Finite Group Databasebeta

← #31790728

Subgroups — #31790728

Order 17536.

462 conjugacy classes of subgroups.

The subgroup lattice has 386 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#121×D137C211yesmaximal, characteristic
2#117×D137C211yesmaximal, characteristic, complemented
3#22577728C211yesmaximal, characteristic, complemented
4#22577727C211yesmaximal, characteristic, complemented
5#22577724C211yesmaximal, characteristic, complemented
6#22577723C211yesmaximal, characteristic, complemented
7#22577720C211yesmaximal, characteristic, complemented
8#22577718C211yesmaximal, characteristic, complemented
9#22577715C211yesmaximal, characteristic, complemented
10#22577444C211yesmaximal, characteristic, complemented
11#22577442C211yesmaximal, characteristic, complemented
12#22577406C211yesmaximal, characteristic, complemented
13#22577406C211yesmaximal, characteristic, complemented
14#22577365C211yesmaximal, characteristic, complemented
15#482×C137C211yesmaximal, characteristic, complemented
16C2×D4×D137—21no—
17C2×D4×D137C2211yescharacteristic
18C22×C4×D137C2211yescharacteristic
19C2×#14555493C2211yescharacteristic
20C2×#14555493C2211yescharacteristic
21#17984063C2211yescharacteristic
22#17984062C2211yescharacteristic
23#17984061—21no—
24#17984060—21no—
25#17984059C2211yescharacteristic
26#17984047C2211yescharacteristic, complemented
27#17984046C2211yescharacteristic
28#17984046C2211yescharacteristic
29#17984045—21no—
30#17984044—21no—
31C4⋊C4×D137C2211yescharacteristic
32C4⋊C4×D137C2211yescharacteristic, complemented
33C4⋊C4×D137C2211yescharacteristic, complemented
34C22⋊C4×D137—21no—
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#17984010C2211yescharacteristic, complemented
44#17984010C2211yescharacteristic, complemented
45#17984009C2211yescharacteristic, complemented
46#17984009C2211yescharacteristic, complemented
47#17984008C2211yescharacteristic, complemented
48#17984008C2211yescharacteristic, complemented
49C22⋊#11764040C2211yescharacteristic, complemented
50#17984000C2211yescharacteristic, complemented
51#17984000C2211yescharacteristic, complemented
52#121×C137C2211yescharacteristic
53#117×C137C2211yescharacteristic, complemented
54#107×C137C2211yescharacteristic, complemented
55#106×C137C2211yescharacteristic, complemented
56#102×C137C2211yescharacteristic, complemented
57#102×C137C2211yescharacteristic, complemented
58C22⋊C8×C137C2211yescharacteristic, complemented
59C23×D137—21no—
60C23×D137D411yescharacteristic
61C2×D548—21no—
62C2×D548C2311yescharacteristic
63C2×C22⋊D137—21no—
64C2×C22⋊D137—21no—
65C2×C22⋊D137—21no—
66D4×D137—22no—
67D4×D137—22no—
68D4×D137—44no—
69C22×Dic137D411yescharacteristic
70C2×C4×D137—22no—
71C2×C4×D137—22no—
72C2×C4×D137—21no—
73C2×C4×D137—21no—
74C2×C4×D137—21no—
75C2×C4×D137D411yescharacteristic
76C2×C4×D137D411yescharacteristic
77C2×C4×D137C2311yescharacteristic
78C2×C4×D137C2311yescharacteristic
79#14555493—22no—
80#14555493—22no—
81C2×#11764040C2311yescharacteristic
82C2×#11764040C2311yescharacteristic
83C4⋊Dic137C2311yescharacteristic
84C4⋊Dic137C2311yescharacteristic
85C4⋊Dic137C2311yescharacteristic
86#14555489—21no—
87#14555489—21no—
88#14555489—21no—
89C22⋊Dic137—21no—
90C22⋊Dic137—21no—
91#14555487—21no—
92#14555487—21no—
93#14555487—21no—
94#14555487—21no—
95C4×Dic137—21no—
96C4×Dic137—21no—
97C4×Dic137—21no—
98C2×D4×C137—21no—
99C2×D4×C137C2311yescharacteristic
100C22×C4×C137C2311yescharacteristic
101C2×C8×C137C2311yescharacteristic
102C2×C8×C137C2311yescharacteristic
103C4⋊C4×C137C2311yescharacteristic
104C4⋊C4×C137C2311yescharacteristic
