Finite Group Databasebeta

← C137×#19408 #46868794

Subgroups — C137×#19408 #46868794

Order 35072.

674 conjugacy classes of subgroups.

The subgroup lattice has 498 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#3206×C137C211yesmaximal, characteristic, complemented
2#3079×C137C211yesmaximal, characteristic, complemented
3#3035×C137C211yesmaximal, characteristic
4#2988×C137C211yesmaximal, characteristic, complemented
5#1680×C137C212yesmaximal, complemented
6#1679×C137C212yesmaximal, complemented
7#1666×C137C212yesmaximal, complemented
8#1665×C137C212yesmaximal, complemented
9#1570×C137C211yesmaximal, characteristic, complemented
10#1547×C137C211yesmaximal, characteristic, complemented
11C2×#332×C137C211yesmaximal, characteristic, complemented
12C2×#135×C137—21no—
13#532×C137C2211yescharacteristic
14C2×C4⋊Q8×C137C2211yescharacteristic
15C4×#41×C137C2211yescharacteristic
16#492×C137C2212yes—
17#487×C137C2212yes—
18#477×C137—21no—
19#476×C137—21no—
20#475×C137C2212yes—
21#474×C137C2212yes—
22#464×C137—21no—
23#463×C137—21no—
24#462×C137C2211yescharacteristic
25#461×C137—21no—
26#461×C137—21no—
27#461×C137C2211yescharacteristic
28#460×C137C2211yescharacteristic
29#459×C137C2211yescharacteristic
30#448×C137—22no—
31C8×Q8×C137C2212yes—
32C8×D4×C137—22no—
33C8×D4×C137C2212yes—
34#431×C137—21no—
35#431×C137C2211yescharacteristic
36#428×C137—21no—
37#423×C137C2211yescharacteristic
38C2×C4⋊C8×C137—21no—
39C2×C4⋊C8×C137C2211yescharacteristic
40C2×C4⋊C8×C137C2211yescharacteristic
41#417×C137—21no—
42#416×C137—21no—
43#334×C137C2212yes—
44#334×C137C2212yes—
45#332×C137C2212yes—
46#332×C137C2212yes—
47#332×C137C2212yes—
48#331×C137C2212yes—
49C2×#41×C137—21no—
50C2×#41×C137D411yescharacteristic
51C22×Q8×C137—21no—
52#135×C137—44no—
53C2×Q16×C137—21no—
54C2×SD16×C137—21no—
55C2×SD16×C137—21no—
56C2×SD16×C137—22no—
57C2×SD16×C137—22no—
58C2×D8×C137—21no—
59C2×M16×C137—21no—
60C22×C8×C137—21no—
61C22×C8×C137—21no—
62C22×C8×C137—22no—
63C4⋊Q8×C137—21no—
64C4⋊Q8×C137—21no—
65C4⋊Q8×C137—21no—
66C4⋊Q8×C137C2311yescharacteristic
67C4⋊Q8×C137C2312yes—
68#127×C137C2311yescharacteristic
69#124×C137—21no—
70#121×C137—21no—
71#121×C137—21no—
72C4×Q8×C137D411yescharacteristic
73C4×Q8×C137C2311yescharacteristic
74C4×D4×C137—21no—
75C4×D4×C137—21no—
76C4×D4×C137—21no—
77C4×D4×C137—21no—
78C4×D4×C137D411yescharacteristic
79C4×D4×C137C2311yescharacteristic
80#117×C137—21no—
81#117×C137D411yescharacteristic
82C2×C4⋊C4×C137—21no—
83C2×C4⋊C4×C137—21no—
84C2×C42×C137—21no—
85C2×C42×C137C2311yescharacteristic
86#107×C137—22no—
87#106×C137—22no—
88C4⋊C8×C137—22no—
89C4⋊C8×C137—21no—
90C4⋊C8×C137—21no—
91C4⋊C8×C137—21no—
92C4⋊C8×C137D411yescharacteristic
93C4⋊C8×C137D411yescharacteristic
94C4⋊C8×C137C2312yes—
95C4⋊C8×C137C2311yescharacteristic
96C4⋊C8×C137C2311yescharacteristic
97C4⋊C8×C137C2312yes—
98#103×C137—22no—
99#103×C137—22no—
100#103×C137—22no—
101#103×C137—21no—
102#103×C137—21no—
103#102×C137—22no—
104#102×C137—21no—
105#102×C137—21no—
106C22⋊C8×C137—22no—
107C22⋊C8×C137D412yes—
108C4×C8×C137—22no—
109C4×C8×C137C2312yes—
110#41×C137—22no—
111#41×C137—42no—
112#41×C137—42no—
113#41×C137SD1614yescomplemented
