Finite Group Databasebeta

← #34019371

Subgroups — #34019371

Order 18688.

566 conjugacy classes of subgroups.

The subgroup lattice has 548 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#22626842C211yesmaximal, characteristic
2#22626832C211yesmaximal, characteristic
3#22625576C211yesmaximal, characteristic, complemented
4#22625569C211yesmaximal, characteristic, complemented
5#22625557C211yesmaximal, characteristic, complemented
6#22625555C211yesmaximal, characteristic, complemented
7#22625545C211yesmaximal, characteristic, complemented
8#22625543C211yesmaximal, characteristic, complemented
9#22625526C211yesmaximal, characteristic, complemented
10#22625519C211yesmaximal, characteristic, complemented
11#22625503C211yesmaximal, characteristic
12#22625295C211yesmaximal, characteristic
13#22625234C211yesmaximal, characteristic
14#22625153C211yesmaximal, characteristic
15#22619972C211yesmaximal, characteristic
16#17989639—21no—
17#17989639—21no—
18#17989638—21no—
19#17989638C2211yescharacteristic
20#17989638C2211yescharacteristic
21#17989637C2211yescharacteristic
22#17989637C2211yescharacteristic
23#17989636C2211yescharacteristic
24#17989636C2211yescharacteristic
25#17989636C2211yescharacteristic
26#17989636C2211yescharacteristic
27Q8×#10596833—21no—
28#17989612—21no—
29#17989612—21no—
30#17989610—21no—
31#17989609C2211yescharacteristic
32#17989606C2211yescharacteristic
33#17989606C2211yescharacteristic
34#17989605—21no—
35#17989605C2211yescharacteristic
36#17989451C2211yescharacteristic
37#17989450C2211yescharacteristic
38#17989449C2211yescharacteristic
39#17989448C2211yescharacteristic
40#17989437C2211yescharacteristic
41#17989437C2211yescharacteristic
42#17989437C2211yescharacteristic
43#17989437C2211yescharacteristic
44#17989437C2211yescharacteristic
45#17989437C2211yescharacteristic
46Q8⋊#10596833C2211yescharacteristic
47Q8⋊#10596833C2211yescharacteristic
48#17989429C2211yescharacteristic
49#17989429C2211yescharacteristic
50#17989428C2211yescharacteristic
51C8×#10596833C2211yescharacteristic
52#17989142C2211yescharacteristic
53C4×Q8×D73C2211yescharacteristic
54#17988924C411yescharacteristic
55#17988924C411yescharacteristic
56#17988879C411yescharacteristic
57#17988798C411yescharacteristic
58#17988797C411yescharacteristic
59#17988744C411yescharacteristic
60#17988744C411yescharacteristic
61#17988681C2211yescharacteristic
62#17988681C2211yescharacteristic
63#17988679C2211yescharacteristic
64#17988679C2211yescharacteristic
65#17988639C2211yescharacteristic
66#477×C73C411yescharacteristic
67#14556658—21no—
68#14556658—21no—
69#14556658—21no—
70#14556658—21no—
71#14556658—21no—
72#14556658—21no—
73#14556658—21no—
