Group #36370473 — subgroups

Subgroups — #36370473

Order 22144.

330 conjugacy classes of subgroups; only the 148 normal subgroups are shown.

normal subgroups only

This group has 330 conjugacy classes of subgroups — too many to draw. Only its 148 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	#26012543	C₂	1	4	yes	maximal, complemented
2	#26012540	C₂	1	4	yes	maximal, complemented
3	#26012516	C₂	1	4	yes	maximal
4	#26012509	C₂	1	2	yes	maximal
5	#26012047	C₂	1	1	yes	maximal, characteristic
6	#19149746	C₂²	1	2	yes	—
7	#19149746	C₂²	1	2	yes	—
8	#19149746	C₂²	1	4	yes	—
9	C₄⋊#13063443	C₂²	1	4	yes	—
10	#19149736	C₂²	1	4	yes	—
11	#19149734	C₂²	1	4	yes	—
12	C₄×#13063443	C₂²	1	4	yes	—
13	C₄×#13063443	C₂²	1	4	yes	—
14	#19149678	C₄	1	2	yes	—
15	#19149671	C₄	1	1	yes	characteristic
16	#19149657	C₄	1	4	yes	—
17	#19149644	C₂²	1	4	yes	—
18	#19149637	C₂²	1	2	yes	—
19	C₄²×D₁₇₃	C₂²	1	1	yes	characteristic
20	#125×C₁₇₃	C₄	1	1	yes	characteristic
21	C₂×#13063443	C₂³	1	4	yes	—
22	C₂×#13063443	C₂³	1	4	yes	—
23	C₂×Dic₃₄₆	C₂×C₄	1	4	yes	—
24	C₂×C₄×D₁₇₃	C₂×C₄	1	2	yes	—
25	C₂×C₄×D₁₇₃	C₂³	1	1	yes	characteristic
26	C₄⋊Dic₁₇₃	C₂×C₄	1	2	yes	—
27	#14583805	C₂×C₄	1	4	yes	—
28	#14583805	C₂×C₄	1	4	yes	—
29	#14583803	C₂×C₄	1	4	yes	—
30	C₄×Dic₁₇₃	C₂×C₄	1	1	yes	characteristic
31	C₄×Dic₁₇₃	C₂³	1	2	yes	—
32	C₄×Dic₁₇₃	C₂³	1	4	yes	—
33	C₄⋊C₄×C₁₇₃	C₂×C₄	1	2	yes	—
34	C₄⋊C₄×C₁₇₃	C₂×C₄	1	4	yes	—
35	C₄²×C₁₇₃	C₂×C₄	1	1	yes	characteristic
36	#13063443	—	2	4	no	—
37	#13063443	—	2	4	no	—
38	C₂²×D₁₇₃	C₂²×C₄	1	1	yes	characteristic
39	Dic₃₄₆	—	2	8	no	—
40	C₂×Dic₁₇₃	C₂²×C₄	1	4	yes	—
41	C₂×Dic₁₇₃	C₂²×C₄	1	2	yes	—
42	C₂×Dic₁₇₃	C₂⁴	1	1	yes	characteristic
43	C₄×D₁₇₃	—	2	4	no	—
44	C₄×D₁₇₃	—	2	2	no	—
45	C₂×C₄×C₁₇₃	C₂²×C₄	1	2	yes	—
46	C₂×C₄×C₁₇₃	C₂²×C₄	1	1	yes	characteristic
47	C₂×C₄×C₁₇₃	C₂²×C₄	1	4	yes	—
48	C₂²×C₁₇₃	C₂³×C₄	1	1	yes	characteristic
