Finite Group Databasebeta

← #31792828

Subgroups — #31792828

Order 17536.

402 conjugacy classes of subgroups.

The subgroup lattice has 372 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#22578121C211yesmaximal, characteristic, complemented
2#22578044C211yesmaximal, characteristic
3#22577764C211yesmaximal, characteristic, complemented
4#22577730C211yesmaximal, characteristic, complemented
5#22577692C211yesmaximal, characteristic, complemented
6#22577659C211yesmaximal, characteristic, complemented
7#22577633C211yesmaximal, characteristic, complemented
8#22577628C211yesmaximal, characteristic, complemented
9#22577568C211yesmaximal, characteristic, complemented
10#22577556C211yesmaximal, characteristic, complemented
11#22577502C211yesmaximal, characteristic, complemented
12#22577477C211yesmaximal, characteristic, complemented
13#22577407C211yesmaximal, characteristic, complemented
14#22577398C211yesmaximal, characteristic, complemented
15#485×C137C211yesmaximal, characteristic, complemented
16C2×#14555511C2211yescharacteristic
17C22×D548C2211yescharacteristic
18C2×#14555500C2211yescharacteristic, complemented
19C2×#14555499C2211yescharacteristic
20C2×#14555499C2211yescharacteristic
21C2×#14555494C2211yescharacteristic
22C2×C8×D137C2211yescharacteristic, complemented
23#17984087C2211yescharacteristic, complemented
24#17984082—21no—
25#17984081—21no—
26#17984077C2211yescharacteristic
27#17984074—21no—
28#17984073—21no—
29#17984068C2211yescharacteristic, complemented
30#17984067C2211yescharacteristic
31#17984063—21no—
32#17984063C2211yescharacteristic
33#17984058—21no—
34C4×Dic274—21no—
35#17984053C2211yescharacteristic, complemented
36#17984050C2211yescharacteristic
37C4×C22⋊D137C2211yescharacteristic
38#17984046C2211yescharacteristic, complemented
39C22⋊C4×D137—21no—
40#17984019C2211yescharacteristic, complemented
41#17984017C2211yescharacteristic, complemented
42#17984012C2211yescharacteristic, complemented
43#17984011C2211yescharacteristic, complemented
44#17984010C2211yescharacteristic
45#17984010C2211yescharacteristic
46#17984008C2211yescharacteristic
47#17984008C2211yescharacteristic
48#17984008C2211yescharacteristic
49#17984008C2211yescharacteristic
50#17984000C2211yescharacteristic, complemented
51#17983998C2211yescharacteristic, complemented
52C22⋊Q8×C137C2211yescharacteristic
53#117×C137C2211yescharacteristic
54#107×C137C2211yescharacteristic, complemented
55#106×C137C2211yescharacteristic, complemented
56#103×C137C2211yescharacteristic, complemented
57#103×C137C2211yescharacteristic, complemented
58C22⋊C8×C137C2211yescharacteristic
59C23×D137—21no—
60#14555511—22no—
61#14555511—41no—
62C2×Dic274—21no—
63C2×Dic274C2311yescharacteristic
64C2×D548—22no—
65C2×D548—21no—
66C2×D548—21no—
67C2×D548—21no—
68C2×D548—41no—
69C2×D548C2311yescharacteristic
70C2×D548C2311yescharacteristic
71C2×C22⋊D137D411yescharacteristic
72C2×C4×D137—21no—
73C2×C4×D137—21no—
74C2×C4×D137C2311yescharacteristic
75#14555500—41no—
76#14555499—22no—
77#14555499—41no—
78#14555494—41no—
79C2×#11764040C2311yescharacteristic
80C2×#11764040C2311yescharacteristic
81C8×D137—41no—
82C4⋊Dic137—21no—
83C4⋊Dic137C2311yescharacteristic
84#14555489—21no—
85#14555489—21no—
86#14555489—21no—
87#14555489D411yescharacteristic
88C22⋊Dic137—21no—
89C22⋊Dic137D411yescharacteristic
90#14555487—21no—
91#14555487—21no—
92#14555487—21no—
93#14555487—21no—
94#14555487—21no—
95#14555487—21no—
96#14555487D411yescharacteristic
97C4×Dic137—21no—
98C4×Dic137—21no—
99C4×Dic137C2311yescharacteristic
100C2×Q8×C137C2311yescharacteristic
101C22×C4×C137C2311yescharacteristic
102C2×C8×C137C2311yescharacteristic
103C2×C8×C137C2311yescharacteristic
104C4⋊C4×C137—21no—
105C4⋊C4×C137C2311yescharacteristic
106C4⋊C4×C137C2311yescharacteristic
107C4⋊C4×C137C2311yescharacteristic
108C22⋊C4×C137—21no—
109C22⋊C4×C137—21no—
110C42×C137—21no—
111C22×D137—21no—
112C22×D137—21no—
113C22×D137—42no—
114C22×D137—42no—
115C22×D137—81no—
116C22×D137C2×D411yescharacteristic
117Dic274—21no—
118Dic274—21no—
119Dic274—42no—
120D548—21no—
121D548—21no—
122D548—21no—
123D548—42no—
124D548—42no—
125D548—81no—
126D548#4112yes—
127C22⋊D137—41no—
128C2×Dic137—21no—
129C2×Dic137—21no—
130C2×Dic137C2×D411yescharacteristic
131C2×Dic137C2×D411yescharacteristic
132C2×Dic137C2×D411yescharacteristic
133C4×D137—21no—
134C4×D137—21no—
135C4×D137—42no—
136C4×D137—81no—
137#11764040—21no—
138#11764040—21no—
139C23×C137C2×D411yescharacteristic
140Q8×C137—21no—
141Q8×C137—21no—
142C2×C4×C137—21no—
143C2×C4×C137—21no—
144C2×C4×C137—21no—
145C2×C4×C137—21no—
146C2×C4×C137—21no—
