Finite Group Databasebeta

← #33098284

Subgroups — #33098284

Order 18688.

648 conjugacy classes of subgroups.

The subgroup lattice has 622 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#22619715C211yesmaximal, characteristic
2#22618864C211yesmaximal, characteristic, complemented
3#22618571C211yesmaximal, characteristic
4#22610873C211yesmaximal, characteristic, complemented
5#22610705C211yesmaximal, characteristic, complemented
6#22610679C211yesmaximal, characteristic, complemented
7#22610676C211yesmaximal, characteristic, complemented
8#22610638C211yesmaximal, characteristic
9#22610582C211yesmaximal, characteristic
10#22610581C211yesmaximal, characteristic
11#22610529C211yesmaximal, characteristic, complemented
12#22610528C211yesmaximal, characteristic, complemented
13#22609949C211yesmaximal, characteristic
14C8×#11924007C211yesmaximal, characteristic, complemented
15C73×#1937C211yesmaximal, characteristic
16#17989140C2211yescharacteristic
17#41×Dic73C2211yescharacteristic
18#17988896C411yescharacteristic, complemented
19#17988892C411yescharacteristic, complemented
20#17988889C411yescharacteristic, complemented
21#17988885C411yescharacteristic, complemented
22#17988880C411yescharacteristic, complemented
23#17988875C411yescharacteristic, complemented
24#17988870C411yescharacteristic, complemented
25#17988865C411yescharacteristic, complemented
26#17988716—21no—
27#17988716—21no—
28#17988711—21no—
29#17988711—21no—
30#17988645—21no—
31#17988645—21no—
32#17988637—21no—
33#17988637—21no—
34C2×#14556527C2211yescharacteristic
35C2×#14556526C2211yescharacteristic
36#17988625C2211yescharacteristic
37#17988623C2211yescharacteristic
38#17988615C2211yescharacteristic
39#17988613C2211yescharacteristic
40#17988613C2211yescharacteristic
41#17988610C2211yescharacteristic
42C2×#14556520C2211yescharacteristic
43C2×#14556518C2211yescharacteristic
44#17988583C2211yescharacteristic
45#17988554C2211yescharacteristic
46#17988545C2211yescharacteristic
47#17988543C2211yescharacteristic
48M16×Dic73C2211yescharacteristic
49C2×C8×Dic73C2211yescharacteristic
50C2×C8×Dic73C2211yescharacteristic
51C4×#11924007C2211yescharacteristic
52#17988351C2211yescharacteristic
53#17988348C2211yescharacteristic
54#17988348C2211yescharacteristic
55#17988347C2211yescharacteristic
56#17988347C2211yescharacteristic
57#17988346C2211yescharacteristic
58#17988342C2211yescharacteristic
59#17988341C2211yescharacteristic
60#424×C73C2211yescharacteristic
61C2×#104×C73C2211yescharacteristic
62#416×C73C2211yescharacteristic
63#408×C73C2211yescharacteristic
64C2×C4×C8×C73C2211yescharacteristic
65#341×C73C2211yescharacteristic
66#339×C73C2211yescharacteristic
67C2×#11924012C2311yescharacteristic
68C2×#11924012C2311yescharacteristic
69C22×#10596831C2311yescharacteristic
70C22×#10596831C2311yescharacteristic
71#14556590C2×C411yescharacteristic
72#14556590C2×C411yescharacteristic
73#14556582C2×C411yescharacteristic
74#14556582C2×C411yescharacteristic
75#14556568—21no—
76#14556568—21no—
77Q8×Dic73C2×C411yescharacteristic
78Q8×Dic73C2×C411yescharacteristic
79C4×Dic146—21no—
80C4×Dic146—21no—
