Group #36353699 — subgroups

Subgroups — #36353699

Order 22144.

188 conjugacy classes of subgroups; only the 81 normal subgroups are shown.

normal subgroups only

This group has 188 conjugacy classes of subgroups — too many to draw. Only its 81 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	#26011514	C₂	1	2	yes	maximal
2	C₂×C₈×Dic₁₇₃	C₂	1	1	yes	maximal, characteristic
3	#26011256	C₂	1	1	yes	maximal, characteristic
4	#26011255	C₂	1	2	yes	maximal
5	#341×C₁₇₃	C₂	1	1	yes	maximal, characteristic
6	C₂×#14583810	C₂²	1	2	yes	—
7	C₂²×#13063436	C₂²	1	1	yes	characteristic
8	C₂×C₄×Dic₁₇₃	C₂²	1	1	yes	characteristic
9	#19149608	—	2	4	no	—
10	C₈×Dic₁₇₃	C₄	1	2	yes	—
11	C₈×Dic₁₇₃	C₄	1	2	yes	—
12	C₂×M₁₆×C₁₇₃	C₂²	1	2	yes	—
13	C₂²×C₈×C₁₇₃	C₂²	1	1	yes	characteristic
14	C₂²×Dic₁₇₃	Q₈	1	2	yes	—
15	#14583810	—	2	2	no	—
16	#14583810	—	2	2	no	—
17	C₂×#13063436	—	2	2	no	—
18	C₂×#13063436	—	2	2	no	—
19	C₂×#13063436	C₂×C₄	1	2	yes	—
20	C₄×Dic₁₇₃	C₂×C₄	1	1	yes	characteristic
21	C₄×Dic₁₇₃	C₂×C₄	1	1	yes	characteristic
22	C₄×Dic₁₇₃	D₄	1	2	yes	—
23	C₂²×C₄×C₁₇₃	C₂³	1	1	yes	characteristic
24	M₁₆×C₁₇₃	—	2	2	no	—
25	M₁₆×C₁₇₃	—	2	2	no	—
26	C₂×C₈×C₁₇₃	—	2	2	no	—
27	C₂×C₈×C₁₇₃	—	2	2	no	—
28	C₂×C₈×C₁₇₃	C₂×C₄	1	2	yes	—
29	C₂×Dic₁₇₃	—	2	4	no	—
30	C₂×Dic₁₇₃	C₄⋊C₄	1	2	yes	—
31	C₂×Dic₁₇₃	C₄⋊C₄	1	2	yes	—
32	#13063436	—	2	2	no	—
33	#13063436	—	4	2	no	—
34	C₂³×C₁₇₃	C₂×Q₈	1	1	yes	characteristic
35	C₂×C₄×C₁₇₃	C₂²×C₄	1	1	yes	characteristic
36	C₂×C₄×C₁₇₃	C₂×D₄	1	1	yes	characteristic
37	C₂×C₄×C₁₇₃	#41	1	2	yes	—
38	C₂×C₄×C₁₇₃	#41	1	2	yes	—
39	C₈×C₁₇₃	—	2	2	no	—
40	C₈×C₁₇₃	—	4	2	no	—
41	C₂²×C₁₇₃	—	2	2	no	—
42	C₂²×C₁₇₃	C₂×C₄⋊C₄	1	1	yes	characteristic
