Finite Group Databasebeta

← #34026744

Subgroups — #34026744

Order 18688.

704 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#22626625C211yesmaximal, characteristic
2#22626624C211yesmaximal, characteristic
3#22626038C212yesmaximal, complemented
4#22625999C212yesmaximal, complemented
5#22625996C212yesmaximal, complemented
6#22625839C211yesmaximal, characteristic
7#22625839C211yesmaximal, characteristic
8#22625810C211yesmaximal, characteristic
9#22625734C212yesmaximal, complemented
10C8×C22⋊#64408C211yesmaximal, characteristic
11#427×D73C211yesmaximal, characteristic
12#17989611—22no—
13#17989611—22no—
14#17989610—21no—
15#17989609—21no—
16#17989606—21no—
17#17989605—21no—
18C2×#14556661C2212yes—
19#17989594C2212yes—
20#17989594C2212yes—
21#17989558C2211yescharacteristic
22#17989558C2211yescharacteristic
23#17989555C2211yescharacteristic
24#17989552C2212yes—
25#17989552C2212yes—
26C2×C8×#64408C2212yes—
27C4×C22⋊#64408C2211yescharacteristic
28#17989463C2212yes—
29#17989457C2212yes—
30#17989457C2212yes—
31#17989457C2212yes—
32#17989457C2212yes—
33#17989452C2212yes—
34C22×C8×D73C2211yescharacteristic
35#117×D73C2211yescharacteristic
36#117×D73C2211yescharacteristic
37#17988625C411yescharacteristic, complemented
38#17988624C2211yescharacteristic
39#17988624C2211yescharacteristic
40#17988623C411yescharacteristic, complemented
41#17988623C411yescharacteristic, complemented
42#17988622C2211yescharacteristic
43#17988622C2211yescharacteristic
44#107×D73C411yescharacteristic, complemented
45#107×D73C411yescharacteristic, complemented
46#106×D73C411yescharacteristic, complemented
47#106×D73C411yescharacteristic, complemented
48#427×C73C411yescharacteristic, complemented
49C2×#11924038C2312yes—
50C2×#11924038C2312yes—
51C2×#11924036C2312yes—
52C2×C22⋊#64408—21no—
53C2×C4×#64408C2312yes—
54#14556661—24no—
55#14556658—22no—
56#14556658—22no—
57C4⋊#10596833—22no—
58C4⋊#10596833—22no—
59#14556648—21no—
60#14556648—21no—
61#14556648D411yescharacteristic
62#14556648D411yescharacteristic
63C22⋊#10596833—21no—
64C22⋊#10596833—21no—
65C22⋊#10596833D412yes—
66#14556646—22no—
67#14556646—22no—
68C4×#10596833—22no—
69C4×#10596833—22no—
70C8×#64408—24no—
71(C2×#64408)⋊C4—21no—
72C22×C4×D73C2311yescharacteristic
73C22×#10596831C2×C411yescharacteristic
74C2×C8×D73—24no—
75C2×C8×D73C2×C411yescharacteristic
76C2×C8×D73C2×C411yescharacteristic
77C2×C8×D73C2311yescharacteristic
78C2×C8×D73C2311yescharacteristic
79#14556550C2×C411yescharacteristic
80#14556550C2×C411yescharacteristic
81#14556549C2311yescharacteristic
82#14556549C2311yescharacteristic
83#14556549C2311yescharacteristic
84#14556549C2311yescharacteristic
85#14556548—21no—
86#14556548—21no—
87#14556547—21no—
88#14556547—21no—
89C4⋊C4×D73C2×C411yescharacteristic
90C4⋊C4×D73C2×C411yescharacteristic
91C4⋊C4×D73C2×C411yescharacteristic
92C4⋊C4×D73C2×C411yescharacteristic
93C22⋊C4×D73—21no—
94C22⋊C4×D73—21no—
95C42×D73—21no—
96C42×D73—21no—
97#14556525C2×C411yescharacteristic
98#14556525C2×C411yescharacteristic
99#14556524C2×C411yescharacteristic
