Group #33006597 — subgroups

Subgroups — #33006597

Order 18688.

184 conjugacy classes of subgroups; only the 69 normal subgroups are shown.

normal subgroups only

This group has 184 conjugacy classes of subgroups — too many to draw. Only its 69 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	#22611812	C₂	1	2	yes	maximal, complemented
2	#22608135	C₂	1	1	yes	maximal, characteristic, complemented
3	#22608115	C₂	1	2	yes	maximal, complemented
4	#22608064	C₂	1	1	yes	maximal, characteristic, complemented
5	C₇₃×#2262	C₂	1	1	yes	maximal, characteristic
6	#17988704	—	2	2	no	—
7	C₂×#14556497	C₂²	1	1	yes	characteristic
8	#17988417	C₂²	1	2	yes	—
9	#17988394	—	2	2	no	—
10	C₁₆×Dic₇₃	C₂²	1	1	yes	characteristic
11	#508×C₇₃	C₂²	1	2	yes	—
12	Q₆₄×C₇₃	C₄	1	1	yes	characteristic, complemented
13	SD₆₄×C₇₃	C₄	1	2	yes	complemented
14	D₃₂×C₇₃	C₄	1	1	yes	characteristic, complemented
15	C₂×C₃₂×C₇₃	C₂²	1	1	yes	characteristic
16	#14556598	—	4	2	no	—
17	C₂×#11924003	D₄	1	1	yes	characteristic
18	#14556527	—	2	2	no	—
19	#14556518	—	4	2	no	—
20	C₈×Dic₇₃	D₄	1	1	yes	characteristic
21	#14556497	—	2	1	no	—
22	#135×C₇₃	—	2	2	no	—
23	Q₃₂×C₇₃	C₂×C₄	1	2	yes	—
24	SD₃₂×C₇₃	—	2	2	no	—
25	D₁₆×C₇₃	C₂×C₄	1	2	yes	—
26	C₂×C₁₆×C₇₃	C₂³	1	1	yes	characteristic
27	C₃₂×C₇₃	C₂×C₄	1	1	yes	characteristic
28	C₃₂×C₇₃	C₂×C₄	1	1	yes	characteristic
29	#11924012	—	4	2	no	—
30	#11924012	—	8	2	no	—
31	C₂×#10596831	—	8	2	no	—
32	C₂×#10596831	D₈	1	1	yes	characteristic
33	C₄×Dic₇₃	D₈	1	1	yes	characteristic
34	#11924003	—	2	1	no	—
35	#41×C₇₃	—	4	2	no	—
36	Q₁₆×C₇₃	—	2	2	no	—
37	SD₁₆×C₇₃	—	4	2	no	—
38	D₈×C₇₃	—	2	2	no	—
39	C₂×C₈×C₇₃	C₂×D₄	1	1	yes	characteristic
40	C₁₆×C₇₃	C₂²×C₄	1	1	yes	characteristic
41	C₁₆×C₇₃	#41	1	1	yes	characteristic
42	Q₈×C₇₃	—	4	2	no	—
43	D₄×C₇₃	—	4	2	no	—
44	D₄×C₇₃	—	8	2	no	—
45	C₂×C₄×C₇₃	—	8	2	no	—
46	C₂×C₄×C₇₃	C₂×D₈	1	1	yes	characteristic
47	C₂×Dic₇₃	—	2	1	no	—
48	C₈×C₇₃	C₄×D₄	1	1	yes	characteristic
49	C₈×C₇₃	#135	1	1	yes	characteristic
50	#10596831	—	8	2	no	—
51	#10596831	D₁₆	1	2	yes	—
52	C₂²×C₇₃	—	8	2	no	—
53	C₂²×C₇₃	#508	1	1	yes	characteristic
54	C₄×C₇₃	—	8	2	no	—
55	C₄×C₇₃	C₄×D₈	1	1	yes	characteristic
56	C₄×C₇₃	C₂×D₁₆	1	1	yes	characteristic
57	Dic₇₃	—	4	1	no	—
58	#13656	—	73	1	no	maximal, Sylow
59	C₂×C₇₃	—	2	1	no	—
60	C₂×C₇₃	—	16	2	no	—
61	C₂×C₇₃	C₄×D₁₆	1	1	yes	characteristic
62	#2262	D₇₃	1	1	yes	characteristic
63	#2178	—	73	2	no	—
64	#1425	—	73	1	no	—
65	#1417	—	73	2	no	—
66	#1397	—	73	1	no	—
67	C₇₃	#13656	1	1	yes	characteristic, complemented, Sylow
68	#508	C₂×D₇₃	1	2	yes	—
69	#443	—	146	2	no	—
70	Q₆₄	Dic₇₃	1	1	yes	characteristic, complemented
71	SD₆₄	Dic₇₃	1	2	yes	complemented
72	D₃₂	Dic₇₃	1	1	yes	characteristic, complemented
73	C₂×C₃₂	—	73	1	no	—
74	C₂×C₃₂	C₂×D₇₃	1	1	yes	characteristic
75	#368	—	73	2	no	—
76	#359	—	146	2	no	—
77	C₄×C₁₆	—	73	1	no	—
78	#135	—	2	2	no	—
79	#131	—	292	2	no	—
80	Q₃₂	C₂×Dic₇₃	1	2	yes	—
81	SD₃₂	—	2	2	no	—
82	D₁₆	C₂×Dic₇₃	1	2	yes	—
83	C₂×C₁₆	—	73	1	no	—
84	C₂×C₁₆	C₂²×D₇₃	1	1	yes	characteristic
85	#108	—	146	2	no	—
86	#104	—	292	2	no	—
87	C₄×C₈	—	73	1	no	—
88	C₃₂	—	146	1	no	—
89	C₃₂	C₂×Dic₇₃	1	1	yes	characteristic
90	C₃₂	C₂×Dic₇₃	1	1	yes	characteristic
91	#41	—	4	2	no	—
92	Q₁₆	—	2	2	no	—
93	SD₁₆	—	4	2	no	—
94	D₈	—	2	2	no	—
95	M₁₆	—	292	2	no	—
96	M₁₆	—	584	2	no	—
97	C₂×C₈	—	584	2	no	—
98	C₂×C₈	—	73	1	no	—
99	C₂×C₈	D₄×D₇₃	1	1	yes	characteristic
100	C₄²	—	73	1	no	—
101	C₁₆	—	146	1	no	—
102	C₁₆	C₂²×Dic₇₃	1	1	yes	characteristic
103	C₁₆	#11924027	1	1	yes	characteristic
104	Q₈	—	4	2	no	—
105	D₄	—	4	2	no	—
106	D₄	—	8	2	no	—
107	C₂×C₄	—	8	2	no	—
108	C₂×C₄	—	146	1	no	—
109	C₂×C₄	D₈×D₇₃	1	1	yes	characteristic
110	C₈	—	584	2	no	—
111	C₈	—	73	2	no	—
112	C₈	D₄×Dic₇₃	1	1	yes	characteristic
113	C₈	#14556610	1	1	yes	characteristic
114	C₂²	—	8	2	no	—
115	C₂²	#17989016	1	1	yes	characteristic
116	C₄	—	8	2	no	—
117	C₄	—	292	1	no	—
118	C₄	D₈×Dic₇₃	1	1	yes	characteristic
119	C₄	D₁₆×D₇₃	1	1	yes	characteristic
120	C₂	—	2	1	no	—
121	C₂	—	16	2	no	—
122	C₂	D₁₆×Dic₇₃	1	1	yes	characteristic

