Finite Group Databasebeta

← C137×#18778 #46868164

Subgroups — C137×#18778 #46868164

Order 35072.

932 conjugacy classes of subgroups.

The subgroup lattice has 476 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#557×C137C211yesmaximal, characteristic, complemented
2#3227×C137C211yesmaximal, characteristic
3#3103×C137C211yesmaximal, characteristic
4#3037×C137C211yesmaximal, characteristic
5#1653×C137C214yesmaximal, complemented
6#1626×C137C214yesmaximal, complemented
7#1549×C137C211yesmaximal, characteristic, complemented
8#1487×C137C211yesmaximal, characteristic, complemented
9#1479×C137C211yesmaximal, characteristic, complemented
10C2×#137×C137—21no—
11#557×C137—22no—
12#557×C137—22no—
13#557×C137—22no—
14#557×C137C2214yes—
15#532×C137C2211yescharacteristic
16C2×C4⋊Q8×C137—21no—
17C2×C4⋊Q8×C137C2211yescharacteristic
18C2×#125×C137—22no—
19C2×#125×C137—21no—
20C2×C4×Q8×C137—21no—
21C2×C4×Q8×C137—21no—
22C2×C4×Q8×C137C2211yescharacteristic
23C2×C4×Q8×C137C2211yescharacteristic
24#498×C137C2212yes—
25#486×C137C2212yes—
26#481×C137—21no—
27#481×C137—22no—
28#476×C137—22no—
29#475×C137C2212yes—
30#464×C137—22no—
31#462×C137C2212yes—
32#460×C137C2212yes—
33#460×C137C2212yes—
34#448×C137—21no—
35C4×SD16×C137—22no—
36#431×C137C2211yescharacteristic
37#431×C137C2211yescharacteristic
38#423×C137C2211yescharacteristic
39#417×C137—21no—
40#332×C137C2214yes—
41Q8⋊C8×C137C2214yes—
42Q8⋊C8×C137C2214yes—
43C2×#41×C137—21no—
44C22×Q8×C137—21no—
45C22×Q8×C137D411yescharacteristic
46C22×Q8×C137D411yescharacteristic
47#137×C137—44no—
48C2×Q16×C137—22no—
49C2×SD16×C137—22no—
50C2×SD16×C137—22no—
51C2×M16×C137—21no—
52C4⋊Q8×C137—24no—
53C4⋊Q8×C137—24no—
54C4⋊Q8×C137—21no—
55C4⋊Q8×C137—21no—
56C4⋊Q8×C137D412yes—
57C4⋊Q8×C137C2311yescharacteristic
58C4⋊Q8×C137C2312yes—
59#127×C137C2311yescharacteristic
60#125×C137—28no—
61#125×C137—24no—
62#125×C137—28no—
63#125×C137—24no—
64#124×C137—21no—
65#121×C137—21no—
66#121×C137—21no—
67C4×Q8×C137—24no—
68C4×Q8×C137—24no—
69C4×Q8×C137—24no—
70C4×Q8×C137—24no—
71C4×Q8×C137—21no—
72C4×Q8×C137—21no—
73C4×Q8×C137—21no—
74C4×Q8×C137—21no—
75C4×Q8×C137D412yes—
76C4×Q8×C137D412yes—
77C4×Q8×C137C2312yes—
78C4×Q8×C137C2312yes—
79C4×D4×C137—21no—
80C4×D4×C137—21no—
81C2×C4⋊C4×C137—22no—
82C2×C4⋊C4×C137—22no—
83C2×C4⋊C4×C137—22no—
84C2×C4⋊C4×C137—21no—
85C2×C4⋊C4×C137—21no—
86C2×C4⋊C4×C137—21no—
87C2×C4⋊C4×C137D411yescharacteristic
88C2×C4⋊C4×C137D411yescharacteristic
89C2×C4⋊C4×C137D412yes—
90C2×C42×C137—22no—
91C2×C42×C137—21no—
92C2×C42×C137—21no—
93C2×C42×C137C2311yescharacteristic
94#106×C137—22no—
95#106×C137—24no—
96C4⋊C8×C137—21no—
97C4⋊C8×C137—21no—
98C4⋊C8×C137C2312yes—
99C4⋊C8×C137C2312yes—
100#103×C137—22no—
101#103×C137—22no—
102#102×C137—22no—
103#102×C137—22no—
104#102×C137—24no—
105C22⋊C8×C137—21no—
106C22⋊C8×C137—21no—
107C4×C8×C137—21no—
108C4×C8×C137C2312yes—
