Group #33000723 — subgroups

Subgroups — #33000723

Order 18688.

238 conjugacy classes of subgroups; only the 87 normal subgroups are shown.

normal subgroups only

This group has 238 conjugacy classes of subgroups — too many to draw. Only its 87 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	#22612211	C₂	1	2	yes	maximal
2	#22612206	C₂	1	1	yes	maximal, characteristic
3	#22609540	C₂	1	2	yes	maximal
4	C₁₆×#10596831	C₂	1	1	yes	maximal, characteristic
5	C₇₃×#2252	C₂	1	1	yes	maximal, characteristic
6	#17988992	C₂²	1	2	yes	—
7	#17988939	—	2	2	no	—
8	#17988937	—	2	2	no	—
9	#17988414	—	2	4	no	—
10	#17988413	—	2	2	no	—
11	C₄×#11924003	C₂²	1	1	yes	characteristic
12	C₈×#10596831	C₂²	1	1	yes	characteristic
13	#500×C₇₃	C₂²	1	2	yes	—
14	#366×C₇₃	—	2	2	no	—
15	C₄×C₁₆×C₇₃	C₂²	1	1	yes	characteristic
16	#14556591	—	2	2	no	—
17	C₂×#11924003	D₄	1	2	yes	—
18	#14556525	—	2	2	no	—
19	#14556525	—	2	2	no	—
20	#14556515	—	4	4	no	—
21	#14556514	—	4	4	no	—
22	C₄×#10596831	D₄	1	1	yes	characteristic
23	C₄×#10596831	D₄	1	1	yes	characteristic
24	C₄⋊Q₈×C₇₃	—	2	2	no	—
25	C₂×C₁₆×C₇₃	D₄	1	2	yes	—
26	#107×C₇₃	—	2	2	no	—
27	#107×C₇₃	—	2	2	no	—
28	C₄×C₈×C₇₃	C₂³	1	1	yes	characteristic
29	C₂×Dic₁₄₆	—	4	2	no	—
30	C₂×#10596831	D₈	1	2	yes	—
31	C₂×#10596831	D₈	1	2	yes	—
32	C₄⋊Dic₇₃	—	4	2	no	—
33	C₄⋊Dic₇₃	—	4	2	no	—
34	#11924003	Q₁₆	1	4	yes	—
35	C₂×Q₈×C₇₃	—	4	2	no	—
36	C₂×C₈×C₇₃	C₂×D₄	1	1	yes	characteristic
37	C₂×C₈×C₇₃	C₂×D₄	1	1	yes	characteristic
38	C₄⋊C₄×C₇₃	—	4	2	no	—
39	C₄⋊C₄×C₇₃	—	4	2	no	—
40	C₄²×C₇₃	C₂×D₄	1	1	yes	characteristic
41	C₁₆×C₇₃	Q₁₆	1	4	yes	—
42	Q₈×C₇₃	—	8	4	no	—
43	C₂×C₄×C₇₃	—	8	2	no	—
44	C₂×C₄×C₇₃	#127	1	1	yes	characteristic
45	C₂×C₄×C₇₃	C₂×D₈	1	1	yes	characteristic
46	C₂×C₄×C₇₃	C₂×D₈	1	1	yes	characteristic
47	C₂×Dic₇₃	—	8	2	no	—
48	Dic₁₄₆	—	8	4	no	—
49	C₈×C₇₃	C₂×Q₁₆	1	2	yes	—
50	C₈×C₇₃	C₂×Q₁₆	1	2	yes	—
51	#10596831	—	2	2	no	—
52	#10596831	—	2	2	no	—
53	C₂²×C₇₃	#493	1	1	yes	characteristic
54	C₄×C₇₃	—	16	2	no	—
55	C₄×C₇₃	#494	1	1	yes	characteristic
