Group #36354533 — subgroups

Subgroups — #36354533

Order 22144.

260 conjugacy classes of subgroups; only the 113 normal subgroups are shown.

normal subgroups only

This group has 260 conjugacy classes of subgroups — too many to draw. Only its 113 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	C₂×#19149590	C₂	1	1	yes	maximal, characteristic
2	#26011361	C₂	1	1	yes	maximal, characteristic
3	#26011253	C₂	1	4	yes	maximal
4	C₂×#125×C₁₇₃	C₂	1	1	yes	maximal, characteristic
5	C₂²×#13063436	C₂²	1	2	yes	—
6	C₂×C₄⋊Dic₁₇₃	—	2	2	no	—
7	C₂×C₄⋊Dic₁₇₃	C₂²	1	2	yes	—
8	#19149590	—	2	2	no	—
9	#125×C₁₇₃	—	2	2	no	—
10	#125×C₁₇₃	C₄	1	4	yes	complemented
11	C₂×C₄⋊C₄×C₁₇₃	—	2	2	no	—
12	C₂×C₄⋊C₄×C₁₇₃	C₂²	1	2	yes	—
13	C₂×C₄²×C₁₇₃	C₂²	1	1	yes	characteristic
14	C₂²×Dic₁₇₃	—	2	2	no	—
15	C₂×#13063436	—	2	2	no	—
16	C₂×#13063436	—	2	2	no	—
17	C₂×#13063436	Q₈	1	4	yes	—
18	C₄⋊Dic₁₇₃	—	2	4	no	—
19	C₄⋊Dic₁₇₃	—	4	4	no	—
20	C₂²×C₄×C₁₇₃	—	2	2	no	—
21	C₂²×C₄×C₁₇₃	D₄	1	2	yes	—
22	C₂²×C₄×C₁₇₃	C₂³	1	1	yes	characteristic
23	C₄⋊C₄×C₁₇₃	—	2	8	no	—
24	C₄⋊C₄×C₁₇₃	—	2	2	no	—
25	C₄⋊C₄×C₁₇₃	C₂×C₄	1	4	yes	—
26	C₄²×C₁₇₃	C₂×C₄	1	2	yes	—
27	C₄²×C₁₇₃	D₄	1	2	yes	—
28	C₂×Dic₁₇₃	—	4	2	no	—
29	C₂×Dic₁₇₃	—	4	4	no	—
30	#13063436	—	2	4	no	—
31	C₂³×C₁₇₃	C₂×D₄	1	1	yes	characteristic
32	C₂×C₄×C₁₇₃	—	2	2	no	—
33	C₂×C₄×C₁₇₃	—	2	2	no	—
34	C₂×C₄×C₁₇₃	—	2	4	no	—
35	C₂×C₄×C₁₇₃	—	4	2	no	—
36	C₂×C₄×C₁₇₃	—	4	2	no	—
37	C₂×C₄×C₁₇₃	C₂²⋊C₄	1	4	yes	—
38	C₂×C₄×C₁₇₃	C₂²×C₄	1	1	yes	characteristic
39	C₂×C₄×C₁₇₃	C₂×D₄	1	1	yes	characteristic
40	C₂×C₄×C₁₇₃	C₂×Q₈	1	2	yes	—
41	C₂×C₄×C₁₇₃	#41	1	2	yes	—
42	C₂²×C₁₇₃	C₂×C₂²⋊C₄	1	1	yes	characteristic
43	C₂²×C₁₇₃	#124	1	1	yes	characteristic
44	C₂²×C₁₇₃	C₄⋊Q₈	1	1	yes	characteristic
45	C₂²×C₁₇₃	AGL(1,ℤ/8)	1	2	yes	—
46	C₂²×C₁₇₃	#137	1	2	yes	—
47	C₄×C₁₇₃	—	2	4	no	—
48	C₄×C₁₇₃	—	4	4	no	—
49	C₄×C₁₇₃	C₄×Q₈	1	2	yes	—
50	C₄×C₁₇₃	C₂²⋊Q₈	1	2	yes	—
51	Dic₁₇₃	—	8	2	no	—
