Group #36353654 — subgroups

Subgroups — #36353654

Order 22144.

344 conjugacy classes of subgroups; only the 117 normal subgroups are shown.

normal subgroups only

This group has 344 conjugacy classes of subgroups — too many to draw. Only its 117 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	C₂²×C₄⋊Dic₁₇₃	C₂	1	1	yes	maximal, characteristic
2	#26011351	C₂	1	1	yes	maximal, characteristic
3	#26011254	C₂	1	4	yes	maximal, complemented
4	C₂×C₂²⋊C₈×C₁₇₃	C₂	1	1	yes	maximal, characteristic
5	C₂³×Dic₁₇₃	—	2	1	no	—
6	C₂×C₄⋊Dic₁₇₃	—	2	2	no	—
7	C₂×C₄⋊Dic₁₇₃	—	2	1	no	—
8	C₂×C₄⋊Dic₁₇₃	C₄	1	4	yes	complemented
9	C₂×C₄⋊Dic₁₇₃	C₂²	1	2	yes	—
10	C₂×C₄⋊Dic₁₇₃	C₂²	1	2	yes	—
11	C₂×C₂²⋊Dic₁₇₃	—	2	1	no	—
12	#19149586	—	2	1	no	—
13	C₂³×C₄×C₁₇₃	C₂²	1	1	yes	characteristic
14	C₂²×C₈×C₁₇₃	C₂²	1	2	yes	—
15	C₂²⋊C₈×C₁₇₃	—	2	2	no	—
16	C₂²×Dic₁₇₃	—	2	4	no	—
17	C₂²×Dic₁₇₃	—	2	2	no	—
18	C₂²×Dic₁₇₃	—	2	2	no	—
19	C₂²×Dic₁₇₃	—	4	4	no	—
20	C₄⋊Dic₁₇₃	—	2	4	no	—
21	C₄⋊Dic₁₇₃	—	2	2	no	—
22	C₄⋊Dic₁₇₃	—	2	4	no	—
23	C₄⋊Dic₁₇₃	C₂×C₄	1	4	yes	—
24	C₂²⋊Dic₁₇₃	—	4	2	no	—
25	C₂⁴×C₁₇₃	D₄	1	1	yes	characteristic
26	C₂²×C₄×C₁₇₃	—	2	4	no	—
27	C₂²×C₄×C₁₇₃	C₂×C₄	1	2	yes	—
28	C₂²×C₄×C₁₇₃	D₄	1	1	yes	characteristic
29	C₂²×C₄×C₁₇₃	D₄	1	2	yes	—
30	C₂²×C₄×C₁₇₃	C₂³	1	1	yes	characteristic
31	C₂×C₈×C₁₇₃	—	2	2	no	—
32	C₂×C₈×C₁₇₃	—	2	4	no	—
33	C₂×Dic₁₇₃	—	2	4	no	—
34	C₂×Dic₁₇₃	—	4	8	no	—
35	C₂×Dic₁₇₃	—	4	4	no	—
36	C₂×Dic₁₇₃	—	4	2	no	—
37	C₂×Dic₁₇₃	—	4	2	no	—
38	C₂×Dic₁₇₃	—	4	2	no	—
39	C₂³×C₁₇₃	—	2	2	no	—
40	C₂³×C₁₇₃	—	2	2	no	—
41	C₂³×C₁₇₃	C₂²⋊C₄	1	2	yes	—
42	C₂³×C₁₇₃	SD₁₆	1	2	yes	—
43	C₂³×C₁₇₃	Q₁₆	1	2	yes	—
44	C₂³×C₁₇₃	C₂×D₄	1	1	yes	characteristic
45	C₂×C₄×C₁₇₃	—	2	2	no	—
46	C₂×C₄×C₁₇₃	—	2	4	no	—
47	C₂×C₄×C₁₇₃	—	2	4	no	—
48	C₂×C₄×C₁₇₃	C₂²⋊C₄	1	2	yes	—
49	C₂×C₄×C₁₇₃	C₂²×C₄	1	1	yes	characteristic
50	C₂×C₄×C₁₇₃	C₂×D₄	1	1	yes	characteristic
51	C₂×C₄×C₁₇₃	#41	1	4	yes	—
52	C₈×C₁₇₃	—	4	2	no	—
53	C₂²×C₁₇₃	—	2	2	no	—
54	C₂²×C₁₇₃	—	2	2	no	—
55	C₂²×C₁₇₃	—	2	4	no	—
56	C₂²×C₁₇₃	—	2	4	no	—
57	C₂²×C₁₇₃	#103	1	4	yes	—
58	C₂²×C₁₇₃	C₂×C₂²⋊C₄	1	1	yes	characteristic
59	C₂²×C₁₇₃	#123	1	2	yes	—
60	C₂²×C₁₇₃	C₂×SD₁₆	1	1	yes	characteristic
61	C₂²×C₁₇₃	C₂×Q₁₆	1	1	yes	characteristic
62	C₂²×C₁₇₃	AGL(1,ℤ/8)	1	2	yes	—
63	C₄×C₁₇₃	—	2	2	no	—
64	C₄×C₁₇₃	#117	1	2	yes	—
65	C₄×C₁₇₃	#123	1	2	yes	—
66	Dic₁₇₃	—	4	4	no	—
