Finite Group Databasebeta

← #41×C22⋊D173 #36367048

Subgroups — #41×C22⋊D173 #36367048

Order 22144.

1648 conjugacy classes of subgroups.

The subgroup lattice has 432 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#41×D173C211yesmaximal, characteristic, complemented
2#26012160C211yesmaximal, characteristic, complemented
3Q8×C22⋊D173C211yesmaximal, characteristic, complemented
4#26012152C211yesmaximal, characteristic, complemented
5#26012148C211yesmaximal, characteristic, complemented
6#26012140C213yesmaximal, complemented
7#26012135C213yesmaximal, complemented
8#26012129C213yesmaximal, complemented
9D4×C22⋊D173C213yesmaximal, complemented
10#26012037C213yesmaximal, complemented
11#26012002C213yesmaximal, complemented
12#26011998C213yesmaximal, complemented
13#41×Dic173C211yesmaximal, characteristic, complemented
14C2×C4×C22⋊D173C213yesmaximal, complemented
15C22×#41×C173C211yesmaximal, characteristic, complemented
16C2×#14583827C2211yescharacteristic, complemented
17C2×#14583826C2213yescomplemented
18C2×#14583825C2213yescomplemented
19#41×D173—28no—
20C2×Q8×D173C2211yescharacteristic, complemented
21C22×C22⋊D173C2213yes—
22C2×D4×D173C2213yescomplemented
23C22×C4×D173C2213yescomplemented
24#19149677C2213yescomplemented
25#19149675C2213yescomplemented
26#19149668C2213yescomplemented
27#19149666C2213yescomplemented
28#19149662C2216yescomplemented
29#19149660C2216yescomplemented
30#19149658C2213yescomplemented
31#19149658C2213yescomplemented
32#19149655C2213yescomplemented
33#19149655C2213yescomplemented
34#19149653C2213yescomplemented
35#19149653C2213yescomplemented
36#19149651C2216yescomplemented
37#19149651C2216yescomplemented
38#19149650C2213yescomplemented
39#19149650C2213yescomplemented
40#19149648C2216yescomplemented
41#19149647C2216yescomplemented
42Q8×Dic173C2211yescharacteristic, complemented
43Q8×Dic173C2211yescharacteristic, complemented
44D4×Dic173C2213yescomplemented
45D4×Dic173C2213yescomplemented
46C4×C22⋊D173C2211yescharacteristic
47C4×C22⋊D173C2213yes—
48C4×C22⋊D173C2213yescomplemented
49C4×C22⋊D173C2213yescomplemented
50C4×C22⋊D173C2213yescomplemented
51C4×C22⋊D173C2213yescomplemented
52C4×C22⋊D173C2213yescomplemented
53C4×C22⋊D173C2213yescomplemented
54C4×C22⋊D173C2216yescomplemented
55#19149638C2213yescomplemented
56C2×#14583805C2213yescomplemented
57C2×C22⋊Dic173C2213yescomplemented
58C2×#14583803C2213yescomplemented
59C2×C4×Dic173C2213yescomplemented
60C2×#41×C173—28no—
61C2×#41×C173C2211yescharacteristic
62C2×#41×C173C2211yescharacteristic, complemented
63C2×#41×C173C2213yescomplemented
64C2×#41×C173C2213yescomplemented
