Finite Group Databasebeta

← C2×#20273744 #27356238

Subgroups — C2×#20273744 #27356238

Order 13568.

1016 conjugacy classes of subgroups.

The subgroup lattice has 538 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#16811220C211yesmaximal, characteristic
2C2×#16810958C211yesmaximal, characteristic, complemented
3C2×#16810941C211yesmaximal, characteristic
4#20273744C218yesmaximal, direct product factor
5C2×#16810359C211yesmaximal, characteristic, complemented
6C2×#16810356C211yesmaximal, characteristic
7C2×#16810349C211yesmaximal, characteristic, complemented
8C2×C53×#400C211yesmaximal, characteristic, complemented
9#16811220—22no—
10#16811220—22no—
11#16811220—22no—
12#16811220C2214yes—
13C2×#13133212—21no—
14C2×#13133207—21no—
15C2×#13133202—21no—
16C2×#13133200—21no—
17C2×#13133196C2211yescharacteristic
18C2×#13133195—21no—
19C2×#13133191C2211yescharacteristic
20C2×C4×C22⋊D53—21no—
21#16810958—22no—
22#16810958C2214yescomplemented
23#16810941—22no—
24#16810941C2214yes—
25C2×#13133150—21no—
26C2×#13133148—21no—
27C2×#13133140—21no—
28C2×#13133137—21no—
29C2×C22⋊#88322C2211yescharacteristic
30C2×C22⋊#88322C2211yescharacteristic
31#16810359C2214yescomplemented
32#16810356C2214yes—
33#16810349C2214yescomplemented
34C22×C4⋊C4×C53C2211yescharacteristic
35C22×C4⋊C4×C53C2211yescharacteristic
36C53×#400—22no—
37C53×#400C2214yescomplemented
38C2×#95×C53—21no—
39C2×#95×C53—21no—
40C2×#95×C53C2211yescharacteristic
41C22×Dic106D411yescharacteristic
42C22×D212D411yescharacteristic
43C22×C22⋊D53—21no—
44C22×C4×D53—21no—
45C22×#88322—21no—
46C22×#88322—21no—
47#13133212—24no—
48#13133212—24no—
49#13133207—24no—
50#13133207—24no—
51#13133202—24no—
52#13133202—24no—
53#13133200—24no—
54#13133200—24no—
55#13133196—21no—
56#13133196—21no—
57#13133196D412yes—
58#13133196C2312yes—
59#13133195—24no—
60#13133195—24no—
61#13133191—21no—
62#13133191—21no—
63#13133191D412yes—
64#13133191C2312yes—
65C4×C22⋊D53—24no—
66C4×C22⋊D53—24no—
67C2×C4⋊Dic53D411yescharacteristic
68C2×C4⋊Dic53D411yescharacteristic
69C2×#11719919—21no—
70C2×#11719919—21no—
71C2×#11719919—21no—
72C2×#11719919—21no—
73C2×C22⋊Dic53—21no—
74C2×#11719921—21no—
75C2×#11719921—21no—
76C2×#11719921—21no—
77C2×#11719921—21no—
78C2×C4×Dic53—21no—
79#13133150—24no—
80#13133148—24no—
81#13133140—24no—
82#13133137—24no—
83C22⋊#88322—21no—
84C22⋊#88322—21no—
85C22⋊#88322C2312yes—
86C22⋊#88322C2312yes—
87C23×C4×C53—21no—
88C23×C4×C53—21no—
