Finite Group Databasebeta

← #33239325

Subgroups — #33239325

Order 18688.

696 conjugacy classes of subgroups.

The subgroup lattice has 524 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#17989024C211yesmaximal, characteristic
2#22618806C211yesmaximal, characteristic, complemented
3#22618778C211yesmaximal, characteristic, complemented
4#22611808C211yesmaximal, characteristic, complemented
5#22611804C211yesmaximal, characteristic, complemented
6#22611798C211yesmaximal, characteristic, complemented
7SD32×Dic73C211yesmaximal, characteristic, complemented
8C4×#14556534C211yesmaximal, characteristic, complemented
9#22611777C211yesmaximal, characteristic, complemented
10C4×#14556532C211yesmaximal, characteristic, complemented
11#22611716C211yesmaximal, characteristic, complemented
12#22611637C211yesmaximal, characteristic, complemented
13#357×D73C211yesmaximal, characteristic, complemented
14#22611529C211yesmaximal, characteristic, complemented
15C73×#2175C211yesmaximal, characteristic, complemented
16C2×#14556614C2211yescharacteristic
17C2×D8×D73C2211yescharacteristic
18#17989069—21no—
19#17989059—21no—
20#17989024C418yescomplemented
21C2×#14556534C2211yescharacteristic
22C2×D1168C2211yescharacteristic
23C2×#14556532C2211yescharacteristic
24C2×#14556530C2211yescharacteristic
25C2×C16×D73C2211yescharacteristic
26#17988697—21no—
27#17988696—21no—
28#17988690—21no—
29#17988685—21no—
30Q16×Dic73C2211yescharacteristic, complemented
31C4×D584C2211yescharacteristic
32C4×D584C2211yescharacteristic, complemented
33D8×Dic73C2211yescharacteristic
34#17988669C2211yescharacteristic
35#17988669C2211yescharacteristic, complemented
36#17988624C2211yescharacteristic
37#107×D73C2211yescharacteristic
38#17988590—21no—
39#17988587—21no—
40#17988541C2211yescharacteristic
41#17988415C2211yescharacteristic, complemented
42#17988414C2211yescharacteristic, complemented
43#17988412C2211yescharacteristic, complemented
44#17988391C2211yescharacteristic, complemented
45#17988389C2211yescharacteristic
46#17988389C2211yescharacteristic, complemented
47#17988388C2211yescharacteristic
48#17988387C2211yescharacteristic
49#17988387C2211yescharacteristic, complemented
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×#11924029—21no—
61C2×D4×D73—21no—
62#14556614—22no—
63#14556614C2×C414yescomplemented
64C2×#11924015—21no—
65C2×D584C2311yescharacteristic
66C2×D584C2311yescharacteristic
67C2×#11924013—21no—
68D8×D73—22no—
69D8×D73C2×C414yescomplemented
70C2×#11924011D411yescharacteristic
71C2×C8×D73C2311yescharacteristic
72Q8×Dic73—21no—
73C4×D292—21no—
74C4×D292—21no—
75C4×D292—41no—
76D4×Dic73—21no—
77#14556553—21no—
78#14556553—21no—
79#14556553—41no—
80C4×C22⋊D73—41no—
81#14556551—41no—
82#14556549—21no—
83#14556548—41no—
84#14556548D411yescharacteristic
85C4⋊C4×D73—21no—
86C4⋊C4×D73—41no—
87C2×#11924005—41no—
88C22⋊C4×D73—41no—
89#14556534C2×C414yescomplemented
90D1168C2×C414yescomplemented
91#14556532C2×C414yescomplemented
92#14556530C2×C414yescomplemented
93C2×#11924003C2311yescharacteristic
94C2×#11924003C2311yescharacteristic
