Finite Group Databasebeta

← #33108072

Subgroups — #33108072

Order 18688.

692 conjugacy classes of subgroups.

The subgroup lattice has 636 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#22619715C211yesmaximal, characteristic
2#22618877C211yesmaximal, characteristic, complemented
3#22618497C211yesmaximal, characteristic
4#22610910C211yesmaximal, characteristic, complemented
5#22610703C211yesmaximal, characteristic, complemented
6#22610681C211yesmaximal, characteristic, complemented
7#22610677C211yesmaximal, characteristic, complemented
8#22610531C211yesmaximal, characteristic, complemented
9#22610528C211yesmaximal, characteristic, complemented
10#22610204C211yesmaximal, characteristic
11#22609787C211yesmaximal, characteristic, complemented
12#22609773C211yesmaximal, characteristic
13#22609771C211yesmaximal, characteristic
14#22609743C211yesmaximal, characteristic
15C73×#1866C211yesmaximal, characteristic
16C2×#14556598C2211yescharacteristic
17#17989140C2211yescharacteristic
18#41×Dic73C2211yescharacteristic
19#17988896C411yescharacteristic, complemented
20#17988892C411yescharacteristic, complemented
21#17988889C411yescharacteristic, complemented
22#17988885C411yescharacteristic, complemented
23#17988880C411yescharacteristic, complemented
24#17988875C411yescharacteristic, complemented
25#17988870C411yescharacteristic, complemented
26#17988865C411yescharacteristic, complemented
27#17988716—21no—
28#17988716—21no—
29#17988714—21no—
30#17988711—21no—
31C2×#14556526C2211yescharacteristic
32#17988625C2211yescharacteristic
33#17988623C2211yescharacteristic
34#17988615—21no—
35#17988615C2211yescharacteristic
36#17988615C2211yescharacteristic
37#17988613—21no—
38#17988613C2211yescharacteristic
39#17988610—21no—
40#17988610C2211yescharacteristic
41C2×#14556520—21no—
42C2×#14556520C2211yescharacteristic
43C2×#14556520C2211yescharacteristic
44C2×#14556518C2211yescharacteristic
45#17988583C2211yescharacteristic
46#17988543C2211yescharacteristic
47#17988542C2211yescharacteristic
48M16×Dic73C2211yescharacteristic
49C2×#14556501C2211yescharacteristic
50C2×C8×Dic73C2211yescharacteristic
51#17988433C2211yescharacteristic
52#17988352C2211yescharacteristic
53#17988351C2211yescharacteristic
54#17988348C2211yescharacteristic
55#17988348C2211yescharacteristic
56#17988347C2211yescharacteristic
57#17988347C2211yescharacteristic
58#17988346C2211yescharacteristic
59#17988342C2211yescharacteristic
60C2×#131×C73C2211yescharacteristic
61C2×#108×C73C2211yescharacteristic
62#424×C73C2211yescharacteristic
63C2×#104×C73C2211yescharacteristic
64#416×C73C2211yescharacteristic
65#339×C73C2211yescharacteristic
66#336×C73C2211yescharacteristic
67#14556598—22no—
68#14556598—41no—
69C2×#11924012—21no—
70C2×#11924012C2311yescharacteristic
71C2×#11924012C2311yescharacteristic
72C22×#10596831—21no—
73C22×#10596831C2311yescharacteristic
74C22×#10596831C2311yescharacteristic
75#14556590C2×C411yescharacteristic
76#14556590C2×C411yescharacteristic
77#14556582C2×C411yescharacteristic
78#14556582C2×C411yescharacteristic
79#14556568—21no—
80#14556568—21no—
81Q8×Dic73C2×C411yescharacteristic
82Q8×Dic73C2×C411yescharacteristic
83C4×Dic146—21no—
