Finite Group Databasebeta

← #33055677

Subgroups — #33055677

Order 18688.

626 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
1#22619013C211yesmaximal, characteristic
2#22619010C211yesmaximal, characteristic
3C4×Q16×D73C211yesmaximal, characteristic
4#22608805C211yesmaximal, characteristic, complemented
5#22608802C211yesmaximal, characteristic, complemented
6#22608800C211yesmaximal, characteristic, complemented
7#22608799C211yesmaximal, characteristic, complemented
8#22608783C211yesmaximal, characteristic, complemented
9#22608782C211yesmaximal, characteristic, complemented
10#22608780C211yesmaximal, characteristic, complemented
11#22608678C211yesmaximal, characteristic
12#22608324C211yesmaximal, characteristic
13#22608324C211yesmaximal, characteristic
14#22608222C211yesmaximal, characteristic
15C73×#1584C211yesmaximal, characteristic, complemented
16C2×Q16×D73C411yescharacteristic
17C4×Q8×D73C2211yescharacteristic
18C4×Q8×D73C2211yescharacteristic
19#17988718C2211yescharacteristic
20#17988717C2211yescharacteristic
21#17988716—21no—
22#17988715—21no—
23#17988714C2211yescharacteristic
24Q8×#10596831C2211yescharacteristic
25C8×Dic146—21no—
26C8×Dic146C2211yescharacteristic
27#17988711—21no—
28#17988711C2211yescharacteristic
29C4×Dic292C2211yescharacteristic
30Q16×Dic73C411yescharacteristic
31#17988681C411yescharacteristic
32C4×#11924018C2211yescharacteristic
33C4×#11924018C2211yescharacteristic
34#17988679C411yescharacteristic
35#17988679C411yescharacteristic
36#17988643—21no—
37#17988642—21no—
38#17988639—21no—
39#17988639C2211yescharacteristic
40#17988639C2211yescharacteristic
41#17988638—21no—
42#17988638C2211yescharacteristic
43#17988638C2211yescharacteristic
44#107×D73C411yescharacteristic
45#103×D73C411yescharacteristic
46#103×D73C411yescharacteristic
47C4×C8×D73C2211yescharacteristic
48#17988340C2211yescharacteristic
49#17988337C2211yescharacteristic
50#17988337C2211yescharacteristic
51#17988319C2211yescharacteristic
52#17988319C2211yescharacteristic
53Q8⋊#10596831C2211yescharacteristic
54Q8⋊#10596831C2211yescharacteristic
55#17988317C2211yescharacteristic
56#17988317C2211yescharacteristic
57#17988309C2211yescharacteristic
58#17988309C2211yescharacteristic
59#17988308C2211yescharacteristic
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×Q8×D73C2×C411yescharacteristic
69C2×Dic292C2×C411yescharacteristic
70C2×#11924018C2×C411yescharacteristic
71C2×#11924018C2×C411yescharacteristic
72Q16×D73—24no—
73C2×C8×D73C2×C411yescharacteristic
74C2×C8×D73C2×C411yescharacteristic
75Q8×Dic73C2×C411yescharacteristic
76Q8×Dic73C2×C411yescharacteristic
77C4×Dic146—21no—
78C4×Dic146—21no—
79C4×Dic146C2311yescharacteristic
80C4×Dic146C2311yescharacteristic
