Finite Group Databasebeta

← C2×#20274777 #27383770

Subgroups — C2×#20274777 #27383770

Order 13568.

1374 conjugacy classes of subgroups.

The subgroup lattice has 540 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C22×#13133224C211yesmaximal, characteristic
2C22×#13133152C211yesmaximal, characteristic, complemented
3C22×#13133150C211yesmaximal, characteristic, complemented
4C2×#16810624C211yesmaximal, characteristic
5#20274777C218yesmaximal, direct product factor
6C2×#16810392C211yesmaximal, characteristic, complemented
7C2×#16810391C211yesmaximal, characteristic, complemented
8C2×C53×#417C211yesmaximal, characteristic
9C2×#13133224—24no—
10C2×#13133224—22no—
11C2×#13133224C418yescomplemented
12C2×#13133224C2214yes—
13C22×#11719917—21no—
14C22×#11719917C2211yescharacteristic
15C22×#11719917C2211yescharacteristic
16C23×#88322—21no—
17C23×#88322C2211yescharacteristic
18C22×C4⋊Dic53C2211yescharacteristic
19C2×#13133152—24no—
20C2×#13133152C2214yescomplemented
21C2×#13133150—21no—
22C2×#13133150—21no—
23C2×#13133150—22no—
24C2×#13133150C2214yescomplemented
25#16810624—22no—
26#16810624C2214yes—
27C2×#13133141—21no—
28C2×#13133138—21no—
29#16810392C2214yescomplemented
30#16810391C2214yescomplemented
31C22×#41×C53C2211yescharacteristic
32C22×M16×C53C2211yescharacteristic
33C22×C4⋊C4×C53C2211yescharacteristic
34C53×#417—22no—
35C53×#417C2214yes—
36C2×#103×C53—21no—
37C2×#102×C53—21no—
38C23×Dic53—21no—
39#13133224—216no—
40#13133224—28no—
41#13133224C2×C4116yescomplemented
42C2×#11719917—28no—
43C2×#11719917—24no—
44C2×#11719917—22no—
45C2×#11719917—21no—
46C2×#11719917—21no—
47C2×#11719917—22no—
48C2×#11719917—22no—
49C2×#11719917C2×C414yes—
50C2×#11719917C2312yes—
51C2×#11719917C2312yes—
52C22×#88322—28no—
53C22×#88322—24no—
54C22×#88322—21no—
55C22×#88322—21no—
56C22×#88322C2×C414yes—
57C22×#88322D411yescharacteristic
58C22×#88322D411yescharacteristic
59C22×#88322D412yes—
60C22×#88322D414yes—
61C22×#88322C2312yes—
62C2×C4⋊Dic53—21no—
63C2×C4⋊Dic53—21no—
64C2×C4⋊Dic53—21no—
65C2×C4⋊Dic53—21no—
66C2×C4⋊Dic53—21no—
67C2×C4⋊Dic53C2312yes—
68#13133152—28no—
69#13133150—24no—
70#13133150—24no—
71#13133141—24no—
72#13133138—24no—
73C2×#41×C53—22no—
74C2×#41×C53—21no—
75C2×#41×C53—21no—
76C2×#41×C53—22no—
77C2×#41×C53C2×C414yes—
78C2×#41×C53D412yes—
79C2×#41×C53Q814yes—
80C2×#41×C53C2312yes—
81C22×Q8×C53D411yescharacteristic
82C22×D4×C53—21no—
83C22×D4×C53D411yescharacteristic
84C23×C4×C53—21no—
85C23×C4×C53—21no—
86C23×C4×C53C2311yescharacteristic
87C2×M16×C53—21no—
88C2×M16×C53—22no—
89C2×M16×C53—22no—
90C2×M16×C53C2312yes—
