Finite Group Databasebeta

← #33240694

Subgroups — #33240694

Order 18688.

534 conjugacy classes of subgroups.

The subgroup lattice has 464 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#22620468C211yesmaximal, characteristic, complemented
2#22618801C211yesmaximal, characteristic
3#22612098C211yesmaximal, characteristic, complemented
4#22612092C211yesmaximal, characteristic, complemented
5#22612086C211yesmaximal, characteristic, complemented
6#22612084C211yesmaximal, characteristic, complemented
7#22612082C211yesmaximal, characteristic, complemented
8#22612068C211yesmaximal, characteristic, complemented
9#22612066C211yesmaximal, characteristic, complemented
10#22611718C211yesmaximal, characteristic, complemented
11#366×D73C211yesmaximal, characteristic, complemented
12#22611688C211yesmaximal, characteristic, complemented
13#22611640C211yesmaximal, characteristic
14#22611635C211yesmaximal, characteristic
15C73×#2228C211yesmaximal, characteristic, complemented
16C2×#14556610C2211yescharacteristic
17#17989217—21no—
18#17989066—21no—
19C2×#14556530C2211yescharacteristic
20C2×C16×D73C2211yescharacteristic
21#17988997—21no—
22#17988994—21no—
23#17988992C2211yescharacteristic
24#17988991C2211yescharacteristic
25#17988988C2211yescharacteristic
26#17988687—21no—
27#17988684—21no—
28C4×Dic292C2211yescharacteristic
29#17988681C2211yescharacteristic
30D8×Dic73C2211yescharacteristic
31#17988626—21no—
32#17988624C2211yescharacteristic
33#107×D73C2211yescharacteristic
34#107×D73C2211yescharacteristic
35#17988585—21no—
36#17988541C2211yescharacteristic
37#17988414C2211yescharacteristic
38#17988413C2211yescharacteristic
39#17988413C2211yescharacteristic
40#17988412C2211yescharacteristic
41#17988412C2211yescharacteristic
42#17988412C2211yescharacteristic
43C4⋊#11924003C2211yescharacteristic
44#17988405C2211yescharacteristic
45#17988405C2211yescharacteristic
46#17988392C2211yescharacteristic
47#17988392C2211yescharacteristic
48#17988390C2211yescharacteristic
49#17988390C2211yescharacteristic
50#17988388C2211yescharacteristic
51#17988388C2211yescharacteristic
52#499×C73C2211yescharacteristic
53C4×D8×C73C2211yescharacteristic
54#367×C73C2211yescharacteristic
55#366×C73C2211yescharacteristic
56C4⋊C16×C73C2211yescharacteristic
57#357×C73C2211yescharacteristic
58#357×C73C2211yescharacteristic
59C2×#11924027—21no—
60#14556610—22no—
61#14556610Q814yescomplemented
62C2×Dic292C2311yescharacteristic
63C2×#11924016—21no—
64C2×#11924011D411yescharacteristic
65C2×C8×D73C2311yescharacteristic
66#14556591—21no—
67#14556590—21no—
68#14556590—41no—
69#14556582—21no—
70#14556581—41no—
71#14556571—41no—
72#14556569—41no—
73C4×Dic146—21no—
74#14556556—21no—
75D4×Dic73—21no—
76C4×C22⋊D73—41no—
77#14556551—41no—
78#14556549—21no—
79#14556548D411yescharacteristic
80#14556547—41no—
81C2×#11924007—41no—
82C4⋊C4×D73—21no—
83C4⋊C4×D73—21no—
