Finite Group Databasebeta

← #12259503

Subgroups — #12259503

Order 1280.

2414 conjugacy classes of subgroups.

The subgroup lattice has 330 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×C4×C5×#41C211yesmaximal, characteristic
2#10615820C212yesmaximal
3Q8×C4⋊Dic5C212yesmaximal, complemented
4#10614181C212yesmaximal, complemented
5#10614068C216yesmaximal, complemented
6#10614056C216yesmaximal, complemented
7#10613971C216yesmaximal, complemented
8#10613793C213yesmaximal, complemented
9#10613791C213yesmaximal, complemented
10C22×C5×#41C2211yescharacteristic
11C4×C5×#41C4116yescomplemented
12C2×C4×C5×Q8C2211yescharacteristic
13C2×C4×C5×Q8C2211yescharacteristic
14C2×C4×C5×D4C2213yes—
15C2×C4×C5×D4C2213yes—
16C2×C5×#117C2213yes—
17C22×C42×C5C2213yes—
18C2×Q8×Dic5C2214yes—
19C2×#4072C2216yes—
20#65510C22112yes—
21#65501C2216yes—
22#65501C2216yes—
23#65501C2216yes—
24#65500C2216yes—
25C4×C22⋊Dic5C2216yes—
26#65274C22112yes—
27C4⋊C4×Dic5C22112yes—
28#65240C22124yes—
29#65226C2216yes—
30#65225C2211yescharacteristic
31#65225C2213yes—
32#65224C2216yes—
33C4×C4⋊Dic5C2216yes—
34C4×C4⋊Dic5C2216yes—
35C4×C4⋊Dic5C2216yes—
36#65221C2216yes—
37#34193—51nomaximal, Sylow
38C2×C5×#41—28no—
39C2×C5×#41C2×C418yes—
40C22×C5×Q8C2311yescharacteristic
41C22×C5×D4C2313yes—
42C23×C4×C5C2313yes—
43C23×C4×C5C2313yes—
44C23×C4×C5C2313yes—
45C4×C5×Q8C2×C418yes—
46C4×C5×Q8C2×C418yes—
47C4×C5×D4C2×C4124yes—
48C4×C5×D4C2×C4124yes—
49C5×#117C2×C4124yes—
50C2×C5×C4⋊C4C2313yes—
51C2×C5×C4⋊C4C2313yes—
52C2×C5×C22⋊C4C2316yes—
53C2×C42×C5C2×C4124yes—
54C2×C42×C5C2311yescharacteristic
55C2×C42×C5C2313yes—
56C2×C42×C5C2313yes—
57C2×C42×C5C2313yes—
58C2×C22⋊Dic5C23112yes—
59C2×C22⋊Dic5C23112yes—
60Q8×Dic5—216no—
61#4072—412no—
62C2×C4⋊Dic5C2316yes—
63C2×C4⋊Dic5C23112yes—
64C2×C4⋊Dic5C2312yes—
65C2×C4⋊Dic5C23112yes—
66C2×C4×Dic5C23112yes—
67C2×C4×Dic5C23112yes—
68#3963C2312yes—
69#3963C2316yes—
70#3963C23112yes—
71#3963C2316yes—
72#3963C23112yes—
73#3963C2312yes—
74C2×C4×#41D511yescharacteristic
75#2354—56no—
76Q8×C4⋊C4—52no—
77#2347—56no—
78#2341—52no—
79#2319—56no—
80#2285—53no—
81#2283—53no—
82#2276—52no—
83C24×C5C2413yes—
84C5×#41—232no—
85C2×C5×Q8C22×C414yes—
86C2×C5×Q8#4118yes—
87C2×C5×D4—212no—
88C2×C5×D4C22×C4112yes—
89C22×C4×C5—212no—
90C22×C4×C5—26no—
91C22×C4×C5—26no—
92C22×C4×C5—46no—
93C22×C4×C5C22×C4112yes—
94C22×C4×C5C22×C4112yes—
95C22×C4×C5C22×C4112yes—
96C22×C4×C5C2411yescharacteristic
97C22×C4×C5C2411yescharacteristic
98C22×C4×C5C2413yes—
99C22×C4×C5C2413yes—
100C22×C4×C5C2413yes—
101C22×C4×C5C2411yescharacteristic
102C22×Dic5C2414yes—
103C22×Dic5C24112yes—
104C5×C4⋊C4C22×C4112yes—
105C5×C4⋊C4C22×C4112yes—
106C5×C22⋊C4C22×C4124yes—
