Finite Group Databasebeta

← #28312928

Subgroups — #28312928

Order 13568.

1486 conjugacy classes of subgroups.

The subgroup lattice has 592 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#16811569C211yesmaximal, characteristic
2C2×#16811567C211yesmaximal, characteristic
3C2×#16811567C211yesmaximal, characteristic
4C2×C4×C4⋊#6157C211yesmaximal, characteristic
5#20289900C214yesmaximal
6#20289895C214yesmaximal
7#20289893C211yesmaximal, characteristic
8#20289893C211yesmaximal, characteristic
9C4×C4⋊C4×D53C211yesmaximal, characteristic
10C22×C4⋊#6157C2211yescharacteristic
11C22×C4⋊#6157C2211yescharacteristic
12C22×C4⋊#6157C2214yes—
13C22×C4⋊#6157C2211yescharacteristic
14C22×C4⋊#6157C2211yescharacteristic
15C22×C4×#6157C2212yes—
16#16811569—24no—
17#16811567—24no—
18#16811567—24no—
19C4×C4⋊#6157—24no—
20C2×(C2×#6157)⋊C4C2212yes—
21C2×(C2×#6157)⋊C4C2212yes—
22C2×(C2×#6157)⋊C4C2214yes—
23C2×(C2×#6157)⋊C4C2212yes—
24C2×(C2×#6157)⋊C4C2214yes—
25C2×(C2×#6157)⋊C4C2214yes—
26C2×C4⋊C4×D53C2212yes—
27C2×C42×D53C2211yescharacteristic
28C2×C42×D53C2211yescharacteristic
29C2×C42×D53C2211yescharacteristic
30C4⋊C4×Dic53C412yescomplemented
31C4×#11719919C411yescharacteristic, complemented
32C4×#11719919C411yescharacteristic, complemented
33C4×C4⋊Dic53C411yescharacteristic, complemented
34#16810474C412yescomplemented
35#95×D53C2212yes—
36C4×C4⋊C4×C53C411yescharacteristic, complemented
37C23×#6157C2312yes—
38C23×#6157C2312yes—
39C23×#6157C2312yes—
40C23×#6157C2312yes—
41C2×C4⋊#6157—22no—
42C2×C4⋊#6157—22no—
43C2×C4⋊#6157—22no—
44C2×C4⋊#6157—22no—
45C2×C4⋊#6157—28no—
46C2×C4⋊#6157—22no—
47C2×C4⋊#6157—22no—
48C2×C4⋊#6157—22no—
49C2×C4⋊#6157—22no—
50C2×C4⋊#6157—28no—
51C2×C4⋊#6157—22no—
52C2×C4⋊#6157—22no—
53C2×C4⋊#6157—44no—
54C2×C4⋊#6157D414yes—
55C2×C4⋊#6157Q814yes—
56C2×C4×#6157—24no—
57C2×C4×#6157—44no—
58(C2×#6157)⋊C4—28no—
59(C2×#6157)⋊C4—24no—
60(C2×#6157)⋊C4—24no—
61(C2×#6157)⋊C4—24no—
62(C2×#6157)⋊C4—28no—
63(C2×#6157)⋊C4—44no—
64C22×C4×D53C2312yes—
65C22×C4×D53C2311yescharacteristic
66C22×C4×D53C2311yescharacteristic
67C22×C4×D53C2312yes—
68C22×C4×D53C2311yescharacteristic
69C2×C4⋊Dic53C2×C412yes—
70C2×#11719919C2×C412yes—
71C2×#11719919C2×C412yes—
72C4⋊C4×D53—24no—
73C4⋊C4×D53—24no—
74C2×C4×Dic53C2×C412yes—
75C2×C4×Dic53C2×C412yes—
76C2×C4×Dic53C2×C411yescharacteristic
77C2×C4×Dic53C2×C411yescharacteristic
78C2×C4×Dic53C2×C411yescharacteristic
79C2×C4×Dic53C2×C411yescharacteristic
80C2×C4×Dic53C2×C411yescharacteristic
81C42×D53—22no—
82C42×D53—22no—
83C42×D53—22no—
84C42×D53—22no—
85C42×D53D414yes—
86C42×D53Q814yes—
87#13133124C2×C412yes—
88#13133124C2×C412yes—
89#13133124C2×C412yes—
90C2×C4⋊C4×C53C2×C412yes—
91C2×C42×C53C2×C411yescharacteristic
