Group #36353848 — subgroups

Subgroups — #36353848

Order 22144.

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

normal subgroups only

This group has 260 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	#26012258	C₂	1	1	yes	maximal, characteristic
2	#26011517	C₂	1	2	yes	maximal, complemented
3	#26011465	C₂	1	1	yes	maximal, characteristic
4	#26011263	C₂	1	2	yes	maximal, complemented
5	#408×C₁₇₃	C₂	1	1	yes	maximal, characteristic
6	#19149686	—	2	4	no	—
7	C₂×#14583810	—	2	2	no	—
8	C₂×#14583810	C₂²	1	2	yes	—
9	#19149638	C₂²	1	2	yes	—
10	#19149608	—	2	4	no	—
11	#19149594	C₄	1	4	yes	—
12	C₂×#41×C₁₇₃	C₂²	1	1	yes	characteristic
13	C₂×M₁₆×C₁₇₃	C₂²	1	1	yes	characteristic
14	C₂²×C₈×C₁₇₃	C₂²	1	1	yes	characteristic
15	C₂²⋊C₈×C₁₇₃	—	2	2	no	—
16	#14583810	—	2	4	no	—
17	#14583810	—	2	2	no	—
18	#14583810	—	2	2	no	—
19	C₂×#13063436	—	2	4	no	—
20	C₂×#13063436	D₄	1	4	yes	—
21	C₄⋊Dic₁₇₃	—	2	2	no	—
22	C₂²⋊Dic₁₇₃	C₂×C₄	1	4	yes	—
23	C₄×Dic₁₇₃	—	2	2	no	—
24	#41×C₁₇₃	—	2	2	no	—
25	#41×C₁₇₃	—	2	2	no	—
26	C₂×Q₈×C₁₇₃	Q₈	1	1	yes	characteristic
27	C₂×D₄×C₁₇₃	C₂×C₄	1	2	yes	—
28	C₂×D₄×C₁₇₃	Q₈	1	1	yes	characteristic
29	C₂²×C₄×C₁₇₃	D₄	1	2	yes	—
30	C₂²×C₄×C₁₇₃	C₂³	1	1	yes	characteristic
31	M₁₆×C₁₇₃	—	2	2	no	—
32	C₂×C₈×C₁₇₃	—	2	2	no	—
33	C₂×C₈×C₁₇₃	—	2	1	no	—
34	C₂×C₈×C₁₇₃	—	2	1	no	—
35	C₂×Dic₁₇₃	—	2	4	no	—
36	#13063436	—	2	4	no	—
37	C₂³×C₁₇₃	C₄⋊C₄	1	2	yes	—
38	C₂³×C₁₇₃	C₂²×C₄	1	1	yes	characteristic
39	Q₈×C₁₇₃	—	2	2	no	—
40	D₄×C₁₇₃	—	2	4	no	—
41	D₄×C₁₇₃	—	2	2	no	—
42	C₂×C₄×C₁₇₃	—	2	2	no	—
43	C₂×C₄×C₁₇₃	—	2	2	no	—
44	C₂×C₄×C₁₇₃	C₄⋊C₄	1	2	yes	—
45	C₂×C₄×C₁₇₃	C₂×D₄	1	2	yes	—
46	C₂×C₄×C₁₇₃	C₂×D₄	1	1	yes	characteristic
47	C₂×C₄×C₁₇₃	C₂×Q₈	1	1	yes	characteristic
48	C₂×C₄×C₁₇₃	#41	1	2	yes	—
49	C₈×C₁₇₃	—	4	1	no	—
50	C₈×C₁₇₃	—	4	1	no	—
51	C₂²×C₁₇₃	—	2	2	no	—
52	C₂²×C₁₇₃	—	4	1	no	—
53	C₂²×C₁₇₃	—	4	2	no	—
54	C₂²×C₁₇₃	C₂×C₄⋊C₄	1	1	yes	characteristic
55	C₂²×C₁₇₃	C₄×D₄	1	2	yes	—
56	C₄×C₁₇₃	—	2	2	no	—
57	C₄×C₁₇₃	#120	1	1	yes	characteristic
58	C₄×C₁₇₃	C₂²⋊Q₈	1	2	yes	—
