Finite Group Databasebeta

← #31790721

Subgroups — #31790721

Order 17536.

414 conjugacy classes of subgroups.

The subgroup lattice has 366 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#22577916C211yesmaximal, characteristic
2#117×D137C211yesmaximal, characteristic, complemented
3#22577743C211yesmaximal, characteristic, complemented
4#22577742C211yesmaximal, characteristic, complemented
5#22577741C211yesmaximal, characteristic, complemented
6#22577730C211yesmaximal, characteristic, complemented
7#22577729C211yesmaximal, characteristic, complemented
8#22577728C211yesmaximal, characteristic, complemented
9#22577727C211yesmaximal, characteristic, complemented
10#22577444C211yesmaximal, characteristic, complemented
11#22577442C211yesmaximal, characteristic, complemented
12#22577403C211yesmaximal, characteristic, complemented
13#22577403C211yesmaximal, characteristic, complemented
14#22577365C211yesmaximal, characteristic, complemented
15#485×C137C211yesmaximal, characteristic, complemented
16C2×#14555511C2211yescharacteristic
17C22×C4×D137C2211yescharacteristic
18C2×#14555493C2211yescharacteristic
19C2×#14555493C2211yescharacteristic
20#17984073—21no—
21#17984064C2211yescharacteristic
22#17984063C2211yescharacteristic
23#17984062C2211yescharacteristic
24#17984058—21no—
25#17984050—21no—
26#17984048—21no—
27#17984047C2211yescharacteristic
28#17984046C2211yescharacteristic
29#17984046C2211yescharacteristic
30#17984046C2211yescharacteristic
31#17984045—21no—
32#17984044—21no—
33C4⋊C4×D137C2211yescharacteristic
34C4⋊C4×D137C2211yescharacteristic
35C22⋊C4×D137—21no—
36C42×D137—21no—
37#17984022C2211yescharacteristic, complemented
38#17984021C2211yescharacteristic, complemented
39#17984021C2211yescharacteristic, complemented
40#17984020C2211yescharacteristic, complemented
41#17984019C2211yescharacteristic, complemented
42#17984019C2211yescharacteristic, complemented
43#17984012C2211yescharacteristic, complemented
44#17984012C2211yescharacteristic, complemented
45#17984010C2211yescharacteristic, complemented
46#17984010C2211yescharacteristic, complemented
47#17984008C2211yescharacteristic, complemented
48#17984008C2211yescharacteristic, complemented
49C22⋊#11764040C2211yescharacteristic
50#17984000C2211yescharacteristic
51#17984000C2211yescharacteristic
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×D137D411yescharacteristic
60#14555511—22no—
61#14555511—22no—
62C2×D548—21no—
63C2×D548—21no—
64C2×D548C2311yescharacteristic
65C2×C22⋊D137—21no—
66C22×Dic137D411yescharacteristic
67C2×C4×D137—22no—
68C2×C4×D137—22no—
69C2×C4×D137—21no—
70C2×C4×D137—21no—
71C2×C4×D137—21no—
72C2×C4×D137D411yescharacteristic
73C2×C4×D137D411yescharacteristic
74C2×C4×D137C2311yescharacteristic
75C2×C4×D137C2311yescharacteristic
76#14555493—22no—
77#14555493—22no—
78C2×#11764040C2311yescharacteristic
79C2×#11764040C2311yescharacteristic
80C4⋊Dic137C2311yescharacteristic
81C4⋊Dic137C2311yescharacteristic
82C4⋊Dic137C2311yescharacteristic
83#14555489—21no—
84#14555489—21no—
85#14555489—21no—
86C22⋊Dic137—21no—
87#14555487—21no—
88#14555487—21no—
89#14555487—21no—
90#14555487—21no—
91#14555487—21no—
92#14555487—21no—
93C4×Dic137—21no—
94C4×Dic137—21no—
95C4×Dic137—21no—
96C4×Dic137—21no—
97C2×Q8×C137C2311yescharacteristic
98C22×C4×C137C2311yescharacteristic
99C2×C8×C137C2311yescharacteristic
