Finite Group Databasebeta

← #31790701

Subgroups — #31790701

Order 17536.

420 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#22577913C211yesmaximal, characteristic
2#117×D137C211yesmaximal, characteristic, complemented
3#22577745C211yesmaximal, characteristic, complemented
4#22577744C211yesmaximal, characteristic, complemented
5#22577740C211yesmaximal, characteristic, complemented
6#22577735C211yesmaximal, characteristic, complemented
7#22577724C211yesmaximal, characteristic, complemented
8#22577720C211yesmaximal, characteristic, complemented
9#22577715C211yesmaximal, characteristic, complemented
10#22577440C211yesmaximal, characteristic, complemented
11#22577438C211yesmaximal, characteristic, complemented
12#22577396C211yesmaximal, characteristic, complemented
13#22577396C211yesmaximal, characteristic, complemented
14C22⋊C8×D137C211yesmaximal, characteristic, complemented
15#482×C137C211yesmaximal, characteristic, complemented
16C2×#14555509C2211yescharacteristic
17C22×C4×D137C2211yescharacteristic
18C2×C8×D137C2211yescharacteristic
19C2×C8×D137C2211yescharacteristic
20#17984070—21no—
21#17984068C2211yescharacteristic
22#17984067C2211yescharacteristic
23#17984059C2211yescharacteristic
24#17984056—21no—
25D4×Dic137—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#17984013C2211yescharacteristic, complemented
44#17984013C2211yescharacteristic, complemented
45#17984011C2211yescharacteristic, complemented
46#17984011C2211yescharacteristic, complemented
47#17984009C2211yescharacteristic, complemented
48#17984009C2211yescharacteristic, complemented
49C22⋊#11764040C2211yescharacteristic
50#17984000C2211yescharacteristic
51#17984000C2211yescharacteristic
52#121×C137C2211yescharacteristic
53#117×C137C2211yescharacteristic
54#107×C137C2211yescharacteristic, complemented
55#106×C137C2211yescharacteristic, complemented
56#102×C137C2211yescharacteristic, complemented
57#102×C137C2211yescharacteristic, complemented
58C22⋊C8×C137C2211yescharacteristic
59C23×D137D411yescharacteristic
60#14555509—22no—
61#14555509—22no—
62C2×Dic274C2311yescharacteristic
63C2×C22⋊D137—21no—
64C2×C22⋊D137—21no—
65C22×Dic137—21no—
66C22×Dic137D411yescharacteristic
67C2×C4×D137—22no—
68C2×C4×D137—22no—
69C2×C4×D137—21no—
70C2×C4×D137—21no—
71C2×C4×D137D411yescharacteristic
72C2×C4×D137D411yescharacteristic
73C2×C4×D137C2311yescharacteristic
74C2×C4×D137C2311yescharacteristic
75C2×#11764040C2311yescharacteristic
76C2×#11764040C2311yescharacteristic
77C8×D137—22no—
78C8×D137—22no—
79C4⋊Dic137—21no—
80C4⋊Dic137C2311yescharacteristic
81C4⋊Dic137C2311yescharacteristic
82C4⋊Dic137C2311yescharacteristic
83#14555489—21no—
84#14555489—21no—
85#14555489—21no—
86C22⋊Dic137—21no—
87C22⋊Dic137—21no—
88C22⋊Dic137—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×D4×C137—21no—
98C2×D4×C137C2311yescharacteristic
99C22×C4×C137C2311yescharacteristic
100C2×C8×C137C2311yescharacteristic
101C2×C8×C137C2311yescharacteristic
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—22no—
110C22×D137—41no—
111C22×D137C2×D411yescharacteristic
112C22×D137C2×D411yescharacteristic
113Dic274—21no—
114Dic274—21no—
115C22⋊D137—42no—
116C22⋊D137—42no—
117C2×Dic137—22no—
118C2×Dic137—21no—
119C2×Dic137—21no—
120C2×Dic137—21no—
121C2×Dic137—21no—
122C2×Dic137—42no—
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#4112yes—
132C4×D137#4112yes—
133#11764040—21no—
134#11764040—21no—
135C23×C137—21no—
136C23×C137C2×D411yescharacteristic
137D4×C137—21no—
138D4×C137—21no—
139D4×C137—42no—
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×C137—41no—
153C22×C137—41no—
154C22×C137—41no—
155C22×C137C22×D411yescharacteristic
156C2×D137—21no—
157C2×D137—21no—
158C2×D137—21no—
159C2×D137—21no—
160C2×D137—44no—
161C2×D137#12312yes—
162C2×D137#13512yes—
163C4×C137—21no—
164C4×C137—41no—
165C4×C137—41no—
166C4×C137—41no—
167C4×C137C2×#4111yescharacteristic
168C4×C137C2×#4111yescharacteristic
169Dic137—21no—
170Dic137—41no—
171Dic137—41no—
172Dic137—41no—
173Dic137—41no—
174Dic137#12312yes—
175C2×C137—41no—
176C2×C137—81no—
177C2×C137C2×#12311yescharacteristic
178C2×C137C2×#13511yescharacteristic
179C2×C137#57511yescharacteristic
180D137—22no—
181D137—22no—
182C137#309111yescharacteristic, complemented, Sylow
183#3091—1371nomaximal, Sylow
184#525—1371no—
185C2×#117—1371no—
186#485—1371no—
187#485—1371no—
188#484—1371no—
189#483—1371no—
190#482—1371no—
191#482D13711yescharacteristic, complemented
192#481—1371no—
193#480—1371no—
194#427—1371no—
195#427—1371no—
196#416—1371no—
197#416—1371no—
198C2×C22⋊C8—1371no—
199C2×#41—1371no—
200C23×C4—1371no—
201C22×C8—1371no—
202C22×C8—1371no—
203#123—2741no—
204C22⋊Q8—1371no—
205C22⋊Q8—1371no—
206#121—1371no—
207#121C2×D13711yescharacteristic
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#103—1371no—
233#103—1371no—
234#102—1371no—
235#102—1371no—
236#102C2×D13711yescharacteristic, complemented
237#102C2×D13711yescharacteristic, complemented
238C22⋊C8—1371no—
239C22⋊C8—1371no—
240C22⋊C8—1371no—
241C22⋊C8C2×D13711yescharacteristic
242C24—1371no—
243#41—2742no—
244#41—2742no—
245C2×Q8—1371no—
246C2×D4—21no—
247C2×D4—2741no—
248C2×D4—2741no—
249C2×D4C22×D13711yescharacteristic
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
261C2×C8—2742no—
262C2×C8—2742no—
263C2×C8—1371no—
264C2×C8—1371no—
265C2×C8C22×D13711yescharacteristic
266C2×C8C22×D13711yescharacteristic
267C4⋊C4—2741no—
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—21no—
292C23—2742no—
293C23—2742no—
294C23—5481no—
295C23—1371no—
296C23—1371no—
297C23D4×D13711yescharacteristic
298Q8—2741no—
299Q8—2741no—
300D4—21no—
301D4—21no—
302D4—42no—
303D4—5482no—
304D4—5482no—
305C2×C4—21no—
306C2×C4—21no—
307C2×C4—21no—
308C2×C4—21no—
309C2×C4—21no—
310C2×C4—2742no—
311C2×C4—2742no—
312C2×C4—2742no—
313C2×C4—2741no—
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—41no—
336C22—41no—
337C22—41no—
338C22—2741no—
339C22—2741no—
340C22—2741no—
341C22—2741no—
342C22—5484no—
343C22—1372no—
344C22—1372no—
345C22C2×D4×D13711yescharacteristic
346C4—21no—
347C4—41no—
348C4—41no—
349C4—41no—
350C4—2741no—
351C4—5481no—
352C4—5481no—
353C4—5481no—
354C4—5481no—
355C4—1372no—
356C4#41×D13711yescharacteristic
357C4#41×D13711yescharacteristic
358C2—41no—
359C2—81no—
360C2—2742no—
361C2—2742no—
362C2#123×D13711yescharacteristic
363C2#135×D13711yescharacteristic
364C2#2257819911yescharacteristic
subgroupG/H
centerC22C2×D4×D137
commutator, cosocleC2×C4×C137C24
FrattiniC2×C4C23×D137
Fitting#482×C137C2
socleC22×C137C22×D4
prime radical#22577913C2
hypercenter#482D137
hypocenterC137#3091
LCSG ▷ C2×C4×C137 ▷ C2×C137 ▷ C137
UCSC1 ◁ C22 ◁ C22×C4 ◁ #482

