Finite Group Databasebeta

← C137×#13609 #46862995

Subgroups — C137×#13609 #46862995

Order 35072.

348 conjugacy classes of subgroups; only the 86 normal subgroups are shown.

normal subgroups only

This group has 348 conjugacy classes of subgroups — too many to draw. Only its 86 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
1#3220×C137C211yesmaximal, characteristic
2#2128×C137C211yesmaximal, characteristic, complemented
3#2127×C137C211yesmaximal, characteristic, complemented
4#1412×C137C212yesmaximal, complemented
5#1408×C137C212yesmaximal, complemented
6#540×C137C411yescharacteristic, complemented
7#535×C137C2211yescharacteristic
8#496×C137C411yescharacteristic, complemented
9#493×C137C2211yescharacteristic
10#490×C137C2211yescharacteristic
11#488×C137C411yescharacteristic, complemented
12#463×C137—21no—
13#459×C137—21no—
14#449×C137—21no—
15#447×C137—21no—
16#433×C137C411yescharacteristic, complemented
17#410×C137—21no—
18C2×#99×C137—21no—
19#354×C137—21no—
20#353×C137—21no—
21#353×C137C2212yes—
22#353×C137C2212yes—
23C2×#41×C137D411yescharacteristic
24C22×D4×C137D411yescharacteristic
25C2×SD16×C137—21no—
26C2×D8×C137—21no—
27C2×D8×C137—21no—
28C2×D8×C137—21no—
29#127×C137C2×C411yescharacteristic, complemented
30#127×C137D411yescharacteristic
31#127×C137C2311yescharacteristic
32#125×C137C2×C411yescharacteristic, complemented
33#124×C137C2×C411yescharacteristic
34#124×C137D411yescharacteristic
35#123×C137—41no—
36#121×C137—21no—
37#121×C137—41no—
38#121×C137—41no—
39#120×C137—41no—
40C4×D4×C137—21no—
41#117×C137—41no—
42#117×C137C2×C411yescharacteristic
43C2×C22⋊C4×C137—41no—
44C4⋊C8×C137—21no—
45#103×C137—21no—
46#102×C137—21no—
47#102×C137—21no—
48#102×C137—21no—
49#99×C137—22no—
50#99×C137—22no—
51#99×C137—41no—
52#99×C137—41no—
53C22⋊C8×C137—21no—
54#97×C137C2×C411yescharacteristic, complemented
55C4×C8×C137C2×C411yescharacteristic, complemented
56C24×C137—21no—
57#41×C137—81no—
58C2×Q8×C137C22⋊C411yescharacteristic
59C2×D4×C137—21no—
60C2×D4×C137—21no—
61C2×D4×C137—21no—
62C2×D4×C137—41no—
63C2×D4×C137—41no—
64C2×D4×C137—41no—
65C2×D4×C137—41no—
66C2×D4×C137—81no—
67C2×D4×C137C22⋊C411yescharacteristic
68C2×D4×C137C2×D411yescharacteristic
69C2×D4×C137C2×D411yescharacteristic
70C22×C4×C137—21no—
71C22×C4×C137—41no—
72C22×C4×C137—81no—
73C22×C4×C137C22⋊C411yescharacteristic
74SD16×C137—81no—
75D8×C137—81no—
76D8×C137—81no—
77D8×C137—81no—
78C2×C8×C137—21no—
79C2×C8×C137—21no—
80C4⋊C4×C137—21no—
81C4⋊C4×C137—21no—
82C4⋊C4×C137—41no—
83C4⋊C4×C137—81no—
84C22⋊C4×C137—21no—
85C22⋊C4×C137—41no—
86C22⋊C4×C137—41no—
87C22⋊C4×C137—41no—
88C22⋊C4×C137—41no—
89C22⋊C4×C137—42no—
90C22⋊C4×C137—42no—
91C22⋊C4×C137—81no—
92C42×C137—81no—
93C42×C137C22⋊C411yescharacteristic
94C42×C137C22×C411yescharacteristic
95C23×C137—21no—
96C23×C137—21no—
97C23×C137—21no—
