Finite Group Databasebeta

← #28328813

Subgroups — #28328813

Order 13568.

2806 conjugacy classes of subgroups.

The subgroup lattice has 610 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#20291224C218yesmaximal, complemented
2#20291223C218yesmaximal, complemented
3#20291195C212yesmaximal, complemented
4#20291192C214yesmaximal, complemented
5#20291159C214yesmaximal
6#20291159C212yesmaximal
7#20291153C212yesmaximal, complemented
8C22×C4⋊C4×D53C211yesmaximal, characteristic
9C2×#13133296C2214yes—
10C2×#13133296C2218yes—
11#16811574C22116yescomplemented
12#16811572C22132yescomplemented
13#16811566C2214yes—
14#16811566C2218yes—
15#16811565C2218yes—
16#16811565C2214yes—
17#16811565C2214yes—
18#16811565C2218yes—
19#16811564C2214yes—
20C4⋊C4×#6157C22116yescomplemented
21C4×C22⋊#6157C2218yes—
22C4×C22⋊#6157C2214yes—
23C4×C22⋊#6157C2214yes—
24C4×C22⋊#6157C2218yes—
25C23×C4×D53C2211yescharacteristic
26C23×C4×D53C2212yes—
27C22×C4⋊Dic53C411yescharacteristic, complemented
28C22×#11719919C412yescomplemented
29C2×C4⋊C4×D53C414yescomplemented
30C2×C4⋊C4×D53C412yescomplemented
31C2×C4⋊C4×D53C414yescomplemented
32C2×C4⋊C4×D53C411yescharacteristic, complemented
33C2×C4⋊C4×D53C411yescharacteristic, complemented
34C2×C4⋊C4×D53C2211yescharacteristic
35C2×C4⋊C4×D53C2212yes—
36C2×C4⋊C4×D53C2211yescharacteristic
37C2×C4⋊C4×D53C2214yes—
38C2×C4⋊C4×D53C2214yes—
39C22×C4⋊C4×C53C411yescharacteristic, complemented
40#13133296—216no—
41#13133296—416no—
42C2×C4⋊#6157C2318yes—
43C2×C4⋊#6157C23116yes—
44C2×C22⋊#6157C2318yes—
45C2×C22⋊#6157C2318yes—
46C2×C22⋊#6157C2318yes—
47C2×C4×#6157C2318yes—
48C2×C4×#6157C23116yes—
49(C2×#6157)⋊C4C23116yes—
50(C2×#6157)⋊C4C2318yes—
51(C2×#6157)⋊C4C2318yes—
52(C2×#6157)⋊C4C23116yes—
53C24×D53C2311yescharacteristic
54C23×Dic53C2×C411yescharacteristic
55C23×Dic53C2×C412yes—
56C22×C4×D53—28no—
57C22×C4×D53—48no—
58C22×C4×D53C2×C411yescharacteristic
59C22×C4×D53C2×C411yescharacteristic
60C22×C4×D53C2×C412yes—
61C22×C4×D53C2×C412yes—
62C22×C4×D53C2×C414yes—
63C22×C4×D53C2×C412yes—
64C22×C4×D53C2×C412yes—
65C22×C4×D53C2×C414yes—
66C22×C4×D53C2312yes—
67C22×C4×D53C2312yes—
68C22×C4×D53C2311yescharacteristic
69C22×C4×D53C2314yes—
70C22×C4×D53C2312yes—
71C22×C4×D53C2314yes—
72C22×C4×D53C2312yes—
73C22×C4×D53C2311yescharacteristic
74C2×C4⋊Dic53C2×C414yes—
75C2×C4⋊Dic53C2×C412yes—
76C2×C4⋊Dic53C2×C414yes—
77C2×C4⋊Dic53C2×C411yescharacteristic
78C2×C4⋊Dic53C2×C411yescharacteristic
79C2×#11719919C2×C418yes—
80C2×#11719919C2×C412yes—
81C2×#11719919C2×C412yes—
82C2×#11719919C2×C418yes—
83C2×#11719919C2×C412yes—
84C2×#11719919C2×C412yes—
85C4⋊C4×D53C2×C414yes—
86C4⋊C4×D53C2×C414yes—
87C4⋊C4×D53C2×C418yes—
88C4⋊C4×D53C2×C418yes—
89C4⋊C4×D53C2×C414yes—
90C4⋊C4×D53C2×C414yes—
91C4⋊C4×D53C2×C418yes—
92C4⋊C4×D53C2×C414yes—
93C4⋊C4×D53C2×C414yes—
94C4⋊C4×D53C2314yes—