107C42×C5C22×C414yes—
108C42×C5C22×C4112yes—
109C42×C5C22×C4112yes—
110C42×C5C22×C4112yes—
111C22⋊Dic5—412no—
112C22⋊Dic5—412no—
113C4⋊Dic5—224no—
114C4⋊Dic5—212no—
115C4⋊Dic5—24no—
116C4⋊Dic5—412no—
117C4×Dic5—224no—
118C4×Dic5—412no—
119C22×#41C2×D511yescharacteristic
120C4×#41Dic5116yescomplemented
121C2×C4×Q8—54no—
122C2×C4×Q8C2×D511yescharacteristic
123C2×C4×Q8C2×D511yescharacteristic
124C2×C4×D4C2×D513yes—
125C2×C4×D4C2×D513yes—
126C2×#117—56no—
127C2×#117C2×D513yes—
128C22×C42C2×D513yes—
129#390—512no—
130#389—512no—
131#388—524no—
132#384—56no—
133#383—56no—
134#382—51no—
135#382—53no—
136#381—56no—
137#380—56no—
138#380—56no—
139#380—56no—
140C4×C4⋊C4—512no—
141C4×C4⋊C4—56no—
142C4×C4⋊C4—56no—
143C4×C4⋊C4—56no—
144C4×C22⋊C4—56no—
145#376—56no—
146C23×C5—26no—
147C23×C5—23no—
148C23×C5—23no—
149C23×C5—26no—
150C23×C5C23×C416yes—
151C23×C5C2511yescharacteristic
152C5×Q8#117116yescomplemented
153C5×D4—224no—
154C2×C4×C5—22no—
155C2×C4×C5—23no—
156C2×C4×C5—23no—
157C2×C4×C5—26no—
158C2×C4×C5—22no—
159C2×C4×C5—224no—
160C2×C4×C5—224no—
161C2×C4×C5—412no—
162C2×C4×C5C23×C412yes—
163C2×C4×C5C23×C412yes—
164C2×C4×C5C23×C416yes—
165C2×C4×C5C23×C416yes—
166C2×C4×C5C23×C416yes—
167C2×C4×C5C23×C412yes—
168C2×C4×C5C2×#41112yes—
169C2×C4×C5H(5,𝔽2)12yes—
170C2×C4×C52−1+412yes—
171C2×Dic5—24no—
172C2×Dic5—212no—
173C2×Dic5—212no—
174C2×Dic5—24no—
175C2×Dic5—212no—
176C2×Dic5—24no—
177C2×#41—28no—
178C2×#41C2×Dic518yes—
179C22×Q8C22×D511yescharacteristic
180C22×D4C22×D513yes—
181C23×C4C22×D513yes—
182C23×C4C22×D513yes—
183C23×C4C22×D513yes—
184C4×Q8—1016no—
185C4×Q8C2×Dic518yes—
186C4×Q8C2×Dic518yes—
187C4×D4C2×Dic5124yes—
188C4×D4C2×Dic5124yes—
189#117—2012no—
190#117C2×Dic5124yes—
191C2×C4⋊C4—56no—
192C2×C4⋊C4—512no—
193C2×C4⋊C4—52no—
194C2×C4⋊C4—512no—
195C2×C4⋊C4C22×D513yes—
196C2×C4⋊C4C22×D513yes—
197C2×C22⋊C4—512no—
198C2×C22⋊C4—512no—
199C2×C22⋊C4C22×D516yes—
200C2×C42—512no—
201C2×C42—512no—
202C2×C42C2×Dic5124yes—
203C2×C42C22×D513yes—
204C2×C42C22×D513yes—
205C2×C42C22×D513yes—
206C2×C42C22×D511yescharacteristic
207#95—52no—
208#95—56no—
209#95—512no—
210#95—52no—
211#95—56no—
212#95—512no—
213C22×C5—26no—
214C22×C5—26no—
215C22×C5—26no—
216C22×C5—46no—
217C22×C5—46no—
218C22×C5C24×C411yescharacteristic
219C22×C5C22×#4112yes—
220C22×C5C2×H(5,𝔽2)11yescharacteristic
221C22×C5C2×2−1+411yescharacteristic
222C22×C5#58512yes—
223C4×C5—22no—
224C4×C5—26no—
225C4×C5—22no—
226C4×C5—26no—
227C4×C5C2×#117112yes—
228C4×C5#52014yes—
229Dic5—412no—
230Dic5—44no—
231C24C23×D513yes—
232#41—232no—
233C2×Q8#70918yes—
234C2×Q8C22×Dic514yes—
235C2×D4—212no—
236C2×D4C22×Dic5112yes—
237C22×C4—212no—
238C22×C4—26no—
