Finite Group Databasebeta

← #31792816

Subgroups — #31792816

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#22578095C211yesmaximal, characteristic, complemented
2#22578037C211yesmaximal, characteristic
3#22577766C211yesmaximal, characteristic, complemented
4#22577729C211yesmaximal, characteristic, complemented
5#22577714C211yesmaximal, characteristic, complemented
6#22577702C211yesmaximal, characteristic, complemented
7#22577652C211yesmaximal, characteristic, complemented
8#22577601C211yesmaximal, characteristic, complemented
9#22577553C211yesmaximal, characteristic, complemented
10#22577552C211yesmaximal, characteristic, complemented
11#22577506C211yesmaximal, characteristic, complemented
12#22577474C211yesmaximal, characteristic, complemented
13#22577406C211yesmaximal, characteristic, complemented
14C2×#17984010C211yesmaximal, characteristic, complemented
15#465×C137C211yesmaximal, characteristic, complemented
16C22×D548C2211yescharacteristic
17C2×D4×D137—21no—
18C2×D4×D137C2211yescharacteristic
19C2×#14555497C2211yescharacteristic
20C2×#14555497C2211yescharacteristic, complemented
21C2×#14555493C2211yescharacteristic, complemented
22#17984084C2211yescharacteristic
23#17984083—21no—
24#17984079—21no—
25#17984079C2211yescharacteristic, complemented
26#17984072—21no—
27#17984064C2211yescharacteristic, complemented
28#17984063C2211yescharacteristic
29#17984061—21no—
30#17984058—21no—
31#17984056—21no—
32Q8×Dic137C2211yescharacteristic, complemented
33#17984050C2211yescharacteristic
34C4×C22⋊D137C2211yescharacteristic
35#17984046C2211yescharacteristic, complemented
36#17984044—21no—
37C2×C4⋊Dic137C2211yescharacteristic, complemented
38#17984021C2211yescharacteristic, complemented
39#17984021C2211yescharacteristic, complemented
40#17984017C2211yescharacteristic, complemented
41#17984010C2211yescharacteristic
42#17984010C2211yescharacteristic
43#17984010C2211yescharacteristic, complemented
44#17984010C2211yescharacteristic, complemented
45#17984009C2211yescharacteristic, complemented
46#17984009C2211yescharacteristic, complemented
47#17984008C2211yescharacteristic
48#17984008C2211yescharacteristic, complemented
49C22⋊#11764040C2211yescharacteristic, complemented
50#17984000C2211yescharacteristic, complemented
51#17983998C2211yescharacteristic, complemented
52C2×SD16×C137C2211yescharacteristic, complemented
53C22×C8×C137C2211yescharacteristic, complemented
54C22⋊Q8×C137C2211yescharacteristic, complemented
55#121×C137C2211yescharacteristic
56#106×C137C2211yescharacteristic, complemented
57#103×C137C2211yescharacteristic, complemented
58#102×C137C2211yescharacteristic, complemented
59C23×D137—21no—
60C23×D137—21no—
61C2×D548—22no—
62C2×D548—21no—
63C2×D548—21no—
64C2×D548—41no—
65C2×D548C2311yescharacteristic
66C2×D548C2311yescharacteristic
67C2×C22⋊D137—21no—
68C2×C22⋊D137—21no—
69C2×C22⋊D137—21no—
70C2×C22⋊D137D411yescharacteristic
71D4×D137—22no—
72D4×D137—41no—
73D4×D137—44no—
74C22×Dic137—21no—
75C2×C4×D137—21no—
76C2×C4×D137C2311yescharacteristic
77#14555497—22no—
78#14555497—41no—
79#14555493—41no—
80C2×#11764040C2311yescharacteristic
81C2×#11764040C2311yescharacteristic
82C4⋊Dic137—21no—
83C4⋊Dic137—21no—
84C4⋊Dic137—21no—
85C4⋊Dic137C2311yescharacteristic
86C4⋊Dic137C2311yescharacteristic
87#14555489—21no—
88#14555489—21no—
89#14555489D411yescharacteristic
90C22⋊Dic137D411yescharacteristic
91#14555487—21no—
92#14555487—21no—
93#14555487—21no—
94#14555487—21no—
95#14555487D411yescharacteristic
96C4×Dic137—21no—
97C4×Dic137—21no—
98C4×Dic137C2311yescharacteristic
99C2×Q8×C137C2311yescharacteristic
100C2×D4×C137—21no—
101C2×D4×C137C2311yescharacteristic
102C22×C4×C137C2311yescharacteristic
103SD16×C137—41no—
104C2×C8×C137—41no—
105C2×C8×C137C2311yescharacteristic
106C2×C8×C137C2311yescharacteristic
107C4⋊C4×C137—21no—
108C4⋊C4×C137C2311yescharacteristic
109C4⋊C4×C137C2311yescharacteristic
110C22⋊C4×C137—21no—
111C22⋊C4×C137—21no—
112C22×D137—21no—
113C22×D137—21no—
114C22×D137—42no—
115C22×D137—42no—
116C22×D137—42no—
117C22×D137—42no—
118C22×D137—81no—
119C22×D137—81no—
120C22×D137C2×D411yescharacteristic
121D548—21no—
122D548—21no—
123D548—21no—
124D548—42no—
125D548—42no—
126D548#4112yes—
127C22⋊D137—22no—
128C22⋊D137—42no—
129C22⋊D137—42no—
130C22⋊D137—81no—
131C2×Dic137—21no—
132C2×Dic137—21no—
133C2×Dic137—42no—
134C2×Dic137C2×D411yescharacteristic
135C2×Dic137C2×D411yescharacteristic
136C2×Dic137C2×D411yescharacteristic
137C4×D137—21no—
138C4×D137—21no—
139C4×D137—42no—
140#11764040—21no—
141#11764040—21no—
142C23×C137—21no—
143C23×C137C2×D411yescharacteristic
144Q8×C137—21no—
145Q8×C137—21no—
146D4×C137—21no—
147D4×C137—21no—
