Finite Group Databasebeta

← C2×#20288331 #28291185

Subgroups — C2×#20288331 #28291185

Order 13568.

7706 conjugacy classes of subgroups.

The subgroup lattice has 724 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#20288331C2132yesmaximal, direct product factor
2C2×#16811349C211yesmaximal, characteristic, complemented
3C2×#16811342C213yesmaximal, complemented
4C2×#16811313C213yesmaximal, complemented
5C2×#16811307C213yesmaximal, complemented
6C2×Q8×C22⋊D53C211yesmaximal, characteristic, complemented
7C2×#16811280C211yesmaximal, characteristic, complemented
8C2×#16811273C213yesmaximal, complemented
9C2×#16811175C213yesmaximal, complemented
10C2×#16811154C213yesmaximal, complemented
11C2×#16811152C213yesmaximal, complemented
12C2×#41×Dic53C211yesmaximal, characteristic, complemented
13C2×C4×#11719947C213yesmaximal, complemented
14C22×Q8×Dic53C211yesmaximal, characteristic, complemented
15C2×C4×Q8×D53C211yesmaximal, characteristic, complemented
16C2×C53×#548C211yesmaximal, characteristic, complemented
17C22×#11719947C2213yes—
18C22×Q8×D53C2211yescharacteristic
19#16811349C22116yescomplemented
20#16811342C22148yescomplemented
21#16811313C22148yescomplemented
22#16811307C22148yescomplemented
23Q8×C22⋊D53C22116yescomplemented
24#16811280C22116yescomplemented
25#16811273C22148yescomplemented
26#16811175C22148yescomplemented
27C2×#13133216C2213yes—
28C2×#13133215C2213yes—
29#16811154C22148yescomplemented
30#16811152C22148yescomplemented
31C2×#13133210C2216yes—
32C2×#13133210C2216yes—
33C2×#13133209C2213yes—
34C2×#13133208C2213yes—
35C2×#13133207C2216yes—
36C2×#13133204C2213yes—
37C2×#13133199C2216yes—
38C2×#13133198C2216yes—
39C2×#13133196C2213yes—
40C2×#13133196C2213yes—
41C2×#13133194C2216yes—
42C2×#13133193C2213yes—
43C2×#13133188C2213yes—
44#41×Dic53C22116yescomplemented
45C4×#11719947C22148yescomplemented
46C2×Q8×Dic53C2213yes—
47C2×Q8×Dic53C2213yes—
48C2×Q8×Dic53C2211yescharacteristic
49C2×Q8×Dic53C2211yescharacteristic
50C2×Q8×Dic53C2211yescharacteristic
51C2×Q8×Dic53C22116yescomplemented
52C2×C4×Dic106C2213yes—
53C2×C4×Dic106C2213yes—
54C2×#13133182C2213yes—
55C2×#13133182C2213yes—
56C4×Q8×D53C22116yescomplemented
57C2×D4×Dic53C2213yes—
58C2×D4×Dic53C2213yes—
59C2×D4×Dic53C2213yes—
60C2×C4×C22⋊D53C2213yes—
61C2×C4×C22⋊D53C2213yes—
62C2×#13133177C2211yescharacteristic
63C2×#13133177C2213yes—
64C2×#13133176C2213yes—
65C2×#13133175C2213yes—
66C2×#13133173C2216yes—
67C2×#13133173C2213yes—
68C2×#13133173C2213yes—
69C22×C4⋊Dic53C2213yes—
70C2×C4⋊C4×D53C2213yes—
71C22×C4×Dic53C2213yes—