81D4×Dic73—21no—
82D4×Dic73—21no—
83D4×Dic73C2×C411yescharacteristic
84D4×Dic73C2×C411yescharacteristic
85#14556550—21no—
86#14556550C2311yescharacteristic
87#14556550C2311yescharacteristic
88C2×#11924007—21no—
89C2×C4×Dic73—21no—
90C2×C4×Dic73C2311yescharacteristic
91C2×C4×Dic73C2311yescharacteristic
92#14556527—41no—
93#14556526—41no—
94#14556525C2×C411yescharacteristic
95#14556525C2×C411yescharacteristic
96#14556524C2×C411yescharacteristic
97#14556524C2×C411yescharacteristic
98#14556523C2×C411yescharacteristic
99#14556523C2×C411yescharacteristic
100#14556522C2×C411yescharacteristic
101#14556522C2×C411yescharacteristic
102#14556520—21no—
103#14556520—21no—
104#14556520—21no—
105#14556520—21no—
106#14556520—21no—
107#14556520—21no—
108#14556520—21no—
109#14556520—21no—
110#14556520C2×C411yescharacteristic
111#14556520C2×C411yescharacteristic
112#14556520C2×C411yescharacteristic
113#14556520C2×C411yescharacteristic
114#14556518—41no—
115#14556515C2×C411yescharacteristic
116#14556515C2×C411yescharacteristic
117#14556512C2×C411yescharacteristic
118#14556512C2×C411yescharacteristic
119C22⋊#10596831—21no—
120C22⋊#10596831—21no—
121#14556502—21no—
122#14556501—21no—
123#14556501—21no—
124C4×#10596831—21no—
125C8×Dic73—21no—
126C8×Dic73—21no—
127C8×Dic73—41no—
128C8×Dic73D411yescharacteristic
129C8×Dic73D411yescharacteristic
130C8×Dic73D411yescharacteristic
131C8×Dic73D411yescharacteristic
132#14556498—21no—
133C2×#41×C73C2311yescharacteristic
134C2×M16×C73C2311yescharacteristic
135C2×M16×C73C2311yescharacteristic
136C22×C8×C73C2311yescharacteristic
137C22×C8×C73C2311yescharacteristic
138#117×C73C2311yescharacteristic
139C2×C42×C73C2311yescharacteristic
140C4⋊C8×C73—21no—
141C4⋊C8×C73—21no—
142#104×C73—41no—
143#103×C73C2×C411yescharacteristic
144#103×C73C2×C411yescharacteristic
145#102×C73C2×C411yescharacteristic
146#102×C73C2×C411yescharacteristic
147C22⋊C8×C73—21no—
148C22⋊C8×C73—21no—
149C4×C8×C73—41no—
150C2×Dic146—21no—
151C2×Dic146—21no—
152C22×Dic73—21no—
153C22×Dic73—21no—
154C22×Dic73—21no—
155C22×Dic73#4111yescharacteristic
156C22×Dic73#4111yescharacteristic
157#11924012—21no—
158#11924012—21no—
159#11924012—21no—
160#11924012—21no—
161C2×#10596831—21no—
162C2×#10596831—21no—
163C2×#10596831—21no—
164C2×#10596831—21no—
165C2×#10596831—21no—
166C2×#10596831—21no—
167C2×#10596831C22×C411yescharacteristic
168C2×#10596831C22×C411yescharacteristic
169C2×#10596831C2×D411yescharacteristic
170C2×#10596831C2×D411yescharacteristic
171C4⋊Dic73—21no—
172C4⋊Dic73—21no—
173C4⋊Dic73—21no—
174C4⋊Dic73—21no—
175C4⋊Dic73C22×C411yescharacteristic
176C4⋊Dic73C22×C411yescharacteristic
177C4⋊Dic73C22×C411yescharacteristic
178C4⋊Dic73C22×C411yescharacteristic
179#11924007—24no—
180#11924007—21no—
181#11924007—21no—
182#11924007—21no—
183#11924007—21no—
184C22⋊Dic73—21no—
185C22⋊Dic73—21no—
186C22⋊Dic73—21no—
187C22⋊Dic73—21no—
188C4×Dic73—21no—
189C4×Dic73—21no—
190C4×Dic73—21no—
191C4×Dic73—21no—
192C4×Dic73—21no—
193C4×Dic73—21no—
194C4×Dic73C22×C411yescharacteristic
195C4×Dic73C22×C411yescharacteristic
196C4×Dic73C2×D411yescharacteristic
197C4×Dic73C2×D411yescharacteristic
