Finite Group Databasebeta

← C137×#9365 #46858751

Subgroups — C137×#9365 #46858751

Order 35072.

802 conjugacy classes of subgroups.

The subgroup lattice has 478 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#3006×C137C211yesmaximal, characteristic
2#2339×C137C211yesmaximal, characteristic, complemented
3#1964×C137C211yesmaximal, characteristic, complemented
4#1789×C137C211yesmaximal, characteristic, complemented
5#1507×C137C211yesmaximal, characteristic, complemented
6#1305×C137C211yesmaximal, characteristic, complemented
7#1297×C137C211yesmaximal, characteristic, complemented
8C22×#41×C137C2211yescharacteristic
9C2×AGL(1,ℤ/8)×C137C412yescomplemented
10C2×#121×C137C2211yescharacteristic
11C2×C4×D4×C137—21no—
12#449×C137—22no—
13#448×C137—21no—
14#447×C137—21no—
15C2×#104×C137—22no—
16#417×C137C412yescomplemented
17C2×#99×C137—22no—
18#407×C137C2211yescharacteristic
19#407×C137C2211yescharacteristic
20#391×C137—21no—
21#386×C137—22no—
22#381×C137—21no—
23#380×C137C2211yescharacteristic
24C4×C22⋊C4×C137C2211yescharacteristic
25C4×C22⋊C4×C137C2211yescharacteristic
26C2×#95×C137—21no—
27#327×C137—21no—
28#327×C137—21no—
29C2×#41×C137—21no—
30C2×#41×C137—21no—
31C2×#41×C137—44no—
32C2×#41×C137C2×C412yes—
33C2×#41×C137D412yes—
34C22×Q8×C137D411yescharacteristic
35C22×D4×C137—21no—
36C22×D4×C137—21no—
37C22×D4×C137C2×C411yescharacteristic, complemented
38C22×D4×C137D411yescharacteristic
39C23×C4×C137—21no—
40C23×C4×C137—22no—
41C23×C4×C137C2311yescharacteristic
42AGL(1,ℤ/8)×C137—44no—
43C2×SD16×C137—22no—
44C2×D8×C137—22no—
45C2×M16×C137—21no—
46C2×M16×C137C2×C412yescomplemented
47#121×C137—21no—
48#121×C137—21no—
49#121×C137—21no—
50#121×C137—21no—
51C4×D4×C137—44no—
52C2×C4⋊C4×C137—21no—
53C2×C4⋊C4×C137C2×C411yescharacteristic, complemented
54C2×C22⋊C4×C137—21no—
55C2×C22⋊C4×C137—22no—
56C2×C22⋊C4×C137—21no—
57C2×C22⋊C4×C137—21no—
58C2×C22⋊C4×C137D412yes—
59C2×C42×C137—21no—
60C2×C42×C137—21no—
61#104×C137—44no—
62#103×C137—22no—
63#102×C137—22no—
64#99×C137—44no—
65C22⋊C8×C137—21no—
66C22⋊C8×C137—21no—
67C22⋊C8×C137—21no—
68C22⋊C8×C137—21no—
69#95×C137—22no—
70#95×C137—22no—
71#95×C137—21no—
72#95×C137—21no—
73#95×C137D411yescharacteristic
74#95×C137D411yescharacteristic
75C24×C137—21no—
76C24×C137—21no—
77C24×C137C2×D411yescharacteristic
78#41×C137—24no—
79#41×C137—24no—
80#41×C137—42no—
81#41×C137—42no—
82#41×C137—44no—
83#41×C137—44no—
84C2×Q8×C137—21no—
85C2×Q8×C137—42no—
86C2×Q8×C137C22⋊C412yes—
87C2×D4×C137—24no—
88C2×D4×C137—22no—
89C2×D4×C137—22no—
90C2×D4×C137—21no—
91C2×D4×C137—21no—
92C2×D4×C137—21no—
93C2×D4×C137—42no—
94C2×D4×C137—44no—
95C2×D4×C137—42no—
96C2×D4×C137—42no—
97C2×D4×C137—41no—
98C2×D4×C137—41no—
99C2×D4×C137C22⋊C412yes—