59	C₄×C₁₇₃	#123	1	1	yes	characteristic
60	Dic₁₇₃	—	4	4	no	—
61	C₂×C₁₇₃	—	2	2	no	—
62	C₂×C₁₇₃	—	2	1	no	—
63	C₂×C₁₇₃	—	4	2	no	—
64	C₂×C₁₇₃	#385	1	1	yes	characteristic
65	C₁₇₃	#1868	1	1	yes	characteristic, complemented, Sylow
66	#1868	—	173	1	no	maximal, Sylow
67	#568	—	173	1	no	—
68	#424	—	173	2	no	—
69	#410	—	173	1	no	—
70	#408	D₁₇₃	1	1	yes	characteristic
71	#344	—	173	2	no	—
72	C₂×#41	C₂×D₁₇₃	1	1	yes	characteristic
73	#131	—	346	4	no	—
74	C₂×M₁₆	—	346	2	no	—
75	C₂×M₁₆	—	173	2	no	—
76	C₂×M₁₆	C₂×D₁₇₃	1	1	yes	characteristic
77	C₂²×C₈	C₂×D₁₇₃	1	1	yes	characteristic
78	#117	—	173	2	no	—
79	C₄⋊C₈	—	346	4	no	—
80	#99	—	173	4	no	—
81	C₂²⋊C₈	—	2	2	no	—
82	#41	—	2	2	no	—
83	#41	—	2	2	no	—
84	C₂×Q₈	Dic₃₄₆	1	1	yes	characteristic
85	C₂×D₄	C₄×D₁₇₃	1	2	yes	—
86	C₂×D₄	Dic₃₄₆	1	1	yes	characteristic
87	C₂²×C₄	C₂²⋊D₁₇₃	1	2	yes	—
88	C₂²×C₄	C₂²×D₁₇₃	1	1	yes	characteristic
89	M₁₆	—	2	2	no	—
90	M₁₆	—	346	4	no	—
91	M₁₆	—	346	2	no	—
92	M₁₆	—	346	2	no	—
93	C₂×C₈	—	2	2	no	—
94	C₂×C₈	—	2	1	no	—
95	C₂×C₈	—	2	1	no	—
96	C₂×C₈	—	346	4	no	—
97	C₂×C₈	—	173	4	no	—
98	C₄⋊C₄	—	346	2	no	—
99	C₂²⋊C₄	—	173	4	no	—
100	C₄²	—	346	2	no	—
101	C₂³	#14583805	1	2	yes	—
102	C₂³	C₂×C₄×D₁₇₃	1	1	yes	characteristic
103	Q₈	—	2	2	no	—
104	D₄	—	2	4	no	—
105	D₄	—	2	2	no	—
106	C₂×C₄	—	2	2	no	—
107	C₂×C₄	—	2	2	no	—
108	C₂×C₄	—	346	4	no	—
109	C₂×C₄	#14583805	1	2	yes	—
110	C₂×C₄	D₄×D₁₇₃	1	2	yes	—
111	C₂×C₄	C₂×C₂²⋊D₁₇₃	1	1	yes	characteristic
112	C₂×C₄	C₂×Dic₃₄₆	1	1	yes	characteristic
113	C₂×C₄	#14583825	1	2	yes	—
114	C₈	—	4	1	no	—
115	C₈	—	4	1	no	—
116	C₈	—	346	4	no	—
117	C₂²	—	2	2	no	—
118	C₂²	—	4	1	no	—
119	C₂²	—	4	2	no	—
120	C₂²	C₂×#14583805	1	1	yes	characteristic
121	C₂²	#19149639	1	2	yes	—
122	C₄	—	2	2	no	—
123	C₄	—	692	4	no	—
124	C₄	#19149647	1	1	yes	characteristic
125	C₄	#19149656	1	2	yes	—
126	C₄	#19149660	1	1	yes	characteristic
127	C₂	—	2	2	no	—
128	C₂	—	2	1	no	—
129	C₂	—	4	2	no	—
130	C₂	#26011363	1	1	yes	characteristic

subgroup

G/H

class size

multiplicity

normal?