92C2×C42×C53C2×C411yescharacteristic
93C2×C42×C53C2×C411yescharacteristic
94#95×C53C2×C412yes—
95C23×D53C2411yescharacteristic
96C22×#6157—22no—
97C22×#6157—22no—
98C22×#6157—24no—
99C22×#6157—22no—
100C22×#6157—22no—
101C22×#6157—24no—
102C22×#6157—24no—
103C22×#6157—24no—
104C22×#6157—22no—
105C22×#6157—22no—
106C22×#6157—22no—
107C22×#6157—22no—
108C22×#6157—24no—
109C22×#6157—22no—
110C22×#6157—22no—
111C22×#6157—24no—
112C22×#6157—24no—
113C22×#6157—24no—
114C22×#6157C2×D414yes—
115C22×#6157C2×Q814yes—
116C22×C4×C53C22×C412yes—
117C22×C4×C53C22×C411yescharacteristic
118C22×C4×C53C22×C411yescharacteristic
119C22×C4×C53C22×C412yes—
120C22×C4×C53C22×C411yescharacteristic
121C22×Dic53C22×C412yes—
122C22×Dic53C22×C411yescharacteristic
123C22×Dic53C22×C411yescharacteristic
124C22×Dic53C22×C412yes—
125C22×Dic53C22×C411yescharacteristic
126C2×C4×D53—22no—
127C2×C4×D53—22no—
128C2×C4×D53—22no—
129C2×C4×D53—22no—
130C2×C4×D53—22no—
131C2×C4×D53—22no—
132C2×C4×D53—22no—
133C2×C4×D53—22no—
134C2×C4×D53—22no—
135C2×C4×D53—22no—
136C2×C4×D53—22no—
137C2×C4×D53—22no—
138C2×C4×D53—22no—
139C2×C4×D53—22no—
140C2×C4×D53C2×D412yes—
141C2×C4×D53C2×D412yes—
142C2×C4×D53C2×D412yes—
143C2×C4×D53C2×D412yes—
144C2×C4×D53C2×Q812yes—
145C2×C4×D53C2×Q812yes—
146C2×C4×D53C2×Q812yes—
147C2×C4×D53C2×Q812yes—
148C2×C4×D53#4114yes—
149C2×C4×D53#4114yes—
150C2×C4×D53#4112yes—
151C2×C4×D53#4112yes—
152C4⋊#6157—28no—
153C4⋊#6157—28no—
154C4⋊#6157—416no—
155C4⋊#6157—44no—
156C4⋊#6157—44no—
157C4×#6157—48no—
158C4⋊C4×C53—24no—
159C42×C53—22no—
160C42×C53—22no—
161C42×C53C4⋊C414yescomplemented
162C4⋊Dic53—24no—
163#11719919—24no—
164#11719919—24no—
165C4×Dic53—22no—
166C4×Dic53—22no—
167C4×Dic53—24no—
168C4×Dic53—24no—
169C4×Dic53C4⋊C414yescomplemented
170C4×Dic53C4⋊C414yescomplemented
171C4×Dic53C4⋊C414yescomplemented
172C23×C53C23×C411yescharacteristic
173C22×D53C22×D411yescharacteristic
174C22×D53C22×D411yescharacteristic
175C22×D53C22×Q811yescharacteristic
176C22×D53C22×Q811yescharacteristic
177C22×D53C2×#4111yescharacteristic
178C22×D53C2×#4111yescharacteristic
179C22×D53C2×#4112yes—
180C22×D53C2×#4112yes—
181C22×D53H(5,𝔽2)12yes—
182C22×D532−1+412yes—
183C2×#6157—24no—
184C2×#6157—24no—
185C2×#6157—28no—
186C2×#6157—28no—
187C2×#6157—42no—
188C2×#6157—42no—
189C2×#6157—42no—
190C2×#6157—42no—
191C2×#6157—44no—
192C2×#6157—44no—
193C2×#6157—42no—
194C2×#6157—42no—
195C2×#6157—42no—
196C2×#6157—42no—
197C2×#6157—42no—
198C2×#6157—42no—
199C2×#6157—44no—
200C2×#6157—44no—
201C2×#6157—44no—
202C2×#6157—44no—
203C2×C4×C53—22no—
204C2×C4×C53—22no—
205C2×C4×C53—22no—
206C2×C4×C53—22no—
207C2×C4×C53—22no—
208C2×C4×C53—22no—
209C2×C4×C53—22no—
210C2×C4×C53C2×C4⋊C412yes—
211C2×C4×C53C2×C4⋊C412yes—