100C2×C8×C137C2311yescharacteristic
101C4⋊C4×C137—21no—
102C4⋊C4×C137C2311yescharacteristic
103C4⋊C4×C137C2311yescharacteristic
104C4⋊C4×C137C2311yescharacteristic
105C22⋊C4×C137—21no—
106C22⋊C4×C137—21no—
107C42×C137—21no—
108C22×D137—22no—
109C22×D137—21no—
110C22×D137—41no—
111C22×D137—41no—
112C22×D137C2×D411yescharacteristic
113C22×D137C2×D411yescharacteristic
114D548—21no—
115D548—21no—
116D548—42no—
117D548—42no—
118C22⋊D137—42no—
119C2×Dic137—22no—
120C2×Dic137—21no—
121C2×Dic137—21no—
122C2×Dic137—21no—
123C2×Dic137C2×D411yescharacteristic
124C2×Dic137C2×D411yescharacteristic
125C4×D137—22no—
126C4×D137—21no—
127C4×D137—21no—
128C4×D137—22no—
129C4×D137—42no—
130C4×D137—42no—
131C4×D137—42no—
132C4×D137#4112yes—
133C4×D137#4112yes—
134#11764040—21no—
135#11764040—21no—
136C23×C137C2×D411yescharacteristic
137Q8×C137—21no—
138Q8×C137—21no—
139C2×C4×C137—21no—
140C2×C4×C137—21no—
141C2×C4×C137—21no—
142C2×C4×C137—21no—
143C2×C4×C137—21no—
144C2×C4×C137—21no—
145C2×C4×C137C2×D411yescharacteristic
146C2×C4×C137C2411yescharacteristic
147C8×C137—21no—
148C8×C137—21no—
149C22×C137—21no—
150C22×C137—21no—
151C22×C137—21no—
152C22×C137C22×D411yescharacteristic
153C2×D137—21no—
154C2×D137—21no—
155C2×D137—21no—
156C2×D137—21no—
157C2×D137—21no—
158C2×D137—44no—
159C2×D137—41no—
160C2×D137—41no—
161C2×D137—41no—
162C2×D137#12312yes—
163C4×C137—21no—
164C4×C137—41no—
165C4×C137—41no—
166C4×C137—41no—
167C4×C137—41no—
168C4×C137C2×#4111yescharacteristic
169C4×C137C2×#4111yescharacteristic
170Dic137—21no—
171Dic137—41no—
172Dic137—41no—
173Dic137—41no—
174Dic137#12312yes—
175C2×C137—41no—
176C2×C137C2×#12311yescharacteristic
177C2×C137#57511yescharacteristic
178C2×C137#57511yescharacteristic
179D137—22no—
180D137—41no—
181D137—81no—
182C137#309311yescharacteristic, complemented, Sylow
183#3093—1371nomaximal, Sylow
184#525—1371no—
185C2×#117—1371no—
186#485—1371no—
187#485—1371no—
188#485—1371no—
189#485D13711yescharacteristic, complemented
190#482—1371no—
191#482—1371no—
192#482—1371no—
193#482—1371no—
194#428—1371no—
195#428—1371no—
196#417—1371no—
197#417—1371no—
198#407—1371no—
199C2×#41—1371no—
200C23×C4—1371no—
201C2×M16—1371no—
202C2×M16—1371no—
203#123—2741no—
204C22⋊Q8—1371no—
205C22⋊Q8C2×D13711yescharacteristic
206#121—1371no—
207#121—1371no—
208#120—2741no—
209C4×D4—2741no—
210C4×D4—2741no—
211#117—2741no—
212#117—2741no—
213#117—1371no—
214#117—1371no—
215#117—1371no—
216#117—1371no—
217#117C2×D13711yescharacteristic
218C2×C4⋊C4—1371no—
219C2×C4⋊C4—1371no—
220C2×C22⋊C4—2741no—
221C2×C42—2741no—
222#107—1371no—
223#107—1371no—
224#107—1371no—
225#107C2×D13711yescharacteristic, complemented
226#106—1371no—
227#106—1371no—
228#106—1371no—
229#106C2×D13711yescharacteristic, complemented
230#103—1371no—
231#103—1371no—
232#103C2×D13711yescharacteristic, complemented
233#103C2×D13711yescharacteristic, complemented
234#102—1371no—
235#102—1371no—
236#102—1371no—
237#102—1371no—
238C22⋊C8—1371no—
239C22⋊C8—1371no—
240C22⋊C8—1371no—
241C22⋊C8C2×D13711yescharacteristic
242C24—1371no—
243#41—2742no—
244#41—2742no—
245C2×Q8C22×D13711yescharacteristic
246C2×D4—2741no—
247C2×D4—2741no—
248C2×D4—2741no—
249C2×D4—1371no—
250C22×C4—2742no—
251C22×C4—2742no—
252C22×C4—2741no—
253C22×C4—2741no—
254C22×C4—2741no—
255C22×C4—1371no—
256C22×C4—1371no—
257C22×C4—1371no—
258C22×C4—1371no—
259C22×C4—1371no—