43	C₂²×C₁₇₃	#125	1	2	yes	—
44	C₄×C₁₇₃	#117	1	1	yes	characteristic
45	C₄×C₁₇₃	#117	1	1	yes	characteristic
46	C₄×C₁₇₃	#124	1	2	yes	—
47	Dic₁₇₃	—	2	2	no	—
48	Dic₁₇₃	#108	1	4	yes	—
49	C₂×C₁₇₃	—	2	2	no	—
50	C₂×C₁₇₃	#382	1	1	yes	characteristic
51	C₂×C₁₇₃	C₂×#108	1	2	yes	—
52	C₁₇₃	#1837	1	1	yes	characteristic, complemented, Sylow
53	#1837	—	173	1	no	maximal, Sylow
54	#423	—	173	2	no	—
55	C₂×C₄×C₈	—	173	1	no	—
56	#341	—	173	2	no	—
57	#341	—	173	1	no	—
58	#341	D₁₇₃	1	1	yes	characteristic
59	C₂×M₁₆	—	173	2	no	—
60	C₂×M₁₆	C₂×D₁₇₃	1	2	yes	—
61	C₂²×C₈	—	173	1	no	—
62	C₂²×C₈	C₂×D₁₇₃	1	1	yes	characteristic
63	C₂×C₄²	—	173	1	no	—
64	C₄⋊C₈	—	346	4	no	—
65	C₄×C₈	—	173	2	no	—
66	C₄×C₈	—	173	2	no	—
67	C₂²×C₄	—	173	2	no	—
68	C₂²×C₄	C₂²×D₁₇₃	1	1	yes	characteristic
69	M₁₆	—	2	2	no	—
70	M₁₆	—	2	2	no	—
71	M₁₆	—	346	2	no	—
72	M₁₆	—	346	2	no	—
73	C₂×C₈	—	2	2	no	—
74	C₂×C₈	—	2	2	no	—
75	C₂×C₈	—	346	2	no	—
76	C₂×C₈	—	346	2	no	—
77	C₂×C₈	—	173	2	no	—
78	C₂×C₈	C₄×D₁₇₃	1	2	yes	—
79	C₄²	—	173	1	no	—
80	C₄²	—	173	1	no	—
81	C₄²	—	173	2	no	—
82	C₂³	Q₈×D₁₇₃	1	1	yes	characteristic
83	C₂×C₄	—	346	4	no	—
84	C₂×C₄	—	173	2	no	—
85	C₂×C₄	—	173	2	no	—
86	C₂×C₄	C₂×C₄×D₁₇₃	1	1	yes	characteristic
87	C₂×C₄	D₄×D₁₇₃	1	1	yes	characteristic
88	C₂×C₄	#14583825	1	2	yes	—
89	C₂×C₄	#14583826	1	2	yes	—
90	C₈	—	2	2	no	—
91	C₈	—	4	2	no	—
92	C₈	—	346	2	no	—
93	C₈	—	692	2	no	—
94	C₂²	—	2	2	no	—
95	C₂²	C₄⋊C₄×D₁₇₃	1	1	yes	characteristic
96	C₂²	#19149669	1	2	yes	—
97	C₄	—	346	2	no	—
98	C₄	—	173	4	no	—
99	C₄	#19149635	1	1	yes	characteristic
100	C₄	#19149636	1	1	yes	characteristic
101	C₄	#19149665	1	2	yes	—
102	C₂	—	2	2	no	—
103	C₂	#26011355	1	1	yes	characteristic
104	C₂	#108×D₁₇₃	1	2	yes	—

