Group #36354412 — subgroups

Subgroups — #36354412

Order 22144.

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

normal subgroups only

This group has 312 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×#19149667	C₂	1	1	yes	maximal, characteristic
2	#26011515	C₂	1	1	yes	maximal, characteristic, complemented
3	C₂×#19149597	C₂	1	2	yes	maximal, complemented
4	C₂×#19149595	C₂	1	2	yes	maximal
5	C₄×M₁₆×C₁₇₃	C₂	1	1	yes	maximal, characteristic, complemented
6	C₂²×Dic₃₄₆	C₂²	1	1	yes	characteristic
7	C₂²×D₆₉₂	C₂²	1	1	yes	characteristic
8	C₂×#14583810	C₂²	1	2	yes	—
9	#19149667	—	2	1	no	—
10	#19149667	—	2	1	no	—
11	#19149667	—	2	2	no	—
12	C₂×#14583803	C₄	1	4	yes	—
13	C₄⋊#13063436	—	2	2	no	—
14	#19149597	—	2	4	no	—
15	#19149595	—	2	4	no	—
16	C₂×M₁₆×C₁₇₃	C₂²	1	2	yes	—
17	C₂×C₄²×C₁₇₃	C₂²	1	1	yes	characteristic
18	#97×C₁₇₃	—	2	1	no	—
19	C₄×C₈×C₁₇₃	—	2	1	no	—
20	C₂³×D₁₇₃	C₂×C₄	1	2	yes	—
21	C₂×Dic₃₄₆	—	2	2	no	—
22	C₂×Dic₃₄₆	—	2	1	no	—
23	C₂×Dic₃₄₆	—	2	1	no	—
24	C₂×Dic₃₄₆	—	2	2	no	—
25	C₂×D₆₉₂	—	2	2	no	—
26	C₂×D₆₉₂	—	2	1	no	—
27	C₂×D₆₉₂	—	2	1	no	—
28	C₂×D₆₉₂	—	2	2	no	—
29	C₂²×Dic₁₇₃	C₂×C₄	1	2	yes	—
30	#14583810	—	2	2	no	—
31	#14583810	—	2	2	no	—
32	C₂×#13063436	—	2	2	no	—
33	#14583803	—	2	4	no	—
34	#14583803	—	2	4	no	—
35	C₂²×C₄×C₁₇₃	C₂×C₄	1	2	yes	—
36	C₂²×C₄×C₁₇₃	C₂³	1	1	yes	characteristic
37	M₁₆×C₁₇₃	—	2	2	no	—
38	M₁₆×C₁₇₃	D₄	1	4	yes	—
39	C₂×C₈×C₁₇₃	—	2	2	no	—
40	C₄²×C₁₇₃	D₄	1	1	yes	characteristic
41	C₄²×C₁₇₃	D₄	1	1	yes	characteristic
42	C₄²×C₁₇₃	D₄	1	2	yes	—
43	C₂²×D₁₇₃	—	2	2	no	—
44	C₂²×D₁₇₃	—	2	4	no	—
