Finite Group Databasebeta

← C22×#16810827 #28250873

Subgroups — C22×#16810827 #28250873

Order 13568.

3222 conjugacy classes of subgroups.

The subgroup lattice has 366 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C23×#11719913C211yesmaximal, characteristic
2C23×#11719924C211yesmaximal, characteristic
3C22×#13133215C211yesmaximal, characteristic, complemented
4C22×#13133179C211yesmaximal, characteristic
5C2×#16810827C2124yesmaximal, direct product factor
6C22×#13133146C211yesmaximal, characteristic, complemented
7C22×#13133140C211yesmaximal, characteristic, complemented
8C22×#103×C53C211yesmaximal, characteristic, complemented
9C23×Dic106C2211yescharacteristic
10C23×D212C2211yescharacteristic
11C23×C4×D53—21no—
12C22×#11719913—28no—
13C22×#11719913C22112yes—
14C22×#11719924—28no—
15C22×#11719924C22112yes—
16C23×#88322C2211yescharacteristic
17C2×#13133215—26no—
18C2×#13133215C22112yescomplemented
19C2×#13133179—26no—
20C2×#13133179C22112yes—
21C22×#11719919—21no—
22C22×#11719919—21no—
23C22×#11719921—21no—
24C22×C4×Dic53C2211yescharacteristic
25#16810827C22164yesdirect product factor
26C2×#13133146C22112yescomplemented
27C2×#13133140C22112yescomplemented
28C23×Q8×C53C2211yescharacteristic
29C23×C8×C53C2211yescharacteristic
30C22×C4⋊C4×C53C2211yescharacteristic
31C2×#103×C53C22112yescomplemented
32C24×D53—21no—
33C22×Dic106—24no—
34C22×Dic106—24no—
35C22×Dic106—23no—
36C22×Dic106C2316yes—
37C22×D212—23no—
38C22×D212—24no—
39C22×D212D418yes—
40C22×D212C2316yes—
41C23×Dic53—21no—
42C23×Dic53D411yescharacteristic
43C23×Dic53D411yescharacteristic
44C22×C4×D53—212no—
45C22×C4×D53—48no—
46C2×#11719913—248no—
47C2×#11719913C23116yes—
48C2×#11719924—248no—
49C2×#11719924C23116yes—
50C22×#88322—24no—
51C22×#88322C2316yes—
52#13133215—224no—
53#13133215C23116yescomplemented
54#13133179—224no—
55#13133179C23116yes—
56C2×#11719919—212no—
57C2×#11719919—212no—
58C2×#11719921—212no—
59C2×C4×Dic53D416yes—
60C2×C4×Dic53C2316yes—
61#13133146C23116yescomplemented
62#13133140C23116yescomplemented
63C22×Q8×C53—23no—
64C22×Q8×C53—24no—
65C22×Q8×C53—24no—
66C22×Q8×C53C2316yes—
67C23×C4×C53—21no—
68C23×C4×C53—21no—
69C23×C4×C53C2311yescharacteristic
70C22×C8×C53—24no—
71C22×C8×C53C2316yes—
72C2×C4⋊C4×C53—23no—
73C2×C4⋊C4×C53C2316yes—
74#103×C53C23116yescomplemented
75C24×C53C2×D411yescharacteristic
76C23×D53—28no—
77C23×D53—26no—
78C23×D53—44no—
79C23×D53—44no—
80C2×Q8×C53—26no—
81C2×Q8×C53—212no—
82C2×Q8×C53—212no—
83C2×Q8×C53—412no—
84C2×Q8×C53C2414yes—
85C22×C4×C53—26no—
86C22×C4×C53—26no—
87C22×C4×C53—44no—
88C22×C4×C53—44no—
89C22×C4×C53C2×D413yes—
90C22×C4×C53C2×D414yes—
91C22×C4×C53#4114yes—
92C22×C4×C53C2413yes—
93C22×Dic53—24no—
94C22×Dic53—26no—
95C22×Dic53—44no—
96C22×Dic53SD1618yescomplemented
97C22×Dic53C2×D416yes—
98C22×Dic53C2×D416yes—
99C2×D212—224no—
100C2×D212—26no—
101C2×D212—212no—
102C2×D212C2×D4124yes—
103C2×D212C2414yes—
104C2×C4×D53—216no—
105C2×C4×D53—448no—
106C2×Dic106—212no—
107C2×Dic106—212no—
108C2×Dic106—26no—
109C2×Dic106—412no—
110C2×Dic106C2414yes—
111C2×C8×C53—212no—
112C2×C8×C53C2414yes—
113C4⋊C4×C53—26no—
114C4⋊C4×C53C2414yes—
115#11719924—232no—
116#11719921—216no—
117#11719919—216no—
