Finite Group Databasebeta

← C2×#20274049 #27365262

Subgroups — C2×#20274049 #27365262

Order 13568.

894 conjugacy classes of subgroups.

The subgroup lattice has 468 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#16811033C211yesmaximal, characteristic, complemented
2C2×#16810993C211yesmaximal, characteristic, complemented
3C2×#16810982C211yesmaximal, characteristic
4#20274049C218yesmaximal, direct product factor
5C2×#16810365C211yesmaximal, characteristic, complemented
6C2×#16810365C211yesmaximal, characteristic, complemented
7C2×#16810363C211yesmaximal, characteristic
8C2×C53×#563C211yesmaximal, characteristic
9C2×#13133217C2211yescharacteristic
10#16811033—22no—
11#16811033C2214yescomplemented
12#16810993—22no—
13#16810993C2214yescomplemented
14#16810982—22no—
15#16810982C2214yes—
16C2×#13133150—21no—
17C2×#13133148—21no—
18C2×#13133141—21no—
19C2×#13133140—21no—
20C2×#13133140—21no—
21C2×#13133138—21no—
22C2×#13133128C2211yescharacteristic
23C2×#13133128C2211yescharacteristic
24C2×#13133128C2211yescharacteristic
25#16810365C2214yescomplemented
26#16810365C2214yescomplemented
27#16810363C2214yes—
28C53×#563—22no—
29C53×#563—22no—
30C53×#563—22no—
31C53×#563C2214yes—
32C2×C4⋊Q8×C53—21no—
33C2×#126×C53—21no—
34C2×#126×C53—21no—
35C2×#125×C53C2211yescharacteristic
36C2×#125×C53C2211yescharacteristic
37C2×#124×C53C2211yescharacteristic
38C2×#123×C53—21no—
39C2×C22⋊Q8×C53—21no—
40C2×C22⋊Q8×C53—21no—
41C22×Dic106—21no—
42C22×#88322—21no—
43C22×#88322—21no—
44C22×#88322—21no—
45#13133217—21no—
46#13133217—21no—
47#13133217—21no—
48#13133217C2312yes—
49C2×C4⋊Dic53—21no—
50C2×C4⋊Dic53—21no—
51#13133150—24no—
52#13133148—24no—
53#13133141—24no—
54#13133140—24no—
55#13133140—24no—
56#13133138—24no—
57#13133128—21no—
58#13133128—21no—
59#13133128—21no—
60#13133128C2312yes—
61#13133128C2312yes—
62#13133128C2312yes—
63C22×Q8×C53—21no—
64C22×Q8×C53D411yescharacteristic
65C22×D4×C53D411yescharacteristic
66C23×C4×C53—21no—
67C4⋊Q8×C53—24no—
68C4⋊Q8×C53—24no—
69#126×C53—24no—
70#126×C53—24no—
71#126×C53—24no—
72#126×C53—24no—
73#125×C53—21no—
74#125×C53—21no—
75#125×C53—21no—
76#125×C53—21no—
77#125×C53D412yes—
78#125×C53D412yes—
79#125×C53C2312yes—
80#125×C53C2312yes—
81#124×C53—21no—
82#124×C53—21no—
83#124×C53D412yes—
84#124×C53C2312yes—
85#123×C53—24no—
86#123×C53—24no—
87C22⋊Q8×C53—24no—
88C22⋊Q8×C53—24no—
89C22⋊Q8×C53—24no—
90C22⋊Q8×C53—24no—
91C2×C4⋊C4×C53—21no—
92C2×C4⋊C4×C53—21no—
93C2×C4⋊C4×C53—21no—
94C2×C4⋊C4×C53—21no—
95C2×C4⋊C4×C53—21no—
96C2×C4⋊C4×C53—21no—
97C2×C4⋊C4×C53D411yescharacteristic
98C2×C4⋊C4×C53D411yescharacteristic
99C2×C4⋊C4×C53D411yescharacteristic
100C2×C4⋊C4×C53D411yescharacteristic
101C2×C22⋊C4×C53—21no—
