Finite Group Databasebeta

← #12278362

Subgroups — #12278362

Order 1280.

7198 conjugacy classes of subgroups; only the 835 normal subgroups are shown.

normal subgroups only

This group has 7198 conjugacy classes of subgroups — too many to draw. Only its 835 normal subgroups are shown.

Dotted edges: containments with no intermediate normal subgroup — omitted non-normal subgroups may lie between.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C22×C5×H(5,𝔽2)C211yesmaximal, characteristic, complemented
2C22×#4083C213yesmaximal, complemented
3#10615827C2112yesmaximal, complemented
4#10615783C2112yesmaximal, complemented
5#10615774C213yesmaximal, complemented
6C25×D5C2211yescharacteristic, complemented
7C2×C5×H(5,𝔽2)—224no—
8C22×C5×#41C2216yescomplemented
9C23×C5×D4C2213yes—
10C23×C5×D4C2216yescomplemented
11C23×C22⋊D5C2216yescomplemented
12C2×#4099C2218yescomplemented
13C22×C22⋊Dic5C2213yescomplemented
14C2×#4083—224no—
15C2×#4083C22124yescomplemented
16C22×#685C2216yescomplemented
17#65536C2216yescomplemented
18#65520C2218yescomplemented
19#65512C22124yescomplemented
20#65509C22124yescomplemented
21#65502C22124yescomplemented
22#65500C2216yescomplemented
23#35346—51nomaximal, Sylow
24C25×C5C2313yes—
25C25×C5C2311yescharacteristic
26C25×C5C2312yescomplemented
27C24×D5—212no—
28C24×D5—48no—
29C24×D5C2316yes—
30C5×H(5,𝔽2)—2128no—
31C2×C5×#41—248no—
32C2×C5×#41—224no—
33C22×C5×Q8C2312yescomplemented
34C22×C5×D4—224no—
35C22×C5×D4—224no—
36C22×C5×D4—248no—
37C22×C5×D4D4124yescomplemented
38C22×C5×D4C23112yes—
39C22×C5×D4C2316yescomplemented
40C23×C4×C5C2313yes—
41C23×C4×C5C2316yescomplemented
42C22×C22⋊D5—224no—
43C22×C22⋊D5—424no—
44C22×C22⋊D5C23124yes—
45C23×Dic5C2316yes—
46#4099—416no—
47C2×C22⋊Dic5—212no—
48C2×C22⋊Dic5C23112yes—
49C2×C22⋊Dic5C23124yescomplemented
50#4083—296no—
51C2×#685—224no—
52C2×#685C23124yes—
53#3963C23124yescomplemented
54C22×H(5,𝔽2)D511yescharacteristic, complemented
55C22×#120—53no—
56#2404—512no—
57#2403—512no—
58#2308—53no—
59C24×C5—212no—
60C24×C5—212no—
61C24×C5—44no—
62C24×C5—412no—
63C24×C5—48no—
64C24×C5C2×D4112yes—
65C24×C5C2×D4112yescomplemented
66C24×C5C2416yes—
67C24×C5C2413yes—
68C23×D5—212no—
69C23×D5—224no—
70C23×D5—224no—
71C23×D5—412no—
72C23×D5—496no—
73C23×D5—416no—
74C23×D5C2414yes—
75C5×#41—2192no—
76C2×C5×Q8—212no—
77C2×C5×D4—2192no—
78C2×C5×D4—296no—
79C2×C5×D4—224no—
80C2×C5×D4—224no—
81C2×C5×D4—224no—
82C2×C5×D4—212no—
83C2×C5×D4C2×D4148yescomplemented
84C22×C4×C5—212no—
85C22×C4×C5—212no—
86C22×C4×C5—224no—
87C22×C4×C5C2×D4112yescomplemented
88C22×C4×C5C2416yes—
89C2×C22⋊D5—248no—
90C2×C22⋊D5—248no—
91C2×C22⋊D5—424no—
92C2×C22⋊D5—496no—
93C22×Dic5—212no—
94C22×Dic5—412no—
95C22×Dic5C24112yes—
96C22⋊Dic5—224no—
97C22⋊Dic5—424no—
98#685—248no—
99C26—51no—
100C2×H(5,𝔽2)—224no—
101C22×#41C2×D516yescomplemented
102C23×D4—56no—
103C23×D4C2×D513yes—
104C23×D4C2×D516yescomplemented
105C2×#120—524no—
106C2×#120—58no—
107C2×#120—1024no—
108C22×C22⋊C4—53no—
109C22×C22⋊C4—56no—
110#393—58no—
111#392—524no—
112#386—524no—
113#386—524no—
114#381—56no—
115#379—56no—
116C23×C5—212no—
117C23×C5—216no—
118C23×C5—212no—
119C23×C5—224no—
120C23×C5—224no—
121C23×C5—424no—
122C23×C5—424no—
123C23×C5—824no—
124C23×C5—824no—
125C23×C5#120116yescomplemented
126C23×C5C22×D4112yes—
127C23×C5C22×D4112yes—
