Finite Group Databasebeta

← C2×#20290586 #28319073

Subgroups — C2×#20290586 #28319073

Order 13568.

1510 conjugacy classes of subgroups.

The subgroup lattice has 586 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×D53⋊(C4⋊C8)C211yesmaximal, characteristic
2C22×#13133280C212yesmaximal
3C2×#16811567C211yesmaximal, characteristic
4#20290586C218yesmaximal, direct product factor
5C2×#16811496C212yesmaximal
6C2×C4×Q8×D53C211yesmaximal, characteristic
7C22×D53⋊C8C2212yes—
8C22×C4⋊#6157C2212yes—
9C2×#13133284—21no—
10D53⋊(C4⋊C8)—22no—
11D53⋊(C4⋊C8)C2214yes—
12C2×C4⋊#88324—21no—
13C2×#13133280—28no—
14C2×#13133280C2218yes—
15#16811567—22no—
16#16811567C2214yes—
17C2×(C2×#6157)⋊C4—21no—
18C2×(C2×#6157)⋊C4—21no—
19#16811496C2218yes—
20C22×Q8×D53C2211yescharacteristic
21C2×Q8×Dic53C411yescharacteristic, complemented
22C2×C4×Dic106—21no—
23C2×C4×Dic106C411yescharacteristic, complemented
24C2×#13133182—21no—
25C2×#13133182C411yescharacteristic, complemented
26C4×Q8×D53—22no—
27C4×Q8×D53—22no—
28C4×Q8×D53C414yescomplemented
29C4×Q8×D53C2214yes—
30C2×C4⋊C4×D53—21no—
31C2×C4⋊C4×D53C2211yescharacteristic
32C2×C42×D53—21no—
33C2×C42×D53C2211yescharacteristic
34C2×C4×Q8×C53C411yescharacteristic, complemented
35C23×#6157—22no—
36C2×D53⋊C8—22no—
37C2×D53⋊C8—24no—
38C2×D53⋊C8—24no—
39C2×D53⋊C8C2314yes—
40C22×#88324—22no—
41C2×C4⋊#6157—24no—
42C2×C4⋊#6157—24no—
43C2×C4⋊#6157—22no—
44C2×C4⋊#6157C2314yes—
45#13133284—24no—
46C4⋊#88324—24no—
47#13133280—216no—
48(C2×#6157)⋊C4—24no—
49(C2×#6157)⋊C4—24no—
50C22×Dic106—21no—
51C22×Dic106C2×C411yescharacteristic
52C2×Q8×D53—21no—
53C2×Q8×D53—21no—
54C2×Q8×D53—22no—
55C2×Q8×D53—22no—
56C2×Q8×D53C2×C412yes—
57C2×Q8×D53D414yes—
58C2×Q8×D53Q814yes—
59C2×Q8×D53C2312yes—
60C22×C4×D53—21no—
61C22×C4×D53—21no—
62C22×C4×D53D411yescharacteristic
63C22×C4×D53D411yescharacteristic
64C22×C4×D53C2311yescharacteristic
65Q8×Dic53—22no—
66Q8×Dic53C2×C414yescomplemented
67C4×Dic106—22no—
68C4×Dic106—24no—
69C4×Dic106—24no—
70C4×Dic106C2×C414yescomplemented
71#13133182—22no—
72#13133182—24no—
73#13133182—24no—
74#13133182C2×C414yescomplemented
75C2×C4⋊Dic53—21no—
76C2×C4⋊Dic53C2×C411yescharacteristic
77C2×#11719919—21no—
78C2×#11719919—21no—
79C2×#11719919—21no—
80C4⋊C4×D53—24no—
81C4⋊C4×D53—24no—
82C4⋊C4×D53—21no—
83C4⋊C4×D53—21no—
84C4⋊C4×D53C2×C412yes—
85C4⋊C4×D53C2312yes—
86C2×C4×Dic53—21no—
87C2×C4×Dic53—21no—
88C2×C4×Dic53—21no—
89C2×C4×Dic53C2×C411yescharacteristic
90C2×C4×Dic53D411yescharacteristic
91C2×C4×Dic53D411yescharacteristic
92C42×D53—24no—
93C42×D53—24no—
94C42×D53C2×C412yes—
95C42×D53D412yes—
96C42×D53D412yes—
97C42×D53C2312yes—
98C22×Q8×C53C2×C411yescharacteristic
99C4×Q8×C53—22no—
100C4×Q8×C53C2×C414yescomplemented
