Finite Group Databasebeta

← #33053771

Subgroups — #33053771

Order 18688.

1014 conjugacy classes of subgroups.

The subgroup lattice has 674 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#22623770C211yesmaximal, characteristic
2#22618972C211yesmaximal, characteristic, complemented
3#22608506C212yesmaximal, complemented
4#22608472C212yesmaximal, complemented
5#22608463C211yesmaximal, characteristic, complemented
6#22608459C211yesmaximal, characteristic, complemented
7#22608450C211yesmaximal, characteristic, complemented
8#22608446C211yesmaximal, characteristic, complemented
9#22608433C212yesmaximal, complemented
10#22608415C212yesmaximal, complemented
11C73×#2353C211yesmaximal, characteristic, complemented
12#17989286—22no—
13#17989283C412yescomplemented
14#17989277C412yescomplemented
15#17989266—22no—
16#17989246C412yescomplemented
17#17989233C412yescomplemented
18#17989184—21no—
19#17989175—21no—
20#17989146—21no—
21#17989121C2212yes—
22#17989098C2212yes—
23C2×#14556570C2211yescharacteristic
24C2×#14556565C2211yescharacteristic
25#17989060—21no—
26#17988665—22no—
27#17988665—22no—
28#17988640C2212yescomplemented
29#17988551C2211yescharacteristic
30#17988551C2211yescharacteristic
31#17988551C2212yescomplemented
32#17988323C2212yescomplemented
33#17988323C2212yescomplemented
34#17988323C2212yescomplemented
35#17988323C2212yescomplemented
36#17988320C2212yes—
37#17988320C2212yes—
38#17988320C2212yes—
39#17988320C2212yes—
40C2×C4×D4×C73C2211yescharacteristic
41#388×C73—21no—
42#387×C73—21no—
43#386×C73—21no—
44#383×C73C2211yescharacteristic, complemented
45#381×C73C2211yescharacteristic
46#380×C73C2211yescharacteristic, complemented
47C4×C22⋊C4×C73—21no—
48C4×C22⋊C4×C73C2211yescharacteristic
49#376×C73C2211yescharacteristic, complemented
50C2×#95×C73C2211yescharacteristic, complemented
51C2×#11924028C2×C412yes—
52C22×Dic146C2×C411yescharacteristic, complemented
53C22×D292C2×C411yescharacteristic, complemented
54C22×C22⋊D73—21no—
55C2×#11924012—21no—
56C2×#11924012—21no—
57C2×#11924012—22no—
58#14556586—22no—
59#14556585—22no—
60#14556584—22no—
61#14556583C2×C412yescomplemented
62#14556581—22no—
63#14556579C2×C412yescomplemented
64#14556577—22no—
65#14556576—22no—
66#14556574—22no—
67#14556574—22no—
68#14556573—22no—
69#14556572—22no—
70#14556570—21no—
71#14556570—21no—
72#14556570C2×C412yes—
73#14556570C2×C412yescomplemented
74#14556570C2312yes—
75#14556569—22no—
76#14556565—21no—
77#14556565—21no—
78#14556565C2×C412yes—
79#14556565C2×C412yescomplemented
80#14556565C2312yes—
81#14556564—22no—
82#14556563—22no—
83#14556563—22no—
84#14556560—22no—
85#14556559—22no—
86C4×Dic146C2×C412yescomplemented
87C4×D292C2×C412yescomplemented
88C4×C22⋊D73—22no—
89C4×C22⋊D73—22no—
90C4×C22⋊D73—22no—
91#14556551—22no—
92#14556551—22no—
93#14556550C2×C412yes—
94#14556548—22no—
95#14556547—22no—
96C2×C4⋊Dic73C2×C411yescharacteristic, complemented
97C2×C4⋊Dic73C2×C411yescharacteristic, complemented
98C2×#11924007—21no—