91C22×C8×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×C53C2312yes—
98#103×C53—24no—
99#102×C53—24no—
100C24×C53—21no—
101C24×C53#4111yescharacteristic
102#41×C53—24no—
103#41×C53—24no—
104#41×C53—24no—
105#41×C53—24no—
106#41×C53—24no—
107#41×C53—22no—
108#41×C53—22no—
109#41×C53C4⋊C418yes—
110#41×C53C22×C414yes—
111#41×C53C2×Q814yes—
112C2×Q8×C53—21no—
113C2×Q8×C53—22no—
114C2×Q8×C53C4⋊C414yes—
115C2×Q8×C53C2×D412yes—
116C2×D4×C53—24no—
117C2×D4×C53—22no—
118C2×D4×C53—24no—
119C2×D4×C53—22no—
120C2×D4×C53—22no—
121C2×D4×C53—21no—
122C2×D4×C53C4⋊C414yes—
123C2×D4×C53C2×D412yes—
124C22×C4×C53—21no—
125C22×C4×C53—24no—
126C22×C4×C53—22no—
127C22×C4×C53—24no—
128C22×C4×C53—22no—
129C22×C4×C53—22no—
130C22×C4×C53—41no—
131C22×C4×C53—41no—
132C22×C4×C53—82no—
133C22×C4×C53C22×C412yes—
134C22×C4×C53C2×D412yes—
135C22×C4×C53C2×D411yescharacteristic
136C22×C4×C53C2×D411yescharacteristic
137C22×C4×C53C2×D411yescharacteristic
138C22×C4×C53C2×D411yescharacteristic
139C22×C4×C53C2×Q812yes—
140C22×C4×C53#4111yescharacteristic
141C22×C4×C53#4112yes—
142C22×C4×C53C2411yescharacteristic
143C22×Dic53—22no—
144C22×Dic53—41no—
145C22×Dic53—41no—
146C22×Dic53—82no—
147M16×C53—22no—
148M16×C53—22no—
149M16×C53—42no—
150C2×C8×C53—22no—
151C2×C8×C53—42no—
152C4⋊C4×C53—22no—
153C4⋊C4×C53—22no—
154C4⋊C4×C53—41no—
155C4⋊C4×C53—41no—
156C4⋊Dic53—22no—
157C4⋊Dic53—22no—
158C4⋊Dic53—41no—
159C4⋊Dic53—41no—
160#11719917—24no—
161#11719917—22no—
162#11719917—216no—
163#11719917—22no—
164#11719917—22no—
165#11719917—42no—
166#11719917C22×C414yes—
167C2×#88322—24no—
168C2×#88322—22no—
169C2×#88322—216no—
170C2×#88322—22no—
171C2×#88322—42no—
172C2×#88322C22⋊C414yes—
173C2×#88322C22⋊C414yes—
174C2×#88322C22⋊C418yes—
175C2×#88322C22×C414yes—
176C2×#88322C2×D412yes—
177C2×#88322C2×D412yes—
178C2×#88322C2×D414yes—
179C23×C53—22no—
180C23×C53—22no—
181C23×C53—22no—
182C23×C53—24no—
183C23×C53—42no—
184C23×C53—42no—
185C23×C53#11712yes—
186C23×C53#12111yescharacteristic
187C23×C53#12111yescharacteristic
188C23×C53C22⋊Q812yes—
189C23×C53C2×#4111yescharacteristic
190Q8×C53—22no—
191Q8×C53—24no—
192Q8×C53C2×C4⋊C414yes—
193D4×C53—24no—
194D4×C53—24no—
195D4×C53—28no—
196D4×C53—22no—
197D4×C53—24no—
198D4×C53C2×C4⋊C414yes—
199C2×C4×C53—22no—
200C2×C4×C53—24no—
201C2×C4×C53—24no—
202C2×C4×C53—24no—
203C2×C4×C53—28no—
204C2×C4×C53—42no—
205C2×C4×C53—82no—
206C2×C4×C53—82no—
207C2×C4×C53—82no—
208C2×C4×C53C2×C22⋊C412yes—
209C2×C4×C53C2×C22⋊C412yes—
210C2×C4×C53C2×C4⋊C412yes—
211C2×C4×C53C2×C4⋊C412yes—
212C2×C4×C53#11712yes—
213C2×C4×C53#12112yes—
214C2×C4×C53#12112yes—