118#11719919—216no—
119C4×Dic53C2×D4112yes—
120C4×Dic53C2414yes—
121C2×#88322—212no—
122C2×#88322C2414yes—
123#11719913—232no—
124C23×C53#12114yes—
125C23×C53C2×SD1614yes—
126C23×C53#13714yes—
127C23×C53C22×D413yes—
128C22×D53—224no—
129C22×D53—24no—
130C22×D53—412no—
131C22×D53—412no—
132C22×D53—432no—
133Q8×C53—24no—
134Q8×C53—24no—
135Q8×C53—412no—
136C2×C4×C53—24no—
137C2×C4×C53—24no—
138C2×C4×C53—412no—
139C2×C4×C53—412no—
140C2×C4×C53#121112yes—
141C2×C4×C53C22×D413yes—
142C2×C4×C53C22×D416yes—
143C2×C4×C53C2×#4116yes—
144C2×C4×C53C2511yescharacteristic
145C2×Dic53—212no—
146C2×Dic53—24no—
147C2×Dic53—412no—
148C2×Dic53C2×SD16124yescomplemented
149C2×Dic53C22×D414yes—
150C2×Dic53C22×D414yes—
151D212—224no—
152D212—24no—
153D212C22×D418yes—
154C4×D53—432no—
155Dic106—24no—
156Dic106—24no—
157Dic106—412no—
158C8×C53—24no—
159#88322—24no—
160C22×#461—531nomaximal, Sylow
161C22×C53#461116yescomplemented
162C22×C53C2×#12116yes—
163C22×C53C22×SD1616yes—
164C22×C53C2×#13716yes—
165C22×C53C23×D411yescharacteristic
166C2×D53—28no—
167C2×D53—448no—
168C2×D53—44no—
169C2×D53—44no—
170C4×C53—44no—
171C4×C53—44no—
172C4×C53C2×#12116yes—
173C4×C53C23×D411yescharacteristic
174C4×C53C22×#4111yescharacteristic
175Dic53—24no—
176Dic53—44no—
177Dic53C22×SD1618yescomplemented
178C23×SD16—531no—
179C23×SD16—531no—
180C22×C4⋊Q8—531no—
181C22×C4×D4—531no—
182C2×#461—5324no—
183C22×C4⋊C8—531no—
184C22×#103—531no—
185C22×#103D5311yescharacteristic, complemented
186C2×C53C2×#461112yescomplemented
187C2×C53C22×#12111yescharacteristic
188C2×C53C23×SD1611yescharacteristic
189C2×C53C22×#13711yescharacteristic
190D53—48no—
191C23×Q8—531no—
192C23×Q8C2×D5311yescharacteristic
193C23×D4—531no—
194C24×C4—1061no—
195C22×SD16—5312no—
196C22×SD16—5312no—
197C22×SD16—1068no—
198C22×SD16—1068no—
199C23×C8—531no—
200C23×C8C2×D5311yescharacteristic
201C2×C4⋊Q8—5312no—
202C2×C4⋊Q8—1066no—
203C2×C4×D4—5312no—
204C2×C4×D4—1066no—
205C22×C4⋊C4—1061no—
206C22×C4⋊C4—1061no—
207C22×C4⋊C4C2×D5311yescharacteristic
208C22×C22⋊C4—1061no—
209C22×C42—531no—
210#461—5364no—
211C2×C4⋊C8—5312no—
212C2×#103—5312no—
213C2×#103C2×D53112yescomplemented
214C53C22×#46111yescharacteristic, complemented, Sylow
215C25—1061no—
216C22×Q8—23no—
217C22×Q8—24no—
218C22×Q8—24no—
219C22×Q8—536no—
220C22×Q8—1063no—
221C22×Q8—1064no—
222C22×Q8—1064no—
223C22×Q8C22×D5316yes—
224C22×D4—538no—
225C22×D4—536no—
226C22×D4—1064no—
227C22×D4—1063no—
228C23×C4—21no—
229C23×C4—21no—
230C23×C4—531no—
231C23×C4—531no—
232C23×C4—10612no—
233C23×C4—1061no—
234C23×C4—2128no—
235C23×C4C22×D5311yescharacteristic
236C2×SD16—5316no—
237C2×SD16—5316no—
238C2×SD16—10648no—
239C2×SD16—10648no—
240C22×C8—24no—
241C22×C8—536no—
242C22×C8—1064no—
243C22×C8C22×D5316yes—
244C4⋊Q8—5316no—
245C4⋊Q8—10624no—
246C4×D4—5316no—
247C4×D4—10624no—
248C2×C4⋊C4—23no—
249C2×C4⋊C4—10612no—
250C2×C4⋊C4—10612no—
251C2×C4⋊C4C22×D5316yes—
252C2×C22⋊C4—10612no—
253C2×C42—536no—
254C2×C42—536no—
255C4⋊C8—5316no—
256#103—5316no—
257#103C22×D53116yescomplemented
258C24—1068no—
259C24—1066no—
260C24—2124no—
261C24—2124no—
262C24D4×D5311yescharacteristic
263C2×Q8—26no—
264C2×Q8—212no—
265C2×Q8—212no—
266C2×Q8—412no—
267C2×Q8—534no—
