Finite Group Databasebeta

← C2×#20289987 #28314129

Subgroups — C2×#20289987 #28314129

Order 13568.

1002 conjugacy classes of subgroups.

The subgroup lattice has 466 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#16811643C211yesmaximal, characteristic
2C2×#16811639C211yesmaximal, characteristic
3#20289987C218yesmaximal, direct product factor
4C2×#16811469C212yesmaximal
5C2×#16811468C212yesmaximal
6C22×#13133207C211yesmaximal, characteristic
7#16811643—22no—
8#16811643C2214yes—
9#16811639—22no—
10#16811639C2214yes—
11C2×#13133284—21no—
12C2×C4⋊#88324—21no—
13C2×C22⋊#88324C2212yes—
14C2×C22⋊#88324C2212yes—
15C2×#13133272—21no—
16C2×C4×#88324—21no—
17#16811469C2218yes—
18#16811468C2218yes—
19C2×#13133207—21no—
20C2×#13133207—21no—
21C2×#13133207—21no—
22C2×#13133207—21no—
23C2×#13133207—22no—
24C2×#13133207—22no—
25C2×#13133207C414yes—
26C2×#13133207C2214yes—
27C22×C4⋊Dic53C2211yescharacteristic
28C22×#11719919C411yescharacteristic
29C22×#11719919C411yescharacteristic
30C22×#11719919C411yescharacteristic
31C22×#11719919C2211yescharacteristic
32C22×C4×Dic53C2211yescharacteristic
33C22×C4⋊C4×C53C411yescharacteristic
34C22×#88324—22no—
35C22×#88324—22no—
36#13133284—24no—
37C4⋊#88324—24no—
38C22⋊#88324—22no—
39C22⋊#88324—22no—
40C22⋊#88324C2314yes—
41C22⋊#88324C2314yes—
42#13133272—24no—
43C4×#88324—24no—
44C23×Dic53C2×C411yescharacteristic
45C23×Dic53C2×C411yescharacteristic
46C23×Dic53C2×C411yescharacteristic
47C23×Dic53C2311yescharacteristic
48#13133207—24no—
49#13133207—24no—
50#13133207—24no—
51#13133207—24no—
52#13133207—24no—
53#13133207—24no—
54#13133207—24no—
55#13133207—24no—
56C2×C4⋊Dic53—21no—
57C2×C4⋊Dic53—21no—
58C2×C4⋊Dic53—21no—
59C2×C4⋊Dic53—21no—
60C2×C4⋊Dic53C2×C412yes—
61C2×C4⋊Dic53C2312yes—
62C2×#11719919—22no—
63C2×#11719919—21no—
64C2×#11719919—21no—
65C2×#11719919—22no—
66C2×#11719919—21no—
67C2×#11719919—21no—
68C2×#11719919—22no—
69C2×#11719919—21no—
70C2×#11719919—21no—
71C2×#11719919—21no—
72C2×#11719919—21no—
73C2×#11719919—21no—
74C2×#11719919—21no—
75C2×#11719919C2×C414yes—
76C2×#11719919C2×C414yes—
77C2×#11719919C2×C414yes—
78C2×#11719919C2×C412yes—
79C2×#11719919C2312yes—
80C2×C4×Dic53C2×C412yes—
81C2×C4×Dic53D411yescharacteristic
82C2×C4×Dic53D411yescharacteristic
83C2×C4×Dic53D411yescharacteristic
84C2×C4×Dic53D411yescharacteristic
85C2×C4×Dic53D412yes—
86C2×C4×Dic53D412yes—
87C2×C4×Dic53C2312yes—
88C23×C4×C53C2×C411yescharacteristic
89C23×C4×C53C2×C411yescharacteristic
90C23×C4×C53C2×C411yescharacteristic
91C2×C4⋊C4×C53—22no—
92C2×C4⋊C4×C53—21no—
93C2×C4⋊C4×C53—21no—
94C2×C4⋊C4×C53C2×C414yes—
95C24×C53C22×C411yescharacteristic
96C22×C4×C53—24no—
97C22×C4×C53—21no—
