Group #34014539 — subgroups

Subgroups — #34014539

Order 18688.

188 conjugacy classes of subgroups; only the 54 normal subgroups are shown.

normal subgroups only

This group has 188 conjugacy classes of subgroups — too many to draw. Only its 54 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	Q₃₂×#64408	C₂	1	2	yes	maximal
2	#22625093	C₂	1	1	yes	maximal, characteristic
3	#22625081	C₂	1	2	yes	maximal
4	#22625037	C₂	1	1	yes	maximal, characteristic
5	Q₆₄×D₇₃	C₂	1	1	yes	maximal, characteristic
6	Q₁₆×#64408	—	2	2	no	—
7	#17989518	C₂²	1	1	yes	characteristic
8	#17989513	C₂²	1	2	yes	—
9	#17989500	—	2	2	no	—
10	C₁₆×#64408	C₂²	1	1	yes	characteristic
11	Q₃₂×D₇₃	C₂²	1	2	yes	—
12	Dic₁₁₆₈	C₄	1	1	yes	characteristic, complemented
13	#17988427	C₄	1	2	yes	complemented
14	C₃₂×D₇₃	C₂²	1	1	yes	characteristic
15	Q₆₄×C₇₃	C₄	1	1	yes	characteristic, complemented
16	Q₈×#64408	—	4	2	no	—
17	#14556664	D₄	1	1	yes	characteristic
18	#14556660	—	2	2	no	—
19	#14556654	—	4	2	no	—
20	C₈×#64408	D₄	1	1	yes	characteristic
21	#14556641	—	2	1	no	—
22	Q₁₆×D₇₃	—	2	2	no	—
23	Dic₅₈₄	C₂×C₄	1	2	yes	—
24	#14556537	—	2	2	no	—
25	C₁₆×D₇₃	C₂³	1	1	yes	characteristic
26	#14556497	C₂×C₄	1	1	yes	characteristic
27	Q₃₂×C₇₃	C₂×C₄	1	2	yes	—
28	C₃₂×C₇₃	C₂×C₄	1	1	yes	characteristic
29	#11924036	D₈	1	1	yes	characteristic
30	C₄⋊#64408	—	4	2	no	—
31	C₄⋊#64408	—	8	2	no	—
32	C₄×#64408	—	8	2	no	—
33	C₄×#64408	D₈	1	1	yes	characteristic
34	#11924031	—	2	1	no	—
35	Q₈×D₇₃	—	4	2	no	—
36	Dic₂₉₂	—	2	2	no	—
37	#11924018	—	4	2	no	—
38	C₈×D₇₃	C₂×D₄	1	1	yes	characteristic
39	#11924003	#41	1	1	yes	characteristic
40	Q₁₆×C₇₃	—	2	2	no	—
41	C₁₆×C₇₃	C₂²×C₄	1	1	yes	characteristic
42	C₂×#64408	—	8	2	no	—
43	C₂×#64408	D₁₆	1	2	yes	—
44	Q₈×C₇₃	—	4	2	no	—
45	C₄×D₇₃	—	8	2	no	—
46	C₄×D₇₃	C₂×D₈	1	1	yes	characteristic
47	Dic₁₄₆	—	4	2	no	—
48	Dic₁₄₆	—	8	2	no	—
49	#10596833	—	2	1	no	—
