Finite Group Databasebeta

← #31792810

Subgroups — #31792810

Order 17536.

450 conjugacy classes of subgroups.

The subgroup lattice has 392 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#22578093C211yesmaximal, characteristic, complemented
2#22577767C211yesmaximal, characteristic, complemented
3#22577724C211yesmaximal, characteristic, complemented
4#22577714C211yesmaximal, characteristic, complemented
5#22577703C211yesmaximal, characteristic, complemented
6#22577647C211yesmaximal, characteristic, complemented
7#22577630C211yesmaximal, characteristic, complemented
8#22577600C211yesmaximal, characteristic, complemented
9#22577554C211yesmaximal, characteristic, complemented
10#22577554C211yesmaximal, characteristic, complemented
11#22577503C211yesmaximal, characteristic, complemented
12#22577475C211yesmaximal, characteristic, complemented
13#22577408C211yesmaximal, characteristic, complemented
14C2×#17984009C211yesmaximal, characteristic, complemented
15D42×C137C211yesmaximal, characteristic
16C2×#14555498C2211yescharacteristic
17C2×#14555498C2211yescharacteristic, complemented
18C2×#14555498C2211yescharacteristic, complemented
19C2×#14555494C2211yescharacteristic, complemented
20C22×#11764040C2211yescharacteristic, complemented
21#17984081—21no—
22#17984080C2211yescharacteristic, complemented
23#17984078—21no—
24#17984069—21no—
25#17984067C2211yescharacteristic, complemented
26#17984067C2211yescharacteristic, complemented
27C4×Dic274C2211yescharacteristic, complemented
28#17984047C2211yescharacteristic, complemented
29#17984044—21no—
30C2×C4⋊Dic137C2211yescharacteristic, complemented
31#17984021C2211yescharacteristic, complemented
32#17984021C2211yescharacteristic, complemented
33#17984019C2211yescharacteristic, complemented
34C4⋊#11764040C2211yescharacteristic, complemented
35#17984011C2211yescharacteristic, complemented
36#17984009C2211yescharacteristic
37#17984009C2211yescharacteristic
38#17984009C2211yescharacteristic
39#17984009C2211yescharacteristic, complemented
40#17984009C2211yescharacteristic, complemented
41#17984009C2211yescharacteristic, complemented
42#17984009C2211yescharacteristic, complemented
43#17984009C2211yescharacteristic, complemented
44#17984009C2211yescharacteristic, complemented
45C22⋊#11764040C2211yescharacteristic, complemented
46C22⋊#11764040C2211yescharacteristic, complemented
47#17983999C2211yescharacteristic, complemented
48C22×D4×C137—21no—
49C22×D4×C137C2211yescharacteristic
50C22×D4×C137C2211yescharacteristic
51#127×C137C2211yescharacteristic
52#121×C137—21no—
53#121×C137C2211yescharacteristic
54#121×C137C2211yescharacteristic
55#120×C137—21no—
56#120×C137—21no—
57C4×D4×C137C2211yescharacteristic
58C4×D4×C137C2211yescharacteristic
59C2×Dic274C2311yescharacteristic
60C22×Dic137—21no—
61#14555498—22no—
62#14555498—41no—
63#14555498—41no—
64#14555494—41no—
65C2×#11764040—41no—
66C2×#11764040C2311yescharacteristic
67C2×#11764040C2311yescharacteristic
68C2×#11764040C2311yescharacteristic
69C2×#11764040C2311yescharacteristic
70C4⋊Dic137—21no—
71C4⋊Dic137C2311yescharacteristic
72C4⋊Dic137C2311yescharacteristic
73C4⋊Dic137C2311yescharacteristic
74#14555489—21no—
75#14555489—21no—
76#14555489—21no—
77#14555489—21no—
78#14555489—21no—
