Finite Group Databasebeta

← #31792639

Subgroups — #31792639

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#22578099C211yesmaximal, characteristic, complemented
2#22577757C211yesmaximal, characteristic, complemented
3#22577736C211yesmaximal, characteristic, complemented
4#22577724C211yesmaximal, characteristic, complemented
5#22577703C211yesmaximal, characteristic, complemented
6#22577637C211yesmaximal, characteristic, complemented
7#22577634C211yesmaximal, characteristic, complemented
8#22577615C211yesmaximal, characteristic, complemented
9#22577548C211yesmaximal, characteristic, complemented
10#22577537C211yesmaximal, characteristic, complemented
11C4×#14555495C211yesmaximal, characteristic, complemented
12D4×#11764040C211yesmaximal, characteristic, complemented
13#22577399C211yesmaximal, characteristic, complemented
14C2×#17984009C211yesmaximal, characteristic, complemented
15#546×C137C211yesmaximal, characteristic, complemented
16C2×#14555497C2211yescharacteristic, complemented
17C2×#14555495C2211yescharacteristic
18C2×#14555495C2211yescharacteristic
19C22×#11764040C2211yescharacteristic
20C22×#11764040C2211yescharacteristic, complemented
21#17984083—21no—
22#17984080C2211yescharacteristic, complemented
23#17984070—21no—
24#17984062C2211yescharacteristic
25#17984062C2211yescharacteristic, complemented
26#17984060—21no—
27C4×D548C2211yescharacteristic, complemented
28#17984048—21no—
29#17984047C2211yescharacteristic, complemented
30C2×C4⋊Dic137C2211yescharacteristic
31#17984021C2211yescharacteristic, complemented
32#17984021C2211yescharacteristic, complemented
33#17984019C2211yescharacteristic, complemented
34C4⋊#11764040C2211yescharacteristic, complemented
35#17984012C2211yescharacteristic, complemented
36#17984012C2211yescharacteristic, complemented
37#17984012C2211yescharacteristic, complemented
38#17984009C2211yescharacteristic
39#17984009C2211yescharacteristic
40#17984009C2211yescharacteristic
41#17984009C2211yescharacteristic
42#17984009C2211yescharacteristic
43#17984009C2211yescharacteristic
44#17984008C2211yescharacteristic, complemented
45C22⋊#11764040C2211yescharacteristic
46C22⋊#11764040C2211yescharacteristic, complemented
47C4×#11764040C2211yescharacteristic, complemented
48C2×#41×C137C2211yescharacteristic
49C22×D4×C137C2211yescharacteristic
50#124×C137C2211yescharacteristic
51#123×C137—21no—
52C22⋊Q8×C137C2211yescharacteristic
53#121×C137—21no—
54#121×C137C2211yescharacteristic
55#120×C137—21no—
56C4×D4×C137C2211yescharacteristic
57C4×D4×C137C2211yescharacteristic
58C2×C22⋊C4×C137—21no—
59C2×D548C2311yescharacteristic
60C2×C22⋊D137—21no—
61C2×C22⋊D137—21no—
62C22×Dic137—21no—
63C2×C4×D137—21no—
64#14555497—41no—
65#14555495—22no—
66#14555495—41no—
67C2×#11764040—41no—
68C2×#11764040—41no—
69C2×#11764040C2311yescharacteristic
70C2×#11764040C2311yescharacteristic
71C2×#11764040C2311yescharacteristic
72C2×#11764040C2311yescharacteristic
73C4⋊Dic137—21no—
74C4⋊Dic137C2311yescharacteristic
75C4⋊Dic137C2311yescharacteristic
76C4⋊Dic137C2311yescharacteristic
77#14555489—21no—
78#14555489—21no—
79C22⋊Dic137—21no—
80#14555487—21no—
81#14555487—21no—
82#14555487—21no—
83C4×Dic137—21no—
84C24×C137—21no—
85#41×C137—22no—
86#41×C137—41no—
87C2×Q8×C137C2311yescharacteristic
88C2×D4×C137—22no—
89C2×D4×C137—21no—
90C2×D4×C137—21no—
91C2×D4×C137—21no—
92C2×D4×C137—41no—
93C2×D4×C137D411yescharacteristic
94C2×D4×C137C2311yescharacteristic
95C2×D4×C137C2311yescharacteristic
96C22×C4×C137—21no—
97C22×C4×C137—21no—
98C22×C4×C137C2311yescharacteristic
99C22×C4×C137C2311yescharacteristic
100C4⋊C4×C137—21no—
101C4⋊C4×C137D411yescharacteristic
102C4⋊C4×C137C2311yescharacteristic
103C22⋊C4×C137—21no—
104C22⋊C4×C137—21no—
105C22⋊C4×C137—21no—
106C22⋊C4×C137—21no—
107C22⋊C4×C137—21no—
108C22⋊C4×C137D411yescharacteristic
109C22⋊C4×C137D411yescharacteristic
110C42×C137C2311yescharacteristic
111C22×D137—21no—
112D548—21no—
113D548—21no—
114C22⋊D137—42no—
115C22⋊D137—81no—
116C2×Dic137—21no—
117C2×Dic137—21no—
118C2×Dic137—21no—
119C2×Dic137—81no—
120C4×D137—81no—
121#11764040—21no—
122#11764040—21no—
123#11764040—21no—
124#11764040—21no—
125C23×C137—21no—
126C23×C137—21no—
127C23×C137—42no—
128C23×C137—42no—
129C23×C137—81no—
130C23×C137C2×D411yescharacteristic
131C23×C137C2×D411yescharacteristic
132Q8×C137—21no—
133Q8×C137—21no—
134D4×C137—22no—
135D4×C137—21no—
136D4×C137—21no—
137D4×C137—21no—
138D4×C137—42no—
139D4×C137—42no—
140D4×C137—81no—
141D4×C137#4112yes—
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×C137—42no—
149C2×C4×C137—81no—
150C2×C4×C137C2×D411yescharacteristic