89C23×C4×C53D411yescharacteristic
90C23×C4×C53D411yescharacteristic
91C23×C4×C53C2311yescharacteristic
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×C53—21no—
98C2×C4⋊C4×C53—21no—
99C2×C4⋊C4×C53—21no—
100C2×C4⋊C4×C53—21no—
101C2×C4⋊C4×C53D412yes—
102C2×C4⋊C4×C53C2312yes—
103C2×C4⋊C4×C53C2312yes—
104#95×C53—24no—
105#95×C53—24no—
106#95×C53—21no—
107#95×C53C2312yes—
108C24×C53C2×D411yescharacteristic
109C23×D53—21no—
110C22×C4×C53—21no—
111C22×C4×C53—21no—
112C22×C4×C53—21no—
113C22×C4×C53—22no—
114C22×C4×C53—22no—
115C22×C4×C53—21no—
116C22×C4×C53—22no—
117C22×C4×C53—22no—
118C22×C4×C53—42no—
119C22×C4×C53—42no—
120C22×C4×C53—42no—
121C22×C4×C53—42no—
122C22×C4×C53—44no—
123C22×C4×C53—41no—
124C22×C4×C53—41no—
125C22×C4×C53—42no—
126C22×C4×C53C2×D412yes—
127C22×C4×C53C2×D411yescharacteristic
128C22×C4×C53C2×D411yescharacteristic
129C22×C4×C53C2×D412yes—
130C22×C4×C53C2×D411yescharacteristic
131C22×C4×C53C2×D411yescharacteristic
132C22×C4×C53C2×D411yescharacteristic
133C22×C4×C53C2411yescharacteristic
134C2×C22⋊D53—22no—
135C2×C22⋊D53—22no—
136C2×C22⋊D53—42no—
137C2×C22⋊D53—42no—
138C22×Dic53—21no—
139C22×Dic53—21no—
140C22×Dic53—21no—
141C2×D212—21no—
142C2×D212—42no—
143C2×D212C2×D412yes—
144C2×C4×D53—24no—
145C2×C4×D53—82no—
146C2×Dic106—21no—
147C2×Dic106—42no—
148C2×Dic106C2×D412yes—
149C4⋊C4×C53—22no—
150C4⋊C4×C53—22no—
151C4⋊C4×C53—22no—
152C4⋊C4×C53—22no—
153C4⋊C4×C53—22no—
154C4⋊C4×C53—22no—
155C4⋊C4×C53—41no—
156C4⋊C4×C53—41no—
157C22⋊Dic53—22no—
158C22⋊Dic53—22no—
159#11719921—24no—
160#11719921—24no—
161#11719921—24no—
162#11719921—22no—
163#11719921—22no—
164C4⋊Dic53—21no—
165C4⋊Dic53—21no—
166C4⋊Dic53C2×D412yes—
167C4⋊Dic53C2×D412yes—
168#11719919—24no—
169#11719919—24no—
170#11719919—24no—
171#11719919—22no—
172#11719919—22no—
173C4×Dic53—24no—
174C2×#88322—22no—
175C2×#88322—22no—
176C2×#88322—42no—
177C2×#88322—42no—
178C23×C53—22no—
179C23×C53—22no—
180C23×C53#12011yescharacteristic
181C23×C53#12011yescharacteristic
182C23×C53#13512yes—
183C23×C53#13712yes—
184C23×C53C22×D411yescharacteristic
185C22×D53—22no—
186C22×D53—42no—
187C22×D53—42no—
188C22×D53—42no—
189C2×C4×C53—22no—
190C2×C4×C53—22no—
191C2×C4×C53—22no—
192C2×C4×C53—22no—
193C2×C4×C53—22no—
194C2×C4×C53—22no—
195C2×C4×C53—41no—
196C2×C4×C53—41no—
197C2×C4×C53—41no—
198C2×C4×C53—41no—
199C2×C4×C53—42no—
200C2×C4×C53—42no—
201C2×C4×C53—42no—
202C2×C4×C53—42no—
203C2×C4×C53—42no—
204C2×C4×C53—42no—
205C2×C4×C53—44no—