100C22×C4×C137—21no—
101C22×C4×C137—21no—
102C22×C4×C137—22no—
103C22×C4×C137—22no—
104C22×C4×C137—21no—
105C22×C4×C137—22no—
106C22×C4×C137—21no—
107C22×C4×C137—42no—
108C22×C4×C137—44no—
109C22×C4×C137—44no—
110C22×C4×C137—44no—
111C22×C4×C137—41no—
112C22×C4×C137—82no—
113C22×C4×C137C22×C411yescharacteristic
114C22×C4×C137C2×D411yescharacteristic
115C22×C4×C137C2×D411yescharacteristic
116C22×C4×C137C2×D411yescharacteristic
117C22×C4×C137#4111yescharacteristic
118C22×C4×C137#4111yescharacteristic
119SD16×C137—44no—
120D8×C137—44no—
121M16×C137—42no—
122M16×C137—42no—
123C2×C8×C137—22no—
124C2×C8×C137—41no—
125C4⋊C4×C137—21no—
126C4⋊C4×C137—21no—
127C4⋊C4×C137—42no—
128C22⋊C4×C137—22no—
129C22⋊C4×C137—42no—
130C22⋊C4×C137—41no—
131C22⋊C4×C137—44no—
132C22⋊C4×C137—41no—
133C22⋊C4×C137—41no—
134C22⋊C4×C137—42no—
135C42×C137—42no—
136C42×C137—42no—
137C23×C137—24no—
138C23×C137—22no—
139C23×C137—21no—
140C23×C137—21no—
141C23×C137—41no—
142C23×C137—41no—
143C23×C137—42no—
144C23×C137—42no—
145C23×C137—41no—
146C23×C137—41no—
147C23×C137—41no—
148C23×C137—81no—
149C23×C137—81no—
150C23×C137C4×D411yescharacteristic
151C23×C137#12011yescharacteristic
152C23×C137#12111yescharacteristic
153Q8×C137—22no—
154Q8×C137—22no—
155Q8×C137—42no—
156D4×C137—24no—
157D4×C137—22no—
158D4×C137—44no—
159D4×C137—44no—
160D4×C137—42no—
161D4×C137—81no—
162D4×C137—82no—
163D4×C137—81no—
164D4×C137—81no—
165D4×C137#9914yescomplemented
166C2×C4×C137—22no—
167C2×C4×C137—22no—
168C2×C4×C137—22no—
169C2×C4×C137—22no—
170C2×C4×C137—22no—
171C2×C4×C137—41no—
172C2×C4×C137—41no—
173C2×C4×C137—41no—
174C2×C4×C137—44no—
175C2×C4×C137—44no—
176C2×C4×C137—41no—
177C2×C4×C137—42no—
178C2×C4×C137—41no—
179C2×C4×C137—41no—
180C2×C4×C137—42no—
181C2×C4×C137—42no—
182C2×C4×C137—41no—
183C2×C4×C137—42no—
184C2×C4×C137—42no—
185C2×C4×C137—41no—
186C2×C4×C137—42no—
187C2×C4×C137—41no—
188C2×C4×C137—88no—
189C2×C4×C137—81no—
190C2×C4×C137—81no—
191C2×C4×C137C2×C22⋊C411yescharacteristic
192C2×C4×C137C4×D411yescharacteristic
193C2×C4×C137#12111yescharacteristic
194C2×C4×C137#12311yescharacteristic
195C8×C137—42no—
196C8×C137—81no—
197C22×C137—21no—
198C22×C137—21no—
199C22×C137—21no—
200C22×C137—21no—
201C22×C137—21no—
202C22×C137—21no—
203C22×C137—24no—
204C22×C137—21no—
205C22×C137—21no—
206C22×C137—44no—
207C22×C137—44no—
208C22×C137—42no—
209C22×C137—42no—
210C22×C137—81no—
211C22×C137—81no—
212C22×C137—81no—
213C22×C137—81no—
214C22×C137—81no—
215C22×C137—162no—
216C22×C137#38611yescharacteristic
217C22×C137#45311yescharacteristic
218C22×C137#45411yescharacteristic
219C4×C137—21no—
220C4×C137—21no—
221C4×C137—21no—
222C4×C137—42no—
223C4×C137—42no—
224C4×C137—81no—
225C4×C137—82no—
226C4×C137—82no—
227C4×C137—81no—
