Finite Group Databasebeta

← #28318848

Subgroups — #28318848

Order 13568.

954 conjugacy classes of subgroups.

The subgroup lattice has 692 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#16811655C211yesmaximal, characteristic
2C2×#16811652C211yesmaximal, characteristic
3#20290514C211yesmaximal, characteristic, complemented
4#20290514C211yesmaximal, characteristic, complemented
5#20290501C211yesmaximal, characteristic, complemented
6#20290457C211yesmaximal, characteristic, complemented
7#20290456C211yesmaximal, characteristic, complemented
8#20290453C211yesmaximal, characteristic, complemented
9#20290452C211yesmaximal, characteristic, complemented
10#20290445C211yesmaximal, characteristic, complemented
11#20290444C211yesmaximal, characteristic, complemented
12#20290432C211yesmaximal, characteristic, complemented
13#20290382C211yesmaximal, characteristic, complemented
14#20290381C211yesmaximal, characteristic, complemented
15#20277175C211yesmaximal, characteristic
16C22×#11719936C2211yescharacteristic
17C22×#11719940C2211yescharacteristic
18C22×D53⋊C8C2211yescharacteristic
19C23×#88324C2211yescharacteristic
20#16811655C418yescomplemented
21#16811652C418yescomplemented
22C2×C4⋊#88324C2211yescharacteristic
23C2×C4⋊#88324C2211yescharacteristic
24C2×#13133274C2211yescharacteristic
25C2×#13133274C2211yescharacteristic
26C2×C22⋊#88324C2211yescharacteristic
27C2×C22⋊#88324C2211yescharacteristic
28C2×#13133272C2211yescharacteristic
29C2×#13133272C2211yescharacteristic
30C2×#13133272C2211yescharacteristic
31C2×#13133272C2211yescharacteristic
32C2×#13133272C2211yescharacteristic
33C2×#13133272C2211yescharacteristic
34C2×C4×#88324C2211yescharacteristic
35C2×C4×#88324C2211yescharacteristic
36#16811489C2211yescharacteristic
37#16811489C2211yescharacteristic
38#16811489C2211yescharacteristic
39#16811489C2211yescharacteristic
40#16811489C2211yescharacteristic
41#16811489C2211yescharacteristic
42#16811489C2211yescharacteristic
43#16811489C2211yescharacteristic
44#16811489C2211yescharacteristic
45#16811489C2211yescharacteristic
46C2×#13133177C2211yescharacteristic
47C2×#13133175C2211yescharacteristic
48C2×#13133173C2211yescharacteristic
49#16810523C411yescharacteristic
50#16810523C411yescharacteristic
51#16810520C411yescharacteristic
52#16810518C411yescharacteristic
53#16810516C411yescharacteristic
54#16810514C411yescharacteristic
55#16810512C411yescharacteristic
56#16810474C2211yescharacteristic
57C22⋊C4×Dic53C2211yescharacteristic
58#16810470C2211yescharacteristic
59C42×Dic53C2211yescharacteristic
60C53×#388C411yescharacteristic
61C2×#11719936—41no—
62C2×#11719936—41no—
63C2×#11719936C2×C414yes—
64C2×#11719940—41no—
65C2×#11719940—41no—
66C2×#11719940C2×C414yes—
67C2×D53⋊C8—41no—
68C2×D53⋊C8—41no—
69C2×D53⋊C8C2×C414yes—
70C22×#88324—41no—