100#14556524C2×C411yescharacteristic
101#14556524C2×C411yescharacteristic
102#14556524C2×C411yescharacteristic
103#14556523C2×C411yescharacteristic
104#14556523C2×C411yescharacteristic
105#14556522C2×C411yescharacteristic
106#14556522C2×C411yescharacteristic
107#14556522C2×C411yescharacteristic
108#14556522C2×C411yescharacteristic
109C22×C8×C73C2×C411yescharacteristic
110#117×C73C2×C411yescharacteristic
111#117×C73C2×C411yescharacteristic
112#107×C73C2×C411yescharacteristic
113#107×C73C2×C411yescharacteristic
114#106×C73C2×C411yescharacteristic
115#106×C73C2×C411yescharacteristic
116C22×#64408—22no—
117#11924038—22no—
118#11924038—22no—
119#11924038—22no—
120#11924038—22no—
121#11924036—22no—
122#11924036—22no—
123C2×#10596833—22no—
124C2×#10596833—22no—
125C2×#10596833C2×D412yes—
126C2×#10596833C2×D412yes—
127C22⋊#64408—24no—
128C4×#64408—22no—
129C4×#64408—22no—
130C23×D73#4111yescharacteristic
131C22×Dic73C2×D411yescharacteristic
132C2×C4×D73—22no—
133C2×C4×D73—22no—
134C2×C4×D73—21no—
135C2×C4×D73—21no—
136C2×C4×D73—21no—
137C2×C4×D73—21no—
138C2×C4×D73C22×C411yescharacteristic
139C2×C4×D73C2×D411yescharacteristic
140C2×C4×D73#4111yescharacteristic
141C2×C4×D73C2411yescharacteristic
142C2×#10596831—22no—
143C2×#10596831C22×C411yescharacteristic
144C2×#10596831C22×C411yescharacteristic
145C8×D73—22no—
146C8×D73—22no—
147C8×D73—22no—
148C8×D73#4114yes—
149C4⋊Dic73C22×C411yescharacteristic
150C4⋊Dic73C22×C411yescharacteristic
151C4⋊Dic73C22×C411yescharacteristic
152C4⋊Dic73C22×C411yescharacteristic
153#11924007—21no—
154#11924007—21no—
155#11924007—21no—
156#11924007—21no—
157C22⋊Dic73—21no—
158C22⋊Dic73—21no—
159#11924005—21no—
160#11924005—21no—
161#11924005—21no—
162#11924005—21no—
163C4×Dic73—21no—
164C4×Dic73—21no—
165C4×Dic73—21no—
166C4×Dic73—21no—
167C4×Dic73—21no—
168C4×Dic73—21no—
169C22×C4×C73C22×C411yescharacteristic
170C2×C8×C73—22no—
171C2×C8×C73C22×C411yescharacteristic
172C2×C8×C73C22×C411yescharacteristic
173C4⋊C4×C73C22×C411yescharacteristic
174C4⋊C4×C73C22×C411yescharacteristic
175C4⋊C4×C73C22×C411yescharacteristic
176C4⋊C4×C73C22×C411yescharacteristic
177C22⋊C4×C73—21no—
178C22⋊C4×C73—21no—
179C42×C73—21no—
180C42×C73—21no—
181C23×C73C4×D411yescharacteristic
182C22×D73—22no—
183C22×D73—22no—
184C22×D73—41no—
185C22×D73C4×D411yescharacteristic
186C22×D73C2×#4111yescharacteristic
187C2×#64408—24no—
188C2×#64408—42no—
189C2×#64408—42no—
190C2×C4×C73—21no—
191C2×C4×C73—21no—
192C2×C4×C73—21no—
193C2×C4×C73—21no—
194C2×C4×C73—21no—
195C2×C4×C73—21no—
196C2×C4×C73C4×D411yescharacteristic
197C2×C4×C73C23×C411yescharacteristic
198C2×Dic73—22no—
199C2×Dic73—21no—
200C2×Dic73—21no—
201C2×Dic73—21no—
202C2×Dic73—21no—
203C2×Dic73C4×D411yescharacteristic
204C2×Dic73C22×D411yescharacteristic
205C4×D73—22no—
206C4×D73—21no—
207C4×D73—21no—
208C4×D73—22no—
209C4×D73—42no—
210C4×D73—42no—
211C4×D73—81no—
212C4×D73—81no—
213C4×D73C2×#4112yes—
214C4×D732−1+412yes—
215#10596833—22no—
216#10596833—22no—
217#10596833—42no—
218#10596833—42no—
219C8×C73—21no—