84C4×Dic146—21no—
85D4×Dic73—21no—
86D4×Dic73—21no—
87D4×Dic73C2×C411yescharacteristic
88D4×Dic73C2×C411yescharacteristic
89#14556550—21no—
90#14556550C2311yescharacteristic
91#14556550C2311yescharacteristic
92C2×C4×Dic73—21no—
93C2×C4×Dic73C2311yescharacteristic
94C2×C4×Dic73C2311yescharacteristic
95#14556526—41no—
96#14556525C2×C411yescharacteristic
97#14556525C2×C411yescharacteristic
98#14556524C2×C411yescharacteristic
99#14556524C2×C411yescharacteristic
100#14556523C2×C411yescharacteristic
101#14556523C2×C411yescharacteristic
102#14556522C2×C411yescharacteristic
103#14556522C2×C411yescharacteristic
104#14556520—21no—
105#14556520—21no—
106#14556520—21no—
107#14556520—21no—
108#14556520—21no—
109#14556520—21no—
110#14556520—21no—
111#14556520—21no—
112#14556520C2×C411yescharacteristic
113#14556520C2×C411yescharacteristic
114#14556520C2×C411yescharacteristic
115#14556520C2×C411yescharacteristic
116#14556520D411yescharacteristic
117#14556520D411yescharacteristic
118#14556520D411yescharacteristic
119#14556520D411yescharacteristic
120#14556518Q814yescomplemented
121#14556515C2×C411yescharacteristic
122#14556515C2×C411yescharacteristic
123#14556512C2×C411yescharacteristic
124#14556512C2×C411yescharacteristic
125#14556502—21no—
126#14556501—21no—
127#14556501—21no—
128#14556501—41no—
129C4×#10596831—21no—
130C8×Dic73—21no—
131C8×Dic73—21no—
132C8×Dic73—41no—
133#14556498—21no—
134#14556498—21no—
135C2×#41×C73C2311yescharacteristic
136#131×C73—22no—
137#131×C73—41no—
138C2×M16×C73—21no—
139C2×M16×C73C2311yescharacteristic
140C2×M16×C73C2311yescharacteristic
141C22×C8×C73—21no—
142C22×C8×C73C2311yescharacteristic
143C22×C8×C73C2311yescharacteristic
144#117×C73C2311yescharacteristic
145C2×C42×C73C2311yescharacteristic
146#108×C73—41no—
147C4⋊C8×C73—21no—
148C4⋊C8×C73—21no—
149#104×C73—22no—
150#103×C73C2×C411yescharacteristic
151#103×C73C2×C411yescharacteristic
152#102×C73C2×C411yescharacteristic
153#102×C73C2×C411yescharacteristic
154C2×Dic146—21no—
155C2×Dic146—21no—
156C22×Dic73—21no—
157C22×Dic73—21no—
158C22×Dic73—21no—
159C22×Dic73#4111yescharacteristic
160C22×Dic73#4111yescharacteristic
161#11924012—21no—
162#11924012—21no—
163#11924012—21no—
164#11924012—42no—
165#11924012C2×Q812yes—
166C2×#10596831—21no—
167C2×#10596831—21no—
168C2×#10596831—21no—
169C2×#10596831—21no—
170C2×#10596831—21no—
171C2×#10596831—21no—
172C2×#10596831—42no—
173C2×#10596831C22×C411yescharacteristic
174C2×#10596831C22×C411yescharacteristic
175C2×#10596831C2×D411yescharacteristic
176C2×#10596831C2×D411yescharacteristic
177C4⋊Dic73—21no—
178C4⋊Dic73—21no—
179C4⋊Dic73—21no—
180C4⋊Dic73—21no—
181C4⋊Dic73C22×C411yescharacteristic
182C4⋊Dic73C22×C411yescharacteristic
183C4⋊Dic73C22×C411yescharacteristic
184C4⋊Dic73C22×C411yescharacteristic
185#11924007—21no—
186#11924007—21no—
187#11924007—21no—
188#11924007—21no—
189C22⋊Dic73—21no—
190C22⋊Dic73—21no—
191C22⋊Dic73—21no—
192C22⋊Dic73—21no—
193C4×Dic73—21no—
194C4×Dic73—21no—
195C4×Dic73—21no—
196C4×Dic73—21no—
197C4×Dic73—21no—
198C4×Dic73C22×C411yescharacteristic
199C4×Dic73C22×C411yescharacteristic
200C4×Dic73C2×D411yescharacteristic
201C4×Dic73C2×D411yescharacteristic
202C4×Dic73C2×Q812yes—
