Finite Group Databasebeta

← #31792297

Subgroups — #31792297

Order 17536.

494 conjugacy classes of subgroups.

The subgroup lattice has 410 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#17984118C211yesmaximal, characteristic, complemented
2C2×SD16×D137C211yesmaximal, characteristic, complemented
3#22578066C211yesmaximal, characteristic, complemented
4#22578041C211yesmaximal, characteristic
5#22577796C211yesmaximal, characteristic, complemented
6#22577651C211yesmaximal, characteristic, complemented
7#22577643C211yesmaximal, characteristic, complemented
8#22577628C211yesmaximal, characteristic, complemented
9#22577606C211yesmaximal, characteristic, complemented
10#22577557C211yesmaximal, characteristic, complemented
11#22577555C211yesmaximal, characteristic, complemented
12#22577544C211yesmaximal, characteristic, complemented
13#22577500C211yesmaximal, characteristic, complemented
14#22577477C211yesmaximal, characteristic, complemented
15#452×C137C211yesmaximal, characteristic, complemented
16C22×Dic274C2211yescharacteristic, complemented
17C2×Q8×D137C2211yescharacteristic
18C2×D4×D137C2211yescharacteristic
19#17984118—24no—
20C2×#14555500C2211yescharacteristic, complemented
21C2×#14555500C2211yescharacteristic, complemented
22C2×#14555499C2211yescharacteristic
23C2×#14555499C2211yescharacteristic
24C2×#14555498C2211yescharacteristic
25C2×#14555498C2211yescharacteristic
26C2×#14555497C2211yescharacteristic
27SD16×D137—24no—
28C2×#14555494C2211yescharacteristic, complemented
29C2×C8×D137C2211yescharacteristic
30#17984087—21no—
31#17984086C2211yescharacteristic, complemented
32#17984077C2211yescharacteristic
33#17984071—21no—
34#17984068C2211yescharacteristic
35#17984066—21no—
36#17984065—21no—
37#17984064C2211yescharacteristic
38#17984062C2211yescharacteristic
39#17984060—21no—
40#17984056—21no—
41#17984053C2211yescharacteristic, complemented
42D4×Dic137C2211yescharacteristic
43C4×C22⋊D137C2211yescharacteristic
44#17984048—21no—
45C2×#14555487—21no—
46#17984021C2211yescharacteristic, complemented
47#17984017C2211yescharacteristic, complemented
48#17984013C2211yescharacteristic, complemented
49#17984013C2211yescharacteristic, complemented
50#17984012C2211yescharacteristic, complemented
51#17984008C2211yescharacteristic
52#17984000C2211yescharacteristic
53#17983998C2211yescharacteristic, complemented
54C2×#41×C137C2211yescharacteristic
55C2×Q16×C137C2211yescharacteristic, complemented
56C2×SD16×C137C2211yescharacteristic
57C22⋊Q8×C137C2211yescharacteristic, complemented
58#103×C137C2211yescharacteristic, complemented
59#102×C137C2211yescharacteristic
60C22⋊C8×C137C2211yescharacteristic, complemented
61C23×D137—21no—
62C2×Dic274—21no—
63C2×Dic274—22no—
64C2×Dic274—21no—
65C2×Dic274—41no—
66C2×Dic274C2311yescharacteristic
67C2×Dic274C2311yescharacteristic
68Q8×D137—22no—
69Q8×D137—41no—
70C2×D548C2311yescharacteristic
71C2×C22⋊D137—21no—
72C2×C22⋊D137—21no—
73C2×C22⋊D137D411yescharacteristic
74D4×D137—22no—
75D4×D137—41no—
76C22×Dic137—21no—
77C22×Dic137—21no—
78C2×C4×D137—21no—
79C2×C4×D137C2311yescharacteristic
80#14555500—22no—
81#14555500—22no—
82#14555499—22no—
83#14555499—41no—
84#14555498—22no—
85#14555498D414yescomplemented
86#14555497—22no—
87#14555494—22no—
88C2×#11764040C2311yescharacteristic
89C2×#11764040C2311yescharacteristic