228C4×C137C2×#9912yes—
229C2×C137—21no—
230C2×C137—21no—
231C2×C137—41no—
232C2×C137—41no—
233C2×C137—44no—
234C2×C137—161no—
235C2×C137#188311yescharacteristic
236C2×C137#189811yescharacteristic
237C2×C137#190611yescharacteristic
238#9365C13711yesmaximal, characteristic, direct product factor, Sylow
239C137#936511yescharacteristic, direct product factor, Sylow
240#3006C2×C13711yescharacteristic
241#2339C2×C13711yescharacteristic, complemented
242#1964C2×C13711yescharacteristic, complemented
243#1789C2×C13711yescharacteristic, complemented
244#1507C2×C13711yescharacteristic, complemented
245#1305C2×C13711yescharacteristic, complemented
246#1297C2×C13711yescharacteristic, complemented
247C22×#41C22×C13711yescharacteristic
248C2×AGL(1,ℤ/8)C4×C13712yescomplemented
249C2×#121C22×C13711yescharacteristic
250C2×C4×D4—21no—
251#449—22no—
252#448—21no—
253#447—21no—
254C2×#104—22no—
255#417C4×C13712yescomplemented
256C2×#99—22no—
257#407C22×C13711yescharacteristic
258#407C22×C13711yescharacteristic
259#391—21no—
260#386—22no—
261#381—21no—
262#380C22×C13711yescharacteristic
263C4×C22⋊C4C22×C13711yescharacteristic
264C4×C22⋊C4C22×C13711yescharacteristic
265C2×#95—21no—
266#327—21no—
267#327—21no—
268C2×#41—21no—
269C2×#41—21no—
270C2×#41—44no—
271C2×#41C2×C4×C13712yes—
272C2×#41D4×C13712yes—
273C22×Q8D4×C13711yescharacteristic
274C22×D4—21no—
275C22×D4—21no—
276C22×D4C2×C4×C13711yescharacteristic, complemented
277C22×D4D4×C13711yescharacteristic
278C23×C4—21no—
279C23×C4—22no—
280C23×C4C23×C13711yescharacteristic
281AGL(1,ℤ/8)—44no—
282C2×SD16—22no—
283C2×D8—22no—
284C2×M16—21no—
285C2×M16C2×C4×C13712yescomplemented
286#121—21no—
287#121—21no—
288#121—21no—
289#121—21no—
290C4×D4—44no—
291C2×C4⋊C4—21no—
292C2×C4⋊C4C2×C4×C13711yescharacteristic, complemented
293C2×C22⋊C4—21no—
294C2×C22⋊C4—22no—
295C2×C22⋊C4—21no—
296C2×C22⋊C4—21no—
297C2×C22⋊C4D4×C13712yes—
298C2×C42—21no—
299C2×C42—21no—
300#104—44no—
301#103—22no—
302#102—22no—
303#99—44no—
304C22⋊C8—21no—
305C22⋊C8—21no—
306C22⋊C8—21no—
307C22⋊C8—21no—
308#95—22no—
309#95—22no—
310#95—21no—
311#95—21no—
312#95D4×C13711yescharacteristic
313#95D4×C13711yescharacteristic
314C24—21no—
315C24—21no—
316C24C2×D4×C13711yescharacteristic
317#41—24no—
318#41—24no—
319#41—42no—
320#41—42no—
321#41—44no—
322#41—44no—
323C2×Q8—21no—
324C2×Q8—42no—
325C2×Q8C22⋊C4×C13712yes—
326C2×D4—24no—
327C2×D4—22no—
328C2×D4—22no—
329C2×D4—21no—
330C2×D4—21no—
331C2×D4—21no—
332C2×D4—42no—
333C2×D4—44no—
334C2×D4—42no—
335C2×D4—42no—
336C2×D4—41no—
337C2×D4—41no—
338C2×D4C22⋊C4×C13712yes—
339C22×C4—21no—
340C22×C4—21no—
341C22×C4—22no—
342C22×C4—22no—
343C22×C4—21no—
344C22×C4—22no—
345C22×C4—21no—
346C22×C4—42no—
347C22×C4—44no—
348C22×C4—44no—
