Group #34005448 — subgroups

Subgroups — #34005448

Order 18688.

228 conjugacy classes of subgroups; only the 76 normal subgroups are shown.

normal subgroups only

This group has 228 conjugacy classes of subgroups — too many to draw. Only its 76 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	#22625494	C₂	1	2	yes	maximal
2	#22624788	C₂	1	4	yes	maximal
3	#22608213	C₂	1	1	yes	maximal, characteristic
4	#17989609	C₂²	1	4	yes	—
5	#17989451	—	2	2	no	—
6	#17989449	—	2	2	no	—
7	#17988638	C₂²	1	2	yes	—
8	C₄×C₈×D₇₃	C₂²	1	1	yes	characteristic
9	#17988309	C₄	1	2	yes	—
10	C₈×#10596831	C₄	1	1	yes	characteristic
11	C₈²×C₇₃	C₄	1	1	yes	characteristic
12	C₄⋊#10596833	—	2	4	no	—
13	#14556648	—	2	4	no	—
14	#14556646	—	2	4	no	—
15	C₂×C₈×D₇₃	Q₈	1	2	yes	—
16	C₄²×D₇₃	C₂³	1	1	yes	characteristic
17	#14556520	—	2	2	no	—
18	#14556503	—	2	2	no	—
19	C₄×#10596831	C₂×C₄	1	2	yes	—
20	C₄×#10596831	C₂×C₄	1	1	yes	characteristic
21	C₈×Dic₇₃	D₄	1	2	yes	—
22	C₄×C₈×C₇₃	C₂×C₄	1	2	yes	—
23	C₄×C₈×C₇₃	C₂×C₄	1	1	yes	characteristic
24	C₂×#10596833	—	4	4	no	—
25	C₂×C₄×D₇₃	C₂×Q₈	1	1	yes	characteristic
26	C₂×C₄×D₇₃	#41	1	2	yes	—
27	C₂×#10596831	—	2	2	no	—
28	C₂×#10596831	C₄⋊C₄	1	2	yes	—
29	C₈×D₇₃	—	2	4	no	—
30	C₄×Dic₇₃	C₂×D₄	1	1	yes	characteristic
31	C₄×Dic₇₃	#41	1	2	yes	—
32	C₂×C₈×C₇₃	—	2	2	no	—
33	C₂×C₈×C₇₃	C₄⋊C₄	1	2	yes	—
34	C₄²×C₇₃	C₂²×C₄	1	1	yes	characteristic
35	C₂²×D₇₃	#125	1	1	yes	characteristic
36	C₂×C₄×C₇₃	C₂×C₄⋊C₄	1	1	yes	characteristic
37	C₂×C₄×C₇₃	#117	1	2	yes	—
38	C₂×Dic₇₃	#124	1	2	yes	—
39	C₂×Dic₇₃	#125	1	1	yes	characteristic
40	C₄×D₇₃	—	2	2	no	—
41	C₄×D₇₃	—	4	2	no	—
42	#10596833	—	8	4	no	—
43	C₈×C₇₃	—	2	2	no	—
44	C₈×C₇₃	—	4	2	no	—
45	#10596831	—	4	2	no	—
46	#10596831	#108	1	4	yes	—
