Group #33006591 — subgroups

Subgroups — #33006591

Order 18688.

186 conjugacy classes of subgroups; only the 61 normal subgroups are shown.

normal subgroups only

This group has 186 conjugacy classes of subgroups — too many to draw. Only its 61 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	#22612240	C₂	1	1	yes	maximal, characteristic
2	#22611811	C₂	1	2	yes	maximal, complemented
3	#22608114	C₂	1	2	yes	maximal, complemented
4	C₃₂×Dic₇₃	C₂	1	1	yes	maximal, characteristic, complemented
5	C₇₃×#1425	C₂	1	1	yes	maximal, characteristic, complemented
6	#17989019	C₂²	1	2	yes	—
7	#17988703	—	2	2	no	—
8	Dic₁₁₆₈	C₄	1	1	yes	characteristic, complemented
9	#17988426	C₄	1	2	yes	complemented
10	D₂₃₃₆	C₄	1	1	yes	characteristic, complemented
11	C₂×#14556497	C₂²	1	1	yes	characteristic
12	#17988393	—	2	2	no	—
13	C₁₆×Dic₇₃	C₂²	1	1	yes	characteristic
14	C₂×C₃₂×C₇₃	C₂²	1	1	yes	characteristic
15	#368×C₇₃	C₂²	1	2	yes	—
16	#14556613	—	2	2	no	—
17	#14556599	—	4	2	no	—
18	Dic₅₈₄	C₂×C₄	1	2	yes	—
19	#14556536	—	2	2	no	—
20	D₁₁₆₈	C₂×C₄	1	2	yes	—
21	C₂×#11924003	D₄	1	1	yes	characteristic
22	#14556517	—	4	2	no	—
23	C₈×Dic₇₃	D₄	1	1	yes	characteristic
24	#14556497	—	2	1	no	—
25	C₂×C₁₆×C₇₃	C₂³	1	1	yes	characteristic
26	#108×C₇₃	—	2	2	no	—
27	C₃₂×C₇₃	C₂×C₄	1	1	yes	characteristic
28	C₃₂×C₇₃	C₂×C₄	1	1	yes	characteristic
29	#11924028	—	4	2	no	—
30	Dic₂₉₂	—	2	2	no	—
31	#11924017	—	4	2	no	—
32	D₅₈₄	—	2	2	no	—
33	#11924011	—	8	2	no	—
34	C₂×#10596831	D₈	1	1	yes	characteristic
35	C₈×D₇₃	—	8	2	no	—
36	C₄×Dic₇₃	D₈	1	1	yes	characteristic
37	#11924003	—	2	1	no	—
38	M₁₆×C₇₃	—	4	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	C₂×C₄×C₇₃	C₂×D₈	1	1	yes	characteristic
43	C₂²⋊D₇₃	—	8	2	no	—
44	C₂×Dic₇₃	—	2	1	no	—
45	D₂₉₂	—	4	2	no	—
46	C₄×D₇₃	—	8	2	no	—
47	Dic₁₄₆	—	4	2	no	—
48	C₈×C₇₃	—	8	2	no	—
49	C₈×C₇₃	C₄×D₄	1	1	yes	characteristic
50	C₈×C₇₃	#135	1	1	yes	characteristic
51	#10596831	D₁₆	1	1	yes	characteristic
52	#10596831	D₁₆	1	1	yes	characteristic
53	C₂²×C₇₃	#508	1	1	yes	characteristic
54	C₂×D₇₃	—	8	2	no	—
55	C₄×C₇₃	C₄×D₈	1	1	yes	characteristic
56	C₄×C₇₃	C₂×D₁₆	1	1	yes	characteristic
57	Dic₇₃	—	2	2	no	—
58	Dic₇₃	—	8	2	no	—
59	#13655	—	73	1	no	maximal, Sylow
60	C₂×C₇₃	—	2	1	no	—
61	C₂×C₇₃	C₄×D₁₆	1	1	yes	characteristic
62	D₇₃	—	16	2	no	—
63	#2262	—	73	1	no	—
64	#2178	—	73	2	no	—
65	#1425	D₇₃	1	1	yes	characteristic, complemented
66	#1417	—	73	2	no	—
67	C₄×C₃₂	—	73	1	no	—
68	C₇₃	#13655	1	1	yes	characteristic, complemented, Sylow
69	#508	—	73	2	no	—
70	#443	—	146	2	no	—
71	Q₆₄	—	73	1	no	—
72	SD₆₄	—	73	2	no	—
73	D₃₂	—	73	1	no	—
74	C₂×C₃₂	—	73	1	no	—
75	C₂×C₃₂	C₂×D₇₃	1	1	yes	characteristic
76	#368	C₂×D₇₃	1	2	yes	—
77	#359	—	146	2	no	—
78	C₄×C₁₆	—	73	1	no	—
79	#135	—	146	2	no	—
80	#131	—	292	2	no	—
81	Q₃₂	—	73	2	no	—
82	SD₃₂	—	146	2	no	—
83	D₁₆	—	73	2	no	—
84	C₂×C₁₆	—	73	1	no	—
85	C₂×C₁₆	C₂²×D₇₃	1	1	yes	characteristic
86	#108	—	2	2	no	—
87	#104	—	292	2	no	—
88	C₄×C₈	—	73	1	no	—
89	C₃₂	—	146	1	no	—
90	C₃₂	C₄×D₇₃	1	1	yes	characteristic
91	C₃₂	C₄×D₇₃	1	1	yes	characteristic
92	#41	—	292	2	no	—
93	Q₁₆	—	146	2	no	—
94	SD₁₆	—	292	2	no	—
95	D₈	—	146	2	no	—
96	M₁₆	—	4	2	no	—
97	M₁₆	—	584	2	no	—
98	C₂×C₈	—	584	2	no	—
99	C₂×C₈	—	73	1	no	—
100	C₂×C₈	D₄×D₇₃	1	1	yes	characteristic
101	C₄²	—	73	1	no	—
102	C₁₆	—	146	1	no	—
103	C₁₆	C₂×C₄×D₇₃	1	1	yes	characteristic
104	C₁₆	#11924029	1	1	yes	characteristic
105	Q₈	—	292	2	no	—
106	D₄	—	292	2	no	—
107	D₄	—	584	2	no	—
108	C₂×C₄	—	584	2	no	—
109	C₂×C₄	—	146	1	no	—
110	C₂×C₄	D₈×D₇₃	1	1	yes	characteristic
111	C₈	—	8	2	no	—
112	C₈	—	73	1	no	—
113	C₈	—	73	1	no	—
114	C₈	#14556553	1	1	yes	characteristic
115	C₈	#14556614	1	1	yes	characteristic
116	C₂²	—	584	2	no	—
117	C₂²	#17989020	1	1	yes	characteristic
118	C₄	—	584	2	no	—
119	C₄	—	146	2	no	—
120	C₄	#17988669	1	1	yes	characteristic
121	C₄	D₁₆×D₇₃	1	1	yes	characteristic
122	C₂	—	2	1	no	—
123	C₂	—	1168	2	no	—
124	C₂	#22611777	1	1	yes	characteristic

