Finite Group Databasebeta

← #3125×C137 #31778136

Subgroups — #3125×C137 #31778136

Order 17536.

384 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#542×C137C211yesmaximal, characteristic, complemented
2#520×C137C211yesmaximal, characteristic, complemented
3#483×C137C211yesmaximal, characteristic, complemented
4#481×C137C211yesmaximal, characteristic, complemented
5#476×C137C211yesmaximal, characteristic, complemented
6#475×C137C211yesmaximal, characteristic, complemented
7#464×C137C211yesmaximal, characteristic, complemented
8#463×C137C211yesmaximal, characteristic, complemented
9#452×C137C211yesmaximal, characteristic, complemented
10#449×C137C211yesmaximal, characteristic
11#442×C137C211yesmaximal, characteristic, complemented
12#441×C137C211yesmaximal, characteristic, complemented
13C4×SD16×C137C211yesmaximal, characteristic, complemented
14C4×SD16×C137C211yesmaximal, characteristic, complemented
15#433×C137C211yesmaximal, characteristic, complemented
16C2×#41×C137C2211yescharacteristic
17C2×Q16×C137C2211yescharacteristic
18C2×SD16×C137C2211yescharacteristic
19C2×SD16×C137C2211yescharacteristic
20C2×D8×C137C2211yescharacteristic
21C4⋊Q8×C137C2211yescharacteristic
22#126×C137—21no—
23#124×C137—21no—
24#124×C137C2211yescharacteristic
25#123×C137—21no—
26C22⋊Q8×C137—21no—
27C22⋊Q8×C137C2211yescharacteristic
28#121×C137C2211yescharacteristic
29C4×Q8×C137C2211yescharacteristic
30C4×Q8×C137C2211yescharacteristic
31C4×Q8×C137C2211yescharacteristic
32C4×D4×C137—21no—
33C4×D4×C137—21no—
34C4×D4×C137C2211yescharacteristic
35C4×D4×C137C2211yescharacteristic
36C4×D4×C137C2211yescharacteristic
37#117×C137—21no—
38#117×C137C2211yescharacteristic
39C2×C4⋊C4×C137—21no—
40C2×C4⋊C4×C137C2211yescharacteristic
41#106×C137C2211yescharacteristic
42#106×C137C2211yescharacteristic
43#106×C137C2211yescharacteristic
44#106×C137C2211yescharacteristic
45C4⋊C8×C137C2211yescharacteristic
46C4⋊C8×C137C2211yescharacteristic
47#103×C137C2211yescharacteristic
48#103×C137C2211yescharacteristic
49#103×C137C2211yescharacteristic
50#103×C137C2211yescharacteristic
51#102×C137C2211yescharacteristic
52#102×C137C2211yescharacteristic
53#102×C137C2211yescharacteristic
54#102×C137C2211yescharacteristic
55C22⋊C8×C137C2211yescharacteristic
56C22⋊C8×C137C2211yescharacteristic
57#97×C137C2211yescharacteristic
58C4×C8×C137C2211yescharacteristic
59#41×C137—22no—
60#41×C137—22no—
61C2×Q8×C137—21no—
62C2×Q8×C137C2311yescharacteristic
63C2×Q8×C137C2311yescharacteristic
64C2×D4×C137—21no—
65C2×D4×C137—21no—
66C2×D4×C137C2311yescharacteristic
67C2×D4×C137C2311yescharacteristic
68C22×C4×C137—21no—
69C22×C4×C137—21no—
70C22×C4×C137—21no—
71C22×C4×C137—21no—
72C22×C4×C137—21no—
73C22×C4×C137C2311yescharacteristic
74Q16×C137—22no—
75SD16×C137—22no—
76SD16×C137—22no—
77D8×C137—22no—
78C2×C8×C137C2311yescharacteristic
79C2×C8×C137C2311yescharacteristic
80C2×C8×C137C2311yescharacteristic
81C2×C8×C137C2311yescharacteristic
82C4⋊C4×C137—21no—
83C4⋊C4×C137—21no—
84C4⋊C4×C137—21no—
85C4⋊C4×C137—21no—
86C4⋊C4×C137—21no—
87C4⋊C4×C137—21no—
88C4⋊C4×C137D411yescharacteristic
89C4⋊C4×C137D411yescharacteristic
90C4⋊C4×C137C2311yescharacteristic
91C4⋊C4×C137C2311yescharacteristic
92C4⋊C4×C137C2311yescharacteristic
93C4⋊C4×C137C2311yescharacteristic
94C22⋊C4×C137—21no—
95C22⋊C4×C137—21no—
96C22⋊C4×C137—21no—
97C22⋊C4×C137—21no—
98C22⋊C4×C137—21no—
99C22⋊C4×C137—21no—
100C22⋊C4×C137—21no—
101C22⋊C4×C137D411yescharacteristic
102C22⋊C4×C137D411yescharacteristic
103C42×C137—21no—
104C42×C137—21no—
105C42×C137—21no—
106C42×C137C2311yescharacteristic
107C42×C137C2311yescharacteristic
108C23×C137—21no—
109C23×C137—21no—
110C23×C137C2×D411yescharacteristic
111Q8×C137—21no—
112Q8×C137—21no—
113Q8×C137—21no—
114Q8×C137—81no—
115Q8×C137#4112yes—
116D4×C137—24no—
117D4×C137—21no—
118D4×C137—21no—
119D4×C137—21no—
120D4×C137—81no—
121D4×C137#4112yes—
122C2×C4×C137—24no—
123C2×C4×C137—21no—
124C2×C4×C137—21no—
125C2×C4×C137—21no—
126C2×C4×C137—21no—
127C2×C4×C137—21no—
128C2×C4×C137—21no—
129C2×C4×C137—21no—
130C2×C4×C137—21no—
131C2×C4×C137—81no—
132C2×C4×C137—81no—
133C2×C4×C137—81no—
134C2×C4×C137—81no—
135C2×C4×C137C2×D411yescharacteristic
136C2×C4×C137C2×D411yescharacteristic
137C2×C4×C137C2×D411yescharacteristic
138C2×C4×C137C2×D411yescharacteristic
139C2×C4×C137C2×D411yescharacteristic
140C2×C4×C137C2411yescharacteristic
141C8×C137—21no—
142C8×C137—21no—
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×C137—41no—
152C22×C137—41no—
153C22×C137—41no—
154C22×C137C22×D411yescharacteristic
155C4×C137—21no—
156C4×C137—22no—
157C4×C137—21no—
158C4×C137—21no—
159C4×C137—21no—
160C4×C137—21no—
161C4×C137—41no—
162C4×C137—41no—
163C4×C137—41no—
164C4×C137—41no—
165C4×C137—41no—
166C4×C137C2×#4111yescharacteristic
167C4×C137C2×#4111yescharacteristic
168C2×C137—41no—
169C2×C137—42no—