215C2×C4×C53#12111yescharacteristic
216C2×C4×C53#12111yescharacteristic
217C2×C4×C53C22⋊Q812yes—
218C2×C4×C53C22⋊Q812yes—
219C2×C4×C53C22⋊Q812yes—
220C2×C4×C53C23×C411yescharacteristic
221C2×C4×C53C22×D411yescharacteristic
222C2×C4×C53C22×D411yescharacteristic
223C2×C4×C53C22×D411yescharacteristic
224C2×C4×C53C22×Q811yescharacteristic
225C2×C4×C53C2×#4111yescharacteristic
226C2×Dic53—42no—
227C2×Dic53—82no—
228C2×Dic53—82no—
229C2×Dic53—82no—
230C8×C53—42no—
231#88322—24no—
232#88322—42no—
233#88322C2×C22⋊C414yes—
234#88322C2×C22⋊C414yes—
235C2×#1816—531nomaximal, Sylow
236C22×C53—24no—
237C22×C53—24no—
238C22×C53—21no—
239C22×C53—21no—
240C22×C53—21no—
241C22×C53—21no—
242C22×C53—24no—
243C22×C53—48no—
244C22×C53—42no—
245C22×C53—42no—
246C22×C53#38012yes—
247C22×C53#38012yes—
248C22×C53#47212yes—
249C22×C53#47312yes—
250C22×C53C2×#11711yescharacteristic
251C22×C53C2×#12111yescharacteristic
252C22×C53C2×C22⋊Q811yescharacteristic
253C4×C53—24no—
254C4×C53—82no—
255C4×C53#38012yes—
256C4×C53#38012yes—
257C4×C53C22×C22⋊C411yescharacteristic
258C4×C53C22×C4⋊C411yescharacteristic
259C4×C53C2×#12111yescharacteristic
260C4×C53C2×C22⋊Q811yescharacteristic
261Dic53—82no—
262C22×#131—531no—
263C22×#108—531no—
264C22×#107—531no—
265C2×#417—531no—
266C2×#417D5311yescharacteristic
267#1816—538no—
268C2×#343—531no—
269C2×#343—531no—
270C2×C53—21no—
271C2×C53—21no—
272C2×C53—22no—
273C2×C53—44no—
274C2×C53#181614yescomplemented
275C2×C53C2×#38011yescharacteristic
276C2×C53C2×#47211yescharacteristic
277C2×C53C2×#47311yescharacteristic
278C22×#41C2×D5311yescharacteristic
279C2×#131—538no—
280C2×#131—534no—
281C2×#131—1062no—
282C2×#131—1064no—
283C22×M16—531no—
284C22×M16—531no—
285C22×M16—1061no—
286C22×M16C2×D5311yescharacteristic
287C23×C8—531no—
288C23×C8—1061no—
289C22×C4⋊C4—531no—
290C22×C4⋊C4C2×D5311yescharacteristic
291C2×#108—534no—
292C2×#108—1064no—
293C2×#107—534no—
294C2×#107—1061no—
295C2×#107—1061no—
296C2×#107—1062no—
297#417—22no—
298#417—534no—
299#417—1062no—
300#417C2×D5314yes—
301C2×#103—21no—
302C2×#103—1061no—
303C2×#102—21no—
304C2×#102—1061no—
305#343—534no—
306#343—534no—
307C53C2×#181611yescharacteristic, complemented, Sylow
308C2×#41—22no—
309C2×#41—21no—
310C2×#41—21no—
311C2×#41—22no—
312C2×#41Dic10614yes—
313C2×#41C4×D5314yes—
314C2×#41C22⋊D5312yes—
315C2×#41C22×D5312yes—
316C22×Q8C22⋊D5311yescharacteristic
317C22×D4—21no—
318C22×D4C22⋊D5311yescharacteristic
319C23×C4—21no—
320C23×C4—21no—
321C23×C4—1061no—
322C23×C4C22×D5311yescharacteristic
323#131—5316no—
324#131—10616no—
325#131—1068no—
326C2×M16—21no—
