Finite Group Databasebeta

← C2×#20289964 #28314020

Subgroups — C2×#20289964 #28314020

Order 13568.

1718 conjugacy classes of subgroups.

The subgroup lattice has 484 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#16811594C212yesmaximal
2#20289964C218yesmaximal, direct product factor
3C2×#16811469C212yesmaximal, complemented
4C2×#16811468C212yesmaximal, complemented
5C22×C4⋊C4×D53C211yesmaximal, characteristic
6C22×#11719936—22no—
7#16811594—24no—
8#16811594C2218yes—
9C2×#13133274—22no—
10C2×C22⋊#88324C2214yes—
11#16811469C2218yescomplemented
12#16811468C2218yescomplemented
13C23×C4×D53C411yescharacteristic
14C23×C4×D53C411yescharacteristic
15C23×C4×D53C2211yescharacteristic
16C22×C4⋊Dic53C2211yescharacteristic
17C22×#11719919C411yescharacteristic
18C22×#11719919C2211yescharacteristic
19C2×C4⋊C4×D53—22no—
20C2×C4⋊C4×D53—22no—
21C2×C4⋊C4×D53—24no—
22C2×C4⋊C4×D53C414yes—
23C2×C4⋊C4×D53C2214yes—
24C22×C4⋊C4×C53C411yescharacteristic
25C2×#11719936—28no—
26C2×#11719936—48no—
27C22×#88324—24no—
28#13133274—28no—
29C22⋊#88324—24no—
30C22⋊#88324C2318yes—
31C24×D53C2×C411yescharacteristic
32C23×Dic53C2×C411yescharacteristic
33C23×Dic53C2×C411yescharacteristic
34C23×Dic53C2311yescharacteristic
35C22×C4×D53—22no—
36C22×C4×D53—21no—
37C22×C4×D53—21no—
38C22×C4×D53—24no—
39C22×C4×D53—24no—
40C22×C4×D53—22no—
41C22×C4×D53—21no—
42C22×C4×D53—21no—
43C22×C4×D53—44no—
44C22×C4×D53—44no—
45C22×C4×D53C2×C414yes—
46C22×C4×D53C2×C414yes—
47C22×C4×D53C2×C412yes—
48C22×C4×D53D412yes—
49C22×C4×D53D412yes—
50C22×C4×D53D414yes—
51C22×C4×D53C2312yes—
52C2×C4⋊Dic53—22no—
53C2×C4⋊Dic53—22no—
54C2×C4⋊Dic53C2×C412yes—
55C2×C4⋊Dic53C2312yes—
56C2×#11719919—22no—
57C2×#11719919—22no—
58C2×#11719919—22no—
59C2×#11719919—22no—
60C2×#11719919C2×C412yes—
61C2×#11719919C2×C414yes—
62C2×#11719919C2312yes—
63C4⋊C4×D53—28no—
64C4⋊C4×D53—28no—
65C4⋊C4×D53—28no—
66C4⋊C4×D53—28no—
67C23×C4×C53C2×C411yescharacteristic
68C23×C4×C53C2×C411yescharacteristic
69C23×C4×C53C2×C411yescharacteristic
70C2×C4⋊C4×C53—22no—
71C2×C4⋊C4×C53—22no—
72C2×C4⋊C4×C53C2×C414yes—
73C24×C53C22×C411yescharacteristic
74C23×D53—22no—
75C23×D53—22no—
76C23×D53—24no—
77C23×D53—24no—
78C23×D53C22⋊C412yes—
79C23×D53C22⋊C411yescharacteristic
80C23×D53C22⋊C411yescharacteristic
81C23×D53C22×C412yes—
82C22×C4×C53—21no—
83C22×C4×C53—21no—
84C22×C4×C53—22no—
85C22×C4×C53—21no—
86C22×C4×C53—21no—
87C22×C4×C53—42no—
88C22×C4×C53—42no—
89C22×C4×C53C22⋊C412yes—
90C22×C4×C53C22⋊C412yes—
91C22×C4×C53M1614yescomplemented
92C22×C4×C53C22×C412yes—
93C22×C4×C53C22×C412yes—
94C22×C4×C53C22×C412yes—
95C22×Dic53—21no—
96C22×Dic53—21no—
97C22×Dic53—22no—
98C22×Dic53—21no—
99C22×Dic53—21no—
100C22×Dic53—42no—
101C22×Dic53—42no—
102C22×Dic53M1614yescomplemented
103C22×Dic53C22×C412yes—