151C2×C4×C137C2×D411yescharacteristic
152C2×C4×C137C2×D411yescharacteristic
153C2×C4×C137C2×D411yescharacteristic
154C2×C4×C137C2411yescharacteristic
155C22×C137—21no—
156C22×C137—21no—
157C22×C137—21no—
158C22×C137—22no—
159C22×C137—21no—
160C22×C137—21no—
161C22×C137—41no—
162C22×C137—41no—
163C22×C137—41no—
164C22×C137—41no—
165C22×C137—41no—
166C22×C137—84no—
167C22×C137#13512yes—
168C22×C137C22×D411yescharacteristic
169C2×D137—41no—
170C2×D137—41no—
171C2×D137—41no—
172C4×C137—21no—
173C4×C137—21no—
174C4×C137—21no—
175C4×C137—21no—
176C4×C137—41no—
177C4×C137—41no—
178C4×C137C2×#4111yescharacteristic
179C4×C137H(5,𝔽2)11yescharacteristic
180Dic137—41no—
181Dic137—41no—
182Dic137—41no—
183C2×C137—22no—
184C2×C137—41no—
185C2×C137—42no—
186C2×C137—81no—
187C2×C137#54611yescharacteristic
188C2×C137C2×#13511yescharacteristic
189C2×C137#57611yescharacteristic
190D137—81no—
191C137#329711yescharacteristic, complemented, Sylow
192#3297—1371nomaximal, Sylow
193#553—1371no—
194#546D13711yescharacteristic, complemented
195#488—1371no—
196#484—1371no—
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×#41C2×D13711yescharacteristic
209C22×D4C2×D13711yescharacteristic
210C2×SD16—1371no—
211C2×D8—1371no—
212C2×D8—1371no—
213C22×C8—1371no—
214C22×C8—1371no—
215#126—2741no—
216#125—1371no—
217#124C2×D13711yescharacteristic
218#123—21no—
219#123—2741no—
220C22⋊Q8C2×D13711yescharacteristic
221#121—21no—
222#121—2741no—
223#121—1371no—
224#121—1371no—
225#121C2×D13711yescharacteristic
226#120—21no—
227C4×D4—2741no—
228C4×D4—1371no—
229C4×D4C2×D13711yescharacteristic
230C4×D4C2×D13711yescharacteristic
231#117—1371no—
232C2×C4⋊C4—1371no—
233C2×C22⋊C4—21no—
234#107—1371no—
235#107—1371no—
236#106—1371no—
237C4⋊C8—1371no—
238#103—1371no—
239#103—1371no—
240#103—1371no—
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⋊C8—1371no—
250C4×C8—1371no—
251C24—21no—
252#41—22no—
253#41—41no—
254C2×Q8C22×D13711yescharacteristic
255C2×D4—22no—
256C2×D4—21no—
257C2×D4—21no—
258C2×D4—21no—
259C2×D4—41no—
260C2×D4—2741no—
261C2×D4—2741no—
262C2×D4—1371no—
263C2×D4C22⋊D13711yescharacteristic
264C2×D4C22×D13711yescharacteristic
265C2×D4C22×D13711yescharacteristic
266C22×C4—21no—
267C22×C4—21no—
268C22×C4—2741no—
269C22×C4—2741no—
270C22×C4C22×D13711yescharacteristic
271C22×C4C22×D13711yescharacteristic
272SD16—5481no—
273D8—2742no—
274D8—5481no—
275C2×C8—5481no—
276C2×C8—5481no—
277C2×C8—1371no—
278C2×C8—1371no—
279C2×C8—1371no—
280C2×C8—1371no—
281C4⋊C4—21no—
282C4⋊C4—2741no—
283C4⋊C4—2741no—
284C4⋊C4—2741no—
285C4⋊C4—1371no—
286C4⋊C4—1371no—
287C4⋊C4—1371no—
288C4⋊C4C22⋊D13711yescharacteristic
289C4⋊C4C22×D13711yescharacteristic
290C22⋊C4—21no—
291C22⋊C4—21no—
292C22⋊C4—21no—
293C22⋊C4—21no—
294C22⋊C4—21no—
295C22⋊C4—2741no—
296C22⋊C4—2741no—
297C22⋊C4—2741no—
298C22⋊C4—2741no—
299C22⋊C4C22⋊D13711yescharacteristic
300C22⋊C4C22⋊D13711yescharacteristic
301C42—2741no—
302C42C22×D13711yescharacteristic
303C23—21no—
304C23—21no—
305C23—42no—
306C23—42no—
307C23—81no—
308C23—2741no—
309C23C2×C22⋊D13711yescharacteristic
310C23C2×C22⋊D13711yescharacteristic
311Q8—21no—
312Q8—21no—
313D4—22no—
314D4—21no—
315D4—21no—
316D4—21no—
317D4—42no—
318D4—42no—
319D4—81no—
320D4—2741no—
321D4—2741no—
322D4—5482no—
323D4—10961no—
324D4#1455550912yes—
325C2×C4—21no—
326C2×C4—21no—
327C2×C4—21no—
328C2×C4—21no—
329C2×C4—21no—
330C2×C4—21no—
331C2×C4—42no—
332C2×C4—81no—
333C2×C4—2741no—
334C2×C4—2741no—
335C2×C4—2741no—
336C2×C4—10961no—
337C2×C4—10961no—
338C2×C4C2×C22⋊D13711yescharacteristic
339C2×C4C2×C22⋊D13711yescharacteristic
340C2×C4C2×C22⋊D13711yescharacteristic
341C2×C4C2×C22⋊D13711yescharacteristic
342C2×C4C23×D13711yescharacteristic
343C8—2741no—
344C8—2741no—
345C8—2741no—
346C8—2741no—
347C22—21no—
348C22—21no—
349C22—21no—
350C22—21no—
351C22—21no—
352C22—22no—
353C22—41no—
354C22—41no—
355C22—41no—
356C22—41no—
357C22—41no—
358C22—84no—
359C22—5481no—
360C22—5481no—
361C22—5481no—
362C22#1798410812yes—
363C22C22×C22⋊D13711yescharacteristic
364C4—21no—
365C4—21no—
366C4—21no—
367C4—21no—
368C4—41no—
369C4—41no—
370C4—5481no—
371C4—5481no—
372C4—5481no—
373C4C2×#1455550911yescharacteristic
374C4#1798413311yescharacteristic
375C2—22no—
376C2—41no—
377C2—42no—
378C2—81no—
379C2—10961no—
380C2#2257804711yescharacteristic
381C2C2×#1798410811yescharacteristic
382C2#2257820611yescharacteristic
subgroupG/H
centerC22C22×C22⋊D137
commutator, cosocleC2×C4×C137C24
FrattiniC2×C4C23×D137
Fitting#546×C137C2
socleC22×C137C22×D4
hypercenter#546D137
hypocenterC137#3297
LCSG ▷ C2×C4×C137 ▷ C2×C137 ▷ C137
UCSC1 ◁ C22 ◁ C4×D4 ◁ #546

