Finite Group Databasebeta

← #34019572

Subgroups — #34019572

Order 18688.

650 conjugacy classes of subgroups.

The subgroup lattice has 510 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#22626863C211yesmaximal, characteristic, complemented
2#22626857C211yesmaximal, characteristic, complemented
3#22625574C212yesmaximal, complemented
4#22625544C212yesmaximal, complemented
5#22625532C212yesmaximal, complemented
6SD16⋊#10596833C212yesmaximal, complemented
7#22625488C211yesmaximal, characteristic, complemented
8#22625266C211yesmaximal, characteristic, complemented
9#22625266C211yesmaximal, characteristic, complemented
10#22625140C211yesmaximal, characteristic, complemented
11#135×Dic73C211yesmaximal, characteristic
12#17989638C2212yes—
13Q8×#10596833C2212yes—
14#17989622—22no—
15#17989622C2212yes—
16#17989618—22no—
17D4×#10596833C2212yes—
18#17989607—21no—
19#17989604—21no—
20#17989604C2211yescharacteristic
21#17989590—21no—
22#17989590C2211yescharacteristic
23#17989590C2211yescharacteristic
24C2×#14556658—21no—
25C4×#11924037C2211yescharacteristic
26#17989451C2212yes—
27#17989450C2212yes—
28Q8⋊#10596833C2212yes—
29Q8⋊#10596833C2212yes—
30D4⋊#10596833C2212yes—
31D4⋊#10596833C2212yes—
32#17989429C2212yes—
33#17989428C2212yes—
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—
50C2×#11924037—21no—
51C2×#11924037—21no—
52C2×#11924037—21no—
53C22×#10596833—22no—
54#14556658—22no—
55#14556658—21no—
56#14556658—21no—
57#14556658D411yescharacteristic
58#14556658D411yescharacteristic
59C4⋊#10596833—22no—
60C4⋊#10596833C2312yes—
61C4⋊#10596833C2312yes—
62C22⋊#10596833—22no—
63C22⋊#10596833—22no—
64C22⋊#10596833D412yes—
65#14556646—22no—
66#14556646—21no—
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—42no—
115#11924037—82no—
116#11924037—81no—
117#11924037—81no—
118C2×#10596833—22no—
119C2×#10596833—22no—
120C2×#10596833—22no—
121C2×#10596833—82no—
122C2×#10596833C2×D412yes—
123C2×#10596833C2×D412yes—
124C22×Dic73—21no—
125C22×Dic73—21no—
126C22×Dic73—21no—
127C22×Dic73—21no—
128C22×Dic73C2×D411yescharacteristic
129C22×Dic73#4111yescharacteristic
130C2×#10596831—41no—
131C2×#10596831C22×C411yescharacteristic
132C2×#10596831C22×C411yescharacteristic
133C4⋊Dic73—21no—
134C4⋊Dic73—21no—
135C4⋊Dic73—21no—
136C4⋊Dic73—21no—
137C4⋊Dic73C22×C411yescharacteristic
138C4⋊Dic73C22×C411yescharacteristic
139C4⋊Dic73C22×C411yescharacteristic
140C4⋊Dic73C22×C411yescharacteristic
141C22⋊Dic73—21no—
142C22⋊Dic73—21no—
143C22⋊Dic73—21no—
144C22⋊Dic73—21no—
145C22⋊Dic73—21no—
146C22⋊Dic73—21no—
147C4×Dic73—21no—
148C4×Dic73—21no—
149C4×Dic73—21no—
150C4×Dic73—21no—
151C4×Dic73—21no—
152C4×Dic73—21no—
153C4×Dic73C22×C411yescharacteristic
154C4×Dic73C2×D411yescharacteristic
155C4×Dic73#4111yescharacteristic
156C4×Dic73C2411yescharacteristic
157#41×C73—22no—
158#41×C73—22no—
159#41×C73—41no—
160#41×C73—41no—
161C2×Q8×C73C22×C411yescharacteristic
162C2×Q8×C73C22×C411yescharacteristic
163C2×D4×C73—21no—
164C2×D4×C73—21no—
165C2×D4×C73C22×C411yescharacteristic
166C2×D4×C73C22×C411yescharacteristic
167C22×C4×C73—21no—
168C22×C4×C73—21no—
169C22×C4×C73C22×C411yescharacteristic
170Q16×C73—22no—
171SD16×C73—22no—
172SD16×C73M1614yescomplemented
173D8×C73—22no—
174C2×C8×C73—22no—
175C2×C8×C73C22×C411yescharacteristic
176C2×C8×C73C22×C411yescharacteristic
177C23×C73—21no—
178C23×C73—21no—
179C23×C73C4×D411yescharacteristic
180Q8×C73—21no—