subgroup

G/H

class size

multiplicity

normal?

properties

#22612240

C₂

yes

maximal, characteristic

#22611811

C₂

yes

maximal, complemented

#22608114

C₂

yes

maximal, complemented

C₃₂×Dic₇₃

C₂

yes

maximal, characteristic, complemented

C₇₃×#1425

C₂

yes

maximal, characteristic, complemented

#17989019

C₂²

yes

—

#17988703

—

Dic₁₁₆₈

C₄

yes

characteristic, complemented

#17988426

C₄

yes

complemented

D₂₃₃₆

C₄

yes

characteristic, complemented

C₂×#14556497

C₂²

yes

characteristic

#17988393

—

C₁₆×Dic₇₃

C₂²

yes

characteristic

C₂×C₃₂×C₇₃

C₂²

yes

characteristic

#368×C₇₃

C₂²

yes

—

#14556613

—

#14556599

—

Dic₅₈₄

C₂×C₄

yes

—

#14556536

—

D₁₁₆₈

C₂×C₄

yes

—

C₂×#11924003

D₄

yes

characteristic

#14556517

—

C₈×Dic₇₃

D₄

yes

characteristic

#14556497

—

C₂×C₁₆×C₇₃

C₂³

yes

characteristic

#108×C₇₃

—

C₃₂×C₇₃

C₂×C₄

yes

characteristic

C₃₂×C₇₃

C₂×C₄

yes

characteristic

#11924028

—

Dic₂₉₂

—

#11924017

—

D₅₈₄

—

#11924011

—

C₂×#10596831

D₈

yes

characteristic

C₈×D₇₃

—

C₄×Dic₇₃

D₈

yes

characteristic

#11924003

—

M₁₆×C₇₃

—

C₂×C₈×C₇₃

C₂×D₄

yes

characteristic

C₁₆×C₇₃

C₂²×C₄

yes

characteristic

C₁₆×C₇₃

#41

yes

characteristic

C₂×C₄×C₇₃

C₂×D₈

yes

characteristic

C₂²⋊D₇₃

—

C₂×Dic₇₃

—

D₂₉₂

—

C₄×D₇₃

—

Dic₁₄₆

—

C₈×C₇₃

—

C₈×C₇₃

C₄×D₄

yes

C₈×C₇₃

yes

D₁₆

yes

characteristic

#10596831

D₁₆

yes

characteristic

C₂²×C₇₃

#508

yes

characteristic

C₂×D₇₃

—

C₄×C₇₃

C₄×D₈

yes

characteristic

C₄×C₇₃

C₂×D₁₆

yes

characteristic

Dic₇₃

—

Dic₇₃

—

#13655

—

maximal, Sylow

C₂×C₇₃

—

C₂×C₇₃

C₄×D₁₆

yes

characteristic

D₇₃

—

#2262

—

#2178

—

#1425

D₇₃

yes

characteristic, complemented

#1417

—

C₄×C₃₂

—

C₇₃

#13655

yes

characteristic, complemented, Sylow

#508

—

#443

—

146

—

Q₆₄

—

SD₆₄

—

D₃₂

—

C₂×C₃₂

—

C₂×C₃₂

C₂×D₇₃

yes

characteristic

#368

C₂×D₇₃

yes

—

#359

—

146

—

C₄×C₁₆

—

#135

—

146

—

#131

—

292

—

Q₃₂

—

SD₃₂

—

146

—

D₁₆

—

C₂×C₁₆

—

C₂×C₁₆

C₂²×D₇₃

yes

characteristic

#108

—

#104

—

292

—

C₄×C₈

—

C₃₂

—

146

—

C₃₂

C₄×D₇₃

yes

characteristic

C₃₂

C₄×D₇₃

yes

characteristic

