Finite Group Databasebeta

← #33239350

Subgroups — #33239350

Order 18688.

682 conjugacy classes of subgroups.

The subgroup lattice has 512 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×SD32×D73C211yesmaximal, characteristic
2#22618794C211yesmaximal, characteristic, complemented
3#22618778C211yesmaximal, characteristic
4#22611797C211yesmaximal, characteristic, complemented
5#22611796C211yesmaximal, characteristic, complemented
6#22611794C211yesmaximal, characteristic, complemented
7#22611786C211yesmaximal, characteristic, complemented
8C4×#14556534C211yesmaximal, characteristic, complemented
9C4×#14556535C211yesmaximal, characteristic, complemented
10#22611780C211yesmaximal, characteristic, complemented
11#22611712C211yesmaximal, characteristic, complemented
12#22611649C211yesmaximal, characteristic, complemented
13#22611645C211yesmaximal, characteristic
14#22611529C211yesmaximal, characteristic, complemented
15C73×#2175C211yesmaximal, characteristic, complemented
16C2×Q16×D73C2211yescharacteristic
17C2×D8×D73C2211yescharacteristic
18#17989063—21no—
19#17989059—21no—
20C2×#14556536C2211yescharacteristic
21C2×#14556535C2211yescharacteristic
22C2×#14556534C2211yescharacteristic
23SD32×D73C418yescomplemented
24C2×#14556530C2211yescharacteristic
25C2×C16×D73C2211yescharacteristic
26#17988697—21no—
27#17988696—21no—
28#17988692—21no—
29#17988691—21no—
30C4×Dic292C2211yescharacteristic, complemented
31Q16×Dic73C2211yescharacteristic, complemented
32#17988681C2211yescharacteristic, complemented
33C4×D584C2211yescharacteristic
34D8×Dic73C2211yescharacteristic
35#17988669C2211yescharacteristic
36#17988624C2211yescharacteristic
37#17988624C2211yescharacteristic
38#17988594—21no—
39#17988590—21no—
40#17988541C2211yescharacteristic
41#17988414C2211yescharacteristic, complemented
42#17988414C2211yescharacteristic, complemented
43#17988413C2211yescharacteristic, complemented
44#17988392C2211yescharacteristic, complemented
45#17988391C2211yescharacteristic, complemented
46#17988390C2211yescharacteristic, complemented
47#17988389C2211yescharacteristic
48#17988388C2211yescharacteristic
49#17988387C2211yescharacteristic
50#17988362C2211yescharacteristic, complemented
51C4×#11924003C2211yescharacteristic, complemented
52C16×Dic73C2211yescharacteristic, complemented
53C2×SD32×C73C2211yescharacteristic
54C4×Q16×C73C2211yescharacteristic, complemented
55C4×D8×C73C2211yescharacteristic
56#366×C73C2211yescharacteristic, complemented
57#358×C73C2211yescharacteristic, complemented
58#357×C73C2211yescharacteristic
59#346×C73C2211yescharacteristic, complemented
60C2×Q8×D73—21no—
61C2×D4×D73—21no—
62C2×Dic292C2311yescharacteristic
63C2×#11924018—21no—
64Q16×D73—22no—
65Q16×D73C2×C414yescomplemented
66C2×D584C2311yescharacteristic
67C2×#11924013—21no—
68D8×D73—22no—
69D8×D73C2×C414yes—
70C2×#11924011D411yescharacteristic
71C2×C8×D73C2311yescharacteristic
72Q8×Dic73—21no—
73C4×Dic146—21no—
74C4×Dic146—41no—
75#14556556—21no—
76#14556556—41no—
77C4×D292—21no—
