Group #36352620 — subgroups

Subgroups — #36352620

Order 22144.

240 conjugacy classes of subgroups; only the 53 normal subgroups are shown.

normal subgroups only

This group has 240 conjugacy classes of subgroups — too many to draw. Only its 53 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	#26012227	C₂	1	1	yes	maximal, characteristic, complemented
2	#26011688	C₂	1	2	yes	maximal, complemented
3	#26011235	C₂	1	1	yes	maximal, characteristic, complemented
4	#26011223	C₂	1	2	yes	maximal, complemented
5	#493×C₁₇₃	C₂	1	1	yes	maximal, characteristic
6	C₂×#14583814	—	2	2	no	—
7	#19149666	—	2	2	no	—
8	#19149660	—	2	2	no	—
9	C₄×Dic₃₄₆	C₂²	1	2	yes	—
10	D₄×Dic₁₇₃	—	2	2	no	—
11	C₄⋊#13063436	C₂²	1	2	yes	—
12	#19149600	—	2	2	no	—
13	C₂×D₈×C₁₇₃	—	2	2	no	—
14	#127×C₁₇₃	C₂²	1	1	yes	characteristic
15	#127×C₁₇₃	C₂²	1	1	yes	characteristic
16	C₄×C₈×C₁₇₃	C₂²	1	1	yes	characteristic
17	C₂×Dic₃₄₆	D₄	1	2	yes	—
18	C₂²×Dic₁₇₃	—	2	2	no	—
19	#14583814	—	4	4	no	—
20	C₂×#13063436	—	2	2	no	—
21	C₄⋊Dic₁₇₃	D₄	1	2	yes	—
22	#14583805	—	2	2	no	—
23	C₂²⋊Dic₁₇₃	—	2	4	no	—
24	C₂²⋊Dic₁₇₃	—	2	2	no	—
25	C₄×Dic₁₇₃	—	2	2	no	—
26	C₂×D₄×C₁₇₃	—	2	2	no	—
27	C₂×D₄×C₁₇₃	—	2	1	no	—
28	C₂×D₄×C₁₇₃	—	2	2	no	—
29	C₂×D₄×C₁₇₃	D₄	1	2	yes	—
30	D₈×C₁₇₃	—	4	4	no	—
31	C₂×C₈×C₁₇₃	—	2	1	no	—
32	C₄²×C₁₇₃	C₂³	1	1	yes	characteristic
33	Dic₃₄₆	—	2	2	no	—
34	Dic₃₄₆	—	2	2	no	—
35	C₂×Dic₁₇₃	—	2	2	no	—
36	C₂×Dic₁₇₃	—	2	2	no	—
37	C₂×Dic₁₇₃	—	8	2	no	—
38	#13063436	—	4	2	no	—
39	C₂³×C₁₇₃	—	2	2	no	—
40	C₂³×C₁₇₃	—	4	1	no	—
41	D₄×C₁₇₃	—	4	2	no	—
42	D₄×C₁₇₃	—	4	2	no	—
43	D₄×C₁₇₃	—	4	2	no	—
44	D₄×C₁₇₃	—	4	2	no	—
45	D₄×C₁₇₃	—	4	2	no	—
46	D₄×C₁₇₃	—	4	2	no	—
47	C₂×C₄×C₁₇₃	C₂×D₄	1	2	yes	—
48	C₂×C₄×C₁₇₃	C₂×D₄	1	1	yes	characteristic
49	C₈×C₁₇₃	—	2	2	no	—
50	C₂²×C₁₇₃	—	4	4	no	—
51	C₂²×C₁₇₃	—	4	2	no	—
52	C₂²×C₁₇₃	—	8	2	no	—
53	C₂²×C₁₇₃	—	8	1	no	—
54	C₂²×C₁₇₃	#120	1	1	yes	characteristic
55	C₄×C₁₇₃	—	2	1	no	—
56	C₄×C₁₇₃	#135	1	2	yes	—
57	C₄×C₁₇₃	AGL(1,ℤ/8)	1	2	yes	—
58	Dic₁₇₃	—	4	2	no	—
59	Dic₁₇₃	—	4	2	no	—
60	C₂×C₁₇₃	—	8	2	no	—
61	C₂×C₁₇₃	—	16	1	no	—
62	C₂×C₁₇₃	#449	1	2	yes	—
63	C₂×C₁₇₃	#453	1	1	yes	characteristic
64	C₁₇₃	#1620	1	1	yes	characteristic, complemented, Sylow
65	#1620	—	173	1	no	maximal, Sylow
66	#559	—	173	1	no	—
67	#493	D₁₇₃	1	1	yes	characteristic
68	#460	—	173	2	no	—
69	#331	—	173	1	no	—
70	Q₈⋊C₈	—	173	2	no	—
71	C₂×SD₁₆	—	346	2	no	—
72	C₂×D₈	—	2	2	no	—
73	#127	C₂×D₁₇₃	1	1	yes	characteristic
74	#127	C₂×D₁₇₃	1	1	yes	characteristic
75	#124	—	346	2	no	—
76	#123	—	346	2	no	—
77	C₄×Q₈	—	173	2	no	—
78	C₄×D₄	—	346	2	no	—
79	C₄⋊C₈	—	173	2	no	—
80	#102	—	346	2	no	—
81	C₄×C₈	C₂×D₁₇₃	1	1	yes	characteristic
82	C₂×Q₈	—	173	2	no	—
83	C₂×D₄	—	2	2	no	—
84	C₂×D₄	—	2	2	no	—
85	C₂×D₄	—	2	1	no	—
86	C₂×D₄	C₂²⋊D₁₇₃	1	2	yes	—
87	C₂²×C₄	—	346	2	no	—
88	SD₁₆	—	692	4	no	—
89	D₈	—	4	4	no	—
90	C₂×C₈	—	2	1	no	—
91	C₂×C₈	—	346	2	no	—
92	C₄⋊C₄	—	346	2	no	—
93	C₄⋊C₄	—	173	2	no	—
94	C₂²⋊C₄	—	346	4	no	—
95	C₂²⋊C₄	—	346	2	no	—
96	C₄²	—	346	2	no	—
97	C₄²	C₂²×D₁₇₃	1	1	yes	characteristic
98	C₂³	—	2	2	no	—
99	C₂³	—	4	1	no	—
100	Q₈	—	346	2	no	—
101	Q₈	—	346	2	no	—
102	D₄	—	4	2	no	—
103	D₄	—	4	2	no	—
104	D₄	—	4	2	no	—
105	D₄	—	4	2	no	—
106	D₄	—	4	2	no	—
107	D₄	—	4	2	no	—
108	C₂×C₄	—	346	2	no	—
109	C₂×C₄	—	346	2	no	—
110	C₂×C₄	—	1384	2	no	—
111	C₂×C₄	D₄×D₁₇₃	1	2	yes	—
112	C₂×C₄	C₂×C₂²⋊D₁₇₃	1	1	yes	characteristic
113	C₈	—	2	2	no	—
114	C₈	—	692	2	no	—
115	C₂²	—	4	2	no	—
116	C₂²	—	4	4	no	—
117	C₂²	—	8	1	no	—
118	C₂²	—	8	2	no	—
119	C₂²	#19149647	1	1	yes	characteristic
120	C₄	—	2	1	no	—
121	C₄	—	692	2	no	—
122	C₄	—	692	2	no	—
123	C₄	#19149698	1	2	yes	—
124	C₄	#19149703	1	2	yes	—
125	C₂	—	8	2	no	—
126	C₂	—	16	1	no	—
127	C₂	#26011633	1	2	yes	—
128	C₂	#26011658	1	1	yes	characteristic