98C22×C4×C53—21no—
99C22×C4×C53—21no—
100C22×C4×C53—21no—
101C22×C4×C53—24no—
102C22×C4×C53—42no—
103C22×C4×C53C22⋊C412yes—
104C22×C4×C53C22⋊C411yescharacteristic
105C22×C4×C53C22⋊C411yescharacteristic
106C22×C4×C53C22×C412yes—
107C22×C4×C53C22×C412yes—
108C22×C4×C53C22×C412yes—
109C22×Dic53—24no—
110C22×Dic53—21no—
111C22×Dic53—21no—
112C22×Dic53—21no—
113C22×Dic53—21no—
114C22×Dic53—24no—
115C22×Dic53—42no—
116C22×Dic53—42no—
117C22×Dic53C22⋊C411yescharacteristic
118C22×Dic53C22⋊C412yes—
119C22×Dic53C22⋊C411yescharacteristic
120C22×Dic53C22×C412yes—
121C22×Dic53C22×C412yes—
122C22×Dic53C22×C412yes—
123C22×Dic53C22×C411yescharacteristic
124C22×Dic53C2×D411yescharacteristic
125C22×Dic53C2×D411yescharacteristic
126C22×Dic53C2×D411yescharacteristic
127C22×Dic53C2×D411yescharacteristic
128C22×Dic53C2411yescharacteristic
129C2×#88324—24no—
130C2×#88324—24no—
131C2×#88324—44no—
132C2×#88324—44no—
133C4⋊C4×C53—24no—
134C4⋊C4×C53—24no—
135C4⋊Dic53—22no—
136C4⋊Dic53—22no—
137C4⋊Dic53—22no—
138C4⋊Dic53—22no—
139#11719919—24no—
140#11719919—24no—
141#11719919—24no—
142#11719919—22no—
143#11719919—22no—
144#11719919—24no—
145#11719919—24no—
146#11719919—24no—
147#11719919—22no—
148#11719919—22no—
149C4×Dic53C22⋊C412yes—
150C4×Dic53C22⋊C412yes—
151C4×Dic53C22⋊C412yes—
152C4×Dic53C22⋊C412yes—
153C4×Dic53C2×D412yes—
154C4×Dic53C2×D412yes—
155C4×Dic53C2×D412yes—
156C4×Dic53C2×D412yes—
157C23×C53—22no—
158C23×C53—22no—
159C23×C53C2×C22⋊C411yescharacteristic
160C23×C53C2×C22⋊C411yescharacteristic
161C23×C53#13114yes—
162C23×C53C23×C411yescharacteristic
163C2×C4×C53—24no—
164C2×C4×C53—24no—
165C2×C4×C53—22no—
166C2×C4×C53—22no—
167C2×C4×C53—22no—
168C2×C4×C53—22no—
169C2×C4×C53—24no—
170C2×C4×C53—42no—
171C2×C4×C53—42no—
172C2×C4×C53—42no—
173C2×C4×C53—42no—
174C2×C4×C53—42no—
175C2×C4×C53C2×C22⋊C412yes—
176C2×C4×C53C2×C22⋊C412yes—
177C2×Dic53—22no—
178C2×Dic53—22no—
179C2×Dic53—22no—
180C2×Dic53—24no—
181C2×Dic53—24no—
182C2×Dic53—22no—
183C2×Dic53—22no—
184C2×Dic53—24no—
185C2×Dic53—41no—
186C2×Dic53—41no—
187C2×Dic53—41no—
188C2×Dic53—41no—
189C2×Dic53—42no—
190C2×Dic53—42no—
191C2×Dic53—42no—
192C2×Dic53—42no—
193C2×Dic53—42no—
194C2×Dic53#10114yes—
195C2×Dic53C2×C22⋊C412yes—
196C2×Dic53C2×C22⋊C411yescharacteristic
197C2×Dic53C2×C22⋊C411yescharacteristic
198C2×Dic53C2×C22⋊C412yes—
199C2×Dic53C22×D411yescharacteristic
200C2×Dic53C22×D411yescharacteristic
201#88324—44no—
202#88324—44no—
203C2×#1470—531nomaximal, Sylow
204C22×C53—21no—
205C22×C53—21no—
206C22×C53—21no—
207C22×C53—21no—
208C22×C53—28no—
209C22×C53#40812yes—
210C22×C53#40812yes—
211C22×C53#41012yes—
212C22×C53C2×#10112yes—
213C22×C53C22×C22⋊C411yescharacteristic
