Finite Group Databasebeta

← C2×#20276369 #27536829

Subgroups — C2×#20276369 #27536829

Order 13568.

1214 conjugacy classes of subgroups.

The subgroup lattice has 558 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#16810607C211yesmaximal, characteristic
2C2×#16810597C211yesmaximal, characteristic
3#20276369C218yesmaximal, direct product factor
4C2×#16810413C214yesmaximal, complemented
5C2×C53×#536C211yesmaximal, characteristic
6C22×#11719917—21no—
7C2×#13133176—21no—
8#16810607—22no—
9#16810607C2214yes—
10C2×#13133136—21no—
11C2×#13133134C2212yes—
12#16810597—22no—
13#16810597C2214yes—
14C2×#13133132—21no—
15C2×#13133132C2212yes—
16#16810413C22116yescomplemented
17C22×#41×C53C2211yescharacteristic
18C23×Q8×C53C411yescharacteristic, complemented
19C53×#536—22no—
20C53×#536—22no—
21C53×#536C414yescomplemented
22C53×#536C2214yes—
23C2×C4⋊Q8×C53—21no—
24C2×#124×C53—21no—
25C2×#123×C53C412yescomplemented
26C2×#123×C53C2212yes—
27C2×C22⋊Q8×C53—21no—
28C2×C22⋊Q8×C53—21no—
29C2×#117×C53C411yescharacteristic, complemented
30C2×#11719917—22no—
31C2×#11719917—24no—
32C2×#11719917—41no—
33C2×#11719917—42no—
34C22×#88322—41no—
35#13133176—24no—
36#13133176—42no—
37C2×C4⋊Dic53—41no—
38C2×C22⋊Dic53—22no—
39C2×C4×Dic53—41no—
40#13133136—24no—
41#13133134—22no—
42#13133134C2314yes—
43#13133132—24no—
44#13133132—22no—
45#13133132C2314yes—
46C2×#41×C53—41no—
47C2×#41×C53—41no—
48C2×#41×C53—42no—
49C2×#41×C53C2×C412yes—
50C2×#41×C53D412yes—
51C2×#41×C53D412yes—
52C2×#41×C53C2312yes—
53C22×Q8×C53—21no—
54C22×Q8×C53—21no—
55C22×Q8×C53—22no—
56C22×Q8×C53—42no—
57C22×Q8×C53—41no—
58C22×Q8×C53—41no—
59C22×Q8×C53C2×C414yescomplemented
60C22×Q8×C53D411yescharacteristic
61C22×D4×C53C2×C411yescharacteristic
62C22×D4×C53D411yescharacteristic
63C22×D4×C53C2311yescharacteristic
64C23×C4×C53C2×C411yescharacteristic
65C23×C4×C53C2×C412yes—
66C23×C4×C53D411yescharacteristic
67C23×C4×C53D411yescharacteristic
68C4⋊Q8×C53—24no—
69C4⋊Q8×C53—24no—
70#124×C53—24no—
71#124×C53—24no—
72#123×C53—24no—
73#123×C53—22no—
74#123×C53—22no—
75#123×C53C2×C414yes—
76#123×C53C2×C418yescomplemented
77#123×C53C2314yes—
78C22⋊Q8×C53—24no—
79C22⋊Q8×C53—24no—
80C22⋊Q8×C53—24no—
81C22⋊Q8×C53—24no—
82#117×C53—22no—
83#117×C53C2×C414yescomplemented
84C2×C4⋊C4×C53—21no—
85C2×C4⋊C4×C53—22no—
86C2×C4⋊C4×C53—22no—
87C2×C22⋊C4×C53—22no—
88C2×C22⋊C4×C53—22no—
89C2×C22⋊C4×C53C2×C412yes—
90C2×C42×C53—21no—
91C24×C53C22⋊C411yescharacteristic
92C24×C53C22×C411yescharacteristic
93C24×C53C2×D411yescharacteristic
94#41×C53—42no—
95#41×C53—42no—
96#41×C53—42no—
97#41×C53—42no—
98#41×C53—42no—
