Group #36353956 — subgroups

Subgroups — #36353956

Order 22144.

324 conjugacy classes of subgroups; only the 93 normal subgroups are shown.

normal subgroups only

This group has 324 conjugacy classes of subgroups — too many to draw. Only its 93 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	C₂×#19149677	C₂	1	1	yes	maximal, characteristic
2	#26011499	C₂	1	2	yes	maximal
3	C₂×#19149598	C₂	1	1	yes	maximal, characteristic
4	#26011250	C₂	1	2	yes	maximal
5	#519×C₁₇₃	C₂	1	1	yes	maximal, characteristic
6	C₂²×Dic₃₄₆	C₂²	1	1	yes	characteristic
7	C₂×#14583810	C₂²	1	2	yes	—
8	#19149677	—	2	2	no	—
9	#19149677	C₄	1	4	yes	complemented
10	C₂×#14583805	—	2	2	no	—
11	C₂×C₄×Dic₁₇₃	C₂²	1	1	yes	characteristic
12	#19149602	—	2	4	no	—
13	#19149598	—	2	2	no	—
14	C₂²×Q₈×C₁₇₃	C₂²	1	1	yes	characteristic
15	C₄×Q₈×C₁₇₃	—	2	4	no	—
16	#117×C₁₇₃	—	2	2	no	—
17	#117×C₁₇₃	C₂²	1	2	yes	—
18	C₂×Dic₃₄₆	—	2	1	no	—
19	C₂×Dic₃₄₆	—	2	2	no	—
20	C₂×Dic₃₄₆	—	2	2	no	—
21	C₂×Dic₃₄₆	C₂×C₄	1	2	yes	—
22	C₂²×Dic₁₇₃	—	2	1	no	—
23	C₂²×Dic₁₇₃	—	2	1	no	—
24	#14583810	—	2	2	no	—
25	#14583810	—	2	2	no	—
26	C₂×#13063436	—	2	2	no	—
27	#14583805	—	2	8	no	—
28	C₄×Dic₁₇₃	—	2	1	no	—
29	C₄×Dic₁₇₃	C₂×C₄	1	2	yes	—
30	C₂×Q₈×C₁₇₃	—	2	2	no	—
31	C₂×Q₈×C₁₇₃	—	2	2	no	—
32	C₂×Q₈×C₁₇₃	C₂×C₄	1	2	yes	—
33	C₂×Q₈×C₁₇₃	D₄	1	2	yes	—
34	C₂²×C₄×C₁₇₃	D₄	1	2	yes	—
35	C₂²×C₄×C₁₇₃	C₂³	1	1	yes	characteristic
36	C₄⋊C₄×C₁₇₃	—	2	4	no	—
37	C₄⋊C₄×C₁₇₃	D₄	1	4	yes	—
38	C₂²⋊C₄×C₁₇₃	—	2	2	no	—
39	C₄²×C₁₇₃	—	2	4	no	—
40	C₄²×C₁₇₃	—	2	2	no	—
41	Dic₃₄₆	—	2	4	no	—
42	Dic₃₄₆	—	4	2	no	—
43	C₂×Dic₁₇₃	—	2	4	no	—
44	C₂×Dic₁₇₃	—	2	2	no	—
45	C₂×Dic₁₇₃	—	2	2	no	—
46	C₂×Dic₁₇₃	—	4	1	no	—
47	C₂×Dic₁₇₃	—	4	1	no	—
48	#13063436	—	4	2	no	—
49	C₂³×C₁₇₃	C₂×D₄	1	1	yes	characteristic
50	Q₈×C₁₇₃	—	2	4	no	—
51	Q₈×C₁₇₃	—	2	4	no	—
52	C₂×C₄×C₁₇₃	—	2	2	no	—
53	C₂×C₄×C₁₇₃	—	2	2	no	—
54	C₂×C₄×C₁₇₃	—	2	4	no	—
55	C₂×C₄×C₁₇₃	C₂²⋊C₄	1	4	yes	—
56	C₂×C₄×C₁₇₃	C₂²×C₄	1	1	yes	characteristic
57	C₂×C₄×C₁₇₃	C₂×D₄	1	2	yes	—
58	C₂×C₄×C₁₇₃	C₂×D₄	1	1	yes	characteristic
59	C₂×C₄×C₁₇₃	#41	1	2	yes	—
60	C₂²×C₁₇₃	—	2	2	no	—
61	C₂²×C₁₇₃	C₂×C₂²⋊C₄	1	1	yes	characteristic
62	C₂²×C₁₇₃	#120	1	1	yes	characteristic
63	C₂²×C₁₇₃	#123	1	1	yes	characteristic
64	C₄×C₁₇₃	—	2	4	no	—
65	C₄×C₁₇₃	—	4	4	no	—
