Finite Group Databasebeta

← #27305667

Subgroups — #27305667

Order 13568.

1778 conjugacy classes of subgroups.

The subgroup lattice has 594 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#16810518C212yesmaximal
2C2×C4×#11719919C212yesmaximal
3C2×C4×#11719919C211yesmaximal, characteristic
4C2×#16810474C211yesmaximal, characteristic
5#20272948C212yesmaximal
6#20272947C212yesmaximal
7#20272943C211yesmaximal, characteristic
8#20272942C211yesmaximal, characteristic
9#20272939C211yesmaximal, characteristic
10#95×Dic53C211yesmaximal, characteristic
11C2×C4×C4⋊C4×C53C211yesmaximal, characteristic
12C22×#11719919C2212yes—
13C22×#11719919C2212yes—
14C22×#11719919C2211yescharacteristic
15C22×#11719919C2211yescharacteristic
16C22×C4×Dic53C2212yes—
17C22×C4×Dic53C2211yescharacteristic
18C22×C4×Dic53C2212yes—
19C22×C4×Dic53C2211yescharacteristic
20#16810518C4116yescomplemented
21C4×#11719919—28no—
22C4×#11719919C418yescomplemented
23#16810474—24no—
24C2×#13133124C2211yescharacteristic
25C2×#13133124C2212yes—
26C2×#13133124C2212yes—
27C2×#13133124C2211yescharacteristic
28C2×#13133124C2212yes—
29C2×#13133124C2212yes—
30C2×#13133124C2212yes—
31C2×#13133124C2212yes—
32C2×#13133124C2211yescharacteristic
33C2×#13133124C2211yescharacteristic
34C22×C4⋊C4×C53C2212yes—
35C22×C42×C53C2212yes—
36C22×C42×C53C2211yescharacteristic
37C4×C4⋊C4×C53—24no—
38C2×#95×C53C2211yescharacteristic
39C2×#95×C53C2211yescharacteristic
40C23×Dic53C2312yes—
41C23×Dic53C2312yes—
42C23×Dic53C2311yescharacteristic
43C23×Dic53C2311yescharacteristic
44C23×Dic53C2311yescharacteristic
45C23×Dic53C2311yescharacteristic
46C2×#11719919—24no—
47C2×#11719919—24no—
48C2×#11719919—24no—
49C2×#11719919—24no—
50C2×#11719919C2×C418yes—
51C2×#11719919C2×C418yes—
52C2×#11719919C2×C418yes—
53C2×#11719919C2×C414yes—
54C2×#11719919C2×C414yes—
55C2×#11719919D414yes—
56C2×#11719919Q814yes—
57C2×C4×Dic53—22no—
58C2×C4×Dic53—24no—
59C2×C4×Dic53—22no—
60C2×C4×Dic53—24no—
61C2×C4×Dic53—24no—
62C2×C4×Dic53—24no—
63C2×C4×Dic53—24no—
64C2×C4×Dic53—42no—
65C2×C4×Dic53C2×C418yes—
66C2×C4×Dic53C2×C414yes—
67C2×C4×Dic53C2×C414yes—
68#13133124—22no—
69#13133124—24no—
70#13133124—22no—
71#13133124—24no—
72#13133124—24no—
73#13133124—42no—
74#13133124C2×C418yes—
75#13133124C2×C418yes—
76#13133124C2×C414yes—
77#13133124C2×C414yes—
78C23×C4×C53C2312yes—
79C23×C4×C53C2311yescharacteristic
80C23×C4×C53C2311yescharacteristic
81C23×C4×C53C2312yes—
82C23×C4×C53C2311yescharacteristic
83C2×C4⋊C4×C53—24no—
84C2×C4⋊C4×C53—24no—
85C2×C4⋊C4×C53C2×C418yes—
86C2×C42×C53—22no—
87C2×C42×C53—24no—
88C2×C42×C53—24no—
89C2×C42×C53—22no—
90C2×C42×C53—42no—
91C2×C42×C53C2×C414yes—
92#95×C53—22no—
93#95×C53D414yes—
94C24×C53C2411yescharacteristic
95C22×C4×C53—21no—
96C22×C4×C53—21no—
97C22×C4×C53—22no—
