Finite Group Databasebeta

← #31790426

Subgroups — #31790426

Order 17536.

492 conjugacy classes of subgroups.

The subgroup lattice has 408 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×SD16×D137C211yesmaximal, characteristic, complemented
2C2×#17984096C211yesmaximal, characteristic, complemented
3#22577976C211yesmaximal, characteristic, complemented
4#22577969C211yesmaximal, characteristic
5#22577650C211yesmaximal, characteristic, complemented
6#22577649C211yesmaximal, characteristic, complemented
7#22577646C211yesmaximal, characteristic, complemented
8#22577635C211yesmaximal, characteristic, complemented
9#22577634C211yesmaximal, characteristic, complemented
10#22577631C211yesmaximal, characteristic, complemented
11#22577630C211yesmaximal, characteristic, complemented
12#22577440C211yesmaximal, characteristic, complemented
13#22577410C211yesmaximal, characteristic, complemented
14#22577406C211yesmaximal, characteristic, complemented
15#468×C137C211yesmaximal, characteristic, complemented
16C2×#14555510C2211yescharacteristic
17C2×Q8×D137C2211yescharacteristic
18C2×D4×D137C2211yescharacteristic
19C2×#14555499C2211yescharacteristic
20C2×#14555498C2211yescharacteristic
21C2×#14555497C2211yescharacteristic
22SD16×D137—24no—
23#17984096—24no—
24C2×#14555493C2211yescharacteristic
25C2×#14555493C2211yescharacteristic, complemented
26C22×#11764040C2211yescharacteristic, complemented
27C2×C8×D137C2211yescharacteristic
28C2×C8×D137C2211yescharacteristic, complemented
29#17984087—21no—
30#17984084—21no—
31#17984074—21no—
32#17984074—21no—
33#17984073—21no—
34#17984070—21no—
35#17984068C2211yescharacteristic, complemented
36#17984067C2211yescharacteristic, complemented
37#17984064C2211yescharacteristic, complemented
38#17984063C2211yescharacteristic
39#17984062C2211yescharacteristic
40#17984059C2211yescharacteristic
41#17984058—21no—
42#17984056—21no—
43#17984046C2211yescharacteristic
44#17984046C2211yescharacteristic
45#17984022C2211yescharacteristic, complemented
46#17984019C2211yescharacteristic, complemented
47#17984019C2211yescharacteristic, complemented
48#17984013C2211yescharacteristic, complemented
49#17984012C2211yescharacteristic, complemented
50#17984011C2211yescharacteristic, complemented
51#17984010C2211yescharacteristic
52#17984009C2211yescharacteristic
53#17984008C2211yescharacteristic
54C2×SD16×C137C2211yescharacteristic
55C2×M16×C137C2211yescharacteristic, complemented
56C22⋊Q8×C137C2211yescharacteristic, complemented
57#121×C137C2211yescharacteristic
58#107×C137C2211yescharacteristic, complemented
59#103×C137C2211yescharacteristic, complemented
60#102×C137C2211yescharacteristic
61C23×D137—21no—
62#14555510—22no—
63#14555510—22no—
64C2×Dic274—21no—
65C2×Dic274D411yescharacteristic
66C2×Dic274C2311yescharacteristic
67Q8×D137—22no—
68Q8×D137—22no—
69C2×D548—21no—
70C2×D548D411yescharacteristic
71C2×D548C2311yescharacteristic
72C2×C22⋊D137—21no—
73C2×C22⋊D137—21no—
74C2×C22⋊D137—21no—
75C2×C22⋊D137D411yescharacteristic
76C2×C22⋊D137D411yescharacteristic
77D4×D137—22no—
78D4×D137—22no—
79C2×C4×D137—21no—
80C2×C4×D137C2311yescharacteristic
81C2×C4×D137C2311yescharacteristic
82#14555499—22no—
83#14555498—22no—
84#14555497—22no—
85#14555493—22no—
86#14555493—22no—
87C2×#11764040—22no—
