Finite Group Databasebeta

← #31790751

Subgroups — #31790751

Order 17536.

386 conjugacy classes of subgroups.

The subgroup lattice has 352 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#22577976C211yesmaximal, characteristic
2#22577884C211yesmaximal, characteristic
3#22577747C211yesmaximal, characteristic, complemented
4#22577745C211yesmaximal, characteristic, complemented
5#22577742C211yesmaximal, characteristic, complemented
6#22577741C211yesmaximal, characteristic, complemented
7#22577738C211yesmaximal, characteristic, complemented
8#22577737C211yesmaximal, characteristic, complemented
9#22577735C211yesmaximal, characteristic, complemented
10#22577442C211yesmaximal, characteristic, complemented
11#22577438C211yesmaximal, characteristic, complemented
12#22577410C211yesmaximal, characteristic, complemented
13#103×D137C211yesmaximal, characteristic, complemented
14#22577368C211yesmaximal, characteristic
15#483×C137C211yesmaximal, characteristic, complemented
16C2×#14555510C2211yescharacteristic
17C2×Q8×D137C2211yescharacteristic
18C2×#14555493C2211yescharacteristic
19C2×C8×D137C2211yescharacteristic
20#17984087—21no—
21#17984074—21no—
22#17984073—21no—
23#17984070—21no—
24#17984068C2211yescharacteristic
25#17984067C2211yescharacteristic
26#17984064C2211yescharacteristic
27#17984053—21no—
28#17984050—21no—
29C4×C22⋊D137—21no—
30C4×C22⋊D137—21no—
31#17984047C2211yescharacteristic
32#17984046C2211yescharacteristic
33#17984046C2211yescharacteristic
34#17984046C2211yescharacteristic
35C4⋊C4×D137C2211yescharacteristic
36C4⋊C4×D137C2211yescharacteristic
37#17984022C2211yescharacteristic
38#17984021C2211yescharacteristic
39#17984021C2211yescharacteristic
40#17984021C2211yescharacteristic
41#17984020C2211yescharacteristic
42#17984019C2211yescharacteristic
43#17984013C2211yescharacteristic
44#17984013C2211yescharacteristic
45#17984012C2211yescharacteristic
46#17984012C2211yescharacteristic
47#17984011C2211yescharacteristic
48#17984011C2211yescharacteristic
49C22⋊#11764040C2211yescharacteristic
50#17984000C2211yescharacteristic
51#17984000C2211yescharacteristic
52C22⋊Q8×C137C2211yescharacteristic
53C2×C4⋊C4×C137C2211yescharacteristic
54#106×C137C2211yescharacteristic
55#106×C137C2211yescharacteristic
56#103×C137C2211yescharacteristic
57#103×C137C2211yescharacteristic
58C22⋊C8×C137C2211yescharacteristic
59#14555510—22no—
60#14555510—22no—
61C2×Dic274—21no—
62C2×Dic274D411yescharacteristic
63C2×Dic274C2311yescharacteristic
64Q8×D137—22no—
65Q8×D137—22no—
66C2×D548D411yescharacteristic
67C2×C22⋊D137D411yescharacteristic
68C2×C22⋊D137D411yescharacteristic
69C2×C4×D137—21no—
70C2×C4×D137—21no—
71C2×C4×D137—21no—
72C2×C4×D137C2311yescharacteristic
73C2×C4×D137C2311yescharacteristic
74#14555493—22no—
75C2×#11764040C2311yescharacteristic
76C2×#11764040C2311yescharacteristic
77C8×D137—22no—
78C4⋊Dic137—21no—
79C4⋊Dic137C2311yescharacteristic
80C4⋊Dic137C2311yescharacteristic
81C4⋊Dic137C2311yescharacteristic
82#14555489—21no—
83#14555489—21no—
84#14555489—21no—
85#14555489—21no—
86C22⋊Dic137—21no—
87C22⋊Dic137—21no—
88#14555487—21no—