198#41×C73—21no—
199#41×C73—21no—
200#41×C73—21no—
201#41×C73—21no—
202C2×Q8×C73C22×C411yescharacteristic
203C2×D4×C73—21no—
204C2×D4×C73C22×C411yescharacteristic
205C22×C4×C73—21no—
206C22×C4×C73—21no—
207C22×C4×C73C2411yescharacteristic
208M16×C73—21no—
209M16×C73—21no—
210M16×C73—21no—
211M16×C73—21no—
212C2×C8×C73—21no—
213C2×C8×C73—21no—
214C2×C8×C73—21no—
215C2×C8×C73—21no—
216C2×C8×C73—21no—
217C2×C8×C73—21no—
218C2×C8×C73C22×C411yescharacteristic
219C2×C8×C73C22×C411yescharacteristic
220C2×C8×C73C2×D411yescharacteristic
221C2×C8×C73C2×D411yescharacteristic
222C4⋊C4×C73C22×C411yescharacteristic
223C4⋊C4×C73C22×C411yescharacteristic
224C22⋊C4×C73—21no—
225C42×C73—21no—
226C42×C73—21no—
227C42×C73—21no—
228C23×C73—21no—
229C23×C73C2×#4111yescharacteristic
230Q8×C73—21no—
231Q8×C73—21no—
232D4×C73—21no—
233D4×C73—21no—
234D4×C73—41no—
235D4×C73—41no—
236C2×C4×C73—22no—
237C2×C4×C73—21no—
238C2×C4×C73—21no—
239C2×C4×C73—21no—
240C2×C4×C73—41no—
241C2×C4×C73—41no—
242C2×C4×C73—41no—
243C2×C4×C73—41no—
244C2×C4×C73C23×C411yescharacteristic
245C2×C4×C73C22×D411yescharacteristic
246C2×C4×C73H(5,𝔽2)11yescharacteristic
247C2×C4×C732−1+411yescharacteristic
248C2×C4×C732−1+411yescharacteristic
249C2×C4×C732−1+411yescharacteristic
250C2×Dic73—22no—
251C2×Dic73—22no—
252C2×Dic73—21no—
253C2×Dic73—21no—
254C2×Dic73—21no—
255C2×Dic73—21no—
256C2×Dic73—41no—
257C2×Dic73—42no—
258C2×Dic73—41no—
259C2×Dic73—41no—
260C2×Dic73—42no—
261C2×Dic73C4×D411yescharacteristic
262C2×Dic73C4×D411yescharacteristic
263C2×Dic73C4×D411yescharacteristic
264C2×Dic73C4×D411yescharacteristic
265Dic146—42no—
266Dic146—42no—
267C8×C73—21no—
268C8×C73—21no—
269C8×C73—21no—
270C8×C73—21no—
271C8×C73—41no—
272C8×C73—41no—
273#10596831—21no—
274#10596831—21no—
275#10596831—21no—
276#10596831—21no—
277#10596831—41no—
278#10596831—41no—
279C22×C73—21no—
280C22×C73—21no—
281C22×C73—41no—
282C22×C73—41no—
283C22×C73—41no—
284C22×C73C2×C4×D411yescharacteristic
285C22×C73#54111yescharacteristic
286C22×C73#54211yescharacteristic
287C4×C73—42no—
288C4×C73—41no—
289C4×C73—41no—
290C4×C73—41no—
291C4×C73#51911yescharacteristic
292C4×C73#52011yescharacteristic
293C4×C73#53611yescharacteristic
294C4×C73#53711yescharacteristic
295Dic73—21no—
296Dic73—21no—
297Dic73—21no—
298Dic73—42no—
299Dic73—41no—
300Dic73—41no—
301Dic73—41no—
302Dic73—41no—
303Dic73#44312yes—
304#22130—731nomaximal, Sylow
305C2×C73—21no—
306C2×C73—21no—
307C2×C73—81no—
308C2×C73#232311yescharacteristic
309C2×C73C2×#44311yescharacteristic
310C2×C73#295511yescharacteristic
311#3076—731no—
312#2960—731no—
313#2922—731no—
314#1986—731no—
315#1950—731no—
316#1945—731no—
317#1945—731no—
318#1937—731no—
319#1937D7311yescharacteristic
320#1929—731no—
321#1929—731no—
322#1922—731no—
323#1922—731no—
324#1806—731no—
325C8×C4⋊C4—731no—
326C73#2213011yescharacteristic, complemented, Sylow
327#533—731no—
328C4×#41—731no—
329#479—731no—
330#479—731no—
331#478—731no—
332#478—731no—
333#477—731no—
334#476—731no—
335#475—731no—
336#474—731no—
337#446—1461no—
338#446—1461no—
339C8×Q8—1461no—
340C8×Q8—1461no—
341#433—1461no—
342#433—1461no—
