Finite Group Databasebeta

← #33069925

Subgroups — #33069925

Order 18688.

608 conjugacy classes of subgroups.

The subgroup lattice has 560 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#22620441C211yesmaximal, characteristic, complemented
2#22620093C211yesmaximal, characteristic
3#41×#10596831C211yesmaximal, characteristic
4#22609333C211yesmaximal, characteristic, complemented
5#22609328C211yesmaximal, characteristic, complemented
6#22609323C211yesmaximal, characteristic, complemented
7#22609318C211yesmaximal, characteristic, complemented
8#22609315C211yesmaximal, characteristic, complemented
9#22609311C211yesmaximal, characteristic, complemented
10#22609308C211yesmaximal, characteristic, complemented
11#22609304C211yesmaximal, characteristic, complemented
12#22608692C211yesmaximal, characteristic, complemented
13#22608642C211yesmaximal, characteristic, complemented
14#22608641C211yesmaximal, characteristic, complemented
15C73×#3116C211yesmaximal, characteristic
16#17989144C2211yescharacteristic
17#17988990C2211yescharacteristic
18#17988990C2211yescharacteristic
19#17988986—21no—
20#17988986—21no—
21#17988983C2211yescharacteristic
22#17988983C2211yescharacteristic
23#17988928—21no—
24#17988882C2211yescharacteristic
25#17988877C2211yescharacteristic
26#17988797C2211yescharacteristic
27#17988792C2211yescharacteristic
28#17988749—21no—
29Q8×#10596831C2211yescharacteristic
30Q8×#10596831C2211yescharacteristic
31#17988692—21no—
32#17988687—21no—
33D4×#10596831—21no—
34D4×#10596831—21no—
35D4×#10596831C2211yescharacteristic
36D4×#10596831C2211yescharacteristic
37#17988644C2211yescharacteristic
38#17988640—21no—
39#17988640C2211yescharacteristic
40#17988623—21no—
41#17988623—21no—
42C2×C4×#10596831—21no—
43C2×C4×#10596831C2211yescharacteristic
44#17988339C2211yescharacteristic
45#17988339C2211yescharacteristic
46#17988337C2211yescharacteristic
47#17988337C2211yescharacteristic
48#17988336C2211yescharacteristic
49#17988336C2211yescharacteristic
50#17988335C2211yescharacteristic
51#17988335C2211yescharacteristic
52#17988334C2211yescharacteristic
53#17988334C2211yescharacteristic
54#17988333C2211yescharacteristic
55#17988333C2211yescharacteristic
56#533×C73C2211yescharacteristic
57C4×#41×C73C2211yescharacteristic
58#485×C73—21no—
59#477×C73C2211yescharacteristic
60#476×C73C2211yescharacteristic
61#462×C73C2211yescharacteristic
62#461×C73C2211yescharacteristic
63#452×C73—21no—
64#441×C73—21no—
65#440×C73—21no—
66#432×C73C2211yescharacteristic
67C2×#11924018—21no—
68C2×#11924016—21no—
69C22×#10596831—21no—
70C22×#10596831—21no—
71C22×#10596831D411yescharacteristic
72C22×#10596831D411yescharacteristic
73#14556591C2311yescharacteristic
74#14556591C2311yescharacteristic
