Finite Group Databasebeta

← C137×#8250 #46857636

Subgroups — C137×#8250 #46857636

Order 35072.

400 conjugacy classes of subgroups.

The subgroup lattice has 340 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#2966×C137C211yesmaximal, characteristic
2#1746×C137C211yesmaximal, characteristic, complemented
3#1524×C137C211yesmaximal, characteristic
4#1500×C137C211yesmaximal, characteristic
5C8×M16×C137C211yesmaximal, characteristic, complemented
6#1302×C137C211yesmaximal, characteristic, complemented
7#1300×C137C211yesmaximal, characteristic, complemented
8#529×C137C2211yescharacteristic
9#446×C137C411yescharacteristic, complemented
10C8×Q8×C137C411yescharacteristic, complemented
11#436×C137C411yescharacteristic, complemented
12#435×C137—21no—
13C8×D4×C137—21no—
14C8×D4×C137C411yescharacteristic, complemented
15#433×C137—21no—
16#432×C137C2211yescharacteristic
17#431×C137C2211yescharacteristic
18#405×C137—21no—
19C4×M16×C137C2211yescharacteristic
20C2×#97×C137C2211yescharacteristic
21C2×C4×C8×C137C2211yescharacteristic
22#376×C137C2211yescharacteristic
23#336×C137—21no—
24#336×C137—21no—
25#330×C137C412yescomplemented
26#329×C137C412yescomplemented
27#328×C137C412yes—
28#327×C137C412yes—
29#322×C137—21no—
30C82×C137—21no—
31C2×M16×C137—21no—
32C2×M16×C137—21no—
33C22×C8×C137—21no—
34C22×C8×C137—21no—
35C22×C8×C137D411yescharacteristic
36C22×C8×C137D411yescharacteristic
37#126×C137—21no—
38#125×C137C2×C411yescharacteristic
39#124×C137C2×C411yescharacteristic
40#123×C137—21no—
41C22⋊Q8×C137C2×C411yescharacteristic
42#121×C137C2×C411yescharacteristic
43C4×Q8×C137C2×C411yescharacteristic
44C4×D4×C137—21no—
45C4×D4×C137C2×C411yescharacteristic
46#117×C137—21no—
47C2×C42×C137—21no—
48C2×C42×C137C2311yescharacteristic
49C4⋊C8×C137—21no—
50C4⋊C8×C137—21no—
51C4⋊C8×C137—21no—
52C4⋊C8×C137C2×C411yescharacteristic
53C4⋊C8×C137C2×C411yescharacteristic
54C22⋊C8×C137—21no—
55C22⋊C8×C137—21no—
56C22⋊C8×C137—21no—
57C22⋊C8×C137C2×C412yes—
58#97×C137—21no—
59#97×C137—21no—
60#97×C137—21no—
61#97×C137C2×C412yes—
62#97×C137C2×C412yes—
63C4×C8×C137—21no—
64C4×C8×C137—21no—
65C4×C8×C137—21no—
66C4×C8×C137C2×C411yescharacteristic
67C4×C8×C137C2×C411yescharacteristic
68C4×C8×C137D411yescharacteristic
69C4×C8×C137D411yescharacteristic
70#95×C137—21no—
71#95×C137C2×C412yes—
72C2×Q8×C137C4211yescharacteristic
73C2×D4×C137—21no—
74C2×D4×C137C4211yescharacteristic
75C22×C4×C137—21no—
76C22×C4×C137—22no—
77C22×C4×C137C22×C411yescharacteristic
78C22×C4×C137C2×D411yescharacteristic
79C22×C4×C137#4111yescharacteristic
80M16×C137—24no—
81M16×C137—42no—
82C2×C8×C137—22no—
83C2×C8×C137—21no—
84C2×C8×C137—21no—
85C2×C8×C137—22no—
86C2×C8×C137—22no—
87C2×C8×C137—41no—
88C2×C8×C137—41no—
89C2×C8×C137—42no—
90C2×C8×C137—41no—
91C2×C8×C137—41no—
92C2×C8×C137C22⋊C411yescharacteristic
93C2×C8×C137C22⋊C411yescharacteristic
94C2×C8×C137C22⋊C411yescharacteristic
95C2×C8×C137C22⋊C411yescharacteristic
96C4⋊C4×C137—21no—
97C4⋊C4×C137—21no—
98C4⋊C4×C137—21no—
99C4⋊C4×C137—21no—
100C4⋊C4×C137C4211yescharacteristic
101C4⋊C4×C137C4211yescharacteristic
102C22⋊C4×C137—21no—
103C22⋊C4×C137—21no—
104C22⋊C4×C137—21no—
105C22⋊C4×C137—21no—
106C22⋊C4×C137—21no—
107C42×C137—21no—
108C42×C137—41no—
109C42×C137—41no—
110C42×C137C22×C411yescharacteristic