95C16×D73—22no—
96#14556525C2311yescharacteristic
97#14556525C2311yescharacteristic
98#14556524—21no—
99#14556522—21no—
100#14556515—21no—
101#14556513—21no—
102#14556513—21no—
103#14556512—21no—
104#14556511—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—
122#11924029—24no—
123#11924029—42no—
124C2×D292—21no—
125C2×D292—21no—
126C2×D292—41no—
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×D73—41no—
135C2×C4×D73—41no—
136C2×C4×D73—41no—
137C2×C4×D73C2×D411yescharacteristic
138#11924015—24no—
139D584—21no—
140D584—21no—
141D584C22×C412yes—
142D584C22×C412yes—
143#11924013—24no—
144#11924011—22no—
145C2×#10596831C2×D411yescharacteristic
146C2×#10596831C2×D411yescharacteristic
147C8×D73C22×C412yes—
148C8×D73#4112yes—
149C4⋊Dic73—21no—
150C4⋊Dic73—21no—
151C4⋊Dic73—41no—
152#11924007—41no—
153#11924007D811yescharacteristic
154#11924007D811yescharacteristic
155C22⋊Dic73—41no—
156#11924005—41no—
157#11924005—41no—
158#11924005—41no—
159#11924005—41no—
160#11924005—41no—
161#11924005D811yescharacteristic
162#11924005D811yescharacteristic
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—41no—
188C22×D73—44no—
189C22×D73—161no—
190C22×D73—161no—
191C22×D73C2×D811yescharacteristic
192Q8×C73—22no—
193Q8×C73—41no—
194D4×C73—22no—
195D4×C73—41no—
196C2×C4×C73—41no—
197C2×C4×C73—41no—
198C2×C4×C73—41no—
199C2×C4×C73—161no—
200C2×C4×C73C2×D811yescharacteristic
201C2×C4×C73C2×D811yescharacteristic
202C2×C4×C73C22×D411yescharacteristic
203C22⋊D73—44no—
204C2×Dic73—41no—
205C2×Dic73—41no—
206C2×Dic73—161no—
207C2×Dic73C2×D811yescharacteristic
208C2×Dic73C2×D811yescharacteristic
209C2×Dic73C2×D811yescharacteristic
210D292—22no—
211D292—22no—
212D292—44no—
213D292—41no—
214D292—41no—
215C4×D73—44no—
216C4×D73—161no—
217C4×D73—161no—
218C4×D73—161no—
219C4×D73—161no—
220C4×D73—161no—
221C4×D73C4×D412yes—
222C4×D73#13512yes—
223C8×C73—21no—
224C8×C73C23×C411yescharacteristic
225C8×C73C2×#4111yescharacteristic
226#10596831—21no—
227#10596831C4×D412yes—
228C22×C73—42no—
229C22×C73—81no—
230C22×C73—81no—
231C22×C73C22×D811yescharacteristic
232C2×D73—21no—
233C2×D73—21no—
234C2×D73—42no—
235C2×D73—42no—
236C2×D73—81no—
237C2×D73—81no—
238C2×D73—81no—
239C2×D73—81no—
240C2×D73—164no—
241C2×D73C4×D812yes—
242C4×C73—21no—
243C4×C73—21no—
244C4×C73—42no—
245C4×C73—81no—
246C4×C73—81no—
247C4×C73C2×C4×D411yescharacteristic
248C4×C73C2×#13511yescharacteristic
249Dic73—21no—
250Dic73—21no—
251Dic73—81no—
252Dic73—81no—
253Dic73C4×D812yes—
254#33454—731nomaximal, Sylow
255C2×C73—82no—
256C2×C73C2×C4×D811yescharacteristic
257C2×C73#341511yescharacteristic
258C2×C73#341611yescharacteristic
259D73—22no—
260D73—82no—
261D73—82no—
262C2×#509—731no—
263#2951—731no—
264#2948—731no—
265#2177—731no—
266#2177—731no—
267#2175—731no—
268#2175D7311yescharacteristic, complemented
269C4×SD32—731no—
270C4×SD32—731no—
271C4×D16—731no—
272C4×D16—731no—
273#2159—731no—
274#2138—731no—
275C2×#357—731no—
276#2108—731no—
277C73#3345411yescharacteristic, complemented, Sylow
278C2×#135—731no—
279C22×D8—731no—
280#520—1461no—
