Finite Group Databasebeta

← #27387849

Subgroups — #27387849

Order 13568.

1598 conjugacy classes of subgroups.

The subgroup lattice has 686 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#16811201C211yesmaximal, characteristic
2C2×#16810597C211yesmaximal, characteristic, complemented
3C2×#16810596C211yesmaximal, characteristic, complemented
4C2×#16810596C211yesmaximal, characteristic, complemented
5#20277331C211yesmaximal, characteristic, complemented
6#20277071C211yesmaximal, characteristic, complemented
7#20275549C218yesmaximal, complemented
8C2×C53×#410C211yesmaximal, characteristic, complemented
9C22×#11719945C2211yescharacteristic
10#16811201—22no—
11#16811201—22no—
12#16811201—22no—
13#16811201—22no—
14C2×#13133196C412yescomplemented
15C2×#13133193C412yescomplemented
16C2×#13133191C412yescomplemented
17C2×#13133189C2214yes—
18C2×#13133188C412yescomplemented
19C2×#13133176C2211yescharacteristic
20C2×#13133176C2211yescharacteristic
21C22×C4⋊Dic53C2211yescharacteristic
22#16810597—24no—
23#16810596—24no—
24#16810596—24no—
25C2×#13133132C2214yes—
26C2×#13133131C2214yes—
27C2×#13133131C2214yes—
28#16810521—22no—
29#16810519—22no—
30#16810481C2214yes—
31C4⋊C4×Dic53—22no—
32C4×C4⋊Dic53—22no—
33C22⋊C4×Dic53C2214yes—
34C22×#41×C53C2211yescharacteristic
35C2×#117×C53C2211yescharacteristic
36C2×#117×C53C2211yescharacteristic
37C53×#410—24no—
38C2×#99×C53C2214yes—
39C2×#11719945—22no—
40C2×#11719945—22no—
41C2×#11719945—21no—
42C2×#11719945—21no—
43C2×#11719945—21no—
44C2×#11719945—21no—
45C2×#11719945—22no—
46C2×#11719945—22no—
47C22×Dic106C2×C411yescharacteristic
48C22×D212C2×C411yescharacteristic
49C22×C22⋊D53C2×C414yes—
50C22×C22⋊D53C2312yes—
51C23×Dic53C2312yes—
52C22×C4×D53C2×C412yes—
53#13133196—24no—
54#13133196—24no—
55#13133193—24no—
56#13133193—24no—
57#13133191—24no—
58#13133191—24no—
59#13133189—28no—
60#13133189—28no—
61#13133188—24no—
62#13133188—24no—
63#13133176—22no—
64#13133176—22no—
65#13133176—22no—
66#13133176D414yescomplemented
67C2×C4⋊Dic53—21no—
68C2×C4⋊Dic53—21no—
69C2×C4⋊Dic53C2×C412yes—
70C2×C4⋊Dic53C2×C412yes—
71C2×C4⋊Dic53D412yes—
72C2×C4⋊Dic53D412yes—
73C2×C4⋊Dic53D412yes—
74C2×C4⋊Dic53D412yes—
75C2×#11719919—24no—
76C2×#11719919C2×C414yes—
77C2×C22⋊Dic53C2×C414yes—
78C2×C22⋊Dic53C2312yes—
79C2×C22⋊Dic53C2312yes—
80C2×#11719921C2×C414yes—
81C2×C4×Dic53—21no—
82C2×C4×Dic53—21no—
83C2×C4×Dic53—24no—
84#13133132—28no—
85#13133131—28no—
86#13133131—28no—
87#13133124—24no—
88#13133124—24no—
89C2×#41×C53—22no—
90C2×#41×C53—22no—
91C2×#41×C53—21no—
92C2×#41×C53—21no—