104C22×Dic53C22×C412yes—
105C22×Dic53C22×C411yescharacteristic
106C22×Dic53C2×D412yes—
107C22×Dic53C2×D412yes—
108C22×Dic53C2411yescharacteristic
109C2×C4×D53—216no—
110C2×C4×D53—28no—
111C2×C4×D53—28no—
112C2×C4×D53—28no—
113C2×C4×D53—24no—
114C2×C4×D53—24no—
115C2×C4×D53—28no—
116C2×C4×D53—28no—
117C2×C4×D53—24no—
118C2×C4×D53—24no—
119C2×C4×D53—44no—
120C2×C4×D53—48no—
121C2×C4×D53—44no—
122C2×C4×D53—44no—
123C2×C4×D53—44no—
124C2×C4×D53—48no—
125C2×C4×D53—44no—
126C2×C4×D53—48no—
127C2×C4×D53C22⋊C414yes—
128C2×C4×D53C22⋊C414yes—
129C2×C4×D53C2×D414yes—
130C2×C4×D53C2×D414yes—
131C2×#88324—28no—
132C2×#88324—48no—
133#11719936—416no—
134C4⋊C4×C53—28no—
135C4⋊Dic53—24no—
136C4⋊Dic53—24no—
137#11719919—28no—
138#11719919—24no—
139#11719919—24no—
140C23×C53—22no—
141C23×C53—22no—
142C23×C53C2×C22⋊C411yescharacteristic
143C23×C53C2×C22⋊C411yescharacteristic
144C23×C53C2×M1612yes—
145C23×C53C2×M1612yes—
146C23×C53C23×C411yescharacteristic
147C22×D53—216no—
148C22×D53—216no—
149C22×D53—22no—
150C22×D53—22no—
151C22×D53—24no—
152C22×D53—22no—
153C22×D53—22no—
154C22×D53—22no—
155C22×D53—22no—
156C22×D53—24no—
157C22×D53—24no—
158C22×D53—24no—
159C22×D53—24no—
160C22×D53#9914yes—
161C22×D53#10114yes—
162C22×D53C2×C22⋊C412yes—
163C22×D53C2×C22⋊C412yes—
164C2×C4×C53—22no—
165C2×C4×C53—22no—
166C2×C4×C53—24no—
167C2×C4×C53—24no—
168C2×C4×C53—24no—
169C2×C4×C53—22no—
170C2×C4×C53—42no—
171C2×C4×C53—42no—
172C2×C4×C53—42no—
173C2×C4×C53—42no—
174C2×C4×C53—42no—
175C2×C4×C53C2×C22⋊C414yes—
176C2×C4×C53C2×M1614yescomplemented
177C2×Dic53—22no—
178C2×Dic53—22no—
179C2×Dic53—24no—
180C2×Dic53—24no—
181C2×Dic53—24no—
182C2×Dic53—22no—
183C2×Dic53—41no—
184C2×Dic53—42no—
185C2×Dic53—41no—
186C2×Dic53—42no—
187C2×Dic53—42no—
188C2×Dic53—42no—
189C2×Dic53C2×C22⋊C412yes—
190C2×Dic53C2×M1614yescomplemented
191C2×Dic53C22×D412yes—
192C4×D53—216no—
193C4×D53—216no—
194C4×D53—28no—
195C4×D53—28no—
196C4×D53—44no—
197C4×D53—44no—
198C4×D53—48no—
199C4×D53—48no—
200#88324—48no—
201C2×#1463—531nomaximal, Sylow
202C22×C53—21no—
203C22×C53—21no—
204C22×C53—21no—
205C22×C53—21no—
206C22×C53—24no—
207C22×C53—24no—
208C22×C53#40712yes—
209C22×C53#40712yes—
210C22×C53C2×#9912yes—
211C22×C53C2×#10112yes—
212C22×C53C22×C22⋊C411yescharacteristic
213C22×C53C22×M1611yescharacteristic
214C22×C53C22×M1611yescharacteristic
215C2×D53—28no—
216C2×D53—28no—
217C2×D53—216no—
218C2×D53—22no—
219C2×D53—22no—
220C2×D53—22no—
221C2×D53—22no—
222C2×D53—28no—
223C2×D53—28no—
224C2×D53C2×#9914yes—
225C2×D53C2×#10114yes—
226C4×C53—24no—
227C4×C53—24no—
228C4×C53—42no—
229Dic53—24no—
230Dic53—22no—
231Dic53—22no—
232Dic53—42no—
233C23×C4⋊C4—531no—
234C2×#407—532no—
235#1463—538no—
236C2×#324—532no—
237C2×#324—532no—
238C2×C53—21no—
239C2×C53—21no—
240C2×C53—22no—
