Finite Group Databasebeta

← #33053827

Subgroups — #33053827

Order 18688.

1010 conjugacy classes of subgroups.

The subgroup lattice has 654 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#22623783C211yesmaximal, characteristic
2#22618960C211yesmaximal, characteristic, complemented
3#22608485C212yesmaximal, complemented
4#22608481C211yesmaximal, characteristic, complemented
5#22608477C211yesmaximal, characteristic, complemented
6#22608472C212yesmaximal, complemented
7#22608442C211yesmaximal, characteristic, complemented
8#22608438C211yesmaximal, characteristic, complemented
9#22608433C212yesmaximal, complemented
10#22608420C212yesmaximal, complemented
11C73×#2353C211yesmaximal, characteristic, complemented
12#17989320—22no—
13#17989307C412yescomplemented
14#17989302C412yescomplemented
15#17989287—22no—
16#17989258C412yescomplemented
17#17989242C412yescomplemented
18#17989175—21no—
19#17989175—21no—
20#17989154—21no—
21#17989144C2211yescharacteristic
22#17989137C2211yescharacteristic
23#17989098C2212yes—
24#17989098C2212yes—
25#17989068—21no—
26#17988659—22no—
27#17988659—22no—
28#17988644C2211yescharacteristic
29#17988644C2211yescharacteristic
30#17988551C2212yes—
31C2×C22⋊#10596831C2212yes—
32#17988323C2212yes—
33#17988323C2212yes—
34#17988323C2212yes—
35#17988323C2212yes—
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
45#381×C73C2211yescharacteristic
46#380×C73C2211yescharacteristic
47C4×C22⋊C4×C73—21no—
48C4×C22⋊C4×C73C2211yescharacteristic
49#376×C73C2211yescharacteristic
50C2×#95×C73C2211yescharacteristic
51C2×#11924028—21no—
52C2×#11924028C2×C412yes—
53C2×#11924028C2×C412yescomplemented
54C2×#11924012—22no—
55C22×#10596831—22no—
56#14556591C2×C411yescharacteristic, complemented
57#14556591C2×C411yescharacteristic, complemented
58#14556591C2×C411yescharacteristic, complemented
59#14556588C2×C411yescharacteristic, complemented
60#14556587—22no—
61#14556584—22no—
62#14556584—22no—
63#14556583—21no—
64#14556579—21no—
65#14556578—22no—
66#14556577—22no—
67#14556577—22no—
68#14556574—22no—
69#14556574—22no—
70#14556573—22no—
71#14556573—22no—
72#14556570—21no—
73#14556570C2×C412yes—
74#14556570C2×C412yescomplemented
75#14556570C2312yes—
76#14556565—21no—
77#14556565C2×C412yes—
78#14556565C2×C412yescomplemented
79#14556565C2312yes—
80#14556564—22no—
81#14556564—22no—
82#14556562—22no—
83#14556562—22no—
84#14556560—22no—
85#14556560—22no—
86C4×Dic146—21no—
87C4×Dic146C2×C412yescomplemented
88#14556556—22no—
89C4×D292—21no—
90C4×D292C2×C412yescomplemented
91#14556553—22no—
92C4×C22⋊D73—22no—
93C4×C22⋊D73—22no—
94C4×C22⋊D73—22no—
95C4×C22⋊D73—22no—
96#14556550—21no—
97#14556550C2×C412yes—
98#14556550C2×C412yescomplemented
99#14556549—22no—
100C4⋊C4×D73—22no—
101C4⋊#10596831—21no—
102C4⋊#10596831—21no—
103C22⋊#10596831—22no—
104C22⋊#10596831—22no—
105C22⋊#10596831C2312yes—
106C22⋊#10596831C2312yes—
