Finite Group Databasebeta

← #34018466

Subgroups — #34018466

Order 18688.

660 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#22626850C211yesmaximal, characteristic, complemented
2#22626846C211yesmaximal, characteristic, complemented
3#22625556C212yesmaximal, complemented
4#22625552C212yesmaximal, complemented
5#22625540C212yesmaximal, complemented
6#22625536C212yesmaximal, complemented
7C2×#17989449C211yesmaximal, characteristic, complemented
8#22625226C211yesmaximal, characteristic, complemented
9#22625226C211yesmaximal, characteristic, complemented
10#22625144C211yesmaximal, characteristic, complemented
11#135×Dic73C211yesmaximal, characteristic
12#17989638C2212yes—
13Q8×#10596833C2212yes—
14#17989622C2212yes—
15#17989618—22no—
16D4×#10596833—22no—
17D4×#10596833C2212yes—
18#17989608—21no—
19#17989603—21no—
20#17989603C2211yescharacteristic
21#17989603C2211yescharacteristic
22#17989590—21no—
23C2×C4⋊#10596833—21no—
24C2×C4⋊#10596833C2211yescharacteristic
25C2×C4⋊#10596833C2211yescharacteristic
26#17989449C2212yes—
27#17989449C2212yes—
28Q8⋊#10596833C2212yes—
29Q8⋊#10596833C2212yes—
30D4⋊#10596833C2212yes—
31D4⋊#10596833C2212yes—
32#17989429C2212yes—
33#17989429C2212yes—
34#41×Dic73C2211yescharacteristic
35#41×Dic73C2211yescharacteristic
36Q16×Dic73C411yescharacteristic
37Q16×Dic73C2211yescharacteristic
38SD16×Dic73C411yescharacteristic
39SD16×Dic73C411yescharacteristic
40SD16×Dic73C2211yescharacteristic
41SD16×Dic73C2211yescharacteristic
42D8×Dic73C411yescharacteristic
43D8×Dic73C2211yescharacteristic
44#17988625C411yescharacteristic
45#17988583C411yescharacteristic
46#17988583C411yescharacteristic
47C2×C8×Dic73C2211yescharacteristic
48C2×#135×C73C411yescharacteristic
49C2×#11924037—22no—
50C22×#10596833—22no—
51C22×#10596833—21no—
52C22×#10596833—21no—
53#14556658—22no—
54C4⋊#10596833—22no—
55C4⋊#10596833—21no—
56C4⋊#10596833—21no—
57C4⋊#10596833—21no—
58C4⋊#10596833D411yescharacteristic
59C4⋊#10596833D411yescharacteristic
60C4⋊#10596833C2312yes—
61C4⋊#10596833C2312yes—
62C22⋊#10596833—22no—
63C22⋊#10596833—22no—
64C22⋊#10596833—21no—
65C22⋊#10596833D412yes—
66#14556646—22no—
67C4×#10596833—22no—
68C4×#10596833C2312yes—
69C4×#10596833C2312yes—
70C22×#10596831C2×C411yescharacteristic
71Q8×Dic73C2×C411yescharacteristic
72Q8×Dic73C2×C411yescharacteristic
73Q8×Dic73C2311yescharacteristic
74Q8×Dic73C2311yescharacteristic
75D4×Dic73—21no—
76D4×Dic73—21no—
77D4×Dic73—21no—
78D4×Dic73—21no—
79D4×Dic73C2×C411yescharacteristic
80D4×Dic73C2×C411yescharacteristic
81D4×Dic73C2311yescharacteristic
82D4×Dic73C2311yescharacteristic
83#14556550—21no—
84#14556550—21no—
85#14556550C2×C411yescharacteristic
86#14556550C2×C411yescharacteristic
87C2×C4×Dic73—21no—
88C2×C4×Dic73—21no—
89C2×C4×Dic73C2311yescharacteristic
90#14556525C2×C411yescharacteristic
91#14556525C2×C411yescharacteristic
92#14556523C2×C411yescharacteristic
93#14556523C2×C411yescharacteristic
94#14556515C2×C411yescharacteristic
95#14556515C2×C411yescharacteristic
96#14556515C2×C411yescharacteristic
97#14556515C2×C411yescharacteristic
98#14556512C2×C411yescharacteristic