99#41×C53—42no—
100#41×C53—42no—
101#41×C53—42no—
102C2×Q8×C53—24no—
103C2×Q8×C53—24no—
104C2×Q8×C53—41no—
105C2×Q8×C53—41no—
106C2×Q8×C53—42no—
107C2×Q8×C53—42no—
108C2×Q8×C53—44no—
109C2×Q8×C53—44no—
110C2×Q8×C53—44no—
111C2×Q8×C53—44no—
112C2×Q8×C53—42no—
113C2×Q8×C53—41no—
114C2×Q8×C53—41no—
115C2×Q8×C53C22⋊C412yes—
116C2×Q8×C53C2×D412yes—
117C2×D4×C53—41no—
118C2×D4×C53—41no—
119C2×D4×C53—41no—
120C2×D4×C53—41no—
121C2×D4×C53—41no—
122C2×D4×C53—41no—
123C2×D4×C53C22⋊C412yes—
124C2×D4×C53C22⋊C412yes—
125C2×D4×C53C22×C411yescharacteristic
126C2×D4×C53C22×C412yes—
127C2×D4×C53C2×D411yescharacteristic
128C2×D4×C53C2×D412yes—
129C2×D4×C53C2×D411yescharacteristic
130C2×D4×C53C2411yescharacteristic
131C22×C4×C53—22no—
132C22×C4×C53—21no—
133C22×C4×C53—21no—
134C22×C4×C53—21no—
135C22×C4×C53—21no—
136C22×C4×C53—22no—
137C22×C4×C53—21no—
138C22×C4×C53—21no—
139C22×C4×C53—22no—
140C22×C4×C53—22no—
141C22×C4×C53—22no—
142C22×C4×C53—22no—
143C22×C4×C53—42no—
144C22×C4×C53—42no—
145C22×C4×C53C22⋊C411yescharacteristic
146C22×C4×C53C22⋊C411yescharacteristic
147C22×C4×C53C22⋊C412yes—
148C22×C4×C53C22⋊C412yes—
149C22×C4×C53C22⋊C412yes—
150C22×C4×C53C22⋊C411yescharacteristic
151C22×C4×C53C22×C412yes—
152C22×C4×C53C22×C414yes—
153C22×C4×C53C2×D412yes—
154C22×C4×C53C2×D411yescharacteristic
155C22×C4×C53C2×D412yes—
156C22×Dic53—42no—
157C4⋊C4×C53—24no—
158C4⋊C4×C53—24no—
159C4⋊C4×C53—28no—
160C4⋊C4×C53—24no—
161C22⋊C4×C53—24no—
162C22⋊C4×C53—28no—
163C22⋊C4×C53—22no—
164C22⋊C4×C53—24no—
165C22⋊C4×C53C22×C414yes—
166C42×C53—24no—
167C22⋊Dic53—24no—
168C22⋊Dic53—42no—
169C4⋊Dic53—44no—
170C4×Dic53—44no—
171#11719917—24no—
172#11719917—42no—
173#11719917—44no—
174C2×#88322—42no—
175C2×#88322—44no—
176C23×C53—21no—
177C23×C53—21no—
178C23×C53—21no—
179C23×C53—21no—
180C23×C53—21no—
181C23×C53—21no—
182C23×C53—42no—
183C23×C53—42no—
184C23×C53—42no—
185C23×C53C2×C22⋊C411yescharacteristic
186C23×C53C2×C22⋊C411yescharacteristic
187C23×C53C2×C22⋊C411yescharacteristic
188C23×C53C2×C22⋊C412yes—
189C23×C53C23×C411yescharacteristic
190C23×C53C22×D411yescharacteristic
191Q8×C53—42no—
192Q8×C53—42no—
193Q8×C53—48no—
194Q8×C53—44no—
195Q8×C53—44no—
196D4×C53—42no—
197D4×C53—42no—
198D4×C53—42no—
199D4×C53—41no—
200D4×C53—41no—
201D4×C53—41no—
202D4×C53—42no—
203D4×C53—41no—
204C2×C4×C53—21no—
205C2×C4×C53—21no—
206C2×C4×C53—21no—
207C2×C4×C53—21no—
208C2×C4×C53—21no—
209C2×C4×C53—21no—
210C2×C4×C53—22no—
211C2×C4×C53—22no—
212C2×C4×C53—22no—
213C2×C4×C53—22no—
214C2×C4×C53—22no—
215C2×C4×C53—22no—
216C2×C4×C53—22no—
217C2×C4×C53—22no—
218C2×C4×C53—22no—
219C2×C4×C53—22no—