78D4×Dic73—21no—
79#14556553—21no—
80C4×C22⋊D73—41no—
81#14556551—41no—
82#14556549—21no—
83#14556549—21no—
84#14556549—41no—
85#14556548—41no—
86#14556548D411yescharacteristic
87C2×#11924005—41no—
88C22⋊C4×D73—41no—
89#14556536C2×C414yescomplemented
90#14556535C2×C414yescomplemented
91#14556534C2×C414yescomplemented
92#14556530—22no—
93C2×#11924003C2311yescharacteristic
94C2×#11924003C2311yescharacteristic
95C16×D73C2×C414yescomplemented
96#14556525C2311yescharacteristic
97#14556525C2311yescharacteristic
98#14556522—21no—
99#14556522—21no—
100#14556516—21no—
101#14556515—21no—
102#14556514—21no—
103#14556513—21no—
104#14556512—21no—
105#14556511—21no—
106#14556502D411yescharacteristic
107#14556501D411yescharacteristic
108C8×Dic73C2311yescharacteristic
109C2×Q16×C73C2311yescharacteristic
110C2×D8×C73C2311yescharacteristic
111C4×Q8×C73—21no—
112C4×D4×C73—21no—
113SD32×C73C2×C414yescomplemented
114C2×C16×C73C2311yescharacteristic
115C2×C16×C73C2311yescharacteristic
116#107×C73C2311yescharacteristic
117#107×C73C2311yescharacteristic
118#103×C73—21no—
119#102×C73—21no—
120C4×C8×C73C2311yescharacteristic
121C23×D73—41no—
122C2×Dic146—21no—
123C2×Dic146—41no—
124Q8×D73—24no—
125Q8×D73—42no—
126C2×D292—21no—
127C2×C22⋊D73—41no—
128D4×D73—24no—
129D4×D73—42no—
130C22×Dic73—41no—
131C2×C4×D73—41no—
132C2×C4×D73—41no—
133C2×C4×D73—41no—
134C2×C4×D73C2×D411yescharacteristic
135Dic292—21no—
136Dic292C22×C412yes—
137#11924018—24no—
138D584—21no—
139D584C22×C412yes—
140#11924013—24no—
141#11924011—22no—
142C2×#10596831C2×D411yescharacteristic
143C2×#10596831C2×D411yescharacteristic
144C8×D73C22×C412yes—
145C8×D73#4112yes—
146C4⋊Dic73—21no—
147C4⋊Dic73—21no—
148C4⋊Dic73—41no—
149#11924007—41no—
150#11924007—41no—
151#11924007D811yescharacteristic
152#11924007D811yescharacteristic
153C22⋊Dic73—41no—
154#11924005—41no—
155#11924005—41no—
156#11924005—41no—
157#11924005—41no—
158#11924005D811yescharacteristic
159#11924005D811yescharacteristic
160C4×Dic73—41no—
161C4×Dic73—41no—
162C4×Dic73—41no—
163C4×Dic73—41no—
164C4×Dic73—41no—
165C4×Dic73C2×D411yescharacteristic
166#11924003—21no—
167#11924003C22×C412yes—
168C2×Q8×C73—21no—
169C2×D4×C73—21no—
170C22×C4×C73—41no—
171Q16×C73—21no—
172Q16×C73C22×C412yes—
173D8×C73—21no—
174D8×C73C22×C412yes—
175C2×C8×C73C2×D411yescharacteristic
176C2×C8×C73C2411yescharacteristic
177C4⋊C4×C73—21no—
178C4⋊C4×C73—21no—
179C4⋊C4×C73—41no—
180C22⋊C4×C73—41no—
181C42×C73—41no—
182C42×C73C2×D411yescharacteristic
183C16×C73—21no—
184C16×C73C22×C412yes—
185C23×C73—41no—
186C22×D73—41no—
187C22×D73—44no—
188C22×D73—161no—
189C22×D73—161no—
190C22×D73C2×D811yescharacteristic
191Q8×C73—22no—
192Q8×C73—41no—
193D4×C73—22no—
194D4×C73—41no—
195C2×C4×C73—41no—
196C2×C4×C73—41no—
197C2×C4×C73—41no—
198C2×C4×C73—161no—
199C2×C4×C73C2×D811yescharacteristic
200C2×C4×C73C2×D811yescharacteristic
201C2×C4×C73C22×D411yescharacteristic
202C22⋊D73—44no—
203C2×Dic73—41no—
204C2×Dic73—41no—
205C2×Dic73—41no—
206C2×Dic73—161no—