99#14556512C2×C411yescharacteristic
100#14556512C2×C411yescharacteristic
101#14556512C2×C411yescharacteristic
102C8×Dic73C2×C411yescharacteristic
103C8×Dic73C2×C411yescharacteristic
104C8×Dic73C2311yescharacteristic
105C8×Dic73C2311yescharacteristic
106C2×#41×C73C2×C411yescharacteristic
107C2×#41×C73C2×C411yescharacteristic
108#135×C73—24no—
109C2×Q16×C73C2×C411yescharacteristic
110C2×SD16×C73C2×C411yescharacteristic
111C2×SD16×C73C2×C411yescharacteristic
112C2×D8×C73C2×C411yescharacteristic
113C22×C8×C73C2×C411yescharacteristic
114#11924037—82no—
115C2×#10596833—22no—
116C2×#10596833—22no—
117C2×#10596833—22no—
118C2×#10596833—82no—
119C2×#10596833—81no—
120C2×#10596833—81no—
121C2×#10596833C2×D412yes—
122C2×#10596833C2×D412yes—
123C22×Dic73—21no—
124C22×Dic73—21no—
125C22×Dic73—21no—
126C22×Dic73—21no—
127C22×Dic73C2×D411yescharacteristic
128C22×Dic73#4111yescharacteristic
129C2×#10596831—22no—
130C2×#10596831C22×C411yescharacteristic
131C2×#10596831C22×C411yescharacteristic
132C4⋊Dic73—21no—
133C4⋊Dic73—21no—
134C4⋊Dic73—21no—
135C4⋊Dic73—21no—
136C4⋊Dic73C22×C411yescharacteristic
137C4⋊Dic73C22×C411yescharacteristic
138C4⋊Dic73C22×C411yescharacteristic
139C4⋊Dic73C22×C411yescharacteristic
140C22⋊Dic73—21no—
141C22⋊Dic73—21no—
142C22⋊Dic73—21no—
143C22⋊Dic73—21no—
144C22⋊Dic73—21no—
145C22⋊Dic73—21no—
146C4×Dic73—21no—
147C4×Dic73—21no—
148C4×Dic73—21no—
149C4×Dic73—21no—
150C4×Dic73—21no—
151C4×Dic73—21no—
152C4×Dic73C22×C411yescharacteristic
153C4×Dic73C2×D411yescharacteristic
154C4×Dic73#4111yescharacteristic
155C4×Dic73C2411yescharacteristic
156#41×C73—22no—
157#41×C73—22no—
158#41×C73—22no—
159#41×C73M1614yescomplemented
160C2×Q8×C73C22×C411yescharacteristic
161C2×Q8×C73C22×C411yescharacteristic
162C2×D4×C73—21no—
163C2×D4×C73—21no—
164C2×D4×C73C22×C411yescharacteristic
165C2×D4×C73C22×C411yescharacteristic
166C22×C4×C73—21no—
167C22×C4×C73—21no—
168C22×C4×C73C22×C411yescharacteristic
169Q16×C73—22no—
170SD16×C73—22no—
171SD16×C73—22no—
172D8×C73—22no—
173C2×C8×C73—22no—
174C2×C8×C73C22×C411yescharacteristic
175C2×C8×C73C22×C411yescharacteristic
176C23×C73—21no—
177C23×C73—21no—
178C23×C73C4×D411yescharacteristic
179Q8×C73—21no—
180Q8×C73—21no—
181Q8×C73—21no—
182Q8×C73C2×M1612yes—
183D4×C73—24no—
184D4×C73—21no—
185D4×C73—21no—
186D4×C73—21no—
187D4×C73—42no—
188D4×C73C2×M1612yes—
189C2×C4×C73—21no—
190C2×C4×C73—24no—
191C2×C4×C73—21no—
192C2×C4×C73—42no—
193C2×C4×C73C4×D411yescharacteristic
194C2×C4×C73C2×M1612yes—
195C2×C4×C73#13112yes—
196C2×C4×C73C23×C411yescharacteristic
197C2×Dic73—21no—
198C2×Dic73—21no—
199C2×Dic73—21no—
200C2×Dic73—21no—
201C2×Dic73—21no—
202C2×Dic73—21no—
203C2×Dic73—41no—
204C2×Dic73—41no—
205C2×Dic73—41no—
206C2×Dic73—41no—
207C2×Dic73—41no—
208C2×Dic73—81no—
209C2×Dic73—81no—
210C2×Dic73C4×D411yescharacteristic
211C2×Dic73C4×D411yescharacteristic
212C2×Dic73C22×D411yescharacteristic
213C2×Dic73C2×#4111yescharacteristic
214#10596833—22no—
215#10596833—22no—
216#10596833—42no—
217#10596833—42no—
218#10596833—42no—
219C8×C73—21no—
220C8×C73—21no—
221#10596831—21no—
