Finite Group Databasebeta

← #31790770

Subgroups — #31790770

Order 17536.

398 conjugacy classes of subgroups.

The subgroup lattice has 364 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#22577983C211yesmaximal, characteristic
2#22577883C211yesmaximal, characteristic, complemented
3#22577743C211yesmaximal, characteristic, complemented
4#22577737C211yesmaximal, characteristic, complemented
5#22577736C211yesmaximal, characteristic, complemented
6#22577729C211yesmaximal, characteristic, complemented
7#22577727C211yesmaximal, characteristic, complemented
8#22577718C211yesmaximal, characteristic, complemented
9#22577717C211yesmaximal, characteristic, complemented
10#22577442C211yesmaximal, characteristic, complemented
11#107×D137C211yesmaximal, characteristic, complemented
12#22577403C211yesmaximal, characteristic, complemented
13#22577398C211yesmaximal, characteristic, complemented
14#22577368C211yesmaximal, characteristic, complemented
15#485×C137C211yesmaximal, characteristic, complemented
16C2×#14555511C2211yescharacteristic
17C2×#14555510C2211yescharacteristic
18C2×#14555493C2211yescharacteristic
19C2×C8×D137C2211yescharacteristic
20#17984066—21no—
21#17984064C2211yescharacteristic
22#17984063—21no—
23#17984063C2211yescharacteristic
24#17984062C2211yescharacteristic
25#17984061—21no—
26#17984061—21no—
27C4×Dic274—21no—
28C4×D548—21no—
29#17984048—21no—
30#17984048—21no—
31#17984046C2211yescharacteristic
32#17984046C2211yescharacteristic
33C2×C4⋊Dic137C2211yescharacteristic
34C4⋊C4×D137C2211yescharacteristic
35C4⋊C4×D137C2211yescharacteristic
36C4⋊C4×D137C2211yescharacteristic
37#17984022C2211yescharacteristic
38#17984022C2211yescharacteristic
39#17984021C2211yescharacteristic
40#17984021C2211yescharacteristic
41#17984021C2211yescharacteristic
42#17984019C2211yescharacteristic
43#17984012C2211yescharacteristic
44#17984012C2211yescharacteristic
45#17984010C2211yescharacteristic
46#17984010C2211yescharacteristic
47#17984008C2211yescharacteristic
48#17984008C2211yescharacteristic
49C22⋊#11764040C2211yescharacteristic
50#17984000C2211yescharacteristic
51#17984000C2211yescharacteristic
52C22⋊Q8×C137C2211yescharacteristic
53#117×C137C2211yescharacteristic
54#107×C137C2211yescharacteristic
55#106×C137C2211yescharacteristic
56#103×C137C2211yescharacteristic
57#103×C137C2211yescharacteristic
58C22⋊C8×C137C2211yescharacteristic
59#14555511—22no—
60#14555511—22no—
61#14555510—22no—
62#14555510—22no—
63C2×Dic274D411yescharacteristic
64C2×D548—21no—
65C2×D548—21no—
66C2×D548D411yescharacteristic
67C2×D548C2311yescharacteristic
68C2×C22⋊D137—21no—
69C2×C22⋊D137D411yescharacteristic
70C2×C22⋊D137D411yescharacteristic
71C22×Dic137—21no—
72C2×C4×D137—21no—
73C2×C4×D137—21no—
74C2×C4×D137—21no—
75C2×C4×D137—21no—
76C2×C4×D137C2311yescharacteristic
77C2×C4×D137C2311yescharacteristic
78#14555493—22no—
79C2×#11764040C2311yescharacteristic
80C2×#11764040C2311yescharacteristic
81C8×D137—22no—
82C4⋊Dic137—21no—
83C4⋊Dic137C2311yescharacteristic
84C4⋊Dic137C2311yescharacteristic
85C4⋊Dic137C2311yescharacteristic
86#14555489—21no—
87#14555489—21no—
88#14555489—21no—
89#14555489—21no—
90#14555487—21no—
91#14555487—21no—
92#14555487—21no—
