Finite Group Databasebeta

← #31792628

Subgroups — #31792628

Order 17536.

408 conjugacy classes of subgroups.

The subgroup lattice has 372 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#22578095C211yesmaximal, characteristic, complemented
2#22578047C211yesmaximal, characteristic
3#22577756C211yesmaximal, characteristic, complemented
4#22577734C211yesmaximal, characteristic, complemented
5#22577726C211yesmaximal, characteristic, complemented
6#22577705C211yesmaximal, characteristic, complemented
7#22577642C211yesmaximal, characteristic, complemented
8#22577629C211yesmaximal, characteristic, complemented
9#22577623C211yesmaximal, characteristic, complemented
10#22577562C211yesmaximal, characteristic, complemented
11SD16×Dic137C211yesmaximal, characteristic, complemented
12#22577466C211yesmaximal, characteristic, complemented
13#22577396C211yesmaximal, characteristic, complemented
14C2×#17984009C211yesmaximal, characteristic, complemented
15#450×C137C211yesmaximal, characteristic, complemented
16C2×#14555509C2211yescharacteristic
17C2×#14555500C2211yescharacteristic, complemented
18C22×#11764040C2211yescharacteristic
19C2×C8×D137C2211yescharacteristic, complemented
20#17984083—21no—
21#17984079—21no—
22#17984079C2211yescharacteristic, complemented
23#17984075C2211yescharacteristic
24#17984072—21no—
25#17984069—21no—
26#17984067C2211yescharacteristic
27#17984064C2211yescharacteristic, complemented
28#17984060—21no—
29#17984059C2211yescharacteristic
30#17984057—21no—
31Q8×Dic137C2211yescharacteristic, complemented
32D4×Dic137C2211yescharacteristic
33C4×C22⋊D137C2211yescharacteristic
34#17984046C2211yescharacteristic, complemented
35#17984044—21no—
36C2×C4⋊Dic137C2211yescharacteristic
37C2×C22⋊Dic137—21no—
38#17984021C2211yescharacteristic, complemented
39#17984020C2211yescharacteristic, complemented
40#17984020C2211yescharacteristic, complemented
41#17984017C2211yescharacteristic, complemented
42#17984013C2211yescharacteristic, complemented
43#17984013C2211yescharacteristic, complemented
44#17984012C2211yescharacteristic, complemented
45#17984011C2211yescharacteristic, complemented
46#17984009C2211yescharacteristic
47#17984009C2211yescharacteristic
48#17984009C2211yescharacteristic
49#17984009C2211yescharacteristic
50#17984000C2211yescharacteristic, complemented
51C8×Dic137C2211yescharacteristic, complemented
52C22×D4×C137C2211yescharacteristic
53C2×SD16×C137C2211yescharacteristic
54C2×SD16×C137C2211yescharacteristic
55C22⋊Q8×C137C2211yescharacteristic
56#102×C137C2211yescharacteristic
57#102×C137C2211yescharacteristic
58C22⋊C8×C137C2211yescharacteristic
59#14555509—22no—
60#14555509—41no—
61C2×Dic274C2311yescharacteristic
62C2×C22⋊D137—21no—
63C2×C22⋊D137—21no—
64C2×C22⋊D137D411yescharacteristic
65C22×Dic137—21no—
66C22×Dic137—21no—
67C22×Dic137—21no—
68C2×C4×D137C2311yescharacteristic
69#14555500—41no—
70C2×#11764040—41no—
71C2×#11764040C2311yescharacteristic
72C2×#11764040C2311yescharacteristic
73C8×D137—41no—
74C4⋊Dic137—21no—
75C4⋊Dic137—21no—
76C4⋊Dic137—21no—
77C4⋊Dic137C2311yescharacteristic
78C4⋊Dic137C2311yescharacteristic
79C4⋊Dic137C2311yescharacteristic
80#14555489—21no—
81#14555489—21no—
82#14555489—21no—
83#14555489D411yescharacteristic
84C22⋊Dic137—21no—
85C22⋊Dic137—21no—
86C22⋊Dic137—21no—
87C22⋊Dic137—21no—
88C22⋊Dic137—21no—
89C22⋊Dic137D411yescharacteristic
90#14555487—21no—
91#14555487—21no—
92#14555487D411yescharacteristic
93C4×Dic137—21no—
94C4×Dic137—21no—
95C4×Dic137C2311yescharacteristic
96C24×C137—21no—
97C2×Q8×C137C2311yescharacteristic
98C2×D4×C137—22no—
99C2×D4×C137—21no—
100C2×D4×C137—41no—
101C2×D4×C137C2311yescharacteristic
102C2×D4×C137C2311yescharacteristic
103C22×C4×C137C2311yescharacteristic
104SD16×C137—22no—
105SD16×C137—41no—
106C2×C8×C137C2311yescharacteristic
107C2×C8×C137C2311yescharacteristic
108C4⋊C4×C137—21no—
109C4⋊C4×C137C2311yescharacteristic
110C22⋊C4×C137—21no—
111C22×D137C2×D411yescharacteristic
112Dic274—21no—
113Dic274—21no—
114C22⋊D137—22no—
115C22⋊D137—42no—
116C22⋊D137—81no—
117C2×Dic137—21no—
118C2×Dic137—21no—
119C2×Dic137—21no—
120C2×Dic137—21no—
121C2×Dic137—42no—
122C2×Dic137—42no—
123C2×Dic137—81no—
124C2×Dic137C2×D411yescharacteristic
125C2×Dic137C2×D411yescharacteristic
126C2×Dic137C2×D411yescharacteristic
127C4×D137—21no—
128C4×D137—21no—
129#11764040—21no—
130#11764040—21no—
131C23×C137—21no—
132C23×C137—21no—
133C23×C137—42no—
134C23×C137—42no—
135C23×C137—81no—
136C23×C137C2×D411yescharacteristic
137Q8×C137—21no—
138Q8×C137—21no—
139D4×C137—21no—
140D4×C137—21no—
141D4×C137—21no—
142D4×C137—42no—
143D4×C137#4112yes—
