Group #34021257 — subgroups

Subgroups — #34021257

Order 18688.

830 conjugacy classes of subgroups; only the 232 normal subgroups are shown.

normal subgroups only

This group has 830 conjugacy classes of subgroups — too many to draw. Only its 232 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	#22625860	C₂	1	8	yes	maximal, complemented
2	#22625595	C₂	1	6	yes	maximal
3	#376×D₇₃	C₂	1	1	yes	maximal, characteristic
4	C₂²×#11924038	C₂²	1	4	yes	—
5	#17989453	C₂²	1	24	yes	—
6	C₂×C₄²×D₇₃	C₂²	1	3	yes	—
7	#17988433	C₄	1	3	yes	—
8	#17988432	C₂²	1	4	yes	—
9	#95×D₇₃	C₄	1	4	yes	—
10	#376×C₇₃	C₄	1	1	yes	characteristic
11	C₂×#11924038	—	2	24	no	—
12	C₂²×#10596833	C₂³	1	8	yes	—
13	C₂²×C₄×D₇₃	C₂×C₄	1	6	yes	—
14	C₂²×C₄×D₇₃	C₂³	1	1	yes	characteristic
15	C₂×C₄×Dic₇₃	C₂×C₄	1	3	yes	—
16	C₂×C₄×Dic₇₃	C₂³	1	6	yes	—
17	C₄²×D₇₃	—	2	12	no	—
18	#14556498	C₂×C₄	1	12	yes	—
19	C₂×C₄²×C₇₃	C₂×C₄	1	3	yes	—
20	#95×C₇₃	C₂×C₄	1	4	yes	—
21	#11924038	—	4	16	no	—
22	C₂×#10596833	—	2	24	no	—
23	C₂³×D₇₃	C₂²×C₄	1	1	yes	characteristic
24	C₂²×Dic₇₃	C₂²×C₄	1	6	yes	—
25	C₂²×Dic₇₃	C₂⁴	1	1	yes	characteristic
26	C₂×C₄×D₇₃	—	2	12	no	—
27	C₂×C₄×D₇₃	—	4	12	no	—
28	C₂×C₄×D₇₃	#41	1	12	yes	—
29	C₄×Dic₇₃	—	2	6	no	—
30	C₄×Dic₇₃	—	2	12	no	—
31	C₂²×C₄×C₇₃	C₂²×C₄	1	6	yes	—
32	C₂²×C₄×C₇₃	C₂²×C₄	1	1	yes	characteristic
33	C₄²×C₇₃	—	2	6	no	—
34	C₂³×C₇₃	C₂³×C₄	1	1	yes	characteristic
35	C₂²×D₇₃	—	2	4	no	—
36	C₂²×D₇₃	#117	1	6	yes	—
37	C₂×C₄×C₇₃	—	2	6	no	—
38	C₂×C₄×C₇₃	—	2	12	no	—
39	C₂×C₄×C₇₃	#117	1	6	yes	—
40	C₂×Dic₇₃	—	2	6	no	—
41	C₂×Dic₇₃	—	2	12	no	—
42	C₂×Dic₇₃	C₂×#41	1	6	yes	—
43	C₄×D₇₃	—	4	24	no	—
44	C₄×D₇₃	#126	1	16	yes	—
45	#10596833	—	4	8	no	—
46	C₂²×C₇₃	C₂×#117	1	3	yes	—
47	C₂²×C₇₃	#568	1	4	yes	—