212C2×C4×C53C2×C4⋊C412yes—
213C2×C4×C53#11714yes—
214C2×C4×C53C4×D412yes—
215C2×C4×C53C4×Q812yes—
216C2×Dic53—22no—
217C2×Dic53—22no—
218C2×Dic53—22no—
219C2×Dic53—22no—
220C2×Dic53—22no—
221C2×Dic53—22no—
222C2×Dic53—22no—
223C2×Dic53C2×C4⋊C412yes—
224C2×Dic53C2×C4⋊C412yes—
225C2×Dic53C2×C4⋊C412yes—
226C2×Dic53#11714yes—
227C2×Dic53C4×D412yes—
228C2×Dic53C4×Q812yes—
229C4×D53—22no—
230C4×D53—22no—
231C4×D53—24no—
232C4×D53—22no—
233C4×D53—22no—
234C4×D53—24no—
235C4×D53—24no—
236C4×D53—24no—
237C4×D53—24no—
238C4×D53—24no—
239C4×D53—42no—
240C4×D53—42no—
241C4×D53—42no—
242C4×D53—42no—
243C4×D53#12114yes—
244C4×D53C22⋊Q814yes—
245C4×D53#12514yes—
246C4×D53C4⋊Q814yes—
247#13909—531nomaximal, Sylow
248C22×C53C22×C4⋊C411yescharacteristic
249C22×C53C2×#11712yes—
250C22×C53C2×C4×D411yescharacteristic
251C22×C53C2×C4×Q811yescharacteristic
252C22×C53#52012yes—
253C2×D53C2×#12111yescharacteristic
254C2×D53C2×C22⋊Q811yescharacteristic
255C2×D53C2×#12511yescharacteristic
256C2×D53C2×C4⋊Q811yescharacteristic
257C2×D53#53211yescharacteristic
258C2×D53#53211yescharacteristic
259C2×D53#53312yes—
260C2×D53#53511yescharacteristic
261C2×D53#53611yescharacteristic
262C2×D53#54412yes—
263C2×D53#54612yes—
264C2×D53#54812yes—
265C2×D53#55212yes—
266C2×D53#55312yes—
267C2×D53#55412yes—
268C2×D53#55612yes—
269C2×D53#55712yes—
270C2×D53#55912yes—
271#6157—44no—
272#6157—44no—
273#6157—82no—
274#6157—82no—
275C4×C53—22no—
276C4×C53—24no—
277C4×C53—22no—
278C4×C53—42no—
279C4×C53#38914yes—
280Dic53—22no—
281Dic53—24no—
282Dic53—22no—
283Dic53—42no—
284Dic53#38914yes—
285C2×#384—531no—
286C2×#382—531no—
287C2×#382—531no—
288C2×C4×C4⋊C4—531no—
289C2×C4×C4⋊C4—531no—
290#1441—534no—
291#1436—534no—
292C4⋊#95—531no—
293C4⋊#95—531no—
294C2×C53#228511yescharacteristic
295C2×C53C2×#38911yescharacteristic
296C2×C53#231711yescharacteristic
297C2×C53#235212yes—
298C2×C53#235512yes—
299D53#254411yescharacteristic
300D53#254511yescharacteristic
301D53#255012yes—
302D53#256511yescharacteristic
303D53#256611yescharacteristic
304D53#257012yes—
305C22×C4⋊C4—531no—
306C22×C4⋊C4—531no—
307C22×C4⋊C4—532no—
308C22×C4⋊C4—531no—
309C22×C4⋊C4—531no—
310C22×C4⋊C4—534no—
311C22×C42—531no—
312C22×C42—531no—
313C22×C42—531no—
314C22×C42—532no—
315#384—1064no—
316#382—1064no—
317#382—1064no—
318C4×C4⋊C4—532no—
319C4×C4⋊C4—531no—
320C4×C4⋊C4—531no—
321C4×C4⋊C4—531no—
322C4×C4⋊C4—532no—
323C4×C4⋊C4—1064no—
324C4×C4⋊C4#615711yescharacteristic, complemented
325C2×#95—532no—
326C2×#95—534no—
327C2×#95—532no—
328C2×#95—534no—
329C2×#95—532no—
330C2×#95—532no—
331C2×#95—534no—
332C53#1390911yescharacteristic, complemented, Sylow
333C23×C4—531no—
334C23×C4—532no—
335C23×C4—531no—
336C23×C4—531no—