95C4⋊C4×D53C2318yes—
96C4⋊C4×D53C2314yes—
97C23×C4×C53C2×C411yescharacteristic
98C23×C4×C53C2×C412yes—
99C2×C4⋊C4×C53C2×C414yes—
100C2×C4⋊C4×C53C2×C412yes—
101C2×C4⋊C4×C53C2×C414yes—
102C2×C4⋊C4×C53C2×C411yescharacteristic
103C2×C4⋊C4×C53C2×C411yescharacteristic
104C24×C53C22×C411yescharacteristic
105C23×D53—24no—
106C23×D53—24no—
107C23×D53—24no—
108C23×D53C22×C411yescharacteristic
109C23×D53C22×C411yescharacteristic
110C23×D53C22×C411yescharacteristic
111C23×D53C2411yescharacteristic
112C23×D53C2411yescharacteristic
113C23×D53C2411yescharacteristic
114C22×#6157C24116yes—
115C22×C4×C53—24no—
116C22×C4×C53—44no—
117C22×C4×C53C22×C414yes—
118C22×C4×C53C22×C411yescharacteristic
119C22×C4×C53C22×C412yes—
120C22×C4×C53C22×C412yes—
121C22×C4×C53C22×C412yes—
122C22×C4×C53C22×C412yes—
123C22×C4×C53C22×C414yes—
124C22×C4×C53C22×C411yescharacteristic
125C22×Dic53—24no—
126C22×Dic53—44no—
127C22×Dic53C22×C411yescharacteristic
128C22×Dic53C22×C412yes—
129C22×Dic53C22×C412yes—
130C22×Dic53C22×C414yes—
131C22×Dic53C22×C412yes—
132C22×Dic53C22×C412yes—
133C22×Dic53C22×C414yes—
134C22×Dic53C22×C411yescharacteristic
135C2×C4×D53—24no—
136C2×C4×D53—216no—
137C2×C4×D53—24no—
138C2×C4×D53—24no—
139C2×C4×D53—28no—
140C2×C4×D53—28no—
141C2×C4×D53—28no—
142C2×C4×D53—28no—
143C2×C4×D53—44no—
144C2×C4×D53—48no—
145C2×C4×D53—44no—
146C2×C4×D53—44no—
147C2×C4×D53—48no—
148C2×C4×D53—44no—
149C2×C4×D53—48no—
150C2×C4×D53C22×C412yes—
151C2×C4×D53C22×C414yes—
152C2×C4×D53C22×C414yes—
153C2×C4×D53C22×C412yes—
154C2×C4×D53C22×C412yes—
155C2×C4×D53C22×C412yes—
156C2×C4×D53C22×C412yes—
157C2×C4×D53C22×C414yes—
158C2×C4×D53C22×C414yes—
159C2×C4×D53C22×C414yes—
160C2×C4×D53C22×C414yes—
161C2×C4×D53C22×C412yes—
162C2×C4×D53#4118yes—
163C2×C4×D53C2412yes—
164C2×C4×D53C2414yes—
165C2×C4×D53C2414yes—
166C2×C4×D53C2412yes—
167C22⋊#6157—216no—
168C22⋊#6157—216no—
169C22⋊#6157—216no—
170C4⋊#6157—216no—
171C4⋊#6157—416no—
172C4×#6157—216no—
173C4×#6157—416no—
174C4⋊C4×C53C22×C418yes—
175C4⋊C4×C53C22×C414yes—
176C4⋊C4×C53C22×C414yes—
177C4⋊Dic53C22×C418yes—
178C4⋊Dic53C22×C414yes—
179C4⋊Dic53C22×C414yes—
180#11719919C22×C418yes—
181#11719919C22×C418yes—
182#11719919C22×C418yes—
183#11719919C22×C418yes—
184C23×C53—22no—
185C23×C53—22no—
186C23×C53—22no—
187C23×C53C23×C411yescharacteristic
188C23×C53C23×C411yescharacteristic
189C23×C53C23×C411yescharacteristic
190C22×D53—232no—
191C22×D53—22no—
192C22×D53—22no—
193C22×D53—22no—
194C22×D53—22no—
195C22×D53—22no—
196C22×D53—22no—
197C22×D53—22no—
198C22×D53—22no—
199C22×D53—22no—
200C22×D53—22no—
201C22×D53—22no—
202C22×D53—22no—
203C22×D53—22no—
204C22×D53—22no—
205C22×D53—22no—
206C22×D53C23×C411yescharacteristic
207C22×D53C23×C411yescharacteristic
208C22×D53C23×C411yescharacteristic
209C22×D53C2×#4114yes—
210C22×D53H(5,𝔽2)14yes—
211C22×D532−1+414yes—
212C22×D53C2511yescharacteristic
