Finite Group Databasebeta

← C137×#11560 #46860946

Subgroups — C137×#11560 #46860946

Order 35072.

534 conjugacy classes of subgroups; only the 102 normal subgroups are shown.

normal subgroups only

This group has 534 conjugacy classes of subgroups — too many to draw. Only its 102 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#2534×C137C211yesmaximal, characteristic
2#1466×C137C211yesmaximal, characteristic, complemented
3#1459×C137C211yesmaximal, characteristic, complemented
4#1317×C137C212yesmaximal, complemented
5#1316×C137C212yesmaximal, complemented
6C2×#123×C137C411yescharacteristic
7C2×#120×C137C411yescharacteristic
8C2×C4×D4×C137C2211yescharacteristic
9C22×C22⋊C4×C137C411yescharacteristic
10#432×C137—21no—
11C2×C22⋊C8×C137—21no—
12#396×C137—21no—
13#394×C137—21no—
14#388×C137—21no—
15#386×C137—21no—
16#381×C137C2211yescharacteristic
17#380×C137C411yescharacteristic
18#379×C137C2211yescharacteristic
19#324×C137—21no—
20#323×C137—21no—
21#323×C137C2212yes—
22#323×C137C2212yes—
23C25×C137C2×C411yescharacteristic
24C22×D4×C137—21no—
25C22×D4×C137C2×C411yescharacteristic
26C23×C4×C137C2×C411yescharacteristic
27C23×C4×C137C2×C411yescharacteristic
28C23×C4×C137D411yescharacteristic
29C22×C8×C137—41no—
30#123×C137—42no—
31#120×C137—42no—
32C4×D4×C137—41no—
33C4×D4×C137—41no—
34C2×C4⋊C4×C137—21no—
35C2×C4⋊C4×C137—21no—
36C2×C4⋊C4×C137D411yescharacteristic
37C2×C22⋊C4×C137—21no—
38C2×C22⋊C4×C137—21no—
39C2×C22⋊C4×C137—21no—
40C2×C22⋊C4×C137—21no—
41C2×C22⋊C4×C137—21no—
42C2×C22⋊C4×C137—21no—
43C2×C22⋊C4×C137—41no—
44C2×C22⋊C4×C137—41no—
45C2×C22⋊C4×C137C2×C411yescharacteristic
46C2×C22⋊C4×C137C2×C411yescharacteristic
47C2×C22⋊C4×C137D411yescharacteristic
48C2×C22⋊C4×C137C2311yescharacteristic
49C2×C42×C137D411yescharacteristic
50C4⋊C8×C137—41no—
51C22⋊C8×C137—22no—
52C22⋊C8×C137—22no—
53C22⋊C8×C137—41no—
54#97×C137—41no—
55#95×C137—21no—
56#95×C137—21no—
57#95×C137—21no—
58C24×C137—21no—
59C24×C137—22no—
60C24×C137—41no—
61C24×C137—42no—
62C24×C137—41no—
63C24×C137—81no—
64C24×C137C22⋊C411yescharacteristic
65C24×C137C22×C411yescharacteristic
66C2×D4×C137—22no—
67C2×D4×C137—22no—
68C2×D4×C137—41no—
69C2×D4×C137—42no—
70C2×D4×C137—42no—
71C2×D4×C137—81no—
72C22×C4×C137—21no—
73C22×C4×C137—21no—
74C22×C4×C137—21no—
75C22×C4×C137—21no—
76C22×C4×C137—21no—
77C22×C4×C137—21no—
78C22×C4×C137—21no—
79C22×C4×C137—22no—
80C22×C4×C137—41no—
81C22×C4×C137—41no—
82C22×C4×C137—81no—
83C22×C4×C137—81no—
84C22×C4×C137C22⋊C411yescharacteristic
85C22×C4×C137C22⋊C411yescharacteristic
86C22×C4×C137C22⋊C411yescharacteristic
87C22×C4×C137C2×D411yescharacteristic
88C22×C4×C137C2×D411yescharacteristic
89C2×C8×C137—42no—
90C2×C8×C137—42no—
91C2×C8×C137—82no—
92C4⋊C4×C137—41no—
93C4⋊C4×C137—42no—
94C4⋊C4×C137—81no—
95C22⋊C4×C137—41no—
96C22⋊C4×C137—41no—
97C22⋊C4×C137—42no—
98C22⋊C4×C137—42no—
99C22⋊C4×C137—42no—
100C22⋊C4×C137—42no—
101C22⋊C4×C137—41no—
102C22⋊C4×C137—41no—
103C22⋊C4×C137—41no—
104C22⋊C4×C137—41no—
105C22⋊C4×C137—81no—
106C42×C137—21no—
107C42×C137—21no—
108C23×C137—21no—
109C23×C137—21no—
110C23×C137—21no—
111C23×C137—21no—
112C23×C137—22no—
113C23×C137—41no—
114C23×C137—41no—
115C23×C137—41no—
116C23×C137—41no—
117C23×C137—41no—
118C23×C137—41no—