74C4⋊#10596833—21no—
75C4⋊#10596833—21no—
76C4⋊#10596833D411yescharacteristic
77C4⋊#10596833D411yescharacteristic
78C4⋊#10596833C2311yescharacteristic
79C4⋊#10596833C2311yescharacteristic
80C4⋊#10596833C2311yescharacteristic
81C4⋊#10596833C2311yescharacteristic
82#14556648—21no—
83#14556648—21no—
84#14556648—21no—
85#14556648D411yescharacteristic
86#14556648D411yescharacteristic
87#14556646—21no—
88#14556646—21no—
89#14556646—21no—
90#14556646—21no—
91#14556646—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×Q8×D73C2×C411yescharacteristic
102C2×Dic292C2×C411yescharacteristic
103C2×Dic292C2×C411yescharacteristic
104C2×#11924018C2×C411yescharacteristic
105C2×#11924018C2×C411yescharacteristic
106#14556591C2×C411yescharacteristic
107#14556582—21no—
108#14556571C2×C411yescharacteristic
109#14556571C2×C411yescharacteristic
110#14556567—21no—
111Q8×Dic73—21no—
112Q8×Dic73C2311yescharacteristic
113C4×Dic146—21no—
114C4×Dic146C2×C411yescharacteristic
115#14556556—21no—
116#14556556C2311yescharacteristic
117#14556556C2311yescharacteristic
118#14556556C2311yescharacteristic
119#14556549—21no—
120C4⋊C4×D73—21no—
121C4⋊C4×D73C2×C411yescharacteristic
122C42×D73—21no—
123C42×D73C2311yescharacteristic
124#14556524C2×C411yescharacteristic
125#14556524C2×C411yescharacteristic
126C4⋊#10596831C2×C411yescharacteristic
127#14556516C2×C411yescharacteristic
128#14556516C2×C411yescharacteristic
129#14556514C2×C411yescharacteristic
130#14556514C2×C411yescharacteristic
131#14556514C2×C411yescharacteristic
132#14556514C2×C411yescharacteristic
133#14556503C2×C411yescharacteristic
134#14556503C2×C411yescharacteristic
135C8×Dic73C2311yescharacteristic
136C8×Dic73C2311yescharacteristic
137C4⋊Q8×C73C2×C411yescharacteristic
138C4×Q8×C73C2×C411yescharacteristic
139#107×C73C2×C411yescharacteristic
140#107×C73C2×C411yescharacteristic
141C4⋊C8×C73C2×C411yescharacteristic
142#103×C73C2×C411yescharacteristic
143#103×C73C2×C411yescharacteristic
144C2×#10596833—21no—
145C2×#10596833—21no—
146C2×#10596833—21no—
147C2×#10596833—21no—
148C2×#10596833—21no—
149C2×#10596833—21no—
150C2×#10596833C2×D411yescharacteristic
151C2×#10596833C2×D411yescharacteristic
152C2×#10596833C2×D411yescharacteristic
153C2×#10596833C2×D411yescharacteristic
154C2×Dic146—21no—
155C2×Dic146C22×C411yescharacteristic
156C2×Dic146C22×C411yescharacteristic
157C2×Dic146C22×C411yescharacteristic
158Q8×D73—22no—
159Q8×D73—41no—
160C2×C4×D73—21no—
161C2×C4×D73—21no—
162C2×C4×D73C22×C411yescharacteristic
163C2×C4×D73C2×D411yescharacteristic
164C2×C4×D73#4111yescharacteristic
165Dic292—22no—
166Dic292—22no—
167#11924018—41no—
168#11924018—41no—
169C2×#10596831C22×C411yescharacteristic