241C2×C53#146314yescomplemented
242C2×C53C2×#40711yescharacteristic
243C2×C53C22×#9911yescharacteristic
244C2×C53C22×#10111yescharacteristic
245D53—24no—
246D53—24no—
247C24×C4—531no—
248C24×C4—531no—
249C24×C4—531no—
250C22×M16—1062no—
251C22×C4⋊C4—531no—
252C22×C4⋊C4—531no—
253C22×C4⋊C4—531no—
254C22×C4⋊C4—534no—
255C22×C4⋊C4—534no—
256C22×C4⋊C4—1062no—
257C22×C4⋊C4—1062no—
258C22×C4⋊C4—1064no—
259C22×C4⋊C4#615711yescharacteristic
260#407—538no—
261#407—1064no—
262C2×C22⋊C8—534no—
263C2×C22⋊C8—1062no—
264#324—538no—
265#324—538no—
266C53C2×#146311yescharacteristic, complemented, Sylow
267C25—531no—
268C23×C4—531no—
269C23×C4—531no—
270C23×C4—532no—
271C23×C4—532no—
272C23×C4—534no—
273C23×C4—532no—
274C23×C4—532no—
275C23×C4—531no—
276C23×C4—534no—
277C23×C4—534no—
278C23×C4—1064no—
279C23×C4—1062no—
280C23×C4—1061no—
281C23×C4—1061no—
282C23×C4—1062no—
283C23×C4—1061no—
284C23×C4—1061no—
285C23×C4—1064no—
286C23×C4—2124no—
287C23×C4—2124no—
288C23×C4C2×#615711yescharacteristic
289C23×C4C2×#615711yescharacteristic
290C23×C4C2×#615711yescharacteristic
291C2×M16—1068no—
292C2×M16—2128no—
293C22×C8—1064no—
294C2×C4⋊C4—22no—
295C2×C4⋊C4—22no—
296C2×C4⋊C4—532no—
297C2×C4⋊C4—532no—
298C2×C4⋊C4—532no—
299C2×C4⋊C4—534no—
300C2×C4⋊C4—532no—
301C2×C4⋊C4—1068no—
302C2×C4⋊C4—1068no—
303C2×C4⋊C4—1068no—
304C2×C4⋊C4—1068no—
305C2×C4⋊C4—1062no—
306C2×C4⋊C4—1062no—
307C2×C4⋊C4—1062no—
308C2×C4⋊C4—1062no—
309C2×C4⋊C4—1062no—
310C2×C4⋊C4—1062no—
311C2×C4⋊C4C2×#615714yes—
312C22⋊C8—538no—
313C22⋊C8—1064no—
314C22⋊C8—1068no—
315C24—531no—
316C24—531no—
317C24—532no—
318C24—532no—
319C24—1064no—
320C24—1064no—
321C24—1062no—
322C24—1062no—
323C24C22×#615711yescharacteristic
324C22×C4—21no—
325C22×C4—21no—
326C22×C4—22no—
327C22×C4—21no—
328C22×C4—21no—
329C22×C4—42no—
330C22×C4—42no—
331C22×C4—532no—
332C22×C4—534no—
333C22×C4—532no—
334C22×C4—532no—
335C22×C4—531no—
336C22×C4—531no—
337C22×C4—534no—
338C22×C4—534no—
339C22×C4—534no—
340C22×C4—534no—
341C22×C4—532no—
342C22×C4—1068no—
343C22×C4—1064no—
344C22×C4—1064no—
345C22×C4—1068no—
346C22×C4—10616no—
347C22×C4—1068no—
348C22×C4—1068no—
349C22×C4—1064no—
350C22×C4—1064no—
351C22×C4—1061no—
352C22×C4—1061no—
353C22×C4—1062no—
354C22×C4—1068no—
355C22×C4—1061no—
356C22×C4—1061no—
357C22×C4—2124no—
358C22×C4—2128no—
359C22×C4—2124no—
360C22×C4—2124no—
361C22×C4—2124no—
362C22×C4—2128no—
363C22×C4—2124no—
364C22×C4—2128no—
365C22×C4—2122no—
366C22×C4—2122no—
367C22×C4#1171993614yescomplemented
368C22×C4C22⋊#615712yes—
369C22×C4C22⋊#615712yes—
370C22×C4C22×#615712yes—
371C22×C4C22×#615712yes—
372C22×C4C22×#615712yes—
373M16—21216no—
374C2×C8—1068no—
375C2×C8—2128no—
376C4⋊C4—28no—
377C4⋊C4—1064no—
378C4⋊C4—1064no—
379C4⋊C4—1064no—
380C4⋊C4—1064no—
381C4⋊C4—1068no—