88C2×#11764040C2311yescharacteristic
89C2×#11764040C2311yescharacteristic
90C8×D137—22no—
91C8×D137D414yescomplemented
92C4⋊Dic137—21no—
93C4⋊Dic137C2311yescharacteristic
94C4⋊Dic137C2311yescharacteristic
95#14555489—21no—
96#14555489—21no—
97C22⋊Dic137—21no—
98C22⋊Dic137—21no—
99#14555487—21no—
100#14555487—21no—
101#14555487—21no—
102#14555487—21no—
103#14555487—21no—
104#14555487—21no—
105C4×Dic137—21no—
106C4×Dic137—21no—
107C2×Q8×C137C2311yescharacteristic
108C2×D4×C137—21no—
109C2×D4×C137C2311yescharacteristic
110C22×C4×C137C2311yescharacteristic
111SD16×C137—22no—
112M16×C137—22no—
113C2×C8×C137C2311yescharacteristic
114C2×C8×C137C2311yescharacteristic
115C4⋊C4×C137—21no—
116C4⋊C4×C137C2311yescharacteristic
117C4⋊C4×C137C2311yescharacteristic
118C22⋊C4×C137—21no—
119C22⋊C4×C137—21no—
120C22×D137—21no—
121C22×D137—42no—
122C22×D137—81no—
123C22×D137—81no—
124C22×D137C2×D411yescharacteristic
125C22×D137C2×D411yescharacteristic
126Dic274—22no—
127Dic274—21no—
128Dic274—21no—
129Dic274—42no—
130D548—22no—
131D548—21no—
132D548—21no—
133D548—81no—
134C22⋊D137—22no—
135C22⋊D137—22no—
136C22⋊D137—42no—
137C22⋊D137—81no—
138C22⋊D137—81no—
139C2×Dic137—21no—
140C2×Dic137—21no—
141C2×Dic137—21no—
142C2×Dic137C2×D411yescharacteristic
143C2×Dic137C2×D411yescharacteristic
144C4×D137—21no—
145C4×D137—21no—
146C4×D137—42no—
147C4×D137C2×D412yes—
148C4×D137#4112yes—
149#11764040—21no—
150#11764040C2×D412yes—
151C23×C137—21no—
152C23×C137C2×D411yescharacteristic
153Q8×C137—21no—
154Q8×C137—21no—
155D4×C137—21no—
156D4×C137—21no—
157D4×C137—81no—
158C2×C4×C137—21no—
159C2×C4×C137—21no—
160C2×C4×C137—21no—
161C2×C4×C137—21no—
162C2×C4×C137—21no—
163C2×C4×C137C2×D411yescharacteristic
164C2×C4×C137C2411yescharacteristic
165C8×C137—21no—
166C8×C137C2×D412yes—
167C22×C137—21no—
168C22×C137—21no—
169C22×C137—21no—
170C22×C137—41no—
171C22×C137—41no—
172C22×C137—41no—
173C22×C137C22×D411yescharacteristic
174C2×D137—21no—
175C2×D137—21no—
176C2×D137—21no—
177C2×D137—21no—
178C2×D137—21no—
179C2×D137—41no—
180C2×D137—41no—
181C2×D137—41no—
182C2×D137—84no—
183C2×D137#12112yes—
184C4×C137—21no—
185C4×C137—41no—
186C4×C137—41no—
187C4×C137—41no—
188C4×C137C22×D411yescharacteristic
189C4×C137C2×#4111yescharacteristic
190Dic137—21no—
191Dic137—41no—
192Dic137—41no—
193Dic137—41no—
194Dic137#12112yes—
195C2×C137—41no—
196C2×C137—81no—
197C2×C137C2×#12111yescharacteristic
198C2×C137#57711yescharacteristic
199C2×C137#57711yescharacteristic
200D137—22no—
201D137—41no—
202D137—81no—
203C137#305711yescharacteristic, complemented, Sylow
204#3057—1371nomaximal, Sylow
205C22×SD16—1371no—
206C2×#131—1371no—
207#536—1371no—
208#535—1371no—
209#468—1371no—
210#468—1371no—
211#468—1371no—
212#468D13711yescharacteristic, complemented
213#465—1371no—
214#465—1371no—
215#465—1371no—
216#465—1371no—
217#427—1371no—
218#419—1371no—
219#418—1371no—
220C2×#41—1371no—
221C22×Q8—1371no—
222C22×D4—1371no—
223C2×SD16—2744no—
224C2×SD16—1371no—
225C2×SD16—1371no—
226C2×SD16—1371no—
227C2×SD16C2×D13711yescharacteristic
228#131—2744no—
229C2×M16—1371no—
230C2×M16—1371no—
231C2×M16C2×D13711yescharacteristic, complemented
232C22×C8—1371no—
233C22×C8—1371no—
234C22×C8—1371no—
235C4⋊Q8—2741no—
236#127—2741no—
237#124—2741no—
238#124—2741no—
239#123—2741no—
240#123—2741no—
