Group #36352639 — subgroups

Subgroups — #36352639

Order 22144.

228 conjugacy classes of subgroups; only the 57 normal subgroups are shown.

normal subgroups only

This group has 228 conjugacy classes of subgroups — too many to draw. Only its 57 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	#26012221	C₂	1	1	yes	maximal, characteristic, complemented
2	#26011688	C₂	1	2	yes	maximal
3	#26011240	C₂	1	1	yes	maximal, characteristic, complemented
4	#26011223	C₂	1	2	yes	maximal, complemented
5	#492×C₁₇₃	C₂	1	1	yes	maximal, characteristic
6	C₂×#14583814	—	2	2	no	—
7	#19149678	—	2	2	no	—
8	#19149677	—	2	2	no	—
9	C₄×Dic₃₄₆	C₂²	1	2	yes	—
10	#19149644	—	2	2	no	—
11	C₄⋊#13063436	C₂²	1	2	yes	—
12	#19149600	—	2	2	no	—
13	C₂×SD₁₆×C₁₇₃	—	2	2	no	—
14	C₄⋊Q₈×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	—	2	2	no	—
18	C₂×Dic₃₄₆	D₄	1	2	yes	—
19	#14583814	—	4	4	no	—
20	C₂×#13063436	—	2	2	no	—
21	C₄⋊Dic₁₇₃	D₄	1	2	yes	—
22	#14583805	—	2	4	no	—
23	#14583805	—	2	2	no	—
24	C₄×Dic₁₇₃	—	2	2	no	—
25	C₄×Dic₁₇₃	—	2	2	no	—
26	C₂×Q₈×C₁₇₃	D₄	1	2	yes	—
27	C₂×D₄×C₁₇₃	—	2	1	no	—
28	C₂×D₄×C₁₇₃	—	2	2	no	—
29	SD₁₆×C₁₇₃	—	4	4	no	—
30	C₂×C₈×C₁₇₃	—	2	1	no	—
31	C₄⋊C₄×C₁₇₃	—	2	2	no	—
32	C₄²×C₁₇₃	C₂³	1	1	yes	characteristic
33	Dic₃₄₆	—	2	2	no	—
34	Dic₃₄₆	—	8	2	no	—
35	Dic₃₄₆	SD₁₆	1	4	yes	complemented
36	C₂×Dic₁₇₃	—	2	2	no	—
37	C₂×Dic₁₇₃	—	2	2	no	—
38	#13063436	—	4	2	no	—
39	C₂³×C₁₇₃	—	4	1	no	—
40	Q₈×C₁₇₃	—	4	2	no	—
41	D₄×C₁₇₃	—	4	2	no	—
42	D₄×C₁₇₃	—	4	2	no	—
43	D₄×C₁₇₃	—	4	2	no	—
44	C₂×C₄×C₁₇₃	—	2	2	no	—
45	C₂×C₄×C₁₇₃	C₂×D₄	1	2	yes	—
46	C₂×C₄×C₁₇₃	C₂×D₄	1	1	yes	characteristic
47	C₈×C₁₇₃	—	2	2	no	—
48	C₂²×C₁₇₃	—	8	2	no	—
49	C₂²×C₁₇₃	—	8	1	no	—
50	C₂²×C₁₇₃	#120	1	1	yes	characteristic
51	C₄×C₁₇₃	—	2	1	no	—