114C2×Q8×C137—21no—
115C2×Q8×C137—22no—
116C2×Q8×C137—41no—
117C2×Q8×C137—81no—
118C2×Q8×C137—81no—
119C2×Q8×C137C2×D411yescharacteristic
120C2×D4×C137—21no—
121C2×D4×C137—21no—
122C2×D4×C137—21no—
123C2×D4×C137—21no—
124C2×D4×C137—41no—
125C2×D4×C137C2×D411yescharacteristic
126C22×C4×C137—21no—
127C22×C4×C137—21no—
128C22×C4×C137—21no—
129C22×C4×C137—41no—
130C22×C4×C137—41no—
131C22×C4×C137C2×D411yescharacteristic
132C22×C4×C137C2×D411yescharacteristic
133C22×C4×C137#4111yescharacteristic
134Q16×C137—42no—
135SD16×C137—42no—
136SD16×C137—42no—
137SD16×C137—82no—
138SD16×C137—82no—
139D8×C137—42no—
140M16×C137—42no—
141C2×C8×C137—22no—
142C2×C8×C137—22no—
143C2×C8×C137—21no—
144C2×C8×C137—21no—
145C2×C8×C137—42no—
146C2×C8×C137—82no—
147C2×C8×C137—81no—
148C2×C8×C137C2×D412yes—
149C2×C8×C137C2×D412yes—
150C4⋊C4×C137—21no—
151C4⋊C4×C137—21no—
152C4⋊C4×C137—22no—
153C4⋊C4×C137—21no—
154C4⋊C4×C137—22no—
155C4⋊C4×C137—21no—
156C4⋊C4×C137—41no—
157C4⋊C4×C137—41no—
158C4⋊C4×C137C2×D411yescharacteristic
159C4⋊C4×C137C2×D411yescharacteristic
160C22⋊C4×C137—21no—
161C22⋊C4×C137—21no—
162C22⋊C4×C137—21no—
163C22⋊C4×C137—41no—
164C22⋊C4×C137—41no—
165C42×C137—21no—
166C42×C137—21no—
167C42×C137—21no—
168C42×C137C2×D411yescharacteristic
169C42×C137C2×D411yescharacteristic
170C42×C137#4111yescharacteristic
171C42×C137C2411yescharacteristic
172C23×C137—21no—
173C23×C137—41no—
174C23×C137#12111yescharacteristic
175Q8×C137—21no—
176Q8×C137—41no—
177Q8×C137—41no—
178Q8×C137—42no—
179Q8×C137—42no—
180Q8×C137—82no—
181Q8×C137C2×SD1612yes—
182D4×C137—24no—
183D4×C137—21no—
184D4×C137—41no—
185D4×C137—41no—
186D4×C137—41no—
187D4×C137—41no—
188D4×C137—81no—
189D4×C137—82no—
190D4×C137C2×SD1612yes—
191C2×C4×C137—21no—
192C2×C4×C137—21no—
193C2×C4×C137—21no—
194C2×C4×C137—24no—
195C2×C4×C137—21no—
196C2×C4×C137—21no—
197C2×C4×C137—41no—
198C2×C4×C137—41no—
199C2×C4×C137—41no—
200C2×C4×C137—41no—
201C2×C4×C137—42no—
202C2×C4×C137—81no—
203C2×C4×C137—82no—
204C2×C4×C137—82no—
205C2×C4×C137#12111yescharacteristic
206C2×C4×C137#12111yescharacteristic
207C2×C4×C137#12111yescharacteristic
208C2×C4×C137C2×SD1612yes—
209C2×C4×C137#13512yes—
210C2×C4×C137C22×D411yescharacteristic
211C2×C4×C137C22×D411yescharacteristic
212C2×C4×C137C2×#4111yescharacteristic
213C8×C137—22no—
214C8×C137—22no—
215C8×C137—41no—
216C8×C137—41no—
217C8×C137—42no—
218C8×C137—42no—
219C22×C137—21no—
220C22×C137—21no—
221C22×C137—22no—
222C22×C137—41no—
223C22×C137—41no—
224C22×C137—81no—
225C22×C137—81no—
226C22×C137—81no—
227C22×C137#46512yes—
228C22×C137C2×#12111yescharacteristic
229C4×C137—21no—
230C4×C137—22no—
231C4×C137—21no—
232C4×C137—21no—
233C4×C137—41no—
234C4×C137—41no—
235C4×C137—82no—
236C4×C137—81no—
237C4×C137#46512yes—
238C4×C137C22×SD1611yescharacteristic
239C4×C137C2×#13511yescharacteristic
240C4×C137#57611yescharacteristic
241C4×C137#57711yescharacteristic
242C2×C137—22no—
243C2×C137—42no—
244C2×C137—161no—
245C2×C137#304511yescharacteristic
246C2×C137C2×#46511yescharacteristic
247C2×C137#306911yescharacteristic
248#19408C13711yesmaximal, characteristic, direct product factor, Sylow