170C2×#10596831C22×C411yescharacteristic
171C4⋊Dic73—21no—
172C4⋊Dic73—21no—
173C4⋊Dic73—21no—
174C4⋊Dic73—21no—
175C4⋊Dic73—21no—
176C4⋊Dic73C22×C411yescharacteristic
177#11924007—21no—
178#11924007—21no—
179#11924007—21no—
180#11924007—21no—
181#11924007—21no—
182#11924005—21no—
183#11924005—21no—
184C4×Dic73—21no—
185C4×Dic73—21no—
186C4×Dic73—21no—
187C4×Dic73—21no—
188C4×Dic73—21no—
189C4×Dic73C2×D411yescharacteristic
190C4×Dic73#4111yescharacteristic
191C4×Dic73C2411yescharacteristic
192C2×Q8×C73—21no—
193C2×Q8×C73C22×C411yescharacteristic
194C2×C8×C73C22×C411yescharacteristic
195C2×C8×C73C22×C411yescharacteristic
196C4⋊C4×C73—21no—
197C4⋊C4×C73—21no—
198C4⋊C4×C73C22×C411yescharacteristic
199C4⋊C4×C73C22×C411yescharacteristic
200C4⋊C4×C73C22×C411yescharacteristic
201C42×C73—21no—
202C42×C73C22×C411yescharacteristic
203C22×D73C4×D411yescharacteristic
204Q8×C73—21no—
205Q8×C73—21no—
206Q8×C73—42no—
207C2×C4×C73—21no—
208C2×C4×C73—21no—
209C2×C4×C73—21no—
210C2×C4×C73—21no—
211C2×C4×C73C4×D411yescharacteristic
212C2×C4×C73C4×D411yescharacteristic
213C2×C4×C73C23×C411yescharacteristic
214C2×Dic73—21no—
215C2×Dic73—21no—
216C2×Dic73—21no—
217C2×Dic73—21no—
218C2×Dic73C4×D411yescharacteristic
219C2×Dic73C22×D411yescharacteristic
220C2×Dic73C2×#4111yescharacteristic
221C4×D73—21no—
222C4×D73—21no—
223C4×D73—21no—
224C4×D73—21no—
225C4×D73—41no—
226C4×D73—42no—
227C4×D73—42no—
228Dic146—21no—
229Dic146—21no—
230Dic146—21no—
231Dic146—21no—
232Dic146—21no—
233Dic146—42no—
234Dic146#13112yes—
235#10596833—21no—
236#10596833—21no—
237#10596833—21no—
238#10596833—21no—
239#10596833—41no—
240#10596833—41no—
241#10596833—41no—
242#10596833—41no—
243#10596833—41no—
244#10596833—41no—
245C8×C73—21no—
246C8×C73—21no—
247#10596831—21no—
248#10596831—21no—
249C22×C73C2×C4×D411yescharacteristic
250C2×D73—21no—
251C2×D73—21no—
252C2×D73—21no—
253C4×C73—21no—
254C4×C73—21no—
255C4×C73—41no—
256C4×C73—41no—
257C4×C73—41no—
258C4×C73—41no—
259C4×C73C2×#13111yescharacteristic
260C4×C73#56811yescharacteristic
261Dic73—22no—
262Dic73—21no—
263Dic73—21no—
264Dic73—41no—
265Dic73—41no—
266Dic73—41no—
267Dic73#57811yescharacteristic
268Dic73#57811yescharacteristic
269#18032—731nomaximal, Sylow
270C2×C73#293311yescharacteristic
271C2×C73#295011yescharacteristic
272C2×C73#295511yescharacteristic
273D73—41no—
274#3104—731no—
275#2986—731no—
276#2984—731no—
277#1593—731no—
278#1591—731no—
279#1587—731no—
280#1587—731no—
281#1584—731no—
282#1584—731no—
283#1579—731no—
284C8×Q16—731no—
285#1572—731no—
286#1497—731no—
287#1479—731no—
288#1454—731no—
289C73#1803211yescharacteristic, complemented, Sylow
290#533—731no—
291C2×C4×Q8—731no—