81#14556556—21no—
82#14556556—21no—
83#14556556C2×C411yescharacteristic
84#14556556C2×C411yescharacteristic
85C4⋊C4×D73—21no—
86C4⋊C4×D73—21no—
87C4⋊C4×D73C2×C411yescharacteristic
88C4⋊C4×D73C2×C411yescharacteristic
89C42×D73—21no—
90C42×D73—21no—
91C42×D73C2311yescharacteristic
92#14556525C2×C411yescharacteristic
93#14556524C2×C411yescharacteristic
94#14556524C2×C411yescharacteristic
95C4⋊#10596831—21no—
96C4⋊#10596831—21no—
97C4⋊#10596831C2311yescharacteristic
98C4⋊#10596831C2311yescharacteristic
99#14556520—21no—
100#14556520—21no—
101#14556520—21no—
102#14556520—21no—
103#14556520—21no—
104#14556520D411yescharacteristic
105#14556520D411yescharacteristic
106#14556516C2×C411yescharacteristic
107#14556516C2×C411yescharacteristic
108#14556515C2×C411yescharacteristic
109#14556515C2×C411yescharacteristic
110#14556514C2×C411yescharacteristic
111#14556514C2×C411yescharacteristic
112#14556503—21no—
113#14556503—21no—
114#14556503—21no—
115#14556503D411yescharacteristic
116#14556503D411yescharacteristic
117#14556502—21no—
118#14556501—21no—
119#14556501—21no—
120C4×#10596831—21no—
121C4×#10596831C2311yescharacteristic
122C4×#10596831C2311yescharacteristic
123C4×#10596831C2311yescharacteristic
124C8×Dic73—21no—
125C8×Dic73—21no—
126C8×Dic73—21no—
127C8×Dic73—21no—
128C8×Dic73C2×C411yescharacteristic
129C8×Dic73C2×C411yescharacteristic
130C2×Q16×C73C2×C411yescharacteristic
131C4×Q8×C73C2311yescharacteristic
132C4×Q8×C73C2311yescharacteristic
133#107×C73C2×C411yescharacteristic
134C4⋊C8×C73—21no—
135C4⋊C8×C73—21no—
136C4⋊C8×C73C2311yescharacteristic
137C4⋊C8×C73C2311yescharacteristic
138#103×C73C2×C411yescharacteristic
139#103×C73C2×C411yescharacteristic
140#97×C73—21no—
141C4×C8×C73—21no—
142C4×C8×C73C2311yescharacteristic
143C4×C8×C73C2311yescharacteristic
144C4×C8×C73C2311yescharacteristic
145C2×Dic146—21no—
146C2×Dic146—21no—
147C2×Dic146C22×C411yescharacteristic
148C2×Dic146C22×C411yescharacteristic
149Q8×D73—22no—
150Q8×D73—22no—
151Q8×D73—22no—
152Q8×D73M1614yes—
153C2×C4×D73—21no—
154C2×C4×D73—21no—
155C2×C4×D73—21no—
156C2×C4×D73—21no—
157C2×C4×D73C22×C411yescharacteristic
158C2×C4×D73C2×D411yescharacteristic
159C2×C4×D73#4111yescharacteristic
160Dic292—22no—
161#11924018—22no—
162#11924018—22no—
163C2×#10596831—21no—
164C2×#10596831—21no—
165C2×#10596831—21no—
166C2×#10596831C22×C411yescharacteristic
167C2×#10596831C22×C411yescharacteristic
168C2×#10596831C2×D411yescharacteristic
169C2×#10596831C2×D411yescharacteristic
170C8×D73—22no—
171C8×D73—22no—
172C4⋊Dic73—21no—
173C4⋊Dic73—21no—
174C4⋊Dic73C22×C411yescharacteristic
175C4⋊Dic73C22×C411yescharacteristic
176#11924007—21no—
177#11924007—21no—
178#11924007—21no—
179#11924007—21no—