343#431—1461no—
344#431—1461no—
345C2×#108—731no—
346C2×#108—731no—
347#427—731no—
348#427—731no—
349#424—731no—
350#424—731no—
351#424—731no—
352#424C2×D7311yescharacteristic
353#423—731no—
354C2×C4⋊C8—731no—
355C2×#104—731no—
356C2×#104C2×D7311yescharacteristic
357#416—731no—
358#416C2×D7311yescharacteristic
359#408—731no—
360#408C2×D7311yescharacteristic
361#405—731no—
362#405—731no—
363C4×M16—731no—
364C2×C4×C8—731no—
365C2×C4×C8—731no—
366C2×C4×C8C2×D7311yescharacteristic
367C4×C4⋊C4—731no—
368#341—731no—
369#341C2×D7311yescharacteristic
370#339—731no—
371#339—731no—
372#339—731no—
373#339—731no—
374#339—731no—
375#339C2×D7311yescharacteristic
376#336—731no—
377#336—731no—
378C2×#41C22×D7311yescharacteristic
379C2×M16—731no—
380C2×M16—731no—
381C2×M16C22×D7311yescharacteristic
382C2×M16C22×D7311yescharacteristic
383C22×C8—731no—
384C22×C8—731no—
385C22×C8C22×D7311yescharacteristic
386C22×C8C22×D7311yescharacteristic
387C4⋊Q8—731no—
388C4⋊Q8—731no—
389#125—731no—
390#125—731no—
391C22⋊Q8—1461no—
392C22⋊Q8—1461no—
393C4×Q8—731no—
394C4×Q8—731no—
395C4×Q8—1461no—
396C4×Q8—1461no—
397C4×D4—731no—
398C4×D4—731no—
399C4×D4—1461no—
400C4×D4—1461no—
401#117—731no—
402#117—731no—
403#117—1461no—
404#117C22×D7311yescharacteristic
405C2×C4⋊C4—1461no—
406C2×C42—731no—
407C2×C42—731no—
408C2×C42—1461no—
409C2×C42C22×D7311yescharacteristic
410#108—2921no—
411#108—2921no—
412#107—731no—
413#107—731no—
414#107—731no—
415#107—731no—
416#106—731no—
417#106—731no—
418#106—731no—
419#106—731no—
420C4⋊C8—21no—
421C4⋊C8—21no—
422C4⋊C8—731no—
423C4⋊C8—731no—
424C4⋊C8—731no—
425C4⋊C8—731no—
426C4⋊C8—1461no—
427C4⋊C8—1461no—
428C4⋊C8—1461no—
429C4⋊C8—1461no—
430C4⋊C8—1461no—
431C4⋊C8—1461no—
432C4⋊C8—1461no—
433C4⋊C8—1461no—
434#104—41no—
435#104—2921no—
436#103—731no—
437#103—731no—
438#103C4×D7311yescharacteristic
439#103C4×D7311yescharacteristic
440#102—731no—
441#102—731no—
442#102C4×D7311yescharacteristic
443#102C4×D7311yescharacteristic
444C22⋊C8—21no—
445C22⋊C8—21no—
446C22⋊C8—1461no—
447C22⋊C8—1461no—
448#97—1461no—
449#97—1461no—
450#97—1461no—
451C4×C8—41no—
452C4×C8—2921no—
453C4×C8—731no—
454C4×C8—731no—
455C4×C8—731no—
456C4×C8—731no—
457C4×C8—1461no—
458C4×C8—1461no—
459C4×C8—1461no—
460#95—1461no—
461#41—21no—
462#41—21no—
463#41—21no—
464#41—21no—
465C2×Q8—1461no—
466C2×Q8—1461no—
467C2×Q8C2×C4×D7311yescharacteristic
468C2×D4—21no—
469C2×D4C2×C4×D7311yescharacteristic
470C22×C4—21no—
471C22×C4—21no—
472C22×C4—731no—
473C22×C4—731no—
474C22×C4—1461no—
475C22×C4—1461no—
476C22×C4—1461no—
477C22×C4C23×D7311yescharacteristic
478M16—21no—
479M16—21no—
480M16—21no—
481M16—21no—
482M16—1461no—
483M16—1461no—
484M16—1461no—
485M16—1461no—
486C2×C8—21no—
487C2×C8—21no—
488C2×C8—21no—
489C2×C8—21no—
490C2×C8—21no—
491C2×C8—21no—
492C2×C8—731no—
493C2×C8—731no—
494C2×C8—731no—
495C2×C8—731no—
496C2×C8—1461no—
497C2×C8—1461no—
498C2×C8—1461no—
499C2×C8—1461no—
500C2×C8—1461no—
501C2×C8—1461no—
502C2×C8C2×C4×D7311yescharacteristic
503C2×C8C2×C4×D7311yescharacteristic
504C2×C8D4×D7311yescharacteristic
505C2×C8D4×D7311yescharacteristic
506C4⋊C4—731no—
507C4⋊C4—731no—