72C22×C4×Dic53C2213yes—
73C2×C42×D53C2213yes—
74C22×#41×C53C2211yescharacteristic
75C23×Q8×C53C2211yescharacteristic
76C53×#548C22116yescomplemented
77C2×#124×C53C2213yes—
78C2×C22⋊Q8×C53C2213yes—
79C2×#121×C53C2213yes—
80C2×C4×Q8×C53C2211yescharacteristic
81C2×C4×D4×C53C2213yes—
82C2×#11719947—224no—
83C2×#11719947C23124yes—
84C22×Dic106C2313yes—
85C22×Dic106C2313yes—
86C2×Q8×D53—28no—
87C2×Q8×D53C2318yes—
88C22×C22⋊D53C2313yes—
89C22×C22⋊D53C2311yescharacteristic
90C23×Dic53C2311yescharacteristic
91C23×Dic53C2313yes—
92C23×Dic53C2313yes—
93C23×Dic53C2313yes—
94C22×C4×D53C2311yescharacteristic
95C22×C4×D53C2313yes—
96C22×C4×D53C2313yes—
97#13133216C23124yes—
98#13133215C23124yes—
99#13133210C23148yes—
100#13133210C23148yes—
101#13133209C23124yes—
102#13133208C23124yes—
103#13133207C23148yes—
104#13133204C23124yes—
105#13133199C23148yes—
106#13133198C23148yes—
107#13133196C23124yes—
108#13133196C23124yes—
109#13133194C23148yes—
110#13133193C23124yes—
111#13133188C23124yes—
112Q8×Dic53C2318yes—
113Q8×Dic53C2318yes—
114Q8×Dic53C23124yes—
115Q8×Dic53C2318yes—
116Q8×Dic53C23124yes—
117C4×Dic106C23124yes—
118C4×Dic106C23124yes—
119#13133182C23124yes—
120#13133182C23124yes—
121D4×Dic53C23124yes—
122D4×Dic53C23124yes—
123D4×Dic53C23124yes—
124C4×C22⋊D53C23124yes—
125C4×C22⋊D53C23124yes—
126#13133177C2318yes—
127#13133177C23124yes—
128#13133176C23124yes—
129#13133175C23124yes—
130#13133173C23124yes—
131#13133173C23124yes—
132#13133173C23148yes—
133C2×C4⋊Dic53C23124yes—
134C2×C4⋊Dic53C2313yes—
135C2×C4⋊Dic53C2313yes—
136C2×C4⋊Dic53C2316yes—
137C2×C4⋊Dic53C2313yes—
138C2×C4⋊Dic53C2313yes—
139C2×#11719919C2316yes—
140C2×#11719919C2311yescharacteristic
141C2×#11719919C2313yes—
142C2×#11719919C2313yes—
143C2×#11719919C2313yes—
144C2×#11719919C2316yes—
145C2×#11719919C2316yes—
146C4⋊C4×D53C23124yes—
147C2×C22⋊Dic53C2313yes—
148C2×C22⋊Dic53C2313yes—
149C2×C22⋊Dic53C2316yes—
150C2×C22⋊Dic53C2316yes—
151C2×#11719921C2311yescharacteristic
152C2×#11719921C2313yes—
153C2×C4×Dic53C2313yes—
154C2×C4×Dic53C2313yes—
155C2×C4×Dic53C2313yes—
156C2×C4×Dic53C2311yescharacteristic
157C2×C4×Dic53C2313yes—
158C2×C4×Dic53C2313yes—
159C2×C4×Dic53C2313yes—
160C2×C4×Dic53C2313yes—
161C2×C4×Dic53C2313yes—
162C2×C4×Dic53C23124yes—
163C2×C4×Dic53C23124yes—
164C42×D53C23124yes—
165C2×#41×C53—28no—
166C2×#41×C53C2318yes—
167C22×Q8×C53—28no—
168C22×Q8×C53C2318yes—
169C22×Q8×C53C2313yes—
170C22×Q8×C53C2311yescharacteristic
171C22×D4×C53C2313yes—
172C22×D4×C53C2313yes—
173C23×C4×C53C2313yes—
174C23×C4×C53C2313yes—
175#124×C53C23124yes—