180#11924007—21no—
181#11924007—21no—
182C4×Dic73—21no—
183C4×Dic73—21no—
184C4×Dic73—21no—
185C4×Dic73—21no—
186C4×Dic73—21no—
187C4×Dic73—21no—
188C4×Dic73C22×C411yescharacteristic
189C4×Dic73C2×D411yescharacteristic
190C4×Dic73#4111yescharacteristic
191C2×Q8×C73C22×C411yescharacteristic
192C2×Q8×C73C22×C411yescharacteristic
193Q16×C73—22no—
194C2×C8×C73—21no—
195C2×C8×C73—21no—
196C2×C8×C73—21no—
197C2×C8×C73C22×C411yescharacteristic
198C2×C8×C73C22×C411yescharacteristic
199C2×C8×C73C2×D411yescharacteristic
200C2×C8×C73C2×D411yescharacteristic
201C4⋊C4×C73—21no—
202C4⋊C4×C73—21no—
203C4⋊C4×C73C22×C411yescharacteristic
204C4⋊C4×C73C22×C411yescharacteristic
205C42×C73—21no—
206C42×C73—21no—
207C42×C73C2411yescharacteristic
208C22×D73C4×D411yescharacteristic
209Q8×C73—21no—
210Q8×C73—21no—
211Q8×C73—21no—
212Q8×C73C2×M1612yes—
213C2×C4×C73—21no—
214C2×C4×C73—21no—
215C2×C4×C73—21no—
216C2×C4×C73—21no—
217C2×C4×C73C23×C411yescharacteristic
218C2×C4×C73C22×D411yescharacteristic
219C2×C4×C73C2×#4111yescharacteristic
220C2×Dic73—21no—
221C2×Dic73—21no—
222C2×Dic73—21no—
223C2×Dic73—21no—
224C2×Dic73C4×D411yescharacteristic
225C2×Dic73C4×D411yescharacteristic
226C2×Dic73C4×D411yescharacteristic
227C4×D73—24no—
228C4×D73—21no—
229C4×D73—21no—
230C4×D73—41no—
231C4×D73—42no—
232C4×D73—81no—
233C4×D73—81no—
234C4×D73C2×M1612yes—
235C4×D73#13112yes—
236Dic146—21no—
237Dic146—24no—
238Dic146—21no—
239Dic146—21no—
240Dic146—42no—
241Dic146C2×M1612yes—
242C8×C73—21no—
243C8×C73—21no—
244C8×C73—21no—
245C8×C73—21no—
246C8×C73—41no—
247C8×C73—41no—
248C8×C73—41no—
249#10596831—21no—
250#10596831—21no—
251#10596831—21no—
252#10596831—21no—
253#10596831—41no—
254#10596831—41no—
255#10596831—41no—
256C22×C73C2×C4×D411yescharacteristic
257C2×D73—21no—
258C2×D73—21no—
259C2×D73#43512yes—
260C4×C73—22no—
261C4×C73—21no—
262C4×C73—41no—
263C4×C73—41no—
264C4×C73—41no—
265C4×C73C22×M1611yescharacteristic
266C4×C73C2×#13111yescharacteristic
267C4×C73#57811yescharacteristic
268C4×C73#57811yescharacteristic
269Dic73—22no—
270Dic73—21no—
271Dic73—21no—
272Dic73—41no—
273Dic73—41no—
274Dic73—41no—
275Dic73#43512yes—
276#17880—731nomaximal, Sylow
277C2×C73C2×#43511yescharacteristic
278C2×C73#295011yescharacteristic
279C2×C73#295711yescharacteristic
280D73—22no—
281#2983—731no—
282#2982—731no—
283C2×C4×Q16—731no—
284#1587—731no—
285#1587—731no—
286#1587—731no—
287#1587—731no—
288#1584—731no—
289#1584—731no—
290#1584—731no—
291#1584D7311yescharacteristic, complemented
292#1564—731no—
293#1479—731no—
294#1479—731no—
295#1452—731no—
296C73#1788011yescharacteristic, complemented, Sylow
297C22×Q16—731no—
298C2×C4×Q8—731no—
299C2×C4×Q8—731no—
300#446—731no—
301#446—731no—
302#446—731no—