107#14556502—21no—
108#14556502—21no—
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
136#11924028—42no—
137#11924028—42no—
138#11924028—82no—
139#11924028—82no—
140C2×Dic146—21no—
141C2×Dic146—21no—
142C2×Dic146C22×C412yes—
143C2×D292—21no—
144C2×D292—21no—
145C2×D292C22×C412yes—
146C2×C22⋊D73—22no—
147C2×C22⋊D73—22no—
148C2×C22⋊D73—22no—
149C2×C22⋊D73—22no—
150C2×C22⋊D73—22no—
151C2×C4×D73—22no—
152C2×C4×D73—22no—
153C2×C4×D73—22no—
154#11924012—82no—
155C2×#10596831—22no—
156C2×#10596831—22no—
157C2×#10596831—82no—
158C4⋊Dic73—21no—
159C4⋊Dic73—21no—
160C4⋊Dic73—21no—
161C4⋊Dic73—21no—
162C4⋊Dic73C22×C412yes—
163C4⋊Dic73C22×C412yes—
164#11924007—22no—
165#11924007—22no—
166#11924007—22no—
167#11924007—22no—
168C22⋊Dic73—22no—
169C22⋊Dic73—22no—
170C22⋊Dic73—22no—
171C22⋊Dic73—22no—
172C22⋊Dic73—22no—
173#11924005—22no—
174#11924005—22no—
175#11924005—22no—
176#11924005—22no—
177C4×Dic73—22no—
178C4×Dic73—22no—
179C4×Dic73—22no—
180C24×C73C2×D411yescharacteristic
181C24×C73C2×D411yescharacteristic
182C2×D4×C73—41no—
183C2×D4×C73—41no—
184C2×D4×C73C22⋊C412yes—
185C2×D4×C73C22⋊C412yes—
186C22×C4×C73—21no—
187C22×C4×C73—21no—
188C22×C4×C73—21no—
189C22×C4×C73—21no—
190C22×C4×C73—42no—
191C22×C4×C73—42no—
192C22×C4×C73—81no—
193C22×C4×C73—81no—
194C22×C4×C73—81no—
195C22×C4×C73C22×C412yes—
196C22×C4×C73C22×C412yes—
197C22×C4×C73C2×D411yescharacteristic
198C22×C4×C73C2×D411yescharacteristic
199C22×C4×C73C2×D411yescharacteristic
200C22×C4×C73C2×D411yescharacteristic
201C22×C4×C73C2×D412yes—
202C22×C4×C73C2×D412yes—
203C22×C4×C73C2411yescharacteristic
204C4⋊C4×C73—81no—
205C4⋊C4×C73C22⋊C412yes—
206C4⋊C4×C73C22⋊C412yes—
207C22⋊C4×C73—42no—
208C22⋊C4×C73—42no—
209C22⋊C4×C73—81no—
210C22⋊C4×C73C22⋊C412yes—
211C22⋊C4×C73C22⋊C412yes—
212C22⋊C4×C73C22⋊C412yes—
213C22⋊C4×C73C22⋊C412yes—
214C42×C73—42no—
215C42×C73—81no—
216C42×C73C22×C411yescharacteristic
217C42×C73C22×C411yescharacteristic
218C42×C73C2×D411yescharacteristic
219C42×C73C2×D411yescharacteristic
220C23×C73—21no—
221C23×C73—21no—
222C23×C73—21no—
223C23×C73—21no—
224C23×C73—41no—
225C23×C73—41no—
226C23×C73—41no—
227C23×C73—41no—
228C23×C73C2×C22⋊C412yes—
229C23×C73C2×C22⋊C412yes—
230C23×C73C22×D411yescharacteristic
231C22×D73—22no—
232D4×C73—42no—
233D4×C73—42no—
234C2×C4×C73—21no—
235C2×C4×C73—21no—
236C2×C4×C73—22no—
237C2×C4×C73—41no—
238C2×C4×C73—41no—
239C2×C4×C73—41no—
240C2×C4×C73—41no—
241C2×C4×C73—42no—
242C2×C4×C73—42no—
243C2×C4×C73—42no—
244C2×C4×C73—42no—
245C2×C4×C73—41no—
246C2×C4×C73—41no—
247C2×C4×C73—41no—
248C2×C4×C73—41no—
249C2×C4×C73—41no—
250C2×C4×C73—41no—
251C2×C4×C73—41no—
252C2×C4×C73—41no—
253C2×C4×C73—41no—
254C2×C4×C73—41no—
255C2×C4×C73—41no—
256C2×C4×C73—84no—
257C2×C4×C73C2×C22⋊C411yescharacteristic
258C2×C4×C73C2×C22⋊C411yescharacteristic
259C2×C4×C73C2×C22⋊C411yescharacteristic
260C2×C4×C73C2×C22⋊C411yescharacteristic
