Finite Group Databasebeta

← #36364291

Subgroups — #36364291

Order 22144.

370 conjugacy classes of subgroups.

The subgroup lattice has 346 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#26012026C211yesmaximal, characteristic, complemented
2#26012020C211yesmaximal, characteristic, complemented
3#26011890C211yesmaximal, characteristic, complemented
4#26011873C211yesmaximal, characteristic, complemented
5#26011846C211yesmaximal, characteristic, complemented
6#26011748C211yesmaximal, characteristic, complemented
7#26011743C211yesmaximal, characteristic
8#26011740C211yesmaximal, characteristic, complemented
9#26011736C211yesmaximal, characteristic, complemented
10#26011600C211yesmaximal, characteristic, complemented
11#26011595C211yesmaximal, characteristic, complemented
12#26011594C211yesmaximal, characteristic, complemented
13#26011588C211yesmaximal, characteristic, complemented
14C4×#14583809C211yesmaximal, characteristic, complemented
15#488×C173C211yesmaximal, characteristic, complemented
16C2×#14583816C2211yescharacteristic
17C2×#14583814C2211yescharacteristic
18C2×#14583813C2211yescharacteristic
19C2×#14583811C2211yescharacteristic
20C2×#14583809C2211yescharacteristic
21C2×#14583809C2211yescharacteristic
22#19149674—21no—
23#19149672—21no—
24#19149671C2211yescharacteristic
25#19149667C2211yescharacteristic
26#19149664—21no—
27#19149661—21no—
28#19149659C2211yescharacteristic
29#19149655C2211yescharacteristic
30#19149654C2211yescharacteristic
31#19149650C2211yescharacteristic
32Q8×Dic173C2211yescharacteristic
33#19149644C2211yescharacteristic
34D4×Dic173C2211yescharacteristic
35#19149641—21no—
36#19149641C2211yescharacteristic
37#19149639—21no—
38#19149637—21no—
39#19149635—21no—
40C42×D173C2211yescharacteristic
41#19149612C2211yescharacteristic
42#19149612C2211yescharacteristic
43#19149610C2211yescharacteristic
44#19149610C2211yescharacteristic
45#19149604C2211yescharacteristic
46#19149604C2211yescharacteristic
47#19149601C2211yescharacteristic
48#19149601C2211yescharacteristic
49#19149589C2211yescharacteristic
50#19149589C2211yescharacteristic
51C4×#13063436C2211yescharacteristic
52#125×C173C2211yescharacteristic
53#124×C173C2211yescharacteristic
54#103×C173C2211yescharacteristic
55#103×C173C2211yescharacteristic
56#102×C173C2211yescharacteristic
57#102×C173C2211yescharacteristic
58C4×C8×C173C2211yescharacteristic
59C2×Dic346C2311yescharacteristic
60C2×D692—21no—
61C2×D692C2311yescharacteristic
62C2×C22⋊D173—21no—
63C22×Dic173—21no—
64C2×C4×D173—21no—
65C2×C4×D173D411yescharacteristic
66C2×C4×D173D411yescharacteristic
67C2×C4×D173C2311yescharacteristic
68#14583816—22no—
69#14583814—22no—
70#14583813—22no—
71#14583811—22no—
72#14583809—22no—
73#14583809—22no—
74C2×#13063436C2311yescharacteristic
75C2×#13063436C2311yescharacteristic
76C4⋊Dic173—21no—
77C4⋊Dic173—21no—
78C4⋊Dic173C2311yescharacteristic
79C4⋊Dic173C2311yescharacteristic
80#14583805—21no—
81#14583805—21no—
82#14583805—21no—
83#14583805—21no—
84C22⋊Dic173—21no—
85C22⋊Dic173—21no—
86#14583803—21no—
87#14583803—21no—
88#14583803—21no—