75#14556583—21no—
76#14556570—21no—
77#14556570—21no—
78C4×Dic146—21no—
79C4×Dic146—21no—
80#14556550—21no—
81#14556524—21no—
82#14556524—21no—
83#14556524—21no—
84#14556524—21no—
85#14556524—21no—
86#14556522—21no—
87#14556522—21no—
88#14556522—21no—
89#14556522—21no—
90#14556522—21no—
91C4⋊#10596831—21no—
92C4⋊#10596831—21no—
93C4⋊#10596831C2311yescharacteristic
94C4⋊#10596831C2311yescharacteristic
95C4⋊#10596831C2311yescharacteristic
96C4⋊#10596831C2311yescharacteristic
97#14556515—21no—
98#14556514—21no—
99#14556514—21no—
100#14556512—21no—
101C22⋊#10596831—21no—
102C22⋊#10596831—21no—
103C22⋊#10596831—21no—
104C22⋊#10596831—21no—
105#14556502—21no—
106C4×#10596831—21no—
107C4×#10596831—21no—
108C4×#10596831—21no—
109C4×#10596831D411yescharacteristic
110C4×#10596831D411yescharacteristic
111C4×#10596831C2311yescharacteristic
112C4×#10596831C2311yescharacteristic
113C2×#41×C73D411yescharacteristic
114C2×Q16×C73—21no—
115C2×SD16×C73—21no—
116C4⋊Q8×C73C2311yescharacteristic
117C4⋊Q8×C73C2311yescharacteristic
118#125×C73—21no—
119C22⋊Q8×C73—21no—
120C22⋊Q8×C73—21no—
121C4×Q8×C73—21no—
122C4×Q8×C73—21no—
123C4×Q8×C73D411yescharacteristic
124C4×Q8×C73C2311yescharacteristic
125C4×D4×C73—21no—
126C4×D4×C73—21no—
127C4×D4×C73D411yescharacteristic
128C4×D4×C73C2311yescharacteristic
129#117×C73—21no—
130#117×C73—21no—
131#117×C73D411yescharacteristic
132C2×C42×C73—21no—
133C2×C42×C73C2311yescharacteristic
134#107×C73—21no—
135#106×C73—21no—
136C4⋊C8×C73C2311yescharacteristic
137C4⋊C8×C73C2311yescharacteristic
138#103×C73—21no—
139#103×C73—21no—
140#103×C73—21no—
141#102×C73—21no—
142C22⋊C8×C73—21no—
143#97×C73—21no—
144C2×Dic146—21no—
145C2×Dic146—21no—
146#11924018—42no—
147#11924016—42no—
148C2×#10596831—22no—
149C2×#10596831—22no—
150C2×#10596831—21no—
151C2×#10596831—21no—
152C2×#10596831—21no—
153C2×#10596831—21no—
154C2×#10596831—42no—
155C2×#10596831—42no—
156C2×#10596831C2×D411yescharacteristic
157C2×#10596831C2×D411yescharacteristic
158C2×#10596831C2×D411yescharacteristic
159C2×#10596831C2×D411yescharacteristic
160C4⋊Dic73—21no—
161C4⋊Dic73—21no—
162C4⋊Dic73—21no—
163C4⋊Dic73—21no—
164#11924007—41no—
165#11924007—41no—
166C22⋊Dic73—41no—
167C4×Dic73—41no—
168#41×C73—22no—
169#41×C73—22no—
170C2×Q8×C73—21no—
171C2×Q8×C73—21no—
172C2×Q8×C73C2×D411yescharacteristic
173C2×D4×C73—21no—
174C2×D4×C73C2×D411yescharacteristic
175C22×C4×C73—21no—
176C22×C4×C73—21no—
177C22×C4×C73—21no—
178C22×C4×C73C2×D411yescharacteristic
179C22×C4×C73C2×D411yescharacteristic
180C22×C4×C73#4111yescharacteristic
181Q16×C73—42no—
182SD16×C73—42no—
183C2×C8×C73—21no—
184C2×C8×C73—21no—
185C4⋊C4×C73—21no—
186C4⋊C4×C73—21no—
187C4⋊C4×C73—21no—
188C4⋊C4×C73—21no—
189C4⋊C4×C73—21no—