249C137#1940811yescharacteristic, direct product factor, Sylow
250#3206C2×C13711yescharacteristic, complemented
251#3079C2×C13711yescharacteristic, complemented
252#3035C2×C13711yescharacteristic
253#2988C2×C13711yescharacteristic, complemented
254#1680C2×C13712yescomplemented
255#1679C2×C13712yescomplemented
256#1666C2×C13712yescomplemented
257#1665C2×C13712yescomplemented
258#1570C2×C13711yescharacteristic, complemented
259#1547C2×C13711yescharacteristic, complemented
260C2×#332C2×C13711yescharacteristic, complemented
261C2×#135—21no—
262#532C22×C13711yescharacteristic
263C2×C4⋊Q8C22×C13711yescharacteristic
264C4×#41C22×C13711yescharacteristic
265#492C22×C13712yes—
266#487C22×C13712yes—
267#477—21no—
268#476—21no—
269#475C22×C13712yes—
270#474C22×C13712yes—
271#464—21no—
272#463—21no—
273#462C22×C13711yescharacteristic
274#461—21no—
275#461—21no—
276#461C22×C13711yescharacteristic
277#460C22×C13711yescharacteristic
278#459C22×C13711yescharacteristic
279#448—22no—
280C8×Q8C22×C13712yes—
281C8×D4—22no—
282C8×D4C22×C13712yes—
283#431—21no—
284#431C22×C13711yescharacteristic
285#428—21no—
286#423C22×C13711yescharacteristic
287C2×C4⋊C8—21no—
288C2×C4⋊C8C22×C13711yescharacteristic
289C2×C4⋊C8C22×C13711yescharacteristic
290#417—21no—
291#416—21no—
292#334C22×C13712yes—
293#334C22×C13712yes—
294#332C22×C13712yes—
295#332C22×C13712yes—
296#332C22×C13712yes—
297#331C22×C13712yes—
298C2×#41—21no—
299C2×#41D4×C13711yescharacteristic
300C22×Q8—21no—
301#135—44no—
302C2×Q16—21no—
303C2×SD16—21no—
304C2×SD16—21no—
305C2×SD16—22no—
306C2×SD16—22no—
307C2×D8—21no—
308C2×M16—21no—
309C22×C8—21no—
310C22×C8—21no—
311C22×C8—22no—
312C4⋊Q8—21no—
313C4⋊Q8—21no—
314C4⋊Q8—21no—
315C4⋊Q8C23×C13712yes—
316C4⋊Q8C23×C13711yescharacteristic
317#127C23×C13711yescharacteristic
318#124—21no—
319#121—21no—
320#121—21no—
321C4×Q8D4×C13711yescharacteristic
322C4×Q8C23×C13711yescharacteristic
323C4×D4—21no—
324C4×D4—21no—
325C4×D4—21no—
326C4×D4—21no—
327C4×D4D4×C13711yescharacteristic
328C4×D4C23×C13711yescharacteristic
329#117—21no—
330#117D4×C13711yescharacteristic
331C2×C4⋊C4—21no—
332C2×C4⋊C4—21no—
333C2×C42—21no—
334C2×C42C23×C13711yescharacteristic
335#107—22no—
336#106—22no—
337C4⋊C8—22no—
338C4⋊C8—21no—
339C4⋊C8—21no—
340C4⋊C8—21no—
341C4⋊C8D4×C13711yescharacteristic
342C4⋊C8D4×C13711yescharacteristic
343C4⋊C8C23×C13712yes—
344C4⋊C8C23×C13712yes—
345C4⋊C8C23×C13711yescharacteristic
346C4⋊C8C23×C13711yescharacteristic
347#103—22no—
348#103—22no—
349#103—22no—
350#103—21no—
351#103—21no—
352#102—22no—
353#102—21no—
354#102—21no—
355C22⋊C8—22no—
356C22⋊C8D4×C13712yes—
357C4×C8—22no—
358C4×C8C23×C13712yes—
359#41—22no—
360#41—42no—
361#41—42no—
362#41SD16×C13714yescomplemented
363C2×Q8—21no—
364C2×Q8—22no—
365C2×Q8—41no—
366C2×Q8—81no—
367C2×Q8—81no—
368C2×Q8C2×D4×C13711yescharacteristic
369C2×D4—21no—
370C2×D4—21no—
371C2×D4—21no—
372C2×D4—21no—
373C2×D4—41no—
374C2×D4C2×D4×C13711yescharacteristic
375C22×C4—21no—
376C22×C4—21no—
377C22×C4—21no—
378C22×C4—41no—
379C22×C4—41no—
380C22×C4C2×D4×C13711yescharacteristic
381C22×C4C2×D4×C13711yescharacteristic
382C22×C4#41×C13711yescharacteristic
383Q16—42no—
384SD16—42no—
