Finite Group Databasebeta

← #34018701

Subgroups — #34018701

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#22626835C211yesmaximal, characteristic
2#22626824C211yesmaximal, characteristic
3#22625556C211yesmaximal, characteristic, complemented
4#22625554C211yesmaximal, characteristic, complemented
5#22625544C211yesmaximal, characteristic, complemented
6#22625542C211yesmaximal, characteristic, complemented
7#22625525C211yesmaximal, characteristic, complemented
8#22625524C211yesmaximal, characteristic, complemented
9#22625518C211yesmaximal, characteristic, complemented
10Q16×#10596833C211yesmaximal, characteristic, complemented
11#22625502C211yesmaximal, characteristic
12#22625273C211yesmaximal, characteristic
13#22625253C211yesmaximal, characteristic
14#22625148C211yesmaximal, characteristic
15#22619965C211yesmaximal, characteristic
16#17989639—21no—
17#17989638C2211yescharacteristic
18#17989637—21no—
19#17989637C2211yescharacteristic
20#17989636C2211yescharacteristic
21Q8×#10596833—21no—
22Q8×#10596833—21no—
23Q8×#10596833C2211yescharacteristic
24Q8×#10596833C2211yescharacteristic
25Q8×#10596833C2211yescharacteristic
26Q8×#10596833C2211yescharacteristic
27Q8×#10596833C2211yescharacteristic
28#17989611—21no—
29#17989611—21no—
30#17989608C2211yescharacteristic
31#17989603—21no—
32#17989603—21no—
33#17989603C2211yescharacteristic
34#17989603C2211yescharacteristic
35C4×#11924037C2211yescharacteristic
36#17989451C2211yescharacteristic
37#17989450C2211yescharacteristic
38#17989449C2211yescharacteristic
39#17989448C2211yescharacteristic
40#17989437C2211yescharacteristic
41#17989437C2211yescharacteristic
42Q8⋊#10596833C2211yescharacteristic
43Q8⋊#10596833C2211yescharacteristic
44Q8⋊#10596833C2211yescharacteristic
45Q8⋊#10596833C2211yescharacteristic
46Q8⋊#10596833C2211yescharacteristic
47Q8⋊#10596833C2211yescharacteristic
48#17989429C2211yescharacteristic
49#17989429C2211yescharacteristic
50#17989428C2211yescharacteristic
51C8×#10596833C2211yescharacteristic
52#17989139C2211yescharacteristic
53C2×Q8×Dic73C2211yescharacteristic
54#17988923C411yescharacteristic
55#17988920C411yescharacteristic
56#17988880C411yescharacteristic
57#17988880C411yescharacteristic
58#17988795C411yescharacteristic
59#17988795C411yescharacteristic
60#17988743C411yescharacteristic
61Q16×Dic73C2211yescharacteristic
62Q16×Dic73C2211yescharacteristic
63#17988679C2211yescharacteristic
64#17988679C2211yescharacteristic
65#17988637C2211yescharacteristic
66#451×C73C411yescharacteristic
67C2×#11924037—21no—
68#14556658—21no—
69#14556658—21no—
70C4⋊#10596833—21no—
71C4⋊#10596833—21no—
72C4⋊#10596833—21no—
73C4⋊#10596833—21no—
74C4⋊#10596833—21no—
75C4⋊#10596833—21no—
76C4⋊#10596833D411yescharacteristic
77C4⋊#10596833D411yescharacteristic
78C4⋊#10596833C2311yescharacteristic
79C4⋊#10596833C2311yescharacteristic
80C4⋊#10596833C2311yescharacteristic
81C4⋊#10596833C2311yescharacteristic
82C22⋊#10596833—21no—
83C22⋊#10596833—21no—
84C22⋊#10596833D411yescharacteristic
85C22⋊#10596833D411yescharacteristic
86#14556646—21no—
87#14556646—21no—
88#14556646—21no—
89#14556646—21no—
90C4×#10596833—21no—
91C4×#10596833—21no—
92C4×#10596833—21no—
93C4×#10596833—21no—
94C4×#10596833—21no—
