Finite Group Databasebeta

← #33242881

Subgroups — #33242881

Order 18688.

546 conjugacy classes of subgroups.

The subgroup lattice has 508 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#22622300C211yesmaximal, characteristic, complemented
2#22612162C211yesmaximal, characteristic, complemented
3#22612136C211yesmaximal, characteristic, complemented
4#22612109C211yesmaximal, characteristic, complemented
5#22612095C211yesmaximal, characteristic, complemented
6#22612025C211yesmaximal, characteristic, complemented
7#22612020C211yesmaximal, characteristic, complemented
8#22611990C211yesmaximal, characteristic, complemented
9#22611871C211yesmaximal, characteristic, complemented
10#22611865C211yesmaximal, characteristic, complemented
11C4×#14556537C211yesmaximal, characteristic, complemented
12D4×#11924003C211yesmaximal, characteristic, complemented
13#22611638C211yesmaximal, characteristic, complemented
14C2×#17988391C211yesmaximal, characteristic, complemented
15C73×#3285C211yesmaximal, characteristic
16#17989277—21no—
17C2×#14556537C2211yescharacteristic
18C2×#14556537C2211yescharacteristic
19C2×#14556535C2211yescharacteristic
20C22×#11924003C2211yescharacteristic
21C22×#11924003C2211yescharacteristic
22#17988989C2211yescharacteristic
23#17988929—21no—
24#17988918—21no—
25#17988895—21no—
26#17988828C2211yescharacteristic
27#17988828C2211yescharacteristic
28C4×Dic292C2211yescharacteristic
29#17988625C2211yescharacteristic
30C2×#14556525C2211yescharacteristic
31#17988414C2211yescharacteristic
32#17988412C2211yescharacteristic
33#17988412C2211yescharacteristic
34C4⋊#11924003C2211yescharacteristic
35#17988391C2211yescharacteristic
36#17988391C2211yescharacteristic
37#17988391C2211yescharacteristic
38#17988391C2211yescharacteristic
39#17988391C2211yescharacteristic
40#17988391C2211yescharacteristic
41#17988390C2211yescharacteristic
42#17988388C2211yescharacteristic
43#17988388C2211yescharacteristic
44#17988388C2211yescharacteristic
45C22⋊#11924003C2211yescharacteristic
46C22⋊#11924003C2211yescharacteristic
47C4×#11924003C2211yescharacteristic
48C2×#135×C73C2211yescharacteristic
49C22×Q16×C73C2211yescharacteristic
50#548×C73—21no—
51#495×C73C2211yescharacteristic
52#467×C73C2211yescharacteristic
53#466×C73C2211yescharacteristic
54#464×C73—21no—
55#452×C73—21no—
56#448×C73—21no—
57C4×Q16×C73C2211yescharacteristic
58C8×D4×C73C2211yescharacteristic
59C2×Dic292C2311yescharacteristic
60#14556584—41no—
61#14556583—21no—
62#14556581—41no—
63#14556572—41no—
64#14556570—21no—
65#14556570—21no—
66C4×Dic146—21no—
67#14556550—21no—
68#14556547—41no—
69C2×C4⋊Dic73—21no—
70#14556537—22no—
71#14556537—41no—
72#14556535—41no—
73C2×#11924003—41no—
74C2×#11924003—41no—
75C2×#11924003C2311yescharacteristic
76C2×#11924003C2311yescharacteristic
77C2×#11924003C2311yescharacteristic
78C2×#11924003C2311yescharacteristic
79#14556525—21no—
80#14556525C2311yescharacteristic
81#14556525C2311yescharacteristic
82#14556525C2311yescharacteristic
83#14556523—21no—
84#14556523—21no—
85#14556516—21no—
86#14556516—21no—
87#14556516—21no—
88C2×#41×C73—21no—
89C22×Q8×C73—21no—
90#135×C73—22no—
91#135×C73—41no—
