Finite Group Databasebeta

← C2×#10608280 #12269573

Subgroups — C2×#10608280 #12269573

Order 1280.

2182 conjugacy classes of subgroups.

The subgroup lattice has 412 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×C5×#391C211yesmaximal, characteristic
2C2×#65518C212yesmaximal
3C2×#65272C211yesmaximal, characteristic
4C2×#65268C211yesmaximal, characteristic
5C2×C4×#683C212yesmaximal
6#10608280C2116yesmaximal, direct product factor
7C2×#64935C212yesmaximal
8C2×#64936C212yesmaximal
9C2×#64941C212yesmaximal
10C2×#64941C212yesmaximal
11C23×C5×Q8C2211yescharacteristic
12C23×Dic10C2211yescharacteristic
13C22×C5×C4⋊C4C2211yescharacteristic
14C22×C42×C5C2211yescharacteristic
15C22×C4⋊Dic5C2211yescharacteristic
16C22×#683C2212yes—
17C22×#683C2212yes—
18C22×#683C2212yes—
19C22×#683C2212yes—
20C22×#683C2212yes—
21C22×C4×Dic5C2212yes—
22C22×C4×Dic5C2212yes—
23C5×#391C2218yes—
24C2×C5×#95C2212yes—
25C2×C5×#95C2212yes—
26#65518C22116yes—
27C2×#3963C2211yescharacteristic
28C2×#3963C2211yescharacteristic
29C2×#3963C2212yes—
30C2×#3963C2212yes—
31C2×#3963C2212yes—
32C2×#3963C2212yes—
33C2×#3963C2212yes—
34#65272C2218yes—
35#65268C2218yes—
36C4×#683C22116yes—
37#64941C22116yes—
38#64941C22116yes—
39#64936C22116yes—
40#64935C22116yes—
41C2×#2586—51nomaximal, Sylow
42C22×C5×Q8—24no—
43C22×C5×Q8—42no—
44C22×C5×Q8—42no—
45C22×C5×Q8C2314yes—
46C23×C4×C5C2311yescharacteristic
47C23×C4×C5C2312yes—
48C23×C4×C5C2311yescharacteristic
49C23×C4×C5C2312yes—
50C23×C4×C5C2311yescharacteristic
51C23×Dic5C2312yes—
52C23×Dic5C2312yes—
53C23×Dic5C2312yes—
54C23×Dic5C2312yes—
55C22×Dic10—28no—
56C22×Dic10—42no—
57C22×Dic10C2314yes—
58C2×C5×C4⋊C4—42no—
59C2×C5×C4⋊C4C2314yes—
60C2×C42×C5—42no—
61C2×C42×C5C2314yes—
62C2×C4⋊Dic5—42no—
63C2×C4⋊Dic5C2314yes—
64C2×#683—28no—
65C2×#683—28no—
66C2×#683—44no—
67C2×#683—44no—
68C2×#683Q8116yescomplemented
69C2×#683C2318yes—
70C2×#683C2318yes—
71C2×#683C2318yes—
72C2×#683C2318yes—
73C2×#683C2318yes—
74C2×C4×Dic5—28no—
75C2×C4×Dic5—44no—
76C2×C4×Dic5C2318yes—
77C2×C4×Dic5C2318yes—
78C5×#95C2318yes—
79C5×#95C2318yes—
80#3963C2318yes—
81#3963C2318yes—
82#3963C2318yes—
83#3963C2318yes—
84#3963C2314yes—
85#3963C2314yes—
86#3963C2318yes—
87#2586—516no—
88C2×#398—51no—
89C2×#395—52no—
90C2×#395—51no—
91C2×#391—52no—
92C2×#391D511yescharacteristic
93C2×#389—52no—
94C2×#387—52no—
95C2×#387—52no—
96C2×C4×C4⋊C4—52no—
97C24×C5C2411yescharacteristic
98C2×C5×Q8—28no—
99C2×C5×Q8—44no—
100C2×C5×Q8—44no—
101C2×C5×Q8—416no—
102C22×C4×C5—24no—
103C22×C4×C5—22no—
104C22×C4×C5—24no—
105C22×C4×C5—22no—
106C22×C4×C5—24no—
107C22×C4×C5—24no—
108C22×C4×C5—24no—
109C22×C4×C5—22no—
110C22×C4×C5—22no—
111C22×C4×C5—24no—
112C22×C4×C5—24no—
113C22×C4×C5C2×Q818yes—
114C22×C4×C5#4114yes—
115C22×C4×C5C2412yes—
116C22×C4×C5C2412yes—
117C22×C4×C5C2414yes—
118C22×C4×C5C2414yes—
119C22×C4×C5C2412yes—
120C22×Dic5—24no—
121C22×Dic5—24no—
122C22×Dic5—24no—
123C22×Dic5—24no—
124C22×Dic5—24no—
125C22×Dic5—24no—
126C22×Dic5—24no—
127C22×Dic5—24no—
128C22×Dic5—24no—
