Finite Group Databasebeta

← #33239301

Subgroups — #33239301

Order 18688.

738 conjugacy classes of subgroups.

The subgroup lattice has 436 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#17989025C211yesmaximal, characteristic
2#22618702C211yesmaximal, characteristic, complemented
3C2×#17988669C211yesmaximal, characteristic, complemented
4#22611806C212yesmaximal, complemented
5#22611794C212yesmaximal, complemented
6#22611782C212yesmaximal, complemented
7#22611777C212yesmaximal, complemented
8#22611646C211yesmaximal, characteristic, complemented
9#22611641C211yesmaximal, characteristic, complemented
10M32×Dic73C211yesmaximal, characteristic, complemented
11C73×#2159C211yesmaximal, characteristic, complemented
12C2×#14556613C2211yescharacteristic
13C22×D584C2211yescharacteristic
14#17989055—21no—
15C2×#14556553—21no—
16#17989025C418yescomplemented
17C2×#14556536C2212yes—
18C2×D1168C2212yes—
19C2×#14556531C2211yescharacteristic
20#17988681C2211yescharacteristic, complemented
21#17988681C2211yescharacteristic, complemented
22#17988674—21no—
23#17988674—21no—
24#17988669—21no—
25#17988669—21no—
26#17988669C2212yes—
27#17988669C2212yes—
28#17988669C2211yescharacteristic, complemented
29#17988669C2211yescharacteristic, complemented
30#17988581—21no—
31C2×#14556511—21no—
32C2×C8×Dic73C2211yescharacteristic
33#17988390C2212yescomplemented
34#17988387C2212yes—
35#17988387C2212yes—
36#17988387C2212yescomplemented
37#17988362C2212yescomplemented
38C16×Dic73C2212yescomplemented
39C2×M32×C73C2211yescharacteristic
40#427×C73C2211yescharacteristic
41C2×#107×C73C2211yescharacteristic
42#367×C73C2212yescomplemented
43#366×C73C2212yescomplemented
44C2×#11924028—21no—
45C22×D292—21no—
46C22×C4×D73—41no—
47#14556613—22no—
48#14556613C2×C414yescomplemented
49C2×Dic292C2311yescharacteristic
50C2×#11924017—21no—
51C2×D584—21no—
52C2×D584—22no—
53C2×D584C2×C414yescomplemented
54C2×D584C2312yes—
55C2×D584C2311yescharacteristic
56C22×#10596831D411yescharacteristic
57#14556556—21no—
58#14556556—21no—
59#14556553—22no—
60#14556553—22no—
61#14556553—21no—
62#14556553—21no—
63#14556553—41no—
64#14556553—41no—
65#14556551—41no—
66#14556551—41no—
67#14556548—41no—
68C2×#11924005—41no—
69C2×C4×Dic73D411yescharacteristic
70C42×D73—41no—
71#14556536C2×C418yescomplemented
72D1168C2×C418yescomplemented
73#14556531—22no—
74C2×#11924003C2312yes—
75#14556514—21no—
76#14556514—21no—
77#14556511—22no—
78#14556511—22no—
79#14556511—21no—
80#14556511—21no—
81C8×Dic73D411yescharacteristic
82C8×Dic73D411yescharacteristic
83C8×Dic73C2311yescharacteristic
84C8×Dic73C2311yescharacteristic
85C22×C8×C73C2311yescharacteristic
86#117×C73—21no—
87C2×C4⋊C4×C73—21no—
88M32×C73C2×C414yescomplemented
89C2×C16×C73C2312yes—
90#107×C73—21no—
91#107×C73C2311yescharacteristic
92#107×C73C2311yescharacteristic
93#107×C73C2312yes—
94#106×C73—21no—
95C23×D73—41no—
96#11924028—24no—
97#11924028—42no—
98C2×Dic146—21no—
99C2×D292—24no—
100C2×D292—22no—
101C2×D292—21no—
102C2×D292—41no—
103C2×D292—42no—
104C2×C22⋊D73—41no—
105C22×Dic73D811yescharacteristic
106C22×Dic73D811yescharacteristic
107C2×C4×D73—41no—
108C2×C4×D73—41no—
109C2×C4×D73—41no—
110C2×C4×D73—42no—
111C2×C4×D73—42no—
112C2×C4×D73—84no—
113Dic292—21no—
114Dic292C22×C412yescomplemented
115#11924017—24no—
116D584—24no—
