Finite Group Databasebeta

← #33240672

Subgroups — #33240672

Order 18688.

556 conjugacy classes of subgroups.

The subgroup lattice has 478 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#22620468C211yesmaximal, characteristic, complemented
2#22618778C211yesmaximal, characteristic
3#22612090C211yesmaximal, characteristic, complemented
4#22612087C211yesmaximal, characteristic, complemented
5#22612085C211yesmaximal, characteristic, complemented
6#22612079C211yesmaximal, characteristic, complemented
7#22612076C211yesmaximal, characteristic, complemented
8#22612068C211yesmaximal, characteristic, complemented
9#22612067C211yesmaximal, characteristic, complemented
10#22611716C211yesmaximal, characteristic, complemented
11#22611714C211yesmaximal, characteristic, complemented
12#22611688C211yesmaximal, characteristic, complemented
13#22611645C211yesmaximal, characteristic
14#357×D73C211yesmaximal, characteristic
15C73×#2228C211yesmaximal, characteristic, complemented
16C2×D8×D73C2211yescharacteristic
17#17989217—21no—
18#17989059—21no—
19C2×#14556530C2211yescharacteristic
20C2×C16×D73C2211yescharacteristic
21#17988997—21no—
22#17988994—21no—
23#17988992C2211yescharacteristic, complemented
24#17988991C2211yescharacteristic, complemented
25#17988988C2211yescharacteristic, complemented
26#17988697—21no—
27#17988696—21no—
28C4×D584C2211yescharacteristic
29D8×Dic73C2211yescharacteristic
30#17988669C2211yescharacteristic
31#17988626—21no—
32#17988624C2211yescharacteristic
33#17988624C2211yescharacteristic
34#107×D73C2211yescharacteristic
35#17988590—21no—
36#17988541C2211yescharacteristic
37#17988415C2211yescharacteristic, complemented
38#17988415C2211yescharacteristic, complemented
39#17988414C2211yescharacteristic, complemented
40#17988412C2211yescharacteristic, complemented
41#17988412C2211yescharacteristic, complemented
42#17988412C2211yescharacteristic, complemented
43C4⋊#11924003C2211yescharacteristic, complemented
44#17988405C2211yescharacteristic, complemented
45#17988405C2211yescharacteristic, complemented
46#17988389C2211yescharacteristic
47#17988389C2211yescharacteristic
48#17988388C2211yescharacteristic
49#17988388C2211yescharacteristic
50#17988387C2211yescharacteristic
51#17988387C2211yescharacteristic
52#499×C73C2211yescharacteristic, complemented
53C4×D8×C73C2211yescharacteristic
54#367×C73C2211yescharacteristic, complemented
55#366×C73C2211yescharacteristic, complemented
56C4⋊C16×C73C2211yescharacteristic, complemented
57#357×C73C2211yescharacteristic
58#357×C73C2211yescharacteristic
59C2×D4×D73—21no—
60C2×D584C2311yescharacteristic
61C2×#11924013—21no—
62D8×D73—22no—
63D8×D73Q814yescomplemented
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×D292—21no—
74D4×Dic73—21no—
75#14556553—21no—
76C4×C22⋊D73—41no—
77#14556551—41no—
