Finite Group Databasebeta

← #548×D173 #36367025

Subgroups — #548×D173 #36367025

Order 22144.

1620 conjugacy classes of subgroups.

The subgroup lattice has 430 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#41×D173C211yesmaximal, characteristic, complemented
2C22×Q8×D173C211yesmaximal, characteristic, complemented
3#26012153C211yesmaximal, characteristic, complemented
4#26012152C211yesmaximal, characteristic, complemented
5#26012151C213yesmaximal, complemented
6#26012150C213yesmaximal, complemented
7#26012149C213yesmaximal, complemented
8#26012148C211yesmaximal, characteristic, complemented
9#26012147C213yesmaximal, complemented
10#124×D173C213yesmaximal, complemented
11C22⋊Q8×D173C213yesmaximal, complemented
12#121×D173C213yesmaximal, complemented
13C4×Q8×D173C211yesmaximal, characteristic, complemented
14C4×D4×D173C213yesmaximal, complemented
15#548×C173C211yesmaximal, characteristic, complemented
16C2×#14583827C2211yescharacteristic, complemented
17C2×#14583826C2213yescomplemented
18C2×#14583825C2213yescomplemented
19#41×D173—28no—
20C22×Dic346C2213yescomplemented
21C2×Q8×D173—28no—
22C2×Q8×D173C2213yes—
23C2×Q8×D173C2211yescharacteristic
24C2×Q8×D173C2211yescharacteristic, complemented
25C2×Q8×D173C2213yescomplemented
26C2×D4×D173C2213yes—
27C2×D4×D173C2213yes—
28C22×C4×D173C2213yes—
29C22×C4×D173C2213yes—
30#19149668C2213yescomplemented
31#19149667C2213yescomplemented
32#19149666C2213yescomplemented
33#19149665C2216yescomplemented
34#19149665C2216yescomplemented
35#19149659C2213yescomplemented
36#19149658C2213yescomplemented
37#19149657C2216yescomplemented
38#19149656C2216yescomplemented
39#19149655C2213yescomplemented
40#19149654C2213yescomplemented
41#19149653C2213yescomplemented
42#19149652C2216yescomplemented
43#19149651C2216yescomplemented
44#19149650C2213yescomplemented
45Q8×Dic173C2211yescharacteristic, complemented
46C4×Dic346C2213yescomplemented
47#19149644C2213yescomplemented
48C4×D692C2213yescomplemented
49D4×Dic173C2213yescomplemented
50#19149641C2213yescomplemented
51C4×C22⋊D173C2213yescomplemented
52C4×C22⋊D173C2213yescomplemented
53#19149639C2213yescomplemented
54#19149639C2213yescomplemented
55C4⋊C4×D173C2213yes—
56C4⋊C4×D173C2213yes—
57C22⋊C4×D173C2211yescharacteristic
58C22⋊C4×D173C2213yes—
59C22⋊C4×D173C2216yes—
60C42×D173C2213yes—
61C2×#41×C173C2211yescharacteristic, complemented
62C22×Q8×C173C2211yescharacteristic, complemented
63#124×C173C2213yescomplemented
64C22⋊Q8×C173C2213yescomplemented
65#121×C173C2213yescomplemented
66C4×Q8×C173C2211yescharacteristic, complemented
67C4×D4×C173C2213yescomplemented
68C23×D173C2313yes—
69C23×D173C2311yescharacteristic
70#14583827—24no—
71#14583826—212no—
72#14583825—212no—
73C2×Dic346—212no—
74C2×Dic346C2316yes—
75C2×Dic346C2313yes—
76C2×Dic346C2313yes—
77Q8×D173—212no—
78Q8×D173—212no—
79Q8×D173—24no—
80Q8×D173D418yescomplemented
81C2×D692C2313yes—
82C2×D692C2313yes—
83C2×C22⋊D173C2313yes—
84C2×C22⋊D173C2313yes—
85C2×C22⋊D173C2316yes—
86D4×D173—212no—
87D4×D173—212no—
88C22×Dic173C2313yes—
89C22×Dic173C2313yes—
90C2×C4×D173—212no—
91C2×C4×D173—212no—
92C2×C4×D173C2311yescharacteristic
93C2×C4×D173C2313yes—
94C2×C4×D173C2313yes—
95C2×C4×D173C2311yescharacteristic
96C2×C4×D173C2313yes—
97C2×C4×D173C2313yes—
98C2×C4×D173C2313yes—
99C2×C4×D173C2313yes—
100C2×C4×D173C2313yes—
101C4⋊Dic173C2313yes—
102C4⋊Dic173C2313yes—
103#14583805C2313yes—