92C2×Q16×C73—22no—
93C2×Q16×C73—21no—
94C2×Q16×C73—41no—
95C2×Q16×C73C2311yescharacteristic
96C2×Q16×C73C2311yescharacteristic
97C2×SD16×C73—21no—
98C2×SD16×C73—21no—
99C2×D8×C73C2311yescharacteristic
100C22×C8×C73C2311yescharacteristic
101C22×C8×C73C2311yescharacteristic
102#124×C73—21no—
103#124×C73—41no—
104C22⋊Q8×C73—21no—
105C22⋊Q8×C73—41no—
106#121×C73—21no—
107#121×C73—41no—
108C4×Q8×C73—21no—
109C4×D4×C73—41no—
110C4×D4×C73D411yescharacteristic
111#107×C73C2311yescharacteristic
112C4⋊C8×C73D411yescharacteristic
113#103×C73—21no—
114#103×C73—21no—
115#102×C73—21no—
116C22⋊C8×C73D411yescharacteristic
117C22⋊C8×C73D411yescharacteristic
118C4×C8×C73C2311yescharacteristic
119C2×Dic146—21no—
120C22×Dic73—41no—
121Dic292—21no—
122Dic292—21no—
123C4⋊Dic73—21no—
124C4⋊Dic73—21no—
125C4⋊Dic73—21no—
126C4⋊Dic73—41no—
127#11924007—41no—
128#11924007—41no—
129#11924007—41no—
130#11924007—41no—
131#11924007—41no—
132C22⋊Dic73—41no—
133C22⋊Dic73—41no—
134C22⋊Dic73—41no—
135C4×Dic73—41no—
136C4×Dic73—41no—
137#11924003—21no—
138#11924003—21no—
139#11924003—21no—
140#11924003—21no—
141#41×C73—42no—
142#41×C73—81no—
143C2×Q8×C73—21no—
144C2×Q8×C73—21no—
145C2×Q8×C73—42no—
146C2×Q8×C73—41no—
147C2×Q8×C73—81no—
148C2×D4×C73—21no—
149C2×D4×C73—41no—
150C2×D4×C73—41no—
151C2×D4×C73D811yescharacteristic
152C22×C4×C73—41no—
153C22×C4×C73—41no—
154C22×C4×C73C2×D411yescharacteristic
155C22×C4×C73C2×D411yescharacteristic
156Q16×C73—21no—
157Q16×C73—21no—
158Q16×C73—21no—
159Q16×C73—42no—
160Q16×C73#4112yes—
161SD16×C73—42no—
162SD16×C73—81no—
163D8×C73—21no—
164D8×C73—21no—
165C2×C8×C73—21no—
166C2×C8×C73—21no—
167C2×C8×C73—21no—
168C2×C8×C73—21no—
169C2×C8×C73C2×D411yescharacteristic
170C2×C8×C73C2×D411yescharacteristic
171C2×C8×C73C2×D411yescharacteristic
172C2×C8×C73C2411yescharacteristic
173C4⋊C4×C73—21no—
174C4⋊C4×C73—41no—
175C4⋊C4×C73—41no—
176C4⋊C4×C73D811yescharacteristic
177C22⋊C4×C73—41no—
178C22⋊C4×C73—41no—
179C22⋊C4×C73—41no—
180C22⋊C4×C73—41no—
181C22⋊C4×C73D811yescharacteristic
182C22⋊C4×C73D811yescharacteristic
183C42×C73—41no—
184C42×C73C2×D411yescharacteristic
185C23×C73—41no—
186C23×C73C2×D811yescharacteristic
187C23×C73C2×D811yescharacteristic
188Q8×C73—22no—
189Q8×C73—41no—
190Q8×C73—41no—
191Q8×C73—41no—
192Q8×C73—82no—
193D4×C73—22no—
194D4×C73—41no—
195D4×C73—41no—
196D4×C73—82no—
197D4×C73—82no—
198C2×C4×C73—21no—
199C2×C4×C73—21no—
200C2×C4×C73—21no—
201C2×C4×C73—21no—
202C2×C4×C73—41no—
203C2×C4×C73—41no—
204C2×C4×C73—41no—
205C2×C4×C73—82no—
206C2×C4×C73—82no—
207C2×C4×C73C2×D811yescharacteristic
208C2×C4×C73C2×D811yescharacteristic
209C2×C4×C73C2×D811yescharacteristic
210C2×C4×C73C2×D811yescharacteristic
211C2×C4×C73C22×D411yescharacteristic
212C2×Dic73—41no—
213C2×Dic73—41no—
214C2×Dic73—41no—
215C2×Dic73—41no—
216C2×Dic73—82no—
217Dic146—41no—
218Dic146—41no—
219C8×C73—21no—
220C8×C73—21no—
221C8×C73—21no—
222C8×C73C2×#4111yescharacteristic
223C8×C73H(5,𝔽2)11yescharacteristic
224C22×C73—21no—
225C22×C73—21no—
226C22×C73—21no—
227C22×C73—21no—
228C22×C73—21no—
229C22×C73—81no—
230C22×C73—81no—
231C22×C73—81no—
232C22×C73#50812yes—
233C22×C73C22×D811yescharacteristic