78#14556549—21no—
79#14556549—21no—
80#14556548D411yescharacteristic
81C4⋊C4×D73—21no—
82C2×#11924005—41no—
83C22⋊C4×D73—41no—
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#14556522—21no—
95#14556522—21no—
96#14556522—21no—
97#14556513—21no—
98#14556512—21no—
99#14556511—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
114C23×D73—41no—
115C2×Dic146—41no—
116C2×Dic146—41no—
117C2×D292—21no—
118C2×C22⋊D73—41no—
119D4×D73—24no—
120D4×D73—42no—
121C22×Dic73—41no—
122C2×C4×D73—41no—
123C2×C4×D73—41no—
124C2×C4×D73—41no—
125C2×C4×D73C2×D411yescharacteristic
126D584—21no—
127D584C2×Q812yes—
128#11924013—24no—
129#11924011—22no—
130C2×#10596831C2×D411yescharacteristic
131C2×#10596831C2×D411yescharacteristic
132C8×D73C2×Q812yes—
133C8×D73#4112yes—
134C4⋊Dic73—21no—
135C4⋊Dic73—21no—
136C4⋊Dic73—21no—
137C4⋊Dic73—41no—
138C4⋊Dic73—41no—
139#11924007—41no—
140#11924007—41no—
141#11924007—41no—
142#11924007D811yescharacteristic
143#11924007D811yescharacteristic
144C22⋊Dic73—41no—
145#11924005—41no—
146#11924005—41no—
147#11924005—41no—
148#11924005—41no—
149#11924005D811yescharacteristic
150#11924005D811yescharacteristic
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×D73—41no—
173C22×D73—44no—
174C22×D73—161no—
175C22×D73—161no—
176C22×D73C2×D811yescharacteristic
177D4×C73—22no—
178D4×C73—41no—
179C2×C4×C73—41no—
180C2×C4×C73—41no—
181C2×C4×C73—41no—
182C2×C4×C73—161no—
183C2×C4×C73C2×D811yescharacteristic
184C2×C4×C73C2×D811yescharacteristic
185C2×C4×C73C22×D411yescharacteristic
186C22⋊D73—44no—
187C2×Dic73—41no—
188C2×Dic73—41no—
189C2×Dic73—41no—
190C2×Dic73—161no—
191C2×Dic73C2×D811yescharacteristic
192C2×Dic73C2×D811yescharacteristic
193C2×Dic73C2×D811yescharacteristic
194D292—22no—
195D292—41no—
196C4×D73—161no—
197C4×D73—161no—
198C4×D73—161no—
199C4×D73C22⋊Q812yes—
200C4×D73#13712yes—
201Dic146—161no—
202Dic146—161no—
203C8×C73—21no—
204C8×C73C22×Q811yescharacteristic
205C8×C73C2×#4111yescharacteristic
206#10596831—21no—
207#10596831C22⋊Q812yes—
208C22×C73—42no—
209C22×C73—81no—
210C22×C73—81no—
211C22×C73C22×D811yescharacteristic
212C2×D73—21no—
213C2×D73—21no—
214C2×D73—42no—
215C2×D73—81no—
216C2×D73—81no—
217C2×D73—164no—
218C2×D73#47412yes—
219C4×C73—21no—
220C4×C73—21no—
221C4×C73—81no—
222C4×C73—81no—
223C4×C73—81no—
224C4×C73C2×C22⋊Q811yescharacteristic
225C4×C73C2×#13711yescharacteristic
226Dic73—21no—
227Dic73—21no—
228Dic73—81no—
229Dic73—81no—
230Dic73—81no—
231Dic73#47412yes—
232#33631—731nomaximal, Sylow
233C2×C73—82no—
234C2×C73C2×#47411yescharacteristic
235C2×C73#341511yescharacteristic
236C2×C73#341611yescharacteristic
237D73—22no—
238D73—82no—
239#3164—731no—
240#2948—731no—
241#2228—731no—
242#2228—731no—
243#2228—731no—
244#2228D7311yescharacteristic, complemented
245#2226—731no—
246#2226—731no—
247#2224—731no—