90C8×D137—22no—
91C4⋊Dic137—21no—
92C4⋊Dic137C2311yescharacteristic
93#14555489—21no—
94#14555489—21no—
95#14555489—21no—
96#14555489—21no—
97#14555489D411yescharacteristic
98C22⋊Dic137—21no—
99C22⋊Dic137D411yescharacteristic
100#14555487—21no—
101#14555487—21no—
102#14555487—21no—
103#14555487—21no—
104#14555487—21no—
105#14555487D411yescharacteristic
106C4×Dic137—21no—
107C4×Dic137C2311yescharacteristic
108#41×C137—22no—
109#41×C137—41no—
110C2×Q8×C137C2311yescharacteristic
111C2×Q8×C137C2311yescharacteristic
112C2×D4×C137—21no—
113C2×D4×C137C2311yescharacteristic
114C22×C4×C137—21no—
115C22×C4×C137C2311yescharacteristic
116Q16×C137—41no—
117SD16×C137—22no—
118C2×C8×C137C2311yescharacteristic
119C2×C8×C137C2311yescharacteristic
120C4⋊C4×C137—21no—
121C4⋊C4×C137C2311yescharacteristic
122C22⋊C4×C137—21no—
123C22×D137—21no—
124C22×D137—42no—
125C22×D137—42no—
126C22×D137—81no—
127C22×D137C2×D411yescharacteristic
128Dic274—21no—
129Dic274—21no—
130Dic274—21no—
131Dic274—42no—
132Dic274—42no—
133Dic274C2×D412yes—
134D548—21no—
135D548—21no—
136C22⋊D137—22no—
137C22⋊D137—42no—
138C22⋊D137—81no—
139C2×Dic137—21no—
140C2×Dic137—21no—
141C2×Dic137—21no—
142C2×Dic137—42no—
143C2×Dic137—81no—
144C2×Dic137C2×D411yescharacteristic
145C2×Dic137C2×D411yescharacteristic
146C2×Dic137C2×D411yescharacteristic
147C4×D137—21no—
148C4×D137—21no—
149C4×D137—42no—
150#11764040—21no—
151#11764040C2×D412yes—
152C23×C137—21no—
153C23×C137C2×D411yescharacteristic
154Q8×C137—21no—
155Q8×C137—21no—
156Q8×C137—21no—
157Q8×C137—21no—
158D4×C137—21no—
159D4×C137—42no—
160D4×C137C2×D412yes—
161C2×C4×C137—21no—
162C2×C4×C137—21no—
163C2×C4×C137—21no—
164C2×C4×C137—21no—
165C2×C4×C137—21no—
166C2×C4×C137—42no—
167C2×C4×C137C2×D411yescharacteristic
168C2×C4×C137C2411yescharacteristic
169C8×C137—21no—
170C8×C137—21no—
171C22×C137—21no—
172C22×C137—22no—
173C22×C137—21no—
174C22×C137—21no—
175C22×C137—41no—
176C22×C137—41no—
177C22×C137C22×D411yescharacteristic
178C2×D137—21no—
179C2×D137—21no—
180C2×D137—41no—
181C2×D137—41no—
182C2×D137—41no—
183C2×D137—84no—
184C2×D137#13712yescomplemented
185C4×C137—21no—
186C4×C137—41no—
187C4×C137—41no—
188C4×C137—41no—
189C4×C137C22×D411yescharacteristic
190C4×C137H(5,𝔽2)11yescharacteristic
191Dic137—21no—
192Dic137—22no—
193Dic137—21no—
194Dic137—21no—
195Dic137—41no—
196Dic137—41no—
197C2×C137—42no—
198C2×C137—41no—
199C2×C137D4211yescharacteristic
200C2×C137C2×#13711yescharacteristic
201C2×C137#57711yescharacteristic
202D137—22no—
203D137—81no—
204C137#326611yescharacteristic, complemented, Sylow
205#3266—1371nomaximal, Sylow
206C2×#137—1371no—
207C22×SD16—1371no—
208D4×Q8—1371no—
209#546—1371no—
210#497—1371no—
211#468—1371no—
212#467—1371no—
213#464—1371no—
214#461—1371no—
215#452D13711yescharacteristic, complemented
216#451—1371no—
217#450—1371no—
218#448—1371no—
219#440—1371no—
220#435—1371no—
221C2×#41C2×D13711yescharacteristic
222C22×Q8—1371no—
223C22×Q8—1371no—
224C22×D4—1371no—
225#137—2744no—
226C2×Q16—1371no—
227C2×Q16—1371no—
228C2×Q16C2×D13711yescharacteristic, complemented
229C2×SD16—2744no—
230C2×SD16—1371no—
231C2×SD16—1371no—
232C2×SD16—1371no—
233C2×SD16—1371no—
234C2×SD16—1371no—
235C2×SD16C2×D13711yescharacteristic
236C2×M16—1371no—
237C22×C8—1371no—
238C4⋊Q8—2741no—
239C4⋊Q8—1371no—
240#124—1371no—
