Finite Group Databasebeta

← #36363698

Subgroups — #36363698

Order 22144.

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

normal subgroups only

This group has 450 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.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#26012022C211yesmaximal, characteristic, complemented
2C2×#19149641C211yesmaximal, characteristic, complemented
3#26011821C211yesmaximal, characteristic, complemented
4#26011794C212yesmaximal, complemented
5#26011705C212yesmaximal, complemented
6#26011644C211yesmaximal, characteristic, complemented
7#26011624C211yesmaximal, characteristic
8#26011600C212yesmaximal, complemented
9#26011590C212yesmaximal, complemented
10#26011545C211yesmaximal, characteristic, complemented
11#480×C173C211yesmaximal, characteristic, complemented
12C22×D692C2211yescharacteristic
13C22×C4×D173—21no—
14C2×#14583815C2212yes—
15C2×#14583811C2212yes—
16#19149672—21no—
17#19149670C2211yescharacteristic, complemented
18#19149666C2211yescharacteristic, complemented
19#19149660—21no—
20#19149658C2211yescharacteristic
21#19149644C2211yescharacteristic, complemented
22D4×Dic173—21no—
23D4×Dic173C2211yescharacteristic, complemented
24#19149641—21no—
25#19149641—21no—
26#19149641C2212yes—
27#19149641C2212yes—
28#19149635—21no—
29C2×#14583803—21no—
30C2×C4×Dic173C2211yescharacteristic
31#19149610C2212yescomplemented
32#19149608C2212yescomplemented
33#19149602C2212yescomplemented
34#19149601C2212yes—
35#19149599C2212yes—
36C22⋊#13063436C2211yescharacteristic
37#19149589C2212yescomplemented
38#121×C173C2211yescharacteristic
39C2×C4⋊C4×C173C2211yescharacteristic
40#107×C173C2212yescomplemented
41#102×C173C2212yescomplemented
42C22⋊C8×C173C2211yescharacteristic
43C23×D173—21no—
44C2×Dic346C2311yescharacteristic
45C2×D692—24no—
46C2×D692—21no—
47C2×D692C2312yes—
48C22×Dic173—21no—
49C22×Dic173D411yescharacteristic
50C22×Dic173D411yescharacteristic
51C2×C4×D173—22no—
52C2×C4×D173—22no—
53C2×C4×D173—44no—
54#14583815—24no—
55#14583811—24no—
56C2×#13063436C2312yes—
57C4⋊Dic173—21no—
58C4⋊Dic173C2311yescharacteristic
59#14583805—21no—
60#14583805—21no—
61#14583805—21no—
62C22⋊Dic173—21no—
63C22⋊Dic173—21no—
64C22⋊Dic173—21no—
65C22⋊Dic173—21no—
66#14583803—22no—
67#14583803—22no—
68C4×Dic173—21no—
69C4×Dic173D411yescharacteristic
70C4×Dic173D411yescharacteristic
71C4×Dic173C2311yescharacteristic
72C4×Dic173C2311yescharacteristic
73C2×D4×C173—21no—
74C2×D4×C173C2311yescharacteristic
75C22×C4×C173—21no—
76C22×C4×C173C2311yescharacteristic
77C2×C8×C173C2312yes—
78C4⋊C4×C173—21no—
79C4⋊C4×C173C2311yescharacteristic
80C4⋊C4×C173C2312yes—
81C22⋊C4×C173—21no—
82C22×D173—24no—
83C22×D173—22no—
84C22×D173—42no—
85C22×D173—42no—
86Dic346—22no—
87D692—24no—
88D692—22no—
89D692#4114yes—
90C2×Dic173—22no—
91C2×Dic173—21no—
92C2×Dic173—21no—
93C2×Dic173—41no—
94C2×Dic173—42no—
95C2×Dic173C2×D411yescharacteristic
96C2×Dic173C2×D411yescharacteristic
97C2×Dic173C2×D411yescharacteristic
98C2×Dic173C2×D411yescharacteristic
99C4×D173—44no—
100C4×D173—44no—
101#13063436—22no—
102C23×C173—21no—
103C23×C173C2×D411yescharacteristic
104D4×C173—22no—
105D4×C173—42no—
106C2×C4×C173—22no—
107C2×C4×C173—22no—
108C2×C4×C173—21no—
109C2×C4×C173—42no—
110C2×C4×C173C2×D411yescharacteristic
111C2×C4×C173C2411yescharacteristic
112C8×C173—22no—
113C22×C173—21no—
114C22×C173—21no—
115C22×C173—41no—
116C22×C173—41no—
117C22×C173—41no—
118C22×C173AGL(1,ℤ/8)12yes—
119C22×C173C22×D411yescharacteristic
120C2×D173—24no—
121C2×D173—48no—
122C2×D173—42no—
123C2×D173—42no—
124C4×C173—21no—
125C4×C173—42no—
126C4×C173—41no—
127C4×C173C2×#4112yes—
128Dic173—21no—
129Dic173—21no—
130Dic173—21no—
131Dic173—21no—
132Dic173—41no—
133Dic173—41no—
134C2×C173—22no—
135C2×C173—81no—
136C2×C173#52511yescharacteristic
137C2×C173C2×AGL(1,ℤ/8)11yescharacteristic
138C2×C173#57511yescharacteristic
