Finite Group Databasebeta

← #27400920

Subgroups — #27400920

Order 13568.

1098 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×#16811342C211yesmaximal, characteristic
2C2×#16810752C211yesmaximal, characteristic, complemented
3C4⋊C4×Dic106C211yesmaximal, characteristic, complemented
4#20275646C211yesmaximal, characteristic
5#20275636C211yesmaximal, characteristic, complemented
6#20275532C211yesmaximal, characteristic
7#20275530C211yesmaximal, characteristic
8#20275515C211yesmaximal, characteristic, complemented
9#20274721C211yesmaximal, characteristic, complemented
10#20274718C211yesmaximal, characteristic, complemented
11#20274713C211yesmaximal, characteristic
12#20274548C211yesmaximal, characteristic, complemented
13#20274547C211yesmaximal, characteristic, complemented
14#102×Dic53C211yesmaximal, characteristic
15C53×#1983C211yesmaximal, characteristic
16#16811342—24no—
17#16811342C418yescomplemented
18C2×#13133215C2211yescharacteristic
19C2×#13133210—21no—
20C2×#13133210—21no—
21C2×#13133204C2211yescharacteristic
22C2×#13133194—21no—
23C2×Q8×Dic53C2211yescharacteristic
24C2×C4×Dic106—21no—
25C2×C4×Dic106C2211yescharacteristic
26C2×C4×Dic106C2211yescharacteristic
27C2×D4×Dic53C2211yescharacteristic
28C2×#13133173—21no—
29#16810752—24no—
30C2×#13133146C2211yescharacteristic
31C2×#13133146C2211yescharacteristic
32C2×#13133142C2211yescharacteristic
33C2×#13133142C2211yescharacteristic
34C2×#13133141C2211yescharacteristic
35C2×#13133141C2211yescharacteristic
36C2×#13133140C2211yescharacteristic
37C2×#13133140C2211yescharacteristic
38C2×#13133138C2211yescharacteristic
39C2×#13133138C2211yescharacteristic
40C2×#13133128C2211yescharacteristic
41C2×#13133127C2211yescharacteristic
42C2×C8×Dic53C2211yescharacteristic
43#16810521—21no—
44#16810520—21no—
45#16810520—21no—
46#16810493C2211yescharacteristic
47#16810492C2211yescharacteristic
48#16810492C2211yescharacteristic
49C4⋊C4×Dic53C2211yescharacteristic
50C4⋊C4×Dic53C2211yescharacteristic
51#16810386C2211yescharacteristic
52#16810386C2211yescharacteristic
53#16810385C2211yescharacteristic
54#16810385C2211yescharacteristic
55C2×#124×C53C2211yescharacteristic
56C2×C4⋊C8×C53C2211yescharacteristic
57C2×#103×C53C2211yescharacteristic
58C2×#103×C53C2211yescharacteristic
59C2×#102×C53C2211yescharacteristic
60C2×#102×C53C2211yescharacteristic
61C4×C4⋊C4×C53C2211yescharacteristic
62C22×Dic106D411yescharacteristic
63C22×Dic106C2311yescharacteristic
64C23×Dic53—21no—
65C22×#88322C2311yescharacteristic
66C22×#88322C2311yescharacteristic
67#13133215—22no—
68#13133215C2×C414yes—
69#13133210—28no—
70#13133210—28no—
71#13133204—22no—
72#13133204C2×C414yes—
73#13133194—28no—