170C2×C137—81no—
171C2×C137#52511yescharacteristic
172C2×C137#57711yescharacteristic
173C2×C137#57711yescharacteristic
174C137#312511yescharacteristic, direct product factor, Sylow
175#3125C13711yesmaximal, characteristic, direct product factor, Sylow
176#542C2×C13711yescharacteristic, complemented
177#520C2×C13711yescharacteristic, complemented
178#483C2×C13711yescharacteristic, complemented
179#481C2×C13711yescharacteristic, complemented
180#476C2×C13711yescharacteristic, complemented
181#475C2×C13711yescharacteristic, complemented
182#464C2×C13711yescharacteristic, complemented
183#463C2×C13711yescharacteristic, complemented
184#452C2×C13711yescharacteristic, complemented
185#449C2×C13711yescharacteristic
186#442C2×C13711yescharacteristic, complemented
187#441C2×C13711yescharacteristic, complemented
188C4×SD16C2×C13711yescharacteristic, complemented
189C4×SD16C2×C13711yescharacteristic, complemented
190#433C2×C13711yescharacteristic, complemented
191C2×#41C22×C13711yescharacteristic
192C2×Q16C22×C13711yescharacteristic
193C2×SD16C22×C13711yescharacteristic
194C2×SD16C22×C13711yescharacteristic
195C2×D8C22×C13711yescharacteristic
196C4⋊Q8C22×C13711yescharacteristic
197#126—21no—
198#124—21no—
199#124C22×C13711yescharacteristic
200#123—21no—
201C22⋊Q8—21no—
202C22⋊Q8C22×C13711yescharacteristic
203#121C22×C13711yescharacteristic
204C4×Q8C22×C13711yescharacteristic
205C4×Q8C22×C13711yescharacteristic
206C4×Q8C22×C13711yescharacteristic
207C4×D4—21no—
208C4×D4—21no—
209C4×D4C22×C13711yescharacteristic
210C4×D4C22×C13711yescharacteristic
211C4×D4C22×C13711yescharacteristic
212#117—21no—
213#117C22×C13711yescharacteristic
214C2×C4⋊C4—21no—
215C2×C4⋊C4C22×C13711yescharacteristic
216#106C22×C13711yescharacteristic
217#106C22×C13711yescharacteristic
218#106C22×C13711yescharacteristic
219#106C22×C13711yescharacteristic
220C4⋊C8C22×C13711yescharacteristic
221C4⋊C8C22×C13711yescharacteristic
222#103C22×C13711yescharacteristic
223#103C22×C13711yescharacteristic
224#103C22×C13711yescharacteristic
225#103C22×C13711yescharacteristic
226#102C22×C13711yescharacteristic
227#102C22×C13711yescharacteristic
228#102C22×C13711yescharacteristic
229#102C22×C13711yescharacteristic
230C22⋊C8C22×C13711yescharacteristic
231C22⋊C8C22×C13711yescharacteristic
232#97C22×C13711yescharacteristic
233C4×C8C22×C13711yescharacteristic
234#41—22no—
235#41—22no—
236C2×Q8—21no—
237C2×Q8C23×C13711yescharacteristic
238C2×Q8C23×C13711yescharacteristic
239C2×D4—21no—
240C2×D4—21no—
241C2×D4C23×C13711yescharacteristic
242C2×D4C23×C13711yescharacteristic
243C22×C4—21no—
244C22×C4—21no—
245C22×C4—21no—
246C22×C4—21no—
247C22×C4—21no—
248C22×C4C23×C13711yescharacteristic
249Q16—22no—
250SD16—22no—
251SD16—22no—
252D8—22no—
253C2×C8C23×C13711yescharacteristic
254C2×C8C23×C13711yescharacteristic
255C2×C8C23×C13711yescharacteristic
256C2×C8C23×C13711yescharacteristic
257C4⋊C4—21no—
258C4⋊C4—21no—
259C4⋊C4—21no—
260C4⋊C4—21no—
261C4⋊C4—21no—
262C4⋊C4—21no—
263C4⋊C4D4×C13711yescharacteristic
264C4⋊C4D4×C13711yescharacteristic
265C4⋊C4C23×C13711yescharacteristic
266C4⋊C4C23×C13711yescharacteristic
267C4⋊C4C23×C13711yescharacteristic
268C4⋊C4C23×C13711yescharacteristic
269C22⋊C4—21no—
270C22⋊C4—21no—
271C22⋊C4—21no—
272C22⋊C4—21no—
273C22⋊C4—21no—
274C22⋊C4—21no—
275C22⋊C4—21no—
276C22⋊C4D4×C13711yescharacteristic
277C22⋊C4D4×C13711yescharacteristic
278C42—21no—
279C42—21no—
280C42—21no—
281C42C23×C13711yescharacteristic
282C42C23×C13711yescharacteristic
283C23—21no—
284C23—21no—
285C23C2×D4×C13711yescharacteristic
286Q8—21no—
287Q8—21no—
288Q8—21no—
289Q8—81no—
290Q8#41×C13712yes—
291D4—24no—
292D4—21no—
293D4—21no—
294D4—21no—
295D4—81no—
296D4#41×C13712yes—
297C2×C4—24no—
298C2×C4—21no—
299C2×C4—21no—
300C2×C4—21no—
301C2×C4—21no—
302C2×C4—21no—
303C2×C4—21no—
304C2×C4—21no—
305C2×C4—21no—
306C2×C4—81no—
307C2×C4—81no—
308C2×C4—81no—
309C2×C4—81no—
310C2×C4C2×D4×C13711yescharacteristic
311C2×C4C2×D4×C13711yescharacteristic
312C2×C4C2×D4×C13711yescharacteristic
313C2×C4C2×D4×C13711yescharacteristic
314C2×C4C2×D4×C13711yescharacteristic
315C2×C4C24×C13711yescharacteristic
316C8—21no—
317C8—21no—
318C8—21no—
319C8—21no—
320C22—21no—
321C22—22no—
322C22—21no—
323C22—21no—
324C22—41no—
325C22—41no—
326C22—41no—
327C22—41no—
328C22—41no—
329C22C22×D4×C13711yescharacteristic
330C4—21no—
331C4—22no—
332C4—21no—
333C4—21no—
334C4—21no—
335C4—21no—
336C4—41no—
337C4—41no—
338C4—41no—
339C4—41no—
340C4—41no—
341C4C2×#41×C13711yescharacteristic
342C4C2×#41×C13711yescharacteristic
343C2—41no—
344C2—42no—
345C2—81no—
346C2#525×C13711yescharacteristic
347C2#577×C13711yescharacteristic
348C2#577×C13711yescharacteristic
subgroupG/H
center, socleC22×C137C22×D4
commutator, Frattini, cosocleC2×C4C24×C137
prime radical#449×C137C2
LCSG ▷ C2×C4 ▷ C2 ▷ C1
UCSC1 ◁ C22×C137 ◁ #117×C137 ◁ G

