Finite Group Databasebeta

← #34021836

Subgroups — #34021836

Order 18688.

790 conjugacy classes of subgroups.

The subgroup lattice has 568 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#22626574C211yesmaximal, characteristic
2#22626496C211yesmaximal, characteristic
3#22625999C212yesmaximal, complemented
4#22625998C212yesmaximal, complemented
5#22625997C212yesmaximal, complemented
6#22625996C212yesmaximal, complemented
7C2×#17989463C211yesmaximal, characteristic
8#22625611C211yesmaximal, characteristic
9#22625611C211yesmaximal, characteristic
10#22625596C211yesmaximal, characteristic
11#427×D73C211yesmaximal, characteristic
12C22×#11924038C2211yescharacteristic
13C22×#11924038C2211yescharacteristic
14C2×D73⋊M16C2211yescharacteristic
15C2×#14556670C2211yescharacteristic
16#17989594C2212yes—
17#17989594C2212yes—
18#17989593—22no—
19#17989592—21no—
20#17989591—21no—
21#17989552C2212yes—
22#17989552C2212yes—
23#17989551—22no—
24C4×#11924038—21no—
25C4×#11924038—21no—
26#17989463C2212yes—
27#17989463C2212yes—
28#17989457C2212yes—
29#17989457C2212yes—
30#17989457C2212yes—
31#17989457C2212yes—
32#17989452C2212yes—
33#17989452C2212yes—
34C22×C8×D73C2211yescharacteristic
35#117×D73C2211yescharacteristic
36#117×D73C2211yescharacteristic
37#17988625C411yescharacteristic, complemented
38#17988624C2211yescharacteristic
39#17988624C2211yescharacteristic
40#17988623C411yescharacteristic, complemented
41#17988623C411yescharacteristic, complemented
42#17988622C2211yescharacteristic
43#17988622C2211yescharacteristic
44#107×D73C411yescharacteristic, complemented
45#107×D73C411yescharacteristic, complemented
46#106×D73C411yescharacteristic, complemented
47#106×D73C411yescharacteristic, complemented
48#427×C73C411yescharacteristic, complemented
49C2×#11924038—21no—
50C2×#11924038—21no—
51C2×#11924038—22no—
52C2×#11924038—22no—
53C2×#11924038—22no—
54C2×#11924038D411yescharacteristic
55C2×#11924038D411yescharacteristic
56C2×#11924038Q814yes—
57C2×#11924038C2312yes—
58C2×#11924038C2312yes—
59D73⋊M16—22no—
60D73⋊M16—22no—
61C2×#11924037D412yes—
62C2×#11924036C2312yes—
63C22×#10596833—21no—
64C22×#10596833—21no—
65#14556670—22no—
66#14556670—22no—
67C2×C4⋊#64408—21no—
68C2×C22⋊#64408—21no—
69C2×C4×#64408C2312yes—
70#14556658—22no—
71#14556658—22no—
72C4⋊#10596833—22no—
73C4⋊#10596833—22no—
74#14556646—22no—
75#14556646—22no—
76C4×#10596833—22no—
77C4×#10596833—22no—
78C22×C4×D73C2311yescharacteristic
79C22×#10596831C2×C411yescharacteristic
80C2×C8×D73—22no—
81C2×C8×D73—22no—
82C2×C8×D73C2×C411yescharacteristic
83C2×C8×D73C2×C411yescharacteristic
84C2×C8×D73C2311yescharacteristic
85C2×C8×D73C2311yescharacteristic
86#14556550C2×C411yescharacteristic
87#14556550C2×C411yescharacteristic
