Finite Group Databasebeta

← C137×#18529 #46867915

Subgroups — C137×#18529 #46867915

Order 35072.

1630 conjugacy classes of subgroups; only the 246 normal subgroups are shown.

normal subgroups only

This group has 1630 conjugacy classes of subgroups — too many to draw. Only its 246 normal subgroups are shown.

Dotted edges: containments with no intermediate normal subgroup — omitted non-normal subgroups may lie between.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×D42×C137C211yesmaximal, characteristic
2#3138×C137C211yesmaximal, characteristic, complemented
3#3110×C137C212yesmaximal, complemented
4#1639×C137C218yesmaximal, complemented
5#1555×C137C211yesmaximal, characteristic, complemented
6#1490×C137C212yesmaximal, complemented
7C23×D4×C137—22no—
8D42×C137—22no—
9D42×C137—24no—
10D42×C137C2214yes—
11#533×C137C2211yescharacteristic, complemented
12C2×#127×C137C2211yescharacteristic
13C2×#121×C137—22no—
14C2×#120×C137—22no—
15C2×C4×D4×C137C2212yes—
16#490×C137—22no—
17#486×C137C2214yescomplemented
18#484×C137—22no—
19#474×C137C2214yescomplemented
20#461×C137C2214yescomplemented
21#450×C137—22no—
22#440×C137—24no—
23#432×C137C2212yescomplemented
24#418×C137—22no—
25C4×M16×C137C2211yescharacteristic, complemented
26#331×C137C2214yescomplemented
27D4⋊C8×C137C2218yescomplemented
28C25×C137—22no—
29C22×D4×C137—24no—
30C22×D4×C137—28no—
31C22×D4×C137—28no—
32C22×D4×C137—22no—
33C22×D4×C137—416no—
34C22×D4×C137D412yes—
35C22×D4×C137D412yes—
36C23×C4×C137—22no—
37C2×SD16×C137—24no—
38C2×M16×C137—21no—
39C4⋊Q8×C137C2312yescomplemented
40#127×C137—22no—
41#127×C137D412yes—
42#127×C137C2312yes—
43#125×C137—21no—
44C22⋊Q8×C137—22no—
45#121×C137—28no—
46#121×C137—28no—
47#120×C137—28no—
48#120×C137—28no—
49C4×Q8×C137—22no—
50C4×D4×C137—22no—
51C4×D4×C137—22no—
52C4×D4×C137D414yes—
53C4×D4×C137C2314yes—
54#117×C137—21no—
55C2×C4⋊C4×C137D412yes—
56C2×C22⋊C4×C137—22no—
57C2×C22⋊C4×C137—22no—
58C2×C42×C137C2311yescharacteristic
59#107×C137—24no—
60C4⋊C8×C137C2314yescomplemented
61#103×C137—24no—
62#102×C137—28no—
63#102×C137—24no—
64C22⋊C8×C137—22no—
65#97×C137—21no—
66#97×C137—22no—
67C4×C8×C137C2312yescomplemented
68C24×C137—22no—
69C24×C137—28no—
70C24×C137—22no—
71C24×C137—416no—
72C24×C137—84no—
73C2×Q8×C137—22no—
74C2×D4×C137—28no—
75C2×D4×C137—216no—
76C2×D4×C137—22no—
77C2×D4×C137—24no—
78C2×D4×C137—22no—
79C2×D4×C137—24no—
80C2×D4×C137—48no—
81C2×D4×C137—44no—
82C2×D4×C137—432no—
83C2×D4×C137—416no—
84C2×D4×C137—416no—
85C2×D4×C137—416no—
86C2×D4×C137C2×D414yes—
87C2×D4×C137C2×D414yes—
88C22×C4×C137—22no—
89C22×C4×C137—24no—
90C22×C4×C137—24no—
91C22×C4×C137—48no—
92C22×C4×C137C2×D412yes—
93C22×C4×C137C2×D411yescharacteristic
94SD16×C137—48no—
95M16×C137—42no—
96C2×C8×C137—22no—
97C2×C8×C137—24no—
98C4⋊C4×C137—22no—
99C4⋊C4×C137—22no—
100C4⋊C4×C137—22no—
101C4⋊C4×C137—42no—
102C4⋊C4×C137C2×D414yes—
103C22⋊C4×C137—28no—
104C22⋊C4×C137—24no—
105C22⋊C4×C137—24no—
106C22⋊C4×C137—41no—
107C42×C137—41no—
108C42×C137C2×D411yescharacteristic
109C42×C137C2×D412yes—
110C42×C137C2411yescharacteristic
111C23×C137—28no—
112C23×C137—24no—
113C23×C137—24no—
114C23×C137—44no—
115C23×C137—48no—
116C23×C137—432no—
117C23×C137—44no—
118C23×C137—44no—
119C23×C137—88no—
120C23×C137—832no—
121C23×C137#12011yescharacteristic
122Q8×C137—44no—
123D4×C137—28no—
124D4×C137—24no—
125D4×C137—44no—
126D4×C137—48no—
127D4×C137—432no—
128D4×C137—48no—
129D4×C137—82no—
130D4×C137AGL(1,ℤ/8)18yes—
131C2×C4×C137—22no—
132C2×C4×C137—24no—
133C2×C4×C137—24no—
134C2×C4×C137—48no—
135C2×C4×C137—44no—
136C2×C4×C137—48no—
137C2×C4×C137—42no—
138C2×C4×C137#12011yescharacteristic
139C2×C4×C137#12012yes—
140C2×C4×C137C22×D411yescharacteristic
141C2×C4×C137C22×D412yes—
142C8×C137—42no—
143C8×C137—44no—
144C22×C137—22no—
145C22×C137—28no—
146C22×C137—416no—
147C22×C137—44no—
148C22×C137—44no—