181Q8×C73—21no—
182Q8×C73—21no—
183Q8×C73C2×M1612yes—
184D4×C73—21no—
185D4×C73—21no—
186D4×C73—21no—
187D4×C73—42no—
188D4×C73—42no—
189D4×C73C2×M1612yes—
190C2×C4×C73—21no—
191C2×C4×C73—21no—
192C2×C4×C73—21no—
193C2×C4×C73—21no—
194C2×C4×C73—42no—
195C2×C4×C73—42no—
196C2×C4×C73C4×D411yescharacteristic
197C2×C4×C73C23×C411yescharacteristic
198C2×Dic73—21no—
199C2×Dic73—21no—
200C2×Dic73—22no—
201C2×Dic73—21no—
202C2×Dic73—21no—
203C2×Dic73—21no—
204C2×Dic73—21no—
205C2×Dic73—41no—
206C2×Dic73—41no—
207C2×Dic73—41no—
208C2×Dic73—41no—
209C2×Dic73—42no—
210C2×Dic73—42no—
211C2×Dic73C4×D411yescharacteristic
212C2×Dic73C4×D411yescharacteristic
213C2×Dic73C22×D411yescharacteristic
214C2×Dic73C2×#4111yescharacteristic
215#10596833—22no—
216#10596833—22no—
217#10596833—42no—
218#10596833—42no—
219#10596833—42no—
220C8×C73—21no—
221C8×C73C2×M1612yes—
222#10596831—21no—
223#10596831—21no—
224C22×C73—21no—
225C22×C73—22no—
226C22×C73—21no—
227C22×C73—21no—
228C22×C73—41no—
229C22×C73—41no—
230C22×C73—41no—
231C22×C73—41no—
232C22×C73—41no—
233C22×C73C2×C4×D411yescharacteristic
234C4×C73—21no—
235C4×C73—22no—
236C4×C73—41no—
237C4×C73C22×M1611yescharacteristic
238C4×C73#56811yescharacteristic
239Dic73—21no—
240Dic73—21no—
241Dic73—41no—
242Dic73—41no—
243Dic73—41no—
244Dic73—41no—
245Dic73#44412yes—
246Dic73#57611yescharacteristic
247Dic73#57811yescharacteristic
248#18061—731nomaximal, Sylow
249C2×C73—41no—
250C2×C73—42no—
251C2×C73—81no—
252C2×C73D4×M1611yescharacteristic
253C2×C73#295111yescharacteristic
254C2×C73C2×#44411yescharacteristic
255#2991—731no—
256#2990—731no—
257C4×#135—731no—
258#1592—732no—
259#1584—732no—
260#1581—732no—
261SD16⋊C8—732no—
262#1567—731no—
263#1489—731no—
264#1489—731no—
265#1451—731no—
266C73#1806111yescharacteristic, complemented, Sylow
267C2×#135#6440811yescharacteristic
268C4×#41—731no—
269C4×#41—731no—
270#446—732no—
271C8×Q8—732no—
272C4×Q16—731no—
273C4×Q16—731no—
274C4×SD16—731no—
275C4×SD16—731no—
276C4×SD16—731no—
277C4×SD16—731no—
278C4×D8—731no—
279C4×D8—731no—
280#436—732no—
281#436—1462no—
282#435—1462no—
283C8×D4—732no—
284#432—1461no—
285#431—731no—
286#431—1461no—
287#427—731no—
288#423—731no—
289#423—731no—
290#423—1461no—
291C2×C4⋊C8—1461no—
292#416—731no—
293#416—731no—
294C4×M16—731no—
295C2×C4×C8—731no—
296#335—732no—
297#335—732no—
298Q8⋊C8—732no—
299Q8⋊C8—732no—
300D4⋊C8—732no—
301D4⋊C8—732no—
302#322—732no—
303#322—732no—
304C2×#41C2×#6440811yescharacteristic
305C2×#41C2×#6440811yescharacteristic
306#135—24no—
307C2×Q16C2×#6440811yescharacteristic
308C2×SD16C2×#6440811yescharacteristic
309C2×SD16C2×#6440811yescharacteristic
310C2×D8C2×#6440811yescharacteristic
311C2×M16—1462no—
312C2×M16—1461no—
313C2×M16—1461no—
314C2×M16—1461no—
315C22×C8—731no—
316C22×C8—1462no—
317C22×C8C2×#6440811yescharacteristic
318C4×Q8—731no—
319C4×Q8—731no—
320C4×Q8—731no—
321C4×Q8—731no—
322C4×D4—731no—
323C4×D4—731no—
324C4×D4—731no—
325C4×D4—731no—
326C4×D4—1461no—
327C4×D4—1461no—
328C4×D4—1461no—
329C4×D4—1461no—
330#117—731no—
331#117—731no—
332#117—1461no—
333#117—1461no—
334C2×C42—731no—
335C2×C42—1461no—
336C2×C42—1461no—
337#107—731no—
338#107—731no—
339#106—731no—
340#106—731no—
341C4⋊C8—731no—
342C4⋊C8—731no—
343C4⋊C8—732no—
344C4⋊C8—732no—
345C4⋊C8—1461no—
346C4⋊C8—1461no—
347C4⋊C8—1462no—
348C4⋊C8—1462no—
349#103—731no—
350#103—731no—
351#103—731no—
352#103—731no—
353#102—731no—
