Finite Group Databasebeta

← #31792637

Subgroups — #31792637

Order 17536.

420 conjugacy classes of subgroups.

The subgroup lattice has 384 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#22578101C211yesmaximal, characteristic, complemented
2#22578044C211yesmaximal, characteristic, complemented
3#22577759C211yesmaximal, characteristic, complemented
4#22577730C211yesmaximal, characteristic, complemented
5#22577706C211yesmaximal, characteristic, complemented
6#22577640C211yesmaximal, characteristic, complemented
7#22577634C211yesmaximal, characteristic, complemented
8#22577621C211yesmaximal, characteristic, complemented
9#22577551C211yesmaximal, characteristic, complemented
10#22577539C211yesmaximal, characteristic, complemented
11#22577485C211yesmaximal, characteristic, complemented
12#22577466C211yesmaximal, characteristic, complemented
13#22577398C211yesmaximal, characteristic, complemented
14C2×#17984008C211yesmaximal, characteristic, complemented
15#484×C137C211yesmaximal, characteristic, complemented
16C2×#14555511C2211yescharacteristic
17C22×D548C2211yescharacteristic
18C2×#14555497C2211yescharacteristic, complemented
19C2×D1096C2211yescharacteristic
20C2×D1096C2211yescharacteristic
21C22×#11764040C2211yescharacteristic
22C2×C8×D137C2211yescharacteristic, complemented
23#17984083—21no—
24#17984079C2211yescharacteristic, complemented
25#17984077C2211yescharacteristic
26#17984073—21no—
27#17984073—21no—
28#17984063—21no—
29#17984063C2211yescharacteristic
30#17984063C2211yescharacteristic, complemented
31#17984062—21no—
32#17984062C2211yescharacteristic
33#17984058—21no—
34#17984050C2211yescharacteristic
35#17984050C2211yescharacteristic, complemented
36C4×C22⋊D137—21no—
37C4×C22⋊D137C2211yescharacteristic
38#17984046C2211yescharacteristic, complemented
39C22⋊C4×D137—21no—
40#17984019C2211yescharacteristic, complemented
41#17984017C2211yescharacteristic, complemented
42#17984012C2211yescharacteristic, complemented
43#17984010C2211yescharacteristic
44#17984010C2211yescharacteristic
45#17984008C2211yescharacteristic
46#17984008C2211yescharacteristic
47#17984008C2211yescharacteristic
48#17984008C2211yescharacteristic
49#17984008C2211yescharacteristic, complemented
50#17984000C2211yescharacteristic, complemented
51C8×Dic137C2211yescharacteristic, complemented
52C22⋊Q8×C137C2211yescharacteristic
53C2×C4⋊C4×C137C2211yescharacteristic
54#107×C137C2211yescharacteristic, complemented
55#107×C137C2211yescharacteristic, complemented
56#103×C137C2211yescharacteristic, complemented
57#103×C137C2211yescharacteristic, complemented
58C22⋊C8×C137C2211yescharacteristic
59C23×D137—21no—
60#14555511—22no—
61#14555511—41no—
62C2×D548—22no—
63C2×D548—21no—
64C2×D548—21no—
65C2×D548—21no—
66C2×D548—21no—
67C2×D548—41no—
68C2×D548C2311yescharacteristic
69C2×D548C2311yescharacteristic
70C2×D548C2311yescharacteristic
71C2×C22⋊D137—21no—
72C2×C22⋊D137D411yescharacteristic
73C2×C4×D137—21no—
74C2×C4×D137—21no—
75C2×C4×D137—21no—
76C2×C4×D137C2311yescharacteristic
77#14555497—41no—
78D1096—22no—
79D1096—41no—
80C2×#11764040—41no—
81C2×#11764040C2311yescharacteristic
82C2×#11764040C2311yescharacteristic
83C8×D137—41no—
84C4⋊Dic137—21no—
85C4⋊Dic137C2311yescharacteristic
86#14555489—21no—
87#14555489D411yescharacteristic
88C22⋊Dic137D411yescharacteristic
89#14555487—21no—
90#14555487—21no—
91#14555487—21no—
92#14555487—21no—
93#14555487—21no—
94#14555487—21no—
95#14555487—21no—
96#14555487—21no—
97#14555487D411yescharacteristic
98C4×Dic137—21no—
99C4×Dic137C2311yescharacteristic
100C2×Q8×C137C2311yescharacteristic
101C22×C4×C137—21no—
102C22×C4×C137C2311yescharacteristic
103C2×C8×C137C2311yescharacteristic
104C2×C8×C137C2311yescharacteristic
105C4⋊C4×C137—21no—
106C4⋊C4×C137—21no—
107C4⋊C4×C137C2311yescharacteristic
108C4⋊C4×C137C2311yescharacteristic
109C4⋊C4×C137C2311yescharacteristic
110C22⋊C4×C137—21no—
111C22×D137—21no—
112C22×D137—21no—
113C22×D137—21no—
114C22×D137—42no—
115C22×D137—42no—
116C22×D137—81no—
117C22×D137C2×D411yescharacteristic
118D548—21no—
119D548—21no—
120D548—21no—
121D548—21no—
122D548—21no—
123D548—42no—
124D548—42no—
125D548—81no—
126D548—81no—
127D548#4112yes—
128C22⋊D137—22no—
129C22⋊D137—42no—
130C2×Dic137—21no—
131C2×Dic137C2×D411yescharacteristic
132C2×Dic137C2×D411yescharacteristic
133C2×Dic137C2×D411yescharacteristic
134C4×D137—21no—
135C4×D137—21no—
136C4×D137—42no—
137C4×D137—81no—
138C4×D137—81no—
139#11764040—21no—
140#11764040—21no—
141C23×C137C2×D411yescharacteristic
142Q8×C137—21no—
143Q8×C137—21no—
144C2×C4×C137—21no—
145C2×C4×C137—21no—
146C2×C4×C137—21no—
147C2×C4×C137—21no—
148C2×C4×C137—21no—
149C2×C4×C137—21no—
150C2×C4×C137—81no—
