Group #32978552 — subgroups

Subgroups — #32978552

Order 18688.

656 conjugacy classes of subgroups; only the 139 normal subgroups are shown.

normal subgroups only

This group has 656 conjugacy classes of subgroups — too many to draw. Only its 139 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	#22610909	C₂	1	3	yes	maximal
2	#22608669	C₂	1	3	yes	maximal
3	C₇₃×#3556	C₂	1	1	yes	maximal, characteristic
4	#17988611	C₂²	1	3	yes	—
5	C₄²×Dic₇₃	C₂²	1	1	yes	characteristic
6	#17988346	—	2	6	no	—
7	#17988333	—	2	6	no	—
8	#564×C₇₃	C₄	1	4	yes	complemented
9	#557×C₇₃	—	2	6	no	—
10	#541×C₇₃	—	2	6	no	—
11	#533×C₇₃	C₂²	1	3	yes	—
12	C₂×#11924012	—	2	3	no	—
13	C₂×C₄×Dic₇₃	—	2	3	no	—
14	C₄⋊#10596831	—	2	6	no	—
15	C₄⋊Q₈×C₇₃	—	2	6	no	—
16	C₄⋊Q₈×C₇₃	D₄	1	6	yes	—
17	#126×C₇₃	—	2	8	no	—
18	#125×C₇₃	—	2	6	no	—
19	#125×C₇₃	—	2	24	no	—
20	#125×C₇₃	C₂×C₄	1	6	yes	—
21	C₂²⋊Q₈×C₇₃	—	2	6	no	—
22	C₄×Q₈×C₇₃	—	2	12	no	—
23	C₄×Q₈×C₇₃	—	2	6	no	—
24	#117×C₇₃	D₄	1	6	yes	—
25	C₂×C₄²×C₇₃	C₂³	1	1	yes	characteristic
26	C₂²×Dic₇₃	—	4	1	no	—
27	#11924012	—	4	6	no	—
28	C₂×#10596831	—	4	3	no	—
29	C₄×Dic₇₃	—	2	6	no	—
30	C₄×Dic₇₃	—	4	3	no	—
31	C₂×Q₈×C₇₃	—	2	6	no	—
32	C₂²×C₄×C₇₃	C₂×D₄	1	3	yes	—
33	C₄⋊C₄×C₇₃	—	2	24	no	—
34	C₄⋊C₄×C₇₃	—	2	12	no	—
35	C₄⋊C₄×C₇₃	—	2	6	no	—
36	C₄⋊C₄×C₇₃	C₂²⋊C₄	1	12	yes	—
37	C₂²⋊C₄×C₇₃	—	2	6	no	—
38	C₄²×C₇₃	—	2	8	no	—
39	C₄²×C₇₃	—	2	6	no	—
40	C₄²×C₇₃	C₂²×C₄	1	1	yes	characteristic
41	C₄²×C₇₃	C₂×D₄	1	3	yes	—
42	C₂³×C₇₃	#120	1	1	yes	characteristic
43	Q₈×C₇₃	—	4	6	no	—
44	Q₈×C₇₃	—	4	6	no	—
45	C₂×C₄×C₇₃	—	2	3	no	—
46	C₂×C₄×C₇₃	—	2	3	no	—
47	C₂×C₄×C₇₃	—	2	12	no	—
48	C₂×C₄×C₇₃	C₂×C₂²⋊C₄	1	3	yes	—
49	C₂×C₄×C₇₃	#120	1	3	yes	—
50	C₂×Dic₇₃	—	4	6	no	—
51	#10596831	—	8	3	no	—
52	C₂²×C₇₃	—	2	3	no	—