Complements

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

NH ≅ G/Ntype
1. #542×C137C2semidirect
2. #520×C137C2semidirect
3. #483×C137C2semidirect
4. #481×C137C2semidirect
5. #476×C137C2semidirect
6. #475×C137C2semidirect
7. #464×C137C2semidirect
8. #463×C137C2semidirect
9. #452×C137C2semidirect
11. #442×C137C2semidirect
12. #441×C137C2semidirect
13. C4×SD16×C137C2semidirect
14. C4×SD16×C137C2semidirect
15. #433×C137C2semidirect
174. C137#3125direct (N × H)
175. #3125C137direct (N × H)
176. #542C2×C137semidirect
177. #520C2×C137semidirect
178. #483C2×C137semidirect
179. #481C2×C137semidirect
180. #476C2×C137semidirect
181. #475C2×C137semidirect
182. #464C2×C137semidirect
183. #463C2×C137semidirect
184. #452C2×C137semidirect
186. #442C2×C137semidirect
187. #441C2×C137semidirect
188. C4×SD16C2×C137semidirect
189. C4×SD16C2×C137semidirect
190. #433C2×C137semidirect

Maximal subgroups

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

subgroupindex
#542×C1372
#520×C1372
#483×C1372
#481×C1372
#476×C1372
#475×C1372
#464×C1372
#463×C1372
#452×C1372
#449×C1372
#442×C1372
#441×C1372
C4×SD16×C1372
#433×C1372
#3125137