222#10596831—21no—
223C22×C73—21no—
224C22×C73—21no—
225C22×C73—22no—
226C22×C73—41no—
227C22×C73—41no—
228C22×C73—41no—
229C22×C73—41no—
230C22×C73—41no—
231C22×C73#43512yes—
232C22×C73C2×C4×D411yescharacteristic
233C4×C73—22no—
234C4×C73—41no—
235C4×C73#43512yes—
236C4×C73C22×M1611yescharacteristic
237C4×C73C2×#13111yescharacteristic
238Dic73—21no—
239Dic73—21no—
240Dic73—21no—
241Dic73—41no—
242Dic73—41no—
243Dic73—41no—
244Dic73—41no—
245Dic73#57711yescharacteristic
246Dic73#57711yescharacteristic
247#17882—731nomaximal, Sylow
248C2×C73—22no—
249C2×C73—42no—
250C2×C73—81no—
251C2×C73C2×#43511yescharacteristic
252C2×C73#295211yescharacteristic
253C2×C73#295711yescharacteristic
254#2989—731no—
255#2988—731no—
256C4×#135—731no—
257#1587—732no—
258#1586—732no—
259#1583—732no—
260#1582—732no—
261C2×#334—731no—
262#1477—731no—
263#1477—731no—
264#1452—731no—
265C73#1788211yescharacteristic, complemented, Sylow
266C2×#135#6440811yescharacteristic
267C4×#41—731no—
268C4×#41—731no—
269#446—732no—
270C8×Q8—732no—
271C4×Q16—731no—
272C4×Q16—731no—
273C4×SD16—731no—
274C4×SD16—731no—
275C4×SD16—731no—
276C4×SD16—731no—
277C4×D8—731no—
278C4×D8—731no—
279#436—732no—
280#435—1462no—
281C8×D4—732no—
282C8×D4—1462no—
283#432—1461no—
284#431—731no—
285#431—731no—
286#431—1461no—
287#427—731no—
288#423—1461no—
289C2×C4⋊C8—731no—
290C2×C4⋊C8—731no—
291C2×C4⋊C8—1461no—
292#416—731no—
293#416—731no—
294C2×C4×C8—731no—
295#334—732no—
296#334—732no—
297Q8⋊C8—732no—
298Q8⋊C8—732no—
299D4⋊C8—732no—
300D4⋊C8—732no—
301#322—732no—
302#322—732no—
303C2×#41C2×#6440811yescharacteristic
304C2×#41C2×#6440811yescharacteristic
305#135—24no—
306C2×Q16C2×#6440811yescharacteristic
307C2×SD16C2×#6440811yescharacteristic
308C2×SD16C2×#6440811yescharacteristic
309C2×D8C2×#6440811yescharacteristic
310C2×M16—1462no—
311C22×C8—731no—
312C22×C8—1462no—
313C22×C8—1461no—
314C22×C8—1461no—
315C22×C8C2×#6440811yescharacteristic
316C4×Q8—731no—
317C4×Q8—731no—
318C4×Q8—731no—
319C4×Q8—731no—
320C4×D4—731no—
321C4×D4—731no—
322C4×D4—731no—
323C4×D4—731no—
324C4×D4—1461no—
325C4×D4—1461no—
326C4×D4—1461no—
327C4×D4—1461no—
328#117—731no—
329#117—731no—
330#117—1461no—
331#117—1461no—
332C2×C42—731no—
333C2×C42—1461no—
334C2×C42—1461no—
335#107—731no—
336#107—731no—
337#106—731no—
338#106—731no—
339C4⋊C8—731no—
340C4⋊C8—731no—
341C4⋊C8—732no—
342C4⋊C8—732no—
343C4⋊C8—1461no—
344C4⋊C8—1461no—
345C4⋊C8—1461no—
346C4⋊C8—1462no—
347C4⋊C8—1462no—
348#103—731no—
349#103—731no—
350#103—731no—
351#103—731no—
352#102—731no—
353#102—731no—
354#102—731no—
355#102—731no—
356C22⋊C8—732no—
357C22⋊C8—1462no—
358C22⋊C8—1462no—
359C22⋊C8—1461no—
360#97—1462no—
361C4×C8—732no—
362C4×C8—731no—
363C4×C8—731no—
364C4×C8—731no—
365C4×C8—731no—
366C4×C8—732no—
367C4×C8—1462no—
368#41—22no—
369#41—22no—
370#41—22no—
371#41#1192403714yescomplemented
372C2×Q8C22×#6440811yescharacteristic
373C2×Q8C22×#6440811yescharacteristic
374C2×D4—21no—
375C2×D4—21no—
376C2×D4C22×#6440811yescharacteristic
377C2×D4C22×#6440811yescharacteristic
378C22×C4—21no—
379C22×C4—21no—
380C22×C4—731no—
381C22×C4—731no—