104#14583805C2313yes—
105#14583805C2316yes—
106C22⋊Dic173C2311yescharacteristic
107C22⋊Dic173C2313yes—
108C22⋊Dic173C2316yes—
109#14583803C2311yescharacteristic
110#14583803C2313yes—
111#14583803C2311yescharacteristic
112#14583803C2313yes—
113#14583803C2316yes—
114#14583803C2316yes—
115C4×Dic173C2313yes—
116C4×Dic173C2313yes—
117C4×Dic173C2313yes—
118#41×C173—24no—
119C2×Q8×C173—24no—
120C2×Q8×C173C2313yes—
121C2×Q8×C173C2311yescharacteristic
122C2×D4×C173C2313yes—
123C2×D4×C173C2313yes—
124C22×C4×C173C2313yes—
125C22×C4×C173C2313yes—
126C4⋊C4×C173C2313yes—
127C4⋊C4×C173C2313yes—
128C22⋊C4×C173C2311yescharacteristic
129C22⋊C4×C173C2313yes—
130C22⋊C4×C173C2316yes—
131C42×C173C2313yes—
132C22×D173—22no—
133C22×D173—22no—
134C22×D173—26no—
135C22×D173—26no—
136C22×D173—26no—
137C22×D173#4114yescomplemented
138C22×D173C2413yes—
139C22×D173C2411yescharacteristic
140C22×D173C2411yescharacteristic
141Dic346—26no—
142Dic346—212no—
143Dic346C2×D4112yescomplemented
144D692—26no—
145D692—26no—
146C22⋊D173—26no—
147C22⋊D173—26no—
148C22⋊D173—212no—
149C2×Dic173—26no—
150C2×Dic173—26no—
151C2×Dic173C2411yescharacteristic
152C2×Dic173C2413yes—
153C2×Dic173C2413yes—
154C2×Dic173C2411yescharacteristic
155C2×Dic173C2413yes—
156C4×D173—22no—
157C4×D173—26no—
158C4×D173—26no—
159C4×D173—22no—
160C4×D173—26no—
161C4×D173—26no—
162C4×D173—26no—
163C4×D173—26no—
164C4×D173C2×D4112yes—
165C23×C173C2413yes—
166C23×C173C2411yescharacteristic
167Q8×C173—26no—
168Q8×C173C2×D414yescomplemented
169D4×C173—26no—
170D4×C173—26no—
171C2×C4×C173—26no—
172C2×C4×C173—26no—
173C2×C4×C173C2411yescharacteristic
174C2×C4×C173C2413yes—
175C2×C4×C173C2413yes—
176C2×C4×C173C2411yescharacteristic
177C2×C4×C173C2413yes—
178C22×C173—21no—
179C22×C173—21no—
180C22×C173—23no—
181C22×C173—23no—
182C22×C173—23no—
183C22×C173C2×#4112yescomplemented
184C22×C173C2511yescharacteristic
185C2×D173—21no—
186C2×D173—21no—
187C2×D173—28no—
188C2×D173—23no—
189C2×D173—23no—
190C2×D173—23no—
191C2×D173—412no—
192C2×D173C22×D412yes—
193C2×D173C2×#4112yes—
194C2×D173C2×#4112yescomplemented
195C2×D1732−1+412yes—
196C4×C173—21no—
197C4×C173—23no—
198C4×C173—23no—
199C4×C173—21no—
200C4×C173C22×D416yes—
201Dic173—21no—
202Dic173—23no—
203Dic173—23no—
204Dic173—21no—
205Dic173C22×D416yes—
206C2×C173—22no—
207C2×C173—43no—
208C2×C173C23×D411yescharacteristic
209C2×C173C22×#4111yescharacteristic
210C2×C173C2×2−1+411yescharacteristic
211D173—22no—
212D173—43no—
213D173#54814yesdirect product factor
214C173C2×#54811yescharacteristic, complemented, Sylow
215C2×#548—1731nomaximal, Sylow
216C22×#41—1731no—
217C23×Q8—1731no—
218#548—1731no—
219#548—1731no—
220#548—1733no—
221#548—1733no—
222#548—1733no—
223#548—1731no—
224#548—1733no—
225#548D17311yescharacteristic, direct product factor
226C2×#124—1733no—
227C2×C22⋊Q8—1733no—
228C2×#121—1733no—
229C2×C4×Q8—1731no—
230C2×C4×D4—1733no—
231C2×#41—3468no—
232C2×#41—1731no—
233C2×#41—1733no—
234C2×#41—1733no—
235C2×#41C2×D17311yescharacteristic, complemented
236C22×Q8—3468no—
237C22×Q8—1733no—
238C22×Q8—1733no—
239C22×Q8—1731no—
240C22×Q8—1731no—
241C22×Q8—1733no—
242C22×Q8C2×D17311yescharacteristic, complemented
243C22×D4—1733no—
244C22×D4—1733no—
245C23×C4—1733no—
246C23×C4—1733no—
247#124—1733no—
248#124—1733no—
249#124—1736no—
250#124—1736no—
251#124—1733no—
252#124C2×D17313yescomplemented
253C22⋊Q8—1733no—
254C22⋊Q8—1733no—