101C2×C4⋊C4×C53—21no—
102C2×C4⋊C4×C53C2×C411yescharacteristic
103C2×C42×C53—21no—
104C2×C42×C53C2×C411yescharacteristic
105C23×D53C2×D411yescharacteristic
106C22×#6157—24no—
107C22×#6157—44no—
108C22×#6157—44no—
109C22×#6157—44no—
110C2×Q8×C53—21no—
111C2×Q8×C53—22no—
112C2×Q8×C53C4⋊C414yes—
113C2×Q8×C53C22×C412yes—
114C22×C4×C53—21no—
115C22×C4×C53—21no—
116C22×C4×C53C22⋊C411yescharacteristic
117C22×C4×C53C22⋊C411yescharacteristic
118C22×C4×C53C22×C411yescharacteristic
119C22×Dic53—21no—
120C22×Dic53—21no—
121C22×Dic53C22⋊C411yescharacteristic
122C22×Dic53C22⋊C411yescharacteristic
123C22×Dic53C2×D411yescharacteristic
124Q8×D53—24no—
125Q8×D53—24no—
126Q8×D53—24no—
127Q8×D53—24no—
128Q8×D53—24no—
129Q8×D53—22no—
130Q8×D53—22no—
131Q8×D53C4⋊C418yes—
132Q8×D53C2×D414yes—
133Q8×D53C2×Q814yes—
134C2×C4×D53—22no—
135C2×C4×D53—24no—
136C2×C4×D53—24no—
137C2×C4×D53—24no—
138C2×C4×D53—24no—
139C2×C4×D53—22no—
140C2×C4×D53—22no—
141C2×C4×D53—22no—
142C2×C4×D53—44no—
143C2×C4×D53C22⋊C412yes—
144C2×C4×D53C22⋊C412yes—
145C2×C4×D53C22×C411yescharacteristic
146C2×C4×D53C2×D412yes—
147C2×C4×D53C2×D412yes—
148C2×C4×D53C2×D412yes—
149C2×C4×D53C2×D411yescharacteristic
150C2×C4×D53C2×D411yescharacteristic
151C2×C4×D53C2×Q812yes—
152C2×C4×D53#4112yes—
153C2×C4×D53#4112yes—
154C2×C4×D53C2411yescharacteristic
155C2×Dic106—21no—
156C2×Dic106—22no—
157C2×Dic106—22no—
158C2×Dic106—24no—
159C2×Dic106—24no—
160C2×Dic106—22no—
161C2×Dic106C4⋊C414yes—
162C2×Dic106C22×C412yes—
163C2×#88324—24no—
164C2×#88324—44no—
165C4⋊#6157—24no—
166C4⋊#6157—24no—
167C4⋊#6157—44no—
168D53⋊C8—24no—
169D53⋊C8—24no—
170D53⋊C8—44no—
171C4⋊C4×C53—24no—
172C4⋊C4×C53—21no—
173C4⋊C4×C53C22×C412yes—
174C42×C53—24no—
175C42×C53C22⋊C412yes—
176C42×C53C22×C412yes—
177C4⋊Dic53—21no—
178C4⋊Dic53—24no—
179C4⋊Dic53C22×C412yes—
180#11719919—24no—
181#11719919—24no—
182#11719919—22no—
183#11719919—22no—
184C4×Dic53—24no—
185C4×Dic53—24no—
186C4×Dic53—24no—
187C4×Dic53C22⋊C412yes—
188C4×Dic53C22⋊C412yes—
189C4×Dic53C22⋊C412yes—
190C4×Dic53C22×C412yes—
191C4×Dic53C2×D412yes—
192C4×Dic53C2×D412yes—
193C23×C53C2×C22⋊C411yescharacteristic
194C22×D53C2×C22⋊C411yescharacteristic
195C22×D53#12112yes—
196C22×D53C22⋊Q812yes—
197C22×D53#13514yes—
198C22×D53#13714yes—
199C22×D53C22×D411yescharacteristic
200C2×#6157—44no—
201C2×#6157—44no—
202C2×#6157—44no—
203C2×#6157—88no—
204Q8×C53—22no—
205Q8×C53—24no—
206Q8×C53C2×C4⋊C414yes—
207C2×C4×C53—22no—
208C2×C4×C53—24no—
209C2×C4×C53—22no—
210C2×C4×C53—22no—
211C2×C4×C53—22no—
212C2×C4×C53—42no—
213C2×C4×C53C2×C22⋊C412yes—
214C2×C4×C53C2×C22⋊C412yes—
215C2×C4×C53C2×C22⋊C411yescharacteristic
216C2×C4×C53C2×C4⋊C412yes—
217C2×C4×C53#11712yes—
218C2×C4×C53C23×C411yescharacteristic