176C22⋊Q8×C53C23124yes—
177#121×C53C23124yes—
178C4×Q8×C53C2318yes—
179C4×D4×C53C23124yes—
180C2×C4⋊C4×C53C2313yes—
181C2×C4⋊C4×C53C2313yes—
182C2×C22⋊C4×C53C2311yescharacteristic
183C2×C22⋊C4×C53C2313yes—
184C2×C22⋊C4×C53C2316yes—
185C2×C42×C53C2313yes—
186C24×C53C2413yes—
187C24×C53C2411yescharacteristic
188C23×D53C2411yescharacteristic
189#41×C53—232no—
190C2×Q8×C53—232no—
191C2×Q8×C53—212no—
192C2×Q8×C53#4118yescomplemented
193C2×Q8×C53C24112yes—
194C2×Q8×C53C2414yes—
195C2×D4×C53—212no—
196C2×D4×C53—212no—
197C2×D4×C53C24112yes—
198C2×D4×C53C24112yes—
199C22×C4×C53—212no—
200C22×C4×C53—212no—
201C22×C4×C53C2411yescharacteristic
202C22×C4×C53C2413yes—
203C22×C4×C53C2413yes—
204C22×C4×C53C2411yescharacteristic
205C22×C4×C53C24112yes—
206C22×C4×C53C24112yes—
207C22×C4×C53C2413yes—
208C2×C22⋊D53—212no—
209C2×C22⋊D53#4118yescomplemented
210C2×C22⋊D53C24112yes—
211C2×C22⋊D53C2414yes—
212C22×Dic53—212no—
213C22×Dic53—212no—
214C22×Dic53—24no—
215C22×Dic53—212no—
216C22×Dic53C2411yescharacteristic
217C22×Dic53C2414yes—
218C22×Dic53C2413yes—
219C22×Dic53C2413yes—
220C22×Dic53C2413yes—
221C22×Dic53C2413yes—
222C22×Dic53C24112yes—
223C22×Dic53C24112yes—
224C22×Dic53C24112yes—
225C22×Dic53C2411yescharacteristic
226C22×Dic53C2411yescharacteristic
227Q8×D53—232no—
228#11719947—296no—
229C2×C4×D53—212no—
230C2×C4×D53—212no—
231C2×C4×D53—24no—
232C2×C4×D53C2414yes—
233C2×C4×D53C24112yes—
234C2×C4×D53C24112yes—
235C2×Dic106—212no—
236C2×Dic106—212no—
237C2×Dic106C24112yes—
238C2×Dic106C24112yes—
239C4⋊C4×C53C24112yes—
240C4⋊C4×C53C24112yes—
241C22⋊C4×C53C2414yes—
242C22⋊C4×C53C24112yes—
243C22⋊C4×C53C24124yes—
244C42×C53C24112yes—
245C22⋊Dic53C24112yes—
246C22⋊Dic53C24112yes—
247C22⋊Dic53C24124yes—
248C22⋊Dic53C24124yes—
249#11719921C2414yes—
250#11719921C24112yes—
251C4⋊Dic53C24112yes—
252C4⋊Dic53C24112yes—
253C4⋊Dic53C24124yes—
254C4⋊Dic53C24112yes—
255C4⋊Dic53C24112yes—
256#11719919C24124yes—
257#11719919C2414yes—
258#11719919C24112yes—
259#11719919C24112yes—
260#11719919C24112yes—
261#11719919C24124yes—
262#11719919C24124yes—
263C4×Dic53C24112yes—
264C4×Dic53C24112yes—
265C4×Dic53C24112yes—
266C4×Dic53C2414yes—
267C4×Dic53C24112yes—
268C4×Dic53C24112yes—
269C4×Dic53C24112yes—
270C4×Dic53C24112yes—
271C4×Dic53C24112yes—
272C23×C53—22no—
273C23×C53—22no—
274C23×C53—26no—
275C23×C53—26no—
276C23×C53—26no—
277C23×C53C2×#4114yes—
278C23×C53C2516yes—
279C23×C53C2512yes—
280C23×C53C2511yescharacteristic
281C22×D53—22no—
282C22×D53—22no—
283C22×D53C2×#4114yes—
284C22×D53C2512yes—
285Q8×C53—224no—
