Group #36353318 — subgroups

Subgroups — #36353318

Order 22144.

200 conjugacy classes of subgroups; only the 105 normal subgroups are shown.

normal subgroups only

This group has 200 conjugacy classes of subgroups — too many to draw. Only its 105 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	#26011446	C₂	1	2	yes	maximal
2	#26011249	C₂	1	1	yes	maximal, characteristic
3	#26011248	C₂	1	1	yes	maximal, characteristic
4	#26011247	C₂	1	2	yes	maximal
5	#405×C₁₇₃	C₂	1	1	yes	maximal, characteristic
6	C₂×#14583810	C₂²	1	2	yes	—
7	#19149638	C₂²	1	2	yes	—
8	#19149589	C₄	1	4	yes	complemented
9	C₈×Dic₁₇₃	C₄	1	4	yes	complemented
10	C₂×M₁₆×C₁₇₃	C₂²	1	1	yes	characteristic
11	C₂²×C₈×C₁₇₃	C₂²	1	1	yes	characteristic
12	#117×C₁₇₃	C₂²	1	1	yes	characteristic
13	#97×C₁₇₃	C₄	1	2	yes	complemented
14	C₄×C₈×C₁₇₃	C₄	1	2	yes	complemented
15	#14583810	—	2	2	no	—
16	#14583810	—	2	2	no	—
17	C₂×#13063436	C₂×C₄	1	4	yes	—
18	C₄⋊Dic₁₇₃	—	2	2	no	—
19	C₂²⋊Dic₁₇₃	—	2	2	no	—
20	C₄×Dic₁₇₃	C₂×C₄	1	4	yes	—
21	C₂²×C₄×C₁₇₃	C₂³	1	1	yes	characteristic
22	M₁₆×C₁₇₃	—	2	2	no	—
23	C₂×C₈×C₁₇₃	C₂×C₄	1	2	yes	—
24	C₂×C₈×C₁₇₃	C₂×C₄	1	2	yes	—
25	C₂×C₈×C₁₇₃	C₂×C₄	1	2	yes	—
26	C₂×C₈×C₁₇₃	C₂×C₄	1	1	yes	characteristic
27	C₂×C₈×C₁₇₃	C₂×C₄	1	1	yes	characteristic
28	C₄⋊C₄×C₁₇₃	—	2	1	no	—
29	C₂²⋊C₄×C₁₇₃	—	2	1	no	—
30	C₄²×C₁₇₃	C₂×C₄	1	2	yes	—
31	C₂×Dic₁₇₃	—	2	4	no	—
32	#13063436	—	2	4	no	—
33	C₂³×C₁₇₃	#41	1	1	yes	characteristic
34	C₂×C₄×C₁₇₃	—	2	2	no	—
35	C₂×C₄×C₁₇₃	C₂²×C₄	1	2	yes	—
36	C₂×C₄×C₁₇₃	C₂²×C₄	1	1	yes	characteristic
37	C₂×C₄×C₁₇₃	#41	1	1	yes	characteristic
38	C₂×C₄×C₁₇₃	#41	1	2	yes	—
39	C₈×C₁₇₃	—	2	2	no	—
40	C₈×C₁₇₃	C₄²	1	2	yes	—
41	C₈×C₁₇₃	C₄²	1	1	yes	characteristic
42	C₈×C₁₇₃	C₄²	1	1	yes	characteristic
43	C₂²×C₁₇₃	—	4	1	no	—
44	C₂²×C₁₇₃	#117	1	2	yes	—
45	C₂²×C₁₇₃	#117	1	1	yes	characteristic
46	C₄×C₁₇₃	—	4	2	no	—
47	C₄×C₁₇₃	C₂×C₄²	1	1	yes	characteristic
48	C₄×C₁₇₃	#117	1	2	yes	—
49	C₄×C₁₇₃	#117	1	1	yes	characteristic
50	Dic₁₇₃	—	4	4	no	—
51	C₂×C₁₇₃	—	2	2	no	—
52	C₂×C₁₇₃	—	2	1	no	—
53	C₂×C₁₇₃	#376	1	1	yes	characteristic
54	C₁₇₃	#1747	1	1	yes	characteristic, complemented, Sylow
55	#1747	—	173	1	no	maximal, Sylow
56	#405	—	173	2	no	—
57	#405	D₁₇₃	1	1	yes	characteristic
58	#338	—	173	1	no	—
59	H(3,ℤ/4)	—	173	1	no	—
60	H(3,ℤ/4)	—	173	2	no	—
61	C₂×M₁₆	—	173	2	no	—
62	C₂×M₁₆	C₂×D₁₇₃	1	1	yes	characteristic
63	C₂²×C₈	C₂×D₁₇₃	1	1	yes	characteristic
64	#117	—	173	2	no	—
65	#117	C₂×D₁₇₃	1	1	yes	characteristic
66	#97	—	173	4	no	—
67	#97	Dic₁₇₃	1	2	yes	complemented
68	C₄×C₈	—	173	4	no	—
69	C₄×C₈	Dic₁₇₃	1	2	yes	complemented
70	C₂²×C₄	C₂²×D₁₇₃	1	1	yes	characteristic
71	M₁₆	—	2	2	no	—
72	M₁₆	—	346	2	no	—
73	M₁₆	—	346	2	no	—
74	C₂×C₈	—	173	4	no	—
75	C₂×C₈	C₄×D₁₇₃	1	2	yes	—
76	C₂×C₈	C₄×D₁₇₃	1	2	yes	—
77	C₂×C₈	C₂×Dic₁₇₃	1	2	yes	—
78	C₂×C₈	C₂×Dic₁₇₃	1	1	yes	characteristic
79	C₂×C₈	C₂×Dic₁₇₃	1	1	yes	characteristic
80	C₄⋊C₄	—	2	1	no	—
81	C₄⋊C₄	—	346	2	no	—
82	C₂²⋊C₄	—	2	1	no	—
83	C₂²⋊C₄	—	346	2	no	—
84	C₄²	—	173	4	no	—
85	C₄²	C₂×Dic₁₇₃	1	2	yes	—
86	C₂³	#14583826	1	1	yes	characteristic
87	C₂×C₄	—	2	2	no	—
88	C₂×C₄	—	346	4	no	—
89	C₂×C₄	C₂×C₄×D₁₇₃	1	2	yes	—
90	C₂×C₄	C₂²×Dic₁₇₃	1	1	yes	characteristic
91	C₂×C₄	#14583826	1	1	yes	characteristic
92	C₂×C₄	#14583826	1	2	yes	—
93	C₈	—	2	2	no	—
94	C₈	—	346	4	no	—
95	C₈	C₄×Dic₁₇₃	1	1	yes	characteristic
96	C₈	C₄×Dic₁₇₃	1	2	yes	—
97	C₈	C₄×Dic₁₇₃	1	1	yes	characteristic
98	C₂²	—	4	1	no	—
99	C₂²	#19149636	1	2	yes	—
100	C₂²	#19149638	1	1	yes	characteristic
101	C₄	—	4	2	no	—
102	C₄	—	692	4	no	—
103	C₄	C₂×C₄×Dic₁₇₃	1	1	yes	characteristic
104	C₄	#19149636	1	2	yes	—
105	C₄	#19149638	1	1	yes	characteristic
106	C₂	—	2	1	no	—
107	C₂	—	2	2	no	—
108	C₂	#26011337	1	1	yes	characteristic