93C2×#41×C53—21no—
94C2×#41×C53—21no—
95C2×#41×C53—22no—
96C2×#41×C53—22no—
97C22×Q8×C53C2×C411yescharacteristic
98C22×D4×C53C2×C411yescharacteristic
99C22×D4×C53C2312yes—
100C23×C4×C53C2×C412yes—
101C23×C4×C53C2311yescharacteristic
102#117×C53—22no—
103#117×C53—22no—
104#117×C53—22no—
105#117×C53D414yescomplemented
106C2×C4⋊C4×C53—21no—
107C2×C4⋊C4×C53—21no—
108C2×C22⋊C4×C53C2312yes—
109C2×C22⋊C4×C53C2312yes—
110C2×C42×C53—21no—
111C2×C42×C53—21no—
112#99×C53—28no—
113C24×C53C22×C412yes—
114C24×C53C2411yescharacteristic
115C23×D53C22×C412yes—
116#41×C53—42no—
117#41×C53—42no—
118#41×C53—42no—
119#41×C53—42no—
120#41×C53—42no—
121#41×C53—42no—
122#41×C53—42no—
123#41×C53—42no—
124C2×Q8×C53—21no—
125C2×Q8×C53—21no—
126C2×Q8×C53—22no—
127C2×Q8×C53—22no—
128C2×D4×C53—22no—
129C2×D4×C53—22no—
130C2×D4×C53—22no—
131C2×D4×C53—22no—
132C2×D4×C53—22no—
133C2×D4×C53—22no—
134C2×D4×C53—21no—
135C2×D4×C53—21no—
136C2×D4×C53—22no—
137C2×D4×C53—22no—
138C22×C4×C53—22no—
139C22×C4×C53—22no—
140C22×C4×C53—22no—
141C22×C4×C53—24no—
142C22×C4×C53—24no—
143C22×C4×C53—22no—
144C22×C4×C53C22×C412yes—
145C22×C4×C53C22×C411yescharacteristic
146C22×C4×C53C22×C411yescharacteristic
147C22×C4×C53C2×D411yescharacteristic
148C22×C4×C53C2×D411yescharacteristic
149C22×C4×C53C2×D411yescharacteristic
150C22×C4×C53C2×D411yescharacteristic
151C22×C4×C53C2×D412yes—
152C22×C4×C53C2×D412yes—
153C22×C4×C53#4112yes—
154C22×C4×C53#4112yes—
155C2×C22⋊D53—22no—
156C2×C22⋊D53—22no—
157C2×C22⋊D53—22no—
158C2×C22⋊D53—22no—
159C2×C22⋊D53—22no—
160C2×C22⋊D53—22no—
161C2×C22⋊D53—28no—
162C2×C22⋊D53—84no—
163C22×Dic53—22no—
164C22×Dic53—22no—
165C22×Dic53—82no—
166C22×Dic53C22×C412yes—
167C22×Dic53C22×C414yes—
168C22×Dic53C2×D414yes—
169C22×Dic53C2×D414yes—
170#11719945—42no—
171#11719945—42no—
172#11719945—42no—
173#11719945—42no—
174#11719945—42no—
175#11719945—42no—
176#11719945—42no—
177#11719945—42no—
178C2×D212—21no—
179C2×D212—21no—
180C2×D212—22no—
181C2×D212—22no—
182C2×C4×D53—22no—
183C2×C4×D53—22no—
184C2×C4×D53—24no—
185C2×C4×D53—24no—
186C2×Dic106—21no—
187C2×Dic106—21no—
188C2×Dic106—22no—
189C2×Dic106—22no—
190C4⋊C4×C53—24no—
191C4⋊C4×C53—42no—
192C22⋊C4×C53—22no—
193C22⋊C4×C53—22no—
194C22⋊C4×C53—22no—
195C22⋊C4×C53C2×D414yes—
196C42×C53—24no—
197C42×C53—42no—
198C22⋊Dic53—22no—
199C22⋊Dic53—28no—
200C22⋊Dic53—22no—
201C22⋊Dic53—22no—
202C22⋊Dic53C2×D414yes—
203#11719921—28no—
204C4⋊Dic53—24no—
205C4⋊Dic53—42no—
206C4⋊Dic53C22⋊C414yes—
207C4⋊Dic53C22⋊C414yes—
208C4⋊Dic53C22⋊C414yes—
209C4⋊Dic53C22⋊C414yes—
210#11719919—28no—
