Finite Group Databasebeta

← #36364389

Subgroups — #36364389

Order 22144.

414 conjugacy classes of subgroups; only the 121 normal subgroups are shown.

normal subgroups only

This group has 414 conjugacy classes of subgroups — too many to draw. Only its 121 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×#19149658C211yesmaximal, characteristic, complemented
2#26011832C212yesmaximal, complemented
3#26011824C211yesmaximal, characteristic, complemented
4#26011807C211yesmaximal, characteristic, complemented
5#26011747C212yesmaximal, complemented
6#26011735C212yesmaximal, complemented
7#26011653C212yesmaximal, complemented
8#26011646C211yesmaximal, characteristic, complemented
9#26011629C211yesmaximal, characteristic, complemented
10#26011455C211yesmaximal, characteristic, complemented
11#525×C173C211yesmaximal, characteristic
12C22×Dic346C2211yescharacteristic
13C2×#14583816C2212yescomplemented
14C2×#14583814C2212yescomplemented
15C2×#14583810C2212yes—
16#19149658—21no—
17#19149658—21no—
18#19149658C2212yes—
19#19149658C2212yes—
20C2×C4⋊Dic173C2211yescharacteristic
21C2×#14583805—21no—
22C2×C22⋊Dic173—21no—
23#19149612C2212yescomplemented
24#19149610C2212yescomplemented
25#19149603C2212yescomplemented
26#19149602C2212yescomplemented
27#19149602C2212yescomplemented
28#19149600C2212yescomplemented
29C22⋊#13063436C2212yes—
30C22⋊#13063436C2211yescharacteristic
31C22⋊#13063436C2211yescharacteristic
32C2×#41×C173C2211yescharacteristic
33C23×C4×C173C2211yescharacteristic
34#123×C173—21no—
35C22⋊Q8×C173C2211yescharacteristic
36C22⋊Q8×C173C2211yescharacteristic
37#121×C173C2211yescharacteristic
38#121×C173C2211yescharacteristic
39#120×C173—21no—
40C4×D4×C173—21no—
41C4×D4×C173—21no—
42#117×C173C2211yescharacteristic
43C2×Dic346—21no—
44C2×Dic346—42no—
45C2×Dic346C2312yes—
46C22×Dic173—21no—
47C22×Dic173—21no—
48#14583816—42no—
49#14583814—42no—
50#14583810—42no—
51C2×#13063436C2312yes—
52C2×#13063436C2312yes—
53C4⋊Dic173—21no—
54C4⋊Dic173C2312yes—
55#14583805—22no—
56#14583805—22no—
57C22⋊Dic173—22no—
58C22⋊Dic173—22no—
59C24×C173D411yescharacteristic
60#41×C173—42no—
61C2×Q8×C173C2311yescharacteristic
62C2×D4×C173—21no—
63C2×D4×C173—21no—
64C2×D4×C173—21no—
65C2×D4×C173C2311yescharacteristic
66C22×C4×C173—21no—
67C22×C4×C173—42no—
68C22×C4×C173D411yescharacteristic
69C22×C4×C173D411yescharacteristic
70C22×C4×C173D411yescharacteristic
71C22×C4×C173C2311yescharacteristic
72C22×C4×C173C2311yescharacteristic
73C22×C4×C173C2311yescharacteristic
74C4⋊C4×C173—21no—
75C4⋊C4×C173—21no—
76C4⋊C4×C173C2311yescharacteristic
77C4⋊C4×C173C2311yescharacteristic
78C22⋊C4×C173—21no—
79C22⋊C4×C173—21no—
80C22⋊C4×C173—21no—
81C22⋊C4×C173—21no—
82C22⋊C4×C173—21no—
83C42×C173—21no—