207C2×Dic73C2×D811yescharacteristic
208C2×Dic73C2×D811yescharacteristic
209C2×Dic73C2×D811yescharacteristic
210D292—22no—
211D292—41no—
212C4×D73—44no—
213C4×D73—161no—
214C4×D73—161no—
215C4×D73C4×D412yes—
216C4×D73#13512yes—
217Dic146—22no—
218Dic146—41no—
219Dic146—44no—
220C8×C73—21no—
221C8×C73C23×C411yescharacteristic
222C8×C73C2×#4111yescharacteristic
223#10596831—21no—
224#10596831C4×D412yes—
225C22×C73—42no—
226C22×C73—81no—
227C22×C73—81no—
228C22×C73C22×D811yescharacteristic
229C2×D73—21no—
230C2×D73—21no—
231C2×D73—42no—
232C2×D73—81no—
233C2×D73—81no—
234C2×D73—164no—
235C2×D73C4×D812yes—
236C4×C73—21no—
237C4×C73—21no—
238C4×C73—42no—
239C4×C73—81no—
240C4×C73—81no—
241C4×C73C2×C4×D411yescharacteristic
242C4×C73C2×#13511yescharacteristic
243Dic73—21no—
244Dic73—21no—
245Dic73—42no—
246Dic73—81no—
247Dic73—81no—
248Dic73C4×D812yes—
249#33457—731nomaximal, Sylow
250C2×C73—82no—
251C2×C73C2×C4×D811yescharacteristic
252C2×C73#341611yescharacteristic
253C2×C73#341611yescharacteristic
254D73—22no—
255D73—82no—
256C22×SD32—731no—
257#2950—731no—
258#2948—731no—
259#2175—731no—
260#2175—731no—
261#2175—731no—
262#2175D7311yescharacteristic, complemented
263C4×SD32—731no—
264C4×SD32—731no—
265C4×SD32—731no—
266C4×SD32—731no—
267#2158—731no—
268#2141—731no—
269#2140—731no—
270#2108—731no—
271C73#3345711yescharacteristic, complemented, Sylow
272C22×Q16—731no—
273C22×D8—731no—
274#519—1461no—
275#518—1461no—
276C2×SD32—731no—
277C2×SD32—731no—
278C2×SD32—738no—
279C2×SD32—731no—
280C2×SD32C2×D7311yescharacteristic
281C2×M32—731no—
282C22×C16—731no—
283#442—1461no—
284#442—1461no—
285#441—1461no—
286#441—1461no—
287C4×Q16—731no—
288C4×Q16—731no—
289C4×Q16—731no—
290C4×Q16C2×D7311yescharacteristic, complemented
291C4×D8—731no—
292C4×D8—731no—
293C4×D8—731no—
294C4×D8C2×D7311yescharacteristic
295#427—731no—
296#427—731no—
297#419—1461no—
298#418—1461no—
299#405—731no—
300#367—731no—
301#367—731no—
302#366—731no—
303#366C2×D7311yescharacteristic, complemented
304#358—731no—
305#358—731no—
306#358—731no—
307#358C2×D7311yescharacteristic, complemented
308#357—731no—
309#357—731no—
310#357—731no—
311#357C2×D7311yescharacteristic
312#346—731no—
313#346C2×D7311yescharacteristic, complemented
314C4×C16—731no—
315C4×C16—731no—
316C22×Q8—1461no—
317C22×D4—1461no—
318C2×Q16—734no—
319C2×Q16—731no—
320C2×Q16—1462no—
321C2×Q16—1461no—
322C2×Q16C22×D7311yescharacteristic
323C2×D8—734no—
324C2×D8—731no—
325C2×D8—1462no—
326C2×D8—1461no—
327C2×D8C22×D7311yescharacteristic
328C2×M16—731no—
329C22×C8—731no—
330C4×Q8—21no—
331C4×Q8—2921no—
332C4×Q8—2921no—
333C4×Q8—1461no—
334C4×Q8—1461no—
335C4×Q8—1461no—
336C4×D4—21no—
337C4×D4—2921no—
338C4×D4—2921no—
339C4×D4—1461no—
340C4×D4—1461no—
341C4×D4—1461no—
342#117—2921no—
343#117—2921no—
344#117—731no—
345#117—1461no—
346#117—1461no—
347C2×C22⋊C4—2921no—
348C2×C22⋊C4—2921no—
349SD32—734no—
350SD32—734no—
351SD32—734no—
352SD32C4×D7314yescomplemented
353M32—1462no—
354C2×C16—731no—
355C2×C16—734no—
356C2×C16—731no—