117D584—22no—
118D584—21no—
119D584C22×C414yes—
120D584C22×C412yescomplemented
121C2×#10596831—22no—
122C2×#10596831C2×D411yescharacteristic
123C2×#10596831C2×D411yescharacteristic
124#11924007—41no—
125#11924007—41no—
126#11924005—42no—
127#11924005—42no—
128#11924005—41no—
129#11924005—41no—
130C4×Dic73—41no—
131C4×Dic73—41no—
132C4×Dic73D811yescharacteristic
133C4×Dic73D811yescharacteristic
134C4×Dic73C2×D411yescharacteristic
135C4×Dic73C2×D411yescharacteristic
136#11924003—22no—
137C22×C4×C73—41no—
138C22×C4×C73C2×D411yescharacteristic
139C2×C8×C73C22×C412yes—
140C2×C8×C73C2×D411yescharacteristic
141C2×C8×C73#4112yes—
142C2×C8×C73C2411yescharacteristic
143C4⋊C4×C73—21no—
144C4⋊C4×C73—21no—
145C4⋊C4×C73—22no—
146C4⋊C4×C73—41no—
147C22⋊C4×C73—41no—
148C42×C73—41no—
149C16×C73C22×C414yescomplemented
150C23×C73C2×D811yescharacteristic
151C22×D73—41no—
152C22×D73—42no—
153C22×D73—44no—
154C22×D73—82no—
155C22×D73—82no—
156C2×C4×C73—41no—
157C2×C4×C73—41no—
158C2×C4×C73—42no—
159C2×C4×C73—82no—
160C2×C4×C73C4×D412yes—
161C2×C4×C73C2×D811yescharacteristic
162C2×C4×C73#13512yes—
163C2×C4×C73C22×D411yescharacteristic
164C22⋊D73—44no—
165C2×Dic73—22no—
166C2×Dic73—41no—
167C2×Dic73—41no—
168C2×Dic73C2×D811yescharacteristic
169C2×Dic73C2×D811yescharacteristic
170C2×Dic73C2×D811yescharacteristic
171C2×Dic73C2×D811yescharacteristic
172D292—24no—
173D292—22no—
174D292—42no—
175D292—41no—
176D292—44no—
177C4×D73—44no—
178C4×D73—82no—
179C4×D73—84no—
180C4×D73—82no—
181C4×D73—84no—
182Dic146—22no—
183Dic146—41no—
184C8×C73C4×D412yes—
185C8×C73C23×C411yescharacteristic
186C8×C73C2×#4111yescharacteristic
187#10596831—21no—
188#10596831—21no—
189C22×C73—21no—
190C22×C73—21no—
191C22×C73C4×D812yes—
192C22×C73C22×D811yescharacteristic
193C2×D73—42no—
194C2×D73—44no—
195C2×D73—88no—
196C2×D73—82no—
197C2×D73—81no—
198C2×D73—81no—
199C2×D73—82no—
200C4×C73—82no—
201C4×C73—81no—
202C4×C73—81no—
203C4×C73C4×D812yes—
204C4×C73C2×C4×D411yescharacteristic
205C4×C73C2×#13511yescharacteristic
206Dic73—21no—
207Dic73—21no—
208Dic73—21no—
209Dic73—42no—
210Dic73#50912yescomplemented
211C4×#509—731nomaximal, Sylow
212C2×C73—22no—
213C2×C73C2×C4×D811yescharacteristic
214C2×C73C2×#50911yescharacteristic
215C2×C73#341411yescharacteristic
216D73—82no—
217D73—84no—
218C2×#509—731no—
219C4×#135—731no—
220C2×C4×D8—731no—
221#2177—732no—
222#2175—732no—
223C4×SD32—732no—
224C4×D16—732no—
225#2159D7311yescharacteristic, complemented
226#2140—731no—
227#2139—731no—
228C4×M32—731no—
229C73C4×#50911yescharacteristic, complemented, Sylow
230C2×#135—731no—
231C22×D8—731no—
232C4×#41—1461no—
233C2×C4×D4—1461no—
234#509—738no—
235C2×SD32—732no—
236C2×D16—732no—
237C2×M32—731no—
238C2×M32C2×D7311yescharacteristic
239C4×Q16—731no—
240C4×Q16—731no—
241C4×SD16—1461no—
242C4×SD16—1461no—
243C4×D8—732no—
244C4×D8—732no—
245C4×D8—731no—
246C4×D8—731no—
247C4×D8—1461no—
248C4×D8—1461no—
249#427C2×D7311yescharacteristic
250C2×#107C2×D7311yescharacteristic
251#416—1461no—
252C2×#102—1461no—
253C2×C4×C8—731no—
254#367C2×D7312yescomplemented
255#366C2×D7312yescomplemented
256#358—732no—
257#357—732no—
258#357—732no—
259#357—732no—
260#346—732no—
261C4×C16—732no—
262C2×#41—1461no—
263C22×D4—1461no—
264C23×C4—2921no—
265#135—734no—
266#135—1462no—
267C2×Q16—731no—
268C2×SD16—1461no—
269C2×D8—734no—
