Finite Group Databasebeta

← C2×#20274032 #27365225

Subgroups — C2×#20274032 #27365225

Order 13568.

1022 conjugacy classes of subgroups.

The subgroup lattice has 512 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#16811033C211yesmaximal, characteristic, complemented
2C2×#16811018C211yesmaximal, characteristic, complemented
3C2×#16810982C211yesmaximal, characteristic, complemented
4#20274032C218yesmaximal, direct product factor
5C2×#16810365C211yesmaximal, characteristic, complemented
6C2×#16810363C211yesmaximal, characteristic, complemented
7C2×#16810363C211yesmaximal, characteristic, complemented
8C2×C53×#562C211yesmaximal, characteristic
9C22×#11719925—21no—
10C22×#11719923—21no—
11C2×#13133217C2211yescharacteristic
12#16811033—22no—
13#16811033C2214yescomplemented
14#16811018—22no—
15#16811018C2214yescomplemented
16#16810982—22no—
17#16810982C2214yescomplemented
18C2×#13133150—21no—
19C2×#13133148—21no—
20C2×#13133141—21no—
21C2×#13133138—21no—
22C2×#13133128C2211yescharacteristic
23C2×#13133128C2211yescharacteristic
24C2×#13133128C2211yescharacteristic
25#16810365C2214yescomplemented
26#16810363C2214yescomplemented
27#16810363C2214yescomplemented
28C53×#562—22no—
29C53×#562—22no—
30C53×#562—22no—
31C53×#562C2214yes—
32C2×#125×C53C2211yescharacteristic
33C2×#124×C53C2211yescharacteristic
34C2×#124×C53C2211yescharacteristic
35C2×#123×C53—21no—
36C2×#123×C53—21no—
37C2×C22⋊Q8×C53—21no—
38C2×C22⋊Q8×C53—21no—
39C2×#121×C53—21no—
40C2×#121×C53—21no—
41C22×Dic106—21no—
42C2×#11719925—24no—
43C2×#11719925—44no—
44C2×#11719923—24no—
45C2×#11719923—44no—
46C22×#88322—21no—
47C22×#88322—21no—
48C22×#88322—21no—
49#13133217—21no—
50#13133217—21no—
51#13133217—21no—
52#13133217C2312yes—
53C2×C4⋊Dic53—21no—
54C2×C4⋊Dic53—21no—
55#13133150—24no—
56#13133148—24no—
57#13133141—24no—
58#13133138—24no—
59#13133128—21no—
60#13133128—21no—
61#13133128—21no—
62#13133128C2312yes—
63#13133128C2312yes—
64#13133128C2312yes—
65C22×Q8×C53D411yescharacteristic
66C22×Q8×C53D411yescharacteristic
67C22×D4×C53—21no—
68C22×D4×C53—21no—
69C22×D4×C53D411yescharacteristic
70C22×D4×C53D411yescharacteristic
71C23×C4×C53—21no—
72C23×C4×C53—21no—
73#125×C53—21no—
74#125×C53—21no—
75#125×C53D412yes—
76#125×C53C2312yes—
77#124×C53—21no—
78#124×C53—21no—
79#124×C53—21no—
80#124×C53—21no—
81#124×C53D412yes—
82#124×C53D412yes—
83#124×C53C2312yes—
84#124×C53C2312yes—
85#123×C53—24no—
86#123×C53—24no—
87#123×C53—24no—
88#123×C53—24no—
89C22⋊Q8×C53—24no—
90C22⋊Q8×C53—24no—
91C22⋊Q8×C53—24no—
92C22⋊Q8×C53—24no—
93#121×C53—24no—
94#121×C53—24no—
95#121×C53—24no—
96#121×C53—24no—
97C2×C4⋊C4×C53—21no—
98C2×C4⋊C4×C53—21no—
99C2×C4⋊C4×C53—21no—
100C2×C4⋊C4×C53—21no—
101C2×C4⋊C4×C53D411yescharacteristic
102C2×C4⋊C4×C53D411yescharacteristic
103C2×C22⋊C4×C53—21no—
104C2×C22⋊C4×C53—21no—
105C2×C22⋊C4×C53—21no—
106C2×C22⋊C4×C53—21no—
107C2×C22⋊C4×C53—21no—
108C2×C22⋊C4×C53—21no—
109C2×C42×C53C2311yescharacteristic