subgroup

G/H

class size

multiplicity

normal?

properties

#26011514

C₂

yes

maximal

C₂×C₈×Dic₁₇₃

C₂

yes

maximal, characteristic

#26011256

C₂

yes

maximal, characteristic

#26011255

C₂

yes

maximal

#341×C₁₇₃

C₂

yes

maximal, characteristic

C₂×#14583810

C₂²

yes

—

C₂²×#13063436

C₂²

yes

characteristic

C₂×C₄×Dic₁₇₃

C₂²

yes

characteristic

#19149608

—

C₈×Dic₁₇₃

C₄

yes

—

C₈×Dic₁₇₃

C₄

yes

—

C₂×M₁₆×C₁₇₃

C₂²

yes

—

C₂²×C₈×C₁₇₃

C₂²

yes

characteristic

C₂²×Dic₁₇₃

Q₈

yes

—

—

—

—

—

C₂×C₄

yes

—

C₄×Dic₁₇₃

C₂×C₄

yes

characteristic

C₄×Dic₁₇₃

C₂×C₄

yes

characteristic

C₄×Dic₁₇₃

D₄

yes

—

C₂²×C₄×C₁₇₃

C₂³

yes

characteristic

M₁₆×C₁₇₃

—

M₁₆×C₁₇₃

—

C₂×C₈×C₁₇₃

—

C₂×C₈×C₁₇₃

—

C₂×C₈×C₁₇₃

C₂×C₄

yes

—

C₂×Dic₁₇₃

—

C₂×Dic₁₇₃

C₄⋊C₄

yes

—

C₂×Dic₁₇₃

C₄⋊C₄

yes

—

#13063436

—

#13063436

—

C₂³×C₁₇₃

C₂×Q₈

yes

characteristic

C₂×C₄×C₁₇₃

C₂²×C₄

yes

characteristic

C₂×C₄×C₁₇₃

C₂×D₄

yes

characteristic

C₂×C₄×C₁₇₃

#41

yes

—

C₂×C₄×C₁₇₃

#41

yes

—

C₈×C₁₇₃

—

C₈×C₁₇₃

—

C₂²×C₁₇₃

—

C₂²×C₁₇₃

C₂×C₄⋊C₄

yes

characteristic

C₂²×C₁₇₃

#125

yes

—

C₄×C₁₇₃

yes

C₄×C₁₇₃

yes

C₄×C₁₇₃

yes

—

Dic₁₇₃

—

Dic₁₇₃

#108

yes

—

C₂×C₁₇₃

—

C₂×C₁₇₃

#382

yes

characteristic

C₂×C₁₇₃

C₂×#108

yes

—

C₁₇₃

#1837

yes

characteristic, complemented, Sylow

#1837

—

173

maximal, Sylow

#423

—

173

—

C₂×C₄×C₈

—

173

—

#341

—

173

—

#341

—

173

—

#341

D₁₇₃

yes

characteristic

C₂×M₁₆

—

173

—

C₂×M₁₆

C₂×D₁₇₃

yes

—

C₂²×C₈

—

173

—

C₂²×C₈

C₂×D₁₇₃

yes

characteristic

C₂×C₄²

—

173

—

C₄⋊C₈

—

346

—

C₄×C₈

—

173

—

C₄×C₈

—

173

—

C₂²×C₄

—

173

—

C₂²×C₄

C₂²×D₁₇₃

yes

characteristic

M₁₆

—

M₁₆

—

M₁₆

—

346

—

M₁₆

—

346

—

C₂×C₈

—

C₂×C₈

—

C₂×C₈

—

346

—

C₂×C₈

—

346

—

C₂×C₈

—

173

—

C₂×C₈

C₄×D₁₇₃

yes

—

C₄²

—

173

—

C₄²

—

173

—

C₄²

—

173

—

C₂³

Q₈×D₁₇₃

yes

characteristic

C₂×C₄

—

346

—

C₂×C₄

—

173

—

C₂×C₄

—

173

—

C₂×C₄

C₂×C₄×D₁₇₃

yes

characteristic

C₂×C₄

D₄×D₁₇₃

yes

characteristic

C₂×C₄

#14583825

yes

—

C₂×C₄

#14583826

yes

—

C₈

—

C₈

—

C₈

—

346

—

C₈

—

692

—

C₂²

—

C₂²

C₄⋊C₄×D₁₇₃

yes

characteristic

C₂²

#19149669

yes

—

C₄

—

346

—

C₄

—

173

—

C₄

#19149635

yes

characteristic

100

C₄

#19149636

yes

characteristic

101

C₄

#19149665

yes

—

102

C₂

—

103

C₂

#26011355

yes

characteristic

104

C₂

#108×D₁₇₃

yes

—

	subgroup	G/H
center	C₂×C₄	D₄×D₁₇₃
commutator	C₂×C₄×C₁₇₃	C₂²×C₄
Frattini	C₂²×C₄	C₂²×D₁₇₃
Fitting	#341×C₁₇₃	C₂
socle	C₂²×C₁₇₃	C₂×C₄⋊C₄
prime radical	C₂³×C₁₇₃	C₂×Q₈
cosocle	C₂²×C₄×C₁₇₃	C₂³
hypercenter	#341	D₁₇₃
hypocenter	C₁₇₃	#1837
LCS	G ▷ C₂×C₄×C₁₇₃ ▷ C₂×C₁₇₃ ▷ C₁₇₃
UCS	C₁ ◁ C₂×C₄ ◁ C₂²×C₄ ◁ #341

subgroup

G/H

center

C₂×C₄

D₄×D₁₇₃

commutator

C₂×C₄×C₁₇₃

C₂²×C₄

Frattini

C₂²×C₄

C₂²×D₁₇₃

Fitting

#341×C₁₇₃

C₂

socle

C₂²×C₁₇₃

C₂×C₄⋊C₄

prime radical

C₂³×C₁₇₃

C₂×Q₈

cosocle

C₂²×C₄×C₁₇₃

C₂³

D₁₇₃

C₁₇₃

G ▷ C₂×C₄×C₁₇₃ ▷ C₂×C₁₇₃ ▷ C₁₇₃

UCS

C₁ ◁ C₂×C₄ ◁ C₂²×C₄ ◁ #341

N	H ≅ G/N	type
52. C₁₇₃	#1837	semidirect

H ≅ G/N

type

52. C₁₇₃

#1837

semidirect

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

subgroup	index
#26011514	2
C₂×C₈×Dic₁₇₃	2
#26011256	2
#26011255	2
#341×C₁₇₃	2
#1837	173

subgroup

index

C₂×C₈×Dic₁₇₃

#341×C₁₇₃

173

Subgroups — #36353699

Conjugacy classes of subgroups

Complements

Maximal subgroups