128C23×C5C22×D416yes—
129C23×C5C2511yescharacteristic
130C22×D5—248no—
131C22×D5—212no—
132C22×D5—212no—
133C22×D5—496no—
134C22×D5—424no—
135C22×D5—4128no—
136C22×D5—424no—
137C22×D5H(5,𝔽2)18yescomplemented
138C5×Q8—216no—
139C5×D4—248no—
140C5×D4—296no—
141C2×C4×C5—248no—
142C2×C4×C5—224no—
143C2×C4×C5—212no—
144C2×C4×C5C22×D4112yes—
145C22⋊D5—496no—
146C2×Dic5—212no—
147C2×Dic5—224no—
148C2×Dic5—424no—
149C25—56no—
150C25—1012no—
151C25—208no—
152C25C22×D513yes—
153C25C22×D511yescharacteristic
154C25C22×D512yescomplemented
155H(5,𝔽2)—2128no—
156C2×#41—248no—
157C2×#41—224no—
158C22×Q8C22×D512yescomplemented
159C22×D4—224no—
160C22×D4—224no—
161C22×D4—248no—
162C22×D4—524no—
163C22×D4—1024no—
164C22×D4—2024no—
165C22×D4C22⋊D5124yescomplemented
166C22×D4C22×D5112yes—
167C22×D4C22×D516yescomplemented
168C23×C4—56no—
169C23×C4C22×D513yes—
170C23×C4C22×D516yescomplemented
171#120—1096no—
172#120—2016no—
173C2×C22⋊C4—512no—
174C2×C22⋊C4—524no—
175C2×C22⋊C4—524no—
176C2×C22⋊C4—1012no—
177C2×C22⋊C4—1024no—
178#95—524no—
179C22×C5—212no—
180C22×C5—224no—
181C22×C5—212no—
182C22×C5—224no—
183C22×C5—416no—
184C22×C5—424no—
185C22×C5—824no—
186C22×C5—816no—
187C22×C5C2×#120112yes—
188C22×C5C23×D413yes—
189C22×C5C2×H(5,𝔽2)14yes—
190C2×D5—224no—
191C2×D5—496no—
192C2×D5—44no—
193C2×D5—412no—
194C4×C5—212no—
195Dic5—412no—
196C24—212no—
197C24—212no—
198C24—44no—
199C24—412no—
200C24—48no—
201C24—54no—
202C24—1024no—
203C24—1012no—
204C24—1024no—
205C24—2012no—
206C24—2096no—
207C24—2016no—
208C24C2×C22⋊D5112yes—
209C24C2×C22⋊D5112yescomplemented
210C24C23×D516yes—
211C24C23×D513yes—
212#41—2192no—
213C2×Q8—212no—
214C2×D4—2192no—
215C2×D4—296no—
216C2×D4—224no—
217C2×D4—224no—
218C2×D4—224no—
219C2×D4—212no—
220C2×D4—1048no—
221C2×D4—1048no—
222C2×D4—2024no—
223C2×D4—2096no—
224C2×D4C2×C22⋊D5148yescomplemented
225C22×C4—212no—
226C22×C4—212no—
227C22×C4—224no—
228C22×C4—512no—
229C22×C4—1012no—
230C22×C4—2012no—
231C22×C4C2×C22⋊D5112yescomplemented
232C22×C4C23×D516yes—
233C22⋊C4—1024no—
234C22⋊C4—1048no—
235C22⋊C4—2024no—
236C2×C5—212no—
237C2×C5—412no—
238C2×C5C22×#12011yescharacteristic
239C2×C5#348416yes—
240D5—48no—
241C23—212no—
242C23—216no—
243C23—224no—
244C23—212no—
245C23—224no—
246C23—424no—
247C23—424no—
248C23—58no—
249C23—824no—
250C23—824no—
251C23—1048no—
252C23—1012no—
253C23—1012no—
254C23—2096no—
255C23—2024no—
256C23—20128no—
257C23—2024no—
258C23#4099116yescomplemented
259C23C22×C22⋊D5112yes—
260C23C22×C22⋊D5112yes—
261C23C22×C22⋊D516yes—
262C23C24×D511yescharacteristic
263Q8—216no—
264D4—248no—
265D4—296no—
266D4—2096no—
267C2×C4—248no—
268C2×C4—224no—
269C2×C4—212no—
270C2×C4—1024no—
271C2×C4—1012no—
272C2×C4—2024no—
273C2×C4C22×C22⋊D5112yes—
274C5#3534611yescharacteristic, complemented, Sylow
275C22—212no—
276C22—224no—
277C22—224no—
278C22—212no—
279C22—416no—
280C22—424no—
281C22—824no—
282C22—816no—
283C22—1024no—
284C22—2096no—
285C22—2012no—
286C22—204no—
287C22C2×#4099112yes—
288C22D5×H(5,𝔽2)14yes—
289C22C23×C22⋊D513yes—
290C4—212no—
291C4—2012no—
292C2—212no—
293C2—412no—
294C2—208no—
295C2#1061918516yes—
296C2C22×#409911yescharacteristic
subgroupG/H
center, FrattiniC23C24×D5
commutator, socle, cosocleC23×C5C25
FittingC22×C5×H(5,𝔽2)C2
hypercenterC22×H(5,𝔽2)D5
hypocenterC5#35346
LCSG ▷ C23×C5 ▷ C5
UCSC1 ◁ C23 ◁ C22×H(5,𝔽2)

