Finite Group Databasebeta

← C137×#18542 #46867928

Subgroups — C137×#18542 #46867928

Order 35072.

918 conjugacy classes of subgroups.

The subgroup lattice has 462 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#556×C137C211yesmaximal, characteristic
2#3165×C137C211yesmaximal, characteristic, complemented
3#3113×C137C211yesmaximal, characteristic, complemented
4#3110×C137C211yesmaximal, characteristic
5#1652×C137C214yesmaximal, complemented
6#1650×C137C214yesmaximal, complemented
7#1556×C137C211yesmaximal, characteristic, complemented
8#1491×C137C211yesmaximal, characteristic, complemented
9#1490×C137C211yesmaximal, characteristic
10#556×C137—22no—
11#556×C137—22no—
12#556×C137—22no—
13#556×C137C2214yes—
14#533×C137C2211yescharacteristic
15C2×C4⋊Q8×C137C2211yescharacteristic
16C2×#126×C137—22no—
17C2×#124×C137—21no—
18C2×C22⋊Q8×C137—21no—
19C2×C4×Q8×C137—21no—
20C2×C4×Q8×C137C2211yescharacteristic
21C2×C4×D4×C137C2211yescharacteristic
22C2×#117×C137—21no—
23#501×C137—22no—
24#500×C137C2212yescomplemented
25#498×C137C2212yescomplemented
26#484×C137—21no—
27#483×C137—21no—
28#475×C137C2212yescomplemented
29#474×C137C2212yes—
30#462×C137C2212yescomplemented
31#461×C137C2212yes—
32#451×C137—21no—
33#450×C137—21no—
34#441×C137—22no—
35#440×C137—22no—
36#432×C137C2211yescharacteristic
37#432×C137C2211yescharacteristic
38#428×C137—22no—
39C4×M16×C137C2211yescharacteristic
40#332×C137C2214yescomplemented
41Q8⋊C8×C137C2214yescomplemented
42D4⋊C8×C137C2214yes—
43C22×Q8×C137—21no—
44C22×Q8×C137D411yescharacteristic
45C22×D4×C137D411yescharacteristic
46C23×C4×C137—21no—
47C2×Q16×C137—22no—
48C2×SD16×C137—22no—
49C2×M16×C137—21no—
50C4⋊Q8×C137—21no—
51C4⋊Q8×C137—21no—
52C4⋊Q8×C137D412yes—
53C4⋊Q8×C137C2311yescharacteristic
54C4⋊Q8×C137C2311yescharacteristic
55C4⋊Q8×C137C2312yes—
56#126×C137—28no—
57#126×C137—28no—
58#125×C137—21no—
59#124×C137—24no—
60#124×C137—24no—
61C22⋊Q8×C137—24no—
62C22⋊Q8×C137—24no—
63C22⋊Q8×C137—21no—
64C22⋊Q8×C137—21no—
65C4×Q8×C137—24no—
66C4×Q8×C137—24no—
67C4×Q8×C137—21no—
68C4×Q8×C137—21no—
69C4×Q8×C137—21no—
70C4×Q8×C137—21no—
71C4×Q8×C137D412yes—
72C4×Q8×C137C2312yes—
73C4×D4×C137—21no—
74C4×D4×C137—21no—
75C4×D4×C137D412yes—
76C4×D4×C137C2312yes—
77#117×C137—24no—
78#117×C137—24no—
79#117×C137—21no—
80C2×C4⋊C4×C137—22no—
81C2×C4⋊C4×C137—21no—
82C2×C4⋊C4×C137—21no—
83C2×C4⋊C4×C137D412yes—
84C2×C4⋊C4×C137D411yescharacteristic
85C2×C4⋊C4×C137D411yescharacteristic
86C2×C22⋊C4×C137—22no—
87C2×C22⋊C4×C137—22no—
88C2×C22⋊C4×C137—21no—
89C2×C42×C137—22no—
90C2×C42×C137—21no—
91C2×C42×C137C2311yescharacteristic
92#107×C137—24no—
93#107×C137—22no—
94#106×C137—24no—
95#106×C137—22no—
96C4⋊C8×C137C2312yes—
97C4⋊C8×C137C2312yes—
98#103×C137—22no—
99#103×C137—22no—
100#103×C137—22no—
101#102×C137—22no—
102C22⋊C8×C137—21no—
103C22⋊C8×C137—21no—
104#97×C137—21no—
105#97×C137—21no—
