Finite Group Databasebeta

← #31791140

Subgroups — #31791140

Order 17536.

404 conjugacy classes of subgroups.

The subgroup lattice has 368 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#22578030C211yesmaximal, characteristic, complemented
2#22577971C211yesmaximal, characteristic
3#22577768C211yesmaximal, characteristic, complemented
4#22577766C211yesmaximal, characteristic, complemented
5#22577764C211yesmaximal, characteristic, complemented
6#22577759C211yesmaximal, characteristic, complemented
7#22577757C211yesmaximal, characteristic, complemented
8#22577751C211yesmaximal, characteristic, complemented
9#22577748C211yesmaximal, characteristic, complemented
10#22577406C211yesmaximal, characteristic, complemented
11#22577403C211yesmaximal, characteristic, complemented
12#22577398C211yesmaximal, characteristic, complemented
13#102×D137C211yesmaximal, characteristic, complemented
14#22577360C211yesmaximal, characteristic, complemented
15#489×C137C211yesmaximal, characteristic, complemented
16C2×#14555511C2211yescharacteristic
17C2×D4×D137C2211yescharacteristic
18C2×#14555493C2211yescharacteristic
19C2×C8×D137C2211yescharacteristic
20#17984087C2211yescharacteristic, complemented
21#17984086—21no—
22#17984085C2211yescharacteristic
23#17984084C2211yescharacteristic
24#17984080C2211yescharacteristic, complemented
25#17984079C2211yescharacteristic, complemented
26#17984078—21no—
27#17984077C2211yescharacteristic
28#17984066—21no—
29#17984064—21no—
30#17984061—21no—
31#17984061—21no—
32#17984058—21no—
33#17984058—21no—
34#17984046C2211yescharacteristic
35#17984045C2211yescharacteristic
36C4⋊C4×D137C2211yescharacteristic
37#17984012C2211yescharacteristic, complemented
38#17984012C2211yescharacteristic, complemented
39#17984010C2211yescharacteristic
40#17984010C2211yescharacteristic
41#17984010C2211yescharacteristic, complemented
42#17984010C2211yescharacteristic, complemented
43#17984009C2211yescharacteristic
44#17984009C2211yescharacteristic
45#17984008C2211yescharacteristic
46#17984008C2211yescharacteristic
47#17984008C2211yescharacteristic, complemented
48#17984008C2211yescharacteristic, complemented
49#17983998C2211yescharacteristic, complemented
50C4×#11764040C2211yescharacteristic, complemented
51C8×Dic137C2211yescharacteristic, complemented
52C4⋊Q8×C137C2211yescharacteristic, complemented
53#124×C137C2211yescharacteristic
54#103×C137C2211yescharacteristic, complemented
55#103×C137C2211yescharacteristic, complemented
56#102×C137C2211yescharacteristic
57#102×C137C2211yescharacteristic
58#97×C137C2211yescharacteristic, complemented
59C23×D137—21no—
60#14555511—22no—
61#14555511—22no—
62C2×Dic274—21no—
63C2×D548—21no—
64C2×D548—21no—
65C2×D548—21no—
66C2×D548C2311yescharacteristic
67C2×D548C2311yescharacteristic
68C2×C22⋊D137—21no—
69C2×C22⋊D137—21no—
70C2×C22⋊D137—21no—
71D4×D137—22no—
72D4×D137—22no—
73C2×C4×D137—21no—
74C2×C4×D137—21no—
75C2×C4×D137C2311yescharacteristic
76#14555493—22no—
77C2×#11764040C2311yescharacteristic
78C2×#11764040C2311yescharacteristic
79C8×D137—22no—
80C4⋊Dic137—21no—
81C4⋊Dic137C2311yescharacteristic
82C4⋊Dic137C2311yescharacteristic
83#14555489—21no—
84#14555489—21no—
85#14555489—21no—
86#14555489—21no—
87#14555489—21no—
88#14555489D411yescharacteristic
89#14555489D411yescharacteristic
90#14555487—21no—
91#14555487—21no—
92#14555487—21no—
93#14555487D411yescharacteristic
94#14555487D411yescharacteristic
95C4×Dic137—21no—
96C4×Dic137C2311yescharacteristic
97C2×Q8×C137—21no—
98C2×Q8×C137C2311yescharacteristic
99C2×D4×C137C2311yescharacteristic
100C2×C8×C137C2311yescharacteristic
101C2×C8×C137C2311yescharacteristic
102C4⋊C4×C137—21no—
103C4⋊C4×C137C2311yescharacteristic
104C4⋊C4×C137C2311yescharacteristic
105C22⋊C4×C137—21no—
106C22⋊C4×C137—21no—
107C42×C137C2311yescharacteristic
108C22×D137—21no—
109C22×D137—21no—
110C22×D137—42no—
111C22×D137—81no—
112C22×D137—81no—
113C22×D137C2×D411yescharacteristic
114Dic274—81no—
115D548—21no—
116D548—21no—
117D548—21no—
118D548—21no—
119D548—42no—
120D548—81no—
121D548—81no—
122C22⋊D137—42no—
123C22⋊D137—81no—
124C22⋊D137—81no—
125C2×Dic137—21no—
126C2×Dic137—21no—
127C2×Dic137C2×D411yescharacteristic
128C2×Dic137C2×D411yescharacteristic
129C2×Dic137C2×D411yescharacteristic
130C4×D137—42no—
131C4×D137—81no—
132C4×D137#4112yes—
133C4×D137#4112yes—
134#11764040—21no—
135#11764040—21no—
136C23×C137—21no—
137Q8×C137—21no—
138Q8×C137—21no—
139Q8×C137—81no—
140D4×C137—21no—
141D4×C137—21no—
142C2×C4×C137—21no—
143C2×C4×C137—21no—
144C2×C4×C137—21no—
