Finite Group Databasebeta

← C137×#9055 #46858441

Subgroups — C137×#9055 #46858441

Order 35072.

500 conjugacy classes of subgroups.

The subgroup lattice has 398 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#3167×C137C211yesmaximal, characteristic
2#2935×C137C211yesmaximal, characteristic
3#1986×C137C211yesmaximal, characteristic, complemented
4#1502×C137C211yesmaximal, characteristic
5#1481×C137C211yesmaximal, characteristic, complemented
6#1279×C137C211yesmaximal, characteristic, complemented
7#1278×C137C211yesmaximal, characteristic, complemented
8C2×#131×C137—21no—
9#533×C137C2211yescharacteristic
10#532×C137C2211yescharacteristic
11#498×C137C411yescharacteristic, complemented
12#492×C137C411yescharacteristic, complemented
13#487×C137C411yescharacteristic, complemented
14#486×C137C411yescharacteristic, complemented
15#465×C137—22no—
16C4×SD16×C137—22no—
17#436×C137—21no—
18#435×C137—22no—
19C8×D4×C137—21no—
20#431×C137C2211yescharacteristic
21C2×C4⋊C8×C137C2211yescharacteristic
22#408×C137—21no—
23C4×M16×C137C2211yescharacteristic
24C2×C4×C8×C137C2211yescharacteristic
25C4×C4⋊C4×C137C2211yescharacteristic
26#339×C137—21no—
27#339×C137—21no—
28#327×C137—22no—
29Q8⋊C8×C137—21no—
30Q8⋊C8×C137—21no—
31C2×#41×C137—21no—
32C2×SD16×C137—22no—
33#131×C137—42no—
34#131×C137—42no—
35C2×M16×C137—22no—
36C2×M16×C137D411yescharacteristic
37C2×M16×C137D411yescharacteristic
38C22×C8×C137—22no—
39C22×C8×C137—21no—
40C22×C8×C137—21no—
41C4⋊Q8×C137C2×C411yescharacteristic
42C4⋊Q8×C137C2×C411yescharacteristic
43C4⋊Q8×C137C2×C411yescharacteristic
44#127×C137C2×C411yescharacteristic
45#125×C137—21no—
46#124×C137—21no—
47C22⋊Q8×C137—21no—
48C22⋊Q8×C137—21no—
49#121×C137—21no—
50#121×C137D412yes—
51C4×Q8×C137—21no—
52C4×Q8×C137—21no—
53C4×D4×C137—21no—
54C4×D4×C137D412yes—
55#117×C137—21no—
56C2×C4⋊C4×C137—21no—
57C2×C42×C137—21no—
58C2×C42×C137C2311yescharacteristic
59#106×C137—22no—
60C4⋊C8×C137—21no—
61C4⋊C8×C137—21no—
62C4⋊C8×C137D411yescharacteristic
63C4⋊C8×C137D411yescharacteristic
64#103×C137—22no—
65#102×C137—22no—
66C22⋊C8×C137—22no—
67C22⋊C8×C137—22no—
68C22⋊C8×C137—21no—
69#97×C137—21no—
70C4×C8×C137—21no—
71C4×C8×C137—21no—
72C4×C8×C137—21no—
73C4×C8×C137C2×C411yescharacteristic
74C4×C8×C137C2×C411yescharacteristic
75#95×C137—21no—
76#41×C137—41no—
77#41×C137—41no—
78#41×C137—41no—
79#41×C137—41no—
80C2×Q8×C137—21no—
81C2×Q8×C137—21no—
82C2×Q8×C137—21no—
83C2×D4×C137—22no—
84C2×D4×C137—21no—
85C2×D4×C137—21no—
86C2×D4×C137C22⋊C412yes—
87C22×C4×C137—22no—
88C22×C4×C137—41no—
89C22×C4×C137C2×D411yescharacteristic
90C22×C4×C137C2×D411yescharacteristic
91C22×C4×C137#4111yescharacteristic
