Group #34000504 — subgroups

Subgroups — #34000504

Order 18688.

184 conjugacy classes of subgroups; only the 62 normal subgroups are shown.

normal subgroups only

This group has 184 conjugacy classes of subgroups — too many to draw. Only its 62 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	#22625493	C₂	1	2	yes	maximal
2	#22611686	C₂	1	1	yes	maximal, characteristic
3	#17989608	—	2	2	no	—
4	#17989450	—	2	2	no	—
5	#17989449	C₄	1	4	yes	—
6	C₂×#14556531	C₄	1	1	yes	characteristic
7	C₂×C₈×Dic₇₃	C₂²	1	1	yes	characteristic
8	#17988405	—	2	2	no	—
9	C₂×M₃₂×C₇₃	C₄	1	1	yes	characteristic
10	C₄⋊#10596833	—	2	4	no	—
11	C₂²⋊#10596833	—	4	2	no	—
12	#14556646	—	4	2	no	—
13	C₂²×#10596831	Q₈	1	1	yes	characteristic
14	C₂×C₄×Dic₇₃	D₄	1	1	yes	characteristic
15	#14556531	—	4	1	no	—
16	C₂×#11924003	—	2	1	no	—
17	C₈×Dic₇₃	C₂×C₄	1	2	yes	—
18	C₈×Dic₇₃	D₄	1	2	yes	—
19	C₂²×C₈×C₇₃	C₂×C₄	1	1	yes	characteristic
20	M₃₂×C₇₃	—	4	1	no	—
21	C₂×C₁₆×C₇₃	—	2	1	no	—
22	C₂×#10596833	—	4	4	no	—
23	C₂²×Dic₇₃	SD₁₆	1	2	yes	—
24	C₂×#10596831	—	4	1	no	—
25	C₂×#10596831	C₄⋊C₄	1	2	yes	—
26	C₄×Dic₇₃	C₂²⋊C₄	1	2	yes	—
27	C₄×Dic₇₃	D₈	1	1	yes	characteristic
28	C₄×Dic₇₃	Q₁₆	1	1	yes	characteristic
29	#11924003	—	4	1	no	—
30	C₂²×C₄×C₇₃	C₄⋊C₄	1	1	yes	characteristic
31	C₂×C₈×C₇₃	—	2	1	no	—
32	C₂×C₈×C₇₃	—	2	1	no	—
33	C₂×C₈×C₇₃	C₄²	1	1	yes	characteristic
34	C₂×C₈×C₇₃	C₂²⋊C₄	1	1	yes	characteristic
35	C₁₆×C₇₃	—	4	1	no	—
36	C₂³×C₇₃	#106	1	1	yes	characteristic
37	C₂×C₄×C₇₃	—	2	1	no	—
38	C₂×C₄×C₇₃	—	2	1	no	—
39	C₂×C₄×C₇₃	#95	1	1	yes	characteristic
40	C₂×C₄×C₇₃	#107	1	1	yes	characteristic
41	C₂×Dic₇₃	—	4	2	no	—
42	C₂×Dic₇₃	#102	1	2	yes	—
43	C₂×Dic₇₃	#103	1	2	yes	—
44	#10596833	—	8	4	no	—
45	C₈×C₇₃	—	2	1	no	—
46	C₈×C₇₃	#100	1	1	yes	characteristic
47	C₈×C₇₃	#101	1	1	yes	characteristic
48	#10596831	—	2	2	no	—
49	C₂²×C₇₃	—	2	1	no	—
50	C₂²×C₇₃	—	2	2	no	—
51	C₂²×C₇₃	#340	1	1	yes	characteristic