subgroup

G/H

class size

multiplicity

normal?

properties

#26012227

C₂

yes

maximal, characteristic, complemented

#26011688

C₂

yes

maximal, complemented

#26011235

C₂

yes

maximal, characteristic, complemented

#26011223

C₂

yes

maximal, complemented

#493×C₁₇₃

C₂

yes

maximal, characteristic

C₂×#14583814

—

#19149666

—

#19149660

—

C₄×Dic₃₄₆

C₂²

yes

—

D₄×Dic₁₇₃

—

C₄⋊#13063436

C₂²

yes

—

#19149600

—

C₂×D₈×C₁₇₃

—

#127×C₁₇₃

C₂²

yes

characteristic

#127×C₁₇₃

C₂²

yes

characteristic

C₄×C₈×C₁₇₃

C₂²

yes

characteristic

C₂×Dic₃₄₆

D₄

yes

—

C₂²×Dic₁₇₃

—

#14583814

—

C₂×#13063436

—

C₄⋊Dic₁₇₃

D₄

yes

—

#14583805

—

C₂²⋊Dic₁₇₃

—

C₂²⋊Dic₁₇₃

—

C₄×Dic₁₇₃

—

C₂×D₄×C₁₇₃

—

C₂×D₄×C₁₇₃

—

C₂×D₄×C₁₇₃

—

C₂×D₄×C₁₇₃

D₄

yes

—

D₈×C₁₇₃

—

C₂×C₈×C₁₇₃

—

C₄²×C₁₇₃

C₂³

yes

characteristic

Dic₃₄₆

—

Dic₃₄₆

—

C₂×Dic₁₇₃

—

C₂×Dic₁₇₃

—

C₂×Dic₁₇₃

—

#13063436

—

C₂³×C₁₇₃

—

C₂³×C₁₇₃

—

D₄×C₁₇₃

—

D₄×C₁₇₃

—

D₄×C₁₇₃

—

D₄×C₁₇₃

—

D₄×C₁₇₃

—

D₄×C₁₇₃

—

C₂×C₄×C₁₇₃

C₂×D₄

yes

—

C₂×C₄×C₁₇₃

C₂×D₄

yes

characteristic

C₈×C₁₇₃

—

C₂²×C₁₇₃

—

C₂²×C₁₇₃

—

C₂²×C₁₇₃

—

C₂²×C₁₇₃

—

C₂²×C₁₇₃

#120

yes

characteristic

C₄×C₁₇₃

—

C₄×C₁₇₃

#135

yes

—

C₄×C₁₇₃

AGL(1,ℤ/8)

