Finite Group Databasebeta

← #31790762

Subgroups — #31790762

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#22577982C211yesmaximal, characteristic
2#22577884C211yesmaximal, characteristic, complemented
3#22577746C211yesmaximal, characteristic, complemented
4#22577745C211yesmaximal, characteristic, complemented
5#22577740C211yesmaximal, characteristic, complemented
6#22577738C211yesmaximal, characteristic, complemented
7#22577726C211yesmaximal, characteristic, complemented
8#22577724C211yesmaximal, characteristic, complemented
9#22577715C211yesmaximal, characteristic, complemented
10#22577444C211yesmaximal, characteristic, complemented
11#107×D137C211yesmaximal, characteristic, complemented
12#22577401C211yesmaximal, characteristic, complemented
13#22577396C211yesmaximal, characteristic, complemented
14#22577368C211yesmaximal, characteristic, complemented
15#480×C137C211yesmaximal, characteristic, complemented
16C2×#14555510C2211yescharacteristic
17C2×#14555509C2211yescharacteristic
18C2×#14555493C2211yescharacteristic
19C2×C8×D137C2211yescharacteristic
20#17984068C2211yescharacteristic
21#17984067C2211yescharacteristic
22#17984065—21no—
23#17984061—21no—
24#17984060—21no—
25#17984059—21no—
26#17984059C2211yescharacteristic
27#17984053—21no—
28#17984050—21no—
29C4×C22⋊D137—21no—
30C4×C22⋊D137—21no—
31#17984047C2211yescharacteristic
32#17984046C2211yescharacteristic
33#17984046C2211yescharacteristic
34C4⋊C4×D137C2211yescharacteristic
35C4⋊C4×D137C2211yescharacteristic
36C4⋊C4×D137C2211yescharacteristic
37#17984022C2211yescharacteristic
38#17984021C2211yescharacteristic
39#17984021C2211yescharacteristic
40#17984021C2211yescharacteristic
41#17984020C2211yescharacteristic
42#17984019C2211yescharacteristic
43#17984013C2211yescharacteristic
44#17984013C2211yescharacteristic
45#17984011C2211yescharacteristic
46#17984011C2211yescharacteristic
47#17984009C2211yescharacteristic
48#17984009C2211yescharacteristic
49C22⋊#11764040C2211yescharacteristic
50#17984000C2211yescharacteristic
51#17984000C2211yescharacteristic
52#121×C137C2211yescharacteristic
53C2×C4⋊C4×C137C2211yescharacteristic
54#107×C137C2211yescharacteristic
55#107×C137C2211yescharacteristic
56#102×C137C2211yescharacteristic
57#102×C137C2211yescharacteristic
58C22⋊C8×C137C2211yescharacteristic
59#14555510—22no—
60#14555510—22no—
61#14555509—22no—
62#14555509—22no—
63C2×Dic274D411yescharacteristic
64C2×Dic274C2311yescharacteristic
65C2×D548D411yescharacteristic
66C2×C22⋊D137—21no—
67C2×C22⋊D137—21no—
68C2×C22⋊D137D411yescharacteristic
69C2×C22⋊D137D411yescharacteristic
70C22×Dic137—21no—
71C2×C4×D137—21no—
72C2×C4×D137—21no—
73C2×C4×D137—21no—
74C2×C4×D137C2311yescharacteristic
75C2×C4×D137C2311yescharacteristic
76#14555493—22no—
77C2×#11764040C2311yescharacteristic
78C2×#11764040C2311yescharacteristic
79C8×D137—22no—
80C4⋊Dic137—21no—
81C4⋊Dic137C2311yescharacteristic
82C4⋊Dic137C2311yescharacteristic
83C4⋊Dic137C2311yescharacteristic
84#14555489—21no—
85#14555489—21no—
86#14555489—21no—
87#14555489—21no—
88C22⋊Dic137—21no—
89C22⋊Dic137—21no—
90C22⋊Dic137—21no—
91#14555487—21no—
92#14555487—21no—
93#14555487—21no—
94C4×Dic137—21no—
95C4×Dic137—21no—
96C4×Dic137—21no—
97C2×D4×C137—21no—
98C2×D4×C137C2311yescharacteristic
99C22×C4×C137—21no—
100C22×C4×C137C2311yescharacteristic
101C2×C8×C137C2311yescharacteristic
102C2×C8×C137C2311yescharacteristic
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—
113D548—22no—
114C22⋊D137—22no—
115C22⋊D137—22no—
116C22⋊D137—42no—
117C22⋊D137—81no—
118C2×Dic137—21no—
119C2×Dic137—21no—
120C2×Dic137—21no—
121C2×Dic137—21no—
122C2×Dic137—42no—
123C2×Dic137C2×D411yescharacteristic
124C2×Dic137C2×D411yescharacteristic
125C4×D137—21no—
126C4×D137—21no—
127C4×D137—81no—
128C4×D137—81no—
129C4×D137—81no—
130C4×D137#4112yes—
131C4×D137#4112yes—
132#11764040—21no—
133#11764040—21no—
134C23×C137—21no—
135C23×C137C2×D411yescharacteristic
136D4×C137—21no—
137D4×C137—21no—
138D4×C137—81no—
139C2×C4×C137—21no—
140C2×C4×C137—21no—
141C2×C4×C137—21no—
142C2×C4×C137—21no—
143C2×C4×C137—21no—
144C2×C4×C137—81no—
145C2×C4×C137C2×D411yescharacteristic
