Finite Group Databasebeta

← #36362412

Subgroups — #36362412

Order 22144.

528 conjugacy classes of subgroups; only the 175 normal subgroups are shown.

normal subgroups only

This group has 528 conjugacy classes of subgroups — too many to draw. Only its 175 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
1C2×D8×D173C211yesmaximal, characteristic
2#26011963C212yesmaximal, complemented
3#26011604C211yesmaximal, characteristic, complemented
4#26011603C211yesmaximal, characteristic, complemented
5#26011602C211yesmaximal, characteristic, complemented
6C4×#14583811C212yesmaximal, complemented
7#26011571C212yesmaximal, complemented
8#26011526C211yesmaximal, characteristic, complemented
9#26011494C212yesmaximal, complemented
10#26011448C211yesmaximal, characteristic, complemented
11#442×C173C211yesmaximal, characteristic, complemented
12C2×D4×D173C2212yes—
13C2×D1384C2211yescharacteristic
14C2×#14583811C2212yes—
15D8×D173C418yescomplemented
16C2×#14583809C2211yescharacteristic
17C2×C8×D173C2211yescharacteristic
18C4×D692C2212yescomplemented
19D4×Dic173C2212yescomplemented
20#19149641C2212yescomplemented
21C4×C22⋊D173—22no—
22#19149639—22no—
23#19149637C2212yes—
24#19149636C2211yescharacteristic
25C2×#14583803—22no—
26C22⋊C4×D173—22no—
27#19149612C2212yescomplemented
28#19149611C2211yescharacteristic, complemented
29#19149601C2212yescomplemented
30#19149600C2212yescomplemented
31#19149599C2212yescomplemented
32#19149589C2211yescharacteristic, complemented
33C4×#13063436C2211yescharacteristic, complemented
34C8×Dic173C2211yescharacteristic, complemented
35C2×D8×C173C2211yescharacteristic
36C4×D4×C173C2212yescomplemented
37#106×C173C2211yescharacteristic, complemented
38#102×C173C2212yescomplemented
39#97×C173C2211yescharacteristic, complemented
40C23×D173—22no—
41C2×D692C2312yes—
42C2×C22⋊D173—22no—
43D4×D173—24no—
44D4×D173C2×C418yescomplemented
45C22×Dic173—22no—
46C2×C4×D173—22no—
47C2×C4×D173—22no—
48C2×C4×D173C2311yescharacteristic
49D1384C2×C414yescomplemented
50#14583811C2×C418yescomplemented
51#14583809—22no—
52C2×#13063436C2311yescharacteristic
53C2×#13063436C2311yescharacteristic
54C8×D173C2×C414yescomplemented
55C4⋊Dic173C2312yes—
56#14583805D412yes—
57C22⋊Dic173—22no—
58#14583803—22no—
59#14583803—22no—
60#14583803—22no—
61#14583803D412yes—
62C4×Dic173—22no—
63C4×Dic173C2311yescharacteristic
64C2×D4×C173C2312yes—
65C22×C4×C173—22no—
66D8×C173C2×C414yescomplemented
67C2×C8×C173C2311yescharacteristic
68C2×C8×C173C2311yescharacteristic
69C4⋊C4×C173C2312yes—
70C22⋊C4×C173—22no—
71C42×C173C2311yescharacteristic
72C22×D173—22no—
73C22×D173—28no—
74C22×D173—82no—
75C22×D173—82no—
76C22×D173C2×D411yescharacteristic
77D692—22no—
78D692C22×C414yes—
79C22⋊D173—28no—
80C2×Dic173—22no—
81C2×Dic173—82no—
82C2×Dic173C2×D412yes—
83C2×Dic173C2×D411yescharacteristic
84C4×D173—82no—
85C4×D173—82no—
86C4×D173C22×C412yes—
87C4×D173#4112yes—
88#13063436—21no—
89#13063436C22×C412yes—
90C23×C173—22no—
91D4×C173—22no—
92D4×C173C22×C414yes—
93C2×C4×C173—22no—
94C2×C4×C173—82no—
95C2×C4×C173C2×D412yes—
96C2×C4×C173C2411yescharacteristic
97C8×C173—21no—
98C8×C173C22×C412yes—
99C22×C173—24no—
100C22×C173—42no—
101C22×C173—42no—
102C22×C173C22×D411yescharacteristic
103C2×D173—22no—
104C2×D173—24no—
105C2×D173—42no—
106C2×D173—42no—
107C2×D173—88no—
108C2×D173C4×D412yes—
109C4×C173—22no—
110C4×C173—42no—
111C4×C173C23×C411yescharacteristic
112C4×C173C2×#4111yescharacteristic
113Dic173—22no—
114Dic173—42no—
115Dic173C4×D412yes—
116C2×C173—44no—
117C2×C173C2×C4×D411yescharacteristic
118C2×C173#57612yes—
119D173—22no—
120D173—44no—
121C173#294811yescharacteristic, complemented, Sylow
122#2948—1731nomaximal, Sylow
123C22×D8—1731no—
124#518—1732no—
125#442—1731no—
126#442—1731no—
127#442—1731no—
128#442D17311yescharacteristic, complemented
129C4×D8—1732no—
130C4×D8—1732no—
131#427—1731no—
132#418—1732no—
133#405—1731no—
134C22×D4—1732no—
135C2×D8—1732no—
136C2×D8—1738no—
137C2×D8—1731no—
138C2×D8C2×D17311yescharacteristic
139C2×M16—1731no—
140C22×C8—1731no—
141C4×D4—3462no—
142C4×D4—3462no—
143C4×D4—1732no—
144C4×D4—1732no—
145C4×D4—1732no—
146C4×D4C2×D17312yescomplemented
147#117—1732no—
148#117—1731no—
149C2×C22⋊C4—3462no—
150C2×C22⋊C4—3462no—
151#107—1732no—
152#106—1731no—
153#106C2×D17311yescharacteristic, complemented
154#102—1732no—
155#102—1732no—
156#102—1732no—
157#102C2×D17312yescomplemented
158#97—1731no—
159#97C2×D17311yescharacteristic, complemented
160C4×C8—1731no—
161C4×C8—1731no—
162C24—3462no—
163C2×D4—3464no—
164C2×D4—3462no—
165C2×D4—1738no—
166C2×D4—1732no—
167C2×D4C22×D17312yes—
168C22×C4—22no—
169C22×C4—3462no—
170C22×C4—3462no—
171C22×C4—3462no—
172C22×C4—1731no—
173D8—1738no—
174D8—1734no—
175D8C4×D17314yescomplemented
176M16—3462no—
177C2×C8—1731no—
178C2×C8—1731no—
179C2×C8—1734no—
180C2×C8C22×D17311yescharacteristic
181C2×C8C22×D17311yescharacteristic
182C4⋊C4—1732no—
183C4⋊C4—1732no—
184C4⋊C4C22×D17312yes—
185C22⋊C4—22no—
186C22⋊C4—3462no—
187C22⋊C4—3462no—
188C22⋊C4—3462no—
189C22⋊C4—3462no—
190C22⋊C4—1732no—
191C42—3462no—
192C42—1731no—
193C42C22×D17311yescharacteristic
194C23—22no—
195C23—3468no—
196C23—3462no—
197C23—13842no—
198C23—13842no—
199C23—1731no—
200D4—22no—
201D4—3468no—
202D4—3462no—
203D4—1734no—
204D4C2×C4×D17314yes—
205C2×C4—22no—
206C2×C4—82no—
207C2×C4—3462no—
208C2×C4—13842no—
209C2×C4—13842no—
210C2×C4—13842no—
211C2×C4—1732no—
212C2×C4—1731no—
213C2×C4—1732no—
214C2×C4—1732no—
215C2×C4D4×D17312yes—
216C2×C4C23×D17311yescharacteristic
217C8—21no—
218C8—3461no—
219C8—1732no—
220C8C2×C4×D17312yes—
221C22—24no—
222C22—42no—
223C22—42no—
224C22—3462no—
225C22—3464no—
226C22—13848no—
227C22—1732no—
228C22—6922no—
229C22—6922no—
230C22C2×D4×D17311yescharacteristic
231C4—22no—
232C4—42no—
233C4—3462no—
234C4—1732no—
235C4—6922no—
236C4C22×C4×D17311yescharacteristic
237C4#41×D17311yescharacteristic
238C2—44no—
239C2—3462no—
240C2—6924no—
241C2C4×D4×D17311yescharacteristic
242C2#2601229512yes—
subgroupG/H
centerC22C2×D4×D173
commutatorC4×C173C23×C4
FrattiniC2×C4C23×D173
Fitting#442×C173C2
socleC22×C173C22×D4
prime radicalC2×D8×D173C2
cosocleC2×C4×C173C24
hypercenter#442D173
hypocenterC173#2948
LCSG ▷ C4×C173 ▷ C2×C173 ▷ C173
UCSC1 ◁ C22 ◁ C42 ◁ #442