88#14556549C2311yescharacteristic
89#14556549C2311yescharacteristic
90#14556549C2311yescharacteristic
91#14556549C2311yescharacteristic
92#14556548—21no—
93#14556548—21no—
94#14556547—21no—
95#14556547—21no—
96C4⋊C4×D73C2×C411yescharacteristic
97C4⋊C4×D73C2×C411yescharacteristic
98C4⋊C4×D73C2×C411yescharacteristic
99C4⋊C4×D73C2×C411yescharacteristic
100C22⋊C4×D73—21no—
101C22⋊C4×D73—21no—
102C42×D73—21no—
103C42×D73—21no—
104#14556525C2×C411yescharacteristic
105#14556525C2×C411yescharacteristic
106#14556524C2×C411yescharacteristic
107#14556524C2×C411yescharacteristic
108#14556524C2×C411yescharacteristic
109#14556524C2×C411yescharacteristic
110#14556523C2×C411yescharacteristic
111#14556523C2×C411yescharacteristic
112#14556522C2×C411yescharacteristic
113#14556522C2×C411yescharacteristic
114#14556522C2×C411yescharacteristic
115#14556522C2×C411yescharacteristic
116C22×C8×C73C2×C411yescharacteristic
117#117×C73C2×C411yescharacteristic
118#117×C73C2×C411yescharacteristic
119#107×C73C2×C411yescharacteristic
120#107×C73C2×C411yescharacteristic
121#106×C73C2×C411yescharacteristic
122#106×C73C2×C411yescharacteristic
123C22×#64408—22no—
124#11924038—24no—
125#11924038—22no—
126#11924038—22no—
127#11924038—22no—
128#11924038—22no—
129#11924038—22no—
130#11924038—42no—
131#11924038C2×Q814yes—
132#11924037—24no—
133#11924036—22no—
134#11924036—22no—
135C2×#10596833—24no—
136C2×#10596833—22no—
137C2×#10596833—22no—
138C2×#10596833—42no—
139C2×#10596833C2×D412yes—
140C2×#10596833C2×D412yes—
141C4⋊#64408—42no—
142C22⋊#64408—42no—
143C4×#64408—22no—
144C4×#64408—22no—
145C23×D73#4111yescharacteristic
146C22×Dic73C2×D411yescharacteristic
147C2×C4×D73—21no—
148C2×C4×D73—21no—
149C2×C4×D73—21no—
150C2×C4×D73—21no—
151C2×C4×D73C22×C411yescharacteristic
152C2×C4×D73C2×D411yescharacteristic
153C2×C4×D73C2×Q812yes—
154C2×C4×D73#4112yes—
155C2×C4×D73#4112yes—
156C2×C4×D73#4112yes—
157C2×C4×D73#4111yescharacteristic
158C2×C4×D73C2411yescharacteristic
159C2×#10596831—22no—
160C2×#10596831C22×C411yescharacteristic
161C2×#10596831C22×C411yescharacteristic
162C8×D73—22no—
163C8×D73—22no—
164C8×D73—21no—
165C8×D73—21no—
166C8×D73—21no—
167C8×D73—21no—
168C4⋊Dic73C22×C411yescharacteristic
169C4⋊Dic73C22×C411yescharacteristic
170C4⋊Dic73C22×C411yescharacteristic
171C4⋊Dic73C22×C411yescharacteristic
172#11924007—21no—
173#11924007—21no—
174#11924007—21no—
175#11924007—21no—
176C22⋊Dic73—21no—
177C22⋊Dic73—21no—
178#11924005—21no—
179#11924005—21no—
180#11924005—21no—
181#11924005—21no—
182C4×Dic73—21no—
183C4×Dic73—21no—
184C4×Dic73—21no—
185C4×Dic73—21no—
186C4×Dic73—21no—
187C4×Dic73—21no—
188C22×C4×C73C22×C411yescharacteristic
189C2×C8×C73—22no—
190C2×C8×C73C22×C411yescharacteristic
191C2×C8×C73C22×C411yescharacteristic