#41

—

292

—

Q₁₆

—

146

—

SD₁₆

—

292

—

D₈

—

146

—

M₁₆

—

M₁₆

—

584

—

C₂×C₈

—

584

—

C₂×C₈

—

100

C₂×C₈

D₄×D₇₃

yes

characteristic

101

C₄²

—

102

C₁₆

—

146

—

103

C₁₆

C₂×C₄×D₇₃

yes

characteristic

104

C₁₆

#11924029

yes

characteristic

105

Q₈

—

292

—

106

D₄

—

292

—

107

D₄

—

584

—

108

C₂×C₄

—

584

—

109

C₂×C₄

—

146

—

110

C₂×C₄

D₈×D₇₃

yes

characteristic

111

C₈

—

112

C₈

—

113

C₈

—

114

C₈

#14556553

yes

characteristic

115

C₈

#14556614

yes

characteristic

116

C₂²

—

584

—

117

C₂²

#17989020

yes

characteristic

118

C₄

—

584

—

119

C₄

—

146

—

120

C₄

#17988669

yes

characteristic

121

C₄

D₁₆×D₇₃

yes

characteristic

122

C₂

—

123

C₂

—

1168

—

124

C₂

#22611777

yes

characteristic

	subgroup	G/H
center	C₄	D₁₆×D₇₃
commutator	C₁₆×C₇₃	C₂²×C₄
Frattini	C₂×C₁₆	C₂²×D₇₃
Fitting	C₇₃×#1425	C₂
socle	C₂×C₇₃	C₄×D₁₆
prime radical	#22612240	C₂
cosocle	C₂×C₁₆×C₇₃	C₂³
hypercenter	#1425	D₇₃
hypocenter	C₇₃	#13655
LCS	G ▷ C₁₆×C₇₃ ▷ C₈×C₇₃ ▷ C₄×C₇₃ ▷ C₂×C₇₃ ▷ C₇₃
UCS	C₁ ◁ C₄ ◁ C₂×C₄ ◁ C₂×C₈ ◁ C₂×C₁₆ ◁ #1425

subgroup

G/H

center

C₄

D₁₆×D₇₃

commutator

C₁₆×C₇₃

C₂²×C₄

Frattini

C₂×C₁₆

C₂²×D₇₃

Fitting

C₇₃×#1425

C₂

socle

C₂×C₇₃

C₄×D₁₆

prime radical

#22612240

C₂

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 ◁ #1425

N	H ≅ G/N	type
2. #22611811	C₂	semidirect
3. #22608114	C₂	semidirect
4. C₃₂×Dic₇₃	C₂	semidirect
5. C₇₃×#1425	C₂	semidirect
8. Dic₁₁₆₈	C₄	semidirect
9. #17988426	C₄	semidirect
10. D₂₃₃₆	C₄	semidirect
65. #1425	D₇₃	semidirect
68. C₇₃	#13655	semidirect

H ≅ G/N

type

2. #22611811

C₂

semidirect

3. #22608114

C₂

semidirect

4. C₃₂×Dic₇₃

C₂

semidirect

5. C₇₃×#1425

C₂

semidirect

8. Dic₁₁₆₈

C₄

semidirect

9. #17988426

C₄

semidirect

10. D₂₃₃₆

C₄

semidirect

65. #1425

D₇₃

semidirect

68. C₇₃

#13655

semidirect

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

subgroup	index
#22612240	2
#22611811	2
#22608114	2
C₃₂×Dic₇₃	2
C₇₃×#1425	2
#13655	73

subgroup

index

#22612240

#22611811

#22608114

C₃₂×Dic₇₃

C₇₃×#1425

#13655

Subgroups — #33006591

Conjugacy classes of subgroups

Complements

Maximal subgroups