properties

#26012258

C₂

yes

maximal, characteristic

#26011517

C₂

yes

maximal, complemented

#26011465

C₂

yes

maximal, characteristic

#26011263

C₂

yes

maximal, complemented

#408×C₁₇₃

C₂

yes

maximal, characteristic

#19149686

—

C₂×#14583810

—

C₂×#14583810

C₂²

yes

—

#19149638

C₂²

yes

—

#19149608

—

#19149594

C₄

yes

—

C₂×#41×C₁₇₃

C₂²

yes

characteristic

C₂×M₁₆×C₁₇₃

C₂²

yes

characteristic

C₂²×C₈×C₁₇₃

C₂²

yes

characteristic

C₂²⋊C₈×C₁₇₃

—

—

—

—

—

D₄

yes

—

C₄⋊Dic₁₇₃

—

C₂²⋊Dic₁₇₃

C₂×C₄

yes

—

C₄×Dic₁₇₃

—

#41×C₁₇₃

—

#41×C₁₇₃

—

C₂×Q₈×C₁₇₃

Q₈

yes

characteristic

C₂×D₄×C₁₇₃

C₂×C₄

yes

—

C₂×D₄×C₁₇₃

Q₈

yes

characteristic

C₂²×C₄×C₁₇₃

D₄

yes

—

C₂²×C₄×C₁₇₃

C₂³

yes

characteristic

M₁₆×C₁₇₃

—

C₂×C₈×C₁₇₃

—

C₂×C₈×C₁₇₃

—

C₂×C₈×C₁₇₃

—

C₂×Dic₁₇₃

—

#13063436

—

C₂³×C₁₇₃

C₄⋊C₄

yes

—

C₂³×C₁₇₃

C₂²×C₄

yes

characteristic

Q₈×C₁₇₃

—

D₄×C₁₇₃

—

D₄×C₁₇₃

—

C₂×C₄×C₁₇₃

—

C₂×C₄×C₁₇₃

—

C₂×C₄×C₁₇₃

C₄⋊C₄

yes

—

C₂×C₄×C₁₇₃

C₂×D₄

yes

—

C₂×C₄×C₁₇₃

C₂×D₄

yes

characteristic

C₂×C₄×C₁₇₃

C₂×Q₈

yes

characteristic

C₂×C₄×C₁₇₃

#41

yes

—

C₈×C₁₇₃

—

C₈×C₁₇₃

—

C₂²×C₁₇₃

—

C₂²×C₁₇₃

—

C₂²×C₁₇₃

—

C₂²×C₁₇₃

C₂×C₄⋊C₄

yes

characteristic

C₂²×C₁₇₃

C₄×D₄

yes

—

C₄×C₁₇₃

—

C₄×C₁₇₃

#120

yes

characteristic

C₄×C₁₇₃

C₂²⋊Q₈

yes

—

C₄×C₁₇₃

#123

yes

characteristic

Dic₁₇₃

—

C₂×C₁₇₃

—

C₂×C₁₇₃

—

C₂×C₁₇₃

—

C₂×C₁₇₃

#385

yes

characteristic

C₁₇₃

#1868

yes

characteristic, complemented, Sylow

#1868

—

173

maximal, Sylow

#568

—

173

—

#424

—

173

—

#410

—

173

—

#408

D₁₇₃

yes

characteristic

#344

—

173

—

C₂×#41

C₂×D₁₇₃

yes

characteristic

#131

—

346

—

C₂×M₁₆

—

346

—

C₂×M₁₆

—

173

—

C₂×M₁₆

C₂×D₁₇₃

yes

characteristic

C₂²×C₈

C₂×D₁₇₃

yes

characteristic

#117

—

173

—

C₄⋊C₈

—

346

—

#99

—

173

—

C₂²⋊C₈

—

#41

—

#41

—

C₂×Q₈

Dic₃₄₆

yes

characteristic

C₂×D₄

C₄×D₁₇₃

yes

—

C₂×D₄

Dic₃₄₆

yes

characteristic