52	C₂×C₁₇₃	#391	1	1	yes	characteristic
53	C₂×C₁₇₃	#417	1	2	yes	—
54	C₂×C₁₇₃	#490	1	1	yes	characteristic
55	C₂×C₁₇₃	#491	1	1	yes	characteristic
56	C₂×C₁₇₃	#501	1	2	yes	—
57	C₁₇₃	#1992	1	1	yes	characteristic, complemented, Sylow
58	#1992	—	173	1	no	maximal, Sylow
59	C₂×#125	D₁₇₃	1	1	yes	characteristic
60	C₂×#97	—	173	1	no	—
61	#384	—	173	1	no	—
62	#340	—	173	4	no	—
63	C₂²×C₈	—	173	2	no	—
64	#125	—	2	2	no	—
65	#125	Dic₁₇₃	1	4	yes	complemented
66	C₂×C₄⋊C₄	—	2	2	no	—
67	C₂×C₄⋊C₄	—	346	2	no	—
68	C₂×C₄⋊C₄	—	173	2	no	—
69	C₂×C₄⋊C₄	C₂×D₁₇₃	1	2	yes	—
70	C₂×C₄²	C₂×D₁₇₃	1	1	yes	characteristic
71	#97	—	346	2	no	—
72	C₂²×C₄	—	2	2	no	—
73	C₂²×C₄	—	346	2	no	—
74	C₂²×C₄	C₂²⋊D₁₇₃	1	2	yes	—
75	C₂²×C₄	C₂²×D₁₇₃	1	1	yes	characteristic
76	C₂×C₈	—	346	2	no	—
77	C₂×C₈	—	346	2	no	—
78	C₂×C₈	—	173	4	no	—
79	C₄⋊C₄	—	2	2	no	—
80	C₄⋊C₄	—	2	8	no	—
81	C₄⋊C₄	—	346	4	no	—
82	C₄⋊C₄	—	692	4	no	—
83	C₄⋊C₄	C₂×Dic₁₇₃	1	4	yes	—
84	C₄²	C₂×Dic₁₇₃	1	2	yes	—
85	C₄²	C₂²⋊D₁₇₃	1	2	yes	—
86	C₂³	C₂×C₂²⋊D₁₇₃	1	1	yes	characteristic
87	C₂×C₄	—	2	2	no	—
88	C₂×C₄	—	2	2	no	—
89	C₂×C₄	—	2	4	no	—
90	C₂×C₄	—	4	2	no	—
91	C₂×C₄	—	4	2	no	—
92	C₂×C₄	—	692	2	no	—
93	C₂×C₄	—	692	4	no	—
94	C₂×C₄	C₂²⋊Dic₁₇₃	1	4	yes	—
95	C₂×C₄	C₂²×Dic₁₇₃	1	1	yes	characteristic
96	C₂×C₄	C₂×C₂²⋊D₁₇₃	1	1	yes	characteristic
97	C₂×C₄	Q₈×D₁₇₃	1	2	yes	—
98	C₂×C₄	#14583827	1	2	yes	—
99	C₈	—	346	4	no	—
100	C₂²	C₂×C₂²⋊Dic₁₇₃	1	1	yes	characteristic
101	C₂²	#19149668	1	1	yes	characteristic
102	C₂²	#19149677	1	1	yes	characteristic
103	C₂²	#19149704	1	2	yes	—
104	C₂²	#19149709	1	2	yes	—
105	C₄	—	2	4	no	—
106	C₄	—	4	4	no	—
107	C₄	—	1384	2	no	—
108	C₄	Q₈×Dic₁₇₃	1	2	yes	—
109	C₄	#19149655	1	2	yes	—
110	C₂	#26011398	1	1	yes	characteristic
111	C₂	#26011492	1	2	yes	—
112	C₂	#26011857	1	1	yes	characteristic
113	C₂	#26011861	1	1	yes	characteristic
114	C₂	#26011901	1	2	yes	—

subgroup

G/H

class size

multiplicity

normal?

