Finite Group Databasebeta

← #36365140

Subgroups — #36365140

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×#19149712C211yesmaximal, characteristic, complemented
2C2×Q16×D173C211yesmaximal, characteristic, complemented
3#26012154C211yesmaximal, characteristic, complemented
4#26012151C211yesmaximal, characteristic
5#26011884C211yesmaximal, characteristic, complemented
6#26011721C211yesmaximal, characteristic, complemented
7#26011716C211yesmaximal, characteristic, complemented
8#26011700C211yesmaximal, characteristic, complemented
9#26011651C211yesmaximal, characteristic, complemented
10#26011648C211yesmaximal, characteristic, complemented
11#26011646C211yesmaximal, characteristic, complemented
12#26011631C211yesmaximal, characteristic, complemented
13#26011597C211yesmaximal, characteristic, complemented
14#26011560C211yesmaximal, characteristic, complemented
15#470×C173C211yesmaximal, characteristic, complemented
16C2×#14583826C2211yescharacteristic
17C22×Dic346C2211yescharacteristic
18C2×Q8×D173C2211yescharacteristic
19C2×Q8×D173C2211yescharacteristic
20#19149712—24no—
21C2×Dic692C2211yescharacteristic
22C2×Dic692C2211yescharacteristic
23C2×#14583816C2211yescharacteristic
24C2×#14583816C2211yescharacteristic
25C2×#14583816C2211yescharacteristic
26Q16×D173—24no—
27C2×#14583815C2211yescharacteristic
28C2×#14583815C2211yescharacteristic
29C2×#14583813C2211yescharacteristic
30C2×C8×D173C2211yescharacteristic
31#19149678—21no—
32#19149677C2211yescharacteristic
33#19149668C2211yescharacteristic
34#19149665—21no—
35#19149659C2211yescharacteristic
36#19149657—21no—
37#19149657—21no—
38#19149656—21no—
39#19149654C2211yescharacteristic
40#19149652—21no—
41#19149644C2211yescharacteristic
42#19149644C2211yescharacteristic
43#19149641—21no—
44C4×C22⋊D173C2211yescharacteristic
45#19149639—21no—
46#19149608C2211yescharacteristic
47#19149602C2211yescharacteristic
48#19149602C2211yescharacteristic
49#19149602C2211yescharacteristic
50#19149599C2211yescharacteristic
51C22⋊#13063436C2211yescharacteristic
52#19149591C2211yescharacteristic
53#19149589C2211yescharacteristic
54C2×Q16×C173C2211yescharacteristic
55C2×M16×C173C2211yescharacteristic
56C22⋊Q8×C173C2211yescharacteristic
57C22⋊Q8×C173C2211yescharacteristic
58#106×C173C2211yescharacteristic
59#103×C173C2211yescharacteristic
60#103×C173C2211yescharacteristic
61#14583826—22no—
62#14583826—41no—
63C2×Dic346—21no—
64C2×Dic346—21no—
65C2×Dic346—22no—
66C2×Dic346—21no—
67C2×Dic346—41no—
68C2×Dic346C2311yescharacteristic
69C2×Dic346C2311yescharacteristic
70C2×Dic346C2311yescharacteristic
71Q8×D173—22no—
72Q8×D173—22no—
73Q8×D173—41no—
74Q8×D173—41no—
75C2×D692—21no—
76C2×D692C2311yescharacteristic
77C2×C22⋊D173—21no—
78C2×C22⋊D173D411yescharacteristic
79C22×Dic173—21no—
80C2×C4×D173—21no—
81C2×C4×D173—21no—
82C2×C4×D173—21no—
83C2×C4×D173C2311yescharacteristic
84Dic692—22no—
85Dic692D414yescomplemented
86#14583816—22no—
87#14583816—22no—
88#14583816—41no—
89#14583815—22no—
90#14583815—22no—
91#14583813—41no—
92C2×#13063436C2311yescharacteristic
93C2×#13063436C2311yescharacteristic
94C8×D173—22no—
95C4⋊Dic173—21no—
96#14583805—21no—
97#14583805—21no—
98#14583805—21no—
99#14583805—21no—
100#14583805D411yescharacteristic
101C22⋊Dic173D411yescharacteristic
102#14583803—21no—
103#14583803—21no—
104#14583803—21no—
105#14583803—21no—
106#14583803D411yescharacteristic
107C4×Dic173—21no—
108C4×Dic173—21no—
109C4×Dic173C2311yescharacteristic
110C2×Q8×C173C2311yescharacteristic
111C2×Q8×C173C2311yescharacteristic
112C22×C4×C173C2311yescharacteristic
113Q16×C173—22no—
114M16×C173—22no—
115C2×C8×C173C2311yescharacteristic
116C2×C8×C173C2311yescharacteristic
117C4⋊C4×C173—21no—
118C4⋊C4×C173—21no—
119C4⋊C4×C173C2311yescharacteristic
120C4⋊C4×C173C2311yescharacteristic
121C22⋊C4×C173—21no—
122C22⋊C4×C173—21no—
123C22×D173—21no—
124C22×D173C2×D411yescharacteristic
125Dic346—21no—
126Dic346—21no—
127Dic346—21no—
128Dic346—21no—
129Dic346—42no—
130Dic346—42no—
131Dic346—42no—
132Dic346C2×D412yes—
133Dic346C2×D412yes—
134D692—21no—
135D692—21no—
136D692—42no—
137C22⋊D173—22no—
138C22⋊D173—42no—
139C2×Dic173—21no—
140C2×Dic173—21no—
141C2×Dic173—21no—
142C2×Dic173—42no—
143C2×Dic173C2×D411yescharacteristic
144C2×Dic173C2×D411yescharacteristic
145C2×Dic173C2×D411yescharacteristic
