Finite Group Databasebeta

← C2×#10605191 #12258561

Subgroups — C2×#10605191 #12258561

Order 1280.

2790 conjugacy classes of subgroups.

The subgroup lattice has 380 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C22×#4072C211yesmaximal, characteristic
2C2×C4×C5×C22⋊C4C211yesmaximal, characteristic
3C2×#65501C211yesmaximal, characteristic
4C2×#65234C214yesmaximal
5C2×#65217C212yesmaximal, complemented
6C2×C4×C4⋊Dic5C212yesmaximal, complemented
7#10605191C2116yesmaximal, direct product factor
8C2×#64940C212yesmaximal, complemented
9C2×#64938C212yesmaximal, complemented
10C24×C4×C5C2211yescharacteristic
11C22×C5×C22⋊C4C2212yes—
12C22×C42×C5C2212yes—
13C22×C22⋊Dic5C2212yes—
14C22×C22⋊Dic5C2212yes—
15C2×#4072C4116yescomplemented
16C2×#4072C2218yes—
17C22×C4⋊Dic5C2212yes—
18C22×C4⋊Dic5C2211yescharacteristic
19C22×C4⋊Dic5C2211yescharacteristic
20C22×#683C2214yes—
21C22×C4×Dic5C2214yes—
22C22×C4×Dic5C2212yes—
23C4×C5×C22⋊C4C2218yes—
24C2×C5×#95C2211yescharacteristic
25C2×C5×#95C2211yescharacteristic
26#65501C2218yes—
27C2×#3963C2214yes—
28C2×#3963C2214yes—
29C2×#3963C2212yes—
30#65234C22132yes—
31C4×C4⋊Dic5C22116yescomplemented
32#65217C22116yescomplemented
33#64940C22116yescomplemented
34#64938C22116yescomplemented
35C2×#2347—51nomaximal, Sylow
36C25×C5C2311yescharacteristic
37C23×C4×C5—28no—
38C23×C4×C5C2×C418yes—
39C23×C4×C5C2312yes—
40C23×C4×C5C2311yescharacteristic
41C23×C4×C5C2314yes—
42C23×C4×C5C2312yes—
43C23×C4×C5C2311yescharacteristic
44C23×Dic5C2314yes—
45C23×Dic5C2312yes—
46C23×Dic5C2312yes—
47C2×C5×C22⋊C4—28no—
48C2×C5×C22⋊C4C2318yes—
49C2×C42×C5—28no—
50C2×C42×C5C2318yes—
51C2×C22⋊Dic5—44no—
52C2×C22⋊Dic5C2×C4116yes—
53C2×C22⋊Dic5C2318yes—
54C2×C22⋊Dic5C2318yes—
55#4072C2×C4164yescomplemented
56C2×C4⋊Dic5—28no—
57C2×C4⋊Dic5C2×C418yes—
58C2×C4⋊Dic5C2×C418yes—
59C2×C4⋊Dic5C2318yes—
60C2×C4⋊Dic5C2314yes—
61C2×C4⋊Dic5C2314yes—
62C2×#683—216no—
63C2×#683C23116yes—
64C2×C4×Dic5—48no—
65C2×C4×Dic5C2×C4116yes—
66C2×C4×Dic5C23116yes—
67C2×C4×Dic5C2318yes—
68C5×#95C2314yes—
69C5×#95C2314yes—
70#3963C2318yes—
71#3963C23116yes—
72#3963C23116yes—
73C22×#117—51no—
74#2347—516no—
75C2×#389—52no—
76C2×#388—54no—
77C2×#383—52no—
78C2×#380—51no—
79C2×C4×C4⋊C4—52no—
80C2×C4×C4⋊C4—52no—
81C2×C4×C22⋊C4D511yescharacteristic
82C24×C5—24no—
83C24×C5—24no—
84C24×C5—22no—
85C24×C5—22no—
86C24×C5C22×C414yes—
87C24×C5C2412yes—
88C24×C5C2411yescharacteristic
89C22×C4×C5—24no—
90C22×C4×C5—24no—
91C22×C4×C5—24no—
92C22×C4×C5—232no—
93C22×C4×C5—216no—
94C22×C4×C5—24no—
95C22×C4×C5—24no—
96C22×C4×C5—24no—
97C22×C4×C5C22×C414yes—
98C22×C4×C5C22×C4116yes—
99C22×C4×C5C22×C414yes—
100C22×C4×C5#4118yes—
101C22×C4×C5#4118yes—
102C22×C4×C5C2414yes—
103C22×C4×C5C2412yes—
104C22×C4×C5C2414yes—
105C22×C4×C5C2412yes—
106C22×Dic5—28no—
107C22×Dic5—24no—
108C22×Dic5—24no—
109C22×Dic5—28no—
110C22×Dic5—28no—
111C22×Dic5—28no—
112C22×Dic5C22×C418yes—
113C22×Dic5C22×C418yes—
114C22×Dic5C2418yes—
115C22×Dic5C2414yes—
116C22×Dic5C2414yes—
117C5×C22⋊C4—216no—