93#14555487—21no—
94#14555487—21no—
95C4×Dic137—21no—
96C4×Dic137—21no—
97C2×Q8×C137C2311yescharacteristic
98C22×C4×C137C2311yescharacteristic
99C2×C8×C137C2311yescharacteristic
100C2×C8×C137C2311yescharacteristic
101C4⋊C4×C137—21no—
102C4⋊C4×C137C2311yescharacteristic
103C4⋊C4×C137C2311yescharacteristic
104C4⋊C4×C137C2311yescharacteristic
105C22⋊C4×C137—21no—
106C22⋊C4×C137—21no—
107C42×C137—21no—
108C22×D137—21no—
109C22×D137C2×D411yescharacteristic
110C22×D137C2×D411yescharacteristic
111Dic274—22no—
112D548—22no—
113D548—21no—
114D548—21no—
115D548—42no—
116D548—81no—
117C22⋊D137—22no—
118C22⋊D137—22no—
119C22⋊D137—81no—
120C2×Dic137—21no—
121C2×Dic137—21no—
122C2×Dic137—21no—
123C2×Dic137—81no—
124C2×Dic137C2×D411yescharacteristic
125C2×Dic137C2×D411yescharacteristic
126C4×D137—21no—
127C4×D137—21no—
128C4×D137—42no—
129C4×D137—81no—
130C4×D137—81no—
131C4×D137—81no—
132C4×D137#4112yes—
133C4×D137#4112yes—
134#11764040—21no—
135#11764040—21no—
136C23×C137C2×D411yescharacteristic
137Q8×C137—21no—
138Q8×C137—21no—
139C2×C4×C137—21no—
140C2×C4×C137—21no—
141C2×C4×C137—21no—
142C2×C4×C137—21no—
143C2×C4×C137—21no—
144C2×C4×C137—21no—
145C2×C4×C137C2×D411yescharacteristic
146C2×C4×C137C2411yescharacteristic
147C8×C137—21no—
148C8×C137—21no—
149C22×C137—21no—
150C22×C137—21no—
151C22×C137—21no—
152C22×C137C22×D411yescharacteristic
153C2×D137—21no—
154C2×D137—21no—
155C2×D137—21no—
156C2×D137—21no—
157C2×D137—21no—
158C2×D137—41no—
159C2×D137—41no—
160C2×D137—41no—
161C2×D137#12312yes—
162C4×C137—21no—
163C4×C137—41no—
164C4×C137—41no—
165C4×C137—41no—
166C4×C137—41no—
167C4×C137C2×#4111yescharacteristic
168C4×C137C2×#4111yescharacteristic
169Dic137—21no—
170Dic137—41no—
171Dic137—41no—
172Dic137—41no—
173Dic137#12312yes—
174C2×C137—41no—
175C2×C137C2×#12311yescharacteristic
176C2×C137#57611yescharacteristic
177C2×C137#57811yescharacteristic
178D137—22no—
179D137—41no—
180D137—81no—
181C137#309811yescharacteristic, complemented, Sylow
182#3098—1371nomaximal, Sylow
183#537—1371no—
184#520—1371no—
185#485—1371no—
186#485D13711yescharacteristic, complemented
187#484—1371no—
188#484—1371no—
189#482—1371no—
190#482—1371no—
191#480—1371no—
192#480—1371no—
193#428—1371no—
194C2×#107—1371no—
195#417—1371no—
196#416—1371no—
197#408—1371no—
198C2×#41—1371no—
199C2×#41—1371no—
200C2×M16—1371no—
201C22×C8—1371no—
202C22⋊Q8—2741no—
203C22⋊Q8—1371no—
204C22⋊Q8C2×D13711yescharacteristic
205#121—2741no—
206#121—2741no—
207#121—2741no—
208#121—1371no—
209#121—1371no—
210C4×Q8—2741no—
211C4×D4—2741no—
212C4×D4—2741no—
213C4×D4—2741no—
214#117—1371no—
215#117—1371no—
216#117C2×D13711yescharacteristic
217C2×C4⋊C4—1371no—
218C2×C4⋊C4—1371no—
219C2×C4⋊C4—1371no—
220C2×C4⋊C4—1371no—
221#107—1371no—
222#107—1371no—
223#107—1371no—
224#107—1371no—
225#107—1371no—
226#107C2×D13711yescharacteristic
227#106—1371no—
228#106C2×D13711yescharacteristic
229#103—1371no—
230#103—1371no—
231#103C2×D13711yescharacteristic
232#103C2×D13711yescharacteristic
233#102—1371no—
234#102—1371no—
235#102—1371no—
236#102—1371no—
237C22⋊C8—1371no—
238C22⋊C8—1371no—
239C22⋊C8—1371no—
240C22⋊C8C2×D13711yescharacteristic
241#41—2742no—
242#41—2742no—
243#41—2742no—