subgroup

G/H

class size

multiplicity

normal?

properties

#26011446

C₂

yes

maximal

#26011249

C₂

yes

maximal, characteristic

#26011248

C₂

yes

maximal, characteristic

#26011247

C₂

yes

maximal

#405×C₁₇₃

C₂

yes

maximal, characteristic

C₂×#14583810

C₂²

yes

—

#19149638

C₂²

yes

—

#19149589

C₄

yes

complemented

C₈×Dic₁₇₃

C₄

yes

complemented

C₂×M₁₆×C₁₇₃

C₂²

yes

characteristic

C₂²×C₈×C₁₇₃

C₂²

yes

characteristic

#117×C₁₇₃

C₂²

yes

characteristic

#97×C₁₇₃

C₄

yes

complemented

C₄×C₈×C₁₇₃

C₄

yes

—

—

C₂×C₄

yes

—

C₄⋊Dic₁₇₃

—

C₂²⋊Dic₁₇₃

—

C₄×Dic₁₇₃

C₂×C₄

yes

—

C₂²×C₄×C₁₇₃

C₂³

yes

characteristic

M₁₆×C₁₇₃

—

C₂×C₈×C₁₇₃

C₂×C₄

yes

—

C₂×C₈×C₁₇₃

C₂×C₄

yes

—

C₂×C₈×C₁₇₃

C₂×C₄

yes

—

C₂×C₈×C₁₇₃

C₂×C₄

yes

characteristic

C₂×C₈×C₁₇₃

C₂×C₄

yes

characteristic

C₄⋊C₄×C₁₇₃

—

C₂²⋊C₄×C₁₇₃

—

C₄²×C₁₇₃

C₂×C₄

yes

—

C₂×Dic₁₇₃

—

#13063436

—

C₂³×C₁₇₃

#41

yes

characteristic

C₂×C₄×C₁₇₃

—

C₂×C₄×C₁₇₃

C₂²×C₄

yes

—

C₂×C₄×C₁₇₃

C₂²×C₄

yes

C₂×C₄×C₁₇₃

yes

C₂×C₄×C₁₇₃

yes

—

C₈×C₁₇₃

—

C₈×C₁₇₃

C₄²

yes

—

C₈×C₁₇₃

C₄²

yes

characteristic

C₈×C₁₇₃

C₄²

yes

characteristic

C₂²×C₁₇₃

—

C₂²×C₁₇₃

#117

yes

—

C₂²×C₁₇₃

#117

yes

characteristic

C₄×C₁₇₃

—

C₄×C₁₇₃

C₂×C₄²

yes

characteristic

C₄×C₁₇₃

#117

yes

—

C₄×C₁₇₃

#117

yes

characteristic

Dic₁₇₃

—

C₂×C₁₇₃

—

C₂×C₁₇₃

—

C₂×C₁₇₃

#376

yes

characteristic

C₁₇₃

#1747

yes

characteristic, complemented, Sylow

#1747

—

173

maximal, Sylow

#405

—

173

—

#405

D₁₇₃

yes

characteristic

#338

—

173

—

H(3,ℤ/4)