properties

C₂×#19149590

C₂

yes

maximal, characteristic

#26011361

C₂

yes

maximal, characteristic

#26011253

C₂

yes

maximal

C₂×#125×C₁₇₃

C₂

yes

maximal, characteristic

C₂²×#13063436

C₂²

yes

—

C₂×C₄⋊Dic₁₇₃

—

C₂×C₄⋊Dic₁₇₃

C₂²

yes

—

#19149590

—

#125×C₁₇₃

—

#125×C₁₇₃

C₄

yes

complemented

C₂×C₄⋊C₄×C₁₇₃

—

C₂×C₄⋊C₄×C₁₇₃

C₂²

yes

—

C₂×C₄²×C₁₇₃

C₂²

yes

characteristic

C₂²×Dic₁₇₃

—

C₂×#13063436

—

C₂×#13063436

—

C₂×#13063436

Q₈

yes

—

C₄⋊Dic₁₇₃

—

C₄⋊Dic₁₇₃

—

C₂²×C₄×C₁₇₃

—

C₂²×C₄×C₁₇₃

D₄

yes

—

C₂²×C₄×C₁₇₃

C₂³

yes

characteristic

C₄⋊C₄×C₁₇₃

—

C₄⋊C₄×C₁₇₃

—

C₄⋊C₄×C₁₇₃

C₂×C₄

yes

—

C₄²×C₁₇₃

C₂×C₄

yes

—

C₄²×C₁₇₃

D₄

yes

—

C₂×Dic₁₇₃

—

C₂×Dic₁₇₃

—

#13063436

—

C₂³×C₁₇₃

C₂×D₄

yes

characteristic

C₂×C₄×C₁₇₃

—

C₂×C₄×C₁₇₃

—

C₂×C₄×C₁₇₃

—

C₂×C₄×C₁₇₃

—

C₂×C₄×C₁₇₃

—

C₂×C₄×C₁₇₃

C₂²⋊C₄

yes

—

C₂×C₄×C₁₇₃

C₂²×C₄

yes

characteristic

C₂×C₄×C₁₇₃

C₂×D₄

yes

characteristic

C₂×C₄×C₁₇₃

C₂×Q₈

yes

—

C₂×C₄×C₁₇₃

#41

yes

—

C₂²×C₁₇₃

C₂×C₂²⋊C₄

yes

characteristic

C₂²×C₁₇₃

#124

yes

characteristic

C₂²×C₁₇₃

C₄⋊Q₈

yes

characteristic

C₂²×C₁₇₃

AGL(1,ℤ/8)