65C22×Q8×C173C2211yescharacteristic, complemented
66C22×D4×C173C2213yescomplemented
67C23×C4×C173C2213yescomplemented
68C23×D173C2313yes—
69#14583827—24no—
70#14583826—212no—
71#14583825—212no—
72C2×Dic346C2313yes—
73Q8×D173—24no—
74C2×D692C2313yes—
75C2×C22⋊D173—212no—
76C2×C22⋊D173C2313yes—
77C2×C22⋊D173C2313yes—
78C2×C22⋊D173C2313yes—
79C2×C22⋊D173C2316yes—
80C2×C22⋊D173C2316yes—
81C2×C22⋊D173C2311yescharacteristic
82D4×D173—212no—
83C22×Dic173C2313yes—
84C22×Dic173C2313yes—
85C22×Dic173C2313yes—
86C2×C4×D173—212no—
87C2×C4×D173C2311yescharacteristic
88C2×C4×D173C2313yes—
89C2×C4×D173C2313yes—
90C2×C4×D173C2313yes—
91C4⋊Dic173C2313yes—
92C4⋊Dic173C2313yes—
93#14583805C2311yescharacteristic
94#14583805C2313yes—
95#14583805C2313yes—
96#14583805C2313yes—
97#14583805C2316yes—
98C22⋊Dic173C2311yescharacteristic
99C22⋊Dic173C2313yes—
100C22⋊Dic173C2313yes—
101C22⋊Dic173C2313yes—
102C22⋊Dic173C2316yes—
103C22⋊Dic173C2316yes—
104#14583803C2311yescharacteristic
105#14583803C2313yes—
106#14583803C2313yes—
107#14583803C2313yes—
108#14583803C2316yes—
109C4×Dic173C2311yescharacteristic
110C4×Dic173C2313yes—
111C4×Dic173C2313yes—
112C4×Dic173C2313yes—
113C24×C173C2313yes—
114#41×C173—24no—
115#41×C173—212no—
116#41×C173—212no—
117#41×C173D418yescomplemented
118C2×Q8×C173—24no—
119C2×Q8×C173C2313yes—
120C2×Q8×C173C2311yescharacteristic
121C2×D4×C173—212no—
122C2×D4×C173C2316yes—
123C2×D4×C173C2313yes—
124C2×D4×C173C2313yes—
125C22×C4×C173—212no—
126C22×C4×C173C2311yescharacteristic
127C22×C4×C173C2313yes—
128C22×C4×C173C2313yes—
129C22×C4×C173C2313yes—
130C22×C4×C173C2313yes—
131C22×C4×C173C2313yes—
132C22×D173—26no—
133C22×D173—26no—
134C22×D173—43no—
135C22×D173C2413yes—
136C22×D173C2411yescharacteristic
137Dic346—26no—
138D692—26no—
139C22⋊D173—212no—
140C22⋊D173—26no—
141C22⋊D173—26no—
142C22⋊D173—26no—
143C22⋊D173—46no—
144C22⋊D173#4114yesdirect product factor
145C2×Dic173—26no—
146C2×Dic173—26no—
147C2×Dic173—43no—
148C2×Dic173C2411yescharacteristic
149C2×Dic173C2413yes—
150C2×Dic173C2411yescharacteristic
151C2×Dic173C2413yes—
152C2×Dic173C2413yes—
153C2×Dic173C2411yescharacteristic
154C4×D173—22no—
155C4×D173—26no—
156C4×D173—26no—
157C4×D173—26no—
158C23×C173—26no—
159C23×C173—26no—
160C23×C173—43no—
161C23×C173C2413yes—
162C23×C173C2413yes—
163C23×C173C2411yescharacteristic
164Q8×C173—26no—
165Q8×C173C2×D414yescomplemented
166D4×C173—212no—
167D4×C173—26no—
168D4×C173C2×D4112yescomplemented
169C2×C4×C173—26no—
170C2×C4×C173—22no—
171C2×C4×C173—26no—
172C2×C4×C173—26no—
173C2×C4×C173—26no—
174C2×C4×C173C2×D4112yescomplemented