66	C₄×C₁₇₃	C₄×D₄	1	2	yes	—
67	C₄×C₁₇₃	#121	1	2	yes	—
68	Dic₁₇₃	—	2	4	no	—
69	Dic₁₇₃	—	4	2	no	—
70	C₂×C₁₇₃	—	2	1	no	—
71	C₂×C₁₇₃	—	2	1	no	—
72	C₂×C₁₇₃	#386	1	1	yes	characteristic
73	C₂×C₁₇₃	#456	1	2	yes	—
74	C₁₇₃	#1886	1	1	yes	characteristic, complemented, Sylow
75	#1886	—	173	1	no	maximal, Sylow
76	C₂×C₄⋊Q₈	—	173	1	no	—
77	#519	D₁₇₃	1	1	yes	characteristic
78	#419	—	173	2	no	—
79	C₂×#101	—	173	1	no	—
80	#339	—	173	2	no	—
81	C₂²×Q₈	—	173	1	no	—
82	C₂²×Q₈	C₂×D₁₇₃	1	1	yes	characteristic
83	C₂×M₁₆	—	173	2	no	—
84	C₄⋊Q₈	—	346	2	no	—
85	C₄⋊Q₈	—	173	4	no	—
86	C₄×Q₈	—	2	4	no	—
87	#117	—	2	2	no	—
88	#117	C₂×D₁₇₃	1	2	yes	—
89	C₂×C₄⋊C₄	—	346	2	no	—
90	C₂×C₄²	—	173	1	no	—
91	#103	—	346	4	no	—
92	#101	—	346	2	no	—
93	C₂×Q₈	—	2	2	no	—
94	C₂×Q₈	—	2	2	no	—
95	C₂×Q₈	—	346	1	no	—
96	C₂×Q₈	—	346	2	no	—
97	C₂×Q₈	—	346	2	no	—
98	C₂×Q₈	—	173	2	no	—
99	C₂×Q₈	C₄×D₁₇₃	1	2	yes	—
100	C₂×Q₈	D₆₉₂	1	2	yes	—
101	C₂²×C₄	—	346	1	no	—
102	C₂²×C₄	—	346	1	no	—
103	C₂²×C₄	C₂²⋊D₁₇₃	1	2	yes	—
104	C₂²×C₄	C₂²×D₁₇₃	1	1	yes	characteristic
105	M₁₆	—	346	2	no	—
106	M₁₆	—	346	2	no	—
107	C₂×C₈	—	346	2	no	—
108	C₄⋊C₄	—	2	4	no	—
109	C₄⋊C₄	—	346	8	no	—
110	C₄⋊C₄	C₂²⋊D₁₇₃	1	4	yes	—
111	C₂²⋊C₄	—	2	2	no	—
112	C₄²	—	2	4	no	—
113	C₄²	—	2	2	no	—
114	C₄²	—	346	1	no	—
115	C₄²	—	173	2	no	—
116	C₂³	C₂×C₂²⋊D₁₇₃	1	1	yes	characteristic
117	Q₈	—	2	4	no	—
118	Q₈	—	2	4	no	—
119	Q₈	—	346	4	no	—
120	Q₈	—	692	2	no	—
121	C₂×C₄	—	2	2	no	—
122	C₂×C₄	—	2	2	no	—
123	C₂×C₄	—	2	4	no	—
124	C₂×C₄	—	346	4	no	—
125	C₂×C₄	—	346	2	no	—
126	C₂×C₄	—	346	2	no	—
127	C₂×C₄	—	692	1	no	—
128	C₂×C₄	—	692	1	no	—
129	C₂×C₄	#14583803	1	4	yes	—
130	C₂×C₄	C₂×C₄×D₁₇₃	1	1	yes	characteristic
131	C₂×C₄	C₂×C₂²⋊D₁₇₃	1	2	yes	—
132	C₂×C₄	C₂×D₆₉₂	1	1	yes	characteristic
133	C₂×C₄	#14583826	1	2	yes	—
134	C₈	—	692	2	no	—
135	C₂²	—	2	2	no	—
136	C₂²	C₂×#14583803	1	1	yes	characteristic
137	C₂²	#19149648	1	1	yes	characteristic
138	C₂²	#19149662	1	1	yes	characteristic
139	C₄	—	2	4	no	—
140	C₄	—	4	4	no	—
141	C₄	—	346	4	no	—
142	C₄	—	692	2	no	—
143	C₄	C₄×C₂²⋊D₁₇₃	1	2	yes	—
144	C₄	#19149653	1	2	yes	—
145	C₂	—	2	1	no	—
146	C₂	—	2	1	no	—
147	C₂	#26011370	1	1	yes	characteristic
148	C₂	#26011674	1	2	yes	—

subgroup

G/H

class size

multiplicity

normal?

