Finite Group Databasebeta

← C2×#20287353 #28263652

Subgroups — C2×#20287353 #28263652

Order 13568.

2058 conjugacy classes of subgroups.

The subgroup lattice has 332 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×Q8×D212C211yesmaximal, characteristic
2#20287353C2116yesmaximal, direct product factor
3C2×#16810973C213yesmaximal, complemented
4C2×#16810908C213yesmaximal, complemented
5C2×#16810825C213yesmaximal
6C2×Q8×#88322C211yesmaximal, characteristic, complemented
7C2×C4×#11719924C213yesmaximal
8C2×C53×#557C211yesmaximal, characteristic, complemented
9C22×Q8×D53—21no—
10C22×#11719924C2213yes—
11Q8×D212—24no—
12Q8×D212C2218yes—
13C2×#13133217C2213yes—
14C2×#13133197—23no—
15C2×C4×D212C2213yes—
16#16810973C22124yescomplemented
17#16810908C22124yescomplemented
18#16810825C22124yes—
19Q8×#88322C2218yescomplemented
20C4×#11719924C22124yes—
21C2×#13133148C2213yes—
22C2×C4⋊#88322C2213yes—
23C2×#13133141C2216yes—
24C2×#13133141C2213yes—
25C2×#13133137C2211yescharacteristic
26C2×C4×#88322C2213yes—
27C53×#557—24no—
28C53×#557C2218yescomplemented
29C2×C4⋊Q8×C53C2213yes—
30C2×#125×C53—23no—
31C2×C4×Q8×C53—21no—
32C2×C4×Q8×C53C2213yes—
33C2×C4×Q8×C53C2211yescharacteristic
34C22×Dic106—23no—
35C2×Q8×D53—28no—
36C2×Q8×D53—48no—
37C22×D212C2311yescharacteristic
38C22×C4×D53—23no—
39C2×#11719924—212no—
40C2×#11719924C23112yes—
41C22×#88322C2311yescharacteristic
42C22×#88322C2313yes—
43#13133217—26no—
44#13133217C23112yes—
45#13133197—224no—
46C4×D212—26no—
47C4×D212C23112yes—
48C2×C4⋊Dic53—23no—
49C2×C4⋊Dic53C2313yes—
50C2×#11719921—23no—
51#13133148C23112yes—
52C4⋊#88322C23112yes—
53#13133141C23124yes—
54#13133141C23112yes—
55#13133137C2314yes—
56C4×#88322C23112yes—
57C22×Q8×C53D411yescharacteristic
58C22×Q8×C53C2313yes—
59C4⋊Q8×C53—26no—
60C4⋊Q8×C53C23112yes—
61#125×C53—224no—
62C4×Q8×C53—26no—
63C4×Q8×C53—28no—
64C4×Q8×C53D414yes—
65C4×Q8×C53C2314yes—
66C4×Q8×C53C23112yes—
67C2×C4⋊C4×C53—23no—
68C2×C4⋊C4×C53—23no—
69C2×C4⋊C4×C53—23no—
70C2×C4⋊C4×C53D413yes—
71C2×C4⋊C4×C53C2311yescharacteristic
72C2×C42×C53—23no—
73C2×C42×C53C2313yes—
74C23×D53—21no—
75C2×Q8×C53—23no—
76C2×Q8×C53—26no—
77C2×Q8×C53—26no—
78C2×Q8×C53SD1618yescomplemented
79C2×Q8×C53C2×D414yes—
80C2×Q8×C53C2416yes—
81C22×C4×C53—21no—
82C22×C4×C53—23no—
83C22×C4×C53C2×D413yes—
84C22×C4×C53C2×D413yes—
85C22×C4×C53C2411yescharacteristic
86C22×Dic53—23no—
87Q8×D53—432no—
88C2×D212—21no—
89C2×D212—22no—
90C2×D212#4114yes—
91C2×D212C2412yes—
92C2×C4×D53—212no—
93C2×C4×D53—412no—
94C2×Dic106—212no—
95C2×Dic106—412no—
96C4⋊C4×C53—212no—
97C4⋊C4×C53—212no—
98C4⋊C4×C53—212no—
99C4⋊C4×C53—21no—
100C4⋊C4×C53C2×D4112yes—
101C4⋊C4×C53C2412yes—
102C42×C53—212no—
103C42×C53C2×D416yes—
104C42×C53C2416yes—
105#11719924—224no—
106#11719921—212no—
107C4⋊Dic53—212no—
108C4⋊Dic53—23no—
109C4⋊Dic53C2416yes—
110C2×#88322—26no—