99C2×C22⋊Dic73—21no—
100C2×#11924005—21no—
101C22⋊C4×D73—22no—
102C4⋊#10596831—22no—
103C22⋊#10596831—21no—
104C22⋊#10596831—21no—
105C22⋊#10596831—22no—
106C22⋊#10596831C2312yes—
107C22⋊#10596831C2312yes—
108C4×#10596831—22no—
109C22×D4×C73D411yescharacteristic
110C23×C4×C73—21no—
111C23×C4×C73C2311yescharacteristic
112C23×C4×C73C2311yescharacteristic
113C4×D4×C73C2×C412yes—
114C4×D4×C73C2×C412yes—
115C4×D4×C73D412yes—
116C4×D4×C73D412yes—
117C2×C4⋊C4×C73—21no—
118C2×C4⋊C4×C73D411yescharacteristic
119C2×C22⋊C4×C73—21no—
120C2×C22⋊C4×C73—21no—
121C2×C22⋊C4×C73—21no—
122C2×C22⋊C4×C73D411yescharacteristic
123C2×C22⋊C4×C73D411yescharacteristic
124C2×C42×C73—21no—
125C2×C42×C73—21no—
126C2×C42×C73C2311yescharacteristic
127#95×C73—21no—
128#95×C73—21no—
129#95×C73—21no—
130#95×C73—21no—
131#95×C73—21no—
132#95×C73C2311yescharacteristic
133#95×C73C2311yescharacteristic
134#95×C73C2311yescharacteristic
135#95×C73C2311yescharacteristic
136C23×D73—21no—
137#11924028—82no—
138C2×Dic146—21no—
139C2×Dic146—81no—
140C2×Dic146C22×C412yes—
141C2×D292—21no—
142C2×D292—81no—
143C2×D292C22×C412yes—
144C2×C22⋊D73—22no—
145C2×C22⋊D73—22no—
146C2×C22⋊D73—22no—
147C2×C22⋊D73—22no—
148C2×C22⋊D73—42no—
149C2×C22⋊D73—42no—
150C22×Dic73—21no—
151C22×Dic73—21no—
152C22×Dic73—21no—
153C2×C4×D73—22no—
154C2×C4×D73—22no—
155#11924012—81no—
156#11924012—81no—
157#11924012—82no—
158C2×#10596831—22no—
159C2×#10596831—22no—
160C4⋊Dic73—21no—
161C4⋊Dic73—21no—
162C4⋊Dic73C22×C412yes—
163C4⋊Dic73C22×C412yes—
164#11924007—22no—
165#11924007—22no—
166#11924007—22no—
167#11924007—22no—
168#11924007—22no—
169#11924007—22no—
170C22⋊Dic73—22no—
171C22⋊Dic73—22no—
172C22⋊Dic73—22no—
173C22⋊Dic73—22no—
174#11924005—22no—
175#11924005—22no—
176#11924005—22no—
177#11924005—22no—
178#11924005—22no—
179#11924005—22no—
180C4×Dic73—22no—
181C4×Dic73—22no—
182C24×C73C2×D411yescharacteristic
183C24×C73C2×D411yescharacteristic
184C2×D4×C73—41no—
185C2×D4×C73—41no—
186C2×D4×C73C22⋊C412yes—
187C2×D4×C73C22⋊C412yes—
188C22×C4×C73—21no—
189C22×C4×C73—21no—
190C22×C4×C73—21no—
191C22×C4×C73—21no—
192C22×C4×C73—42no—
193C22×C4×C73—42no—
194C22×C4×C73—81no—
195C22×C4×C73—81no—
196C22×C4×C73—81no—
197C22×C4×C73C22×C411yescharacteristic
198C22×C4×C73C22×C411yescharacteristic
199C22×C4×C73C22×C412yes—
200C22×C4×C73C2×D411yescharacteristic
201C22×C4×C73C2×D411yescharacteristic
202C22×C4×C73C2×D411yescharacteristic
203C22×C4×C73C2×D411yescharacteristic
204C22×C4×C73C2×D411yescharacteristic
205C22×C4×C73C2×D411yescharacteristic
206C22×C4×C73C2×D412yes—
207C22×C4×C73C2411yescharacteristic
208C4⋊C4×C73—81no—
209C4⋊C4×C73C22⋊C412yes—
210C4⋊C4×C73C22⋊C412yes—
211C22⋊C4×C73—42no—
212C22⋊C4×C73—42no—
213C22⋊C4×C73—81no—
214C22⋊C4×C73C22⋊C412yes—
215C22⋊C4×C73C22⋊C412yes—
216C22⋊C4×C73C22⋊C412yes—
217C22⋊C4×C73C22⋊C412yes—
218C42×C73—42no—
219C42×C73—81no—
220C42×C73C22×C412yes—
221C42×C73C2×D412yes—
222C23×C73—21no—
223C23×C73—21no—
224C23×C73—21no—
225C23×C73—21no—
