Finite Group Databasebeta

← #31791043

Subgroups — #31791043

Order 17536.

394 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#22578016C211yesmaximal, characteristic, complemented
2#22577875C211yesmaximal, characteristic
3#22577747C211yesmaximal, characteristic, complemented
4#22577746C211yesmaximal, characteristic, complemented
5#22577700C211yesmaximal, characteristic, complemented
6#22577609C211yesmaximal, characteristic, complemented
7#22577608C211yesmaximal, characteristic, complemented
8#22577555C211yesmaximal, characteristic
9#22577555C211yesmaximal, characteristic
10#22577502C211yesmaximal, characteristic, complemented
11#22577500C211yesmaximal, characteristic, complemented
12#22577494C211yesmaximal, characteristic, complemented
13#22577490C211yesmaximal, characteristic, complemented
14#22577463C211yesmaximal, characteristic, complemented
15#478×C137C211yesmaximal, characteristic, complemented
16C2×D4×D137C2211yescharacteristic
17C2×#14555499C2211yescharacteristic
18C2×#14555499C2211yescharacteristic
19C2×#14555498C2211yescharacteristic
20C2×#14555498C2211yescharacteristic
21#17984088C2211yescharacteristic, complemented
22#17984083—21no—
23#17984083—21no—
24#17984079—21no—
25#17984079—21no—
26#17984068C2211yescharacteristic
27#17984068C2211yescharacteristic
28#17984053C2211yescharacteristic, complemented
29#17984053C2211yescharacteristic, complemented
30C4×D548C2211yescharacteristic
31D4×Dic137C2211yescharacteristic
32#17984050C2211yescharacteristic
33C4×C22⋊D137—21no—
34#17984048—21no—
35#17984046C2211yescharacteristic
36#17984045C2211yescharacteristic
37C2×#14555487—21no—
38C22⋊C4×D137—21no—
39#17984021C2211yescharacteristic, complemented
40#17984019C2211yescharacteristic, complemented
41C4⋊#11764040C2211yescharacteristic, complemented
42#17984013C2211yescharacteristic, complemented
43#17984013C2211yescharacteristic, complemented
44#17984011C2211yescharacteristic, complemented
45#17984011C2211yescharacteristic, complemented
46#17984008C2211yescharacteristic
47#17984008C2211yescharacteristic
48#17984000C2211yescharacteristic
49#17984000C2211yescharacteristic
50#17983998C2211yescharacteristic, complemented
51C8×Dic137C2211yescharacteristic, complemented
52#125×C137C2211yescharacteristic, complemented
53C4×D4×C137C2211yescharacteristic
54#107×C137C2211yescharacteristic, complemented
55#106×C137C2211yescharacteristic, complemented
56C4⋊C8×C137C2211yescharacteristic, complemented
57#102×C137C2211yescharacteristic
58#102×C137C2211yescharacteristic
59C23×D137—21no—
60C2×Dic274C2311yescharacteristic
61C2×Dic274C2311yescharacteristic
62C2×D548C2311yescharacteristic
63C2×C22⋊D137—21no—
64D4×D137—22no—
65D4×D137—22no—
66C22×Dic137—21no—
67C2×C4×D137—21no—
68C2×C4×D137—21no—
69C2×C4×D137C2311yescharacteristic
70#14555499—22no—
71#14555499—22no—
72#14555498—22no—
73#14555498—22no—
74C2×#11764040C2311yescharacteristic
75C2×#11764040C2311yescharacteristic
76C4⋊Dic137—21no—
77C4⋊Dic137—21no—
78C4⋊Dic137—21no—
79C4⋊Dic137—21no—
80C4⋊Dic137C2311yescharacteristic
81#14555489—21no—
82#14555489—21no—
83#14555489D411yescharacteristic
84#14555489D411yescharacteristic
85C22⋊Dic137—21no—
86#14555487—21no—
87#14555487—21no—
88#14555487—21no—
89#14555487—21no—
90#14555487—21no—
91#14555487D411yescharacteristic
92#14555487D411yescharacteristic
93C4×Dic137—21no—
94C4×Dic137—21no—
95C4×Dic137—21no—
96C4×Dic137C2311yescharacteristic
97C2×D4×C137C2311yescharacteristic
98C22×C4×C137—21no—
99C2×C8×C137C2311yescharacteristic
100C2×C8×C137C2311yescharacteristic
101C4⋊C4×C137—21no—
102C4⋊C4×C137—21no—
103C4⋊C4×C137C2311yescharacteristic
104C4⋊C4×C137C2311yescharacteristic
105C4⋊C4×C137C2311yescharacteristic
106C22⋊C4×C137—21no—
107C42×C137C2311yescharacteristic
108C22×D137—21no—
109C22×D137—24no—
110C22×D137—81no—
111C22×D137—81no—
112C22×D137C2×D411yescharacteristic
113Dic274—21no—
114Dic274—21no—
115Dic274—21no—
116Dic274—21no—
117D548—21no—
118D548#4112yes—
119C22⋊D137—24no—
120C2×Dic137—21no—
121C2×Dic137—21no—
122C2×Dic137—21no—
123C2×Dic137—81no—
124C2×Dic137C2×D411yescharacteristic
125C2×Dic137C2×D411yescharacteristic
126C2×Dic137C2×D411yescharacteristic
127C4×D137—21no—
128C4×D137—21no—
129C4×D137—81no—
130C4×D137—81no—
131#11764040—21no—
132#11764040—21no—
133C23×C137—21no—
134D4×C137—21no—
135D4×C137#4112yes—
136C2×C4×C137—21no—
137C2×C4×C137—21no—
138C2×C4×C137—21no—