508C4⋊C4—731no—
509C4⋊C4—731no—
510C4⋊C4—1461no—
511C4⋊C4—1461no—
512C4⋊C4—1461no—
513C4⋊C4—1461no—
514C4⋊C4—1461no—
515C4⋊C4—1461no—
516C4⋊C4—1461no—
517C4⋊C4—1461no—
518C4⋊C4—1464no—
519C4⋊C4C2×C4×D7311yescharacteristic
520C4⋊C4C2×C4×D7311yescharacteristic
521C22⋊C4—21no—
522C22⋊C4—1461no—
523C22⋊C4—1461no—
524C22⋊C4—1461no—
525C22⋊C4—1461no—
526C42—21no—
527C42—21no—
528C42—21no—
529C42—731no—
530C42—731no—
531C42—731no—
532C42—731no—
533C42—1461no—
534C42—1461no—
535C42—1461no—
536C42—1461no—
537C42—1461no—
538C42—1461no—
539C23—21no—
540C23#41×D7311yescharacteristic
541Q8—21no—
542Q8—21no—
543Q8—2922no—
544Q8—2922no—
545D4—21no—
546D4—21no—
547D4—41no—
548D4—41no—
549C2×C4—22no—
550C2×C4—21no—
551C2×C4—21no—
552C2×C4—21no—
553C2×C4—41no—
554C2×C4—41no—
555C2×C4—41no—
556C2×C4—41no—
557C2×C4—2921no—
558C2×C4—2921no—
559C2×C4—2922no—
560C2×C4—2922no—
561C2×C4—2921no—
562C2×C4—731no—
563C2×C4—731no—
564C2×C4—731no—
565C2×C4—731no—
566C2×C4—1462no—
567C2×C4—1461no—
568C2×C4—1461no—
569C2×C4—1461no—
570C2×C4—1461no—
571C2×C4—1462no—
572C2×C4C22×C4×D7311yescharacteristic
573C2×C4C2×D4×D7311yescharacteristic
574C2×C4#1455663611yescharacteristic
575C2×C4#1455663811yescharacteristic
576C2×C4#1455663811yescharacteristic
577C2×C4#1455663811yescharacteristic
578C8—21no—
579C8—21no—
580C8—21no—
581C8—21no—
582C8—41no—
583C8—41no—
584C8—2921no—
585C8—2921no—
586C8—1461no—
587C8—1461no—
588C8—1461no—
589C8—1461no—
590C22—21no—
591C22—21no—
592C22—41no—
593C22—41no—
594C22—41no—
595C22C4×D4×D7311yescharacteristic
596C22#1798919411yescharacteristic
597C22#1798920511yescharacteristic
598C4—42no—
599C4—41no—
600C4—41no—
601C4—41no—
602C4—2922no—
603C4—2921no—
604C4—2921no—
605C4—2921no—
606C4—2921no—
607C4—732no—
608C4—1461no—
609C4—1461no—
610C4—1461no—
611C4#1798906411yescharacteristic
612C4#1798906611yescharacteristic
613C4#1798915711yescharacteristic
614C4#1798916411yescharacteristic
615C2—21no—
616C2—21no—
617C2—81no—
618C2#2261260611yescharacteristic
619C2#443×D7311yescharacteristic
620C2#2261882911yescharacteristic
subgroupG/H
centerC2×C4C2×D4×D73
commutatorC2×C4×C73C23×C4
FrattiniC22×C4C23×D73
FittingC73×#1937C2
socleC22×C73C2×C4×D4
prime radicalC2×#41×C73C23
cosocleC22×C4×C73C24
hypercenter#1937D73
hypocenterC73#22130
LCSG ▷ C2×C4×C73 ▷ C2×C73 ▷ C73
UCSC1 ◁ C2×C4 ◁ C22×C8 ◁ #1937

Complements

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

NH ≅ G/Ntype
2. #22618864C2semidirect
4. #22610873C2semidirect
5. #22610705C2semidirect
6. #22610679C2semidirect
7. #22610676C2semidirect
11. #22610529C2semidirect
12. #22610528C2semidirect
14. C8×#11924007C2semidirect
18. #17988896C4semidirect
19. #17988892C4semidirect
20. #17988889C4semidirect
21. #17988885C4semidirect
22. #17988880C4semidirect
23. #17988875C4semidirect
24. #17988870C4semidirect
25. #17988865C4semidirect
326. C73#22130semidirect

Maximal subgroups

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

subgroupindex
#226197152
#226188642
#226185712
#226108732
#226107052
#226106792
#226106762
#226106382
#226105822
#226105812
#226105292
#226105282
#226099492
C8×#119240072
C73×#19372
#2213073