Group #36355640 — subgroups

Subgroups — #36355640

Order 22144.

170 conjugacy classes of subgroups; only the 47 normal subgroups are shown.

normal subgroups only

This group has 170 conjugacy classes of subgroups — too many to draw. Only its 47 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	#26011929	C₂	1	2	yes	maximal, complemented
2	#26011868	C₂	1	1	yes	maximal, characteristic, complemented
3	#26011609	C₂	1	1	yes	maximal, characteristic, complemented
4	#26011305	C₂	1	2	yes	maximal, complemented
5	#364×C₁₇₃	C₂	1	1	yes	maximal, characteristic, complemented
6	#19149701	C₂²	1	1	yes	characteristic
7	C₂×D₁₃₈₄	—	2	1	no	—
8	C₂×D₁₃₈₄	C₂²	1	2	yes	—
9	#19149685	—	2	1	no	—
10	#19149676	—	2	1	no	—
11	#19149624	—	2	2	no	—
12	D₂₇₆₈	—	2	2	no	—
13	#19149615	C₂²	1	1	yes	characteristic
14	#19149607	—	2	1	no	—
15	M₃₂×C₁₇₃	C₂²	1	2	yes	—
16	C₄×C₈×C₁₇₃	C₂²	1	1	yes	characteristic
17	#14583826	—	2	1	no	—
18	C₂×D₆₉₂	—	2	2	no	—
19	C₂×D₆₉₂	—	4	2	no	—
20	Dic₆₉₂	D₄	1	1	yes	characteristic, complemented
21	#14583815	—	2	1	no	—
22	D₁₃₈₄	—	2	2	no	—
23	D₁₃₈₄	—	2	2	no	—
24	D₁₃₈₄	—	4	1	no	—
25	D₁₃₈₄	—	4	1	no	—
26	D₁₃₈₄	D₄	1	1	yes	characteristic, complemented
27	#14583810	—	2	1	no	—
28	#14583809	—	4	1	no	—
29	C₈×D₁₇₃	—	4	1	no	—
30	C₂×C₈×C₁₇₃	D₄	1	1	yes	characteristic
31	C₂×C₈×C₁₇₃	C₂³	1	1	yes	characteristic
32	C₄²×C₁₇₃	D₄	1	1	yes	characteristic
33	C₁₆×C₁₇₃	—	2	2	no	—
34	C₂²×D₁₇₃	—	4	2	no	—
35	Dic₃₄₆	—	2	1	no	—
36	D₆₉₂	—	2	1	no	—
37	D₆₉₂	—	4	2	no	—
38	D₆₉₂	—	4	2	no	—
39	D₆₉₂	—	8	4	no	—
40	C₂²⋊D₁₇₃	—	4	1	no	—
41	C₄×D₁₇₃	—	4	1	no	—
42	#13063436	—	4	1	no	—
43	C₂×C₄×C₁₇₃	—	2	1	no	—
44	C₂×C₄×C₁₇₃	C₂×D₄	1	1	yes	characteristic
45	C₈×C₁₇₃	D₈	1	1	yes	characteristic
46	C₈×C₁₇₃	D₈	1	1	yes	characteristic
47	C₈×C₁₇₃	C₂×D₄	1	1	yes	characteristic
48	C₈×C₁₇₃	#41	1	1	yes	characteristic
49	C₂²×C₁₇₃	AGL(1,ℤ/8)	1	1	yes	characteristic
50	C₂×D₁₇₃	—	4	1	no	—
51	C₂×D₁₇₃	—	8	4	no	—
52	C₂×D₁₇₃	—	8	2	no	—
53	C₄×C₁₇₃	—	2	2	no	—
54	C₄×C₁₇₃	#121	1	1	yes	characteristic
55	C₄×C₁₇₃	C₂×D₈	1	1	yes	characteristic
56	Dic₁₇₃	—	4	1	no	—
57	C₂×C₁₇₃	—	2	1	no	—
58	C₂×C₁₇₃	#459	1	1	yes	characteristic
59	D₁₇₃	—	8	1	no	—
60	D₁₇₃	—	16	2	no	—
61	C₁₇₃	#2213	1	1	yes	characteristic, complemented, Sylow
62	#2213	—	173	1	no	maximal, Sylow
63	#509	—	173	2	no	—
64	#493	—	173	1	no	—
65	#443	—	173	1	no	—
66	#364	D₁₇₃	1	1	yes	characteristic, complemented
67	#361	—	173	2	no	—
68	#135	—	173	1	no	—
69	C₂×D₈	—	346	1	no	—
70	C₂×D₈	—	173	2	no	—
71	#131	—	346	1	no	—
72	#127	—	346	1	no	—
73	SD₃₂	—	346	2	no	—
74	D₁₆	—	346	2	no	—
75	M₃₂	C₂×D₁₇₃	1	2	yes	—
76	#108	—	173	1	no	—
77	#104	—	346	1	no	—
78	C₄×C₈	C₂×D₁₇₃	1	1	yes	characteristic
79	#41	—	346	1	no	—
80	C₂×D₄	—	346	2	no	—
81	C₂×D₄	—	692	2	no	—
82	Q₁₆	—	173	1	no	—
83	SD₁₆	—	346	1	no	—
84	D₈	—	346	2	no	—
85	D₈	—	346	2	no	—
86	D₈	—	173	1	no	—
87	D₈	—	692	1	no	—
88	D₈	—	692	1	no	—
89	M₁₆	—	346	1	no	—
90	M₁₆	—	692	1	no	—
91	C₂×C₈	—	692	1	no	—
92	C₂×C₈	D₆₉₂	1	1	yes	characteristic
93	C₂×C₈	C₂²×D₁₇₃	1	1	yes	characteristic
94	C₄²	D₆₉₂	1	1	yes	characteristic
95	C₁₆	—	2	2	no	—
96	C₂³	—	692	2	no	—
97	Q₈	—	346	1	no	—
98	D₄	—	346	1	no	—
99	D₄	—	1384	4	no	—
100	D₄	—	692	1	no	—
101	D₄	—	692	2	no	—
102	D₄	—	692	2	no	—
103	C₂×C₄	—	2	1	no	—
104	C₂×C₄	—	692	1	no	—
105	C₂×C₄	C₂×D₆₉₂	1	1	yes	characteristic
106	C₈	—	692	1	no	—
107	C₈	D₁₃₈₄	1	1	yes	characteristic
108	C₈	D₁₃₈₄	1	1	yes	characteristic
109	C₈	D₄×D₁₇₃	1	1	yes	characteristic
110	C₈	#14583827	1	1	yes	characteristic
111	C₂²	—	1384	4	no	—
112	C₂²	—	1384	2	no	—
113	C₂²	—	692	1	no	—
114	C₂²	#19149707	1	1	yes	characteristic
115	C₄	—	2	2	no	—
116	C₄	—	692	1	no	—
117	C₄	#19149654	1	1	yes	characteristic
118	C₄	C₂×D₁₃₈₄	1	1	yes	characteristic
119	C₂	—	2	1	no	—
120	C₂	—	1384	1	no	—
121	C₂	—	2768	2	no	—
122	C₂	#26011687	1	1	yes	characteristic