220C2×C4×C53—22no—
221C2×C4×C53—22no—
222C2×C4×C53—24no—
223C2×C4×C53—24no—
224C2×C4×C53—44no—
225C2×C4×C53—44no—
226C2×C4×C53—42no—
227C2×C4×C53—42no—
228C2×C4×C53—44no—
229C2×C4×C53#9912yes—
230C2×C4×C53#9911yescharacteristic
231C2×C4×C53#9911yescharacteristic
232C2×C4×C53C2×C22⋊C412yes—
233C2×C4×C53C2×C22⋊C412yes—
234C2×C4×C53C2×C22⋊C411yescharacteristic
235C2×C4×C53C2×C22⋊C412yes—
236C2×C4×C53C22×D411yescharacteristic
237C2×Dic53—44no—
238C2×Dic53—84no—
239#88322—44no—
240C2×#2123—531nomaximal, Sylow
241C22×C53—22no—
242C22×C53—21no—
243C22×C53—21no—
244C22×C53—21no—
245C22×C53—21no—
246C22×C53—22no—
247C22×C53—41no—
248C22×C53—42no—
249C22×C53—41no—
250C22×C53—41no—
251C22×C53—41no—
252C22×C53—42no—
253C22×C53—42no—
254C22×C53—42no—
255C22×C53—42no—
256C22×C53—81no—
257C22×C53—81no—
258C22×C53C2×#9911yescharacteristic
259C22×C53C2×#9911yescharacteristic
260C22×C53C22×C22⋊C411yescharacteristic
261C4×C53—21no—
262C4×C53—22no—
263C4×C53—21no—
264C4×C53—22no—
265C4×C53—44no—
266C4×C53—44no—
267C4×C53C2×#9912yes—
268C4×C53C2×#9912yes—
269Dic53—84no—
270C2×#536D5311yescharacteristic
271C2×#413—531no—
272C2×#410—531no—
273#2123—538no—
274C2×#352—534no—
275C2×C53—21no—
276C2×C53—21no—
277C2×C53—41no—
278C2×C53—41no—
279C2×C53—41no—
280C2×C53—41no—
281C2×C53—42no—
282C2×C53#212312yescomplemented
283C2×C53C22×#9911yescharacteristic
284C22×#41C2×D5311yescharacteristic
285C23×Q8Dic5311yescharacteristic, complemented
286C22×M16—1061no—
287#536—22no—
288#536—22no—
289#536Dic5314yescomplemented
290#536C2×D5314yes—
291C2×C4⋊Q8—21no—
292C2×#124—21no—
293C2×#123Dic5312yescomplemented
294C2×#123C2×D5312yes—
295C2×C22⋊Q8—21no—
296C2×C22⋊Q8—21no—
297C2×#117—1061no—
298C2×#117Dic5311yescharacteristic, complemented
299#413—534no—
300#413—1062no—
301C2×#101—1061no—
302C2×#100—532no—
303#410—534no—
304#410—1062no—
305C2×#99—532no—
306C2×#99—1061no—
307#352—5316no—
308C53C2×#212311yescharacteristic, complemented, Sylow
309C2×#41—41no—
310C2×#41—41no—
311C2×#41—42no—
312C2×#41C2×Dic5312yes—
313C2×#41C22⋊D5312yes—
314C2×#41C22⋊D5312yes—
315C2×#41C22×D5312yes—
316C22×Q8—21no—
317C22×Q8—22no—
318C22×Q8—21no—
319C22×Q8—42no—
320C22×Q8—41no—
321C22×Q8—41no—
322C22×Q8C2×Dic5314yescomplemented
323C22×Q8C22⋊D5311yescharacteristic
324C22×D4C2×Dic5311yescharacteristic
325C22×D4C22⋊D5311yescharacteristic
326C22×D4C22×D5311yescharacteristic
327C23×C4C2×Dic5311yescharacteristic
328C23×C4C2×Dic5312yes—
329C23×C4C22⋊D5311yescharacteristic
330C23×C4C22⋊D5311yescharacteristic
331C2×M16—1062no—
332C2×M16—1064no—
333C2×M16—2121no—
334C2×M16—2122no—
335C22×C8—2121no—