52	C₄×C₇₃	—	2	1	no	—
53	C₄×C₇₃	#338	1	1	yes	characteristic
54	C₄×C₇₃	#343	1	1	yes	characteristic
55	Dic₇₃	—	2	2	no	—
56	Dic₇₃	—	2	2	no	—
57	#7246	—	73	1	no	maximal, Sylow
58	C₂×C₇₃	—	4	1	no	—
59	C₂×C₇₃	#1292	1	1	yes	characteristic
60	C₂×C₇₃	#1389	1	2	yes	—
61	#2150	—	73	1	no	—
62	#1568	—	73	2	no	—
63	C₇₃	#7246	1	1	yes	characteristic, complemented, Sylow
64	C₂×M₃₂	—	73	1	no	—
65	C₂×M₃₂	#64408	1	1	yes	characteristic
66	#432	—	146	2	no	—
67	C₂×C₄×C₈	—	73	1	no	—
68	C₄⋊C₁₆	—	146	2	no	—
69	#335	—	146	2	no	—
70	#334	—	73	4	no	—
71	C₂²×C₈	—	73	1	no	—
72	C₂²×C₈	C₂×#64408	1	1	yes	characteristic
73	C₂×C₄²	—	73	1	no	—
74	M₃₂	—	4	1	no	—
75	M₃₂	—	292	1	no	—
76	C₂×C₁₆	—	2	1	no	—
77	C₂×C₁₆	—	146	1	no	—
78	C₄⋊C₈	—	146	4	no	—
79	C₂²⋊C₈	—	292	2	no	—
80	#97	—	292	2	no	—
81	C₄×C₈	—	73	2	no	—
82	C₄×C₈	—	73	2	no	—
83	C₂²×C₄	—	73	2	no	—
84	C₂²×C₄	C₄⋊#64408	1	1	yes	characteristic
85	C₂×C₈	—	2	1	no	—
86	C₂×C₈	—	2	1	no	—
87	C₂×C₈	—	292	4	no	—
88	C₂×C₈	—	292	1	no	—
89	C₂×C₈	—	73	2	no	—
90	C₂×C₈	C₄×#64408	1	1	yes	characteristic
91	C₂×C₈	C₂²⋊#64408	1	1	yes	characteristic
92	C₄²	—	73	2	no	—
93	C₄²	—	73	1	no	—
94	C₄²	—	73	1	no	—
95	C₁₆	—	4	1	no	—
96	C₁₆	—	292	1	no	—
97	C₂³	#14556659	1	1	yes	characteristic
98	C₂×C₄	—	2	1	no	—
99	C₂×C₄	—	2	1	no	—
100	C₂×C₄	—	292	2	no	—
101	C₂×C₄	—	73	2	no	—
102	C₂×C₄	—	73	2	no	—
103	C₂×C₄	(C₂×#64408)⋊C₄	1	1	yes	characteristic
104	C₂×C₄	#14556660	1	1	yes	characteristic
105	C₈	—	2	1	no	—
106	C₈	—	584	4	no	—
107	C₈	—	146	2	no	—
108	C₈	#14556651	1	1	yes	characteristic
109	C₈	#14556652	1	1	yes	characteristic
110	C₂²	—	2	1	no	—
111	C₂²	—	2	2	no	—
112	C₂²	#17989461	1	1	yes	characteristic
113	C₄	—	2	1	no	—
114	C₄	—	146	2	no	—
115	C₄	—	146	2	no	—
116	C₄	#17989456	1	1	yes	characteristic
117	C₄	#17989467	1	1	yes	characteristic
118	C₂	—	4	1	no	—
119	C₂	#22624801	1	1	yes	characteristic
120	C₂	#22625012	1	2	yes	—