175C2×C4×C173C2411yescharacteristic
176C2×C4×C173C2413yes—
177C2×C4×C173C2413yes—
178C2×C4×C173C2411yescharacteristic
179C22×C173—21no—
180C22×C173—23no—
181C22×C173—21no—
182C22×C173—23no—
183C22×C173—23no—
184C22×C173—23no—
185C22×C173—23no—
186C22×C173—412no—
187C22×C173C22×D416yes—
188C22×C173C2×#4112yescomplemented
189C22×C173C2511yescharacteristic
190C2×D173—21no—
191C2×D173—21no—
192C2×D173—23no—
193C2×D173—23no—
194C2×D173—23no—
195C2×D173—412no—
196C2×D173C2×#4112yescomplemented
197C4×C173—21no—
198C4×C173—23no—
199C4×C173C22×D412yes—
200C4×C173C22×D416yes—
201Dic173—21no—
202Dic173—23no—
203Dic173—23no—
204Dic173—21no—
205Dic173—23no—
206Dic173C2×#4112yescomplemented
207C2×C173—26no—
208C2×C173—22no—
209C2×C173—43no—
210C2×C173C23×D411yescharacteristic
211C2×C173C22×#4111yescharacteristic
212C2×C173#58511yescharacteristic
213D173—22no—
214D173—43no—
215C173D4×#4111yescharacteristic, complemented, Sylow
216D4×#41—1731nomaximal, Sylow
217C22×#41—1731no—
218C22×#41D17311yescharacteristic, complemented
219#550—1731no—
220D4×Q8—1731no—
221#548—1731no—
222#548—1731no—
223#547—1733no—
224#546—1733no—
225#546—1733no—
226D42—1733no—
227#532—1733no—
228#525—1733no—
229#525—1733no—
230C4×#41—1731no—
231C2×C4×D4—1733no—
232C2×#41—28no—
233C2×#41—3468no—
234C2×#41—1731no—
235C2×#41—1733no—
236C2×#41—1733no—
237C2×#41C2×D17311yescharacteristic
238C2×#41C2×D17313yescomplemented
239C2×#41C2×D17313yescomplemented
240C2×#41C2×D17311yescharacteristic, complemented
241C22×Q8—1731no—
242C22×Q8C2×D17311yescharacteristic, complemented
243C22×D4—1733no—
244C22×D4—1733no—
245C22×D4C2×D17313yescomplemented
246C23×C4—1733no—
247C23×C4C2×D17313yescomplemented
248C4⋊Q8—1733no—
249#127—1733no—
250#124—1733no—
251#124—1733no—
252#123—1736no—
253#123—1736no—
254C22⋊Q8—1733no—
255C22⋊Q8—1733no—
256C22⋊Q8—1733no—
257C22⋊Q8—1733no—
258#121—1733no—
259#121—1733no—
260#121—1736no—
261#121—1736no—
262#121—1733no—
263#121—1733no—
264#120—1736no—
265#120—1736no—
266C4×Q8—1731no—
267C4×Q8—1731no—
268C4×D4—1733no—
269C4×D4—1733no—
270C4×D4—1731no—
271C4×D4—1733no—
272C4×D4—1733no—
273C4×D4—1733no—
274C4×D4—1733no—
275C4×D4—1733no—
276C4×D4—1733no—
277C4×D4—1733no—
278C4×D4—1736no—
279#117—1733no—
280C2×C4⋊C4—1733no—
281C2×C22⋊C4—1733no—
282C2×C22⋊C4—1733no—
283C2×C42—1733no—
284C24—1733no—
285C24C22×D17313yes—
286#41—24no—
287#41—212no—
288#41—212no—
289#41—3464no—
290#41—34612no—
291#41—34612no—
292#41C22⋊D17318yesdirect product factor
293C2×Q8—24no—
294C2×Q8—3464no—
295C2×Q8—1733no—
296C2×Q8C22×D17313yes—
297C2×Q8C22×D17311yescharacteristic
298C2×D4—212no—
299C2×D4—34612no—