241C22⋊Q8—1371no—
242C22⋊Q8—1371no—
243C22⋊Q8—1371no—
244C22⋊Q8C2×D13711yescharacteristic, complemented
245#121—1371no—
246#121—1371no—
247#121—1371no—
248#121C2×D13711yescharacteristic
249#120—2741no—
250#120—2741no—
251#117—1371no—
252#117—1371no—
253#107—1371no—
254#107C2×D13711yescharacteristic, complemented
255#106—1371no—
256#106—1371no—
257#103—1371no—
258#103—1371no—
259#103—1371no—
260#103C2×D13711yescharacteristic, complemented
261#102—1371no—
262#102—1371no—
263#102—1371no—
264#102C2×D13711yescharacteristic
265C24—2741no—
266#41—2742no—
267#41—2742no—
268C2×Q8—2741no—
269C2×Q8—2742no—
270C2×Q8—2742no—
271C2×Q8—1371no—
272C2×Q8—1371no—
273C2×Q8C22×D13711yescharacteristic
274C2×D4—21no—
275C2×D4—2741no—
276C2×D4—2741no—
277C2×D4—2741no—
278C2×D4—2742no—
279C2×D4—2742no—
280C2×D4—2741no—
281C2×D4—1371no—
282C2×D4—1371no—
283C2×D4—1371no—
284C2×D4—1371no—
285C2×D4C22×D13711yescharacteristic
286C22×C4—2741no—
287C22×C4—1371no—
288C22×C4—1371no—
289C22×C4C22×D13711yescharacteristic
290SD16—22no—
291SD16—2742no—
292SD16—2742no—
293SD16—2742no—
294M16—22no—
295M16—2742no—
296M16—2742no—
297C2×C8—2742no—
298C2×C8—2742no—
299C2×C8—1371no—
300C2×C8—1371no—
301C2×C8—1374no—
302C2×C8C22×D13711yescharacteristic
303C2×C8C22×D13711yescharacteristic
304C4⋊C4—21no—
305C4⋊C4—2741no—
306C4⋊C4—2741no—
307C4⋊C4—2741no—
308C4⋊C4—1371no—
309C4⋊C4—1371no—
310C4⋊C4C22×D13711yescharacteristic
311C4⋊C4C22×D13711yescharacteristic
312C22⋊C4—21no—
313C22⋊C4—21no—
314C22⋊C4—2741no—
315C22⋊C4—2741no—
316C22⋊C4—2741no—
317C22⋊C4—2741no—
318C22⋊C4—2741no—
319C22⋊C4—2741no—
320C22⋊C4—2741no—
321C22⋊C4—2741no—
322C42—2741no—
323C42—2741no—
324C23—21no—
325C23—2741no—
326C23—5482no—
327C23—10961no—
328C23—10961no—
329C23—1371no—
330C23—1371no—
331C23D4×D13711yescharacteristic
332Q8—21no—
333Q8—21no—
334Q8—2742no—
335Q8—2741no—
336Q8—2741no—
337Q8—5482no—
338D4—21no—
339D4—21no—
340D4—81no—
341D4—2742no—
342D4—2742no—
343D4—2742no—
344D4—2741no—
345D4—2741no—
346D4—5482no—
347D4—10961no—
348D4—10961no—
349D4—10961no—
350C2×C4—21no—
351C2×C4—21no—
352C2×C4—21no—
353C2×C4—21no—
354C2×C4—21no—
355C2×C4—2741no—
356C2×C4—2741no—
357C2×C4—2741no—
358C2×C4—2741no—
359C2×C4—2741no—
360C2×C4—5482no—
361C2×C4—1371no—
362C2×C4—1371no—
363C2×C4—1372no—
364C2×C4—1372no—
365C2×C4D4×D13711yescharacteristic
366C2×C4C23×D13711yescharacteristic
367C8—21no—
368C8—2741no—
369C8—1372no—
370C8D4×D13712yes—
371C22—21no—
372C22—21no—
373C22—21no—
374C22—41no—
375C22—41no—
376C22—41no—
377C22—2741no—
378C22—2741no—
379C22—2741no—
380C22—2741no—
381C22—2741no—
382C22—5481no—
383C22—5481no—
384C22—5481no—
385C22—10964no—
386C22—1372no—
387C22C2×D4×D13711yescharacteristic
388C4—21no—
389C4—41no—
390C4—41no—
391C4—41no—
392C4—2741no—
393C4—5481no—
394C4—5481no—
395C4—5481no—
396C4—1372no—
397C4C2×D4×D13711yescharacteristic
398C4#41×D13711yescharacteristic
399C2—41no—
400C2—81no—
401C2—2742no—
402C2—5481no—
403C2—10961no—
404C2#121×D13711yescharacteristic
405C2#2257821211yescharacteristic
406C2#2257821311yescharacteristic
subgroupG/H
centerC22C2×D4×D137
commutator, cosocleC2×C4×C137C24
FrattiniC2×C4C23×D137
Fitting#468×C137C2
socleC22×C137C22×D4
prime radical#22577969C2
hypercenter#468D137
hypocenterC137#3057
LCSG ▷ C2×C4×C137 ▷ C2×C137 ▷ C137
UCSC1 ◁ C22 ◁ C22×C4 ◁ #468