50	C₈×C₇₃	C₄×D₄	1	1	yes	characteristic
51	#10596831	#135	1	1	yes	characteristic
52	C₂×D₇₃	C₂×D₁₆	1	1	yes	characteristic
53	#64408	—	2	2	no	—
54	#64408	—	16	2	no	—
55	C₄×C₇₃	—	8	2	no	—
56	C₄×C₇₃	C₄×D₈	1	1	yes	characteristic
57	Dic₇₃	—	8	2	no	—
58	Dic₇₃	#508	1	1	yes	characteristic
59	#13653	—	73	1	no	maximal, Sylow
60	C₂×C₇₃	C₄×D₁₆	1	1	yes	characteristic
61	D₇₃	#2264	1	2	yes	—
62	C₂×Q₆₄	—	73	1	no	—
63	C₄×Q₃₂	—	73	2	no	—
64	#1424	—	73	1	no	—
65	#1416	—	73	2	no	—
66	#1397	—	73	1	no	—
67	C₇₃	#13653	1	1	yes	characteristic, complemented, Sylow
68	C₂×Q₃₂	—	73	2	no	—
69	C₄×Q₁₆	—	146	2	no	—
70	Q₆₄	—	73	2	no	—
71	Q₆₄	—	73	1	no	—
72	Q₆₄	#64408	1	1	yes	characteristic, complemented
73	C₂×C₃₂	—	73	1	no	—
74	C₂×C₃₂	—	73	1	no	—
75	#366	—	73	2	no	—
76	#358	—	146	2	no	—
77	C₄×C₁₆	—	73	1	no	—
78	C₂×Q₁₆	—	146	2	no	—
79	C₄×Q₈	—	292	2	no	—
80	Q₃₂	—	73	2	no	—
81	Q₃₂	—	146	2	no	—
82	Q₃₂	C₂×#64408	1	2	yes	—
83	C₂×C₁₆	—	73	1	no	—
84	C₂×C₁₆	—	73	1	no	—
85	#107	—	146	2	no	—
86	#103	—	292	2	no	—
87	C₄×C₈	—	73	1	no	—
88	C₃₂	—	73	1	no	—
89	C₃₂	—	146	1	no	—
90	C₃₂	C₂×#64408	1	1	yes	characteristic
91	C₂×Q₈	—	292	2	no	—
92	Q₁₆	—	2	2	no	—
93	Q₁₆	—	292	2	no	—
94	Q₁₆	—	146	2	no	—
95	C₂×C₈	—	73	1	no	—
96	C₂×C₈	—	73	1	no	—
97	C₄⋊C₄	—	292	2	no	—
98	C₄⋊C₄	—	584	2	no	—
99	C₄²	—	584	2	no	—
100	C₄²	—	73	1	no	—
101	C₁₆	—	73	1	no	—
102	C₁₆	—	146	1	no	—
103	C₁₆	C₂²×#64408	1	1	yes	characteristic
104	Q₈	—	4	2	no	—
105	Q₈	—	292	2	no	—
106	Q₈	—	584	2	no	—
107	C₂×C₄	—	584	2	no	—
108	C₂×C₄	—	584	2	no	—
109	C₂×C₄	—	73	1	no	—
110	C₂×C₄	—	73	2	no	—
111	C₈	—	73	1	no	—
112	C₈	—	146	1	no	—
113	C₈	D₄×#64408	1	1	yes	characteristic
114	C₂²	—	73	1	no	—
115	C₄	—	8	2	no	—
116	C₄	—	584	2	no	—
117	C₄	—	73	1	no	—
118	C₄	—	1168	2	no	—
119	C₄	—	146	2	no	—
120	C₄	D₈×#64408	1	1	yes	characteristic
121	C₂	—	73	2	no	—
122	C₂	D₁₆×#64408	1	1	yes	characteristic