properties

C₂×#19149677

C₂

yes

maximal, characteristic

#26011499

C₂

yes

maximal

C₂×#19149598

C₂

yes

maximal, characteristic

#26011250

C₂

yes

maximal

#519×C₁₇₃

C₂

yes

maximal, characteristic

C₂²×Dic₃₄₆

C₂²

yes

characteristic

C₂×#14583810

C₂²

yes

—

#19149677

—

#19149677

C₄

yes

complemented

C₂×#14583805

—

C₂×C₄×Dic₁₇₃

C₂²

yes

characteristic

#19149602

—

#19149598

—

C₂²×Q₈×C₁₇₃

C₂²

yes

characteristic

C₄×Q₈×C₁₇₃

—

#117×C₁₇₃

—

#117×C₁₇₃

C₂²

yes

—

C₂×Dic₃₄₆

—

C₂×Dic₃₄₆

—

C₂×Dic₃₄₆

—

C₂×Dic₃₄₆

C₂×C₄

yes

—

C₂²×Dic₁₇₃

—

C₂²×Dic₁₇₃

—

—

—

—

—

C₄×Dic₁₇₃

—

C₄×Dic₁₇₃

C₂×C₄

yes

—

C₂×Q₈×C₁₇₃

—

C₂×Q₈×C₁₇₃

—

C₂×Q₈×C₁₇₃

C₂×C₄

yes

—

C₂×Q₈×C₁₇₃

D₄

yes

—

C₂²×C₄×C₁₇₃

D₄

yes

—

C₂²×C₄×C₁₇₃

C₂³

yes

characteristic

C₄⋊C₄×C₁₇₃

—

C₄⋊C₄×C₁₇₃

D₄

yes

—

C₂²⋊C₄×C₁₇₃

—

C₄²×C₁₇₃

—

C₄²×C₁₇₃

—

Dic₃₄₆

—

Dic₃₄₆

—

C₂×Dic₁₇₃

—

C₂×Dic₁₇₃

—

C₂×Dic₁₇₃

—

C₂×Dic₁₇₃

—

C₂×Dic₁₇₃

—

#13063436

—

C₂³×C₁₇₃

C₂×D₄

yes

characteristic

Q₈×C₁₇₃

—

Q₈×C₁₇₃

—

C₂×C₄×C₁₇₃

—

C₂×C₄×C₁₇₃

—

C₂×C₄×C₁₇₃

—

C₂×C₄×C₁₇₃

C₂²⋊C₄

yes

—

C₂×C₄×C₁₇₃

C₂²×C₄

yes

characteristic

C₂×C₄×C₁₇₃

C₂×D₄

yes

—

C₂×C₄×C₁₇₃

C₂×D₄

yes

characteristic

C₂×C₄×C₁₇₃

#41

yes

—

C₂²×C₁₇₃

—

C₂²×C₁₇₃

C₂×C₂²⋊C₄

yes

C₂²×C₁₇₃

yes

C₂²×C₁₇₃

yes

C₄×C₁₇₃

—

C₄×C₁₇₃

—

C₄×C₁₇₃

C₄×D₄

yes

—

C₄×C₁₇₃

#121

yes

—

Dic₁₇₃

—

Dic₁₇₃

—

C₂×C₁₇₃

—

C₂×C₁₇₃

—

C₂×C₁₇₃

#386

yes

characteristic

C₂×C₁₇₃

#456

yes

—

C₁₇₃

#1886

yes

characteristic, complemented, Sylow

#1886

—

173

maximal, Sylow

C₂×C₄⋊Q₈

—

173

—

#519

D₁₇₃

yes

characteristic

#419

—

173

—

C₂×#101

—

173

—

#339

—

173

—

C₂²×Q₈

—

173

—

C₂²×Q₈

C₂×D₁₇₃

yes

characteristic

C₂×M₁₆

—

173

—

C₄⋊Q₈

—

346

—

C₄⋊Q₈

—

173

—

C₄×Q₈

—

#117

—

#117

C₂×D₁₇₃

yes

—

C₂×C₄⋊C₄

—

346

—

C₂×C₄²

—

173

—

#103

—

346

—

#101

—

346

—

C₂×Q₈

—

C₂×Q₈

—

C₂×Q₈

—

346

—

C₂×Q₈

—

346

—

C₂×Q₈

—

346

—

C₂×Q₈

—

173

—

C₂×Q₈

C₄×D₁₇₃

yes

—

100

C₂×Q₈

D₆₉₂

yes

—

101

C₂²×C₄

—

346

—

102

C₂²×C₄

—

346

