Finite Group Databasebeta

← #36366890

Subgroups — #36366890

Order 22144.

1556 conjugacy classes of subgroups.

The subgroup lattice has 350 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#41×D173C212yesmaximal, complemented
2#26012068C214yesmaximal, complemented
3#26012066C212yesmaximal, complemented
4#26012061C212yesmaximal, complemented
5#26012059C212yesmaximal, complemented
6#26012054C212yesmaximal, complemented
7#26012049C214yesmaximal, complemented
8#26012041C211yesmaximal, characteristic, complemented
9#26012037C212yesmaximal, complemented
10#26012000C218yesmaximal, complemented
11C2×#19149636C211yesmaximal, characteristic, complemented
12#532×C173C211yesmaximal, characteristic, complemented
13C2×#14583827C2212yes—
14C2×#14583826C2214yes—
15C2×#14583826C2212yescomplemented
16C2×#14583825C2212yes—
17C2×#14583825C2214yescomplemented
18#41×D173—216no—
19C2×Q8×D173C2212yescomplemented
20C2×D4×D173C2214yes—
21C2×D4×D173C2212yescomplemented
22C22×C4×D173C2211yescharacteristic
23C22×C4×D173C2214yescomplemented
24#19149679C2211yescharacteristic, complemented
25#19149677C2212yescomplemented
26#19149676C2211yescharacteristic, complemented
27#19149675C2212yescomplemented
28#19149668C2212yescomplemented
29#19149667C2212yescomplemented
30#19149666C2212yescomplemented
31#19149664C2218yescomplemented
32#19149661C2218yescomplemented
33#19149657C2218yescomplemented
34#19149656C2218yescomplemented
35#19149653C2214yescomplemented
36#19149652C2218yescomplemented
37#19149651C2218yescomplemented
38#19149651C2218yescomplemented
39#19149649C2218yescomplemented
40#19149647C2218yescomplemented
41C4×D692C2214yescomplemented
42C4×C22⋊D173C2218yescomplemented
43#19149636C2214yescomplemented
44#19149636C2212yescomplemented
45#19149636C2212yescomplemented
46C2×#14583805C2212yescomplemented
47C2×#14583803C2212yes—
48C2×C4×Dic173C2211yescharacteristic, complemented
49C2×#41×C173C2212yescomplemented
50C4⋊Q8×C173C2211yescharacteristic, complemented
51#127×C173C2211yescharacteristic, complemented
52#124×C173C2212yescomplemented
53#121×C173C2214yescomplemented
54C4×D4×C173C2214yescomplemented
55C2×C42×C173C2211yescharacteristic, complemented
56C23×D173C2314yes—
57C23×D173C2311yescharacteristic
58#14583827—28no—
59#14583826—216no—
60#14583826—28no—
61#14583825—216no—
62#14583825—28no—
63C2×Dic346C2314yes—
64C2×Dic346C2312yes—
65Q8×D173—28no—
66C2×D692C2314yes—
67C2×D692C2312yes—
68C2×D692C2314yes—
69C2×C22⋊D173C2314yes—
70C2×C22⋊D173C2314yes—
71C2×C22⋊D173C2314yes—
72C2×C22⋊D173C2318yes—
73D4×D173—216no—
74D4×D173—28no—
75C22×Dic173C2314yes—
76C22×Dic173C2312yes—
77C22×Dic173C2311yescharacteristic
78C2×C4×D173—216no—
79C2×C4×D173D418yescomplemented
80C2×C4×D173C2314yes—
81C2×C4×D173C2314yes—
82C2×C4×D173C2314yes—
83C2×C4×D173C2312yes—
84C2×C4×D173C2312yes—
85C4⋊Dic173C2314yes—
86#14583805C2314yes—
87#14583805C2314yes—
88#14583805C2318yes—
89C22⋊Dic173C2318yes—
90#14583803C2314yes—
91#14583803C2314yes—
92#14583803C2318yes—
93#14583803C2318yes—
94C4×Dic173C2312yes—
95C4×Dic173C2312yes—
96#41×C173—28no—
97C2×Q8×C173C2312yes—
98C2×D4×C173C2314yes—
99C2×D4×C173C2312yes—
100C2×D4×C173C2314yes—
101C22×C4×C173C2314yes—
102C22×C4×C173C2312yes—
103C22×C4×C173C2311yescharacteristic
104C4⋊C4×C173C2314yes—
105C22⋊C4×C173C2318yes—
106C42×C173C2311yescharacteristic
107C42×C173C2312yes—
108C42×C173C2311yescharacteristic
109C22×D173—28no—
110C22×D173—42no—
111C22×D173—48no—
112C22×D173C2×D414yes—
113C22×D173C2414yes—
114C22×D173C2412yes—
115Dic346—24no—
116Dic346—28no—
117D692—28no—
118D692—24no—
119D692—44no—
120C22⋊D173—28no—
121C22⋊D173—28no—
122C22⋊D173—28no—
123C22⋊D173—48no—
124C2×Dic173—28no—
125C2×Dic173—42no—
126C2×Dic173C2×D414yes—
127C2×Dic173C2414yes—
128C2×Dic173C2414yes—
129C2×Dic173C2412yes—
130C4×D173—28no—
131C4×D173—28no—
132C4×D173—28no—
133C4×D173C2×D418yes—
134C4×D173C2×D418yes—
135C23×C173C2414yes—
136C23×C173C2411yescharacteristic
137Q8×C173—24no—
138D4×C173—28no—
139D4×C173—24no—
140D4×C173—44no—
141C2×C4×C173—28no—
142C2×C4×C173—42no—
143C2×C4×C173C2×D414yes—
144C2×C4×C173C2414yes—
145C2×C4×C173C2412yes—
146C2×C4×C173C2412yes—