49	C₂×D₁₇₃	—	2	1	no	—
50	C₂×D₁₇₃	—	2	2	no	—
51	C₄×C₁₇₃	—	2	1	no	—
52	C₄×C₁₇₃	—	2	2	no	—
53	C₄×C₁₇₃	—	2	4	no	—
54	Dic₁₇₃	—	2	4	no	—
55	Dic₁₇₃	#131	1	4	yes	—
56	Dic₁₇₃	2₋¹⁺⁴	1	2	yes	—
57	C₂×C₁₇₃	#519	1	1	yes	characteristic
58	C₂×C₁₇₃	C₂×#131	1	2	yes	—
59	D₁₇₃	—	4	1	no	—
60	C₁₇₃	#2986	1	1	yes	characteristic, complemented, Sylow
61	#2986	—	173	1	no	maximal, Sylow
62	#533	—	173	1	no	—
63	#446	—	173	4	no	—
64	C₈×Q₈	—	173	4	no	—
65	#433	—	173	4	no	—
66	#431	—	173	2	no	—
67	C₄⋊Q₈	—	173	2	no	—
68	#125	—	173	1	no	—
69	#125	#10621924	1	1	yes	characteristic
70	C₂²⋊Q₈	—	173	4	no	—
71	C₄×Q₈	—	173	4	no	—
72	#117	—	173	2	no	—
73	C₂×C₄²	—	173	1	no	—
74	C₄⋊C₈	—	173	2	no	—
75	C₄⋊C₈	—	173	2	no	—
76	C₄⋊C₈	—	173	4	no	—
77	C₄⋊C₈	—	173	4	no	—
78	C₂²⋊C₈	—	173	4	no	—
79	#97	—	173	4	no	—
80	C₄×C₈	—	173	4	no	—
81	C₄×C₈	—	173	4	no	—
82	C₂×Q₈	—	173	4	no	—
83	C₂²×C₄	—	173	2	no	—
84	C₂²×C₄	—	173	1	no	—
85	C₂×C₈	—	173	4	no	—
86	C₂×C₈	—	173	4	no	—
87	C₄⋊C₄	—	173	2	no	—
88	C₄⋊C₄	—	173	4	no	—
89	C₄⋊C₄	—	173	4	no	—
90	C₄⋊C₄	C₂×#10621924	1	2	yes	—
91	C₄⋊C₄	C₂×#10621924	1	4	yes	—
92	C₂²⋊C₄	—	173	4	no	—
93	C₄²	—	173	2	no	—
94	C₄²	—	173	4	no	—
95	C₄²	—	173	1	no	—
96	C₄²	C₂×#10621924	1	1	yes	characteristic
97	C₂³	—	173	1	no	—
98	Q₈	—	346	8	no	—
99	C₂×C₄	—	346	2	no	—
100	C₂×C₄	—	346	4	no	—
101	C₂×C₄	—	173	4	no	—
102	C₂×C₄	—	173	2	no	—
103	C₂×C₄	—	173	1	no	—
104	C₂×C₄	C₂²×#10621924	1	1	yes	characteristic
105	C₂×C₄	C₂²×#10621924	1	2	yes	—
106	C₂×C₄	C₂²×#10621924	1	4	yes	—
107	C₈	—	346	4	no	—
108	C₈	—	346	4	no	—
109	C₂²	—	346	1	no	—
110	C₂²	—	346	2	no	—
111	C₂²	C₂³×#10621924	1	1	yes	characteristic
112	C₄	—	2	1	no	—
113	C₄	—	2	2	no	—
114	C₄	—	2	4	no	—
115	C₄	—	346	4	no	—
116	C₄	—	173	2	no	—
117	C₄	—	173	4	no	—
118	C₂	—	692	1	no	—
119	C₂	#26012559	1	1	yes	characteristic
120	C₂	D₁₇₃⋊#131	1	2	yes	—