129C22×Dic5C2×Q818yes—
130C22×Dic5C2×Q818yes—
131C22×Dic5#4118yes—
132C22×Dic5C2414yes—
133C22×Dic5C2414yes—
134C22×Dic5C2414yes—
135C22×Dic5C2414yes—
136C2×Dic10—216no—
137C2×Dic10—44no—
138C2×Dic10—416no—
139C5×C4⋊C4—44no—
140C42×C5—44no—
141C4⋊Dic5—44no—
142#683—216no—
143#683—216no—
144#683—48no—
145#683—48no—
146#683C2×Q8132yescomplemented
147C4×Dic5—216no—
148C4×Dic5—48no—
149C23×Q8—51no—
150C23×Q8C2×D511yescharacteristic
151C22×C4⋊C4—52no—
152C22×C4⋊C4—52no—
153C22×C4⋊C4—52no—
154C22×C4⋊C4—52no—
155C22×C4⋊C4—52no—
156C22×C4⋊C4—51no—
157C22×C4⋊C4C2×D511yescharacteristic
158C22×C42—52no—
159C22×C42—52no—
160C22×C42C2×D511yescharacteristic
161#398—58no—
162#395—516no—
163#395—58no—
164#391—516no—
165#391C2×D518yes—
166#389—516no—
167#387—516no—
168#387—516no—
169C4×C4⋊C4—516no—
170C2×#95—52no—
171C2×#95—51no—
172C2×#95—52no—
173C2×#95—52no—
174C2×#95—51no—
175C2×#95—52no—
176C2×#95—52no—
177C2×#95C2×D512yes—
178C2×#95C2×D512yes—
179C23×C5C22×Q814yes—
180C23×C5C2×#4112yes—
181C23×C5C2×#4114yes—
182C23×C5H(5,𝔽2)12yes—
183C23×C5H(5,𝔽2)12yes—
184C23×C5C2511yescharacteristic
185C5×Q8—416no—
186C2×C4×C5—28no—
187C2×C4×C5—216no—
188C2×C4×C5—24no—
189C2×C4×C5—24no—
190C2×C4×C5—22no—
191C2×C4×C5—22no—
192C2×C4×C5—24no—
193C2×C4×C5—24no—
194C2×C4×C5—24no—
195C2×C4×C5—24no—
196C2×C4×C5—24no—
197C2×C4×C5—22no—
198C2×C4×C5—22no—
199C2×C4×C5—48no—
200C2×C4×C5—44no—
201C2×C4×C5—44no—
202C2×C4×C5C22×Q818yes—
203C2×C4×C5C2×#4114yes—
204C2×Dic5—216no—
205C2×Dic5—216no—
206C2×Dic5—216no—
207C2×Dic5—24no—
208C2×Dic5—24no—
209C2×Dic5—24no—
210C2×Dic5—24no—
211C2×Dic5—24no—
212C2×Dic5—24no—
213C2×Dic5—24no—
214C2×Dic5—24no—
215C2×Dic5—24no—
216C2×Dic5—48no—
217C2×Dic5C22×Q818yes—
218C2×Dic5C22×Q818yes—
219C2×Dic5C2×#4118yes—
220Dic10—416no—
221C22×Q8—24no—
222C22×Q8—42no—
223C22×Q8—42no—
224C22×Q8—54no—
225C22×Q8—108no—
226C22×Q8—202no—
227C22×Q8C22×D514yes—
228C23×C4—52no—
229C23×C4—52no—
230C23×C4—52no—
231C23×C4—52no—
232C23×C4C22×D511yescharacteristic
233C23×C4C22×D512yes—
234C23×C4C22×D511yescharacteristic
235C23×C4C22×D511yescharacteristic
236C23×C4C22×D512yes—
237C2×C4⋊C4—42no—
238C2×C4⋊C4—516no—
239C2×C4⋊C4—58no—
240C2×C4⋊C4—58no—
241C2×C4⋊C4—58no—
242C2×C4⋊C4—58no—
243C2×C4⋊C4—54no—
244C2×C4⋊C4—58no—
245C2×C4⋊C4—108no—
246C2×C4⋊C4—108no—
247C2×C4⋊C4—204no—
248C2×C4⋊C4—204no—
249C2×C4⋊C4—202no—
250C2×C4⋊C4C22×D514yes—
251C2×C42—42no—
252C2×C42—58no—
253C2×C42—58no—
254C2×C42—108no—
255C2×C42—204no—
256C2×C42C22×D514yes—
257#95—58no—
258#95—58no—
259#95—58no—
260#95—54no—
261#95—54no—
262#95—58no—
263#95—58no—
264#95C22×D518yes—
265#95C22×D518yes—
266C22×C5#53318yes—
267C22×C5#53814yes—
268C22×C5#55114yes—
269C22×C5#55418yes—
270C22×C5Q8214yes—
271C22×C5C23×Q812yes—
272C22×C5C22×#4111yescharacteristic
273C22×C5C22×#4112yes—
274C22×C5C2×H(5,𝔽2)11yescharacteristic
275C22×C5C2×H(5,𝔽2)11yescharacteristic
276C4×C5—24no—
277C4×C5—28no—
278C4×C5—44no—
279C4×C5—42no—
280C4×C5—42no—
281Dic5—28no—
282Dic5—28no—
283Dic5—28no—
284Dic5—44no—
285C24C23×D511yescharacteristic
