Finite Group Databasebeta

← C2×#20290849 #28320313

Subgroups — C2×#20290849 #28320313

Order 13568.

1878 conjugacy classes of subgroups.

The subgroup lattice has 396 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#16811627C211yesmaximal, characteristic, complemented
2C22×#13133281C214yesmaximal, complemented
3C2×C42×#6157C211yesmaximal, characteristic, complemented
4#20290849C218yesmaximal, direct product factor
5C2×#16811171C211yesmaximal, characteristic
6C22×#11719936C2212yes—
7C22×C4×#6157—22no—
8C22×C4×#6157C2212yes—
9#16811627—22no—
10#16811627C2214yescomplemented
11C2×C4⋊#88324—22no—
12C2×#13133281—216no—
13C2×#13133281C22116yescomplemented
14C42×#6157—22no—
15C42×#6157C2214yescomplemented
16C22×#11719947C2212yes—
17#16811171—22no—
18#16811171—22no—
19#16811171C414yescomplemented
20#16811171C2214yes—
21C2×#13133217C411yescharacteristic, complemented
22C2×#13133204C412yescomplemented
23C2×#13133188—22no—
24C2×D4×Dic53—22no—
25C2×C42×D53C2211yescharacteristic
26C2×#127×C53C411yescharacteristic, complemented
27C23×#6157—22no—
28C2×#11719936—28no—
29C2×#11719936—22no—
30C2×#11719936C2314yes—
31C22×#88324—22no—
32C2×C4×#6157—216no—
33C2×C4×#6157—22no—
34C2×C4×#6157—28no—
35C2×C4×#6157—24no—
36C2×C4×#6157D418yescomplemented
37C2×C4×#6157C2314yes—
38C4⋊#88324—28no—
39#13133281—232no—
40C2×#11719947—28no—
41C2×#11719947—22no—
42C2×#11719947—22no—
43C2×#11719947—28no—
44C2×#11719947C2×C414yes—
45C2×#11719947C2314yes—
46C22×Dic106C2×C412yes—
47C22×C22⋊D53—22no—
48C23×Dic53—22no—
49C22×C4×D53D412yes—
50C22×C4×D53C2311yescharacteristic
51#13133217—22no—
52#13133217C2×C414yescomplemented
53#13133204—24no—
54#13133204C2×C418yescomplemented
55#13133188—28no—
56#13133188—28no—
57D4×Dic53—28no—
58D4×Dic53—28no—
59C2×C4⋊Dic53—22no—
60C2×C22⋊Dic53—24no—
61C2×C4×Dic53C2×C411yescharacteristic
62C2×C4×Dic53D412yes—
63C42×D53C2×C412yes—
64C42×D53D412yes—
65C42×D53D412yes—
66C42×D53C2312yes—
67C22×D4×C53—22no—
68C22×D4×C53C2×C412yes—
69#127×C53—22no—
70#127×C53C2×C414yescomplemented
71C2×C42×C53C2×C411yescharacteristic
72C24×C53—22no—
73C23×D53C2×D411yescharacteristic
74C22×#6157—216no—
75C22×#6157—28no—
76C22×#6157—24no—
77C2×D4×C53—22no—
78C2×D4×C53—28no—
79C2×D4×C53—28no—
80C2×D4×C53—48no—
81C2×D4×C53C22×C414yes—
82C22×C4×C53C22⋊C412yes—
83C22×C4×C53C22×C411yescharacteristic
84C2×C22⋊D53—24no—
85C2×C22⋊D53—24no—
86C2×C22⋊D53—48no—
87C22×Dic53—22no—
88C22×Dic53—24no—
89C22×Dic53—24no—
90C22×Dic53—44no—
91C22×Dic53—44no—
92C22×Dic53C22⋊C412yes—
93C22×Dic53C2×D411yescharacteristic
94#11719947—216no—
95#11719947—28no—
96#11719947—28no—
97#11719947—48no—
98#11719947—44no—
99#11719947—44no—
100C2×C4×D53—28no—
101C2×C4×D53C22⋊C414yes—
102C2×C4×D53C22×C411yescharacteristic
103C2×C4×D53C2×D414yes—
104C2×C4×D53C2×D414yes—
105C2×C4×D53C2×D411yescharacteristic
106C2×C4×D53C2×D411yescharacteristic
107C2×C4×D53#4114yes—
108C2×C4×D53C2411yescharacteristic
109C2×Dic106—28no—
110C2×Dic106—22no—
111C2×Dic106C22×C414yes—
112C2×#88324—24no—
113C2×#88324—44no—
114C4×#6157—232no—
115C4×#6157—28no—
116C4×#6157—24no—
117C4×#6157C2×D418yescomplemented
118#11719936—28no—
119#11719936—44no—
120C42×C53C22⋊C412yes—
121C42×C53C22×C412yes—
122C22⋊Dic53—216no—
123C4⋊Dic53—28no—
124C4×Dic53C22⋊C414yes—