241#123—2741no—
242C22⋊Q8—2741no—
243C22⋊Q8—2741no—
244C22⋊Q8—1371no—
245C22⋊Q8—1371no—
246C22⋊Q8C2×D13711yescharacteristic, complemented
247#121—2741no—
248#121—1371no—
249#120—2741no—
250C4×Q8—1371no—
251C4×D4—2741no—
252C4×D4—1371no—
253C4×D4—1371no—
254C2×C22⋊C4—2741no—
255#107—1371no—
256C4⋊C8—1371no—
257#103—1371no—
258#103—1371no—
259#103—1371no—
260#103C2×D13711yescharacteristic, complemented
261#102—1371no—
262#102C2×D13711yescharacteristic
263C22⋊C8—1371no—
264C22⋊C8C2×D13711yescharacteristic, complemented
265#97—1371no—
266C24—2741no—
267#41—22no—
268#41—41no—
269C2×Q8—2741no—
270C2×Q8—2742no—
271C2×Q8—2741no—
272C2×Q8—2742no—
273C2×Q8—5481no—
274C2×Q8—5481no—
275C2×Q8—1371no—
276C2×Q8—1371no—
277C2×Q8C22×D13711yescharacteristic
278C2×Q8C22×D13711yescharacteristic
279C2×D4—21no—
280C2×D4—2742no—
281C2×D4—2741no—
282C2×D4—2741no—
283C2×D4—5481no—
284C2×D4—1371no—
285C2×D4—1371no—
286C2×D4C22×D13711yescharacteristic
287C22×C4—21no—
288C22×C4—2741no—
289C22×C4—2741no—
290C22×C4—2741no—
291C22×C4—1371no—
292C22×C4C22×D13711yescharacteristic
293Q16—41no—
294Q16—2742no—
295Q16—2742no—
296SD16—22no—
297SD16—2742no—
298SD16—2742no—
299SD16—2742no—
300SD16—5481no—
301SD16—1374no—
302M16—2742no—
303C2×C8—2742no—
304C2×C8—1371no—
305C2×C8—1371no—
306C2×C8C22×D13711yescharacteristic
307C2×C8C22×D13711yescharacteristic
308C4⋊C4—21no—
309C4⋊C4—2741no—
310C4⋊C4—2741no—
311C4⋊C4—2741no—
312C4⋊C4—2741no—
313C4⋊C4—2741no—
314C4⋊C4—1371no—
315C4⋊C4—1371no—
316C4⋊C4C22×D13711yescharacteristic
317C22⋊C4—21no—
318C22⋊C4—2741no—
319C22⋊C4—2741no—
320C22⋊C4—2741no—
321C22⋊C4—2741no—
322C22⋊C4—2741no—
323C22⋊C4—2741no—
324C22⋊C4—1371no—
325C22⋊C4—1371no—
326C42—2741no—
327C42—1371no—
328C23—21no—
329C23—2741no—
330C23—5482no—
331C23—5482no—
332C23—10961no—
333C23—1371no—
334C23D4×D13711yescharacteristic
335Q8—21no—
336Q8—21no—
337Q8—21no—
338Q8—21no—
339Q8—2741no—
340Q8—2741no—
341Q8—2741no—
342Q8—5482no—
343Q8—5482no—
344Q8—1372no—
345D4—21no—
346D4—42no—
347D4—2742no—
348D4—2741no—
349D4—2741no—
350D4—5482no—
351D4—10961no—
352D4D4×D13712yes—
353C2×C4—21no—
354C2×C4—21no—
355C2×C4—21no—
356C2×C4—21no—
357C2×C4—21no—
358C2×C4—42no—
359C2×C4—2741no—
360C2×C4—2741no—
361C2×C4—2741no—
362C2×C4—2741no—
363C2×C4—2741no—
364C2×C4—5482no—
365C2×C4—5482no—
366C2×C4—10961no—
367C2×C4—1371no—
368C2×C4—1371no—
369C2×C4—1371no—
370C2×C4D4×D13711yescharacteristic
371C2×C4C23×D13711yescharacteristic
372C8—21no—
373C8—21no—
374C8—2741no—
375C8—1372no—
376C22—21no—
377C22—21no—
378C22—21no—
379C22—22no—
380C22—41no—
381C22—41no—
382C22—2741no—
383C22—2741no—
384C22—5481no—
385C22—5481no—
386C22—5481no—
387C22—10964no—
388C22—1372no—
389C22C2×D4×D13711yescharacteristic
390C4—21no—
391C4—41no—
392C4—41no—
393C4—41no—
394C4—2741no—
395C4—2741no—
396C4—2741no—
397C4—2742no—
398C4—5481no—
399C4—5481no—
400C4C2×D4×D13711yescharacteristic
401C4#1798413311yescharacteristic
402C2—41no—
403C2—42no—
404C2—2742no—
405C2—10961no—
406C2#2257803511yescharacteristic
407C2#137×D13711yescharacteristic
408C2#2257821011yescharacteristic
subgroupG/H
centerC22C2×D4×D137
commutator, cosocleC2×C4×C137C24
FrattiniC2×C4C23×D137
Fitting#452×C137C2
socleC22×C137C22×D4
prime radical#22578041C2
hypercenter#452D137
hypocenterC137#3266
LCSG ▷ C2×C4×C137 ▷ C2×C137 ▷ C137
UCSC1 ◁ C22 ◁ C22×C4 ◁ #452