149C22×C137—48no—
150C22×C137—44no—
151C22×C137—88no—
152C22×C137—832no—
153C22×C137#45512yes—
154C22×C137C2×#12011yescharacteristic
155C4×C137—22no—
156C4×C137—21no—
157C4×C137—21no—
158C4×C137—44no—
159C4×C137—82no—
160C4×C137C2×AGL(1,ℤ/8)14yes—
161C2×C137—22no—
162C2×C137—48no—
163C2×C137—84no—
164C2×C137#300312yes—
165C2×C137C2×#45511yescharacteristic
166#18529C13711yesmaximal, characteristic, direct product factor, Sylow
167C137#1852911yescharacteristic, direct product factor, Sylow
168C2×D42C2×C13711yescharacteristic
169#3138C2×C13711yescharacteristic, complemented
170#3110C2×C13712yescomplemented
171#1639C2×C13718yescomplemented
172#1555C2×C13711yescharacteristic, complemented
173#1490C2×C13712yescomplemented
174C23×D4—22no—
175D42—22no—
176D42—24no—
177D42C22×C13714yes—
178#533C22×C13711yescharacteristic, complemented
179C2×#127C22×C13711yescharacteristic
180C2×#121—22no—
181C2×#120—22no—
182C2×C4×D4C22×C13712yes—
183#490—22no—
184#486C22×C13714yescomplemented
185#484—22no—
186#474C22×C13714yescomplemented
187#461C22×C13714yescomplemented
188#450—22no—
189#440—24no—
190#432C22×C13712yescomplemented
191#418—22no—
192C4×M16C22×C13711yescharacteristic, complemented
193#331C22×C13714yescomplemented
194D4⋊C8C22×C13718yescomplemented
195C25—22no—
196C22×D4—24no—
197C22×D4—28no—
198C22×D4—28no—
199C22×D4—22no—
200C22×D4—416no—
201C22×D4D4×C13712yes—
202C22×D4D4×C13712yes—
203C23×C4—22no—
204C2×SD16—24no—
205C2×M16—21no—
206C4⋊Q8C23×C13712yescomplemented
207#127—22no—
208#127D4×C13712yes—
209#127C23×C13712yes—
210#125—21no—
211C22⋊Q8—22no—
212#121—28no—
213#121—28no—
214#120—28no—
215#120—28no—
216C4×Q8—22no—
217C4×D4—22no—
218C4×D4—22no—
219C4×D4D4×C13714yes—
220C4×D4C23×C13714yes—
221#117—21no—
222C2×C4⋊C4D4×C13712yes—
223C2×C22⋊C4—22no—
224C2×C22⋊C4—22no—
225C2×C42C23×C13711yescharacteristic
226#107—24no—
227C4⋊C8C23×C13714yescomplemented
228#103—24no—
229#102—28no—
230#102—24no—
231C22⋊C8—22no—
232#97—21no—
233#97—22no—
234C4×C8C23×C13712yescomplemented
235C24—22no—
236C24—28no—
237C24—22no—
238C24—416no—
239C24—84no—
240C2×Q8—22no—
241C2×D4—28no—
242C2×D4—216no—
243C2×D4—22no—
244C2×D4—24no—
245C2×D4—22no—
246C2×D4—24no—
247C2×D4—48no—
248C2×D4—44no—
249C2×D4—432no—
250C2×D4—416no—
251C2×D4—416no—
252C2×D4—416no—
253C2×D4C2×D4×C13714yes—
254C2×D4C2×D4×C13714yes—
255C22×C4—22no—
256C22×C4—24no—
257C22×C4—24no—
258C22×C4—48no—
259C22×C4C2×D4×C13712yes—
260C22×C4C2×D4×C13711yescharacteristic
261SD16—48no—
262M16—42no—
263C2×C8—22no—
264C2×C8—24no—
265C4⋊C4—22no—
266C4⋊C4—22no—
267C4⋊C4—22no—
268C4⋊C4—42no—
269C4⋊C4C2×D4×C13714yes—
270C22⋊C4—28no—
271C22⋊C4—24no—
272C22⋊C4—24no—
273C22⋊C4—41no—
274C42—41no—
275C42C2×D4×C13711yescharacteristic
276C42C2×D4×C13712yes—
277C42C24×C13711yescharacteristic
278C23—28no—
279C23—24no—
280C23—24no—
281C23—44no—
282C23—48no—
283C23—432no—
284C23—44no—
285C23—44no—
286C23—88no—
287C23—832no—
288C23#120×C13711yescharacteristic
289Q8—44no—
290D4—28no—
291D4—24no—
292D4—44no—
293D4—48no—
294D4—432no—
295D4—48no—
296D4—82no—
297D4AGL(1,ℤ/8)×C13718yes—
298C2×C4—22no—
299C2×C4—24no—
300C2×C4—24no—
301C2×C4—48no—
302C2×C4—44no—
303C2×C4—48no—
304C2×C4—42no—
305C2×C4#120×C13711yescharacteristic
306C2×C4#120×C13712yes—
307C2×C4C22×D4×C13711yescharacteristic
308C2×C4C22×D4×C13712yes—
309C8—42no—
310C8—44no—
311C22—22no—
312C22—28no—
313C22—416no—
314C22—44no—
315C22—44no—
316C22—48no—
317C22—44no—
318C22—88no—
319C22—832no—
320C22#455×C13712yes—
321C22C2×#120×C13711yescharacteristic
322C4—22no—
323C4—21no—
324C4—21no—
325C4—44no—
326C4—82no—
327C4C2×AGL(1,ℤ/8)×C13714yes—
328C2—22no—
329C2—48no—
330C2—84no—
331C2#3003×C13712yes—
332C2C2×#455×C13711yescharacteristic
subgroupG/H
center, socleC22×C137C2×#120
commutator, Frattini, cosocleC42C24×C137
prime radicalC2×D42×C137C2
LCSG ▷ C42 ▷ C22 ▷ C1
UCSC1 ◁ C22×C137 ◁ C2×C42×C137 ◁ G