53	C₂²×C₇₃	#379	1	1	yes	characteristic
54	C₄×C₇₃	—	2	3	no	—
55	C₄×C₇₃	—	4	12	no	—
56	C₄×C₇₃	#454	1	3	yes	—
57	C₄×C₇₃	#456	1	3	yes	—
58	Dic₇₃	—	4	4	no	—
59	#8533	—	73	1	no	maximal, Sylow
60	C₂×C₇₃	—	4	1	no	—
61	C₂×C₇₃	#1793	1	3	yes	—
62	#3556	D₇₃	1	1	yes	characteristic
63	#1994	—	73	3	no	—
64	#1561	—	73	3	no	—
65	C₇₃	#8533	1	1	yes	characteristic, complemented, Sylow
66	#564	Dic₇₃	1	4	yes	complemented
67	#557	—	2	6	no	—
68	#541	—	2	6	no	—
69	#533	C₂×D₇₃	1	3	yes	—
70	#423	—	73	3	no	—
71	C₄³	—	73	1	no	—
72	#339	—	146	6	no	—
73	#332	—	146	6	no	—
74	C₂×M₁₆	—	146	3	no	—
75	C₄⋊Q₈	—	2	6	no	—
76	C₄⋊Q₈	C₂²⋊D₇₃	1	6	yes	—
77	#126	—	2	8	no	—
78	#125	—	2	6	no	—
79	#125	—	2	24	no	—
80	#125	C₂×Dic₇₃	1	6	yes	—
81	C₂²⋊Q₈	—	2	6	no	—
82	C₄×Q₈	—	2	12	no	—
83	C₄×Q₈	—	2	6	no	—
84	#117	C₂²⋊D₇₃	1	6	yes	—
85	C₂×C₄²	—	146	3	no	—
86	C₂×C₄²	C₂²×D₇₃	1	1	yes	characteristic
87	C₄⋊C₈	—	146	6	no	—
88	C₂×Q₈	—	2	6	no	—
89	C₂²×C₄	—	292	1	no	—
90	C₂²×C₄	C₂×C₂²⋊D₇₃	1	3	yes	—
91	M₁₆	—	292	6	no	—
92	C₂×C₈	—	292	3	no	—
93	C₄⋊C₄	—	2	24	no	—
94	C₄⋊C₄	—	2	12	no	—
95	C₄⋊C₄	—	2	6	no	—
96	C₄⋊C₄	C₂²⋊Dic₇₃	1	12	yes	—
97	C₂²⋊C₄	—	2	6	no	—
98	C₄²	—	2	8	no	—
99	C₄²	—	2	6	no	—
100	C₄²	—	292	3	no	—
101	C₄²	—	146	6	no	—
102	C₄²	C₂²×Dic₇₃	1	1	yes	characteristic
103	C₄²	C₂×C₂²⋊D₇₃	1	3	yes	—
104	C₂³	#14556560	1	1	yes	characteristic
105	Q₈	—	4	6	no	—
106	Q₈	—	4	6	no	—
107	C₂×C₄	—	2	3	no	—
108	C₂×C₄	—	2	3	no	—
109	C₂×C₄	—	2	12	no	—
110	C₂×C₄	—	292	6	no	—
111	C₂×C₄	C₂×C₂²⋊Dic₇₃	1	3	yes	—
112	C₂×C₄	#14556560	1	3	yes	—
113	C₈	—	584	3	no	—
114	C₂²	—	2	3	no	—
115	C₂²	#17988443	1	1	yes	characteristic
116	C₄	—	2	3	no	—
117	C₄	—	4	12	no	—
118	C₄	—	292	4	no	—
119	C₄	#17988760	1	3	yes	—
120	C₄	#17988770	1	3	yes	—
121	C₂	—	4	1	no	—
122	C₂	#22609890	1	3	yes	—

