Group #33002367 — subgroups

Subgroups — #33002367

Order 18688.

310 conjugacy classes of subgroups; only the 163 normal subgroups are shown.

normal subgroups only

This group has 310 conjugacy classes of subgroups — too many to draw. Only its 163 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	#22610095	C₂	1	2	yes	maximal
2	#22608023	C₂	1	4	yes	maximal
3	C₂×C₄×C₁₆×C₇₃	C₂	1	1	yes	maximal, characteristic
4	C₂×#14556525	C₂²	1	4	yes	—
5	C₂×#14556523	—	2	2	no	—
6	#17988453	—	2	2	no	—
7	C₂²×C₁₆×C₇₃	C₂²	1	2	yes	—
8	C₂×C₄×C₈×C₇₃	C₂²	1	1	yes	characteristic
9	C₄×C₁₆×C₇₃	C₄	1	4	yes	complemented
10	C₂×C₄⋊Dic₇₃	—	2	4	no	—
11	#14556525	—	2	8	no	—
12	#14556523	—	4	4	no	—
13	#14556498	—	4	4	no	—
14	C₂²×C₈×C₇₃	D₄	1	1	yes	characteristic
15	C₂²×C₈×C₇₃	C₂³	1	1	yes	characteristic
16	C₂×C₄²×C₇₃	D₄	1	1	yes	characteristic
17	C₂×C₁₆×C₇₃	—	2	4	no	—
18	C₂×C₁₆×C₇₃	C₂×C₄	1	4	yes	—
19	C₄×C₈×C₇₃	C₂×C₄	1	2	yes	—
20	C₄×C₈×C₇₃	Q₈	1	2	yes	—
21	C₂²×Dic₇₃	—	4	4	no	—
22	C₄⋊Dic₇₃	—	4	4	no	—
23	C₄⋊Dic₇₃	—	4	4	no	—
24	C₂²×C₄×C₇₃	D₈	1	2	yes	—
25	C₂²×C₄×C₇₃	C₂×D₄	1	1	yes	characteristic
26	C₂×C₈×C₇₃	C₄⋊C₄	1	2	yes	—
27	C₂×C₈×C₇₃	SD₁₆	1	4	yes	—
28	C₂×C₈×C₇₃	C₂²×C₄	1	1	yes	characteristic
29	C₂×C₈×C₇₃	C₂×Q₈	1	1	yes	characteristic
30	C₂×C₈×C₇₃	#41	1	4	yes	—
31	C₄²×C₇₃	C₄⋊C₄	1	2	yes	—
32	C₄²×C₇₃	Q₁₆	1	2	yes	—
33	C₁₆×C₇₃	—	2	4	no	—
34	C₂³×C₇₃	C₂×D₈	1	1	yes	characteristic
35	C₂×C₄×C₇₃	—	2	4	no	—
36	C₂×C₄×C₇₃	#107	1	4	yes	—
37	C₂×C₄×C₇₃	C₂×C₄⋊C₄	1	1	yes	characteristic
38	C₂×C₄×C₇₃	#124	1	2	yes	—
39	C₂×C₄×C₇₃	C₂×SD₁₆	1	2	yes	—
40	C₂×C₄×C₇₃	C₂×Q₁₆	1	1	yes	characteristic
41	C₂×Dic₇₃	—	8	4	no	—
42	C₂×Dic₇₃	—	8	8	no	—
43	C₈×C₇₃	#106	1	4	yes	—
44	C₈×C₇₃	#117	1	2	yes	—
45	C₈×C₇₃	#125	1	2	yes	—
46	C₂²×C₇₃	C₂×#107	1	1	yes	characteristic
47	C₂²×C₇₃	#486	1	2	yes	—
48	C₂²×C₇₃	#508	1	4	yes	—
49	C₄×C₇₃	—	2	4	no	—
50	C₄×C₇₃	#382	1	1	yes	characteristic
51	C₄×C₇₃	C₂×#106	1	1	yes	characteristic
52	C₄×C₇₃	#487	1	2	yes	—