146C2×C4×C137C2411yescharacteristic
147C8×C137—21no—
148C8×C137—21no—
149C22×C137—21no—
150C22×C137—21no—
151C22×C137—21no—
152C22×C137—41no—
153C22×C137—41no—
154C22×C137—41no—
155C22×C137C22×D411yescharacteristic
156C2×D137—21no—
157C2×D137—21no—
158C2×D137—21no—
159C2×D137—21no—
160C2×D137—21no—
161C2×D137#12312yes—
162C4×C137—21no—
163C4×C137—41no—
164C4×C137—41no—
165C4×C137—41no—
166C4×C137C2×#4111yescharacteristic
167C4×C137C2×#4111yescharacteristic
168Dic137—21no—
169Dic137—41no—
170Dic137—41no—
171Dic137—41no—
172Dic137—41no—
173Dic137#12312yes—
174C2×C137—41no—
175C2×C137—81no—
176C2×C137C2×#12311yescharacteristic
177C2×C137#57611yescharacteristic
178C2×C137#57811yescharacteristic
179D137—22no—
180D137—41no—
181C137#309811yescharacteristic, complemented, Sylow
182#3098—1371nomaximal, Sylow
183#537—1371no—
184#520—1371no—
185#485—1371no—
186#485—1371no—
187#484—1371no—
188#484—1371no—
189#482—1371no—
190#482—1371no—
191#480—1371no—
192#480D13711yescharacteristic, complemented
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⋊Q8—1371no—
205#121—2741no—
206#121—2741no—
207#121—2741no—
208#121—1371no—
209#121C2×D13711yescharacteristic
210C4×Q8—2741no—
211C4×D4—2741no—
212C4×D4—2741no—
213C4×D4—2741no—
214#117—1371no—
215#117—1371no—
216#117—1371no—
217C2×C4⋊C4—1371no—
218C2×C4⋊C4—1371no—
219C2×C4⋊C4—1371no—
220C2×C4⋊C4C2×D13711yescharacteristic
221#107—1371no—
222#107—1371no—
223#107—1371no—
224#107—1371no—
225#107C2×D13711yescharacteristic
226#107C2×D13711yescharacteristic
227#106—1371no—
228#106—1371no—
229#103—1371no—
230#103—1371no—
231#103—1371no—
232#103—1371no—
233#102—1371no—
234#102—1371no—
235#102C2×D13711yescharacteristic
236#102C2×D13711yescharacteristic
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×Q8—1371no—
247C2×D4—21no—
248C2×D4—2741no—
249C2×D4—2741no—
250C2×D4—1371no—
251C2×D4—1371no—
252C2×D4—1371no—
253C2×D4C22×D13711yescharacteristic
254C22×C4—21no—
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—2741no—
282C22⋊C4—2741no—
283C22⋊C4—2741no—
284C22⋊C4—2741no—
285C22⋊C4—2741no—
286C22⋊C4—2741no—
287C42—2741no—
288C42—2741no—
289C42—2741no—
290C23—21no—
291C23—1371no—
292C23—1371no—
293C23D4×D13711yescharacteristic
294Q8—2742no—
295Q8—2741no—
296Q8—2741no—
297D4—21no—
298D4—21no—
299D4—81no—
300D4—2742no—
301D4—2742no—
302D4—2742no—
303D4—5482no—
304D4—10961no—
305C2×C4—21no—
306C2×C4—21no—
307C2×C4—21no—
308C2×C4—21no—
309C2×C4—21no—
310C2×C4—81no—
311C2×C4—2741no—
312C2×C4—2741no—
313C2×C4—2741no—
314C2×C4—2741no—
315C2×C4—2741no—
316C2×C4—2741no—
317C2×C4—5482no—
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—41no—
335C22—41no—
336C22—41no—
337C22—2741no—
338C22—2741no—
339C22—2741no—
340C22—2741no—
341C22—2741no—
342C22—1372no—
343C22C2×D4×D13711yescharacteristic
344C4—21no—
345C4—41no—
346C4—41no—
347C4—41no—
348C4—2741no—
349C4—5481no—
350C4—5481no—
351C4—5481no—
352C4—5481no—
353C4—1372no—
354C4#41×D13711yescharacteristic
355C4#41×D13711yescharacteristic
356C2—41no—
357C2—81no—
358C2—2742no—
359C2—5481no—
360C2#123×D13711yescharacteristic
361C2#2257820411yescharacteristic
362C2#2257821911yescharacteristic
subgroupG/H
centerC22C2×D4×D137
commutator, cosocleC2×C4×C137C24
FrattiniC2×C4C23×D137
Fitting#480×C137C2
socleC22×C137C22×D4
prime radical#22577982C2
hypercenter#480D137
hypocenterC137#3098
LCSG ▷ C2×C4×C137 ▷ C2×C137 ▷ C137
UCSC1 ◁ C22 ◁ C22×C4 ◁ #480

Complements

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

NH ≅ G/Ntype
2. #22577884C2semidirect
3. #22577746C2semidirect
4. #22577745C2semidirect
5. #22577740C2semidirect
6. #22577738C2semidirect
7. #22577726C2semidirect
8. #22577724C2semidirect
9. #22577715C2semidirect
10. #22577444C2semidirect
11. #107×D137C2semidirect
12. #22577401C2semidirect
13. #22577396C2semidirect
14. #22577368C2semidirect
15. #480×C137C2semidirect
181. C137#3098semidirect
192. #480D137semidirect

Maximal subgroups

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

subgroupindex
#225779822
#225778842
#225777462
#225777452
#225777402
#225777382
#225777262
#225777242
#225777152
#225774442
#107×D1372
#225774012
#225773962
#225773682
#480×C1372
#3098137