48	C₂×D₇₃	—	2	12	no	—
49	C₂×D₇₃	#383	1	4	yes	—
50	C₄×C₇₃	—	4	6	no	—
51	C₄×C₇₃	#383	1	4	yes	—
52	Dic₇₃	—	4	6	no	—
53	Dic₇₃	C₂×#126	1	4	yes	—
54	#20773	—	73	1	no	maximal, Sylow
55	C₂×C₇₃	C₂×#383	1	1	yes	characteristic
56	C₂×C₇₃	#2921	1	6	yes	—
57	D₇₃	—	2	4	no	—
58	C₂×#376	—	73	1	no	—
59	#1840	—	73	8	no	—
60	#1728	—	73	6	no	—
61	C₇₃	#20773	1	1	yes	characteristic, complemented, Sylow
62	C₂²×M₁₆	—	73	4	no	—
63	C₂²×C₄²	—	73	3	no	—
64	#376	—	73	3	no	—
65	#376	—	73	4	no	—
66	#376	#64408	1	1	yes	characteristic
67	C₂×#95	—	73	4	no	—
68	#336	—	73	24	no	—
69	C₂³×C₄	—	73	6	no	—
70	C₂³×C₄	—	73	1	no	—
71	C₂×M₁₆	—	146	24	no	—
72	C₂²×C₈	—	73	8	no	—
73	C₂×C₄²	—	73	6	no	—
74	C₂×C₄²	—	73	3	no	—
75	C₂×C₄²	—	146	12	no	—
76	C₂×C₄²	C₂×#64408	1	3	yes	—
77	#95	—	73	12	no	—
78	#95	C₂×#64408	1	4	yes	—
79	C₂⁴	—	73	1	no	—
80	C₂²×C₄	—	292	12	no	—
81	C₂²×C₄	—	73	1	no	—
82	C₂²×C₄	—	73	12	no	—
83	C₂²×C₄	—	73	6	no	—
84	C₂²×C₄	—	146	12	no	—
85	C₂²×C₄	C₂²×#64408	1	6	yes	—
86	C₂²×C₄	C₂²×#64408	1	1	yes	characteristic
87	M₁₆	—	292	16	no	—
88	C₂×C₈	—	146	24	no	—
89	C₄²	—	2	6	no	—
90	C₄²	—	146	12	no	—
91	C₄²	—	146	6	no	—
92	C₂³	—	73	6	no	—
93	C₂³	—	146	4	no	—
94	C₂³	C₂³×#64408	1	1	yes	characteristic
95	C₂×C₄	—	2	6	no	—
96	C₂×C₄	—	2	12	no	—
97	C₂×C₄	—	292	24	no	—
98	C₂×C₄	—	73	6	no	—
99	C₂×C₄	—	73	16	no	—
100	C₂×C₄	—	146	6	no	—
101	C₂×C₄	—	146	12	no	—
102	C₂×C₄	#14556670	1	6	yes	—
103	C₈	—	292	8	no	—
104	C₂²	—	73	4	no	—
105	C₂²	—	146	12	no	—
106	C₂²	C₂×#14556670	1	3	yes	—
107	C₂²	#17989666	1	4	yes	—
108	C₄	—	4	6	no	—
109	C₄	—	292	6	no	—
110	C₄	—	73	4	no	—
111	C₄	#17989532	1	4	yes	—
112	C₂	—	146	4	no	—
113	C₂	C₂×#17989532	1	1	yes	characteristic
114	C₂	#22626612	1	6	yes	—