subgroup

G/H

class size

multiplicity

normal?

properties

Q₃₂×#64408

C₂

yes

maximal

#22625093

C₂

yes

maximal, characteristic

#22625081

C₂

yes

maximal

#22625037

C₂

yes

maximal, characteristic

Q₆₄×D₇₃

C₂

yes

maximal, characteristic

Q₁₆×#64408

—

#17989518

C₂²

yes

characteristic

#17989513

C₂²

yes

—

#17989500

—

C₁₆×#64408

C₂²

yes

characteristic

Q₃₂×D₇₃

C₂²

yes

—

Dic₁₁₆₈

C₄

yes

characteristic, complemented

#17988427

C₄

yes

complemented

C₃₂×D₇₃

C₂²

yes

characteristic

Q₆₄×C₇₃

C₄

yes

characteristic, complemented

Q₈×#64408

—

#14556664

D₄

yes

—

—

D₄

yes

characteristic

#14556641

—

Q₁₆×D₇₃

—

Dic₅₈₄

C₂×C₄

yes

—

#14556537

—

C₁₆×D₇₃

C₂³

yes

characteristic

#14556497

C₂×C₄

yes

characteristic

Q₃₂×C₇₃

C₂×C₄

yes

—

C₃₂×C₇₃

C₂×C₄

yes

characteristic

#11924036

D₈

yes

—

—

—

D₈

yes

characteristic

#11924031

—

Q₈×D₇₃

—

Dic₂₉₂

—

#11924018

—

C₈×D₇₃

C₂×D₄

yes

yes

Q₁₆×C₇₃

—

C₁₆×C₇₃

C₂²×C₄

yes

characteristic

C₂×#64408

—

C₂×#64408

D₁₆

yes

—

Q₈×C₇₃

—

C₄×D₇₃

—

C₄×D₇₃

C₂×D₈

yes

characteristic

Dic₁₄₆

—

Dic₁₄₆

—

#10596833

—

C₈×C₇₃

C₄×D₄

yes

yes

C₂×D₇₃

C₂×D₁₆

yes

characteristic

#64408

—

#64408

—

C₄×C₇₃

—

C₄×C₇₃

C₄×D₈

yes

characteristic

Dic₇₃

—

Dic₇₃

yes

—

C₂×C₇₃

C₄×D₁₆

yes

characteristic

D₇₃

#2264

yes

—

C₂×Q₆₄

—

C₄×Q₃₂

—

#1424

—

#1416

—

#1397

—

C₇₃

#13653

yes

characteristic, complemented, Sylow

C₂×Q₃₂

—

C₄×Q₁₆

—

146

—

Q₆₄

—

Q₆₄

—

Q₆₄

#64408

yes

characteristic, complemented

C₂×C₃₂

—

C₂×C₃₂

—

#366

—

#358

—

146

—

C₄×C₁₆

—

C₂×Q₁₆

—

146

—

C₄×Q₈

—

292

—

Q₃₂

—

Q₃₂

—

146

—

Q₃₂

C₂×#64408

yes

—

C₂×C₁₆

—

C₂×C₁₆

—

#107

—

146

—

#103

—

292

—

C₄×C₈

—

C₃₂

—

C₃₂

—

146

—

C₃₂

C₂×#64408

yes

characteristic

C₂×Q₈

—

292

—

Q₁₆

—

Q₁₆

—

292

—

Q₁₆

—

146

—

C₂×C₈

—

C₂×C₈

—

C₄⋊C₄

—

292

—

C₄⋊C₄

—

584

—

C₄²

—

584

—

100

C₄²

—

101

C₁₆

—

102

C₁₆

—

146

—

103

C₁₆

C₂²×#64408

yes

characteristic

104

Q₈

—

105

Q₈

—

292

—

106

Q₈

—

584

—

107

C₂×C₄

—

584

—

108

C₂×C₄

—

584

—

109

C₂×C₄

—

110

C₂×C₄

—

111

C₈

—

112

C₈

—

146

—

113

C₈

D₄×#64408

yes

characteristic

114

C₂²

—

115

C₄

—

116

C₄

—

584

—

117

C₄

—

118

C₄

—

1168

—

119

C₄

—

146

—

120

C₄

D₈×#64408

yes

characteristic

121

C₂

—

122

C₂

D₁₆×#64408

yes

characteristic

	subgroup	G/H
center	C₂	D₁₆×#64408
commutator	C₁₆×C₇₃	C₂²×C₄
Frattini	C₁₆	C₂²×#64408
Fitting	Q₆₄×C₇₃	C₄
socle	C₂×C₇₃	C₄×D₁₆
prime radical	C₂×D₇₃	C₂×D₁₆
cosocle	C₁₆×D₇₃	C₂³
hypercenter	Q₆₄	#64408
hypocenter	C₇₃	#13653
LCS	G ▷ C₁₆×C₇₃ ▷ C₈×C₇₃ ▷ C₄×C₇₃ ▷ C₂×C₇₃ ▷ C₇₃
UCS	C₁ ◁ C₂ ◁ C₄ ◁ C₈ ◁ C₁₆ ◁ Q₆₄

subgroup

G/H

center

C₂

D₁₆×#64408

commutator

C₁₆×C₇₃

C₂²×C₄

Frattini

C₁₆

C₂²×#64408

Fitting

Q₆₄×C₇₃

C₄

socle

C₂×C₇₃

C₄×D₁₆

prime radical

C₂×D₇₃

C₂×D₁₆

cosocle

C₁₆×D₇₃

C₂³

Q₆₄

C₇₃

G ▷ C₁₆×C₇₃ ▷ C₈×C₇₃ ▷ C₄×C₇₃ ▷ C₂×C₇₃ ▷ C₇₃

UCS

C₁ ◁ C₂ ◁ C₄ ◁ C₈ ◁ C₁₆ ◁ Q₆₄

N	H ≅ G/N	type
12. Dic₁₁₆₈	C₄	semidirect
13. #17988427	C₄	semidirect
15. Q₆₄×C₇₃	C₄	semidirect
67. C₇₃	#13653	semidirect
72. Q₆₄	#64408	semidirect

H ≅ G/N

type

12. Dic₁₁₆₈

C₄

semidirect

13. #17988427

C₄

semidirect

15. Q₆₄×C₇₃

C₄

semidirect

67. C₇₃

#13653

semidirect

72. Q₆₄

#64408

semidirect

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

subgroup	index
Q₃₂×#64408	2
#22625093	2
#22625081	2
#22625037	2
Q₆₄×D₇₃	2
#13653	73

subgroup

index

Q₃₂×#64408

Q₆₄×D₇₃

Subgroups — #34014539

Conjugacy classes of subgroups

Complements

Maximal subgroups