281#518—1461no—
282#509—738no—
283C2×SD32—731no—
284C2×SD32C2×D7311yescharacteristic
285C2×D16—731no—
286C2×D16—731no—
287C2×M32—731no—
288C22×C16—731no—
289#442—1461no—
290#442—1461no—
291#440—1461no—
292#440—1461no—
293C4×Q16—731no—
294C4×Q16C2×D7311yescharacteristic, complemented
295C4×D8—731no—
296C4×D8—731no—
297C4×D8—731no—
298C4×D8—731no—
299C4×D8—731no—
300C4×D8C2×D7311yescharacteristic
301#427—731no—
302C2×#107—731no—
303#418—1461no—
304#417—1461no—
305#405—731no—
306#367—731no—
307#367—731no—
308#366—731no—
309#366C2×D7311yescharacteristic, complemented
310#358—731no—
311#358C2×D7311yescharacteristic, complemented
312#357—731no—
313#357—731no—
314#357—731no—
315#357—731no—
316#357—731no—
317#357C2×D7311yescharacteristic
318#346—731no—
319#346C2×D7311yescharacteristic, complemented
320C4×C16—731no—
321C4×C16—731no—
322C2×#41—1461no—
323C22×D4—1461no—
324#135—734no—
325#135—1462no—
326C2×Q16C22×D7311yescharacteristic
327C2×SD16—1461no—
328C2×D8—734no—
329C2×D8—731no—
330C2×D8—731no—
331C2×D8—1462no—
332C2×D8—1461no—
333C2×D8C22×D7311yescharacteristic
334C2×M16—731no—
335C22×C8—731no—
336C4×Q8—21no—
337C4×Q8—1461no—
338C4×D4—21no—
339C4×D4—2921no—
340C4×D4—2921no—
341C4×D4—2921no—
342C4×D4—2921no—
343C4×D4—1461no—
344C4×D4—1461no—
345C4×D4—1461no—
346C4×D4—1461no—
347C4×D4—1461no—
348#117—2921no—
349#117—731no—
350#117—1461no—
351C2×C4⋊C4—2921no—
352C2×C4⋊C4—1461no—
353C2×C22⋊C4—2921no—
354C2×C22⋊C4—2921no—
355SD32—734no—
356SD32C4×D7314yescomplemented
357D16—734no—
358D16—734no—
359M32—734no—
360C2×C16—731no—
361C2×C16—731no—
362C2×C16—1462no—
363C2×C16C22×D7311yescharacteristic
364C2×C16C22×D7311yescharacteristic
365#107—731no—
366#107—731no—
367#107—1461no—
368#107C22×D7311yescharacteristic
369#107C22×D7311yescharacteristic
370#106—1461no—
371#103—21no—
372#103—1461no—
373#102—21no—
374#102—1461no—
375#102—1461no—
376#102—1461no—
377#102—1461no—
378#102—1461no—
379#97—731no—
380#97—731no—
381C4×C8—731no—
382C4×C8C22×D7311yescharacteristic
383C24—2921no—
384#41—2922no—
385#41—1464no—
386C2×Q8—21no—
387C2×D4—21no—
388C2×D4—2921no—
389C2×D4—2922no—
390C2×D4—2921no—
391C2×D4—1464no—
392C2×D4—1461no—
393C2×D4—1461no—
394C22×C4—41no—
395C22×C4—2921no—
396C22×C4—2921no—
397C22×C4—2921no—
398C22×C4—2921no—
399C22×C4—2921no—
400C22×C4—2921no—
401C22×C4—2921no—
402C22×C4—731no—
403Q16—21no—
404Q16C2×C4×D7312yes—
405SD16—1464no—
406D8—21no—
407D8—732no—
408D8—732no—
409D8—1464no—
410D8—1461no—
411D8—1461no—
412D8C2×C4×D7312yes—
413M16—1462no—
414C2×C8—731no—
415C2×C8—732no—
416C2×C8—732no—
417C2×C8—731no—
418C2×C8D4×D7311yescharacteristic
419C2×C8C23×D7311yescharacteristic
420C4⋊C4—21no—
421C4⋊C4—21no—
422C4⋊C4—41no—
423C4⋊C4—2921no—
424C4⋊C4—2921no—
425C4⋊C4—731no—
426C4⋊C4—731no—
427C4⋊C4—1461no—
428C4⋊C4—1461no—
429C22⋊C4—41no—
430C22⋊C4—2921no—
431C22⋊C4—2921no—
432C22⋊C4—2921no—
433C22⋊C4—2921no—
434C22⋊C4—2921no—
435C22⋊C4—2921no—
436C22⋊C4—731no—
437C22⋊C4—731no—
438C42—41no—
439C42—2921no—
440C42—2921no—
441C42—731no—
442C42D4×D7311yescharacteristic
443C16—21no—
444C16—732no—
445C16—1461no—