98C23×C137—41no—
99C23×C137—41no—
100C23×C137—41no—
101C23×C137—161no—
102C23×C137#9911yescharacteristic
103Q8×C137—41no—
104D4×C137—41no—
105D4×C137—41no—
106D4×C137—41no—
107D4×C137—41no—
108D4×C137—41no—
109D4×C137—81no—
110D4×C137—81no—
111D4×C137—161no—
112D4×C137—161no—
113D4×C137—161no—
114D4×C137—161no—
115C2×C4×C137—21no—
116C2×C4×C137—21no—
117C2×C4×C137—21no—
118C2×C4×C137—41no—
119C2×C4×C137—41no—
120C2×C4×C137—41no—
121C2×C4×C137—41no—
122C2×C4×C137—82no—
123C2×C4×C137—82no—
124C2×C4×C137—161no—
125C2×C4×C137—162no—
126C2×C4×C137#9911yescharacteristic
127C2×C4×C137C2×C22⋊C411yescharacteristic
128C8×C137—41no—
129C8×C137—41no—
130C22×C137—21no—
131C22×C137—41no—
132C22×C137—41no—
133C22×C137—41no—
134C22×C137—41no—
135C22×C137—81no—
136C22×C137—81no—
137C22×C137—81no—
138C22×C137—81no—
139C22×C137—81no—
140C22×C137—82no—
141C22×C137—81no—
142C22×C137—161no—
143C22×C137C2×#9911yescharacteristic
144C4×C137—21no—
145C4×C137—21no—
146C4×C137—41no—
147C4×C137—41no—
148C4×C137—41no—
149C4×C137—81no—
150C4×C137—162no—
151C4×C137—162no—
152C2×C137—21no—
153C2×C137—41no—
154C2×C137—81no—
155C2×C137—81no—
156C2×C137—81no—
157C2×C137—161no—
158C2×C137#212711yescharacteristic
159#13609C13711yesmaximal, characteristic, direct product factor, Sylow
160C137#1360911yescharacteristic, direct product factor, Sylow
161#3220C2×C13711yescharacteristic
162#2128C2×C13711yescharacteristic, complemented
163#2127C2×C13711yescharacteristic, complemented
164#1412C2×C13712yescomplemented
165#1408C2×C13712yescomplemented
166#540C4×C13711yescharacteristic, complemented
167#535C22×C13711yescharacteristic
168#496C4×C13711yescharacteristic, complemented
169#493C22×C13711yescharacteristic
170#490C22×C13711yescharacteristic
171#488C4×C13711yescharacteristic, complemented
172#463—21no—
173#459—21no—
174#449—21no—
175#447—21no—
176#433C4×C13711yescharacteristic, complemented
177#410—21no—
178C2×#99—21no—
179#354—21no—
180#353—21no—
181#353C22×C13712yes—
182#353C22×C13712yes—
183C2×#41D4×C13711yescharacteristic
184C22×D4D4×C13711yescharacteristic
185C2×SD16—21no—
186C2×D8—21no—
187C2×D8—21no—
188C2×D8—21no—
189#127C2×C4×C13711yescharacteristic, complemented
190#127D4×C13711yescharacteristic
191#127C23×C13711yescharacteristic
192#125C2×C4×C13711yescharacteristic, complemented
193#124C2×C4×C13711yescharacteristic
194#124D4×C13711yescharacteristic
195#123—41no—
196#121—21no—
197#121—41no—
198#121—41no—
199#120—41no—
200C4×D4—21no—
201#117—41no—
202#117C2×C4×C13711yescharacteristic
203C2×C22⋊C4—41no—
204C4⋊C8—21no—
205#103—21no—
206#102—21no—
207#102—21no—
208#102—21no—
209#99—22no—
210#99—22no—
211#99—41no—
212#99—41no—
213C22⋊C8—21no—
214#97C2×C4×C13711yescharacteristic, complemented
215C4×C8C2×C4×C13711yescharacteristic, complemented
216C24—21no—
217#41—81no—
218C2×Q8C22⋊C4×C13711yescharacteristic