125C4×Dic53C22⋊C412yes—
126C4×Dic53C22×C412yes—
127C4×Dic53C2×D414yes—
128C23×C53—24no—
129C23×C53—44no—
130C23×C53—48no—
131C23×C53C2×C22⋊C411yescharacteristic
132C22×D53—24no—
133C22×D53C2×C22⋊C411yescharacteristic
134C22×D53#12114yes—
135C22×D53C22×D411yescharacteristic
136C2×#6157—216no—
137C2×#6157—232no—
138C2×#6157—28no—
139D4×C53—28no—
140D4×C53—44no—
141D4×C53—416no—
142C2×C4×C53#10418yescomplemented
143C2×C4×C53C2×C22⋊C411yescharacteristic
144C2×C4×C53C2×C22⋊C414yes—
145C2×C4×C53C4×D414yes—
146C2×C4×C53C23×C411yescharacteristic
147C22⋊D53—48no—
148C22⋊D53—48no—
149C2×Dic53—24no—
150C2×Dic53—24no—
151C2×Dic53—44no—
152C2×Dic53—44no—
153C2×Dic53—44no—
154C2×Dic53—48no—
155C2×Dic53C2×C22⋊C414yes—
156C2×Dic53C2×C22⋊C411yescharacteristic
157C2×Dic53#12412yes—
158C2×Dic53#12712yes—
159C2×Dic53C22×D411yescharacteristic
160C4×D53—28no—
161C4×D53—28no—
162C4×D53C4×D414yes—
163C4×D53#12114yes—
164C4×D53#12412yes—
165C4×D53#12712yes—
166C4×D53C22×D412yes—
167C4×D53C2×#4112yes—
168Dic106—28no—
169Dic106—44no—
170#88324—44no—
171C2×#1962—531nomaximal, Sylow
172C22×C53—48no—
173C22×C53—44no—
174C22×C53—88no—
175C22×C53#39012yes—
176C22×C53C2×#10414yes—
177C22×C53C22×C22⋊C411yescharacteristic
178C2×D53—22no—
179C2×D53—22no—
180C2×D53—28no—
181C2×D53#39012yes—
182C2×D53C2×#12112yes—
183#6157—216no—
184C4×C53#39012yes—
185C4×C53C2×#10418yescomplemented
186C4×C53C2×C4×D412yes—
187Dic53—24no—
188Dic53—44no—
189Dic53#39012yes—
190Dic53C2×#12411yescharacteristic
191Dic53C2×#12711yescharacteristic
192C2×#532—531no—
193C2×#423—531no—
194C22×#104—534no—
195C2×C43—531no—
196#1962—538no—
197C2×C53—84no—
198C2×C53#196214yescomplemented
199C2×C53C2×#39011yescharacteristic
200C2×C53C22×#10412yes—
201D53—22no—
202D53—22no—
203C22×#41—532no—
204C22×M16—532no—
205#532—534no—
206#532—534no—
207#532—1062no—
208#532—1062no—
209C2×C4⋊Q8—531no—
210C2×#127#615711yescharacteristic, complemented
211C2×#124—532no—
212C2×#121—1062no—
213C2×C4×D4—1062no—
214C22×C42—531no—
215C22×C42—532no—
216C22×C42—1062no—
217#423—534no—
218#423—1062no—
219C2×C4⋊C8—1062no—
220C2×#104—5316no—
221C2×#104—10616no—
222C43—534no—
223C43—1062no—
224C53C2×#196211yescharacteristic, complemented, Sylow
225C2×#41—534no—
226C2×#41—534no—
227C2×#41—1062no—
228C2×#41—1062no—
229C2×#41—1068no—
230C2×#41—1068no—
231C22×Q8—532no—
232C22×D4—22no—
233C22×D4—1062no—
234C22×D4C2×#615712yes—
235C23×C4—531no—
236C23×C4—532no—
237C23×C4—1062no—
238C23×C4—1062no—
239C2×M16—534no—
240C2×M16—1062no—
241C2×M16—1068no—
242C22×C8—1062no—
243C4⋊Q8—534no—
244C4⋊Q8—1062no—
245#127—22no—
246#127C2×#615714yescomplemented
247#124—538no—
248#124—1064no—
249#121—1068no—
250#121—1068no—
251C4×D4—1068no—
252C4×D4—1068no—
253C2×C4⋊C4—1062no—
254C2×C22⋊C4—1064no—
255C2×C42—532no—
256C2×C42—531no—
257C2×C42—532no—
258C2×C42—532no—
259C2×C42—532no—
260C2×C42—532no—
261C2×C42—534no—
262C2×C42—538no—
263C2×C42—1068no—
264C2×C42—1062no—
265C2×C42—1064no—
266C2×C42—10616no—
267C2×C42C2×#615711yescharacteristic
268C4⋊C8—1068no—
269#104—10632no—
270C24—22no—
271C24—531no—
272#41—1068no—
273#41—10616no—
274#41—1068no—
275#41—2128no—
276#41—2124no—
277#41—2124no—
278C2×Q8—534no—
279C2×Q8—1062no—
280C2×Q8—1068no—
281C2×D4—28no—
282C2×D4—22no—
283C2×D4—28no—
284C2×D4—48no—
285C2×D4—1064no—