89#14555487—21no—
90#14555487—21no—
91#14555487—21no—
92#14555487—21no—
93C4×Dic137—21no—
94C4×Dic137—21no—
95C4×Dic137—21no—
96C4×Dic137—21no—
97C2×Q8×C137C2311yescharacteristic
98C22×C4×C137—21no—
99C22×C4×C137C2311yescharacteristic
100C2×C8×C137C2311yescharacteristic
101C2×C8×C137C2311yescharacteristic
102C4⋊C4×C137—21no—
103C4⋊C4×C137—21no—
104C4⋊C4×C137C2311yescharacteristic
105C4⋊C4×C137C2311yescharacteristic
106C4⋊C4×C137C2311yescharacteristic
107C22⋊C4×C137—21no—
108C22×D137C2×D411yescharacteristic
109C22×D137C2×D411yescharacteristic
110Dic274—22no—
111Dic274—21no—
112Dic274—21no—
113Dic274—42no—
114D548—22no—
115C22⋊D137—22no—
116C22⋊D137—22no—
117C2×Dic137—21no—
118C2×Dic137—21no—
119C2×Dic137—21no—
120C2×Dic137—21no—
121C2×Dic137C2×D411yescharacteristic
122C2×Dic137C2×D411yescharacteristic
123C4×D137—21no—
124C4×D137—21no—
125C4×D137—42no—
126C4×D137—81no—
127C4×D137—81no—
128C4×D137#4112yes—
129C4×D137#4112yes—
130#11764040—21no—
131#11764040—21no—
132C23×C137C2×D411yescharacteristic
133Q8×C137—21no—
134Q8×C137—21no—
135C2×C4×C137—21no—
136C2×C4×C137—21no—
137C2×C4×C137—21no—
138C2×C4×C137—21no—
139C2×C4×C137—21no—
140C2×C4×C137—21no—
141C2×C4×C137—81no—
142C2×C4×C137C2×D411yescharacteristic
143C2×C4×C137C2411yescharacteristic
144C8×C137—21no—
145C8×C137—21no—
146C22×C137—21no—
147C22×C137—21no—
148C22×C137—21no—
149C22×C137C22×D411yescharacteristic
150C2×D137—21no—
151C2×D137—21no—
152C2×D137—21no—
153C2×D137—21no—
154C2×D137—21no—
155C2×D137#12312yes—
156C4×C137—21no—
157C4×C137—41no—
158C4×C137—41no—
159C4×C137—41no—
160C4×C137—41no—
161C4×C137C2×#4111yescharacteristic
162C4×C137C2×#4111yescharacteristic
163Dic137—21no—
164Dic137—41no—
165Dic137—41no—
166Dic137—41no—
167Dic137—41no—
168Dic137#12312yes—
169C2×C137—41no—
170C2×C137C2×#12311yescharacteristic
171C2×C137#57711yescharacteristic
172C2×C137#57811yescharacteristic
173D137—22no—
174D137—41no—
175C137#309711yescharacteristic, complemented, Sylow
176#3097—1371nomaximal, Sylow
177#536—1371no—
178#520—1371no—
179#485—1371no—
180#485—1371no—
181#485—1371no—
182#485—1371no—
183#484—1371no—
184#484—1371no—
185#483—1371no—
186#483D13711yescharacteristic, complemented
187#428—1371no—
188#427—1371no—
189#419—1371no—
190C2×#103—1371no—
191#408—1371no—
192C2×#41—1371no—
193C22×Q8—1371no—
194C2×M16—1371no—
195C22×C8—1371no—
196C4⋊Q8—2741no—
197#124—2741no—
198#123—2741no—
199#123—2741no—
200C22⋊Q8—1371no—
201C22⋊Q8—1371no—
202C22⋊Q8—1371no—
203C22⋊Q8C2×D13711yescharacteristic
204C4×Q8—2741no—
205C4×D4—2741no—
206C4×D4—2741no—
207C4×D4—2741no—
208#117—1371no—
209#117—1371no—
210#117—1371no—
211#117—1371no—
212C2×C4⋊C4—1371no—
213C2×C4⋊C4—1371no—
214C2×C4⋊C4C2×D13711yescharacteristic
215#107—1371no—
216#107—1371no—
217#107—1371no—
218#107—1371no—
219#106—1371no—
220#106—1371no—
221#106C2×D13711yescharacteristic
222#106C2×D13711yescharacteristic
223#103—1371no—
224#103—1371no—
225#103—1371no—
226#103—1371no—
227#103—1371no—
228#103—1371no—
229#103C2×D13711yescharacteristic
230#103C2×D13711yescharacteristic
231C22⋊C8—1371no—
232C22⋊C8—1371no—
233C22⋊C8—1371no—
234C22⋊C8C2×D13711yescharacteristic