47	C₂²×C₇₃	#382	1	1	yes	characteristic
48	C₂×D₇₃	—	2	2	no	—
49	C₂×D₇₃	—	2	1	no	—
50	C₄×C₇₃	—	2	2	no	—
51	C₄×C₇₃	C₂×#108	1	2	yes	—
52	Dic₇₃	—	2	2	no	—
53	Dic₇₃	—	2	1	no	—
54	#8797	—	73	1	no	maximal, Sylow
55	C₂×C₇₃	#1833	1	2	yes	—
56	C₂×C₇₃	#1837	1	1	yes	characteristic
57	D₇₃	—	4	1	no	—
58	#1568	—	73	2	no	—
59	#1450	—	73	1	no	—
60	#1287	—	73	4	no	—
61	C₇₃	#8797	1	1	yes	characteristic, complemented, Sylow
62	#432	—	73	4	no	—
63	#431	—	73	2	no	—
64	C₂×C₄×C₈	—	73	1	no	—
65	#335	—	146	2	no	—
66	#334	—	146	2	no	—
67	#322	—	73	2	no	—
68	C₈²	—	73	1	no	—
69	C₈²	#64408	1	1	yes	characteristic
70	C₂²×C₈	—	73	2	no	—
71	C₂×C₄²	—	73	1	no	—
72	C₄⋊C₈	—	146	2	no	—
73	C₄⋊C₈	—	146	4	no	—
74	C₂²⋊C₈	—	146	2	no	—
75	C₂²⋊C₈	—	146	4	no	—
76	#97	—	146	4	no	—
77	C₄×C₈	—	73	2	no	—
78	C₄×C₈	—	73	1	no	—
79	C₄×C₈	—	73	2	no	—
80	C₄×C₈	C₂×#64408	1	2	yes	—
81	C₄×C₈	C₂×#64408	1	1	yes	characteristic
82	C₂²×C₄	—	73	2	no	—
83	C₂²×C₄	—	73	1	no	—
84	C₂×C₈	—	2	2	no	—
85	C₂×C₈	—	292	4	no	—
86	C₂×C₈	—	73	2	no	—
87	C₂×C₈	—	146	4	no	—
88	C₂×C₈	—	146	2	no	—
89	C₂×C₈	C₄⋊#64408	1	2	yes	—
90	C₄²	—	73	1	no	—
91	C₄²	—	73	2	no	—
92	C₄²	C₂²×#64408	1	1	yes	characteristic
93	C₂³	—	73	1	no	—
94	C₂×C₄	—	292	2	no	—
95	C₂×C₄	—	73	2	no	—
96	C₂×C₄	—	73	1	no	—
97	C₂×C₄	—	146	2	no	—
98	C₂×C₄	C₂×C₄⋊#64408	1	1	yes	characteristic
99	C₂×C₄	#14556670	1	2	yes	—
100	C₈	—	2	2	no	—
101	C₈	—	4	2	no	—
102	C₈	—	292	2	no	—
103	C₈	—	584	4	no	—
104	C₈	—	73	4	no	—
105	C₂²	—	146	2	no	—
106	C₂²	—	146	1	no	—
107	C₂²	#17989531	1	1	yes	characteristic
108	C₄	—	2	2	no	—
109	C₄	—	146	2	no	—
110	C₄	—	146	1	no	—
111	C₄	D₇₃⋊#108	1	2	yes	—
112	C₂	—	292	1	no	—
113	C₂	#22625850	1	2	yes	—
114	C₂	#22625856	1	1	yes	characteristic