234C4×C73—21no—
235C4×C73—21no—
236C4×C73—21no—
237C4×C73—21no—
238C4×C73—42no—
239C4×C73—81no—
240C4×C73—81no—
241C4×C73#54611yescharacteristic
242C4×C73#57711yescharacteristic
243Dic73—81no—
244Dic73—81no—
245Dic73—81no—
246Dic73—81no—
247#33867—731nomaximal, Sylow
248C2×C73—22no—
249C2×C73—41no—
250C2×C73—161no—
251C2×C73#329411yescharacteristic
252C2×C73C2×#50811yescharacteristic
253C2×C73#341711yescharacteristic
254#3336—731no—
255#3285D7311yescharacteristic
256#2242—731no—
257#2237—731no—
258#2232—731no—
259#2229—731no—
260#2217—731no—
261#2216—731no—
262#2211—731no—
263#2189—731no—
264#2188—731no—
265C4×Q32—731no—
266D4×C16—731no—
267#2138—731no—
268C2×#358—731no—
269C73#3386711yescharacteristic, complemented, Sylow
270C2×#135C2×D7311yescharacteristic
271C22×Q16C2×D7311yescharacteristic
272#552—1461no—
273#548—21no—
274C2×Q32—731no—
275C2×Q32—731no—
276C2×SD32—731no—
277C22×C16—731no—
278C22×C16—731no—
279#499—731no—
280#495C2×D7311yescharacteristic
281#485—1461no—
282#483—1461no—
283#479—1461no—
284#467—731no—
285#467—731no—
286#467C2×D7311yescharacteristic
287#466C2×D7311yescharacteristic
288#464—21no—
289#452—21no—
290#448—21no—
291C4×Q16—731no—
292C4×Q16C2×D7311yescharacteristic
293C8×D4C2×D7311yescharacteristic
294#427—731no—
295C2×#107—731no—
296#367—731no—
297#366—731no—
298#366—731no—
299C4⋊C16—731no—
300#358—731no—
301#358—731no—
302#358—731no—
303#358—731no—
304#358—731no—
305#358—731no—
306#358—731no—
307#357—731no—
308#357—731no—
309#357—731no—
310C22⋊C16—731no—
311C22⋊C16—731no—
312C4×C16—731no—
313C2×#41—21no—
314C22×Q8—21no—
315#135—22no—
316#135—41no—
317C2×Q16—22no—
318C2×Q16—21no—
319C2×Q16—41no—
320C2×Q16—731no—
321C2×Q16C22×D7311yescharacteristic
322C2×Q16C22×D7311yescharacteristic
323C2×SD16—21no—
324C2×SD16—21no—
325C2×D8C22×D7311yescharacteristic
326C22×C8C22×D7311yescharacteristic
327C22×C8C22×D7311yescharacteristic
328#126—2921no—
329#125—2921no—
330#125—1461no—
331#124—21no—
332#124—41no—
333#123—2921no—
334C22⋊Q8—21no—
335C22⋊Q8—41no—
336C22⋊Q8—1461no—
337C22⋊Q8—1461no—
338#121—21no—
339#121—41no—
340C4×Q8—21no—
341C4×Q8—1461no—
342C4×D4—41no—
343C4×D4C22⋊D7311yescharacteristic
344#117—2921no—
345#117—1461no—
346C2×C4⋊C4—1461no—
347Q32—2921no—
348Q32—1462no—
349SD32—2921no—
350C2×C16—2921no—
351C2×C16—2921no—
352C2×C16—731no—
353C2×C16—731no—
354C2×C16—731no—
355C2×C16—731no—
356#107—731no—
357#107—731no—
358#107—731no—
359#107—1461no—
360#107C22×D7311yescharacteristic
361#106—1461no—
362#106—1461no—
363C4⋊C8C22⋊D7311yescharacteristic
364#103—21no—
365#103—21no—
366#103—1461no—
367#103—1461no—
368#103—1461no—
369#102—21no—
370C22⋊C8C22⋊D7311yescharacteristic
371C22⋊C8C22⋊D7311yescharacteristic
372C4×C8C22×D7311yescharacteristic
373#41—42no—
374#41—81no—
375C2×Q8—21no—
376C2×Q8—21no—
377C2×Q8—42no—
378C2×Q8—41no—
379C2×Q8—81no—
380C2×Q8—1461no—
381C2×D4—21no—
382C2×D4—41no—
383C2×D4—41no—
384C2×D4#1192401311yescharacteristic
385C22×C4—41no—
386C22×C4—41no—
387C22×C4—2921no—
388C22×C4C2×C22⋊D7311yescharacteristic
389C22×C4C2×C22⋊D7311yescharacteristic
390Q16—21no—
391Q16—21no—
392Q16—21no—
393Q16—42no—
394Q16—1461no—
395Q16—1461no—