260C22×C4C22×D13711yescharacteristic
261M16—2742no—
262M16—2742no—
263C2×C8—1371no—
264C2×C8—1371no—
265C2×C8C22×D13711yescharacteristic
266C2×C8C22×D13711yescharacteristic
267C4⋊C4—21no—
268C4⋊C4—2741no—
269C4⋊C4—2741no—
270C4⋊C4—2741no—
271C4⋊C4—1371no—
272C4⋊C4—1371no—
273C4⋊C4—1371no—
274C4⋊C4C22×D13711yescharacteristic
275C4⋊C4C22×D13711yescharacteristic
276C4⋊C4C22×D13711yescharacteristic
277C22⋊C4—21no—
278C22⋊C4—21no—
279C22⋊C4—2741no—
280C22⋊C4—2741no—
281C22⋊C4—2741no—
282C22⋊C4—2741no—
283C22⋊C4—2741no—
284C22⋊C4—2741no—
285C22⋊C4—2741no—
286C42—21no—
287C42—2741no—
288C42—2741no—
289C42—2741no—
290C42—2741no—
291C23—2742no—
292C23—2741no—
293C23—5481no—
294C23—5481no—
295C23—1371no—
296C23—1371no—
297C23D4×D13711yescharacteristic
298Q8—21no—
299Q8—21no—
300D4—2741no—
301D4—2741no—
302D4—5482no—
303D4—5482no—
304D4—5482no—
305C2×C4—21no—
306C2×C4—21no—
307C2×C4—21no—
308C2×C4—21no—
309C2×C4—21no—
310C2×C4—21no—
311C2×C4—2742no—
312C2×C4—2742no—
313C2×C4—2742no—
314C2×C4—2741no—
315C2×C4—2741no—
316C2×C4—2741no—
317C2×C4—2741no—
318C2×C4—2741no—
319C2×C4—5482no—
320C2×C4—5482no—
321C2×C4—5482no—
322C2×C4—1371no—
323C2×C4—1371no—
324C2×C4—1372no—
325C2×C4—1372no—
326C2×C4D4×D13711yescharacteristic
327C2×C4C23×D13711yescharacteristic
328C8—21no—
329C8—21no—
330C8—2741no—
331C8—2741no—
332C22—21no—
333C22—21no—
334C22—21no—
335C22—2741no—
336C22—2741no—
337C22—2741no—
338C22—2741no—
339C22—2741no—
340C22—5484no—
341C22—5481no—
342C22—5481no—
343C22—5481no—
344C22—1372no—
345C22C2×D4×D13711yescharacteristic
346C4—21no—
347C4—41no—
348C4—41no—
349C4—41no—
350C4—41no—
351C4—2741no—
352C4—5481no—
353C4—5481no—
354C4—5481no—
355C4—1372no—
356C4#41×D13711yescharacteristic
357C4#41×D13711yescharacteristic
358C2—41no—
359C2—2742no—
360C2—5481no—
361C2—10961no—
362C2#123×D13711yescharacteristic
363C2#2257820011yescharacteristic
364C2#2257820211yescharacteristic
subgroupG/H
centerC22C2×D4×D137
commutator, cosocleC2×C4×C137C24
FrattiniC2×C4C23×D137
Fitting#485×C137C2
socleC22×C137C22×D4
prime radical#22577916C2
hypercenter#485D137
hypocenterC137#3093
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
2. #117×D137C2semidirect
3. #22577743C2semidirect
4. #22577742C2semidirect
5. #22577741C2semidirect
6. #22577730C2semidirect
7. #22577729C2semidirect
8. #22577728C2semidirect
9. #22577727C2semidirect
10. #22577444C2semidirect
11. #22577442C2semidirect
12. #22577403C2semidirect
13. #22577403C2semidirect
14. #22577365C2semidirect
15. #485×C137C2semidirect
37. #17984022C22semidirect
38. #17984021C22semidirect
39. #17984021C22semidirect
40. #17984020C22semidirect
41. #17984019C22semidirect
42. #17984019C22semidirect
43. #17984012C22semidirect
44. #17984012C22semidirect
45. #17984010C22semidirect
46. #17984010C22semidirect
47. #17984008C22semidirect
48. #17984008C22semidirect
54. #107×C137C22semidirect
55. #106×C137C22semidirect
56. #103×C137C22semidirect
57. #103×C137C22semidirect
182. C137#3093semidirect
189. #485D137semidirect
225. #107C2×D137semidirect
229. #106C2×D137semidirect
232. #103C2×D137semidirect
233. #103C2×D137semidirect

Maximal subgroups

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

subgroupindex
#225779162
#117×D1372
#225777432
#225777422
#225777412
#225777302
#225777292
#225777282
#225777272
#225774442
#225774422
#225774032
#225773652
#485×C1372
#3093137