Finite Group Databasebeta

← #36365137

Subgroups — #36365137

Order 22144.

472 conjugacy classes of subgroups.

The subgroup lattice has 396 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#19149711C211yesmaximal, characteristic, complemented
2C22×#14583816C211yesmaximal, characteristic, complemented
3#26012154C211yesmaximal, characteristic, complemented
4#26012152C211yesmaximal, characteristic
5#26011884C211yesmaximal, characteristic, complemented
6#26011748C211yesmaximal, characteristic, complemented
7#26011722C211yesmaximal, characteristic, complemented
8#26011716C211yesmaximal, characteristic, complemented
9#26011699C211yesmaximal, characteristic, complemented
10#26011656C211yesmaximal, characteristic, complemented
11#26011645C211yesmaximal, characteristic, complemented
12#26011632C211yesmaximal, characteristic, complemented
13#26011595C211yesmaximal, characteristic, complemented
14#26011559C211yesmaximal, characteristic, complemented
15#451×C173C211yesmaximal, characteristic, complemented
16C2×#14583827C2211yescharacteristic
17C22×Dic346C2211yescharacteristic
18C2×Q8×D173C2211yescharacteristic
19#19149711—24no—
20C2×#14583816—24no—
21C2×#14583816C2211yescharacteristic
22C2×#14583816C2211yescharacteristic
23C2×#14583816C2211yescharacteristic
24C2×#14583816C2211yescharacteristic
25C2×#14583815C2211yescharacteristic
26C2×#14583815C2211yescharacteristic
27C2×#14583813C2211yescharacteristic
28C2×#14583809C2211yescharacteristic
29C22×#13063436C2211yescharacteristic
30#19149678—21no—
31#19149677C2211yescharacteristic
32#19149668—21no—
33#19149668C2211yescharacteristic
34#19149659C2211yescharacteristic
35#19149657—21no—
36#19149656—21no—
37#19149655—21no—
38#19149655C2211yescharacteristic
39#19149653—21no—
40#19149653C2211yescharacteristic
41Q8×Dic173C2211yescharacteristic
42#19149644C2211yescharacteristic
43C4×C22⋊D173—21no—
44C4×C22⋊D173C2211yescharacteristic
45#19149639—21no—
46#19149610C2211yescharacteristic
47#19149608C2211yescharacteristic
48#19149604C2211yescharacteristic
49#19149604C2211yescharacteristic
50#19149603C2211yescharacteristic
51#19149599C2211yescharacteristic
52#19149591C2211yescharacteristic
53#19149589C2211yescharacteristic
54C22×Q8×C173C2211yescharacteristic
55C2×Q16×C173C2211yescharacteristic
56C2×Q16×C173C2211yescharacteristic
57C22⋊Q8×C173C2211yescharacteristic
58#103×C173C2211yescharacteristic
59#103×C173C2211yescharacteristic
60C22⋊C8×C173C2211yescharacteristic
61#14583827—22no—
62#14583827—41no—
63C2×Dic346—21no—
64C2×Dic346—22no—
65C2×Dic346—21no—
66C2×Dic346—41no—
67C2×Dic346C2311yescharacteristic
68C2×Dic346C2311yescharacteristic
69Q8×D173—22no—
70Q8×D173—41no—
71C2×D692—21no—
72C2×D692C2311yescharacteristic
73C2×C22⋊D173—21no—
74C2×C22⋊D173D411yescharacteristic
75C22×Dic173—21no—
76C2×C4×D173—21no—
77C2×C4×D173—21no—
78C2×C4×D173C2311yescharacteristic
79#14583816—22no—
80#14583816—22no—
81#14583816—22no—
82#14583816D414yescomplemented
83#14583815—22no—
84#14583815—41no—
85#14583813—22no—
86#14583809—22no—
87C2×#13063436—22no—
88C2×#13063436C2311yescharacteristic
89C2×#13063436C2311yescharacteristic
90C4⋊Dic173—21no—
91C4⋊Dic173—21no—
92C4⋊Dic173C2311yescharacteristic
93#14583805—21no—
94#14583805—21no—
95#14583805—21no—
96#14583805—21no—
97#14583805D411yescharacteristic
98C22⋊Dic173D411yescharacteristic
99#14583803—21no—
100#14583803—21no—
101#14583803—21no—
102#14583803—21no—
103#14583803—21no—
104#14583803D411yescharacteristic