190C4⋊C4×C73—21no—
191C4⋊C4×C73—41no—
192C4⋊C4×C73—41no—
193C4⋊C4×C73C2×D411yescharacteristic
194C4⋊C4×C73C2×D411yescharacteristic
195C22⋊C4×C73—21no—
196C22⋊C4×C73—21no—
197C22⋊C4×C73—21no—
198C22⋊C4×C73—41no—
199C42×C73—21no—
200C42×C73—21no—
201C42×C73—21no—
202C42×C73—41no—
203C42×C73C2×D411yescharacteristic
204C42×C73C2×D411yescharacteristic
205C42×C73#4111yescharacteristic
206C42×C73C2411yescharacteristic
207C23×C73—21no—
208C23×C73#12111yescharacteristic
209Q8×C73—21no—
210Q8×C73—41no—
211Q8×C73—41no—
212Q8×C73—81no—
213Q8×C73#13512yes—
214D4×C73—24no—
215D4×C73—21no—
216D4×C73#13512yes—
217C2×C4×C73—24no—
218C2×C4×C73—21no—
219C2×C4×C73—21no—
220C2×C4×C73—21no—
221C2×C4×C73—21no—
222C2×C4×C73—21no—
223C2×C4×C73—21no—
224C2×C4×C73—21no—
225C2×C4×C73—41no—
226C2×C4×C73—42no—
227C2×C4×C73—41no—
228C2×C4×C73—41no—
229C2×C4×C73—41no—
230C2×C4×C73—41no—
231C2×C4×C73#12111yescharacteristic
232C2×C4×C73#12111yescharacteristic
233C2×C4×C73#12111yescharacteristic
234C2×C4×C73C22×D411yescharacteristic
235C2×C4×C73C22×D411yescharacteristic
236C2×C4×C73C2×#4111yescharacteristic
237C2×Dic73—41no—
238C2×Dic73—41no—
239Dic146—41no—
240Dic146—41no—
241Dic146—81no—
242C8×C73—41no—
243C8×C73—41no—
244#10596831—21no—
245#10596831—21no—
246#10596831—41no—
247#10596831—41no—
248#10596831—41no—
249#10596831—41no—
250#10596831#13712yes—
251#10596831#13712yes—
252C22×C73—21no—
253C22×C73—21no—
254C22×C73—22no—
255C22×C73—21no—
256C22×C73—41no—
257C22×C73—41no—
258C22×C73C2×#12111yescharacteristic
259C4×C73—21no—
260C4×C73—21no—
261C4×C73—22no—
262C4×C73—21no—
263C4×C73—21no—
264C4×C73—41no—
265C4×C73—41no—
266C4×C73—81no—
267C4×C73—81no—
268C4×C73C2×#13511yescharacteristic
269C4×C73C2×#13511yescharacteristic
270C4×C73C2×#13711yescharacteristic
271C4×C73C2×#13711yescharacteristic
272Dic73—81no—
273Dic73—81no—
274#19273—731nomaximal, Sylow
275C2×C73—41no—
276C2×C73—42no—
277C2×C73#303911yescharacteristic
278C2×C73#305011yescharacteristic
279C2×C73#306711yescharacteristic
280#3160—731no—
281#3116—731no—
282#3116D7311yescharacteristic
283C8×#41—731no—
284#1676—731no—
285#1675—731no—
286#1674—731no—
287#1673—731no—
288#1672—731no—
289#1672—731no—
290#1671—731no—
291#1671—731no—
292#1568—731no—
293#1554—731no—
294#1554—731no—
295C73#1927311yescharacteristic, complemented, Sylow
296#533—731no—
297#533C2×D7311yescharacteristic
298C4×#41C2×D7311yescharacteristic
299#500—731no—
300#500—731no—
301#499—1461no—
302#499—1461no—
303#498—731no—
304#498—731no—
305#485—21no—
306#485—1461no—
307#477—731no—
308#477C2×D7311yescharacteristic
309#476—731no—
310#476C2×D7311yescharacteristic
311#462—731no—
312#462C2×D7311yescharacteristic
313#461—731no—
314#461C2×D7311yescharacteristic
315#452—21no—