336C4⋊Q8—24no—
337C4⋊Q8—24no—
338#124—24no—
339#124—24no—
340#123—22no—
341#123—22no—
342#123—24no—
343#123C2×Dic5314yes—
344#123C2×Dic5318yescomplemented
345#123C22×D5314yes—
346C22⋊Q8—24no—
347C22⋊Q8—24no—
348C22⋊Q8—24no—
349C22⋊Q8—24no—
350#117—22no—
351#117—1064no—
352#117—2122no—
353#117C2×Dic5314yescomplemented
354C2×C4⋊C4—22no—
355C2×C4⋊C4—22no—
356C2×C4⋊C4—21no—
357C2×C4⋊C4—2121no—
358C2×C22⋊C4—22no—
359C2×C22⋊C4—22no—
360C2×C22⋊C4—1062no—
361C2×C22⋊C4C2×Dic5312yes—
362C2×C42—21no—
363C2×C42—2121no—
364#101—1064no—
365#100—534no—
366#100—1062no—
367#99—534no—
368#99—1062no—
369#99—1064no—
370C24C22⋊Dic5311yescharacteristic
371C24C22×Dic5311yescharacteristic
372C24C2×C22⋊D5311yescharacteristic
373#41—42no—
374#41—42no—
375#41—42no—
376#41—42no—
377#41—42no—
378#41—42no—
379#41—42no—
380#41—42no—
381C2×Q8—24no—
382C2×Q8—24no—
383C2×Q8—41no—
384C2×Q8—41no—
385C2×Q8—44no—
386C2×Q8—44no—
387C2×Q8—42no—
388C2×Q8—42no—
389C2×Q8—44no—
390C2×Q8—44no—
391C2×Q8—42no—
392C2×Q8—41no—
393C2×Q8—41no—
394C2×Q8C22⋊Dic5312yes—
395C2×Q8C2×C22⋊D5312yes—
396C2×D4—41no—
397C2×D4—41no—
398C2×D4—41no—
399C2×D4—41no—
400C2×D4—41no—
401C2×D4—41no—
402C2×D4C22⋊Dic5312yes—
403C2×D4C22⋊Dic5312yes—
404C2×D4C22×Dic5311yescharacteristic
405C2×D4C22×Dic5312yes—
406C2×D4C2×C22⋊D5311yescharacteristic
407C2×D4C2×C22⋊D5312yes—
408C2×D4C2×C22⋊D5311yescharacteristic
409C2×D4C23×D5311yescharacteristic
410C22×C4—21no—
411C22×C4—21no—
412C22×C4—22no—
413C22×C4—21no—
414C22×C4—22no—
415C22×C4—21no—
416C22×C4—22no—
417C22×C4—21no—
418C22×C4—21no—
419C22×C4—22no—
420C22×C4—22no—
421C22×C4—22no—
422C22×C4—42no—
423C22×C4—42no—
424C22×C4—2122no—
425C22×C4C22⋊Dic5311yescharacteristic
426C22×C4C22⋊Dic5311yescharacteristic
427C22×C4C22⋊Dic5312yes—
428C22×C4C22⋊Dic5312yes—
429C22×C4C22⋊Dic5312yes—
430C22×C4C22⋊Dic5311yescharacteristic
431C22×C4C22×Dic5312yes—
432C22×C4C22×Dic5314yes—
433C22×C4C2×C22⋊D5312yes—
434C22×C4C2×C22⋊D5311yescharacteristic
435C22×C4C2×C22⋊D5312yes—
436M16—1064no—
437M16—2122no—
438M16—2124no—
439C2×C8—2122no—
440C2×C8—2124no—
441C4⋊C4—24no—
442C4⋊C4—24no—
443C4⋊C4—28no—
444C4⋊C4—24no—
445C4⋊C4—2124no—
446C22⋊C4—24no—
447C22⋊C4—28no—
448C22⋊C4—22no—
449C22⋊C4—24no—
450C22⋊C4—1064no—
451C22⋊C4—2122no—
452C22⋊C4C22×Dic5314yes—
453C42—24no—
454C42—2124no—
455C23—21no—
456C23—21no—
457C23—21no—
458C23—21no—
459C23—21no—
460C23—21no—
461C23—42no—
462C23—42no—
463C23—42no—
464C23C2×C22⋊Dic5311yescharacteristic
465C23C2×C22⋊Dic5312yes—
466C23C2×C22⋊Dic5311yescharacteristic
467C23C2×C22⋊Dic5311yescharacteristic
468C23C23×Dic5311yescharacteristic
469C23C22×C22⋊D5311yescharacteristic
