Finite Group Databasebeta

← #31790808

Subgroups — #31790808

Order 17536.

396 conjugacy classes of subgroups.

The subgroup lattice has 350 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#22577939C211yesmaximal, characteristic, complemented
2#41×Dic137C211yesmaximal, characteristic, complemented
3#22577741C211yesmaximal, characteristic, complemented
4#22577728C211yesmaximal, characteristic, complemented
5#22577708C211yesmaximal, characteristic, complemented
6#22577701C211yesmaximal, characteristic, complemented
7#22577627C211yesmaximal, characteristic, complemented
8#22577620C211yesmaximal, characteristic, complemented
9#22577548C211yesmaximal, characteristic
10Q16×Dic137C211yesmaximal, characteristic, complemented
11SD16×Dic137C211yesmaximal, characteristic, complemented
12#22577488C211yesmaximal, characteristic, complemented
13#22577483C211yesmaximal, characteristic, complemented
14#22577453C211yesmaximal, characteristic, complemented
15#452×C137C211yesmaximal, characteristic, complemented
16C2×#14555497C2211yescharacteristic
17C2×#14555495C2211yescharacteristic
18#17984083—21no—
19#17984079C2211yescharacteristic
20#17984077C2211yescharacteristic
21#17984072—21no—
22#17984062C2211yescharacteristic
23Q8×Dic137C2211yescharacteristic
24Q8×Dic137C2211yescharacteristic
25Q8×Dic137C2211yescharacteristic
26D4×Dic137—21no—
27D4×Dic137—21no—
28D4×Dic137C2211yescharacteristic
29D4×Dic137C2211yescharacteristic
30#17984050C2211yescharacteristic
31#17984048—21no—
32#17984047—21no—
33#17984047C2211yescharacteristic
34#17984046—21no—
35C2×C4×Dic137—21no—
36C2×C4×Dic137C2211yescharacteristic
37#17984022C2211yescharacteristic
38#17984021C2211yescharacteristic
39#17984020C2211yescharacteristic
40#17984019C2211yescharacteristic
41#17984017C2211yescharacteristic
42#17984017C2211yescharacteristic
43#17984012C2211yescharacteristic
44#17984012C2211yescharacteristic
45#17984012C2211yescharacteristic
46#17984010C2211yescharacteristic
47#17984010C2211yescharacteristic
48#17984009C2211yescharacteristic
49C22⋊#11764040C2211yescharacteristic
50C8×Dic137C2211yescharacteristic
51C8×Dic137C2211yescharacteristic
52C2×#41×C137C2211yescharacteristic
53C2×Q16×C137C2211yescharacteristic
54C2×SD16×C137C2211yescharacteristic
55C22⋊Q8×C137C2211yescharacteristic
56#103×C137C2211yescharacteristic
57#102×C137C2211yescharacteristic
58C22⋊C8×C137C2211yescharacteristic
59C2×D548C2311yescharacteristic
60C2×C22⋊D137—21no—
61C22×Dic137—21no—
62C22×Dic137—21no—
63C22×Dic137D411yescharacteristic
64C22×Dic137D411yescharacteristic
65C2×C4×D137—21no—
66#14555497—22no—
67#14555495—22no—
68C2×#11764040C2311yescharacteristic
69C2×#11764040C2311yescharacteristic
70C4⋊Dic137—21no—
71C4⋊Dic137—21no—
72C4⋊Dic137—21no—
73C4⋊Dic137—21no—
74C4⋊Dic137C2311yescharacteristic
75C4⋊Dic137C2311yescharacteristic
76C4⋊Dic137C2311yescharacteristic
77#14555489—21no—
78C22⋊Dic137—21no—
79C22⋊Dic137—21no—
80C22⋊Dic137—21no—
81#14555487—21no—
82#14555487—21no—
83#14555487—21no—
84#14555487—21no—
85C4×Dic137—21no—
86C4×Dic137—21no—
87C4×Dic137—21no—
88C4×Dic137—21no—
89C4×Dic137D411yescharacteristic