382C22×C4—1461no—
383C22×C4—1461no—
384C22×C4—1461no—
385C22×C4—1461no—
386C22×C4C22×#6440811yescharacteristic
387Q16—22no—
388SD16—22no—
389SD16—22no—
390D8—22no—
391M16—5842no—
392C2×C8—22no—
393C2×C8—5841no—
394C2×C8—5841no—
395C2×C8—5842no—
396C2×C8—732no—
397C2×C8—732no—
398C2×C8—731no—
399C2×C8—731no—
400C2×C8—1462no—
401C2×C8—1462no—
402C2×C8—1462no—
403C2×C8—1462no—
404C2×C8C22×#6440811yescharacteristic
405C2×C8C22×#6440811yescharacteristic
406C4⋊C4—731no—
407C4⋊C4—731no—
408C4⋊C4—731no—
409C4⋊C4—731no—
410C4⋊C4—1461no—
411C4⋊C4—1461no—
412C4⋊C4—1461no—
413C4⋊C4—1461no—
414C22⋊C4—1461no—
415C22⋊C4—1461no—
416C22⋊C4—1461no—
417C22⋊C4—1461no—
418C22⋊C4—1461no—
419C22⋊C4—1461no—
420C42—731no—
421C42—731no—
422C42—731no—
423C42—731no—
424C42—1461no—
425C42—1461no—
426C42—1461no—
427C42—1461no—
428C42—1461no—
429C42—1461no—
430C23—21no—
431C23—21no—
432C23D4×#6440811yescharacteristic
433Q8—21no—
434Q8—21no—
435Q8—21no—
436Q8C2×#1192403712yes—
437D4—24no—
438D4—21no—
439D4—21no—
440D4—21no—
441D4—42no—
442D4C2×#1192403712yes—
443C2×C4—21no—
444C2×C4—24no—
445C2×C4—21no—
446C2×C4—42no—
447C2×C4—2921no—
448C2×C4—2921no—
449C2×C4—2921no—
450C2×C4—2921no—
451C2×C4—2921no—
452C2×C4—5841no—
453C2×C4—5841no—
454C2×C4—731no—
455C2×C4—731no—
456C2×C4—731no—
457C2×C4—731no—
458C2×C4—1461no—
459C2×C4—1461no—
460C2×C4—1461no—
461C2×C4—1461no—
462C2×C4—1461no—
463C2×C4—1461no—
464C2×C4D4×#6440811yescharacteristic
465C2×C4C2×#1192403712yes—
466C2×C4#1455667812yes—
467C2×C4C23×#6440811yescharacteristic
468C8—21no—
469C8—21no—
470C8—2922no—
471C8—2922no—
472C8—2922no—
473C8—1462no—
474C8—1462no—
475C8—1461no—
476C8—1461no—
477C22—21no—
478C22—21no—
479C22—22no—
480C22—41no—
481C22—41no—
482C22—41no—
483C22—41no—
484C22—41no—
485C22#1798961812yes—
486C22C2×D4×#6440811yescharacteristic
487C4—22no—
488C4—41no—
489C4—2921no—
490C4—2921no—
491C4—2921no—
492C4—2921no—
493C4—731no—
494C4—731no—
495C4—1461no—
496C4—1461no—
497C4—1461no—
498C4#1798961812yes—
499C4C22×#1192403711yescharacteristic
500C4C2×#1455667811yescharacteristic
501C2—22no—
502C2—42no—
503C2—81no—
504C2C2×#1798961811yescharacteristic
505C2#2262669411yescharacteristic
506C2#2262670811yescharacteristic
subgroupG/H
centerC22C2×D4×#64408
commutatorC2×C4×C73C23×C4
FrattiniC2×C4C23×#64408
Fitting, prime radicalC2×#135×C73C4
socleC22×C73C2×C4×D4
cosocleC4×Dic73C24
hypercenterC2×#135#64408
hypocenterC73#17882
LCSG ▷ C2×C4×C73 ▷ C2×C73 ▷ C73
UCSC1 ◁ C22 ◁ C22×C4 ◁ C2×#135

Complements

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

NH ≅ G/Ntype
1. #22626850C2semidirect
2. #22626846C2semidirect
3. #22625556C2semidirect
4. #22625552C2semidirect
5. #22625540C2semidirect
6. #22625536C2semidirect
7. C2×#17989449C2semidirect
8. #22625226C2semidirect
9. #22625226C2semidirect
10. #22625144C2semidirect
159. #41×C73M16semidirect
265. C73#17882semidirect
371. #41#11924037semidirect

Maximal subgroups

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

subgroupindex
#226268502
#226268462
#226255562
#226255522
#226255402
#226255362
C2×#179894492
#226252262
#226251442
#135×Dic732
#1788273