102C2×C22⋊C4×C53—21no—
103C2×C22⋊C4×C53—21no—
104C2×C22⋊C4×C53—21no—
105C2×C22⋊C4×C53—21no—
106C2×C42×C53—21no—
107C2×C42×C53C2311yescharacteristic
108C24×C53—21no—
109C2×Q8×C53—24no—
110C2×Q8×C53—21no—
111C2×Q8×C53—42no—
112C2×Q8×C53—82no—
113C2×Q8×C53C2×D412yes—
114C2×D4×C53—21no—
115C2×D4×C53—42no—
116C2×D4×C53C2×D412yes—
117C22×C4×C53—21no—
118C22×C4×C53—21no—
119C22×C4×C53—21no—
120C22×C4×C53—21no—
121C22×C4×C53—21no—
122C22×C4×C53—24no—
123C22×C4×C53—82no—
124C22×C4×C53C2×D411yescharacteristic
125C22×C4×C53C2×D411yescharacteristic
126C22×C4×C53C2×D411yescharacteristic
127C22×Dic53—41no—
128C2×Dic106—22no—
129C2×Dic106—44no—
130C2×Dic106—41no—
131C4⋊C4×C53—24no—
132C4⋊C4×C53—24no—
133C4⋊C4×C53—22no—
134C4⋊C4×C53—22no—
135C4⋊C4×C53—22no—
136C4⋊C4×C53—22no—
137C4⋊C4×C53—21no—
138C4⋊C4×C53—21no—
139C4⋊C4×C53—21no—
140C4⋊C4×C53—21no—
141C4⋊C4×C53—22no—
142C4⋊C4×C53—22no—
143C4⋊C4×C53—22no—
144C4⋊C4×C53—22no—
145C4⋊C4×C53C2×D412yes—
146C4⋊C4×C53C2×D412yes—
147C4⋊C4×C53C2×D412yes—
148C4⋊C4×C53C2×D412yes—
149C22⋊C4×C53—24no—
150C22⋊C4×C53—24no—
151C22⋊C4×C53—24no—
152C22⋊C4×C53—22no—
153C22⋊C4×C53—22no—
154C22⋊C4×C53—22no—
155C22⋊C4×C53—22no—
156C42×C53—24no—
157C42×C53C2×D411yescharacteristic
158C42×C53C2×D411yescharacteristic
159C42×C53C2×D411yescharacteristic
160C42×C53C2411yescharacteristic
161C4⋊Dic53—22no—
162C4⋊Dic53—22no—
163C4⋊Dic53—41no—
164C4⋊Dic53—41no—
165C2×#88322—24no—
166C2×#88322—22no—
167C2×#88322—22no—
168C2×#88322—22no—
169C2×#88322—42no—
170C2×#88322—42no—
171C23×C53—22no—
172C23×C53—42no—
173C23×C53—42no—
174C23×C53—42no—
175C23×C53#12011yescharacteristic
176Q8×C53—42no—
177Q8×C53—81no—
178Q8×C53—84no—
179D4×C53—42no—
180D4×C53—81no—
181C2×C4×C53—22no—
182C2×C4×C53—22no—
183C2×C4×C53—22no—
184C2×C4×C53—22no—
185C2×C4×C53—22no—
186C2×C4×C53—22no—
187C2×C4×C53—22no—
188C2×C4×C53—22no—
189C2×C4×C53—42no—
190C2×C4×C53—42no—
191C2×C4×C53—42no—
192C2×C4×C53—42no—
193C2×C4×C53—42no—
194C2×C4×C53—84no—
195C2×C4×C53#12011yescharacteristic
196C2×C4×C53#12011yescharacteristic
197C2×C4×C53#12011yescharacteristic
198C2×C4×C53C22×D411yescharacteristic
199C2×C4×C53C22×D411yescharacteristic
200C2×C4×C53C22×D411yescharacteristic
201C2×Dic53—42no—
202C2×Dic53—82no—
203Dic106—44no—
204Dic106—82no—
205#88322—24no—
206#88322—42no—
207#88322—42no—
208C2×#1664—531nomaximal, Sylow
209C22×C53—42no—
210C22×C53—42no—
211C22×C53—42no—
212C22×C53—84no—
213C22×C53#45412yes—
214C22×C53#45512yes—
215C22×C53#45612yes—
216C22×C53C2×#12011yescharacteristic
217C4×C53—21no—
218C4×C53—21no—
219C4×C53—21no—
220C4×C53—21no—
221C4×C53—21no—
222C4×C53—21no—
223C4×C53—42no—
224C4×C53—42no—
225C4×C53—42no—
226C4×C53—42no—
227C4×C53—42no—
228Dic53—82no—