357C2×C16C22×D7311yescharacteristic
358C2×C16C22×D7311yescharacteristic
359#107—731no—
360#107—731no—
361#107C22×D7311yescharacteristic
362#107C22×D7311yescharacteristic
363#106—1461no—
364#106—1461no—
365#103—21no—
366#103—1461no—
367#103—1461no—
368#103—1461no—
369#102—21no—
370#102—1461no—
371#102—1461no—
372#102—1461no—
373#97—731no—
374#97—731no—
375C4×C8—731no—
376C4×C8C22×D7311yescharacteristic
377C24—2921no—
378C2×Q8—21no—
379C2×Q8—2921no—
380C2×Q8—2922no—
381C2×Q8—1464no—
382C2×Q8—1461no—
383C2×D4—21no—
384C2×D4—2922no—
385C2×D4—2921no—
386C2×D4—1464no—
387C2×D4—1461no—
388C22×C4—41no—
389C22×C4—2921no—
390C22×C4—2921no—
391C22×C4—2921no—
392C22×C4—2921no—
393C22×C4—731no—
394Q16—21no—
395Q16—732no—
396Q16—1464no—
397Q16—1461no—
398Q16C2×C4×D7312yes—
399D8—21no—
400D8—732no—
401D8—1464no—
402D8—1461no—
403D8C2×C4×D7312yes—
404M16—1462no—
405C2×C8—731no—
406C2×C8—732no—
407C2×C8—732no—
408C2×C8—731no—
409C2×C8D4×D7311yescharacteristic
410C2×C8C23×D7311yescharacteristic
411C4⋊C4—21no—
412C4⋊C4—21no—
413C4⋊C4—41no—
414C4⋊C4—2921no—
415C4⋊C4—2921no—
416C4⋊C4—2921no—
417C4⋊C4—731no—
418C4⋊C4—731no—
419C4⋊C4—1461no—
420C4⋊C4—1461no—
421C22⋊C4—41no—
422C22⋊C4—2921no—
423C22⋊C4—2921no—
424C22⋊C4—2921no—
425C22⋊C4—2921no—
426C22⋊C4—2921no—
427C22⋊C4—731no—
428C22⋊C4—731no—
429C42—41no—
430C42—2921no—
431C42—2921no—
432C42—2921no—
433C42—2921no—
434C42—2921no—
435C42—731no—
436C42D4×D7311yescharacteristic
437C16—21no—
438C16—732no—
439C16—1461no—
440C16C2×C4×D7312yes—
441C23—41no—
442C23—2924no—
443C23—2921no—
444C23—731no—
445C23—11681no—
446C23—11681no—
447Q8—22no—
448Q8—41no—
449Q8—2924no—
450Q8—2921no—
451Q8—1462no—
452D4—22no—
453D4—41no—
454D4—2924no—
455D4—2921no—
456D4—1462no—
457C2×C4—41no—
458C2×C4—41no—
459C2×C4—41no—
460C2×C4—161no—
461C2×C4—2924no—
462C2×C4—2921no—
463C2×C4—2921no—
464C2×C4—2921no—
465C2×C4—731no—
466C2×C4—732no—
467C2×C4—732no—
468C2×C4—731no—
469C2×C4—731no—
470C2×C4—11681no—
471C2×C4—11681no—
472C2×C4—11681no—
473C2×C4D8×D7311yescharacteristic
474C2×C4D8×D7311yescharacteristic
475C2×C4C2×D4×D7311yescharacteristic
476C8—21no—
477C8—732no—
478C8—1461no—
479C8C22×C4×D7311yescharacteristic
480C8#41×D7311yescharacteristic
481C22—42no—
482C22—81no—
483C22—81no—
484C22—2922no—
485C22—5841no—
486C22—5841no—
487C22—732no—
488C22—11684no—
489C22—1461no—
490C22—1461no—
491C22C2×D8×D7311yescharacteristic
492C4—21no—
493C4—21no—
494C4—42no—
495C4—81no—
496C4—81no—
497C4—2922no—
498C4—5841no—
499C4—5841no—
500C4—732no—
501C4—1461no—
502C4—1461no—
503C4C4×D4×D7311yescharacteristic
504C4#135×D7311yescharacteristic
505C2—82no—
506C2—5842no—
507C2—1462no—
508C2C4×D8×D7311yescharacteristic
509C2#2262307611yescharacteristic
510C2#2262307711yescharacteristic
subgroupG/H
centerC22C2×D8×D73
commutatorC8×C73C23×C4
FrattiniC2×C8C23×D73
FittingC73×#2175C2
socleC22×C73C22×D8
prime radicalC2×D8×D73C22
cosocleC2×C8×C73C24
hypercenter#2175D73
hypocenterC73#33457
LCSG ▷ C8×C73 ▷ C4×C73 ▷ C2×C73 ▷ C73
UCSC1 ◁ C22 ◁ C42 ◁ C4×C8 ◁ #2175