206C2×C4×C53—42no—
207C2×C4×C53—42no—
208C2×C4×C53—81no—
209C2×C4×C53—84no—
210C2×C4×C53—82no—
211C2×C4×C53—82no—
212C2×C4×C53#12011yescharacteristic
213C2×C4×C53#12011yescharacteristic
214C2×C4×C53C22×D411yescharacteristic
215C2×C4×C53C22×D411yescharacteristic
216C22⋊D53—42no—
217C22⋊D53—42no—
218C22⋊D53—44no—
219C2×Dic53—22no—
220C2×Dic53—22no—
221C2×Dic53—22no—
222C2×Dic53—42no—
223C2×Dic53—42no—
224C2×Dic53—42no—
225D212—42no—
226D212—81no—
227C4×D53—84no—
228Dic106—42no—
229Dic106—81no—
230#88322—42no—
231#88322—42no—
232C2×#1615—531nomaximal, Sylow
233C22×C53—24no—
234C22×C53—24no—
235C22×C53—21no—
236C22×C53—21no—
237C22×C53—21no—
238C22×C53—21no—
239C22×C53#45212yes—
240C22×C53#45212yes—
241C22×C53#45412yes—
242C22×C53#45812yes—
243C22×C53C2×#12011yescharacteristic
244C22×C53C2×#13511yescharacteristic
245C22×C53C2×#13711yescharacteristic
246C2×D53—42no—
247C2×D53—42no—
248C2×D53—42no—
249C2×D53—84no—
250C4×C53—21no—
251C4×C53—21no—
252C4×C53—21no—
253C4×C53—21no—
254C4×C53—42no—
255C4×C53—42no—
256C4×C53—82no—
257C4×C53—82no—
258Dic53—42no—
259Dic53—42no—
260Dic53—42no—
261C2×#539—531no—
262C2×#484—531no—
263C2×#481—531no—
264C2×#400D5311yescharacteristic, complemented
265#1615—538no—
266C2×#328—531no—
267C2×#327—531no—
268C2×#324—531no—
269C2×C53—21no—
270C2×C53—21no—
271C2×C53—22no—
272C2×C53#161514yescomplemented
273C2×C53C2×#45211yescharacteristic
274C2×C53C2×#45411yescharacteristic
275C2×C53C2×#45811yescharacteristic
276D53—82no—
277#539—534no—
278#539—1062no—
279#539—1062no—
280#539—1062no—
281C2×#126—1061no—
282C2×#125—1061no—
283C2×#123—1061no—
284C2×#123—1061no—
285C2×C22⋊Q8—531no—
286C2×C22⋊Q8—1061no—
287C2×#121—531no—
288C2×C4×D4—1061no—
289C22×C4⋊C4C2×D5311yescharacteristic
290C22×C4⋊C4C2×D5311yescharacteristic
291#484—534no—
292#484—1062no—
293#481—534no—
294#481—1062no—
295C2×#107—1061no—
296C2×#106—1061no—
297C2×#103—1061no—
298C2×#102—1061no—
299C2×C22⋊C8—531no—
300C2×C22⋊C8—531no—
301#400—22no—
302#400C2×D5314yescomplemented
303C2×#95—21no—
304C2×#95—21no—
305C2×#95C2×D5311yescharacteristic
306#328—534no—
307#327—534no—
308#324—534no—
309C53C2×#161511yescharacteristic, complemented, Sylow
310C22×Q8—531no—
311C22×D4—531no—
312C22×D4—1061no—
313C23×C4—21no—
314C23×C4—21no—
315C23×C4—1061no—
316C23×C4C22⋊D5311yescharacteristic
317C23×C4C22⋊D5311yescharacteristic
318C23×C4C22×D5311yescharacteristic
319C22×C8—1061no—
320C22×C8—1061no—
321#126—1064no—
322#126—1064no—
