Group #33000659 — subgroups

Subgroups — #33000659

Order 18688.

258 conjugacy classes of subgroups; only the 91 normal subgroups are shown.

normal subgroups only

This group has 258 conjugacy classes of subgroups — too many to draw. Only its 91 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	#22612213	C₂	1	1	yes	maximal, characteristic
2	#22612184	C₂	1	1	yes	maximal, characteristic
3	#22612159	C₂	1	2	yes	maximal
4	#22609528	C₂	1	2	yes	maximal
5	C₇₃×#1311	C₂	1	1	yes	maximal, characteristic
6	C₂×Dic₅₈₄	—	2	2	no	—
7	#17988992	C₂²	1	2	yes	—
8	#17988985	—	2	2	no	—
9	#17988967	C₂²	1	2	yes	—
10	#17988938	—	2	2	no	—
11	#17988415	—	2	2	no	—
12	#17988392	—	2	4	no	—
13	C₄×C₁₆×C₇₃	C₂²	1	1	yes	characteristic
14	C₄×C₁₆×C₇₃	C₂²	1	1	yes	characteristic
15	C₈²×C₇₃	C₂²	1	1	yes	characteristic
16	C₂×Dic₂₉₂	—	2	2	no	—
17	C₂×Dic₂₉₂	—	2	2	no	—
18	#14556591	—	2	2	no	—
19	#14556591	—	2	2	no	—
20	Dic₅₈₄	—	4	4	no	—
21	#14556525	—	2	2	no	—
22	#14556525	—	2	2	no	—
23	#14556523	—	4	4	no	—
24	#14556516	—	4	4	no	—
25	C₂×C₁₆×C₇₃	D₄	1	2	yes	—
26	C₂×C₁₆×C₇₃	D₄	1	2	yes	—
27	C₄×C₈×C₇₃	D₄	1	2	yes	—
28	C₄×C₈×C₇₃	C₂³	1	1	yes	characteristic
29	C₂×Dic₁₄₆	—	4	2	no	—
30	C₂×Dic₁₄₆	—	4	2	no	—
31	Dic₂₉₂	—	4	4	no	—
32	Dic₂₉₂	—	4	4	no	—
33	C₄⋊Dic₇₃	—	4	2	no	—
34	C₄⋊Dic₇₃	—	4	2	no	—
35	C₄⋊Dic₇₃	—	4	2	no	—
36	C₄⋊Dic₇₃	—	4	2	no	—
37	C₂×C₈×C₇₃	D₈	1	4	yes	—
38	C₂×C₈×C₇₃	C₂×D₄	1	1	yes	characteristic
39	C₂×C₈×C₇₃	C₂×D₄	1	1	yes	characteristic
40	C₄²×C₇₃	C₂×D₄	1	1	yes	characteristic
41	C₁₆×C₇₃	—	2	2	no	—
42	C₁₆×C₇₃	SD₁₆	1	4	yes	—
43	C₂×C₄×C₇₃	#127	1	1	yes	characteristic
44	C₂×C₄×C₇₃	C₂×D₈	1	2	yes	—
45	C₂×Dic₇₃	—	8	2	no	—
46	C₂×Dic₇₃	—	8	2	no	—
47	Dic₁₄₆	—	8	2	no	—
48	Dic₁₄₆	—	8	2	no	—
49	Dic₁₄₆	—	8	4	no	—
50	C₈×C₇₃	—	2	4	no	—
51	C₈×C₇₃	C₂×SD₁₆	1	2	yes	—
52	C₈×C₇₃	#135	1	2	yes	—
53	C₂²×C₇₃	#493	1	1	yes	characteristic
54	C₄×C₇₃	#492	1	1	yes	characteristic