Complements

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

NH ≅ G/Ntype
1. #22578099C2semidirect
2. #22577757C2semidirect
3. #22577736C2semidirect
4. #22577724C2semidirect
5. #22577703C2semidirect
6. #22577637C2semidirect
7. #22577634C2semidirect
8. #22577615C2semidirect
9. #22577548C2semidirect
10. #22577537C2semidirect
11. C4×#14555495C2semidirect
12. D4×#11764040C2semidirect
13. #22577399C2semidirect
14. C2×#17984009C2semidirect
15. #546×C137C2semidirect
16. C2×#14555497C22semidirect
20. C22×#11764040C22semidirect
22. #17984080C22semidirect
25. #17984062C22semidirect
27. C4×D548C22semidirect
29. #17984047C22semidirect
31. #17984021C22semidirect
32. #17984021C22semidirect
33. #17984019C22semidirect
34. C4⋊#11764040C22semidirect
35. #17984012C22semidirect
36. #17984012C22semidirect
37. #17984012C22semidirect
44. #17984008C22semidirect
46. C22⋊#11764040C22semidirect
47. C4×#11764040C22semidirect
191. C137#3297semidirect
194. #546D137semidirect

Maximal subgroups

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

subgroupindex
#225780992
#225777572
#225777362
#225777242
#225777032
#225776372
#225776342
#225776152
#225775482
#225775372
C4×#145554952
D4×#117640402
#225773992
C2×#179840092
#546×C1372
#3297137