56	C₄×C₇₃	#494	1	1	yes	characteristic
57	C₄×C₇₃	#508	1	2	yes	—
58	C₄×C₇₃	#508	1	2	yes	—
59	Dic₇₃	—	16	2	no	—
60	#12522	—	73	1	no	maximal, Sylow
61	C₂×C₇₃	#1713	1	1	yes	characteristic
62	C₂×C₇₃	#2249	1	1	yes	characteristic
63	C₂×C₇₃	#2249	1	1	yes	characteristic
64	#2254	—	73	2	no	—
65	#2252	—	73	1	no	—
66	#2252	D₇₃	1	1	yes	characteristic
67	#1710	—	73	2	no	—
68	C₈×C₁₆	—	73	1	no	—
69	C₇₃	#12522	1	1	yes	characteristic, complemented, Sylow
70	#500	—	73	2	no	—
71	#500	C₂×D₇₃	1	2	yes	—
72	#487	—	146	2	no	—
73	#487	—	146	2	no	—
74	#367	—	146	4	no	—
75	#366	—	2	2	no	—
76	#366	—	146	2	no	—
77	C₄×C₁₆	—	73	1	no	—
78	C₄×C₁₆	C₂×D₇₃	1	1	yes	characteristic
79	C₈²	—	73	1	no	—
80	C₄⋊Q₈	—	2	2	no	—
81	C₄⋊Q₈	—	146	2	no	—
82	C₂×C₁₆	—	73	2	no	—
83	C₂×C₁₆	C₂²⋊D₇₃	1	2	yes	—
84	#107	—	2	2	no	—
85	#107	—	2	2	no	—
86	#107	—	146	2	no	—
87	#107	—	146	2	no	—
88	#103	—	292	4	no	—
89	#103	—	292	4	no	—
90	C₄×C₈	—	73	1	no	—
91	C₄×C₈	—	73	1	no	—
92	C₄×C₈	C₂²×D₇₃	1	1	yes	characteristic
93	C₂×Q₈	—	4	2	no	—
94	C₂×Q₈	—	292	2	no	—
95	C₂×C₈	—	73	2	no	—
96	C₂×C₈	—	73	2	no	—
97	C₂×C₈	D₄×D₇₃	1	1	yes	characteristic
98	C₂×C₈	C₂×C₂²⋊D₇₃	1	1	yes	characteristic
99	C₄⋊C₄	—	4	2	no	—
100	C₄⋊C₄	—	4	2	no	—
101	C₄⋊C₄	—	292	2	no	—
102	C₄⋊C₄	—	292	2	no	—
103	C₄²	D₄×D₇₃	1	1	yes	characteristic
104	C₁₆	—	73	4	no	—
105	C₁₆	#11924018	1	4	yes	—
106	Q₈	—	8	4	no	—
107	Q₈	—	584	4	no	—
108	C₂×C₄	—	8	2	no	—
109	C₂×C₄	—	584	2	no	—
110	C₂×C₄	#14556587	1	1	yes	characteristic
111	C₂×C₄	D₈×D₇₃	1	1	yes	characteristic
112	C₂×C₄	D₈×D₇₃	1	1	yes	characteristic
113	C₈	—	146	2	no	—
114	C₈	—	146	2	no	—
115	C₈	Q₁₆×D₇₃	1	2	yes	—
116	C₈	C₂×#11924018	1	2	yes	—
117	C₂²	#17988962	1	1	yes	characteristic
118	C₄	—	16	2	no	—
119	C₄	—	1168	2	no	—
120	C₄	#17988965	1	1	yes	characteristic
121	C₄	#17988966	1	1	yes	characteristic
122	C₄	#17989016	1	2	yes	—
123	C₄	#17989020	1	2	yes	—
124	C₂	#22609548	1	1	yes	characteristic
125	C₂	#22612189	1	1	yes	characteristic
126	C₂	#22612193	1	1	yes	characteristic