subgroup

G/H

class size

multiplicity

normal?

properties

#22611812

C₂

yes

maximal, complemented

#22608135

C₂

yes

maximal, characteristic, complemented

#22608115

C₂

yes

maximal, complemented

#22608064

C₂

yes

maximal, characteristic, complemented

C₇₃×#2262

C₂

yes

maximal, characteristic

#17988704

—

C₂×#14556497

C₂²

yes

characteristic

#17988417

C₂²

yes

—

#17988394

—

C₁₆×Dic₇₃

C₂²

yes

characteristic

#508×C₇₃

C₂²

yes

—

Q₆₄×C₇₃

C₄

yes

characteristic, complemented

SD₆₄×C₇₃

C₄

yes

complemented

D₃₂×C₇₃

C₄

yes

characteristic, complemented

C₂×C₃₂×C₇₃

C₂²

yes

characteristic

#14556598

—

C₂×#11924003

D₄

yes

characteristic

#14556527

—

#14556518

—

C₈×Dic₇₃

D₄

yes

characteristic

#14556497

—

#135×C₇₃

—

Q₃₂×C₇₃

C₂×C₄

yes

—

SD₃₂×C₇₃

—

D₁₆×C₇₃

C₂×C₄

yes

—

C₂×C₁₆×C₇₃

C₂³

yes

characteristic

C₃₂×C₇₃

C₂×C₄

yes

characteristic

C₃₂×C₇₃

C₂×C₄

yes

—

—

—

D₈

yes

characteristic

C₄×Dic₇₃

D₈

yes

characteristic

#11924003

—

#41×C₇₃

—

Q₁₆×C₇₃

—

SD₁₆×C₇₃

—

D₈×C₇₃

—

C₂×C₈×C₇₃

C₂×D₄

yes

characteristic

C₁₆×C₇₃

C₂²×C₄

yes

characteristic

C₁₆×C₇₃

#41

yes

characteristic

Q₈×C₇₃

—

D₄×C₇₃

—

D₄×C₇₃

—

C₂×C₄×C₇₃

—

C₂×C₄×C₇₃

C₂×D₈

yes

characteristic

C₂×Dic₇₃

—

C₈×C₇₃

C₄×D₄

yes

C₈×C₇₃

yes

—

D₁₆

yes

—

C₂²×C₇₃

—

C₂²×C₇₃

#508

yes

characteristic

C₄×C₇₃

—

C₄×C₇₃

C₄×D₈

yes

characteristic

C₄×C₇₃

C₂×D₁₆

yes

characteristic

Dic₇₃

—

#13656

—

maximal, Sylow

C₂×C₇₃

—

C₂×C₇₃

—

C₂×C₇₃

C₄×D₁₆

yes

characteristic

#2262

D₇₃

yes

—

—

—

—

C₇₃

#13656

yes

characteristic, complemented, Sylow

#508

C₂×D₇₃

yes

—

#443

—

146

—

Q₆₄

Dic₇₃

yes

characteristic, complemented

SD₆₄

Dic₇₃

yes

complemented

D₃₂

Dic₇₃

yes

characteristic, complemented

C₂×C₃₂

—

C₂×C₃₂

C₂×D₇₃

yes

characteristic

#368

—

#359

—

146

—

C₄×C₁₆

—

#135

—

#131

—

292

—

Q₃₂

C₂×Dic₇₃

yes

—

SD₃₂

—

D₁₆

C₂×Dic₇₃

yes

—

C₂×C₁₆

—

C₂×C₁₆

C₂²×D₇₃

yes

characteristic

#108

—

146

—

#104

—

292

—

C₄×C₈

—

C₃₂

—

146

—

C₃₂

C₂×Dic₇₃

yes

characteristic

C₃₂

C₂×Dic₇₃

yes

characteristic

#41

—

Q₁₆

—

SD₁₆

—