95C4×#10596833—21no—
96C4×#10596833—21no—
97C4×#10596833C2311yescharacteristic
98C4×#10596833C2311yescharacteristic
99C4×#10596833C2311yescharacteristic
100C4×#10596833C2311yescharacteristic
101C2×#11924018C2×C411yescharacteristic
102C2×#11924018C2×C411yescharacteristic
103#14556590C2×C411yescharacteristic
104#14556589C2×C411yescharacteristic
105#14556581—21no—
106#14556570C2×C411yescharacteristic
107#14556568—21no—
108Q8×Dic73—21no—
109Q8×Dic73—21no—
110Q8×Dic73C2×C411yescharacteristic
111Q8×Dic73C2×C411yescharacteristic
112Q8×Dic73C2311yescharacteristic
113Q8×Dic73C2311yescharacteristic
114Q8×Dic73C2311yescharacteristic
115#14556556—21no—
116#14556556C2311yescharacteristic
117#14556547—21no—
118C2×C4⋊Dic73—21no—
119C2×C4⋊Dic73C2×C411yescharacteristic
120C2×C4×Dic73—21no—
121C2×C4×Dic73C2311yescharacteristic
122#14556525C2×C411yescharacteristic
123#14556525C2×C411yescharacteristic
124#14556524C2×C411yescharacteristic
125#14556524C2×C411yescharacteristic
126#14556520C2×C411yescharacteristic
127#14556520C2×C411yescharacteristic
128#14556516C2×C411yescharacteristic
129#14556516C2×C411yescharacteristic
130#14556515C2×C411yescharacteristic
131#14556515C2×C411yescharacteristic
132#14556515C2×C411yescharacteristic
133#14556515C2×C411yescharacteristic
134C22⋊#10596831C2×C411yescharacteristic
135C8×Dic73C2311yescharacteristic
136C8×Dic73C2311yescharacteristic
137C22×Q8×C73C2×C411yescharacteristic
138C2×Q16×C73C2×C411yescharacteristic
139C2×Q16×C73C2×C411yescharacteristic
140C22⋊Q8×C73C2×C411yescharacteristic
141#103×C73C2×C411yescharacteristic
142#103×C73C2×C411yescharacteristic
143C22⋊C8×C73C2×C411yescharacteristic
144#11924037—42no—
145C2×#10596833—21no—
146C2×#10596833—21no—
147C2×#10596833—21no—
148C2×#10596833—21no—
149C2×#10596833—21no—
150C2×#10596833—21no—
151C2×#10596833C2×D411yescharacteristic
152C2×#10596833C2×D411yescharacteristic
153C2×#10596833C2×D411yescharacteristic
154C2×#10596833C2×D411yescharacteristic
155C2×Dic146—21no—
156C2×Dic146C22×C411yescharacteristic
157C22×Dic73—21no—
158C22×Dic73C2×D411yescharacteristic
159C22×Dic73#4111yescharacteristic
160#11924018—22no—
161#11924018—22no—
162C2×#10596831C22×C411yescharacteristic
163C2×#10596831C22×C411yescharacteristic
164C4⋊Dic73—21no—
165C4⋊Dic73—21no—
166C4⋊Dic73—21no—
167C4⋊Dic73—21no—
168C4⋊Dic73—21no—
169C4⋊Dic73—21no—
170C4⋊Dic73C22×C411yescharacteristic
171C4⋊Dic73C22×C411yescharacteristic
172C4⋊Dic73C22×C411yescharacteristic
173#11924007—21no—
174#11924007—21no—
175#11924007—21no—
176#11924007—21no—
177#11924007—21no—
178C22⋊Dic73—21no—
179C4×Dic73—21no—
180C4×Dic73—21no—
181C4×Dic73—21no—
182C4×Dic73—21no—
183C4×Dic73—21no—
184C4×Dic73—21no—
185C4×Dic73C22×C411yescharacteristic
186C4×Dic73C2×D411yescharacteristic
187C4×Dic73#4111yescharacteristic
188C4×Dic73C2411yescharacteristic
189C2×Q8×C73—22no—
190C2×Q8×C73—22no—
191C2×Q8×C73—21no—
192C2×Q8×C73C22×C411yescharacteristic
193C2×Q8×C73C22×C411yescharacteristic
194C2×Q8×C73C22×C411yescharacteristic
195C22×C4×C73—21no—
196C22×C4×C73C22×C411yescharacteristic
197Q16×C73—22no—
198Q16×C73—22no—
199C2×C8×C73C22×C411yescharacteristic
200C2×C8×C73C22×C411yescharacteristic
201C4⋊C4×C73—21no—
202C4⋊C4×C73C22×C411yescharacteristic