Complements

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

NH ≅ G/Ntype
2. #3138×C137C2semidirect
3. #3110×C137C2semidirect
4. #1639×C137C2semidirect
5. #1555×C137C2semidirect
6. #1490×C137C2semidirect
11. #533×C137C22semidirect
17. #486×C137C22semidirect
19. #474×C137C22semidirect
20. #461×C137C22semidirect
23. #432×C137C22semidirect
25. C4×M16×C137C22semidirect
26. #331×C137C22semidirect
27. D4⋊C8×C137C22semidirect
39. C4⋊Q8×C137C23semidirect
60. C4⋊C8×C137C23semidirect
67. C4×C8×C137C23semidirect
166. #18529C137direct (N × H)
167. C137#18529direct (N × H)
169. #3138C2×C137semidirect
170. #3110C2×C137semidirect
171. #1639C2×C137semidirect
172. #1555C2×C137semidirect
173. #1490C2×C137semidirect
178. #533C22×C137semidirect
184. #486C22×C137semidirect
186. #474C22×C137semidirect
187. #461C22×C137semidirect
190. #432C22×C137semidirect
192. C4×M16C22×C137semidirect
193. #331C22×C137semidirect
194. D4⋊C8C22×C137semidirect
206. C4⋊Q8C23×C137semidirect
227. C4⋊C8C23×C137semidirect
234. C4×C8C23×C137semidirect

Maximal subgroups

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

subgroupindex
C2×D42×C1372
#3138×C1372
#3110×C1372
#1639×C1372
#1555×C1372
#1490×C1372
#18529137