268C2×Q8—1066no—
269C2×Q8—10612no—
270C2×Q8—10612no—
271C2×Q8—21212no—
272C2×Q8C23×D5314yes—
273C2×D4—5324no—
274C2×D4—534no—
275C2×D4—10624no—
276C2×D4—1066no—
277C2×D4—10612no—
278C22×C4—26no—
279C22×C4—26no—
280C22×C4—44no—
281C22×C4—44no—
282C22×C4—538no—
283C22×C4—536no—
284C22×C4—536no—
285C22×C4—10616no—
286C22×C4—1064no—
287C22×C4—1066no—
288C22×C4—21248no—
289C22×C4—2124no—
290C22×C4#1171994514yes—
291C22×C4D4×D5313yes—
292C22×C4D4×D5314yes—
293C22×C4C23×D5313yes—
294SD16—10632no—
295SD16—10632no—
296C2×C8—212no—
297C2×C8—534no—
298C2×C8—10612no—
299C2×C8C23×D5314yes—
300C4⋊C4—26no—
301C4⋊C4—10616no—
302C4⋊C4—10616no—
303C4⋊C4C23×D5314yes—
304C22⋊C4—10616no—
305C42—5312no—
306C42—534no—
307C23—10624no—
308C23—1064no—
309C23—21232no—
310C23—21212no—
311C23—21212no—
312C23#1313319014yes—
313C23SD16×D5314yes—
314C23#1313324914yes—
315C23C2×D4×D5313yes—
316Q8—24no—
317Q8—24no—
318Q8—412no—
319Q8—1064no—
320Q8—1064no—
321Q8—21212no—
322D4—538no—
323D4—10624no—
324D4—1064no—
325C2×C4—24no—
326C2×C4—24no—
327C2×C4—412no—
328C2×C4—412no—
329C2×C4—5324no—
330C2×C4—534no—
331C2×C4—534no—
332C2×C4—10612no—
333C2×C4—1064no—
334C2×C4—21232no—
335C2×C4—21212no—
336C2×C4#13133190112yes—
337C2×C4C2×D4×D5313yes—
338C2×C4C2×D4×D5316yes—
339C2×C4C2×#1171994516yes—
340C2×C4C24×D5311yescharacteristic
341C8—24no—
342C8—1064no—
343C22—1068no—
344C22—2124no—
345C22—2124no—
346C22—21248no—
347C22#16810827116yesdirect product factor
348C22C2×#1313319016yes—
349C22C2×SD16×D5316yes—
350C22C2×#1313324916yes—
351C22C22×D4×D5311yescharacteristic
352C4—44no—
353C4—44no—
354C4—538no—
355C4—1064no—
356C4—2124no—
357C4C2×#1313319016yes—
358C4C22×D4×D5311yescharacteristic
359C4C22×#1171994511yescharacteristic
360C2—2128no—
361C2C2×#16810827112yesdirect product factor
362C2C22×#1313319011yescharacteristic
363C2C22×SD16×D5311yescharacteristic
364C2C22×#1313324911yescharacteristic
subgroupG/H
centerC24D4×D53
commutator, cosocleC2×C4×C53C25
FrattiniC2×C4C24×D53
FittingC22×#103×C53C2
socleC24×C53C2×D4
prime radicalC23×D212C22
hypercenterC22×#103D53
hypocenterC53C22×#461
LCSG ▷ C2×C4×C53 ▷ C2×C53 ▷ C53
UCSC1 ◁ C24 ◁ C23×C4 ◁ C22×#103

Complements

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

NH ≅ G/Ntype
3. C22×#13133215C2semidirect
5. C2×#16810827C2direct (N × H)
6. C22×#13133146C2semidirect
7. C22×#13133140C2semidirect
8. C22×#103×C53C2semidirect
18. C2×#13133215C22semidirect
25. #16810827C22direct (N × H)
26. C2×#13133146C22semidirect
27. C2×#13133140C22semidirect
31. C2×#103×C53C22semidirect
53. #13133215C23semidirect
61. #13133146C23semidirect
62. #13133140C23semidirect
74. #103×C53C23semidirect
96. C22×Dic53SD16semidirect
148. C2×Dic53C2×SD16semidirect
161. C22×C53#461semidirect
177. Dic53C22×SD16semidirect
185. C22×#103D53semidirect
186. C2×C53C2×#461semidirect
213. C2×#103C2×D53semidirect
214. C53C22×#461semidirect
257. #103C22×D53semidirect
347. C22#16810827direct (N × H)
361. C2C2×#16810827direct (N × H)

Maximal subgroups

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

subgroupindex
C23×#117199132
C23×#117199242
C22×#131332152
C22×#131331792
C2×#168108272
C22×#131331462
C22×#131331402
C22×#103×C532
C22×#46153