Finite Group Databasebeta

← #31793371

Subgroups — #31793371

Order 17536.

374 conjugacy classes of subgroups.

The subgroup lattice has 342 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#22578121C211yesmaximal, characteristic, complemented
2#22578053C211yesmaximal, characteristic
3#22577807C211yesmaximal, characteristic, complemented
4#22577740C211yesmaximal, characteristic, complemented
5#22577739C211yesmaximal, characteristic, complemented
6#22577725C211yesmaximal, characteristic, complemented
7#22577687C211yesmaximal, characteristic, complemented
8#22577681C211yesmaximal, characteristic, complemented
9#22577648C211yesmaximal, characteristic, complemented
10#22577647C211yesmaximal, characteristic, complemented
11#22577494C211yesmaximal, characteristic, complemented
12#22577478C211yesmaximal, characteristic, complemented
13#22577446C211yesmaximal, characteristic, complemented
14#22577442C211yesmaximal, characteristic, complemented
15#482×C137C211yesmaximal, characteristic, complemented
16C2×#14555509—21no—
17C2×#14555498C2211yescharacteristic
18C2×#14555494C2211yescharacteristic
19C2×#14555493C2211yescharacteristic
20#17984087C2211yescharacteristic
21#17984087C2211yescharacteristic
22#17984082—21no—
23#17984081—21no—
24#17984074—21no—
25#17984070—21no—
26#17984069—21no—
27#17984068C2211yescharacteristic
28#17984067C2211yescharacteristic
29#17984065—21no—
30#17984059C2211yescharacteristic
31C4×Dic274—21no—
32#17984053C2211yescharacteristic
33D4×Dic137C2211yescharacteristic
34C4×C22⋊D137C2211yescharacteristic
35#17984046C2211yescharacteristic
36C2×C4⋊Dic137C2211yescharacteristic
37C4⋊C4×D137C2211yescharacteristic
38#17984022C2211yescharacteristic
39#17984022C2211yescharacteristic
40#17984021C2211yescharacteristic
41#17984021C2211yescharacteristic
42#17984021C2211yescharacteristic
43#17984019C2211yescharacteristic
44#17984019C2211yescharacteristic
45#17984017C2211yescharacteristic
46#17984013C2211yescharacteristic
47#17984013C2211yescharacteristic
48#17984011C2211yescharacteristic
49#17984009C2211yescharacteristic
50#17984000C2211yescharacteristic
51C8×Dic137C2211yescharacteristic
52#121×C137C2211yescharacteristic
53#117×C137C2211yescharacteristic
54#107×C137C2211yescharacteristic
55#106×C137C2211yescharacteristic
56#102×C137C2211yescharacteristic
57#102×C137C2211yescharacteristic
58C22⋊C8×C137C2211yescharacteristic
59#14555509—44no—
60C2×Dic274—21no—
61C2×Dic274—21no—
62C2×Dic274C2311yescharacteristic
63C2×C22⋊D137—21no—
64C2×C22⋊D137D411yescharacteristic
65C22×Dic137—21no—
66C22×Dic137—21no—
67C2×C4×D137—21no—
68C2×C4×D137C2311yescharacteristic
69#14555498—22no—
70#14555494—22no—
71#14555493—22no—
72C2×#11764040C2311yescharacteristic
73C2×#11764040C2311yescharacteristic
74C4⋊Dic137—21no—
75C4⋊Dic137—21no—
76C4⋊Dic137C2311yescharacteristic
77C4⋊Dic137C2311yescharacteristic
78C4⋊Dic137C2311yescharacteristic
79#14555489—21no—
80#14555489—21no—
81#14555489—21no—
82#14555489—21no—
83#14555489—21no—
84#14555489D411yescharacteristic
85C22⋊Dic137—21no—
86C22⋊Dic137—21no—
87C22⋊Dic137—21no—
88C22⋊Dic137D411yescharacteristic