354#102—731no—
355#102—731no—
356#102—731no—
357C22⋊C8—732no—
358C22⋊C8—1462no—
359C22⋊C8—1462no—
360#97—1461no—
361#97—1462no—
362C4×C8—732no—
363C4×C8—731no—
364C4×C8—731no—
365C4×C8—731no—
366C4×C8—731no—
367C4×C8—732no—
368C4×C8—1462no—
369#41—22no—
370#41—22no—
371#41—41no—
372#41—41no—
373C2×Q8C22×#6440811yescharacteristic
374C2×Q8C22×#6440811yescharacteristic
375C2×D4—21no—
376C2×D4—21no—
377C2×D4C22×#6440811yescharacteristic
378C2×D4C22×#6440811yescharacteristic
379C22×C4—21no—
380C22×C4—21no—
381C22×C4—731no—
382C22×C4—731no—
383C22×C4—1461no—
384C22×C4—1461no—
385C22×C4—1461no—
386C22×C4—1461no—
387C22×C4C22×#6440811yescharacteristic
388Q16—22no—
389SD16—22no—
390SD16#1192403714yescomplemented
391D8—22no—
392M16—2922no—
393M16—5841no—
394M16—5841no—
395M16—5842no—
396C2×C8—22no—
397C2×C8—2921no—
398C2×C8—5842no—
399C2×C8—732no—
400C2×C8—732no—
401C2×C8—731no—
402C2×C8—731no—
403C2×C8—1462no—
404C2×C8—1462no—
405C2×C8—1462no—
406C2×C8C22×#6440811yescharacteristic
407C2×C8C22×#6440811yescharacteristic
408C4⋊C4—731no—
409C4⋊C4—731no—
410C4⋊C4—731no—
411C4⋊C4—731no—
412C4⋊C4—1461no—
413C4⋊C4—1461no—
414C4⋊C4—1461no—
415C4⋊C4—1461no—
416C22⋊C4—1461no—
417C22⋊C4—1461no—
418C22⋊C4—1461no—
419C22⋊C4—1461no—
420C22⋊C4—1461no—
421C22⋊C4—1461no—
422C42—731no—
423C42—731no—
424C42—731no—
425C42—731no—
426C42—1461no—
427C42—1461no—
428C42—1461no—
429C42—1461no—
430C42—1461no—
431C42—1461no—
432C23—21no—
433C23—21no—
434C23D4×#6440811yescharacteristic
435Q8—21no—
436Q8—21no—
437Q8—21no—
438Q8C2×#1192403712yes—
439D4—21no—
440D4—21no—
441D4—21no—
442D4—42no—
443D4—42no—
444D4C2×#1192403712yes—
445C2×C4—21no—
446C2×C4—21no—
447C2×C4—21no—
448C2×C4—21no—
449C2×C4—42no—
450C2×C4—42no—
451C2×C4—2921no—
452C2×C4—2921no—
453C2×C4—2921no—
454C2×C4—2921no—
455C2×C4—2922no—
456C2×C4—2922no—
457C2×C4—731no—
458C2×C4—731no—
459C2×C4—731no—
460C2×C4—731no—
461C2×C4—1461no—
462C2×C4—1461no—
463C2×C4—1461no—
464C2×C4—1461no—
465C2×C4—1462no—
466C2×C4—1461no—
467C2×C4—1461no—
468C2×C4D4×#6440811yescharacteristic
469C2×C4C23×#6440811yescharacteristic
470C8—21no—
471C8—2922no—
472C8—2922no—
473C8—2922no—
474C8—1462no—
475C8—1462no—
476C8—1461no—
477C8—1461no—
478C8C2×#1192403712yes—
479C22—21no—
480C22—22no—
481C22—21no—
482C22—21no—
483C22—41no—
484C22—41no—
485C22—41no—
486C22—41no—
487C22—41no—
488C22C2×D4×#6440811yescharacteristic
489C4—21no—
490C4—22no—
491C4—41no—
492C4—2921no—
493C4—2921no—
494C4—2921no—
495C4—2921no—
496C4—731no—
497C4—731no—
498C4—732no—
499C4—1461no—
500C4—1461no—
501C4C22×#1192403711yescharacteristic
502C4#1798966511yescharacteristic
503C2—41no—
504C2—42no—
505C2—81no—
506C2D4×#1192403711yescharacteristic
507C2#2262669311yescharacteristic
508C2D73⋊#44411yescharacteristic
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#18061
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. #22626863C2semidirect
2. #22626857C2semidirect
3. #22625574C2semidirect
4. #22625544C2semidirect
5. #22625532C2semidirect
6. SD16⋊#10596833C2semidirect
7. #22625488C2semidirect
8. #22625266C2semidirect
9. #22625266C2semidirect
10. #22625140C2semidirect
172. SD16×C73M16semidirect
266. C73#18061semidirect
390. SD16#11924037semidirect

Maximal subgroups

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

subgroupindex
#226268632
#226268572
#226255742
#226255442
#226255322
SD16⋊#105968332
#226254882
#226252662
#226251402
#135×Dic732
#1806173