90C4×Dic137D411yescharacteristic
91C4×Dic137C2311yescharacteristic
92C4×Dic137C2311yescharacteristic
93#41×C137—22no—
94#41×C137—22no—
95C2×Q8×C137C2311yescharacteristic
96C2×Q8×C137C2311yescharacteristic
97C2×D4×C137—21no—
98C2×D4×C137C2311yescharacteristic
99C22×C4×C137—21no—
100C22×C4×C137C2311yescharacteristic
101Q16×C137—22no—
102SD16×C137—22no—
103C2×C8×C137C2311yescharacteristic
104C2×C8×C137C2311yescharacteristic
105C4⋊C4×C137—21no—
106C4⋊C4×C137C2311yescharacteristic
107C22⋊C4×C137—21no—
108C22×D137—21no—
109D548—21no—
110D548—21no—
111C22⋊D137—42no—
112C2×Dic137—22no—
113C2×Dic137—21no—
114C2×Dic137—21no—
115C2×Dic137—21no—
116C2×Dic137—41no—
117C2×Dic137—42no—
118C2×Dic137—42no—
119C2×Dic137C2×D411yescharacteristic
120C2×Dic137C2×D411yescharacteristic
121C2×Dic137C2×D411yescharacteristic
122C2×Dic137C2×D411yescharacteristic
123C4×D137—42no—
124#11764040—21no—
125#11764040—21no—
126C23×C137—21no—
127C23×C137C2×D411yescharacteristic
128Q8×C137—21no—
129Q8×C137—21no—
130Q8×C137—21no—
131Q8×C137#4112yes—
132D4×C137—24no—
133D4×C137—21no—
134D4×C137#4112yes—
135C2×C4×C137—24no—
136C2×C4×C137—21no—
137C2×C4×C137—21no—
138C2×C4×C137—21no—
139C2×C4×C137—21no—
140C2×C4×C137—21no—
141C2×C4×C137C2×D411yescharacteristic
142C2×C4×C137C2411yescharacteristic
143C8×C137—21no—
144C8×C137—21no—
145C22×C137—21no—
146C22×C137—22no—
147C22×C137—21no—
148C22×C137—21no—
149C22×C137—41no—
150C22×C137—41no—
151C22×C137C22×D411yescharacteristic
152C2×D137—41no—
153C2×D137—41no—
154C2×D137—41no—
155C4×C137—21no—
156C4×C137—22no—
157C4×C137—41no—
158C4×C137—41no—
159C4×C137C2×#4111yescharacteristic
160C4×C137C2×#4111yescharacteristic
161Dic137—21no—
162Dic137—21no—
163Dic137—21no—
164Dic137—41no—
165Dic137—41no—
166Dic137—41no—
167Dic137#13512yes—
168C2×C137—41no—
169C2×C137—42no—
170C2×C137#52511yescharacteristic
171C2×C137C2×#13511yescharacteristic
172C2×C137#57511yescharacteristic
173D137—81no—
174C137#310211yescharacteristic, complemented, Sylow
175#3102—1371nomaximal, Sylow
176#529—1371no—
177C4×#41—1371no—
178#485—1371no—
179#482—1371no—
180#479—1371no—
181#478—1371no—
182#464—1371no—
183#463—1371no—
184#452D13711yescharacteristic, complemented
185#449—1371no—
186C4×Q16—1371no—
187C4×SD16—1371no—
188C4×SD16—1371no—
189C4×D8—1371no—
190#431—1371no—
191C2×#41C2×D13711yescharacteristic
192C2×Q16C2×D13711yescharacteristic
193C2×SD16—1371no—
194C2×SD16C2×D13711yescharacteristic
195C2×D8—1371no—
196#126—2741no—
197#125—1371no—
198#124—1371no—
199#123—2741no—
200C22⋊Q8C2×D13711yescharacteristic
201#121—1371no—
202C4×Q8—1371no—
203C4×Q8—1371no—
204C4×Q8—1371no—
205C4×D4—2741no—
206C4×D4—2741no—
207C4×D4—2741no—
208C4×D4—1371no—
209C4×D4—1371no—
210C4×D4—1371no—
211#117—2741no—
212#117—2741no—
213#117—1371no—
214C2×C42—2741no—
215C2×C42—1371no—
216#107—1371no—
217#107—1371no—
218#106—1371no—
219#106—1371no—
220C4⋊C8—1371no—
221C4⋊C8—1371no—
222#103—1371no—
223#103—1371no—
224#103—1371no—
225#103C2×D13711yescharacteristic
226#102—1371no—
227#102—1371no—
228#102—1371no—
229#102C2×D13711yescharacteristic
230C22⋊C8—1371no—
231C22⋊C8C2×D13711yescharacteristic
232C4×C8—1371no—
