Finite Group Databasebeta

← #36363850

Subgroups — #36363850

Order 22144.

382 conjugacy classes of subgroups.

The subgroup lattice has 350 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#26012094C211yesmaximal, characteristic
2#26011968C211yesmaximal, characteristic, complemented
3#26011815C211yesmaximal, characteristic, complemented
4#26011801C211yesmaximal, characteristic, complemented
5#26011797C211yesmaximal, characteristic, complemented
6#26011690C211yesmaximal, characteristic, complemented
7#26011690C211yesmaximal, characteristic, complemented
8#26011632C211yesmaximal, characteristic, complemented
9#26011629C211yesmaximal, characteristic, complemented
10#26011590C211yesmaximal, characteristic, complemented
11#26011588C211yesmaximal, characteristic, complemented
12#26011582C211yesmaximal, characteristic, complemented
13#26011578C211yesmaximal, characteristic, complemented
14#26011549C211yesmaximal, characteristic, complemented
15#482×C173C211yesmaximal, characteristic, complemented
16C22×Dic346C2211yescharacteristic
17C2×#14583815C2211yescharacteristic
18C2×#14583815C2211yescharacteristic
19C2×#14583814C2211yescharacteristic
20C2×#14583814C2211yescharacteristic
21#19149675C2211yescharacteristic
22#19149670C2211yescharacteristic
23#19149663—21no—
24#19149661—21no—
25#19149653C2211yescharacteristic
26#19149651—21no—
27#19149650—21no—
28C4×Dic346—21no—
29C4×Dic346—21no—
30#19149644C2211yescharacteristic
31#19149644C2211yescharacteristic
32#19149644C2211yescharacteristic
33#19149644C2211yescharacteristic
34D4×Dic173C2211yescharacteristic
35#19149641C2211yescharacteristic
36#19149638C2211yescharacteristic
37#19149635—21no—
38#19149635—21no—
39#19149612C2211yescharacteristic
40#19149610C2211yescharacteristic
41#19149608C2211yescharacteristic
42#19149608C2211yescharacteristic
43#19149604C2211yescharacteristic
44#19149604C2211yescharacteristic
45#19149602C2211yescharacteristic
46#19149602C2211yescharacteristic
47#19149599C2211yescharacteristic
48#19149599C2211yescharacteristic
49C22⋊#13063436C2211yescharacteristic
50#19149589C2211yescharacteristic
51C8×Dic173C2211yescharacteristic
52#121×C173C2211yescharacteristic
53#117×C173C2211yescharacteristic
54#107×C173C2211yescharacteristic
55#106×C173C2211yescharacteristic
56#102×C173C2211yescharacteristic
57#102×C173C2211yescharacteristic
58C22⋊C8×C173C2211yescharacteristic
59C2×Dic346—22no—
60C2×Dic346—22no—
61C2×Dic346—21no—
62C2×Dic346C2311yescharacteristic
63C2×Dic346C2311yescharacteristic
64C2×D692C2311yescharacteristic
65C2×C22⋊D173—21no—
66C2×C22⋊D173—21no—
67C2×C22⋊D173—21no—
68C22×Dic173—21no—
69C22×Dic173—21no—
70C2×C4×D173—21no—
71#14583815—22no—
72#14583815—22no—
73#14583814—22no—
74#14583814—22no—
75C2×#13063436C2311yescharacteristic
76C2×#13063436C2311yescharacteristic
77C4⋊Dic173—21no—
78C4⋊Dic173D411yescharacteristic
79C4⋊Dic173D411yescharacteristic
80C4⋊Dic173C2311yescharacteristic
81#14583805—21no—
82#14583805—21no—
83#14583805—21no—
84#14583805—21no—
85#14583805—21no—
86C22⋊Dic173—21no—
87C22⋊Dic173D411yescharacteristic
88C22⋊Dic173D411yescharacteristic