84Dic346—22no—
85Dic346—22no—
86Dic346—42no—
87C2×Dic173—22no—
88C2×Dic173—22no—
89C2×Dic173—42no—
90C2×Dic173—42no—
91#13063436—22no—
92#13063436—22no—
93C23×C173—22no—
94C23×C173—21no—
95C23×C173—41no—
96C23×C173—41no—
97C23×C173C2×D411yescharacteristic
98C23×C173C2×D411yescharacteristic
99C23×C173C2×D411yescharacteristic
100Q8×C173—22no—
101D4×C173—22no—
102D4×C173—42no—
103D4×C173—42no—
104D4×C173—42no—
105C2×C4×C173—22no—
106C2×C4×C173—22no—
107C2×C4×C173—22no—
108C2×C4×C173—21no—
109C2×C4×C173—21no—
110C2×C4×C173—21no—
111C2×C4×C173—41no—
112C2×C4×C173—41no—
113C2×C4×C173—41no—
114C2×C4×C173—42no—
115C2×C4×C173C2×D411yescharacteristic
116C2×C4×C173C2×D411yescharacteristic
117C2×C4×C173C2×D411yescharacteristic
118C2×C4×C173C2411yescharacteristic
119C22×C173—21no—
120C22×C173—21no—
121C22×C173—21no—
122C22×C173—21no—
123C22×C173—21no—
124C22×C173—21no—
125C22×C173—21no—
126C22×C173—21no—
127C22×C173—44no—
128C22×C173—41no—
129C22×C173—41no—
130C22×C173—41no—
131C22×C173#13712yes—
132C22×C173C22×D411yescharacteristic
133C4×C173—21no—
134C4×C173—21no—
135C4×C173—21no—
136C4×C173—41no—
137C4×C173—41no—
138C4×C173—41no—
139C4×C173H(5,𝔽2)12yes—
140Dic173—42no—
141Dic173—42no—
142C2×C173—22no—
143C2×C173—41no—
144C2×C173—41no—
145C2×C173—81no—
146C2×C173#53411yescharacteristic
147C2×C173C2×#13711yescharacteristic
148C2×C173#57511yescharacteristic
149C173#319611yescharacteristic, complemented, Sylow
150#3196—1731nomaximal, Sylow
151#525D17311yescharacteristic
152C2×C22⋊Q8—1731no—
153#485—1732no—
154#484—1731no—
155#481—1731no—
156#470—1732no—
157#468—1732no—
158#452—1732no—
159#451—1731no—
160#448—1731no—
161#407—1731no—
162C2×#41C2×D17311yescharacteristic
163C22×Q8—1731no—
164C23×C4C2×D17311yescharacteristic
165C2×Q16—1732no—
166C2×SD16—1732no—
167C2×M16—1732no—
168#123—21no—
169C22⋊Q8—3461no—
170C22⋊Q8—3461no—
171C22⋊Q8—1732no—
172C22⋊Q8—1732no—
173C22⋊Q8C2×D17311yescharacteristic
174C22⋊Q8C2×D17311yescharacteristic
175#121C2×D17311yescharacteristic
176#121C2×D17311yescharacteristic
177#120—21no—
178C4×D4—21no—
179C4×D4—21no—
180#117C2×D17311yescharacteristic
181C2×C4⋊C4—3461no—
182C2×C4⋊C4—1731no—
183C2×C22⋊C4—3461no—
184#107—1732no—
185#106—1732no—
186#103—1732no—
187#103—1732no—
188#103—1732no—
189#102—1732no—
190C22⋊C8—1731no—
191C22⋊C8—1731no—
192C22⋊C8—1732no—
193C24C22⋊D17311yescharacteristic
194#41—42no—
195C2×Q8—3461no—
196C2×Q8—1732no—
197C2×Q8—6922no—
198C2×Q8C22×D17311yescharacteristic
199C2×D4—21no—
200C2×D4—21no—
201C2×D4—21no—
202C2×D4C22×D17311yescharacteristic
203C22×C4—21no—
204C22×C4—42no—
205C22×C4—3461no—
206C22×C4—3461no—
207C22×C4C22⋊D17311yescharacteristic
208C22×C4C22⋊D17311yescharacteristic
209C22×C4C22⋊D17311yescharacteristic