233C4×C8—1371no—
234#41—22no—
235#41—22no—
236C2×Q8C22×D13711yescharacteristic
237C2×Q8C22×D13711yescharacteristic
238C2×D4—21no—
239C2×D4—2741no—
240C2×D4—1371no—
241C2×D4C22×D13711yescharacteristic
242C22×C4—21no—
243C22×C4—2741no—
244C22×C4—2741no—
245C22×C4—2741no—
246C22×C4—1371no—
247C22×C4—1371no—
248C22×C4C22×D13711yescharacteristic
249Q16—22no—
250SD16—22no—
251SD16—2742no—
252D8—2742no—
253C2×C8—1371no—
254C2×C8—1371no—
255C2×C8C22×D13711yescharacteristic
256C2×C8C22×D13711yescharacteristic
257C4⋊C4—21no—
258C4⋊C4—2741no—
259C4⋊C4—2741no—
260C4⋊C4—2741no—
261C4⋊C4—2741no—
262C4⋊C4—2741no—
263C4⋊C4—1371no—
264C4⋊C4—1371no—
265C4⋊C4—1371no—
266C4⋊C4C22×D13711yescharacteristic
267C22⋊C4—21no—
268C22⋊C4—2741no—
269C22⋊C4—2741no—
270C22⋊C4—2741no—
271C22⋊C4—2741no—
272C22⋊C4—2741no—
273C22⋊C4—2741no—
274C22⋊C4—2741no—
275C42—2741no—
276C42—2741no—
277C42—2741no—
278C42—2741no—
279C42—1371no—
280C42—1371no—
281C42—1371no—
282C42—1371no—
283C23—21no—
284C23—2741no—
285C23D4×D13711yescharacteristic
286Q8—21no—
287Q8—21no—
288Q8—21no—
289Q8#1455550912yes—
290D4—24no—
291D4—21no—
292D4—2741no—
293D4—2741no—
294D4—5482no—
295D4#1455550912yes—
296C2×C4—24no—
297C2×C4—21no—
298C2×C4—21no—
299C2×C4—21no—
300C2×C4—21no—
301C2×C4—21no—
302C2×C4—2742no—
303C2×C4—2741no—
304C2×C4—2741no—
305C2×C4—2741no—
306C2×C4—5482no—
307C2×C4—5481no—
308C2×C4—5482no—
309C2×C4—5482no—
310C2×C4—1371no—
311C2×C4—1371no—
312C2×C4—1371no—
313C2×C4—1371no—
314C2×C4D4×D13711yescharacteristic
315C2×C4C23×D13711yescharacteristic
316C8—21no—
317C8—21no—
318C8—2741no—
319C8—2741no—
320C22—21no—
321C22—21no—
322C22—21no—
323C22—22no—
324C22—41no—
325C22—41no—
326C22—5481no—
327C22—5481no—
328C22—5481no—
329C22C2×D4×D13711yescharacteristic
330C4—21no—
331C4—22no—
332C4—41no—
333C4—41no—
334C4—2741no—
335C4—2741no—
336C4—2741no—
337C4—5481no—
338C4—5481no—
339C4—5481no—
340C4—1372no—
341C4C2×#1455550911yescharacteristic
342C4C2×#1455550911yescharacteristic
343C2—41no—
344C2—42no—
345C2—10961no—
346C2#2257790911yescharacteristic
347C2#135×D13711yescharacteristic
348C2#2257820211yescharacteristic
subgroupG/H
centerC22C2×D4×D137
commutator, cosocleC2×C4×C137C24
FrattiniC2×C4C23×D137
Fitting#452×C137C2
socleC22×C137C22×D4
prime radical#22577548C2
hypercenter#452D137
hypocenterC137#3102
LCSG ▷ C2×C4×C137 ▷ C2×C137 ▷ C137
UCSC1 ◁ C22 ◁ C22×C4 ◁ #452

Complements

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

NH ≅ G/Ntype
1. #22577939C2semidirect
2. #41×Dic137C2semidirect
3. #22577741C2semidirect
4. #22577728C2semidirect
5. #22577708C2semidirect
6. #22577701C2semidirect
7. #22577627C2semidirect
8. #22577620C2semidirect
10. Q16×Dic137C2semidirect
11. SD16×Dic137C2semidirect
12. #22577488C2semidirect
13. #22577483C2semidirect
14. #22577453C2semidirect
15. #452×C137C2semidirect
174. C137#3102semidirect
184. #452D137semidirect

Maximal subgroups

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

subgroupindex
#225779392
#41×Dic1372
#225777412
#225777282
#225777082
#225777012
#225776272
#225776202
#225775482
Q16×Dic1372
SD16×Dic1372
#225774882
#225774832
#225774532
#452×C1372
#3102137