Finite Group Databasebeta

← #33105438

Subgroups — #33105438

Order 18688.

740 conjugacy classes of subgroups.

The subgroup lattice has 562 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C4×#14556613C211yesmaximal, characteristic
2#22618596C211yesmaximal, characteristic, complemented
3#22618496C211yesmaximal, characteristic, complemented
4#22610642C212yesmaximal, complemented
5#22610531C212yesmaximal, complemented
6#22610257C212yesmaximal
7#22610069C211yesmaximal, characteristic, complemented
8#22609952C211yesmaximal, characteristic, complemented
9#22609952C211yesmaximal, characteristic, complemented
10#22609775C212yesmaximal, complemented
11C73×#1770C211yesmaximal, characteristic, complemented
12C2×#14556613C2211yescharacteristic
13C2×#14556597C2212yes—
14C4×#11924028C2211yescharacteristic
15C4×#11924028C2211yescharacteristic
16C4×Dic292C411yescharacteristic, complemented
17C4×Dic292C411yescharacteristic, complemented
18C4×#11924017C411yescharacteristic, complemented
19C4×#11924017C411yescharacteristic, complemented
20C4×#11924017C411yescharacteristic, complemented
21C4×#11924017C411yescharacteristic, complemented
22C4×D584C411yescharacteristic, complemented
23C4×D584C411yescharacteristic, complemented
24#17988647—22no—
25#17988643—21no—
26#17988639—21no—
27#17988625C2211yescharacteristic
28#17988613C2212yes—
29#17988612—22no—
30#17988611C2211yescharacteristic
31#17988611C2211yescharacteristic
32#17988609—21no—
33C4⋊C8×D73—21no—
34C2×#14556519C2212yes—
35C2×#14556519C2212yes—
36#17988584C2211yescharacteristic
37#17988584C2211yescharacteristic
38#17988553C2212yes—
39#17988542C2212yes—
40#17988352C2212yes—
41#17988352C2212yes—
42#17988347C2212yes—
43#17988347C2212yes—
44C2×C4⋊C8×C73C2211yescharacteristic
45C2×#97×C73C2212yes—
46C2×C4×C8×C73C2211yescharacteristic
47C4×C4⋊C4×C73C2211yescharacteristic
48#336×C73C2212yes—
49C2×#11924028C2311yescharacteristic
50C2×#11924028C2311yescharacteristic
51#14556613—22no—
52#14556613—22no—
53C2×Dic292C2×C411yescharacteristic
54C2×#11924017C2×C411yescharacteristic
55C2×#11924017C2×C411yescharacteristic
56C2×D584C2×C411yescharacteristic
57#14556597—24no—
58#14556597—42no—
59C2×#11924012C2312yes—
60C2×#11924012C2312yes—
61C2×#11924011—22no—
62C2×C8×D73—22no—
63C4×Dic146C2×C411yescharacteristic
64C4×Dic146C2×C411yescharacteristic
65C4×Dic146C2×C411yescharacteristic
66C4×Dic146C2×C411yescharacteristic
67C4×D292C2×C411yescharacteristic
68C4×D292C2×C411yescharacteristic
69C4×D292C2×C411yescharacteristic
70C4×D292C2×C411yescharacteristic
71C4×C22⋊D73—21no—
72C4×C22⋊D73—21no—
73C4×C22⋊D73—21no—
74C4×C22⋊D73—21no—
75#14556550C2311yescharacteristic
76#14556550C2311yescharacteristic
77#14556548—21no—
78#14556548—21no—
79C42×D73—21no—
80C42×D73—21no—
81#14556525C2×C411yescharacteristic
