Finite Group Databasebeta

← #33056136

Subgroups — #33056136

Order 18688.

586 conjugacy classes of subgroups.

The subgroup lattice has 550 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#22619968C211yesmaximal, characteristic
2#22619027C211yesmaximal, characteristic
3#22619010C211yesmaximal, characteristic
4#22608802C211yesmaximal, characteristic, complemented
5#22608800C211yesmaximal, characteristic, complemented
6#22608783C211yesmaximal, characteristic, complemented
7#22608747C211yesmaximal, characteristic, complemented
8#22608747C211yesmaximal, characteristic, complemented
9Q16×#10596831C211yesmaximal, characteristic, complemented
10#22608737C211yesmaximal, characteristic, complemented
11#22608710C211yesmaximal, characteristic
12#22608376C211yesmaximal, characteristic
13#22608350C211yesmaximal, characteristic
14#22608227C211yesmaximal, characteristic
15C73×#1584C211yesmaximal, characteristic, complemented
16#17989141C2211yescharacteristic
17C4×Q8×D73C2211yescharacteristic
18#17988924C411yescharacteristic
19#17988921C411yescharacteristic
20#17988881C411yescharacteristic
21#17988880C411yescharacteristic
22#17988796C411yescharacteristic
23#17988795C411yescharacteristic
24#17988744C411yescharacteristic
25#17988741C411yescharacteristic
26#17988717C2211yescharacteristic
27#17988715—21no—
28#17988714—21no—
29Q8×#10596831C2211yescharacteristic
30Q8×#10596831C2211yescharacteristic
31C8×Dic146—21no—
32C8×Dic146C2211yescharacteristic
33#17988711—21no—
34#17988711C2211yescharacteristic
35#17988711C2211yescharacteristic
36C4×Dic292C2211yescharacteristic
37C4×#11924018C2211yescharacteristic
38C4×#11924018C2211yescharacteristic
39#17988649—21no—
40#17988647—21no—
41#17988643C2211yescharacteristic
42#17988639—21no—
43#17988639C2211yescharacteristic
44#17988638—21no—
45#17988638C2211yescharacteristic
46#17988638C2211yescharacteristic
47C4×#11924011C2211yescharacteristic
48#17988340C2211yescharacteristic
49#17988337C2211yescharacteristic
50#17988337C2211yescharacteristic
51#17988319C2211yescharacteristic
52#17988319C2211yescharacteristic
53Q8⋊#10596831C2211yescharacteristic
54Q8⋊#10596831C2211yescharacteristic
55#17988317C2211yescharacteristic
56#17988317C2211yescharacteristic
57#17988309C2211yescharacteristic
58#17988309C2211yescharacteristic
59C8×#10596831C2211yescharacteristic
60#446×C73C2211yescharacteristic
61C8×Q8×C73C2211yescharacteristic
62C4×Q16×C73C2211yescharacteristic
63#335×C73C2211yescharacteristic
64Q8⋊C8×C73C2211yescharacteristic
65Q8⋊C8×C73C2211yescharacteristic
66#322×C73C2211yescharacteristic
67C2×Q8×D73C2×C411yescharacteristic
68C2×Dic292C2×C411yescharacteristic
69C2×#11924018C2×C411yescharacteristic
70C2×#11924018C2×C411yescharacteristic
71C2×#11924011—21no—
72#14556590C2×C411yescharacteristic