210C22×C4C22×D17311yescharacteristic
211C22×C4C22×D17311yescharacteristic
212C22×C4C22×D17311yescharacteristic
213Q16—6922no—
214SD16—6922no—
215M16—6922no—
216C2×C8—1732no—
217C2×C8—1732no—
218C4⋊C4—21no—
219C4⋊C4—21no—
220C4⋊C4—3462no—
221C4⋊C4—3462no—
222C4⋊C4—3461no—
223C4⋊C4—1732no—
224C4⋊C4C22×D17311yescharacteristic
225C4⋊C4C22×D17311yescharacteristic
226C22⋊C4—21no—
227C22⋊C4—21no—
228C22⋊C4—21no—
229C22⋊C4—21no—
230C22⋊C4—21no—
231C22⋊C4—3462no—
232C22⋊C4—3462no—
233C42—21no—
234C23—22no—
235C23—21no—
236C23—41no—
237C23—41no—
238C23C2×C22⋊D17311yescharacteristic
239C23C2×C22⋊D17311yescharacteristic
240C23C2×C22⋊D17311yescharacteristic
241Q8—22no—
242Q8—3462no—
243Q8—3462no—
244Q8—6922no—
245D4—22no—
246D4—42no—
247D4—42no—
248D4—42no—
249C2×C4—22no—
250C2×C4—22no—
251C2×C4—22no—
252C2×C4—21no—
253C2×C4—21no—
254C2×C4—21no—
255C2×C4—41no—
256C2×C4—41no—
257C2×C4—41no—
258C2×C4—42no—
259C2×C4—3462no—
260C2×C4—3462no—
261C2×C4—6922no—
262C2×C4—6922no—
263C2×C4C2×C22⋊D17311yescharacteristic
264C2×C4C2×C22⋊D17311yescharacteristic
265C2×C4C2×C22⋊D17311yescharacteristic
266C2×C4C23×D17311yescharacteristic
267C8—3462no—
268C8—3462no—
269C22—21no—
270C22—21no—
271C22—21no—
272C22—21no—
273C22—21no—
274C22—21no—
275C22—21no—
276C22—21no—
277C22—44no—
278C22—41no—
279C22—41no—
280C22—41no—
281C22#1914970912yes—
282C22C22×C22⋊D17311yescharacteristic
283C4—21no—
284C4—21no—
285C4—21no—
286C4—41no—
287C4—41no—
288C4—41no—
289C4—6922no—
290C4—6922no—
291C4#1914972412yes—
292C2—22no—
293C2—41no—
294C2—41no—
295C2—81no—
296C2#2601205011yescharacteristic
297C2C2×#1914970911yescharacteristic
298C2#2601228911yescharacteristic
subgroupG/H
centerC22C22×C22⋊D173
commutator, cosocleC2×C4×C173C24
FrattiniC2×C4C23×D173
Fitting, prime radical#525×C173C2
socleC22×C173C22×D4
hypercenter#525D173
hypocenterC173#3196
LCSG ▷ C2×C4×C173 ▷ C2×C173 ▷ C173
UCSC1 ◁ C22 ◁ C23×C4 ◁ #525

Complements

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

NH ≅ G/Ntype
1. C2×#19149658C2semidirect
2. #26011832C2semidirect
3. #26011824C2semidirect
4. #26011807C2semidirect
5. #26011747C2semidirect
6. #26011735C2semidirect
7. #26011653C2semidirect
8. #26011646C2semidirect
9. #26011629C2semidirect
10. #26011455C2semidirect
13. C2×#14583816C22semidirect
14. C2×#14583814C22semidirect
23. #19149612C22semidirect
24. #19149610C22semidirect
25. #19149603C22semidirect
26. #19149602C22semidirect
27. #19149602C22semidirect
28. #19149600C22semidirect
149. C173#3196semidirect

Maximal subgroups

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

subgroupindex
C2×#191496582
#260118322
#260118242
#260118072
#260117472
#260117352
#260116532
#260116462
#260116292
#260114552
#525×C1732
#3196173