286Q8×C53C2×#41116yescomplemented
287D4×C53—224no—
288D4×C53—224no—
289C2×C4×C53—26no—
290C2×C4×C53—22no—
291C2×C4×C53—26no—
292C2×C4×C53—22no—
293C2×C4×C53—224no—
294C2×C4×C53—224no—
295C2×C4×C53C2×#41112yes—
296C2×C4×C53C2516yes—
297C2×C4×C53C2512yes—
298C2×C4×C53C2512yes—
299C2×C4×C53C2516yes—
300C2×C4×C53C2516yes—
301C22⋊D53—224no—
302C22⋊D53C2×#41116yescomplemented
303C2×Dic53—26no—
304C2×Dic53—26no—
305C2×Dic53—26no—
306C2×Dic53—224no—
307C2×Dic53—22no—
308C2×Dic53—26no—
309C2×Dic53—224no—
310C2×Dic53—28no—
311C2×Dic53—224no—
312C2×Dic53C2×#4114yes—
313C2×Dic532−1+414yes—
314C2×Dic53C2516yes—
315C2×Dic53C2516yes—
316C2×Dic53C2516yes—
317C2×Dic53C2512yes—
318C2×Dic53C2516yes—
319C2×Dic53C2512yes—
320C2×Dic53C2512yes—
321C4×D53—224no—
322C4×D53—224no—
323C4×D53—28no—
324Dic106—224no—
325Dic106—224no—
326C2×#3482—531nomaximal, Sylow
327C22×C53—22no—
328C22×C53—22no—
329C22×C53—28no—
330C22×C53—26no—
331C22×C53—26no—
332C22×C53—26no—
333C22×C53—412no—
334C22×C53C22×#4114yes—
335C22×C53C22×#4112yes—
336C22×C53C22×#4112yes—
337C22×C53C2×2−1+412yes—
338C22×C53C2611yescharacteristic
339C2×D53—22no—
340C2×D53—22no—
341C2×D53—28no—
342C2×D53C22×#4114yes—
343C4×C53—26no—
344C4×C53—22no—
345C4×C53—26no—
346C4×C53—22no—
347C4×C53C22×#41112yes—
348Dic53—26no—
349Dic53—26no—
350Dic53—26no—
351Dic53—22no—
352Dic53—26no—
353Dic53C22×#4114yes—
354Dic53C2×2−1+414yes—
355#3482—5332no—
356C2×#557—531no—
357C2×#556—533no—
358C2×#552—533no—
359C2×#552—533no—
360C2×D4×Q8—531no—
361C2×#548—531no—
362C2×#548D5311yescharacteristic, complemented
363C2×#547—533no—
364C2×#533—533no—
365C2×#529—533no—
366C2×#529—533no—
367C2×C4×#41—531no—
368C2×C4×#41—533no—
369C22×C4×Q8—531no—
370C22×C4×Q8—531no—
371C2×C53—24no—
372C2×C53—46no—
373C2×C53#348214yescomplemented
374C2×C53C23×#4111yescharacteristic
375C2×C53C23×#4111yescharacteristic
376C2×C53C22×2−1+411yescharacteristic
377D53—24no—
378C22×#41—533no—
379C22×#41C2×D5311yescharacteristic
380C23×Q8—531no—
381C23×Q8C2×D5311yescharacteristic
382#557—5316no—
383#556—5348no—
384#552—5348no—
385#552—5348no—
386D4×Q8—5316no—
387#548—5316no—
388#548C2×D53116yescomplemented
389#547—5348no—
390#533—5348no—
391C2×C4⋊Q8—533no—
392C2×C4⋊Q8—533no—
393#529—5348no—
394#529—5348no—
395C2×#126—536no—
396C2×#126—536no—
397C2×#125—533no—
398C2×#125—536no—
399C2×#125—533no—
400C2×#124—533no—
401C2×#124C2×D5313yes—
402C2×#123—536no—
403C2×#123—536no—
404C2×C22⋊Q8—533no—
405C2×C22⋊Q8—533no—
406C2×C22⋊Q8—533no—
407C2×C22⋊Q8—536no—
408C2×C22⋊Q8C2×D5313yes—
409C2×#121—533no—
410C2×#121C2×D5313yes—