213C2×#6157—216no—
214C2×#6157—216no—
215C2×#6157—216no—
216C2×C4×C53—22no—
217C2×C4×C53—28no—
218C2×C4×C53—24no—
219C2×C4×C53—24no—
220C2×C4×C53—42no—
221C2×C4×C53—44no—
222C2×C4×C53—42no—
223C2×C4×C53#11714yescomplemented
224C2×C4×C53C23×C414yes—
225C2×C4×C53C23×C414yes—
226C2×C4×C53C23×C412yes—
227C2×C4×C53C23×C412yes—
228C2×Dic53—22no—
229C2×Dic53—28no—
230C2×Dic53—24no—
231C2×Dic53—24no—
232C2×Dic53—42no—
233C2×Dic53—44no—
234C2×Dic53—42no—
235C2×Dic53#11714yescomplemented
236C2×Dic53C23×C412yes—
237C2×Dic53C23×C414yes—
238C2×Dic53C23×C414yes—
239C2×Dic53C23×C412yes—
240C4×D53—24no—
241C4×D53—28no—
242C4×D53—28no—
243C4×D53—24no—
244C4×D53—28no—
245C4×D53—28no—
246C4×D53—24no—
247C4×D53—24no—
248C4×D53—24no—
249C4×D53—24no—
250C4×D53—44no—
251C4×D53—44no—
252C4×D53—44no—
253C4×D53—44no—
254C4×D53#11718yescomplemented
255C4×D53C2×#4118yes—
256#34178—531nomaximal, Sylow
257C22×C53—28no—
258C22×C53—21no—
259C22×C53—21no—
260C22×C53—21no—
261C22×C53—21no—
262C22×C53—21no—
263C22×C53—21no—
264C22×C53C2×#11712yes—
265C22×C53#51812yes—
266C22×C53#51912yes—
267C22×C53C24×C411yescharacteristic
268C2×D53—28no—
269C2×D53—28no—
270C2×D53—28no—
271C2×D53—21no—
272C2×D53—21no—
273C2×D53—21no—
274C2×D53—21no—
275C2×D53—21no—
276C2×D53—21no—
277C2×D53—28no—
278C2×D53—28no—
279C2×D53—28no—
280C2×D53C2×#11712yes—
281C2×D53#51812yes—
282C2×D53#51912yes—
283C2×D53C22×#4112yes—
284C2×D53C2×H(5,𝔽2)12yes—
285C2×D53C2×2−1+412yes—
286#6157—416no—
287C4×C53—22no—
288C4×C53—24no—
289C4×C53—24no—
290C4×C53C2×#11714yes—
291Dic53—22no—
292Dic53—24no—
293Dic53—24no—
294Dic53C2×#11714yes—
295C23×C4⋊C4—531no—
296#2345—538no—
297#2344—538no—
298#2316—532no—
299#2313—534no—
300#2280—532no—
301#2280—534no—
302#2274—532no—
303C2×C53—22no—
304C2×C53—22no—
305C2×C53—22no—
306C2×C53C22×#11711yescharacteristic
307C2×C53C2×#51811yescharacteristic
308C2×C53C2×#51911yescharacteristic
309D53—22no—
310D53—22no—
311D53—22no—
312D53#349214yes—
313C24×C4—531no—
314C24×C4—532no—
315C2×#117—534no—
316C2×#117—538no—
317C22×C4⋊C4—532no—
318C22×C4⋊C4—531no—
319C22×C4⋊C4—532no—
320C22×C4⋊C4—531no—
321C22×C4⋊C4—534no—
322C22×C4⋊C4—534no—
323C22×C4⋊C4—532no—
324C22×C4⋊C4—534no—
325C22×C4⋊C4—531no—
326C22×C4⋊C4—531no—
327C22×C4⋊C4—534no—
328C22×C4⋊C4—531no—
329C22×C4⋊C4#615711yescharacteristic, complemented
330#389—5316no—
331#387—5332no—
332#381—534no—
333#381—538no—
334#380—538no—
335#380—534no—
336#380—538no—
337#380—534no—
338#379—534no—
339C4×C4⋊C4—5316no—
340C4×C22⋊C4—538no—
341C4×C22⋊C4—534no—
342C4×C22⋊C4—534no—
343C4×C22⋊C4—538no—
344C53#3417811yescharacteristic, complemented, Sylow
345C25—531no—
346C23×C4—531no—
347C23×C4—531no—
348C23×C4—532no—
349C23×C4—534no—
350C23×C4—532no—
351C23×C4—532no—
352C23×C4—531no—
353C23×C4—531no—
354C23×C4—532no—
355C23×C4—532no—
356C23×C4—531no—
357C23×C4—532no—