45	C₂²×D₁₇₃	—	4	4	no	—
46	Dic₃₄₆	—	4	2	no	—
47	Dic₃₄₆	—	4	1	no	—
48	Dic₃₄₆	—	4	1	no	—
49	D₆₉₂	—	4	2	no	—
50	D₆₉₂	—	4	1	no	—
51	D₆₉₂	—	4	1	no	—
52	C₂×Dic₁₇₃	—	2	2	no	—
53	C₂×Dic₁₇₃	—	2	4	no	—
54	#13063436	—	4	2	no	—
55	C₂³×C₁₇₃	C₂²×C₄	1	1	yes	characteristic
56	C₂×C₄×C₁₇₃	—	4	2	no	—
57	C₂×C₄×C₁₇₃	C₂²⋊C₄	1	2	yes	—
58	C₂×C₄×C₁₇₃	C₂²⋊C₄	1	2	yes	—
59	C₂×C₄×C₁₇₃	C₂×D₄	1	1	yes	characteristic
60	C₂×C₄×C₁₇₃	C₂×D₄	1	2	yes	—
61	C₂×C₄×C₁₇₃	C₂×D₄	1	1	yes	characteristic
62	C₂×C₄×C₁₇₃	#41	1	2	yes	—
63	C₈×C₁₇₃	—	2	4	no	—
64	C₂²×C₁₇₃	—	2	2	no	—
65	C₂²×C₁₇₃	C₂×C₂²⋊C₄	1	1	yes	characteristic
66	C₂²×C₁₇₃	C₄×D₄	1	2	yes	—
67	C₂×D₁₇₃	—	4	2	no	—
68	C₂×D₁₇₃	—	4	4	no	—
69	C₂×D₁₇₃	—	8	4	no	—
70	C₄×C₁₇₃	—	2	2	no	—
71	C₄×C₁₇₃	—	2	2	no	—
72	C₄×C₁₇₃	#121	1	2	yes	—
73	C₄×C₁₇₃	#124	1	1	yes	characteristic
74	C₄×C₁₇₃	#127	1	1	yes	characteristic
75	Dic₁₇₃	—	4	2	no	—
76	C₂×C₁₇₃	—	2	2	no	—
77	C₂×C₁₇₃	#390	1	1	yes	characteristic
78	C₂×C₁₇₃	#413	1	2	yes	—
79	D₁₇₃	—	8	2	no	—
80	C₁₇₃	#1967	1	1	yes	characteristic, complemented, Sylow
81	#1967	—	173	1	no	maximal, Sylow
82	C₂×#124	—	173	1	no	—
83	#423	—	173	1	no	—
84	C₂×#101	—	173	2	no	—
85	C₂×#100	—	173	2	no	—
86	C₄×M₁₆	D₁₇₃	1	1	yes	characteristic, complemented
87	C₂²×Q₈	—	173	1	no	—
88	C₂²×D₄	—	173	1	no	—
89	C₂×M₁₆	—	173	2	no	—
90	C₂×M₁₆	C₂×D₁₇₃	1	2	yes	—
91	#124	—	346	1	no	—
92	#124	—	346	1	no	—
93	#124	—	346	2	no	—
94	C₂×C₂²⋊C₄	—	173	4	no	—
95	C₂×C₄²	C₂×D₁₇₃	1	1	yes	characteristic
96	C₄⋊C₈	—	346	2	no	—
97	#101	—	346	4	no	—
98	#100	—	346	4	no	—
99	#97	—	2	1	no	—
100	C₄×C₈	—	2	1	no	—
101	C₂⁴	—	173	2	no	—