105C4⋊C4×C137C2311yescharacteristic
106C22⋊C4×C137—21no—
107C22⋊C4×C137—21no—
108C42×C137—21no—
109C22×D137—22no—
110C22×D137—22no—
111C22×D137—21no—
112C22×D137—41no—
113C22×D137—42no—
114C22×D137—42no—
115C22×D137—42no—
116C22×D137C2×D411yescharacteristic
117C22×D137C2×D411yescharacteristic
118D548—21no—
119D548—21no—
120D548—42no—
121C22⋊D137—42no—
122C22⋊D137—42no—
123C22⋊D137—42no—
124C2×Dic137—22no—
125C2×Dic137—21no—
126C2×Dic137—21no—
127C2×Dic137—21no—
128C2×Dic137C2×D411yescharacteristic
129C2×Dic137C2×D411yescharacteristic
130C4×D137—22no—
131C4×D137—21no—
132C4×D137—21no—
133C4×D137—22no—
134C4×D137—42no—
135C4×D137—42no—
136C4×D137—42no—
137C4×D137#4112yes—
138C4×D137#4112yes—
139#11764040—21no—
140#11764040—21no—
141C23×C137—21no—
142C23×C137C2×D411yescharacteristic
143D4×C137—21no—
144D4×C137—21no—
145D4×C137—42no—
146C2×C4×C137—21no—
147C2×C4×C137—21no—
148C2×C4×C137—21no—
149C2×C4×C137—21no—
150C2×C4×C137—21no—
151C2×C4×C137C2×D411yescharacteristic
152C2×C4×C137C2411yescharacteristic
153C8×C137—21no—
154C8×C137—21no—
155C22×C137—21no—
156C22×C137—21no—
157C22×C137—21no—
158C22×C137—41no—
159C22×C137—41no—
160C22×C137—41no—
161C22×C137C22×D411yescharacteristic
162C2×D137—21no—
163C2×D137—21no—
164C2×D137—21no—
165C2×D137—21no—
166C2×D137—44no—
167C2×D137—41no—
168C2×D137—41no—
169C2×D137—41no—
170C2×D137—84no—
171C2×D137#12312yes—
172C2×D137AGL(1,ℤ/8)12yes—
173C4×C137—21no—
174C4×C137—41no—
175C4×C137—41no—
176C4×C137—41no—
177C4×C137C2×#4111yescharacteristic
178C4×C137C2×#4111yescharacteristic
179Dic137—21no—
180Dic137—41no—
181Dic137—41no—
182Dic137—41no—
183Dic137#12312yes—
184C2×C137—41no—
185C2×C137—81no—
186C2×C137C2×#12311yescharacteristic
187C2×C137C2×AGL(1,ℤ/8)11yescharacteristic
188C2×C137#57511yescharacteristic
189D137—22no—
190D137—22no—
191D137—81no—
192C137#309411yescharacteristic, complemented, Sylow
193#3094—1371nomaximal, Sylow
194C2×#121—1371no—
195C2×#117—1371no—
196#482—1371no—
197#482—1371no—
198#482—1371no—
199#482D13711yescharacteristic, complemented
200#481—1371no—
201#481—1371no—
202#480—1371no—
203#480—1371no—
204#428—1371no—
205#428—1371no—
206#418—1371no—
207#418—1371no—
208#407—1371no—
209C22×D4—2741no—
210C22×D4—1371no—
211C23×C4—1371no—
212C2×M16—1371no—
213C2×M16—1371no—
214#121—2741no—
215#121—2741no—
216#121—1371no—
217#121—1371no—
218#121—1371no—
219#121C2×D13711yescharacteristic
220#117—2741no—
221#117—2741no—
222#117—1371no—
223#117—1371no—
224#117—1371no—
225#117C2×D13711yescharacteristic, complemented
226C2×C4⋊C4—1371no—
227C2×C4⋊C4—1371no—
228C2×C4⋊C4—1371no—
229C2×C22⋊C4—2741no—
230C2×C22⋊C4—2741no—
231C2×C42—2741no—
232#107—1371no—
233#107—1371no—
234#107—1371no—
235#107C2×D13711yescharacteristic, complemented
236#106—1371no—
237#106—1371no—
238#106—1371no—
239#106C2×D13711yescharacteristic, complemented
240#102—1371no—
241#102—1371no—
242#102—1371no—
243#102—1371no—
244#102—1371no—
245#102—1371no—
246#102C2×D13711yescharacteristic, complemented
247#102C2×D13711yescharacteristic, complemented
248C22⋊C8—1371no—
249C22⋊C8—1371no—
250C22⋊C8—1371no—
251C22⋊C8C2×D13711yescharacteristic, complemented
252C24—2741no—
253C24—1371no—
254C2×D4—21no—
255C2×D4—2741no—
256C2×D4—2741no—
257C2×D4—2741no—
258C2×D4—2742no—
259C2×D4—2742no—
260C2×D4—2741no—
261C2×D4—5484no—
262C2×D4—1371no—
263C2×D4C22×D13711yescharacteristic
264C22×C4—2742no—
265C22×C4—2742no—
266C22×C4—2741no—
267C22×C4—2741no—
268C22×C4—2741no—
269C22×C4—1371no—
270C22×C4—1371no—
271C22×C4—1371no—
272C22×C4—1371no—
273C22×C4—1371no—
274C22×C4C22×D13711yescharacteristic
275M16—2742no—
276M16—2742no—
277C2×C8—1371no—