92SD16×C137—44no—
93M16×C137—21no—
94M16×C137—21no—
95M16×C137—21no—
96M16×C137—44no—
97M16×C137SD1612yescomplemented
98C2×C8×C137—21no—
99C2×C8×C137—21no—
100C2×C8×C137—21no—
101C2×C8×C137—21no—
102C2×C8×C137—21no—
103C2×C8×C137—21no—
104C2×C8×C137—42no—
105C2×C8×C137—44no—
106C2×C8×C137—41no—
107C2×C8×C137—41no—
108C2×C8×C137—41no—
109C4⋊C4×C137—21no—
110C4⋊C4×C137—22no—
111C4⋊C4×C137—21no—
112C4⋊C4×C137—21no—
113C4⋊C4×C137—21no—
114C4⋊C4×C137—21no—
115C4⋊C4×C137—41no—
116C4⋊C4×C137—41no—
117C4⋊C4×C137—41no—
118C4⋊C4×C137C22⋊C412yes—
119C22⋊C4×C137—22no—
120C22⋊C4×C137—22no—
121C22⋊C4×C137—41no—
122C42×C137—41no—
123C42×C137—41no—
124C42×C137—41no—
125C42×C137C22×C411yescharacteristic
126C42×C137C2×D411yescharacteristic
127C42×C137C2×D411yescharacteristic
128C42×C137#4111yescharacteristic
129C23×C137—22no—
130C23×C137#12111yescharacteristic
131Q8×C137—42no—
132Q8×C137—41no—
133Q8×C137—41no—
134Q8×C137—41no—
135Q8×C137—41no—
136D4×C137—42no—
137D4×C137—42no—
138D4×C137—42no—
139D4×C137—41no—
140D4×C137—41no—
141D4×C137—41no—
142D4×C137—41no—
143C2×C4×C137—21no—
144C2×C4×C137—21no—
145C2×C4×C137—22no—
146C2×C4×C137—41no—
147C2×C4×C137—42no—
148C2×C4×C137—42no—
149C2×C4×C137—42no—
150C2×C4×C137—41no—
151C2×C4×C137—41no—
152C2×C4×C137—81no—
153C2×C4×C137—81no—
154C2×C4×C137C2×C22⋊C411yescharacteristic
155C2×C4×C137C4×D411yescharacteristic
156C2×C4×C137C4×D411yescharacteristic
157C2×C4×C137#12011yescharacteristic
158C2×C4×C137#12111yescharacteristic
159C2×C4×C137#12311yescharacteristic
160C2×C4×C137C2×SD1611yescharacteristic
161C2×C4×C137#13511yescharacteristic
162C2×C4×C137AGL(1,ℤ/8)11yescharacteristic
163C2×C4×C137#13711yescharacteristic
164C8×C137—22no—
165C8×C137—22no—
166C8×C137—41no—
167C8×C137—41no—
168C8×C137—41no—
169C8×C137—41no—
170C8×C137—81no—
171C22×C137—21no—
172C22×C137—21no—
173C22×C137—42no—
174C22×C137—42no—
175C22×C137—42no—
176C22×C137#38611yescharacteristic
177C22×C137#46111yescharacteristic
178C22×C137#46311yescharacteristic
179C4×C137—21no—
180C4×C137—21no—
181C4×C137—21no—
182C4×C137—42no—
183C4×C137—82no—
184C4×C137—81no—
185C4×C137—81no—
186C4×C137C4×SD1611yescharacteristic
187C4×C137#44011yescharacteristic
188C4×C137#44311yescharacteristic
189C4×C137#44411yescharacteristic
190C4×C137#45011yescharacteristic
191C4×C137#45211yescharacteristic
192C2×C137—21no—
193C2×C137—21no—
194C2×C137—82no—
195C2×C137#187611yescharacteristic
196C2×C137#187711yescharacteristic
197C2×C137#188411yescharacteristic
198#9055C13711yesmaximal, characteristic, direct product factor, Sylow
199C137#905511yescharacteristic, direct product factor, Sylow