71C22×#88324—41no—
72C22×#88324C2×C414yes—
73C22×#88324C2311yescharacteristic
74C22×#88324C2311yescharacteristic
75C22×#88324C2311yescharacteristic
76C22×#88324C2311yescharacteristic
77C22×#88324C2311yescharacteristic
78C22×#88324C2311yescharacteristic
79C22×#88324C2311yescharacteristic
80C22×#88324C2311yescharacteristic
81C4⋊#88324C2×C414yes—
82C4⋊#88324C2×C414yes—
83#13133274C2×C414yes—
84#13133274C2×C414yes—
85C22⋊#88324C2×C414yes—
86C22⋊#88324C2×C414yes—
87#13133272—41no—
88#13133272—41no—
89#13133272—41no—
90#13133272—41no—
91#13133272C2×C414yes—
92#13133272C2×C414yes—
93C4×#88324—41no—
94C4×#88324—41no—
95C22×C22⋊D53C2×C411yescharacteristic
96C23×Dic53C2311yescharacteristic
97C22×C4×D53C2311yescharacteristic
98#13133177C2×C414yes—
99#13133177C2×C414yes—
100#13133175—41no—
101#13133175—41no—
102#13133173—41no—
103#13133173—41no—
104C2×C4⋊Dic53C2×C411yescharacteristic
105C2×#11719919C2×C411yescharacteristic
106C2×#11719919C2×C411yescharacteristic
107C2×#11719919C2×C411yescharacteristic
108C2×#11719919C2×C411yescharacteristic
109C2×#11719919C2×C411yescharacteristic
110C2×C22⋊Dic53C2×C411yescharacteristic
111C2×C22⋊Dic53C2×C411yescharacteristic
112C2×#11719921C2×C411yescharacteristic
113C2×#11719921C2×C411yescharacteristic
114C2×#11719921C2×C411yescharacteristic
115C2×#11719921C2×C411yescharacteristic
116C2×#11719921C2×C411yescharacteristic
117C2×C4×Dic53C2×C411yescharacteristic
118C2×C4×Dic53C2×C411yescharacteristic
119C2×C4×Dic53C2×C411yescharacteristic
120C2×C4×Dic53C2311yescharacteristic
121C2×C4×Dic53C2311yescharacteristic
122C2×C4×Dic53C2311yescharacteristic
123C2×C4×Dic53C2311yescharacteristic
124C2×C4×Dic53C2311yescharacteristic
125#13133124C2×C411yescharacteristic
126#13133124C2×C411yescharacteristic
127#13133124C2×C411yescharacteristic
128#13133124C2×C411yescharacteristic
129C2×C4⋊C4×C53C2×C411yescharacteristic
130C2×C22⋊C4×C53C2×C411yescharacteristic
131C2×C22⋊C4×C53C2×C411yescharacteristic
132C2×C22⋊C4×C53C2×C411yescharacteristic
133C2×C42×C53C2×C411yescharacteristic
134#95×C53C2×C411yescharacteristic
135#95×C53C2×C411yescharacteristic
136C24×C53C22×C411yescharacteristic
137C23×D53C22×C411yescharacteristic
138C22×C4×C53C22×C411yescharacteristic
139C22×C4×C53C22×C411yescharacteristic
140C22×C4×C53C22×C411yescharacteristic
141C22×C4×C53C22×C411yescharacteristic
142C22×C4×C53C22×C411yescharacteristic
143C22×C4×C53C22×C411yescharacteristic
144C2×C22⋊D53—41no—
145C2×C22⋊D53—41no—
146C2×C22⋊D53C4214yes—
147C22×Dic53—21no—
148C22×Dic53—21no—
149C22×Dic53—21no—
150C22×Dic53—21no—
151C22×Dic53C22×C411yescharacteristic
152C22×Dic53C22×C411yescharacteristic
153C22×Dic53C22×C411yescharacteristic
154C22×Dic53C22×C411yescharacteristic
155C22×Dic53C22×C411yescharacteristic
156C22×Dic53C22×C411yescharacteristic
157C22×Dic53C22×C412yes—