53	Dic₇₃	—	16	4	no	—
54	#12889	—	73	1	no	maximal, Sylow
55	C₂×C₇₃	#1834	1	1	yes	characteristic
56	C₂×C₇₃	#2158	1	2	yes	—
57	C₂×C₇₃	#2242	1	4	yes	—
58	C₂×C₄×C₁₆	D₇₃	1	1	yes	characteristic
59	#1846	—	73	2	no	—
60	#1380	—	73	4	no	—
61	C₇₃	#12889	1	1	yes	characteristic, complemented, Sylow
62	C₂²×C₁₆	C₂×D₇₃	1	2	yes	—
63	C₂×#107	—	73	4	no	—
64	C₂×#106	—	146	2	no	—
65	C₂×C₄×C₈	C₂×D₇₃	1	1	yes	characteristic
66	#382	—	146	2	no	—
67	C₄×C₁₆	Dic₇₃	1	4	yes	complemented
68	C₂²×C₈	D₂₉₂	1	1	yes	characteristic
69	C₂²×C₈	C₂²×D₇₃	1	1	yes	characteristic
70	C₂×C₄⋊C₄	—	146	4	no	—
71	C₂×C₄²	D₂₉₂	1	1	yes	characteristic
72	C₂×C₁₆	—	2	4	no	—
73	C₂×C₁₆	C₂×Dic₇₃	1	4	yes	—
74	#107	—	146	8	no	—
75	#106	—	292	4	no	—
76	C₄×C₈	Dic₁₄₆	1	2	yes	—
77	C₄×C₈	C₂×Dic₇₃	1	2	yes	—
78	#95	—	292	4	no	—
79	C₂²×C₄	—	292	4	no	—
80	C₂²×C₄	D₅₈₄	1	2	yes	—
81	C₂²×C₄	C₂×D₂₉₂	1	1	yes	characteristic
82	C₂×C₈	C₄⋊Dic₇₃	1	2	yes	—
83	C₂×C₈	#11924017	1	4	yes	—
84	C₂×C₈	C₂²×Dic₇₃	1	1	yes	characteristic
85	C₂×C₈	C₂×Dic₁₄₆	1	1	yes	characteristic
86	C₂×C₈	#11924028	1	4	yes	—
87	C₄⋊C₄	—	292	4	no	—
88	C₄⋊C₄	—	292	4	no	—
89	C₄²	C₄⋊Dic₇₃	1	2	yes	—
90	C₄²	Dic₂₉₂	1	2	yes	—
91	C₁₆	—	2	4	no	—
92	C₂³	C₂×D₅₈₄	1	1	yes	characteristic
93	C₂×C₄	—	2	4	no	—
94	C₂×C₄	—	584	4	no	—
95	C₂×C₄	—	584	8	no	—
96	C₂×C₄	#14556525	1	4	yes	—
97	C₂×C₄	C₂×C₄⋊Dic₇₃	1	1	yes	characteristic
98	C₂×C₄	#14556579	1	2	yes	—
99	C₂×C₄	C₂×#11924017	1	2	yes	—
100	C₂×C₄	C₂×Dic₂₉₂	1	1	yes	characteristic
101	C₈	#14556523	1	4	yes	—
102	C₈	#14556550	1	2	yes	—
103	C₈	#14556583	1	2	yes	—
104	C₂²	C₂×#14556525	1	1	yes	characteristic
105	C₂²	#17988934	1	2	yes	—
106	C₂²	#17989019	1	4	yes	—
107	C₄	—	2	4	no	—
108	C₄	—	1168	4	no	—
109	C₄	#17988453	1	1	yes	characteristic
110	C₄	C₂×#14556523	1	1	yes	characteristic
111	C₄	#17988938	1	2	yes	—
112	C₂	#22610059	1	1	yes	characteristic
113	C₂	#22611715	1	2	yes	—
114	C₂	#22612163	1	4	yes	—

subgroup

G/H

class size

multiplicity

normal?