89#14583803—21no—
90#14583803—21no—
91C4×Dic173—21no—
92C4×Dic173—21no—
93C4×Dic173—21no—
94C4×Dic173—21no—
95C4×Dic173C2311yescharacteristic
96C4×Dic173C2311yescharacteristic
97C2×D4×C173—21no—
98C2×D4×C173C2311yescharacteristic
99C22×C4×C173C2311yescharacteristic
100C2×C8×C173C2311yescharacteristic
101C2×C8×C173C2311yescharacteristic
102C4⋊C4×C173C2311yescharacteristic
103C4⋊C4×C173C2311yescharacteristic
104C4⋊C4×C173C2311yescharacteristic
105C22⋊C4×C173—21no—
106C22⋊C4×C173—21no—
107C42×C173—21no—
108C22×D173—21no—
109Dic346—24no—
110Dic346—21no—
111Dic346—21no—
112Dic346#4112yes—
113Dic346#4112yes—
114D692—21no—
115D692—21no—
116C22⋊D173—81no—
117C22⋊D173—81no—
118C22⋊D173—81no—
119C2×Dic173—24no—
120C2×Dic173—21no—
121C2×Dic173—21no—
122C2×Dic173—21no—
123C2×Dic173—81no—
124C2×Dic173C2×D411yescharacteristic
125C2×Dic173C2×D411yescharacteristic
126C2×Dic173C2×D411yescharacteristic
127C2×Dic173C2×D411yescharacteristic
128C4×D173—81no—
129#13063436—21no—
130#13063436—21no—
131C23×C173—21no—
132C23×C173C2×D411yescharacteristic
133D4×C173—21no—
134D4×C173—21no—
135D4×C173—81no—
136C2×C4×C173—21no—
137C2×C4×C173—21no—
138C2×C4×C173—21no—
139C2×C4×C173—21no—
140C2×C4×C173—21no—
141C2×C4×C173C2×D411yescharacteristic
142C2×C4×C173C2411yescharacteristic
143C8×C173—21no—
144C8×C173—21no—
145C22×C173—21no—
146C22×C173—21no—
147C22×C173—21no—
148C22×C173—41no—
149C22×C173—41no—
150C22×C173—41no—
151C22×C173C22×D411yescharacteristic
152C2×D173—41no—
153C2×D173—41no—
154C2×D173—41no—
155C4×C173—21no—
156C4×C173—41no—
157C4×C173—41no—
158C4×C173—41no—
159C4×C173C2×#4111yescharacteristic
160C4×C173C2×#4111yescharacteristic
161Dic173—22no—
162Dic173—22no—
163Dic173—21no—
164Dic173—21no—
165Dic173—21no—
166Dic173—21no—
167Dic173—41no—
168C2×C173—41no—
169C2×C173—81no—
170C2×C173#52511yescharacteristic
171C2×C173#57711yescharacteristic
172C2×C173#57711yescharacteristic
173D173—81no—
174C173#312611yescharacteristic, complemented, Sylow
175#3126—1731nomaximal, Sylow
176#540—1731no—
177#519—1731no—
178#482—1731no—
179#482D17311yescharacteristic, complemented
180#479—1731no—
181#479—1731no—
182#460—1731no—
183#460—1731no—
184#448—1731no—
185#448—1731no—
186#440—1731no—
187#440—1731no—
188C4×SD16—1731no—
189C4×SD16—1731no—
190#433—1731no—
191C22×Q8—1731no—
192C2×SD16—1731no—
193C2×SD16—1731no—
194C2×SD16—1731no—
195C2×SD16—1731no—
196#127—1731no—
197#125—1731no—
198#123—3461no—
199#123—3461no—
200#121—3461no—
201#121—3461no—
202#121—1731no—
203#121C2×D17311yescharacteristic
204C4×Q8—3461no—
205C4×Q8—3461no—
206C4×Q8—1731no—
207C4×Q8—1731no—
208C4×Q8—1731no—
209C4×Q8—1731no—
210C4×D4—1731no—
211C4×D4—1731no—
212#117—3461no—
213#117—3461no—
214#117—1731no—
215#117C2×D17311yescharacteristic
216#107—1731no—
217#107C2×D17311yescharacteristic
218#106—1731no—
219#106C2×D17311yescharacteristic
220C4⋊C8—1731no—
221C4⋊C8—1731no—
222#103—1731no—
223#103—1731no—
224#103—1731no—
225#103—1731no—
226#102—1731no—
227#102—1731no—
228#102C2×D17311yescharacteristic
229#102C2×D17311yescharacteristic
230C22⋊C8—1731no—
231C22⋊C8C2×D17311yescharacteristic
232#97—1731no—