286C2×Q8—28no—
287C2×Q8—44no—
288C2×Q8—44no—
289C2×Q8—416no—
290C2×Q8—1016no—
291C2×Q8—204no—
292C2×Q8—2016no—
293C22×C4—24no—
294C22×C4—22no—
295C22×C4—22no—
296C22×C4—24no—
297C22×C4—24no—
298C22×C4—22no—
299C22×C4—22no—
300C22×C4—24no—
301C22×C4—24no—
302C22×C4—24no—
303C22×C4—24no—
304C22×C4—58no—
305C22×C4—58no—
306C22×C4—58no—
307C22×C4—54no—
308C22×C4—54no—
309C22×C4—54no—
310C22×C4—54no—
311C22×C4—104no—
312C22×C4—104no—
313C22×C4—104no—
314C22×C4—104no—
315C22×C4—104no—
316C22×C4—104no—
317C22×C4—104no—
318C22×C4—104no—
319C22×C4—104no—
320C22×C4#70914yes—
321C22×C4Q8×D518yes—
322C22×C4C23×D512yes—
323C22×C4C23×D514yes—
324C22×C4C23×D512yes—
325C22×C4C23×D512yes—
326C22×C4C23×D514yes—
327C4⋊C4—44no—
328C4⋊C4—532no—
329C4⋊C4—1016no—
330C4⋊C4—1016no—
331C4⋊C4—208no—
332C4⋊C4—208no—
333C4⋊C4—204no—
334C42—44no—
335C42—1016no—
336C42—208no—
337C2×C5#258618yescomplemented
338C2×C5C2×#53312yes—
339C2×C5C2×#53811yescharacteristic
340C2×C5C2×#55111yescharacteristic
341C2×C5C2×#55412yes—
342C2×C5C2×Q8211yescharacteristic
343C23C2×#70912yes—
344C23#414412yes—
345C23#414412yes—
346C23C2×Q8×D514yes—
347C23D5×#4114yes—
348C23C24×D511yescharacteristic
349Q8—416no—
350Q8—2016no—
351C2×C4—28no—
352C2×C4—216no—
353C2×C4—24no—
354C2×C4—22no—
355C2×C4—22no—
356C2×C4—24no—
357C2×C4—24no—
358C2×C4—22no—
359C2×C4—22no—
360C2×C4—24no—
361C2×C4—24no—
362C2×C4—24no—
363C2×C4—24no—
364C2×C4—48no—
365C2×C4—44no—
366C2×C4—44no—
367C2×C4—58no—
368C2×C4—58no—
369C2×C4—58no—
370C2×C4—1016no—
371C2×C4—1016no—
372C2×C4—1016no—
373C2×C4—104no—
374C2×C4—104no—
375C2×C4—104no—
376C2×C4—104no—
377C2×C4—104no—
378C2×C4—104no—
379C2×C4—104no—
380C2×C4—104no—
381C2×C4—104no—
382C2×C4—208no—
383C2×C4C2×#70914yes—
384C2×C4C2×Q8×D518yes—
385C5C2×#258611yescharacteristic, complemented, Sylow
386C22#6582814yes—
387C22#6591218yes—
388C22#6597018yes—
389C22#6601314yes—
390C22#6605614yes—
391C22C22×#70911yescharacteristic
392C22C2×#414411yescharacteristic
393C22C2×#414411yescharacteristic
394C22C22×Q8×D512yes—
395C22C2×D5×#4112yes—
396C4—28no—
397C4—24no—
398C4—42no—
399C4—42no—
400C4—44no—
401C4—108no—
402C4—108no—
403C4—108no—
404C4—204no—
405C2#1060828018yesdirect product factor
406C2C2×#6582811yescharacteristic
407C2C2×#6591212yes—
408C2C2×#6597012yes—
409C2C2×#6601311yescharacteristic
410C2C2×#6605611yescharacteristic
subgroupG/H
centerC24C23×D5
commutator, cosocleC23×C5C25
FrattiniC23C24×D5
FittingC2×C5×#391C2
socle, prime radicalC24×C5C24
hypercenterC2×#391D5
hypocenterC5C2×#2586
LCSG ▷ C23×C5 ▷ C5
UCSC1 ◁ C24 ◁ C2×#391

Complements

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

NH ≅ G/Ntype
6. #10608280C2direct (N × H)
68. C2×#683Q8semidirect
146. #683C2×Q8semidirect
337. C2×C5#2586semidirect
385. C5C2×#2586semidirect
405. C2#10608280direct (N × H)

Maximal subgroups

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

subgroupindex
C2×C5×#3912
C2×#655182
C2×#652722
C2×#652682
C2×C4×#6832
#106082802
C2×#649352
C2×#649362
C2×#649412
C2×#25865