158C22×Dic53C22×C412yes—
159C22×Dic53C2411yescharacteristic
160C2×C4×D53—21no—
161C2×C4×D53—21no—
162C2×C4×D53—21no—
163C2×C4×D53—21no—
164C2×C4×D53C22×C412yes—
165C2×C4×D53C22×C412yes—
166#11719940—44no—
167C2×#88324—21no—
168C2×#88324—21no—
169C2×#88324—21no—
170C2×#88324—21no—
171C2×#88324—21no—
172C2×#88324—21no—
173C2×#88324—21no—
174C2×#88324—21no—
175C2×#88324—21no—
176C2×#88324—21no—
177C2×#88324—21no—
178C2×#88324—21no—
179C2×#88324—21no—
180C2×#88324—21no—
181C2×#88324—21no—
182C2×#88324—21no—
183C2×#88324—44no—
184C2×#88324C22×C412yes—
185C2×#88324C22×C412yes—
186C2×#88324C22×C412yes—
187C2×#88324C22×C412yes—
188C2×#88324C22×C412yes—
189C2×#88324C22×C412yes—
190C2×#88324C22×C412yes—
191C2×#88324C22×C412yes—
192#11719936—44no—
193D53⋊C8—44no—
194C4⋊C4×C53—41no—
195C22⋊C4×C53—41no—
196C22⋊C4×C53—41no—
197C22⋊C4×C53C4214yes—
198C42×C53—41no—
199C22⋊Dic53—41no—
200C22⋊Dic53—41no—
201#11719921—41no—
202#11719921—41no—
203#11719921—41no—
204#11719921—41no—
205#11719921C4214yes—
206C4⋊Dic53—41no—
207#11719919—41no—
208#11719919—41no—
209#11719919—41no—
210#11719919—41no—
211#11719919C4214yes—
212C4×Dic53—21no—
213C4×Dic53—21no—
214C4×Dic53—21no—
215C4×Dic53—21no—
216C4×Dic53—21no—
217C4×Dic53—21no—
218C4×Dic53—21no—
219C4×Dic53—21no—
220C4×Dic53—41no—
221C4×Dic53—41no—
222C4×Dic53—41no—
223C4×Dic53C22×C412yes—
224C4×Dic53C22×C412yes—
225C23×C53—21no—
226C23×C53—21no—
227C23×C53—21no—
228C23×C53—21no—
229C23×C53—21no—
230C23×C53—21no—
231C23×C53C2×C4212yes—
232C23×C53C23×C411yescharacteristic
233C22×D53—21no—
234C22×D53—21no—
235C22×D53—21no—
236C22×D53—21no—
237C22×D53—21no—
238C22×D53—21no—
239C22×D53C2×C4212yes—
240C2×C4×C53—21no—
241C2×C4×C53—21no—
242C2×C4×C53—21no—
243C2×C4×C53—21no—
244C2×C4×C53—21no—
245C2×C4×C53—21no—
246C2×C4×C53—21no—
247C2×C4×C53—21no—
248C2×C4×C53—21no—
249C2×C4×C53—21no—
250C2×C4×C53—21no—
251C2×C4×C53—21no—
252C2×C4×C53—21no—
253C2×C4×C53—21no—
254C2×C4×C53—21no—
255C2×C4×C53—21no—
256C2×C4×C53C2×C4212yes—
257C2×C4×C53C2×C4212yes—
258C22⋊D53—44no—
259C2×Dic53—21no—
260C2×Dic53—21no—
261C2×Dic53—21no—
262C2×Dic53—21no—
263C2×Dic53—22no—
264C2×Dic53—22no—
265C2×Dic53—22no—
266C2×Dic53—22no—
267C2×Dic53—21no—
268C2×Dic53—21no—
269C2×Dic53—21no—
270C2×Dic53—21no—
271C2×Dic53—21no—
272C2×Dic53—21no—
273C2×Dic53—21no—
274C2×Dic53—21no—
275C2×Dic53—21no—
276C2×Dic53—21no—
277C2×Dic53—21no—
278C2×Dic53—21no—
279C2×Dic53C2×C4212yes—
280C2×Dic53C2×C4212yes—
281C2×Dic53C23×C411yescharacteristic
282C2×Dic53C23×C411yescharacteristic
283C2×Dic53H(5,𝔽2)11yescharacteristic
284C2×Dic53H(5,𝔽2)11yescharacteristic
285C2×Dic53H(5,𝔽2)11yescharacteristic
286C2×Dic532−1+411yescharacteristic
287C4×D53—22no—
288C4×D53—22no—
289C4×D53—22no—
290C4×D53—22no—
291#88324—22no—
292#88324—22no—
293#88324—22no—
294#88324—22no—
