Finite Group Databasebeta

← #36365130

Subgroups — #36365130

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×#19149708C211yesmaximal, characteristic, complemented
2C22×Dic692C211yesmaximal, characteristic, complemented
3#26012154C211yesmaximal, characteristic, complemented
4#26012147C211yesmaximal, characteristic
5#26011882C211yesmaximal, characteristic, complemented
6#26011750C211yesmaximal, characteristic, complemented
7#26011712C211yesmaximal, characteristic, complemented
8#26011700C211yesmaximal, characteristic, complemented
9#26011654C211yesmaximal, characteristic, complemented
10#26011648C211yesmaximal, characteristic, complemented
11#26011646C211yesmaximal, characteristic, complemented
12#26011629C211yesmaximal, characteristic, complemented
13#26011597C211yesmaximal, characteristic, complemented
14#26011562C211yesmaximal, characteristic, complemented
15#465×C173C211yesmaximal, characteristic, complemented
16C2×#14583825C2211yescharacteristic
17C22×Dic346C2211yescharacteristic
18C22×Dic346C2211yescharacteristic
19C2×Q8×D173C2211yescharacteristic
20#19149708—24no—
21C2×Dic692—24no—
22C2×Dic692C2211yescharacteristic
23C2×Dic692C2211yescharacteristic
24C2×Dic692C2211yescharacteristic
25C2×Dic692C2211yescharacteristic
26C2×#14583816C2211yescharacteristic
27C2×#14583816C2211yescharacteristic
28C2×#14583814C2211yescharacteristic
29C2×#14583814C2211yescharacteristic
30C2×#14583809C2211yescharacteristic
31#19149678—21no—
32#19149677C2211yescharacteristic
33#19149666C2211yescharacteristic
34#19149665—21no—
35#19149659C2211yescharacteristic
36#19149659C2211yescharacteristic
37#19149657—21no—
38#19149656—21no—
39#19149656—21no—
40#19149651—21no—
41#19149644C2211yescharacteristic
42#19149644C2211yescharacteristic
43D4×Dic173—21no—
44C4×C22⋊D173C2211yescharacteristic
45#19149639—21no—
46#19149608C2211yescharacteristic
47#19149602C2211yescharacteristic
48#19149602C2211yescharacteristic
49#19149602C2211yescharacteristic
50#19149602C2211yescharacteristic
51C22⋊#13063436C2211yescharacteristic
52#19149591C2211yescharacteristic
53#19149589C2211yescharacteristic
54C2×SD16×C173C2211yescharacteristic
55C22×C8×C173C2211yescharacteristic
56C22⋊Q8×C173C2211yescharacteristic
57#121×C173C2211yescharacteristic
58#106×C173C2211yescharacteristic
59#103×C173C2211yescharacteristic
60#102×C173C2211yescharacteristic
61#14583825—22no—
62#14583825—41no—
63C2×Dic346—21no—
64C2×Dic346—22no—
65C2×Dic346—22no—
66C2×Dic346—21no—
67C2×Dic346—21no—
68C2×Dic346—41no—
69C2×Dic346—41no—
70C2×Dic346C2311yescharacteristic
71C2×Dic346C2311yescharacteristic
72C2×Dic346C2311yescharacteristic
73C2×Dic346C2311yescharacteristic
74Q8×D173—22no—
75Q8×D173—41no—
76C2×C22⋊D173—21no—
77C2×C22⋊D173D411yescharacteristic
78C22×Dic173—21no—
79C22×Dic173—21no—
80C22×Dic173—21no—
81C2×C4×D173—21no—
82C2×C4×D173C2311yescharacteristic
83Dic692—22no—
84Dic692—22no—
85Dic692—22no—
86Dic692D414yescomplemented
87#14583816—22no—
88#14583816—41no—
89#14583814—22no—
90#14583814—41no—
91#14583809—22no—
92C2×#13063436C2311yescharacteristic
93C2×#13063436C2311yescharacteristic
94C4⋊Dic173—21no—
95C4⋊Dic173—21no—
96#14583805—21no—
97#14583805—21no—
98#14583805—21no—
99#14583805—21no—