111C42×C137C22×C411yescharacteristic
112C42×C137C2×D411yescharacteristic
113C42×C137#4111yescharacteristic
114C23×C137—21no—
115C23×C137C4×D411yescharacteristic
116Q8×C137—41no—
117D4×C137—41no—
118D4×C137—42no—
119C2×C4×C137—22no—
120C2×C4×C137—21no—
121C2×C4×C137—21no—
122C2×C4×C137—21no—
123C2×C4×C137—21no—
124C2×C4×C137—21no—
125C2×C4×C137—41no—
126C2×C4×C137—41no—
127C2×C4×C137—41no—
128C2×C4×C137—42no—
129C2×C4×C137—42no—
130C2×C4×C137—42no—
131C2×C4×C137C2×C4211yescharacteristic
132C2×C4×C137C2×C22⋊C411yescharacteristic
133C2×C4×C137#11711yescharacteristic
134C2×C4×C137C4×D411yescharacteristic
135C2×C4×C137C4×D411yescharacteristic
136C2×C4×C137C4×D411yescharacteristic
137C8×C137—24no—
138C8×C137—21no—
139C8×C137—21no—
140C8×C137—21no—
141C8×C137—21no—
142C8×C137—42no—
143C8×C137—42no—
144C8×C137—41no—
145C8×C137—41no—
146C22×C137—21no—
147C22×C137—21no—
148C22×C137—41no—
149C22×C137—41no—
150C22×C137—41no—
151C22×C137C4×C22⋊C411yescharacteristic
152C22×C137#44312yes—
153C4×C137—21no—
154C4×C137—21no—
155C4×C137—21no—
156C4×C137—21no—
157C4×C137—21no—
158C4×C137—41no—
159C4×C137—41no—
160C4×C137—41no—
161C4×C137—82no—
162C4×C137#42111yescharacteristic
163C4×C137#42111yescharacteristic
164C2×C137—22no—
165C2×C137—81no—
166C2×C137#175711yescharacteristic
167C2×C137#175911yescharacteristic
168C2×C137#176411yescharacteristic
169#8250C13711yesmaximal, characteristic, direct product factor, Sylow
170C137#825011yescharacteristic, direct product factor, Sylow
171#2966C2×C13711yescharacteristic
172#1746C2×C13711yescharacteristic, complemented
173#1524C2×C13711yescharacteristic
174#1500C2×C13711yescharacteristic
175C8×M16C2×C13711yescharacteristic, complemented
176#1302C2×C13711yescharacteristic, complemented
177#1300C2×C13711yescharacteristic, complemented
178#529C22×C13711yescharacteristic
179#446C4×C13711yescharacteristic, complemented
180C8×Q8C4×C13711yescharacteristic, complemented
181#436C4×C13711yescharacteristic, complemented
182#435—21no—
183C8×D4—21no—
184C8×D4C4×C13711yescharacteristic, complemented
185#433—21no—
186#432C22×C13711yescharacteristic
187#431C22×C13711yescharacteristic
188#405—21no—
189C4×M16C22×C13711yescharacteristic
190C2×#97C22×C13711yescharacteristic
191C2×C4×C8C22×C13711yescharacteristic
192#376C22×C13711yescharacteristic
193#336—21no—
194#336—21no—
195#330C4×C13712yescomplemented
196#329C4×C13712yescomplemented
197#328C4×C13712yes—
198#327C4×C13712yes—
199#322—21no—
200C82—21no—
201C2×M16—21no—
202C2×M16—21no—
203C22×C8—21no—
204C22×C8—21no—
205C22×C8D4×C13711yescharacteristic
206C22×C8D4×C13711yescharacteristic
207#126—21no—
208#125C2×C4×C13711yescharacteristic
209#124C2×C4×C13711yescharacteristic
210#123—21no—
211C22⋊Q8C2×C4×C13711yescharacteristic
212#121C2×C4×C13711yescharacteristic
213C4×Q8C2×C4×C13711yescharacteristic
214C4×D4—21no—
215C4×D4C2×C4×C13711yescharacteristic
216#117—21no—
217C2×C42—21no—
218C2×C42C23×C13711yescharacteristic
219C4⋊C8—21no—
220C4⋊C8—21no—
221C4⋊C8—21no—
222C4⋊C8C2×C4×C13711yescharacteristic
223C4⋊C8C2×C4×C13711yescharacteristic
224C22⋊C8—21no—
225C22⋊C8—21no—
226C22⋊C8—21no—
227C22⋊C8C2×C4×C13712yes—
228#97—21no—
229#97—21no—
230#97—21no—
231#97C2×C4×C13712yes—
232#97C2×C4×C13712yes—
233C4×C8—21no—
234C4×C8—21no—
235C4×C8—21no—