139C2×C4×C137—81no—
140C2×C4×C137C2×D411yescharacteristic
141C2×C4×C137C2×D411yescharacteristic
142C2×C4×C137C2411yescharacteristic
143C8×C137—21no—
144C8×C137—21no—
145C22×C137—22no—
146C22×C137—41no—
147C22×C137—41no—
148C22×C137C22×D411yescharacteristic
149C2×D137—21no—
150C2×D137—22no—
151C2×D137—21no—
152C2×D137—21no—
153C2×D137—41no—
154C2×D137—41no—
155C2×D137—84no—
156C4×C137—21no—
157C4×C137—21no—
158C4×C137—41no—
159C4×C137—41no—
160C4×C137—41no—
161C4×C137C2×#4111yescharacteristic
162C4×C137C2×#4111yescharacteristic
163Dic137—21no—
164Dic137—21no—
165Dic137—21no—
166Dic137—41no—
167Dic137—41no—
168Dic137—41no—
169C2×C137—42no—
170C2×C137#52511yescharacteristic
171C2×C137#57711yescharacteristic
172C2×C137#57711yescharacteristic
173D137—41no—
174D137—42no—
175C137#312311yescharacteristic, complemented, Sylow
176#3123—1371nomaximal, Sylow
177#541—1371no—
178#518—1371no—
179#485—1371no—
180#485—1371no—
181#478—1371no—
182#478D13711yescharacteristic, complemented
183#461—1371no—
184#461—1371no—
185#450—1371no—
186#450—1371no—
187#440—1371no—
188#440—1371no—
189C4×SD16—1371no—
190C4×SD16—1371no—
191#433—1371no—
192C22×D4—1371no—
193C2×SD16—1371no—
194C2×SD16—1371no—
195C2×SD16—1371no—
196C2×SD16—1371no—
197C4⋊Q8—1371no—
198#126—2741no—
199#126—2741no—
200#125—2741no—
201#125—2741no—
202#125C2×D13711yescharacteristic, complemented
203C22⋊Q8—1371no—
204C22⋊Q8—1371no—
205C4×Q8—1371no—
206C4×Q8—1371no—
207C4×D4—2741no—
208C4×D4—2741no—
209C4×D4—1371no—
210C4×D4—1371no—
211C4×D4—1371no—
212C4×D4C2×D13711yescharacteristic
213#117—1371no—
214#117—1371no—
215C2×C22⋊C4—2741no—
216C2×C22⋊C4—2741no—
217#107—1371no—
218#107C2×D13711yescharacteristic, complemented
219#106—1371no—
220#106C2×D13711yescharacteristic, complemented
221C4⋊C8—1371no—
222C4⋊C8C2×D13711yescharacteristic, complemented
223#103—1371no—
224#103—1371no—
225#103—1371no—
226#103—1371no—
227#102—1371no—
228#102—1371no—
229#102C2×D13711yescharacteristic
230#102C2×D13711yescharacteristic
231C22⋊C8—1371no—
232C22⋊C8—1371no—
233#97—1371no—
234C4×C8—1371no—
235C24—2741no—
236C2×Q8—1371no—
237C2×Q8—1371no—
238C2×D4—2742no—
239C2×D4—2742no—
240C2×D4—2741no—
241C2×D4—1371no—
242C2×D4C22×D13711yescharacteristic
243C22×C4—21no—
244C22×C4—2741no—
245C22×C4—2741no—
246C22×C4—2741no—
247C22×C4—1371no—
248SD16—2742no—
249SD16—2742no—
250SD16—2742no—
251SD16—2742no—
252C2×C8—1371no—
253C2×C8—1371no—
254C2×C8C22×D13711yescharacteristic
255C2×C8C22×D13711yescharacteristic
256C4⋊C4—21no—
257C4⋊C4—21no—
258C4⋊C4—2741no—
259C4⋊C4—2741no—
260C4⋊C4—2741no—
261C4⋊C4—2741no—
262C4⋊C4—2741no—
263C4⋊C4—2741no—
264C4⋊C4—1371no—
265C4⋊C4—1371no—
266C4⋊C4—1371no—
267C4⋊C4C22×D13711yescharacteristic
268C4⋊C4C22×D13711yescharacteristic
269C4⋊C4C22×D13711yescharacteristic
270C22⋊C4—21no—
271C22⋊C4—2741no—
272C22⋊C4—2741no—
273C22⋊C4—2741no—
274C22⋊C4—2741no—
275C22⋊C4—2741no—
276C22⋊C4—2741no—
277C22⋊C4—1371no—
278C22⋊C4—1371no—
279C42—2741no—
280C42—2741no—
281C42—2741no—
282C42—1371no—
283C42C22×D13711yescharacteristic
284C23—21no—
285C23—2744no—
286C23—2741no—
287C23—10961no—
288C23—10961no—
289C23—1371no—
290Q8—2741no—
291Q8—2741no—
292Q8—2741no—
293Q8—2741no—
294D4—21no—
295D4—2744no—
296D4—2741no—
297D4—1372no—
298D4#1455551112yes—
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—10961no—
309C2×C4—10961no—
310C2×C4—10961no—
311C2×C4—1371no—
312C2×C4—1371no—
313C2×C4—1371no—
314C2×C4D4×D13711yescharacteristic
315C2×C4D4×D13711yescharacteristic
316C2×C4C23×D13711yescharacteristic
317C8—21no—
318C8—21no—
319C8—2741no—
320C8—2741no—
321C22—22no—
322C22—41no—
323C22—41no—
324C22—2741no—
325C22—2741no—
326C22—2741no—
327C22—2742no—
328C22—5481no—
329C22—5481no—
330C22—10964no—
331C22C2×D4×D13711yescharacteristic
332C4—21no—
333C4—21no—
334C4—41no—
335C4—41no—
336C4—41no—
337C4—2741no—
338C4—2741no—
339C4—2741no—
340C4—5481no—
341C4—5481no—
342C4—5481no—
343C4#41×D13711yescharacteristic
344C4C2×#1455551111yescharacteristic
345C2—42no—
346C2—5481no—
347C2—5482no—
348C2#2257791611yescharacteristic
349C2#2257821011yescharacteristic
350C2#2257821311yescharacteristic
subgroupG/H
centerC22C2×D4×D137
commutator, cosocleC2×C4×C137C24
FrattiniC2×C4C23×D137
Fitting#478×C137C2
socleC22×C137C22×D4
prime radicalC2×D4×D137C22
hypercenter#478D137
hypocenterC137#3123
LCSG ▷ C2×C4×C137 ▷ C2×C137 ▷ C137
UCSC1 ◁ C22 ◁ C42 ◁ #478

