Finite Group Databasebeta

← #36367704

Subgroups — #36367704

Order 22144.

1058 conjugacy classes of subgroups.

The subgroup lattice has 378 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#26012240C212yesmaximal, complemented
2#26012239C212yesmaximal, complemented
3#26012237C214yesmaximal, complemented
4#26012233C211yesmaximal, characteristic, complemented
5#26012232C214yesmaximal, complemented
6#26012231C212yesmaximal, complemented
7#26012042C212yesmaximal, complemented
8#26012040C211yesmaximal, characteristic, complemented
9#26012026C212yesmaximal, complemented
10#26012024C214yesmaximal, complemented
11#26012004C214yesmaximal, complemented
12#26011999C212yesmaximal, complemented
13#563×C173C211yesmaximal, characteristic, complemented
14C2×#14583827C2212yes—
15C2×#14583827C2211yescharacteristic, complemented
16C2×#14583825C2211yescharacteristic
17C22×C4×D173C2212yescomplemented
18#19149677C2211yescharacteristic, complemented
19#19149676C2212yes—
20#19149675C2211yescharacteristic
21#19149674C2214yescomplemented
22#19149673C2214yescomplemented
23#19149669C2214yescomplemented
24#19149668C2212yes—
25#19149668C2212yescomplemented
26#19149667C2212yes—
27#19149667C2211yescharacteristic, complemented
28#19149665C2214yes—
29#19149664C2214yes—
30#19149664C2214yescomplemented
31#19149664C2214yescomplemented
32#19149664C2214yescomplemented
33#19149662C2212yescomplemented
34#19149661C2214yescomplemented
35#19149659C2212yescomplemented
36#19149657C2214yescomplemented
37#19149657C2214yescomplemented
38#19149655C2214yescomplemented
39#19149654C2212yes—
40#19149654C2214yes—
41#19149654C2214yescomplemented
42#19149654C2214yescomplemented
43#19149653C2214yes—
44#19149652C2214yes—
45#19149652C2214yes—
46#19149652C2214yescomplemented
47#19149649C2214yes—
48#19149649C2214yescomplemented
49#19149649C2212yescomplemented
50#19149647C2212yes—
51#19149644C2212yescomplemented
52#19149641C2212yes—
53#19149641C2214yescomplemented
54#19149641C2214yescomplemented
55#19149639C2214yes—
56#19149639C2214yescomplemented
57#19149637C2214yescomplemented
58#19149635C2212yescomplemented
59C42×D173C2212yescomplemented
60C42×D173C2211yescharacteristic, complemented
61C4⋊Q8×C173C2212yescomplemented
62#126×C173C2214yescomplemented
63#125×C173C2212yescomplemented
64#124×C173C2211yescharacteristic
65#123×C173C2212yes—
66C22⋊Q8×C173C2214yescomplemented
67C23×D173C2312yes—
68#14583827—44no—
69#14583827—42no—
70#14583825—42no—
71C2×Dic346C2312yes—
72C2×Dic346C2311yescharacteristic
73C2×D692C2312yes—
74C2×D692C2314yes—
75C2×D692C2314yes—
76C2×D692C2312yes—
77C2×D692C2314yes—
78C2×D692C2312yes—
79C2×D692C2311yescharacteristic
80C2×C22⋊D173C2314yes—
81C2×C22⋊D173C2314yes—
82C2×C22⋊D173C2312yes—
83C22×Dic173C2312yes—
84C2×C4×D173—44no—
85C2×C4×D173C2312yes—
86C2×C4×D173C2311yescharacteristic
87C2×C4×D173C2314yes—
88C2×C4×D173C2314yes—
89C2×C4×D173C2312yes—
90C2×C4×D173C2314yes—
91C4⋊Dic173C2314yes—
92#14583805C2314yes—
93#14583805C2314yes—
94#14583805C2314yes—
95#14583805C2314yes—
96C22⋊Dic173C2312yes—
97#14583803C2314yes—
98#14583803C2314yes—
99#14583803C2314yes—
100#14583803C2314yes—
101#14583803C2314yes—
102#14583803C2314yes—
103#14583803C2314yes—
104#14583803C2314yes—
105#14583803C2314yes—
106C4×Dic173C2314yes—
107C4×Dic173C2312yes—
108C4×Dic173C2312yes—
109C4×Dic173C2311yescharacteristic
110C2×Q8×C173C2311yescharacteristic
111C2×Q8×C173C2312yes—
112C2×D4×C173C2311yescharacteristic
113C22×C4×C173C2312yes—
114C4⋊C4×C173C2314yes—
115C4⋊C4×C173C2314yes—
116C4⋊C4×C173C2314yes—
117C4⋊C4×C173C2312yes—
118C4⋊C4×C173C2312yes—
119C22⋊C4×C173C2312yes—
120C22⋊C4×C173C2314yes—
121C22⋊C4×C173C2314yes—
122C42×C173C2312yes—
123C42×C173C2311yescharacteristic
124C22×D173—42no—
125C22×D173—44no—
126C22×D173C2411yescharacteristic
127C22×D173C2412yes—
128C22×D173C2414yes—
129Dic346—42no—
130Dic346—41no—
131D692—44no—
132D692—44no—
133D692—42no—
134D692—44no—
135D692—41no—
136D692—42no—
137D692—42no—
138C22⋊D173—44no—
139C22⋊D173—44no—
140C22⋊D173—42no—
141C2×Dic173—24no—
142C2×Dic173C2412yes—
143C2×Dic173C2412yes—
144C2×Dic173C2414yes—
145C2×Dic173C2411yescharacteristic
146C4×D173—28no—
147C4×D173—22no—
148C4×D173—24no—
149C4×D173—44no—
150C4×D173—42no—
151C4×D173—44no—
152C23×C173C2412yes—
153Q8×C173—42no—
154Q8×C173—41no—