211#11719919—84no—
212C4×Dic53—24no—
213C4×Dic53—42no—
214C4×Dic53—84no—
215C23×C53—22no—
216C23×C53—22no—
217C23×C53—22no—
218C23×C53—41no—
219C23×C53—41no—
220C23×C53—42no—
221C23×C53—42no—
222C23×C53C23×C411yescharacteristic
223C23×C53C22×D411yescharacteristic
224C23×C53C22×D411yescharacteristic
225C23×C53C22×D411yescharacteristic
226C23×C53C22×D411yescharacteristic
227C23×C53C2×#4111yescharacteristic
228C23×C53C2×#4111yescharacteristic
229C22×D53—22no—
230C22×D53—22no—
231C22×D53—22no—
232C22×D53—42no—
233C22×D53—42no—
234Q8×C53—42no—
235Q8×C53—42no—
236D4×C53—42no—
237D4×C53—42no—
238D4×C53—42no—
239D4×C53—42no—
240D4×C53—42no—
241D4×C53—42no—
242C2×C4×C53—22no—
243C2×C4×C53—22no—
244C2×C4×C53—22no—
245C2×C4×C53—24no—
246C2×C4×C53—42no—
247C2×C4×C53—42no—
248C2×C4×C53—44no—
249C2×C4×C53—44no—
250C2×C4×C53—42no—
251C2×C4×C53—42no—
252C2×C4×C53—42no—
253C2×C4×C53C2×C22⋊C411yescharacteristic
254C2×C4×C53C2×C22⋊C411yescharacteristic
255C2×C4×C53C2×C22⋊C411yescharacteristic
256C2×C4×C53C2×C22⋊C411yescharacteristic
257C2×C4×C53C4×D412yes—
258C2×C4×C53C4×D412yes—
259C2×C4×C53C4×D412yes—
260C2×C4×C53C4×D412yes—
261C2×C4×C53#12112yes—
262C2×C4×C53#12112yes—
263C2×C4×C53#12112yes—
264C2×C4×C53#12112yes—
265C2×C4×C53#12412yes—
266C2×C4×C53#12412yes—
267C2×C4×C53#12712yes—
268C2×C4×C53#12712yes—
269C22⋊D53—42no—
270C22⋊D53—42no—
271C22⋊D53—42no—
272C22⋊D53—42no—
273C22⋊D53—88no—
274C2×Dic53—22no—
275C2×Dic53—22no—
276C2×Dic53—22no—
277C2×Dic53—24no—
278C2×Dic53—44no—
279C2×Dic53—44no—
280C2×Dic53—44no—
281C2×Dic53—42no—
282C2×Dic53—42no—
283C2×Dic53—42no—
284C2×Dic53—42no—
285C2×Dic53—42no—
286C2×Dic53C2×C22⋊C418yes—
287D212—42no—
288D212—42no—
289C4×D53—44no—
290C4×D53—44no—
291Dic106—42no—
292Dic106—42no—
293#21214—531nomaximal, Sylow
294C22×C53—21no—
295C22×C53—21no—
296C22×C53—21no—
297C22×C53—21no—
298C22×C53—21no—
299C22×C53—21no—
300C22×C53—41no—
301C22×C53—41no—
302C22×C53—41no—
303C22×C53—41no—
304C22×C53—42no—
305C22×C53—42no—
306C22×C53—42no—
307C22×C53—82no—
308C22×C53—82no—
309C22×C53C22×C22⋊C411yescharacteristic
310C22×C53C2×C4×D411yescharacteristic
311C22×C53C2×C4×D411yescharacteristic
312C22×C53C2×#12111yescharacteristic
313C22×C53C2×#12111yescharacteristic
314C22×C53C2×#12411yescharacteristic
315C22×C53C2×#12711yescharacteristic
316C2×D53—42no—
317C2×D53—42no—
318C2×D53—42no—
319C2×D53—82no—
320C2×D53—82no—
321C4×C53—44no—
322C4×C53—42no—
323C4×C53—82no—
324C4×C53#39012yes—
325C4×C53#39012yes—
326C4×C53#39012yes—
327C4×C53#39012yes—
328Dic53—28no—
329Dic53—44no—
330Dic53—42no—