144C2×C4×C137—21no—
145C2×C4×C137—21no—
146C2×C4×C137—21no—
147C2×C4×C137—21no—
148C2×C4×C137C2×D411yescharacteristic
149C2×C4×C137C2411yescharacteristic
150C8×C137—21no—
151C8×C137—21no—
152C22×C137—21no—
153C22×C137—21no—
154C22×C137—22no—
155C22×C137—41no—
156C22×C137—41no—
157C22×C137—41no—
158C22×C137—41no—
159C22×C137—41no—
160C22×C137—84no—
161C22×C137#13512yes—
162C22×C137C22×D411yescharacteristic
163C2×D137—21no—
164C2×D137—21no—
165C2×D137—21no—
166C4×C137—21no—
167C4×C137—41no—
168C4×C137—41no—
169C4×C137C2×#4111yescharacteristic
170C4×C137H(5,𝔽2)11yescharacteristic
171Dic137—21no—
172Dic137—21no—
173Dic137—21no—
174Dic137—41no—
175Dic137—41no—
176Dic137—41no—
177Dic137—41no—
178C2×C137—22no—
179C2×C137—42no—
180C2×C137—81no—
181C2×C137#54611yescharacteristic
182C2×C137C2×#13511yescharacteristic
183C2×C137#57711yescharacteristic
184D137—41no—
185C137#329611yescharacteristic, complemented, Sylow
186#3296—1371nomaximal, Sylow
187#552—1371no—
188#546—1371no—
189#488—1371no—
190#483—1371no—
191#482—1371no—
192#478—1371no—
193#467—1371no—
194#465—1371no—
195#464—1371no—
196#452—1371no—
197#450D13711yescharacteristic, complemented
198C4×SD16—1371no—
199C8×D4—1371no—
200#416—1371no—
201C2×#102—1371no—
202C2×#41—1371no—
203C22×D4C2×D13711yescharacteristic
204C2×Q16—1371no—
205C2×SD16C2×D13711yescharacteristic
206C2×SD16C2×D13711yescharacteristic
207C22×C8—1371no—
208C22×C8—1371no—
209#126—2741no—
210#125—2741no—
211#125—1371no—
212#124—1371no—
213#123—2741no—
214#123—2741no—
215C22⋊Q8—1371no—
216C22⋊Q8—1371no—
217C22⋊Q8C2×D13711yescharacteristic
218#121—2741no—
219#121—1371no—
220#120—2741no—
221C4×Q8—1371no—
222C4×D4—1371no—
223C4×D4—1371no—
224#117—2741no—
225#117—1371no—
226C2×C4⋊C4—1371no—
227C2×C22⋊C4—2741no—
228#107—1371no—
229#106—1371no—
230#106—1371no—
231C4⋊C8—1371no—
232#103—1371no—
233#103—1371no—
234#103—1371no—
235#103—1371no—
236#102—1371no—
237#102—1371no—
238#102—1371no—
239#102—1371no—
240#102C2×D13711yescharacteristic
241#102C2×D13711yescharacteristic
242C22⋊C8—1371no—
243C22⋊C8C2×D13711yescharacteristic
244C4×C8—1371no—
245C24—21no—
246#41—2742no—
247#41—5481no—
248C2×Q8—1371no—
249C2×Q8C22×D13711yescharacteristic
250C2×D4—22no—
251C2×D4—21no—
252C2×D4—41no—
253C2×D4—2741no—
254C2×D4—2741no—
255C2×D4—1371no—
256C2×D4C22×D13711yescharacteristic
257C2×D4C22×D13711yescharacteristic
258C22×C4—2741no—
259C22×C4—2741no—
260C22×C4—2741no—
261C22×C4—1371no—
262C22×C4C22×D13711yescharacteristic
263Q16—5481no—
264SD16—22no—
265SD16—41no—
266C2×C8—5481no—
267C2×C8—5481no—
268C2×C8—1371no—
269C2×C8—1371no—
270C2×C8C22×D13711yescharacteristic
271C2×C8C22×D13711yescharacteristic
272C4⋊C4—21no—
273C4⋊C4—2741no—
274C4⋊C4—2741no—
275C4⋊C4—2741no—
276C4⋊C4—2741no—
277C4⋊C4—2741no—
278C4⋊C4—2741no—
279C4⋊C4—1371no—
280C4⋊C4—1371no—
281C4⋊C4—1371no—
282C4⋊C4—1371no—
283C4⋊C4C22×D13711yescharacteristic
284C22⋊C4—21no—
285C22⋊C4—2741no—
286C22⋊C4—2741no—
287C22⋊C4—2741no—
288C22⋊C4—2741no—
289C22⋊C4—2741no—
290C22⋊C4—2741no—
291C22⋊C4—2741no—
292C22⋊C4—1371no—
293C22⋊C4—1371no—
294C42—2741no—
295C42—2741no—
296C42—1371no—
297C23—21no—
298C23—21no—
299C23—42no—
300C23—42no—
301C23—81no—
302C23—1371no—
303C23D4×D13711yescharacteristic
304Q8—21no—
305Q8—21no—
306Q8—2741no—
307Q8—2741no—
308D4—21no—
309D4—21no—
310D4—21no—
311D4—42no—
312D4—2742no—
313D4—5482no—
314D4—10961no—
315D4#1455550912yes—
316C2×C4—21no—
317C2×C4—21no—
318C2×C4—21no—
319C2×C4—21no—
320C2×C4—2741no—
321C2×C4—2741no—
322C2×C4—2741no—
323C2×C4—2741no—
324C2×C4—2741no—
325C2×C4—2741no—
326C2×C4—5482no—
327C2×C4—5482no—
328C2×C4—10961no—
329C2×C4—1371no—
330C2×C4—1371no—
331C2×C4—1371no—
332C2×C4D4×D13711yescharacteristic
333C2×C4C23×D13711yescharacteristic
334C8—21no—
335C8—21no—
336C8—2741no—
337C8—2741no—
338C22—21no—
339C22—21no—
340C22—22no—
341C22—41no—
342C22—41no—
343C22—41no—
344C22—41no—
345C22—41no—
346C22—84no—
347C22—2741no—
348C22—2741no—
349C22—2741no—
350C22#1798410912yes—
351C22C2×D4×D13711yescharacteristic
352C4—21no—
353C4—41no—
354C4—41no—
355C4—2741no—
356C4—2741no—
357C4—2741no—
358C4—5481no—
359C4—5481no—
360C4—5481no—
361C4—5481no—
362C4C2×#1455550911yescharacteristic
363C4#1798413311yescharacteristic
364C2—22no—
365C2—42no—
366C2—81no—
367C2—5481no—
368C2#2257804011yescharacteristic
369C2C2×#1798410911yescharacteristic
370C2#2257821311yescharacteristic
subgroupG/H
centerC22C2×D4×D137
commutator, cosocleC2×C4×C137C24
FrattiniC2×C4C23×D137
Fitting#450×C137C2
socleC22×C137C22×D4
prime radical#22578047C2
hypercenter#450D137
hypocenterC137#3296
LCSG ▷ C2×C4×C137 ▷ C2×C137 ▷ C137
UCSC1 ◁ C22 ◁ C22×C4 ◁ #450