411C4×#41—5316no—
412C4×#41—5348no—
413C2×C4×Q8—531no—
414C2×C4×Q8—533no—
415C2×C4×Q8—531no—
416C2×C4×Q8—533no—
417C2×C4×Q8—531no—
418C2×C4×Q8—5316no—
419C2×C4×Q8—5316no—
420C2×C4×Q8—533no—
421C2×C4×Q8—533no—
422C2×C4×Q8—533no—
423C2×C4×Q8—533no—
424C2×C4×Q8C2×D5311yescharacteristic
425C2×C4×D4—533no—
426C2×C4×D4—533no—
427C2×C4×D4—533no—
428C2×C4×D4—533no—
429C2×C4×D4—533no—
430C2×C4×D4—531no—
431C2×C4×D4—533no—
432C2×C4×D4C2×D5313yes—
433C2×#117—533no—
434C2×#117—533no—
435C2×#117—533no—
436C2×#117—533no—
437C2×#117—536no—
438C22×C4⋊C4—533no—
439C22×C4⋊C4—533no—
440C22×C42—533no—
441C22×C42—533no—
442C22×C42—533no—
443C53C2×#348211yescharacteristic, complemented, Sylow
444C2×#41—28no—
445C2×#41—5324no—
446C2×#41—10624no—
447C2×#41C22×D5318yes—
448C22×Q8—28no—
449C22×Q8—538no—
450C22×Q8—533no—
451C22×Q8—533no—
452C22×Q8—1068no—
453C22×Q8C22×D5313yes—
454C22×Q8C22×D5318yes—
455C22×Q8C22×D5311yescharacteristic
456C22×D4—531no—
457C22×D4—533no—
458C22×D4C22×D5313yes—
459C22×D4C22×D5313yes—
460C23×C4—533no—
461C23×C4—533no—
462C23×C4—533no—
463C23×C4—531no—
464C23×C4—533no—
465C23×C4—533no—
466C23×C4—531no—
467C23×C4C22×D5313yes—
468C23×C4C22×D5313yes—
469C4⋊Q8—5324no—
470C4⋊Q8—5324no—
471#126—5348no—
472#126—5348no—
473#125—5324no—
474#125—5348no—
475#125—5324no—
476#124—5324no—
477#124C22×D53124yes—
478#123—5348no—
479#123—5348no—
480C22⋊Q8—5324no—
481C22⋊Q8—5324no—
482C22⋊Q8—5324no—
483C22⋊Q8—5348no—
484C22⋊Q8C22×D53124yes—
485#121—5324no—
486#121C22×D53124yes—
487C4×Q8—538no—
488C4×Q8—5324no—
489C4×Q8—538no—
490C4×Q8—5324no—
491C4×Q8—538no—
492C4×Q8—5324no—
493C4×Q8—5324no—
494C4×Q8—5324no—
495C4×Q8—5324no—
496C4×Q8C22×D5318yes—
497C4×D4—5324no—
498C4×D4—5324no—
499C4×D4—5324no—
500C4×D4—5324no—
501C4×D4—5324no—
502C4×D4—538no—
503C4×D4—5324no—
504C4×D4C22×D53124yes—
505#117—5324no—
506#117—5324no—
507#117—5324no—
508#117—5324no—
509#117—5348no—
510C2×C4⋊C4—533no—
511C2×C4⋊C4—533no—
512C2×C4⋊C4—536no—
513C2×C4⋊C4—536no—
514C2×C4⋊C4—5324no—
515C2×C4⋊C4—531no—
516C2×C4⋊C4—533no—
517C2×C4⋊C4—533no—
518C2×C4⋊C4—533no—
519C2×C4⋊C4—533no—
520C2×C4⋊C4—533no—
521C2×C4⋊C4—536no—
522C2×C4⋊C4—536no—
523C2×C4⋊C4—5324no—
524C2×C4⋊C4C22×D5313yes—
525C2×C4⋊C4C22×D5313yes—
526C2×C22⋊C4—533no—
527C2×C22⋊C4—536no—
528C2×C22⋊C4—533no—
529C2×C22⋊C4—536no—
530C2×C22⋊C4—531no—
531C2×C22⋊C4—533no—
532C2×C22⋊C4C22×D5311yescharacteristic
533C2×C22⋊C4C22×D5313yes—
534C2×C22⋊C4C22×D5316yes—
535C2×C42—533no—
536C2×C42—533no—
537C2×C42—533no—
538C2×C42—533no—
539C2×C42—533no—
540C2×C42—533no—
541C2×C42—533no—
542C2×C42—531no—