203#41×C73—21no—
204#41×C73—21no—
205#41×C73C2×Q812yes—
206#41×C73#4112yes—
207C2×Q8×C73C22×C411yescharacteristic
208C2×D4×C73—21no—
209C2×D4×C73C22×C411yescharacteristic
210C22×C4×C73—21no—
211C22×C4×C73—21no—
212C22×C4×C73C2411yescharacteristic
213M16×C73—21no—
214M16×C73—21no—
215M16×C73—21no—
216M16×C73—21no—
217M16×C73—42no—
218C2×C8×C73—21no—
219C2×C8×C73—21no—
220C2×C8×C73—21no—
221C2×C8×C73—21no—
222C2×C8×C73—21no—
223C2×C8×C73—21no—
224C2×C8×C73—42no—
225C2×C8×C73C22×C411yescharacteristic
226C2×C8×C73C22×C411yescharacteristic
227C2×C8×C73C2×D411yescharacteristic
228C2×C8×C73C2×D411yescharacteristic
229C4⋊C4×C73C22×C411yescharacteristic
230C4⋊C4×C73C22×C411yescharacteristic
231C22⋊C4×C73—21no—
232C42×C73—21no—
233C42×C73—21no—
234C42×C73—21no—
235C23×C73—21no—
236C23×C73C2×#4111yescharacteristic
237Q8×C73—21no—
238Q8×C73C4×Q812yes—
239D4×C73—22no—
240D4×C73—22no—
241D4×C73—21no—
242D4×C73C4×Q812yes—
243C2×C4×C73—21no—
244C2×C4×C73—22no—
245C2×C4×C73—22no—
246C2×C4×C73—21no—
247C2×C4×C73—21no—
248C2×C4×C73—41no—
249C2×C4×C73—41no—
250C2×C4×C73—41no—
251C2×C4×C73C23×C411yescharacteristic
252C2×C4×C73C22×D411yescharacteristic
253C2×C4×C73C22×Q811yescharacteristic
254C2×C4×C73C2×#4111yescharacteristic
255C2×C4×C73H(5,𝔽2)11yescharacteristic
256C2×C4×C732−1+411yescharacteristic
257C2×Dic73—21no—
258C2×Dic73—21no—
259C2×Dic73—21no—
260C2×Dic73—21no—
261C2×Dic73—22no—
262C2×Dic73—41no—
263C2×Dic73—41no—
264C2×Dic73—42no—
265C2×Dic73—41no—
266C2×Dic73—41no—
267C2×Dic73—81no—
268C2×Dic73C4×D411yescharacteristic
269C2×Dic73C4×D411yescharacteristic
270C2×Dic73C4×D411yescharacteristic
271C2×Dic73C4×D411yescharacteristic
272Dic146—42no—
273Dic146—81no—
274C8×C73—21no—
275C8×C73—21no—
276C8×C73—21no—
277C8×C73—21no—
278C8×C73—41no—
279C8×C73—41no—
280#10596831—21no—
281#10596831—21no—
282#10596831—21no—
283#10596831—21no—
284#10596831—22no—
285#10596831—41no—
286C22×C73—21no—
287C22×C73—21no—
288C22×C73—22no—
289C22×C73—41no—
290C22×C73—41no—
291C22×C73C2×C4×D411yescharacteristic
292C22×C73#55211yescharacteristic
293C22×C73#55411yescharacteristic
294C4×C73—22no—
295C4×C73—41no—
296C4×C73—41no—
297C4×C73—81no—
298C4×C73C2×C4×Q811yescharacteristic
299C4×C73#52011yescharacteristic
300C4×C73#54611yescharacteristic
301C4×C73D4×Q811yescharacteristic
302Dic73—21no—
303Dic73—21no—
304Dic73—21no—
305Dic73—21no—
306Dic73—42no—
307Dic73—41no—
308Dic73—41no—
309Dic73—41no—
310Dic73—41no—
311#22984—731nomaximal, Sylow
312C2×C73—21no—
313C2×C73—21no—
314C2×C73—42no—
315C2×C73#236011yescharacteristic
316C2×C73#295511yescharacteristic
317C2×C73#295711yescharacteristic
318#3076—731no—
319#2962—731no—
320#2906—731no—
321#1995—731no—
322#1950—731no—
323#1946—731no—
324#1945—731no—
325#1922—731no—
326#1922—731no—
327#1866—731no—
328#1866D7311yescharacteristic
329#1768—731no—
330#1765—731no—
331#1764—731no—
332#1759—731no—
333C73#2298411yescharacteristic, complemented, Sylow
334C2×#131—731no—
335C2×#131C2×D7311yescharacteristic
336#533—731no—
337C4×#41—731no—
338#479—731no—
339#479—731no—
340#478—731no—
341#478—731no—
342#477—731no—
343#476—731no—
344#475—731no—
345#474—731no—
346#446—1461no—
347#446—1461no—
348#446—1461no—
349C8×Q8—1461no—