properties

#22610095

C₂

yes

maximal

#22608023

C₂

yes

maximal

C₂×C₄×C₁₆×C₇₃

C₂

yes

maximal, characteristic

C₂×#14556525

C₂²

yes

—

C₂×#14556523

—

#17988453

—

C₂²×C₁₆×C₇₃

C₂²

yes

—

C₂×C₄×C₈×C₇₃

C₂²

yes

characteristic

C₄×C₁₆×C₇₃

C₄

yes

complemented

C₂×C₄⋊Dic₇₃

—

#14556525

—

#14556523

—

#14556498

—

C₂²×C₈×C₇₃

D₄

yes

characteristic

C₂²×C₈×C₇₃

C₂³

yes

characteristic

C₂×C₄²×C₇₃

D₄

yes

characteristic

C₂×C₁₆×C₇₃

—

C₂×C₁₆×C₇₃

C₂×C₄

yes

—

C₄×C₈×C₇₃

C₂×C₄

yes

—

C₄×C₈×C₇₃

Q₈

yes

—

C₂²×Dic₇₃

—

C₄⋊Dic₇₃

—

C₄⋊Dic₇₃

—

C₂²×C₄×C₇₃

D₈

yes

—

C₂²×C₄×C₇₃

C₂×D₄

yes

characteristic

C₂×C₈×C₇₃

C₄⋊C₄

yes

—

C₂×C₈×C₇₃

SD₁₆

yes

—

C₂×C₈×C₇₃

C₂²×C₄

yes

characteristic

C₂×C₈×C₇₃

C₂×Q₈

yes

characteristic

C₂×C₈×C₇₃

#41

yes

—

C₄²×C₇₃

C₄⋊C₄

yes

—

C₄²×C₇₃

Q₁₆

yes

—

C₁₆×C₇₃

—

C₂³×C₇₃

C₂×D₈

yes

characteristic

C₂×C₄×C₇₃

—

C₂×C₄×C₇₃

#107

yes

—

C₂×C₄×C₇₃

C₂×C₄⋊C₄

yes

characteristic

C₂×C₄×C₇₃

#124

yes

—

C₂×C₄×C₇₃

C₂×SD₁₆

yes

—

C₂×C₄×C₇₃

C₂×Q₁₆

yes

characteristic

C₂×Dic₇₃

—

C₂×Dic₇₃

—

C₈×C₇₃

#106

yes

—

C₈×C₇₃

#117

yes

—

C₈×C₇₃

#125

yes

—

C₂²×C₇₃

C₂×#107

yes

characteristic

C₂²×C₇₃

#486

yes

—

C₂²×C₇₃

#508

yes

—

C₄×C₇₃

—

C₄×C₇₃

yes

C₄×C₇₃

yes

C₄×C₇₃

yes

—

Dic₇₃

—

—

C₂×C₇₃

yes

C₂×C₇₃

yes

—

C₂×C₇₃

#2242

yes

—

C₂×C₄×C₁₆

D₇₃

yes

characteristic

#1846

—

#1380

—

C₇₃

#12889

yes

characteristic, complemented, Sylow

C₂²×C₁₆

C₂×D₇₃

yes

—

C₂×#107

—

C₂×#106

—

146

—

C₂×C₄×C₈

C₂×D₇₃

yes

characteristic

#382

—

146

—

C₄×C₁₆

Dic₇₃

yes

complemented

C₂²×C₈

D₂₉₂

yes

characteristic

C₂²×C₈

C₂²×D₇₃

yes

characteristic

C₂×C₄⋊C₄

—

146

—

C₂×C₄²

D₂₉₂

yes

characteristic

C₂×C₁₆

—

C₂×C₁₆

C₂×Dic₇₃

yes

—

#107

—

146

—

#106

—

292

—

C₄×C₈

Dic₁₄₆

yes

—

C₄×C₈

C₂×Dic₇₃

yes

—

#95

—

292

—

C₂²×C₄

—

292

—

C₂²×C₄

D₅₈₄

yes

—

C₂²×C₄

C₂×D₂₉₂

yes

characteristic

C₂×C₈

C₄⋊Dic₇₃

yes

—

C₂×C₈