C₂²×C₄

C₂²⋊D₁₇₃

yes

—

C₂²×C₄

C₂²×D₁₇₃

yes

characteristic

M₁₆

—

M₁₆

—

346

—

M₁₆

—

346

—

M₁₆

—

346

—

C₂×C₈

—

C₂×C₈

—

C₂×C₈

—

C₂×C₈

—

346

—

C₂×C₈

—

173

—

C₄⋊C₄

—

346

—

C₂²⋊C₄

—

173

—

100

C₄²

—

346

—

101

C₂³

#14583805

yes

—

102

C₂³

C₂×C₄×D₁₇₃

yes

characteristic

103

Q₈

—

104

D₄

—

105

D₄

—

106

C₂×C₄

—

107

C₂×C₄

—

108

C₂×C₄

—

346

—

109

C₂×C₄

#14583805

yes

—

110

C₂×C₄

D₄×D₁₇₃

yes

—

111

C₂×C₄

C₂×C₂²⋊D₁₇₃

yes

characteristic

112

C₂×C₄

C₂×Dic₃₄₆

yes

characteristic

113

C₂×C₄

#14583825

yes

—

114

C₈

—

115

C₈

—

116

C₈

—

346

—

117

C₂²

—

118

C₂²

—

119

C₂²

—

120

C₂²

C₂×#14583805

yes

characteristic

121

C₂²

#19149639

yes

—

122

C₄

—

123

C₄

—

692

—

124

C₄

#19149647

yes

characteristic

125

C₄

#19149656

yes

—

126

C₄

#19149660

yes

characteristic

127

C₂

—

128

C₂

—

129

C₂

—

130

C₂

#26011363

yes

characteristic

	subgroup	G/H
center	C₄	#19149647
commutator	C₂³×C₁₇₃	C₂²×C₄
Frattini	C₂²×C₄	C₂²×D₁₇₃
Fitting	#408×C₁₇₃	C₂
socle	C₂×C₁₇₃	#385
prime radical	C₂×#41×C₁₇₃	C₂²
cosocle	C₂²×C₄×C₁₇₃	C₂³
hypercenter	#408	D₁₇₃
hypocenter	C₁₇₃	#1868
LCS	G ▷ C₂³×C₁₇₃ ▷ C₂×C₁₇₃ ▷ C₁₇₃
UCS	C₁ ◁ C₄ ◁ C₂²×C₄ ◁ #408

subgroup

G/H

center

C₄

#19149647

commutator

C₂³×C₁₇₃

C₂²×C₄

Frattini

C₂²×C₄

C₂²×D₁₇₃

Fitting

#408×C₁₇₃

C₂

socle

C₂×C₁₇₃

#385

prime radical

C₂×#41×C₁₇₃

C₂²

cosocle

C₂²×C₄×C₁₇₃

C₂³

D₁₇₃

C₁₇₃

G ▷ C₂³×C₁₇₃ ▷ C₂×C₁₇₃ ▷ C₁₇₃

UCS

C₁ ◁ C₄ ◁ C₂²×C₄ ◁ #408

N	H ≅ G/N	type
2. #26011517	C₂	semidirect
4. #26011263	C₂	semidirect
65. C₁₇₃	#1868	semidirect

H ≅ G/N

type

2. #26011517

C₂

semidirect

4. #26011263

C₂

semidirect

65. C₁₇₃

#1868

semidirect

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

subgroup	index
#26012258	2
#26011517	2
#26011465	2
#26011263	2
#408×C₁₇₃	2
#1868	173

subgroup

index

#408×C₁₇₃

173

Subgroups — #36353848

Conjugacy classes of subgroups

Complements

Maximal subgroups