214C22×C53C2×#13112yes—
215C4×C53—22no—
216C4×C53—22no—
217C4×C53—42no—
218C4×C53—42no—
219Dic53—21no—
220Dic53—21no—
221Dic53—21no—
222Dic53—21no—
223Dic53—22no—
224Dic53—42no—
225Dic53—42no—
226Dic53C2×#10114yes—
227C22×#125—531no—
228C2×#432—531no—
229C2×#431—531no—
230#1470—538no—
231C2×#324—532no—
232C2×#324—532no—
233C2×C53—22no—
234C2×C53—22no—
235C2×C53#147014yescomplemented
236C2×C53C2×#40811yescharacteristic
237C2×C53C2×#41011yescharacteristic
238C2×C53C22×#10111yescharacteristic
239C2×#125—534no—
240C2×#125—534no—
241C2×#125—1061no—
242C2×#125—1061no—
243C2×#125—1061no—
244C2×#125—1061no—
245C2×#125—1062no—
246C2×#125—1062no—
247C22×C4⋊C4—531no—
248C22×C4⋊C4—531no—
249C22×C4⋊C4—531no—
250C22×C4⋊C4—531no—
251C22×C4⋊C4—531no—
252C22×C4⋊C4#615711yescharacteristic
253C22×C42—531no—
254#432—534no—
255#432—1062no—
256#431—534no—
257#431—1062no—
258C2×C4⋊C8—1061no—
259C2×C4⋊C8—1061no—
260C2×C22⋊C8—532no—
261C2×C22⋊C8—532no—
262C2×#97—1061no—
263C2×C4×C8—1061no—
264#324—538no—
265#324—538no—
266C53C2×#147011yescharacteristic, complemented, Sylow
267C23×C4—531no—
268C23×C4—531no—
269C23×C4—531no—
270C23×C4—531no—
271C23×C4C2×#615711yescharacteristic
272C23×C4C2×#615711yescharacteristic
273C23×C4C2×#615711yescharacteristic
274C22×C8—1062no—
275C22×C8—1062no—
276#125—1064no—
277#125—1064no—
278#125—1064no—
279#125—1064no—
280#125—1064no—
281#125—1064no—
282#125—1064no—
283#125—1064no—
284C2×C4⋊C4—22no—
285C2×C4⋊C4—21no—
286C2×C4⋊C4—21no—
287C2×C4⋊C4—532no—
288C2×C4⋊C4—532no—
289C2×C4⋊C4—534no—
290C2×C4⋊C4—534no—
291C2×C4⋊C4—532no—
292C2×C4⋊C4—534no—
293C2×C4⋊C4—532no—
294C2×C4⋊C4—1062no—
295C2×C4⋊C4—1062no—
296C2×C4⋊C4—1061no—
297C2×C4⋊C4—1061no—
298C2×C4⋊C4—1061no—
299C2×C4⋊C4—1061no—
300C2×C4⋊C4—1061no—
301C2×C4⋊C4—1061no—
302C2×C4⋊C4—1061no—
303C2×C4⋊C4—1061no—
304C2×C4⋊C4—1061no—
305C2×C4⋊C4—1061no—
306C2×C4⋊C4—1061no—
307C2×C4⋊C4—1061no—
308C2×C4⋊C4—1062no—
309C2×C4⋊C4—1061no—
310C2×C4⋊C4—1061no—
311C2×C4⋊C4C2×#615714yes—
312C2×C42—531no—
313C2×C42—531no—
314C2×C42—531no—
315C2×C42—531no—
316C2×C42—532no—
317C2×C42—532no—
318C2×C42—532no—
319C2×C42—532no—
320C4⋊C8—1064no—
321C4⋊C8—1064no—
322C22⋊C8—534no—
323C22⋊C8—534no—
324C22⋊C8—1062no—
325C22⋊C8—1062no—
326#97—1064no—
327C4×C8—1064no—
328C24C22×#615711yescharacteristic
329C22×C4—24no—
330C22×C4—21no—
331C22×C4—21no—
332C22×C4—21no—
333C22×C4—21no—
334C22×C4—24no—
335C22×C4—42no—
336C22×C4—531no—
337C22×C4—532no—
338C22×C4—531no—
339C22×C4—531no—
340C22×C4—531no—
341C22×C4—532no—
342C22×C4—531no—
343C22×C4—531no—
344C22×C4—531no—
345C22×C4—531no—
346C22×C4—532no—
347C22×C4—532no—
348C22×C4—1064no—
349C22×C4—1061no—
350C22×C4—1061no—