145C2×C4×C137C2×D411yescharacteristic
146C2×C4×C137C2×D411yescharacteristic
147C2×C4×C137C2411yescharacteristic
148C8×C137—21no—
149C8×C137—21no—
150C22×C137—41no—
151C22×C137—41no—
152C22×C137—41no—
153C22×C137C22×D411yescharacteristic
154C2×D137—21no—
155C2×D137—21no—
156C2×D137—41no—
157C2×D137—41no—
158C2×D137—41no—
159C2×D137—41no—
160C2×D137—41no—
161C2×D137—41no—
162C2×D137—84no—
163C2×D137#12412yes—
164C4×C137—21no—
165C4×C137—21no—
166C4×C137—41no—
167C4×C137—41no—
168C4×C137—41no—
169C4×C137C2×#4111yescharacteristic
170C4×C137C2×#4111yescharacteristic
171Dic137—21no—
172Dic137—21no—
173Dic137—41no—
174Dic137—41no—
175Dic137#12412yes—
176C2×C137—81no—
177C2×C137C2×#12411yescharacteristic
178C2×C137#57611yescharacteristic
179C2×C137#57711yescharacteristic
180D137—22no—
181D137—81no—
182D137—81no—
183C137#313611yescharacteristic, complemented, Sylow
184#3136—1371nomaximal, Sylow
185#544—1371no—
186#535—1371no—
187#490—1371no—
188#490—1371no—
189#489—1371no—
190#489D13711yescharacteristic, complemented
191#488—1371no—
192#488—1371no—
193#486—1371no—
194#486—1371no—
195#418—1371no—
196#417—1371no—
197#416—1371no—
198C2×#102—1371no—
199#405—1371no—
200C2×#41—1371no—
201C22×D4—1371no—
202C2×M16—1371no—
203C22×C8—1371no—
204C4⋊Q8—2741no—
205C4⋊Q8—1371no—
206C4⋊Q8C2×D13711yescharacteristic, complemented
207#127—1371no—
208#127—1371no—
209#125—2741no—
210#125—1371no—
211#125—1371no—
212#124—1371no—
213#124C2×D13711yescharacteristic
214C22⋊Q8—2741no—
215C22⋊Q8—2741no—
216#121—2741no—
217#121—2741no—
218#120—2741no—
219#120—2741no—
220#117—1371no—
221#117—1371no—
222C2×C4⋊C4—1371no—
223#103—1371no—
224#103—1371no—
225#103C2×D13711yescharacteristic, complemented
226#103C2×D13711yescharacteristic, complemented
227#102—1371no—
228#102—1371no—
229#102—1371no—
230#102—1371no—
231#102—1371no—
232#102—1371no—
233#102—1371no—
234#102—1371no—
235#102—1371no—
236#102—1371no—
237#102C2×D13711yescharacteristic
238#102C2×D13711yescharacteristic
239#97—1371no—
240#97C2×D13711yescharacteristic, complemented
241C4×C8—1371no—
242C4×C8—1371no—
243C24—2741no—
244#41—2742no—
245#41—2742no—
246C2×Q8—21no—
247C2×Q8—2741no—
248C2×Q8C22×D13711yescharacteristic
249C2×D4—2741no—
250C2×D4—2742no—
251C2×D4—2742no—
252C2×D4—2741no—
253C2×D4—2741no—
254C2×D4—2741no—
255C2×D4—2741no—
256C2×D4—2741no—
257C2×D4—1371no—
258C2×D4—1371no—
259C2×D4C22×D13711yescharacteristic
260C22×C4—2741no—
261C22×C4—2741no—
262C22×C4—1371no—
263M16—2742no—
264C2×C8—2742no—
265C2×C8—1371no—
266C2×C8—1371no—
267C2×C8C22×D13711yescharacteristic
268C2×C8C22×D13711yescharacteristic
269C4⋊C4—21no—
270C4⋊C4—2741no—
271C4⋊C4—2741no—
272C4⋊C4—2741no—
273C4⋊C4—2741no—
274C4⋊C4—2741no—
275C4⋊C4—2741no—
276C4⋊C4—1371no—
277C4⋊C4—1371no—
278C4⋊C4—1371no—
279C4⋊C4—1371no—
280C4⋊C4C22×D13711yescharacteristic
281C4⋊C4C22×D13711yescharacteristic
282C22⋊C4—21no—
283C22⋊C4—21no—
284C22⋊C4—2741no—
285C22⋊C4—2741no—
286C22⋊C4—2741no—
287C22⋊C4—1371no—
288C22⋊C4—1371no—
289C42—2741no—
290C42—1371no—
291C42C22×D13711yescharacteristic
292C23—21no—
293C23—2741no—
294C23—2741no—
295C23—5482no—
296C23—10961no—
297C23—10961no—
298C23—1371no—
299Q8—21no—
300Q8—21no—
301Q8—81no—
302Q8—10961no—
303D4—21no—
304D4—21no—
305D4—2741no—
306D4—2741no—
307D4—2741no—
308D4—2741no—
309D4—5482no—
310D4—5482no—
311D4—10961no—
312D4—10961no—
313D4—10961no—
314D4—10961no—
315C2×C4—21no—
316C2×C4—21no—
317C2×C4—21no—
318C2×C4—2741no—
319C2×C4—2741no—
320C2×C4—5482no—
321C2×C4—10961no—
322C2×C4—1371no—
323C2×C4—1371no—
324C2×C4—1371no—
325C2×C4—1372no—
326C2×C4—1372no—
327C2×C4D4×D13711yescharacteristic
328C2×C4D4×D13711yescharacteristic
329C2×C4C23×D13711yescharacteristic
330C8—21no—
331C8—21no—
332C8—2741no—
333C8—2741no—
334C22—41no—
335C22—41no—
336C22—41no—
337C22—2741no—
338C22—2741no—
339C22—5481no—
340C22—5481no—
341C22—5481no—
342C22—5481no—
343C22—5481no—
344C22—5481no—
345C22—10964no—
346C22—1372no—
347C22C2×D4×D13711yescharacteristic
348C4—21no—
349C4—21no—
350C4—41no—
351C4—41no—
352C4—41no—
353C4—2741no—
354C4—2741no—
355C4—5481no—
356C4—5481no—
357C4—1372no—
358C4#41×D13711yescharacteristic
359C4#41×D13711yescharacteristic
360C2—81no—
361C2—2742no—
362C2—10961no—
363C2—10961no—
364C2#124×D13711yescharacteristic
365C2#2257820711yescharacteristic
366C2#2257821211yescharacteristic
subgroupG/H
centerC22C2×D4×D137
commutator, cosocleC2×C4×C137C24
FrattiniC2×C4C23×D137
Fitting#489×C137C2
socleC22×C137C22×D4
prime radical#22577971C2
hypercenter#489D137
hypocenterC137#3136
LCSG ▷ C2×C4×C137 ▷ C2×C137 ▷ C137
UCSC1 ◁ C22 ◁ C42 ◁ #489