102	C₂×Q₈	—	346	1	no	—
103	C₂×Q₈	—	346	2	no	—
104	C₂×Q₈	—	346	1	no	—
105	C₂×Q₈	—	346	2	no	—
106	C₂×D₄	—	346	1	no	—
107	C₂×D₄	—	346	2	no	—
108	C₂×D₄	—	346	1	no	—
109	C₂×D₄	—	346	2	no	—
110	C₂²×C₄	—	173	2	no	—
111	C₂²×C₄	C₄×D₁₇₃	1	2	yes	—
112	C₂²×C₄	C₂²×D₁₇₃	1	1	yes	characteristic
113	M₁₆	—	2	2	no	—
114	M₁₆	—	346	2	no	—
115	M₁₆	—	346	2	no	—
116	M₁₆	D₆₉₂	1	4	yes	—
117	C₂×C₈	—	2	2	no	—
118	C₂×C₈	—	346	2	no	—
119	C₂²⋊C₄	—	346	4	no	—
120	C₂²⋊C₄	—	346	4	no	—
121	C₄²	C₂²⋊D₁₇₃	1	1	yes	characteristic
122	C₄²	C₂²⋊D₁₇₃	1	1	yes	characteristic
123	C₄²	D₆₉₂	1	2	yes	—
124	C₂³	—	346	4	no	—
125	C₂³	—	346	2	no	—
126	C₂³	—	692	4	no	—
127	C₂³	C₂×C₄×D₁₇₃	1	1	yes	characteristic
128	Q₈	—	692	2	no	—
129	Q₈	—	692	1	no	—
130	Q₈	—	692	1	no	—
131	D₄	—	692	2	no	—
132	D₄	—	692	1	no	—
133	D₄	—	692	1	no	—
134	C₂×C₄	—	4	2	no	—
135	C₂×C₄	—	346	2	no	—
136	C₂×C₄	—	346	4	no	—
137	C₂×C₄	#14583803	1	2	yes	—
138	C₂×C₄	#14583803	1	2	yes	—
139	C₂×C₄	C₂×C₂²⋊D₁₇₃	1	1	yes	characteristic
140	C₂×C₄	C₂×D₆₉₂	1	2	yes	—
141	C₂×C₄	C₂×D₆₉₂	1	1	yes	characteristic
142	C₂×C₄	#14583826	1	2	yes	—
143	C₈	—	2	4	no	—
144	C₈	—	692	2	no	—
145	C₂²	—	2	2	no	—
146	C₂²	—	1384	4	no	—
147	C₂²	—	692	2	no	—
148	C₂²	—	692	4	no	—
149	C₂²	C₂×#14583803	1	1	yes	characteristic
150	C₂²	C₄×D₆₉₂	1	2	yes	—
151	C₄	—	2	2	no	—
152	C₄	—	2	2	no	—
153	C₄	—	692	2	no	—
154	C₄	#19149653	1	2	yes	—
155	C₄	#19149667	1	1	yes	characteristic
156	C₄	#19149676	1	1	yes	characteristic
157	C₂	—	2	2	no	—
158	C₂	—	1384	2	no	—
159	C₂	#26011393	1	1	yes	characteristic
160	C₂	#26011473	1	2	yes	—