Complements

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

NH ≅ G/Ntype
1. C2×SD16×D137C2semidirect
2. C2×#17984096C2semidirect
3. #22577976C2semidirect
5. #22577650C2semidirect
6. #22577649C2semidirect
7. #22577646C2semidirect
8. #22577635C2semidirect
9. #22577634C2semidirect
10. #22577631C2semidirect
11. #22577630C2semidirect
12. #22577440C2semidirect
13. #22577410C2semidirect
14. #22577406C2semidirect
15. #468×C137C2semidirect
25. C2×#14555493C22semidirect
26. C22×#11764040C22semidirect
28. C2×C8×D137C22semidirect
35. #17984068C22semidirect
36. #17984067C22semidirect
37. #17984064C22semidirect
45. #17984022C22semidirect
46. #17984019C22semidirect
47. #17984019C22semidirect
48. #17984013C22semidirect
49. #17984012C22semidirect
50. #17984011C22semidirect
55. C2×M16×C137C22semidirect
56. C22⋊Q8×C137C22semidirect
58. #107×C137C22semidirect
59. #103×C137C22semidirect
91. C8×D137D4semidirect
203. C137#3057semidirect
212. #468D137semidirect
231. C2×M16C2×D137semidirect
244. C22⋊Q8C2×D137semidirect
254. #107C2×D137semidirect
260. #103C2×D137semidirect

Maximal subgroups

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

subgroupindex
C2×SD16×D1372
C2×#179840962
#225779762
#225779692
#225776502
#225776492
#225776462
#225776352
#225776342
#225776312
#225776302
#225774402
#225774102
#225774062
#468×C1372
#3057137