Finite Group Databasebeta

← #28320582

Subgroups — #28320582

Order 13568.

1114 conjugacy classes of subgroups.

The subgroup lattice has 616 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#16811626C211yesmaximal, characteristic
2C2×C4×#11719940C211yesmaximal, characteristic
3#20290909C214yesmaximal, complemented
4#20290908C212yesmaximal, complemented
5#20290907C212yesmaximal, complemented
6#20290421C214yesmaximal
7C2×#16811175C211yesmaximal, characteristic
8C22×#11719940C2212yes—
9C22×#11719940C2212yes—
10#16811626—21no—
11#16811626—21no—
12#16811626C2212yes—
13#16811626C2212yes—
14C2×#13133284—21no—
15C2×C4⋊#88324—21no—
16C4×#11719940—21no—
17C4×#11719940—21no—
18C4×#11719940C2212yes—
19C4×#11719940C2212yes—
20C2×#13133272—21no—
21C2×C4×#88324—21no—
22#16811503C2218yes—
23#16811503C2218yes—
24#16811175—21no—
25#16811175—21no—
26#16811175—21no—
27#16811175—21no—
28#16811175—21no—
29#16811175—21no—
30#16811175—21no—
31#16811175—21no—
32C2×#13133216C2211yescharacteristic
33C2×#13133215C2211yescharacteristic
34C2×#13133207C2212yes—
35C2×#13133196C411yescharacteristic
36C2×#13133194C412yes—
37C2×Q8×Dic53C411yescharacteristic
38C2×#13133182C412yes—
39C2×#13133182C411yescharacteristic
40C2×#13133173C2212yes—
41C22×C4×Dic53C2211yescharacteristic
42C2×C22⋊Q8×C53C411yescharacteristic
43C2×#11719940—22no—
44C2×#11719940—24no—
45C2×#11719940—22no—
46C2×#11719940—28no—
47C2×#11719940Q818yescomplemented
48C2×#11719940C2314yes—
49C2×#11719940C2314yes—
50C22×#88324—22no—
51C22×#88324—22no—
52#13133284—22no—
53#13133284—22no—
54C4⋊#88324—22no—
55C4⋊#88324—22no—
56#13133272—22no—
57#13133272—22no—
58C4×#88324—22no—
59C4×#88324—22no—
60C22×Dic106C2×C412yes—
61C22×Dic106C2×C411yescharacteristic
62C23×Dic53C2×C411yescharacteristic
63C23×Dic53C2311yescharacteristic
64#13133216—21no—
65#13133216—21no—
66#13133216—21no—
67#13133216—21no—
68#13133215—21no—
69#13133215—21no—
70#13133215—21no—
71#13133215—21no—
72#13133207—22no—
73#13133207—22no—
74#13133207—22no—
75#13133207—22no—
76#13133196—21no—
77#13133196—21no—
78#13133196—21no—
79#13133196—21no—
80#13133194—22no—
81#13133194—22no—
82#13133194—22no—
83#13133194—22no—
84Q8×Dic53—21no—
85Q8×Dic53—21no—
86Q8×Dic53—21no—
87Q8×Dic53—21no—
88#13133182—21no—
89#13133182—21no—
90#13133182—21no—
91#13133182—21no—
92#13133182—22no—
93#13133182—22no—
94#13133182—22no—
95#13133182—22no—
96#13133173—22no—
97#13133173—22no—
98#13133173—22no—
99#13133173—22no—
100C2×C4⋊Dic53C2×C412yes—
101C2×C4⋊Dic53C2×C411yescharacteristic
102C2×#11719919C2×C412yes—
103C2×#11719919C2×C412yes—
104C2×#11719919C2×C412yes—
105C2×#11719919C2312yes—
106C2×#11719919C2312yes—
107C2×C22⋊Dic53C2×C412yes—
108C2×C4×Dic53C2×C412yes—
109C2×C4×Dic53C2×C412yes—
110C2×C4×Dic53C2×C411yescharacteristic
111C2×C4×Dic53C2×C411yescharacteristic
112C2×C4×Dic53D411yescharacteristic