385SD16—42no—
386SD16—82no—
387SD16—82no—
388D8—42no—
389M16—42no—
390C2×C8—22no—
391C2×C8—22no—
392C2×C8—21no—
393C2×C8—21no—
394C2×C8—42no—
395C2×C8—82no—
396C2×C8—81no—
397C2×C8C2×D4×C13712yes—
398C2×C8C2×D4×C13712yes—
399C4⋊C4—21no—
400C4⋊C4—21no—
401C4⋊C4—22no—
402C4⋊C4—21no—
403C4⋊C4—22no—
404C4⋊C4—21no—
405C4⋊C4—41no—
406C4⋊C4—41no—
407C4⋊C4C2×D4×C13711yescharacteristic
408C4⋊C4C2×D4×C13711yescharacteristic
409C22⋊C4—21no—
410C22⋊C4—21no—
411C22⋊C4—21no—
412C22⋊C4—41no—
413C22⋊C4—41no—
414C42—21no—
415C42—21no—
416C42—21no—
417C42C2×D4×C13711yescharacteristic
418C42C2×D4×C13711yescharacteristic
419C42#41×C13711yescharacteristic
420C42C24×C13711yescharacteristic
421C23—21no—
422C23—41no—
423C23#121×C13711yescharacteristic
424Q8—21no—
425Q8—41no—
426Q8—41no—
427Q8—42no—
428Q8—42no—
429Q8—82no—
430Q8C2×SD16×C13712yes—
431D4—24no—
432D4—21no—
433D4—41no—
434D4—41no—
435D4—41no—
436D4—41no—
437D4—81no—
438D4—82no—
439D4C2×SD16×C13712yes—
440C2×C4—21no—
441C2×C4—24no—
442C2×C4—21no—
443C2×C4—21no—
444C2×C4—21no—
445C2×C4—21no—
446C2×C4—41no—
447C2×C4—41no—
448C2×C4—41no—
449C2×C4—41no—
450C2×C4—42no—
451C2×C4—81no—
452C2×C4—82no—
453C2×C4—82no—
454C2×C4#121×C13711yescharacteristic
455C2×C4#121×C13711yescharacteristic
456C2×C4#121×C13711yescharacteristic
457C2×C4C2×SD16×C13712yes—
458C2×C4#135×C13712yes—
459C2×C4C22×D4×C13711yescharacteristic
460C2×C4C22×D4×C13711yescharacteristic
461C2×C4C2×#41×C13711yescharacteristic
462C8—22no—
463C8—22no—
464C8—41no—
465C8—41no—
466C8—42no—
467C8—42no—
468C22—21no—
469C22—21no—
470C22—22no—
471C22—41no—
472C22—41no—
473C22—81no—
474C22—81no—
475C22—81no—
476C22#465×C13712yes—
477C22C2×#121×C13711yescharacteristic
478C4—21no—
479C4—22no—
480C4—21no—
481C4—21no—
482C4—41no—
483C4—41no—
484C4—82no—
485C4—81no—
486C4#465×C13712yes—
487C4C22×SD16×C13711yescharacteristic
488C4C2×#135×C13711yescharacteristic
489C4#576×C13711yescharacteristic
490C4#577×C13711yescharacteristic
491C2—22no—
492C2—42no—
493C2—161no—
494C2#3045×C13711yescharacteristic
495C2C2×#465×C13711yescharacteristic
496C2#3069×C13711yescharacteristic
subgroupG/H
center, socleC22×C137C2×#121
commutator, Frattini, cosocleC42C24×C137
prime radical#3035×C137C2
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. #3206×C137C2semidirect
2. #3079×C137C2semidirect
4. #2988×C137C2semidirect
5. #1680×C137C2semidirect
6. #1679×C137C2semidirect
7. #1666×C137C2semidirect
8. #1665×C137C2semidirect
9. #1570×C137C2semidirect
10. #1547×C137C2semidirect
11. C2×#332×C137C2semidirect
113. #41×C137SD16semidirect
248. #19408C137direct (N × H)
249. C137#19408direct (N × H)
250. #3206C2×C137semidirect
251. #3079C2×C137semidirect
253. #2988C2×C137semidirect
254. #1680C2×C137semidirect
255. #1679C2×C137semidirect
256. #1666C2×C137semidirect
257. #1665C2×C137semidirect
258. #1570C2×C137semidirect
259. #1547C2×C137semidirect
260. C2×#332C2×C137semidirect
362. #41SD16×C137semidirect

Maximal subgroups

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

subgroupindex
#3206×C1372
#3079×C1372
#3035×C1372
#2988×C1372
#1680×C1372
#1679×C1372
#1666×C1372
#1665×C1372
#1570×C1372
#1547×C1372
C2×#332×C1372
#19408137