D₈

—

M₁₆

—

292

—

M₁₆

—

584

—

C₂×C₈

—

584

—

C₂×C₈

—

C₂×C₈

D₄×D₇₃

yes

characteristic

100

C₄²

—

101

C₁₆

—

146

—

102

C₁₆

C₂²×Dic₇₃

yes

characteristic

103

C₁₆

#11924027

yes

characteristic

104

Q₈

—

105

D₄

—

106

D₄

—

107

C₂×C₄

—

108

C₂×C₄

—

146

—

109

C₂×C₄

D₈×D₇₃

yes

characteristic

110

C₈

—

584

—

111

C₈

—

112

C₈

D₄×Dic₇₃

yes

characteristic

113

C₈

#14556610

yes

characteristic

114

C₂²

—

115

C₂²

#17989016

yes

characteristic

116

C₄

—

117

C₄

—

292

—

118

C₄

D₈×Dic₇₃

yes

characteristic

119

C₄

D₁₆×D₇₃

yes

characteristic

120

C₂

—

121

C₂

—

122

C₂

D₁₆×Dic₇₃

yes

characteristic

	subgroup	G/H
center	C₄	D₁₆×D₇₃
commutator	C₁₆×C₇₃	C₂²×C₄
Frattini	C₂×C₁₆	C₂²×D₇₃
Fitting, prime radical	C₇₃×#2262	C₂
socle	C₂×C₇₃	C₄×D₁₆
cosocle	C₂×C₁₆×C₇₃	C₂³
hypercenter	#2262	D₇₃
hypocenter	C₇₃	#13656
LCS	G ▷ C₁₆×C₇₃ ▷ C₈×C₇₃ ▷ C₄×C₇₃ ▷ C₂×C₇₃ ▷ C₇₃
UCS	C₁ ◁ C₄ ◁ C₂×C₄ ◁ C₂×C₈ ◁ C₂×C₁₆ ◁ #2262

subgroup

G/H

center

C₄

D₁₆×D₇₃

commutator

C₁₆×C₇₃

C₂²×C₄

Frattini

C₂×C₁₆

C₂²×D₇₃

Fitting, prime radical

C₇₃×#2262

C₂

socle

C₂×C₇₃

C₄×D₁₆

cosocle

C₂×C₁₆×C₇₃

C₂³

D₇₃

C₇₃

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

UCS

C₁ ◁ C₄ ◁ C₂×C₄ ◁ C₂×C₈ ◁ C₂×C₁₆ ◁ #2262

N	H ≅ G/N	type
1. #22611812	C₂	semidirect
2. #22608135	C₂	semidirect
3. #22608115	C₂	semidirect
4. #22608064	C₂	semidirect
12. Q₆₄×C₇₃	C₄	semidirect
13. SD₆₄×C₇₃	C₄	semidirect
14. D₃₂×C₇₃	C₄	semidirect
67. C₇₃	#13656	semidirect
70. Q₆₄	Dic₇₃	semidirect
71. SD₆₄	Dic₇₃	semidirect
72. D₃₂	Dic₇₃	semidirect

H ≅ G/N

type

1. #22611812

C₂

semidirect

2. #22608135

C₂

semidirect

3. #22608115

C₂

semidirect

4. #22608064

C₂

semidirect

12. Q₆₄×C₇₃

C₄

semidirect

13. SD₆₄×C₇₃

C₄

semidirect

14. D₃₂×C₇₃

C₄

semidirect

67. C₇₃

#13656

semidirect

70. Q₆₄

Dic₇₃

semidirect

71. SD₆₄

Dic₇₃

semidirect

72. D₃₂

Dic₇₃

semidirect

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

subgroup	index
#22611812	2
#22608135	2
#22608115	2
#22608064	2
C₇₃×#2262	2
#13656	73

subgroup

index

C₇₃×#2262

Subgroups — #33006597

Conjugacy classes of subgroups

Complements

Maximal subgroups