89#14555487—21no—
90#14555487—21no—
91#14555487D411yescharacteristic
92C4×Dic137—21no—
93C4×Dic137—21no—
94C4×Dic137C2311yescharacteristic
95C2×D4×C137—21no—
96C2×D4×C137C2311yescharacteristic
97C22×C4×C137C2311yescharacteristic
98C2×C8×C137C2311yescharacteristic
99C2×C8×C137C2311yescharacteristic
100C4⋊C4×C137C2311yescharacteristic
101C4⋊C4×C137C2311yescharacteristic
102C4⋊C4×C137C2311yescharacteristic
103C22⋊C4×C137—21no—
104C22⋊C4×C137—21no—
105C42×C137—21no—
106C22×D137C2×D411yescharacteristic
107Dic274—21no—
108Dic274—21no—
109Dic274—42no—
110Dic274—42no—
111C22⋊D137—22no—
112C22⋊D137—42no—
113C2×Dic137—21no—
114C2×Dic137—21no—
115C2×Dic137—21no—
116C2×Dic137—21no—
117C2×Dic137—42no—
118C2×Dic137—42no—
119C2×Dic137C2×D411yescharacteristic
120C2×Dic137C2×D411yescharacteristic
121C2×Dic137C2×D411yescharacteristic
122C4×D137—21no—
123C4×D137—21no—
124C4×D137—42no—
125#11764040—21no—
126#11764040#4112yescomplemented
127C23×C137—21no—
128C23×C137C2×D411yescharacteristic
129D4×C137—21no—
130D4×C137—21no—
131D4×C137—42no—
132C2×C4×C137—21no—
133C2×C4×C137—21no—
134C2×C4×C137—21no—
135C2×C4×C137—21no—
136C2×C4×C137—21no—
137C2×C4×C137C2×D411yescharacteristic
138C2×C4×C137C2411yescharacteristic
139C8×C137—21no—
140C8×C137—21no—
141C22×C137—21no—
142C22×C137—21no—
143C22×C137—21no—
144C22×C137—41no—
145C22×C137—41no—
146C22×C137—41no—
147C22×C137C22×D411yescharacteristic
148C2×D137—21no—
149C2×D137—21no—
150C2×D137—21no—
151C4×C137—21no—
152C4×C137—41no—
153C4×C137—41no—
154C4×C137—41no—
155C4×C137C2×#4111yescharacteristic
156C4×C1372−1+411yescharacteristic
157Dic137—21no—
158Dic137—21no—
159Dic137—41no—
160Dic137—41no—
161Dic137—41no—
162Dic137—41no—
163Dic137AGL(1,ℤ/8)12yes—
164C2×C137—41no—
165C2×C137—81no—
166C2×C137#54711yescharacteristic
167C2×C137C2×AGL(1,ℤ/8)11yescharacteristic
168C2×C137#57811yescharacteristic
169D137—41no—
170C137#335311yescharacteristic, complemented, Sylow
171#3353—1371nomaximal, Sylow
172#556—1371no—
173#547—1371no—
174#500—1371no—
175#484—1371no—
176#484—1371no—
177#482—1371no—
178#482D13711yescharacteristic, complemented
179#475—1371no—
180#474—1371no—
181#468—1371no—
182#468—1371no—
183C4×SD16—1371no—
184#436—1371no—
185#428—1371no—
186#428—1371no—
187C2×#41—2741no—
188C2×SD16—1371no—
189C2×M16—1371no—
190C2×M16—1371no—
191C4⋊Q8—1371no—
192C4⋊Q8—1371no—
193#126—2741no—
194#126—2741no—
195#124—2741no—
196#123—2741no—
197#123—2741no—
198C22⋊Q8—2741no—
199C22⋊Q8—1371no—
200C22⋊Q8—1371no—
201#121—1371no—
202#121C2×D13711yescharacteristic
203C4×Q8—2741no—
204C4×Q8—1371no—
205C4×D4—1371no—
206C4×D4—1371no—
207#117—1371no—
208#117C2×D13711yescharacteristic
209C2×C4⋊C4—1371no—
210C2×C4⋊C4—1371no—
211#107—1371no—
212#107—1371no—
213#107—1371no—
214#107—1371no—
215#107—1371no—
216#107C2×D13711yescharacteristic
217#106—1371no—
218#106—1371no—
219#106C2×D13711yescharacteristic
220C4⋊C8—1371no—
221#103—1371no—
222#103—1371no—
223#103—1371no—
224#102—1371no—
225#102C2×D13711yescharacteristic
226#102C2×D13711yescharacteristic
227C22⋊C8—1371no—
228C22⋊C8C2×D13711yescharacteristic