192C4⋊C4×C73C22×C411yescharacteristic
193C4⋊C4×C73C22×C411yescharacteristic
194C4⋊C4×C73C22×C411yescharacteristic
195C4⋊C4×C73C22×C411yescharacteristic
196C22⋊C4×C73—21no—
197C22⋊C4×C73—21no—
198C42×C73—21no—
199C42×C73—21no—
200C23×C73C4×D411yescharacteristic
201C22×D73—41no—
202C22×D73—41no—
203C22×D73C4×D411yescharacteristic
204C22×D73#12512yes—
205C22×D73#12612yes—
206C22×D73C2×#4111yescharacteristic
207C2×#64408—42no—
208C2×#64408—42no—
209C2×#64408—42no—
210C2×C4×C73—21no—
211C2×C4×C73—21no—
212C2×C4×C73—21no—
213C2×C4×C73—21no—
214C2×C4×C73#11712yes—
215C2×C4×C73C4×D411yescharacteristic
216C2×C4×C73C4×Q812yes—
217C2×C4×C73C23×C411yescharacteristic
218C2×Dic73—21no—
219C2×Dic73—21no—
220C2×Dic73—21no—
221C2×Dic73—21no—
222C2×Dic73C4×D411yescharacteristic
223C2×Dic73C22⋊Q812yes—
224C2×Dic73#12312yes—
225C2×Dic73C22×D411yescharacteristic
226C4×D73—81no—
227C4×D73—81no—
228C4×D73—81no—
229C4×D73—81no—
230C4×D73#11712yes—
231C4×D73C4×Q812yes—
232C4×D73C22⋊Q812yes—
233C4×D73#12312yes—
234C4×D73#12512yes—
235C4×D73#12612yes—
236C4×D73C22×Q811yescharacteristic
237C4×D73C2×#4111yescharacteristic
238C4×D73C2×#4111yescharacteristic
239C4×D73C2×#4111yescharacteristic
240#10596833—24no—
241#10596833—22no—
242#10596833—22no—
243#10596833—42no—
244C8×C73—21no—
245C8×C73—21no—
246#10596831—21no—
247#10596831—21no—
248C22×C73—21no—
249C22×C73—21no—
250C22×C73#38712yes—
251C22×C73C2×C4×D411yescharacteristic
252C2×D73—21no—
253C2×D73—21no—
254C2×D73—21no—
255C2×D73—21no—
256C2×D73—42no—
257C2×D73—42no—
258C2×D73#38712yes—
259C2×D73C2×#12511yescharacteristic
260C2×D73C2×#12611yescharacteristic
261#64408—82no—
262C4×C73—41no—
263C4×C73—41no—
264C4×C73—41no—
265C4×C73—41no—
266C4×C73#38712yes—
267C4×C73C2×#11711yescharacteristic
268C4×C73C2×C4×Q811yescharacteristic
269Dic73—41no—
270Dic73—41no—
271Dic73—41no—
272Dic73—41no—
273Dic73#38712yes—
274Dic73C2×C22⋊Q811yescharacteristic
275Dic73C2×#12311yescharacteristic
276#21043—731nomaximal, Sylow
277C2×C73—22no—
278C2×C73C2×#38711yescharacteristic
279C2×C73#295711yescharacteristic
280C2×C73#295711yescharacteristic
281D73—21no—
282D73—21no—
283D73—22no—
284C2×#427—731no—
285#2905—731no—
286#2873—731no—
287#1922—732no—
288#1922—732no—
289#1922—732no—
290#1922—732no—
291C2×#341—731no—
292#1733—731no—
293#1733—731no—
294#1729—731no—
295C73#2104311yescharacteristic, complemented, Sylow
296C22×M16—731no—
297C22×M16—731no—
298C22×M16—731no—
299C23×C8—731no—
300C2×#117—731no—
301C2×#117—731no—
302C2×#117—731no—
303#427—731no—
304#427—731no—
305#427—731no—
306#427—731no—
307#427—731no—
308#427—731no—
309#427—731no—
310#427#6440811yescharacteristic, complemented
311C2×#107—731no—
312C2×#107—731no—
313C2×#106—731no—
314C2×#106—731no—