261C2×C4×C73C2×C22⋊C412yes—
262C2×C4×C73C2×C22⋊C412yes—
263C2×C4×C73C23×C411yescharacteristic
264C2×C4×C73C22×D411yescharacteristic
265C22⋊D73—42no—
266C22⋊D73—42no—
267C22⋊D73—82no—
268C22⋊D73—82no—
269C22⋊D73—82no—
270C22⋊D73—82no—
271C2×Dic73—22no—
272C2×Dic73—22no—
273C2×Dic73—22no—
274D292—42no—
275D292—42no—
276D292—81no—
277C4×D73—44no—
278C4×D73—82no—
279C4×D73—82no—
280Dic146—42no—
281Dic146—42no—
282Dic146—81no—
283#10596831—42no—
284#10596831—42no—
285C22×C73—21no—
286C22×C73—21no—
287C22×C73—22no—
288C22×C73—22no—
289C22×C73—22no—
290C22×C73—22no—
291C22×C73—22no—
292C22×C73—22no—
293C22×C73—42no—
294C22×C73—42no—
295C22×C73—81no—
296C22×C73—81no—
297C22×C73C22×C22⋊C411yescharacteristic
298C22×C73#57511yescharacteristic
299C22×C73#57511yescharacteristic
300C2×D73—42no—
301C2×D73—42no—
302C2×D73—42no—
303C4×C73—21no—
304C4×C73—21no—
305C4×C73—22no—
306C4×C73—22no—
307C4×C73—21no—
308C4×C73—21no—
309C4×C73—21no—
310C4×C73—21no—
311C4×C73—42no—
312C4×C73—81no—
313C4×C73—81no—
314C4×C73—81no—
315Dic73—42no—
316Dic73—42no—
317Dic73—42no—
318#17629—731nomaximal, Sylow
319C2×C73—21no—
320C2×C73—21no—
321C2×C73—42no—
322C2×C73—42no—
323C2×C73#288311yescharacteristic
324C2×C73#289611yescharacteristic
325C2×C73#290111yescharacteristic
326D73—82no—
327#3506—731no—
328#2973—731no—
329#2353D7311yescharacteristic, complemented
330#1516—732no—
331#1515—731no—
332#1514—731no—
333#1513—732no—
334#1506—731no—
335#1505—731no—
336#1504—732no—
337#1501—732no—
338C73#1762911yescharacteristic, complemented, Sylow
339#559—1462no—
340#556—732no—
341#555—732no—
342#553—1462no—
343#550—732no—
344#547—732no—
345#538—1461no—
346#538—1461no—
347#535—1461no—
348#533—731no—
349#532—731no—
350#525—732no—
351#525—732no—
352#520—1461no—
353C2×C4×D4C2×D7311yescharacteristic
354#435—1462no—
355#435—1462no—
356#432—731no—
357#432—731no—
358#407—732no—
359C2×C22⋊C8—732no—
360#388—21no—
361#387—21no—
362#386—21no—
363#383C2×D7311yescharacteristic
364#381C2×D7311yescharacteristic
365#380C2×D7311yescharacteristic
366C4×C22⋊C4—21no—
367C4×C22⋊C4C2×D7311yescharacteristic
368#376C2×D7311yescharacteristic
369C2×#95C2×D7311yescharacteristic
370#328—732no—
371#328—732no—
372#328—732no—
373#328—732no—
374#327—732no—
375#327—732no—
376#327—732no—
377#327—732no—
378C2×#41—732no—
379C2×#41—732no—
380C2×#41—1461no—
381C22×D4C22⋊D7311yescharacteristic
382C23×C4—21no—
383C23×C4C22×D7311yescharacteristic
384C23×C4C22×D7311yescharacteristic
385C2×M16—1462no—
386C22×C8—1462no—
387C4⋊Q8—731no—
388C4⋊Q8—731no—
389C4⋊Q8—731no—
390#127—731no—
391#127—1462no—
392#126—1462no—
393#126—1462no—
394#125—1461no—
395#124—1462no—
396#124—1462no—
397#124—1462no—
398#124—1461no—
399#123—1462no—
400#123—1462no—
401#123—1462no—
402#123—1462no—
403C22⋊Q8—732no—
404C22⋊Q8—732no—
405C22⋊Q8—732no—
406C22⋊Q8—1461no—
407#121—732no—
408#121—732no—
409#121—732no—