52	C₄×C₁₇₃	—	4	2	no	—
53	C₄×C₁₇₃	C₂×SD₁₆	1	2	yes	—
54	C₄×C₁₇₃	#137	1	2	yes	—
55	Dic₁₇₃	—	2	4	no	—
56	Dic₁₇₃	—	4	2	no	—
57	C₂×C₁₇₃	—	16	1	no	—
58	C₂×C₁₇₃	#448	1	2	yes	—
59	C₂×C₁₇₃	#453	1	1	yes	characteristic
60	C₁₇₃	#1623	1	1	yes	characteristic, complemented, Sylow
61	#1623	—	173	1	no	maximal, Sylow
62	Q₈²	—	173	1	no	—
63	#492	D₁₇₃	1	1	yes	characteristic
64	#460	—	173	2	no	—
65	#333	—	173	1	no	—
66	Q₈⋊C₈	—	173	2	no	—
67	C₂×SD₁₆	—	2	2	no	—
68	C₂×SD₁₆	—	346	2	no	—
69	C₄⋊Q₈	—	346	2	no	—
70	C₄⋊Q₈	—	346	2	no	—
71	C₄⋊Q₈	C₂×D₁₇₃	1	1	yes	characteristic
72	#127	C₂×D₁₇₃	1	1	yes	characteristic
73	C₄×Q₈	—	346	2	no	—
74	C₄×Q₈	—	173	2	no	—
75	C₄⋊C₈	—	173	2	no	—
76	#102	—	346	2	no	—
77	C₄×C₈	C₂×D₁₇₃	1	1	yes	characteristic
78	C₂×Q₈	—	346	2	no	—
79	C₂×Q₈	—	173	2	no	—
80	C₂×Q₈	C₂²⋊D₁₇₃	1	2	yes	—
81	C₂×D₄	—	2	2	no	—
82	C₂×D₄	—	2	1	no	—
83	SD₁₆	—	4	4	no	—
84	SD₁₆	—	692	4	no	—
85	C₂×C₈	—	2	1	no	—
86	C₂×C₈	—	346	2	no	—
87	C₄⋊C₄	—	2	2	no	—
88	C₄⋊C₄	—	346	4	no	—
89	C₄⋊C₄	—	346	2	no	—
90	C₄⋊C₄	—	173	2	no	—
91	C₄²	—	346	2	no	—
92	C₄²	—	346	2	no	—
93	C₄²	C₂²×D₁₇₃	1	1	yes	characteristic
94	C₂³	—	4	1	no	—
95	Q₈	—	4	2	no	—
96	Q₈	—	346	2	no	—
97	Q₈	—	1384	2	no	—
98	Q₈	—	173	4	no	—
99	D₄	—	4	2	no	—
100	D₄	—	4	2	no	—
101	D₄	—	4	2	no	—
102	C₂×C₄	—	2	2	no	—
103	C₂×C₄	—	346	2	no	—
104	C₂×C₄	—	346	2	no	—
105	C₂×C₄	D₄×D₁₇₃	1	2	yes	—
106	C₂×C₄	C₂×C₂²⋊D₁₇₃	1	1	yes	characteristic
107	C₈	—	2	2	no	—
108	C₈	—	692	2	no	—
109	C₂²	—	8	1	no	—
110	C₂²	—	8	2	no	—
111	C₂²	#19149647	1	1	yes	characteristic
112	C₄	—	2	1	no	—
113	C₄	—	4	2	no	—
114	C₄	—	346	4	no	—
115	C₄	—	692	2	no	—
116	C₄	SD₁₆×D₁₇₃	1	2	yes	—
117	C₄	#19149708	1	2	yes	—
118	C₂	—	16	1	no	—
119	C₂	#26011628	1	2	yes	—
120	C₂	#26011658	1	1	yes	characteristic