Complements

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

NH ≅ G/Ntype
1. #22578030C2semidirect
3. #22577768C2semidirect
4. #22577766C2semidirect
5. #22577764C2semidirect
6. #22577759C2semidirect
7. #22577757C2semidirect
8. #22577751C2semidirect
9. #22577748C2semidirect
10. #22577406C2semidirect
11. #22577403C2semidirect
12. #22577398C2semidirect
13. #102×D137C2semidirect
14. #22577360C2semidirect
15. #489×C137C2semidirect
20. #17984087C22semidirect
24. #17984080C22semidirect
25. #17984079C22semidirect
37. #17984012C22semidirect
38. #17984012C22semidirect
41. #17984010C22semidirect
42. #17984010C22semidirect
47. #17984008C22semidirect
48. #17984008C22semidirect
49. #17983998C22semidirect
50. C4×#11764040C22semidirect
51. C8×Dic137C22semidirect
52. C4⋊Q8×C137C22semidirect
54. #103×C137C22semidirect
55. #103×C137C22semidirect
58. #97×C137C22semidirect
183. C137#3136semidirect
190. #489D137semidirect
206. C4⋊Q8C2×D137semidirect
225. #103C2×D137semidirect
226. #103C2×D137semidirect
240. #97C2×D137semidirect

Maximal subgroups

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

subgroupindex
#225780302
#225779712
#225777682
#225777662
#225777642
#225777592
#225777572
#225777512
#225777482
#225774062
#225774032
#225773982
#102×D1372
#225773602
#489×C1372
#3136137