255C22⋊Q8—1736no—
256C22⋊Q8—1733no—
257C22⋊Q8—1736no—
258C22⋊Q8C2×D17313yescomplemented
259#121—1733no—
260#121—1736no—
261#121—1733no—
262#121—1733no—
263#121—1736no—
264#121C2×D17313yescomplemented
265C4×Q8—1731no—
266C4×Q8—1733no—
267C4×Q8—1733no—
268C4×Q8C2×D17311yescharacteristic, complemented
269C4×D4—1733no—
270C4×D4—1733no—
271C4×D4—1733no—
272C4×D4—1733no—
273C4×D4—1733no—
274C4×D4—1733no—
275C4×D4—1733no—
276C4×D4C2×D17313yescomplemented
277C2×C4⋊C4—1733no—
278C2×C4⋊C4—1733no—
279C2×C22⋊C4—1731no—
280C2×C22⋊C4—1733no—
281C2×C22⋊C4—1736no—
282C2×C42—1733no—
283C24—1733no—
284C24—1731no—
285#41—24no—
286#41—34612no—
287#41—34612no—
288#41—3464no—
289C2×Q8—24no—
290C2×Q8—34612no—
291C2×Q8—34612no—
292C2×Q8—3464no—
293C2×Q8—34612no—
294C2×Q8—1738no—
295C2×Q8—1733no—
296C2×Q8—1736no—
297C2×Q8—1733no—
298C2×Q8C22×D17313yes—
299C2×Q8C22×D17311yescharacteristic
300C2×D4—34612no—
301C2×D4—34612no—
302C2×D4—1733no—
303C2×D4—1733no—
304C2×D4—1736no—
305C2×D4—1733no—
306C2×D4—1733no—
307C2×D4C22×D17313yes—
308C2×D4C22×D17313yes—
309C22×C4—34612no—
310C22×C4—34612no—
311C22×C4—1733no—
312C22×C4—1733no—
313C22×C4—1731no—
314C22×C4—1733no—
315C22×C4—1733no—
316C22×C4—1731no—
317C22×C4—1733no—
318C22×C4—1733no—
319C22×C4—1733no—
320C22×C4—1733no—
321C22×C4—1733no—
322C22×C4C22×D17313yes—
323C22×C4C22×D17313yes—
324C4⋊C4—1733no—
325C4⋊C4—1733no—
326C4⋊C4—1733no—
327C4⋊C4—1736no—
328C4⋊C4—1733no—
329C4⋊C4C22×D17313yes—
330C4⋊C4C22×D17313yes—
331C22⋊C4—1731no—
332C22⋊C4—1733no—
333C22⋊C4—1736no—
334C22⋊C4—1731no—
335C22⋊C4—1731no—
336C22⋊C4—1733no—
337C22⋊C4—1733no—
338C22⋊C4—1736no—
339C22⋊C4—1736no—
340C22⋊C4C22×D17311yescharacteristic
341C22⋊C4C22×D17313yes—
342C22⋊C4C22×D17316yes—
343C42—1733no—
344C42—1733no—
345C42—1733no—
346C42C22×D17313yes—
347C23—3462no—
348C23—3462no—
349C23—3466no—
350C23—3466no—
351C23—3466no—
352C23—1733no—
353C23—1734no—
354C23—1731no—
355C23—1731no—
356C23C23×D17313yes—
357C23C23×D17311yescharacteristic
358Q8—26no—
359Q8—34612no—
360Q8—3466no—
361Q8—17312no—
362Q8D4×D17314yescomplemented
363D4—26no—
364D4—26no—
365D4—3466no—
366D4—3466no—
367D4—3466no—
368D4—34612no—
369D4—3466no—
370C2×C4—26no—
371C2×C4—26no—
372C2×C4—3462no—
373C2×C4—3466no—
374C2×C4—3462no—
375C2×C4—3466no—
376C2×C4—3466no—
377C2×C4—3466no—
378C2×C4—3466no—
379C2×C4—3466no—
380C2×C4—3466no—
381C2×C4—3466no—
382C2×C4—1731no—
383C2×C4—1733no—
384C2×C4—1733no—
385C2×C4—1731no—
386C2×C4—1733no—
387C2×C4—17312no—
388C2×C4C23×D17311yescharacteristic
389C2×C4C23×D17313yes—
390C2×C4C23×D17313yes—
391C2×C4C23×D17311yescharacteristic
392C2×C4C23×D17313yes—
393C22—21no—
394C22—21no—
395C22—23no—
396C22—23no—
397C22—23no—
398C22—3468no—
399C22—3461no—
400C22—3461no—
401C22—3463no—
402C22—3463no—
403C22—3463no—
404C22—1732no—
405C22—1732no—
406C22—1732no—
407C22—1732no—
408C22—69212no—
409C22#41×D17312yescomplemented
410C22C24×D17311yescharacteristic
411C4—21no—
412C4—23no—
413C4—23no—
414C4—21no—
415C4—3461no—
416C4—3463no—
417C4—3463no—
418C4—3461no—
419C4—1736no—
420C4C2×D4×D17316yes—
421C2—22no—
422C2—43no—
423C2—3462no—
424C2—1734no—
425C2—6923no—
426C2C22×D4×D17311yescharacteristic
427C2C2×#41×D17311yescharacteristic
428C22−1+4×D17311yescharacteristic
subgroupG/H
center, FrattiniC22C24×D173
commutator, socle, cosocleC22×C173C25
Fitting#548×C173C2
hypercenter#548D173
hypocenterC173C2×#548
LCSG ▷ C22×C173 ▷ C173
UCSC1 ◁ C22 ◁ #548