84#14556530—22no—
85C2×#11924003C2311yescharacteristic
86C2×#11924003C2311yescharacteristic
87C16×D73—22no—
88#14556525—21no—
89#14556525C2311yescharacteristic
90#14556525C2311yescharacteristic
91#14556525C2311yescharacteristic
92#14556524—21no—
93#14556524—21no—
94#14556524—21no—
95#14556522—21no—
96#14556522—21no—
97#14556516—21no—
98#14556514—21no—
99#14556512—21no—
100#14556502D411yescharacteristic
101#14556501D411yescharacteristic
102C8×Dic73C2311yescharacteristic
103C2×D8×C73C2311yescharacteristic
104#125×C73—21no—
105C4×D4×C73—21no—
106C2×C16×C73C2311yescharacteristic
107C2×C16×C73C2311yescharacteristic
108#107×C73C2311yescharacteristic
109#107×C73C2311yescharacteristic
110#107×C73C2311yescharacteristic
111#106×C73—21no—
112#102×C73—21no—
113C4×C8×C73C2311yescharacteristic
114#11924027—24no—
115#11924027—42no—
116C2×Dic146—21no—
117C2×Dic146—41no—
118C2×Dic146—41no—
119C2×C22⋊D73—41no—
120C22×Dic73—41no—
121C22×Dic73—41no—
122C2×C4×D73—41no—
123C2×C4×D73—41no—
124C2×C4×D73C2×D411yescharacteristic
125Dic292—21no—
126Dic292C2×Q812yes—
127#11924016—24no—
128#11924011—22no—
129C2×#10596831C2×D411yescharacteristic
130C2×#10596831C2×D411yescharacteristic
131C8×D73C2×Q812yes—
132C8×D73#4112yes—
133C4⋊Dic73—21no—
134C4⋊Dic73—21no—
135C4⋊Dic73—21no—
136C4⋊Dic73—41no—
137C4⋊Dic73—41no—
138#11924007—41no—
139#11924007—41no—
140#11924007—41no—
141#11924007—41no—
142#11924007—41no—
143#11924007—41no—
144#11924007D811yescharacteristic
145#11924007D811yescharacteristic
146C22⋊Dic73—41no—
147#11924005—41no—
148#11924005D811yescharacteristic
149#11924005D811yescharacteristic
150C4×Dic73—41no—
151C4×Dic73—41no—
152C4×Dic73—41no—
153C4×Dic73C2×D411yescharacteristic
154#11924003—21no—
155#11924003—21no—
156C2×D4×C73—21no—
157C22×C4×C73—41no—
158D8×C73—21no—
159D8×C73C2×Q812yes—
160C2×C8×C73C2×D411yescharacteristic
161C2×C8×C73C2411yescharacteristic
162C4⋊C4×C73—21no—
163C4⋊C4×C73—21no—
164C4⋊C4×C73—21no—
165C4⋊C4×C73—41no—
166C4⋊C4×C73—41no—
167C22⋊C4×C73—41no—
168C42×C73C2×D411yescharacteristic
169C16×C73—21no—
170C16×C73—21no—
171C23×C73—41no—
172C22×D73C2×D811yescharacteristic
173D4×C73—22no—
174D4×C73—41no—
175C2×C4×C73—41no—
176C2×C4×C73—41no—
177C2×C4×C73—41no—
178C2×C4×C73—161no—
179C2×C4×C73C2×D811yescharacteristic
180C2×C4×C73C2×D811yescharacteristic
181C2×C4×C73C22×D411yescharacteristic
182C22⋊D73—44no—
183C2×Dic73—41no—
184C2×Dic73—44no—
185C2×Dic73—41no—
186C2×Dic73—41no—
187C2×Dic73—41no—
188C2×Dic73—161no—
189C2×Dic73C2×D811yescharacteristic
190C2×Dic73C2×D811yescharacteristic
191C2×Dic73C2×D811yescharacteristic
192C4×D73—161no—
193C4×D73—161no—
194C4×D73C22⋊Q812yes—
195C4×D73#13712yes—
196Dic146—22no—
197Dic146—41no—
198Dic146—161no—
199Dic146—161no—
200C8×C73—21no—
201C8×C73C22×Q811yescharacteristic
202C8×C73C2×#4111yescharacteristic
203#10596831—21no—
204#10596831C22⋊Q812yes—
205C22×C73—42no—
206C22×C73—81no—