248#2224—731no—
249#2159—731no—
250#2158—731no—
251#2151—731no—
252#2140—731no—
253C2×#357—731no—
254C73#3363111yescharacteristic, complemented, Sylow
255C22×D8—731no—
256#544—1461no—
257#518—1461no—
258C2×M32—731no—
259C22×C16—731no—
260#501—1461no—
261#501—1461no—
262#500—731no—
263#500—731no—
264#499—731no—
265#499C2×D7311yescharacteristic, complemented
266#442—1461no—
267#442—1461no—
268C4×D8—731no—
269C4×D8—731no—
270C4×D8—731no—
271C4×D8C2×D7311yescharacteristic
272#428—1461no—
273#427—731no—
274#427—731no—
275C2×#107—731no—
276#418—1461no—
277#405—731no—
278#367—731no—
279#367—731no—
280#367—731no—
281#367C2×D7311yescharacteristic, complemented
282#366—731no—
283#366—731no—
284#366—731no—
285#366C2×D7311yescharacteristic, complemented
286C4⋊C16—731no—
287C4⋊C16—731no—
288C4⋊C16—731no—
289C4⋊C16C2×D7311yescharacteristic, complemented
290#357—731no—
291#357—731no—
292#357—731no—
293#357—731no—
294#357—731no—
295#357—731no—
296#357C2×D7311yescharacteristic
297#357C2×D7311yescharacteristic
298C22×D4—1461no—
299C2×D8—734no—
300C2×D8—731no—
301C2×D8—1462no—
302C2×D8—1461no—
303C2×D8C22×D7311yescharacteristic
304C2×M16—731no—
305C22×C8—731no—
306C4⋊Q8—2921no—
307C4⋊Q8—1461no—
308C4⋊Q8—1461no—
309#125—21no—
310#125—2921no—
311#125—1461no—
312C22⋊Q8—2921no—
313C22⋊Q8—2921no—
314C4×D4—21no—
315C4×D4—2921no—
316C4×D4—2921no—
317C4×D4—1461no—
318C4×D4—1461no—
319C4×D4—1461no—
320#117—731no—
321#117—1461no—
322#117—1461no—
323C2×C4⋊C4—1461no—
324C2×C22⋊C4—2921no—
325C2×C22⋊C4—2921no—
326M32—1462no—
327C2×C16—731no—
328C2×C16—731no—
329C2×C16—1462no—
330C2×C16C22×D7311yescharacteristic
331C2×C16C22×D7311yescharacteristic
332#107—731no—
333#107—731no—
334#107—731no—
335#107—1461no—
336#107—1461no—
337#107—1461no—
338#107C22×D7311yescharacteristic
339#107C22×D7311yescharacteristic
340#107C22×D7311yescharacteristic
341#106—21no—
342#106—1461no—
343#106—1461no—
344#106—1461no—
345#102—21no—
346#102—1461no—
347#102—1461no—
348#102—1461no—
349#97—731no—
350#97—731no—
351C4×C8—731no—
352C4×C8C22×D7311yescharacteristic
353C24—2921no—
354C2×Q8—2921no—
355C2×Q8—2921no—
356C2×D4—21no—
357C2×D4—2922no—
358C2×D4—2921no—
359C2×D4—1464no—
360C2×D4—1461no—
361C22×C4—41no—
362C22×C4—2921no—
363C22×C4—2921no—
364C22×C4—2921no—
365C22×C4—2921no—
366C22×C4—731no—
367D8—21no—
368D8—732no—
369D8—1464no—
370D8—1461no—
371D8Q8×D7312yes—
372M16—1462no—
373C2×C8—731no—
374C2×C8—732no—
375C2×C8—732no—
376C2×C8—731no—
377C2×C8D4×D7311yescharacteristic
378C2×C8C23×D7311yescharacteristic
379C4⋊C4—21no—
380C4⋊C4—21no—
381C4⋊C4—21no—
382C4⋊C4—41no—
383C4⋊C4—41no—
384C4⋊C4—2921no—
385C4⋊C4—2921no—
386C4⋊C4—2921no—
387C4⋊C4—2921no—
388C4⋊C4—2921no—
389C4⋊C4—731no—
390C4⋊C4—731no—
391C4⋊C4—1461no—
392C4⋊C4—1461no—
393C4⋊C4—1461no—
394C22⋊C4—41no—
395C22⋊C4—2921no—
396C22⋊C4—2921no—