100#14583805D411yescharacteristic
101C22⋊Dic173—21no—
102C22⋊Dic173—21no—
103C22⋊Dic173D411yescharacteristic
104#14583803—21no—
105#14583803—21no—
106#14583803D411yescharacteristic
107C4×Dic173—21no—
108C4×Dic173—21no—
109C4×Dic173C2311yescharacteristic
110C2×Q8×C173C2311yescharacteristic
111C2×D4×C173—21no—
112C2×D4×C173C2311yescharacteristic
113C22×C4×C173C2311yescharacteristic
114SD16×C173—22no—
115C2×C8×C173—22no—
116C2×C8×C173C2311yescharacteristic
117C2×C8×C173C2311yescharacteristic
118C4⋊C4×C173—21no—
119C4⋊C4×C173C2311yescharacteristic
120C4⋊C4×C173C2311yescharacteristic
121C22⋊C4×C173—21no—
122C22⋊C4×C173—21no—
123C22×D173C2×D411yescharacteristic
124Dic346—21no—
125Dic346—21no—
126Dic346—21no—
127Dic346—21no—
128Dic346—21no—
129Dic346—21no—
130Dic346—42no—
131Dic346—42no—
132Dic346—42no—
133Dic346C2×D412yes—
134Dic346C2×D412yes—
135C22⋊D173—22no—
136C22⋊D173—42no—
137C2×Dic173—21no—
138C2×Dic173—21no—
139C2×Dic173—21no—
140C2×Dic173—21no—
141C2×Dic173—42no—
142C2×Dic173—42no—
143C2×Dic173—42no—
144C2×Dic173C2×D411yescharacteristic
145C2×Dic173C2×D411yescharacteristic
146C2×Dic173C2×D411yescharacteristic
147C4×D173—21no—
148C4×D173—21no—
149C4×D173—42no—
150#13063436—21no—
151#13063436—21no—
152C23×C173—21no—
153C23×C173C2×D411yescharacteristic
154Q8×C173—21no—
155Q8×C173—21no—
156D4×C173—21no—
157D4×C173—21no—
158D4×C173—42no—
159C2×C4×C173—21no—
160C2×C4×C173—21no—
161C2×C4×C173—21no—
162C2×C4×C173—21no—
163C2×C4×C173—21no—
164C2×C4×C173C2×D411yescharacteristic
165C2×C4×C173C2411yescharacteristic
166C8×C173—21no—
167C8×C173C2×D412yes—
168C22×C173—21no—
169C22×C173—21no—
170C22×C173—41no—
171C22×C173—41no—
172C22×C173—41no—
173C22×C173#13712yescomplemented
174C22×C173C22×D411yescharacteristic
175C2×D173—21no—
176C2×D173—21no—
177C2×D173—21no—
178C4×C173—21no—
179C4×C173—41no—
180C4×C173—41no—
181C4×C173—41no—
182C4×C173C22×D411yescharacteristic
183C4×C173H(5,𝔽2)11yescharacteristic
184Dic173—21no—
185Dic173—22no—
186Dic173—22no—
187Dic173—21no—
188Dic173—21no—
189Dic173—41no—
190Dic173—41no—
191C2×C173—22no—
192C2×C173—81no—
193C2×C173D4211yescharacteristic
194C2×C173C2×#13711yescharacteristic
195C2×C173#57811yescharacteristic
196D173—41no—
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#465D17311yescharacteristic, complemented
206#464—1731no—
207#462—1731no—
208#452—1731no—
209#451—1731no—
210#451—1731no—
211#448—1731no—
212#441—1731no—
213#435—1731no—
214C2×#41—1731no—
215C22×Q8—1731no—
216C22×Q8—1731no—
217C22×Q8—1731no—
218#137—3464no—
219C2×Q16—3464no—
220C2×Q16—1731no—
221C2×Q16—1731no—
222C2×Q16—1731no—
223C2×Q16—1731no—
224C2×Q16—1731no—
225C2×Q16—1731no—
226C2×SD16—1731no—
227C2×SD16—1731no—
228C2×SD16C2×D17311yescharacteristic
229C2×M16—1731no—
230C22×C8C2×D17311yescharacteristic
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#121C2×D17311yescharacteristic
243C4×Q8—1731no—
244C4×Q8—1731no—
245C4×D4—3461no—
246C4×D4—3461no—
247C4×D4—1731no—
248#106C2×D17311yescharacteristic
249C4⋊C8—1731no—
250#103—1731no—
251#103—1731no—
252#103—1731no—
253#103—1731no—
254#103C2×D17311yescharacteristic
255#102C2×D17311yescharacteristic
256C22⋊C8—1731no—
257C22⋊C8—1731no—
258#97—1731no—
259#41—3462no—
260#41—6921no—