109#41×C137—42no—
110#41×C137—42no—
111C2×Q8×C137—21no—
112C2×Q8×C137—21no—
113C2×Q8×C137—22no—
114C2×Q8×C137—22no—
115C2×Q8×C137—22no—
116C2×Q8×C137—21no—
117C2×Q8×C137—42no—
118C2×Q8×C137—44no—
119C2×Q8×C137SD1614yescomplemented
120C2×Q8×C137C2×D412yes—
121C2×Q8×C137C2×D412yes—
122C2×D4×C137—21no—
123C2×D4×C137—21no—
124C2×D4×C137—21no—
125C2×D4×C137—41no—
126C22×C4×C137—21no—
127C22×C4×C137—21no—
128C22×C4×C137—21no—
129C22×C4×C137—21no—
130C22×C4×C137—22no—
131C22×C4×C137—41no—
132C22×C4×C137C2×D411yescharacteristic
133C22×C4×C137C2×D411yescharacteristic
134C22×C4×C137C2×D411yescharacteristic
135Q16×C137—44no—
136SD16×C137—44no—
137SD16×C137—44no—
138M16×C137—42no—
139C2×C8×C137—22no—
140C2×C8×C137—22no—
141C2×C8×C137—22no—
142C4⋊C4×C137—28no—
143C4⋊C4×C137—28no—
144C4⋊C4×C137—28no—
145C4⋊C4×C137—22no—
146C4⋊C4×C137—22no—
147C4⋊C4×C137—22no—
148C4⋊C4×C137—21no—
149C4⋊C4×C137—21no—
150C4⋊C4×C137—22no—
151C4⋊C4×C137—22no—
152C4⋊C4×C137—22no—
153C4⋊C4×C137—22no—
154C4⋊C4×C137—21no—
155C4⋊C4×C137—21no—
156C4⋊C4×C137C2×D412yes—
157C4⋊C4×C137C2×D412yes—
158C4⋊C4×C137C2×D414yes—
159C22⋊C4×C137—41no—
160C22⋊C4×C137—41no—
161C42×C137—28no—
162C42×C137—22no—
163C42×C137—22no—
164C42×C137—22no—
165C42×C137—22no—
166C42×C137C2×D411yescharacteristic
167C42×C137C2×D411yescharacteristic
168C42×C137C2×D411yescharacteristic
169C42×C137C2411yescharacteristic
170C23×C137—41no—
171C23×C137#12011yescharacteristic
172Q8×C137—22no—
173Q8×C137—24no—
174Q8×C137—22no—
175Q8×C137—22no—
176Q8×C137—42no—
177Q8×C137—44no—
178Q8×C137—44no—
179Q8×C137—41no—
180Q8×C137—41no—
181Q8×C137C2×SD1614yes—
182D4×C137—42no—
183D4×C137—41no—
184D4×C137—41no—
185D4×C137—42no—
186D4×C137—82no—
187C2×C4×C137—22no—
188C2×C4×C137—24no—
189C2×C4×C137—22no—
190C2×C4×C137—22no—
191C2×C4×C137—22no—
192C2×C4×C137—22no—
193C2×C4×C137—24no—
194C2×C4×C137—41no—
195C2×C4×C137—42no—
196C2×C4×C137—42no—
197C2×C4×C137—42no—
198C2×C4×C137—44no—
199C2×C4×C137—41no—
200C2×C4×C137—82no—
201C2×C4×C137#12011yescharacteristic
202C2×C4×C137#12011yescharacteristic
203C2×C4×C137#12011yescharacteristic
204C2×C4×C137C2×SD1612yes—
205C2×C4×C137#13712yes—
206C2×C4×C137C22×D411yescharacteristic
207C2×C4×C137C22×D411yescharacteristic
208C2×C4×C137C22×D411yescharacteristic
209C8×C137—42no—
210C8×C137—42no—
211C8×C137—42no—
212C22×C137—21no—
213C22×C137—21no—
214C22×C137—81no—
215C22×C137—81no—
216C22×C137—81no—
217C22×C137#44812yes—
218C22×C137C2×#12011yescharacteristic
219C4×C137—24no—
220C4×C137—21no—
221C4×C137—21no—
222C4×C137—21no—
223C4×C137—42no—
224C4×C137—44no—
225C4×C137—42no—
226C4×C137—42no—
227C4×C137—81no—
228C4×C137#44812yes—
229C4×C137C22×SD1611yescharacteristic
230C4×C137C2×#13711yescharacteristic
231C4×C137#57512yes—
232C2×C137—22no—
233C2×C137—161no—
234C2×C137C2×#44811yescharacteristic
235C2×C137#300511yescharacteristic