351C22×C4—1061no—
352C22×C4—1061no—
353C22×C4—1064no—
354C22×C4—2122no—
355C22×C4—2122no—
356C22×C4C22⋊#615712yes—
357C22×C4C22⋊#615711yescharacteristic
358C22×C4C22⋊#615711yescharacteristic
359C22×C4C22×#615712yes—
360C22×C4C22×#615712yes—
361C22×C4C22×#615712yes—
362C2×C8—1064no—
363C2×C8—1064no—
364C2×C8—2124no—
365C2×C8—2124no—
366C4⋊C4—24no—
367C4⋊C4—24no—
368C4⋊C4—1064no—
369C4⋊C4—1064no—
370C4⋊C4—1064no—
371C4⋊C4—1062no—
372C4⋊C4—1062no—
373C4⋊C4—1064no—
374C4⋊C4—1062no—
375C4⋊C4—1062no—
376C4⋊C4—1062no—
377C4⋊C4—1062no—
378C4⋊C4—1062no—
379C4⋊C4—1062no—
380C4⋊C4—1064no—
381C4⋊C4—1064no—
382C42—532no—
383C42—532no—
384C42—532no—
385C42—532no—
386C42—532no—
387C42—532no—
388C42—532no—
389C42—532no—
390C23—22no—
391C23—22no—
392C23C2×C22⋊#615711yescharacteristic
393C23C2×C22⋊#615711yescharacteristic
394C23#1313330414yes—
395C23C23×#615711yescharacteristic
396C2×C4—24no—
397C2×C4—22no—
398C2×C4—22no—
399C2×C4—24no—
400C2×C4—22no—
401C2×C4—22no—
402C2×C4—24no—
403C2×C4—42no—
404C2×C4—42no—
405C2×C4—42no—
406C2×C4—42no—
407C2×C4—42no—
408C2×C4—534no—
409C2×C4—532no—
410C2×C4—531no—
411C2×C4—531no—
412C2×C4—532no—
413C2×C4—531no—
414C2×C4—531no—
415C2×C4—1064no—
416C2×C4—1064no—
417C2×C4—1062no—
418C2×C4—1062no—
419C2×C4—1062no—
420C2×C4—1062no—
421C2×C4—1062no—
422C2×C4—1064no—
423C2×C4—2121no—
424C2×C4—2121no—
425C2×C4—2121no—
426C2×C4—2121no—
427C2×C4—2122no—
428C2×C4—2122no—
429C2×C4—2122no—
430C2×C4—2122no—
431C2×C4—2122no—
432C2×C4C2×C22⋊#615712yes—
433C2×C4C2×C22⋊#615712yes—
434C8—2124no—
435C8—2124no—
436C22—28no—
437C22—21no—
438C22—21no—
439C22—21no—
440C22—21no—
441C22#1681159612yes—
442C22#1681159612yes—
443C22#1681160212yes—
444C22D53⋊#10112yes—
445C22C22×C22⋊#615711yescharacteristic
446C22C2×#1313330412yes—
447C4—22no—
448C4—22no—
449C4—42no—
450C4—42no—
451C4—534no—
452C4—1061no—
453C4—1061no—
454C4—1061no—
455C4—1061no—
456C4—1062no—
457C4—2122no—
458C4—2122no—
459C2—22no—
460C2—22no—
461C2#2028998714yesdirect product factor
462C2C2×#1681159611yescharacteristic
463C2C2×#1681160211yescharacteristic
464C2C2×D53⋊#10111yescharacteristic
subgroupG/H
centerC23C2×C22⋊#6157
commutatorC23×C53C23×C4
FrattiniC23C23×#6157
FittingC22×C4⋊C4×C53C4
socleC23×C53C2×C22⋊C4
prime radicalC24×C53C22×C4
cosocleC22×Dic53C24
hypercenterC22×C4⋊C4#6157
hypocenterC53C2×#1470
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
3. #20289987C2direct (N × H)
235. C2×C53#1470semidirect
266. C53C2×#1470semidirect
461. C2#20289987direct (N × H)

Maximal subgroups

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

subgroupindex
C2×#168116432
C2×#168116392
#202899872
C2×#168114692
C2×#168114682
C22×#131332072
C2×#147053