331Dic53—82no—
332C2×#537—531no—
333C2×#410—531no—
334C2×#410—531no—
335C2×#410—531no—
336C2×#410D5311yescharacteristic, complemented
337#2316—531no—
338#2274—531no—
339#1964—538no—
340C2×C53—21no—
341C2×C53—21no—
342C2×C53—21no—
343C2×C53—21no—
344C2×C53—21no—
345C2×C53—21no—
346C2×C53—82no—
347C2×C53C2×#39011yescharacteristic
348C2×C53#288311yescharacteristic
349C2×C53#288411yescharacteristic
350D53—82no—
351C22×#41—531no—
352C22×#41C2×D5311yescharacteristic
353#537—1062no—
354#537—1062no—
355#537—1062no—
356#537—1062no—
357C2×C22⋊Q8—532no—
358C2×C22⋊Q8—532no—
359C2×#121—532no—
360C2×#121—534no—
361C2×#121—532no—
362C2×#117—531no—
363C2×#117—531no—
364C2×#117C2×D5311yescharacteristic
365C2×#117C2×D5311yescharacteristic
366C22×C4⋊C4—531no—
367#410—24no—
368#410—1064no—
369#410—1064no—
370#410—1064no—
371C2×#99—534no—
372C2×#99—534no—
373C2×#99—534no—
374C2×#99C2×D5314yes—
375#389—1062no—
376#389—1062no—
377#380—534no—
378C4×C4⋊C4—1062no—
379C4×C4⋊C4—1062no—
380C4×C22⋊C4—534no—
381C53#2121411yescharacteristic, complemented, Sylow
382C2×#41—22no—
383C2×#41—21no—
384C2×#41—21no—
385C2×#41—21no—
386C2×#41—21no—
387C2×#41—22no—
388C2×#41—22no—
389C2×#41—22no—
390C2×#41—1062no—
391C2×#41—1061no—
392C2×#41—1061no—
393C2×#41—1061no—
394C2×#41—1061no—
395C2×#41—1062no—
396C2×#41—1062no—
397C2×#41—1062no—
398C22×Q8—531no—
399C22×Q8C4×D5311yescharacteristic
400C22×D4—532no—
401C22×D4—534no—
402C22×D4—531no—
403C22×D4C4×D5311yescharacteristic
404C22×D4C22×D5312yes—
405C23×C4—532no—
406C23×C4—532no—
407C23×C4C4×D5312yes—
408C23×C4C22×D5311yescharacteristic
409C22⋊Q8—1064no—
410C22⋊Q8—1064no—
411C22⋊Q8—1064no—
412C22⋊Q8—1064no—
413#121—1064no—
414#121—1064no—
415#121—1064no—
416#121—1064no—
417#121—1068no—
418#121—1068no—
419#117—22no—
420#117—22no—
421#117—22no—
422#117—534no—
423#117—1062no—
424#117—1062no—
425#117—1062no—
426#117C22⋊D5314yescomplemented
427C2×C4⋊C4—21no—
428C2×C4⋊C4—21no—
429C2×C4⋊C4—532no—
430C2×C4⋊C4—532no—
431C2×C4⋊C4—532no—
432C2×C4⋊C4—532no—
433C2×C4⋊C4—532no—
434C2×C4⋊C4—532no—
435C2×C4⋊C4—534no—
436C2×C4⋊C4—1061no—
437C2×C4⋊C4—1064no—
438C2×C4⋊C4—1061no—
439C2×C22⋊C4—532no—
440C2×C22⋊C4—534no—
441C2×C22⋊C4—534no—
442C2×C22⋊C4—532no—
443C2×C22⋊C4C22×D5312yes—
444C2×C22⋊C4C22×D5312yes—
445C2×C42—21no—
446C2×C42—21no—
447C2×C42—1061no—
448C2×C42—1064no—
449C2×C42—1061no—
450#99—28no—
451#99—1068no—
452#99—1068no—
453#99—1068no—
454#95—1064no—
455#95—1064no—
456C24—532no—
457C24C2×C4×D5312yes—
458C24C23×D5311yescharacteristic
459#41—42no—
460#41—42no—
461#41—42no—
462#41—42no—
463#41—42no—