236C2×C137#302011yescharacteristic
237#18778C13711yesmaximal, characteristic, direct product factor, Sylow
238C137#1877811yescharacteristic, direct product factor, Sylow
239C2×#557C2×C13711yescharacteristic, complemented
240#3227C2×C13711yescharacteristic
241#3103C2×C13711yescharacteristic
242#3037C2×C13711yescharacteristic
243#1653C2×C13714yescomplemented
244#1626C2×C13714yescomplemented
245#1549C2×C13711yescharacteristic, complemented
246#1487C2×C13711yescharacteristic, complemented
247#1479C2×C13711yescharacteristic, complemented
248C2×#137—21no—
249#557—22no—
250#557—22no—
251#557—22no—
252#557C22×C13714yes—
253#532C22×C13711yescharacteristic
254C2×C4⋊Q8—21no—
255C2×C4⋊Q8C22×C13711yescharacteristic
256C2×#125—22no—
257C2×#125—21no—
258C2×C4×Q8—21no—
259C2×C4×Q8—21no—
260C2×C4×Q8C22×C13711yescharacteristic
261C2×C4×Q8C22×C13711yescharacteristic
262#498C22×C13712yes—
263#486C22×C13712yes—
264#481—21no—
265#481—22no—
266#476—22no—
267#475C22×C13712yes—
268#464—22no—
269#462C22×C13712yes—
270#460C22×C13712yes—
271#460C22×C13712yes—
272#448—21no—
273C4×SD16—22no—
274#431C22×C13711yescharacteristic
275#431C22×C13711yescharacteristic
276#423C22×C13711yescharacteristic
277#417—21no—
278#332C22×C13714yes—
279Q8⋊C8C22×C13714yes—
280Q8⋊C8C22×C13714yes—
281C2×#41—21no—
282C22×Q8—21no—
283C22×Q8D4×C13711yescharacteristic
284C22×Q8D4×C13711yescharacteristic
285#137—44no—
286C2×Q16—22no—
287C2×SD16—22no—
288C2×SD16—22no—
289C2×M16—21no—
290C4⋊Q8—24no—
291C4⋊Q8—24no—
292C4⋊Q8—21no—
293C4⋊Q8—21no—
294C4⋊Q8D4×C13712yes—
295C4⋊Q8C23×C13711yescharacteristic
296C4⋊Q8C23×C13712yes—
297#127C23×C13711yescharacteristic
298#125—28no—
299#125—24no—
300#125—28no—
301#125—24no—
302#124—21no—
303#121—21no—
304#121—21no—
305C4×Q8—24no—
306C4×Q8—24no—
307C4×Q8—24no—
308C4×Q8—24no—
309C4×Q8—21no—
310C4×Q8—21no—
311C4×Q8—21no—
312C4×Q8—21no—
313C4×Q8D4×C13712yes—
314C4×Q8D4×C13712yes—
315C4×Q8C23×C13712yes—
316C4×Q8C23×C13712yes—
317C4×D4—21no—
318C4×D4—21no—
319C2×C4⋊C4—22no—
320C2×C4⋊C4—22no—
321C2×C4⋊C4—22no—
322C2×C4⋊C4—21no—
323C2×C4⋊C4—21no—
324C2×C4⋊C4—21no—
325C2×C4⋊C4D4×C13711yescharacteristic
326C2×C4⋊C4D4×C13711yescharacteristic
327C2×C4⋊C4D4×C13712yes—
328C2×C42—22no—
329C2×C42—21no—
330C2×C42—21no—
331C2×C42C23×C13711yescharacteristic
332#106—22no—
333#106—24no—
334C4⋊C8—21no—
335C4⋊C8—21no—
336C4⋊C8C23×C13712yes—
337C4⋊C8C23×C13712yes—
338#103—22no—
339#103—22no—
340#102—22no—
341#102—22no—
342#102—24no—
343C22⋊C8—21no—
344C22⋊C8—21no—
345C4×C8—21no—
346C4×C8C23×C13712yes—
347#41—42no—
348#41—42no—
349C2×Q8—21no—
350C2×Q8—21no—
351C2×Q8—22no—
352C2×Q8—22no—
353C2×Q8—22no—
354C2×Q8—21no—
355C2×Q8—42no—
356C2×Q8—44no—
357C2×Q8SD16×C13714yescomplemented
358C2×Q8C2×D4×C13712yes—
359C2×Q8C2×D4×C13712yes—
360C2×D4—21no—
361C2×D4—21no—
362C2×D4—21no—
363C2×D4—41no—
364C22×C4—21no—
365C22×C4—21no—
366C22×C4—21no—
367C22×C4—21no—