350C2×#108—731no—
351C2×#108C2×D7311yescharacteristic
352#427—731no—
353#427—731no—
354#424—731no—
355#424—731no—
356#424—731no—
357#424—1461no—
358#424—1461no—
359#424C2×D7311yescharacteristic
360#423—731no—
361#423—1461no—
362C2×C4⋊C8—731no—
363C2×C4⋊C8—731no—
364C2×C4⋊C8—1461no—
365C2×#104—731no—
366C2×#104C2×D7311yescharacteristic
367#416—731no—
368#416C2×D7311yescharacteristic
369#405—731no—
370#405—731no—
371C4×M16—731no—
372C2×#97—731no—
373C2×C4×C8—731no—
374#376—731no—
375#341—731no—
376#341—731no—
377#339—731no—
378#339—731no—
379#339—731no—
380#339—731no—
381#339—731no—
382#339C2×D7311yescharacteristic
383#336—731no—
384#336C2×D7311yescharacteristic
385C2×#41C22×D7311yescharacteristic
386#131—22no—
387#131—41no—
388#131—2921no—
389#131—1462no—
390C2×M16—21no—
391C2×M16—731no—
392C2×M16—731no—
393C2×M16—1461no—
394C2×M16C22×D7311yescharacteristic
395C2×M16C22×D7311yescharacteristic
396C22×C8—21no—
397C22×C8—731no—
398C22×C8—731no—
399C22×C8—1461no—
400C22×C8C22×D7311yescharacteristic
401C22×C8C22×D7311yescharacteristic
402C4⋊Q8—731no—
403C4⋊Q8—731no—
404#125—731no—
405#125—731no—
406C22⋊Q8—1461no—
407C22⋊Q8—1461no—
408C4×Q8—731no—
409C4×Q8—731no—
410C4×Q8—1461no—
411C4×Q8—1461no—
412C4×D4—731no—
413C4×D4—731no—
414C4×D4—1461no—
415C4×D4—1461no—
416#117—731no—
417#117—731no—
418#117—1461no—
419#117C22×D7311yescharacteristic
420C2×C42—731no—
421C2×C42—731no—
422C2×C42—1461no—
423C2×C42C22×D7311yescharacteristic
424#108—41no—
425#108—2921no—
426#107—731no—
427#107—731no—
428#107—731no—
429#107—731no—
430#106—731no—
431#106—731no—
432#106—731no—
433#106—731no—
434C4⋊C8—21no—
435C4⋊C8—21no—
436C4⋊C8—731no—
437C4⋊C8—731no—
438C4⋊C8—731no—
439C4⋊C8—731no—
440C4⋊C8—731no—
441C4⋊C8—731no—
442C4⋊C8—731no—
443C4⋊C8—731no—
444C4⋊C8—1461no—
445C4⋊C8—1461no—
446C4⋊C8—1461no—
447C4⋊C8—1461no—
448C4⋊C8—1461no—
449C4⋊C8—1461no—
450C4⋊C8—1461no—
451C4⋊C8—1461no—
452#104—22no—
453#104—734no—
454#103—731no—
455#103—731no—
456#103C4×D7311yescharacteristic
457#103C4×D7311yescharacteristic
458#102—731no—
459#102—731no—
460#102C4×D7311yescharacteristic
461#102C4×D7311yescharacteristic
462#97—2921no—
463#97—1461no—
464#97—1461no—
465#97—1461no—
466C4×C8—2921no—
467C4×C8—1461no—
468C4×C8—1461no—
469C4×C8—1461no—
470#95—1461no—
471#95—1461no—
472#41—21no—
473#41—21no—
474#41Q8×D7312yes—
475#41#1192402712yes—
476C2×Q8—1461no—
477C2×Q8—1461no—
478C2×Q8C2×C4×D7311yescharacteristic
479C2×D4—21no—
480C2×D4C2×C4×D7311yescharacteristic
481C22×C4—21no—
482C22×C4—21no—
483C22×C4—731no—
484C22×C4—731no—
485C22×C4—1461no—
486C22×C4—1461no—
487C22×C4—1461no—
488C22×C4C23×D7311yescharacteristic
489M16—21no—
490M16—21no—
491M16—21no—
492M16—21no—
493M16—42no—
494M16—2922no—
495M16—732no—
496M16—1461no—
497M16—1461no—
498M16—1461no—
499C2×C8—21no—
500C2×C8—21no—
501C2×C8—21no—
502C2×C8—21no—
503C2×C8—21no—
504C2×C8—21no—
505C2×C8—42no—
506C2×C8—2922no—
507C2×C8—731no—
508C2×C8—731no—
509C2×C8—731no—
510C2×C8—731no—
511C2×C8—1461no—
512C2×C8—1461no—
513C2×C8—1461no—
514C2×C8—1461no—
515C2×C8—1461no—
516C2×C8—1461no—
517C2×C8C2×C4×D7311yescharacteristic
518C2×C8C2×C4×D7311yescharacteristic
519C2×C8D4×D7311yescharacteristic
520C2×C8D4×D7311yescharacteristic