278C2×C8—1371no—
279C2×C8C22×D13711yescharacteristic
280C2×C8C22×D13711yescharacteristic
281C4⋊C4—2741no—
282C4⋊C4—2741no—
283C4⋊C4—2741no—
284C4⋊C4—1371no—
285C4⋊C4—1371no—
286C4⋊C4—1371no—
287C4⋊C4C22×D13711yescharacteristic
288C4⋊C4C22×D13711yescharacteristic
289C4⋊C4C22×D13711yescharacteristic
290C22⋊C4—21no—
291C22⋊C4—21no—
292C22⋊C4—2741no—
293C22⋊C4—2741no—
294C22⋊C4—2741no—
295C22⋊C4—2741no—
296C22⋊C4—2741no—
297C22⋊C4—2741no—
298C42—21no—
299C42—2741no—
300C42—2741no—
301C42—2741no—
302C23—21no—
303C23—2742no—
304C23—2742no—
305C23—2741no—
306C23—5481no—
307C23—5482no—
308C23—5482no—
309C23—5482no—
310C23—1371no—
311C23—1371no—
312C23D4×D13711yescharacteristic
313D4—21no—
314D4—21no—
315D4—42no—
316D4—2741no—
317D4—2741no—
318D4—5482no—
319D4—5482no—
320D4—5482no—
321D4—5482no—
322C2×C4—21no—
323C2×C4—21no—
324C2×C4—21no—
325C2×C4—21no—
326C2×C4—21no—
327C2×C4—2742no—
328C2×C4—2742no—
329C2×C4—2742no—
330C2×C4—2741no—
331C2×C4—2741no—
332C2×C4—2741no—
333C2×C4—2741no—
334C2×C4—2741no—
335C2×C4—5482no—
336C2×C4—5482no—
337C2×C4—5482no—
338C2×C4—1371no—
339C2×C4—1371no—
340C2×C4—1372no—
341C2×C4—1372no—
342C2×C4D4×D13711yescharacteristic
343C2×C4C23×D13711yescharacteristic
344C8—21no—
345C8—21no—
346C8—2741no—
347C8—2741no—
348C22—21no—
349C22—21no—
350C22—21no—
351C22—41no—
352C22—41no—
353C22—41no—
354C22—2741no—
355C22—2741no—
356C22—2741no—
357C22—2741no—
358C22—5484no—
359C22—5481no—
360C22—5481no—
361C22—5481no—
362C22—10964no—
363C22—1372no—
364C22—1372no—
365C22C2×D4×D13711yescharacteristic
366C4—21no—
367C4—41no—
368C4—41no—
369C4—41no—
370C4—2741no—
371C4—5481no—
372C4—5481no—
373C4—5481no—
374C4—1372no—
375C4#41×D13711yescharacteristic
376C4#41×D13711yescharacteristic
377C2—41no—
378C2—81no—
379C2—2742no—
380C2—2742no—
381C2—10961no—
382C2#123×D13711yescharacteristic
383C2AGL(1,ℤ/8)×D13711yescharacteristic
384C2#2257820011yescharacteristic
subgroupG/H
centerC22C2×D4×D137
commutator, cosocleC2×C4×C137C24
FrattiniC2×C4C23×D137
Fitting#482×C137C2
socleC22×C137C22×D4
prime radical#121×D137C2
hypercenter#482D137
hypocenterC137#3094
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. #22577728C2semidirect
4. #22577727C2semidirect
5. #22577724C2semidirect
6. #22577723C2semidirect
7. #22577720C2semidirect
8. #22577718C2semidirect
9. #22577715C2semidirect
10. #22577444C2semidirect
11. #22577442C2semidirect
12. #22577406C2semidirect
13. #22577406C2semidirect
14. #22577365C2semidirect
15. #482×C137C2semidirect
26. #17984047C22semidirect
32. C4⋊C4×D137C22semidirect
33. C4⋊C4×D137C22semidirect
37. #17984022C22semidirect
38. #17984021C22semidirect
39. #17984021C22semidirect
40. #17984020C22semidirect
41. #17984019C22semidirect
42. #17984019C22semidirect
43. #17984010C22semidirect
44. #17984010C22semidirect
45. #17984009C22semidirect
46. #17984009C22semidirect
47. #17984008C22semidirect
48. #17984008C22semidirect
49. C22⋊#11764040C22semidirect
50. #17984000C22semidirect
51. #17984000C22semidirect
53. #117×C137C22semidirect
54. #107×C137C22semidirect
55. #106×C137C22semidirect
56. #102×C137C22semidirect
57. #102×C137C22semidirect
58. C22⋊C8×C137C22semidirect
192. C137#3094semidirect
199. #482D137semidirect
225. #117C2×D137semidirect
235. #107C2×D137semidirect
239. #106C2×D137semidirect
246. #102C2×D137semidirect
247. #102C2×D137semidirect
251. C22⋊C8C2×D137semidirect

Maximal subgroups

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

subgroupindex
#121×D1372
#117×D1372
#225777282
#225777272
#225777242
#225777232
#225777202
#225777182
#225777152
#225774442
#225774422
#225774062
#225773652
#482×C1372
#3094137