337C23×C4—532no—
338C23×C4—532no—
339C23×C4—532no—
340C23×C4—532no—
341C23×C4—532no—
342C2×C4⋊C4—534no—
343C2×C4⋊C4—534no—
344C2×C4⋊C4—532no—
345C2×C4⋊C4—532no—
346C2×C4⋊C4—532no—
347C2×C4⋊C4—1064no—
348C2×C4⋊C4—1064no—
349C2×C4⋊C4—1062no—
350C2×C4⋊C4—1062no—
351C2×C4⋊C4—1062no—
352C2×C4⋊C4—1062no—
353C2×C4⋊C4—1062no—
354C2×C4⋊C4—1062no—
355C2×C4⋊C4—1062no—
356C2×C4⋊C4—1062no—
357C2×C4⋊C4—1068no—
358C2×C4⋊C4—1062no—
359C2×C4⋊C4—1062no—
360C2×C4⋊C4—1068no—
361C2×C4⋊C4—2124no—
362C2×C4⋊C4C2×#615712yes—
363C2×C42—531no—
364C2×C42—531no—
365C2×C42—531no—
366C2×C42—534no—
367C2×C42—534no—
368C2×C42—532no—
369C2×C42—532no—
370C2×C42—531no—
371C2×C42—531no—
372C2×C42—1062no—
373C2×C42—1062no—
374C2×C42—1062no—
375C2×C42—1062no—
376C2×C42—1064no—
377C2×C42—2124no—
378C2×C42C2×#615711yescharacteristic
379C2×C42C2×#615711yescharacteristic
380C2×C42C2×#615711yescharacteristic
381#95—532no—
382#95—532no—
383#95—532no—
384#95—1064no—
385#95—1064no—
386#95—1064no—
387#95—1068no—
388#95—1068no—
389#95—2124no—
390#95C2×#615712yes—
391C24—531no—
392C22×C4—531no—
393C22×C4—532no—
394C22×C4—532no—
395C22×C4—532no—
396C22×C4—532no—
397C22×C4—532no—
398C22×C4—532no—
399C22×C4—531no—
400C22×C4—532no—
401C22×C4—531no—
402C22×C4—532no—
403C22×C4—532no—
404C22×C4—532no—
405C22×C4—532no—
406C22×C4—534no—
407C22×C4—534no—
408C22×C4—534no—
409C22×C4—534no—
410C22×C4—532no—
411C22×C4—1062no—
412C22×C4—1062no—
413C22×C4—1062no—
414C22×C4—1062no—
415C22×C4—1062no—
416C22×C4—1062no—
417C22×C4—1062no—
418C22×C4—1062no—
419C22×C4—1062no—
420C22×C4—1062no—
421C22×C4—1062no—
422C22×C4—1062no—
423C22×C4—1062no—
424C22×C4—1062no—
425C22×C4—1062no—
426C22×C4—1062no—
427C22×C4—1064no—
428C22×C4—1064no—
429C22×C4—1064no—
430C22×C4—1062no—
431C22×C4—1062no—
432C22×C4—1064no—
433C22×C4—1062no—
434C22×C4—1062no—
435C22×C4—1062no—
436C22×C4—1064no—
437C22×C4—1064no—
438C22×C4—1062no—
439C22×C4—1064no—
440C22×C4—1062no—
441C22×C4—1062no—
442C22×C4—1064no—
443C22×C4C22×#615712yes—
444C22×C4C22×#615711yescharacteristic
445C22×C4C22×#615711yescharacteristic
446C22×C4C22×#615712yes—
447C22×C4C22×#615711yescharacteristic
448C4⋊C4—24no—
449C4⋊C4—1064no—
450C4⋊C4—1064no—
451C4⋊C4—1068no—
452C4⋊C4—1068no—
453C4⋊C4—1064no—
454C4⋊C4—2124no—
455C4⋊C4—2124no—
456C4⋊C4—21216no—
457C42—22no—
458C42—22no—
459C42—534no—
460C42—534no—
461C42—534no—
462C42—1064no—
463C42—1064no—
464C42—1062no—
465C42—1062no—
466C42—2128no—
467C42C4⋊#615714yescomplemented
468C23—531no—
469C23—531no—
470C23—532no—
471C23—532no—
472C23—531no—
473C23—531no—
474C23—531no—
475C23—531no—
476C23—532no—
477C23—532no—
478C23C23×#615711yescharacteristic
479C2×C4—22no—
480C2×C4—22no—
481C2×C4—22no—
482C2×C4—22no—
483C2×C4—22no—
484C2×C4—22no—