200#3167C2×C13711yescharacteristic
201#2935C2×C13711yescharacteristic
202#1986C2×C13711yescharacteristic, complemented
203#1502C2×C13711yescharacteristic
204#1481C2×C13711yescharacteristic, complemented
205#1279C2×C13711yescharacteristic, complemented
206#1278C2×C13711yescharacteristic, complemented
207C2×#131—21no—
208#533C22×C13711yescharacteristic
209#532C22×C13711yescharacteristic
210#498C4×C13711yescharacteristic, complemented
211#492C4×C13711yescharacteristic, complemented
212#487C4×C13711yescharacteristic, complemented
213#486C4×C13711yescharacteristic, complemented
214#465—22no—
215C4×SD16—22no—
216#436—21no—
217#435—22no—
218C8×D4—21no—
219#431C22×C13711yescharacteristic
220C2×C4⋊C8C22×C13711yescharacteristic
221#408—21no—
222C4×M16C22×C13711yescharacteristic
223C2×C4×C8C22×C13711yescharacteristic
224C4×C4⋊C4C22×C13711yescharacteristic
225#339—21no—
226#339—21no—
227#327—22no—
228Q8⋊C8—21no—
229Q8⋊C8—21no—
230C2×#41—21no—
231C2×SD16—22no—
232#131—42no—
233#131—42no—
234C2×M16—22no—
235C2×M16D4×C13711yescharacteristic
236C2×M16D4×C13711yescharacteristic
237C22×C8—22no—
238C22×C8—21no—
239C22×C8—21no—
240C4⋊Q8C2×C4×C13711yescharacteristic
241C4⋊Q8C2×C4×C13711yescharacteristic
242C4⋊Q8C2×C4×C13711yescharacteristic
243#127C2×C4×C13711yescharacteristic
244#125—21no—
245#124—21no—
246C22⋊Q8—21no—
247C22⋊Q8—21no—
248#121—21no—
249#121D4×C13712yes—
250C4×Q8—21no—
251C4×Q8—21no—
252C4×D4—21no—
253C4×D4D4×C13712yes—
254#117—21no—
255C2×C4⋊C4—21no—
256C2×C42—21no—
257C2×C42C23×C13711yescharacteristic
258#106—22no—
259C4⋊C8—21no—
260C4⋊C8—21no—
261C4⋊C8D4×C13711yescharacteristic
262C4⋊C8D4×C13711yescharacteristic
263#103—22no—
264#102—22no—
265C22⋊C8—22no—
266C22⋊C8—22no—
267C22⋊C8—21no—
268#97—21no—
269C4×C8—21no—
270C4×C8—21no—
271C4×C8—21no—
272C4×C8C2×C4×C13711yescharacteristic
273C4×C8C2×C4×C13711yescharacteristic
274#95—21no—
275#41—41no—
276#41—41no—
277#41—41no—
278#41—41no—
279C2×Q8—21no—
280C2×Q8—21no—
281C2×Q8—21no—
282C2×D4—22no—
283C2×D4—21no—
284C2×D4—21no—
285C2×D4C22⋊C4×C13712yes—
286C22×C4—22no—
287C22×C4—41no—
288C22×C4C2×D4×C13711yescharacteristic
289C22×C4C2×D4×C13711yescharacteristic
290C22×C4#41×C13711yescharacteristic
291SD16—44no—
292M16—21no—
293M16—21no—
294M16—21no—
295M16—44no—
296M16SD16×C13712yescomplemented
297C2×C8—21no—
298C2×C8—21no—
299C2×C8—21no—
300C2×C8—21no—
301C2×C8—21no—
302C2×C8—21no—
303C2×C8—42no—
304C2×C8—44no—
305C2×C8—41no—
306C2×C8—41no—
307C2×C8—41no—
308C4⋊C4—21no—
309C4⋊C4—22no—
310C4⋊C4—21no—
311C4⋊C4—21no—
312C4⋊C4—21no—
313C4⋊C4—21no—
314C4⋊C4—41no—
315C4⋊C4—41no—
316C4⋊C4—41no—
317C4⋊C4C22⋊C4×C13712yes—
318C22⋊C4—22no—
319C22⋊C4—22no—