110C24×C53—21no—
111C24×C53—21no—
112C2×Q8×C53—21no—
113C2×Q8×C53—21no—
114C2×Q8×C53—42no—
115C2×Q8×C53—42no—
116C2×Q8×C53C2×D412yes—
117C2×Q8×C53C2×D412yes—
118C2×D4×C53—24no—
119C2×D4×C53—24no—
120C2×D4×C53—21no—
121C2×D4×C53—21no—
122C2×D4×C53—42no—
123C2×D4×C53—42no—
124C2×D4×C53—82no—
125C2×D4×C53—82no—
126C2×D4×C53C2×D412yes—
127C2×D4×C53C2×D412yes—
128C22×C4×C53—21no—
129C22×C4×C53—21no—
130C22×C4×C53—21no—
131C22×C4×C53—21no—
132C22×C4×C53—24no—
133C22×C4×C53—24no—
134C22×C4×C53—82no—
135C22×C4×C53—82no—
136C22×C4×C53C2×D411yescharacteristic
137C22×C4×C53C2×D411yescharacteristic
138C22×C4×C53C2×D411yescharacteristic
139C22×Dic53—41no—
140C2×Dic106—22no—
141C2×Dic106—42no—
142C2×Dic106—42no—
143C2×Dic106—41no—
144C4⋊C4×C53—24no—
145C4⋊C4×C53—24no—
146C4⋊C4×C53—22no—
147C4⋊C4×C53—22no—
148C4⋊C4×C53—21no—
149C4⋊C4×C53—21no—
150C4⋊C4×C53—22no—
151C4⋊C4×C53—22no—
152C4⋊C4×C53C2×D412yes—
153C4⋊C4×C53C2×D412yes—
154C22⋊C4×C53—24no—
155C22⋊C4×C53—24no—
156C22⋊C4×C53—22no—
157C22⋊C4×C53—22no—
158C22⋊C4×C53—22no—
159C22⋊C4×C53—22no—
160C22⋊C4×C53—22no—
161C22⋊C4×C53—22no—
162C22⋊C4×C53—22no—
163C22⋊C4×C53—22no—
164C42×C53C2×D411yescharacteristic
165C42×C53C2×D411yescharacteristic
166C42×C53C2×D411yescharacteristic
167C42×C53C2411yescharacteristic
168#11719925—48no—
169#11719923—48no—
170C4⋊Dic53—22no—
171C4⋊Dic53—22no—
172C4⋊Dic53—41no—
173C4⋊Dic53—41no—
174C2×#88322—24no—
175C2×#88322—24no—
176C2×#88322—22no—
177C2×#88322—22no—
178C2×#88322—22no—
179C2×#88322—42no—
180C23×C53—22no—
181C23×C53—22no—
182C23×C53—42no—
183C23×C53—42no—
184C23×C53—42no—
185C23×C53—42no—
186C23×C53—42no—
187C23×C53—42no—
188C23×C53#12011yescharacteristic
189Q8×C53—42no—
190Q8×C53—42no—
191Q8×C53—81no—
192Q8×C53—81no—
193D4×C53—42no—
194D4×C53—42no—
195D4×C53—81no—
196D4×C53—81no—
197D4×C53—84no—
198D4×C53—84no—
199C2×C4×C53—22no—
200C2×C4×C53—22no—
201C2×C4×C53—22no—
202C2×C4×C53—22no—
203C2×C4×C53—22no—
204C2×C4×C53—22no—
205C2×C4×C53—22no—
206C2×C4×C53—42no—
207C2×C4×C53—42no—
208C2×C4×C53—42no—
209C2×C4×C53—42no—
210C2×C4×C53—84no—
211C2×C4×C53—84no—
212C2×C4×C53#12011yescharacteristic
213C2×C4×C53#12011yescharacteristic
214C2×C4×C53#12011yescharacteristic
215C2×C4×C53C22×D411yescharacteristic
216C2×C4×C53C22×D411yescharacteristic
217C2×C4×C53C22×D411yescharacteristic
218C2×Dic53—42no—
219C2×Dic53—82no—
220Dic106—42no—
221Dic106—42no—
222Dic106—82no—
223#88322—24no—
224#88322—24no—
225#88322—42no—
226C2×#1661—531nomaximal, Sylow
227C22×C53—42no—
228C22×C53—42no—
229C22×C53—42no—
230C22×C53—42no—
231C22×C53—42no—
232C22×C53—42no—
233C22×C53—84no—
234C22×C53—84no—
235C22×C53#45312yes—
236C22×C53#45512yes—
237C22×C53#45612yes—
238C22×C53C2×#12011yescharacteristic
239C4×C53—21no—
240C4×C53—21no—
241C4×C53—21no—
242C4×C53—21no—
243C4×C53—21no—
244C4×C53—21no—
245C4×C53—42no—
246C4×C53—42no—
247C4×C53—42no—
248C4×C53—42no—
249Dic53—82no—
250C2×#562D5311yescharacteristic