105C4×Dic173—21no—
106C4×Dic173—21no—
107C4×Dic173C2311yescharacteristic
108C2×Q8×C173—21no—
109C2×Q8×C173—22no—
110C2×Q8×C173—41no—
111C2×Q8×C173C2311yescharacteristic
112C2×Q8×C173C2311yescharacteristic
113C2×Q8×C173C2311yescharacteristic
114C22×C4×C173—21no—
115C22×C4×C173C2311yescharacteristic
116Q16×C173—22no—
117Q16×C173—41no—
118C2×C8×C173C2311yescharacteristic
119C2×C8×C173C2311yescharacteristic
120C4⋊C4×C173—21no—
121C4⋊C4×C173C2311yescharacteristic
122C22⋊C4×C173—21no—
123C22×D173—21no—
124C22×D173C2×D411yescharacteristic
125Dic346—21no—
126Dic346—21no—
127Dic346—21no—
128Dic346—42no—
129Dic346—42no—
130Dic346C2×D412yes—
131D692—21no—
132D692—21no—
133D692—42no—
134C22⋊D173—22no—
135C22⋊D173—42no—
136C2×Dic173—21no—
137C2×Dic173—21no—
138C2×Dic173—21no—
139C2×Dic173—42no—
140C2×Dic173C2×D411yescharacteristic
141C2×Dic173C2×D411yescharacteristic
142C2×Dic173C2×D411yescharacteristic
143C4×D173—21no—
144C4×D173—21no—
145C4×D173—42no—
146C4×D173—42no—
147#13063436—21no—
148#13063436C2×D412yes—
149C23×C173C2×D411yescharacteristic
150Q8×C173—21no—
151Q8×C173—21no—
152Q8×C173—21no—
153Q8×C173—21no—
154Q8×C173—21no—
155Q8×C173—42no—
156Q8×C173C2×D412yes—
157C2×C4×C173—21no—
158C2×C4×C173—21no—
159C2×C4×C173—21no—
160C2×C4×C173—21no—
161C2×C4×C173—21no—
162C2×C4×C173—21no—
163C2×C4×C173—42no—
164C2×C4×C173C2×D411yescharacteristic
165C2×C4×C173C2411yescharacteristic
166C8×C173—21no—
167C8×C173—21no—
168C22×C173—21no—
169C22×C173—21no—
170C22×C173#13712yescomplemented
171C22×C173C22×D411yescharacteristic
172C2×D173—21no—
173C2×D173—21no—
174C2×D173—21no—
175C2×D173—41no—
176C2×D173—41no—
177C2×D173—41no—
178C4×C173—21no—
179C4×C173—22no—
180C4×C173—41no—
181C4×C173—41no—
182C4×C173—41no—
183C4×C173C22×D411yescharacteristic
184C4×C173H(5,𝔽2)11yescharacteristic
185Dic173—21no—
186Dic173—22no—
187Dic173—21no—
188Dic173—21no—
189Dic173—41no—
190Dic173—41no—
191C2×C173—22no—
192C2×C173D4211yescharacteristic
193C2×C173C2×#13711yescharacteristic
194C2×C173#57811yescharacteristic
195D173—41no—
196D173—81no—
197C173#327011yescharacteristic, complemented, Sylow
198#3270—1731nomaximal, Sylow
199C2×#137—1731no—
200C22×Q16—1731no—
201D4×Q8—1731no—
202#548—1731no—
203#497—1731no—
204#470—1731no—
205#465—1731no—
206#464—1731no—
207#462—1731no—
208#452—1731no—
209#451—1731no—
210#451D17311yescharacteristic, complemented
211#448—1731no—
212#441—1731no—
213#435—1731no—
214C2×#41—1731no—
215C22×Q8—1731no—
216C22×Q8—1731no—
217C22×Q8C2×D17311yescharacteristic
218#137—3464no—
219C2×Q16—3464no—
220C2×Q16—1731no—
221C2×Q16—1731no—
222C2×Q16—1731no—
223C2×Q16—1731no—
224C2×Q16C2×D17311yescharacteristic
225C2×Q16C2×D17311yescharacteristic
226C2×SD16—1731no—
227C2×SD16—1731no—
228C2×SD16—1731no—
229C2×M16—1731no—
230C22×C8—1731no—
231C4⋊Q8—3461no—
232C4⋊Q8—1731no—
233#124—3461no—
234#124—1731no—
235C22⋊Q8—3461no—
236C22⋊Q8—3461no—
237C22⋊Q8—3461no—
238C22⋊Q8—1731no—
239C22⋊Q8—1731no—
240C22⋊Q8C2×D17311yescharacteristic
241#121—3461no—
242#121—1731no—
243C4×Q8—1731no—
244C4×Q8—1731no—
245C4×D4—3461no—
246C4×D4—3461no—
247C4×D4—1731no—
248#106—1731no—
249C4⋊C8—1731no—
250#103—1731no—
251#103—1731no—
252#103—1731no—
253#103C2×D17311yescharacteristic
254#103C2×D17311yescharacteristic
255#102—1731no—
256C22⋊C8—1731no—
257C22⋊C8C2×D17311yescharacteristic
258#97—1731no—
259#41—3462no—
260#41—6921no—
261C2×Q8—21no—
262C2×Q8—22no—
263C2×Q8—41no—
264C2×Q8—3461no—
265C2×Q8—3462no—