yes

—

Dic₁₇₃

—

Dic₁₇₃

—

C₂×C₁₇₃

—

C₂×C₁₇₃

—

C₂×C₁₇₃

#449

yes

—

C₂×C₁₇₃

#453

yes

characteristic

C₁₇₃

#1620

yes

characteristic, complemented, Sylow

#1620

—

173

maximal, Sylow

#559

—

173

—

#493

D₁₇₃

yes

characteristic

#460

—

173

—

#331

—

173

—

Q₈⋊C₈

—

173

—

C₂×SD₁₆

—

346

—

C₂×D₈

—

#127

C₂×D₁₇₃

yes

characteristic

#127

C₂×D₁₇₃

yes

characteristic

#124

—

346

—

#123

—

346

—

C₄×Q₈

—

173

—

C₄×D₄

—

346

—

C₄⋊C₈

—

173

—

#102

—

346

—

C₄×C₈

C₂×D₁₇₃

yes

characteristic

C₂×Q₈

—

173

—

C₂×D₄

—

C₂×D₄

—

C₂×D₄

—

C₂×D₄

C₂²⋊D₁₇₃

yes

—

C₂²×C₄

—

346

—

SD₁₆

—

692

—

D₈

—

C₂×C₈

—

C₂×C₈

—

346

—

C₄⋊C₄

—

346

—

C₄⋊C₄

—

173

—

C₂²⋊C₄

—

346

—

C₂²⋊C₄

—

346

—

C₄²

—

346

—

C₄²

C₂²×D₁₇₃

yes

characteristic

C₂³

—

C₂³

—

100

Q₈

—

346

—

101

Q₈

—

346

—

102

D₄

—

103

D₄

—

104

D₄

—

105

D₄

—

106

D₄

—

107

D₄

—

108

C₂×C₄

—

346

—

109

C₂×C₄

—

346

—

110

C₂×C₄

—

1384

—

111

C₂×C₄

D₄×D₁₇₃

yes

—

112

C₂×C₄

C₂×C₂²⋊D₁₇₃

yes

characteristic

113

C₈

—

114

C₈

—

692

—

115

C₂²

—

116

C₂²

—

117

C₂²

—

118

C₂²

—

119

C₂²

#19149647

yes

characteristic

120

C₄

—

121

C₄

—

692

—

122

C₄

—

692

—

123

C₄

#19149698

yes

—

124

C₄

#19149703

yes

—

125

C₂

—

126

C₂

—

127

C₂

#26011633

yes

—

128

C₂

#26011658

yes

characteristic

	subgroup	G/H
center	C₂²	#19149647
commutator, cosocle	C₄²×C₁₇₃	C₂³
Frattini	C₄²	C₂²×D₁₇₃
Fitting, prime radical	#493×C₁₇₃	C₂
socle	C₂²×C₁₇₃	#120
hypercenter	#493	D₁₇₃
hypocenter	C₁₇₃	#1620
LCS	G ▷ C₄²×C₁₇₃ ▷ C₂²×C₁₇₃ ▷ C₁₇₃
UCS	C₁ ◁ C₂² ◁ C₄² ◁ #493

subgroup

G/H

center

C₂²

#19149647

commutator, cosocle

C₄²×C₁₇₃

C₂³

Frattini

C₄²

C₂²×D₁₇₃

Fitting, prime radical

#493×C₁₇₃

C₂

C₂²×C₁₇₃

D₁₇₃

C₁₇₃

G ▷ C₄²×C₁₇₃ ▷ C₂²×C₁₇₃ ▷ C₁₇₃

UCS

C₁ ◁ C₂² ◁ C₄² ◁ #493

N	H ≅ G/N	type
1. #26012227	C₂	semidirect
2. #26011688	C₂	semidirect
3. #26011235	C₂	semidirect
4. #26011223	C₂	semidirect
64. C₁₇₃	#1620	semidirect

H ≅ G/N

type

1. #26012227

C₂

semidirect

2. #26011688

C₂

semidirect

3. #26011235

C₂

semidirect

4. #26011223

C₂

semidirect

64. C₁₇₃

#1620

semidirect

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

subgroup	index
#26012227	2
#26011688	2
#26011235	2
#26011223	2
#493×C₁₇₃	2
#1620	173

subgroup

index

#493×C₁₇₃

173

Subgroups — #36352620

Conjugacy classes of subgroups

Complements

Maximal subgroups