67	Dic₁₇₃	—	8	2	no	—
68	C₂×C₁₇₃	—	2	4	no	—
69	C₂×C₁₇₃	#381	1	1	yes	characteristic
70	C₂×C₁₇₃	C₂×#103	1	1	yes	characteristic
71	C₂×C₁₇₃	#418	1	1	yes	characteristic
72	C₂×C₁₇₃	#481	1	2	yes	—
73	C₂×C₁₇₃	#484	1	2	yes	—
74	C₁₇₃	#1824	1	1	yes	characteristic, complemented, Sylow
75	#1824	—	173	1	no	maximal, Sylow
76	C₂²×C₄⋊C₄	—	173	1	no	—
77	C₂×C₂²⋊C₈	D₁₇₃	1	1	yes	characteristic
78	#380	—	173	1	no	—
79	#340	—	173	4	no	—
80	C₂³×C₄	—	346	1	no	—
81	C₂³×C₄	C₂×D₁₇₃	1	1	yes	characteristic
82	C₂²×C₈	C₂×D₁₇₃	1	2	yes	—
83	C₂×C₄⋊C₄	—	346	2	no	—
84	C₂×C₄⋊C₄	—	346	1	no	—
85	C₂×C₄⋊C₄	—	173	2	no	—
86	C₂×C₄⋊C₄	—	173	4	no	—
87	C₂×C₄⋊C₄	—	173	2	no	—
88	C₂×C₂²⋊C₄	—	346	1	no	—
89	C₂²⋊C₈	—	2	2	no	—
90	#95	—	346	1	no	—
91	C₂⁴	D₆₉₂	1	1	yes	characteristic
92	C₂²×C₄	—	2	4	no	—
93	C₂²×C₄	—	346	2	no	—
94	C₂²×C₄	—	346	4	no	—
95	C₂²×C₄	—	346	2	no	—
96	C₂²×C₄	—	692	4	no	—
97	C₂²×C₄	C₄×D₁₇₃	1	2	yes	—
98	C₂²×C₄	C₂²⋊D₁₇₃	1	2	yes	—
99	C₂²×C₄	D₆₉₂	1	1	yes	characteristic
100	C₂²×C₄	C₂²×D₁₇₃	1	1	yes	characteristic
101	C₂×C₈	—	2	2	no	—
102	C₂×C₈	—	2	4	no	—
103	C₄⋊C₄	—	346	2	no	—
104	C₄⋊C₄	—	346	4	no	—
105	C₄⋊C₄	—	346	4	no	—
106	C₄⋊C₄	—	173	4	no	—
107	C₂²⋊C₄	—	692	2	no	—
108	C₂³	—	2	2	no	—
109	C₂³	—	2	2	no	—
110	C₂³	#14583803	1	2	yes	—
111	C₂³	#14583815	1	2	yes	—
112	C₂³	Dic₆₉₂	1	2	yes	—
113	C₂³	C₂×D₆₉₂	1	1	yes	characteristic
114	C₂×C₄	—	2	2	no	—
115	C₂×C₄	—	2	4	no	—
116	C₂×C₄	—	2	4	no	—
117	C₂×C₄	—	346	4	no	—
118	C₂×C₄	—	692	2	no	—
119	C₂×C₄	—	692	2	no	—
120	C₂×C₄	—	692	2	no	—
121	C₂×C₄	—	692	8	no	—
122	C₂×C₄	—	692	4	no	—
123	C₂×C₄	#14583803	1	2	yes	—
124	C₂×C₄	C₂×C₄×D₁₇₃	1	1	yes	characteristic
125	C₂×C₄	C₂×C₂²⋊D₁₇₃	1	1	yes	characteristic
126	C₂×C₄	#14583825	1	4	yes	—
127	C₈	—	4	2	no	—
128	C₂²	—	2	2	no	—
129	C₂²	—	2	4	no	—
130	C₂²	—	2	4	no	—
131	C₂²	—	2	2	no	—
132	C₂²	#19149604	1	4	yes	—
133	C₂²	C₂×#14583803	1	1	yes	characteristic
134	C₂²	#19149663	1	2	yes	—
135	C₂²	C₂×#14583815	1	1	yes	characteristic
136	C₂²	C₂×Dic₆₉₂	1	1	yes	characteristic
137	C₂²	#19149707	1	2	yes	—
138	C₄	—	2	2	no	—
139	C₄	—	1384	2	no	—
140	C₄	—	692	4	no	—
141	C₄	#19149635	1	2	yes	—
142	C₄	#19149660	1	2	yes	—
143	C₂	—	2	4	no	—
144	C₂	#26011352	1	1	yes	characteristic
145	C₂	C₂×#19149604	1	1	yes	characteristic
146	C₂	#26011497	1	1	yes	characteristic
147	C₂	#26011810	1	2	yes	—
148	C₂	#26011827	1	2	yes	—