239C22×C4—26no—
240C22×C4—46no—
241C22×C4—54no—
242C22×C4—512no—
243C22×C4C22×Dic5112yes—
244C22×C4C22×Dic5112yes—
245C22×C4C22×Dic5112yes—
246C22×C4C23×D511yescharacteristic
247C22×C4C23×D511yescharacteristic
248C22×C4C23×D513yes—
249C22×C4C23×D513yes—
250C22×C4C23×D513yes—
251C22×C4C23×D511yescharacteristic
252C4⋊C4—1024no—
253C4⋊C4—1012no—
254C4⋊C4—104no—
255C4⋊C4—2012no—
256C4⋊C4C22×Dic5112yes—
257C4⋊C4C22×Dic5112yes—
258C22⋊C4—2012no—
259C22⋊C4—2012no—
260C22⋊C4C22×Dic5124yes—
261C42—1024no—
262C42—2012no—
263C42C22×Dic514yes—
264C42C22×Dic5112yes—
265C42C22×Dic5112yes—
266C42C22×Dic5112yes—
267C2×C5—46no—
268C2×C5C22×#11711yescharacteristic
269C2×C5C2×#52011yescharacteristic
270C2×C5#342811yescharacteristic
271C2×C5#351712yes—
272C2×C5#351812yes—
273C23—26no—
274C23—23no—
275C23—23no—
276C23—26no—
277C23C23×Dic516yes—
278C23C24×D511yescharacteristic
279Q8#4072116yescomplemented
280D4—224no—
281C2×C4—22no—
282C2×C4—23no—
283C2×C4—23no—
284C2×C4—26no—
285C2×C4—22no—
286C2×C4—224no—
287C2×C4—224no—
288C2×C4—412no—
289C2×C4—104no—
290C2×C4—1012no—
291C2×C4—1012no—
292C2×C4—104no—
293C2×C4—104no—
294C2×C4—1012no—
295C2×C4C2×#709112yes—
296C2×C4#414912yes—
297C2×C4#415012yes—
298C2×C4C23×Dic512yes—
299C2×C4C23×Dic512yes—
300C2×C4C23×Dic516yes—
301C2×C4C23×Dic516yes—
302C2×C4C23×Dic516yes—
303C2×C4C23×Dic512yes—
304C5#3419311yescharacteristic, complemented, Sylow
305C22—26no—
306C22—26no—
307C22—26no—
308C22—46no—
309C22—46no—
310C22C22×#70912yes—
311C22C2×#414911yescharacteristic
312C22C2×#415011yescharacteristic
313C22#6628312yes—
314C22C24×Dic511yescharacteristic
315C4—26no—
316C4—22no—
317C4—26no—
318C4—22no—
319C4—204no—
320C4—2012no—
321C4C2×#4072112yes—
322C4#6616114yes—
323C2—46no—
324C2#1061773312yes—
325C2#1061775412yes—
326C2C22×#407211yescharacteristic
327C2C2×#6616111yescharacteristic
328C2#1061916111yescharacteristic
subgroupG/H
center, FrattiniC23C24×D5
commutatorC22×C5C24×C4
FittingC2×C4×C5×#41C2
socle, cosocleC23×C5C25
prime radicalC22×C5×#41C22
hypercenterC2×C4×#41D5
hypocenterC5#34193
LCSG ▷ C22×C5 ▷ C5
UCSC1 ◁ C23 ◁ C2×C4×#41

Complements

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

NH ≅ G/Ntype
3. Q8×C4⋊Dic5C2semidirect
4. #10614181C2semidirect
5. #10614068C2semidirect
6. #10614056C2semidirect
7. #10613971C2semidirect
8. #10613793C2semidirect
9. #10613791C2semidirect
11. C4×C5×#41C4semidirect
120. C4×#41Dic5semidirect
152. C5×Q8#117semidirect
279. Q8#4072semidirect
304. C5#34193semidirect

Maximal subgroups

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

subgroupindex
C2×C4×C5×#412
#106158202
Q8×C4⋊Dic52
#106141812
#106140682
#106140562
#106139712
#106137932
#106137912
#341935