226C23×C73—41no—
227C23×C73—41no—
228C23×C73—41no—
229C23×C73—41no—
230C23×C73C2×C22⋊C411yescharacteristic
231C23×C73C2×C22⋊C412yes—
232C23×C73C2×C22⋊C411yescharacteristic
233C23×C73C22×D411yescharacteristic
234C22×D73—22no—
235C22×D73—42no—
236C22×D73—81no—
237C22×D73—81no—
238D4×C73—42no—
239D4×C73—42no—
240C2×C4×C73—21no—
241C2×C4×C73—21no—
242C2×C4×C73—22no—
243C2×C4×C73—41no—
244C2×C4×C73—41no—
245C2×C4×C73—41no—
246C2×C4×C73—41no—
247C2×C4×C73—41no—
248C2×C4×C73—41no—
249C2×C4×C73—42no—
250C2×C4×C73—41no—
251C2×C4×C73—42no—
252C2×C4×C73—41no—
253C2×C4×C73—41no—
254C2×C4×C73—41no—
255C2×C4×C73—41no—
256C2×C4×C73—41no—
257C2×C4×C73—41no—
258C2×C4×C73—41no—
259C2×C4×C73—41no—
260C2×C4×C73—41no—
261C2×C4×C73—41no—
262C2×C4×C73—41no—
263C2×C4×C73—41no—
264C2×C4×C73—84no—
265C2×C4×C73C2×C22⋊C412yes—
266C2×C4×C73C2×C22⋊C412yes—
267C2×C4×C73C2×C22⋊C412yes—
268C2×C4×C73C2×C22⋊C411yescharacteristic
269C2×C4×C73C2×C22⋊C411yescharacteristic
270C2×C4×C73C23×C411yescharacteristic
271C2×C4×C73C22×D411yescharacteristic
272C22⋊D73—44no—
273C22⋊D73—42no—
274C22⋊D73—42no—
275C22⋊D73—82no—
276C22⋊D73—82no—
277C2×Dic73—22no—
278C2×Dic73—22no—
279C2×Dic73—22no—
280C2×Dic73—42no—
281C2×Dic73—81no—
282C2×Dic73—81no—
283D292—42no—
284D292—81no—
285C4×D73—82no—
286C4×D73—82no—
287Dic146—42no—
288Dic146—81no—
289#10596831—42no—
290#10596831—42no—
291C22×C73—21no—
292C22×C73—21no—
293C22×C73—21no—
294C22×C73—22no—
295C22×C73—21no—
296C22×C73—21no—
297C22×C73—21no—
298C22×C73—22no—
299C22×C73—21no—
300C22×C73—22no—
301C22×C73—21no—
302C22×C73—42no—
303C22×C73—42no—
304C22×C73—81no—
305C22×C73—81no—
306C22×C73C22×C22⋊C411yescharacteristic
307C22×C73#57511yescharacteristic
308C22×C73#57511yescharacteristic
309C2×D73—42no—
310C2×D73—42no—
311C2×D73—42no—
312C2×D73—162no—
313C4×C73—21no—
314C4×C73—21no—
315C4×C73—21no—
316C4×C73—22no—
317C4×C73—21no—
318C4×C73—22no—
319C4×C73—22no—
320C4×C73—42no—
321C4×C73—81no—
322C4×C73—81no—
323C4×C73—81no—
324Dic73—42no—
325Dic73—42no—
326Dic73—42no—
327#17623—731nomaximal, Sylow
328C2×C73—21no—
329C2×C73—21no—
330C2×C73—41no—
331C2×C73—42no—
332C2×C73—41no—
333C2×C73#288311yescharacteristic
334C2×C73#289611yescharacteristic
335C2×C73#290111yescharacteristic
336D73—82no—
337#3505—731no—
338#2975—731no—
339#2353D7311yescharacteristic, complemented
340#1521—732no—
341#1513—732no—
342#1511—731no—
343#1510—731no—
344#1508—731no—
345#1507—731no—
346#1504—732no—
347#1500—732no—
348C73#1762311yescharacteristic, complemented, Sylow
349#553—732no—
350#553—1462no—
351#552—732no—
352#551—1462no—
353#548—732no—
354#546—732no—
355#539—1461no—
356#538—1461no—
357#534—1461no—
358#529—732no—
359#525—732no—
360C2×C22⋊Q8—731no—
361C2×#121—731no—
362#518—1461no—
363C2×C4×D4C2×D7311yescharacteristic
364#436—1462no—
365#436—1462no—
366#431—732no—
367#407—731no—
368#407—731no—
369#407—732no—
370#388—21no—
371#387—21no—
372#386—21no—
373#383C2×D7311yescharacteristic, complemented
374#381C2×D7311yescharacteristic
