Group #36351399 — subgroups

Subgroups — #36351399

Order 22144.

172 conjugacy classes of subgroups; only the 85 normal subgroups are shown.

normal subgroups only

This group has 172 conjugacy classes of subgroups — too many to draw. Only its 85 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	C₄×#14583810	C₂	1	1	yes	maximal, characteristic
2	C₂×C₄×#13063436	C₂	1	1	yes	maximal, characteristic
3	#431×C₁₇₃	C₂	1	1	yes	maximal, characteristic
4	C₂×#14583810	C₄	1	2	yes	—
5	C₂²×#13063436	—	2	1	no	—
6	#19149590	—	2	1	no	—
7	C₄×#13063436	—	2	1	no	—
8	C₄×#13063436	—	2	1	no	—
9	C₄×#13063436	C₄	1	2	yes	—
10	C₂×C₄²×C₁₇₃	C₂²	1	1	yes	characteristic
11	C₄⋊C₈×C₁₇₃	C₄	1	2	yes	—
12	C₂²⋊C₈×C₁₇₃	—	2	1	no	—
13	C₄×C₈×C₁₇₃	—	2	1	no	—
14	#14583810	—	2	2	no	—
15	#14583810	C₈	1	4	yes	complemented
16	C₂×#13063436	—	2	4	no	—
17	C₂×#13063436	—	2	2	no	—
18	C₂×#13063436	—	2	2	no	—
19	C₂²×C₄×C₁₇₃	C₂×C₄	1	1	yes	characteristic
20	C₂²×C₄×C₁₇₃	D₄	1	1	yes	characteristic
21	C₂²×C₄×C₁₇₃	Q₈	1	1	yes	characteristic
22	C₂×C₈×C₁₇₃	—	2	2	no	—
23	C₄²×C₁₇₃	C₂×C₄	1	1	yes	characteristic
24	C₄²×C₁₇₃	C₂×C₄	1	1	yes	characteristic
25	C₄²×C₁₇₃	D₄	1	1	yes	characteristic
26	C₄²×C₁₇₃	D₄	1	1	yes	characteristic
27	#13063436	—	2	4	no	—
28	#13063436	—	2	4	no	—
29	C₂³×C₁₇₃	C₄⋊C₄	1	1	yes	characteristic
30	C₂×C₄×C₁₇₃	—	2	2	no	—
31	C₂×C₄×C₁₇₃	—	2	2	no	—
32	C₂×C₄×C₁₇₃	C₄²	1	1	yes	characteristic
33	C₂×C₄×C₁₇₃	C₂²⋊C₄	1	1	yes	characteristic
34	C₂×C₄×C₁₇₃	C₂²⋊C₄	1	1	yes	characteristic
35	C₂×C₄×C₁₇₃	C₂²⋊C₄	1	1	yes	characteristic
36	C₂×C₄×C₁₇₃	C₄⋊C₄	1	1	yes	characteristic
37	C₂×C₄×C₁₇₃	C₄⋊C₄	1	1	yes	characteristic
38	C₂×C₄×C₁₇₃	C₂×C₈	1	2	yes	—
39	C₂×C₄×C₁₇₃	M₁₆	1	2	yes	—
40	C₈×C₁₇₃	—	4	2	no	—
41	C₂²×C₁₇₃	—	2	1	no	—
42	C₂²×C₁₇₃	—	2	1	no	—
43	C₂²×C₁₇₃	#95	1	1	yes	characteristic
44	C₂²×C₁₇₃	C₄⋊C₈	1	2	yes	—
45	C₄×C₁₇₃	—	2	1	no	—
46	C₄×C₁₇₃	—	2	1	no	—
47	C₄×C₁₇₃	C₄×C₈	1	1	yes	characteristic
48	C₄×C₁₇₃	#97	1	1	yes	characteristic
49	C₄×C₁₇₃	C₂²⋊C₈	1	2	yes	—