98C22×C4×C53—21no—
99C22×C4×C53—21no—
100C22×C4×C53—22no—
101C22×C4×C53—22no—
102C22×C4×C53—22no—
103C22×C4×C53—21no—
104C22×C4×C53—22no—
105C22×C4×C53—22no—
106C22×C4×C53—22no—
107C22×C4×C53—21no—
108C22×C4×C53—22no—
109C22×C4×C53—22no—
110C22×C4×C53—22no—
111C22×C4×C53—22no—
112C22×C4×C53—22no—
113C22×C4×C53—22no—
114C22×C4×C53C22×C414yes—
115C22×C4×C53C22×C412yes—
116C22×C4×C53C22×C414yes—
117C22×C4×C53C22×C412yes—
118C22×C4×C53C22×C412yes—
119C22×C4×C53C22×C414yes—
120C22×C4×C53C2×D412yes—
121C22×C4×C53C2×D412yes—
122C22×C4×C53C2×D414yes—
123C22×C4×C53C2×Q812yes—
124C22×C4×C53#4112yes—
125C22×C4×C53#4114yes—
126C22×Dic53—21no—
127C22×Dic53—21no—
128C22×Dic53—21no—
129C22×Dic53—21no—
130C22×Dic53—22no—
131C22×Dic53—21no—
132C22×Dic53—21no—
133C22×Dic53—22no—
134C22×Dic53—22no—
135C22×Dic53—22no—
136C22×Dic53—22no—
137C22×Dic53—22no—
138C22×Dic53—22no—
139C22×Dic53—22no—
140C22×Dic53—22no—
141C22×Dic53—22no—
142C22×Dic53—22no—
143C22×Dic53—21no—
144C22×Dic53—22no—
145C22×Dic53—22no—
146C22×Dic53—22no—
147C22×Dic53—22no—
148C22×Dic53—21no—
149C22×Dic53C22×C412yes—
150C22×Dic53C22×C414yes—
151C22×Dic53C22×C414yes—
152C22×Dic53C22×C412yes—
153C22×Dic53C22×C414yes—
154C22×Dic53C22×C412yes—
155C22×Dic53C22×C412yes—
156C22×Dic53C22×C414yes—
157C22×Dic53C2×D412yes—
158C22×Dic53C2×D412yes—
159C22×Dic53C2×Q812yes—
160C22×Dic53C2×Q812yes—
161C22×Dic53#4114yes—
162C4⋊C4×C53—216no—
163C42×C53—28no—
164C42×C53—48no—
165#11719919—216no—
166#11719919—216no—
167#11719919C42116yescomplemented
168#11719919C4⋊C4116yescomplemented
169C4×Dic53—28no—
170C4×Dic53—28no—
171C4×Dic53—216no—
172C4×Dic53—48no—
173C23×C53C23×C411yescharacteristic
174C23×C53C23×C412yes—
175C23×C53C22×D411yescharacteristic
176C23×C53C22×D411yescharacteristic
177C23×C53C22×Q811yescharacteristic
178C23×C53C2×#4112yes—
179C23×C53C2×#4111yescharacteristic
180C23×C53C2×#4112yes—
181C23×C53H(5,𝔽2)11yescharacteristic
182C23×C532−1+412yes—
183C23×C532−1+411yescharacteristic
184C2×C4×C53—22no—
185C2×C4×C53—24no—
186C2×C4×C53—22no—
187C2×C4×C53—22no—
188C2×C4×C53—24no—
189C2×C4×C53—22no—
190C2×C4×C53—24no—
191C2×C4×C53—22no—
192C2×C4×C53—24no—
193C2×C4×C53—22no—
194C2×C4×C53—24no—
195C2×C4×C53—22no—
196C2×C4×C53—24no—
197C2×C4×C53—24no—
198C2×C4×C53—22no—
199C2×C4×C53—24no—
200C2×C4×C53—22no—
201C2×C4×C53—24no—
202C2×C4×C53—24no—
203C2×C4×C53—22no—
204C2×C4×C53—24no—
205C2×C4×C53—24no—
206C2×C4×C53—24no—
207C2×C4×C53—42no—
208C2×C4×C53—42no—
209C2×C4×C53—42no—
210C2×C4×C53—42no—
211C2×C4×C53C2×C4214yes—
212C2×C4×C53C2×C4⋊C414yes—
213C2×C4×C53C4×D414yes—
214C2×C4×C53C4×D418yes—
215C2×Dic53—22no—
216C2×Dic53—22no—
217C2×Dic53—24no—
218C2×Dic53—22no—
219C2×Dic53—24no—
220C2×Dic53—22no—
221C2×Dic53—22no—
222C2×Dic53—22no—
223C2×Dic53—22no—
224C2×Dic53—22no—
225C2×Dic53—22no—