147C2×C4×C173C2411yescharacteristic
148C2×C4×C173C2411yescharacteristic
149C22×C173—22no—
150C22×C173—24no—
151C22×C173—28no—
152C22×C173C22×D412yes—
153C22×C173C2511yescharacteristic
154C2×D173—24no—
155C2×D173—24no—
156C2×D173—28no—
157C2×D173—44no—
158C2×D173—416no—
159C2×D173C22×D414yes—
160C4×C173—24no—
161C4×C173—22no—
162C4×C173—22no—
163C4×C173C22×D412yes—
164C4×C173C22×D412yes—
165Dic173—24no—
166Dic173—24no—
167Dic173C22×D414yes—
168C2×C173—22no—
169C2×C173—44no—
170C2×C173C23×D411yescharacteristic
171C2×C173#58512yes—
172D173—24no—
173D173—44no—
174C173#344911yescharacteristic, complemented, Sylow
175#3449—1731nomaximal, Sylow
176C22×#41—1732no—
177#537—1734no—
178#536—1732no—
179#536—1732no—
180#535—1732no—
181#535—1732no—
182#534—1734no—
183#532—1732no—
184#532—1731no—
185#532D17311yescharacteristic, complemented
186#525—1738no—
187C2×#117—1731no—
188C2×#41—34616no—
189C2×#41—1732no—
190C2×#41—1732no—
191C2×#41—1734no—
192C2×#41—1734no—
193C2×#41—1732no—
194C2×#41C2×D17312yescomplemented
195C22×Q8—1732no—
196C22×D4—1734no—
197C22×D4—1732no—
198C23×C4—1734no—
199C23×C4—1731no—
200C4⋊Q8—1732no—
201C4⋊Q8—1731no—
202C4⋊Q8C2×D17311yescharacteristic, complemented
203#127—1732no—
204#127—1731no—
205#127C2×D17311yescharacteristic, complemented
206#124—1732no—
207#124—1732no—
208#124—1732no—
209#124C2×D17312yescomplemented
210#123—1738no—
211#123—1738no—
212C22⋊Q8—1738no—
213C22⋊Q8—1738no—
214#121—1738no—
215#121—1738no—
216#121—1734no—
217#121—1738no—
218#121C2×D17314yescomplemented
219#120—1738no—
220#120—1738no—
221C4×D4—1738no—
222C4×D4—1734no—
223C4×D4C2×D17314yescomplemented
224#117—1734no—
225#117—1732no—
226#117—1732no—
227C2×C4⋊C4—1732no—
228C2×C22⋊C4—1732no—
229C2×C42—1731no—
230C2×C42C2×D17311yescharacteristic, complemented
231C24—1734no—
232C24—1731no—
233#41—28no—
234#41—34616no—
235#41—34616no—
236#41—3468no—
237#41—3468no—
238#41—3468no—
239C2×Q8—3468no—
240C2×Q8—1734no—
241C2×Q8—1732no—
242C2×Q8C22×D17312yes—
243C2×D4—34616no—
244C2×D4—3468no—
245C2×D4—1734no—
246C2×D4—1734no—
247C2×D4—1734no—
248C2×D4—1738no—
249C2×D4—1734no—
250C2×D4—1732no—
251C2×D4—1734no—
252C2×D4C22×D17314yes—
253C2×D4C22×D17312yes—
254C2×D4C22×D17314yes—
255C22×C4—34616no—
256C22×C4—1734no—
257C22×C4—1734no—
258C22×C4—1734no—
259C22×C4—1734no—
260C22×C4—1731no—
261C22×C4—1738no—
262C22×C4—1732no—
263C22×C4—1732no—
264C22×C4—1732no—
265C22×C4C22×D17314yes—
266C22×C4C22×D17312yes—
267C22×C4C22×D17311yescharacteristic
268C4⋊C4—1734no—
269C4⋊C4—1734no—
270C4⋊C4—1738no—
271C4⋊C4—1734no—
272C4⋊C4C22×D17314yes—
273C22⋊C4—1738no—
274C22⋊C4—1734no—
275C22⋊C4—1734no—
276C22⋊C4—1738no—
277C22⋊C4—1738no—
278C22⋊C4C22×D17318yes—
279C42—1732no—
280C42—1732no—
281C42C22×D17311yescharacteristic
282C42C22×D17311yescharacteristic
283C42C22×D17312yes—
284C23—3468no—
285C23—1734no—
286C23—1732no—
287C23—1734no—
288C23—6922no—
289C23—6928no—
290C23C23×D17314yes—
291C23C23×D17311yescharacteristic
292Q8—24no—
293Q8—3468no—
294Q8—3464no—
295D4—28no—
296D4—24no—
297D4—44no—
298D4—3468no—
299D4—3468no—
300D4—3468no—
301D4—3468no—
302D4—3464no—
303D4—6924no—
304D4—6928no—
305C2×C4—28no—
306C2×C4—42no—
307C2×C4—3468no—
308C2×C4—3468no—
309C2×C4—3468no—
310C2×C4—3468no—
311C2×C4—1734no—
312C2×C4—1734no—
313C2×C4—1734no—
314C2×C4—1732no—
315C2×C4—1738no—
316C2×C4—1738no—
317C2×C4—6922no—
318C2×C4D4×D17314yes—
319C2×C4C23×D17312yes—
320C2×C4C23×D17312yes—
321C2×C4C23×D17314yes—
322C2×C4C23×D17311yescharacteristic
323C2×C4C23×D17311yescharacteristic
324C22—22no—
325C22—24no—
326C22—28no—
327C22—3464no—
328C22—3464no—
329C22—3468no—
330C22—1734no—
331C22—6924no—
332C22—69216no—
333C22C2×D4×D17312yes—
334C22C24×D17311yescharacteristic
335C4—24no—
336C4—22no—
337C4—22no—
338C4—3464no—
339C4—3464no—
340C4—1734no—
341C4C2×D4×D17312yes—
342C4C2×D4×D17312yes—
343C2—22no—
344C2—44no—
345C2—3464no—
346C2—6924no—
347C2C22×D4×D17311yescharacteristic
348C2#2601232712yes—
subgroupG/H
centerC2×C4C23×D173
commutator, socle, cosocleC22×C173C25
FrattiniC22C24×D173
Fitting#532×C173C2
hypercenter#532D173
hypocenterC173#3449
LCSG ▷ C22×C173 ▷ C173
UCSC1 ◁ C2×C4 ◁ #532