Complements

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

NH ≅ G/Ntype
1. #22578016C2semidirect
3. #22577747C2semidirect
4. #22577746C2semidirect
5. #22577700C2semidirect
6. #22577609C2semidirect
7. #22577608C2semidirect
10. #22577502C2semidirect
11. #22577500C2semidirect
12. #22577494C2semidirect
13. #22577490C2semidirect
14. #22577463C2semidirect
15. #478×C137C2semidirect
21. #17984088C22semidirect
28. #17984053C22semidirect
29. #17984053C22semidirect
39. #17984021C22semidirect
40. #17984019C22semidirect
41. C4⋊#11764040C22semidirect
42. #17984013C22semidirect
43. #17984013C22semidirect
44. #17984011C22semidirect
45. #17984011C22semidirect
50. #17983998C22semidirect
51. C8×Dic137C22semidirect
52. #125×C137C22semidirect
54. #107×C137C22semidirect
55. #106×C137C22semidirect
56. C4⋊C8×C137C22semidirect
175. C137#3123semidirect
182. #478D137semidirect
202. #125C2×D137semidirect
218. #107C2×D137semidirect
220. #106C2×D137semidirect
222. C4⋊C8C2×D137semidirect

Maximal subgroups

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

subgroupindex
#225780162
#225778752
#225777472
#225777462
#225777002
#225776092
#225776082
#225775552
#225775022
#225775002
#225774942
#225774902
#225774632
#478×C1372
#3123137