73#14556589C2×C411yescharacteristic
74#14556581—21no—
75#14556571C2×C411yescharacteristic
76#14556569—21no—
77#14556567C2×C411yescharacteristic
78Q8×Dic73C2×C411yescharacteristic
79C4×Dic146—21no—
80C4×Dic146—21no—
81C4×Dic146C2311yescharacteristic
82C4×Dic146C2311yescharacteristic
83#14556556—21no—
84#14556556C2×C411yescharacteristic
85#14556548—21no—
86C4⋊C4×D73—21no—
87C4⋊C4×D73C2×C411yescharacteristic
88C42×D73—21no—
89C42×D73C2311yescharacteristic
90#14556525C2×C411yescharacteristic
91#14556524C2×C411yescharacteristic
92#14556524C2×C411yescharacteristic
93C4⋊#10596831—21no—
94C4⋊#10596831—21no—
95C4⋊#10596831C2311yescharacteristic
96C4⋊#10596831C2311yescharacteristic
97#14556520—21no—
98#14556520—21no—
99#14556520—21no—
100#14556520—21no—
101#14556520C2×C411yescharacteristic
102#14556520C2×C411yescharacteristic
103#14556520D411yescharacteristic
104#14556520D411yescharacteristic
105#14556516C2×C411yescharacteristic
106#14556516C2×C411yescharacteristic
107#14556515C2×C411yescharacteristic
108#14556515C2×C411yescharacteristic
109#14556514C2×C411yescharacteristic
110#14556514C2×C411yescharacteristic
111#14556503—21no—
112#14556503—21no—
113#14556503C2×C411yescharacteristic
114#14556503C2×C411yescharacteristic
115#14556503D411yescharacteristic
116#14556503D411yescharacteristic
117#14556501—21no—
118#14556501—21no—
119#14556501—21no—
120C4×#10596831—21no—
121C4×#10596831—21no—
122C4×#10596831C2311yescharacteristic
123C4×#10596831C2311yescharacteristic
124C4×#10596831C2311yescharacteristic
125C8×Dic73—21no—
126C8×Dic73—21no—
127C8×Dic73—21no—
128C8×Dic73—21no—
129C2×Q16×C73C2×C411yescharacteristic
130C4×Q8×C73C2311yescharacteristic
131C4×Q8×C73C2311yescharacteristic
132#107×C73C2×C411yescharacteristic
133C4⋊C8×C73—21no—
134C4⋊C8×C73—21no—
135C4⋊C8×C73C2311yescharacteristic
136C4⋊C8×C73C2311yescharacteristic
137#103×C73C2×C411yescharacteristic
138#103×C73C2×C411yescharacteristic
139#97×C73—21no—
140C4×C8×C73—21no—
141C4×C8×C73C2311yescharacteristic
142C4×C8×C73C2311yescharacteristic
143C4×C8×C73C2311yescharacteristic
144C2×Dic146—21no—
145C2×Dic146—21no—
146C2×Dic146C22×C411yescharacteristic
147C2×Dic146C22×C411yescharacteristic
148Q8×D73—22no—
149Q8×D73—22no—
150C2×C4×D73—21no—
151C2×C4×D73—21no—
152C2×C4×D73C22×C411yescharacteristic
153C2×C4×D73C2×D411yescharacteristic
154C2×C4×D73#4111yescharacteristic
155Dic292—22no—
156#11924018—22no—
157#11924018—22no—
158#11924011—42no—
159C2×#10596831—21no—
160C2×#10596831—21no—
161C2×#10596831—21no—
162C2×#10596831C22×C411yescharacteristic
163C2×#10596831C22×C411yescharacteristic
164C2×#10596831C2×D411yescharacteristic
165C2×#10596831C2×D411yescharacteristic
166C4⋊Dic73—21no—
167C4⋊Dic73—21no—
168C4⋊Dic73C22×C411yescharacteristic
169C4⋊Dic73C22×C411yescharacteristic