286C2×D4—1064no—
287C2×D4—2128no—
288C2×D4C22×#615714yes—
289C22×C4—531no—
290C22×C4—531no—
291C22×C4—531no—
292C22×C4—534no—
293C22×C4—534no—
294C22×C4—531no—
295C22×C4—531no—
296C22×C4—532no—
297C22×C4—534no—
298C22×C4—534no—
299C22×C4—1064no—
300C22×C4—1064no—
301C22×C4—1064no—
302C22×C4—1068no—
303C22×C4—10616no—
304C22×C4—1068no—
305C22×C4—1062no—
306C22×C4—2124no—
307C22×C4—2124no—
308C22×C4C22⋊#615712yes—
309C22×C4C22×#615711yescharacteristic
310M16—1068no—
311M16—2124no—
312C2×C8—1064no—
313C2×C8—2124no—
314C4⋊C4—1068no—
315C22⋊C4—10616no—
316C42—534no—
317C42—534no—
318C42—532no—
319C42—532no—
320C42—538no—
321C42—1068no—
322C42—1064no—
323C42—10632no—
324C42C22⋊#615712yes—
325C42C22×#615712yes—
326C23—24no—
327C23—44no—
328C23—48no—
329C23—534no—
330C23—531no—
331C23—531no—
332C23—1064no—
333C23C2×C22⋊#615711yescharacteristic
334Q8—1068no—
335Q8—2124no—
336D4—28no—
337D4—44no—
338D4—416no—
339D4—2128no—
340D4—2128no—
341C2×C4—532no—
342C2×C4—532no—
343C2×C4—531no—
344C2×C4—531no—
345C2×C4—534no—
346C2×C4—532no—
347C2×C4—532no—
348C2×C4—534no—
349C2×C4—532no—
350C2×C4—532no—
351C2×C4—534no—
352C2×C4—1068no—
353C2×C4—10616no—
354C2×C4—10632no—
355C2×C4—1064no—
356C2×C4—1068no—
357C2×C4—1068no—
358C2×C4—1064no—
359C2×C4—2124no—
360C2×C4—2124no—
361C2×C4—2128no—
362C2×C4—2124no—
363C2×C4#1313328118yescomplemented
364C2×C4C2×C22⋊#615711yescharacteristic
365C2×C4C2×C22⋊#615714yes—
366C2×C4D4×#615714yes—
367C2×C4C23×#615711yescharacteristic
368C8—2124no—
369C22—48no—
370C22—44no—
371C22—88no—
372C22—532no—
373C22—532no—
374C22—1062no—
375C22—1062no—
376C22—1068no—
377C22#1681157512yes—
378C22C2×#1313328114yes—
379C22C22×C22⋊#615711yescharacteristic
380C4—531no—
381C4—531no—
382C4—532no—
383C4—10616no—
384C4—1064no—
385C4—2124no—
386C4#1681157512yes—
387C4C2×#1313328118yescomplemented
388C4C2×D4×#615712yes—
389C2—84no—
390C2—1062no—
391C2—1062no—
392C2#2029084914yesdirect product factor
393C2C2×#1681157511yescharacteristic
394C2C22×#1313328112yes—
subgroupG/H
centerC23C2×C22⋊#6157
commutatorC2×C4×C53C23×C4
FrattiniC2×C4C23×#6157
FittingC2×#127×C53C4
socleC23×C53C2×C22⋊C4
prime radicalC2×#16811171C2
cosocleC2×C4×D53C24
hypercenterC2×#127#6157
hypocenterC53C2×#1962
LCSG ▷ C2×C4×C53 ▷ C2×C53 ▷ C53
UCSC1 ◁ C23 ◁ C2×C42 ◁ C2×#127

Complements

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

NH ≅ G/Ntype
1. C2×#16811627C2semidirect
2. C22×#13133281C2semidirect
3. C2×C42×#6157C2semidirect
4. #20290849C2direct (N × H)
10. #16811627C22semidirect
13. C2×#13133281C22semidirect
15. C42×#6157C22semidirect
19. #16811171C4semidirect
21. C2×#13133217C4semidirect
22. C2×#13133204C4semidirect
26. C2×#127×C53C4semidirect
36. C2×C4×#6157D4semidirect
52. #13133217C2×C4semidirect
54. #13133204C2×C4semidirect
70. #127×C53C2×C4semidirect
117. C4×#6157C2×D4semidirect
142. C2×C4×C53#104semidirect
185. C4×C53C2×#104semidirect
198. C2×C53#1962semidirect
210. C2×#127#6157semidirect
224. C53C2×#1962semidirect
246. #127C2×#6157semidirect
363. C2×C4#13133281semidirect
387. C4C2×#13133281semidirect
392. C2#20290849direct (N × H)

Maximal subgroups

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

subgroupindex
C2×#168116272
C22×#131332812
C2×C42×#61572
#202908492
C2×#168111712
C2×#196253