subgroup

G/H

class size

multiplicity

normal?

properties

#22625860

C₂

yes

maximal, complemented

#22625595

C₂

yes

maximal

#376×D₇₃

C₂

yes

maximal, characteristic

C₂²×#11924038

C₂²

yes

—

#17989453

C₂²

yes

—

C₂×C₄²×D₇₃

C₂²

yes

—

#17988433

C₄

yes

—

#17988432

C₂²

yes

—

#95×D₇₃

C₄

yes

—

#376×C₇₃

C₄

yes

characteristic

C₂×#11924038

—

C₂²×#10596833

C₂³

yes

—

C₂²×C₄×D₇₃

C₂×C₄

yes

—

C₂²×C₄×D₇₃

C₂³

yes

characteristic

C₂×C₄×Dic₇₃

C₂×C₄

yes

—

C₂×C₄×Dic₇₃

C₂³

yes

—

C₄²×D₇₃

—

#14556498

C₂×C₄

yes

—

C₂×C₄²×C₇₃

C₂×C₄

yes

—

#95×C₇₃

C₂×C₄

yes

—

#11924038

—

C₂×#10596833

—

C₂³×D₇₃

C₂²×C₄

yes

characteristic

C₂²×Dic₇₃

C₂²×C₄

yes

—

C₂²×Dic₇₃

C₂⁴

yes

characteristic

C₂×C₄×D₇₃

—

C₂×C₄×D₇₃

—

C₂×C₄×D₇₃

#41

yes

—

C₄×Dic₇₃

—

C₄×Dic₇₃

—

C₂²×C₄×C₇₃

C₂²×C₄

yes

—

C₂²×C₄×C₇₃

C₂²×C₄

yes

characteristic

C₄²×C₇₃

—

C₂³×C₇₃

C₂³×C₄

yes

characteristic

C₂²×D₇₃

—

C₂²×D₇₃

#117

yes

—

C₂×C₄×C₇₃

—

C₂×C₄×C₇₃

—

C₂×C₄×C₇₃

#117

yes

—

C₂×Dic₇₃

—

C₂×Dic₇₃

—

C₂×Dic₇₃

C₂×#41

yes

—

C₄×D₇₃

—

C₄×D₇₃

#126

yes

—

#10596833

—

C₂²×C₇₃

C₂×#117

yes

—

C₂²×C₇₃

#568

yes

—

C₂×D₇₃

—

C₂×D₇₃

#383

yes

—

C₄×C₇₃

—

C₄×C₇₃

#383

yes

—

Dic₇₃

—

Dic₇₃

C₂×#126

yes

—

—

C₂×C₇₃

yes

C₂×C₇₃

yes

—

D₇₃

—

C₂×#376

—

#1840

—

#1728

—

C₇₃

#20773

yes

characteristic, complemented, Sylow

C₂²×M₁₆

—

C₂²×C₄²

—

—

—

yes

C₂×#95

—

#336

—

C₂³×C₄

—

C₂³×C₄

—

C₂×M₁₆

—

146

—

C₂²×C₈

—

C₂×C₄²

—

C₂×C₄²

—

C₂×C₄²

—

146

—

C₂×C₄²

C₂×#64408

yes

—

#95

—

#95

C₂×#64408

yes

—

C₂⁴

—

C₂²×C₄

—

292

—

C₂²×C₄

—

C₂²×C₄

—

C₂²×C₄

—

C₂²×C₄

—

146

—

C₂²×C₄

C₂²×#64408

yes

—

C₂²×C₄

C₂²×#64408

yes

characteristic

M₁₆

—

292

—

C₂×C₈

—

146

—

C₄²

—

C₄²

—

146

—

C₄²

—

146

—

C₂³

—

C₂³

—

146

—

C₂³

C₂³×#64408

yes

characteristic

C₂×C₄

—

C₂×C₄

—

C₂×C₄

—

292

—

C₂×C₄

—

C₂×C₄

—

100

C₂×C₄

—

146

—

101

C₂×C₄

—

146

—

102

C₂×C₄

#14556670

yes

—

103

C₈

—

292

—

104

C₂²

—

105

C₂²

—

146

—

106

C₂²

C₂×#14556670

yes

—

107

C₂²

#17989666

yes

—

108

C₄

—

109

C₄

—

292

—

110

C₄

—

111

C₄

#17989532

yes

—

112

C₂

—

146

—

113

C₂

C₂×#17989532

yes

characteristic

114

C₂

#22626612

yes

—

	subgroup	G/H
center, Frattini	C₂³	C₂³×#64408
commutator, socle	C₂³×C₇₃	C₂³×C₄
Fitting	#376×C₇₃	C₄
prime radical	C₂³×D₇₃	C₂²×C₄
cosocle	C₂²×Dic₇₃	C₂⁴
hypercenter	#376	#64408
hypocenter	C₇₃	#20773
LCS	G ▷ C₂³×C₇₃ ▷ C₇₃
UCS	C₁ ◁ C₂³ ◁ #376

subgroup

G/H

center, Frattini

C₂³

C₂³×#64408

commutator, socle

C₂³×C₇₃

C₂³×C₄

Fitting

#376×C₇₃

C₄

prime radical

C₂³×D₇₃

C₂²×C₄

cosocle

C₂²×Dic₇₃

C₂⁴

C₇₃

G ▷ C₂³×C₇₃ ▷ C₇₃

C₁ ◁ C₂³ ◁ #376

N	H ≅ G/N	type
1. #22625860	C₂	semidirect
61. C₇₃	#20773	semidirect

H ≅ G/N

type

1. #22625860

C₂

semidirect

61. C₇₃

#20773

semidirect

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

subgroup	index
#22625860	2
#22625595	2
#376×D₇₃	2
#20773	73

subgroup

index

#22625860

#22625595

#376×D₇₃

#20773

Subgroups — #34021257

Conjugacy classes of subgroups

Complements

Maximal subgroups