219C2×Dic53—22no—
220C2×Dic53—24no—
221C2×Dic53—22no—
222C2×Dic53—22no—
223C2×Dic53—22no—
224C2×Dic53—42no—
225C2×Dic53C2×C22⋊C412yes—
226C2×Dic53C2×C22⋊C412yes—
227C2×Dic53C2×C22⋊C411yescharacteristic
228C2×Dic53#12112yes—
229C2×Dic53C22⋊Q812yes—
230C2×Dic53C22×D411yescharacteristic
231C4×D53—24no—
232C4×D53—24no—
233C4×D53—24no—
234C4×D53—24no—
235C4×D53—28no—
236C4×D53—24no—
237C4×D53—24no—
238C4×D53—44no—
239C4×D53—44no—
240C4×D53C2×C4⋊C412yes—
241C4×D53#11712yes—
242C4×D53#12112yes—
243C4×D53#12112yes—
244C4×D53C22⋊Q812yes—
245C4×D53C22⋊Q812yes—
246C4×D53C22×D411yescharacteristic
247C4×D53C22×Q811yescharacteristic
248C4×D53C2×#4111yescharacteristic
249C4×D53C2×#4111yescharacteristic
250Dic106—22no—
251Dic106—24no—
252Dic106—24no—
253Dic106—24no—
254Dic106—28no—
255Dic106C2×C4⋊C414yes—
256#88324—44no—
257C2×#1805—531nomaximal, Sylow
258C22×C53#38012yes—
259C22×C53#41612yes—
260C22×C53#41912yes—
261C22×C53C22×C22⋊C411yescharacteristic
262C2×D53#38012yes—
263C2×D53#41612yes—
264C2×D53#41912yes—
265C2×D53#46418yes—
266C2×D53#47918yes—
267C2×D53C2×#12111yescharacteristic
268C2×D53C2×C22⋊Q811yescharacteristic
269C2×D53C2×#13512yes—
270C2×D53C2×#13712yes—
271#6157—84no—
272C4×C53—22no—
273C4×C53—22no—
274C4×C53—24no—
275C4×C53—42no—
276C4×C53#38012yes—
277C4×C53C22×C4⋊C411yescharacteristic
278C4×C53C2×#11711yescharacteristic
279Dic53—22no—
280Dic53—22no—
281Dic53—24no—
282Dic53—42no—
283Dic53#38012yes—
284Dic53C2×#12111yescharacteristic
285Dic53C2×C22⋊Q811yescharacteristic
286C22×C4×Q8—531no—
287C22×C4⋊C8—531no—
288C22×#103—532no—
289C2×#382—531no—
290#1805—538no—
291C2×#340—532no—
292C2×C53#180514yescomplemented
293C2×C53C2×#38011yescharacteristic
294C2×C53C2×#41611yescharacteristic
295C2×C53C2×#41911yescharacteristic
296D53#180514yescomplemented
297D53C2×#46412yes—
298D53C2×#47912yes—
299C23×Q8—531no—
300C23×C8—532no—
301C2×C4×Q8—531no—
302C2×C4×Q8—534no—
303C2×C4×Q8—534no—
304C2×C4×Q8—531no—
305C2×C4×Q8—531no—
306C2×C4×Q8—1061no—
307C2×C4×Q8—1061no—
308C2×C4×Q8—1062no—
309C2×C4×Q8—1062no—
310C2×C4×Q8#615711yescharacteristic, complemented
311C22×C4⋊C4—531no—
312C22×C4⋊C4—532no—
313C22×C4⋊C4—1061no—
314C22×C42—531no—
315C22×C42—1061no—
316C2×C4⋊C8—534no—
317C2×C4⋊C8—1061no—
318C2×C4⋊C8—1062no—
319C2×C4⋊C8—1061no—
320C2×#103—538no—
321C2×#103—1068no—
322#382—534no—
323#382—1062no—
324C2×#95—1061no—
325C2×#95—1061no—
326#340—538no—
327C53C2×#180511yescharacteristic, complemented, Sylow
328C22×Q8—534no—
329C22×Q8—534no—
330C22×Q8—532no—
331C22×Q8—532no—
332C22×Q8—531no—
333C22×Q8—1061no—
334C22×Q8—1061no—
335C22×Q8—1061no—
336C22×Q8—1062no—
337C22×Q8—1062no—
338C22×Q8C2×#615711yescharacteristic
339C23×C4—531no—
340C23×C4—531no—
341C23×C4—531no—
342C23×C4—1061no—