233C4×C8—1731no—
234C2×Q8—3462no—
235C2×Q8—3462no—
236C2×Q8—3461no—
237C2×Q8—1731no—
238C2×Q8—1731no—
239C2×D4—21no—
240C2×D4—3461no—
241C2×D4—3461no—
242C2×D4—3461no—
243C2×D4—1731no—
244C2×D4C22×D17311yescharacteristic
245C22×C4—3461no—
246C22×C4—3461no—
247C22×C4—3461no—
248C22×C4C22×D17311yescharacteristic
249SD16—3462no—
250SD16—3462no—
251SD16—3462no—
252SD16—3462no—
253C2×C8—1731no—
254C2×C8—1731no—
255C2×C8C22×D17311yescharacteristic
256C2×C8C22×D17311yescharacteristic
257C4⋊C4—3461no—
258C4⋊C4—3461no—
259C4⋊C4—3461no—
260C4⋊C4—3461no—
261C4⋊C4—3461no—
262C4⋊C4—3461no—
263C4⋊C4—1731no—
264C4⋊C4—1731no—
265C4⋊C4—1731no—
266C4⋊C4C22×D17311yescharacteristic
267C4⋊C4C22×D17311yescharacteristic
268C4⋊C4C22×D17311yescharacteristic
269C22⋊C4—21no—
270C22⋊C4—21no—
271C22⋊C4—3461no—
272C22⋊C4—3461no—
273C22⋊C4—3461no—
274C22⋊C4—1731no—
275C22⋊C4—1731no—
276C42—21no—
277C42—3461no—
278C42—3461no—
279C42—3461no—
280C42—3461no—
281C42—1731no—
282C42—1731no—
283C23—21no—
284C23—3461no—
285C23D4×D17311yescharacteristic
286Q8—3464no—
287Q8—3461no—
288Q8—3461no—
289Q8—1732no—
290Q8—1732no—
291D4—21no—
292D4—21no—
293D4—81no—
294D4—3461no—
295D4—3461no—
296D4—13841no—
297D4—13841no—
298D4—13841no—
299C2×C4—21no—
300C2×C4—21no—
301C2×C4—21no—
302C2×C4—21no—
303C2×C4—21no—
304C2×C4—3464no—
305C2×C4—3461no—
306C2×C4—3461no—
307C2×C4—3461no—
308C2×C4—13841no—
309C2×C4—13841no—
310C2×C4—1731no—
311C2×C4—1731no—
312C2×C4—1731no—
313C2×C4—1731no—
314C2×C4D4×D17311yescharacteristic
315C2×C4C23×D17311yescharacteristic
316C8—21no—
317C8—21no—
318C8—3461no—
319C8—3461no—
320C22—21no—
321C22—21no—
322C22—21no—
323C22—41no—
324C22—41no—
325C22—41no—
326C22—6921no—
327C22—6921no—
328C22—6921no—
329C22C2×D4×D17311yescharacteristic
330C4—21no—
331C4—41no—
332C4—41no—
333C4—41no—
334C4—3462no—
335C4—3462no—
336C4—3461no—
337C4—3461no—
338C4—3461no—
339C4—3461no—
340C4—6921no—
341C4#41×D17311yescharacteristic
342C4#41×D17311yescharacteristic
343C2—41no—
344C2—81no—
345C2—13841no—
346C2#2601199911yescharacteristic
347C2#2601229811yescharacteristic
348C2#2601230111yescharacteristic
subgroupG/H
centerC22C2×D4×D173
commutator, cosocleC2×C4×C173C24
FrattiniC2×C4C23×D173
Fitting#482×C173C2
socleC22×C173C22×D4
prime radical#26012094C2
hypercenter#482D173
hypocenterC173#3126
LCSG ▷ C2×C4×C173 ▷ C2×C173 ▷ C173
UCSC1 ◁ C22 ◁ C22×C4 ◁ #482

Complements

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

NH ≅ G/Ntype
2. #26011968C2semidirect
3. #26011815C2semidirect
4. #26011801C2semidirect
5. #26011797C2semidirect
6. #26011690C2semidirect
7. #26011690C2semidirect
8. #26011632C2semidirect
9. #26011629C2semidirect
10. #26011590C2semidirect
11. #26011588C2semidirect
12. #26011582C2semidirect
13. #26011578C2semidirect
14. #26011549C2semidirect
15. #482×C173C2semidirect
174. C173#3126semidirect
179. #482D173semidirect

Maximal subgroups

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

subgroupindex
#260120942
#260119682
#260118152
#260118012
#260117972
#260116902
#260116322
#260116292
#260115902
#260115882
#260115822
#260115782
#260115492
#482×C1732
#3126173