151C2×C4×C137C2×D411yescharacteristic
152C2×C4×C137C2411yescharacteristic
153C8×C137—21no—
154C8×C137—21no—
155C22×C137—21no—
156C22×C137—21no—
157C22×C137#13512yes—
158C22×C137C22×D411yescharacteristic
159C2×D137—21no—
160C2×D137—22no—
161C2×D137—21no—
162C2×D137—21no—
163C2×D137—41no—
164C2×D137—41no—
165C2×D137—41no—
166C2×D137—41no—
167C2×D137—41no—
168C2×D137—41no—
169C2×D137—41no—
170C2×D137—41no—
171C2×D137—84no—
172C4×C137—21no—
173C4×C137—41no—
174C4×C137—41no—
175C4×C137—41no—
176C4×C137—41no—
177C4×C137C2×#4111yescharacteristic
178C4×C137H(5,𝔽2)11yescharacteristic
179Dic137—21no—
180Dic137—21no—
181Dic137—21no—
182Dic137—41no—
183C2×C137—22no—
184C2×C137#54611yescharacteristic
185C2×C137C2×#13511yescharacteristic
186C2×C137#57611yescharacteristic
187D137—41no—
188D137—42no—
189D137—81no—
190D137—81no—
191C137#329711yescharacteristic, complemented, Sylow
192#3297—1371nomaximal, Sylow
193#553—1371no—
194#546—1371no—
195#488—1371no—
196#484D13711yescharacteristic, complemented
197#482—1371no—
198#478—1371no—
199#466—1371no—
200#465—1371no—
201#463—1371no—
202#449—1371no—
203#447—1371no—
204C4×D8—1371no—
205C8×D4—1371no—
206#416—1371no—
207C2×#102—1371no—
208C2×#41—1371no—
209C22×D4—1371no—
210C2×SD16—1371no—
211C2×D8—1371no—
212C2×D8—1371no—
213C22×C8—1371no—
214C22×C8—1371no—
215#126—2741no—
216#125—1371no—
217#124—1371no—
218#123—2741no—
219#123—2741no—
220C22⋊Q8C2×D13711yescharacteristic
221#121—2741no—
222#121—2741no—
223#121—1371no—
224#121—1371no—
225#121—1371no—
226#120—2741no—
227C4×D4—2741no—
228C4×D4—1371no—
229C4×D4—1371no—
230C4×D4—1371no—
231#117—1371no—
232C2×C4⋊C4C2×D13711yescharacteristic
233C2×C22⋊C4—2741no—
234#107C2×D13711yescharacteristic, complemented
235#107C2×D13711yescharacteristic, complemented
236#106—1371no—
237C4⋊C8—1371no—
238#103—1371no—
239#103C2×D13711yescharacteristic, complemented
240#103C2×D13711yescharacteristic, complemented
241#102—1371no—
242#102—1371no—
243#102—1371no—
244#102—1371no—
245#102—1371no—
246#102—1371no—
247#102—1371no—
248C22⋊C8—1371no—
249C22⋊C8C2×D13711yescharacteristic
250C4×C8—1371no—
251C24—2741no—
252#41—2742no—
253#41—5481no—
254C2×Q8C22×D13711yescharacteristic
255C2×D4—2742no—
256C2×D4—2741no—
257C2×D4—2741no—
258C2×D4—2741no—
259C2×D4—2741no—
260C2×D4—2741no—
261C2×D4—5481no—
262C2×D4—1371no—
263C2×D4—1371no—
264C2×D4—1371no—
265C2×D4—1371no—
266C22×C4—21no—
267C22×C4—2741no—
268C22×C4—2741no—
269C22×C4—2741no—
270C22×C4—1371no—
271C22×C4C22×D13711yescharacteristic
272SD16—5481no—
273D8—2742no—
274D8—5481no—
275C2×C8—5481no—
276C2×C8—5481no—
277C2×C8—1371no—
278C2×C8—1371no—
279C2×C8C22×D13711yescharacteristic
280C2×C8C22×D13711yescharacteristic
281C4⋊C4—21no—
282C4⋊C4—21no—
283C4⋊C4—2741no—
284C4⋊C4—2741no—
285C4⋊C4—1371no—
286C4⋊C4—1371no—
287C4⋊C4C22×D13711yescharacteristic
288C4⋊C4C22×D13711yescharacteristic
289C4⋊C4C22×D13711yescharacteristic
290C22⋊C4—21no—
291C22⋊C4—2741no—
292C22⋊C4—2741no—
293C22⋊C4—2741no—
294C22⋊C4—2741no—
295C22⋊C4—2741no—
296C22⋊C4—2741no—
297C22⋊C4—2741no—
298C22⋊C4—2741no—
299C22⋊C4—1371no—
300C22⋊C4—1371no—
301C42—2741no—
302C42—1371no—
303C23—2741no—
304C23—2741no—
305C23—2741no—
306C23—5482no—
307C23—5482no—
308C23—10961no—
309C23—1371no—
310C23D4×D13711yescharacteristic
311Q8—21no—
312Q8—21no—
313D4—2742no—
314D4—2741no—
315D4—2741no—
316D4—2741no—
317D4—2741no—
318D4—2741no—
319D4—5482no—
320D4—5482no—
321D4—5482no—
322D4—10961no—
323D4—10961no—
324D4—1372no—
325C2×C4—21no—
326C2×C4—21no—
327C2×C4—21no—
328C2×C4—21no—
329C2×C4—21no—
330C2×C4—21no—
331C2×C4—81no—
332C2×C4—2741no—
333C2×C4—2741no—
334C2×C4—2741no—
335C2×C4—5482no—
336C2×C4—10961no—
337C2×C4—10961no—
338C2×C4—1371no—
339C2×C4—1371no—
340C2×C4—1371no—
341C2×C4D4×D13711yescharacteristic
342C2×C4C23×D13711yescharacteristic
343C8—21no—
344C8—21no—
345C8—2741no—
346C8—2741no—
347C22—21no—
348C22—21no—
349C22—2741no—
350C22—2741no—
351C22—2741no—
352C22—2742no—
353C22—5481no—
354C22—5481no—
355C22—5481no—
356C22—5481no—
357C22—5481no—
358C22—5481no—
359C22—5481no—
360C22—5481no—
361C22—10964no—
362C22#1798411112yes—
363C22C2×D4×D13711yescharacteristic
364C4—21no—
365C4—41no—
366C4—41no—
367C4—41no—
368C4—41no—
369C4—2741no—
370C4—2741no—
371C4—2741no—
372C4—5481no—
373C4#41×D13711yescharacteristic
374C4#1798413411yescharacteristic
375C2—22no—
376C2—5481no—
377C2—5482no—
378C2—10961no—
379C2—10961no—
380C2#2257804511yescharacteristic
381C2C2×#1798411111yescharacteristic
382C2#2257820711yescharacteristic
subgroupG/H
centerC22C2×D4×D137
commutator, cosocleC2×C4×C137C24
FrattiniC2×C4C23×D137
Fitting#484×C137C2
socleC22×C137C22×D4
hypercenter#484D137
hypocenterC137#3297
LCSG ▷ C2×C4×C137 ▷ C2×C137 ▷ C137
UCSC1 ◁ C22 ◁ C22×C4 ◁ #484