343C23×C4—1061no—
344C23×C4—1062no—
345C22×C8—534no—
346C22×C8—1064no—
347C22×C8—1062no—
348C22×C8—1062no—
349C22×C8—1064no—
350C4×Q8—22no—
351C4×Q8—534no—
352C4×Q8—534no—
353C4×Q8—534no—
354C4×Q8—1064no—
355C4×Q8—1064no—
356C4×Q8—1064no—
357C4×Q8—1064no—
358C4×Q8—1062no—
359C4×Q8—1062no—
360C4×Q8—1062no—
361C4×Q8C2×#615714yescomplemented
362C2×C4⋊C4—21no—
363C2×C4⋊C4—531no—
364C2×C4⋊C4—532no—
365C2×C4⋊C4—532no—
366C2×C4⋊C4—534no—
367C2×C4⋊C4—1061no—
368C2×C4⋊C4—1061no—
369C2×C4⋊C4—1064no—
370C2×C4⋊C4—1064no—
371C2×C4⋊C4—1061no—
372C2×C4⋊C4—1061no—
373C2×C4⋊C4—1061no—
374C2×C4⋊C4—1061no—
375C2×C4⋊C4—1064no—
376C2×C4⋊C4—1064no—
377C2×C4⋊C4—1062no—
378C2×C4⋊C4C2×#615711yescharacteristic
379C2×C42—21no—
380C2×C42—531no—
381C2×C42—531no—
382C2×C42—531no—
383C2×C42—532no—
384C2×C42—532no—
385C2×C42—532no—
386C2×C42—532no—
387C2×C42—1064no—
388C2×C42—1064no—
389C2×C42—1061no—
390C2×C42—1061no—
391C2×C42—1061no—
392C2×C42C2×#615711yescharacteristic
393C4⋊C8—1064no—
394C4⋊C8—1064no—
395#103—10616no—
396#95—1064no—
397#95—1064no—
398C24—531no—
399C2×Q8—21no—
400C2×Q8—22no—
401C2×Q8—538no—
402C2×Q8—534no—
403C2×Q8—534no—
404C2×Q8—534no—
405C2×Q8—532no—
406C2×Q8—1062no—
407C2×Q8—1062no—
408C2×Q8—1064no—
409C2×Q8—1064no—
410C2×Q8—1064no—
411C2×Q8—1064no—
412C2×Q8—1064no—
413C2×Q8—1064no—
414C2×Q8—1064no—
415C2×Q8—1062no—
416C2×Q8—1062no—
417C2×Q8—1061no—
418C2×Q8—1062no—
419C2×Q8C4⋊#615714yes—
420C2×Q8C22×#615712yes—
421C22×C4—21no—
422C22×C4—21no—
423C22×C4—531no—
424C22×C4—531no—
425C22×C4—531no—
426C22×C4—532no—
427C22×C4—532no—
428C22×C4—532no—
429C22×C4—532no—
430C22×C4—531no—
431C22×C4—531no—
432C22×C4—531no—
433C22×C4—531no—
434C22×C4—532no—
435C22×C4—532no—
436C22×C4—532no—
437C22×C4—532no—
438C22×C4—1062no—
439C22×C4—1062no—
440C22×C4—1064no—
441C22×C4—1064no—
442C22×C4—1064no—
443C22×C4—1064no—
444C22×C4—1062no—
445C22×C4—1062no—
446C22×C4—1061no—
447C22×C4—1061no—
448C22×C4—1064no—
449C22×C4—2124no—
450C22×C4—2124no—
451C22×C4—2124no—
452C22×C4—2124no—
453C22×C4C22⋊#615711yescharacteristic
454C22×C4C22⋊#615711yescharacteristic
455C22×C4C22×#615711yescharacteristic
456C2×C8—1064no—
457C2×C8—1064no—
458C2×C8—1064no—
459C2×C8—2124no—
460C2×C8—2124no—
461C4⋊C4—24no—
462C4⋊C4—21no—
463C4⋊C4—532no—
464C4⋊C4—1062no—
465C4⋊C4—1062no—
466C4⋊C4—1064no—
467C4⋊C4—1064no—
468C4⋊C4—1064no—
469C4⋊C4—1064no—
470C4⋊C4—1061no—
471C4⋊C4—1064no—
472C4⋊C4—2124no—
473C4⋊C4C22×#615712yes—
474C42—24no—
475C42—532no—
476C42—532no—
477C42—532no—
478C42—532no—
479C42—532no—
480C42—532no—
481C42—1064no—
482C42—1064no—
483C42—1064no—
484C42C22⋊#615712yes—
485C42C22×#615712yes—
486C23—532no—
487C23—532no—
488C23—534no—
489C23—534no—
490C23—531no—
491C23—531no—