148D4×C137—42no—
149C2×C4×C137—21no—
150C2×C4×C137—21no—
151C2×C4×C137—21no—
152C2×C4×C137—21no—
153C2×C4×C137—21no—
154C2×C4×C137C2×D411yescharacteristic
155C2×C4×C137C2411yescharacteristic
156C8×C137—21no—
157C8×C137—21no—
158C22×C137—21no—
159C22×C137—21no—
160C22×C137—41no—
161C22×C137—41no—
162C22×C137—41no—
163C22×C137AGL(1,ℤ/8)12yes—
164C22×C137C22×D411yescharacteristic
165C2×D137—21no—
166C2×D137—22no—
167C2×D137—21no—
168C2×D137—21no—
169C2×D137—41no—
170C2×D137—41no—
171C2×D137—41no—
172C2×D137—41no—
173C2×D137—41no—
174C2×D137—84no—
175C2×D137—84no—
176C4×C137—21no—
177C4×C137—41no—
178C4×C137—41no—
179C4×C137—41no—
180C4×C137C2×#4111yescharacteristic
181C4×C137H(5,𝔽2)11yescharacteristic
182Dic137—21no—
183Dic137—21no—
184Dic137—21no—
185Dic137—41no—
186Dic137—41no—
187C2×C137—22no—
188C2×C137—81no—
189C2×C137#54611yescharacteristic
190C2×C137C2×AGL(1,ℤ/8)11yescharacteristic
191C2×C137#57711yescharacteristic
192D137—41no—
193D137—42no—
194D137—81no—
195C137#331011yescharacteristic, complemented, Sylow
196#3310—1371nomaximal, Sylow
197#552—1371no—
198D42—1371no—
199#490—1371no—
200#482—1371no—
201#480—1371no—
202#478—1371no—
203#468—1371no—
204#465D13711yescharacteristic, complemented
205#460—1371no—
206#450—1371no—
207#450—1371no—
208#440—1371no—
209#435—1371no—
210#418—1371no—
211C2×#102—1371no—
212C22×D4—2741no—
213C22×D4—1371no—
214C22×D4—1371no—
215C2×SD16—1371no—
216C2×SD16—1371no—
217C2×SD16C2×D13711yescharacteristic, complemented
218C2×M16—1371no—
219C22×C8C2×D13711yescharacteristic, complemented
220#127—1371no—
221#126—2741no—
222#125—2741no—
223#125—1371no—
224#123—2741no—
225C22⋊Q8—1371no—
226C22⋊Q8C2×D13711yescharacteristic, complemented
227#121—2741no—
228#121—1371no—
229#121C2×D13711yescharacteristic
230#120—2741no—
231#120—2741no—
232C4×Q8—1371no—
233C4×D4—1371no—
234C4×D4—1371no—
235#117—2741no—
236#117—1371no—
237C2×C4⋊C4—1371no—
238#107—1371no—
239#107—1371no—
240#106C2×D13711yescharacteristic, complemented
241C4⋊C8—1371no—
242#103C2×D13711yescharacteristic, complemented
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#102C2×D13711yescharacteristic, complemented
252C22⋊C8—1371no—
253C22⋊C8—1371no—
254#97—1371no—
255C24—2741no—
256C24—2741no—
257C2×Q8C22×D13711yescharacteristic
258C2×D4—21no—
259C2×D4—2742no—
260C2×D4—2741no—
261C2×D4—2742no—
262C2×D4—2741no—
263C2×D4—2741no—
264C2×D4—2741no—
265C2×D4—2741no—
266C2×D4—5481no—
267C2×D4—5481no—
268C2×D4—5484no—
269C2×D4—1371no—
270C2×D4—1371no—
271C2×D4—1371no—
272C2×D4C22×D13711yescharacteristic
273C22×C4—2741no—
274C22×C4—2741no—
275C22×C4—1371no—
276C22×C4C22×D13711yescharacteristic
277SD16—41no—
278SD16—2742no—
279SD16—5481no—
280M16—5481no—
281C2×C8—41no—
282C2×C8—1371no—
283C2×C8—1371no—
284C2×C8C22×D13711yescharacteristic
285C2×C8C22×D13711yescharacteristic
286C4⋊C4—21no—
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—2741no—
300C22⋊C4—2741no—
301C22⋊C4—2741no—
302C22⋊C4—2741no—
303C22⋊C4—1371no—
304C22⋊C4—1371no—
305C42—2741no—
306C42—2741no—
307C42—1371no—
308C23—21no—
309C23—2741no—
310C23—2741no—
311C23—5482no—
312C23—5482no—
313C23—5482no—
314C23—5482no—
315C23—10961no—
316C23—10961no—
317C23—1371no—
318C23D4×D13711yescharacteristic
319Q8—21no—
320Q8—21no—
321D4—21no—
322D4—21no—
323D4—42no—
324D4—2742no—
325D4—2741no—
326D4—2741no—
327D4—2741no—
328D4—5482no—
329D4—5482no—
330D4—5482no—
331D4—5482no—
332D4—10961no—
333D4—1372no—
334C2×C4—21no—
335C2×C4—21no—
336C2×C4—21no—
337C2×C4—21no—
338C2×C4—21no—
339C2×C4—2741no—
340C2×C4—2741no—
341C2×C4—2741no—
342C2×C4—2741no—
343C2×C4—5482no—
344C2×C4—5482no—
345C2×C4—1371no—
346C2×C4—1371no—
347C2×C4—1371no—
348C2×C4D4×D13711yescharacteristic
349C2×C4C23×D13711yescharacteristic
350C8—21no—
351C8—21no—
352C8—2741no—
353C8—2741no—
354C22—21no—
355C22—21no—
356C22—41no—
357C22—41no—
358C22—41no—
359C22—2741no—
360C22—2741no—
361C22—2741no—
362C22—2742no—
363C22—5481no—
364C22—5481no—
365C22—5481no—
366C22—5481no—
367C22—5481no—
368C22—10964no—
369C22—10964no—
370C22#1798411512yes—
371C22C2×D4×D13711yescharacteristic
372C4—21no—
373C4—41no—
374C4—41no—
375C4—41no—
376C4—2741no—
377C4—2741no—
378C4—2741no—
379C4—5481no—
380C4—5481no—
381C4#41×D13711yescharacteristic
382C4#1798413311yescharacteristic
383C2—22no—
384C2—81no—
385C2—5481no—
386C2—5482no—
387C2—10961no—
388C2#2257804211yescharacteristic
389C2C2×#1798411511yescharacteristic
390C2#2257821011yescharacteristic
subgroupG/H
centerC22C2×D4×D137
commutator, cosocleC2×C4×C137C24
FrattiniC2×C4C23×D137
Fitting#465×C137C2
socleC22×C137C22×D4
prime radical#22578037C2
hypercenter#465D137
hypocenterC137#3310
LCSG ▷ C2×C4×C137 ▷ C2×C137 ▷ C137
UCSC1 ◁ C22 ◁ C22×C4 ◁ #465