300C2×D4—34612no—
301C2×D4—1733no—
302C2×D4—1733no—
303C2×D4—1733no—
304C2×D4—1736no—
305C2×D4—1731no—
306C2×D4—1736no—
307C2×D4—1733no—
308C2×D4C22×D17316yes—
309C2×D4C22×D17313yes—
310C2×D4C22×D17313yes—
311C22×C4—212no—
312C22×C4—34612no—
313C22×C4—1733no—
314C22×C4—1733no—
315C22×C4—1733no—
316C22×C4—1733no—
317C22×C4—1731no—
318C22×C4—1733no—
319C22×C4—1733no—
320C22×C4C22×D17313yes—
321C22×C4C22×D17313yes—
322C22×C4C22×D17313yes—
323C22×C4C22×D17311yescharacteristic
324C22×C4C22×D17313yes—
325C22×C4C22×D17313yes—
326C4⋊C4—1731no—
327C4⋊C4—1733no—
328C4⋊C4—1733no—
329C4⋊C4—1733no—
330C4⋊C4—1733no—
331C4⋊C4—1733no—
332C4⋊C4—1736no—
333C22⋊C4—1731no—
334C22⋊C4—1733no—
335C22⋊C4—1733no—
336C22⋊C4—1733no—
337C22⋊C4—1736no—
338C22⋊C4—1736no—
339C22⋊C4—1731no—
340C22⋊C4—1733no—
341C22⋊C4—1733no—
342C22⋊C4—1733no—
343C22⋊C4—1736no—
344C42—1733no—
345C42—1733no—
346C42—1731no—
347C42—1733no—
348C23—26no—
349C23—26no—
350C23—43no—
351C23—3466no—
352C23—3466no—
353C23—1733no—
354C23—1731no—
355C23—6923no—
356C23C23×D17313yes—
357C23C23×D17313yes—
358C23C23×D17311yescharacteristic
359Q8—26no—
360Q8—3466no—
361Q8C2×C22⋊D17314yescomplemented
362D4—212no—
363D4—26no—
364D4—3466no—
365D4—34612no—
366D4—3466no—
367D4—3466no—
368D4—3466no—
369D4—1734no—
370D4—6926no—
371D4C2×C22⋊D173112yescomplemented
372C2×C4—26no—
373C2×C4—22no—
374C2×C4—26no—
375C2×C4—26no—
376C2×C4—26no—
377C2×C4—3466no—
378C2×C4—3466no—
379C2×C4—3462no—
380C2×C4—3466no—
381C2×C4—3466no—
382C2×C4—3466no—
383C2×C4—1731no—
384C2×C4—1733no—
385C2×C4—1733no—
386C2×C4—1731no—
387C2×C4—1733no—
388C2×C4—1731no—
389C2×C4—6923no—
390C2×C4C2×C22⋊D173112yescomplemented
391C2×C4C23×D17311yescharacteristic
392C2×C4C23×D17313yes—
393C2×C4C23×D17313yes—
394C2×C4C23×D17311yescharacteristic
395C22—21no—
396C22—23no—
397C22—23no—
398C22—21no—
399C22—23no—
400C22—23no—
401C22—23no—
402C22—412no—
403C22—3461no—
404C22—3461no—
405C22—3463no—
406C22—3463no—
407C22—3463no—
408C22—1732no—
409C22—69212no—
410C22C22×C22⋊D17316yes—
411C22#41×D17312yescomplemented
412C22C24×D17311yescharacteristic
413C4—21no—
414C4—23no—
415C4—3461no—
416C4—3463no—
417C4—3463no—
418C4—3461no—
419C4—3463no—
420C4—1732no—
421C4C22×C22⋊D17312yes—
422C4C22×C22⋊D17316yes—
423C2—22no—
424C2—26no—
425C2—43no—
426C2—3462no—
427C2—6923no—
428C2C23×C22⋊D17311yescharacteristic
429C2C2×#41×D17311yescharacteristic
430C2#2601232711yescharacteristic
subgroupG/H
centerC2×C4C23×D173
commutator, socle, cosocleC22×C173C25
FrattiniC22C24×D173
FittingC22×#41×C173C2
hypercenterC22×#41D173
hypocenterC173D4×#41
LCSG ▷ C22×C173 ▷ C173
UCSC1 ◁ C2×C4 ◁ C22×#41