295#88324—22no—
296#88324—22no—
297#88324—22no—
298#88324—22no—
299#20563—531nomaximal, Sylow
300C22×C53—22no—
301C22×C53—22no—
302C22×C53—22no—
303C22×C53—41no—
304C22×C53—41no—
305C22×C53—41no—
306C22×C53—41no—
307C22×C53C22×C4211yescharacteristic
308C22×C53#51811yescharacteristic
309C22×C53#52011yescharacteristic
310C22×C53#56811yescharacteristic
311C22×C53#56811yescharacteristic
312C22×C53#56811yescharacteristic
313C22×C53#56811yescharacteristic
314C2×D53—22no—
315C2×D53—22no—
316C2×D53—22no—
317C2×D53—41no—
318C2×D53—41no—
319C2×D53—41no—
320C2×D53—41no—
321C4×C53—22no—
322C4×C53—22no—
323C4×C53—41no—
324C4×C53—41no—
325C4×C53—41no—
326C4×C53—41no—
327Dic53—22no—
328Dic53—22no—
329Dic53—41no—
330Dic53—41no—
331Dic53—41no—
332Dic53—41no—
333Dic53#51811yescharacteristic
334Dic53#51811yescharacteristic
335Dic53#52011yescharacteristic
336Dic53#52011yescharacteristic
337C2×#435—531no—
338C2×#435—531no—
339#2293—531no—
340#1771—531no—
341#1771—531no—
342#1767—531no—
343#1753—531no—
344#1753—531no—
345#1752—531no—
346#1752—531no—
347#1750—531no—
348#1750—531no—
349#1745—531no—
350#1729—531no—
351#1729—531no—
352C2×C53—42no—
353C2×C53#227511yescharacteristic
354C2×C53#287311yescharacteristic
355C2×C53#287511yescharacteristic
356C2×C53#298011yescharacteristic
357C2×C53#298011yescharacteristic
358C2×C53#299511yescharacteristic
359C2×C53#299511yescharacteristic
360D53—42no—
361C22×M16—531no—
362C22×M16—531no—
363C23×C8—531no—
364C23×C8—531no—
365C2×C4×D4—531no—
366C2×#117—531no—
367C2×#117—531no—
368#435—538no—
369#435—538no—
370C2×C4⋊C8—531no—
371C2×C4⋊C8—531no—
372C2×C22⋊C8—531no—
373C2×C22⋊C8—531no—
374C2×C22⋊C8—531no—
375C2×C22⋊C8—531no—
376C2×#97—531no—
377C2×#97—531no—
378C2×#97—531no—
379C2×#97—531no—
380C2×#97—531no—
381C2×#97—531no—
382C2×C4×C8—531no—
383C2×C4×C8—531no—
384#390—531no—
385#390—531no—
386#389—531no—
387#389—531no—
388#388—531no—
389#388—531no—
390#388—531no—
391#388#615711yescharacteristic
392C4×C4⋊C4—531no—
393C4×C22⋊C4—531no—
394C4×C22⋊C4—531no—
395C43—531no—
396#336—531no—
397#336—531no—
398#336—531no—
399#336—531no—
400#336—531no—
401#336—531no—
402#336—531no—
403#336—531no—
404#336—531no—
405#336—531no—
406C53#2056311yescharacteristic, complemented, Sylow
407C22×D4—531no—
408C23×C4—531no—
409C23×C4—531no—
410C2×M16—534no—
411C2×M16—534no—
412C2×M16—2121no—
413C2×M16—2121no—
414C2×M16—2121no—
415C2×M16—2121no—
416C22×C8—531no—
417C22×C8—531no—
418C22×C8—531no—
419C22×C8—531no—
420C22×C8—531no—
421C22×C8—531no—
422C22×C8—531no—
423C22×C8—531no—
424C22×C8—534no—
425C22×C8—534no—
426C22×C8—2121no—
427C22×C8—2121no—
428C22×C8—2121no—
429C22×C8—2121no—
430C4×D4—534no—
431C4×D4—534no—
432#117—2121no—
433#117—2121no—
434#117—2121no—
435#117—2121no—
436C2×C4⋊C4—531no—
437C2×C4⋊C4—531no—
438C2×C4⋊C4—531no—