226C2×Dic53—24no—
227C2×Dic53—24no—
228C2×Dic53—24no—
229C2×Dic53—22no—
230C2×Dic53—24no—
231C2×Dic53—24no—
232C2×Dic53—24no—
233C2×Dic53—22no—
234C2×Dic53—42no—
235C2×Dic53—42no—
236C2×Dic53—42no—
237C2×Dic53—42no—
238C2×Dic53—42no—
239C2×Dic53—42no—
240C2×Dic53—42no—
241C2×Dic53—42no—
242C2×Dic53—42no—
243C2×Dic53—42no—
244C2×Dic53C2×C4214yes—
245C2×Dic53C2×C4214yes—
246C2×Dic53C2×C4⋊C414yes—
247C2×Dic53C2×C4⋊C414yes—
248C2×Dic53#11714yes—
249C2×Dic53C4×D418yes—
250C2×Dic53C4×Q818yes—
251#13821—531nomaximal, Sylow
252C22×C53C22×C4211yescharacteristic
253C22×C53C22×C4⋊C411yescharacteristic
254C22×C53C2×#11711yescharacteristic
255C22×C53C2×C4×D412yes—
256C22×C53C2×C4×D411yescharacteristic
257C22×C53C2×C4×D412yes—
258C22×C53C2×C4×Q812yes—
259C22×C53C4×#4111yescharacteristic
260C22×C53C4×#4112yes—
261C22×C53#51912yes—
262C22×C53#51911yescharacteristic
263C22×C53#52012yes—
264C22×C53#52011yescharacteristic
265C22×C53D4211yescharacteristic
266C22×C53#54712yes—
267C22×C53#54711yescharacteristic
268C22×C53#54812yes—
269C22×C53D4×Q811yescharacteristic
270C22×C53#55212yes—
271C22×C53#55312yes—
272C22×C53#55411yescharacteristic
273C22×C53#55612yes—
274C22×C53#55712yes—
275C4×C53—24no—
276C4×C53—24no—
277C4×C53—28no—
278C4×C53—24no—
279C4×C53—44no—
280Dic53—24no—
281Dic53—24no—
282Dic53—42no—
283Dic53—44no—
284Dic53—44no—
285Dic53C4×C4⋊C418yescomplemented
286C2×#389—532no—
287C2×C4×C4⋊C4—531no—
288C2×C4×C4⋊C4—532no—
289C2×C4×C4⋊C4—531no—
290C2×C4×C4⋊C4D5311yescharacteristic
291#1436—532no—
292#1436—532no—
293C4⋊#95—531no—
294#1434—531no—
295#1434—531no—
296C4×#95—531no—
297C2×C53C2×C4×C4⋊C411yescharacteristic
298C2×C53C42×D411yescharacteristic
299C2×C53#227611yescharacteristic
300C2×C53#234411yescharacteristic
301C2×C53#234711yescharacteristic
302C2×C53D4×C4⋊C411yescharacteristic
303C2×C53#235111yescharacteristic
304C2×C53#236112yes—
305C2×C53#236212yes—
306C2×C53#236512yes—
307C2×C53#237012yes—
308C22×C4⋊C4—531no—
309C22×C4⋊C4—531no—
310C22×C4⋊C4—532no—
311C22×C4⋊C4—532no—
312C22×C4⋊C4C2×D5312yes—
313C22×C42—531no—
314C22×C42—532no—
315C22×C42—531no—
316C22×C42—532no—
317C22×C42C2×D5312yes—
318C22×C42C2×D5311yescharacteristic
319#389—5316no—
320C4×C4⋊C4—24no—
321C4×C4⋊C4—538no—
322C4×C4⋊C4—1068no—
323C4×C4⋊C4—1064no—
324C2×#95—532no—
325C2×#95—532no—
326C2×#95—531no—
327C2×#95—531no—
328C2×#95—531no—
329C2×#95—532no—
330C2×#95—531no—
331C2×#95—532no—
332C2×#95—532no—
333C2×#95—532no—
334C2×#95C2×D5311yescharacteristic
335C2×#95C2×D5311yescharacteristic
336C53#1382111yescharacteristic, complemented, Sylow
337C23×C4—531no—
338C23×C4—531no—
339C23×C4—531no—
340C23×C4—532no—
341C23×C4—532no—
342C23×C4—531no—
343C23×C4C22×D5312yes—
344C23×C4C22×D5311yescharacteristic
345C23×C4C22×D5312yes—
346C23×C4C22×D5311yescharacteristic