219C2×D4—21no—
220C2×D4—21no—
221C2×D4—21no—
222C2×D4—41no—
223C2×D4—41no—
224C2×D4—41no—
225C2×D4—41no—
226C2×D4—81no—
227C2×D4C22⋊C4×C13711yescharacteristic
228C2×D4C2×D4×C13711yescharacteristic
229C2×D4C2×D4×C13711yescharacteristic
230C22×C4—21no—
231C22×C4—41no—
232C22×C4—81no—
233C22×C4C22⋊C4×C13711yescharacteristic
234SD16—81no—
235D8—81no—
236D8—81no—
237D8—81no—
238C2×C8—21no—
239C2×C8—21no—
240C4⋊C4—21no—
241C4⋊C4—21no—
242C4⋊C4—41no—
243C4⋊C4—81no—
244C22⋊C4—21no—
245C22⋊C4—41no—
246C22⋊C4—41no—
247C22⋊C4—41no—
248C22⋊C4—41no—
249C22⋊C4—42no—
250C22⋊C4—42no—
251C22⋊C4—81no—
252C42—81no—
253C42C22⋊C4×C13711yescharacteristic
254C42C22×C4×C13711yescharacteristic
255C23—21no—
256C23—21no—
257C23—21no—
258C23—41no—
259C23—41no—
260C23—41no—
261C23—161no—
262C23#99×C13711yescharacteristic
263Q8—41no—
264D4—41no—
265D4—41no—
266D4—41no—
267D4—41no—
268D4—41no—
269D4—81no—
270D4—81no—
271D4—161no—
272D4—161no—
273D4—161no—
274D4—161no—
275C2×C4—21no—
276C2×C4—21no—
277C2×C4—21no—
278C2×C4—41no—
279C2×C4—41no—
280C2×C4—41no—
281C2×C4—41no—
282C2×C4—82no—
283C2×C4—82no—
284C2×C4—161no—
285C2×C4—162no—
286C2×C4#99×C13711yescharacteristic
287C2×C4C2×C22⋊C4×C13711yescharacteristic
288C8—41no—
289C8—41no—
290C22—21no—
291C22—41no—
292C22—41no—
293C22—41no—
294C22—41no—
295C22—81no—
296C22—81no—
297C22—81no—
298C22—81no—
299C22—81no—
300C22—82no—
301C22—81no—
302C22—161no—
303C22C2×#99×C13711yescharacteristic
304C4—21no—
305C4—21no—
306C4—41no—
307C4—41no—
308C4—41no—
309C4—81no—
310C4—162no—
311C4—162no—
312C2—21no—
313C2—41no—
314C2—81no—
315C2—81no—
316C2—81no—
317C2—161no—
318C2#2127×C13711yescharacteristic
subgroupG/H
center, socleC2×C137#2127
commutatorC42C22×C4×C137
Frattini, cosocle#127C23×C137
prime radical#3220×C137C2
LCSG ▷ C42 ▷ C2×C4 ▷ C22 ▷ C2 ▷ C1
UCSC1 ◁ C2×C137 ◁ C22×C137 ◁ C22×C4×C137 ◁ #535×C137 ◁ G

Complements

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

NH ≅ G/Ntype
2. #2128×C137C2semidirect
3. #2127×C137C2semidirect
4. #1412×C137C2semidirect
5. #1408×C137C2semidirect
6. #540×C137C4semidirect
8. #496×C137C4semidirect
11. #488×C137C4semidirect
16. #433×C137C4semidirect
29. #127×C137C2×C4semidirect
32. #125×C137C2×C4semidirect
54. #97×C137C2×C4semidirect
55. C4×C8×C137C2×C4semidirect
159. #13609C137direct (N × H)
160. C137#13609direct (N × H)
162. #2128C2×C137semidirect
163. #2127C2×C137semidirect
164. #1412C2×C137semidirect
165. #1408C2×C137semidirect
166. #540C4×C137semidirect
168. #496C4×C137semidirect
171. #488C4×C137semidirect
176. #433C4×C137semidirect
189. #127C2×C4×C137semidirect
192. #125C2×C4×C137semidirect
214. #97C2×C4×C137semidirect
215. C4×C8C2×C4×C137semidirect

Maximal subgroups

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

subgroupindex
#3220×C1372
#2128×C1372
#2127×C1372
#1412×C1372
#1408×C1372
#13609137