74Q8×Dic53—22no—
75Q8×Dic53C2×C414yes—
76C4×Dic106—22no—
77C4×Dic106—44no—
78C4×Dic106C2×C414yes—
79C4×Dic106C2×C414yes—
80C4×Dic106D414yes—
81D4×Dic53—22no—
82D4×Dic53C2×C414yes—
83#13133173—28no—
84C2×C4⋊Dic53—21no—
85C2×C4⋊Dic53—21no—
86C2×C4⋊Dic53—21no—
87C2×C4⋊Dic53D411yescharacteristic
88C2×C4⋊Dic53C2311yescharacteristic
89C2×C4⋊Dic53C2311yescharacteristic
90C2×C4⋊Dic53C2311yescharacteristic
91C2×#11719919—21no—
92C2×#11719919—21no—
93C2×#11719919—21no—
94C2×#11719919—21no—
95C2×#11719919—21no—
96C2×#11719919—21no—
97C2×#11719919—21no—
98C2×#11719919D411yescharacteristic
99C2×#11719919D411yescharacteristic
100C2×C22⋊Dic53—21no—
101C2×C22⋊Dic53—21no—
102C2×C22⋊Dic53—21no—
103C2×C4×Dic53—21no—
104C2×C4×Dic53—21no—
105C2×C4×Dic53—21no—
106C2×C4×Dic53—21no—
107C2×C4×Dic53C2311yescharacteristic
108C2×C4×Dic53C2311yescharacteristic
109#13133146—22no—
110#13133146—22no—
111#13133142—41no—
112#13133142—41no—
113#13133141—22no—
114#13133141—41no—
115#13133140—22no—
116#13133140—22no—
117#13133138—22no—
118#13133138—41no—
119#13133128—22no—
120#13133127—22no—
121C8×Dic53—22no—
122#13133124—21no—
123#13133124—21no—
124C22×Q8×C53C2311yescharacteristic
125C22×D4×C53C2311yescharacteristic
126C22×C8×C53C2311yescharacteristic
127C22×C8×C53C2311yescharacteristic
128#124×C53—22no—
129#124×C53C2×C414yes—
130C2×C4⋊C4×C53C2311yescharacteristic
131C2×C4⋊C4×C53C2311yescharacteristic
132C2×C22⋊C4×C53—21no—
133C2×C22⋊C4×C53—21no—
134C2×C42×C53—21no—
135C2×C42×C53C2311yescharacteristic
136C4⋊C8×C53—22no—
137#103×C53—22no—
138#103×C53—41no—
139#102×C53—22no—
140#102×C53—41no—
141#95×C53—21no—
142C24×C53—21no—
143C2×Q8×C53—21no—
144C2×Q8×C53—21no—
145C2×Q8×C53—21no—
146C2×Q8×C53—21no—
147C2×Q8×C53—21no—
148C2×Q8×C53C22×C412yes—
149C2×D4×C53—21no—
150C2×D4×C53—21no—
151C2×D4×C53—21no—
152C2×D4×C53—21no—
153C2×D4×C53—21no—
154C2×D4×C53C22×C412yes—
155C22×C4×C53—21no—
156C22×C4×C53—21no—
157C22×C4×C53—21no—
158C22×C4×C53C2×D411yescharacteristic
159C22×C4×C53C2×D411yescharacteristic
160C22×C4×C53C2411yescharacteristic
161C22×Dic53—21no—
162C22×Dic53—21no—
163C22×Dic53—24no—
164C22×Dic53—21no—
165C22×Dic53—21no—
166C22×Dic53—42no—
167C22×Dic53—42no—
168C22×Dic53C2×D411yescharacteristic
169C22×Dic53C2×D411yescharacteristic
170C22×Dic53C2×D411yescharacteristic
171C22×Dic53C2×D411yescharacteristic
172C2×Dic106—22no—
173C2×Dic106—22no—
174C2×Dic106—21no—
175C2×Dic106—21no—
176C2×Dic106—21no—
177C2×Dic106—21no—
178C2×Dic106—21no—
179C2×Dic106C22⋊C414yes—
180C2×Dic106C22×C412yes—
181C2×C8×C53—21no—
182C2×C8×C53—21no—
183C2×C8×C53—21no—
184C2×C8×C53—21no—
185C2×C8×C53—21no—
186C2×C8×C53—21no—
187C4⋊C4×C53—21no—
188C4⋊C4×C53—21no—
189C4⋊C4×C53—21no—