—

103

C₂²×C₄

C₂²⋊D₁₇₃

yes

—

104

C₂²×C₄

C₂²×D₁₇₃

yes

characteristic

105

M₁₆

—

346

—

106

M₁₆

—

346

—

107

C₂×C₈

—

346

—

108

C₄⋊C₄

—

109

C₄⋊C₄

—

346

—

110

C₄⋊C₄

C₂²⋊D₁₇₃

yes

—

111

C₂²⋊C₄

—

112

C₄²

—

113

C₄²

—

114

C₄²

—

346

—

115

C₄²

—

173

—

116

C₂³

C₂×C₂²⋊D₁₇₃

yes

characteristic

117

Q₈

—

118

Q₈

—

119

Q₈

—

346

—

120

Q₈

—

692

—

121

C₂×C₄

—

122

C₂×C₄

—

123

C₂×C₄

—

124

C₂×C₄

—

346

—

125

C₂×C₄

—

346

—

126

C₂×C₄

—

346

—

127

C₂×C₄

—

692

—

128

C₂×C₄

—

692

—

129

C₂×C₄

#14583803

yes

—

130

C₂×C₄

C₂×C₄×D₁₇₃

yes

characteristic

131

C₂×C₄

C₂×C₂²⋊D₁₇₃

yes

—

132

C₂×C₄

C₂×D₆₉₂

yes

characteristic

133

C₂×C₄

#14583826

yes

—

134

C₈

—

692

—

135

C₂²

—

136

C₂²

C₂×#14583803

yes

characteristic

137

C₂²

#19149648

yes

characteristic

138

C₂²

#19149662

yes

characteristic

139

C₄

—

140

C₄

—

141

C₄

—

346

—

142

C₄

—

692

—

143

C₄

C₄×C₂²⋊D₁₇₃

yes

—

144

C₄

#19149653

yes

—

145

C₂

—

146

C₂

—

147

C₂

#26011370

yes

characteristic

148

C₂

#26011674

yes

—

	subgroup	G/H
center	C₂²	#19149648
commutator	C₂×C₄×C₁₇₃	C₂²×C₄
Frattini	C₂²×C₄	C₂²×D₁₇₃
Fitting	#519×C₁₇₃	C₂
socle	C₂²×C₁₇₃	#120
prime radical	C₂³×C₁₇₃	C₂×D₄
cosocle	C₂²×C₄×C₁₇₃	C₂³
hypercenter	#519	D₁₇₃
hypocenter	C₁₇₃	#1886
LCS	G ▷ C₂×C₄×C₁₇₃ ▷ C₂×C₁₇₃ ▷ C₁₇₃
UCS	C₁ ◁ C₂² ◁ C₂²×C₄ ◁ #519

subgroup

G/H

center

C₂²

#19149648

commutator

C₂×C₄×C₁₇₃

C₂²×C₄

Frattini

C₂²×C₄

C₂²×D₁₇₃

Fitting

#519×C₁₇₃

C₂

socle

C₂²×C₁₇₃

#120

prime radical

C₂³×C₁₇₃

C₂×D₄

cosocle

C₂²×C₄×C₁₇₃

C₂³

D₁₇₃

C₁₇₃

G ▷ C₂×C₄×C₁₇₃ ▷ C₂×C₁₇₃ ▷ C₁₇₃

UCS

C₁ ◁ C₂² ◁ C₂²×C₄ ◁ #519

N	H ≅ G/N	type
9. #19149677	C₄	semidirect
74. C₁₇₃	#1886	semidirect

H ≅ G/N

type

9. #19149677

C₄

semidirect

74. C₁₇₃

#1886

semidirect

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

subgroup	index
C₂×#19149677	2
#26011499	2
C₂×#19149598	2
#26011250	2
#519×C₁₇₃	2
#1886	173

subgroup

index

#519×C₁₇₃

173

Subgroups — #36353956

Conjugacy classes of subgroups

Complements

Maximal subgroups