229C4×C8—1371no—
230#41—5484no—
231C2×Q8—2741no—
232C2×Q8—2741no—
233C2×Q8—1371no—
234C2×D4—21no—
235C2×D4—2741no—
236C2×D4—1371no—
237C2×D4C22×D13711yescharacteristic
238C22×C4—2741no—
239C22×C4—2741no—
240C22×C4—2741no—
241C22×C4—1371no—
242C22×C4C22×D13711yescharacteristic
243SD16—2742no—
244M16—2742no—
245M16—2742no—
246C2×C8—1371no—
247C2×C8—1371no—
248C2×C8C22×D13711yescharacteristic
249C2×C8C22×D13711yescharacteristic
250C4⋊C4—2741no—
251C4⋊C4—2741no—
252C4⋊C4—2741no—
253C4⋊C4—2741no—
254C4⋊C4—2741no—
255C4⋊C4—2741no—
256C4⋊C4—2741no—
257C4⋊C4—1371no—
258C4⋊C4—1371no—
259C4⋊C4—1371no—
260C4⋊C4—1371no—
261C4⋊C4C22×D13711yescharacteristic
262C4⋊C4C22×D13711yescharacteristic
263C4⋊C4C22×D13711yescharacteristic
264C22⋊C4—21no—
265C22⋊C4—21no—
266C22⋊C4—2741no—
267C22⋊C4—2741no—
268C22⋊C4—2741no—
269C22⋊C4—2741no—
270C22⋊C4—2741no—
271C22⋊C4—1371no—
272C22⋊C4—1371no—
273C42—21no—
274C42—2741no—
275C42—2741no—
276C42—1371no—
277C23—21no—
278C23—1371no—
279C23D4×D13711yescharacteristic
280Q8—2741no—
281Q8—2741no—
282Q8—5482no—
283Q8—5482no—
284D4—21no—
285D4—21no—
286D4—42no—
287D4—2742no—
288D4—5482no—
289C2×C4—21no—
290C2×C4—21no—
291C2×C4—21no—
292C2×C4—21no—
293C2×C4—21no—
294C2×C4—2741no—
295C2×C4—2741no—
296C2×C4—2741no—
297C2×C4—2741no—
298C2×C4—2741no—
299C2×C4—2741no—
300C2×C4—5482no—
301C2×C4—5482no—
302C2×C4—5482no—
303C2×C4—1371no—
304C2×C4—1371no—
305C2×C4—1371no—
306C2×C4D4×D13711yescharacteristic
307C2×C4C23×D13711yescharacteristic
308C8—21no—
309C8—21no—
310C8—2741no—
311C8—1372no—
312C22—21no—
313C22—21no—
314C22—21no—
315C22—41no—
316C22—41no—
317C22—41no—
318C22—2741no—
319C22—2741no—
320C22—2741no—
321C22C2×D4×D13711yescharacteristic
322C4—21no—
323C4—41no—
324C4—41no—
325C4—41no—
326C4—2741no—
327C4—2741no—
328C4—5481no—
329C4—5481no—
330C4—5481no—
331C4—5481no—
332C4—1372no—
333C4#41×D13711yescharacteristic
334C4#1798413511yescharacteristic
335C2—41no—
336C2—81no—
337C2—5481no—
338C2#2257805111yescharacteristic
339C2AGL(1,ℤ/8)×D13711yescharacteristic
340C2#2257821711yescharacteristic
subgroupG/H
centerC22C2×D4×D137
commutator, cosocleC2×C4×C137C24
FrattiniC2×C4C23×D137
Fitting#482×C137C2
socleC22×C137C22×D4
prime radical#22578053C2
hypercenter#482D137
hypocenterC137#3353
LCSG ▷ C2×C4×C137 ▷ C2×C137 ▷ C137
UCSC1 ◁ C22 ◁ C22×C4 ◁ #482

Complements

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

NH ≅ G/Ntype
1. #22578121C2semidirect
3. #22577807C2semidirect
4. #22577740C2semidirect
5. #22577739C2semidirect
6. #22577725C2semidirect
7. #22577687C2semidirect
8. #22577681C2semidirect
9. #22577648C2semidirect
10. #22577647C2semidirect
11. #22577494C2semidirect
12. #22577478C2semidirect
13. #22577446C2semidirect
14. #22577442C2semidirect
15. #482×C137C2semidirect
126. #11764040#41semidirect
170. C137#3353semidirect
178. #482D137semidirect

Maximal subgroups

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

subgroupindex
#225781212
#225780532
#225778072
#225777402
#225777392
#225777252
#225776872
#225776812
#225776482
#225776472
#225774942
#225774782
#225774462
#225774422
#482×C1372
#3353137