190C4⋊C4×C53#4112yes—
191C22⋊C4×C53—24no—
192C22⋊C4×C53—24no—
193C42×C53—42no—
194C42×C53C22×C412yes—
195C42×C53C2×D412yes—
196C22⋊Dic53—24no—
197C22⋊Dic53—24no—
198C22⋊Dic53—24no—
199C4⋊Dic53—21no—
200C4⋊Dic53—24no—
201C4⋊Dic53—21no—
202C4⋊Dic53—21no—
203C4⋊Dic53—21no—
204C4⋊Dic53—21no—
205C4⋊Dic53—42no—
206C4⋊Dic53—81no—
207C4⋊Dic53C22⋊C414yes—
208C4⋊Dic53C22×C412yes—
209#11719919—24no—
210#11719919—24no—
211#11719919—24no—
212#11719919—42no—
213#11719919—42no—
214#11719919—81no—
215#11719919—81no—
216#11719919C22⋊C414yes—
217#11719919C22⋊C414yes—
218C4×Dic53—24no—
219C4×Dic53—24no—
220C4×Dic53—42no—
221C4×Dic53—42no—
222C4×Dic53C22×C412yes—
223C4×Dic53C22×C412yes—
224C4×Dic53C2×D412yes—
225C4×Dic53C2×D412yes—
226C2×#88322—21no—
227C2×#88322—21no—
228C2×#88322—21no—
229C2×#88322—21no—
230C2×#88322—21no—
231C2×#88322#4112yes—
232C23×C53—22no—
233C23×C53—41no—
234C23×C53—41no—
235C23×C53—41no—
236C23×C53—41no—
237C23×C53—41no—
238C23×C53—41no—
239C23×C53C22×D411yescharacteristic
240Q8×C53—22no—
241Q8×C53—22no—
242Q8×C53—41no—
243Q8×C53—41no—
244D4×C53—22no—
245D4×C53—22no—
246D4×C53—41no—
247D4×C53—41no—
248C2×C4×C53—21no—
249C2×C4×C53—21no—
250C2×C4×C53—21no—
251C2×C4×C53—21no—
252C2×C4×C53—22no—
253C2×C4×C53—22no—
254C2×C4×C53—41no—
255C2×C4×C53—41no—
256C2×C4×C53—41no—
257C2×C4×C53—41no—
258C2×C4×C53—41no—
259C2×C4×C53—41no—
260C2×C4×C53—41no—
261C2×C4×C53C2×C22⋊C412yes—
262C2×C4×C53C2×C22⋊C412yes—
263C2×C4×C53C23×C411yescharacteristic
264C2×C4×C53C22×D411yescharacteristic
265C2×C4×C53C2×#4111yescharacteristic
266C2×C4×C53C2×#4111yescharacteristic
267C2×C4×C53H(5,𝔽2)11yescharacteristic
268C2×C4×C532−1+411yescharacteristic
269C2×Dic53—21no—
270C2×Dic53—21no—
271C2×Dic53—21no—
272C2×Dic53—21no—
273C2×Dic53—21no—
274C2×Dic53—21no—
275C2×Dic53—22no—
276C2×Dic53—22no—
277C2×Dic53—41no—
278C2×Dic53—41no—
279C2×Dic53—41no—
280C2×Dic53—41no—
281C2×Dic53—48no—
282C2×Dic53—41no—
283C2×Dic53—41no—
284C2×Dic53—41no—
285C2×Dic53—41no—
286C2×Dic53—41no—
287C2×Dic53—41no—
288C2×Dic53C2×C22⋊C412yes—
289C2×Dic53C2×C22⋊C412yes—
290C2×Dic53C2×C22⋊C412yes—
291C2×Dic53C2×C22⋊C412yes—
292C2×Dic53AGL(1,ℤ/8)12yes—
293C2×Dic53#13712yes—
294Dic106—28no—
295Dic106—22no—
296Dic106—22no—
297Dic106—41no—
298Dic106—41no—
299C8×C53—41no—
300C8×C53—41no—
301#88322—22no—
302#88322—41no—
303#22544—531nomaximal, Sylow
304C22×C53—42no—
305C22×C53—42no—
306C22×C53—42no—
307C22×C53—81no—
308C22×C53—81no—
309C22×C53—81no—
310C22×C53—81no—
311C22×C53C22×C22⋊C411yescharacteristic
312C22×C53#54811yescharacteristic
313C22×C53#55011yescharacteristic
314C22×C53C2×AGL(1,ℤ/8)11yescharacteristic
315C22×C53C2×#13711yescharacteristic
316C22×C53#57711yescharacteristic