543C2×C42—533no—
544C2×C42—5324no—
545C2×C42—5324no—
546C2×C42—5324no—
547C2×C42C22×D5313yes—
548C24—531no—
549C24C23×D5313yes—
550C24C23×D5311yescharacteristic
551#41—232no—
552#41—10696no—
553C2×Q8—232no—
554C2×Q8—212no—
555C2×Q8—5312no—
556C2×Q8—5312no—
557C2×Q8—10612no—
558C2×Q8—10632no—
559C2×Q8—10612no—
560C2×Q8#1171994718yescomplemented
561C2×Q8C23×D53112yes—
562C2×Q8C23×D5314yes—
563C2×D4—212no—
564C2×D4—212no—
565C2×D4—538no—
566C2×D4—534no—
567C2×D4—5312no—
568C2×D4—10612no—
569C2×D4C23×D53112yes—
570C2×D4C23×D53112yes—
571C22×C4—212no—
572C22×C4—212no—
573C22×C4—531no—
574C22×C4—531no—
575C22×C4—534no—
576C22×C4—5312no—
577C22×C4—5312no—
578C22×C4—5312no—
579C22×C4—531no—
580C22×C4—533no—
581C22×C4—533no—
582C22×C4—533no—
583C22×C4—533no—
584C22×C4—534no—
585C22×C4—5312no—
586C22×C4—5312no—
587C22×C4—1064no—
588C22×C4—10612no—
589C22×C4—1064no—
590C22×C4—10612no—
591C22×C4—10612no—
592C22×C4—10612no—
593C22×C4—10612no—
594C22×C4C23×D53112yes—
595C22×C4C23×D53112yes—
596C22×C4C23×D5311yescharacteristic
597C22×C4C23×D5311yescharacteristic
598C22×C4C23×D5313yes—
599C22×C4C23×D5313yes—
600C22×C4C23×D5313yes—
601C4⋊C4—5312no—
602C4⋊C4—5312no—
603C4⋊C4—5324no—
604C4⋊C4—5324no—
605C4⋊C4—534no—
606C4⋊C4—5312no—
607C4⋊C4—5312no—
608C4⋊C4—5312no—
609C4⋊C4—5312no—
610C4⋊C4—5312no—
611C4⋊C4—5324no—
612C4⋊C4—5324no—
613C4⋊C4C23×D53112yes—
614C4⋊C4C23×D53112yes—
615C22⋊C4—5312no—
616C22⋊C4—5324no—
617C22⋊C4—5312no—
618C22⋊C4—5324no—
619C22⋊C4—534no—
620C22⋊C4—5312no—
621C22⋊C4C23×D5314yes—
622C22⋊C4C23×D53112yes—
623C22⋊C4C23×D53124yes—
624C42—534no—
625C42—5312no—
626C42—5312no—
627C42—5312no—
628C42—5312no—
629C42—5312no—
630C42—5312no—
631C42—5312no—
632C42—5312no—
633C42C23×D53112yes—
634C23—22no—
635C23—22no—
636C23—26no—
637C23—26no—
638C23—26no—
639C23—534no—
640C23—532no—
641C23—1062no—
642C23—1062no—
643C23#41×D5314yes—
644C23C24×D5316yes—
645C23C24×D5312yes—
646C23C24×D5311yescharacteristic
647Q8—224no—
648Q8—10624no—
649Q8—10624no—
650Q8C2×#11719947116yescomplemented
651D4—224no—
652D4—224no—
653D4—5316no—
654D4—10624no—
655C2×C4—26no—
656C2×C4—22no—
657C2×C4—26no—
658C2×C4—22no—
659C2×C4—224no—
660C2×C4—224no—
661C2×C4—534no—
662C2×C4—534no—
663C2×C4—532no—
664C2×C4—532no—
665C2×C4—532no—
666C2×C4—536no—
667C2×C4—536no—
668C2×C4—536no—
669C2×C4—536no—
670C2×C4—1068no—
671C2×C4—10624no—
672C2×C4—1068no—
673C2×C4—10624no—
674C2×C4—10624no—
675C2×C4—1066no—
676C2×C4—1066no—
677C2×C4—10624no—
678C2×C4—1066no—
679C2×C4—1062no—
680C2×C4—1066no—
681C2×C4—10624no—
682C2×C4C2×#11719947112yes—