82#14556525C2×C411yescharacteristic
83#14556523C2×C411yescharacteristic
84#14556523C2×C411yescharacteristic
85C4⋊#10596831—21no—
86C4⋊#10596831—21no—
87C4⋊#10596831—21no—
88C4⋊#10596831—21no—
89#14556520—22no—
90#14556520—22no—
91#14556519—24no—
92#14556519—42no—
93#14556516C2×C411yescharacteristic
94#14556516C2×C411yescharacteristic
95#14556516C2×C411yescharacteristic
96#14556516C2×C411yescharacteristic
97#14556513C2×C411yescharacteristic
98#14556513C2×C411yescharacteristic
99#14556513C2×C411yescharacteristic
100#14556513C2×C411yescharacteristic
101#14556503—22no—
102#14556503—22no—
103#14556501—22no—
104C8×Dic73—22no—
105C22×C8×C73C2311yescharacteristic
106C22×C8×C73C2311yescharacteristic
107C22×C8×C73C2312yes—
108C2×C4⋊C4×C73—21no—
109C2×C42×C73C2312yes—
110C2×C42×C73C2311yescharacteristic
111C4⋊C8×C73D411yescharacteristic
112C4⋊C8×C73D412yes—
113C4⋊C8×C73D411yescharacteristic
114#97×C73—42no—
115C4×C8×C73C2×C411yescharacteristic
116C4×C8×C73C2×C411yescharacteristic
117C4×C8×C73C2×C411yescharacteristic
118C4×C8×C73C2×C411yescharacteristic
119#95×C73—21no—
120#11924028—21no—
121#11924028—21no—
122#11924028—21no—
123#11924028—21no—
124#11924028—21no—
125#11924028—21no—
126#11924028#4112yes—
127#11924028#4112yes—
128C2×Dic146C22×C411yescharacteristic
129C2×Dic146C22×C411yescharacteristic
130C2×D292C22×C411yescharacteristic
131C2×D292C22×C411yescharacteristic
132C2×C22⋊D73—21no—
133C2×C22⋊D73—21no—
134C2×C4×D73—21no—
135C2×C4×D73—21no—
136C2×C4×D73—21no—
137C2×C4×D73—21no—
138C2×C4×D73—21no—
139C2×C4×D73—21no—
140Dic292—22no—
141#11924017—22no—
142#11924017—22no—
143D584—22no—
144#11924012—22no—
145#11924012—22no—
146#11924012—22no—
147#11924012—22no—
148#11924011—44no—
149C2×#10596831—22no—
150C2×#10596831—22no—
151C8×D73—44no—
152C4⋊Dic73C22×C411yescharacteristic
153C4⋊Dic73C22×C411yescharacteristic
154C4⋊Dic73C22×C411yescharacteristic
155C4⋊Dic73C22×C411yescharacteristic
156#11924007—21no—
157#11924007—21no—
158#11924007—21no—
159#11924007—21no—
160C22⋊Dic73—21no—
161C22⋊Dic73—21no—
162#11924005—21no—
163#11924005—21no—
164#11924005—21no—
165#11924005—21no—
166C4×Dic73—21no—
167C4×Dic73—21no—
168C4×Dic73—21no—
169C4×Dic73—21no—
170C4×Dic73—21no—
171C4×Dic73—21no—
172C22×C4×C73—22no—
173C22×C4×C73#4111yescharacteristic
174C22×C4×C73#4111yescharacteristic
175C22×C4×C73C2411yescharacteristic
176C2×C8×C73—22no—
177C2×C8×C73—21no—
178C2×C8×C73—21no—
179C2×C8×C73—21no—
180C2×C8×C73—22no—
181C2×C8×C73—21no—
182C2×C8×C73C22×C411yescharacteristic
183C2×C8×C73C22×C411yescharacteristic
184C2×C8×C73C22×C411yescharacteristic
185C2×C8×C73C22×C411yescharacteristic
186C2×C8×C73C2×D412yes—
187C2×C8×C73C2×D412yes—
188C4⋊C4×C73—42no—
189C42×C73—22no—
190C42×C73—22no—
191C42×C73C22×C411yescharacteristic
192C42×C73C22×C411yescharacteristic
