Finite Group Databasebeta

← #37544788

Subgroups — #37544788

Order 23168.

396 conjugacy classes of subgroups; only the 109 normal subgroups are shown.

normal subgroups only

This group has 396 conjugacy classes of subgroups — too many to draw. Only its 109 normal subgroups are shown.

Dotted edges: containments with no intermediate normal subgroup — omitted non-normal subgroups may lie between.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#26041890C211yesmaximal, characteristic, complemented
2#26041847C211yesmaximal, characteristic
3#26041653C211yesmaximal, characteristic, complemented
4#26041644C211yesmaximal, characteristic, complemented
5#26041633C211yesmaximal, characteristic, complemented
6#26041503C212yesmaximal, complemented
7#26041477C212yesmaximal, complemented
8#26041440C212yesmaximal, complemented
9#26041428C212yesmaximal, complemented
10#26041341C211yesmaximal, characteristic, complemented
11#497×C181C211yesmaximal, characteristic, complemented
12C2×#15668450C2211yescharacteristic
13C2×D4×D181C2211yescharacteristic
14C2×#15668436C2212yes—
15C2×#15668436C2212yescomplemented
16#19153813C2211yescharacteristic
17#19153812C2211yescharacteristic
18#19153811—22no—
19#19153808C2211yescharacteristic, complemented
20#19153806—22no—
21#19153805C2211yescharacteristic
22#19153792C2212yes—
23#19153789—22no—
24#19153786—22no—
25Q8×Dic181C2212yescomplemented
26#19153773C2211yescharacteristic
27#19153745C2212yescomplemented
28#19153740C2212yescomplemented
29#19153738C2212yes—
30#19153738C2212yescomplemented
31#19153737C2212yes—
32#19153728C2212yes—
33C4×#13089678C2211yescharacteristic, complemented
34C2×Q16×C181C2212yescomplemented
35C2×SD16×C181C2212yes—
36C4⋊Q8×C181C2211yescharacteristic, complemented
37#124×C181C2211yescharacteristic
38#97×C181C2211yescharacteristic, complemented
39C23×D181—21no—
40#15668450—42no—
41C2×D724—21no—
42C2×D724—22no—
43C2×D724C2311yescharacteristic
44C2×D724C2311yescharacteristic
45C2×C22⋊D181—21no—
46C2×C22⋊D181—22no—
47D4×D181—42no—
48C2×C4×D181—21no—
49C2×C4×D181C2311yescharacteristic
50#15668436—42no—
51#15668436—42no—
52C2×#13089678C2312yes—
53C4⋊Dic181—22no—
54C4⋊Dic181C2312yes—
55#15668428—22no—
56#15668428—22no—
57#15668428D412yes—
58#15668426—22no—
59#15668426—22no—
60#15668426D412yes—
61C4×Dic181—22no—
62C4×Dic181C2311yescharacteristic
63C2×Q8×C181C2311yescharacteristic
64C2×Q8×C181C2312yes—
65C2×D4×C181C2311yescharacteristic
66Q16×C181—42no—
67SD16×C181—42no—
68C2×C8×C181C2312yes—
69C4⋊C4×C181—22no—
70C22⋊C4×C181—22no—
71C42×C181C2311yescharacteristic
72C22×D181—21no—
73C22×D181—21no—
74C22×D181—42no—
75C22×D181—82no—
76C22×D181C2×D411yescharacteristic
77D724—22no—
78D724—22no—
79D724—42no—
80D724—82no—
81C22⋊D181—42no—
82C22⋊D181—82no—
83C2×Dic181—22no—
84C2×Dic181C2×D412yes—
85C2×Dic181C2×D411yescharacteristic
86C4×D181—22no—
87C4×D181—42no—
88#13089678—22no—
89C23×C181—21no—
90Q8×C181—22no—
91Q8×C181—22no—
92Q8×C181—22no—
93D4×C181—22no—
94C2×C4×C181—21no—
95C2×C4×C181—22no—
96C2×C4×C181C2×D412yes—
97C2×C4×C181C2411yescharacteristic
98C8×C181—22no—
99C22×C181—41no—
100C22×C181—42no—
101C22×C181C22×D411yescharacteristic
102C2×D181—21no—
103C2×D181—22no—
104C2×D181—41no—
105C2×D181—41no—
106C2×D181—42no—
107C2×D181—42no—
108C2×D181—84no—
109C4×C181—22no—
110C4×C181—42no—
111C4×C181—41no—
112C4×C181H(5,𝔽2)12yes—
113Dic181—21no—
114Dic181—22no—
115Dic181—42no—
116C2×C181—81no—
117C2×C181#53511yescharacteristic
118C2×C181#57712yes—
119D181—41no—
120D181—81no—