270C2×D8—732no—
271C2×D8—731no—
272C2×D8—1461no—
273C2×D8—1462no—
274C22×C8—731no—
275C22×C8C22×D7311yescharacteristic
276C4×Q8—1461no—
277C4×Q8—1461no—
278C4×D4—2921no—
279C4×D4—2921no—
280C4×D4—2921no—
281C4×D4—2921no—
282C4×D4—1462no—
283C4×D4—1462no—
284C4×D4—1461no—
285C4×D4—1461no—
286#117—21no—
287#117—2921no—
288C2×C4⋊C4—21no—
289C2×C22⋊C4—2921no—
290C2×C42—2921no—
291C2×C42—731no—
292SD32—738no—
293D16—738no—
294M32—1462no—
295M32C4×D7314yescomplemented
296C2×C16—732no—
297C2×C16C22×D7312yes—
298#107—21no—
299#107C22×D7311yescharacteristic
300#107C22×D7311yescharacteristic
301#107C22×D7312yes—
302#106—21no—
303#103—1461no—
304#103—1461no—
305#102—1462no—
306#102—1462no—
307#102—1461no—
308#102—1461no—
309C4×C8—731no—
310C4×C8—731no—
311C4×C8—731no—
312C4×C8—731no—
313C24—2921no—
314#41—2922no—
315#41—1464no—
316C2×Q8—1461no—
317C2×D4—2921no—
318C2×D4—2921no—
319C2×D4—2922no—
320C2×D4—1464no—
321C2×D4—1462no—
322C2×D4—1461no—
323C22×C4—41no—
324C22×C4—2921no—
325C22×C4—2921no—
326C22×C4—2922no—
327C22×C4—2922no—
328C22×C4—2921no—
329C22×C4—5844no—
330C22×C4—731no—
331C22×C4—731no—
332C22×C4D4×D7311yescharacteristic
333Q16—732no—
334Q16—1461no—
335SD16—1464no—
336D8—734no—
337D8—732no—
338D8—1464no—
339D8—1462no—
340D8—1461no—
341C2×C8—731no—
342C2×C8—731no—
343C2×C8—1462no—
344C2×C8C2×C4×D7312yes—
345C2×C8D4×D7311yescharacteristic
346C2×C8#1192402912yes—
347C2×C8C23×D7311yescharacteristic
348C4⋊C4—22no—
349C4⋊C4—21no—
350C4⋊C4—21no—
351C4⋊C4—41no—
352C4⋊C4—2921no—
353C4⋊C4—2921no—
354C22⋊C4—41no—
355C22⋊C4—2922no—
356C22⋊C4—2922no—
357C22⋊C4—2921no—
358C22⋊C4—2921no—
359C42—41no—
360C42—2921no—
361C42—2921no—
362C42—731no—
363C42—731no—
364C42—731no—
365C42—731no—
366C16—1462no—
367C16C2×C4×D7314yescomplemented
368C23—2924no—
369C23—2921no—
370C23—2922no—
371C23—5842no—
372C23—5842no—
373C23D8×D7311yescharacteristic
374Q8—2921no—
375Q8—1462no—
376D4—2924no—
377D4—2924no—
378D4—2922no—
379D4—2921no—
380D4—1464no—
381D4—1462no—
382C2×C4—41no—
383C2×C4—41no—
384C2×C4—42no—
385C2×C4—82no—
386C2×C4—2921no—
387C2×C4—2924no—
388C2×C4—2921no—
389C2×C4—5844no—
390C2×C4—5842no—
391C2×C4—5844no—
392C2×C4—5842no—
393C2×C4—731no—
394C2×C4—731no—
395C2×C4—731no—
396C2×C4—731no—
397C2×C4—1462no—
398C2×C4#1455655312yes—
399C2×C4D8×D7311yescharacteristic
400C2×C4#1455661412yes—
401C2×C4C2×D4×D7311yescharacteristic
402C8—1461no—
403C8—1461no—
404C8#1455655312yes—
405C8C22×C4×D7311yescharacteristic
406C8C2×#1192402911yescharacteristic
407C22—21no—
408C22—21no—
409C22—2922no—
410C22—2924no—
411C22—5848no—
412C22—5842no—
413C22—5841no—
414C22—5841no—
415C22—5842no—
416C22#1798866912yes—
417C22C2×D8×D7311yescharacteristic
418C4—82no—
419C4—81no—
420C4—81no—
421C4—2922no—
422C4—732no—
423C4—1461no—
424C4—1461no—
425C4—1461no—
426C4#1798866912yes—
427C4C2×#1455655311yescharacteristic
428C4C2×#1455661411yescharacteristic
429C2—22no—
430C2—5844no—
431C2—5842no—
432C2C2×#1798866911yescharacteristic
433C2#509×D7311yescharacteristic
434C2#2262306611yescharacteristic
subgroupG/H
centerC22C2×D8×D73
commutatorC8×C73C23×C4
FrattiniC2×C8C23×D73
FittingC73×#2159C2
socleC22×C73C22×D8
prime radicalC2×#17989025C2
cosocleC2×C8×C73C24
hypercenter#2159D73
hypocenterC73C4×#509
LCSG ▷ C8×C73 ▷ C4×C73 ▷ C2×C73 ▷ C73
UCSC1 ◁ C22 ◁ C22×C4 ◁ C22×C8 ◁ #2159