170#11924007—21no—
171#11924007—21no—
172#11924007—21no—
173#11924007—21no—
174#11924007—21no—
175#11924007—21no—
176#11924007—21no—
177#11924007—21no—
178#11924005—21no—
179#11924005—21no—
180C4×Dic73—21no—
181C4×Dic73—21no—
182C4×Dic73—21no—
183C4×Dic73—21no—
184C4×Dic73C22×C411yescharacteristic
185C4×Dic73C2×D411yescharacteristic
186C4×Dic73#4111yescharacteristic
187C2×Q8×C73C22×C411yescharacteristic
188C2×Q8×C73C22×C411yescharacteristic
189Q16×C73—22no—
190C2×C8×C73—21no—
191C2×C8×C73—21no—
192C2×C8×C73—21no—
193C2×C8×C73C22×C411yescharacteristic
194C2×C8×C73C22×C411yescharacteristic
195C2×C8×C73C2×D411yescharacteristic
196C2×C8×C73C2×D411yescharacteristic
197C4⋊C4×C73—21no—
198C4⋊C4×C73—21no—
199C4⋊C4×C73C22×C411yescharacteristic
200C4⋊C4×C73C22×C411yescharacteristic
201C42×C73—21no—
202C42×C73—21no—
203C42×C73C2411yescharacteristic
204C22×D73C4×D411yescharacteristic
205Q8×C73—21no—
206Q8×C73—21no—
207Q8×C73—21no—
208Q8×C73#13112yes—
209C2×C4×C73—21no—
210C2×C4×C73—21no—
211C2×C4×C73—21no—
212C2×C4×C73—21no—
213C2×C4×C73C23×C411yescharacteristic
214C2×C4×C73C22×D411yescharacteristic
215C2×C4×C73C2×#4111yescharacteristic
216C2×Dic73—21no—
217C2×Dic73—21no—
218C2×Dic73—21no—
219C2×Dic73—21no—
220C2×Dic73C4×D411yescharacteristic
221C2×Dic73C4×D411yescharacteristic
222C2×Dic73C4×D411yescharacteristic
223C4×D73—24no—
224C4×D73—21no—
225C4×D73—21no—
226C4×D73—21no—
227C4×D73—21no—
228C4×D73—41no—
229C4×D73—81no—
230Dic146—24no—
231Dic146—21no—
232Dic146—21no—
233Dic146—21no—
234Dic146—81no—
235Dic146#13112yes—
236C8×C73—21no—
237C8×C73—21no—
238C8×C73—21no—
239C8×C73—21no—
240C8×C73—41no—
241C8×C73—41no—
242C8×C73—41no—
243#10596831—22no—
244#10596831—21no—
245#10596831—21no—
246#10596831—21no—
247#10596831—21no—
248#10596831—41no—
249#10596831—41no—
250C22×C73C2×C4×D411yescharacteristic
251C2×D73—21no—
252C2×D73—21no—
253C2×D73—21no—
254C4×C73—22no—
255C4×C73—21no—
256C4×C73—41no—
257C4×C73—41no—
258C4×C73—41no—
259C4×C73C2×#13111yescharacteristic
260C4×C73C2×#13111yescharacteristic
261C4×C73#57811yescharacteristic
262C4×C73#57811yescharacteristic
263Dic73—21no—
264Dic73—22no—
265Dic73—21no—
266Dic73—21no—
267Dic73—41no—
268Dic73—41no—
269Dic73—41no—
270#17925—731nomaximal, Sylow
271C2×C73#292911yescharacteristic
272C2×C73#295011yescharacteristic
273C2×C73#295611yescharacteristic
274D73—41no—
275#3104—731no—
276#2985—731no—
277#2982—731no—
278#1587—731no—
279#1587—731no—
280#1584—731no—
281#1584D7311yescharacteristic, complemented
282#1579—731no—
283#1579—731no—
284C8×Q16—731no—
285C8×Q16—731no—
286#1572—731no—
287#1491—731no—
288#1485—731no—
289#1453—731no—
290C73#1792511yescharacteristic, complemented, Sylow
291#533—731no—
292C2×C4×Q8—731no—
293#484—731no—
294#484—731no—
295#477—731no—
296#477—731no—