—

173

—

H(3,ℤ/4)

—

173

—

C₂×M₁₆

—

173

—

C₂×M₁₆

C₂×D₁₇₃

yes

characteristic

C₂²×C₈

C₂×D₁₇₃

yes

characteristic

#117

—

173

—

#117

C₂×D₁₇₃

yes

characteristic

#97

—

173

—

#97

Dic₁₇₃

yes

complemented

C₄×C₈

—

173

—

C₄×C₈

Dic₁₇₃

yes

complemented

C₂²×C₄

C₂²×D₁₇₃

yes

characteristic

M₁₆

—

M₁₆

—

346

—

M₁₆

—

346

—

C₂×C₈

—

173

—

C₂×C₈

C₄×D₁₇₃

yes

—

C₂×C₈

C₄×D₁₇₃

yes

—

C₂×C₈

C₂×Dic₁₇₃

yes

—

C₂×C₈

C₂×Dic₁₇₃

yes

characteristic

C₂×C₈

C₂×Dic₁₇₃

yes

characteristic

C₄⋊C₄

—

C₄⋊C₄

—

346

—

C₂²⋊C₄

—

C₂²⋊C₄

—

346

—

C₄²

—

173

—

C₄²

C₂×Dic₁₇₃

yes

—

C₂³

#14583826

yes

characteristic

C₂×C₄

—

C₂×C₄

—

346

—

C₂×C₄

C₂×C₄×D₁₇₃

yes

—

C₂×C₄

C₂²×Dic₁₇₃

yes

C₂×C₄

yes

C₂×C₄

yes

—

C₈

—

C₈

—

346

—

C₈

C₄×Dic₁₇₃

yes

characteristic

C₈

C₄×Dic₁₇₃

yes

—

C₈

C₄×Dic₁₇₃

yes

characteristic

C₂²

—

C₂²

#19149636

yes

—

100

C₂²

#19149638

yes

characteristic

101

C₄

—

102

C₄

—

692

—

103

C₄

C₂×C₄×Dic₁₇₃

yes

characteristic

104

C₄

#19149636

yes

—

105

C₄

#19149638

yes

characteristic

106

C₂

—

107

C₂

—

108

C₂

#26011337

yes

characteristic

	subgroup	G/H
center	C₈	C₄×Dic₁₇₃
commutator	C₄×C₁₇₃	C₂×C₄²
Frattini	C₂²×C₄	C₂²×D₁₇₃
Fitting	#405×C₁₇₃	C₂
socle	C₂×C₁₇₃	#376
prime radical	C₂³×C₁₇₃	#41
cosocle	C₂²×C₄×C₁₇₃	C₂³
hypercenter	#405	D₁₇₃
hypocenter	C₁₇₃	#1747
LCS	G ▷ C₄×C₁₇₃ ▷ C₁₇₃
UCS	C₁ ◁ C₈ ◁ #405

subgroup

G/H

center

C₈

C₄×Dic₁₇₃

commutator

C₄×C₁₇₃

C₂×C₄²

Frattini

C₂²×C₄

C₂²×D₁₇₃

Fitting

#405×C₁₇₃

C₂

C₂×C₁₇₃

C₂³×C₁₇₃

C₂²×C₄×C₁₇₃

C₂³

D₁₇₃

C₁₇₃

G ▷ C₄×C₁₇₃ ▷ C₁₇₃

C₁ ◁ C₈ ◁ #405

N	H ≅ G/N	type
8. #19149589	C₄	semidirect
9. C₈×Dic₁₇₃	C₄	semidirect
13. #97×C₁₇₃	C₄	semidirect
14. C₄×C₈×C₁₇₃	C₄	semidirect
54. C₁₇₃	#1747	semidirect
67. #97	Dic₁₇₃	semidirect
69. C₄×C₈	Dic₁₇₃	semidirect

H ≅ G/N

type

8. #19149589

C₄

semidirect

9. C₈×Dic₁₇₃

C₄

semidirect

13. #97×C₁₇₃

C₄

semidirect

14. C₄×C₈×C₁₇₃

C₄

semidirect

54. C₁₇₃

#1747

semidirect

67. #97

Dic₁₇₃

semidirect

69. C₄×C₈

Dic₁₇₃

semidirect

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

subgroup	index
#26011446	2
#26011249	2
#26011248	2
#26011247	2
#405×C₁₇₃	2
#1747	173

subgroup

index

#405×C₁₇₃

173

Subgroups — #36353318

Conjugacy classes of subgroups

Complements

Maximal subgroups