220C8×C73#11712yes—
221#10596831—21no—
222#10596831#11712yes—
223C22×C73—21no—
224C22×C73—21no—
225C22×C73—21no—
226C22×C73C2×C4×D411yescharacteristic
227C2×D73—21no—
228C2×D73—21no—
229C2×D73—21no—
230C2×D73—21no—
231C2×D73—44no—
232C2×D73#44312yes—
233C2×D73#55212yes—
234#64408—44no—
235C4×C73—21no—
236C4×C73—41no—
237C4×C73—41no—
238C4×C73—41no—
239C4×C73—41no—
240C4×C73C2×#11711yescharacteristic
241C4×C73#51911yescharacteristic
242Dic73—21no—
243Dic73—41no—
244Dic73—41no—
245Dic73—41no—
246Dic73—41no—
247Dic73#54712yes—
248#22606—731nomaximal, Sylow
249C2×C73—41no—
250C2×C73#234411yescharacteristic
251C2×C73C2×#44311yescharacteristic
252C2×C73#295711yescharacteristic
253D73—22no—
254D73—22no—
255#2924—731no—
256#2924—731no—
257C2×#427—731no—
258#1945—732no—
259#1922—732no—
260#1922—732no—
261#1826—731no—
262#1826—731no—
263#1810—731no—
264#1772—732no—
265C8×C22⋊C4—731no—
266C73#2260611yescharacteristic, complemented, Sylow
267C23×C8—731no—
268C2×#117—731no—
269C2×#117—731no—
270#433—1462no—
271#433—1462no—
272#432—1461no—
273#432—1461no—
274#431—1461no—
275#431—1461no—
276C2×#108—732no—
277#427—731no—
278#427—731no—
279#427—731no—
280#427—731no—
281#427—731no—
282#427—731no—
283#427—731no—
284#427#6440811yescharacteristic, complemented
285C2×#107—731no—
286C2×#107—731no—
287C2×#106—731no—
288C2×#106—731no—
289#424—732no—
290#424—732no—
291#407—731no—
292#407—731no—
293C2×C22⋊C8—731no—
294#405—732no—
295#405—732no—
296C2×C4×C8—732no—
297C4×C22⋊C4—731no—
298#341—732no—
299#339—732no—
300#339—732no—
301#339—732no—
302#339—732no—
303#336—732no—
304C23×C4—731no—
305C2×M16—732no—
306C2×M16—732no—
307C22×C8—731no—
308C22×C8—732no—
309C22×C8—731no—
310C22×C8—731no—
311C22×C8—731no—
312C22×C8—731no—
313C22×C8—1464no—
314C22×C8C2×#6440811yescharacteristic
315#117—731no—
316#117—731no—
317#117—731no—
318#117—731no—
319#117—731no—
320#117—731no—
321#117—1461no—
322#117—1461no—
323#117—1461no—
324#117—1461no—
325#117C2×#6440811yescharacteristic
326#117C2×#6440811yescharacteristic
327C2×C4⋊C4—731no—
328C2×C4⋊C4—731no—
329C2×C4⋊C4—731no—
330C2×C4⋊C4—731no—
331C2×C22⋊C4—1461no—
332C2×C22⋊C4—1461no—
333C2×C22⋊C4—1461no—
334C2×C42—732no—
335C2×C42—1461no—
336C2×C42—1461no—
337#108—1464no—
338#107—731no—
339#107—731no—
340#107—731no—
341#107—731no—
342#107—731no—
343#107—731no—
344#107C2×#6440811yescharacteristic
345#107C2×#6440811yescharacteristic
346#106—731no—
347#106—731no—
348#106—731no—
349#106—731no—
350#106—731no—
351#106—731no—
352#106C2×#6440811yescharacteristic
353#106C2×#6440811yescharacteristic
354C4⋊C8—1462no—
355C4⋊C8—1462no—
356C4⋊C8—1462no—
357C4⋊C8—1462no—
358C22⋊C8—731no—
359C22⋊C8—731no—
360C22⋊C8—732no—
361C22⋊C8—1461no—
362C22⋊C8—1461no—
363C22⋊C8—1461no—
364C22⋊C8—1461no—
365#97—1462no—
366#97—1462no—
367C4×C8—1464no—
368C4×C8—1462no—
369C4×C8—1462no—
370#95—1461no—
371C24—731no—
372C22×C4—731no—
373C22×C4—731no—
374C22×C4—731no—
375C22×C4—731no—
376C22×C4—731no—
377C22×C4—1461no—
378C22×C4—1461no—
379C22×C4—1461no—
380C22×C4—1461no—
381C22×C4—1462no—
382C22×C4—1462no—