subgroup

G/H

class size

multiplicity

normal?

properties

#22610909

C₂

yes

maximal

#22608669

C₂

yes

maximal

C₇₃×#3556

C₂

yes

maximal, characteristic

#17988611

C₂²

yes

—

C₄²×Dic₇₃

C₂²

yes

characteristic

#17988346

—

#17988333

—

#564×C₇₃

C₄

yes

complemented

#557×C₇₃

—

#541×C₇₃

—

#533×C₇₃

C₂²

yes

—

C₂×#11924012

—

C₂×C₄×Dic₇₃

—

C₄⋊#10596831

—

C₄⋊Q₈×C₇₃

—

C₄⋊Q₈×C₇₃

D₄

yes

—

#126×C₇₃

—

#125×C₇₃

—

#125×C₇₃

—

#125×C₇₃

C₂×C₄

yes

—

C₂²⋊Q₈×C₇₃

—

C₄×Q₈×C₇₃

—

C₄×Q₈×C₇₃

—

#117×C₇₃

D₄

yes

—

C₂×C₄²×C₇₃

C₂³

yes

characteristic

C₂²×Dic₇₃

—

#11924012

—

C₂×#10596831

—

C₄×Dic₇₃

—

C₄×Dic₇₃

—

C₂×Q₈×C₇₃

—

C₂²×C₄×C₇₃

C₂×D₄

yes

—

C₄⋊C₄×C₇₃

—

C₄⋊C₄×C₇₃

—

C₄⋊C₄×C₇₃

—

C₄⋊C₄×C₇₃

C₂²⋊C₄

yes

—

C₂²⋊C₄×C₇₃

—

C₄²×C₇₃

—

C₄²×C₇₃

—

C₄²×C₇₃

C₂²×C₄

yes

characteristic

C₄²×C₇₃

C₂×D₄

yes

—

C₂³×C₇₃

#120

yes

characteristic

Q₈×C₇₃

—

Q₈×C₇₃

—

C₂×C₄×C₇₃

—

C₂×C₄×C₇₃

—

C₂×C₄×C₇₃

—

C₂×C₄×C₇₃

C₂×C₂²⋊C₄

yes

—

C₂×C₄×C₇₃

#120

yes

—

C₂×Dic₇₃

—

#10596831

—

C₂²×C₇₃

—

C₂²×C₇₃

#379

yes

characteristic

C₄×C₇₃

—

C₄×C₇₃

—

C₄×C₇₃

#454

yes

—

C₄×C₇₃

#456

yes

—

Dic₇₃

—

#8533

—

maximal, Sylow

C₂×C₇₃

—

C₂×C₇₃

#1793

yes

—

#3556

D₇₃

yes

characteristic

#1994

—

#1561

—

C₇₃

#8533

yes

characteristic, complemented, Sylow

#564

Dic₇₃

yes

—

—

C₂×D₇₃

yes

—

#423

—

C₄³

—

#339

—

146

—

#332

—

146

—

C₂×M₁₆

—

146

—

C₄⋊Q₈

—

C₄⋊Q₈

C₂²⋊D₇₃

yes

—

—

—

—

C₂×Dic₇₃

yes

—

C₂²⋊Q₈

—

C₄×Q₈

—

C₄×Q₈

—

#117

C₂²⋊D₇₃

yes

—

C₂×C₄²

—

146

—

C₂×C₄²

C₂²×D₇₃

yes

characteristic

C₄⋊C₈

—

146

—

C₂×Q₈

—

C₂²×C₄

—

292

—

C₂²×C₄

C₂×C₂²⋊D₇₃

yes

—

M₁₆

—

292

—

C₂×C₈

—

292

—

C₄⋊C₄

—

C₄⋊C₄

—

C₄⋊C₄

—

C₄⋊C₄

C₂²⋊Dic₇₃

yes

—

C₂²⋊C₄

—

C₄²

—

C₄²

—

100

C₄²

—

292

—

101

C₄²

—

146

—

102

C₄²

C₂²×Dic₇₃

yes

characteristic

103

C₄²

C₂×C₂²⋊D₇₃

yes

—

104

C₂³

#14556560

yes

characteristic

105

Q₈

—

106

Q₈

—

107

C₂×C₄

—

108

C₂×C₄

—

109

C₂×C₄

—

110

C₂×C₄

—

292

—

111

C₂×C₄

C₂×C₂²⋊Dic₇₃

yes

—

112

C₂×C₄

#14556560

yes

—

113

C₈

—

584

—

114

C₂²

—

115

C₂²

#17988443

yes

characteristic

116

C₄

—

117

C₄

—

118

C₄

—

292

—

119

C₄

#17988760

yes

—

120

C₄

#17988770

yes

—

121

C₂

—

122

C₂

#22609890

yes

—

	subgroup	G/H
center	C₂²	#17988443
commutator	C₄²×C₇₃	C₂²×C₄
Frattini	C₂×C₄²	C₂²×D₇₃
Fitting	C₇₃×#3556	C₂
socle	C₂²×C₇₃	#379
prime radical	C₂³×C₇₃	#120
cosocle	C₂×C₄²×C₇₃	C₂³
hypercenter	#3556	D₇₃
hypocenter	C₇₃	#8533
LCS	G ▷ C₄²×C₇₃ ▷ C₂²×C₇₃ ▷ C₇₃
UCS	C₁ ◁ C₂² ◁ C₂×C₄² ◁ #3556

subgroup

G/H

center

C₂²

#17988443

commutator

C₄²×C₇₃

C₂²×C₄

Frattini

C₂×C₄²

C₂²×D₇₃

Fitting

C₇₃×#3556

C₂

C₂²×C₇₃

C₂³×C₇₃

C₂×C₄²×C₇₃

C₂³

D₇₃

C₇₃

G ▷ C₄²×C₇₃ ▷ C₂²×C₇₃ ▷ C₇₃

UCS

C₁ ◁ C₂² ◁ C₂×C₄² ◁ #3556

N	H ≅ G/N	type
8. #564×C₇₃	C₄	semidirect
65. C₇₃	#8533	semidirect
66. #564	Dic₇₃	semidirect

H ≅ G/N

type

8. #564×C₇₃

C₄

semidirect

65. C₇₃

#8533

semidirect

66. #564

Dic₇₃

semidirect

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

subgroup	index
#22610909	2
#22608669	2
C₇₃×#3556	2
#8533	73

subgroup

index

#22610909

#22608669

C₇₃×#3556

#8533

Subgroups — #32978552

Conjugacy classes of subgroups

Complements

Maximal subgroups