396Q16#1192402712yes—
397SD16—42no—
398SD16—81no—
399D8—21no—
400D8—21no—
401C2×C8—21no—
402C2×C8—21no—
403C2×C8—21no—
404C2×C8—21no—
405C2×C8C2×C22⋊D7311yescharacteristic
406C2×C8C2×C22⋊D7311yescharacteristic
407C2×C8C2×C22⋊D7311yescharacteristic
408C2×C8C23×D7311yescharacteristic
409C4⋊C4—21no—
410C4⋊C4—41no—
411C4⋊C4—41no—
412C4⋊C4—2921no—
413C4⋊C4—2921no—
414C4⋊C4—2921no—
415C4⋊C4—2921no—
416C4⋊C4—2921no—
417C4⋊C4—2921no—
418C4⋊C4—1461no—
419C4⋊C4—1461no—
420C4⋊C4—1461no—
421C4⋊C4#1192401311yescharacteristic
422C22⋊C4—41no—
423C22⋊C4—41no—
424C22⋊C4—41no—
425C22⋊C4—41no—
426C22⋊C4—2921no—
427C22⋊C4—2921no—
428C22⋊C4—2921no—
429C22⋊C4#1192401311yescharacteristic
430C22⋊C4#1192401311yescharacteristic
431C42—41no—
432C42—2921no—
433C42—2921no—
434C42C2×C22⋊D7311yescharacteristic
435C16—1461no—
436C16—1461no—
437C16—1461no—
438C16—1461no—
439C23—41no—
440C23C2×#1192401311yescharacteristic
441C23C2×#1192401311yescharacteristic
442Q8—22no—
443Q8—41no—
444Q8—41no—
445Q8—41no—
446Q8—82no—
447Q8—2921no—
448Q8—2921no—
449D4—22no—
450D4—41no—
451D4—41no—
452D4—82no—
453D4—82no—
454C2×C4—21no—
455C2×C4—21no—
456C2×C4—21no—
457C2×C4—21no—
458C2×C4—41no—
459C2×C4—41no—
460C2×C4—41no—
461C2×C4—82no—
462C2×C4—82no—
463C2×C4—2921no—
464C2×C4—2921no—
465C2×C4—2921no—
466C2×C4—2921no—
467C2×C4—5842no—
468C2×C4C2×#1192401311yescharacteristic
469C2×C4C2×#1192401311yescharacteristic
470C2×C4C2×#1192401311yescharacteristic
471C2×C4C2×#1192401311yescharacteristic
472C2×C4C22×C22⋊D7311yescharacteristic
473C8—21no—
474C8—21no—
475C8—21no—
476C8C2×#1192402711yescharacteristic
477C8#1455663611yescharacteristic
478C22—21no—
479C22—21no—
480C22—21no—
481C22—21no—
482C22—21no—
483C22—81no—
484C22—81no—
485C22—81no—
486C22#1798901712yes—
487C22C22×#1192401311yescharacteristic
488C4—21no—
489C4—21no—
490C4—21no—
491C4—21no—
492C4—42no—
493C4—81no—
494C4—81no—
495C4—5841no—
496C4—5841no—
497C4—5841no—
498C4—5841no—
499C4#1798923111yescharacteristic
500C4#1798939811yescharacteristic
501C2—22no—
502C2—41no—
503C2—161no—
504C2#2262174411yescharacteristic
505C2C2×#1798901711yescharacteristic
506C2#2262308211yescharacteristic
subgroupG/H
centerC22C22×#11924013
commutator, cosocleC2×C8×C73C24
FrattiniC2×C8C23×D73
Fitting, prime radicalC73×#3285C2
socleC22×C73C22×D8
hypercenter#3285D73
hypocenterC73#33867
LCSG ▷ C2×C8×C73 ▷ C4×C73 ▷ C2×C73 ▷ C73
UCSC1 ◁ C22 ◁ C4×D4 ◁ C8×D4 ◁ #3285

Complements

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

NH ≅ G/Ntype
1. #22622300C2semidirect
2. #22612162C2semidirect
3. #22612136C2semidirect
4. #22612109C2semidirect
5. #22612095C2semidirect
6. #22612025C2semidirect
7. #22612020C2semidirect
8. #22611990C2semidirect
9. #22611871C2semidirect
10. #22611865C2semidirect
11. C4×#14556537C2semidirect
12. D4×#11924003C2semidirect
13. #22611638C2semidirect
14. C2×#17988391C2semidirect
269. C73#33867semidirect

Maximal subgroups

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

subgroupindex
#226223002
#226121622
#226121362
#226121092
#226120952
#226120252
#226120202
#226119902
#226118712
#226118652
C4×#145565372
D4×#119240032
#226116382
C2×#179883912
C73×#32852
#3386773