439C2×C4⋊C4—531no—
440C2×C4⋊C4—531no—
441C2×C4⋊C4—531no—
442C2×C4⋊C4C2×#615711yescharacteristic
443C2×C22⋊C4—531no—
444C2×C22⋊C4—531no—
445C2×C22⋊C4—531no—
446C2×C22⋊C4—531no—
447C2×C22⋊C4—531no—
448C2×C22⋊C4—531no—
449C2×C22⋊C4—531no—
450C2×C22⋊C4C2×#615711yescharacteristic
451C2×C22⋊C4C2×#615711yescharacteristic
452C2×C22⋊C4C2×#615711yescharacteristic
453C2×C42—531no—
454C2×C42—531no—
455C2×C42—531no—
456C2×C42—531no—
457C2×C42—531no—
458C2×C42—531no—
459C2×C42—531no—
460C2×C42—531no—
461C2×C42C2×#615711yescharacteristic
462C4⋊C8—534no—
463C4⋊C8—534no—
464C22⋊C8—534no—
465C22⋊C8—534no—
466C22⋊C8—534no—
467C22⋊C8—534no—
468#97—534no—
469#97—534no—
470#97—2121no—
471#97—2121no—
472#97—2121no—
473#97—2121no—
474C4×C8—2121no—
475C4×C8—2121no—
476#95—531no—
477#95—531no—
478#95—531no—
479#95—531no—
480#95C2×#615711yescharacteristic
481#95C2×#615711yescharacteristic
482C24—531no—
483C24C22×#615711yescharacteristic
484C2×D4—534no—
485C2×D4—2121no—
486C2×D4—2121no—
487C22×C4—531no—
488C22×C4—532no—
489C22×C4—532no—
490C22×C4—531no—
491C22×C4—531no—
492C22×C4—532no—
493C22×C4—532no—
494C22×C4—531no—
495C22×C4—531no—
496C22×C4—531no—
497C22×C4—531no—
498C22×C4—1061no—
499C22×C4—1061no—
500C22×C4—1061no—
501C22×C4—1061no—
502C22×C4—1061no—
503C22×C4—1061no—
504C22×C4—1061no—
505C22×C4—1061no—
506C22×C4C22×#615711yescharacteristic
507C22×C4C22×#615711yescharacteristic
508C22×C4C22×#615711yescharacteristic
509C22×C4C22×#615711yescharacteristic
510C22×C4C22×#615711yescharacteristic
511C22×C4C22×#615711yescharacteristic
512M16—2124no—
513M16—2124no—
514C2×C8—532no—
515C2×C8—532no—
516C2×C8—532no—
517C2×C8—532no—
518C2×C8—532no—
519C2×C8—532no—
520C2×C8—532no—
521C2×C8—532no—
522C2×C8—1061no—
523C2×C8—1061no—
524C2×C8—1061no—
525C2×C8—1061no—
526C2×C8—1061no—
527C2×C8—1061no—
528C2×C8—1061no—
529C2×C8—1061no—
530C2×C8—1061no—
531C2×C8—1061no—
532C2×C8—1061no—
533C2×C8—1061no—
534C2×C8—1061no—
535C2×C8—1061no—
536C2×C8—1061no—
537C2×C8—1061no—
538C2×C8—2124no—
539C2×C8—2124no—
540C4⋊C4—41no—
541C4⋊C4—534no—
542C4⋊C4—2121no—
543C4⋊C4—2121no—
544C4⋊C4—2121no—
545C4⋊C4—2121no—
546C4⋊C4—2121no—
547C22⋊C4—41no—
548C22⋊C4—41no—
549C22⋊C4—534no—
550C22⋊C4—2121no—
551C22⋊C4—2121no—
552C22⋊C4—2121no—
553C22⋊C4—2121no—
554C22⋊C4—2121no—
555C22⋊C4—2121no—
556C22⋊C4C4×#615714yes—
557C42—41no—
558C42—532no—
559C42—532no—
560C42—1061no—
561C42—1061no—
562C42—1061no—
563C42—1061no—
564C42—1061no—
565C42—1061no—
566C42—1061no—
567C42—1061no—
568C42—2121no—
569C42—2121no—
570C42—2121no—
571C23—21no—
572C23—21no—
573C23—21no—
574C23—21no—
575C23—21no—
576C23—21no—
577C23—532no—
578C23—1061no—
579C23—1061no—
580C23—1061no—
581C23—1061no—
582C23—1061no—
583C23—1061no—
584C23C2×C4×#615712yes—
585C23C23×#615711yescharacteristic
586D4—2124no—
587C2×C4—21no—
588C2×C4—21no—