297#462—731no—
298#462—731no—
299#451—731no—
300#451—731no—
301#446—731no—
302#446—1461no—
303#446—1461no—
304#446C2×D7311yescharacteristic
305C8×Q8—731no—
306C8×Q8—731no—
307C8×Q8—731no—
308C8×Q8—731no—
309C8×Q8—731no—
310C8×Q8—1461no—
311C8×Q8—1461no—
312C8×Q8C2×D7311yescharacteristic
313C4×Q16—731no—
314C4×Q16—731no—
315C4×Q16—731no—
316C4×Q16C2×D7311yescharacteristic
317#433—1461no—
318#433—1461no—
319#432—731no—
320#431—731no—
321#431—731no—
322#431—731no—
323#431—1461no—
324#431—1461no—
325C4×M16—731no—
326#335—731no—
327#335C2×D7311yescharacteristic
328#334—731no—
329#334—731no—
330Q8⋊C8—731no—
331Q8⋊C8—731no—
332Q8⋊C8—731no—
333Q8⋊C8—731no—
334Q8⋊C8—731no—
335Q8⋊C8—731no—
336Q8⋊C8C2×D7311yescharacteristic
337Q8⋊C8C2×D7311yescharacteristic
338#322—731no—
339#322—731no—
340#322C2×D7311yescharacteristic
341C82—731no—
342C22×Q8—731no—
343C2×Q16—731no—
344C2×Q16—731no—
345C2×Q16—731no—
346C2×Q16C4×D7311yescharacteristic
347C2×M16—1461no—
348C4⋊Q8—731no—
349C4⋊Q8—731no—
350#125—1461no—
351C22⋊Q8—731no—
352C22⋊Q8—731no—
353C22⋊Q8—1461no—
354C4×Q8—731no—
355C4×Q8—731no—
356C4×Q8—731no—
357C4×Q8—731no—
358C4×Q8—1461no—
359C4×Q8—1461no—
360C4×Q8—1461no—
361C4×Q8C22×D7311yescharacteristic
362C4×Q8C22×D7311yescharacteristic
363#117—1461no—
364C2×C4⋊C4—731no—
365C2×C4⋊C4—1461no—
366C2×C42—731no—
367C2×C42—1461no—
368#107—731no—
369#107—731no—
370#107—731no—
371#107C4×D7311yescharacteristic
372C4⋊C8—21no—
373C4⋊C8—21no—
374C4⋊C8—731no—
375C4⋊C8—731no—
376C4⋊C8—731no—
377C4⋊C8—731no—
378C4⋊C8—731no—
379C4⋊C8—731no—
380C4⋊C8—1461no—
381C4⋊C8—1461no—
382C4⋊C8—1461no—
383C4⋊C8—1461no—
384C4⋊C8—1461no—
385C4⋊C8—1461no—
386C4⋊C8C22×D7311yescharacteristic
387C4⋊C8C22×D7311yescharacteristic
388#103—731no—
389#103—731no—
390#103—731no—
391#103—731no—
392#103—731no—
393#103—731no—
394#103C4×D7311yescharacteristic
395#103C4×D7311yescharacteristic
396C22⋊C8—731no—
397C22⋊C8—731no—
398C22⋊C8—731no—
399C22⋊C8—731no—
400C22⋊C8—1461no—
401C22⋊C8—1461no—
402#97—21no—
403#97—1461no—
404#97—1461no—
405#97—1461no—
406C4×C8—21no—
407C4×C8—731no—
408C4×C8—731no—
409C4×C8—731no—
410C4×C8—1461no—
411C4×C8—1461no—
412C4×C8—1461no—
413C4×C8—1461no—
414C4×C8—1461no—
415C4×C8—1461no—
416C4×C8C22×D7311yescharacteristic
417C4×C8C22×D7311yescharacteristic
418C4×C8C22×D7311yescharacteristic
419C2×Q8—731no—
420C2×Q8—731no—
421C2×Q8—1461no—
422C2×Q8—1461no—
423C2×Q8—1462no—
424C2×Q8—1462no—
425C2×Q8C2×C4×D7311yescharacteristic
426C2×Q8C2×C4×D7311yescharacteristic
427C22×C4—731no—
428C22×C4—731no—
429C22×C4—731no—
430C22×C4—1461no—
431C22×C4—1461no—
432Q16—22no—
433Q16—1462no—
434Q16—1462no—
435Q16—1462no—
436M16—2922no—
437C2×C8—21no—
438C2×C8—21no—
439C2×C8—21no—
440C2×C8—731no—
441C2×C8—731no—
442C2×C8—731no—
443C2×C8—731no—
444C2×C8—1461no—
445C2×C8—1461no—