yes

—

C₂²×C₁₇₃

#137

yes

—

C₄×C₁₇₃

—

C₄×C₁₇₃

—

C₄×C₁₇₃

C₄×Q₈

yes

—

C₄×C₁₇₃

C₂²⋊Q₈

yes

—

Dic₁₇₃

—

C₂×C₁₇₃

#391

yes

characteristic

C₂×C₁₇₃

#417

yes

—

C₂×C₁₇₃

yes

C₂×C₁₇₃

yes

C₂×C₁₇₃

yes

—

C₁₇₃

#1992

yes

characteristic, complemented, Sylow

#1992

—

173

maximal, Sylow

C₂×#125

D₁₇₃

yes

characteristic

C₂×#97

—

173

—

#384

—

173

—

#340

—

173

—

C₂²×C₈

—

173

—

#125

—

#125

Dic₁₇₃

yes

complemented

C₂×C₄⋊C₄

—

C₂×C₄⋊C₄

—

346

—

C₂×C₄⋊C₄

—

173

—

C₂×C₄⋊C₄

C₂×D₁₇₃

yes

—

C₂×C₄²

C₂×D₁₇₃

yes

characteristic

#97

—

346

—

C₂²×C₄

—

C₂²×C₄

—

346

—

C₂²×C₄

C₂²⋊D₁₇₃

yes

—

C₂²×C₄

C₂²×D₁₇₃

yes

characteristic

C₂×C₈

—

346

—

C₂×C₈

—

346

—

C₂×C₈

—

173

—

C₄⋊C₄

—

C₄⋊C₄

—

C₄⋊C₄

—

346

—

C₄⋊C₄

—

692

—

C₄⋊C₄

C₂×Dic₁₇₃

yes

—

C₄²

C₂×Dic₁₇₃

yes

—

C₄²

C₂²⋊D₁₇₃

yes

—

C₂³

C₂×C₂²⋊D₁₇₃

yes

characteristic

C₂×C₄

—

C₂×C₄

—

C₂×C₄

—

C₂×C₄

—

C₂×C₄

—

C₂×C₄

—

692

—

C₂×C₄

—

692

—

C₂×C₄

C₂²⋊Dic₁₇₃

yes

—

C₂×C₄

C₂²×Dic₁₇₃

yes

characteristic

C₂×C₄

C₂×C₂²⋊D₁₇₃

yes

characteristic

C₂×C₄

Q₈×D₁₇₃

yes

—

C₂×C₄

#14583827

yes

—

C₈

—

346

—

100

C₂²

C₂×C₂²⋊Dic₁₇₃

yes

characteristic

101

C₂²

#19149668

yes

characteristic

102

C₂²

#19149677

yes

characteristic

103

C₂²

#19149704

yes

—

104

C₂²

#19149709

yes

—

105

C₄

—

106

C₄

—

107

C₄

—

1384

—

108

C₄

Q₈×Dic₁₇₃

yes

—

109

C₄

#19149655

yes

—

110

C₂

#26011398

yes

characteristic

111

C₂

#26011492

yes

—

112

C₂

#26011857

yes

characteristic

113

C₂

#26011861

yes

characteristic

114

C₂

#26011901

yes

—

	subgroup	G/H
center	C₂³	C₂×C₂²⋊D₁₇₃
commutator	C₂×C₄×C₁₇₃	C₂²×C₄
Frattini	C₂²×C₄	C₂²×D₁₇₃
Fitting	C₂×#125×C₁₇₃	C₂
socle, prime radical	C₂³×C₁₇₃	C₂×D₄
cosocle	C₂²×C₄×C₁₇₃	C₂³
hypercenter	C₂×#125	D₁₇₃
hypocenter	C₁₇₃	#1992
LCS	G ▷ C₂×C₄×C₁₇₃ ▷ C₂×C₁₇₃ ▷ C₁₇₃
UCS	C₁ ◁ C₂³ ◁ C₂×C₄² ◁ C₂×#125

subgroup

G/H

center

C₂³

C₂×C₂²⋊D₁₇₃

commutator

C₂×C₄×C₁₇₃

C₂²×C₄

Frattini

C₂²×C₄

C₂²×D₁₇₃

Fitting

C₂×#125×C₁₇₃

C₂

socle, prime radical

C₂³×C₁₇₃

C₂×D₄

cosocle

C₂²×C₄×C₁₇₃

C₂³

D₁₇₃

C₁₇₃

G ▷ C₂×C₄×C₁₇₃ ▷ C₂×C₁₇₃ ▷ C₁₇₃

UCS

C₁ ◁ C₂³ ◁ C₂×C₄² ◁ C₂×#125

N	H ≅ G/N	type
10. #125×C₁₇₃	C₄	semidirect
57. C₁₇₃	#1992	semidirect
65. #125	Dic₁₇₃	semidirect

H ≅ G/N

type

10. #125×C₁₇₃

C₄

semidirect

57. C₁₇₃

#1992

semidirect

65. #125

Dic₁₇₃

semidirect

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

subgroup	index
C₂×#19149590	2
#26011361	2
#26011253	2
C₂×#125×C₁₇₃	2
#1992	173

subgroup

index

C₂×#125×C₁₇₃

173

Subgroups — #36354533

Conjugacy classes of subgroups

Complements

Maximal subgroups