119C23×C137—41no—
120C23×C137—42no—
121C23×C137—42no—
122C23×C137—41no—
123C23×C137—41no—
124C23×C137—41no—
125C23×C137—82no—
126C23×C137—84no—
127C23×C137—84no—
128C23×C137—82no—
129C23×C137#10012yes—
130C23×C137C2×C22⋊C411yescharacteristic
131C23×C137#13112yes—
132D4×C137—44no—
133D4×C137—84no—
134C2×C4×C137—21no—
135C2×C4×C137—21no—
136C2×C4×C137—21no—
137C2×C4×C137—21no—
138C2×C4×C137—21no—
139C2×C4×C137—21no—
140C2×C4×C137—41no—
141C2×C4×C137—42no—
142C2×C4×C137—41no—
143C2×C4×C137—41no—
144C2×C4×C137—41no—
145C2×C4×C137—41no—
146C2×C4×C137—41no—
147C2×C4×C137—41no—
148C2×C4×C137—41no—
149C2×C4×C137—41no—
150C2×C4×C137—41no—
151C2×C4×C137—41no—
152C2×C4×C137—41no—
153C2×C4×C137—41no—
154C2×C4×C137—81no—
155C2×C4×C137—81no—
156C2×C4×C137—81no—
157C2×C4×C137—82no—
158C2×C4×C137#9911yescharacteristic
159C2×C4×C137#9911yescharacteristic
160C8×C137—82no—
161C8×C137—82no—
162C22×C137—21no—
163C22×C137—21no—
164C22×C137—22no—
165C22×C137—22no—
166C22×C137—42no—
167C22×C137—42no—
168C22×C137—42no—
169C22×C137—42no—
170C22×C137—42no—
171C22×C137—41no—
172C22×C137—41no—
173C22×C137—41no—
174C22×C137—41no—
175C22×C137—42no—
176C22×C137—81no—
177C22×C137—81no—
178C22×C137—81no—
179C22×C137—84no—
180C22×C137—84no—
181C22×C137—81no—
182C22×C137—81no—
183C22×C137#40811yescharacteristic
184C22×C137C2×#9911yescharacteristic
185C22×C137C2×#10011yescharacteristic
186C4×C137—21no—
187C4×C137—21no—
188C4×C137—41no—
189C4×C137—41no—
190C4×C137—41no—
191C4×C137—41no—
192C4×C137—81no—
193C4×C137—81no—
194C4×C137—81no—
195C2×C137—21no—
196C2×C137—21no—
197C2×C137—42no—
198C2×C137—42no—
199C2×C137—82no—
200C2×C137#147111yescharacteristic
201C2×C137#212011yescharacteristic
202C2×C137#212111yescharacteristic
203#11560C13711yesmaximal, characteristic, direct product factor, Sylow
204C137#1156011yescharacteristic, direct product factor, Sylow
205#2534C2×C13711yescharacteristic
206#1466C2×C13711yescharacteristic, complemented
207#1459C2×C13711yescharacteristic, complemented
208#1317C2×C13712yescomplemented
209#1316C2×C13712yescomplemented
210C2×#123C4×C13711yescharacteristic
211C2×#120C4×C13711yescharacteristic
212C2×C4×D4C22×C13711yescharacteristic
213C22×C22⋊C4C4×C13711yescharacteristic
214#432—21no—
215C2×C22⋊C8—21no—
216#396—21no—
217#394—21no—
218#388—21no—
219#386—21no—
220#381C22×C13711yescharacteristic
221#380C4×C13711yescharacteristic
222#379C22×C13711yescharacteristic
223#324—21no—
224#323—21no—
225#323C22×C13712yes—
226#323C22×C13712yes—
227C25C2×C4×C13711yescharacteristic
228C22×D4—21no—
229C22×D4C2×C4×C13711yescharacteristic
230C23×C4C2×C4×C13711yescharacteristic
231C23×C4C2×C4×C13711yescharacteristic
232C23×C4D4×C13711yescharacteristic
233C22×C8—41no—
234#123—42no—
235#120—42no—
236C4×D4—41no—
237C4×D4—41no—
238C2×C4⋊C4—21no—
239C2×C4⋊C4—21no—
240C2×C4⋊C4D4×C13711yescharacteristic
241C2×C22⋊C4—21no—
242C2×C22⋊C4—21no—
243C2×C22⋊C4—21no—
244C2×C22⋊C4—21no—
245C2×C22⋊C4—21no—
246C2×C22⋊C4—21no—
247C2×C22⋊C4—41no—
248C2×C22⋊C4—41no—
249C2×C22⋊C4C2×C4×C13711yescharacteristic
250C2×C22⋊C4C2×C4×C13711yescharacteristic
251C2×C22⋊C4D4×C13711yescharacteristic
252C2×C22⋊C4C23×C13711yescharacteristic
253C2×C42D4×C13711yescharacteristic
254C4⋊C8—41no—
255C22⋊C8—22no—
256C22⋊C8—22no—