subgroup

G/H

class size

multiplicity

normal?

properties

#22625493

C₂

yes

maximal

#22611686

C₂

yes

maximal, characteristic

#17989608

—

#17989450

—

#17989449

C₄

yes

—

C₂×#14556531

C₄

yes

characteristic

C₂×C₈×Dic₇₃

C₂²

yes

characteristic

#17988405

—

C₂×M₃₂×C₇₃

C₄

yes

—

—

—

Q₈

yes

characteristic

C₂×C₄×Dic₇₃

D₄

yes

characteristic

#14556531

—

C₂×#11924003

—

C₈×Dic₇₃

C₂×C₄

yes

—

C₈×Dic₇₃

D₄

yes

—

C₂²×C₈×C₇₃

C₂×C₄

yes

characteristic

M₃₂×C₇₃

—

C₂×C₁₆×C₇₃

—

C₂×#10596833

—

C₂²×Dic₇₃

SD₁₆

yes

—

C₂×#10596831

—

C₂×#10596831

C₄⋊C₄

yes

—

C₄×Dic₇₃

C₂²⋊C₄

yes

—

C₄×Dic₇₃

D₈

yes

characteristic

C₄×Dic₇₃

Q₁₆

yes

characteristic

#11924003

—

C₂²×C₄×C₇₃

C₄⋊C₄

yes

characteristic

C₂×C₈×C₇₃

—

C₂×C₈×C₇₃

—

C₂×C₈×C₇₃

C₄²

yes

characteristic

C₂×C₈×C₇₃

C₂²⋊C₄

yes

characteristic

C₁₆×C₇₃

—

C₂³×C₇₃

#106

yes

characteristic

C₂×C₄×C₇₃

—

C₂×C₄×C₇₃

—

C₂×C₄×C₇₃

yes

C₂×C₄×C₇₃

yes

C₂×Dic₇₃

—

C₂×Dic₇₃

#102

yes

—

C₂×Dic₇₃

#103

yes

—

#10596833

—

C₈×C₇₃

—

C₈×C₇₃

yes

C₈×C₇₃

yes

—

C₂²×C₇₃

—

C₂²×C₇₃

—

C₂²×C₇₃

#340

yes

characteristic

C₄×C₇₃

—

C₄×C₇₃

yes

C₄×C₇₃

yes

Dic₇₃

—

Dic₇₃

—

#7246

—

maximal, Sylow

C₂×C₇₃

—

C₂×C₇₃

#1292

yes

characteristic

C₂×C₇₃

#1389

yes

—

#2150

—

#1568

—

C₇₃

#7246

yes

characteristic, complemented, Sylow

C₂×M₃₂

—

C₂×M₃₂

#64408

yes

characteristic

#432

—

146

—

C₂×C₄×C₈

—

C₄⋊C₁₆

—

146

—

#335

—

146

—

#334

—

C₂²×C₈

—

C₂²×C₈

C₂×#64408

yes

characteristic

C₂×C₄²

—

M₃₂

—

M₃₂

—

292

—

C₂×C₁₆

—

C₂×C₁₆

—

146

—

C₄⋊C₈

—

146

—

C₂²⋊C₈

—

292

—

#97

—

292

—

C₄×C₈

—

C₄×C₈

—

C₂²×C₄

—

C₂²×C₄

C₄⋊#64408

yes

characteristic

C₂×C₈

—

C₂×C₈

—

C₂×C₈

—

292

—

C₂×C₈

—

292

—

C₂×C₈

—

C₂×C₈

yes

C₂×C₈

yes

C₄²

—

C₄²

—

C₄²

—

C₁₆

—

C₁₆

—

292

—

C₂³

#14556659

yes

characteristic

C₂×C₄

—

C₂×C₄

—

100

C₂×C₄

—

292

—

101

C₂×C₄

—

102

C₂×C₄

—

103

C₂×C₄

(C₂×#64408)⋊C₄

yes

characteristic

104

C₂×C₄

#14556660

yes

characteristic

105

C₈

—

106

C₈

—

584

—

107

C₈

—

146

—

108

C₈

#14556651

yes

characteristic

109

C₈

#14556652

yes

characteristic

110

C₂²

—

111

C₂²

—

112

C₂²

#17989461

yes

characteristic

113

C₄

—

114

C₄

—

146

—

115

C₄

—

146

—

116

C₄

#17989456

yes

characteristic

117

C₄

#17989467

yes

characteristic

118

C₂

—

119

C₂

#22624801

yes

characteristic

120

C₂

#22625012

yes

—

	subgroup	G/H
center	C₂²	#17989461
commutator	C₂×C₈×C₇₃	C₄²
Frattini	C₂²×C₈	C₂×#64408
Fitting	C₂×M₃₂×C₇₃	C₄
socle	C₂²×C₇₃	#340
prime radical	C₂³×C₇₃	#106
cosocle	C₂×C₈×Dic₇₃	C₂²
hypercenter	C₂×M₃₂	#64408
hypocenter	C₇₃	#7246
LCS	G ▷ C₂×C₈×C₇₃ ▷ C₄×C₇₃ ▷ C₂×C₇₃ ▷ C₇₃
UCS	C₁ ◁ C₂² ◁ C₂²×C₄ ◁ C₂²×C₈ ◁ C₂×M₃₂

subgroup

G/H

center

C₂²

#17989461

commutator

C₂×C₈×C₇₃

C₄²

Frattini

C₂²×C₈

C₂×#64408

Fitting

C₂×M₃₂×C₇₃

C₄

C₂²×C₇₃

C₂³×C₇₃

C₂×C₈×Dic₇₃

C₂²

C₂×M₃₂

C₇₃

G ▷ C₂×C₈×C₇₃ ▷ C₄×C₇₃ ▷ C₂×C₇₃ ▷ C₇₃

UCS

C₁ ◁ C₂² ◁ C₂²×C₄ ◁ C₂²×C₈ ◁ C₂×M₃₂

N	H ≅ G/N	type
63. C₇₃	#7246	semidirect

H ≅ G/N

type

63. C₇₃

#7246

semidirect

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

subgroup	index
#22625493	2
#22611686	2
#7246	73

subgroup

index

#22625493

#22611686

#7246

Subgroups — #34000504

Conjugacy classes of subgroups

Complements

Maximal subgroups