397C22⋊C4—2921no—
398C22⋊C4—2921no—
399C22⋊C4—2921no—
400C22⋊C4—731no—
401C22⋊C4—731no—
402C42—2921no—
403C42—2921no—
404C42—731no—
405C42D4×D7311yescharacteristic
406C16—21no—
407C16—21no—
408C16—1461no—
409C16—1461no—
410C23—41no—
411C23—2924no—
412C23—2921no—
413C23—731no—
414C23—11681no—
415C23—11681no—
416Q8—11681no—
417Q8—11681no—
418D4—22no—
419D4—41no—
420D4—2924no—
421D4—2921no—
422D4—1462no—
423C2×C4—41no—
424C2×C4—41no—
425C2×C4—41no—
426C2×C4—161no—
427C2×C4—2921no—
428C2×C4—2921no—
429C2×C4—2921no—
430C2×C4—731no—
431C2×C4—732no—
432C2×C4—732no—
433C2×C4—731no—
434C2×C4—731no—
435C2×C4—11681no—
436C2×C4—11681no—
437C2×C4—11681no—
438C2×C4—11681no—
439C2×C4D8×D7311yescharacteristic
440C2×C4D8×D7311yescharacteristic
441C2×C4C2×D4×D7311yescharacteristic
442C8—21no—
443C8—732no—
444C8—1461no—
445C8C2×Q8×D7311yescharacteristic
446C8#41×D7311yescharacteristic
447C22—42no—
448C22—81no—
449C22—81no—
450C22—2922no—
451C22—5841no—
452C22—5841no—
453C22—732no—
454C22—11684no—
455C22—1461no—
456C22—1461no—
457C22C2×D8×D7311yescharacteristic
458C4—21no—
459C4—21no—
460C4—81no—
461C4—81no—
462C4—81no—
463C4—5841no—
464C4—5841no—
465C4—5841no—
466C4—732no—
467C4—1461no—
468C4—1461no—
469C4C22⋊Q8×D7311yescharacteristic
470C4#137×D7311yescharacteristic
471C2—82no—
472C2—5842no—
473C2—1462no—
474C2#474×D7311yescharacteristic
475C2#2262307111yescharacteristic
476C2#2262307411yescharacteristic
subgroupG/H
centerC22C2×D8×D73
commutator, cosocleC2×C8×C73C24
FrattiniC2×C8C23×D73
FittingC73×#2228C2
socleC22×C73C22×D8
prime radicalC2×D8×D73C22
hypercenter#2228D73
hypocenterC73#33631
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. #22612090C2semidirect
4. #22612087C2semidirect
5. #22612085C2semidirect
6. #22612079C2semidirect
7. #22612076C2semidirect
8. #22612068C2semidirect
9. #22612067C2semidirect
10. #22611716C2semidirect
11. #22611714C2semidirect
12. #22611688C2semidirect
15. C73×#2228C2semidirect
23. #17988992C22semidirect
24. #17988991C22semidirect
25. #17988988C22semidirect
37. #17988415C22semidirect
38. #17988415C22semidirect
39. #17988414C22semidirect
40. #17988412C22semidirect
41. #17988412C22semidirect
42. #17988412C22semidirect
43. C4⋊#11924003C22semidirect
44. #17988405C22semidirect
45. #17988405C22semidirect
52. #499×C73C22semidirect
54. #367×C73C22semidirect
55. #366×C73C22semidirect
56. C4⋊C16×C73C22semidirect
63. D8×D73Q8semidirect
244. #2228D73semidirect
254. C73#33631semidirect
265. #499C2×D73semidirect
281. #367C2×D73semidirect
285. #366C2×D73semidirect
289. C4⋊C16C2×D73semidirect

Maximal subgroups

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

subgroupindex
#226204682
#226187782
#226120902
#226120872
#226120852
#226120792
#226120762
#226120682
#226120672
#226117162
#226117142
#226116882
#226116452
#357×D732
C73×#22282
#3363173