113C2×C4×Dic53D411yescharacteristic
114C2×C4×Dic53D411yescharacteristic
115C2×C4×Dic53D411yescharacteristic
116C2×C4×Dic53D411yescharacteristic
117C2×C4×Dic53D411yescharacteristic
118C2×C4×Dic53D411yescharacteristic
119C2×C4×Dic53D411yescharacteristic
120C2×C4×Dic53C2311yescharacteristic
121C2×C4×Dic53C2311yescharacteristic
122C22×Q8×C53C2×C411yescharacteristic
123C23×C4×C53C2×C411yescharacteristic
124C22⋊Q8×C53—21no—
125C22⋊Q8×C53—21no—
126C22⋊Q8×C53—21no—
127C22⋊Q8×C53—21no—
128C2×C4⋊C4×C53C2×C412yes—
129C2×C4⋊C4×C53C2×C411yescharacteristic
130C2×C22⋊C4×C53C2×C412yes—
131C24×C53C22×C411yescharacteristic
132C2×Q8×C53—21no—
133C2×Q8×C53—21no—
134C2×Q8×C53—42no—
135C22×C4×C53—42no—
136C22×C4×C53C22⋊C411yescharacteristic
137C22×C4×C53C22⋊C411yescharacteristic
138C22×C4×C53C22⋊C411yescharacteristic
139C22×C4×C53C22⋊C411yescharacteristic
140C22×C4×C53C22×C411yescharacteristic
141C22×C4×C53C22×C412yes—
142C22×C4×C53C22×C412yes—
143C22×C4×C53C22×C411yescharacteristic
144C22×Dic53—42no—
145C22×Dic53C22⋊C411yescharacteristic
146C22×Dic53C22⋊C411yescharacteristic
147C22×Dic53C22⋊C411yescharacteristic
148C22×Dic53C22⋊C411yescharacteristic
149C22×Dic53C22×C411yescharacteristic
150C22×Dic53C22×C412yes—
151C22×Dic53C22×C412yes—
152C22×Dic53C22×C411yescharacteristic
153C22×Dic53C22×C411yescharacteristic
154C22×Dic53C2×D411yescharacteristic
155C22×Dic53C2×D411yescharacteristic
156C22×Dic53C2×D411yescharacteristic
157C22×Dic53C2×D411yescharacteristic
158C22×Dic53C2×Q814yes—
159C22×Dic53#4114yes—
160C22×Dic53C2411yescharacteristic
161C2×Dic106—21no—
162C2×Dic106—21no—
163C2×Dic106—22no—
164C2×Dic106—22no—
165C2×Dic106—44no—
166C2×Dic106—42no—
167#11719940—28no—
168#11719940—24no—
169#11719940—28no—
170#11719940—44no—
171#11719940C2×Q818yes—
172C2×#88324—28no—
173C2×#88324—24no—
174C2×#88324—24no—
175C2×#88324—44no—
176C4⋊C4×C53—22no—
177C4⋊C4×C53—22no—
178C4⋊C4×C53—21no—
179C4⋊C4×C53—21no—
180C22⋊C4×C53—22no—
181C22⋊C4×C53—22no—
182C22⋊Dic53—22no—
183C22⋊Dic53—22no—
184C4⋊Dic53—21no—
185C4⋊Dic53—21no—
186C4⋊Dic53—22no—
187C4⋊Dic53—22no—
188#11719919—22no—
189#11719919—22no—
190#11719919—22no—
191#11719919—22no—
192#11719919—22no—
193#11719919—22no—
194#11719919—22no—
195#11719919—22no—
196#11719919—22no—
197#11719919—22no—
198C4×Dic53—22no—
199C4×Dic53—22no—
200C4×Dic53—22no—
201C4×Dic53—22no—
202C4×Dic53C22⋊C411yescharacteristic
203C4×Dic53C22⋊C411yescharacteristic
204C4×Dic53C22⋊C411yescharacteristic
205C4×Dic53C22⋊C411yescharacteristic
206C4×Dic53C22⋊C411yescharacteristic
207C4×Dic53C22⋊C411yescharacteristic
208C4×Dic53C22⋊C411yescharacteristic
209C4×Dic53C22⋊C411yescharacteristic
210C4×Dic53C2×D411yescharacteristic
211C4×Dic53C2×D411yescharacteristic
212C4×Dic53C2×D411yescharacteristic
213C4×Dic53C2×D411yescharacteristic
214C4×Dic53C2×D411yescharacteristic