368C22×C4—22no—
369C22×C4—41no—
370C22×C4C2×D4×C13711yescharacteristic
371C22×C4C2×D4×C13711yescharacteristic
372C22×C4C2×D4×C13711yescharacteristic
373Q16—44no—
374SD16—44no—
375SD16—44no—
376M16—42no—
377C2×C8—22no—
378C2×C8—22no—
379C2×C8—22no—
380C4⋊C4—28no—
381C4⋊C4—28no—
382C4⋊C4—28no—
383C4⋊C4—22no—
384C4⋊C4—22no—
385C4⋊C4—22no—
386C4⋊C4—21no—
387C4⋊C4—21no—
388C4⋊C4—22no—
389C4⋊C4—22no—
390C4⋊C4—22no—
391C4⋊C4—22no—
392C4⋊C4—21no—
393C4⋊C4—21no—
394C4⋊C4C2×D4×C13712yes—
395C4⋊C4C2×D4×C13712yes—
396C4⋊C4C2×D4×C13714yes—
397C22⋊C4—41no—
398C22⋊C4—41no—
399C42—28no—
400C42—22no—
401C42—22no—
402C42—22no—
403C42—22no—
404C42C2×D4×C13711yescharacteristic
405C42C2×D4×C13711yescharacteristic
406C42C2×D4×C13711yescharacteristic
407C42C24×C13711yescharacteristic
408C23—41no—
409C23#120×C13711yescharacteristic
410Q8—24no—
411Q8—22no—
412Q8—22no—
413Q8—22no—
414Q8—42no—
415Q8—44no—
416Q8—44no—
417Q8—41no—
418Q8—41no—
419Q8C2×SD16×C13714yes—
420D4—42no—
421D4—41no—
422D4—41no—
423D4—42no—
424D4—82no—
425C2×C4—24no—
426C2×C4—22no—
427C2×C4—22no—
428C2×C4—22no—
429C2×C4—22no—
430C2×C4—22no—
431C2×C4—24no—
432C2×C4—41no—
433C2×C4—42no—
434C2×C4—44no—
435C2×C4—42no—
436C2×C4—42no—
437C2×C4—41no—
438C2×C4—82no—
439C2×C4#120×C13711yescharacteristic
440C2×C4#120×C13711yescharacteristic
441C2×C4#120×C13711yescharacteristic
442C2×C4C2×SD16×C13712yes—
443C2×C4#137×C13712yes—
444C2×C4C22×D4×C13711yescharacteristic
445C2×C4C22×D4×C13711yescharacteristic
446C2×C4C22×D4×C13711yescharacteristic
447C8—42no—
448C8—42no—
449C8—42no—
450C22—21no—
451C22—21no—
452C22—81no—
453C22—81no—
454C22—81no—
455C22#448×C13712yes—
456C22C2×#120×C13711yescharacteristic
457C4—24no—
458C4—21no—
459C4—21no—
460C4—21no—
461C4—42no—
462C4—44no—
463C4—42no—
464C4—42no—
465C4—81no—
466C4#448×C13712yes—
467C4C22×SD16×C13711yescharacteristic
468C4C2×#137×C13711yescharacteristic
469C4#575×C13712yes—
470C2—22no—
471C2—161no—
472C2C2×#448×C13711yescharacteristic
473C2#3005×C13711yescharacteristic
474C2#3020×C13711yescharacteristic
subgroupG/H
center, socleC22×C137C2×#120
commutator, Frattini, cosocleC42C24×C137
prime radical#532×C137C22
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
1. C2×#557×C137C2semidirect
5. #1653×C137C2semidirect
6. #1626×C137C2semidirect
7. #1549×C137C2semidirect
8. #1487×C137C2semidirect
9. #1479×C137C2semidirect
119. C2×Q8×C137SD16semidirect
237. #18778C137direct (N × H)
238. C137#18778direct (N × H)
239. C2×#557C2×C137semidirect
243. #1653C2×C137semidirect
244. #1626C2×C137semidirect
245. #1549C2×C137semidirect
246. #1487C2×C137semidirect
247. #1479C2×C137semidirect
357. C2×Q8SD16×C137semidirect

Maximal subgroups

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

subgroupindex
C2×#557×C1372
#3227×C1372
#3103×C1372
#3037×C1372
#1653×C1372
#1626×C1372
#1549×C1372
#1487×C1372
#1479×C1372
#18778137