subgroup

G/H

class size

multiplicity

normal?

properties

C₂×#19149667

C₂

yes

maximal, characteristic

#26011515

C₂

yes

maximal, characteristic, complemented

C₂×#19149597

C₂

yes

maximal, complemented

C₂×#19149595

C₂

yes

maximal

C₄×M₁₆×C₁₇₃

C₂

yes

maximal, characteristic, complemented

C₂²×Dic₃₄₆

C₂²

yes

characteristic

C₂²×D₆₉₂

C₂²

yes

characteristic

C₂×#14583810

C₂²

yes

—

—

—

—

C₄

yes

—

C₄⋊#13063436

—

#19149597

—

#19149595

—

C₂×M₁₆×C₁₇₃

C₂²

yes

—

C₂×C₄²×C₁₇₃

C₂²

yes

characteristic

#97×C₁₇₃

—

C₄×C₈×C₁₇₃

—

C₂³×D₁₇₃

C₂×C₄

yes

—

C₂×Dic₃₄₆

—

C₂×Dic₃₄₆

—

C₂×Dic₃₄₆

—

C₂×Dic₃₄₆

—

C₂×D₆₉₂

—

C₂×D₆₉₂

—

C₂×D₆₉₂

—

C₂×D₆₉₂

—

C₂²×Dic₁₇₃

C₂×C₄

yes

—

—

—

—

—

—

C₂²×C₄×C₁₇₃

C₂×C₄

yes

—

C₂²×C₄×C₁₇₃

C₂³

yes

characteristic

M₁₆×C₁₇₃

—

M₁₆×C₁₇₃

D₄

yes

—

C₂×C₈×C₁₇₃

—

C₄²×C₁₇₃

D₄

yes

characteristic

C₄²×C₁₇₃

D₄

yes

characteristic

C₄²×C₁₇₃

D₄

yes

—

C₂²×D₁₇₃

—

C₂²×D₁₇₃

—

C₂²×D₁₇₃

—

Dic₃₄₆

—

Dic₃₄₆

—

Dic₃₄₆

—

D₆₉₂

—

D₆₉₂

—

D₆₉₂

—

C₂×Dic₁₇₃

—

C₂×Dic₁₇₃

—

#13063436

—

C₂³×C₁₇₃

C₂²×C₄

yes

characteristic

C₂×C₄×C₁₇₃

—

C₂×C₄×C₁₇₃

C₂²⋊C₄

yes

—

C₂×C₄×C₁₇₃

C₂²⋊C₄

yes

—

C₂×C₄×C₁₇₃

C₂×D₄

yes

characteristic

C₂×C₄×C₁₇₃

C₂×D₄

yes

—

C₂×C₄×C₁₇₃

C₂×D₄

yes

characteristic

C₂×C₄×C₁₇₃

#41

yes

—

C₈×C₁₇₃

—

C₂²×C₁₇₃

—

C₂²×C₁₇₃

C₂×C₂²⋊C₄

yes

characteristic

C₂²×C₁₇₃

C₄×D₄

yes

—

C₂×D₁₇₃

—

C₂×D₁₇₃

—

C₂×D₁₇₃

—

C₄×C₁₇₃

—

C₄×C₁₇₃

—

C₄×C₁₇₃

#121

yes

—

C₄×C₁₇₃

yes

C₄×C₁₇₃

yes

Dic₁₇₃

—

C₂×C₁₇₃

—

C₂×C₁₇₃

#390

yes

characteristic

C₂×C₁₇₃

#413

yes

—

D₁₇₃

—

C₁₇₃

#1967

yes

characteristic, complemented, Sylow

#1967

—

173

maximal, Sylow

C₂×#124

—

173

—

#423

—

173

—

C₂×#101

—

173

—

C₂×#100

—

173

—

C₄×M₁₆

D₁₇₃

yes

characteristic, complemented

C₂²×Q₈

—

173

—

C₂²×D₄

—

173

—

C₂×M₁₆

—

173

—

C₂×M₁₆

C₂×D₁₇₃

yes

—

#124

—

346

—

#124

—

346

—

#124

—

346

—

C₂×C₂²⋊C₄

—

173

—

C₂×C₄²

C₂×D₁₇₃

yes

characteristic

C₄⋊C₈

—

346

—

#101

—

346

—

#100

—

346

—

#97

—

100

C₄×C₈

—

101

C₂⁴

—

173

—

102

C₂×Q₈

—

346

—

103

C₂×Q₈

—

346

—

104

C₂×Q₈

—

346

—

105

C₂×Q₈

—

346

—

106

C₂×D₄

—

346

—

107

C₂×D₄

—

346

—

108

C₂×D₄

—

346

—

109

C₂×D₄

—

346

—

110

C₂²×C₄

—

173

—

111

C₂²×C₄

C₄×D₁₇₃

yes

—

112

C₂²×C₄

C₂²×D₁₇₃

yes

characteristic

113

M₁₆

—