292#484—731no—
293#484—731no—
294#477—731no—
295#477#6440811yescharacteristic
296#462—731no—
297#462—731no—
298#451—731no—
299#451—731no—
300#446—731no—
301#446—731no—
302#446—731no—
303#446—731no—
304#446—1461no—
305#446—1461no—
306#446—1461no—
307C8×Q8—731no—
308C8×Q8—731no—
309C8×Q8—731no—
310C8×Q8—731no—
311C8×Q8—1461no—
312C4×Q16—731no—
313C4×Q16—731no—
314C4×Q16—731no—
315C4×Q16—731no—
316#433—1461no—
317#433—1461no—
318#432—731no—
319#432—1461no—
320#431—731no—
321#431—731no—
322#431—731no—
323#431—731no—
324#431—1461no—
325#335—731no—
326#335—731no—
327#334—731no—
328#334—731no—
329Q8⋊C8—731no—
330Q8⋊C8—731no—
331Q8⋊C8—731no—
332Q8⋊C8—731no—
333Q8⋊C8—731no—
334Q8⋊C8—731no—
335Q8⋊C8—731no—
336Q8⋊C8—731no—
337#322—731no—
338#322—731no—
339#322—731no—
340C82—731no—
341C22×Q8—731no—
342C2×Q16—731no—
343C2×Q16—731no—
344C2×Q16—731no—
345C2×Q16—731no—
346C4⋊Q8—731no—
347C4⋊Q8C2×#6440811yescharacteristic
348#125—1461no—
349C22⋊Q8—731no—
350C22⋊Q8—731no—
351C22⋊Q8—1461no—
352C4×Q8—731no—
353C4×Q8—731no—
354C4×Q8—731no—
355C4×Q8—731no—
356C4×Q8—731no—
357C4×Q8—1461no—
358C4×Q8—1461no—
359C4×Q8—1461no—
360C4×Q8C2×#6440811yescharacteristic
361#117—1461no—
362C2×C4⋊C4—731no—
363C2×C4⋊C4—1461no—
364C2×C42—731no—
365C2×C42—1461no—
366#107—731no—
367#107—731no—
368#107C2×#6440811yescharacteristic
369#107C2×#6440811yescharacteristic
370C4⋊C8—731no—
371C4⋊C8—731no—
372C4⋊C8—731no—
373C4⋊C8—731no—
374C4⋊C8—731no—
375C4⋊C8—731no—
376C4⋊C8—731no—
377C4⋊C8—1461no—
378C4⋊C8—1461no—
379C4⋊C8—1461no—
380C4⋊C8—1461no—
381C4⋊C8—1461no—
382C4⋊C8—1461no—
383C4⋊C8—1461no—
384C4⋊C8—1461no—
385C4⋊C8—1461no—
386C4⋊C8C2×#6440811yescharacteristic
387#103—731no—
388#103—731no—
389#103—731no—
390#103—731no—
391#103—731no—
392#103—731no—
393#103C2×#6440811yescharacteristic
394#103C2×#6440811yescharacteristic
395C22⋊C8—731no—
396C22⋊C8—731no—
397C22⋊C8—731no—
398C22⋊C8—731no—
399C22⋊C8—1461no—
400C22⋊C8—1461no—
401C22⋊C8—1461no—
402#97—1461no—
403#97—1461no—
404#97—1461no—
405#97—1461no—
406#97—1461no—
407C4×C8—731no—
408C4×C8—731no—
409C4×C8—731no—
410C4×C8—731no—
411C4×C8—731no—
412C4×C8—731no—
413C4×C8—1461no—
414C4×C8—1461no—
415C4×C8—1461no—
416C4×C8—1461no—
417C4×C8—1461no—
418C2×Q8—21no—
419C2×Q8—2921no—
420C2×Q8—731no—
421C2×Q8—731no—
422C2×Q8—731no—
423C2×Q8—1461no—
424C2×Q8—1462no—
425C2×Q8C22×#6440811yescharacteristic
426C22×C4—731no—
427C22×C4—731no—
428C22×C4—731no—
429C22×C4—1461no—
430C22×C4—1461no—
431Q16—2921no—
432Q16—2921no—
433Q16—1462no—
434Q16—1462no—
435C2×C8—731no—
436C2×C8—731no—
437C2×C8—731no—
438C2×C8—731no—
439C2×C8—731no—
440C2×C8—731no—
441C2×C8—1461no—
442C2×C8—1461no—
443C2×C8—1461no—
444C2×C8—1461no—
445C2×C8—1461no—