50	C₄×C₁₇₃	#104	1	2	yes	—
51	C₄×C₁₇₃	#108	1	2	yes	—
52	C₂×C₁₇₃	—	2	2	no	—
53	C₂×C₁₇₃	#336	1	1	yes	characteristic
54	C₂×C₁₇₃	#339	1	1	yes	characteristic
55	C₂×C₁₇₃	#341	1	1	yes	characteristic
56	C₁₇₃	#1278	1	1	yes	characteristic, complemented, Sylow
57	#1278	—	173	1	no	maximal, Sylow
58	#431	D₁₇₃	1	1	yes	characteristic
59	C₄×M₁₆	—	173	1	no	—
60	C₂×C₄×C₈	—	173	1	no	—
61	C₂×M₁₆	—	173	2	no	—
62	C₂²×C₈	—	346	1	no	—
63	C₂×C₄²	C₂×D₁₇₃	1	1	yes	characteristic
64	C₄⋊C₈	Dic₁₇₃	1	2	yes	—
65	C₂²⋊C₈	—	2	1	no	—
66	#97	—	346	1	no	—
67	C₄×C₈	—	2	1	no	—
68	C₄×C₈	—	346	1	no	—
69	C₄×C₈	—	346	1	no	—
70	C₄×C₈	—	173	2	no	—
71	C₂²×C₄	C₄×D₁₇₃	1	1	yes	characteristic
72	C₂²×C₄	C₂²⋊D₁₇₃	1	1	yes	characteristic
73	C₂²×C₄	Dic₃₄₆	1	1	yes	characteristic
74	M₁₆	—	346	2	no	—
75	M₁₆	—	173	4	no	—
76	C₂×C₈	—	2	2	no	—
77	C₂×C₈	—	346	4	no	—
78	C₂×C₈	—	346	2	no	—
79	C₂×C₈	—	346	2	no	—
80	C₄²	C₄×D₁₇₃	1	1	yes	characteristic
81	C₄²	C₂×Dic₁₇₃	1	1	yes	characteristic
82	C₄²	C₂²⋊D₁₇₃	1	1	yes	characteristic
83	C₄²	D₆₉₂	1	1	yes	characteristic
84	C₂³	#14583805	1	1	yes	characteristic
85	C₂×C₄	—	2	2	no	—
86	C₂×C₄	—	2	2	no	—
87	C₂×C₄	C₄×Dic₁₇₃	1	1	yes	characteristic
88	C₂×C₄	#14583803	1	1	yes	characteristic
89	C₂×C₄	#14583803	1	1	yes	characteristic
90	C₂×C₄	C₂²⋊Dic₁₇₃	1	1	yes	characteristic
91	C₂×C₄	#14583805	1	1	yes	characteristic
92	C₂×C₄	C₄⋊Dic₁₇₃	1	1	yes	characteristic
93	C₂×C₄	C₈×D₁₇₃	1	2	yes	—
94	C₂×C₄	#14583809	1	2	yes	—
95	C₈	—	4	2	no	—
96	C₈	—	346	4	no	—
97	C₈	—	346	4	no	—
98	C₂²	—	2	1	no	—
99	C₂²	—	2	1	no	—
100	C₂²	#19149586	1	1	yes	characteristic
101	C₂²	#19149608	1	2	yes	—
102	C₄	—	2	1	no	—
103	C₄	—	2	1	no	—
104	C₄	C₈×Dic₁₇₃	1	1	yes	characteristic
105	C₄	#19149589	1	1	yes	characteristic
106	C₄	#19149591	1	2	yes	—
107	C₄	#19149606	1	2	yes	—
108	C₄	#19149614	1	2	yes	—
109	C₂	—	2	2	no	—
110	C₂	#26011245	1	1	yes	characteristic
111	C₂	#26011250	1	1	yes	characteristic
112	C₂	#26011255	1	1	yes	characteristic