470Q8—42no—
471Q8—42no—
472Q8—48no—
473Q8—44no—
474Q8—44no—
475D4—42no—
476D4—41no—
477D4—41no—
478D4—41no—
479D4—42no—
480D4—41no—
481D4—42no—
482D4—42no—
483C2×C4—21no—
484C2×C4—21no—
485C2×C4—21no—
486C2×C4—21no—
487C2×C4—21no—
488C2×C4—21no—
489C2×C4—22no—
490C2×C4—22no—
491C2×C4—22no—
492C2×C4—22no—
493C2×C4—22no—
494C2×C4—22no—
495C2×C4—22no—
496C2×C4—22no—
497C2×C4—22no—
498C2×C4—22no—
499C2×C4—22no—
500C2×C4—22no—
501C2×C4—24no—
502C2×C4—24no—
503C2×C4—44no—
504C2×C4—44no—
505C2×C4—42no—
506C2×C4—42no—
507C2×C4—44no—
508C2×C4—4244no—
509C2×C4—2124no—
510C2×C4#1313313212yes—
511C2×C4#1313313211yescharacteristic
512C2×C4#1313313211yescharacteristic
513C2×C4C2×C22⋊Dic5312yes—
514C2×C4C2×C22⋊Dic5311yescharacteristic
515C2×C4C2×C22⋊Dic5312yes—
516C2×C4C2×C22⋊Dic5312yes—
517C2×C4C22×C22⋊D5311yescharacteristic
518C8—2124no—
519C22—22no—
520C22—21no—
521C22—21no—
522C22—21no—
523C22—22no—
524C22—21no—
525C22—41no—
526C22—41no—
527C22—42no—
528C22—41no—
529C22—41no—
530C22—42no—
531C22—42no—
532C22—42no—
533C22—42no—
534C22—81no—
535C22—81no—
536C22C2×#1313313211yescharacteristic
537C22C2×#1313313211yescharacteristic
538C22C22×C22⋊Dic5311yescharacteristic
539C4—21no—
540C4—21no—
541C4—22no—
542C4—22no—
543C4—44no—
544C4—44no—
545C4—4244no—
546C4C2×#1313313212yes—
547C4C2×#1313313212yes—
548C2—21no—
549C2—21no—
550C2—41no—
551C2—41no—
552C2—41no—
553C2—41no—
554C2—42no—
555C2#2027636912yesdirect product factor
556C2C22×#1313313211yescharacteristic
subgroupG/H
centerC22C2×#13133132
commutatorC23×C53C23×C4
FrattiniC2×D4C23×D53
FittingC2×C53×#536C2
socleC22×C53C2×#99
prime radicalC22×#41×C53C22
cosocleC2×D4×C53C24
hypercenterC2×#536D53
hypocenterC53C2×#2123
LCSG ▷ C23×C53 ▷ C22×C53 ▷ C2×C53 ▷ C53
UCSC1 ◁ C22 ◁ C22×C4 ◁ C22×#41 ◁ C2×#536

Complements

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

NH ≅ G/Ntype
3. #20276369C2direct (N × H)
4. C2×#16810413C2semidirect
16. #16810413C22semidirect
18. C23×Q8×C53C4semidirect
21. C53×#536C4semidirect
25. C2×#123×C53C4semidirect
29. C2×#117×C53C4semidirect
59. C22×Q8×C53C2×C4semidirect
76. #123×C53C2×C4semidirect
83. #117×C53C2×C4semidirect
282. C2×C53#2123semidirect
285. C23×Q8Dic53semidirect
289. #536Dic53semidirect
293. C2×#123Dic53semidirect
298. C2×#117Dic53semidirect
308. C53C2×#2123semidirect
322. C22×Q8C2×Dic53semidirect
344. #123C2×Dic53semidirect
353. #117C2×Dic53semidirect
555. C2#20276369direct (N × H)

Maximal subgroups

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

subgroupindex
C2×#168106072
C2×#168105972
#202763692
C2×#168104132
C2×C53×#5362
C2×#212353