375#380C2×D7311yescharacteristic, complemented
376C4×C22⋊C4—21no—
377C4×C22⋊C4C2×D7311yescharacteristic
378#376C2×D7311yescharacteristic, complemented
379C2×#95C2×D7311yescharacteristic, complemented
380#328—732no—
381#328—732no—
382#328—732no—
383#328—732no—
384#327—732no—
385#327—732no—
386#327—732no—
387#327—732no—
388C2×#41—732no—
389C22×Q8—731no—
390C22×D4—731no—
391C22×D4—1461no—
392C22×D4C22⋊D7311yescharacteristic
393C23×C4—21no—
394C23×C4C22×D7311yescharacteristic
395C23×C4C22×D7311yescharacteristic
396C2×M16—1461no—
397C2×M16—1461no—
398C2×M16—1462no—
399#126—1462no—
400#126—1462no—
401#126—1462no—
402#125—732no—
403#125—1462no—
404#124—732no—
405#124—1462no—
406#123—1462no—
407#123—1462no—
408#123—1462no—
409#123—1462no—
410#123—1462no—
411C22⋊Q8—732no—
412C22⋊Q8—732no—
413C22⋊Q8—732no—
414C22⋊Q8—1461no—
415C22⋊Q8—1461no—
416C22⋊Q8—1462no—
417#121—732no—
418#121—732no—
419#121—732no—
420#121—1462no—
421#121—1462no—
422#121—1461no—
423#121—1461no—
424#121—1462no—
425#120—1462no—
426#120—1462no—
427C4×Q8—732no—
428C4×D4—732no—
429C4×D4—1462no—
430C4×D4—1462no—
431C4×D4—1462no—
432C4×D4—1462no—
433C4×D4—1462no—
434C4×D4C4×D7312yes—
435C4×D4C4×D7312yes—
436C4×D4D29212yes—
437C4×D4D29212yes—
438#117—732no—
439#117—1462no—
440#117—1462no—
441C2×C4⋊C4—21no—
442C2×C4⋊C4—731no—
443C2×C4⋊C4—731no—
444C2×C4⋊C4—1461no—
445C2×C4⋊C4C22⋊D7311yescharacteristic
446C2×C22⋊C4—21no—
447C2×C22⋊C4—21no—
448C2×C22⋊C4—21no—
449C2×C22⋊C4—1461no—
450C2×C22⋊C4—1461no—
451C2×C22⋊C4—1462no—
452C2×C22⋊C4C22⋊D7311yescharacteristic
453C2×C22⋊C4C22⋊D7311yescharacteristic
454C2×C42—21no—
455C2×C42—21no—
456C2×C42C22×D7311yescharacteristic
457C4⋊C8—1462no—
458C22⋊C8—732no—
459C22⋊C8—732no—
460C22⋊C8—1461no—
461C22⋊C8—1461no—
462C22⋊C8—1462no—
463C4×C8—1462no—
464#95—21no—
465#95—21no—
466#95—21no—
467#95—21no—
468#95—21no—
469#95C22×D7311yescharacteristic
470#95C22×D7311yescharacteristic
471#95C22×D7311yescharacteristic
472#95C22×D7311yescharacteristic
473C24—1461no—
474C24C2×C22⋊D7311yescharacteristic
475C24C2×C22⋊D7311yescharacteristic
476#41—5842no—
477C2×Q8—5841no—
478C2×Q8—732no—
479C2×Q8—1461no—
480C2×D4—41no—
481C2×D4—41no—
482C2×D4—2922no—
483C2×D4—2922no—
484C2×D4—5841no—
485C2×D4—732no—
486C2×D4—1462no—
487C2×D4—1462no—
488C2×D4—1462no—
489C2×D4—1462no—
490C2×D4—1461no—
491C2×D4#1192400512yes—
492C2×D4#1192400512yes—
493C22×C4—21no—
494C22×C4—21no—
495C22×C4—21no—
496C22×C4—21no—
497C22×C4—42no—
498C22×C4—42no—
499C22×C4—81no—
500C22×C4—81no—
501C22×C4—81no—
502C22×C4—1461no—
503C22×C4—1461no—
504C22×C4—1461no—
505C22×C4—1462no—
506C22×C4—1462no—
507C22×C4C2×C4×D7311yescharacteristic
508C22×C4C2×C4×D7311yescharacteristic
509C22×C4C2×C4×D7312yes—
510C22×C4C2×C22⋊D7311yescharacteristic
511C22×C4C2×C22⋊D7311yescharacteristic
512C22×C4C2×C22⋊D7311yescharacteristic
513C22×C4C2×C22⋊D7311yescharacteristic
514C22×C4C2×D29211yescharacteristic
515C22×C4C2×D29211yescharacteristic
516C22×C4C2×D29212yes—
517C22×C4C23×D7311yescharacteristic