235#41—2742no—
236#41—2742no—
237C2×Q8—2741no—
238C2×Q8—2742no—
239C2×Q8—2742no—
240C2×Q8—1371no—
241C2×Q8—1371no—
242C2×Q8C22×D13711yescharacteristic
243C2×D4—1371no—
244C2×D4—1371no—
245C2×D4—1371no—
246C22×C4—21no—
247C22×C4—2741no—
248C22×C4—2741no—
249C22×C4—2741no—
250C22×C4—1371no—
251C22×C4—1371no—
252C22×C4C22×D13711yescharacteristic
253M16—2742no—
254C2×C8—2742no—
255C2×C8—1371no—
256C2×C8—1371no—
257C2×C8C22×D13711yescharacteristic
258C2×C8C22×D13711yescharacteristic
259C4⋊C4—21no—
260C4⋊C4—21no—
261C4⋊C4—2741no—
262C4⋊C4—2741no—
263C4⋊C4—2741no—
264C4⋊C4—2741no—
265C4⋊C4—2741no—
266C4⋊C4—1371no—
267C4⋊C4—1371no—
268C4⋊C4—1371no—
269C4⋊C4C22×D13711yescharacteristic
270C4⋊C4C22×D13711yescharacteristic
271C4⋊C4C22×D13711yescharacteristic
272C22⋊C4—21no—
273C22⋊C4—2741no—
274C22⋊C4—2741no—
275C22⋊C4—2741no—
276C22⋊C4—2741no—
277C22⋊C4—2741no—
278C22⋊C4—2741no—
279C22⋊C4—2741no—
280C42—2741no—
281C42—2741no—
282C42—2741no—
283C42—2741no—
284C23—1371no—
285C23—1371no—
286C23D4×D13711yescharacteristic
287Q8—21no—
288Q8—21no—
289Q8—2742no—
290Q8—2741no—
291Q8—2741no—
292Q8—5482no—
293D4—2742no—
294D4—2742no—
295D4—2742no—
296C2×C4—21no—
297C2×C4—21no—
298C2×C4—21no—
299C2×C4—21no—
300C2×C4—21no—
301C2×C4—21no—
302C2×C4—81no—
303C2×C4—2741no—
304C2×C4—2741no—
305C2×C4—2741no—
306C2×C4—2741no—
307C2×C4—2741no—
308C2×C4—2741no—
309C2×C4—5482no—
310C2×C4—10961no—
311C2×C4—10961no—
312C2×C4—1371no—
313C2×C4—1371no—
314C2×C4—1372no—
315C2×C4—1372no—
316C2×C4D4×D13711yescharacteristic
317C2×C4C23×D13711yescharacteristic
318C8—21no—
319C8—21no—
320C8—2741no—
321C8—2741no—
322C22—21no—
323C22—21no—
324C22—21no—
325C22—2741no—
326C22—2741no—
327C22—2741no—
328C22—2741no—
329C22—2741no—
330C22—1372no—
331C22C2×D4×D13711yescharacteristic
332C4—21no—
333C4—41no—
334C4—41no—
335C4—41no—
336C4—41no—
337C4—2741no—
338C4—5481no—
339C4—5481no—
340C4—5481no—
341C4—5481no—
342C4—1372no—
343C4#41×D13711yescharacteristic
344C4#41×D13711yescharacteristic
345C2—41no—
346C2—2742no—
347C2—5481no—
348C2#123×D13711yescharacteristic
349C2#2257820911yescharacteristic
350C2#2257821911yescharacteristic
subgroupG/H
centerC22C2×D4×D137
commutator, cosocleC2×C4×C137C24
FrattiniC2×C4C23×D137
Fitting#483×C137C2
socleC22×C137C22×D4
prime radicalC2×#14555510C22
hypercenter#483D137
hypocenterC137#3097
LCSG ▷ C2×C4×C137 ▷ C2×C137 ▷ C137
UCSC1 ◁ C22 ◁ C22×C4 ◁ #483

Complements

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

NH ≅ G/Ntype
3. #22577747C2semidirect
4. #22577745C2semidirect
5. #22577742C2semidirect
6. #22577741C2semidirect
7. #22577738C2semidirect
8. #22577737C2semidirect
9. #22577735C2semidirect
10. #22577442C2semidirect
11. #22577438C2semidirect
12. #22577410C2semidirect
13. #103×D137C2semidirect
15. #483×C137C2semidirect
175. C137#3097semidirect
186. #483D137semidirect

Maximal subgroups

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

subgroupindex
#225779762
#225778842
#225777472
#225777452
#225777422
#225777412
#225777382
#225777372
#225777352
#225774422
#225774382
#225774102
#103×D1372
#225773682
#483×C1372
#3097137