Complements

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

NH ≅ G/Ntype
2. #22618702C2semidirect
3. C2×#17988669C2semidirect
4. #22611806C2semidirect
5. #22611794C2semidirect
6. #22611782C2semidirect
7. #22611777C2semidirect
8. #22611646C2semidirect
9. #22611641C2semidirect
10. M32×Dic73C2semidirect
11. C73×#2159C2semidirect
16. #17989025C4semidirect
20. #17988681C22semidirect
21. #17988681C22semidirect
28. #17988669C22semidirect
29. #17988669C22semidirect
33. #17988390C22semidirect
36. #17988387C22semidirect
37. #17988362C22semidirect
38. C16×Dic73C22semidirect
42. #367×C73C22semidirect
43. #366×C73C22semidirect
48. #14556613C2×C4semidirect
53. C2×D584C2×C4semidirect
71. #14556536C2×C4semidirect
72. D1168C2×C4semidirect
88. M32×C73C2×C4semidirect
114. Dic292C22×C4semidirect
120. D584C22×C4semidirect
149. C16×C73C22×C4semidirect
210. Dic73#509semidirect
225. #2159D73semidirect
229. C73C4×#509semidirect
254. #367C2×D73semidirect
255. #366C2×D73semidirect
295. M32C4×D73semidirect
367. C16C2×C4×D73semidirect

Maximal subgroups

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

subgroupindex
C2×#179890252
#226187022
C2×#179886692
#226118062
#226117942
#226117822
#226117772
#226116462
#226116412
M32×Dic732
C73×#21592
C4×#50973