111C2×#88322—22no—
112C2×#88322C2412yes—
113C2×#88322C2416yes—
114C23×C53C22×D411yescharacteristic
115C22×D53—24no—
116C22×D53—22no—
117C22×D53—42no—
118C22×D53—42no—
119Q8×C53—26no—
120Q8×C53—26no—
121Q8×C53—46no—
122Q8×C53C2×SD16116yescomplemented
123C2×C4×C53—26no—
124C2×C4×C53—26no—
125C2×C4×C53—22no—
126C2×C4×C53—46no—
127C2×C4×C53—42no—
128C2×C4×C53C2×SD16112yes—
129C2×C4×C53C22×D416yes—
130C2×C4×C53C22×D411yescharacteristic
131C2×C4×C53C22×D416yes—
132C2×C4×C53C2×#4112yes—
133C2×C4×C532−1+412yes—
134C2×C4×C53C2511yescharacteristic
135C2×Dic53—26no—
136C2×Dic53—46no—
137D212—24no—
138D212—22no—
139D212C2×#4114yes—
140C4×D53—424no—
141Dic106—424no—
142#88322—26no—
143#88322—22no—
144C2×#3359—531nomaximal, Sylow
145C22×C53#54812yes—
146C22×C53C22×SD1612yes—
147C22×C53#57812yes—
148C22×C53C23×D411yescharacteristic
149C2×D53—24no—
150C2×D53—48no—
151C2×D53—42no—
152C2×D53—42no—
153C4×C53—26no—
154C4×C53—46no—
155C4×C53—42no—
156C4×C53#54812yes—
157C4×C53C22×SD16112yes—
158C4×C53C22×#4111yescharacteristic
159C4×C53C2×2−1+411yescharacteristic
160Dic53—46no—
161C2×#557D5311yescharacteristic, complemented
162C2×D4×Q8—531no—
163#3359—5316no—
164C2×#487—533no—
165C2×#475—533no—
166C2×#461—533no—
167C2×C8×Q8—531no—
168C2×C4×SD16—533no—
169C2×C53#335914yescomplemented
170C2×C53C2×#54811yescharacteristic
171C2×C53C23×SD1611yescharacteristic
172C2×C53C2×#57811yescharacteristic
173D53—44no—
174C23×Q8—1061no—
175C22×SD16—533no—
176#557—24no—
177#557C2×D5318yescomplemented
178D4×Q8—538no—
179D4×Q8—1064no—
180C2×C4⋊Q8—533no—
181C2×C4⋊Q8C2×D5313yes—
182C2×#125—23no—
183C2×C22⋊Q8—1063no—
184C2×C4×Q8—21no—
185C2×C4×Q8C2×D5313yes—
186C2×C4×Q8C2×D5311yescharacteristic
187C2×C4×D4—533no—
188#487—5324no—
189#475—5324no—
190#461—5324no—
191C8×Q8—538no—
192C4×SD16—5324no—
193C2×#106—533no—
194C2×C4⋊C8—533no—
195C2×#103—536no—
196C2×#103—533no—
197C2×#102—531no—
198C2×C4×C8—533no—
199C53C2×#335911yescharacteristic, complemented, Sylow
200C22×Q8—1068no—
201C22×Q8—1063no—
202C22×Q8—2128no—
203C22×Q8C22⋊D5311yescharacteristic
204C22×Q8C22×D5313yes—
205C22×D4—531no—
206C23×C4—1063no—
207C2×SD16—5312no—
208C2×SD16—10612no—
209C22×C8—533no—
210C22×C8—531no—
211C4⋊Q8—26no—
212C4⋊Q8—5312no—
213C4⋊Q8—1066no—
214C4⋊Q8C22×D53112yes—
215#125—224no—
216C22⋊Q8—10624no—
217C4×Q8—28no—
218C4×Q8—26no—
219C4×Q8C22⋊D5314yes—
220C4×Q8C22×D53112yes—
221C4×Q8C22×D5314yes—
222C4×D4—5312no—
223C4×D4—1066no—
224C2×C4⋊C4—23no—
225C2×C4⋊C4—23no—
226C2×C4⋊C4—23no—
227C2×C4⋊C4—533no—
228C2×C4⋊C4—1063no—
229C2×C4⋊C4C22⋊D5313yes—
230C2×C4⋊C4C22×D5311yescharacteristic
231C2×C22⋊C4—1063no—
232C2×C42—23no—
233C2×C42C22×D5313yes—
234#106—5312no—
235C4⋊C8—5312no—
236#103—5312no—
237#103—5324no—
238#102—534no—
239C4×C8—5312no—
240C24—1061no—
241C2×Q8—23no—
242C2×Q8—26no—
243C2×Q8—26no—
244C2×Q8—10612no—