147C2×C4×C137—21no—
148C2×C4×C137C2×D411yescharacteristic
149C2×C4×C137C2411yescharacteristic
150C8×C137—21no—
151C8×C137—21no—
152C22×C137—21no—
153C22×C137—21no—
154C22×C137—21no—
155C22×C137C22×D411yescharacteristic
156C2×D137—21no—
157C2×D137—22no—
158C2×D137—21no—
159C2×D137—21no—
160C2×D137—41no—
161C2×D137—41no—
162C2×D137—41no—
163C2×D137—41no—
164C2×D137—41no—
165C2×D137—84no—
166C4×C137—21no—
167C4×C137—41no—
168C4×C137—41no—
169C4×C137—41no—
170C4×C137—41no—
171C4×C137C2×#4111yescharacteristic
172C4×C137H(5,𝔽2)11yescharacteristic
173Dic137—21no—
174Dic137—21no—
175Dic137—21no—
176Dic137—41no—
177Dic137—41no—
178C2×C137—41no—
179C2×C137#54611yescharacteristic
180C2×C137#57511yescharacteristic
181C2×C137#57711yescharacteristic
182D137—41no—
183D137—42no—
184D137—81no—
185C137#331111yescharacteristic, complemented, Sylow
186#3311—1371nomaximal, Sylow
187#556—1371no—
188#546—1371no—
189#489—1371no—
190#485D13711yescharacteristic, complemented
191#482—1371no—
192#476—1371no—
193#470—1371no—
194#465—1371no—
195#464—1371no—
196#452—1371no—
197#450—1371no—
198#440—1371no—
199#435—1371no—
200#418—1371no—
201#416—1371no—
202C2×#41—1371no—
203C22×D4—1371no—
204C2×Q16—1371no—
205C2×SD16—1371no—
206C2×SD16—1371no—
207C2×M16—1371no—
208C22×C8—1371no—
209C4⋊Q8—1371no—
210#126—2741no—
211#126—2741no—
212#124—2741no—
213#124—1371no—
214#123—2741no—
215C22⋊Q8—1371no—
216C22⋊Q8—1371no—
217C22⋊Q8C2×D13711yescharacteristic
218#121—2741no—
219#121—1371no—
220#120—2741no—
221C4×Q8—2741no—
222C4×Q8—1371no—
223C4×D4—1371no—
224C4×D4—1371no—
225#117—1371no—
226#117C2×D13711yescharacteristic
227C2×C22⋊C4—2741no—
228#107C2×D13711yescharacteristic, complemented
229#106—1371no—
230#106C2×D13711yescharacteristic, complemented
231C4⋊C8—1371no—
232#103—1371no—
233#103—1371no—
234#103C2×D13711yescharacteristic, complemented
235#103C2×D13711yescharacteristic, complemented
236#102—1371no—
237#102—1371no—
238#102—1371no—
239#102—1371no—
240#102—1371no—
241#102—1371no—
242C22⋊C8—1371no—
243C22⋊C8C2×D13711yescharacteristic
244#97—1371no—
245C24—2741no—
246#41—2742no—
247#41—5481no—
248C2×Q8—2741no—
249C2×Q8—1371no—
250C2×Q8C22×D13711yescharacteristic
251C2×D4—2742no—
252C2×D4—2741no—
253C2×D4—2741no—
254C2×D4—2741no—
255C2×D4—5481no—
256C2×D4—1371no—
257C2×D4—1371no—
258C2×D4—1371no—
259C22×C4—2741no—
260C22×C4—2741no—
261C22×C4—1371no—
262C22×C4C22×D13711yescharacteristic
263Q16—5481no—
264SD16—2742no—
265SD16—5481no—
266M16—5481no—
267C2×C8—5481no—
268C2×C8—1371no—
269C2×C8—1371no—
270C2×C8C22×D13711yescharacteristic
271C2×C8C22×D13711yescharacteristic
272C4⋊C4—21no—
273C4⋊C4—2741no—
274C4⋊C4—2741no—
275C4⋊C4—2741no—
276C4⋊C4—2741no—
277C4⋊C4—1371no—
278C4⋊C4—1371no—
279C4⋊C4C22×D13711yescharacteristic
280C4⋊C4C22×D13711yescharacteristic
281C4⋊C4C22×D13711yescharacteristic
282C22⋊C4—21no—
283C22⋊C4—21no—
284C22⋊C4—2741no—
285C22⋊C4—2741no—
286C22⋊C4—2741no—
287C22⋊C4—2741no—
288C22⋊C4—2741no—
289C22⋊C4—2741no—
290C22⋊C4—2741no—
291C22⋊C4—1371no—
292C22⋊C4—1371no—
293C42—21no—
294C42—2741no—
295C42—2741no—
296C42—1371no—
297C23—2741no—
298C23—2741no—
299C23—5482no—
300C23—5482no—
301C23—10961no—
302C23—1371no—
303C23D4×D13711yescharacteristic
304Q8—21no—
305Q8—21no—
306Q8—2741no—
307Q8—2741no—
308Q8—5482no—
309D4—2741no—
310D4—2741no—
311D4—2741no—
312D4—5482no—
313D4—5482no—
314D4—5481no—
315D4—10961no—
316D4—1372no—
317C2×C4—21no—
318C2×C4—21no—
319C2×C4—21no—
320C2×C4—21no—
321C2×C4—21no—
322C2×C4—21no—
323C2×C4—2741no—
324C2×C4—2741no—
325C2×C4—2741no—
326C2×C4—2741no—
327C2×C4—5482no—
328C2×C4—10961no—
329C2×C4—1371no—
330C2×C4—1371no—
331C2×C4—1371no—
332C2×C4D4×D13711yescharacteristic
333C2×C4C23×D13711yescharacteristic
334C8—21no—
335C8—21no—
336C8—2741no—
337C8—2741no—
338C22—21no—
339C22—21no—
340C22—21no—
341C22—2741no—
342C22—2741no—
343C22—2741no—
344C22—2742no—
345C22—5481no—
346C22—5481no—
347C22—5481no—
348C22—5481no—
349C22—5481no—
350C22—10964no—
351C22C2×D4×D13711yescharacteristic
352C4—21no—
353C4—41no—
354C4—41no—
355C4—41no—
356C4—41no—
357C4—2741no—
358C4—2741no—
359C4—2741no—
360C4—5481no—
361C4—5481no—
362C4#41×D13711yescharacteristic
363C4#1798413411yescharacteristic
364C2—41no—
365C2—5481no—
366C2—5482no—
367C2—10961no—
368C2#2257804511yescharacteristic
369C2#2257820211yescharacteristic
370C2#2257821011yescharacteristic
subgroupG/H
centerC22C2×D4×D137
commutator, cosocleC2×C4×C137C24
FrattiniC2×C4C23×D137
Fitting#485×C137C2
socleC22×C137C22×D4
prime radical#22578044C2
hypercenter#485D137
hypocenterC137#3311
LCSG ▷ C2×C4×C137 ▷ C2×C137 ▷ C137
UCSC1 ◁ C22 ◁ C22×C4 ◁ #485

