Finite Group Databasebeta

← C2×#10615319 #12980201

Subgroups — C2×#10615319 #12980201

Order 1280.

2238 conjugacy classes of subgroups.

The subgroup lattice has 340 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×C5×#550C211yesmaximal, characteristic
2C2×#65904C211yesmaximal, characteristic, complemented
3#10615319C2116yesmaximal, direct product factor
4C2×#65364C213yesmaximal, complemented
5C2×#65365C213yesmaximal, complemented
6C2×Q8×#168C211yesmaximal, characteristic, complemented
7C2×#65308C213yesmaximal, complemented
8C2×C4×#687C213yesmaximal, complemented
9C22×C5×#41—21no—
10C5×#550—24no—
11C5×#550C2218yes—
12C2×C5×#127C2213yes—
13C2×C5×#121—23no—
14C2×C4×C5×Q8C2211yescharacteristic
15C2×C4×C5×D4C2213yes—
16C2×Q8×Dic5—21no—
17C22×#687C2213yes—
18C22×#687C2216yes—
19#65904—24no—
20#65904C2218yescomplemented
21C2×#4036—23no—
22C2×#4015C2213yes—
23C2×C4×Dic10C2213yes—
24C2×#3964C2213yes—
25#65365C22124yescomplemented
26#65364C22124yescomplemented
27Q8×#168C2218yescomplemented
28#65308C22124yescomplemented
29C4×#687C22124yescomplemented
30C2×#3942C2213yes—
31C2×#3940C2211yescharacteristic
32C2×C4⋊#168C2213yes—
33C2×C4×#168C2213yes—
34C2×#3392—51nomaximal, Sylow
35C2×C5×#41—28no—
36C2×C5×#41—48no—
37C22×C5×Q8D411yescharacteristic
38C22×C5×D4—23no—
39C22×C5×D4—23no—
40C22×C5×D4C2313yes—
41C23×C4×C5—23no—
42C22×Dic10C2313yes—
43C5×#127—26no—
44C5×#127C23112yes—
45C5×#121—224no—
46C4×C5×Q8D414yes—
47C4×C5×Q8C2314yes—
48C4×C5×D4—26no—
49C4×C5×D4C23112yes—
50C2×C5×C4⋊C4D413yes—
51C2×C5×C4⋊C4C2311yescharacteristic
52C2×C5×C22⋊C4—23no—
53C2×C42×C5C2313yes—
54Q8×Dic5—28no—
55C2×#687—212no—
56C2×#687—412no—
57C2×#687C23112yes—
58C2×#687C23124yes—
59C2×C4⋊Dic5—23no—
60C2×C4⋊Dic5—23no—
61C2×C4⋊Dic5C2311yescharacteristic
62C2×#683—23no—
63C2×C4×Dic5—23no—
64C22×#168C2313yes—
65C22×#168C2311yescharacteristic
66#4036—224no—
67#4015—26no—
68#4015C23112yes—
69C4×Dic10—26no—
70C4×Dic10C23112yes—
71#3964C23112yes—
72#3942C23112yes—
73#3940C2314yes—
74C4⋊#168C23112yes—
75C4×#168C23112yes—
76C2×#557—51no—
77C2×#550D511yescharacteristic
78#3392—516no—
79C2×#492—53no—
80C2×#460—53no—
81C2×#461—53no—
82C2×C8×Q8—51no—
83C2×C4×SD16—53no—
84C24×C5—23no—
85C5×#41—432no—
86C2×C5×Q8SD1618yescomplemented
87C2×C5×Q8C2×D414yes—
88C2×C5×D4—23no—
89C2×C5×D4—212no—
90C2×C5×D4—212no—
91C2×C5×D4—26no—
92C2×C5×D4—26no—
93C2×C5×D4—412no—
94C2×C5×D4—86no—
95C2×C5×D4C2416yes—
96C22×C4×C5—21no—
97C22×C4×C5—212no—
98C22×C4×C5—412no—
99C22×C4×C5C2×D413yes—
100C22×C4×C5C2×D413yes—
101C22×C4×C5C2411yescharacteristic
102C22×Dic5—23no—
103C22×Dic5—21no—
104C2×Dic10—26no—
105C2×Dic10—26no—
106C2×Dic10—23no—
107C2×Dic10C2416yes—
108C5×C4⋊C4—21no—
109C5×C4⋊C4C2×D4112yes—
110C5×C4⋊C4C2412yes—
111C5×C22⋊C4—212no—
112C42×C5C2×D416yes—