382C23—22no—
383C23—22no—
384C23—534no—
385C23—534no—
386C23—532no—
387C23—532no—
388C23—1064no—
389C23—1064no—
390C23—1064no—
391C23—1064no—
392C23—10616no—
393C23—10616no—
394C23—1064no—
395C23—1062no—
396C23—1062no—
397C23—1062no—
398C23—1062no—
399C23—1062no—
400C23—1062no—
401C23C2×C22⋊#615711yescharacteristic
402C23C2×C22⋊#615711yescharacteristic
403C23D53⋊M1612yes—
404C23C2×#1171993612yes—
405C23C23×#615711yescharacteristic
406C2×C4—24no—
407C2×C4—24no—
408C2×C4—24no—
409C2×C4—22no—
410C2×C4—22no—
411C2×C4—22no—
412C2×C4—42no—
413C2×C4—42no—
414C2×C4—42no—
415C2×C4—42no—
416C2×C4—42no—
417C2×C4—532no—
418C2×C4—532no—
419C2×C4—534no—
420C2×C4—1068no—
421C2×C4—1068no—
422C2×C4—10616no—
423C2×C4—10616no—
424C2×C4—1064no—
425C2×C4—1064no—
426C2×C4—1064no—
427C2×C4—1062no—
428C2×C4—1062no—
429C2×C4—1062no—
430C2×C4—2124no—
431C2×C4—2124no—
432C2×C4—2128no—
433C2×C4—2128no—
434C2×C4—2121no—
435C2×C4—2122no—
436C2×C4—2121no—
437C2×C4—2122no—
438C2×C4—2122no—
439C2×C4—2122no—
440C2×C4C2×C22⋊#615714yes—
441C2×C4C2×#1171993614yescomplemented
442C8—2128no—
443C22—24no—
444C22—24no—
445C22—21no—
446C22—21no—
447C22—21no—
448C22—21no—
449C22—534no—
450C22—534no—
451C22—10616no—
452C22—1068no—
453C22—1068no—
454C22—1068no—
455C22—1068no—
456C22—1062no—
457C22—1062no—
458C22—1062no—
459C22—1062no—
460C22#1681159412yes—
461C22#1681159412yes—
462C22D53⋊#9912yes—
463C22D53⋊#10112yes—
464C22C22×C22⋊#615711yescharacteristic
465C22C2×D53⋊M1611yescharacteristic
466C22C22×#1171993611yescharacteristic
467C4—24no—
468C4—24no—
469C4—42no—
470C4—1064no—
471C4—1062no—
472C4—1062no—
473C4—2122no—
474C2—22no—
475C2—21no—
476C2—21no—
477C2—1064no—
478C2—1064no—
479C2#2028996414yesdirect product factor
480C2C2×#1681159411yescharacteristic
481C2C2×D53⋊#9911yescharacteristic
482C2C2×D53⋊#10111yescharacteristic
subgroupG/H
centerC23C2×C22⋊#6157
commutatorC23×C53C23×C4
FrattiniC23C23×#6157
FittingC22×C4⋊C4×C53C4
socleC23×C53C2×C22⋊C4
prime radicalC24×D53C2×C4
cosocleC22×Dic53C24
hypercenterC22×C4⋊C4#6157
hypocenterC53C2×#1463
LCSG ▷ C23×C53 ▷ C2×C53 ▷ C53
UCSC1 ◁ C23 ◁ C23×C4 ◁ C22×C4⋊C4

Complements

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

NH ≅ G/Ntype
2. #20289964C2direct (N × H)
3. C2×#16811469C2semidirect
4. C2×#16811468C2semidirect
11. #16811469C22semidirect
12. #16811468C22semidirect
91. C22×C4×C53M16semidirect
102. C22×Dic53M16semidirect
176. C2×C4×C53C2×M16semidirect
190. C2×Dic53C2×M16semidirect
241. C2×C53#1463semidirect
266. C53C2×#1463semidirect
367. C22×C4#11719936semidirect
441. C2×C4C2×#11719936semidirect
479. C2#20289964direct (N × H)

Maximal subgroups

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

subgroupindex
C2×#168115942
#202899642
C2×#168114692
C2×#168114682
C22×C4⋊C4×D532
C2×#146353