Complements

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

NH ≅ G/Ntype
1. #22578121C2semidirect
3. #22577764C2semidirect
4. #22577730C2semidirect
5. #22577692C2semidirect
6. #22577659C2semidirect
7. #22577633C2semidirect
8. #22577628C2semidirect
9. #22577568C2semidirect
10. #22577556C2semidirect
11. #22577502C2semidirect
12. #22577477C2semidirect
13. #22577407C2semidirect
14. #22577398C2semidirect
15. #485×C137C2semidirect
18. C2×#14555500C22semidirect
22. C2×C8×D137C22semidirect
23. #17984087C22semidirect
29. #17984068C22semidirect
35. #17984053C22semidirect
38. #17984046C22semidirect
40. #17984019C22semidirect
41. #17984017C22semidirect
42. #17984012C22semidirect
43. #17984011C22semidirect
50. #17984000C22semidirect
51. #17983998C22semidirect
54. #107×C137C22semidirect
55. #106×C137C22semidirect
56. #103×C137C22semidirect
57. #103×C137C22semidirect
185. C137#3311semidirect
190. #485D137semidirect
228. #107C2×D137semidirect
230. #106C2×D137semidirect
234. #103C2×D137semidirect
235. #103C2×D137semidirect

Maximal subgroups

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

subgroupindex
#225781212
#225780442
#225777642
#225777302
#225776922
#225776592
#225776332
#225776282
#225775682
#225775562
#225775022
#225774772
#225774072
#225773982
#485×C1372
#3311137