317C22×C53#57811yescharacteristic
318C4×C53—22no—
319C4×C53—22no—
320C4×C53—42no—
321C4×C53—42no—
322C4×C53—81no—
323C4×C53C4×#4111yescharacteristic
324C4×C53#52011yescharacteristic
325C4×C53#54811yescharacteristic
326C4×C53#55011yescharacteristic
327Dic53—22no—
328Dic53—22no—
329Dic53—22no—
330Dic53—42no—
331Dic53—42no—
332Dic53—81no—
333Dic53—81no—
334Dic53#41714yes—
335C2×#556—531no—
336C2×#446—531no—
337Q8×C4⋊C4—531no—
338#1983—531no—
339#1983D5311yescharacteristic
340#1981—531no—
341#1959—531no—
342#1959—531no—
343#1956—531no—
344#1803—531no—
345#1803—531no—
346#1802—531no—
347#1763—531no—
348#1763—531no—
349C4×#102—531no—
350C2×C53—82no—
351C2×C53#234111yescharacteristic
352C2×C53C2×#41711yescharacteristic
353C2×C53#289811yescharacteristic
354C2×C53#336711yescharacteristic
355C2×C53#336911yescharacteristic
356C2×C53#339711yescharacteristic
357C2×C53#339811yescharacteristic
358#556—538no—
359#556—1064no—
360C2×C4⋊Q8—531no—
361C2×#126—1061no—
362C2×#126—1061no—
363C2×#124—531no—
364C2×#124C2×D5311yescharacteristic
365C2×C22⋊Q8—1061no—
366C2×C4×Q8—531no—
367C2×C4×Q8—531no—
368C2×C4×Q8—531no—
369C2×C4×Q8—1061no—
370C2×C4×D4—531no—
371C2×#117—1061no—
372#446—1064no—
373C2×C4⋊C8—531no—
374C2×C4⋊C8—531no—
375C2×C4⋊C8C2×D5311yescharacteristic
376C2×#103—531no—
377C2×#103—531no—
378C2×#103—531no—
379C2×#103—531no—
380C2×#103—531no—
381C2×#103—531no—
382C2×#103C2×D5311yescharacteristic
383C2×#103C2×D5311yescharacteristic
384C2×#102—531no—
385C2×#102—531no—
386C2×#102C2×D5311yescharacteristic
387C2×#102C2×D5311yescharacteristic
388C2×#97—531no—
389C2×#97—531no—
390C2×C4×C8—531no—
391#389—1061no—
392#389—1061no—
393#389—1061no—
394#384—531no—
395#384—531no—
396#384—531no—
397C4×C4⋊C4—531no—
398C4×C4⋊C4—531no—
399C4×C4⋊C4C2×D5311yescharacteristic
400#340—531no—
401#340—531no—
402#340—531no—
403#340—531no—
404C53#2254411yescharacteristic, complemented, Sylow
405C22×Q8—531no—
406C22×Q8—531no—
407C22×Q8C22×D5311yescharacteristic
408C22×D4C22×D5311yescharacteristic
409C23×C4—1061no—
410C22×C8—531no—
411C22×C8—531no—
412C22×C8C22×D5311yescharacteristic
413C22×C8C22×D5311yescharacteristic
414C4⋊Q8—534no—
415C4⋊Q8—1062no—
416#126—1068no—
417#126—1068no—
418#124—22no—
419#124—534no—
420#124—1062no—
421#124C4×D5314yes—
422C22⋊Q8—1068no—
423C4×Q8—534no—
424C4×Q8—534no—
425C4×Q8—534no—
426C4×Q8—534no—
427C4×Q8—1062no—
428C4×Q8—1062no—
429C4×Q8—2124no—
430C4×D4—534no—
431C4×D4—1062no—
432#117—1068no—
433C2×C4⋊C4—531no—
434C2×C4⋊C4—531no—
435C2×C4⋊C4—531no—
436C2×C4⋊C4—531no—
437C2×C4⋊C4—531no—
438C2×C4⋊C4—531no—
439C2×C4⋊C4—1061no—
440C2×C4⋊C4—1061no—
441C2×C4⋊C4—1061no—
442C2×C4⋊C4—1061no—
443C2×C4⋊C4—1061no—