446C2×C8—1461no—
447C2×C8C2×C4×D7311yescharacteristic
448C2×C8C2×C4×D7311yescharacteristic
449C2×C8D4×D7311yescharacteristic
450C2×C8D4×D7311yescharacteristic
451C4⋊C4—21no—
452C4⋊C4—21no—
453C4⋊C4—731no—
454C4⋊C4—731no—
455C4⋊C4—1461no—
456C4⋊C4—1461no—
457C4⋊C4—1461no—
458C4⋊C4—1461no—
459C4⋊C4—1461no—
460C4⋊C4—1461no—
461C4⋊C4—1461no—
462C4⋊C4—1461no—
463C4⋊C4—1461no—
464C4⋊C4—1461no—
465C4⋊C4C2×C4×D7311yescharacteristic
466C4⋊C4C2×C4×D7311yescharacteristic
467C22⋊C4—1461no—
468C22⋊C4—1461no—
469C42—21no—
470C42—21no—
471C42—731no—
472C42—731no—
473C42—731no—
474C42—1461no—
475C42—1461no—
476C42—1461no—
477C42—1461no—
478C42C23×D7311yescharacteristic
479C23—731no—
480Q8—21no—
481Q8—21no—
482Q8—21no—
483Q8—5841no—
484Q8—732no—
485Q8—1464no—
486Q8—1461no—
487Q8—1461no—
488Q8—1461no—
489Q8#1455659912yes—
490C2×C4—21no—
491C2×C4—21no—
492C2×C4—21no—
493C2×C4—21no—
494C2×C4—2921no—
495C2×C4—5841no—
496C2×C4—731no—
497C2×C4—731no—
498C2×C4—731no—
499C2×C4—1464no—
500C2×C4—1461no—
501C2×C4—1461no—
502C2×C4—1461no—
503C2×C4—1461no—
504C2×C4—1461no—
505C2×C4—1461no—
506C2×C4—1461no—
507C2×C4—1461no—
508C2×C4C22×C4×D7311yescharacteristic
509C2×C4C2×D4×D7311yescharacteristic
510C2×C4#41×D7311yescharacteristic
511C8—21no—
512C8—21no—
513C8—21no—
514C8—21no—
515C8—41no—
516C8—41no—
517C8—41no—
518C8—2921no—
519C8—2921no—
520C8—1462no—
521C8—1461no—
522C8—1461no—
523C8—1461no—
524C8—1461no—
525C22—1461no—
526C22—1461no—
527C22—1461no—
528C22C4×D4×D7311yescharacteristic
529C4—21no—
530C4—22no—
531C4—41no—
532C4—41no—
533C4—41no—
534C4—2921no—
535C4—2921no—
536C4—2921no—
537C4—1461no—
538C4—1461no—
539C4—1461no—
540C4—1462no—
541C4#131×D7311yescharacteristic
542C4C2×#1455659911yescharacteristic
543C4#1798940011yescharacteristic
544C4#1798940111yescharacteristic
545C2—2921no—
546C2#2261862211yescharacteristic
547C2#2261879411yescharacteristic
548C2#2261883911yescharacteristic
subgroupG/H
centerC2×C4C2×D4×D73
commutatorC2×C4×C73C23×C4
FrattiniC42C23×D73
FittingC73×#1584C2
socleC22×C73C2×C4×D4
prime radicalC22×D73C4×D4
cosocleC42×C73C24
hypercenter#1584D73
hypocenterC73#17925
LCSG ▷ C2×C4×C73 ▷ C2×C73 ▷ C73
UCSC1 ◁ C2×C4 ◁ C4×C8 ◁ #1584

Complements

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

NH ≅ G/Ntype
4. #22608802C2semidirect
5. #22608800C2semidirect
6. #22608783C2semidirect
7. #22608747C2semidirect
8. #22608747C2semidirect
9. Q16×#10596831C2semidirect
10. #22608737C2semidirect
15. C73×#1584C2semidirect
281. #1584D73semidirect
290. C73#17925semidirect

Maximal subgroups

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

subgroupindex
#226199682
#226190272
#226190102
#226088022
#226088002
#226087832
#226087472
Q16×#105968312
#226087372
#226087102
#226083762
#226083502
#226082272
C73×#15842
#1792573