55	C₄×C₇₃	#495	1	1	yes	characteristic
56	C₄×C₇₃	#510	1	4	yes	—
57	Dic₇₃	—	16	2	no	—
58	Dic₇₃	—	16	2	no	—
59	#12507	—	73	1	no	maximal, Sylow
60	C₂×C₇₃	#1706	1	1	yes	characteristic
61	C₂×C₇₃	#2251	1	2	yes	—
62	#2254	—	73	1	no	—
63	#2247	—	73	1	no	—
64	#2241	—	73	2	no	—
65	#1707	—	73	2	no	—
66	#1311	D₇₃	1	1	yes	characteristic
67	C₇₃	#12507	1	1	yes	characteristic, complemented, Sylow
68	C₂×Q₃₂	—	146	2	no	—
69	#500	—	73	2	no	—
70	#498	—	146	2	no	—
71	#494	—	73	2	no	—
72	#487	—	146	2	no	—
73	#367	—	146	2	no	—
74	#358	—	146	4	no	—
75	C₄×C₁₆	C₂×D₇₃	1	1	yes	characteristic
76	C₄×C₁₆	C₂×D₇₃	1	1	yes	characteristic
77	C₈²	C₂×D₇₃	1	1	yes	characteristic
78	C₂×Q₁₆	—	146	2	no	—
79	C₂×Q₁₆	—	146	2	no	—
80	C₄⋊Q₈	—	146	2	no	—
81	C₄⋊Q₈	—	146	2	no	—
82	Q₃₂	—	292	4	no	—
83	C₂×C₁₆	D₂₉₂	1	2	yes	—
84	C₂×C₁₆	D₂₉₂	1	2	yes	—
85	#107	—	146	2	no	—
86	#107	—	146	2	no	—
87	#106	—	292	4	no	—
88	#103	—	292	4	no	—
89	C₄×C₈	D₂₉₂	1	2	yes	—
90	C₄×C₈	C₂²×D₇₃	1	1	yes	characteristic
91	C₂×Q₈	—	292	2	no	—
92	C₂×Q₈	—	292	2	no	—
93	Q₁₆	—	292	4	no	—
94	Q₁₆	—	292	4	no	—
95	C₂×C₈	D₅₈₄	1	4	yes	—
96	C₂×C₈	C₂×D₂₉₂	1	1	yes	characteristic
97	C₂×C₈	C₂×D₂₉₂	1	1	yes	characteristic
98	C₄⋊C₄	—	292	2	no	—
99	C₄⋊C₄	—	292	2	no	—
100	C₄⋊C₄	—	292	2	no	—
101	C₄⋊C₄	—	292	2	no	—
102	C₄²	C₂×D₂₉₂	1	1	yes	characteristic
103	C₁₆	—	2	2	no	—
104	C₁₆	#11924017	1	4	yes	—
105	Q₈	—	584	2	no	—
106	Q₈	—	584	2	no	—
107	Q₈	—	584	4	no	—
108	C₂×C₄	—	584	2	no	—
109	C₂×C₄	—	584	2	no	—
110	C₂×C₄	#14556588	1	1	yes	characteristic
111	C₂×C₄	C₂×D₅₈₄	1	2	yes	—
112	C₈	—	2	4	no	—
113	C₈	C₂×#11924017	1	2	yes	—
114	C₈	#14556613	1	2	yes	—
115	C₂²	#17988964	1	1	yes	characteristic
116	C₄	—	1168	2	no	—
117	C₄	—	1168	2	no	—
118	C₄	#17988960	1	1	yes	characteristic
119	C₄	#17988971	1	1	yes	characteristic
120	C₄	#17989030	1	4	yes	—
121	C₂	#22609523	1	1	yes	characteristic
122	C₂	#22612202	1	2	yes	—