Complements

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

NH ≅ G/Ntype
1. C22×C5×H(5,𝔽2)C2semidirect
2. C22×#4083C2semidirect
3. #10615827C2semidirect
4. #10615783C2semidirect
5. #10615774C2semidirect
6. C25×D5C22semidirect
8. C22×C5×#41C22semidirect
10. C23×C5×D4C22semidirect
11. C23×C22⋊D5C22semidirect
12. C2×#4099C22semidirect
13. C22×C22⋊Dic5C22semidirect
15. C2×#4083C22semidirect
16. C22×#685C22semidirect
17. #65536C22semidirect
18. #65520C22semidirect
19. #65512C22semidirect
20. #65509C22semidirect
21. #65502C22semidirect
22. #65500C22semidirect
26. C25×C5C23semidirect
33. C22×C5×Q8C23semidirect
37. C22×C5×D4D4semidirect
39. C22×C5×D4C23semidirect
41. C23×C4×C5C23semidirect
49. C2×C22⋊Dic5C23semidirect
53. #3963C23semidirect
54. C22×H(5,𝔽2)D5semidirect
65. C24×C5C2×D4semidirect
83. C2×C5×D4C2×D4semidirect
87. C22×C4×C5C2×D4semidirect
101. C22×#41C2×D5semidirect
104. C23×D4C2×D5semidirect
125. C23×C5#120semidirect
137. C22×D5H(5,𝔽2)semidirect
154. C25C22×D5semidirect
158. C22×Q8C22×D5semidirect
165. C22×D4C22⋊D5semidirect
167. C22×D4C22×D5semidirect
170. C23×C4C22×D5semidirect
209. C24C2×C22⋊D5semidirect
224. C2×D4C2×C22⋊D5semidirect
231. C22×C4C2×C22⋊D5semidirect
258. C23#4099semidirect
274. C5#35346semidirect

Maximal subgroups

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

subgroupindex
C22×C5×H(5,𝔽2)2
C22×#40832
#106158272
#106157832
#106157742
#353465