347C23×C4C22×D5311yescharacteristic
348C2×C4⋊C4—24no—
349C2×C4⋊C4—24no—
350C2×C4⋊C4—534no—
351C2×C4⋊C4—534no—
352C2×C4⋊C4—538no—
353C2×C4⋊C4—534no—
354C2×C4⋊C4—534no—
355C2×C4⋊C4—538no—
356C2×C4⋊C4—538no—
357C2×C4⋊C4—1064no—
358C2×C4⋊C4—1064no—
359C2×C4⋊C4—1064no—
360C2×C4⋊C4—1064no—
361C2×C4⋊C4C4×D5318yes—
362C2×C42—22no—
363C2×C42—22no—
364C2×C42—24no—
365C2×C42—24no—
366C2×C42—42no—
367C2×C42—534no—
368C2×C42—534no—
369C2×C42—538no—
370C2×C42—1064no—
371C2×C42—1064no—
372C2×C42—1064no—
373C2×C42—1062no—
374C2×C42—1062no—
375C2×C42—1064no—
376C2×C42—1064no—
377C2×C42—2122no—
378C2×C42C4×D5314yes—
379#95—22no—
380#95—534no—
381#95—534no—
382#95—538no—
383#95—538no—
384#95—1064no—
385#95—1062no—
386#95—1064no—
387#95—1064no—
388#95—1062no—
389#95—2122no—
390#95C22⋊D5314yes—
391C24C23×D5311yescharacteristic
392C22×C4—22no—
393C22×C4—21no—
394C22×C4—21no—
395C22×C4—22no—
396C22×C4—22no—
397C22×C4—21no—
398C22×C4—21no—
399C22×C4—22no—
400C22×C4—21no—
401C22×C4—22no—
402C22×C4—22no—
403C22×C4—22no—
404C22×C4—21no—
405C22×C4—22no—
406C22×C4—22no—
407C22×C4—22no—
408C22×C4—22no—
409C22×C4—22no—
410C22×C4—22no—
411C22×C4—532no—
412C22×C4—534no—
413C22×C4—532no—
414C22×C4—532no—
415C22×C4—534no—
416C22×C4—532no—
417C22×C4—532no—
418C22×C4—534no—
419C22×C4—532no—
420C22×C4—532no—
421C22×C4—532no—
422C22×C4—534no—
423C22×C4—534no—
424C22×C4—1061no—
425C22×C4—1061no—
426C22×C4—1061no—
427C22×C4—1061no—
428C22×C4—1062no—
429C22×C4—1061no—
430C22×C4—1061no—
431C22×C4—1062no—
432C22×C4—1062no—
433C22×C4—1062no—
434C22×C4—1062no—
435C22×C4—1062no—
436C22×C4—1062no—
437C22×C4—1062no—
438C22×C4—1062no—
439C22×C4—1062no—
440C22×C4—1062no—
441C22×C4—1061no—
442C22×C4—1062no—
443C22×C4—1062no—
444C22×C4—1062no—
445C22×C4—1062no—
446C22×C4—1061no—
447C22×C4C2×C4×D5314yes—
448C22×C4C2×C4×D5312yes—
449C22×C4C2×C4×D5314yes—
450C22×C4C2×C4×D5312yes—
451C22×C4C2×C4×D5312yes—
452C22×C4C2×C4×D5314yes—
453C22×C4#1171994512yes—
454C22×C4#1171994514yes—
455C22×C4D4×D5312yes—
456C22×C4Q8×D5312yes—
457C22×C4C2×C22⋊D5312yes—
458C22×C4C2×C22⋊D5314yes—
459C4⋊C4—216no—
460C4⋊C4—5316no—
461C4⋊C4—5316no—
462C4⋊C4—10616no—
463C4⋊C4—10616no—
464C42—28no—
465C42—48no—
466C42—1068no—
467C42—10616no—
468C42—1068no—
469C42—2128no—
470C23C22×C4×D5311yescharacteristic
471C23C22×C4×D5312yes—
472C23C2×D4×D5311yescharacteristic
473C23C22×C22⋊D5311yescharacteristic
474C23C2×Q8×D5311yescharacteristic
475C23#41×D5312yes—
476C23C2×#1171994511yescharacteristic
477C23C2×#1171994512yes—
478C23#1313326211yescharacteristic
479C23#1313326412yes—
480C23#1313326511yescharacteristic
481C2×C4—24no—
482C2×C4—24no—
483C2×C4—22no—
484C2×C4—24no—
485C2×C4—22no—
486C2×C4—24no—
487C2×C4—24no—
488C2×C4—22no—
489C2×C4—22no—
490C2×C4—24no—
491C2×C4—22no—
492C2×C4—22no—