683C2×C4C24×D5312yes—
684C2×C4C24×D5312yes—
685C2×C4C24×D5316yes—
686C2×C4C24×D5316yes—
687C2×C4C24×D5316yes—
688C22—28no—
689C22—22no—
690C22—22no—
691C22—26no—
692C22—26no—
693C22—26no—
694C22—412no—
695C22—534no—
696C22—1068no—
697C22—1062no—
698C22—1062no—
699C22C2×#41×D5314yes—
700C22C2×#41×D5312yes—
701C22C22×#1171994712yes—
702C222−1+4×D5312yes—
703C22C25×D5311yescharacteristic
704C4—26no—
705C4—22no—
706C4—26no—
707C4—22no—
708C4—534no—
709C4—534no—
710C4—1066no—
711C4—1066no—
712C4—1066no—
713C4—1062no—
714C4—1066no—
715C4C22×#11719947112yes—
716C2—24no—
717C2—46no—
718C2—1064no—
719C2#2028833114yesdirect product factor
720C2C22×#41×D5311yescharacteristic
721C2C23×#1171994711yescharacteristic
722C2C2×2−1+4×D5311yescharacteristic
subgroupG/H
centerC23C24×D53
commutator, cosocleC22×C53C26
FrattiniC22C25×D53
FittingC2×C53×#548C2
socleC23×C53C25
hypercenterC2×#548D53
hypocenterC53C2×#3482
LCSG ▷ C22×C53 ▷ C53
UCSC1 ◁ C23 ◁ C2×#548

Complements

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

NH ≅ G/Ntype
1. #20288331C2direct (N × H)
2. C2×#16811349C2semidirect
3. C2×#16811342C2semidirect
4. C2×#16811313C2semidirect
5. C2×#16811307C2semidirect
6. C2×Q8×C22⋊D53C2semidirect
7. C2×#16811280C2semidirect
8. C2×#16811273C2semidirect
9. C2×#16811175C2semidirect
10. C2×#16811154C2semidirect
11. C2×#16811152C2semidirect
12. C2×#41×Dic53C2semidirect
13. C2×C4×#11719947C2semidirect
14. C22×Q8×Dic53C2semidirect
15. C2×C4×Q8×D53C2semidirect
16. C2×C53×#548C2semidirect
19. #16811349C22semidirect
20. #16811342C22semidirect
21. #16811313C22semidirect
22. #16811307C22semidirect
23. Q8×C22⋊D53C22semidirect
24. #16811280C22semidirect
25. #16811273C22semidirect
26. #16811175C22semidirect
29. #16811154C22semidirect
30. #16811152C22semidirect
44. #41×Dic53C22semidirect
45. C4×#11719947C22semidirect
51. C2×Q8×Dic53C22semidirect
56. C4×Q8×D53C22semidirect
76. C53×#548C22semidirect
192. C2×Q8×C53#41semidirect
209. C2×C22⋊D53#41semidirect
286. Q8×C53C2×#41semidirect
302. C22⋊D53C2×#41semidirect
362. C2×#548D53semidirect
373. C2×C53#3482semidirect
388. #548C2×D53semidirect
443. C53C2×#3482semidirect
560. C2×Q8#11719947semidirect
650. Q8C2×#11719947semidirect
719. C2#20288331direct (N × H)

Maximal subgroups

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

subgroupindex
#202883312
C2×#168113492
C2×#168113422
C2×#168113132
C2×#168113072
C2×Q8×C22⋊D532
C2×#168112802
C2×#168112732
C2×#168111752
C2×#168111542
C2×#168111522
C2×#41×Dic532
C2×C4×#117199472
C22×Q8×Dic532
C2×C4×Q8×D532
C2×C53×#5482
C2×#348253