316#452—1461no—
317C8×Q8—731no—
318C8×Q8—731no—
319#441—21no—
320#441—1461no—
321#440—21no—
322#440—1461no—
323C8×D4—731no—
324C8×D4—731no—
325C8×D4—1461no—
326C8×D4—1461no—
327#432—731no—
328#432C2×D7311yescharacteristic
329#431—731no—
330#431—1461no—
331#427—1461no—
332#427—1461no—
333C2×C4×C8—731no—
334C2×C4×C8—1461no—
335#335—731no—
336#335—731no—
337#334—731no—
338#334—731no—
339#333—731no—
340#333—731no—
341#333—731no—
342#333—731no—
343#332—731no—
344#332—731no—
345#332—731no—
346#332—731no—
347C2×#41C22⋊D7311yescharacteristic
348C2×Q16—21no—
349C2×Q16—1461no—
350C2×SD16—21no—
351C2×SD16—1461no—
352C22×C8—731no—
353C22×C8—731no—
354C22×C8—1461no—
355C22×C8—1461no—
356C4⋊Q8—731no—
357C4⋊Q8—731no—
358C4⋊Q8C22×D7311yescharacteristic
359C4⋊Q8C22×D7311yescharacteristic
360#125—21no—
361#125—1461no—
362C22⋊Q8—21no—
363C22⋊Q8—21no—
364C22⋊Q8—1461no—
365C22⋊Q8—1461no—
366C4×Q8—21no—
367C4×Q8—21no—
368C4×Q8—1461no—
369C4×Q8—1461no—
370C4×Q8C22⋊D7311yescharacteristic
371C4×Q8C22×D7311yescharacteristic
372C4×D4—21no—
373C4×D4—21no—
374C4×D4C22⋊D7311yescharacteristic
375C4×D4C22×D7311yescharacteristic
376#117—21no—
377#117—21no—
378#117—1461no—
379#117C22⋊D7311yescharacteristic
380C2×C42—21no—
381C2×C42C22×D7311yescharacteristic
382#107—21no—
383#107—1461no—
384#107—1461no—
385#107—1461no—
386#107—1461no—
387#107—1461no—
388#106—21no—
389#106—1461no—
390#106—1461no—
391#106—1461no—
392#106—1461no—
393#106—1461no—
394C4⋊C8—731no—
395C4⋊C8—731no—
396C4⋊C8—731no—
397C4⋊C8—731no—
398C4⋊C8—1461no—
399C4⋊C8—1461no—
400C4⋊C8C22×D7311yescharacteristic
401C4⋊C8C22×D7311yescharacteristic
402#103—21no—
403#103—21no—
404#103—21no—
405#103—1461no—
406#103—1461no—
407#103—1461no—
408#102—21no—
409#102—1461no—
410C22⋊C8—21no—
411C22⋊C8—1461no—
412C22⋊C8—1461no—
413C22⋊C8—1461no—
414C22⋊C8—1461no—
415#97—21no—
416#97—1461no—
417C4×C8—731no—
418C4×C8—731no—
419C4×C8—731no—
420C4×C8—731no—
421C4×C8—1461no—
422C4×C8—1461no—
423C4×C8—1461no—
424#41—22no—
425#41—22no—
426C2×Q8—21no—
427C2×Q8—21no—
428C2×Q8—1461no—
429C2×Q8—1461no—
430C2×Q8C2×C22⋊D7311yescharacteristic
431C2×D4—21no—
432C2×D4C2×C22⋊D7311yescharacteristic
433C22×C4—21no—
434C22×C4—21no—
435C22×C4—21no—
436C22×C4D4×D7311yescharacteristic
437C22×C4C2×C22⋊D7311yescharacteristic
438C22×C4#1192402711yescharacteristic
439Q16—42no—
440Q16—2922no—
441SD16—42no—
442SD16—2922no—
443C2×C8—21no—
444C2×C8—21no—
445C2×C8—2922no—
446C2×C8—2922no—
447C2×C8—731no—
448C2×C8—731no—
449C2×C8—731no—
450C2×C8—731no—
451C2×C8—1461no—
452C2×C8—1461no—
453C2×C8—1461no—
454C2×C8—1461no—
455C2×C8—1462no—
456C2×C8—1462no—
457C4⋊C4—21no—
458C4⋊C4—21no—
459C4⋊C4—21no—
460C4⋊C4—21no—
461C4⋊C4—21no—
462C4⋊C4—21no—
463C4⋊C4—41no—
464C4⋊C4—41no—