493C2×C4—24no—
494C2×C4—24no—
495C2×C4—22no—
496C2×C4—22no—
497C2×C4—24no—
498C2×C4—22no—
499C2×C4—22no—
500C2×C4—24no—
501C2×C4—24no—
502C2×C4—24no—
503C2×C4—24no—
504C2×C4—42no—
505C2×C4—42no—
506C2×C4—42no—
507C2×C4—42no—
508C2×C4—534no—
509C2×C4—534no—
510C2×C4—534no—
511C2×C4—538no—
512C2×C4—538no—
513C2×C4—534no—
514C2×C4—534no—
515C2×C4—1062no—
516C2×C4—1062no—
517C2×C4—1062no—
518C2×C4—1062no—
519C2×C4—1062no—
520C2×C4—1064no—
521C2×C4—1064no—
522C2×C4—1062no—
523C2×C4—1062no—
524C2×C4—1062no—
525C2×C4—1062no—
526C2×C4—1064no—
527C2×C4—1064no—
528C2×C4—1064no—
529C2×C4—1062no—
530C2×C4—1064no—
531C2×C4—1064no—
532C2×C4—1064no—
533C2×C4—1062no—
534C2×C4—2122no—
535C2×C4—2122no—
536C2×C4—2122no—
537C2×C4—2122no—
538C2×C4—2122no—
539C2×C4—2122no—
540C2×C4—2122no—
541C2×C4—2122no—
542C2×C4—2122no—
543C2×C4—2122no—
544C2×C4C42×D5314yes—
545C2×C4C4⋊C4×D5314yes—
546C2×C4C4×C22⋊D5314yes—
547C2×C4C4×C22⋊D5318yes—
548C22C2×C42×D5311yescharacteristic
549C22C2×C4⋊C4×D5311yescharacteristic
550C22#117×D5311yescharacteristic
551C22C4×D4×D5312yes—
552C22C2×C4×C22⋊D5312yes—
553C22C2×C4×C22⋊D5311yescharacteristic
554C22C4×Q8×D5312yes—
555C22C4×#1171994512yes—
556C22C4×#1171994511yescharacteristic
557C22#1681109912yes—
558C22#1681110011yescharacteristic
559C22#1681110212yes—
560C22#1681110411yescharacteristic
561C22D4×C22⋊D5311yescharacteristic
562C22#1681127212yes—
563C22#1681127411yescharacteristic
564C22#1681128212yes—
565C22Q8×C22⋊D5311yescharacteristic
566C22#1681130612yes—
567C22#1681132212yes—
568C22#1681132811yescharacteristic
569C22#1681134512yes—
570C22#1681135112yes—
571C4—24no—
572C4—24no—
573C4—24no—
574C4—28no—
575C4—44no—
576C4—538no—
577C4—1064no—
578C4—1064no—
579C4—2122no—
580C4—2124no—
581C4—2124no—
582C2C4×C4⋊C4×D5311yescharacteristic
583C2C42×C22⋊D5311yescharacteristic
584C2#2027707911yescharacteristic
585C2#2027758511yescharacteristic
586C2#2027760911yescharacteristic
587C2C4⋊C4×C22⋊D5311yescharacteristic
588C2#2027763511yescharacteristic
589C2#2027773212yes—
590C2#2027774312yes—
591C2#2027776912yes—
592C2#2027781212yes—
subgroupG/H
center, FrattiniC24C23×D53
commutatorC22×C53C22×C42
FittingC2×C4×C4⋊C4×C53C2
socle, prime radical, cosocleC24×C53C24
hypercenterC2×C4×C4⋊C4D53
hypocenterC53#13821
LCSG ▷ C22×C53 ▷ C53
UCSC1 ◁ C24 ◁ C2×C4×C4⋊C4

Complements

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

NH ≅ G/Ntype
20. #16810518C4semidirect
22. C4×#11719919C4semidirect
167. #11719919C42semidirect
168. #11719919C4⋊C4semidirect
285. Dic53C4×C4⋊C4semidirect
336. C53#13821semidirect

Maximal subgroups

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

subgroupindex
C2×#168105182
C2×C4×#117199192
C2×#168104742
#202729482
#202729472
#202729432
#202729422
#202729392
#95×Dic532
C2×C4×C4⋊C4×C532
#1382153