446C16C2×C4×D7312yes—
447C23—41no—
448C23—2924no—
449C23—2921no—
450C23—2921no—
451C23—731no—
452C23—11681no—
453C23—11681no—
454Q8—22no—
455Q8—41no—
456D4—22no—
457D4—41no—
458D4—2924no—
459D4—2924no—
460D4—2921no—
461D4—2921no—
462D4—1462no—
463D4—1462no—
464C2×C4—41no—
465C2×C4—41no—
466C2×C4—41no—
467C2×C4—161no—
468C2×C4—2924no—
469C2×C4—2921no—
470C2×C4—2921no—
471C2×C4—731no—
472C2×C4—732no—
473C2×C4—732no—
474C2×C4—731no—
475C2×C4—731no—
476C2×C4—11681no—
477C2×C4—11681no—
478C2×C4—11681no—
479C2×C4—11681no—
480C2×C4—11681no—
481C2×C4—11681no—
482C2×C4D8×D7311yescharacteristic
483C2×C4D8×D7311yescharacteristic
484C2×C4C2×D4×D7311yescharacteristic
485C8—21no—
486C8—732no—
487C8—1461no—
488C8C22×C4×D7311yescharacteristic
489C8#41×D7311yescharacteristic
490C22—42no—
491C22—81no—
492C22—81no—
493C22—2922no—
494C22—2922no—
495C22—5841no—
496C22—5841no—
497C22—5841no—
498C22—5841no—
499C22—732no—
500C22—11684no—
501C22—1461no—
502C22—1461no—
503C22C2×D8×D7311yescharacteristic
504C4—21no—
505C4—21no—
506C4—42no—
507C4—81no—
508C4—81no—
509C4—5841no—
510C4—5841no—
511C4—732no—
512C4—1461no—
513C4—1461no—
514C4C4×D4×D7311yescharacteristic
515C4#135×D7311yescharacteristic
516C2—82no—
517C2—5842no—
518C2—5842no—
519C2—1462no—
520C2C4×D8×D7311yescharacteristic
521C2#2262307111yescharacteristic
522C2#2262307611yescharacteristic
subgroupG/H
centerC22C2×D8×D73
commutatorC8×C73C23×C4
FrattiniC2×C8C23×D73
FittingC73×#2175C2
socleC22×C73C22×D8
prime radicalC2×#17989024C2
cosocleC2×C8×C73C24
hypercenter#2175D73
hypocenterC73#33454
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. #22618806C2semidirect
3. #22618778C2semidirect
4. #22611808C2semidirect
5. #22611804C2semidirect
6. #22611798C2semidirect
7. SD32×Dic73C2semidirect
8. C4×#14556534C2semidirect
9. #22611777C2semidirect
10. C4×#14556532C2semidirect
11. #22611716C2semidirect
12. #22611637C2semidirect
13. #357×D73C2semidirect
14. #22611529C2semidirect
15. C73×#2175C2semidirect
20. #17989024C4semidirect
30. Q16×Dic73C22semidirect
32. C4×D584C22semidirect
35. #17988669C22semidirect
41. #17988415C22semidirect
42. #17988414C22semidirect
43. #17988412C22semidirect
44. #17988391C22semidirect
46. #17988389C22semidirect
49. #17988387C22semidirect
50. #17988362C22semidirect
51. C4×#11924003C22semidirect
52. C16×Dic73C22semidirect
54. C4×Q16×C73C22semidirect
56. #366×C73C22semidirect
57. #358×C73C22semidirect
59. #346×C73C22semidirect
63. #14556614C2×C4semidirect
69. D8×D73C2×C4semidirect
89. #14556534C2×C4semidirect
90. D1168C2×C4semidirect
91. #14556532C2×C4semidirect
92. #14556530C2×C4semidirect
113. SD32×C73C2×C4semidirect
268. #2175D73semidirect
277. C73#33454semidirect
294. C4×Q16C2×D73semidirect
309. #366C2×D73semidirect
311. #358C2×D73semidirect
319. #346C2×D73semidirect
356. SD32C4×D73semidirect

Maximal subgroups

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

subgroupindex
C2×#179890242
#226188062
#226187782
#226118082
#226118042
#226117982
SD32×Dic732
C4×#145565342
#226117772
C4×#145565322
#226117162
#226116372
#357×D732
#226115292
C73×#21752
#3345473