464#41—42no—
465#41—42no—
466#41—42no—
467#41—2122no—
468#41—2122no—
469#41—2122no—
470#41—2122no—
471#41—2122no—
472#41—2122no—
473#41—2122no—
474#41—2122no—
475C2×Q8—21no—
476C2×Q8—21no—
477C2×Q8—22no—
478C2×Q8—22no—
479C2×Q8—1061no—
480C2×Q8—1061no—
481C2×Q8—1062no—
482C2×Q8—1062no—
483C2×D4—22no—
484C2×D4—22no—
485C2×D4—22no—
486C2×D4—22no—
487C2×D4—22no—
488C2×D4—22no—
489C2×D4—21no—
490C2×D4—21no—
491C2×D4—22no—
492C2×D4—22no—
493C2×D4—1068no—
494C2×D4—1062no—
495C2×D4—1062no—
496C2×D4—1062no—
497C2×D4—1062no—
498C2×D4—1062no—
499C2×D4—1062no—
500C2×D4—1061no—
501C2×D4—1061no—
502C2×D4—1062no—
503C2×D4—1062no—
504C2×D4—4244no—
505C22×C4—22no—
506C22×C4—22no—
507C22×C4—22no—
508C22×C4—24no—
509C22×C4—24no—
510C22×C4—22no—
511C22×C4—534no—
512C22×C4—534no—
513C22×C4—534no—
514C22×C4—532no—
515C22×C4—1062no—
516C22×C4—1062no—
517C22×C4—1062no—
518C22×C4—1064no—
519C22×C4—1064no—
520C22×C4—1062no—
521C22×C4—4242no—
522C22×C4C2×C4×D5312yes—
523C22×C4C2×C4×D5311yescharacteristic
524C22×C4C2×C4×D5311yescharacteristic
525C22×C4#1171994512yes—
526C22×C4D4×D5311yescharacteristic
527C22×C4D4×D5311yescharacteristic
528C22×C4D4×D5311yescharacteristic
529C22×C4D4×D5311yescharacteristic
530C22×C4D4×D5312yes—
531C22×C4#1171994712yes—
532C22×C4C2×C22⋊D5312yes—
533C4⋊C4—24no—
534C4⋊C4—42no—
535C4⋊C4—534no—
536C4⋊C4—534no—
537C4⋊C4—534no—
538C4⋊C4—534no—
539C4⋊C4—1068no—
540C4⋊C4—1064no—
541C4⋊C4—4244no—
542C4⋊C4—2122no—
543C22⋊C4—22no—
544C22⋊C4—22no—
545C22⋊C4—22no—
546C22⋊C4—534no—
547C22⋊C4—1068no—
548C22⋊C4—1068no—
549C22⋊C4—1062no—
550C22⋊C4—1062no—
551C22⋊C4—1062no—
552C22⋊C4C2×C22⋊D5314yes—
553C42—24no—
554C42—42no—
555C42—1064no—
556C42—4244no—
557C42—2122no—
558C23—22no—
559C23—22no—
560C23—22no—
561C23—41no—
562C23—41no—
563C23—42no—
564C23—42no—
565C23—1062no—
566C23—1062no—
567C23—1062no—
568C23—2122no—
569C23—2122no—
570C23C22×C4×D5311yescharacteristic
571C23C2×D4×D5311yescharacteristic
572C23C2×D4×D5311yescharacteristic
573C23C2×D4×D5311yescharacteristic
574C23C22×C22⋊D5311yescharacteristic
575C23C2×#1171994711yescharacteristic
576C23C2×#1171994511yescharacteristic
577Q8—42no—
578Q8—42no—
579Q8—2122no—
580Q8—2122no—
581D4—42no—
582D4—42no—
583D4—42no—
584D4—42no—
585D4—42no—
586D4—42no—
587D4—4248no—
588D4—2122no—
589D4—2122no—
590D4—2122no—
591D4—2122no—
592D4—2122no—
593D4—2122no—
594C2×C4—22no—
595C2×C4—22no—
596C2×C4—22no—
597C2×C4—24no—
598C2×C4—42no—
599C2×C4—42no—
600C2×C4—44no—
601C2×C4—44no—
602C2×C4—42no—
603C2×C4—42no—
604C2×C4—42no—
605C2×C4—538no—
606C2×C4—1062no—
607C2×C4—1062no—