113C42×C5C2416yes—
114#687—224no—
115#687—424no—
116C4⋊Dic5—21no—
117C4⋊Dic5—212no—
118C4⋊Dic5—212no—
119C4⋊Dic5C2412yes—
120#683—212no—
121C4×Dic5—212no—
122C2×#168—26no—
123C2×#168#4114yescomplemented
124C2×#168C2416yes—
125C2×#168C2412yes—
126C22×#41—21no—
127C22×SD16—53no—
128C22×SD16—56no—
129#557—58no—
130#557—104no—
131#550—24no—
132#550C2×D518yes—
133C2×C4⋊Q8—53no—
134C2×#127C2×D513yes—
135C2×#125—103no—
136C2×#121—23no—
137C2×C4×Q8—53no—
138C2×C4×Q8—101no—
139C2×C4×Q8C2×D511yescharacteristic
140C2×C4×D4C2×D513yes—
141#492—524no—
142#461—524no—
143#460—524no—
144C8×Q8—58no—
145C4×SD16—524no—
146C2×#106—51no—
147C2×C4⋊C8—53no—
148C2×#103—53no—
149C2×#102—53no—
150C2×C4×C8—53no—
151C23×C5—26no—
152C23×C5—46no—
153C23×C5—46no—
154C23×C5—46no—
155C23×C5C22×D411yescharacteristic
156C5×Q8C2×SD16116yescomplemented
157C5×D4—26no—
158C5×D4—26no—
159C5×D4—46no—
160C5×D4—424no—
161C5×D4—812no—
162C2×C4×C5—26no—
163C2×C4×C5—22no—
164C2×C4×C5—42no—
165C2×C4×C5—424no—
166C2×C4×C5C2×SD16112yes—
167C2×C4×C5C22×D416yes—
168C2×C4×C5C22×D411yescharacteristic
169C2×C4×C5C22×D416yes—
170C2×C4×C5C2×#4112yes—
171C2×C4×C5H(5,𝔽2)12yes—
172C2×C4×C5C2511yescharacteristic
173C2×Dic5—22no—
174C2×Dic5—26no—
175C2×Dic5—46no—
176C2×Dic5—42no—
177Dic10—26no—
178Dic10—26no—
179Dic10—46no—
180#168—26no—
181#168C2×#4114yescomplemented
182C2×#41—28no—
183C2×#41—48no—
184C22×Q8—53no—
185C22×Q8C22⋊D511yescharacteristic
186C22×D4—23no—
187C22×D4—23no—
188C22×D4C22×D513yes—
189C23×C4—23no—
190C2×SD16—512no—
191C2×SD16—524no—
192C2×SD16—1012no—
193C2×SD16—2012no—
194C22×C8—51no—
195C22×C8—53no—
196C4⋊Q8—512no—
197C4⋊Q8—106no—
198#127—26no—
199#127C22×D5112yes—
200#125—1024no—
201#121—224no—
202C4×Q8—512no—
203C4×Q8—108no—
204C4×Q8—106no—
205C4×Q8C22⋊D514yes—
206C4×Q8C22×D514yes—
207C4×D4—26no—
208C4×D4C22×D5112yes—
209C2×C4⋊C4—51no—
210C2×C4⋊C4—103no—
211C2×C4⋊C4—103no—
212C2×C4⋊C4—103no—
213C2×C4⋊C4C22⋊D513yes—
214C2×C4⋊C4C22×D511yescharacteristic
215C2×C22⋊C4—23no—
216C2×C42—103no—
217C2×C42C22×D513yes—
218#106—54no—
219C4⋊C8—512no—
220#103—512no—
221#102—512no—
222C4×C8—512no—
223C22×C5—46no—
224C22×C5—46no—
225C22×C5—46no—
226C22×C5—812no—
227C22×C5#55012yes—
228C22×C5C22×SD1612yes—
229C22×C5#57712yes—
230C22×C5C23×D411yescharacteristic
231C4×C5—26no—
232C4×C5—42no—
233C4×C5#55012yes—
234C4×C5C22×SD16112yes—
235C4×C5C22×#4111yescharacteristic
236C4×C5C2×H(5,𝔽2)11yescharacteristic
237Dic5—46no—
238Dic5—42no—
239C24—23no—
240#41—432no—
241C2×Q8—56no—
242C2×Q8—103no—
243C2×Q8—106no—
244C2×Q8—106no—
245C2×Q8#68718yescomplemented
246C2×Q8C2×C22⋊D514yes—
247C2×D4—26no—
248C2×D4—26no—
249C2×D4—23no—
250C2×D4—212no—
251C2×D4—212no—
252C2×D4—412no—
253C2×D4—86no—
254C2×D4C23×D516yes—
255C22×C4—21no—
256C22×C4—212no—