121D181—81no—
122C181#322211yescharacteristic, complemented, Sylow
123#3222—1811nomaximal, Sylow
124#541—1811no—
125#535—1811no—
126#497D18111yescharacteristic, complemented
127#492—1811no—
128#490—1811no—
129#488—1811no—
130#464—1812no—
131#460—1812no—
132#452—1812no—
133#450—1812no—
134#433—1811no—
135C2×#41—1811no—
136C22×D4—1811no—
137C2×Q16C2×D18112yescomplemented
138C2×SD16—1812no—
139C2×SD16—1812no—
140C2×SD16C2×D18112yes—
141C4⋊Q8C2×D18111yescharacteristic, complemented
142#127—1811no—
143#127—1811no—
144#126—3622no—
145#125—3622no—
146#125—1811no—
147#124—1811no—
148#124C2×D18111yescharacteristic
149C22⋊Q8—1812no—
150#121—3622no—
151#120—3622no—
152C4×Q8—1812no—
153#117—1811no—
154C4⋊C8—1812no—
155#103—1812no—
156#102—1812no—
157#102—1812no—
158#102—1812no—
159C22⋊C8—1812no—
160#97C2×D18111yescharacteristic, complemented
161C4×C8—1811no—
162C24—3621no—
163#41—7242no—
164C2×Q8C22×D18111yescharacteristic
165C2×Q8C22×D18112yes—
166C2×D4—3621no—
167C2×D4—3622no—
168C2×D4—3621no—
169C2×D4—3622no—
170C2×D4—1811no—
171C2×D4—1811no—
172C2×D4—7242no—
173C2×D4C22×D18111yescharacteristic
174C22×C4—3621no—
175C22×C4—1811no—
176Q16—42no—
177SD16—42no—
178SD16—7242no—
179SD16—7242no—
180C2×C8—1812no—
181C2×C8C22×D18112yes—
182C4⋊C4—22no—
183C4⋊C4—3622no—
184C4⋊C4—3622no—
185C4⋊C4—3622no—
186C4⋊C4—1812no—
187C4⋊C4—1812no—
188C22⋊C4—22no—
189C22⋊C4—3622no—
190C22⋊C4—3622no—
191C22⋊C4—1812no—
192C42—3622no—
193C42—1811no—
194C42C22×D18111yescharacteristic
195C23—21no—
196C23—3621no—
197C23—3621no—
198C23—14482no—
199C23—1811no—
200C23—7242no—
201Q8—22no—
202Q8—22no—
203Q8—22no—
204D4—22no—
205D4—3622no—
206D4—3622no—
207D4—14482no—
208D4—14482no—
209D4—7242no—
210D4—7242no—
211C2×C4—22no—
212C2×C4—21no—
213C2×C4—3622no—
214C2×C4—3622no—
215C2×C4—1812no—
216C2×C4—1811no—
217C2×C4—7242no—
218C2×C4D4×D18112yes—
219C2×C4C23×D18111yescharacteristic
220C8—22no—
221C8—3622no—
222C22—42no—
223C22—41no—
224C22—3622no—
225C22—3621no—
226C22—14484no—
227C22—7242no—
228C22—7242no—
229C22—7241no—
230C22—7241no—
231C22C2×D4×D18111yescharacteristic
232C4—22no—
233C4—41no—
234C4—42no—
235C4—3622no—
236C4—3621no—
237C4—7242no—
238C4#1915386112yes—
239C2—81no—
240C2—14481no—
241C2—14481no—
242C2—7241no—
243C2#2604184211yescharacteristic
244C2#2604208812yes—
subgroupG/H
centerC22C2×D4×D181
commutator, cosocleC2×C4×C181C24
FrattiniC2×C4C23×D181
Fitting#497×C181C2
socleC22×C181C22×D4
prime radical#26041847C2
hypercenter#497D181
hypocenterC181#3222
LCSG ▷ C2×C4×C181 ▷ C2×C181 ▷ C181
UCSC1 ◁ C22 ◁ C42 ◁ #497

Complements

Splittings G = N ⋊ H with N normal; H ≅ G/N.

NH ≅ G/Ntype
1. #26041890C2semidirect
3. #26041653C2semidirect
4. #26041644C2semidirect
5. #26041633C2semidirect
6. #26041503C2semidirect
7. #26041477C2semidirect
8. #26041440C2semidirect
9. #26041428C2semidirect
10. #26041341C2semidirect
11. #497×C181C2semidirect
15. C2×#15668436C22semidirect
19. #19153808C22semidirect
25. Q8×Dic181C22semidirect
27. #19153745C22semidirect
28. #19153740C22semidirect
30. #19153738C22semidirect
33. C4×#13089678C22semidirect
34. C2×Q16×C181C22semidirect
36. C4⋊Q8×C181C22semidirect
38. #97×C181C22semidirect
122. C181#3222semidirect
126. #497D181semidirect
137. C2×Q16C2×D181semidirect
141. C4⋊Q8C2×D181semidirect
160. #97C2×D181semidirect

Maximal subgroups

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

subgroupindex
#260418902
#260418472
#260416532
#260416442
#260416332
#260415032
#260414772
#260414402
#260414282
#260413412
#497×C1812
#3222181