215C4×Dic53C2×D411yescharacteristic
216C4×Dic53C2×D411yescharacteristic
217C4×Dic53C2×D411yescharacteristic
218C23×C53—41no—
219C23×C53—41no—
220C23×C53C2×C22⋊C411yescharacteristic
221C23×C53C2×C22⋊C411yescharacteristic
222C23×C53C4×Q814yes—
223C23×C53C23×C411yescharacteristic
224Q8×C53—42no—
225Q8×C53—42no—
226C2×C4×C53—22no—
227C2×C4×C53—22no—
228C2×C4×C53—22no—
229C2×C4×C53—22no—
230C2×C4×C53—22no—
231C2×C4×C53—41no—
232C2×C4×C53—41no—
233C2×C4×C53—41no—
234C2×C4×C53—42no—
235C2×C4×C53—42no—
236C2×C4×C53C2×C22⋊C411yescharacteristic
237C2×C4×C53C2×C22⋊C411yescharacteristic
238C2×C4×C53C2×C22⋊C411yescharacteristic
239C2×C4×C53C2×C22⋊C411yescharacteristic
240C2×Dic53—22no—
241C2×Dic53—22no—
242C2×Dic53—22no—
243C2×Dic53—22no—
244C2×Dic53—22no—
245C2×Dic53—41no—
246C2×Dic53—41no—
247C2×Dic53—41no—
248C2×Dic53—42no—
249C2×Dic53—42no—
250C2×Dic53C2×C22⋊C411yescharacteristic
251C2×Dic53C2×C22⋊C411yescharacteristic
252C2×Dic53C2×C22⋊C411yescharacteristic
253C2×Dic53C2×C22⋊C411yescharacteristic
254C2×Dic53C2×C22⋊C411yescharacteristic
255C2×Dic53C2×C22⋊C411yescharacteristic
256C2×Dic53C4×Q814yes—
257C2×Dic53C22⋊Q814yes—
258C2×Dic53C22⋊Q814yes—
259C2×Dic53#12412yes—
260C2×Dic53#12412yes—
261C2×Dic53C4⋊Q812yes—
262C2×Dic53C4⋊Q812yes—
263C2×Dic53C22×D411yescharacteristic
264C2×Dic53C22×D411yescharacteristic
265C2×Dic53C22×Q812yes—
266C2×Dic53C2×#4112yes—
267Dic106—44no—
268Dic106—42no—
269Dic106—42no—
270Dic106—44no—
271#88324—28no—
272#88324—44no—
273#21337—531nomaximal, Sylow
274C22×C53—21no—
275C22×C53—21no—
276C22×C53—21no—
277C22×C53—21no—
278C22×C53—21no—
279C22×C53—21no—
280C22×C53—44no—
281C22×C53#39112yes—
282C22×C53#39112yes—
283C22×C53C22×C22⋊C411yescharacteristic
284C22×C53C2×C4×Q812yes—
285C4×C53—21no—
286C4×C53—21no—
287C4×C53—21no—
288C4×C53—42no—
289C4×C53—42no—
290C4×C53#41312yes—
291Dic53—21no—
292Dic53—21no—
293Dic53—21no—
294Dic53—42no—
295Dic53—42no—
296Dic53#39112yes—
297Dic53#39112yes—
298Dic53#41312yes—
299Dic53C2×C22⋊Q812yes—
300Dic53C2×#12411yescharacteristic
301Dic53C2×C4⋊Q811yescharacteristic
302C2×#533—531no—
303C2×#423—531no—
304C2×C4×M16—531no—
305#1998—534no—
306#1997—532no—
307#1997—532no—
308#1742—534no—
309C2×C53—22no—
310C2×C53—22no—
311C2×C53—22no—
312C2×C53C2×#39111yescharacteristic
313C2×C53C2×#41311yescharacteristic
314C2×C53C2×#41311yescharacteristic
315C22×M16—532no—
316C22×M16—532no—
317#533—1061no—
318#533—1061no—
319#533—1061no—
320#533—1061no—
321#533—1061no—
322#533—1061no—
323#533—1061no—
324#533—1061no—
325C2×C4⋊Q8—531no—
326C2×C4⋊Q8—531no—
327C2×#125—532no—
328C2×C22⋊Q8—531no—
329C2×C22⋊Q8—532no—
330C2×C22⋊Q8#615711yescharacteristic
331C2×C4×Q8—531no—
332C2×C4×Q8—531no—
333C2×C4×Q8—532no—
334C2×#117—532no—
335C22×C42—531no—
336#423—532no—
337#423—532no—
338#423—1061no—
339#423—1061no—
340C2×C4⋊C8—1061no—
341C2×C4⋊C8—1061no—