521C4⋊C4—731no—
522C4⋊C4—731no—
523C4⋊C4—731no—
524C4⋊C4—731no—
525C4⋊C4—1461no—
526C4⋊C4—1461no—
527C4⋊C4—1461no—
528C4⋊C4—1461no—
529C4⋊C4—1461no—
530C4⋊C4—1461no—
531C4⋊C4—1461no—
532C4⋊C4—1461no—
533C4⋊C4C2×C4×D7311yescharacteristic
534C4⋊C4C2×C4×D7311yescharacteristic
535C22⋊C4—21no—
536C22⋊C4—1461no—
537C22⋊C4—1461no—
538C22⋊C4—1461no—
539C22⋊C4—1461no—
540C42—21no—
541C42—21no—
542C42—21no—
543C42—731no—
544C42—731no—
545C42—731no—
546C42—731no—
547C42—732no—
548C42—1461no—
549C42—1461no—
550C42—1461no—
551C42—1461no—
552C42—1461no—
553C23—21no—
554C23#41×D7311yescharacteristic
555Q8—21no—
556Q8—2922no—
557Q8—5841no—
558Q8#1455655612yes—
559D4—22no—
560D4—22no—
561D4—21no—
562D4#1455655612yes—
563C2×C4—21no—
564C2×C4—22no—
565C2×C4—22no—
566C2×C4—21no—
567C2×C4—21no—
568C2×C4—41no—
569C2×C4—41no—
570C2×C4—41no—
571C2×C4—2921no—
572C2×C4—2921no—
573C2×C4—2922no—
574C2×C4—2921no—
575C2×C4—2921no—
576C2×C4—5841no—
577C2×C4—731no—
578C2×C4—731no—
579C2×C4—731no—
580C2×C4—731no—
581C2×C4—1462no—
582C2×C4—1461no—
583C2×C4—1461no—
584C2×C4—1461no—
585C2×C4—1461no—
586C2×C4C22×C4×D7311yescharacteristic
587C2×C4C2×D4×D7311yescharacteristic
588C2×C4C2×Q8×D7311yescharacteristic
589C2×C4C2×#1192402711yescharacteristic
590C2×C4#1455663611yescharacteristic
591C2×C4#1455663811yescharacteristic
592C8—21no—
593C8—21no—
594C8—21no—
595C8—21no—
596C8—41no—
597C8—41no—
598C8—2921no—
599C8—1461no—
600C8—1461no—
601C8—1462no—
602C8—1461no—
603C8—1461no—
604C22—21no—
605C22—21no—
606C22—22no—
607C22—41no—
608C22—41no—
609C22C4×D4×D7311yescharacteristic
610C22#1798927411yescharacteristic
611C22#1798929111yescharacteristic
612C4—22no—
613C4—41no—
614C4—41no—
615C4—81no—
616C4—2922no—
617C4—2921no—
618C4—2921no—
619C4—2921no—
620C4—2921no—
621C4—1461no—
622C4—1461no—
623C4—1461no—
624C4—1461no—
625C4C2×#1455655611yescharacteristic
626C4#1798906611yescharacteristic
627C4#1798922411yescharacteristic
628C4#1798925011yescharacteristic
629C2—21no—
630C2—21no—
631C2—42no—
632C2#2261293111yescharacteristic
633C2#2261883111yescharacteristic
634C2#2261884311yescharacteristic
subgroupG/H
centerC2×C4C2×D4×D73
commutatorC2×C4×C73C23×C4
FrattiniC22×C4C23×D73
FittingC73×#1866C2
socleC22×C73C2×C4×D4
prime radicalC2×#41×C73C23
cosocleC22×C4×C73C24
hypercenter#1866D73
hypocenterC73#22984
LCSG ▷ C2×C4×C73 ▷ C2×C73 ▷ C73
UCSC1 ◁ C2×C4 ◁ C22×C8 ◁ #1866

Complements

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

NH ≅ G/Ntype
2. #22618877C2semidirect
4. #22610910C2semidirect
5. #22610703C2semidirect
6. #22610681C2semidirect
7. #22610677C2semidirect
8. #22610531C2semidirect
9. #22610528C2semidirect
11. #22609787C2semidirect
19. #17988896C4semidirect
20. #17988892C4semidirect
21. #17988889C4semidirect
22. #17988885C4semidirect
23. #17988880C4semidirect
24. #17988875C4semidirect
25. #17988870C4semidirect
26. #17988865C4semidirect
120. #14556518Q8semidirect
333. C73#22984semidirect

Maximal subgroups

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

subgroupindex
#226197152
#226188772
#226184972
#226109102
#226107032
#226106812
#226106772
#226105312
#226105282
#226102042
#226097872
#226097732
#226097712
#226097432
C73×#18662
#2298473