118C42×C5—216no—
119C22⋊Dic5—48no—
120C22⋊Dic5C22×C4132yes—
121C4⋊Dic5—216no—
122C4⋊Dic5C22×C4116yes—
123C4⋊Dic5C22×C4116yes—
124#683—232no—
125C4×Dic5—416no—
126C4×Dic5C22×C4132yes—
127C24×C4C2×D511yescharacteristic
128C2×#117—516no—
129C2×#117—58no—
130C22×C4⋊C4—51no—
131C22×C4⋊C4—51no—
132C22×C4⋊C4—54no—
133C22×C4⋊C4—52no—
134C22×C22⋊C4—52no—
135C22×C22⋊C4—52no—
136C22×C22⋊C4C2×D512yes—
137C22×C42—52no—
138C22×C42—54no—
139C22×C42C2×D512yes—
140#389—516no—
141#388—532no—
142#383—516no—
143#380—58no—
144C4×C4⋊C4—516no—
145C4×C4⋊C4—516no—
146C4×C22⋊C4C2×D518yes—
147C2×#95—52no—
148C2×#95—54no—
149C2×#95—54no—
150C2×#95C2×D511yescharacteristic
151C2×#95C2×D511yescharacteristic
152C23×C5—28no—
153C23×C5—28no—
154C23×C5—28no—
155C23×C5—24no—
156C23×C5—24no—
157C23×C5—22no—
158C23×C5—22no—
159C23×C5—48no—
160C23×C5—48no—
161C23×C5C23×C414yes—
162C23×C5C23×C412yes—
163C23×C5C2×#4114yes—
164C23×C5C2×#4114yes—
165C23×C5H(5,𝔽2)12yes—
166C23×C52−1+412yes—
167C23×C5C2511yescharacteristic
168C2×C4×C5—28no—
169C2×C4×C5—216no—
170C2×C4×C5—24no—
171C2×C4×C5—24no—
172C2×C4×C5—24no—
173C2×C4×C5—232no—
174C2×C4×C5—24no—
175C2×C4×C5—24no—
176C2×C4×C5—24no—
177C2×C4×C5—48no—
178C2×C4×C5#117116yes—
179C2×C4×C5C23×C414yes—
180C2×C4×C5C23×C414yes—
181C2×C4×C5C2×#4118yes—
182C2×C4×C5C2×#4118yes—
183C2×Dic5—216no—
184C2×Dic5—216no—
185C2×Dic5—28no—
186C2×Dic5—24no—
187C2×Dic5—24no—
188C2×Dic5—28no—
189C2×Dic5—28no—
190C2×Dic5—28no—
191C2×Dic5—416no—
192C2×Dic5C23×C418yes—
193C2×Dic5C23×C418yes—
194C25C22×D511yescharacteristic
195C23×C4—28no—
196C23×C4—52no—
197C23×C4—52no—
198C23×C4—54no—
199C23×C4C4×D518yes—
200C23×C4C22×D512yes—
201C23×C4C22×D514yes—
202C23×C4C22×D511yescharacteristic
203C23×C4C22×D512yes—
204C23×C4C22×D511yescharacteristic
205#117—564no—
206C2×C4⋊C4—58no—
207C2×C4⋊C4—58no—
208C2×C4⋊C4—54no—
209C2×C4⋊C4—54no—
210C2×C4⋊C4—58no—
211C2×C4⋊C4—516no—
212C2×C4⋊C4—1016no—
213C2×C4⋊C4—108no—
214C2×C22⋊C4—28no—
215C2×C22⋊C4—516no—
216C2×C22⋊C4—58no—
217C2×C22⋊C4—58no—
218C2×C22⋊C4—204no—
219C2×C22⋊C4C22×D518yes—
220C2×C42—28no—
221C2×C42—516no—
222C2×C42—58no—
223C2×C42—516no—
224C2×C42—208no—
225C2×C42C22×D518yes—
226#95—58no—
227#95—516no—
228#95—516no—
229#95C22×D514yes—
230#95C22×D514yes—
231C22×C5—24no—
232C22×C5—24no—
233C22×C5—24no—
234C22×C5—416no—
235C22×C5—44no—
236C22×C5—44no—
237C22×C5C2×#11714yes—
238C22×C5C4×#4114yes—
239C22×C5#52014yes—
240C22×C5#55318yes—
241C22×C5#55618yes—
242C22×C5C24×C411yescharacteristic
243C22×C5C22×#4112yes—
244C22×C5C22×#4112yes—
245C22×C5C2×H(5,𝔽2)11yescharacteristic
246C22×C5C2×2−1+411yescharacteristic
247C4×C5—24no—
248C4×C5—28no—
249C4×C5—44no—
250C4×C5C2×#11718yes—
251Dic5—28no—
252Dic5—28no—
253Dic5—48no—
254C24—24no—
255C24—24no—
256C24—22no—
257C24—22no—
258C24C2×C4×D514yes—
259C24C23×D512yes—
260C24C23×D511yescharacteristic
261C22×C4—24no—
262C22×C4—24no—
263C22×C4—24no—
264C22×C4—216no—
265C22×C4—232no—
266C22×C4—24no—
267C22×C4—24no—
268C22×C4—24no—
269C22×C4—58no—
270C22×C4—58no—