303#446—1461no—
304#446—1461no—
305#446C2×D7311yescharacteristic
306C8×Q8—731no—
307C8×Q8—731no—
308C8×Q8—731no—
309C8×Q8—1461no—
310C8×Q8—1461no—
311C8×Q8C2×D7311yescharacteristic
312C4×Q16—731no—
313C4×Q16—731no—
314C4×Q16—731no—
315C4×Q16—731no—
316C4×Q16—731no—
317C4×Q16—731no—
318C4×Q16—731no—
319C4×Q16C2×D7311yescharacteristic
320#432—1461no—
321#432—1461no—
322#431—731no—
323#431—731no—
324#431—731no—
325#431—731no—
326#431—1461no—
327#431—1461no—
328C2×#107—731no—
329C2×#103—731no—
330C2×#103—731no—
331C2×C4×C8—731no—
332#335—731no—
333#335C2×D7311yescharacteristic
334#334—731no—
335#334—731no—
336Q8⋊C8—731no—
337Q8⋊C8—731no—
338Q8⋊C8—731no—
339Q8⋊C8—731no—
340Q8⋊C8—731no—
341Q8⋊C8—731no—
342Q8⋊C8C2×D7311yescharacteristic
343Q8⋊C8C2×D7311yescharacteristic
344#322—731no—
345#322—731no—
346#322—731no—
347#322C2×D7311yescharacteristic
348C22×Q8—731no—
349C22×Q8—731no—
350C2×Q16—731no—
351C2×Q16—731no—
352C2×Q16—731no—
353C2×Q16—1464no—
354C2×Q16C4×D7311yescharacteristic
355C22×C8—731no—
356C22×C8—731no—
357C4×Q8—731no—
358C4×Q8—731no—
359C4×Q8—731no—
360C4×Q8—731no—
361C4×Q8—731no—
362C4×Q8—731no—
363C4×Q8—1461no—
364C4×Q8—1461no—
365C4×Q8—1461no—
366C4×Q8—1461no—
367C4×Q8C22×D7311yescharacteristic
368C4×Q8C22×D7311yescharacteristic
369C2×C4⋊C4—731no—
370C2×C4⋊C4—731no—
371C2×C4⋊C4—1461no—
372C2×C4⋊C4—1461no—
373C2×C42—731no—
374C2×C42—1461no—
375C2×C42—1461no—
376#107—731no—
377#107—731no—
378#107—731no—
379#107C4×D7311yescharacteristic
380C4⋊C8—21no—
381C4⋊C8—21no—
382C4⋊C8—731no—
383C4⋊C8—731no—
384C4⋊C8—731no—
385C4⋊C8—731no—
386C4⋊C8—1461no—
387C4⋊C8—1461no—
388C4⋊C8—1461no—
389C4⋊C8—1461no—
390C4⋊C8—1461no—
391C4⋊C8—1461no—
392C4⋊C8—1461no—
393C4⋊C8C22×D7311yescharacteristic
394C4⋊C8C22×D7311yescharacteristic
395#103—731no—
396#103—731no—
397#103—731no—
398#103—731no—
399#103—731no—
400#103—731no—
401#103C4×D7311yescharacteristic
402#103C4×D7311yescharacteristic
403C22⋊C8—731no—
404C22⋊C8—731no—
405C22⋊C8—1461no—
406C22⋊C8—1461no—
407C22⋊C8—1461no—
408#97—21no—
409#97—1461no—
410#97—1461no—
411#97—1461no—
412C4×C8—21no—
413C4×C8—731no—
414C4×C8—731no—
415C4×C8—731no—
416C4×C8—731no—
417C4×C8—731no—
418C4×C8—1461no—
419C4×C8—1461no—
420C4×C8—1461no—
421C4×C8—1461no—
422C4×C8—1461no—
423C4×C8C22×D7311yescharacteristic
424C4×C8C22×D7311yescharacteristic
425C4×C8C22×D7311yescharacteristic
426C2×Q8—734no—
427C2×Q8—731no—
428C2×Q8—731no—
429C2×Q8—1461no—
430C2×Q8—1461no—
431C2×Q8—1462no—
432C2×Q8—1462no—
433C2×Q8—1462no—
434C2×Q8C2×C4×D7311yescharacteristic
435C2×Q8C2×C4×D7311yescharacteristic
436C22×C4—731no—
437C22×C4—731no—
438C22×C4—731no—
439C22×C4—1461no—
440C22×C4—1461no—
441C22×C4—1461no—
442C22×C4—1461no—
443Q16—22no—
444Q16—1462no—