79C22⋊Dic137—21no—
80C22⋊Dic137—21no—
81C22⋊Dic137—21no—
82C4×Dic137—21no—
83C4×Dic137—21no—
84C24×C137—21no—
85C24×C137—21no—
86C2×D4×C137—22no—
87C2×D4×C137—22no—
88C2×D4×C137—21no—
89C2×D4×C137—21no—
90C2×D4×C137—21no—
91C2×D4×C137—21no—
92C2×D4×C137—21no—
93C2×D4×C137—21no—
94C2×D4×C137—41no—
95C2×D4×C137—41no—
96C2×D4×C137—44no—
97C2×D4×C137D411yescharacteristic
98C2×D4×C137C2311yescharacteristic
99C2×D4×C137C2311yescharacteristic
100C2×D4×C137C2311yescharacteristic
101C22×C4×C137—21no—
102C22×C4×C137C2311yescharacteristic
103C22×C4×C137C2311yescharacteristic
104C4⋊C4×C137D411yescharacteristic
105C4⋊C4×C137C2311yescharacteristic
106C22⋊C4×C137—21no—
107C22⋊C4×C137—21no—
108C22⋊C4×C137—21no—
109C22⋊C4×C137D411yescharacteristic
110C22⋊C4×C137D411yescharacteristic
111C42×C137C2311yescharacteristic
112Dic274—21no—
113Dic274—21no—
114C2×Dic137—21no—
115C2×Dic137—21no—
116C2×Dic137—21no—
117C2×Dic137—21no—
118C2×Dic137—42no—
119#11764040—21no—
120#11764040—21no—
121#11764040—21no—
122#11764040—21no—
123C23×C137—21no—
124C23×C137—21no—
125C23×C137—21no—
126C23×C137—42no—
127C23×C137—42no—
128C23×C137—42no—
129C23×C137—42no—
130C23×C137—81no—
131C23×C137—81no—
132C23×C137C2×D411yescharacteristic
133C23×C137C2×D411yescharacteristic
134D4×C137—22no—
135D4×C137—21no—
136D4×C137—21no—
137D4×C137—21no—
138D4×C137—21no—
139D4×C137—21no—
140D4×C137—42no—
141D4×C137—42no—
142D4×C137—42no—
143D4×C137—42no—
144D4×C137—42no—
145D4×C137—81no—
146D4×C137#4112yes—
147C2×C4×C137—21no—
148C2×C4×C137—21no—
149C2×C4×C137—21no—
150C2×C4×C137—21no—
151C2×C4×C137—21no—
152C2×C4×C137—42no—
153C2×C4×C137C2×D411yescharacteristic
154C2×C4×C137C2×D411yescharacteristic
155C2×C4×C137C2×D411yescharacteristic
156C2×C4×C137C2×D411yescharacteristic
157C2×C4×C137C2411yescharacteristic
158C22×C137—21no—
159C22×C137—21no—
160C22×C137—21no—
161C22×C137—22no—
162C22×C137—21no—
163C22×C137—21no—
164C22×C137—41no—
165C22×C137—41no—
166C22×C137—41no—
167C22×C137—41no—
168C22×C137—41no—
169C22×C137—41no—
170C22×C137—41no—
171C22×C137—41no—
172C22×C137—84no—
173C22×C137—84no—
174C22×C137AGL(1,ℤ/8)12yes—
175C22×C137C22×D411yescharacteristic
176C4×C137—21no—
177C4×C137—21no—
178C4×C137—21no—
179C4×C137—21no—
180C4×C137—41no—
181C4×C137C2×#4111yescharacteristic
182C4×C137H(5,𝔽2)11yescharacteristic
183Dic137—41no—
184Dic137—41no—
185Dic137—41no—
186Dic137—41no—
187C2×C137—22no—
188C2×C137—41no—
189C2×C137—42no—
190C2×C137—81no—
191C2×C137—81no—
192C2×C137#54611yescharacteristic
193C2×C137C2×AGL(1,ℤ/8)11yescharacteristic
194C2×C137#57711yescharacteristic
195C137#331011yescharacteristic, complemented, Sylow
196#3310—1371nomaximal, Sylow
197#552—1371no—
198D42D13711yescharacteristic
199#490—1371no—
200#482—1371no—
201#480—1371no—
202#478—1371no—
203#468—1371no—
204#465—1371no—
205#460—1371no—
206#450—1371no—
207#450—1371no—
208#440—1371no—
209#435—1371no—
210#418—1371no—
211C2×#102—1371no—
212C22×D4—21no—
213C22×D4C2×D13711yescharacteristic
214C22×D4C2×D13711yescharacteristic
215C2×SD16—1371no—
216C2×SD16—1371no—
217C2×SD16—1371no—
218C2×M16—1371no—
219C22×C8—1371no—
220#127C2×D13711yescharacteristic
221#126—2741no—
222#125—2741no—
223#125—1371no—
224#123—2741no—
225C22⋊Q8—1371no—
226C22⋊Q8—1371no—
227#121—21no—
228#121C2×D13711yescharacteristic