315#424—732no—
316#424—732no—
317#424—1462no—
318#423—1461no—
319#423—1461no—
320#405—732no—
321#405—732no—
322#405—1462no—
323C4×M16—1461no—
324C4×M16—1461no—
325#341—732no—
326#341—732no—
327#339—732no—
328#339—732no—
329#339—732no—
330#339—732no—
331#336—732no—
332#336—732no—
333C23×C4—731no—
334C2×M16—732no—
335C2×M16—731no—
336C2×M16—731no—
337C2×M16—734no—
338C2×M16—732no—
339C2×M16—732no—
340C2×M16—1462no—
341C2×M16—1462no—
342C2×M16—1461no—
343C2×M16—1462no—
344C2×M16—1462no—
345C2×M16—1462no—
346C2×M16—1461no—
347C22×C8—732no—
348C22×C8—731no—
349C22×C8—731no—
350C22×C8—731no—
351C22×C8—731no—
352C22×C8—731no—
353C22×C8—1462no—
354C22×C8—1462no—
355C22×C8—1461no—
356C22×C8—1461no—
357C22×C8C2×#6440811yescharacteristic
358#117—731no—
359#117—731no—
360#117—731no—
361#117—731no—
362#117—731no—
363#117—731no—
364#117—1461no—
365#117—1461no—
366#117—1461no—
367#117—1461no—
368#117—1462no—
369#117—1462no—
370#117C2×#6440811yescharacteristic
371#117C2×#6440811yescharacteristic
372C2×C4⋊C4—731no—
373C2×C4⋊C4—731no—
374C2×C4⋊C4—731no—
375C2×C4⋊C4—731no—
376C2×C4⋊C4—1461no—
377C2×C22⋊C4—1461no—
378C2×C22⋊C4—1461no—
379C2×C22⋊C4—1461no—
380C2×C42—732no—
381C2×C42—1461no—
382C2×C42—1461no—
383#107—731no—
384#107—731no—
385#107—731no—
386#107—731no—
387#107—731no—
388#107—731no—
389#107C2×#6440811yescharacteristic
390#107C2×#6440811yescharacteristic
391#106—731no—
392#106—731no—
393#106—731no—
394#106—731no—
395#106—731no—
396#106—731no—
397#106C2×#6440811yescharacteristic
398#106C2×#6440811yescharacteristic
399C4⋊C8—1462no—
400C4⋊C8—1462no—
401C4⋊C8—1462no—
402C4⋊C8—1462no—
403#97—1462no—
404#97—1462no—
405C4×C8—1462no—
406C4×C8—1462no—
407C24—731no—
408C22×C4—731no—
409C22×C4—732no—
410C22×C4—732no—
411C22×C4—731no—
412C22×C4—731no—
413C22×C4—732no—
414C22×C4—732no—
415C22×C4—731no—
416C22×C4—731no—
417C22×C4—1461no—
418C22×C4—1461no—
419C22×C4—1461no—
420C22×C4—1461no—
421C22×C4—1462no—
422C22×C4C22×#6440811yescharacteristic
423M16—2922no—
424M16—734no—
425M16—1462no—
426M16—1462no—
427M16—1464no—
428M16—1462no—
429M16—1462no—
430M16—1462no—
431M16—1464no—
432C2×C8—22no—
433C2×C8—2922no—
434C2×C8—732no—
435C2×C8—732no—
436C2×C8—731no—
437C2×C8—731no—
438C2×C8—1461no—
439C2×C8—1461no—
440C2×C8—1461no—
441C2×C8—1461no—
442C2×C8—1462no—
443C2×C8—1462no—
444C2×C8—1462no—
445C2×C8—1462no—
446C2×C8—1464no—
447C2×C8—1462no—
448C2×C8—1462no—
449C2×C8—1462no—
450C2×C8C22×#6440811yescharacteristic
451C2×C8C22×#6440811yescharacteristic
452C4⋊C4—2922no—
453C4⋊C4—731no—
454C4⋊C4—731no—
455C4⋊C4—731no—
456C4⋊C4—731no—
457C4⋊C4—1461no—
458C4⋊C4—1461no—
459C4⋊C4—1461no—
460C4⋊C4—1461no—
461C4⋊C4C22×#6440811yescharacteristic
462C4⋊C4C22×#6440811yescharacteristic
463C4⋊C4C22×#6440811yescharacteristic