358C23×C4—534no—
359C23×C4—532no—
360C23×C4—534no—
361C23×C4—532no—
362C23×C4—532no—
363C23×C4—534no—
364C23×C4—1068no—
365C23×C4—2128no—
366C23×C4C2×#615711yescharacteristic
367C23×C4C2×#615712yes—
368#117—10616no—
369#117—21216no—
370C2×C4⋊C4—532no—
371C2×C4⋊C4—532no—
372C2×C4⋊C4—538no—
373C2×C4⋊C4—532no—
374C2×C4⋊C4—532no—
375C2×C4⋊C4—538no—
376C2×C4⋊C4—534no—
377C2×C4⋊C4—532no—
378C2×C4⋊C4—534no—
379C2×C4⋊C4—531no—
380C2×C4⋊C4—531no—
381C2×C4⋊C4—534no—
382C2×C4⋊C4—534no—
383C2×C4⋊C4—538no—
384C2×C4⋊C4—534no—
385C2×C4⋊C4—534no—
386C2×C4⋊C4—538no—
387C2×C4⋊C4—538no—
388C2×C4⋊C4—534no—
389C2×C4⋊C4—534no—
390C2×C4⋊C4—534no—
391C2×C4⋊C4—538no—
392C2×C4⋊C4—534no—
393C2×C4⋊C4—538no—
394C2×C4⋊C4—5316no—
395C2×C4⋊C4C2×#615714yes—
396C2×C4⋊C4C2×#615712yes—
397C2×C4⋊C4C2×#615714yes—
398C2×C4⋊C4C2×#615711yescharacteristic
399C2×C4⋊C4C2×#615711yescharacteristic
400C2×C22⋊C4—538no—
401C2×C22⋊C4—538no—
402C2×C22⋊C4—538no—
403C2×C42—538no—
404C2×C42—5316no—
405#95—5316no—
406#95—538no—
407#95—538no—
408#95—5316no—
409C24—531no—
410C24—531no—
411C24—531no—
412C24—531no—
413C24—531no—
414C24—531no—
415C24—1064no—
416C24—1064no—
417C24—1064no—
418C24C22×#615711yescharacteristic
419C22×C4—24no—
420C22×C4—44no—
421C22×C4—531no—
422C22×C4—538no—
423C22×C4—532no—
424C22×C4—532no—
425C22×C4—532no—
426C22×C4—532no—
427C22×C4—531no—
428C22×C4—534no—
429C22×C4—532no—
430C22×C4—532no—
431C22×C4—534no—
432C22×C4—532no—
433C22×C4—532no—
434C22×C4—532no—
435C22×C4—534no—
436C22×C4—534no—
437C22×C4—532no—
438C22×C4—532no—
439C22×C4—532no—
440C22×C4—534no—
441C22×C4—534no—
442C22×C4—534no—
443C22×C4—534no—
444C22×C4—534no—
445C22×C4—534no—
446C22×C4—5316no—
447C22×C4—1064no—
448C22×C4—1068no—
449C22×C4—1068no—
450C22×C4—1068no—
451C22×C4—1068no—
452C22×C4—10616no—
453C22×C4—1064no—
454C22×C4—1064no—
455C22×C4—1064no—
456C22×C4—2128no—
457C22×C4—2124no—
458C22×C4—2128no—
459C22×C4—2124no—
460C22×C4—2124no—
461C22×C4—2128no—
462C22×C4—2124no—
463C22×C4—2124no—
464C22×C4C22×#615711yescharacteristic
465C22×C4C22×#615714yes—
466C22×C4C22×#615712yes—
467C22×C4C22×#615712yes—
468C22×C4C22×#615712yes—
469C22×C4C22×#615712yes—
470C22×C4C22×#615714yes—
471C22×C4C22×#615711yescharacteristic
472C4⋊C4—538no—
473C4⋊C4—538no—
474C4⋊C4—538no—
475C4⋊C4—538no—
476C4⋊C4—538no—
477C4⋊C4—534no—
478C4⋊C4—534no—
479C4⋊C4—10616no—
480C4⋊C4—21216no—
481C4⋊C4C22×#615718yes—
482C4⋊C4C22×#615714yes—
483C4⋊C4C22×#615714yes—
484C22⋊C4—10616no—
485C22⋊C4—10616no—
486C22⋊C4—10616no—
487C42—10616no—
488C42—21216no—
489C23—22no—
490C23—22no—
491C23—22no—
492C23—531no—
493C23—534no—
494C23—534no—
495C23—534no—
496C23—531no—
497C23—531no—
498C23—531no—
499C23—1062no—
500C23—1062no—
501C23—1062no—
502C23—1062no—
503C23—1062no—
504C23—1062no—
505C23—1062no—
506C23—1062no—
507C23—1062no—
508C23—1062no—
509C23—1062no—
510C23—1062no—