518M16—5842no—
519M16—5841no—
520M16—5841no—
521C2×C8—1462no—
522C2×C8—1462no—
523C4⋊C4—81no—
524C4⋊C4—732no—
525C4⋊C4—732no—
526C4⋊C4—1462no—
527C4⋊C4—1462no—
528C4⋊C4—1462no—
529C4⋊C4—1462no—
530C4⋊C4—1462no—
531C4⋊C4—1461no—
532C4⋊C4—1461no—
533C4⋊C4—1462no—
534C4⋊C4#1192400512yes—
535C4⋊C4#1192400512yes—
536C22⋊C4—42no—
537C22⋊C4—42no—
538C22⋊C4—81no—
539C22⋊C4—1462no—
540C22⋊C4—1462no—
541C22⋊C4—1462no—
542C22⋊C4—1462no—
543C22⋊C4—1462no—
544C22⋊C4—1462no—
545C22⋊C4—1462no—
546C22⋊C4—1462no—
547C22⋊C4—1462no—
548C22⋊C4—1462no—
549C22⋊C4#1192400512yes—
550C22⋊C4#1192400512yes—
551C22⋊C4#1192400512yes—
552C22⋊C4#1192400512yes—
553C42—42no—
554C42—81no—
555C42—1462no—
556C42—1462no—
557C42C2×C4×D7312yes—
558C42C2×D29212yes—
559C23—21no—
560C23—21no—
561C23—21no—
562C23—21no—
563C23—41no—
564C23—41no—
565C23—41no—
566C23—41no—
567C23—2922no—
568C23—5841no—
569C23—5841no—
570C23—1462no—
571C23C2×#1192400511yescharacteristic
572C23C2×#1192400512yes—
573C23C2×#1192400511yescharacteristic
574C23C22×C22⋊D7311yescharacteristic
575Q8—2922no—
576Q8—5841no—
577D4—42no—
578D4—42no—
579D4—2922no—
580D4—2924no—
581D4—2922no—
582D4—2922no—
583D4—5841no—
584D4—5842no—
585D4—5842no—
586C2×C4—21no—
587C2×C4—21no—
588C2×C4—22no—
589C2×C4—41no—
590C2×C4—41no—
591C2×C4—41no—
592C2×C4—42no—
593C2×C4—41no—
594C2×C4—41no—
595C2×C4—41no—
596C2×C4—41no—
597C2×C4—41no—
598C2×C4—42no—
599C2×C4—41no—
600C2×C4—41no—
601C2×C4—41no—
602C2×C4—41no—
603C2×C4—41no—
604C2×C4—41no—
605C2×C4—41no—
606C2×C4—41no—
607C2×C4—41no—
608C2×C4—41no—
609C2×C4—41no—
610C2×C4—84no—
611C2×C4—2922no—
612C2×C4—5842no—
613C2×C4—5842no—
614C2×C4—5841no—
615C2×C4—5841no—
616C2×C4—1462no—
617C2×C4—1462no—
618C2×C4—1462no—
619C2×C4C2×#1192400512yes—
620C2×C4C2×#1192400512yes—
621C2×C4C2×#1192400512yes—
622C2×C4C2×#1192400511yescharacteristic
623C2×C4C2×#1192400511yescharacteristic
624C2×C4C22×C4×D7311yescharacteristic
625C2×C4C22×D29211yescharacteristic
626C8—2922no—
627C8—2922no—
628C22—21no—
629C22—21no—
630C22—21no—
631C22—22no—
632C22—21no—
633C22—21no—
634C22—21no—
635C22—22no—
636C22—21no—
637C22—22no—
638C22—21no—
639C22—42no—
640C22—42no—
641C22—81no—
642C22—81no—
643C22—2922no—
644C22—2922no—
645C22—2922no—
646C22—11682no—
647C22C22×#1192400511yescharacteristic
648C22#1798938511yescharacteristic
649C22#1798938511yescharacteristic
650C4—21no—
651C4—21no—
652C4—22no—
653C4—21no—
654C4—21no—
655C4—22no—
656C4—22no—
657C4—42no—
658C4—81no—
659C4—81no—
660C4—81no—
661C4—2922no—
662C4—2922no—
663C4—2922no—
664C2—21no—
665C2—21no—
666C2—42no—
667C2—41no—
668C2—41no—
669C2—5842no—
670C2#2261839111yescharacteristic
671C2#2261844811yescharacteristic
672C2#2261847911yescharacteristic
subgroupG/H
centerC22C22×#11924005
commutatorC2×C4×C73C23×C4
FrattiniC22×C4C23×D73
FittingC73×#2353C2
socleC22×C73C22×C22⋊C4
prime radical#22623770C2
cosocleC22×C4×C73C24
hypercenter#2353D73
hypocenterC73#17623
LCSG ▷ C2×C4×C73 ▷ C22×C73 ▷ C73
UCSC1 ◁ C22 ◁ C2×C4×D4 ◁ #2353