464C4⋊C4C22×#6440811yescharacteristic
465C22⋊C4—21no—
466C22⋊C4—21no—
467C22⋊C4—2922no—
468C22⋊C4—1461no—
469C22⋊C4—1461no—
470C22⋊C4—1461no—
471C22⋊C4—1461no—
472C22⋊C4—1461no—
473C22⋊C4—1461no—
474C42—21no—
475C42—21no—
476C42—1461no—
477C42—1461no—
478C42—1461no—
479C42—1461no—
480C42—1462no—
481C42—1462no—
482C42—1461no—
483C42—1461no—
484C23—2921no—
485C23—2921no—
486C23—732no—
487C23—732no—
488C23—731no—
489C23—731no—
490C23D4×#6440811yescharacteristic
491C2×C4—21no—
492C2×C4—21no—
493C2×C4—21no—
494C2×C4—21no—
495C2×C4—2922no—
496C2×C4—2922no—
497C2×C4—2922no—
498C2×C4—5841no—
499C2×C4—5841no—
500C2×C4—5841no—
501C2×C4—5841no—
502C2×C4—731no—
503C2×C4—731no—
504C2×C4—732no—
505C2×C4—732no—
506C2×C4—731no—
507C2×C4—731no—
508C2×C4—732no—
509C2×C4—731no—
510C2×C4—731no—
511C2×C4—732no—
512C2×C4—732no—
513C2×C4—732no—
514C2×C4—732no—
515C2×C4—732no—
516C2×C4—1461no—
517C2×C4—1461no—
518C2×C4—1461no—
519C2×C4—1461no—
520C2×C4#1455667012yes—
521C2×C4D4×#6440811yescharacteristic
522C2×C4Q8×#6440812yes—
523C2×C4C23×#6440811yescharacteristic
524C8—21no—
525C8—21no—
526C8—2922no—
527C8—1462no—
528C8—1462no—
529C8—1464no—
530C8—1461no—
531C8—1461no—
532C22—21no—
533C22—21no—
534C22—2922no—
535C22—2922no—
536C22—731no—
537C22—731no—
538C22—732no—
539C22—1461no—
540C22—1461no—
541C22—1461no—
542C22—1461no—
543C22#1798953612yes—
544C22C2×D4×#6440811yescharacteristic
545C4—41no—
546C4—41no—
547C4—41no—
548C4—41no—
549C4—2921no—
550C4—2921no—
551C4—2921no—
552C4—2921no—
553C4—5842no—
554C4—731no—
555C4—731no—
556C4—732no—
557C4#1798953612yes—
558C4C2×#1455667011yescharacteristic
559C4C2×Q8×#6440811yescharacteristic
560C2—22no—
561C2—1461no—
562C2—1461no—
563C2—1462no—
564C2C2×#1798953611yescharacteristic
565C2#2262670811yescharacteristic
566C2#2262671011yescharacteristic
subgroupG/H
centerC22C2×D4×#64408
commutatorC2×C4×C73C23×C4
FrattiniC2×C4C23×#64408
Fitting#427×C73C4
socleC22×C73C2×C4×D4
prime radicalC23×D73#41
cosocleC2×C4×D73C24
hypercenter#427#64408
hypocenterC73#21043
LCSG ▷ C2×C4×C73 ▷ C2×C73 ▷ C73
UCSC1 ◁ C22 ◁ C22×C4 ◁ #427

Complements

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

NH ≅ G/Ntype
3. #22625999C2semidirect
4. #22625998C2semidirect
5. #22625997C2semidirect
6. #22625996C2semidirect
37. #17988625C4semidirect
40. #17988623C4semidirect
41. #17988623C4semidirect
44. #107×D73C4semidirect
45. #107×D73C4semidirect
46. #106×D73C4semidirect
47. #106×D73C4semidirect
48. #427×C73C4semidirect
295. C73#21043semidirect
310. #427#64408semidirect

Maximal subgroups

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

subgroupindex
#226265742
#226264962
#226259992
#226259982
#226259972
#226259962
C2×#179894632
#226256112
#226255962
#427×D732
#2104373