subgroup

G/H

class size

multiplicity

normal?

properties

#26011929

C₂

yes

maximal, complemented

#26011868

C₂

yes

maximal, characteristic, complemented

#26011609

C₂

yes

maximal, characteristic, complemented

#26011305

C₂

yes

maximal, complemented

#364×C₁₇₃

C₂

yes

maximal, characteristic, complemented

#19149701

C₂²

yes

characteristic

C₂×D₁₃₈₄

—

C₂×D₁₃₈₄

C₂²

yes

—

#19149685

—

#19149676

—

#19149624

—

D₂₇₆₈

—

#19149615

C₂²

yes

characteristic

#19149607

—

M₃₂×C₁₇₃

C₂²

yes

—

C₄×C₈×C₁₇₃

C₂²

yes

characteristic

#14583826

—

C₂×D₆₉₂

—

C₂×D₆₉₂

—

Dic₆₉₂

D₄

yes

characteristic, complemented

#14583815

—

D₁₃₈₄

—

D₁₃₈₄

—

D₁₃₈₄

—

D₁₃₈₄

—

D₁₃₈₄

D₄

yes

characteristic, complemented

#14583810

—

#14583809

—

C₈×D₁₇₃

—

C₂×C₈×C₁₇₃

D₄

yes

characteristic

C₂×C₈×C₁₇₃

C₂³

yes

characteristic

C₄²×C₁₇₃

D₄

yes

characteristic

C₁₆×C₁₇₃

—

C₂²×D₁₇₃

—

Dic₃₄₆

—

D₆₉₂

—

D₆₉₂

—

D₆₉₂

—

D₆₉₂

—

C₂²⋊D₁₇₃

—

C₄×D₁₇₃

—

#13063436

—

C₂×C₄×C₁₇₃

—

C₂×C₄×C₁₇₃

C₂×D₄

yes

characteristic

C₈×C₁₇₃

D₈

yes

characteristic

C₈×C₁₇₃

D₈

yes

characteristic

C₈×C₁₇₃

C₂×D₄

yes

C₈×C₁₇₃

yes

C₂²×C₁₇₃

yes

C₂×D₁₇₃

—

C₂×D₁₇₃

—

C₂×D₁₇₃

—

C₄×C₁₇₃

—

C₄×C₁₇₃

#121

yes

characteristic

C₄×C₁₇₃

C₂×D₈

yes

characteristic

Dic₁₇₃

—

C₂×C₁₇₃

—

C₂×C₁₇₃

#459

yes

characteristic

D₁₇₃

—

D₁₇₃

—

C₁₇₃

#2213

yes

characteristic, complemented, Sylow

#2213

—

173

maximal, Sylow

#509

—

173

—

#493

—

173

—

#443

—

173

—

#364

D₁₇₃

yes

characteristic, complemented

#361

—

173

—

#135

—

173

—

C₂×D₈

—

346

—

C₂×D₈

—

173

—

#131

—

346

—

#127

—

346

—

SD₃₂

—

346

—

D₁₆

—

346

—

M₃₂

C₂×D₁₇₃

yes

—

#108

—

173

—

#104

—

346

—

C₄×C₈

C₂×D₁₇₃

yes

characteristic

#41

—

346

—

C₂×D₄

—

346

—

C₂×D₄

—

692

—

Q₁₆

—

173

—

SD₁₆

—

346

—

D₈

—

346

—

D₈

—

346

—

D₈

—

173

—

D₈

—

692

—

D₈

—

692

—

M₁₆

—

346

—

M₁₆

—

692

—

C₂×C₈

—

692

—

C₂×C₈

D₆₉₂

yes

characteristic