266C2×Q8—3461no—
267C2×Q8—3462no—
268C2×Q8—1731no—
269C2×Q8—1731no—
270C2×Q8—6921no—
271C2×Q8—6921no—
272C2×Q8C22×D17311yescharacteristic
273C2×Q8C22×D17311yescharacteristic
274C2×Q8C22×D17311yescharacteristic
275C2×D4—3461no—
276C2×D4—3461no—
277C2×D4—1731no—
278C2×D4—1731no—
279C22×C4—21no—
280C22×C4—3461no—
281C22×C4—3461no—
282C22×C4—3461no—
283C22×C4—1731no—
284C22×C4C22×D17311yescharacteristic
285Q16—22no—
286Q16—41no—
287Q16—3462no—
288Q16—3462no—
289Q16—3462no—
290Q16—1734no—
291SD16—3462no—
292SD16—3462no—
293SD16—6921no—
294M16—3462no—
295C2×C8—3462no—
296C2×C8—1731no—
297C2×C8—1731no—
298C2×C8C22×D17311yescharacteristic
299C2×C8C22×D17311yescharacteristic
300C4⋊C4—21no—
301C4⋊C4—3461no—
302C4⋊C4—3461no—
303C4⋊C4—3461no—
304C4⋊C4—3461no—
305C4⋊C4—3461no—
306C4⋊C4—3461no—
307C4⋊C4—1731no—
308C4⋊C4—1731no—
309C4⋊C4C22×D17311yescharacteristic
310C22⋊C4—21no—
311C22⋊C4—3461no—
312C22⋊C4—3461no—
313C22⋊C4—3461no—
314C22⋊C4—3461no—
315C22⋊C4—3461no—
316C22⋊C4—1731no—
317C22⋊C4—1731no—
318C42—3461no—
319C42—3461no—
320C42—1731no—
321C23—3461no—
322C23—1731no—
323C23D4×D17311yescharacteristic
324Q8—21no—
325Q8—21no—
326Q8—21no—
327Q8—21no—
328Q8—21no—
329Q8—42no—
330Q8—3461no—
331Q8—3461no—
332Q8—3461no—
333Q8—1732no—
334Q8—6922no—
335Q8—6922no—
336Q8D4×D17312yes—
337D4—3462no—
338D4—3461no—
339D4—3461no—
340D4—6922no—
341D4—6922no—
342C2×C4—21no—
343C2×C4—21no—
344C2×C4—21no—
345C2×C4—21no—
346C2×C4—21no—
347C2×C4—21no—
348C2×C4—42no—
349C2×C4—3461no—
350C2×C4—3461no—
351C2×C4—3461no—
352C2×C4—3461no—
353C2×C4—3461no—
354C2×C4—1731no—
355C2×C4—1731no—
356C2×C4—1731no—
357C2×C4—6922no—
358C2×C4—6922no—
359C2×C4—6922no—
360C2×C4D4×D17311yescharacteristic
361C2×C4C23×D17311yescharacteristic
362C8—21no—
363C8—21no—
364C8—3461no—
365C8—1732no—
366C22—21no—
367C22—21no—
368C22—3461no—
369C22—3461no—
370C22—3461no—
371C22—6921no—
372C22—6921no—
373C22—6921no—
374C22#1914971112yescomplemented
375C22C2×D4×D17311yescharacteristic
376C4—21no—
377C4—22no—
378C4—41no—
379C4—41no—
380C4—41no—
381C4—3461no—
382C4—3461no—
383C4—3461no—
384C4—3462no—
385C4—6921no—
386C4—6921no—
387C4C2×D4×D17311yescharacteristic
388C4#1914972411yescharacteristic
389C2—22no—
390C2—13841no—
391C2—6921no—
392C2#2601212311yescharacteristic
393C2C2×#1914971111yescharacteristic
394C2#2601230711yescharacteristic
subgroupG/H
centerC22C2×D4×D173
commutator, cosocleC2×C4×C173C24
FrattiniC2×C4C23×D173
Fitting#451×C173C2
socleC22×C173C22×D4
prime radical#26012152C2
hypercenter#451D173
hypocenterC173#3270
LCSG ▷ C2×C4×C173 ▷ C2×C173 ▷ C173
UCSC1 ◁ C22 ◁ C22×C4 ◁ #451

Complements

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

NH ≅ G/Ntype
1. C2×#19149711C2semidirect
2. C22×#14583816C2semidirect
3. #26012154C2semidirect
5. #26011884C2semidirect
6. #26011748C2semidirect
7. #26011722C2semidirect
8. #26011716C2semidirect
9. #26011699C2semidirect
10. #26011656C2semidirect
11. #26011645C2semidirect
12. #26011632C2semidirect
13. #26011595C2semidirect
14. #26011559C2semidirect
15. #451×C173C2semidirect
82. #14583816D4semidirect
170. C22×C173#137semidirect
197. C173#3270semidirect
210. #451D173semidirect
374. C22#19149711semidirect

Maximal subgroups

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

subgroupindex
C2×#191497112
C22×#145838162
#260121542
#260121522
#260118842
#260117482
#260117222
#260117162
#260116992
#260116562
#260116452
#260116322
#260115952
#260115592
#451×C1732
#3270173