#11924017

yes

—

C₂×C₈

C₂²×Dic₇₃

yes

characteristic

C₂×C₈

C₂×Dic₁₄₆

yes

characteristic

C₂×C₈

#11924028

yes

—

C₄⋊C₄

—

292

—

C₄⋊C₄

—

292

—

C₄²

C₄⋊Dic₇₃

yes

—

C₄²

Dic₂₉₂

yes

—

C₁₆

—

C₂³

C₂×D₅₈₄

yes

characteristic

C₂×C₄

—

C₂×C₄

—

584

—

C₂×C₄

—

584

—

C₂×C₄

#14556525

yes

—

C₂×C₄

C₂×C₄⋊Dic₇₃

yes

characteristic

C₂×C₄

#14556579

yes

—

C₂×C₄

C₂×#11924017

yes

—

100

C₂×C₄

C₂×Dic₂₉₂

yes

characteristic

101

C₈

#14556523

yes

—

102

C₈

#14556550

yes

—

103

C₈

#14556583

yes

—

104

C₂²

C₂×#14556525

yes

characteristic

105

C₂²

#17988934

yes

—

106

C₂²

#17989019

yes

—

107

C₄

—

108

C₄

—

1168

—

109

C₄

#17988453

yes

characteristic

110

C₄

C₂×#14556523

yes

characteristic

111

C₄

#17988938

yes

—

112

C₂

#22610059

yes

characteristic

113

C₂

#22611715

yes

—

114

C₂

#22612163

yes

—

	subgroup	G/H
center	C₂³	C₂×D₅₈₄
commutator	C₂×C₈×C₇₃	C₂²×C₄
Frattini	C₂²×C₈	C₂²×D₇₃
Fitting	C₂×C₄×C₁₆×C₇₃	C₂
socle, prime radical	C₂³×C₇₃	C₂×D₈
cosocle	C₂²×C₈×C₇₃	C₂³
hypercenter	C₂×C₄×C₁₆	D₇₃
hypocenter	C₇₃	#12889
LCS	G ▷ C₂×C₈×C₇₃ ▷ C₄×C₇₃ ▷ C₂×C₇₃ ▷ C₇₃
UCS	C₁ ◁ C₂³ ◁ C₂×C₄² ◁ C₂×C₄×C₈ ◁ C₂×C₄×C₁₆

subgroup

G/H

center

C₂³

C₂×D₅₈₄

commutator

C₂×C₈×C₇₃

C₂²×C₄

Frattini

C₂²×C₈

C₂²×D₇₃

Fitting

C₂×C₄×C₁₆×C₇₃

C₂

socle, prime radical

C₂³×C₇₃

C₂×D₈

cosocle

C₂²×C₈×C₇₃

C₂³

hypercenter

C₂×C₄×C₁₆

D₇₃

hypocenter

C₇₃

#12889

LCS

G ▷ C₂×C₈×C₇₃ ▷ C₄×C₇₃ ▷ C₂×C₇₃ ▷ C₇₃

UCS

C₁ ◁ C₂³ ◁ C₂×C₄² ◁ C₂×C₄×C₈ ◁ C₂×C₄×C₁₆

N	H ≅ G/N	type
9. C₄×C₁₆×C₇₃	C₄	semidirect
61. C₇₃	#12889	semidirect
67. C₄×C₁₆	Dic₇₃	semidirect

H ≅ G/N

type

9. C₄×C₁₆×C₇₃

C₄

semidirect

61. C₇₃

#12889

semidirect

67. C₄×C₁₆

Dic₇₃

semidirect

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

subgroup	index
#22610095	2
#22608023	2
C₂×C₄×C₁₆×C₇₃	2
#12889	73

subgroup

index

#22610095

#22608023

C₂×C₄×C₁₆×C₇₃

#12889

Subgroups — #33002367

Conjugacy classes of subgroups

Complements

Maximal subgroups