271C22×C4—54no—
272C22×C4—54no—
273C22×C4—58no—
274C22×C4—108no—
275C22×C4—104no—
276C22×C4—104no—
277C22×C4—108no—
278C22×C4—108no—
279C22×C4—108no—
280C22×C4C2×C4×D514yes—
281C22×C4C2×C4×D5116yes—
282C22×C4C2×C4×D514yes—
283C22×C4#70918yes—
284C22×C4#71118yes—
285C22×C4C23×D514yes—
286C22×C4C23×D512yes—
287C22×C4C23×D514yes—
288C22×C4C23×D512yes—
289C4⋊C4—516no—
290C4⋊C4—516no—
291C4⋊C4—1032no—
292C4⋊C4—1016no—
293C22⋊C4—216no—
294C22⋊C4—532no—
295C22⋊C4—208no—
296C42—216no—
297C42—532no—
298C42—2016no—
299C2×C5—44no—
300C2×C5#234718yescomplemented
301C2×C5C22×#11711yescharacteristic
302C2×C5C2×C4×#4111yescharacteristic
303C2×C5C2×#52011yescharacteristic
304C2×C5C2×#55312yes—
305C2×C5C2×#55612yes—
306C23—28no—
307C23—28no—
308C23—28no—
309C23—24no—
310C23—24no—
311C23—22no—
312C23—22no—
313C23—48no—
314C23—48no—
315C23C22×C4×D514yes—
316C23C22×C4×D512yes—
317C23C2×#70914yes—
318C23C2×#71114yes—
319C23#414912yes—
320C23#415012yes—
321C23C24×D511yescharacteristic
322C2×C4—28no—
323C2×C4—216no—
324C2×C4—24no—
325C2×C4—24no—
326C2×C4—24no—
327C2×C4—232no—
328C2×C4—24no—
329C2×C4—24no—
330C2×C4—24no—
331C2×C4—48no—
332C2×C4—58no—
333C2×C4—58no—
334C2×C4—1016no—
335C2×C4—1016no—
336C2×C4—108no—
337C2×C4—104no—
338C2×C4—104no—
339C2×C4—108no—
340C2×C4—108no—
341C2×C4—108no—
342C2×C4—2016no—
343C2×C4#4023116yes—
344C2×C4C22×C4×D514yes—
345C2×C4C22×C4×D514yes—
346C2×C4C2×#70918yes—
347C2×C4C2×#71118yes—
348C5C2×#234711yescharacteristic, complemented, Sylow
349C22—24no—
350C22—24no—
351C22—24no—
352C22—44no—
353C22—44no—
354C22—416no—
355C22C4×#70914yes—
356C22C2×#402314yes—
357C22#6585714yes—
358C22#6587618yes—
359C22#6589918yes—
360C22C23×C4×D511yescharacteristic
361C22C22×#70912yes—
362C22C22×#71112yes—
363C22C2×#414911yescharacteristic
364C22C2×#415011yescharacteristic
365C4—28no—
366C4—24no—
367C4—44no—
368C4—108no—
369C4—108no—
370C4—208no—
371C4C2×#402318yes—
372C2—44no—
373C2#1060519118yesdirect product factor
374C2C2×C4×#70911yescharacteristic
375C2C22×#402311yescharacteristic
376C2C2×#6585711yescharacteristic
377C2C2×#6589912yes—
378C2C2×#6587612yes—
subgroupG/H
centerC24C23×D5
commutatorC22×C5C24×C4
FrattiniC23C24×D5
FittingC2×C4×C5×C22⋊C4C2
socleC24×C5C24
prime radicalC25×C5C23
cosocleC23×C5C25
hypercenterC2×C4×C22⋊C4D5
hypocenterC5C2×#2347
LCSG ▷ C22×C5 ▷ C5
UCSC1 ◁ C24 ◁ C2×C4×C22⋊C4

Complements

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

NH ≅ G/Ntype
5. C2×#65217C2semidirect
6. C2×C4×C4⋊Dic5C2semidirect
7. #10605191C2direct (N × H)
8. C2×#64940C2semidirect
9. C2×#64938C2semidirect
15. C2×#4072C4semidirect
31. C4×C4⋊Dic5C22semidirect
32. #65217C22semidirect
33. #64940C22semidirect
34. #64938C22semidirect
55. #4072C2×C4semidirect
300. C2×C5#2347semidirect
348. C5C2×#2347semidirect
373. C2#10605191direct (N × H)

Maximal subgroups

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

subgroupindex
C22×#40722
C2×C4×C5×C22⋊C42
C2×#655012
C2×#652342
C2×#652172
C2×C4×C4⋊Dic52
#106051912
C2×#649402
C2×#649382
C2×#23475