511C23—10632no—
512C23—1062no—
513C23—1062no—
514C23—1062no—
515C23C23×#615711yescharacteristic
516C23C23×#615711yescharacteristic
517C23C23×#615711yescharacteristic
518C2×C4—22no—
519C2×C4—28no—
520C2×C4—24no—
521C2×C4—24no—
522C2×C4—42no—
523C2×C4—44no—
524C2×C4—42no—
525C2×C4—534no—
526C2×C4—532no—
527C2×C4—538no—
528C2×C4—538no—
529C2×C4—532no—
530C2×C4—534no—
531C2×C4—534no—
532C2×C4—1064no—
533C2×C4—1064no—
534C2×C4—1068no—
535C2×C4—1068no—
536C2×C4—1064no—
537C2×C4—1064no—
538C2×C4—1068no—
539C2×C4—1068no—
540C2×C4—1064no—
541C2×C4—1064no—
542C2×C4—1062no—
543C2×C4—1068no—
544C2×C4—1064no—
545C2×C4—1064no—
546C2×C4—10616no—
547C2×C4—10616no—
548C2×C4—10616no—
549C2×C4—2124no—
550C2×C4—2124no—
551C2×C4—2124no—
552C2×C4—2124no—
553C2×C4—2122no—
554C2×C4—2124no—
555C2×C4—2122no—
556C2×C4#1313329614yescomplemented
557C2×C4C23×#615712yes—
558C2×C4C23×#615714yes—
559C2×C4C23×#615714yes—
560C2×C4C23×#615712yes—
561C22—28no—
562C22—21no—
563C22—21no—
564C22—21no—
565C22—21no—
566C22—21no—
567C22—21no—
568C22—532no—
569C22—532no—
570C22—532no—
571C22—532no—
572C22—532no—
573C22—532no—
574C22—1068no—
575C22—1068no—
576C22—1068no—
577C22—1068no—
578C22—1068no—
579C22—1068no—
580C22—1061no—
581C22—1061no—
582C22—1061no—
583C22—1061no—
584C22—1061no—
585C22—1061no—
586C22C2×#1313329612yes—
587C22#1681169012yes—
588C22#1681169112yes—
589C22C24×#615711yescharacteristic
590C4—22no—
591C4—24no—
592C4—24no—
593C4—534no—
594C4—1062no—
595C4—1064no—
596C4—1064no—
597C4—21216no—
598C4C2×#1313329614yes—
599C2—22no—
600C2—22no—
601C2—22no—
602C2—534no—
603C2—1062no—
604C2—1062no—
605C2—1062no—
606C2C22×#1313329611yescharacteristic
607C2C2×#1681169011yescharacteristic
608C2C2×#1681169111yescharacteristic
subgroupG/H
center, FrattiniC22C24×#6157
commutator, socleC22×C53C24×C4
FittingC22×C4⋊C4×C53C4
prime radicalC24×D53C23
cosocleC22×D53C25
hypercenterC22×C4⋊C4#6157
hypocenterC53#34178
LCSG ▷ C22×C53 ▷ C53
UCSC1 ◁ C22 ◁ C22×C4⋊C4

Complements

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

NH ≅ G/Ntype
1. #20291224C2semidirect
2. #20291223C2semidirect
3. #20291195C2semidirect
4. #20291192C2semidirect
7. #20291153C2semidirect
11. #16811574C22semidirect
12. #16811572C22semidirect
20. C4⋊C4×#6157C22semidirect
27. C22×C4⋊Dic53C4semidirect
28. C22×#11719919C4semidirect
29. C2×C4⋊C4×D53C4semidirect
30. C2×C4⋊C4×D53C4semidirect
31. C2×C4⋊C4×D53C4semidirect
32. C2×C4⋊C4×D53C4semidirect
33. C2×C4⋊C4×D53C4semidirect
39. C22×C4⋊C4×C53C4semidirect
223. C2×C4×C53#117semidirect
235. C2×Dic53#117semidirect
254. C4×D53#117semidirect
329. C22×C4⋊C4#6157semidirect
344. C53#34178semidirect
556. C2×C4#13133296semidirect

Maximal subgroups

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

subgroupindex
#202912242
#202912232
#202911952
#202911922
#202911592
#202911532
C22×C4⋊C4×D532
#3417853