106#97×C137—21no—
107C4×C8×C137C2312yes—
108C24×C137—21no—
109C2×Q8×C137—21no—
110C2×Q8×C137—22no—
111C2×Q8×C137—22no—
112C2×Q8×C137—21no—
113C2×Q8×C137—21no—
114C2×Q8×C137—42no—
115C2×Q8×C137—44no—
116C2×Q8×C137C2×D412yes—
117C2×D4×C137—21no—
118C2×D4×C137—42no—
119C2×D4×C137C2×D412yes—
120C22×C4×C137—21no—
121C22×C4×C137—21no—
122C22×C4×C137—21no—
123C22×C4×C137—22no—
124C22×C4×C137—22no—
125C22×C4×C137—22no—
126C22×C4×C137—44no—
127C22×C4×C137C2×D411yescharacteristic
128C22×C4×C137C2×D411yescharacteristic
129C22×C4×C137C2×D411yescharacteristic
130Q16×C137—44no—
131SD16×C137—44no—
132M16×C137—24no—
133C2×C8×C137—22no—
134C2×C8×C137—22no—
135C2×C8×C137—22no—
136C4⋊C4×C137—28no—
137C4⋊C4×C137—22no—
138C4⋊C4×C137—22no—
139C4⋊C4×C137—21no—
140C4⋊C4×C137—21no—
141C4⋊C4×C137—22no—
142C4⋊C4×C137—22no—
143C4⋊C4×C137—22no—
144C4⋊C4×C137—21no—
145C4⋊C4×C137—21no—
146C4⋊C4×C137—21no—
147C4⋊C4×C137—21no—
148C4⋊C4×C137—41no—
149C4⋊C4×C137—41no—
150C4⋊C4×C137C2×D412yes—
151C4⋊C4×C137C2×D412yes—
152C4⋊C4×C137C2×D414yes—
153C22⋊C4×C137—28no—
154C22⋊C4×C137—28no—
155C22⋊C4×C137—22no—
156C22⋊C4×C137—22no—
157C22⋊C4×C137—41no—
158C42×C137—28no—
159C42×C137—22no—
160C42×C137—22no—
161C42×C137—41no—
162C42×C137C2×D411yescharacteristic
163C42×C137C2×D411yescharacteristic
164C42×C137C2×D411yescharacteristic
165C42×C137C2411yescharacteristic
166C23×C137—22no—
167C23×C137—42no—
168C23×C137—42no—
169C23×C137—42no—
170C23×C137#12011yescharacteristic
171Q8×C137—22no—
172Q8×C137—22no—
173Q8×C137—42no—
174Q8×C137—44no—
175Q8×C137—44no—
176Q8×C137—42no—
177Q8×C137—42no—
178D4×C137—22no—
179D4×C137—22no—
180D4×C137—42no—
181C2×C4×C137—22no—
182C2×C4×C137—22no—
183C2×C4×C137—22no—
184C2×C4×C137—22no—
185C2×C4×C137—22no—
186C2×C4×C137—22no—
187C2×C4×C137—24no—
188C2×C4×C137—44no—
189C2×C4×C137—42no—
190C2×C4×C137—42no—
191C2×C4×C137—44no—
192C2×C4×C137—42no—
193C2×C4×C137—44no—
194C2×C4×C137—41no—
195C2×C4×C137—41no—
196C2×C4×C137#12011yescharacteristic
197C2×C4×C137#12011yescharacteristic
198C2×C4×C137#12011yescharacteristic
199C2×C4×C137C22×D411yescharacteristic
200C2×C4×C137C22×D411yescharacteristic
201C2×C4×C137C22×D411yescharacteristic
202C8×C137—24no—
203C8×C137—42no—
204C8×C137—42no—
205C22×C137—21no—
206C22×C137—21no—
207C22×C137—42no—
208C22×C137—42no—
209C22×C137—42no—
210C22×C137—84no—
211C22×C137#45612yes—
212C22×C137C2×#12011yescharacteristic
213C4×C137—21no—
214C4×C137—21no—
215C4×C137—21no—
216C4×C137—21no—
217C4×C137—42no—
218C4×C137—42no—
219C4×C137—44no—
220C4×C137—42no—
221C4×C137—81no—
222C4×C137—81no—
223C4×C137#57512yes—
224C4×C137#57512yes—
225C2×C137—22no—
226C2×C137—82no—
227C2×C137#300511yescharacteristic
228C2×C137#300511yescharacteristic
229C2×C137C2×#45611yescharacteristic
230#18542C13711yesmaximal, characteristic, direct product factor, Sylow
231C137#1854211yescharacteristic, direct product factor, Sylow