410#121—1462no—
411#121—1462no—
412#121—1461no—
413#121—1462no—
414#121—1462no—
415#120—1462no—
416#120—1462no—
417C4×Q8—732no—
418C4×Q8—1462no—
419C4×Q8—1461no—
420C4×D4—732no—
421C4×D4—1462no—
422C4×D4—1462no—
423C4×D4—1462no—
424C4×D4—1462no—
425C4×D4—1462no—
426C4×D4—1461no—
427C4×D4C4×D7312yes—
428C4×D4C4×D7312yes—
429C4×D4D29212yes—
430C4×D4D29212yes—
431#117—732no—
432#117—732no—
433#117—1461no—
434#117—1462no—
435C2×C4⋊C4—21no—
436C2×C4⋊C4—1462no—
437C2×C4⋊C4C22⋊D7311yescharacteristic
438C2×C22⋊C4—21no—
439C2×C22⋊C4—21no—
440C2×C22⋊C4—21no—
441C2×C22⋊C4C22⋊D7311yescharacteristic
442C2×C22⋊C4C22⋊D7311yescharacteristic
443C2×C42—21no—
444C2×C42—21no—
445C2×C42C22×D7311yescharacteristic
446C4⋊C8—1461no—
447C4⋊C8—1461no—
448C22⋊C8—732no—
449C22⋊C8—732no—
450C22⋊C8—1462no—
451C22⋊C8—1462no—
452#97—1461no—
453#97—1461no—
454#95—21no—
455#95—21no—
456#95—21no—
457#95—21no—
458#95—21no—
459#95C22×D7311yescharacteristic
460#95C22×D7311yescharacteristic
461#95C22×D7311yescharacteristic
462#95C22×D7311yescharacteristic
463C24C2×C22⋊D7311yescharacteristic
464C24C2×C22⋊D7311yescharacteristic
465#41—2922no—
466#41—2922no—
467#41—5842no—
468#41—5842no—
469C2×Q8—732no—
470C2×Q8—1461no—
471C2×Q8—1461no—
472C2×D4—41no—
473C2×D4—41no—
474C2×D4—732no—
475C2×D4—1462no—
476C2×D4—1462no—
477C2×D4—1462no—
478C2×D4—1462no—
479C2×D4—1462no—
480C2×D4—1461no—
481C2×D4—1461no—
482C2×D4#1192400512yes—
483C2×D4#1192400512yes—
484C22×C4—21no—
485C22×C4—21no—
486C22×C4—21no—
487C22×C4—21no—
488C22×C4—42no—
489C22×C4—42no—
490C22×C4—81no—
491C22×C4—81no—
492C22×C4—81no—
493C22×C4—1462no—
494C22×C4—1462no—
495C22×C4—1462no—
496C22×C4C2×C4×D7312yes—
497C22×C4C2×C4×D7312yes—
498C22×C4C2×C22⋊D7311yescharacteristic
499C22×C4C2×C22⋊D7311yescharacteristic
500C22×C4C2×C22⋊D7311yescharacteristic
501C22×C4C2×C22⋊D7311yescharacteristic
502C22×C4C2×D29212yes—
503C22×C4C2×D29212yes—
504C22×C4C23×D7311yescharacteristic
505M16—5842no—
506C2×C8—5842no—
507C2×C8—1462no—
508C2×C8—1462no—
509C4⋊C4—81no—
510C4⋊C4—732no—
511C4⋊C4—732no—
512C4⋊C4—1462no—
513C4⋊C4—1462no—
514C4⋊C4—1462no—
515C4⋊C4—1461no—
516C4⋊C4—1461no—
517C4⋊C4—1461no—
518C4⋊C4—1461no—
519C4⋊C4—1462no—
520C4⋊C4#1192400512yes—
521C4⋊C4#1192400512yes—
522C22⋊C4—42no—
523C22⋊C4—42no—
524C22⋊C4—81no—
525C22⋊C4—1462no—
526C22⋊C4—1462no—
527C22⋊C4—1462no—
528C22⋊C4—1462no—
529C22⋊C4—1462no—
530C22⋊C4—1462no—
531C22⋊C4—1462no—
532C22⋊C4—1462no—
533C22⋊C4—1462no—
534C22⋊C4#1192400512yes—
535C22⋊C4#1192400512yes—
536C22⋊C4#1192400512yes—
537C22⋊C4#1192400512yes—
538C42—42no—
539C42—81no—
540C42—1462no—
541C42—1462no—
542C42—1462no—
543C42C2×C4×D7311yescharacteristic
544C42C2×C4×D7311yescharacteristic
545C42C2×D29211yescharacteristic
546C42C2×D29211yescharacteristic
547C23—21no—
548C23—21no—
549C23—21no—
550C23—21no—
551C23—41no—
552C23—41no—
553C23—41no—
554C23—41no—
555C23—1462no—
556C23C2×#1192400512yes—