subgroup

G/H

class size

multiplicity

normal?

properties

C₄×#14583810

C₂

yes

maximal, characteristic

C₂×C₄×#13063436

C₂

yes

maximal, characteristic

#431×C₁₇₃

C₂

yes

maximal, characteristic

C₂×#14583810

C₄

yes

—

—

—

—

—

C₄

yes

—

C₂×C₄²×C₁₇₃

C₂²

yes

characteristic

C₄⋊C₈×C₁₇₃

C₄

yes

—

C₂²⋊C₈×C₁₇₃

—

C₄×C₈×C₁₇₃

—

#14583810

—

#14583810

C₈

yes

—

—

—

C₂²×C₄×C₁₇₃

C₂×C₄

yes

characteristic

C₂²×C₄×C₁₇₃

D₄

yes

characteristic

C₂²×C₄×C₁₇₃

Q₈

yes

characteristic

C₂×C₈×C₁₇₃

—

C₄²×C₁₇₃

C₂×C₄

yes

characteristic

C₄²×C₁₇₃

C₂×C₄

yes

characteristic

C₄²×C₁₇₃

D₄

yes

characteristic

C₄²×C₁₇₃

D₄

yes

characteristic

#13063436

—

#13063436

—

C₂³×C₁₇₃

C₄⋊C₄

yes

characteristic

C₂×C₄×C₁₇₃

—

C₂×C₄×C₁₇₃

—

C₂×C₄×C₁₇₃

C₄²

yes

characteristic

C₂×C₄×C₁₇₃

C₂²⋊C₄

yes

characteristic

C₂×C₄×C₁₇₃

C₂²⋊C₄

yes

characteristic

C₂×C₄×C₁₇₃

C₂²⋊C₄

yes

characteristic

C₂×C₄×C₁₇₃

C₄⋊C₄

yes

characteristic

C₂×C₄×C₁₇₃

C₄⋊C₄

yes

characteristic

C₂×C₄×C₁₇₃

C₂×C₈

yes

—

C₂×C₄×C₁₇₃

M₁₆

yes

—

C₈×C₁₇₃

—

C₂²×C₁₇₃

—

C₂²×C₁₇₃

—

C₂²×C₁₇₃

#95

yes

characteristic

C₂²×C₁₇₃

C₄⋊C₈

yes

—

C₄×C₁₇₃

—

C₄×C₁₇₃

—

C₄×C₁₇₃

C₄×C₈

yes

characteristic

C₄×C₁₇₃

#97

yes

characteristic

C₄×C₁₇₃

C₂²⋊C₈

yes

—

C₄×C₁₇₃

#104

yes

—

C₄×C₁₇₃

#108

yes

—

C₂×C₁₇₃

—

C₂×C₁₇₃

yes

C₂×C₁₇₃

yes

C₂×C₁₇₃

yes

C₁₇₃

yes

characteristic, complemented, Sylow

#1278

—

173

maximal, Sylow

#431

D₁₇₃

yes

characteristic

C₄×M₁₆

—

173

—

C₂×C₄×C₈

—

173

—

C₂×M₁₆

—

173

—

C₂²×C₈

—

346

—

C₂×C₄²

C₂×D₁₇₃

yes

characteristic

C₄⋊C₈

Dic₁₇₃

yes

—

C₂²⋊C₈

—

#97

—

346

—

C₄×C₈

—

C₄×C₈

—

346

—

C₄×C₈

—

346

—

C₄×C₈

—

173

—

C₂²×C₄

C₄×D₁₇₃

yes

characteristic

C₂²×C₄

C₂²⋊D₁₇₃

yes

characteristic

C₂²×C₄

Dic₃₄₆

yes

characteristic

M₁₆

—

346

—

M₁₆

—

173

—

C₂×C₈

—

C₂×C₈

—

346

—

C₂×C₈

—

346

—

C₂×C₈

—

346

—

C₄²

C₄×D₁₇₃

yes

characteristic

C₄²

C₂×Dic₁₇₃

yes

characteristic

C₄²

C₂²⋊D₁₇₃

yes

characteristic

C₄²

D₆₉₂

yes

characteristic

C₂³

#14583805

yes

characteristic

C₂×C₄

—

C₂×C₄

—

C₂×C₄

C₄×Dic₁₇₃

yes

C₂×C₄

yes

C₂×C₄

yes

C₂×C₄

C₂²⋊Dic₁₇₃

yes

characteristic

C₂×C₄

#14583805

yes

characteristic

C₂×C₄

C₄⋊Dic₁₇₃

yes

characteristic

C₂×C₄

C₈×D₁₇₃

yes

—

C₂×C₄

#14583809

yes

—

C₈

—

C₈

—

346

—

C₈

—

346

—

C₂²

—

C₂²

—

100

C₂²

#19149586

yes

characteristic

101

C₂²

#19149608

yes

—

102

C₄

—

103

C₄

—

104

C₄

C₈×Dic₁₇₃

yes

characteristic

105

C₄

#19149589

yes

characteristic

106

C₄

#19149591

yes

—

107

C₄

#19149606

yes

—

108

C₄

#19149614

yes

—

109

C₂

—

110

C₂

#26011245

yes

characteristic

111

C₂

#26011250

yes

characteristic

112

C₂

#26011255

yes

characteristic

	subgroup	G/H
center	C₄²	C₂²⋊D₁₇₃
commutator	C₄×C₁₇₃	C₄×C₈
Frattini	C₂×C₄²	C₂×D₁₇₃
Fitting	#431×C₁₇₃	C₂
socle	C₂²×C₁₇₃	#95
prime radical	C₂³×C₁₇₃	C₄⋊C₄
cosocle	C₂×C₄²×C₁₇₃	C₂²
hypercenter	#431	D₁₇₃
hypocenter	C₁₇₃	#1278
LCS	G ▷ C₄×C₁₇₃ ▷ C₂×C₁₇₃ ▷ C₁₇₃
UCS	C₁ ◁ C₄² ◁ C₂×C₄² ◁ #431

subgroup

G/H

center

C₄²

C₂²⋊D₁₇₃

commutator

C₄×C₁₇₃

C₄×C₈

Frattini

C₂×C₄²

C₂×D₁₇₃

Fitting

#431×C₁₇₃

C₂

socle

C₂²×C₁₇₃

#95

prime radical

C₂³×C₁₇₃

C₄⋊C₄

cosocle

C₂×C₄²×C₁₇₃

C₂²

D₁₇₃

C₁₇₃

G ▷ C₄×C₁₇₃ ▷ C₂×C₁₇₃ ▷ C₁₇₃

UCS

C₁ ◁ C₄² ◁ C₂×C₄² ◁ #431

N	H ≅ G/N	type
15. #14583810	C₈	semidirect
56. C₁₇₃	#1278	semidirect

H ≅ G/N

type

15. #14583810

C₈

semidirect

56. C₁₇₃

#1278

semidirect

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

subgroup	index
C₄×#14583810	2
C₂×C₄×#13063436	2
#431×C₁₇₃	2
#1278	173

subgroup

index

C₄×#14583810

C₂×C₄×#13063436

#431×C₁₇₃

#1278

173

Subgroups — #36351399

Conjugacy classes of subgroups

Complements

Maximal subgroups