232C2×#556C2×C13711yescharacteristic
233#3165C2×C13711yescharacteristic, complemented
234#3113C2×C13711yescharacteristic, complemented
235#3110C2×C13711yescharacteristic
236#1652C2×C13714yescomplemented
237#1650C2×C13714yescomplemented
238#1556C2×C13711yescharacteristic, complemented
239#1491C2×C13711yescharacteristic, complemented
240#1490C2×C13711yescharacteristic
241#556—22no—
242#556—22no—
243#556—22no—
244#556C22×C13714yes—
245#533C22×C13711yescharacteristic
246C2×C4⋊Q8C22×C13711yescharacteristic
247C2×#126—22no—
248C2×#124—21no—
249C2×C22⋊Q8—21no—
250C2×C4×Q8—21no—
251C2×C4×Q8C22×C13711yescharacteristic
252C2×C4×D4C22×C13711yescharacteristic
253C2×#117—21no—
254#501—22no—
255#500C22×C13712yescomplemented
256#498C22×C13712yescomplemented
257#484—21no—
258#483—21no—
259#475C22×C13712yescomplemented
260#474C22×C13712yes—
261#462C22×C13712yescomplemented
262#461C22×C13712yes—
263#451—21no—
264#450—21no—
265#441—22no—
266#440—22no—
267#432C22×C13711yescharacteristic
268#432C22×C13711yescharacteristic
269#428—22no—
270C4×M16C22×C13711yescharacteristic
271#332C22×C13714yescomplemented
272Q8⋊C8C22×C13714yescomplemented
273D4⋊C8C22×C13714yes—
274C22×Q8—21no—
275C22×Q8D4×C13711yescharacteristic
276C22×D4D4×C13711yescharacteristic
277C23×C4—21no—
278C2×Q16—22no—
279C2×SD16—22no—
280C2×M16—21no—
281C4⋊Q8—21no—
282C4⋊Q8—21no—
283C4⋊Q8D4×C13712yes—
284C4⋊Q8C23×C13711yescharacteristic
285C4⋊Q8C23×C13711yescharacteristic
286C4⋊Q8C23×C13712yes—
287#126—28no—
288#126—28no—
289#125—21no—
290#124—24no—
291#124—24no—
292C22⋊Q8—24no—
293C22⋊Q8—24no—
294C22⋊Q8—21no—
295C22⋊Q8—21no—
296C4×Q8—24no—
297C4×Q8—24no—
298C4×Q8—21no—
299C4×Q8—21no—
300C4×Q8—21no—
301C4×Q8—21no—
302C4×Q8D4×C13712yes—
303C4×Q8C23×C13712yes—
304C4×D4—21no—
305C4×D4—21no—
306C4×D4D4×C13712yes—
307C4×D4C23×C13712yes—
308#117—24no—
309#117—24no—
310#117—21no—
311C2×C4⋊C4—22no—
312C2×C4⋊C4—21no—
313C2×C4⋊C4—21no—
314C2×C4⋊C4D4×C13711yescharacteristic
315C2×C4⋊C4D4×C13711yescharacteristic
316C2×C4⋊C4D4×C13712yes—
317C2×C22⋊C4—22no—
318C2×C22⋊C4—22no—
319C2×C22⋊C4—21no—
320C2×C42—22no—
321C2×C42—21no—
322C2×C42C23×C13711yescharacteristic
323#107—24no—
324#107—22no—
325#106—24no—
326#106—22no—
327C4⋊C8C23×C13712yes—
328C4⋊C8C23×C13712yes—
329#103—22no—
330#103—22no—
331#103—22no—
332#102—22no—
333C22⋊C8—21no—
334C22⋊C8—21no—
335#97—21no—
336#97—21no—
337#97—21no—
338C4×C8C23×C13712yes—
339C24—21no—
340C2×Q8—21no—
341C2×Q8—22no—
342C2×Q8—22no—
343C2×Q8—21no—
344C2×Q8—21no—
345C2×Q8—42no—
346C2×Q8—44no—
347C2×Q8C2×D4×C13712yes—
348C2×D4—21no—
349C2×D4—42no—
350C2×D4C2×D4×C13712yes—
351C22×C4—21no—
352C22×C4—21no—
353C22×C4—21no—
354C22×C4—22no—
355C22×C4—22no—
356C22×C4—22no—
357C22×C4—44no—
358C22×C4C2×D4×C13711yescharacteristic
359C22×C4C2×D4×C13711yescharacteristic
360C22×C4C2×D4×C13711yescharacteristic
361Q16—44no—