89#14583803—21no—
90#14583803—21no—
91#14583803—21no—
92C4×Dic173—21no—
93C4×Dic173—21no—
94C4×Dic173D411yescharacteristic
95C4×Dic173D411yescharacteristic
96C4×Dic173C2311yescharacteristic
97C2×Q8×C173C2311yescharacteristic
98C2×D4×C173C2311yescharacteristic
99C2×C8×C173C2311yescharacteristic
100C2×C8×C173C2311yescharacteristic
101C4⋊C4×C173—21no—
102C4⋊C4×C173—21no—
103C4⋊C4×C173C2311yescharacteristic
104C4⋊C4×C173C2311yescharacteristic
105C22⋊C4×C173—21no—
106C22⋊C4×C173—21no—
107C42×C173C2311yescharacteristic
108C22×D173—21no—
109C22×D173C2×D411yescharacteristic
110Dic346—21no—
111Dic346—21no—
112D692—21no—
113D692—21no—
114D692—42no—
115C22⋊D173—42no—
116C2×Dic173—21no—
117C2×Dic173—21no—
118C2×Dic173—21no—
119C2×Dic173—42no—
120C2×Dic173C2×D411yescharacteristic
121C2×Dic173C2×D411yescharacteristic
122C2×Dic173C2×D411yescharacteristic
123C4×D173—21no—
124C4×D173—21no—
125C4×D173—41no—
126C4×D173—41no—
127C4×D173—42no—
128#13063436#4112yes—
129#13063436#4112yes—
130C23×C173—21no—
131Q8×C173—21no—
132Q8×C173—21no—
133D4×C173—21no—
134D4×C173—21no—
135C2×C4×C173—21no—
136C2×C4×C173—21no—
137C2×C4×C173—21no—
138C2×C4×C173C2×D411yescharacteristic
139C2×C4×C173C2×D411yescharacteristic
140C2×C4×C173C2411yescharacteristic
141C8×C173—21no—
142C8×C173—21no—
143C22×C173—41no—
144C22×C173—41no—
145C22×C173—41no—
146C22×C173C22×D411yescharacteristic
147C2×D173—21no—
148C2×D173—21no—
149C2×D173—21no—
150C2×D173—41no—
151C2×D173—41no—
152C2×D173—41no—
153C4×C173—21no—
154C4×C173—21no—
155C4×C173—41no—
156C4×C173—41no—
157C4×C173—41no—
158C4×C173C2×#4111yescharacteristic
159C4×C173C2×#4111yescharacteristic
160Dic173—21no—
161Dic173—21no—
162Dic173—21no—
163Dic173—41no—
164Dic173—41no—
165Dic173—41no—
166C2×C173—81no—
167C2×C173#53211yescharacteristic
168C2×C173#57511yescharacteristic
169C2×C173#57511yescharacteristic
170D173—41no—
171D173—81no—
172C173#318311yescharacteristic, complemented, Sylow
173#3183—1731nomaximal, Sylow
174#529—1731no—
175#529—1731no—
176#499—1731no—
177#495—1731no—
178#488—1731no—
179#488D17311yescharacteristic, complemented
180#470—1731no—
181#469—1731no—
182#468—1731no—
183#468—1731no—
184#442—1731no—
185#441—1731no—
186#440—1731no—
187#440—1731no—
188C4×M16—1731no—
189C2×Q16—1731no—
190C2×SD16—1731no—
191C2×SD16—1731no—
192C2×D8—1731no—
193C2×M16—1731no—
194C2×M16—1731no—
195#126—3461no—
196#126—3461no—
197#125—1731no—
198#125C2×D17311yescharacteristic
199#124—1731no—
200#124C2×D17311yescharacteristic
201#123—3461no—
202#123—3461no—
203C22⋊Q8—1731no—
204C22⋊Q8—1731no—
205#121—1731no—
206#121—1731no—
207C4×Q8—1731no—
208C4×Q8—1731no—
209C4×D4—3461no—
210C4×D4—3461no—
211C4×D4—1731no—
212C4×D4—1731no—
213#117—3461no—
214#117—3461no—
215C2×C42—1731no—
216#107—1731no—
217#107—1731no—
218#106—1731no—
219#106—1731no—
220#103—1731no—
221#103—1731no—
222#103C2×D17311yescharacteristic
223#103C2×D17311yescharacteristic
224#102—1731no—
225#102—1731no—
226#102C2×D17311yescharacteristic
227#102C2×D17311yescharacteristic
228#97—1731no—
229#97—1731no—
230C4×C8—1731no—
231C4×C8C2×D17311yescharacteristic