C₂×C₈

C₂²×D₁₇₃

yes

characteristic

C₄²

D₆₉₂

yes

characteristic

C₁₆

—

C₂³

—

692

—

Q₈

—

346

—

D₄

—

346

—

D₄

—

1384

—

100

D₄

—

692

—

101

D₄

—

692

—

102

D₄

—

692

—

103

C₂×C₄

—

104

C₂×C₄

—

692

—

105

C₂×C₄

C₂×D₆₉₂

yes

characteristic

106

C₈

—

692

—

107

C₈

D₁₃₈₄

yes

characteristic

108

C₈

D₁₃₈₄

yes

characteristic

109

C₈

D₄×D₁₇₃

yes

characteristic

110

C₈

#14583827

yes

characteristic

111

C₂²

—

1384

—

112

C₂²

—

1384

—

113

C₂²

—

692

—

114

C₂²

#19149707

yes

characteristic

115

C₄

—

116

C₄

—

692

—

117

C₄

#19149654

yes

characteristic

118

C₄

C₂×D₁₃₈₄

yes

characteristic

119

C₂

—

120

C₂

—

1384

—

121

C₂

—

2768

—

122

C₂

#26011687

yes

characteristic

	subgroup	G/H
center	C₂	#26011687
commutator, cosocle	C₂×C₈×C₁₇₃	C₂³
Frattini	C₂×C₈	C₂²×D₁₇₃
Fitting	#364×C₁₇₃	C₂
socle	C₂×C₁₇₃	#459
hypercenter	#364	D₁₇₃
hypocenter	C₁₇₃	#2213
LCS	G ▷ C₂×C₈×C₁₇₃ ▷ C₄×C₁₇₃ ▷ C₂×C₁₇₃ ▷ C₁₇₃
UCS	C₁ ◁ C₂ ◁ C₂×C₄ ◁ C₄×C₈ ◁ #364

subgroup

G/H

center

C₂

#26011687

commutator, cosocle

C₂×C₈×C₁₇₃

C₂³

Frattini

C₂×C₈

C₂²×D₁₇₃

Fitting

#364×C₁₇₃

C₂

C₂×C₁₇₃

D₁₇₃

C₁₇₃

G ▷ C₂×C₈×C₁₇₃ ▷ C₄×C₁₇₃ ▷ C₂×C₁₇₃ ▷ C₁₇₃

UCS

C₁ ◁ C₂ ◁ C₂×C₄ ◁ C₄×C₈ ◁ #364

N	H ≅ G/N	type
1. #26011929	C₂	semidirect
2. #26011868	C₂	semidirect
3. #26011609	C₂	semidirect
4. #26011305	C₂	semidirect
5. #364×C₁₇₃	C₂	semidirect
20. Dic₆₉₂	D₄	semidirect
26. D₁₃₈₄	D₄	semidirect
61. C₁₇₃	#2213	semidirect
66. #364	D₁₇₃	semidirect

H ≅ G/N

type

1. #26011929

C₂

semidirect

2. #26011868

C₂

semidirect

3. #26011609

C₂

semidirect

4. #26011305

C₂

semidirect

5. #364×C₁₇₃

C₂

semidirect

20. Dic₆₉₂

D₄

semidirect

26. D₁₃₈₄

D₄

semidirect

61. C₁₇₃

#2213

semidirect

66. #364

D₁₇₃

semidirect

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

subgroup	index
#26011929	2
#26011868	2
#26011609	2
#26011305	2
#364×C₁₇₃	2
#2213	173

subgroup

index

#364×C₁₇₃

173

Subgroups — #36355640

Conjugacy classes of subgroups

Complements

Maximal subgroups