193C42×C73C2×D411yescharacteristic
194C42×C73C2×D411yescharacteristic
195C23×C73C2×#4111yescharacteristic
196C22×D73—21no—
197C22×D73—21no—
198C2×C4×C73—22no—
199C2×C4×C73—41no—
200C2×C4×C73—42no—
201C2×C4×C73—42no—
202C2×C4×C73—42no—
203C2×C4×C73C4×D411yescharacteristic
204C2×C4×C73C4×D411yescharacteristic
205C2×C4×C73C4×D411yescharacteristic
206C2×C4×C73C4×D411yescharacteristic
207C2×C4×C73C23×C411yescharacteristic
208C2×C4×C73C22×D411yescharacteristic
209C2×C4×C73C2×#4111yescharacteristic
210C2×C4×C73C2×#4111yescharacteristic
211C2×C4×C73H(5,𝔽2)11yescharacteristic
212C2×C4×C732−1+411yescharacteristic
213C22⋊D73—22no—
214C22⋊D73—22no—
215C22⋊D73—41no—
216C22⋊D73—41no—
217C2×Dic73—21no—
218C2×Dic73—21no—
219C2×Dic73—21no—
220C2×Dic73—21no—
221C2×Dic73—21no—
222C2×Dic73—21no—
223D292—21no—
224D292—21no—
225D292—21no—
226D292#11712yes—
227C4×D73—22no—
228C4×D73—22no—
229C4×D73—41no—
230C4×D73—41no—
231C4×D73—42no—
232C4×D73—42no—
233C4×D73—81no—
234C4×D73—81no—
235Dic146—21no—
236Dic146—21no—
237Dic146—21no—
238Dic146#11712yes—
239C8×C73—21no—
240C8×C73—21no—
241C8×C73—21no—
242C8×C73—21no—
243C8×C73—22no—
244C8×C73—22no—
245#10596831—42no—
246#10596831—42no—
247C22×C73—21no—
248C22×C73—21no—
249C22×C73C2×C4×D411yescharacteristic
250C22×C73#55211yescharacteristic
251C22×C73#55311yescharacteristic
252C2×D73—22no—
253C2×D73—41no—
254C2×D73—41no—
255C2×D73—41no—
256C2×D73—41no—
257C2×D73—41no—
258C4×C73—21no—
259C4×C73—21no—
260C4×C73—21no—
261C4×C73—82no—
262C4×C73#44412yes—
263C4×C73C2×#11711yescharacteristic
264C4×C73#52011yescharacteristic
265C4×C73#54611yescharacteristic
266C4×C73#54811yescharacteristic
267Dic73—22no—
268Dic73—41no—
269Dic73—41no—
270Dic73—41no—
271Dic73—41no—
272Dic73—41no—
273#22782—731nomaximal, Sylow
274C2×C73—21no—
275C2×C73—21no—
276C2×C73#235211yescharacteristic
277C2×C73C2×#44411yescharacteristic
278C2×C73#295711yescharacteristic
279D73—42no—
280D73—81no—
281C4×#135—731no—
282#2925—731no—
283#2906—731no—
284#1938—732no—
285#1922—732no—
286#1876—732no—
287#1837—731no—
288#1806—731no—
289#1806—731no—
290#1770D7311yescharacteristic, complemented
291#1765—732no—
292C73#2278211yescharacteristic, complemented, Sylow
293C2×#135—731no—
294C2×#131—732no—
295C4×#41—731no—
296C4×#41—731no—
297C4×Q16—731no—
298C4×Q16—731no—
299C4×SD16—731no—
300C4×SD16—731no—
301C4×SD16—731no—
302C4×SD16—731no—
303C4×D8—731no—
304C4×D8—731no—
305#433—1462no—
306#432—1461no—
307#431—1461no—
308#427—731no—
309#424—732no—
310#424—1462no—
311#423—731no—
312#423—731no—
313#423—1461no—
314C2×C4⋊C8—1461no—
315C2×C4⋊C8C2×D7311yescharacteristic
316C2×#104—732no—
317C2×#104—732no—
318#416—731no—
319#416—731no—
320#408—732no—
321#405—732no—
322C2×#97C2×D7312yes—
323C2×C4×C8C2×D7311yescharacteristic
324C4×C4⋊C4C2×D7311yescharacteristic
325#341—732no—