327C2×M16—22no—
328C2×M16—22no—
329C2×M16—534no—
330C2×M16—532no—
331C2×M16—532no—
332C2×M16—1068no—
333C2×M16—1064no—
334C2×M16—1062no—
335C2×M16—1061no—
336C2×M16—1061no—
337C2×M16—1062no—
338C2×M16—1062no—
339C2×M16C22×D5312yes—
340C22×C8—21no—
341C22×C8—534no—
342C22×C8—531no—
343C22×C8—531no—
344C22×C8—532no—
345C22×C8—534no—
346C22×C8—532no—
347C22×C8—1068no—
348C22×C8—1064no—
349C22×C8—1061no—
350C22×C8—1061no—
351C2×C4⋊C4—21no—
352C2×C4⋊C4—21no—
353C2×C4⋊C4—21no—
354C2×C4⋊C4—21no—
355C2×C4⋊C4—21no—
356C2×C4⋊C4—532no—
357C2×C4⋊C4—1061no—
358C2×C4⋊C4—1061no—
359C2×C4⋊C4—1061no—
360C2×C4⋊C4—1061no—
361C2×C4⋊C4—1061no—
362C2×C4⋊C4C22×D5312yes—
363#108—1068no—
364#107—1064no—
365#107—1064no—
366#103—24no—
367#103—1064no—
368#102—24no—
369#102—1064no—
370C24—21no—
371C24#1171994711yescharacteristic
372#41—24no—
373#41—24no—
374#41—24no—
375#41—24no—
376#41—24no—
377#41—22no—
378#41—22no—
379#41#1171991918yes—
380#41C2×Dic10614yes—
381#41C2×C4×D5314yes—
382C2×Q8—21no—
383C2×Q8—22no—
384C2×Q8#1171991914yes—
385C2×Q8C2×C22⋊D5312yes—
386C2×D4—24no—
387C2×D4—22no—
388C2×D4—24no—
389C2×D4—22no—
390C2×D4—22no—
391C2×D4—21no—
392C2×D4#1171991914yes—
393C2×D4C2×C22⋊D5312yes—
394C22×C4—21no—
395C22×C4—24no—
396C22×C4—22no—
397C22×C4—24no—
398C22×C4—22no—
399C22×C4—22no—
400C22×C4—41no—
401C22×C4—41no—
402C22×C4—82no—
403C22×C4—1062no—
404C22×C4—4242no—
405C22×C4—2121no—
406C22×C4—2121no—
407C22×C4C2×Dic10612yes—
408C22×C4C2×C4×D5312yes—
409C22×C4D4×D5312yes—
410C22×C4D4×D5311yescharacteristic
411C22×C4D4×D5311yescharacteristic
412C22×C4#1171994711yescharacteristic
413C22×C4#1171994712yes—
414C22×C4C2×C22⋊D5311yescharacteristic
415C22×C4C2×C22⋊D5311yescharacteristic
416C22×C4C23×D5311yescharacteristic
417M16—22no—
418M16—22no—
419M16—42no—
420M16—534no—
421M16—10616no—
422M16—1064no—
423M16—1062no—
424M16—1062no—
425M16—1062no—
426M16—2122no—
427C2×C8—22no—
428C2×C8—42no—
429C2×C8—534no—
430C2×C8—534no—
431C2×C8—538no—
432C2×C8—534no—
433C2×C8—532no—
434C2×C8—532no—
435C2×C8—534no—
436C2×C8—10616no—
437C2×C8—1064no—
438C2×C8—1062no—
439C2×C8—1062no—
440C2×C8—2122no—
441C4⋊C4—22no—
442C4⋊C4—22no—
443C4⋊C4—41no—
444C4⋊C4—41no—
445C4⋊C4—1062no—
446C4⋊C4—1062no—
447C4⋊C4—2121no—
448C4⋊C4—2121no—
449C23—22no—
450C23—22no—
451C23—22no—
452C23—24no—
453C23—42no—
454C23—42no—
455C23#1313317312yes—
456C23#1313318911yescharacteristic
457C23#1313318911yescharacteristic
458C23#1313319412yes—
459C23C2×#1171994711yescharacteristic
460Q8—22no—
461Q8—24no—
462Q8C2×#1171991914yes—
463D4—24no—
464D4—24no—
465D4—28no—
466D4—22no—
467D4—24no—