207C22×C73—81no—
208C22×C73C22×D811yescharacteristic
209C2×D73—21no—
210C2×D73—21no—
211C2×D73#47412yes—
212C4×C73—21no—
213C4×C73—21no—
214C4×C73—81no—
215C4×C73—81no—
216C4×C73—81no—
217C4×C73C2×C22⋊Q811yescharacteristic
218C4×C73C2×#13711yescharacteristic
219Dic73—21no—
220Dic73—21no—
221Dic73—42no—
222Dic73—81no—
223Dic73—81no—
224Dic73—81no—
225Dic73#47412yes—
226#33633—731nomaximal, Sylow
227C2×C73—82no—
228C2×C73C2×#47411yescharacteristic
229C2×C73#341511yescharacteristic
230C2×C73#341711yescharacteristic
231D73—22no—
232#3164—731no—
233#2951—731no—
234#2229—731no—
235#2229—731no—
236#2228—731no—
237#2228D7311yescharacteristic, complemented
238#2227—731no—
239#2227—731no—
240#2224—731no—
241#2224—731no—
242#2159—731no—
243C2×#366—731no—
244#2151—731no—
245#2139—731no—
246#2138—731no—
247C73#3363311yescharacteristic, complemented, Sylow
248C2×#135—731no—
249#544—1461no—
250#520—1461no—
251C2×M32—731no—
252C22×C16—731no—
253#501—1461no—
254#501—1461no—
255#500—731no—
256#500—731no—
257#499—731no—
258#499C2×D7311yescharacteristic
259#440—1461no—
260#440—1461no—
261C4×Q16—731no—
262C4×Q16—731no—
263C4×D8—731no—
264C4×D8C2×D7311yescharacteristic
265#428—1461no—
266#427—731no—
267C2×#107—731no—
268C2×#107—731no—
269#417—1461no—
270#405—731no—
271#367—731no—
272#367C2×D7311yescharacteristic
273#366—731no—
274#366—731no—
275#366—731no—
276#366—731no—
277#366—731no—
278#366C2×D7311yescharacteristic
279C4⋊C16—731no—
280C4⋊C16—731no—
281C4⋊C16—731no—
282C4⋊C16C2×D7311yescharacteristic
283#358—731no—
284#358—731no—
285#358—731no—
286#358—731no—
287#357—731no—
288#357—731no—
289#357C2×D7311yescharacteristic
290#357C2×D7311yescharacteristic
291C2×#41—1461no—
292#135—734no—
293#135—1462no—
294C2×Q16—731no—
295C2×SD16—1461no—
296C2×D8C22×D7311yescharacteristic
297C2×M16—731no—
298C22×C8—731no—
299C4⋊Q8—2921no—
300C4⋊Q8—1461no—
301C4⋊Q8—1461no—
302#125—21no—
303#125—2921no—
304#125—1461no—
305C22⋊Q8—2921no—
306C22⋊Q8—2921no—
307C4×Q8—1461no—
308C4×Q8—1461no—
309C4×D4—21no—
310C4×D4—2921no—
311C4×D4—2921no—
312C4×D4—1461no—
313#117—2921no—
314#117—731no—
315#117—1461no—
316C2×C4⋊C4—2921no—
317C2×C4⋊C4—1461no—
318C2×C4⋊C4—1461no—
319M32—1462no—
320C2×C16—731no—
321C2×C16—731no—
322C2×C16—1462no—
323C2×C16C22×D7311yescharacteristic
324C2×C16C22×D7311yescharacteristic
325#107—731no—
326#107—731no—
327#107—731no—
328#107—1461no—
329#107—1461no—
330#107—1461no—
331#107—1461no—
332#107C22×D7311yescharacteristic
333#107C22×D7311yescharacteristic
334#107C22×D7311yescharacteristic
335#106—21no—
336#106—1461no—
337#106—1461no—
338#103—1461no—
339#103—1461no—
340#102—21no—
341#102—1461no—
342#97—731no—
343#97—731no—
344C4×C8—731no—
345C4×C8C22×D7311yescharacteristic
346#41—2922no—
347#41—1464no—
348C2×Q8—2921no—
349C2×Q8—2921no—
350C2×Q8—1461no—
351C2×D4—21no—
352C2×D4—2921no—
353C22×C4—41no—
354C22×C4—2921no—
355C22×C4—2921no—
356C22×C4—2921no—