236C4×C8C2×C4×C13711yescharacteristic
237C4×C8C2×C4×C13711yescharacteristic
238C4×C8D4×C13711yescharacteristic
239C4×C8D4×C13711yescharacteristic
240#95—21no—
241#95C2×C4×C13712yes—
242C2×Q8C42×C13711yescharacteristic
243C2×D4—21no—
244C2×D4C42×C13711yescharacteristic
245C22×C4—21no—
246C22×C4—22no—
247C22×C4C22×C4×C13711yescharacteristic
248C22×C4C2×D4×C13711yescharacteristic
249C22×C4#41×C13711yescharacteristic
250M16—24no—
251M16—42no—
252C2×C8—22no—
253C2×C8—21no—
254C2×C8—21no—
255C2×C8—22no—
256C2×C8—22no—
257C2×C8—41no—
258C2×C8—41no—
259C2×C8—42no—
260C2×C8—41no—
261C2×C8—41no—
262C2×C8C22⋊C4×C13711yescharacteristic
263C2×C8C22⋊C4×C13711yescharacteristic
264C2×C8C22⋊C4×C13711yescharacteristic
265C2×C8C22⋊C4×C13711yescharacteristic
266C4⋊C4—21no—
267C4⋊C4—21no—
268C4⋊C4—21no—
269C4⋊C4—21no—
270C4⋊C4C42×C13711yescharacteristic
271C4⋊C4C42×C13711yescharacteristic
272C22⋊C4—21no—
273C22⋊C4—21no—
274C22⋊C4—21no—
275C22⋊C4—21no—
276C22⋊C4—21no—
277C42—21no—
278C42—41no—
279C42—41no—
280C42C22×C4×C13711yescharacteristic
281C42C22×C4×C13711yescharacteristic
282C42C2×D4×C13711yescharacteristic
283C42#41×C13711yescharacteristic
284C23—21no—
285C23C4×D4×C13711yescharacteristic
286Q8—41no—
287D4—41no—
288D4—42no—
289C2×C4—22no—
290C2×C4—21no—
291C2×C4—21no—
292C2×C4—21no—
293C2×C4—21no—
294C2×C4—21no—
295C2×C4—41no—
296C2×C4—41no—
297C2×C4—41no—
298C2×C4—42no—
299C2×C4—42no—
300C2×C4—42no—
301C2×C4C2×C42×C13711yescharacteristic
302C2×C4C2×C22⋊C4×C13711yescharacteristic
303C2×C4#117×C13711yescharacteristic
304C2×C4C4×D4×C13711yescharacteristic
305C2×C4C4×D4×C13711yescharacteristic
306C2×C4C4×D4×C13711yescharacteristic
307C8—24no—
308C8—21no—
309C8—21no—
310C8—21no—
311C8—21no—
312C8—42no—
313C8—42no—
314C8—41no—
315C8—41no—
316C22—21no—
317C22—21no—
318C22—41no—
319C22—41no—
320C22—41no—
321C22C4×C22⋊C4×C13711yescharacteristic
322C22#443×C13712yes—
323C4—21no—
324C4—21no—
325C4—21no—
326C4—21no—
327C4—21no—
328C4—41no—
329C4—41no—
330C4—41no—
331C4—82no—
332C4#421×C13711yescharacteristic
333C4#421×C13711yescharacteristic
334C2—22no—
335C2—81no—
336C2#1757×C13711yescharacteristic
337C2#1759×C13711yescharacteristic
338C2#1764×C13711yescharacteristic
subgroupG/H
centerC2×C4×C137C2×C22⋊C4
commutatorC2×C4C2×C42×C137
Frattini, cosocleC2×C42C23×C137
socleC22×C137C4×C22⋊C4
prime radical#121×C137C2×C4
LCSG ▷ C2×C4 ▷ C22 ▷ C1
UCSC1 ◁ C2×C4×C137 ◁ C2×C4×C8×C137 ◁ G

Complements

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

NH ≅ G/Ntype
2. #1746×C137C2semidirect
5. C8×M16×C137C2semidirect
6. #1302×C137C2semidirect
7. #1300×C137C2semidirect
9. #446×C137C4semidirect
10. C8×Q8×C137C4semidirect
11. #436×C137C4semidirect
14. C8×D4×C137C4semidirect
25. #330×C137C4semidirect
26. #329×C137C4semidirect
169. #8250C137direct (N × H)
170. C137#8250direct (N × H)
172. #1746C2×C137semidirect
175. C8×M16C2×C137semidirect
176. #1302C2×C137semidirect
177. #1300C2×C137semidirect
179. #446C4×C137semidirect
180. C8×Q8C4×C137semidirect
181. #436C4×C137semidirect
184. C8×D4C4×C137semidirect
195. #330C4×C137semidirect
196. #329C4×C137semidirect

Maximal subgroups

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

subgroupindex
#2966×C1372
#1746×C1372
#1524×C1372
#1500×C1372
C8×M16×C1372
#1302×C1372
#1300×C1372
#8250137