326#341—732no—
327#339—732no—
328#339—732no—
329#336C2×D7312yes—
330C2×#41—731no—
331C2×#41—731no—
332#135—1462no—
333#135—1462no—
334C2×Q16—731no—
335C2×SD16—731no—
336C2×SD16—731no—
337C2×D8—731no—
338#131—2922no—
339#131—1464no—
340C2×M16—732no—
341C2×M16—732no—
342C2×M16—1462no—
343C22×C8—1462no—
344C22×C8C22×D7311yescharacteristic
345C22×C8C22×D7311yescharacteristic
346C22×C8C22×D7312yes—
347C4×Q8—731no—
348C4×Q8—731no—
349C4×Q8—731no—
350C4×Q8—731no—
351C4×D4—731no—
352C4×D4—731no—
353C4×D4—731no—
354C4×D4—731no—
355C4×D4—1461no—
356C4×D4—1461no—
357C4×D4—1461no—
358C4×D4—1461no—
359#117—731no—
360#117—731no—
361#117—1461no—
362#117—1461no—
363C2×C4⋊C4—21no—
364C2×C42—1461no—
365C2×C42—1461no—
366C2×C42C22×D7312yes—
367C2×C42C22×D7311yescharacteristic
368#107—731no—
369#107—731no—
370#106—731no—
371#106—731no—
372C4⋊C8—1462no—
373C4⋊C8—1462no—
374C4⋊C8—1461no—
375C4⋊C8—1461no—
376C4⋊C8—1461no—
377C4⋊C8—1461no—
378C4⋊C8D29211yescharacteristic
379C4⋊C8D29211yescharacteristic
380C4⋊C8D29212yes—
381#104—2922no—
382#104—1464no—
383#103—731no—
384#103—731no—
385#103—731no—
386#103—731no—
387#102—731no—
388#102—731no—
389#102—731no—
390#102—731no—
391C22⋊C8—1462no—
392C22⋊C8—1462no—
393#97—42no—
394#97—1462no—
395C4×C8—1462no—
396C4×C8C4×D7311yescharacteristic
397C4×C8C4×D7311yescharacteristic
398C4×C8C4×D7311yescharacteristic
399C4×C8C4×D7311yescharacteristic
400#95—21no—
401#41—732no—
402#41—732no—
403#41—1461no—
404#41—1461no—
405#41—1461no—
406#41—1461no—
407#41—1461no—
408#41—1461no—
409C2×Q8—731no—
410C2×Q8—731no—
411C2×D4—731no—
412C2×D4—731no—
413C2×D4—1461no—
414C2×D4—1461no—
415C22×C4—22no—
416C22×C4—1461no—
417C22×C4—1461no—
418C22×C4—1461no—
419C22×C4—1461no—
420C22×C4—1461no—
421C22×C4—1461no—
422C22×C4#1192402811yescharacteristic
423C22×C4#1192402811yescharacteristic
424C22×C4C23×D7311yescharacteristic
425Q16—1462no—
426SD16—1462no—
427SD16—1462no—
428D8—1462no—
429M16—2924no—
430M16—1462no—
431M16—1462no—
432M16—1462no—
433M16—1462no—
434C2×C8—22no—
435C2×C8—21no—
436C2×C8—21no—
437C2×C8—21no—
438C2×C8—22no—
439C2×C8—21no—
440C2×C8—2924no—
441C2×C8—1462no—
442C2×C8—1462no—
443C2×C8C2×C4×D7311yescharacteristic
444C2×C8C2×C4×D7311yescharacteristic
445C2×C8C2×C4×D7311yescharacteristic
446C2×C8C2×C4×D7311yescharacteristic
447C2×C8C2×D29212yes—
448C2×C8C2×D29212yes—
449C4⋊C4—42no—
450C4⋊C4—731no—
451C4⋊C4—731no—
452C4⋊C4—731no—
453C4⋊C4—731no—
454C4⋊C4—1461no—
455C4⋊C4—1461no—
456C4⋊C4—1461no—
457C4⋊C4—1461no—
458C22⋊C4—1461no—
459C22⋊C4—1461no—
460C22⋊C4—1461no—
461C22⋊C4—1461no—
462C22⋊C4—1461no—
463C22⋊C4—1461no—
464C42—22no—
465C42—22no—
466C42—1461no—
467C42—1461no—
468C42—1461no—
469C42—1461no—
470C42—1461no—
471C42—1461no—
472C42C2×C4×D7311yescharacteristic
473C42C2×C4×D7311yescharacteristic
474C42C2×D29211yescharacteristic