323#125—1064no—
324#125—1064no—
325#123—1064no—
326#123—1064no—
327#123—1064no—
328#123—1064no—
329C22⋊Q8—532no—
330C22⋊Q8—532no—
331C22⋊Q8—1064no—
332C22⋊Q8—1064no—
333C22⋊Q8—1061no—
334C22⋊Q8—1061no—
335#121—532no—
336#121—532no—
337#121—1061no—
338#121—1061no—
339C4×D4—1064no—
340C4×D4—1064no—
341C2×C4⋊C4—21no—
342C2×C4⋊C4—21no—
343C2×C4⋊C4—21no—
344C2×C4⋊C4—21no—
345C2×C4⋊C4—21no—
346C2×C4⋊C4—21no—
347C2×C4⋊C4—21no—
348C2×C4⋊C4—21no—
349C2×C4⋊C4—21no—
350C2×C4⋊C4—531no—
351C2×C4⋊C4—531no—
352C2×C4⋊C4—1061no—
353C2×C4⋊C4—1061no—
354C2×C4⋊C4—1061no—
355C2×C4⋊C4—1061no—
356C2×C4⋊C4C22⋊D5312yes—
357C2×C4⋊C4C22×D5312yes—
358C2×C4⋊C4C22×D5312yes—
359C2×C22⋊C4—1061no—
360C2×C22⋊C4—1061no—
361C2×C22⋊C4—1061no—
362C2×C22⋊C4—1061no—
363C2×C22⋊C4—1061no—
364C2×C42—1061no—
365#107—1064no—
366#106—1064no—
367#103—1064no—
368#102—1064no—
369C22⋊C8—532no—
370C22⋊C8—532no—
371C22⋊C8—1061no—
372C22⋊C8—1061no—
373#95—24no—
374#95—24no—
375#95—21no—
376#95C22×D5312yes—
377C24—1061no—
378C24C2×C22⋊D5311yescharacteristic
379C2×Q8—532no—
380C2×Q8—1061no—
381C2×Q8—2122no—
382C2×D4—532no—
383C2×D4—1062no—
384C2×D4—1062no—
385C2×D4—1061no—
386C2×D4—2122no—
387C2×D4—2122no—
388C2×D4—2122no—
389C22×C4—21no—
390C22×C4—21no—
391C22×C4—21no—
392C22×C4—22no—
393C22×C4—22no—
394C22×C4—21no—
395C22×C4—22no—
396C22×C4—22no—
397C22×C4—42no—
398C22×C4—42no—
399C22×C4—42no—
400C22×C4—42no—
401C22×C4—44no—
402C22×C4—41no—
403C22×C4—41no—
404C22×C4—42no—
405C22×C4—1064no—
406C22×C4—1061no—
407C22×C4—1061no—
408C22×C4—1061no—
409C22×C4—4242no—
410C22×C4D4×D5311yescharacteristic
411C22×C4D4×D5311yescharacteristic
412C22×C4D4×D5311yescharacteristic
413C22×C4D4×D5311yescharacteristic
414C22×C4C2×C22⋊D5312yes—
415C22×C4C2×C22⋊D5312yes—
416C22×C4C2×C22⋊D5311yescharacteristic
417C22×C4C23×D5311yescharacteristic
418C2×C8—1062no—
419C2×C8—1062no—
420C2×C8—2122no—
421C2×C8—2122no—
422C4⋊C4—22no—
423C4⋊C4—22no—
424C4⋊C4—22no—
425C4⋊C4—22no—
426C4⋊C4—22no—
427C4⋊C4—22no—
428C4⋊C4—41no—
429C4⋊C4—41no—
430C4⋊C4—532no—
431C4⋊C4—532no—
432C4⋊C4—1064no—
433C4⋊C4—1064no—
434C4⋊C4—1064no—
435C4⋊C4—1061no—
436C4⋊C4—1061no—
437C4⋊C4—1062no—
438C4⋊C4—1062no—
439C22⋊C4—1064no—
440C22⋊C4—1064no—
441C22⋊C4—1064no—
442C22⋊C4—1062no—
443C22⋊C4—1062no—
444C22⋊C4—1062no—
445C22⋊C4—1062no—
446C42—1064no—
447C23—22no—
448C23—22no—
449C23—1062no—
450C23—2122no—
451C23—2122no—
452C23—2122no—
453C23#1313318511yescharacteristic
454C23#1313318511yescharacteristic