Complements

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

NH ≅ G/Ntype
1. #22578095C2semidirect
3. #22577766C2semidirect
4. #22577729C2semidirect
5. #22577714C2semidirect
6. #22577702C2semidirect
7. #22577652C2semidirect
8. #22577601C2semidirect
9. #22577553C2semidirect
10. #22577552C2semidirect
11. #22577506C2semidirect
12. #22577474C2semidirect
13. #22577406C2semidirect
14. C2×#17984010C2semidirect
15. #465×C137C2semidirect
20. C2×#14555497C22semidirect
21. C2×#14555493C22semidirect
25. #17984079C22semidirect
27. #17984064C22semidirect
32. Q8×Dic137C22semidirect
35. #17984046C22semidirect
37. C2×C4⋊Dic137C22semidirect
38. #17984021C22semidirect
39. #17984021C22semidirect
40. #17984017C22semidirect
43. #17984010C22semidirect
44. #17984010C22semidirect
45. #17984009C22semidirect
46. #17984009C22semidirect
48. #17984008C22semidirect
49. C22⋊#11764040C22semidirect
50. #17984000C22semidirect
51. #17983998C22semidirect
52. C2×SD16×C137C22semidirect
53. C22×C8×C137C22semidirect
54. C22⋊Q8×C137C22semidirect
56. #106×C137C22semidirect
57. #103×C137C22semidirect
58. #102×C137C22semidirect
195. C137#3310semidirect
204. #465D137semidirect
217. C2×SD16C2×D137semidirect
219. C22×C8C2×D137semidirect
226. C22⋊Q8C2×D137semidirect
240. #106C2×D137semidirect
242. #103C2×D137semidirect
251. #102C2×D137semidirect

Maximal subgroups

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

subgroupindex
#225780952
#225780372
#225777662
#225777292
#225777142
#225777022
#225776522
#225776012
#225775532
#225775522
#225775062
#225774742
#225774062
C2×#179840102
#465×C1372
#3310137