subgroup

G/H

class size

multiplicity

normal?

properties

#22612211

C₂

yes

maximal

#22612206

C₂

yes

maximal, characteristic

#22609540

C₂

yes

maximal

C₁₆×#10596831

C₂

yes

maximal, characteristic

C₇₃×#2252

C₂

yes

maximal, characteristic

#17988992

C₂²

yes

—

—

—

—

—

C₂²

yes

characteristic

C₈×#10596831

C₂²

yes

characteristic

#500×C₇₃

C₂²

yes

—

#366×C₇₃

—

C₄×C₁₆×C₇₃

C₂²

yes

characteristic

#14556591

—

C₂×#11924003

D₄

yes

—

—

—

—

—

D₄

yes

characteristic

C₄×#10596831

D₄

yes

characteristic

C₄⋊Q₈×C₇₃

—

C₂×C₁₆×C₇₃

D₄

yes

—

#107×C₇₃

—

#107×C₇₃

—

C₄×C₈×C₇₃

C₂³

yes

characteristic

C₂×Dic₁₄₆

—

C₂×#10596831

D₈

yes

—

C₂×#10596831

D₈

yes

—

C₄⋊Dic₇₃

—

C₄⋊Dic₇₃

—

#11924003

Q₁₆

yes

—

C₂×Q₈×C₇₃

—

C₂×C₈×C₇₃

C₂×D₄

yes

characteristic

C₂×C₈×C₇₃

C₂×D₄

yes

characteristic

C₄⋊C₄×C₇₃

—

C₄⋊C₄×C₇₃

—

C₄²×C₇₃

C₂×D₄

yes

characteristic

C₁₆×C₇₃

Q₁₆

yes

—

Q₈×C₇₃

—

C₂×C₄×C₇₃

—

C₂×C₄×C₇₃

#127

yes

characteristic

C₂×C₄×C₇₃

C₂×D₈

yes

characteristic

C₂×C₄×C₇₃

C₂×D₈

yes

characteristic

C₂×Dic₇₃

—

Dic₁₄₆

—

C₈×C₇₃

C₂×Q₁₆

yes

—

C₈×C₇₃

C₂×Q₁₆

yes

—

#10596831

—

#10596831

—

C₂²×C₇₃

#493

yes

characteristic

C₄×C₇₃

—

C₄×C₇₃

yes

C₄×C₇₃

yes

C₄×C₇₃

yes

—

C₄×C₇₃

#508

yes

—

Dic₇₃

—

—

C₂×C₇₃

yes

C₂×C₇₃

yes

C₂×C₇₃

yes

—

—

D₇₃

yes

characteristic

#1710

—

C₈×C₁₆

—

C₇₃

#12522

yes

characteristic, complemented, Sylow

#500

—

#500

C₂×D₇₃

yes

—

#487

—

146

—

#487

—

146

—

#367

—

146

—

#366

—

#366

—

146

—

C₄×C₁₆

—

C₄×C₁₆

C₂×D₇₃

yes

characteristic

C₈²

—

C₄⋊Q₈

—

C₄⋊Q₈

—

146

—

C₂×C₁₆

—

C₂×C₁₆

C₂²⋊D₇₃

yes

—

#107

—

#107

—

#107

—

146

—

#107

—

146

—

#103

—

292

—

#103

—

292

—

C₄×C₈

—

C₄×C₈

—

C₄×C₈

C₂²×D₇₃

yes

characteristic

C₂×Q₈

—

C₂×Q₈

—

292

—

C₂×C₈

—

C₂×C₈

—

C₂×C₈

D₄×D₇₃

yes

characteristic

C₂×C₈

C₂×C₂²⋊D₇₃

yes

characteristic

C₄⋊C₄

—

100

C₄⋊C₄

—

101

C₄⋊C₄

—

292

—

102

C₄⋊C₄

—

292

—

103

C₄²

D₄×D₇₃

yes

characteristic

104

C₁₆

—

105

C₁₆

#11924018

yes

—

106

Q₈

—

107

Q₈

—

584

—

108

C₂×C₄

—

109

C₂×C₄

—

584

—

110

C₂×C₄

#14556587

yes

characteristic

111

C₂×C₄

D₈×D₇₃

yes

characteristic

112

C₂×C₄

D₈×D₇₃

yes

characteristic

113

C₈

—

146

—

114

C₈

—

146

—

115

C₈

Q₁₆×D₇₃

yes

—

116

C₈

C₂×#11924018

yes

—

117

C₂²

#17988962

yes

characteristic

118

C₄

—

119

C₄

—

1168

—

120

C₄

#17988965

yes

characteristic

121

C₄

#17988966

yes

characteristic

122

C₄

#17989016

yes

—

123

C₄

#17989020

yes

—

124

C₂

#22609548

yes

characteristic

125

C₂

#22612189

yes

characteristic

126

C₂

#22612193

yes

characteristic

	subgroup	G/H
center	C₂²	#17988962
commutator, cosocle	C₄×C₈×C₇₃	C₂³
Frattini	C₄×C₈	C₂²×D₇₃
Fitting	C₇₃×#2252	C₂
socle, prime radical	C₂²×C₇₃	#493
hypercenter	#2252	D₇₃
hypocenter	C₇₃	#12522
LCS	G ▷ C₄×C₈×C₇₃ ▷ C₂×C₄×C₇₃ ▷ C₂×C₇₃ ▷ C₇₃
UCS	C₁ ◁ C₂² ◁ C₄² ◁ C₄×C₈ ◁ #2252

subgroup

G/H

center

C₂²

#17988962

commutator, cosocle

C₄×C₈×C₇₃

C₂³

Frattini

C₄×C₈

C₂²×D₇₃

Fitting

C₇₃×#2252

C₂

C₂²×C₇₃

D₇₃

C₇₃

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

UCS

C₁ ◁ C₂² ◁ C₄² ◁ C₄×C₈ ◁ #2252

N	H ≅ G/N	type
69. C₇₃	#12522	semidirect

H ≅ G/N

type

69. C₇₃

#12522

semidirect

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

subgroup	index
#22612211	2
#22612206	2
#22609540	2
C₁₆×#10596831	2
C₇₃×#2252	2
#12522	73

subgroup

index

#22612211

#22612206

#22609540

C₁₆×#10596831

C₇₃×#2252

#12522

Subgroups — #33000723

Conjugacy classes of subgroups

Complements

Maximal subgroups