Complements

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

NH ≅ G/Ntype
1. C2×#17984118C2semidirect
2. C2×SD16×D137C2semidirect
3. #22578066C2semidirect
5. #22577796C2semidirect
6. #22577651C2semidirect
7. #22577643C2semidirect
8. #22577628C2semidirect
9. #22577606C2semidirect
10. #22577557C2semidirect
11. #22577555C2semidirect
12. #22577544C2semidirect
13. #22577500C2semidirect
14. #22577477C2semidirect
15. #452×C137C2semidirect
16. C22×Dic274C22semidirect
20. C2×#14555500C22semidirect
21. C2×#14555500C22semidirect
28. C2×#14555494C22semidirect
31. #17984086C22semidirect
41. #17984053C22semidirect
46. #17984021C22semidirect
47. #17984017C22semidirect
48. #17984013C22semidirect
49. #17984013C22semidirect
50. #17984012C22semidirect
53. #17983998C22semidirect
55. C2×Q16×C137C22semidirect
57. C22⋊Q8×C137C22semidirect
58. #103×C137C22semidirect
60. C22⋊C8×C137C22semidirect
85. #14555498D4semidirect
184. C2×D137#137semidirect
204. C137#3266semidirect
215. #452D137semidirect
228. C2×Q16C2×D137semidirect
246. C22⋊Q8C2×D137semidirect
260. #103C2×D137semidirect
264. C22⋊C8C2×D137semidirect

Maximal subgroups

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

subgroupindex
C2×#179841182
C2×SD16×D1372
#225780662
#225780412
#225777962
#225776512
#225776432
#225776282
#225776062
#225775572
#225775552
#225775442
#225775002
#225774772
#452×C1372
#3266137