257C22×C4—412no—
258C22×C4—101no—
259C22×C4—103no—
260C22×C4C2×C22⋊D513yes—
261C22×C4C2×C22⋊D513yes—
262C22×C4C23×D511yescharacteristic
263SD16—1024no—
264SD16—2024no—
265C2×C8—54no—
266C2×C8—52no—
267C2×C8—56no—
268C2×C8—106no—
269C4⋊C4—21no—
270C4⋊C4—52no—
271C4⋊C4—101no—
272C4⋊C4—1012no—
273C4⋊C4—1012no—
274C4⋊C4—1012no—
275C4⋊C4C2×C22⋊D5112yes—
276C4⋊C4C23×D512yes—
277C22⋊C4—212no—
278C42—1012no—
279C42C2×C22⋊D516yes—
280C42C23×D516yes—
281C2×C5—86no—
282C2×C5#339214yescomplemented
283C2×C5C2×#55011yescharacteristic
284C2×C5C23×SD1611yescharacteristic
285C2×C5C2×#57711yescharacteristic
286C23—26no—
287C23—46no—
288C23—46no—
289C23—46no—
290C23C22×C22⋊D511yescharacteristic
291Q8—106no—
292Q8—106no—
293Q8—206no—
294Q8C2×#687116yescomplemented
295D4—26no—
296D4—26no—
297D4—424no—
298D4—46no—
299D4—812no—
300C2×C4—26no—
301C2×C4—22no—
302C2×C4—42no—
303C2×C4—424no—
304C2×C4—102no—
305C2×C4—106no—
306C2×C4—206no—
307C2×C4—202no—
308C2×C4C2×#687112yes—
309C2×C4D5×#4112yes—
310C2×C4#414912yes—
311C2×C4C22×C22⋊D516yes—
312C2×C4C22×C22⋊D516yes—
313C2×C4C22×C22⋊D511yescharacteristic
314C2×C4C24×D511yescharacteristic
315C8—54no—
316C8—106no—
317C5C2×#339211yescharacteristic, complemented, Sylow
318C22—46no—
319C22—46no—
320C22—46no—
321C22—812no—
322C22C22×#68712yes—
323C22#6616812yes—
324C22#6617012yes—
325C22C23×C22⋊D511yescharacteristic
326C4—26no—
327C4—42no—
328C4—206no—
329C4—202no—
330C4C22×#687112yes—
331C4#6616812yes—
332C4C2×D5×#4111yescharacteristic
333C4C2×#414911yescharacteristic
334C2—86no—
335C2#1061531914yesdirect product factor
336C2C23×#68711yescharacteristic
337C2C2×#6616811yescharacteristic
338C2C2×#6617011yescharacteristic
subgroupG/H
centerC23C22×C22⋊D5
commutator, cosocleC2×C4×C5C25
FrattiniC2×C4C24×D5
Fitting, prime radicalC2×C5×#550C2
socleC23×C5C22×D4
hypercenterC2×#550D5
hypocenterC5C2×#3392
LCSG ▷ C2×C4×C5 ▷ C2×C5 ▷ C5
UCSC1 ◁ C23 ◁ C2×C4×Q8 ◁ C2×#550

Complements

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

NH ≅ G/Ntype
2. C2×#65904C2semidirect
3. #10615319C2direct (N × H)
4. C2×#65364C2semidirect
5. C2×#65365C2semidirect
6. C2×Q8×#168C2semidirect
7. C2×#65308C2semidirect
8. C2×C4×#687C2semidirect
20. #65904C22semidirect
25. #65365C22semidirect
26. #65364C22semidirect
27. Q8×#168C22semidirect
28. #65308C22semidirect
29. C4×#687C22semidirect
86. C2×C5×Q8SD16semidirect
123. C2×#168#41semidirect
156. C5×Q8C2×SD16semidirect
181. #168C2×#41semidirect
245. C2×Q8#687semidirect
282. C2×C5#3392semidirect
294. Q8C2×#687semidirect
317. C5C2×#3392semidirect
335. C2#10615319direct (N × H)

Maximal subgroups

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

subgroupindex
C2×C5×#5502
C2×#659042
#106153192
C2×#653642
C2×#653652
C2×Q8×#1682
C2×#653082
C2×C4×#6872
C2×#33925