362SD16—44no—
363M16—24no—
364C2×C8—22no—
365C2×C8—22no—
366C2×C8—22no—
367C4⋊C4—28no—
368C4⋊C4—22no—
369C4⋊C4—22no—
370C4⋊C4—21no—
371C4⋊C4—21no—
372C4⋊C4—22no—
373C4⋊C4—22no—
374C4⋊C4—22no—
375C4⋊C4—21no—
376C4⋊C4—21no—
377C4⋊C4—21no—
378C4⋊C4—21no—
379C4⋊C4—41no—
380C4⋊C4—41no—
381C4⋊C4C2×D4×C13712yes—
382C4⋊C4C2×D4×C13712yes—
383C4⋊C4C2×D4×C13714yes—
384C22⋊C4—28no—
385C22⋊C4—28no—
386C22⋊C4—22no—
387C22⋊C4—22no—
388C22⋊C4—41no—
389C42—28no—
390C42—22no—
391C42—22no—
392C42—41no—
393C42C2×D4×C13711yescharacteristic
394C42C2×D4×C13711yescharacteristic
395C42C2×D4×C13711yescharacteristic
396C42C24×C13711yescharacteristic
397C23—22no—
398C23—42no—
399C23—42no—
400C23—42no—
401C23#120×C13711yescharacteristic
402Q8—22no—
403Q8—22no—
404Q8—42no—
405Q8—44no—
406Q8—44no—
407Q8—42no—
408Q8—42no—
409D4—22no—
410D4—22no—
411D4—42no—
412C2×C4—22no—
413C2×C4—22no—
414C2×C4—22no—
415C2×C4—22no—
416C2×C4—22no—
417C2×C4—22no—
418C2×C4—24no—
419C2×C4—44no—
420C2×C4—42no—
421C2×C4—42no—
422C2×C4—44no—
423C2×C4—42no—
424C2×C4—44no—
425C2×C4—41no—
426C2×C4—41no—
427C2×C4#120×C13711yescharacteristic
428C2×C4#120×C13711yescharacteristic
429C2×C4#120×C13711yescharacteristic
430C2×C4C22×D4×C13711yescharacteristic
431C2×C4C22×D4×C13711yescharacteristic
432C2×C4C22×D4×C13711yescharacteristic
433C8—24no—
434C8—42no—
435C8—42no—
436C22—21no—
437C22—21no—
438C22—42no—
439C22—42no—
440C22—42no—
441C22—84no—
442C22#456×C13712yes—
443C22C2×#120×C13711yescharacteristic
444C4—21no—
445C4—21no—
446C4—21no—
447C4—21no—
448C4—42no—
449C4—42no—
450C4—44no—
451C4—42no—
452C4—81no—
453C4—81no—
454C4#575×C13712yes—
455C4#575×C13712yes—
456C2—22no—
457C2—82no—
458C2#3005×C13711yescharacteristic
459C2#3005×C13711yescharacteristic
460C2C2×#456×C13711yescharacteristic
subgroupG/H
center, socleC22×C137C2×#120
commutator, Frattini, cosocleC42C24×C137
prime radicalC22×D4×C137D4
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. #3165×C137C2semidirect
3. #3113×C137C2semidirect
5. #1652×C137C2semidirect
6. #1650×C137C2semidirect
7. #1556×C137C2semidirect
8. #1491×C137C2semidirect
24. #500×C137C22semidirect
25. #498×C137C22semidirect
28. #475×C137C22semidirect
30. #462×C137C22semidirect
40. #332×C137C22semidirect
41. Q8⋊C8×C137C22semidirect
230. #18542C137direct (N × H)
231. C137#18542direct (N × H)
233. #3165C2×C137semidirect
234. #3113C2×C137semidirect
236. #1652C2×C137semidirect
237. #1650C2×C137semidirect
238. #1556C2×C137semidirect
239. #1491C2×C137semidirect
255. #500C22×C137semidirect
256. #498C22×C137semidirect
259. #475C22×C137semidirect
261. #462C22×C137semidirect
271. #332C22×C137semidirect
272. Q8⋊C8C22×C137semidirect

Maximal subgroups

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

subgroupindex
C2×#556×C1372
#3165×C1372
#3113×C1372
#3110×C1372
#1652×C1372
#1650×C1372
#1556×C1372
#1491×C1372
#1490×C1372
#18542137