444C2×C4⋊C4—1061no—
445C2×C4⋊C4—1061no—
446C2×C4⋊C4—1061no—
447C2×C4⋊C4—1061no—
448C2×C4⋊C4—1061no—
449C2×C4⋊C4C22×D5311yescharacteristic
450C2×C4⋊C4C22×D5311yescharacteristic
451C2×C22⋊C4—21no—
452C2×C22⋊C4—21no—
453C2×C22⋊C4—1061no—
454C2×C22⋊C4—1061no—
455C2×C22⋊C4—1061no—
456C2×C42—21no—
457C2×C42—531no—
458C2×C42—531no—
459C2×C42—1061no—
460C2×C42—1061no—
461C2×C42—1061no—
462C2×C42—1061no—
463C2×C42C22×D5311yescharacteristic
464C4⋊C8—22no—
465C4⋊C8—1062no—
466C4⋊C8—1062no—
467#103—22no—
468#103—41no—
469#103—1062no—
470#103—1062no—
471#103—1062no—
472#103—2121no—
473#103—2121no—
474#103—2121no—
475#102—22no—
476#102—41no—
477#102—1062no—
478#102—2121no—
479#97—1062no—
480#97—1062no—
481C4×C8—1062no—
482#95—21no—
483#95—1061no—
484#95—1061no—
485C24—21no—
486C2×Q8—21no—
487C2×Q8—21no—
488C2×Q8—21no—
489C2×Q8—21no—
490C2×Q8—21no—
491C2×Q8—534no—
492C2×Q8—532no—
493C2×Q8—1062no—
494C2×Q8—1062no—
495C2×Q8—1061no—
496C2×Q8—1061no—
497C2×Q8—1061no—
498C2×Q8—1061no—
499C2×Q8—1061no—
500C2×Q8C2×C4×D5312yes—
501C2×D4—21no—
502C2×D4—21no—
503C2×D4—21no—
504C2×D4—21no—
505C2×D4—21no—
506C2×D4C2×C4×D5312yes—
507C22×C4—21no—
508C22×C4—21no—
509C22×C4—21no—
510C22×C4—531no—
511C22×C4—531no—
512C22×C4—531no—
513C22×C4—531no—
514C22×C4—1064no—
515C22×C4—1061no—
516C22×C4—1061no—
517C22×C4—1061no—
518C22×C4—1061no—
519C22×C4—2122no—
520C22×C4—2122no—
521C22×C4D4×D5311yescharacteristic
522C22×C4D4×D5311yescharacteristic
523C22×C4C23×D5311yescharacteristic
524C2×C8—21no—
525C2×C8—21no—
526C2×C8—21no—
527C2×C8—21no—
528C2×C8—21no—
529C2×C8—21no—
530C2×C8—532no—
531C2×C8—1061no—
532C2×C8—1061no—
533C2×C8—1061no—
534C2×C8—1061no—
535C2×C8—1061no—
536C4⋊C4—21no—
537C4⋊C4—21no—
538C4⋊C4—21no—
539C4⋊C4—534no—
540C4⋊C4—534no—
541C4⋊C4—534no—
542C4⋊C4—532no—
543C4⋊C4—1064no—
544C4⋊C4—1064no—
545C4⋊C4—1064no—
546C4⋊C4—1064no—
547C4⋊C4—1061no—
548C4⋊C4—1061no—
549C4⋊C4—1061no—
550C4⋊C4—1061no—
551C4⋊C4—1061no—
552C4⋊C4—4241no—
553C4⋊C4—4241no—
554C4⋊C4—4241no—
555C4⋊C4—2122no—
556C4⋊C4—2122no—
557C4⋊C4—2122no—
558C4⋊C4#1171994712yes—
559C22⋊C4—24no—
560C22⋊C4—24no—
561C22⋊C4—1064no—
562C22⋊C4—1064no—
563C22⋊C4—1064no—
564C42—42no—
565C42—532no—
566C42—532no—
567C42—532no—
568C42—532no—
569C42—1064no—
570C42—1064no—
571C42—2122no—
572C42—2122no—
573C42C2×C4×D5312yes—
574C42D4×D5312yes—
575C23—22no—
576C23—41no—
577C23—41no—
578C23—41no—
579C23—41no—
580C23—41no—
581C23—41no—
582C23C2×D4×D5311yescharacteristic
583Q8—22no—
584Q8—22no—
585Q8—41no—
586Q8—41no—
587Q8—1068no—
588Q8—1062no—
589Q8—1062no—
590Q8—2121no—
591Q8—2121no—
592D4—22no—
593D4—22no—
594D4—41no—
595D4—41no—
596C2×C4—21no—
597C2×C4—21no—
598C2×C4—21no—
599C2×C4—21no—