229#121C2×D13711yescharacteristic
230#120—21no—
231#120—21no—
232C4×Q8—1371no—
233C4×D4C2×D13711yescharacteristic
234C4×D4C2×D13711yescharacteristic
235#117—2741no—
236#117—1371no—
237C2×C4⋊C4—1371no—
238#107—1371no—
239#107—1371no—
240#106—1371no—
241C4⋊C8—1371no—
242#103—1371no—
243#102—1371no—
244#102—1371no—
245#102—1371no—
246#102—1371no—
247#102—1371no—
248#102—1371no—
249#102—1371no—
250#102—1371no—
251#102—1371no—
252C22⋊C8—1371no—
253C22⋊C8—1371no—
254#97—1371no—
255C24—21no—
256C24—21no—
257C2×Q8—1371no—
258C2×D4—22no—
259C2×D4—22no—
260C2×D4—21no—
261C2×D4—21no—
262C2×D4—21no—
263C2×D4—21no—
264C2×D4—21no—
265C2×D4—21no—
266C2×D4—41no—
267C2×D4—41no—
268C2×D4—44no—
269C2×D4C22⋊D13711yescharacteristic
270C2×D4C22×D13711yescharacteristic
271C2×D4C22×D13711yescharacteristic
272C2×D4C22×D13711yescharacteristic
273C22×C4—21no—
274C22×C4—2741no—
275C22×C4C22×D13711yescharacteristic
276C22×C4C22×D13711yescharacteristic
277SD16—2742no—
278SD16—5481no—
279SD16—5481no—
280M16—5481no—
281C2×C8—5481no—
282C2×C8—1371no—
283C2×C8—1371no—
284C2×C8—1371no—
285C2×C8—1371no—
286C4⋊C4—2741no—
287C4⋊C4—2741no—
288C4⋊C4—2741no—
289C4⋊C4—2741no—
290C4⋊C4—2741no—
291C4⋊C4—2741no—
292C4⋊C4—1371no—
293C4⋊C4—1371no—
294C4⋊C4—1371no—
295C4⋊C4C22⋊D13711yescharacteristic
296C4⋊C4C22×D13711yescharacteristic
297C22⋊C4—21no—
298C22⋊C4—21no—
299C22⋊C4—21no—
300C22⋊C4—2741no—
301C22⋊C4—2741no—
302C22⋊C4—2741no—
303C22⋊C4C22⋊D13711yescharacteristic
304C22⋊C4C22⋊D13711yescharacteristic
305C42—2741no—
306C42—2741no—
307C42C22×D13711yescharacteristic
308C23—21no—
309C23—21no—
310C23—21no—
311C23—42no—
312C23—42no—
313C23—42no—
314C23—42no—
315C23—81no—
316C23—81no—
317C23C2×C22⋊D13711yescharacteristic
318C23C2×C22⋊D13711yescharacteristic
319Q8—2741no—
320Q8—2741no—
321D4—22no—
322D4—21no—
323D4—21no—
324D4—21no—
325D4—21no—
326D4—21no—
327D4—42no—
328D4—42no—
329D4—42no—
330D4—42no—
331D4—42no—
332D4—81no—
333D4#1455550912yes—
334C2×C4—21no—
335C2×C4—21no—
336C2×C4—21no—
337C2×C4—21no—
338C2×C4—21no—
339C2×C4—42no—
340C2×C4—2741no—
341C2×C4—2741no—
342C2×C4—2741no—
343C2×C4—2741no—
344C2×C4—5482no—
345C2×C4C2×C22⋊D13711yescharacteristic
346C2×C4C2×C22⋊D13711yescharacteristic
347C2×C4C2×C22⋊D13711yescharacteristic
348C2×C4C2×C22⋊D13711yescharacteristic
349C2×C4C23×D13711yescharacteristic
350C8—2741no—
351C8—2741no—
352C8—2741no—
353C8—2741no—
354C22—21no—
355C22—21no—
356C22—21no—
357C22—21no—
358C22—21no—
359C22—22no—
360C22—41no—
361C22—41no—
362C22—41no—
363C22—41no—
364C22—41no—
365C22—41no—
366C22—41no—
367C22—41no—
368C22—84no—
369C22—84no—
370C22#1798411412yes—
371C22C22×C22⋊D13711yescharacteristic
372C4—21no—
373C4—21no—
374C4—21no—
375C4—21no—
376C4—41no—
377C4—5481no—
378C4—5481no—
379C4—5481no—
380C4—5481no—
381C4C2×#1455550911yescharacteristic
382C4#1798413311yescharacteristic
383C2—22no—
384C2—41no—
385C2—42no—
386C2—81no—
387C2—81no—
388C2#2257804711yescharacteristic
389C2C2×#1798411411yescharacteristic
390C2#2257821411yescharacteristic
subgroupG/H
centerC22C22×C22⋊D137
commutator, cosocleC2×C4×C137C24
FrattiniC2×C4C23×D137
Fitting, prime radicalD42×C137C2
socleC22×C137C22×D4
hypercenterD42D137
hypocenterC137#3310
LCSG ▷ C2×C4×C137 ▷ C2×C137 ▷ C137
UCSC1 ◁ C22 ◁ C4×D4 ◁ D42