114

M₁₆

—

346

—

115

M₁₆

—

346

—

116

M₁₆

D₆₉₂

yes

—

117

C₂×C₈

—

118

C₂×C₈

—

346

—

119

C₂²⋊C₄

—

346

—

120

C₂²⋊C₄

—

346

—

121

C₄²

C₂²⋊D₁₇₃

yes

characteristic

122

C₄²

C₂²⋊D₁₇₃

yes

characteristic

123

C₄²

D₆₉₂

yes

—

124

C₂³

—

346

—

125

C₂³

—

346

—

126

C₂³

—

692

—

127

C₂³

C₂×C₄×D₁₇₃

yes

characteristic

128

Q₈

—

692

—

129

Q₈

—

692

—

130

Q₈

—

692

—

131

D₄

—

692

—

132

D₄

—

692

—

133

D₄

—

692

—

134

C₂×C₄

—

135

C₂×C₄

—

346

—

136

C₂×C₄

—

346

—

137

C₂×C₄

#14583803

yes

—

138

C₂×C₄

#14583803

yes

—

139

C₂×C₄

C₂×C₂²⋊D₁₇₃

yes

characteristic

140

C₂×C₄

C₂×D₆₉₂

yes

—

141

C₂×C₄

C₂×D₆₉₂

yes

characteristic

142

C₂×C₄

#14583826

yes

—

143

C₈

—

144

C₈

—

692

—

145

C₂²

—

146

C₂²

—

1384

—

147

C₂²

—

692

—

148

C₂²

—

692

—

149

C₂²

C₂×#14583803

yes

characteristic

150

C₂²

C₄×D₆₉₂

yes

—

151

C₄

—

152

C₄

—

153

C₄

—

692

—

154

C₄

#19149653

yes

—

155

C₄

#19149667

yes

characteristic

156

C₄

#19149676

yes

characteristic

157

C₂

—

158

C₂

—

1384

—

159

C₂

#26011393

yes

characteristic

160

C₂

#26011473

yes

—

	subgroup	G/H
center	C₂²	C₂×#14583803
commutator	C₂³×C₁₇₃	C₂²×C₄
Frattini	C₂²×C₄	C₂²×D₁₇₃
Fitting	C₄×M₁₆×C₁₇₃	C₂
socle	C₂²×C₁₇₃	C₂×C₂²⋊C₄
prime radical	C₂²×D₆₉₂	C₂²
cosocle	C₂²×C₄×C₁₇₃	C₂³
hypercenter	C₄×M₁₆	D₁₇₃
hypocenter	C₁₇₃	#1967
LCS	G ▷ C₂³×C₁₇₃ ▷ C₂×C₁₇₃ ▷ C₁₇₃
UCS	C₁ ◁ C₂² ◁ C₂×C₄² ◁ C₄×M₁₆

subgroup

G/H

center

C₂²

C₂×#14583803

commutator

C₂³×C₁₇₃

C₂²×C₄

Frattini

C₂²×C₄

C₂²×D₁₇₃

Fitting

C₄×M₁₆×C₁₇₃

C₂

socle

C₂²×C₁₇₃

C₂×C₂²⋊C₄

prime radical

C₂²×D₆₉₂

C₂²

cosocle

C₂²×C₄×C₁₇₃

C₂³

hypercenter

C₄×M₁₆

D₁₇₃

hypocenter

C₁₇₃

#1967

LCS

G ▷ C₂³×C₁₇₃ ▷ C₂×C₁₇₃ ▷ C₁₇₃

UCS

C₁ ◁ C₂² ◁ C₂×C₄² ◁ C₄×M₁₆

N	H ≅ G/N	type
2. #26011515	C₂	semidirect
3. C₂×#19149597	C₂	semidirect
5. C₄×M₁₆×C₁₇₃	C₂	semidirect
80. C₁₇₃	#1967	semidirect
86. C₄×M₁₆	D₁₇₃	semidirect

H ≅ G/N

type

2. #26011515

C₂

semidirect

3. C₂×#19149597

C₂

semidirect

5. C₄×M₁₆×C₁₇₃

C₂

semidirect

80. C₁₇₃

#1967

semidirect

86. C₄×M₁₆

D₁₇₃

semidirect

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

subgroup	index
C₂×#19149667	2
#26011515	2
C₂×#19149597	2
C₂×#19149595	2
C₄×M₁₆×C₁₇₃	2
#1967	173

subgroup

index

C₄×M₁₆×C₁₇₃

173

Subgroups — #36354412

Conjugacy classes of subgroups

Complements

Maximal subgroups