232C2×Q8—1731no—
233C2×Q8C22×D17311yescharacteristic
234C2×D4—3461no—
235C2×D4—3461no—
236C2×D4—1731no—
237C2×D4C22×D17311yescharacteristic
238C22×C4—3461no—
239C22×C4—3461no—
240C22×C4—1731no—
241C22×C4—1731no—
242C22×C4—1731no—
243Q16—3462no—
244SD16—3462no—
245SD16—3462no—
246D8—3462no—
247M16—3462no—
248M16—3462no—
249C2×C8—1731no—
250C2×C8—1731no—
251C2×C8C22×D17311yescharacteristic
252C2×C8C22×D17311yescharacteristic
253C4⋊C4—21no—
254C4⋊C4—21no—
255C4⋊C4—3461no—
256C4⋊C4—3461no—
257C4⋊C4—3461no—
258C4⋊C4—3461no—
259C4⋊C4—3461no—
260C4⋊C4—3461no—
261C4⋊C4—1731no—
262C4⋊C4—1731no—
263C4⋊C4C22×D17311yescharacteristic
264C4⋊C4C22×D17311yescharacteristic
265C22⋊C4—21no—
266C22⋊C4—21no—
267C22⋊C4—3461no—
268C22⋊C4—3461no—
269C22⋊C4—3461no—
270C22⋊C4—3461no—
271C22⋊C4—3461no—
272C22⋊C4—3461no—
273C22⋊C4—3461no—
274C22⋊C4—3461no—
275C42—3461no—
276C42—3461no—
277C42—1731no—
278C42—1731no—
279C42—1731no—
280C42C22×D17311yescharacteristic
281C23—21no—
282C23—3461no—
283C23—1731no—
284Q8—21no—
285Q8—21no—
286Q8—3461no—
287Q8—3461no—
288D4—21no—
289D4—21no—
290D4—3461no—
291D4—3461no—
292D4—6922no—
293D4—6922no—
294C2×C4—21no—
295C2×C4—21no—
296C2×C4—21no—
297C2×C4—3461no—
298C2×C4—3461no—
299C2×C4—3461no—
300C2×C4—3461no—
301C2×C4—3461no—
302C2×C4—1731no—
303C2×C4—1731no—
304C2×C4—1731no—
305C2×C4—6921no—
306C2×C4—6921no—
307C2×C4—6922no—
308C2×C4—6922no—
309C2×C4D4×D17311yescharacteristic
310C2×C4D4×D17311yescharacteristic
311C2×C4C23×D17311yescharacteristic
312C8—21no—
313C8—21no—
314C8—1732no—
315C8—1732no—
316C22—41no—
317C22—41no—
318C22—41no—
319C22—3461no—
320C22—3461no—
321C22—3461no—
322C22—6921no—
323C22—6921no—
324C22—6921no—
325C22C2×D4×D17311yescharacteristic
326C4—21no—
327C4—21no—
328C4—41no—
329C4—41no—
330C4—41no—
331C4—3461no—
332C4—3461no—
333C4—3461no—
334C4—6921no—
335C4—6921no—
336C4—6921no—
337C4#41×D17311yescharacteristic
338C4#41×D17311yescharacteristic
339C2—81no—
340C2—13841no—
341C2—6921no—
342C2#2601204011yescharacteristic
343C2#2601228811yescharacteristic
344C2#2601228811yescharacteristic
subgroupG/H
centerC22C2×D4×D173
commutator, cosocleC2×C4×C173C24
FrattiniC2×C4C23×D173
Fitting#488×C173C2
socleC22×C173C22×D4
prime radical#26011743C2
hypercenter#488D173
hypocenterC173#3183
LCSG ▷ C2×C4×C173 ▷ C2×C173 ▷ C173
UCSC1 ◁ C22 ◁ C42 ◁ #488

Complements

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

NH ≅ G/Ntype
1. #26012026C2semidirect
2. #26012020C2semidirect
3. #26011890C2semidirect
4. #26011873C2semidirect
5. #26011846C2semidirect
6. #26011748C2semidirect
8. #26011740C2semidirect
9. #26011736C2semidirect
10. #26011600C2semidirect
11. #26011595C2semidirect
12. #26011594C2semidirect
13. #26011588C2semidirect
14. C4×#14583809C2semidirect
15. #488×C173C2semidirect
172. C173#3183semidirect
179. #488D173semidirect

Maximal subgroups

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

subgroupindex
#260120262
#260120202
#260118902
#260118732
#260118462
#260117482
#260117432
#260117402
#260117362
#260116002
#260115952
#260115942
#260115882
C4×#145838092
#488×C1732
#3183173