Complements

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

NH ≅ G/Ntype
1. #22578093C2semidirect
2. #22577767C2semidirect
3. #22577724C2semidirect
4. #22577714C2semidirect
5. #22577703C2semidirect
6. #22577647C2semidirect
7. #22577630C2semidirect
8. #22577600C2semidirect
9. #22577554C2semidirect
10. #22577554C2semidirect
11. #22577503C2semidirect
12. #22577475C2semidirect
13. #22577408C2semidirect
14. C2×#17984009C2semidirect
17. C2×#14555498C22semidirect
18. C2×#14555498C22semidirect
19. C2×#14555494C22semidirect
20. C22×#11764040C22semidirect
22. #17984080C22semidirect
25. #17984067C22semidirect
26. #17984067C22semidirect
27. C4×Dic274C22semidirect
28. #17984047C22semidirect
30. C2×C4⋊Dic137C22semidirect
31. #17984021C22semidirect
32. #17984021C22semidirect
33. #17984019C22semidirect
34. C4⋊#11764040C22semidirect
35. #17984011C22semidirect
39. #17984009C22semidirect
40. #17984009C22semidirect
41. #17984009C22semidirect
42. #17984009C22semidirect
43. #17984009C22semidirect
44. #17984009C22semidirect
45. C22⋊#11764040C22semidirect
46. C22⋊#11764040C22semidirect
47. #17983999C22semidirect
195. C137#3310semidirect

Maximal subgroups

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

subgroupindex
#225780932
#225777672
#225777242
#225777142
#225777032
#225776472
#225776302
#225776002
#225775542
#225775032
#225774752
#225774082
C2×#179840092
D42×C1372
#3310137