Complements

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

NH ≅ G/Ntype
2. #26011963C2semidirect
3. #26011604C2semidirect
4. #26011603C2semidirect
5. #26011602C2semidirect
6. C4×#14583811C2semidirect
7. #26011571C2semidirect
8. #26011526C2semidirect
9. #26011494C2semidirect
10. #26011448C2semidirect
11. #442×C173C2semidirect
15. D8×D173C4semidirect
18. C4×D692C22semidirect
19. D4×Dic173C22semidirect
20. #19149641C22semidirect
27. #19149612C22semidirect
28. #19149611C22semidirect
29. #19149601C22semidirect
30. #19149600C22semidirect
31. #19149599C22semidirect
32. #19149589C22semidirect
33. C4×#13063436C22semidirect
34. C8×Dic173C22semidirect
36. C4×D4×C173C22semidirect
37. #106×C173C22semidirect
38. #102×C173C22semidirect
39. #97×C173C22semidirect
44. D4×D173C2×C4semidirect
49. D1384C2×C4semidirect
50. #14583811C2×C4semidirect
54. C8×D173C2×C4semidirect
66. D8×C173C2×C4semidirect
121. C173#2948semidirect
128. #442D173semidirect
146. C4×D4C2×D173semidirect
153. #106C2×D173semidirect
157. #102C2×D173semidirect
159. #97C2×D173semidirect
175. D8C4×D173semidirect

Maximal subgroups

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

subgroupindex
C2×D8×D1732
#260119632
#260116042
#260116032
#260116022
C4×#145838112
#260115712
#260115262
#260114942
#260114482
#442×C1732
#2948173