Complements

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

NH ≅ G/Ntype
1. #22578101C2semidirect
2. #22578044C2semidirect
3. #22577759C2semidirect
4. #22577730C2semidirect
5. #22577706C2semidirect
6. #22577640C2semidirect
7. #22577634C2semidirect
8. #22577621C2semidirect
9. #22577551C2semidirect
10. #22577539C2semidirect
11. #22577485C2semidirect
12. #22577466C2semidirect
13. #22577398C2semidirect
14. C2×#17984008C2semidirect
15. #484×C137C2semidirect
18. C2×#14555497C22semidirect
22. C2×C8×D137C22semidirect
24. #17984079C22semidirect
30. #17984063C22semidirect
35. #17984050C22semidirect
38. #17984046C22semidirect
40. #17984019C22semidirect
41. #17984017C22semidirect
42. #17984012C22semidirect
49. #17984008C22semidirect
50. #17984000C22semidirect
51. C8×Dic137C22semidirect
54. #107×C137C22semidirect
55. #107×C137C22semidirect
56. #103×C137C22semidirect
57. #103×C137C22semidirect
191. C137#3297semidirect
196. #484D137semidirect
234. #107C2×D137semidirect
235. #107C2×D137semidirect
239. #103C2×D137semidirect
240. #103C2×D137semidirect

Maximal subgroups

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

subgroupindex
#225781012
#225780442
#225777592
#225777302
#225777062
#225776402
#225776342
#225776212
#225775512
#225775392
#225774852
#225774662
#225773982
C2×#179840082
#484×C1372
#3297137