251C2×#501—531no—
252C2×#497—531no—
253C2×#489—531no—
254#1661—538no—
255C2×#330—531no—
256C2×#329—531no—
257C2×#329—531no—
258C2×C53—82no—
259C2×C53—82no—
260C2×C53#166114yescomplemented
261C2×C53C2×#45311yescharacteristic
262C2×C53C2×#45511yescharacteristic
263C2×C53C2×#45611yescharacteristic
264C22×Q16—1061no—
265C22×SD16—1061no—
266#562—22no—
267#562—22no—
268#562—22no—
269#562C2×D5314yes—
270C2×C4⋊Q8—531no—
271C2×#125C2×D5311yescharacteristic
272C2×#124C2×D5311yescharacteristic
273C2×#124C2×D5311yescharacteristic
274C2×#123—21no—
275C2×#123—21no—
276C2×C22⋊Q8—21no—
277C2×C22⋊Q8—21no—
278C2×#121—21no—
279C2×#121—21no—
280#501—534no—
281#501—1062no—
282#497—534no—
283#497—1062no—
284#489—534no—
285#489—1062no—
286C2×#107—1061no—
287C2×#106—1061no—
288C2×#103—1061no—
289C2×#102—1061no—
290C2×#97—531no—
291C2×#97—531no—
292C2×#97—531no—
293#330—534no—
294#329—534no—
295#329—534no—
296C53C2×#166111yescharacteristic, complemented, Sylow
297C22×Q8—1061no—
298C22×Q8C22⋊D5311yescharacteristic
299C22×Q8C22⋊D5311yescharacteristic
300C22×D4—21no—
301C22×D4—21no—
302C22×D4C22⋊D5311yescharacteristic
303C22×D4C22⋊D5311yescharacteristic
304C23×C4—21no—
305C23×C4—21no—
306C2×Q16—1064no—
307C2×Q16—2124no—
308C2×SD16—1064no—
309C2×SD16—2124no—
310C22×C8—1061no—
311C22×C8—1061no—
312C22×C8—1061no—
313C4⋊Q8—532no—
314C4⋊Q8—1061no—
315C4⋊Q8—1061no—
316C4⋊Q8—1061no—
317#125—21no—
318#125—21no—
319#125C22⋊D5312yes—
320#125C22×D5312yes—
321#124—21no—
322#124—21no—
323#124—21no—
324#124—21no—
325#124C22⋊D5312yes—
326#124C22⋊D5312yes—
327#124C22×D5312yes—
328#124C22×D5312yes—
329#123—24no—
330#123—24no—
331#123—24no—
332#123—24no—
333C22⋊Q8—24no—
334C22⋊Q8—24no—
335C22⋊Q8—24no—
336C22⋊Q8—24no—
337#121—24no—
338#121—24no—
339#121—24no—
340#121—24no—
341C2×C4⋊C4—21no—
342C2×C4⋊C4—21no—
343C2×C4⋊C4—21no—
344C2×C4⋊C4—21no—
345C2×C4⋊C4—1061no—
346C2×C4⋊C4—1061no—
347C2×C4⋊C4C22⋊D5311yescharacteristic
348C2×C4⋊C4C22⋊D5311yescharacteristic
349C2×C22⋊C4—21no—
350C2×C22⋊C4—21no—
351C2×C22⋊C4—21no—
352C2×C22⋊C4—21no—
353C2×C22⋊C4—21no—
354C2×C22⋊C4—21no—
355C2×C42C22×D5311yescharacteristic
356#107—1064no—
357#106—1064no—
358#103—1064no—
359#102—1064no—
360#97—532no—
361#97—532no—
362#97—532no—
363#97—1061no—
364#97—1061no—
365#97—1061no—
366C24—21no—
367C24—21no—
368C2×Q8—21no—
369C2×Q8—21no—
370C2×Q8—42no—
371C2×Q8—42no—
372C2×Q8—1062no—
373C2×Q8—2122no—
374C2×Q8—2122no—
375C2×Q8—2121no—
376C2×Q8C2×C22⋊D5312yes—
377C2×Q8C2×C22⋊D5312yes—
378C2×D4—24no—
379C2×D4—24no—
380C2×D4—21no—
381C2×D4—21no—
382C2×D4—42no—
383C2×D4—42no—
384C2×D4—82no—
385C2×D4—82no—
386C2×D4C2×C22⋊D5312yes—
387C2×D4C2×C22⋊D5312yes—
388C22×C4—21no—
389C22×C4—21no—
390C22×C4—21no—
391C22×C4—21no—
392C22×C4—24no—
393C22×C4—24no—
394C22×C4—82no—
395C22×C4—82no—
396C22×C4—2121no—
397C22×C4C2×C22⋊D5311yescharacteristic
398C22×C4C2×C22⋊D5311yescharacteristic
399C22×C4C2×C22⋊D5311yescharacteristic
400Q16—2128no—
401SD16—2128no—