261C2×Q8—3461no—
262C2×Q8—3462no—
263C2×Q8—3462no—
264C2×Q8—3461no—
265C2×Q8—3461no—
266C2×Q8—3462no—
267C2×Q8—1731no—
268C2×Q8—1731no—
269C2×Q8—1731no—
270C2×Q8—1731no—
271C2×Q8—6921no—
272C2×Q8—6921no—
273C2×Q8—6921no—
274C2×Q8C22×D17311yescharacteristic
275C2×D4—21no—
276C2×D4—3461no—
277C2×D4—1731no—
278C2×D4C22×D17311yescharacteristic
279C22×C4—3461no—
280C22×C4—3461no—
281C22×C4—3461no—
282C22×C4—3461no—
283C22×C4—1731no—
284C22×C4C22×D17311yescharacteristic
285Q16—3462no—
286Q16—3462no—
287Q16—3462no—
288Q16—3462no—
289Q16—1734no—
290Q16—6921no—
291SD16—22no—
292SD16—3462no—
293SD16—6921no—
294M16—3462no—
295C2×C8—22no—
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⋊C4C22×D17311yescharacteristic
309C4⋊C4C22×D17311yescharacteristic
310C22⋊C4—21no—
311C22⋊C4—21no—
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—21no—
322C23—1731no—
323C23D4×D17311yescharacteristic
324Q8—21no—
325Q8—21no—
326Q8—3461no—
327Q8—3461no—
328Q8—3461no—
329Q8—3461no—
330Q8—3461no—
331Q8—3461no—
332Q8—1732no—
333Q8—1732no—
334Q8—6922no—
335Q8—6922no—
336Q8—6922no—
337D4—21no—
338D4—21no—
339D4—42no—
340D4—3462no—
341D4—6922no—
342C2×C4—21no—
343C2×C4—21no—
344C2×C4—21no—
345C2×C4—21no—
346C2×C4—21no—
347C2×C4—3461no—
348C2×C4—3461no—
349C2×C4—3461no—
350C2×C4—3461no—
351C2×C4—3461no—
352C2×C4—3461no—
353C2×C4—1731no—
354C2×C4—1731no—
355C2×C4—1731no—
356C2×C4—6922no—
357C2×C4—6922no—
358C2×C4—6922no—
359C2×C4—6922no—
360C2×C4D4×D17311yescharacteristic
361C2×C4C23×D17311yescharacteristic
362C8—21no—
363C8—3461no—
364C8—3461no—
365C8D4×D17312yes—
366C22—21no—
367C22—21no—
368C22—41no—
369C22—41no—
370C22—41no—
371C22—3461no—
372C22—3461no—
373C22—3461no—
374C22#1914970812yescomplemented
375C22C2×D4×D17311yescharacteristic
376C4—21no—
377C4—41no—
378C4—41no—
379C4—41no—
380C4—3461no—
381C4—3461no—
382C4—3461no—
383C4—3462no—
384C4—3462no—
385C4—6921no—
386C4—6921no—
387C4C2×D4×D17311yescharacteristic
388C4#1914972511yescharacteristic
389C2—22no—
390C2—81no—
391C2—6921no—
392C2#2601212511yescharacteristic
393C2C2×#1914970811yescharacteristic
394C2#2601230511yescharacteristic
subgroupG/H
centerC22C2×D4×D173
commutator, cosocleC2×C4×C173C24
FrattiniC2×C4C23×D173
Fitting#465×C173C2
socleC22×C173C22×D4
prime radical#26012147C2
hypercenter#465D173
hypocenterC173#3270
LCSG ▷ C2×C4×C173 ▷ C2×C173 ▷ C173
UCSC1 ◁ C22 ◁ C22×C4 ◁ #465

Complements

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

NH ≅ G/Ntype
1. C2×#19149708C2semidirect
2. C22×Dic692C2semidirect
3. #26012154C2semidirect
5. #26011882C2semidirect
6. #26011750C2semidirect
7. #26011712C2semidirect
8. #26011700C2semidirect
9. #26011654C2semidirect
10. #26011648C2semidirect
11. #26011646C2semidirect
12. #26011629C2semidirect
13. #26011597C2semidirect
14. #26011562C2semidirect
15. #465×C173C2semidirect
86. Dic692D4semidirect
173. C22×C173#137semidirect
197. C173#3270semidirect
205. #465D173semidirect
374. C22#19149708semidirect

Maximal subgroups

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

subgroupindex
C2×#191497082
C22×Dic6922
#260121542
#260121472
#260118822
#260117502
#260117122
#260117002
#260116542
#260116482
#260116462
#260116292
#260115972
#260115622
#465×C1732
#3270173