257C22⋊C8—41no—
258#97—41no—
259#95—21no—
260#95—21no—
261#95—21no—
262C24—21no—
263C24—22no—
264C24—41no—
265C24—42no—
266C24—41no—
267C24—81no—
268C24C22⋊C4×C13711yescharacteristic
269C24C22×C4×C13711yescharacteristic
270C2×D4—22no—
271C2×D4—22no—
272C2×D4—41no—
273C2×D4—42no—
274C2×D4—42no—
275C2×D4—81no—
276C22×C4—21no—
277C22×C4—21no—
278C22×C4—21no—
279C22×C4—21no—
280C22×C4—21no—
281C22×C4—21no—
282C22×C4—21no—
283C22×C4—22no—
284C22×C4—41no—
285C22×C4—41no—
286C22×C4—81no—
287C22×C4—81no—
288C22×C4C22⋊C4×C13711yescharacteristic
289C22×C4C22⋊C4×C13711yescharacteristic
290C22×C4C22⋊C4×C13711yescharacteristic
291C22×C4C2×D4×C13711yescharacteristic
292C22×C4C2×D4×C13711yescharacteristic
293C2×C8—42no—
294C2×C8—42no—
295C2×C8—82no—
296C4⋊C4—41no—
297C4⋊C4—42no—
298C4⋊C4—81no—
299C22⋊C4—41no—
300C22⋊C4—41no—
301C22⋊C4—42no—
302C22⋊C4—42no—
303C22⋊C4—42no—
304C22⋊C4—42no—
305C22⋊C4—41no—
306C22⋊C4—41no—
307C22⋊C4—41no—
308C22⋊C4—41no—
309C22⋊C4—81no—
310C42—21no—
311C42—21no—
312C23—21no—
313C23—21no—
314C23—21no—
315C23—21no—
316C23—22no—
317C23—41no—
318C23—41no—
319C23—41no—
320C23—41no—
321C23—41no—
322C23—41no—
323C23—41no—
324C23—42no—
325C23—42no—
326C23—41no—
327C23—41no—
328C23—41no—
329C23—82no—
330C23—84no—
331C23—84no—
332C23—82no—
333C23#100×C13712yes—
334C23C2×C22⋊C4×C13711yescharacteristic
335C23#131×C13712yes—
336D4—44no—
337D4—84no—
338C2×C4—21no—
339C2×C4—21no—
340C2×C4—21no—
341C2×C4—21no—
342C2×C4—21no—
343C2×C4—21no—
344C2×C4—41no—
345C2×C4—42no—
346C2×C4—41no—
347C2×C4—41no—
348C2×C4—41no—
349C2×C4—41no—
350C2×C4—41no—
351C2×C4—41no—
352C2×C4—41no—
353C2×C4—41no—
354C2×C4—41no—
355C2×C4—41no—
356C2×C4—41no—
357C2×C4—41no—
358C2×C4—81no—
359C2×C4—81no—
360C2×C4—81no—
361C2×C4—82no—
362C2×C4#99×C13711yescharacteristic
363C2×C4#99×C13711yescharacteristic
364C8—82no—
365C8—82no—
366C22—21no—
367C22—21no—
368C22—22no—
369C22—22no—
370C22—42no—
371C22—42no—
372C22—42no—
373C22—42no—
374C22—42no—
375C22—41no—
376C22—41no—
377C22—41no—
378C22—41no—
379C22—42no—
380C22—81no—
381C22—81no—
382C22—81no—
383C22—84no—
384C22—84no—
385C22—81no—
386C22—81no—
387C22#408×C13711yescharacteristic
388C22C2×#99×C13711yescharacteristic
389C22C2×#100×C13711yescharacteristic
390C4—21no—
391C4—21no—
392C4—41no—
393C4—41no—
394C4—41no—
395C4—41no—
396C4—81no—
397C4—81no—
398C4—81no—
399C2—21no—
400C2—21no—
401C2—42no—
402C2—42no—
403C2—82no—
404C2#1471×C13711yescharacteristic
405C2#2120×C13711yescharacteristic
406C2#2121×C13711yescharacteristic
subgroupG/H
center, socleC22×C137C2×#99
commutatorC24C22×C4×C137
Frattini, cosocleC2×C22⋊C4C23×C137
prime radicalC2×#120×C137C4
LCSG ▷ C24 ▷ C22 ▷ C2 ▷ C1
UCSC1 ◁ C22×C137 ◁ C22×C4×C137 ◁ C2×C4×D4×C137 ◁ G

Complements

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

NH ≅ G/Ntype
2. #1466×C137C2semidirect
3. #1459×C137C2semidirect
4. #1317×C137C2semidirect
5. #1316×C137C2semidirect
203. #11560C137direct (N × H)
204. C137#11560direct (N × H)
206. #1466C2×C137semidirect
207. #1459C2×C137semidirect
208. #1317C2×C137semidirect
209. #1316C2×C137semidirect

Maximal subgroups

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

subgroupindex
#2534×C1372
#1466×C1372
#1459×C1372
#1317×C1372
#1316×C1372
#11560137