468D4C2×#1171991914yes—
469C2×C4—22no—
470C2×C4—24no—
471C2×C4—24no—
472C2×C4—24no—
473C2×C4—28no—
474C2×C4—42no—
475C2×C4—82no—
476C2×C4—82no—
477C2×C4—82no—
478C2×C4—4242no—
479C2×C4—4242no—
480C2×C4—4242no—
481C2×C4—2122no—
482C2×C4C22⋊C4×D5312yes—
483C2×C4C22⋊C4×D5312yes—
484C2×C4C2×#1171991912yes—
485C2×C4C2×#1171991912yes—
486C2×C4#1313317312yes—
487C2×C4#1313318912yes—
488C2×C4#1313318912yes—
489C2×C4#1313318911yescharacteristic
490C2×C4#1313318911yescharacteristic
491C2×C4#1313319412yes—
492C2×C4#1313319412yes—
493C2×C4#1313319412yes—
494C2×C4C22×C4×D5311yescharacteristic
495C2×C4C2×D4×D5311yescharacteristic
496C2×C4C2×D4×D5311yescharacteristic
497C2×C4C22×C22⋊D5311yescharacteristic
498C2×C4C22×Dic10611yescharacteristic
499C2×C4C2×#1171994711yescharacteristic
500C8—42no—
501C8—534no—
502C8—534no—
503C8—1064no—
504C8—2122no—
505C22—21no—
506C22—21no—
507C22—21no—
508C22—21no—
509C22—24no—
510C22—24no—
511C22—24no—
512C22—42no—
513C22—42no—
514C22—48no—
515C22#1681048112yes—
516C22#1681048112yes—
517C22#1681089112yes—
518C22#1681089612yes—
519C22C2×#1313317311yescharacteristic
520C22C2×#1313318911yescharacteristic
521C22C2×#1313319411yescharacteristic
522C4—24no—
523C4—82no—
524C4—4242no—
525C4#1681048112yes—
526C4#1681048112yes—
527C4C2×C22⋊C4×D5311yescharacteristic
528C4C22×#1171991911yescharacteristic
529C4C2×#1313318911yescharacteristic
530C4C2×#1313319411yescharacteristic
531C2—21no—
532C2—21no—
533C2—22no—
534C2—44no—
535C2#2027477714yesdirect product factor
536C2C2×#1681048111yescharacteristic
537C2C2×#1681089111yescharacteristic
538C2C2×#1681089611yescharacteristic
subgroupG/H
centerC23#13133189
commutatorC2×C4×C53C23×C4
FrattiniC22×C4C23×D53
FittingC2×C53×#417C2
socleC23×C53#121
prime radicalC22×#41×C53C22
cosocleC22×C4×C53C24
hypercenterC2×#417D53
hypocenterC53C2×#1816
LCSG ▷ C2×C4×C53 ▷ C2×C53 ▷ C53
UCSC1 ◁ C23 ◁ C23×C4 ◁ C2×#417

Complements

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

NH ≅ G/Ntype
2. C22×#13133152C2semidirect
3. C22×#13133150C2semidirect
5. #20274777C2direct (N × H)
6. C2×#16810392C2semidirect
7. C2×#16810391C2semidirect
11. C2×#13133224C4semidirect
20. C2×#13133152C22semidirect
24. C2×#13133150C22semidirect
29. #16810392C22semidirect
30. #16810391C22semidirect
41. #13133224C2×C4semidirect
274. C2×C53#1816semidirect
307. C53C2×#1816semidirect
535. C2#20274777direct (N × H)

Maximal subgroups

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

subgroupindex
C22×#131332242
C22×#131331522
C22×#131331502
C2×#168106242
#202747772
C2×#168103922
C2×#168103912
C2×C53×#4172
C2×#181653