383C22×C4—1462no—
384C22×C4C22×#6440811yescharacteristic
385M16—1462no—
386M16—1462no—
387M16—1462no—
388M16—1462no—
389C2×C8—22no—
390C2×C8—731no—
391C2×C8—734no—
392C2×C8—732no—
393C2×C8—732no—
394C2×C8—731no—
395C2×C8—1462no—
396C2×C8—1462no—
397C2×C8—1462no—
398C2×C8—1462no—
399C2×C8—1462no—
400C2×C8—1462no—
401C2×C8—1462no—
402C2×C8—1462no—
403C2×C8C22×#6440811yescharacteristic
404C2×C8C22×#6440811yescharacteristic
405C4⋊C4—731no—
406C4⋊C4—731no—
407C4⋊C4—731no—
408C4⋊C4—731no—
409C4⋊C4—1461no—
410C4⋊C4—1461no—
411C4⋊C4—1461no—
412C4⋊C4—1461no—
413C4⋊C4C22×#6440811yescharacteristic
414C4⋊C4C22×#6440811yescharacteristic
415C4⋊C4C22×#6440811yescharacteristic
416C4⋊C4C22×#6440811yescharacteristic
417C22⋊C4—21no—
418C22⋊C4—21no—
419C22⋊C4—1461no—
420C22⋊C4—1461no—
421C22⋊C4—1461no—
422C22⋊C4—1461no—
423C22⋊C4—1464no—
424C22⋊C4—1461no—
425C22⋊C4—1461no—
426C42—21no—
427C42—21no—
428C42—1461no—
429C42—1461no—
430C42—1461no—
431C42—1461no—
432C42—1462no—
433C42—1462no—
434C42—1461no—
435C42—1461no—
436C23—2921no—
437C23—731no—
438C23—731no—
439C23—1462no—
440C23—1462no—
441C23D4×#6440811yescharacteristic
442C2×C4—21no—
443C2×C4—21no—
444C2×C4—21no—
445C2×C4—21no—
446C2×C4—21no—
447C2×C4—21no—
448C2×C4—2922no—
449C2×C4—2922no—
450C2×C4—2922no—
451C2×C4—2922no—
452C2×C4—5841no—
453C2×C4—5841no—
454C2×C4—731no—
455C2×C4—732no—
456C2×C4—732no—
457C2×C4—731no—
458C2×C4—1462no—
459C2×C4—1462no—
460C2×C4—1464no—
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—732no—
474C8—1462no—
475C8—1462no—
476C8—1461no—
477C8#1455667012yes—
478C22—21no—
479C22—21no—
480C22—21no—
481C22—2924no—
482C22—732no—
483C22—732no—
484C22—1461no—
485C22—1461no—
486C22—1461no—
487C22—1461no—
488C22C2×D4×#6440811yescharacteristic
489C4—21no—
490C4—41no—
491C4—41no—
492C4—41no—
493C4—41no—
494C4—2924no—
495C4—2921no—
496C4—2921no—
497C4—2921no—
498C4—2921no—
499C4—732no—
500C4—1461no—
501C4C2×#1455667011yescharacteristic
502C4#1798965511yescharacteristic
503C2—41no—
504C2—1462no—
505C2—1462no—
506C2#2262643911yescharacteristic
507C2D73⋊#44311yescharacteristic
508C2#2262671011yescharacteristic
subgroupG/H
centerC22C2×D4×#64408
commutatorC2×C4×C73C23×C4
FrattiniC2×C4C23×#64408
Fitting#427×C73C4
socleC22×C73C2×C4×D4
prime radicalC23×D73#41
cosocleC2×C4×D73C24
hypercenter#427#64408
hypocenterC73#22606
LCSG ▷ C2×C4×C73 ▷ C2×C73 ▷ C73
UCSC1 ◁ C22 ◁ C22×C4 ◁ #427

Complements

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

NH ≅ G/Ntype
3. #22626038C2semidirect
4. #22625999C2semidirect
5. #22625996C2semidirect
9. #22625734C2semidirect
37. #17988625C4semidirect
40. #17988623C4semidirect
41. #17988623C4semidirect
44. #107×D73C4semidirect
45. #107×D73C4semidirect
46. #106×D73C4semidirect
47. #106×D73C4semidirect
48. #427×C73C4semidirect
266. C73#22606semidirect
284. #427#64408semidirect

Maximal subgroups

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

subgroupindex
#226266252
#226266242
#226260382
#226259992
#226259962
#226258392
#226258102
#226257342
C8×C22⋊#644082
#427×D732
#2260673