Complements

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

NH ≅ G/Ntype
2. #22618972C2semidirect
3. #22608506C2semidirect
4. #22608472C2semidirect
5. #22608463C2semidirect
6. #22608459C2semidirect
7. #22608450C2semidirect
8. #22608446C2semidirect
9. #22608433C2semidirect
10. #22608415C2semidirect
11. C73×#2353C2semidirect
13. #17989283C4semidirect
14. #17989277C4semidirect
16. #17989246C4semidirect
17. #17989233C4semidirect
28. #17988640C22semidirect
31. #17988551C22semidirect
32. #17988323C22semidirect
33. #17988323C22semidirect
34. #17988323C22semidirect
35. #17988323C22semidirect
44. #383×C73C22semidirect
46. #380×C73C22semidirect
49. #376×C73C22semidirect
50. C2×#95×C73C22semidirect
52. C22×Dic146C2×C4semidirect
53. C22×D292C2×C4semidirect
61. #14556583C2×C4semidirect
63. #14556579C2×C4semidirect
73. #14556570C2×C4semidirect
79. #14556565C2×C4semidirect
86. C4×Dic146C2×C4semidirect
87. C4×D292C2×C4semidirect
96. C2×C4⋊Dic73C2×C4semidirect
97. C2×C4⋊Dic73C2×C4semidirect
339. #2353D73semidirect
348. C73#17623semidirect
373. #383C2×D73semidirect
375. #380C2×D73semidirect
378. #376C2×D73semidirect
379. C2×#95C2×D73semidirect

Maximal subgroups

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

subgroupindex
#226237702
#226189722
#226085062
#226084722
#226084632
#226084592
#226084502
#226084462
#226084332
#226084152
C73×#23532
#1762373