Complements

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

NH ≅ G/Ntype
1. C2×#41×D173C2semidirect
2. C22×Q8×D173C2semidirect
3. #26012153C2semidirect
4. #26012152C2semidirect
5. #26012151C2semidirect
6. #26012150C2semidirect
7. #26012149C2semidirect
8. #26012148C2semidirect
9. #26012147C2semidirect
10. #124×D173C2semidirect
11. C22⋊Q8×D173C2semidirect
12. #121×D173C2semidirect
13. C4×Q8×D173C2semidirect
14. C4×D4×D173C2semidirect
15. #548×C173C2semidirect
16. C2×#14583827C22semidirect
17. C2×#14583826C22semidirect
18. C2×#14583825C22semidirect
20. C22×Dic346C22semidirect
24. C2×Q8×D173C22semidirect
25. C2×Q8×D173C22semidirect
30. #19149668C22semidirect
31. #19149667C22semidirect
32. #19149666C22semidirect
33. #19149665C22semidirect
34. #19149665C22semidirect
35. #19149659C22semidirect
36. #19149658C22semidirect
37. #19149657C22semidirect
38. #19149656C22semidirect
39. #19149655C22semidirect
40. #19149654C22semidirect
41. #19149653C22semidirect
42. #19149652C22semidirect
43. #19149651C22semidirect
44. #19149650C22semidirect
45. Q8×Dic173C22semidirect
46. C4×Dic346C22semidirect
47. #19149644C22semidirect
48. C4×D692C22semidirect
49. D4×Dic173C22semidirect
50. #19149641C22semidirect
51. C4×C22⋊D173C22semidirect
52. C4×C22⋊D173C22semidirect
53. #19149639C22semidirect
54. #19149639C22semidirect
61. C2×#41×C173C22semidirect
62. C22×Q8×C173C22semidirect
63. #124×C173C22semidirect
64. C22⋊Q8×C173C22semidirect
65. #121×C173C22semidirect
66. C4×Q8×C173C22semidirect
67. C4×D4×C173C22semidirect
80. Q8×D173D4semidirect
137. C22×D173#41semidirect
143. Dic346C2×D4semidirect
168. Q8×C173C2×D4semidirect
183. C22×C173C2×#41semidirect
194. C2×D173C2×#41semidirect
213. D173#548direct (N × H)
214. C173C2×#548semidirect
225. #548D173direct (N × H)
235. C2×#41C2×D173semidirect
242. C22×Q8C2×D173semidirect
252. #124C2×D173semidirect
258. C22⋊Q8C2×D173semidirect
264. #121C2×D173semidirect
268. C4×Q8C2×D173semidirect
276. C4×D4C2×D173semidirect
362. Q8D4×D173semidirect
409. C22#41×D173semidirect

Maximal subgroups

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

subgroupindex
C2×#41×D1732
C22×Q8×D1732
#260121532
#260121522
#260121512
#260121502
#260121492
#260121482
#260121472
#124×D1732
C22⋊Q8×D1732
#121×D1732
C4×Q8×D1732
C4×D4×D1732
#548×C1732
C2×#548173