155D4×C173—41no—
156C2×C4×C173—42no—
157C2×C4×C173C2414yes—
158C2×C4×C173C2414yes—
159C2×C4×C173C2412yes—
160C2×C4×C173C2412yes—
161C2×C4×C173C2411yescharacteristic
162C22×C173—22no—
163C22×C173—24no—
164C22×C173C2511yescharacteristic
165C2×D173—24no—
166C2×D173—21no—
167C2×D173—22no—
168C2×D173—24no—
169C2×D173—22no—
170C2×D173—24no—
171C2×D173—24no—
172C2×D173—48no—
173C4×C173—24no—
174C4×C173—21no—
175C4×C173—22no—
176C4×C173—24no—
177C4×C173—22no—
178Dic173—22no—
179Dic173—22no—
180Dic173—24no—
181Dic173H(5,𝔽2)12yes—
182C2×C173—42no—
183C2×C173C2×H(5,𝔽2)11yescharacteristic
184C2×C173#58512yes—
185D173—41no—
186D173—42no—
187D173—44no—
188C173#352911yescharacteristic, complemented, Sylow
189#3529—1731nomaximal, Sylow
190#563—1732no—
191#563D17311yescharacteristic, complemented
192#562—1732no—
193#562—1734no—
194H(3,𝔽4)—1731no—
195#560—1734no—
196#560—1732no—
197#532—1732no—
198#532—1731no—
199#529—1734no—
200#529—1732no—
201#525—1732no—
202#525—1734no—
203C2×#41—1732no—
204C2×#41—1731no—
205C2×#41—1731no—
206C23×C4—1732no—
207C4⋊Q8—1731no—
208C4⋊Q8C2×D17312yescomplemented
209#127—1731no—
210#127—1732no—
211#126—1734no—
212#126—1734no—
213#126C2×D17314yescomplemented
214#125—1734no—
215#125C2×D17312yescomplemented
216#124—1732no—
217#124—1732no—
218#124—1734no—
219#124—1732no—
220#124—1731no—
221#124C2×D17311yescharacteristic
222#123—1732no—
223#123—1734no—
224#123—1734no—
225#123—1734no—
226#123—1734no—
227#123—1734no—
228#123C2×D17312yes—
229C22⋊Q8—1734no—
230C22⋊Q8—1734no—
231C22⋊Q8—1734no—
232C22⋊Q8—1732no—
233C22⋊Q8C2×D17314yescomplemented
234#121—1734no—
235#121—1734no—
236#121—1732no—
237#121—1734no—
238#121—1734no—
239#121—1734no—
240#121—1734no—
241#121—1734no—
242#120—1732no—
243#120—1734no—
244#120—1732no—
245#120—1734no—
246C4×Q8—1732no—
247C4×D4—1732no—
248C4×D4—1734no—
249C4×D4—1734no—
250C4×D4—1734no—
251C4×D4—1734no—
252#117—1734no—
253#117—1732no—
254C2×C42—1732no—
255C2×C42—1731no—
256C24—1732no—
257#41—6922no—
258#41—6924no—
259#41—6922no—
260C2×Q8—1732no—
261C2×Q8—1731no—
262C2×Q8C22×D17312yes—
263C2×Q8C22×D17311yescharacteristic
264C2×D4—1734no—
265C2×D4—1734no—
266C2×D4—1732no—
267C2×D4—1732no—
268C2×D4—1734no—
269C2×D4—1734no—
270C2×D4—1732no—
271C2×D4—1734no—
272C2×D4—1731no—
273C2×D4—1732no—
274C2×D4C22×D17311yescharacteristic
275C22×C4—1732no—
276C22×C4—1731no—
277C22×C4—1732no—
278C22×C4—1734no—
279C22×C4—1734no—
280C22×C4—1732no—
281C22×C4—1734no—
282C22×C4—6924no—
283C22×C4C22×D17312yes—
284C4⋊C4—1734no—
285C4⋊C4—1734no—
286C4⋊C4—1734no—
287C4⋊C4—1734no—
288C4⋊C4—1734no—
289C4⋊C4C22×D17314yes—
290C4⋊C4C22×D17314yes—
291C4⋊C4C22×D17314yes—
292C4⋊C4C22×D17312yes—
293C4⋊C4C22×D17312yes—
294C22⋊C4—1732no—
295C22⋊C4—1734no—
296C22⋊C4—1734no—
297C22⋊C4—1734no—
298C22⋊C4—1734no—
299C22⋊C4—1734no—
300C22⋊C4—1734no—
301C22⋊C4—1734no—
302C22⋊C4—1734no—
303C22⋊C4—1734no—
304C22⋊C4C22×D17312yes—
305C22⋊C4C22×D17314yes—
306C22⋊C4C22×D17314yes—
307C42—1734no—
308C42—1732no—
309C42—1732no—
310C42—1731no—
311C42C22×D17311yescharacteristic
312C42C22×D17312yes—
313C23—1734no—
314C23—1732no—
315C23—1731no—
316C23—6922no—
317C23—6924no—
318C23C23×D17312yes—
319Q8—42no—
320Q8—41no—
321Q8—6922no—
322Q8—6921no—
323D4—41no—
324D4—6924no—
325D4—6924no—
326D4—6922no—
327D4—6924no—
328D4—6921no—
329D4—6922no—
330D4—6922no—
331D4—6924no—
332D4—6924no—
333D4—6922no—
334C2×C4—42no—
335C2×C4—3468no—
336C2×C4—3462no—
337C2×C4—3464no—
338C2×C4—3464no—
339C2×C4—1734no—
340C2×C4—1732no—
341C2×C4—1732no—
342C2×C4—1731no—
343C2×C4—6924no—
344C2×C4—6922no—
345C2×C4—6924no—
346C2×C4C23×D17312yes—
347C2×C4C23×D17311yescharacteristic
348C2×C4C23×D17314yes—
349C2×C4C23×D17312yes—
350C2×C4C23×D17314yes—
351C22—22no—
352C22—24no—
353C22—3464no—
354C22—3462no—
355C22—3461no—
356C22—3462no—
357C22—3464no—
358C22—3464no—
359C22—3464no—
360C22—6928no—
361C22C24×D17311yescharacteristic
362C4—24no—
363C4—21no—
364C4—22no—
365C4—24no—
366C4—22no—
367C4—3462no—
368C4—3462no—
369C4—3464no—
370C4—1732no—
371C2—42no—
372C2—6921no—
373C2—6922no—
374C2—6924no—
375C2H(5,𝔽2)×D17311yescharacteristic
376C2#2601232812yes—
subgroupG/H
center, FrattiniC22C24×D173
commutator, socle, cosocleC22×C173C25
Fitting#563×C173C2
hypercenter#563D173
hypocenterC173#3529
LCSG ▷ C22×C173 ▷ C173
UCSC1 ◁ C22 ◁ #563