608C2×C4—1062no—
609C2×C4—1064no—
610C2×C4—2124no—
611C2×C4—2124no—
612C2×C4—2124no—
613C2×C4—2122no—
614C2×C4—2122no—
615C2×C4—2122no—
616C2×C4—2122no—
617C2×C4—2122no—
618C2×C4—2124no—
619C2×C4—2124no—
620C2×C4C22⋊C4×D5311yescharacteristic
621C2×C4C22⋊C4×D5311yescharacteristic
622C2×C4C22⋊C4×D5311yescharacteristic
623C2×C4C22⋊C4×D5311yescharacteristic
624C2×C4#1313317712yes—
625C2×C4#1313317712yes—
626C2×C4C4×C22⋊D5312yes—
627C2×C4C4×C22⋊D5312yes—
628C2×C4#1313318912yes—
629C2×C4#1313318912yes—
630C2×C4#1313319012yes—
631C2×C4#1313319012yes—
632C2×C4#1313320312yes—
633C2×C4#1313320312yes—
634C2×C4#1313321312yes—
635C2×C4#1313321312yes—
636C22—21no—
637C22—21no—
638C22—21no—
639C22—21no—
640C22—21no—
641C22—21no—
642C22—41no—
643C22—41no—
644C22—41no—
645C22—41no—
646C22—42no—
647C22—42no—
648C22—42no—
649C22—82no—
650C22—82no—
651C22—4242no—
652C22—4242no—
653C22—2122no—
654C22—2122no—
655C22—2122no—
656C22C2×C22⋊C4×D5311yescharacteristic
657C22C2×#1313317711yescharacteristic
658C22C2×C4×C22⋊D5311yescharacteristic
659C22C2×#1313318911yescharacteristic
660C22C2×#1313319011yescharacteristic
661C22C2×#1313320311yescharacteristic
662C22C2×#1313321311yescharacteristic
663C4—44no—
664C4—42no—
665C4—82no—
666C4—1068no—
667C4—4242no—
668C4—2124no—
669C4—2122no—
670C4#1681052312yes—
671C4#1681052312yes—
672C4#1681052312yes—
673C4#1681052312yes—
674C2—21no—
675C2—21no—
676C2—21no—
677C2—21no—
678C2—21no—
679C2—21no—
680C2—82no—
681C2—4242no—
682C2C2×#1681052311yescharacteristic
683C2#2028317411yescharacteristic
684C2#2028317711yescharacteristic
subgroupG/H
centerC22C2×C22⋊C4×D53
commutatorC23×C53C23×C4
FrattiniC24C23×D53
FittingC2×C53×#410C2
socleC22×C53C22×C22⋊C4
prime radicalC2×#16811201C2
cosocleC24×C53C24
hypercenterC2×#410D53
hypocenterC53#21214
LCSG ▷ C23×C53 ▷ C2×C53 ▷ C53
UCSC1 ◁ C22 ◁ C23×C4 ◁ C2×#410

Complements

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

NH ≅ G/Ntype
2. C2×#16810597C2semidirect
3. C2×#16810596C2semidirect
4. C2×#16810596C2semidirect
5. #20277331C2semidirect
6. #20277071C2semidirect
7. #20275549C2semidirect
8. C2×C53×#410C2semidirect
14. C2×#13133196C4semidirect
15. C2×#13133193C4semidirect
16. C2×#13133191C4semidirect
18. C2×#13133188C4semidirect
66. #13133176D4semidirect
105. #117×C53D4semidirect
336. C2×#410D53semidirect
381. C53#21214semidirect
426. #117C22⋊D53semidirect

Maximal subgroups

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

subgroupindex
C2×#168112012
C2×#168105972
C2×#168105962
#202773312
#202770712
#202755492
C2×C53×#4102
#2121453