465C4⋊C4—2921no—
466C4⋊C4—2921no—
467C4⋊C4—1461no—
468C4⋊C4—1461no—
469C4⋊C4—1461no—
470C4⋊C4—1461no—
471C4⋊C4C2×C22⋊D7311yescharacteristic
472C4⋊C4C2×C22⋊D7311yescharacteristic
473C22⋊C4—21no—
474C22⋊C4—21no—
475C22⋊C4—21no—
476C22⋊C4—41no—
477C22⋊C4—2921no—
478C42—21no—
479C42—21no—
480C42—21no—
481C42—41no—
482C42—2921no—
483C42D4×D7311yescharacteristic
484C42C2×C22⋊D7311yescharacteristic
485C42#1192402711yescharacteristic
486C42C23×D7311yescharacteristic
487C23—21no—
488C23#1455656311yescharacteristic
489Q8—21no—
490Q8—41no—
491Q8—41no—
492Q8—81no—
493Q8—2921no—
494Q8—2921no—
495Q8—5841no—
496Q8#1455661112yes—
497D4—24no—
498D4—21no—
499D4#1455661112yes—
500C2×C4—24no—
501C2×C4—21no—
502C2×C4—21no—
503C2×C4—21no—
504C2×C4—21no—
505C2×C4—21no—
506C2×C4—21no—
507C2×C4—21no—
508C2×C4—41no—
509C2×C4—42no—
510C2×C4—41no—
511C2×C4—41no—
512C2×C4—41no—
513C2×C4—41no—
514C2×C4—2921no—
515C2×C4—2921no—
516C2×C4#1455656311yescharacteristic
517C2×C4#1455656311yescharacteristic
518C2×C4#1455656311yescharacteristic
519C2×C4C2×D4×D7311yescharacteristic
520C2×C4C22×C22⋊D7311yescharacteristic
521C2×C4C2×#1192402711yescharacteristic
522C8—41no—
523C8—41no—
524C8—2921no—
525C8—2921no—
526C8—2921no—
527C8—2921no—
528C8—732no—
529C8—732no—
530C8—1461no—
531C8—1461no—
532C22—21no—
533C22—21no—
534C22—21no—
535C22—22no—
536C22—41no—
537C22—41no—
538C22C2×#1455656311yescharacteristic
539C4—21no—
540C4—21no—
541C4—21no—
542C4—21no—
543C4—22no—
544C4—41no—
545C4—41no—
546C4—81no—
547C4—81no—
548C4—5841no—
549C4—5841no—
550C4C2×#1455661111yescharacteristic
551C4C2×#1455661111yescharacteristic
552C4#137×D7311yescharacteristic
553C4#137×D7311yescharacteristic
554C2—41no—
555C2—42no—
556C2#2261941611yescharacteristic
557C2#2261950911yescharacteristic
558C2#2261964811yescharacteristic
subgroupG/H
centerC22C2×#14556563
commutator, cosocleC42×C73C24
FrattiniC42C23×D73
FittingC73×#3116C2
socleC22×C73C2×#121
prime radicalC2×#41×C73D4
hypercenter#3116D73
hypocenterC73#19273
LCSG ▷ C42×C73 ▷ C22×C73 ▷ C73
UCSC1 ◁ C22 ◁ C2×C42 ◁ #3116

Complements

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

NH ≅ G/Ntype
1. #22620441C2semidirect
4. #22609333C2semidirect
5. #22609328C2semidirect
6. #22609323C2semidirect
7. #22609318C2semidirect
8. #22609315C2semidirect
9. #22609311C2semidirect
10. #22609308C2semidirect
11. #22609304C2semidirect
12. #22608692C2semidirect
13. #22608642C2semidirect
14. #22608641C2semidirect
295. C73#19273semidirect

Maximal subgroups

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

subgroupindex
#226204412
#226200932
#41×#105968312
#226093332
#226093282
#226093232
#226093182
#226093152
#226093112
#226093082
#226093042
#226086922
#226086422
#226086412
C73×#31162
#1927373