342C4×M16—532no—
343C4×M16—532no—
344C4×M16—1061no—
345C4×M16—1061no—
346C2×#97—1061no—
347C2×C4×C8—1061no—
348#343—538no—
349#343—538no—
350C53#2133711yescharacteristic, complemented, Sylow
351C22×Q8—532no—
352C22×Q8—531no—
353C22×Q8C2×#615711yescharacteristic
354C23×C4—531no—
355C23×C4—531no—
356C23×C4C2×#615711yescharacteristic
357C2×M16—534no—
358C2×M16—538no—
359C2×M16—534no—
360C2×M16—1062no—
361C2×M16—1064no—
362C2×M16—1062no—
363C2×M16—1068no—
364C22×C8—1062no—
365C22×C8—1062no—
366C4⋊Q8—1061no—
367C4⋊Q8—1061no—
368C4⋊Q8—1061no—
369C4⋊Q8—1061no—
370C4⋊Q8—1061no—
371C4⋊Q8—1061no—
372C4⋊Q8—1061no—
373C4⋊Q8—1061no—
374#125—1062no—
375#125—1062no—
376#125—1062no—
377#125—1062no—
378C22⋊Q8—21no—
379C22⋊Q8—21no—
380C22⋊Q8—21no—
381C22⋊Q8—21no—
382C22⋊Q8—1062no—
383C22⋊Q8—1062no—
384C22⋊Q8—1062no—
385C22⋊Q8—1062no—
386C22⋊Q8—1061no—
387C22⋊Q8—1061no—
388C22⋊Q8—1061no—
389C22⋊Q8—1061no—
390C4×Q8—1062no—
391C4×Q8—1062no—
392C4×Q8—1062no—
393C4×Q8—1062no—
394C4×Q8—1061no—
395C4×Q8—1061no—
396C4×Q8—1061no—
397C4×Q8—1061no—
398C4×Q8—1061no—
399C4×Q8—1061no—
400C4×Q8—1061no—
401C4×Q8—1061no—
402#117—1062no—
403#117—1062no—
404#117—1062no—
405#117—1062no—
406C2×C4⋊C4—532no—
407C2×C4⋊C4—532no—
408C2×C4⋊C4—532no—
409C2×C4⋊C4—532no—
410C2×C4⋊C4—532no—
411C2×C4⋊C4—532no—
412C2×C4⋊C4—531no—
413C2×C4⋊C4C2×#615712yes—
414C2×C4⋊C4C2×#615711yescharacteristic
415C2×C22⋊C4—532no—
416C2×C22⋊C4C2×#615712yes—
417C2×C42—531no—
418C2×C42—531no—
419C2×C42—531no—
420C2×C42—531no—
421C2×C42—531no—
422C2×C42—531no—
423C2×C42—531no—
424C2×C42—531no—
425C2×C42—531no—
426C2×C42—531no—
427C2×C42—531no—
428C2×C42—531no—
429C2×C42—532no—
430C2×C42—532no—
431C4⋊C8—1062no—
432C4⋊C8—1062no—
433C4⋊C8—1062no—
434C4⋊C8—1062no—
435#97—1062no—
436#97—1062no—
437C4×C8—1062no—
438C4×C8—1062no—
439C24C22×#615711yescharacteristic
440C2×Q8—21no—
441C2×Q8—21no—
442C2×Q8—42no—
443C2×Q8—1062no—
444C2×Q8—1062no—
445C2×Q8—1061no—
446C2×Q8—1061no—
447C2×Q8—2122no—
448C2×Q8—2124no—
449C22×C4—42no—
450C22×C4—531no—
451C22×C4—531no—
452C22×C4—531no—
453C22×C4—531no—
454C22×C4—534no—
455C22×C4—534no—
456C22×C4—531no—
457C22×C4—531no—
458C22×C4—531no—
459C22×C4—531no—
460C22×C4—531no—
461C22×C4—531no—
462C22×C4—531no—
463C22×C4—531no—
464C22×C4—532no—
465C22×C4—532no—
466C22×C4—2122no—
467C22×C4C22⋊#615711yescharacteristic
468C22×C4C22⋊#615711yescharacteristic
469C22×C4C22⋊#615711yescharacteristic
470C22×C4C22⋊#615711yescharacteristic
471C22×C4C22×#615711yescharacteristic
472C22×C4C22×#615712yes—
473C22×C4C22×#615712yes—
474C22×C4C22×#615711yescharacteristic
475M16—538no—
476M16—1068no—
477M16—1064no—
478M16—1068no—
479M16—2124no—
480C2×C8—1064no—
481C2×C8—1068no—
482C2×C8—1064no—
483C2×C8—2124no—
484C4⋊C4—22no—
485C4⋊C4—22no—
486C4⋊C4—21no—
487C4⋊C4—21no—
488C4⋊C4—1062no—
489C4⋊C4—1062no—
490C4⋊C4—1062no—
491C4⋊C4—1062no—