455C23#1313323712yes—
456C23#1313324812yes—
457C23C22×C22⋊D5311yescharacteristic
458Q8—4241no—
459Q8—2122no—
460D4—4241no—
461D4—2122no—
462D4—2124no—
463D4—2122no—
464D4—2122no—
465C2×C4—22no—
466C2×C4—22no—
467C2×C4—22no—
468C2×C4—22no—
469C2×C4—22no—
470C2×C4—22no—
471C2×C4—41no—
472C2×C4—41no—
473C2×C4—41no—
474C2×C4—41no—
475C2×C4—42no—
476C2×C4—42no—
477C2×C4—42no—
478C2×C4—42no—
479C2×C4—42no—
480C2×C4—42no—
481C2×C4—44no—
482C2×C4—42no—
483C2×C4—42no—
484C2×C4—81no—
485C2×C4—84no—
486C2×C4—82no—
487C2×C4—82no—
488C2×C4—1062no—
489C2×C4—1062no—
490C2×C4—1062no—
491C2×C4—4244no—
492C2×C4—2122no—
493C2×C4—2122no—
494C2×C4—2122no—
495C2×C4#1313318511yescharacteristic
496C2×C4#1313318511yescharacteristic
497C2×C4C2×D4×D5311yescharacteristic
498C2×C4C2×D4×D5311yescharacteristic
499C8—2122no—
500C8—2122no—
501C22—24no—
502C22—24no—
503C22—21no—
504C22—21no—
505C22—21no—
506C22—21no—
507C22—4244no—
508C22—2122no—
509C22—2122no—
510C22—2122no—
511C22#1681078312yes—
512C22#1681078312yes—
513C22#1681079512yes—
514C22#1681081212yes—
515C22C2×#1313318511yescharacteristic
516C22C2×#1313323711yescharacteristic
517C22C2×#1313324811yescharacteristic
518C4—21no—
519C4—21no—
520C4—21no—
521C4—21no—
522C4—42no—
523C4—42no—
524C4—82no—
525C4—82no—
526C4—2122no—
527C4—2122no—
528C4—2122no—
529C2—21no—
530C2—22no—
531C2—21no—
532C2—4242no—
533C2#2027374414yesdirect product factor
534C2C2×#1681078311yescharacteristic
535C2C2×#1681079511yescharacteristic
536C2C2×#1681081211yescharacteristic
subgroupG/H
centerC23#13133185
commutator, cosocleC22×C4×C53C24
FrattiniC22×C4C23×D53
FittingC2×C53×#400C2
socleC23×C53#120
prime radicalC2×#13133191C22
hypercenterC2×#400D53
hypocenterC53C2×#1615
LCSG ▷ C22×C4×C53 ▷ C22×C53 ▷ C53
UCSC1 ◁ C23 ◁ C23×C4 ◁ C2×#400

Complements

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

NH ≅ G/Ntype
2. C2×#16810958C2semidirect
4. #20273744C2direct (N × H)
5. C2×#16810359C2semidirect
7. C2×#16810349C2semidirect
8. C2×C53×#400C2semidirect
22. #16810958C22semidirect
31. #16810359C22semidirect
33. #16810349C22semidirect
37. C53×#400C22semidirect
264. C2×#400D53semidirect
272. C2×C53#1615semidirect
302. #400C2×D53semidirect
309. C53C2×#1615semidirect
533. C2#20273744direct (N × H)

Maximal subgroups

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

subgroupindex
C2×#168112202
C2×#168109582
C2×#168109412
#202737442
C2×#168103592
C2×#168103562
C2×#168103492
C2×C53×#4002
C2×#161553