244#41—2742no—
245C2×Q8—1371no—
246C2×Q8C22×D13711yescharacteristic
247C2×D4—2741no—
248C2×D4—2741no—
249C2×D4—2741no—
250C2×D4—1371no—
251C2×D4—1371no—
252C2×D4—1371no—
253C2×D4—1371no—
254C22×C4—2741no—
255C22×C4—2741no—
256C22×C4—2741no—
257C22×C4—2741no—
258C22×C4—2741no—
259C22×C4—1371no—
260C22×C4—1371no—
261C22×C4C22×D13711yescharacteristic
262M16—2742no—
263C2×C8—2742no—
264C2×C8—1371no—
265C2×C8—1371no—
266C2×C8C22×D13711yescharacteristic
267C2×C8C22×D13711yescharacteristic
268C4⋊C4—21no—
269C4⋊C4—2741no—
270C4⋊C4—2741no—
271C4⋊C4—2741no—
272C4⋊C4—2741no—
273C4⋊C4—2741no—
274C4⋊C4—1371no—
275C4⋊C4—1371no—
276C4⋊C4—1371no—
277C4⋊C4C22×D13711yescharacteristic
278C4⋊C4C22×D13711yescharacteristic
279C4⋊C4C22×D13711yescharacteristic
280C22⋊C4—21no—
281C22⋊C4—21no—
282C22⋊C4—2741no—
283C22⋊C4—2741no—
284C22⋊C4—2741no—
285C22⋊C4—2741no—
286C22⋊C4—2741no—
287C42—21no—
288C42—2741no—
289C42—2741no—
290C23—2741no—
291C23—1371no—
292C23—1371no—
293C23D4×D13711yescharacteristic
294Q8—21no—
295Q8—21no—
296Q8—2742no—
297D4—2742no—
298D4—2742no—
299D4—2742no—
300D4—2741no—
301D4—2741no—
302D4—5482no—
303D4—10961no—
304D4—10961no—
305C2×C4—21no—
306C2×C4—21no—
307C2×C4—21no—
308C2×C4—21no—
309C2×C4—21no—
310C2×C4—21no—
311C2×C4—2741no—
312C2×C4—2741no—
313C2×C4—2741no—
314C2×C4—2741no—
315C2×C4—2741no—
316C2×C4—5482no—
317C2×C4—10961no—
318C2×C4—10961no—
319C2×C4—10961no—
320C2×C4—10961no—
321C2×C4—1371no—
322C2×C4—1371no—
323C2×C4—1372no—
324C2×C4—1372no—
325C2×C4D4×D13711yescharacteristic
326C2×C4C23×D13711yescharacteristic
327C8—21no—
328C8—21no—
329C8—2741no—
330C8—2741no—
331C22—21no—
332C22—21no—
333C22—21no—
334C22—2741no—
335C22—2741no—
336C22—2741no—
337C22—2741no—
338C22—2741no—
339C22—5481no—
340C22—5481no—
341C22—5481no—
342C22—1372no—
343C22C2×D4×D13711yescharacteristic
344C4—21no—
345C4—41no—
346C4—41no—
347C4—41no—
348C4—41no—
349C4—2741no—
350C4—5481no—
351C4—5481no—
352C4—5481no—
353C4—1372no—
354C4#41×D13711yescharacteristic
355C4#41×D13711yescharacteristic
356C2—41no—
357C2—2742no—
358C2—5481no—
359C2—10961no—
360C2#123×D13711yescharacteristic
361C2#2257820511yescharacteristic
362C2#2257821911yescharacteristic
subgroupG/H
centerC22C2×D4×D137
commutator, cosocleC2×C4×C137C24
FrattiniC2×C4C23×D137
Fitting#485×C137C2
socleC22×C137C22×D4
prime radical#22577983C2
hypercenter#485D137
hypocenterC137#3098
LCSG ▷ C2×C4×C137 ▷ C2×C137 ▷ C137
UCSC1 ◁ C22 ◁ C22×C4 ◁ #485

Complements

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

NH ≅ G/Ntype
2. #22577883C2semidirect
3. #22577743C2semidirect
4. #22577737C2semidirect
5. #22577736C2semidirect
6. #22577729C2semidirect
7. #22577727C2semidirect
8. #22577718C2semidirect
9. #22577717C2semidirect
10. #22577442C2semidirect
11. #107×D137C2semidirect
12. #22577403C2semidirect
13. #22577398C2semidirect
14. #22577368C2semidirect
15. #485×C137C2semidirect
181. C137#3098semidirect
186. #485D137semidirect

Maximal subgroups

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

subgroupindex
#225779832
#225778832
#225777432
#225777372
#225777362
#225777292
#225777272
#225777182
#225777172
#225774422
#107×D1372
#225774032
#225773982
#225773682
#485×C1372
#3098137