589C2×C4—21no—
590C2×C4—21no—
591C2×C4—21no—
592C2×C4—21no—
593C2×C4—21no—
594C2×C4—21no—
595C2×C4—21no—
596C2×C4—21no—
597C2×C4—21no—
598C2×C4—21no—
599C2×C4—21no—
600C2×C4—21no—
601C2×C4—21no—
602C2×C4—21no—
603C2×C4—531no—
604C2×C4—531no—
605C2×C4—531no—
606C2×C4—531no—
607C2×C4—531no—
608C2×C4—531no—
609C2×C4—532no—
610C2×C4—532no—
611C2×C4—1062no—
612C2×C4—1062no—
613C2×C4—1062no—
614C2×C4—1062no—
615C2×C4—1062no—
616C2×C4—1062no—
617C2×C4—1062no—
618C2×C4—1062no—
619C2×C4—1061no—
620C2×C4—1061no—
621C2×C4—1061no—
622C2×C4—1061no—
623C2×C4—1061no—
624C2×C4—1061no—
625C2×C4—1061no—
626C2×C4—1061no—
627C2×C4—1061no—
628C2×C4—1061no—
629C2×C4—1061no—
630C2×C4—1061no—
631C2×C4—1061no—
632C2×C4—1061no—
633C2×C4—1061no—
634C2×C4—1061no—
635C2×C4C2×C4×#615712yes—
636C2×C4C2×C4×#615712yes—
637C8—1062no—
638C8—1062no—
639C8—1062no—
640C8—1062no—
641C8—1062no—
642C8—1062no—
643C8—1062no—
644C8—1062no—
645C22—22no—
646C22—22no—
647C22—22no—
648C22—41no—
649C22—41no—
650C22—41no—
651C22—41no—
652C22—1062no—
653C22—1062no—
654C22—1062no—
655C22—2121no—
656C22—2121no—
657C22—2121no—
658C22—2121no—
659C22C22×C4×#615711yescharacteristic
660C22#1681169011yescharacteristic
661C22#1681169211yescharacteristic
662C22#1681170111yescharacteristic
663C22#1681170211yescharacteristic
664C22#1681170211yescharacteristic
665C22#1681170211yescharacteristic
666C4—22no—
667C4—22no—
668C4—41no—
669C4—41no—
670C4—41no—
671C4—41no—
672C4—531no—
673C4—531no—
674C4—531no—
675C4—531no—
676C4—1062no—
677C4—1062no—
678C4—2121no—
679C4—2121no—
680C4—2121no—
681C4—2121no—
682C2—42no—
683C2—2122no—
684C2#2029115411yescharacteristic
685C2#2029128011yescharacteristic
686C2#2029128611yescharacteristic
687C2#2029159911yescharacteristic
688C2#2029160111yescharacteristic
689C2#2029166911yescharacteristic
690C2#2029167011yescharacteristic
subgroupG/H
center, FrattiniC23C23×#6157
commutatorC22×C53C22×C42
FittingC53×#388C4
socleC23×C53C23×C4
prime radicalC22×C22⋊D53C2×C4
cosocleC22×Dic53C24
hypercenter#388#6157
hypocenterC53#20563
LCSG ▷ C22×C53 ▷ C53
UCSC1 ◁ C23 ◁ #388

Complements

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

NH ≅ G/Ntype
3. #20290514C2semidirect
4. #20290514C2semidirect
5. #20290501C2semidirect
6. #20290457C2semidirect
7. #20290456C2semidirect
8. #20290453C2semidirect
9. #20290452C2semidirect
10. #20290445C2semidirect
11. #20290444C2semidirect
12. #20290432C2semidirect
13. #20290382C2semidirect
14. #20290381C2semidirect
20. #16811655C4semidirect
21. #16811652C4semidirect
406. C53#20563semidirect

Maximal subgroups

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

subgroupindex
C2×#168116552
C2×#168116522
#202905142
#202905012
#202904572
#202904562
#202904532
#202904522
#202904452
#202904442
#202904322
#202903822
#202903812
#202771752
#2056353