245C2×Q8—21232no—
246C2×Q8—21212no—
247C2×Q8#1171992418yescomplemented
248C2×Q8C2×C22⋊D5314yes—
249C2×Q8C23×D5316yes—
250C2×D4—534no—
251C2×D4—532no—
252C2×D4—1061no—
253C2×D4—1062no—
254C22×C4—21no—
255C22×C4—23no—
256C22×C4—1063no—
257C22×C4—10612no—
258C22×C4—21212no—
259C22×C4C2×C22⋊D5313yes—
260C22×C4C2×C22⋊D5313yes—
261C22×C4C23×D5311yescharacteristic
262SD16—10624no—
263C2×C8—532no—
264C2×C8—536no—
265C2×C8—1066no—
266C2×C8—1062no—
267C4⋊C4—212no—
268C4⋊C4—212no—
269C4⋊C4—212no—
270C4⋊C4—21no—
271C4⋊C4—536no—
272C4⋊C4—1063no—
273C4⋊C4—10612no—
274C4⋊C4C2×C22⋊D53112yes—
275C4⋊C4C23×D5312yes—
276C22⋊C4—10612no—
277C42—212no—
278C42C2×C22⋊D5316yes—
279C42C23×D5316yes—
280C23—1064no—
281C23—1062no—
282C23—2122no—
283C23—2122no—
284C23C22×C22⋊D5311yescharacteristic
285Q8—26no—
286Q8—26no—
287Q8—46no—
288Q8—21224no—
289Q8C2×#11719924116yescomplemented
290D4—534no—
291D4—1064no—
292D4—1062no—
293C2×C4—26no—
294C2×C4—22no—
295C2×C4—26no—
296C2×C4—46no—
297C2×C4—42no—
298C2×C4—1066no—
299C2×C4—2126no—
300C2×C4—21224no—
301C2×C4C2×#11719924112yes—
302C2×C4C22×C22⋊D5316yes—
303C2×C4C22×C22⋊D5311yescharacteristic
304C2×C4C22×C22⋊D5316yes—
305C2×C4#41×D5312yes—
306C2×C4#1313326412yes—
307C2×C4C24×D5311yescharacteristic
308C8—1066no—
309C8—1062no—
310C22—1064no—
311C22—2128no—
312C22—2122no—
313C22—2122no—
314C22#1681128012yes—
315C22C22×#1171992412yes—
316C22#1681143812yes—
317C22C23×C22⋊D5311yescharacteristic
318C4—26no—
319C4—46no—
320C4—42no—
321C4—2126no—
322C4#1681128012yes—
323C4C22×#11719924112yes—
324C4C2×#41×D5311yescharacteristic
325C4C2×#1313326411yescharacteristic
326C2—2124no—
327C2#2028735314yesdirect product factor
328C2C2×#1681128011yescharacteristic
329C2C23×#1171992411yescharacteristic
330C2C2×#1681143811yescharacteristic
subgroupG/H
centerC23C22×C22⋊D53
commutator, cosocleC2×C4×C53C25
FrattiniC2×C4C24×D53
FittingC2×C53×#557C2
socleC23×C53C22×D4
prime radicalC22×D212C23
hypercenterC2×#557D53
hypocenterC53C2×#3359
LCSG ▷ C2×C4×C53 ▷ C2×C53 ▷ C53
UCSC1 ◁ C23 ◁ C2×C4×Q8 ◁ C2×#557

Complements

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

NH ≅ G/Ntype
2. #20287353C2direct (N × H)
3. C2×#16810973C2semidirect
4. C2×#16810908C2semidirect
6. C2×Q8×#88322C2semidirect
8. C2×C53×#557C2semidirect
16. #16810973C22semidirect
17. #16810908C22semidirect
19. Q8×#88322C22semidirect
28. C53×#557C22semidirect
78. C2×Q8×C53SD16semidirect
122. Q8×C53C2×SD16semidirect
161. C2×#557D53semidirect
169. C2×C53#3359semidirect
177. #557C2×D53semidirect
199. C53C2×#3359semidirect
247. C2×Q8#11719924semidirect
289. Q8C2×#11719924semidirect
327. C2#20287353direct (N × H)

Maximal subgroups

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

subgroupindex
C2×Q8×D2122
#202873532
C2×#168109732
C2×#168109082
C2×#168108252
C2×Q8×#883222
C2×C4×#117199242
C2×C53×#5572
C2×#335953