557C23C2×#1192400512yes—
558C23C22×C22⋊D7311yescharacteristic
559Q8—2922no—
560Q8—2922no—
561Q8—5841no—
562D4—42no—
563D4—42no—
564D4—2922no—
565D4—2922no—
566D4—2922no—
567D4—2922no—
568D4—5841no—
569D4—5842no—
570D4—5842no—
571D4—5842no—
572D4—5842no—
573C2×C4—21no—
574C2×C4—21no—
575C2×C4—22no—
576C2×C4—41no—
577C2×C4—41no—
578C2×C4—41no—
579C2×C4—42no—
580C2×C4—42no—
581C2×C4—41no—
582C2×C4—42no—
583C2×C4—42no—
584C2×C4—41no—
585C2×C4—41no—
586C2×C4—41no—
587C2×C4—41no—
588C2×C4—41no—
589C2×C4—41no—
590C2×C4—41no—
591C2×C4—41no—
592C2×C4—41no—
593C2×C4—41no—
594C2×C4—41no—
595C2×C4—84no—
596C2×C4—2924no—
597C2×C4—5842no—
598C2×C4—5842no—
599C2×C4—1462no—
600C2×C4—1462no—
601C2×C4—1462no—
602C2×C4C2×#1192400511yescharacteristic
603C2×C4C2×#1192400512yes—
604C2×C4C2×#1192400512yes—
605C2×C4C2×#1192400511yescharacteristic
606C2×C4C2×#1192400511yescharacteristic
607C2×C4C2×#1192400511yescharacteristic
608C2×C4C22×C4×D7311yescharacteristic
609C2×C4C22×D29211yescharacteristic
610C8—2922no—
611C8—2922no—
612C22—21no—
613C22—21no—
614C22—22no—
615C22—22no—
616C22—22no—
617C22—22no—
618C22—22no—
619C22—22no—
620C22—42no—
621C22—42no—
622C22—81no—
623C22—81no—
624C22—2922no—
625C22—2922no—
626C22—2922no—
627C22C22×#1192400511yescharacteristic
628C22#1798938511yescharacteristic
629C22#1798938511yescharacteristic
630C4—21no—
631C4—21no—
632C4—22no—
633C4—22no—
634C4—21no—
635C4—21no—
636C4—21no—
637C4—21no—
638C4—42no—
639C4—81no—
640C4—81no—
641C4—81no—
642C4—2922no—
643C4—2922no—
644C4—2922no—
645C2—21no—
646C2—21no—
647C2—42no—
648C2—42no—
649C2—5842no—
650C2#2261839111yescharacteristic
651C2#2261844811yescharacteristic
652C2#2261847911yescharacteristic
subgroupG/H
centerC22C22×#11924005
commutatorC2×C4×C73C23×C4
FrattiniC22×C4C23×D73
FittingC73×#2353C2
socleC22×C73C22×C22⋊C4
prime radical#22623783C2
cosocleC22×C4×C73C24
hypercenter#2353D73
hypocenterC73#17629
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. #22618960C2semidirect
3. #22608485C2semidirect
4. #22608481C2semidirect
5. #22608477C2semidirect
6. #22608472C2semidirect
7. #22608442C2semidirect
8. #22608438C2semidirect
9. #22608433C2semidirect
10. #22608420C2semidirect
11. C73×#2353C2semidirect
13. #17989307C4semidirect
14. #17989302C4semidirect
16. #17989258C4semidirect
17. #17989242C4semidirect
53. C2×#11924028C2×C4semidirect
56. #14556591C2×C4semidirect
57. #14556591C2×C4semidirect
58. #14556591C2×C4semidirect
59. #14556588C2×C4semidirect
74. #14556570C2×C4semidirect
78. #14556565C2×C4semidirect
87. C4×Dic146C2×C4semidirect
90. C4×D292C2×C4semidirect
98. #14556550C2×C4semidirect
329. #2353D73semidirect
338. C73#17629semidirect

Maximal subgroups

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

subgroupindex
#226237832
#226189602
#226084852
#226084812
#226084772
#226084722
#226084422
#226084382
#226084332
#226084202
C73×#23532
#1762973