subgroup

G/H

class size

multiplicity

normal?

properties

C₂²×C₄⋊Dic₁₇₃

C₂

yes

maximal, characteristic

#26011351

C₂

yes

maximal, characteristic

#26011254

C₂

yes

maximal, complemented

C₂×C₂²⋊C₈×C₁₇₃

C₂

yes

maximal, characteristic

C₂³×Dic₁₇₃

—

C₂×C₄⋊Dic₁₇₃

—

C₂×C₄⋊Dic₁₇₃

—

C₂×C₄⋊Dic₁₇₃

C₄

yes

complemented

C₂×C₄⋊Dic₁₇₃

C₂²

yes

—

C₂×C₄⋊Dic₁₇₃

C₂²

yes

—

C₂×C₂²⋊Dic₁₇₃

—

#19149586

—

C₂³×C₄×C₁₇₃

C₂²

yes

characteristic

C₂²×C₈×C₁₇₃

C₂²

yes

—

C₂²⋊C₈×C₁₇₃

—

C₂²×Dic₁₇₃

—

C₂²×Dic₁₇₃

—

C₂²×Dic₁₇₃

—

C₂²×Dic₁₇₃

—

C₄⋊Dic₁₇₃

—

C₄⋊Dic₁₇₃

—

C₄⋊Dic₁₇₃

—

C₄⋊Dic₁₇₃

C₂×C₄

yes

—

C₂²⋊Dic₁₇₃

—

C₂⁴×C₁₇₃

D₄

yes

characteristic

C₂²×C₄×C₁₇₃

—

C₂²×C₄×C₁₇₃

C₂×C₄

yes

—

C₂²×C₄×C₁₇₃

D₄

yes

characteristic

C₂²×C₄×C₁₇₃

D₄

yes

—

C₂²×C₄×C₁₇₃

C₂³

yes

characteristic

C₂×C₈×C₁₇₃

—

C₂×C₈×C₁₇₃

—

C₂×Dic₁₇₃

—

C₂×Dic₁₇₃

—

C₂×Dic₁₇₃

—

C₂×Dic₁₇₃

—

C₂×Dic₁₇₃

—

C₂×Dic₁₇₃

—

C₂³×C₁₇₃

—

C₂³×C₁₇₃

—

C₂³×C₁₇₃

C₂²⋊C₄

yes

—

C₂³×C₁₇₃

SD₁₆

yes

—

C₂³×C₁₇₃

Q₁₆

yes

—

C₂³×C₁₇₃

C₂×D₄

yes

characteristic

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

characteristic

C₂×C₄×C₁₇₃

#41

yes

—

C₈×C₁₇₃

—

C₂²×C₁₇₃

—

C₂²×C₁₇₃

—

C₂²×C₁₇₃

—

C₂²×C₁₇₃

—

C₂²×C₁₇₃

#103

yes

—

C₂²×C₁₇₃

C₂×C₂²⋊C₄

yes

characteristic

C₂²×C₁₇₃

#123

yes

—

C₂²×C₁₇₃

C₂×SD₁₆

yes

characteristic

C₂²×C₁₇₃

C₂×Q₁₆

yes

characteristic

C₂²×C₁₇₃

AGL(1,ℤ/8)