475C42C2×D29211yescharacteristic
476C23—1461no—
477C23—1461no—
478C23C2×#1192402811yescharacteristic
479Q8—732no—
480Q8—1461no—
481Q8—1461no—
482Q8—1461no—
483D4—2921no—
484D4—2921no—
485D4—732no—
486D4—1462no—
487D4—1462no—
488D4—1461no—
489D4—1461no—
490D4—1461no—
491C2×C4—22no—
492C2×C4—41no—
493C2×C4—42no—
494C2×C4—42no—
495C2×C4—42no—
496C2×C4—2922no—
497C2×C4—2922no—
498C2×C4—2921no—
499C2×C4—2921no—
500C2×C4—5841no—
501C2×C4—5841no—
502C2×C4—1461no—
503C2×C4—1461no—
504C2×C4—1461no—
505C2×C4—1461no—
506C2×C4—1461no—
507C2×C4—1461no—
508C2×C4—1462no—
509C2×C4—1462no—
510C2×C4C4×D29211yescharacteristic
511C2×C4C4×D29211yescharacteristic
512C2×C4C4×D29211yescharacteristic
513C2×C4C4×D29211yescharacteristic
514C2×C4C22×C4×D7311yescharacteristic
515C2×C4C22×D29211yescharacteristic
516C2×C4#41×D7311yescharacteristic
517C2×C4#41×D7311yescharacteristic
518C2×C4#1455663611yescharacteristic
519C2×C4#1455663911yescharacteristic
520C8—22no—
521C8—22no—
522C8—21no—
523C8—21no—
524C8—21no—
525C8—21no—
526C8—2922no—
527C8—2922no—
528C22—21no—
529C22—21no—
530C22—2921no—
531C22—2921no—
532C22—2921no—
533C22—2921no—
534C22—2921no—
535C22—1462no—
536C22C2×C4×D29211yescharacteristic
537C22#1798927811yescharacteristic
538C22#1798928711yescharacteristic
539C4—21no—
540C4—21no—
541C4—21no—
542C4—82no—
543C4—2921no—
544C4—2921no—
545C4—2921no—
546C4—2921no—
547C4—2921no—
548C4—1462no—
549C4#1798871012yes—
550C4#117×D7311yescharacteristic
551C4#1798906811yescharacteristic
552C4#1798923211yescharacteristic
553C4#1798924911yescharacteristic
554C2—21no—
555C2—21no—
556C2—2922no—
557C2—5841no—
558C2#2261286311yescharacteristic
559C2C2×#1798871011yescharacteristic
560C2#2261884911yescharacteristic
subgroupG/H
centerC2×C4C22×D292
commutatorC2×C4×C73C23×C4
FrattiniC22×C4C23×D73
FittingC73×#1770C2
socleC22×C73C2×C4×D4
prime radicalC2×#14556613C22
cosocleC22×C4×C73C24
hypercenter#1770D73
hypocenterC73#22782
LCSG ▷ C2×C4×C73 ▷ C2×C73 ▷ C73
UCSC1 ◁ C2×C4 ◁ C2×C4⋊C8 ◁ #1770

Complements

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

NH ≅ G/Ntype
2. #22618596C2semidirect
3. #22618496C2semidirect
4. #22610642C2semidirect
5. #22610531C2semidirect
7. #22610069C2semidirect
8. #22609952C2semidirect
9. #22609952C2semidirect
10. #22609775C2semidirect
11. C73×#1770C2semidirect
16. C4×Dic292C4semidirect
17. C4×Dic292C4semidirect
18. C4×#11924017C4semidirect
19. C4×#11924017C4semidirect
20. C4×#11924017C4semidirect
21. C4×#11924017C4semidirect
22. C4×D584C4semidirect
23. C4×D584C4semidirect
290. #1770D73semidirect
292. C73#22782semidirect

Maximal subgroups

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

subgroupindex
C4×#145566132
#226185962
#226184962
#226106422
#226105312
#226102572
#226100692
#226099522
#226097752
C73×#17702
#2278273