subgroup

G/H

class size

multiplicity

normal?

properties

#22625494

C₂

yes

maximal

#22624788

C₂

yes

maximal

#22608213

C₂

yes

maximal, characteristic

#17989609

C₂²

yes

—

#17989451

—

#17989449

—

#17988638

C₂²

yes

—

C₄×C₈×D₇₃

C₂²

yes

characteristic

#17988309

C₄

yes

—

C₈×#10596831

C₄

yes

characteristic

C₈²×C₇₃

C₄

yes

—

—

—

C₂×C₈×D₇₃

Q₈

yes

—

C₄²×D₇₃

C₂³

yes

—

—

C₂×C₄

yes

—

C₄×#10596831

C₂×C₄

yes

characteristic

C₈×Dic₇₃

D₄

yes

—

C₄×C₈×C₇₃

C₂×C₄

yes

—

C₄×C₈×C₇₃

C₂×C₄

yes

characteristic

C₂×#10596833

—

C₂×C₄×D₇₃

C₂×Q₈

yes

characteristic

C₂×C₄×D₇₃

#41

yes

—

C₂×#10596831

—

C₂×#10596831

C₄⋊C₄

yes

—

C₈×D₇₃

—

C₄×Dic₇₃

C₂×D₄

yes

characteristic

C₄×Dic₇₃

#41

yes

—

C₂×C₈×C₇₃

—

C₂×C₈×C₇₃

C₄⋊C₄

yes

—

C₄²×C₇₃

C₂²×C₄

yes

characteristic

C₂²×D₇₃

#125

yes

characteristic

C₂×C₄×C₇₃

C₂×C₄⋊C₄

yes

characteristic

C₂×C₄×C₇₃

#117

yes

—

C₂×Dic₇₃

#124

yes

—

C₂×Dic₇₃

#125

yes

characteristic

C₄×D₇₃

—

C₄×D₇₃

—

#10596833

—

C₈×C₇₃

—

C₈×C₇₃

—

#10596831

—

#10596831

#108

yes

—

C₂²×C₇₃

#382

yes

characteristic

C₂×D₇₃

—

C₂×D₇₃

—

C₄×C₇₃

—

C₄×C₇₃

C₂×#108

yes

—

Dic₇₃

—

Dic₇₃

—

#8797

—

maximal, Sylow

C₂×C₇₃

#1833

yes

—

C₂×C₇₃

#1837

yes

characteristic

D₇₃

—

#1568

—

#1450

—

#1287

—

C₇₃

#8797

yes

characteristic, complemented, Sylow

#432

—

#431

—

C₂×C₄×C₈

—

#335

—

146

—

#334

—

146

—

#322

—

C₈²

—

C₈²

#64408

yes

characteristic

C₂²×C₈

—

C₂×C₄²

—

C₄⋊C₈

—

146

—

C₄⋊C₈

—

146

—

C₂²⋊C₈

—

146

—

C₂²⋊C₈

—

146

—

#97

—

146

—

C₄×C₈

—

C₄×C₈

—

C₄×C₈

—

C₄×C₈

C₂×#64408

yes

—

C₄×C₈

C₂×#64408

yes

characteristic

C₂²×C₄

—

C₂²×C₄

—

C₂×C₈

—

C₂×C₈

—

292

—

C₂×C₈

—

C₂×C₈

—

146

—

C₂×C₈

—

146

—

C₂×C₈

C₄⋊#64408

yes

—

C₄²

—

C₄²

—

C₄²

C₂²×#64408

yes

characteristic

C₂³

—

C₂×C₄

—

292

—

C₂×C₄

—

C₂×C₄

—

C₂×C₄

—

146

—

C₂×C₄

C₂×C₄⋊#64408

yes

characteristic

C₂×C₄

#14556670

yes

—

100

C₈

—

101

C₈

—

102

C₈

—

292

—

103

C₈

—

584

—

104

C₈

—

105

C₂²

—

146

—

106

C₂²

—

146

—

107

C₂²

#17989531

yes

characteristic

108

C₄

—

109

C₄

—

146

—

110

C₄

—

146

—

111

C₄

D₇₃⋊#108

yes

—

112

C₂

—

292

—

113

C₂

#22625850

yes

—

114

C₂

#22625856

yes

characteristic

	subgroup	G/H
center	C₂²	#17989531
commutator	C₄²×C₇₃	C₂²×C₄
Frattini	C₄²	C₂²×#64408
Fitting	C₈²×C₇₃	C₄
socle	C₂²×C₇₃	#382
prime radical	C₂²×D₇₃	#125
cosocle	C₄²×D₇₃	C₂³
hypercenter	C₈²	#64408
hypocenter	C₇₃	#8797
LCS	G ▷ C₄²×C₇₃ ▷ C₂²×C₇₃ ▷ C₇₃
UCS	C₁ ◁ C₂² ◁ C₄² ◁ C₈²

subgroup

G/H

center

C₂²

#17989531

commutator

C₄²×C₇₃

C₂²×C₄

Frattini

C₄²

C₂²×#64408

Fitting

C₈²×C₇₃

C₄

C₂²×C₇₃

C₂²×D₇₃

C₄²×D₇₃

C₂³

C₈²

C₇₃

G ▷ C₄²×C₇₃ ▷ C₂²×C₇₃ ▷ C₇₃

UCS

C₁ ◁ C₂² ◁ C₄² ◁ C₈²

N	H ≅ G/N	type
61. C₇₃	#8797	semidirect

H ≅ G/N

type

61. C₇₃

#8797

semidirect

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

subgroup	index
#22625494	2
#22624788	2
#22608213	2
#8797	73

subgroup

index

Subgroups — #34005448

Conjugacy classes of subgroups

Complements

Maximal subgroups