yes

—

C₄×C₁₇₃

—

C₄×C₁₇₃

#117

yes

—

C₄×C₁₇₃

#123

yes

—

Dic₁₇₃

—

Dic₁₇₃

—

C₂×C₁₇₃

—

C₂×C₁₇₃

yes

C₂×C₁₇₃

yes

C₂×C₁₇₃

yes

C₂×C₁₇₃

yes

—

C₂×C₁₇₃

#484

yes

—

C₁₇₃

#1824

yes

characteristic, complemented, Sylow

#1824

—

173

maximal, Sylow

C₂²×C₄⋊C₄

—

173

—

C₂×C₂²⋊C₈

D₁₇₃

yes

characteristic

#380

—

173

—

#340

—

173

—

C₂³×C₄

—

346

—

C₂³×C₄

C₂×D₁₇₃

yes

characteristic

C₂²×C₈

C₂×D₁₇₃

yes

—

C₂×C₄⋊C₄

—

346

—

C₂×C₄⋊C₄

—

346

—

C₂×C₄⋊C₄

—

173

—

C₂×C₄⋊C₄

—

173

—

C₂×C₄⋊C₄

—

173

—

C₂×C₂²⋊C₄

—

346

—

C₂²⋊C₈

—

#95

—

346

—

C₂⁴

D₆₉₂

yes

characteristic

C₂²×C₄

—

C₂²×C₄

—

346

—

C₂²×C₄

—

346

—

C₂²×C₄

—

346

—

C₂²×C₄

—

692

—

C₂²×C₄

C₄×D₁₇₃

yes

—

C₂²×C₄

C₂²⋊D₁₇₃

yes

—

C₂²×C₄

D₆₉₂

yes

characteristic

100

C₂²×C₄

C₂²×D₁₇₃

yes

characteristic

101

C₂×C₈

—

102

C₂×C₈

—

103

C₄⋊C₄

—

346

—

104

C₄⋊C₄

—

346

—

105

C₄⋊C₄

—

346

—

106

C₄⋊C₄

—

173

—

107

C₂²⋊C₄

—

692

—

108

C₂³

—

109

C₂³

—

110

C₂³

#14583803

yes

—

111

C₂³

#14583815

yes

—

112

C₂³

Dic₆₉₂

yes

—

113

C₂³

C₂×D₆₉₂

yes

characteristic

114

C₂×C₄

—

115

C₂×C₄

—

116

C₂×C₄

—

117

C₂×C₄

—

346

—

118

C₂×C₄

—

692

—

119

C₂×C₄

—

692

—

120

C₂×C₄

—

692

—

121

C₂×C₄

—

692

—

122

C₂×C₄

—

692

—

123

C₂×C₄

#14583803

yes

—

124

C₂×C₄

C₂×C₄×D₁₇₃

yes

characteristic

125

C₂×C₄

C₂×C₂²⋊D₁₇₃

yes

characteristic

126

C₂×C₄

#14583825

yes

—

127

C₈

—

128

C₂²

—

129

C₂²

—

130

C₂²

—

131

C₂²

—

132

C₂²

#19149604

yes

—

133

C₂²

C₂×#14583803

yes