485C2×C4—22no—
486C2×C4—532no—
487C2×C4—532no—
488C2×C4—532no—
489C2×C4—534no—
490C2×C4—534no—
491C2×C4—534no—
492C2×C4—534no—
493C2×C4—532no—
494C2×C4—532no—
495C2×C4—534no—
496C2×C4—1062no—
497C2×C4—1064no—
498C2×C4—1062no—
499C2×C4—1062no—
500C2×C4—1064no—
501C2×C4—1062no—
502C2×C4—1064no—
503C2×C4—1064no—
504C2×C4—1064no—
505C2×C4—1064no—
506C2×C4—1062no—
507C2×C4—1062no—
508C2×C4—1062no—
509C2×C4—1062no—
510C2×C4—1064no—
511C2×C4—1064no—
512C2×C4—1068no—
513C2×C4—1068no—
514C2×C4—1062no—
515C2×C4—1062no—
516C2×C4—1062no—
517C2×C4—2122no—
518C2×C4—2122no—
519C2×C4—2122no—
520C2×C4—2122no—
521C2×C4—2122no—
522C2×C4—2122no—
523C2×C4—2124no—
524C2×C4—2122no—
525C2×C4—2122no—
526C2×C4—2122no—
527C2×C4—2122no—
528C2×C4—2124no—
529C2×C4—2124no—
530C2×C4—2124no—
531C2×C4—2122no—
532C2×C4—2122no—
533C2×C4—2122no—
534C2×C4—2122no—
535C2×C4—2124no—
536C2×C4—2124no—
537C2×C4C2×C4⋊#615712yes—
538C2×C4C2×C4⋊#615712yes—
539C2×C4C2×C4⋊#615712yes—
540C2×C4#1313329614yes—
541C2×C4D4×#615712yes—
542C2×C4Q8×#615712yes—
543C22—531no—
544C22—531no—
545C22—532no—
546C22—531no—
547C22—531no—
548C22—531no—
549C22—531no—
550C22—531no—
551C22—531no—
552C22—532no—
553C22—532no—
554C22—532no—
555C22—532no—
556C22—532no—
557C22—532no—
558C22—532no—
559C22—532no—
560C22—532no—
561C22C22×C4⋊#615711yescharacteristic
562C22C2×#1313329612yes—
563C22C2×D4×#615711yescharacteristic
564C22C2×Q8×#615711yescharacteristic
565C22#1681169212yes—
566C4—22no—
567C4—22no—
568C4—24no—
569C4—42no—
570C4—534no—
571C4—1062no—
572C4—1062no—
573C4—1064no—
574C4—4242no—
575C4—4242no—
576C4—2124no—
577C4—2124no—
578C4—2122no—
579C4#1681157414yes—
580C2—531no—
581C2—531no—
582C2—531no—
583C2—531no—
584C2—532no—
585C2—532no—
586C2#2029116411yescharacteristic
587C2C2×#1681157411yescharacteristic
588C2#2029119611yescharacteristic
589C2#2029123112yes—
590C2#2029123412yes—
subgroupG/H
center, FrattiniC23C23×#6157
commutator, socleC23×C53C23×C4
FittingC4×C4⋊C4×C53C4
prime radical, cosocleC23×D53C24
hypercenterC4×C4⋊C4#6157
hypocenterC53#13909
LCSG ▷ C23×C53 ▷ C53
UCSC1 ◁ C23 ◁ C4×C4⋊C4

Complements

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

NH ≅ G/Ntype
30. C4⋊C4×Dic53C4semidirect
31. C4×#11719919C4semidirect
32. C4×#11719919C4semidirect
33. C4×C4⋊Dic53C4semidirect
34. #16810474C4semidirect
36. C4×C4⋊C4×C53C4semidirect
161. C42×C53C4⋊C4semidirect
169. C4×Dic53C4⋊C4semidirect
170. C4×Dic53C4⋊C4semidirect
171. C4×Dic53C4⋊C4semidirect
324. C4×C4⋊C4#6157semidirect
332. C53#13909semidirect
467. C42C4⋊#6157semidirect

Maximal subgroups

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

subgroupindex
C2×#168115692
C2×#168115672
C2×C4×C4⋊#61572
#202899002
#202898952
#202898932
C4×C4⋊C4×D532
#1390953