229C2×#563D5311yescharacteristic
230C2×#501—531no—
231C2×#491—531no—
232C2×#489—531no—
233#1664—538no—
234C2×#330—531no—
235C2×#330—531no—
236C2×#329—531no—
237C2×C53—82no—
238C2×C53#166414yescomplemented
239C2×C53C2×#45511yescharacteristic
240C2×C53C2×#45411yescharacteristic
241C2×C53C2×#45611yescharacteristic
242#563—22no—
243#563—22no—
244#563—22no—
245#563C2×D5314yes—
246C2×C4⋊Q8—21no—
247C2×C4⋊Q8—531no—
248C2×#126—21no—
249C2×#126—21no—
250C2×#125C2×D5311yescharacteristic
251C2×#125C2×D5311yescharacteristic
252C2×#124C2×D5311yescharacteristic
253C2×#123—21no—
254C2×C22⋊Q8—21no—
255C2×C22⋊Q8—21no—
256#501—534no—
257#501—1062no—
258#491—534no—
259#491—1062no—
260#489—534no—
261#489—1062no—
262C2×#107—1061no—
263C2×#106—1061no—
264C2×#103—1061no—
265C2×#103—1061no—
266C2×#103—1061no—
267C2×#102—1061no—
268C2×#97—531no—
269C2×#97—531no—
270C2×#97—531no—
271#330—534no—
272#330—534no—
273#329—534no—
274C53C2×#166411yescharacteristic, complemented, Sylow
275C22×Q8—21no—
276C22×Q8—1061no—
277C22×Q8C22⋊D5311yescharacteristic
278C22×D4C22⋊D5311yescharacteristic
279C23×C4—21no—
280C22×C8—1061no—
281C22×C8—1061no—
282C22×C8—1061no—
283C4⋊Q8—24no—
284C4⋊Q8—24no—
285C4⋊Q8—532no—
286C4⋊Q8—1061no—
287C4⋊Q8—1061no—
288C4⋊Q8—1061no—
289#126—24no—
290#126—24no—
291#126—24no—
292#126—24no—
293#125—21no—
294#125—21no—
295#125—21no—
296#125—21no—
297#125C22⋊D5312yes—
298#125C22⋊D5312yes—
299#125C22×D5312yes—
300#125C22×D5312yes—
301#124—21no—
302#124—21no—
303#124C22⋊D5312yes—
304#124C22×D5312yes—
305#123—24no—
306#123—24no—
307C22⋊Q8—24no—
308C22⋊Q8—24no—
309C22⋊Q8—24no—
310C22⋊Q8—24no—
311C2×C4⋊C4—21no—
312C2×C4⋊C4—21no—
313C2×C4⋊C4—21no—
314C2×C4⋊C4—21no—
315C2×C4⋊C4—21no—
316C2×C4⋊C4—21no—
317C2×C4⋊C4—1061no—
318C2×C4⋊C4—1061no—
319C2×C4⋊C4C22⋊D5311yescharacteristic
320C2×C4⋊C4C22⋊D5311yescharacteristic
321C2×C4⋊C4C22⋊D5311yescharacteristic
322C2×C4⋊C4C22⋊D5311yescharacteristic
323C2×C22⋊C4—21no—
324C2×C22⋊C4—21no—
325C2×C22⋊C4—21no—
326C2×C22⋊C4—21no—
327C2×C22⋊C4—21no—
328C2×C42—21no—
329C2×C42C22×D5311yescharacteristic
330#107—1064no—
331#106—1064no—
332#103—1064no—
333#103—1064no—
334#103—1064no—
335#102—1064no—
336#97—532no—
337#97—532no—
338#97—532no—
339#97—1061no—
340#97—1061no—
341#97—1061no—
342C24—21no—
343C2×Q8—24no—
344C2×Q8—21no—
345C2×Q8—42no—
346C2×Q8—82no—
347C2×Q8—1062no—
348C2×Q8—2124no—
349C2×Q8—2121no—
350C2×Q8C2×C22⋊D5312yes—
351C2×D4—21no—
352C2×D4—42no—
353C2×D4C2×C22⋊D5312yes—
354C22×C4—21no—
355C22×C4—21no—
356C22×C4—21no—
357C22×C4—21no—
358C22×C4—21no—
359C22×C4—24no—
360C22×C4—82no—
361C22×C4—2121no—
362C22×C4C2×C22⋊D5311yescharacteristic
363C22×C4C2×C22⋊D5311yescharacteristic
364C22×C4C2×C22⋊D5311yescharacteristic
365C2×C8—1064no—
