₃	#13648	semidirect

H ≅ G/N

type

1. #22612225

C₂

semidirect

2. #22608129

C₂

semidirect

3. #22608073

C₂

semidirect

4. #22608065

C₂

semidirect

69. C₇₃

#13648

semidirect

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

subgroup	index
#22612225	2
#22608129	2
#22608073	2
#22608065	2
C₇₃×#2170	2
#13648	73

subgroup

index

C₇₃×#2170

Subgroups — #33006553

Conjugacy classes of subgroups

Complements

Maximal subgroups