Complements

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

NH ≅ G/Ntype
1. #26012240C2semidirect
2. #26012239C2semidirect
3. #26012237C2semidirect
4. #26012233C2semidirect
5. #26012232C2semidirect
6. #26012231C2semidirect
7. #26012042C2semidirect
8. #26012040C2semidirect
9. #26012026C2semidirect
10. #26012024C2semidirect
11. #26012004C2semidirect
12. #26011999C2semidirect
13. #563×C173C2semidirect
15. C2×#14583827C22semidirect
17. C22×C4×D173C22semidirect
18. #19149677C22semidirect
21. #19149674C22semidirect
22. #19149673C22semidirect
23. #19149669C22semidirect
25. #19149668C22semidirect
27. #19149667C22semidirect
30. #19149664C22semidirect
31. #19149664C22semidirect
32. #19149664C22semidirect
33. #19149662C22semidirect
34. #19149661C22semidirect
35. #19149659C22semidirect
36. #19149657C22semidirect
37. #19149657C22semidirect
38. #19149655C22semidirect
41. #19149654C22semidirect
42. #19149654C22semidirect
46. #19149652C22semidirect
48. #19149649C22semidirect
49. #19149649C22semidirect
51. #19149644C22semidirect
53. #19149641C22semidirect
54. #19149641C22semidirect
56. #19149639C22semidirect
57. #19149637C22semidirect
58. #19149635C22semidirect
59. C42×D173C22semidirect
60. C42×D173C22semidirect
61. C4⋊Q8×C173C22semidirect
62. #126×C173C22semidirect
63. #125×C173C22semidirect
66. C22⋊Q8×C173C22semidirect
188. C173#3529semidirect
191. #563D173semidirect
208. C4⋊Q8C2×D173semidirect
213. #126C2×D173semidirect
215. #125C2×D173semidirect
233. C22⋊Q8C2×D173semidirect

Maximal subgroups

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

subgroupindex
#260122402
#260122392
#260122372
#260122332
#260122322
#260122312
#260120422
#260120402
#260120262
#260120242
#260120042
#260119992
#563×C1732
#3529173