445Q16—1462no—
446Q16—1462no—
447C2×C8—21no—
448C2×C8—21no—
449C2×C8—21no—
450C2×C8—731no—
451C2×C8—731no—
452C2×C8—731no—
453C2×C8—731no—
454C2×C8—1462no—
455C2×C8—1462no—
456C2×C8—1461no—
457C2×C8—1461no—
458C2×C8—1461no—
459C2×C8C2×C4×D7311yescharacteristic
460C2×C8C2×C4×D7311yescharacteristic
461C2×C8D4×D7311yescharacteristic
462C2×C8D4×D7311yescharacteristic
463C4⋊C4—21no—
464C4⋊C4—21no—
465C4⋊C4—731no—
466C4⋊C4—731no—
467C4⋊C4—1461no—
468C4⋊C4—1461no—
469C4⋊C4—1461no—
470C4⋊C4—1461no—
471C4⋊C4—1461no—
472C4⋊C4—1461no—
473C4⋊C4—1461no—
474C4⋊C4—1461no—
475C4⋊C4C2×C4×D7311yescharacteristic
476C4⋊C4C2×C4×D7311yescharacteristic
477C42—21no—
478C42—21no—
479C42—731no—
480C42—731no—
481C42—731no—
482C42—1461no—
483C42—1461no—
484C42—1461no—
485C42—1461no—
486C42—1461no—
487C42—1461no—
488C42C23×D7311yescharacteristic
489C23—731no—
490Q8—21no—
491Q8—21no—
492Q8—21no—
493Q8—2922no—
494Q8—732no—
495Q8—1464no—
496Q8—1461no—
497Q8—1461no—
498Q8—1461no—
499Q8M16×D7312yes—
500C2×C4—21no—
501C2×C4—21no—
502C2×C4—21no—
503C2×C4—21no—
504C2×C4—2921no—
505C2×C4—2922no—
506C2×C4—5841no—
507C2×C4—5841no—
508C2×C4—731no—
509C2×C4—732no—
510C2×C4—732no—
511C2×C4—731no—
512C2×C4—731no—
513C2×C4—1464no—
514C2×C4—1461no—
515C2×C4—1461no—
516C2×C4—1461no—
517C2×C4—1461no—
518C2×C4—1461no—
519C2×C4—1461no—
520C2×C4C22×C4×D7311yescharacteristic
521C2×C4C2×D4×D7311yescharacteristic
522C2×C4#41×D7311yescharacteristic
523C8—21no—
524C8—21no—
525C8—21no—
526C8—21no—
527C8—41no—
528C8—41no—
529C8—41no—
530C8—2921no—
531C8—2921no—
532C8—2921no—
533C8—1461no—
534C8—1461no—
535C8—1461no—
536C8—1461no—
537C22—732no—
538C22—1461no—
539C22—1461no—
540C22C4×D4×D7311yescharacteristic
541C4—21no—
542C4—22no—
543C4—41no—
544C4—41no—
545C4—41no—
546C4—2921no—
547C4—2921no—
548C4—2921no—
549C4—732no—
550C4—1461no—
551C4—1461no—
552C4—1462no—
553C4C2×M16×D7311yescharacteristic
554C4#131×D7311yescharacteristic
555C4#1798940011yescharacteristic
556C4#1798940111yescharacteristic
557C2—1462no—
558C2#435×D7311yescharacteristic
559C2#2261879411yescharacteristic
560C2#2261884511yescharacteristic
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#17880
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. #22608805C2semidirect
5. #22608802C2semidirect
6. #22608800C2semidirect
7. #22608799C2semidirect
8. #22608783C2semidirect
9. #22608782C2semidirect
10. #22608780C2semidirect
15. C73×#1584C2semidirect
291. #1584D73semidirect
296. C73#17880semidirect

Maximal subgroups

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

subgroupindex
#226190132
#226190102
C4×Q16×D732
#226088052
#226088022
#226088002
#226087992
#226087832
#226087822
#226087802
#226086782
#226083242
#226082222
C73×#15842
#1788073