Complements

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

NH ≅ G/Ntype
2. #22618794C2semidirect
4. #22611797C2semidirect
5. #22611796C2semidirect
6. #22611794C2semidirect
7. #22611786C2semidirect
8. C4×#14556534C2semidirect
9. C4×#14556535C2semidirect
10. #22611780C2semidirect
11. #22611712C2semidirect
12. #22611649C2semidirect
14. #22611529C2semidirect
15. C73×#2175C2semidirect
23. SD32×D73C4semidirect
30. C4×Dic292C22semidirect
31. Q16×Dic73C22semidirect
32. #17988681C22semidirect
41. #17988414C22semidirect
42. #17988414C22semidirect
43. #17988413C22semidirect
44. #17988392C22semidirect
45. #17988391C22semidirect
46. #17988390C22semidirect
50. #17988362C22semidirect
51. C4×#11924003C22semidirect
52. C16×Dic73C22semidirect
54. C4×Q16×C73C22semidirect
56. #366×C73C22semidirect
57. #358×C73C22semidirect
59. #346×C73C22semidirect
65. Q16×D73C2×C4semidirect
89. #14556536C2×C4semidirect
90. #14556535C2×C4semidirect
91. #14556534C2×C4semidirect
95. C16×D73C2×C4semidirect
113. SD32×C73C2×C4semidirect
262. #2175D73semidirect
271. C73#33457semidirect
290. C4×Q16C2×D73semidirect
303. #366C2×D73semidirect
307. #358C2×D73semidirect
313. #346C2×D73semidirect
352. SD32C4×D73semidirect

Maximal subgroups

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

subgroupindex
C2×SD32×D732
#226187942
#226187782
#226117972
#226117962
#226117942
#226117862
C4×#145565342
C4×#145565352
#226117802
#226117122
#226116492
#226116452
#226115292
C73×#21752
#3345773