subgroup

G/H

class size

multiplicity

normal?

properties

#26012543

C₂

yes

maximal, complemented

#26012540

C₂

yes

maximal, complemented

#26012516

C₂

yes

maximal

#26012509

C₂

yes

maximal

#26012047

C₂

yes

maximal, characteristic

#19149746

C₂²

yes

—

#19149746

C₂²

yes

—

#19149746

C₂²

yes

—

C₄⋊#13063443

C₂²

yes

—

#19149736

C₂²

yes

—

#19149734

C₂²

yes

—

C₄×#13063443

C₂²

yes

—

C₄×#13063443

C₂²

yes

—

#19149678

C₄

yes

—

#19149671

C₄

yes

characteristic

#19149657

C₄

yes

—

#19149644

C₂²

yes

—

#19149637

C₂²

yes

—

C₄²×D₁₇₃

C₂²

yes

characteristic

#125×C₁₇₃

C₄

yes

characteristic

C₂×#13063443

C₂³

yes

—

C₂×#13063443

C₂³

yes

—

C₂×Dic₃₄₆

C₂×C₄

yes

—

C₂×C₄×D₁₇₃

C₂×C₄

yes

—

C₂×C₄×D₁₇₃

C₂³

yes

characteristic

C₄⋊Dic₁₇₃

C₂×C₄

yes

—

#14583805

C₂×C₄

yes

—

#14583805

C₂×C₄

yes

—

#14583803

C₂×C₄

yes

—

C₄×Dic₁₇₃

C₂×C₄

yes

characteristic

C₄×Dic₁₇₃

C₂³

yes

—

C₄×Dic₁₇₃

C₂³

yes

—

C₄⋊C₄×C₁₇₃

C₂×C₄

yes

—

C₄⋊C₄×C₁₇₃

C₂×C₄

yes

—

C₄²×C₁₇₃

C₂×C₄

yes

characteristic

#13063443

—

#13063443

—

C₂²×D₁₇₃

C₂²×C₄

yes

characteristic

Dic₃₄₆

—

C₂×Dic₁₇₃

C₂²×C₄

yes

—

C₂×Dic₁₇₃

C₂²×C₄

yes

—

C₂×Dic₁₇₃

C₂⁴

yes

characteristic

C₄×D₁₇₃

—

C₄×D₁₇₃

—

C₂×C₄×C₁₇₃

C₂²×C₄

yes

—

C₂×C₄×C₁₇₃

C₂²×C₄

yes

characteristic

C₂×C₄×C₁₇₃

C₂²×C₄

yes

—

C₂²×C₁₇₃

C₂³×C₄

yes

characteristic

C₂×D₁₇₃

—

C₂×D₁₇₃

—

C₄×C₁₇₃

—

C₄×C₁₇₃

—

C₄×C₁₇₃

—

Dic₁₇₃

—

Dic₁₇₃

#131

yes

—

Dic₁₇₃

2₋¹⁺⁴

yes

—

C₂×C₁₇₃

#519

yes

characteristic

C₂×C₁₇₃

C₂×#131

yes

—

D₁₇₃

—

C₁₇₃

#2986

yes

characteristic, complemented, Sylow

#2986

—

173

maximal, Sylow

#533

—

173

—

#446

—

173

—

C₈×Q₈

—

173

—

#433

—

173

—

#431

—

173

—

C₄⋊Q₈

—

173

—

#125

—

173

—

#125

#10621924

yes

characteristic

C₂²⋊Q₈

—

173

—

C₄×Q₈

—

173

—

#117

—

173

—

C₂×C₄²

—

173

—

C₄⋊C₈

—

173

—

C₄⋊C₈

—

173

—

C₄⋊C₈

—

173

—

C₄⋊C₈

—

173

—

C₂²⋊C₈

—

173

—

#97

—

173

—

C₄×C₈

—

173

—

C₄×C₈

—

173

—

C₂×Q₈

—

173

—

C₂²×C₄

—

173

—

C₂²×C₄

—

173

—

C₂×C₈

—

173

—

C₂×C₈

—

173

—

C₄⋊C₄

—

173

—

C₄⋊C₄

—

173

—

C₄⋊C₄

—

173

—

C₄⋊C₄

C₂×#10621924

yes

—

C₄⋊C₄

C₂×#10621924

yes

—

C₂²⋊C₄

—

173

—

C₄²

—

173

—

C₄²

—

173

—

C₄²

—

173

—

C₄²

C₂×#10621924

yes

characteristic

C₂³

—

173

—

Q₈

—

346

—

C₂×C₄

—

346

—

100

C₂×C₄

—

346

—

101

C₂×C₄

—

173

—

102

C₂×C₄

—

173

—

103

C₂×C₄

—

173

—

104

C₂×C₄

C₂²×#10621924

yes

characteristic

105

C₂×C₄

C₂²×#10621924

yes

—

106

C₂×C₄

C₂²×#10621924

yes

—

107

C₈

—

346

—

108

C₈

—

346

—

109

C₂²

—

346

—

110

C₂²

—

346

—

111

C₂²

C₂³×#10621924

yes

characteristic

112

C₄

—

113

C₄

—

114

C₄

—

115

C₄

—

346

—

116

C₄

—

173

—

117

C₄

—

173

—

118

C₂

—

692

—

119

C₂

#26012559

yes

characteristic

120

C₂

D₁₇₃⋊#131

yes

—

	subgroup	G/H
center, Frattini	C₂²	C₂³×#10621924
commutator, socle	C₂²×C₁₇₃	C₂³×C₄
Fitting	#125×C₁₇₃	C₄
prime radical	C₂²×D₁₇₃	C₂²×C₄
cosocle	C₂×Dic₁₇₃	C₂⁴
hypercenter	#125	#10621924
hypocenter	C₁₇₃	#2986
LCS	G ▷ C₂²×C₁₇₃ ▷ C₁₇₃
UCS	C₁ ◁ C₂² ◁ #125

subgroup

G/H

center, Frattini

C₂²

C₂³×#10621924

commutator, socle

C₂²×C₁₇₃

C₂³×C₄

Fitting

#125×C₁₇₃

C₄

prime radical

C₂²×D₁₇₃

C₂²×C₄

cosocle

C₂×Dic₁₇₃

C₂⁴

C₁₇₃

G ▷ C₂²×C₁₇₃ ▷ C₁₇₃

C₁ ◁ C₂² ◁ #125

N	H ≅ G/N	type
1. #26012543	C₂	semidirect
2. #26012540	C₂	semidirect
60. C₁₇₃	#2986	semidirect

H ≅ G/N

type

1. #26012543

C₂

semidirect

2. #26012540

C₂

semidirect

60. C₁₇₃

#2986

semidirect

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

subgroup	index
#26012543	2
#26012540	2
#26012516	2
#26012509	2
#26012047	2
#2986	173

subgroup

index

173

Subgroups — #36370473

Conjugacy classes of subgroups

Complements

Maximal subgroups