446C2×C8—1461no—
447C2×C8C22×#6440811yescharacteristic
448C2×C8C22×#6440811yescharacteristic
449C4⋊C4—21no—
450C4⋊C4—21no—
451C4⋊C4—731no—
452C4⋊C4—1461no—
453C4⋊C4—1461no—
454C4⋊C4—1461no—
455C4⋊C4—1461no—
456C4⋊C4—1461no—
457C4⋊C4—1461no—
458C4⋊C4—1461no—
459C4⋊C4—1461no—
460C4⋊C4—1461no—
461C4⋊C4—1461no—
462C4⋊C4C22×#6440811yescharacteristic
463C4⋊C4C22×#6440811yescharacteristic
464C4⋊C4C22×#6440811yescharacteristic
465C22⋊C4—1461no—
466C22⋊C4—1461no—
467C42—21no—
468C42—731no—
469C42—731no—
470C42—731no—
471C42—1461no—
472C42—1461no—
473C42—1461no—
474C42—1461no—
475C42—1461no—
476C42C22×#6440811yescharacteristic
477C23—731no—
478Q8—21no—
479Q8—21no—
480Q8—42no—
481Q8—2922no—
482Q8—732no—
483Q8—1461no—
484Q8—1461no—
485Q8—1461no—
486Q8—1461no—
487Q8—1461no—
488C2×C4—21no—
489C2×C4—21no—
490C2×C4—21no—
491C2×C4—21no—
492C2×C4—2921no—
493C2×C4—2922no—
494C2×C4—2922no—
495C2×C4—731no—
496C2×C4—731no—
497C2×C4—731no—
498C2×C4—1461no—
499C2×C4—1461no—
500C2×C4—1461no—
501C2×C4—1461no—
502C2×C4—1461no—
503C2×C4—1461no—
504C2×C4—1461no—
505C2×C4—1461no—
506C2×C4D4×#6440811yescharacteristic
507C2×C4D4×#6440811yescharacteristic
508C2×C4C23×#6440811yescharacteristic
509C8—21no—
510C8—21no—
511C8—2921no—
512C8—2921no—
513C8—2921no—
514C8—2921no—
515C8—2921no—
516C8—2921no—
517C8—1461no—
518C8—1461no—
519C8—1461no—
520C8—1461no—
521C8—1461no—
522C8—1461no—
523C22—1461no—
524C22—1461no—
525C22—1461no—
526C22C2×D4×#6440811yescharacteristic
527C4—21no—
528C4—21no—
529C4—41no—
530C4—41no—
531C4—41no—
532C4—41no—
533C4—2921no—
534C4—2921no—
535C4—2921no—
536C4—731no—
537C4—731no—
538C4—1462no—
539C4—1461no—
540C4—1461no—
541C4D73⋊#13111yescharacteristic
542C4#1798966711yescharacteristic
543C2—2921no—
544C2#2262666611yescharacteristic
545C2#2262669211yescharacteristic
546C2#2262670311yescharacteristic
subgroupG/H
centerC22C2×D4×#64408
commutatorC2×C4×C73C23×C4
FrattiniC2×C4C23×#64408
Fitting#477×C73C4
socleC22×C73C2×C4×D4
prime radicalC22×D73C4×D4
cosocleC4×Dic73C24
hypercenter#477#64408
hypocenterC73#18032
LCSG ▷ C2×C4×C73 ▷ C2×C73 ▷ C73
UCSC1 ◁ C22 ◁ C42 ◁ #477

Complements

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

NH ≅ G/Ntype
3. #22625576C2semidirect
4. #22625569C2semidirect
5. #22625557C2semidirect
6. #22625555C2semidirect
7. #22625545C2semidirect
8. #22625543C2semidirect
9. #22625526C2semidirect
10. #22625519C2semidirect
289. C73#18032semidirect

Maximal subgroups

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

subgroupindex
#226268422
#226268322
#226255762
#226255692
#226255572
#226255552
#226255452
#226255432
#226255262
#226255192
#226255032
#226252952
#226252342
#226251532
#226199722
#1803273