357C22×C4—2921no—
358C22×C4—731no—
359Q16—732no—
360Q16—1461no—
361SD16—1464no—
362D8—21no—
363D8Q8×D7312yes—
364M16—1462no—
365C2×C8—731no—
366C2×C8—732no—
367C2×C8—732no—
368C2×C8—731no—
369C2×C8D4×D7311yescharacteristic
370C2×C8C23×D7311yescharacteristic
371C4⋊C4—21no—
372C4⋊C4—21no—
373C4⋊C4—21no—
374C4⋊C4—41no—
375C4⋊C4—41no—
376C4⋊C4—2921no—
377C4⋊C4—2921no—
378C4⋊C4—2921no—
379C4⋊C4—2921no—
380C4⋊C4—2921no—
381C4⋊C4—2921no—
382C4⋊C4—2921no—
383C4⋊C4—2921no—
384C4⋊C4—731no—
385C4⋊C4—731no—
386C4⋊C4—1461no—
387C4⋊C4—1461no—
388C4⋊C4—1461no—
389C22⋊C4—41no—
390C22⋊C4—2921no—
391C22⋊C4—2921no—
392C22⋊C4—731no—
393C22⋊C4—731no—
394C42—2921no—
395C42—2921no—
396C42—2921no—
397C42—731no—
398C42D4×D7311yescharacteristic
399C16—21no—
400C16—21no—
401C16—1461no—
402C16—1461no—
403C23—41no—
404C23—731no—
405Q8—2921no—
406Q8—11681no—
407Q8—11681no—
408Q8—1462no—
409D4—22no—
410D4—41no—
411D4—2924no—
412C2×C4—41no—
413C2×C4—41no—
414C2×C4—41no—
415C2×C4—161no—
416C2×C4—2921no—
417C2×C4—2924no—
418C2×C4—2921no—
419C2×C4—2921no—
420C2×C4—2921no—
421C2×C4—731no—
422C2×C4—732no—
423C2×C4—732no—
424C2×C4—731no—
425C2×C4—731no—
426C2×C4—11681no—
427C2×C4—11681no—
428C2×C4—11681no—
429C2×C4D8×D7311yescharacteristic
430C2×C4D8×D7311yescharacteristic
431C2×C4C2×D4×D7311yescharacteristic
432C8—21no—
433C8—732no—
434C8—1461no—
435C8C2×Q8×D7311yescharacteristic
436C8#41×D7311yescharacteristic
437C22—42no—
438C22—81no—
439C22—81no—
440C22—732no—
441C22—1461no—
442C22—1461no—
443C22C2×D8×D7311yescharacteristic
444C4—21no—
445C4—21no—
446C4—81no—
447C4—81no—
448C4—81no—
449C4—2922no—
450C4—5841no—
451C4—5841no—
452C4—5841no—
453C4—732no—
454C4—1461no—
455C4—1461no—
456C4C22⋊Q8×D7311yescharacteristic
457C4#137×D7311yescharacteristic
458C2—82no—
459C2—1462no—
460C2#474×D7311yescharacteristic
461C2#2262307111yescharacteristic
462C2#2262308111yescharacteristic
subgroupG/H
centerC22C2×D8×D73
commutator, cosocleC2×C8×C73C24
FrattiniC2×C8C23×D73
FittingC73×#2228C2
socleC22×C73C22×D8
prime radicalC2×#14556610C22
hypercenter#2228D73
hypocenterC73#33633
LCSG ▷ C2×C8×C73 ▷ C4×C73 ▷ C2×C73 ▷ C73
UCSC1 ◁ C22 ◁ C42 ◁ C4×C8 ◁ #2228

Complements

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

NH ≅ G/Ntype
1. #22620468C2semidirect
3. #22612098C2semidirect
4. #22612092C2semidirect
5. #22612086C2semidirect
6. #22612084C2semidirect
7. #22612082C2semidirect
8. #22612068C2semidirect
9. #22612066C2semidirect
10. #22611718C2semidirect
11. #366×D73C2semidirect
12. #22611688C2semidirect
15. C73×#2228C2semidirect
61. #14556610Q8semidirect
237. #2228D73semidirect
247. C73#33633semidirect

Maximal subgroups

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

subgroupindex
#226204682
#226188012
#226120982
#226120922
#226120862
#226120842
#226120822
#226120682
#226120662
#226117182
#366×D732
#226116882
#226116402
#226116352
C73×#22282
#3363373