139D173—44no—
140C173#311411yescharacteristic, complemented, Sylow
141#3114—1731nomaximal, Sylow
142#529—1731no—
143C2×C4×D4—1731no—
144#483—1731no—
145#480D17311yescharacteristic, complemented
146#478—1732no—
147#463—1732no—
148#450—1731no—
149#447—1731no—
150#442—1732no—
151#440—1732no—
152#432—1731no—
153C22×D4—1731no—
154C23×C4—3461no—
155C2×SD16—1732no—
156C2×D8—1732no—
157#126—3461no—
158#125—1731no—
159#124—1731no—
160#123—3461no—
161C22⋊Q8—1731no—
162#121C2×D17311yescharacteristic
163C4×Q8—1731no—
164C4×D4—3461no—
165C4×D4—3461no—
166C4×D4—3461no—
167C4×D4—1731no—
168C4×D4—1732no—
169C4×D4—1732no—
170#117—3461no—
171C2×C4⋊C4C2×D17311yescharacteristic
172C2×C22⋊C4—3461no—
173C2×C42—1731no—
174#107C2×D17312yescomplemented
175#106—1732no—
176C4⋊C8—1732no—
177#103—1732no—
178#102—1732no—
179#102—1732no—
180#102C2×D17312yescomplemented
181C22⋊C8—1731no—
182C22⋊C8C2×D17311yescharacteristic
183#97—1732no—
184C24—3461no—
185C2×Q8—1731no—
186C2×D4—21no—
187C2×D4—3464no—
188C2×D4—3461no—
189C2×D4—1732no—
190C2×D4C22×D17311yescharacteristic
191C22×C4—21no—
192C22×C4—3462no—
193C22×C4—3462no—
194C22×C4—3461no—
195C22×C4—1731no—
196C22×C4—1731no—
197C22×C4—6924no—
198C22×C4C22×D17311yescharacteristic
199SD16—3464no—
200D8—3464no—
201C2×C8—1732no—
202C2×C8C22×D17312yes—
203C4⋊C4—21no—
204C4⋊C4—3461no—
205C4⋊C4—3461no—
206C4⋊C4—3461no—
207C4⋊C4—3461no—
208C4⋊C4—1731no—
209C4⋊C4C22×D17311yescharacteristic
210C4⋊C4C22×D17312yes—
211C22⋊C4—21no—
212C22⋊C4—3462no—
213C22⋊C4—3462no—
214C22⋊C4—3461no—
215C22⋊C4—3461no—
216C22⋊C4—3461no—
217C22⋊C4—3461no—
218C42—3461no—
219C42—1731no—
220C42—1731no—
221C42—1731no—
222C42—1731no—
223C23—21no—
224C23—3464no—
225C23—3462no—
226C23—6922no—
227C23—6922no—
228C23D4×D17311yescharacteristic
229Q8—3462no—
230D4—22no—
231D4—42no—
232D4—3464no—
233D4—3462no—
234D4—1734no—
235C2×C4—22no—
236C2×C4—21no—
237C2×C4—22no—
238C2×C4—42no—
239C2×C4—3462no—
240C2×C4—3461no—
241C2×C4—3461no—
242C2×C4—1731no—
243C2×C4—1731no—
244C2×C4—1731no—
245C2×C4—1731no—
246C2×C4—6924no—
247C2×C4—6924no—
248C2×C4—6921no—
249C2×C4—6922no—
250C2×C4D4×D17311yescharacteristic
251C2×C4C23×D17311yescharacteristic
252C8—22no—
253C8—3462no—
254C22—21no—
255C22—21no—
256C22—41no—
257C22—41no—
258C22—41no—
259C22—3464no—
260C22—6928no—
261C22—6922no—
262C22—6922no—
263C22#1914970312yes—
264C22C2×D4×D17311yescharacteristic
265C4—21no—
266C4—42no—
267C4—41no—
268C4—3461no—
269C4—3461no—
270C4—3461no—
271C4—3461no—
272C4—6921no—
273C4—6921no—
274C4#41×D17312yes—
275C2—22no—
276C2—81no—
277C2—6924no—
278C2#2601199911yescharacteristic
279C2C2×#1914970311yescharacteristic
280C2#2601229011yescharacteristic
subgroupG/H
centerC22C2×D4×D173
commutator, cosocleC2×C4×C173C24
FrattiniC2×C4C23×D173
Fitting#480×C173C2
socleC22×C173C22×D4
prime radical#26011624C2
hypercenter#480D173
hypocenterC173#3114
LCSG ▷ C2×C4×C173 ▷ C2×C173 ▷ C173
UCSC1 ◁ C22 ◁ C22×C4 ◁ #480

Complements

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

NH ≅ G/Ntype
1. #26012022C2semidirect
2. C2×#19149641C2semidirect
3. #26011821C2semidirect
4. #26011794C2semidirect
5. #26011705C2semidirect
6. #26011644C2semidirect
8. #26011600C2semidirect
9. #26011590C2semidirect
10. #26011545C2semidirect
11. #480×C173C2semidirect
17. #19149670C22semidirect
18. #19149666C22semidirect
21. #19149644C22semidirect
23. D4×Dic173C22semidirect
31. #19149610C22semidirect
32. #19149608C22semidirect
33. #19149602C22semidirect
37. #19149589C22semidirect
40. #107×C173C22semidirect
41. #102×C173C22semidirect
140. C173#3114semidirect
145. #480D173semidirect
174. #107C2×D173semidirect
180. #102C2×D173semidirect

Maximal subgroups

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

subgroupindex
#260120222
C2×#191496412
#260118212
#260117942
#260117052
#260116442
#260116242
#260116002
#260115902
#260115452
#480×C1732
#3114173