Complements

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

NH ≅ G/Ntype
1. C2×#41×D173C2semidirect
2. #26012068C2semidirect
3. #26012066C2semidirect
4. #26012061C2semidirect
5. #26012059C2semidirect
6. #26012054C2semidirect
7. #26012049C2semidirect
8. #26012041C2semidirect
9. #26012037C2semidirect
10. #26012000C2semidirect
11. C2×#19149636C2semidirect
12. #532×C173C2semidirect
15. C2×#14583826C22semidirect
17. C2×#14583825C22semidirect
19. C2×Q8×D173C22semidirect
21. C2×D4×D173C22semidirect
23. C22×C4×D173C22semidirect
24. #19149679C22semidirect
25. #19149677C22semidirect
26. #19149676C22semidirect
27. #19149675C22semidirect
28. #19149668C22semidirect
29. #19149667C22semidirect
30. #19149666C22semidirect
31. #19149664C22semidirect
32. #19149661C22semidirect
33. #19149657C22semidirect
34. #19149656C22semidirect
35. #19149653C22semidirect
36. #19149652C22semidirect
37. #19149651C22semidirect
38. #19149651C22semidirect
39. #19149649C22semidirect
40. #19149647C22semidirect
41. C4×D692C22semidirect
42. C4×C22⋊D173C22semidirect
43. #19149636C22semidirect
44. #19149636C22semidirect
45. #19149636C22semidirect
46. C2×#14583805C22semidirect
48. C2×C4×Dic173C22semidirect
49. C2×#41×C173C22semidirect
50. C4⋊Q8×C173C22semidirect
51. #127×C173C22semidirect
52. #124×C173C22semidirect
53. #121×C173C22semidirect
54. C4×D4×C173C22semidirect
55. C2×C42×C173C22semidirect
79. C2×C4×D173D4semidirect
174. C173#3449semidirect
185. #532D173semidirect
194. C2×#41C2×D173semidirect
202. C4⋊Q8C2×D173semidirect
205. #127C2×D173semidirect
209. #124C2×D173semidirect
218. #121C2×D173semidirect
223. C4×D4C2×D173semidirect
230. C2×C42C2×D173semidirect

Maximal subgroups

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

subgroupindex
C2×#41×D1732
#260120682
#260120662
#260120612
#260120592
#260120542
#260120492
#260120412
#260120372
#260120002
C2×#191496362
#532×C1732
#3449173