Complements

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

NH ≅ G/Ntype
1. C2×#41×D173C2semidirect
2. #26012160C2semidirect
3. Q8×C22⋊D173C2semidirect
4. #26012152C2semidirect
5. #26012148C2semidirect
6. #26012140C2semidirect
7. #26012135C2semidirect
8. #26012129C2semidirect
9. D4×C22⋊D173C2semidirect
10. #26012037C2semidirect
11. #26012002C2semidirect
12. #26011998C2semidirect
13. #41×Dic173C2semidirect
14. C2×C4×C22⋊D173C2semidirect
15. C22×#41×C173C2semidirect
16. C2×#14583827C22semidirect
17. C2×#14583826C22semidirect
18. C2×#14583825C22semidirect
20. C2×Q8×D173C22semidirect
22. C2×D4×D173C22semidirect
23. C22×C4×D173C22semidirect
24. #19149677C22semidirect
25. #19149675C22semidirect
26. #19149668C22semidirect
27. #19149666C22semidirect
28. #19149662C22semidirect
29. #19149660C22semidirect
30. #19149658C22semidirect
31. #19149658C22semidirect
32. #19149655C22semidirect
33. #19149655C22semidirect
34. #19149653C22semidirect
35. #19149653C22semidirect
36. #19149651C22semidirect
37. #19149651C22semidirect
38. #19149650C22semidirect
39. #19149650C22semidirect
40. #19149648C22semidirect
41. #19149647C22semidirect
42. Q8×Dic173C22semidirect
43. Q8×Dic173C22semidirect
44. D4×Dic173C22semidirect
45. D4×Dic173C22semidirect
48. C4×C22⋊D173C22semidirect
49. C4×C22⋊D173C22semidirect
50. C4×C22⋊D173C22semidirect
51. C4×C22⋊D173C22semidirect
52. C4×C22⋊D173C22semidirect
53. C4×C22⋊D173C22semidirect
54. C4×C22⋊D173C22semidirect
55. #19149638C22semidirect
56. C2×#14583805C22semidirect
57. C2×C22⋊Dic173C22semidirect
58. C2×#14583803C22semidirect
59. C2×C4×Dic173C22semidirect
62. C2×#41×C173C22semidirect
63. C2×#41×C173C22semidirect
64. C2×#41×C173C22semidirect
65. C22×Q8×C173C22semidirect
66. C22×D4×C173C22semidirect
67. C23×C4×C173C22semidirect
117. #41×C173D4semidirect
144. C22⋊D173#41direct (N × H)
165. Q8×C173C2×D4semidirect
168. D4×C173C2×D4semidirect
174. C2×C4×C173C2×D4semidirect
188. C22×C173C2×#41semidirect
196. C2×D173C2×#41semidirect
206. Dic173C2×#41semidirect
215. C173D4×#41semidirect
218. C22×#41D173semidirect
238. C2×#41C2×D173semidirect
239. C2×#41C2×D173semidirect
240. C2×#41C2×D173semidirect
242. C22×Q8C2×D173semidirect
245. C22×D4C2×D173semidirect
247. C23×C4C2×D173semidirect
292. #41C22⋊D173direct (N × H)
361. Q8C2×C22⋊D173semidirect
371. D4C2×C22⋊D173semidirect
390. C2×C4C2×C22⋊D173semidirect
411. 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
#260121602
Q8×C22⋊D1732
#260121522
#260121482
#260121402
#260121352
#260121292
D4×C22⋊D1732
#260120372
#260120022
#260119982
#41×Dic1732
C2×C4×C22⋊D1732
C22×#41×C1732
D4×#41173