characteristic

134

C₂²

#19149663

yes

—

135

C₂²

C₂×#14583815

yes

characteristic

136

C₂²

C₂×Dic₆₉₂

yes

characteristic

137

C₂²

#19149707

yes

—

138

C₄

—

139

C₄

—

1384

—

140

C₄

—

692

—

141

C₄

#19149635

yes

—

142

C₄

#19149660

yes

—

143

C₂

—

144

C₂

#26011352

yes

characteristic

145

C₂

C₂×#19149604

yes

characteristic

146

C₂

#26011497

yes

characteristic

147

C₂

#26011810

yes

—

148

C₂

#26011827

yes

—

	subgroup	G/H
center	C₂³	C₂×D₆₉₂
commutator	C₂×C₄×C₁₇₃	C₂²×C₄
Frattini	C₂²×C₄	C₂²×D₁₇₃
Fitting	C₂×C₂²⋊C₈×C₁₇₃	C₂
socle	C₂³×C₁₇₃	C₂×D₄
prime radical	C₂⁴×C₁₇₃	D₄
cosocle	C₂²×C₄×C₁₇₃	C₂³
hypercenter	C₂×C₂²⋊C₈	D₁₇₃
hypocenter	C₁₇₃	#1824
LCS	G ▷ C₂×C₄×C₁₇₃ ▷ C₂×C₁₇₃ ▷ C₁₇₃
UCS	C₁ ◁ C₂³ ◁ C₂³×C₄ ◁ C₂×C₂²⋊C₈

subgroup

G/H

center

C₂³

C₂×D₆₉₂

commutator

C₂×C₄×C₁₇₃

C₂²×C₄

Frattini

C₂²×C₄

C₂²×D₁₇₃

Fitting

C₂×C₂²⋊C₈×C₁₇₃

C₂

socle

C₂³×C₁₇₃

C₂×D₄

prime radical

C₂⁴×C₁₇₃

D₄

cosocle

C₂²×C₄×C₁₇₃

C₂³

hypercenter

C₂×C₂²⋊C₈

D₁₇₃

hypocenter

C₁₇₃

#1824

LCS

G ▷ C₂×C₄×C₁₇₃ ▷ C₂×C₁₇₃ ▷ C₁₇₃

UCS

C₁ ◁ C₂³ ◁ C₂³×C₄ ◁ C₂×C₂²⋊C₈

N	H ≅ G/N	type
3. #26011254	C₂	semidirect
8. C₂×C₄⋊Dic₁₇₃	C₄	semidirect
74. C₁₇₃	#1824	semidirect

H ≅ G/N

type

3. #26011254

C₂

semidirect

8. C₂×C₄⋊Dic₁₇₃

C₄

semidirect

74. C₁₇₃

#1824

semidirect

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

subgroup	index
C₂²×C₄⋊Dic₁₇₃	2
#26011351	2
#26011254	2
C₂×C₂²⋊C₈×C₁₇₃	2
#1824	173

subgroup

index

C₂²×C₄⋊Dic₁₇₃

#26011351

#26011254

C₂×C₂²⋊C₈×C₁₇₃

#1824

173

Subgroups — #36353654

Conjugacy classes of subgroups

Complements

Maximal subgroups