402C2×C8—1064no—
403C2×C8—1064no—
404C2×C8—1062no—
405C2×C8—1062no—
406C2×C8—1062no—
407C2×C8—2122no—
408C4⋊C4—24no—
409C4⋊C4—24no—
410C4⋊C4—22no—
411C4⋊C4—22no—
412C4⋊C4—21no—
413C4⋊C4—21no—
414C4⋊C4—22no—
415C4⋊C4—22no—
416C4⋊C4—1062no—
417C4⋊C4—1062no—
418C4⋊C4—2121no—
419C4⋊C4—2121no—
420C4⋊C4C2×C22⋊D5312yes—
421C4⋊C4C2×C22⋊D5312yes—
422C22⋊C4—24no—
423C22⋊C4—24no—
424C22⋊C4—22no—
425C22⋊C4—22no—
426C22⋊C4—22no—
427C22⋊C4—22no—
428C22⋊C4—22no—
429C22⋊C4—22no—
430C22⋊C4—22no—
431C22⋊C4—22no—
432C42C2×C22⋊D5311yescharacteristic
433C42C2×C22⋊D5311yescharacteristic
434C42C2×C22⋊D5311yescharacteristic
435C42C23×D5311yescharacteristic
436C23—22no—
437C23—22no—
438C23—42no—
439C23—42no—
440C23—42no—
441C23—42no—
442C23—42no—
443C23—42no—
444C23#1313318611yescharacteristic
445Q8—42no—
446Q8—42no—
447Q8—81no—
448Q8—81no—
449Q8—4242no—
450Q8—2122no—
451Q8—2122no—
452D4—42no—
453D4—42no—
454D4—81no—
455D4—81no—
456D4—84no—
457D4—84no—
458C2×C4—22no—
459C2×C4—22no—
460C2×C4—22no—
461C2×C4—22no—
462C2×C4—22no—
463C2×C4—22no—
464C2×C4—22no—
465C2×C4—42no—
466C2×C4—42no—
467C2×C4—42no—
468C2×C4—42no—
469C2×C4—84no—
470C2×C4—84no—
471C2×C4—4242no—
472C2×C4—2122no—
473C2×C4#1313318611yescharacteristic
474C2×C4#1313318611yescharacteristic
475C2×C4#1313318611yescharacteristic
476C2×C4C22×C22⋊D5311yescharacteristic
477C2×C4C22×C22⋊D5311yescharacteristic
478C2×C4C22×C22⋊D5311yescharacteristic
479C8—1064no—
480C8—1064no—
481C8—2122no—
482C22—42no—
483C22—42no—
484C22—42no—
485C22—42no—
486C22—42no—
487C22—42no—
488C22—84no—
489C22—84no—
490C22#1681079112yes—
491C22#1681080112yes—
492C22#1681080612yes—
493C22C2×#1313318611yescharacteristic
494C4—21no—
495C4—21no—
496C4—21no—
497C4—21no—
498C4—21no—
499C4—21no—
500C4—42no—
501C4—42no—
502C4—42no—
503C4—42no—
504C4—4242no—
505C2—82no—
506C2—82no—
507C2#2027403214yesdirect product factor
508C2C2×#1681079111yescharacteristic
509C2C2×#1681080111yescharacteristic
510C2C2×#1681080611yescharacteristic
subgroupG/H
centerC23#13133186
commutator, cosocleC42×C53C24
FrattiniC42C23×D53
Fitting, prime radicalC2×C53×#562C2
socleC23×C53#120
hypercenterC2×#562D53
hypocenterC53C2×#1661
LCSG ▷ C42×C53 ▷ C22×C53 ▷ C53
UCSC1 ◁ C23 ◁ C2×C42 ◁ C2×#562

Complements

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

NH ≅ G/Ntype
1. C2×#16811033C2semidirect
2. C2×#16811018C2semidirect
3. C2×#16810982C2semidirect
4. #20274032C2direct (N × H)
5. C2×#16810365C2semidirect
6. C2×#16810363C2semidirect
7. C2×#16810363C2semidirect
13. #16811033C22semidirect
15. #16811018C22semidirect
17. #16810982C22semidirect
25. #16810365C22semidirect
26. #16810363C22semidirect
27. #16810363C22semidirect
260. C2×C53#1661semidirect
296. C53C2×#1661semidirect
507. C2#20274032direct (N × H)

Maximal subgroups

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

subgroupindex
C2×#168110332
C2×#168110182
C2×#168109822
#202740322
C2×#168103652
C2×#168103632
C2×C53×#5622
C2×#166153