146C4×D173—21no—
147C4×D173—21no—
148C4×D173—42no—
149C4×D173—42no—
150C4×D173—42no—
151#13063436—21no—
152#13063436—21no—
153C23×C173C2×D411yescharacteristic
154Q8×C173—21no—
155Q8×C173—21no—
156Q8×C173—21no—
157Q8×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—21no—
164C2×C4×C173C2×D411yescharacteristic
165C2×C4×C173C2411yescharacteristic
166C8×C173—21no—
167C8×C173C2×D412yes—
168C22×C173—21no—
169C22×C173—21no—
170C22×C173—21no—
171C22×C173C22×D411yescharacteristic
172C2×D173—21no—
173C2×D173—21no—
174C2×D173—41no—
175C2×D173—41no—
176C2×D173—41no—
177C2×D173#13712yescomplemented
178C4×C173—21no—
179C4×C173—41no—
180C4×C173—41no—
181C4×C173—41no—
182C4×C173—41no—
183C4×C173C22×D411yescharacteristic
184C4×C173H(5,𝔽2)11yescharacteristic
185Dic173—21no—
186Dic173—22no—
187Dic173—22no—
188Dic173—21no—
189Dic173—21no—
190Dic173—41no—
191C2×C173—41no—
192C2×C173D4211yescharacteristic
193C2×C173C2×#13711yescharacteristic
194C2×C173#57811yescharacteristic
195D173—22no—
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#470D17311yescharacteristic, complemented
205#465—1731no—
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×Q16C2×D17311yescharacteristic
226C2×SD16—1731no—
227C2×SD16—1731no—
228C2×SD16—1731no—
229C2×M16C2×D17311yescharacteristic
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⋊Q8C2×D17311yescharacteristic
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#106C2×D17311yescharacteristic
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⋊C8—1731no—
258#97—1731no—
259#41—3462no—
260#41—6921no—
261C2×Q8—3461no—
262C2×Q8—3461no—
263C2×Q8—3462no—
264C2×Q8—3461no—
265C2×Q8—3462no—
266C2×Q8—3462no—
267C2×Q8—1731no—
268C2×Q8—1731no—
269C2×Q8—1731no—
270C2×Q8—6921no—
271C2×Q8—6921no—
272C2×Q8—6921no—
273C2×Q8C22×D17311yescharacteristic
274C2×Q8C22×D17311yescharacteristic
275C2×D4—3461no—
276C2×D4—3461no—
277C2×D4—1731no—
278C2×D4—1731no—
279C22×C4—3461no—
280C22×C4—3461no—
281C22×C4—3461no—
282C22×C4—3461no—
283C22×C4—1731no—
284C22×C4C22×D17311yescharacteristic
285Q16—22no—
286Q16—3462no—
287Q16—3462no—
288Q16—3462no—
289Q16—1734no—
290Q16—6921no—
291SD16—3462no—
292SD16—3462no—
293SD16—6921no—
294M16—22no—
295C2×C8—3462no—
296C2×C8—1731no—
297C2×C8—1731no—
298C2×C8C22×D17311yescharacteristic
299C2×C8C22×D17311yescharacteristic
300C4⋊C4—21no—
301C4⋊C4—21no—
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—3461no—
322C23—1731no—
323C23D4×D17311yescharacteristic
324Q8—21no—
325Q8—21no—
326Q8—21no—
327Q8—21no—
328Q8—3461no—
329Q8—3461no—
330Q8—3461no—
331Q8—3461no—
332Q8—1732no—
333Q8—1732no—
334Q8—6922no—
335Q8—6922no—
336Q8—6922no—
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—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—21no—
369C22—3461no—
370C22—3461no—
371C22—1732no—
372C22—6921no—
373C22—6921no—
374C22—6921no—
375C22C2×D4×D17311yescharacteristic
376C4—21no—
377C4—41no—
378C4—41no—
379C4—41no—
380C4—41no—
381C4—3461no—
382C4—3461no—
383C4—3461no—
384C4—3462no—
385C4—3462no—
386C4—6921no—
387C4C2×D4×D17311yescharacteristic
388C4#1914972511yescharacteristic
389C2—41no—
390C2—3462no—
391C2—13841no—
392C2#2601212511yescharacteristic
393C2#137×D17311yescharacteristic
394C2#2601230711yescharacteristic
subgroupG/H
centerC22C2×D4×D173
commutator, cosocleC2×C4×C173C24
FrattiniC2×C4C23×D173
Fitting#470×C173C2
socleC22×C173C22×D4
prime radical#26012151C2
hypercenter#470D173
hypocenterC173#3270
LCSG ▷ C2×C4×C173 ▷ C2×C173 ▷ C173
UCSC1 ◁ C22 ◁ C22×C4 ◁ #470

Complements

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

NH ≅ G/Ntype
1. C2×#19149712C2semidirect
2. C2×Q16×D173C2semidirect
3. #26012154C2semidirect
5. #26011884C2semidirect
6. #26011721C2semidirect
7. #26011716C2semidirect
8. #26011700C2semidirect
9. #26011651C2semidirect
10. #26011648C2semidirect
11. #26011646C2semidirect
12. #26011631C2semidirect
13. #26011597C2semidirect
14. #26011560C2semidirect
15. #470×C173C2semidirect
85. Dic692D4semidirect
177. C2×D173#137semidirect
197. C173#3270semidirect
204. #470D173semidirect

Maximal subgroups

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

subgroupindex
C2×#191497122
C2×Q16×D1732
#260121542
#260121512
#260118842
#260117212
#260117162
#260117002
#260116512
#260116482
#260116462
#260116312
#260115972
#260115602
#470×C1732
#3270173