492C23C2×C22⋊#615711yescharacteristic
493Q8—24no—
494Q8—22no—
495Q8—534no—
496Q8—1064no—
497Q8—1064no—
498Q8—1068no—
499Q8—1064no—
500Q8—1062no—
501Q8C2×C4⋊#615714yes—
502C2×C4—24no—
503C2×C4—22no—
504C2×C4—22no—
505C2×C4—22no—
506C2×C4—22no—
507C2×C4—42no—
508C2×C4—532no—
509C2×C4—532no—
510C2×C4—531no—
511C2×C4—531no—
512C2×C4—532no—
513C2×C4—532no—
514C2×C4—532no—
515C2×C4—532no—
516C2×C4—532no—
517C2×C4—531no—
518C2×C4—531no—
519C2×C4—532no—
520C2×C4—531no—
521C2×C4—531no—
522C2×C4—532no—
523C2×C4—532no—
524C2×C4—1064no—
525C2×C4—1064no—
526C2×C4—1064no—
527C2×C4—1064no—
528C2×C4—1064no—
529C2×C4—1064no—
530C2×C4—1068no—
531C2×C4—1064no—
532C2×C4—1062no—
533C2×C4—1062no—
534C2×C4—1062no—
535C2×C4—1062no—
536C2×C4—4248no—
537C2×C4—2124no—
538C2×C4—2124no—
539C2×C4—2124no—
540C2×C4—2124no—
541C2×C4—2124no—
542C2×C4—2122no—
543C2×C4C2×C22⋊#615712yes—
544C2×C4C2×C22⋊#615712yes—
545C2×C4C2×C22⋊#615711yescharacteristic
546C2×C4C2×C4⋊#615712yes—
547C2×C4#1313329612yes—
548C2×C4C23×#615711yescharacteristic
549C8—2124no—
550C22—532no—
551C22—532no—
552C22—532no—
553C22—532no—
554C22—538no—
555C22—538no—
556C22—531no—
557C22—531no—
558C22—532no—
559C22#1681156512yes—
560C22#1681161312yes—
561C22#1681161612yes—
562C22C22×C22⋊#615711yescharacteristic
563C4—22no—
564C4—22no—
565C4—24no—
566C4—42no—
567C4—531no—
568C4—531no—
569C4—532no—
570C4—1062no—
571C4—1062no—
572C4—1064no—
573C4—4244no—
574C4—2122no—
575C4#1681156512yes—
576C4C22×C4⋊#615711yescharacteristic
577C4C2×#1313329611yescharacteristic
578C2—532no—
579C2—532no—
580C2—534no—
581C2#2029058614yesdirect product factor
582C2C2×#1681156511yescharacteristic
583C2C2×#1681161311yescharacteristic
584C2C2×#1681161611yescharacteristic
subgroupG/H
centerC23C2×C22⋊#6157
commutatorC2×C4×C53C23×C4
FrattiniC2×C4C23×#6157
FittingC2×C4×Q8×C53C4
socleC23×C53C2×C22⋊C4
prime radicalC23×D53C2×D4
cosocleC2×C4×D53C24
hypercenterC2×C4×Q8#6157
hypocenterC53C2×#1805
LCSG ▷ C2×C4×C53 ▷ C2×C53 ▷ C53
UCSC1 ◁ C23 ◁ C2×C42 ◁ C2×C4×Q8

Complements

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

NH ≅ G/Ntype
4. #20290586C2direct (N × H)
21. C2×Q8×Dic53C4semidirect
23. C2×C4×Dic106C4semidirect
25. C2×#13133182C4semidirect
28. C4×Q8×D53C4semidirect
34. C2×C4×Q8×C53C4semidirect
66. Q8×Dic53C2×C4semidirect
70. C4×Dic106C2×C4semidirect
74. #13133182C2×C4semidirect
100. C4×Q8×C53C2×C4semidirect
292. C2×C53#1805semidirect
296. D53#1805semidirect
310. C2×C4×Q8#6157semidirect
327. C53C2×#1805semidirect
361. C4×Q8C2×#6157semidirect
581. C2#20290586direct (N × H)

Maximal subgroups

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

subgroupindex
C2×D53⋊(C4⋊C8)2
C22×#131332802
C2×#168115672
#202905862
C2×#168114962
C2×C4×Q8×D532
C2×#180553