203C22⋊C4×C73—21no—
204C23×C73C4×D411yescharacteristic
205Q8×C73—24no—
206Q8×C73—21no—
207Q8×C73—21no—
208Q8×C73—21no—
209Q8×C73—21no—
210Q8×C73#13112yes—
211Q8×C73#13112yes—
212C2×C4×C73—24no—
213C2×C4×C73—21no—
214C2×C4×C73—21no—
215C2×C4×C73—21no—
216C2×C4×C73—21no—
217C2×C4×C73—21no—
218C2×C4×C73—21no—
219C2×C4×C73C4×D411yescharacteristic
220C2×C4×C73C23×C411yescharacteristic
221C2×Dic73—21no—
222C2×Dic73—21no—
223C2×Dic73—21no—
224C2×Dic73—21no—
225C2×Dic73—21no—
226C2×Dic73—21no—
227C2×Dic73—41no—
228C2×Dic73—81no—
229C2×Dic73C4×D411yescharacteristic
230C2×Dic73C4×D411yescharacteristic
231C2×Dic73C22×D411yescharacteristic
232C2×Dic73C2×#4111yescharacteristic
233Dic146—21no—
234Dic146—21no—
235Dic146—81no—
236#10596833—22no—
237#10596833—21no—
238#10596833—21no—
239#10596833—21no—
240#10596833—21no—
241#10596833—41no—
242#10596833—41no—
243#10596833—41no—
244#10596833—41no—
245#10596833—41no—
246C8×C73—21no—
247C8×C73—21no—
248#10596831—21no—
249#10596831—21no—
250C22×C73—21no—
251C22×C73—21no—
252C22×C73—21no—
253C22×C73C2×C4×D411yescharacteristic
254C4×C73—22no—
255C4×C73—22no—
256C4×C73—21no—
257C4×C73—41no—
258C4×C73—41no—
259C4×C73C2×#13111yescharacteristic
260C4×C73C2×#13111yescharacteristic
261Dic73—21no—
262Dic73—21no—
263Dic73—21no—
264Dic73—41no—
265Dic73—41no—
266Dic73—41no—
267Dic73—41no—
268Dic73#57811yescharacteristic
269Dic73#57811yescharacteristic
270#17925—731nomaximal, Sylow
271C2×C73—41no—
272C2×C73#292911yescharacteristic
273C2×C73#295011yescharacteristic
274C2×C73#295611yescharacteristic
275#3104—731no—
276#2985—731no—
277#2982—731no—
278#1587—731no—
279#1587—731no—
280#1584—731no—
281#1584—731no—
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#6440811yescharacteristic
301#446—731no—
302#446—731no—
303#446—1461no—
304#446—1461no—
305C8×Q8—731no—
306C8×Q8—731no—
307C8×Q8—731no—
308C8×Q8—731no—
309C8×Q8—731no—
310C8×Q8—731no—
311C8×Q8—1461no—
312C8×Q8—1461no—
313C4×Q16—731no—
314C4×Q16—731no—
315C4×Q16—731no—
316C4×Q16—731no—
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#335—731no—
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⋊C8—731no—
337Q8⋊C8—731no—
338#322—731no—
339#322—731no—
340#322—731no—
341C82—731no—
342C22×Q8C2×#6440811yescharacteristic
343C2×Q16—731no—
344C2×Q16—731no—
345C2×Q16C2×#6440811yescharacteristic
346C2×Q16C2×#6440811yescharacteristic
347C2×M16—1461no—
348C4⋊Q8—731no—
349C4⋊Q8—731no—
350#125—1461no—
351C22⋊Q8—731no—
352C22⋊Q8—1461no—
353C22⋊Q8C2×#6440811yescharacteristic
354C4×Q8—731no—
355C4×Q8—731no—
356C4×Q8—731no—
357C4×Q8—731no—
358C4×Q8—731no—
359C4×Q8—731no—
360C4×Q8—1461no—
361C4×Q8—1461no—
362C4×Q8—1461no—
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#107—731no—
372C4⋊C8—731no—
373C4⋊C8—731no—
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⋊C8—1461no—
387C4⋊C8—1461no—
388#103—731no—
389#103—731no—
390#103—731no—
391#103—731no—
392#103—731no—
393#103—731no—
394#103C2×#6440811yescharacteristic
395#103C2×#6440811yescharacteristic
396C22⋊C8—731no—
397C22⋊C8—731no—
398C22⋊C8—731no—
399C22⋊C8—1461no—
400C22⋊C8—1461no—
401C22⋊C8C2×#6440811yescharacteristic
402#97—1461no—
403#97—1461no—
404#97—1461no—
405#97—1461no—
406C4×C8—731no—
407C4×C8—731no—