320C22⋊C4—41no—
321C42—41no—
322C42—41no—
323C42—41no—
324C42C22×C4×C13711yescharacteristic
325C42C2×D4×C13711yescharacteristic
326C42C2×D4×C13711yescharacteristic
327C42#41×C13711yescharacteristic
328C23—22no—
329C23#121×C13711yescharacteristic
330Q8—42no—
331Q8—41no—
332Q8—41no—
333Q8—41no—
334Q8—41no—
335D4—42no—
336D4—42no—
337D4—42no—
338D4—41no—
339D4—41no—
340D4—41no—
341D4—41no—
342C2×C4—21no—
343C2×C4—21no—
344C2×C4—22no—
345C2×C4—41no—
346C2×C4—42no—
347C2×C4—42no—
348C2×C4—42no—
349C2×C4—41no—
350C2×C4—41no—
351C2×C4—81no—
352C2×C4—81no—
353C2×C4C2×C22⋊C4×C13711yescharacteristic
354C2×C4C4×D4×C13711yescharacteristic
355C2×C4C4×D4×C13711yescharacteristic
356C2×C4#120×C13711yescharacteristic
357C2×C4#121×C13711yescharacteristic
358C2×C4#123×C13711yescharacteristic
359C2×C4C2×SD16×C13711yescharacteristic
360C2×C4#135×C13711yescharacteristic
361C2×C4AGL(1,ℤ/8)×C13711yescharacteristic
362C2×C4#137×C13711yescharacteristic
363C8—22no—
364C8—22no—
365C8—41no—
366C8—41no—
367C8—41no—
368C8—41no—
369C8—81no—
370C22—21no—
371C22—21no—
372C22—42no—
373C22—42no—
374C22—42no—
375C22#386×C13711yescharacteristic
376C22#461×C13711yescharacteristic
377C22#463×C13711yescharacteristic
378C4—21no—
379C4—21no—
380C4—21no—
381C4—42no—
382C4—82no—
383C4—81no—
384C4—81no—
385C4C4×SD16×C13711yescharacteristic
386C4#440×C13711yescharacteristic
387C4#443×C13711yescharacteristic
388C4#444×C13711yescharacteristic
389C4#450×C13711yescharacteristic
390C4#452×C13711yescharacteristic
391C2—21no—
392C2—21no—
393C2—82no—
394C2#1876×C13711yescharacteristic
395C2#1877×C13711yescharacteristic
396C2#1884×C13711yescharacteristic
subgroupG/H
centerC2×C4×C137#120
commutatorC42C22×C4×C137
Frattini, cosocleC2×C42C23×C137
socleC22×C137#386
prime radical#532×C137C22
LCSG ▷ C42 ▷ C22 ▷ C1
UCSC1 ◁ C2×C4×C137 ◁ C2×C42×C137 ◁ G

Complements

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

NH ≅ G/Ntype
3. #1986×C137C2semidirect
5. #1481×C137C2semidirect
6. #1279×C137C2semidirect
7. #1278×C137C2semidirect
11. #498×C137C4semidirect
12. #492×C137C4semidirect
13. #487×C137C4semidirect
14. #486×C137C4semidirect
97. M16×C137SD16semidirect
198. #9055C137direct (N × H)
199. C137#9055direct (N × H)
202. #1986C2×C137semidirect
204. #1481C2×C137semidirect
205. #1279C2×C137semidirect
206. #1278C2×C137semidirect
210. #498C4×C137semidirect
211. #492C4×C137semidirect
212. #487C4×C137semidirect
213. #486C4×C137semidirect
296. M16SD16×C137semidirect

Maximal subgroups

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

subgroupindex
#3167×C1372
#2935×C1372
#1986×C1372
#1502×C1372
#1481×C1372
#1279×C1372
#1278×C1372
#9055137