Complements

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

NH ≅ G/Ntype
1. #22578095C2semidirect
3. #22577756C2semidirect
4. #22577734C2semidirect
5. #22577726C2semidirect
6. #22577705C2semidirect
7. #22577642C2semidirect
8. #22577629C2semidirect
9. #22577623C2semidirect
10. #22577562C2semidirect
11. SD16×Dic137C2semidirect
12. #22577466C2semidirect
13. #22577396C2semidirect
14. C2×#17984009C2semidirect
15. #450×C137C2semidirect
17. C2×#14555500C22semidirect
19. C2×C8×D137C22semidirect
22. #17984079C22semidirect
27. #17984064C22semidirect
31. Q8×Dic137C22semidirect
34. #17984046C22semidirect
38. #17984021C22semidirect
39. #17984020C22semidirect
40. #17984020C22semidirect
41. #17984017C22semidirect
42. #17984013C22semidirect
43. #17984013C22semidirect
44. #17984012C22semidirect
45. #17984011C22semidirect
50. #17984000C22semidirect
51. C8×Dic137C22semidirect
185. C137#3296semidirect
197. #450D137semidirect

Maximal subgroups

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

subgroupindex
#225780952
#225780472
#225777562
#225777342
#225777262
#225777052
#225776422
#225776292
#225776232
#225775622
SD16×Dic1372
#225774662
#225773962
C2×#179840092
#450×C1372
#3296137