492C4⋊C4—1062no—
493C4⋊C4—1062no—
494C4⋊C4—1062no—
495C4⋊C4—1062no—
496C4⋊C4—1062no—
497C4⋊C4—1062no—
498C4⋊C4—1062no—
499C4⋊C4—1062no—
500C4⋊C4—1061no—
501C4⋊C4—1061no—
502C22⋊C4—22no—
503C22⋊C4—22no—
504C22⋊C4—1062no—
505C22⋊C4—1062no—
506C42—531no—
507C42—531no—
508C42—531no—
509C42—531no—
510C42—531no—
511C42—531no—
512C42—531no—
513C42—531no—
514C42—531no—
515C42—531no—
516C42—531no—
517C42—531no—
518C42—531no—
519C42—531no—
520C42—531no—
521C42—531no—
522C42—1062no—
523C42—1062no—
524C42—1062no—
525C42—1062no—
526C23—41no—
527C23—41no—
528C23C2×C22⋊#615711yescharacteristic
529C23C2×C22⋊#615711yescharacteristic
530C23Q8×#615714yes—
531C23C23×#615711yescharacteristic
532Q8—42no—
533Q8—42no—
534Q8—2122no—
535Q8—2124no—
536Q8—2124no—
537Q8—2122no—
538C2×C4—22no—
539C2×C4—22no—
540C2×C4—22no—
541C2×C4—22no—
542C2×C4—22no—
543C2×C4—41no—
544C2×C4—41no—
545C2×C4—41no—
546C2×C4—42no—
547C2×C4—42no—
548C2×C4—534no—
549C2×C4—534no—
550C2×C4—532no—
551C2×C4—532no—
552C2×C4—532no—
553C2×C4—532no—
554C2×C4—531no—
555C2×C4—531no—
556C2×C4—531no—
557C2×C4—531no—
558C2×C4—531no—
559C2×C4—534no—
560C2×C4—532no—
561C2×C4—532no—
562C2×C4—531no—
563C2×C4—531no—
564C2×C4—531no—
565C2×C4—1062no—
566C2×C4—1062no—
567C2×C4—1062no—
568C2×C4—1062no—
569C2×C4—1062no—
570C2×C4—2121no—
571C2×C4—2121no—
572C2×C4—2121no—
573C2×C4—2122no—
574C2×C4—2122no—
575C2×C4C2×C22⋊#615711yescharacteristic
576C2×C4C2×C22⋊#615711yescharacteristic
577C2×C4C2×C22⋊#615711yescharacteristic
578C2×C4C2×C22⋊#615711yescharacteristic
579C8—1068no—
580C8—2124no—
581C22—21no—
582C22—21no—
583C22—21no—
584C22—21no—
585C22—21no—
586C22—21no—
587C22—44no—
588C22(C2×#6157)⋊Q812yes—
589C22(C2×#6157)⋊Q812yes—
590C22C22×C22⋊#615711yescharacteristic
591C22C2×Q8×#615712yes—
592C4—21no—
593C4—21no—
594C4—21no—
595C4—42no—
596C4—42no—
597C4—532no—
598C4—531no—
599C4—531no—
600C4—532no—
601C4—532no—
602C4—532no—
603C4—1061no—
604C4—1061no—
605C4—1061no—
606C4—2122no—
607C4—2122no—
608C4#1681161012yes—
609C2—22no—
610C2—22no—
611C2—22no—
612C2C2×(C2×#6157)⋊Q811yescharacteristic
613C2D53⋊#41311yescharacteristic
614C2C2×#1681161011yescharacteristic
subgroupG/H
centerC22C22×C22⋊#6157
commutatorC23×C53C23×C4
FrattiniC23C23×#6157
FittingC2×C22⋊Q8×C53C4
socleC22×C53C22×C22⋊C4
prime radicalC24×C53C22×C4
cosocleC22×Dic53C24
hypercenterC2×C22⋊Q8#6157
hypocenterC53#21337
LCSG ▷ C23×C53 ▷ C2×C53 ▷ C53
UCSC1 ◁ C22 ◁ C23×C4 ◁ C2×C22⋊Q8

Complements

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

NH ≅ G/Ntype
3. #20290909C2semidirect
4. #20290908C2semidirect
5. #20290907C2semidirect
47. C2×#11719940Q8semidirect
350. C53#21337semidirect

Maximal subgroups

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

subgroupindex
C2×#168116262
C2×C4×#117199402
#202909092
#202909082
#202909072
#202904212
C2×#168111752
#2133753