408C4×C8—731no—
409C4×C8—731no—
410C4×C8—731no—
411C4×C8—731no—
412C4×C8—1461no—
413C4×C8—1461no—
414C4×C8—1461no—
415C4×C8—1461no—
416C4×C8—1461no—
417C4×C8—1461no—
418C4×C8—1461no—
419C2×Q8—22no—
420C2×Q8—22no—
421C2×Q8—21no—
422C2×Q8—731no—
423C2×Q8—1461no—
424C2×Q8C22×#6440811yescharacteristic
425C2×Q8C22×#6440811yescharacteristic
426C2×Q8C22×#6440811yescharacteristic
427C22×C4—21no—
428C22×C4—731no—
429C22×C4—731no—
430C22×C4—1461no—
431C22×C4C22×#6440811yescharacteristic
432Q16—22no—
433Q16—22no—
434Q16—1462no—
435Q16—1462no—
436M16—2922no—
437C2×C8—731no—
438C2×C8—731no—
439C2×C8—731no—
440C2×C8—731no—
441C2×C8—731no—
442C2×C8—731no—
443C2×C8—1461no—
444C2×C8—1461no—
445C2×C8—1461no—
446C2×C8—1461no—
447C2×C8—1461no—
448C2×C8—1461no—
449C2×C8C22×#6440811yescharacteristic
450C2×C8C22×#6440811yescharacteristic
451C4⋊C4—21no—
452C4⋊C4—731no—
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⋊C4—1461no—
466C4⋊C4C22×#6440811yescharacteristic
467C22⋊C4—21no—
468C22⋊C4—1461no—
469C42—731no—
470C42—731no—
471C42—731no—
472C42—731no—
473C42—1461no—
474C42—1461no—
475C42—1461no—
476C42—1461no—
477C42—1461no—
478C42—1461no—
479C23D4×#6440811yescharacteristic
480Q8—24no—
481Q8—21no—
482Q8—21no—
483Q8—21no—
484Q8—21no—
485Q8—5841no—
486Q8—1461no—
487Q8—1461no—
488Q8#1455667812yes—
489Q8#1455667812yes—
490C2×C4—24no—
491C2×C4—21no—
492C2×C4—21no—
493C2×C4—21no—
494C2×C4—21no—
495C2×C4—21no—
496C2×C4—21no—
497C2×C4—2921no—
498C2×C4—5841no—
499C2×C4—731no—
500C2×C4—731no—
501C2×C4—731no—
502C2×C4—731no—
503C2×C4—1461no—
504C2×C4—1461no—
505C2×C4—1461no—
506C2×C4—1461no—
507C2×C4—1461no—
508C2×C4—1461no—
509C2×C4D4×#6440811yescharacteristic
510C2×C4C23×#6440811yescharacteristic
511C8—21no—
512C8—21no—
513C8—2921no—
514C8—2921no—
515C8—2921no—
516C8—2921no—
517C8—2921no—
518C8—1461no—
519C8—1461no—
520C8—1461no—
521C8—1461no—
522C8—1462no—
523C8—1461no—
524C8—1461no—
525C22—21no—
526C22—21no—
527C22—21no—
528C22C2×D4×#6440811yescharacteristic
529C4—22no—
530C4—22no—
531C4—21no—
532C4—41no—
533C4—41no—
534C4—2921no—
535C4—2921no—
536C4—2921no—
537C4—2921no—
538C4—731no—
539C4—731no—
540C4—1461no—
541C4—1461no—
542C4—1461no—
543C4C2×#1455667811yescharacteristic
544C4C2×#1455667811yescharacteristic
545C2—41no—
546C2#2262664611yescharacteristic
547C2#2262669211yescharacteristic
548C2#2262670611yescharacteristic
subgroupG/H
centerC22C2×D4×#64408
commutatorC2×C4×C73C23×C4
FrattiniC2×C4C23×#64408
Fitting#451×C73C4
socleC22×C73C2×C4×D4
prime radicalC23×C73C4×D4
cosocleC4×Dic73C24
hypercenter#451#64408
hypocenterC73#17925
LCSG ▷ C2×C4×C73 ▷ C2×C73 ▷ C73
UCSC1 ◁ C22 ◁ C22×C4 ◁ #451

Complements

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

NH ≅ G/Ntype
3. #22625556C2semidirect
4. #22625554C2semidirect
5. #22625544C2semidirect
6. #22625542C2semidirect
7. #22625525C2semidirect
8. #22625524C2semidirect
9. #22625518C2semidirect
10. Q16×#10596833C2semidirect
290. C73#17925semidirect

Maximal subgroups

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

subgroupindex
#226268352
#226268242
#226255562
#226255542
#226255442
#226255422
#226255252
#226255242
#226255182
Q16×#105968332
#226255022
#226252732
#226252532
#226251482
#226199652
#1792573