Complements

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

NH ≅ G/Ntype
2. #117×D137C2semidirect
3. #22577745C2semidirect
4. #22577744C2semidirect
5. #22577740C2semidirect
6. #22577735C2semidirect
7. #22577724C2semidirect
8. #22577720C2semidirect
9. #22577715C2semidirect
10. #22577440C2semidirect
11. #22577438C2semidirect
12. #22577396C2semidirect
13. #22577396C2semidirect
14. C22⋊C8×D137C2semidirect
15. #482×C137C2semidirect
37. #17984022C22semidirect
38. #17984021C22semidirect
39. #17984021C22semidirect
40. #17984020C22semidirect
41. #17984019C22semidirect
42. #17984019C22semidirect
43. #17984013C22semidirect
44. #17984013C22semidirect
45. #17984011C22semidirect
46. #17984011C22semidirect
47. #17984009C22semidirect
48. #17984009C22semidirect
54. #107×C137C22semidirect
55. #106×C137C22semidirect
56. #102×C137C22semidirect
57. #102×C137C22semidirect
182. C137#3091semidirect
191. #482D137semidirect
225. #107C2×D137semidirect
229. #106C2×D137semidirect
236. #102C2×D137semidirect
237. #102C2×D137semidirect

Maximal subgroups

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

subgroupindex
#225779132
#117×D1372
#225777452
#225777442
#225777402
#225777352
#225777242
#225777202
#225777152
#225774402
#225774382
#225773962
C22⋊C8×D1372
#482×C1372
#3091137