subgroup

G/H

class size

multiplicity

normal?

properties

#26012221

C₂

yes

maximal, characteristic, complemented

#26011688

C₂

yes

maximal

#26011240

C₂

yes

maximal, characteristic, complemented

#26011223

C₂

yes

maximal, complemented

#492×C₁₇₃

C₂

yes

maximal, characteristic

C₂×#14583814

—

#19149678

—

#19149677

—

C₄×Dic₃₄₆

C₂²

yes

—

#19149644

—

C₄⋊#13063436

C₂²

yes

—

#19149600

—

C₂×SD₁₆×C₁₇₃

—

C₄⋊Q₈×C₁₇₃

C₂²

yes

characteristic

#127×C₁₇₃

C₂²

yes

characteristic

C₄×C₈×C₁₇₃

C₂²

yes

characteristic

C₂×Dic₃₄₆

—

C₂×Dic₃₄₆

D₄

yes

—

#14583814

—

C₂×#13063436

—

C₄⋊Dic₁₇₃

D₄

yes

—

#14583805

—

#14583805

—

C₄×Dic₁₇₃

—

C₄×Dic₁₇₃

—

C₂×Q₈×C₁₇₃

D₄

yes

—

C₂×D₄×C₁₇₃

—

C₂×D₄×C₁₇₃

—

SD₁₆×C₁₇₃

—

C₂×C₈×C₁₇₃

—

C₄⋊C₄×C₁₇₃

—

C₄²×C₁₇₃

C₂³

yes

characteristic

Dic₃₄₆

—

Dic₃₄₆

—

Dic₃₄₆

SD₁₆

yes

complemented

C₂×Dic₁₇₃

—

C₂×Dic₁₇₃

—

#13063436

—

C₂³×C₁₇₃

—

Q₈×C₁₇₃

—

D₄×C₁₇₃

—

D₄×C₁₇₃

—

D₄×C₁₇₃

—

C₂×C₄×C₁₇₃

—

C₂×C₄×C₁₇₃

C₂×D₄

yes

—

C₂×C₄×C₁₇₃

C₂×D₄

yes

characteristic

C₈×C₁₇₃

—

C₂²×C₁₇₃

—

C₂²×C₁₇₃

—

C₂²×C₁₇₃

#120

yes

characteristic

C₄×C₁₇₃

—

C₄×C₁₇₃

—

C₄×C₁₇₃

C₂×SD₁₆

yes

—

C₄×C₁₇₃

#137

yes

—

Dic₁₇₃

—

Dic₁₇₃

—

C₂×C₁₇₃

—

C₂×C₁₇₃

#448

yes

—

C₂×C₁₇₃

#453

yes

characteristic

C₁₇₃

#1623

yes

characteristic, complemented, Sylow

#1623

—

173

maximal, Sylow

Q₈²

—

173

—

#492

D₁₇₃

yes

characteristic

#460

—

173

—

#333

—

173

—

Q₈⋊C₈

—

173

—

C₂×SD₁₆

—

C₂×SD₁₆

—

346

—

C₄⋊Q₈

—

346

—

C₄⋊Q₈

—

346

—

C₄⋊Q₈

C₂×D₁₇₃

yes

characteristic

#127

C₂×D₁₇₃

yes

characteristic

C₄×Q₈

—

346

—

C₄×Q₈

—

173

—

C₄⋊C₈

—

173

—

#102

—

346

—

C₄×C₈

C₂×D₁₇₃

yes

characteristic

C₂×Q₈

—

346

—

C₂×Q₈

—

173

—

C₂×Q₈

C₂²⋊D₁₇₃

yes

—

C₂×D₄

—

C₂×D₄

—

SD₁₆

—

SD₁₆

—

692

—

C₂×C₈

—

C₂×C₈

—

346

—

C₄⋊C₄

—

C₄⋊C₄

—

346

—

C₄⋊C₄

—

346

—

C₄⋊C₄

—

173

—

C₄²

—

346

—

C₄²

—

346

—

C₄²

C₂²×D₁₇₃

yes

characteristic

C₂³

—

Q₈

—

Q₈

—

346

—

Q₈

—

1384

—

Q₈

—

173

—

D₄

—

100

D₄

—

101

D₄

—

102

C₂×C₄

—

103

C₂×C₄

—

346

—

104

C₂×C₄

—

346

—

105

C₂×C₄

D₄×D₁₇₃

yes

—

106

C₂×C₄

C₂×C₂²⋊D₁₇₃

yes

characteristic

107

C₈

—

108

C₈

—

692

—

109

C₂²

—

110

C₂²

—

111

C₂²

#19149647

yes

characteristic

112

C₄

—

113

C₄

—

114

C₄

—

346

—

115

C₄

—

692

—

116

C₄

SD₁₆×D₁₇₃

yes

—

117

C₄

#19149708

yes

—

118

C₂

—

119

C₂

#26011628

yes

—

120

C₂

#26011658

yes

characteristic

	subgroup	G/H
center	C₂²	#19149647
commutator, cosocle	C₄²×C₁₇₃	C₂³
Frattini	C₄²	C₂²×D₁₇₃
Fitting	#492×C₁₇₃	C₂
socle	C₂²×C₁₇₃	#120
prime radical	#127×C₁₇₃	C₂²
hypercenter	#492	D₁₇₃
hypocenter	C₁₇₃	#1623
LCS	G ▷ C₄²×C₁₇₃ ▷ C₂²×C₁₇₃ ▷ C₁₇₃
UCS	C₁ ◁ C₂² ◁ C₄² ◁ #492

subgroup

G/H

center

C₂²

#19149647

commutator, cosocle

C₄²×C₁₇₃

C₂³

Frattini

C₄²

C₂²×D₁₇₃

Fitting

#492×C₁₇₃

C₂

socle

C₂²×C₁₇₃

#120

prime radical

#127×C₁₇₃

C₂²

D₁₇₃

C₁₇₃

G ▷ C₄²×C₁₇₃ ▷ C₂²×C₁₇₃ ▷ C₁₇₃

UCS

C₁ ◁ C₂² ◁ C₄² ◁ #492

N	H ≅ G/N	type
1. #26012221	C₂	semidirect
3. #26011240	C₂	semidirect
4. #26011223	C₂	semidirect
35. Dic₃₄₆	SD₁₆	semidirect
60. C₁₇₃	#1623	semidirect

H ≅ G/N

type

1. #26012221

C₂

semidirect

3. #26011240

C₂

semidirect

4. #26011223

C₂

semidirect

35. Dic₃₄₆

SD₁₆

semidirect

60. C₁₇₃

#1623

semidirect

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

subgroup	index
#26012221	2
#26011688	2
#26011240	2
#26011223	2
#492×C₁₇₃	2
#1623	173

subgroup

index

#492×C₁₇₃

173

Subgroups — #36352639

Conjugacy classes of subgroups

Complements

Maximal subgroups