subgroup

G/H

class size

multiplicity

normal?

properties

#22612213

C₂

yes

maximal, characteristic

#22612184

C₂

yes

maximal, characteristic

#22612159

C₂

yes

maximal

#22609528

C₂

yes

maximal

C₇₃×#1311

C₂

yes

maximal, characteristic

C₂×Dic₅₈₄

—

#17988992

C₂²

yes

—

#17988985

—

#17988967

C₂²

yes

—

#17988938

—

#17988415

—

#17988392

—

C₄×C₁₆×C₇₃

C₂²

yes

characteristic

C₄×C₁₆×C₇₃

C₂²

yes

characteristic

C₈²×C₇₃

C₂²

yes

characteristic

C₂×Dic₂₉₂

—

C₂×Dic₂₉₂

—

#14556591

—

#14556591

—

Dic₅₈₄

—

—

—

—

—

C₂×C₁₆×C₇₃

D₄

yes

—

C₂×C₁₆×C₇₃

D₄

yes

—

C₄×C₈×C₇₃

D₄

yes

—

C₄×C₈×C₇₃

C₂³

yes

characteristic

C₂×Dic₁₄₆

—

C₂×Dic₁₄₆

—

Dic₂₉₂

—

Dic₂₉₂

—

C₄⋊Dic₇₃

—

C₄⋊Dic₇₃

—

C₄⋊Dic₇₃

—

C₄⋊Dic₇₃

—

C₂×C₈×C₇₃

D₈

yes

—

C₂×C₈×C₇₃

C₂×D₄

yes

characteristic

C₂×C₈×C₇₃

C₂×D₄

yes

characteristic

C₄²×C₇₃

C₂×D₄

yes

characteristic

C₁₆×C₇₃

—

C₁₆×C₇₃

SD₁₆

yes

—

C₂×C₄×C₇₃

#127

yes

characteristic

C₂×C₄×C₇₃

C₂×D₈

yes

—

C₂×Dic₇₃

—

C₂×Dic₇₃

—

Dic₁₄₆

—

Dic₁₄₆

—

Dic₁₄₆

—

C₈×C₇₃

—

C₈×C₇₃

C₂×SD₁₆

yes

—

C₈×C₇₃

#135

yes

—

C₂²×C₇₃

yes

C₄×C₇₃

yes

C₄×C₇₃

yes

C₄×C₇₃

yes

—

Dic₇₃

—

Dic₇₃

—

—

C₂×C₇₃

yes

C₂×C₇₃

yes

—

—

—

—

—

D₇₃

yes

characteristic

C₇₃

#12507

yes

characteristic, complemented, Sylow

C₂×Q₃₂

—

146

—

#500

—

#498

—

146

—

#494

—

#487

—

146

—

#367

—

146

—

#358

—

146

—

C₄×C₁₆

C₂×D₇₃

yes

characteristic

C₄×C₁₆

C₂×D₇₃

yes

characteristic

C₈²

C₂×D₇₃

yes

characteristic

C₂×Q₁₆

—

146

—

C₂×Q₁₆

—

146

—

C₄⋊Q₈

—

146

—

C₄⋊Q₈

—

146

—

Q₃₂

—

292

—

C₂×C₁₆

D₂₉₂

yes

—

C₂×C₁₆

D₂₉₂

yes

—

#107

—

146

—

#107

—

146

—

#106

—

292

—

#103

—

292

—

C₄×C₈

D₂₉₂

yes

—

C₄×C₈

C₂²×D₇₃

yes

characteristic

C₂×Q₈

—

292

—

C₂×Q₈

—

292

—

Q₁₆

—

292

—

Q₁₆

—

292

—

C₂×C₈

D₅₈₄

yes

—

C₂×C₈

C₂×D₂₉₂

yes

characteristic

C₂×C₈

C₂×D₂₉₂

yes

characteristic

C₄⋊C₄

—

292

—

C₄⋊C₄

—

292

—

100

C₄⋊C₄

—

292

—

101

C₄⋊C₄

—

292

—

102

C₄²

C₂×D₂₉₂

yes

characteristic

103

C₁₆

—

104

C₁₆

#11924017

yes

—

105

Q₈

—

584

—

106

Q₈

—

584

—

107

Q₈

—

584

—

108

C₂×C₄

—

584

—

109

C₂×C₄

—

584

—

110

C₂×C₄

#14556588

yes

characteristic

111

C₂×C₄

C₂×D₅₈₄

yes

—

112

C₈

—

113

C₈

C₂×#11924017

yes

—

114

C₈

#14556613

yes

—

115

C₂²

#17988964

yes

characteristic

116

C₄

—

1168

—

117

C₄

—

1168

—

118

C₄

#17988960

yes

characteristic

119

C₄

#17988971

yes

characteristic

120

C₄

#17989030

yes

—

121

C₂

#22609523

yes

characteristic

122

C₂

#22612202

yes

—

	subgroup	G/H
center	C₂²	#17988964
commutator, cosocle	C₄×C₈×C₇₃	C₂³
Frattini	C₄×C₈	C₂²×D₇₃
Fitting	C₇₃×#1311	C₂
socle, prime radical	C₂²×C₇₃	#493
hypercenter	#1311	D₇₃
hypocenter	C₇₃	#12507
LCS	G ▷ C₄×C₈×C₇₃ ▷ C₂×C₄×C₇₃ ▷ C₂×C₇₃ ▷ C₇₃
UCS	C₁ ◁ C₂² ◁ C₄² ◁ C₈² ◁ #1311

subgroup

G/H

center

C₂²

#17988964

commutator, cosocle

C₄×C₈×C₇₃

C₂³

Frattini

C₄×C₈

C₂²×D₇₃

Fitting

C₇₃×#1311

C₂

C₂²×C₇₃

D₇₃

C₇₃

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

UCS

C₁ ◁ C₂² ◁ C₄² ◁ C₈² ◁ #1311

N	H ≅ G/N	type
67. C₇₃	#12507	semidirect

H ≅ G/N

type

67. C₇₃

#12507

semidirect

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

subgroup	index
#22612213	2
#22612184	2
#22612159	2
#22609528	2
C₇₃×#1311	2
#12507	73

subgroup

index

C₇₃×#1311

Subgroups — #33000659

Conjugacy classes of subgroups

Complements

Maximal subgroups