366C2×C8—1062no—
367C2×C8—1062no—
368C2×C8—1062no—
369C2×C8—2122no—
370C2×C8—2122no—
371C4⋊C4—24no—
372C4⋊C4—24no—
373C4⋊C4—22no—
374C4⋊C4—22no—
375C4⋊C4—22no—
376C4⋊C4—22no—
377C4⋊C4—21no—
378C4⋊C4—21no—
379C4⋊C4—21no—
380C4⋊C4—21no—
381C4⋊C4—22no—
382C4⋊C4—22no—
383C4⋊C4—22no—
384C4⋊C4—22no—
385C4⋊C4—1062no—
386C4⋊C4—1062no—
387C4⋊C4—2121no—
388C4⋊C4—2121no—
389C4⋊C4C2×C22⋊D5312yes—
390C4⋊C4C2×C22⋊D5312yes—
391C4⋊C4C2×C22⋊D5312yes—
392C4⋊C4C2×C22⋊D5312yes—
393C22⋊C4—24no—
394C22⋊C4—24no—
395C22⋊C4—24no—
396C22⋊C4—22no—
397C22⋊C4—22no—
398C22⋊C4—22no—
399C22⋊C4—22no—
400C42—24no—
401C42C2×C22⋊D5311yescharacteristic
402C42C2×C22⋊D5311yescharacteristic
403C42C2×C22⋊D5311yescharacteristic
404C42C23×D5311yescharacteristic
405C23—22no—
406C23—42no—
407C23—42no—
408C23—42no—
409C23#1313318611yescharacteristic
410Q8—42no—
411Q8—81no—
412Q8—84no—
413Q8—4242no—
414Q8—2124no—
415D4—42no—
416D4—81no—
417C2×C4—22no—
418C2×C4—22no—
419C2×C4—22no—
420C2×C4—22no—
421C2×C4—22no—
422C2×C4—22no—
423C2×C4—22no—
424C2×C4—22no—
425C2×C4—42no—
426C2×C4—42no—
427C2×C4—42no—
428C2×C4—42no—
429C2×C4—42no—
430C2×C4—84no—
431C2×C4—4242no—
432C2×C4—2122no—
433C2×C4#1313318611yescharacteristic
434C2×C4#1313318611yescharacteristic
435C2×C4#1313318611yescharacteristic
436C2×C4C22×C22⋊D5311yescharacteristic
437C2×C4C22×C22⋊D5311yescharacteristic
438C2×C4C22×C22⋊D5311yescharacteristic
439C8—1064no—
440C8—2122no—
441C8—2122no—
442C22—42no—
443C22—42no—
444C22—42no—
445C22—84no—
446C22#1681079612yes—
447C22#1681080112yes—
448C22#1681080612yes—
449C22C2×#1313318611yescharacteristic
450C4—21no—
451C4—21no—
452C4—21no—
453C4—21no—
454C4—21no—
455C4—21no—
456C4—42no—
457C4—42no—
458C4—42no—
459C4—42no—
460C4—42no—
461C4—4242no—
462C2—82no—
463C2#2027404914yesdirect product factor
464C2C2×#1681080111yescharacteristic
465C2C2×#1681079611yescharacteristic
466C2C2×#1681080611yescharacteristic
subgroupG/H
centerC23#13133186
commutator, cosocleC42×C53C24
FrattiniC42C23×D53
FittingC2×C53×#563C2
socleC23×C53#120
prime radicalC22×D4×C53D4
hypercenterC2×#563D53
hypocenterC53C2×#1664
LCSG ▷ C42×C53 ▷ C22×C53 ▷ C53
UCSC1 ◁ C23 ◁ C2×C42 ◁ C2×#563

Complements

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

NH ≅ G/Ntype
1. C2×#16811033C2semidirect
2. C2×#16810993C2semidirect
4. #20274049C2direct (N × H)
5. C2×#16810365C2semidirect
6. C2×#16810365C2semidirect
11. #16811033C22semidirect
13. #16810993C22semidirect
25. #16810365C22semidirect
26. #16810365C22semidirect
238. C2×C53#1664semidirect
274. C53C2×#1664semidirect
463. C2#20274049direct (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×#168109932
C2×#168109822
#202740492
C2×#168103652
C2×#168103632
C2×C53×#5632
C2×#166453