600C2×C4—22no—
601C2×C4—22no—
602C2×C4—41no—
603C2×C4—41no—
604C2×C4—41no—
605C2×C4—41no—
606C2×C4—41no—
607C2×C4—41no—
608C2×C4—41no—
609C2×C4—532no—
610C2×C4—532no—
611C2×C4—532no—
612C2×C4—532no—
613C2×C4—532no—
614C2×C4—532no—
615C2×C4—1061no—
616C2×C4—1061no—
617C2×C4—1061no—
618C2×C4—1061no—
619C2×C4—1061no—
620C2×C4—1061no—
621C2×C4—1062no—
622C2×C4—1062no—
623C2×C4—2128no—
624C2×C4—2121no—
625C2×C4—2121no—
626C2×C4—2121no—
627C2×C4—2121no—
628C2×C4—2121no—
629C2×C4—2121no—
630C2×C4—2121no—
631C2×C4—2121no—
632C2×C4—2121no—
633C2×C4—2121no—
634C2×C4C22⋊C4×D5312yes—
635C2×C4C22⋊C4×D5312yes—
636C2×C4C22×C4×D5311yescharacteristic
637C2×C4C2×D4×D5311yescharacteristic
638C2×C4#41×D5311yescharacteristic
639C2×C4C2×#1171994711yescharacteristic
640C2×C4#1313326211yescharacteristic
641C2×C4#1313326411yescharacteristic
642C8—41no—
643C8—41no—
644C8—1062no—
645C8—2121no—
646C22—42no—
647C22—42no—
648C22—42no—
649C22—81no—
650C22—81no—
651C22—81no—
652C22—81no—
653C22C2×C22⋊C4×D5311yescharacteristic
654C22#1681127911yescharacteristic
655C22#1681129111yescharacteristic
656C22AGL(1,ℤ/8)×D5311yescharacteristic
657C22#137×D5311yescharacteristic
658C22#1681143011yescharacteristic
659C22#1681143711yescharacteristic
660C4—22no—
661C4—22no—
662C4—42no—
663C4—42no—
664C4—81no—
665C4—534no—
666C4—1062no—
667C4—1062no—
668C4—1062no—
669C4—4241no—
670C4—4241no—
671C4—2122no—
672C4—2122no—
673C4C4×#1171994711yescharacteristic
674C4#1681110211yescharacteristic
675C4#1681128111yescharacteristic
676C4#1681129311yescharacteristic
677C2—82no—
678C2#2027755411yescharacteristic
679C2#417×D5311yescharacteristic
680C2#2028324411yescharacteristic
681C2#2028742811yescharacteristic
682C2#2028745011yescharacteristic
683C2#2028773811yescharacteristic
684C2#2028774911yescharacteristic
subgroupG/H
centerC23C2×D4×D53
commutatorC2×C4×C53C23×C4
FrattiniC22×C4C23×D53
FittingC53×#1983C2
socleC23×C53C22×D4
prime radicalC22×D4×C53C23
cosocleC22×C4×C53C24
hypercenter#1983D53
hypocenterC53#22544
LCSG ▷ C2×C4×C53 ▷ C2×C53 ▷ C53
UCSC1 ◁ C23 ◁ C2×C42 ◁ #1983

Complements

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

NH ≅ G/Ntype
2. C2×#16810752C2semidirect
3. C4⋊C4×Dic106C2semidirect
5. #20275636C2semidirect
8. #20275515C2semidirect
9. #20274721C2semidirect
10. #20274718C2semidirect
12. #20274548C2semidirect
13. #20274547C2semidirect
17. #16811342C4semidirect
404. C53#22544semidirect

Maximal subgroups

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

subgroupindex
C2×#168113422
C2×#168107522
C4⋊C4×Dic1062
#202756462
#202756362
#202755322
#202755302
#202755152
#202747212
#202747182
#202747132
#202745482
#202745472
#102×Dic532
C53×#19832
#2254453