349C22×C4—44no—
350C22×C4—41no—
351C22×C4—82no—
352C22×C4C22×C4×C13711yescharacteristic
353C22×C4C2×D4×C13711yescharacteristic
354C22×C4C2×D4×C13711yescharacteristic
355C22×C4C2×D4×C13711yescharacteristic
356C22×C4#41×C13711yescharacteristic
357C22×C4#41×C13711yescharacteristic
358SD16—44no—
359D8—44no—
360M16—42no—
361M16—42no—
362C2×C8—22no—
363C2×C8—41no—
364C4⋊C4—21no—
365C4⋊C4—21no—
366C4⋊C4—42no—
367C22⋊C4—22no—
368C22⋊C4—42no—
369C22⋊C4—41no—
370C22⋊C4—44no—
371C22⋊C4—41no—
372C22⋊C4—41no—
373C22⋊C4—42no—
374C42—42no—
375C42—42no—
376C23—24no—
377C23—22no—
378C23—21no—
379C23—21no—
380C23—41no—
381C23—41no—
382C23—42no—
383C23—42no—
384C23—41no—
385C23—41no—
386C23—41no—
387C23—81no—
388C23—81no—
389C23C4×D4×C13711yescharacteristic
390C23#120×C13711yescharacteristic
391C23#121×C13711yescharacteristic
392Q8—22no—
393Q8—22no—
394Q8—42no—
395D4—24no—
396D4—22no—
397D4—44no—
398D4—44no—
399D4—42no—
400D4—81no—
401D4—82no—
402D4—81no—
403D4—81no—
404D4#99×C13714yescomplemented
405C2×C4—22no—
406C2×C4—22no—
407C2×C4—22no—
408C2×C4—22no—
409C2×C4—22no—
410C2×C4—41no—
411C2×C4—41no—
412C2×C4—41no—
413C2×C4—44no—
414C2×C4—44no—
415C2×C4—41no—
416C2×C4—42no—
417C2×C4—41no—
418C2×C4—41no—
419C2×C4—42no—
420C2×C4—42no—
421C2×C4—41no—
422C2×C4—42no—
423C2×C4—42no—
424C2×C4—41no—
425C2×C4—42no—
426C2×C4—41no—
427C2×C4—88no—
428C2×C4—81no—
429C2×C4—81no—
430C2×C4C2×C22⋊C4×C13711yescharacteristic
431C2×C4C4×D4×C13711yescharacteristic
432C2×C4#121×C13711yescharacteristic
433C2×C4#123×C13711yescharacteristic
434C8—42no—
435C8—81no—
436C22—21no—
437C22—21no—
438C22—21no—
439C22—21no—
440C22—21no—
441C22—21no—
442C22—24no—
443C22—21no—
444C22—21no—
445C22—44no—
446C22—44no—
447C22—42no—
448C22—42no—
449C22—81no—
450C22—81no—
451C22—81no—
452C22—81no—
453C22—81no—
454C22—162no—
455C22#386×C13711yescharacteristic
456C22#453×C13711yescharacteristic
457C22#454×C13711yescharacteristic
458C4—21no—
459C4—21no—
460C4—21no—
461C4—42no—
462C4—42no—
463C4—82no—
464C4—81no—
465C4—82no—
466C4—81no—
467C4C2×#99×C13712yes—
468C2—21no—
469C2—21no—
470C2—41no—
471C2—41no—
472C2—44no—
473C2—161no—
474C2#1883×C13711yescharacteristic
475C2#1898×C13711yescharacteristic
476C2#1906×C13711yescharacteristic
subgroupG/H
center, socleC22×C137#386
commutatorC22×C4C22×C4×C137
Frattini, cosocleC23×C4C23×C137
prime radical#3006×C137C2
LCSG ▷ C22×C4 ▷ C22 ▷ C1
UCSC1 ◁ C22×C137 ◁ C23×C4×C137 ◁ G

Complements

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

NH ≅ G/Ntype
2. #2339×C137C2semidirect
3. #1964×C137C2semidirect
4. #1789×C137C2semidirect
5. #1507×C137C2semidirect
6. #1305×C137C2semidirect
7. #1297×C137C2semidirect
9. C2×AGL(1,ℤ/8)×C137C4semidirect
16. #417×C137C4semidirect
37. C22×D4×C137C2×C4semidirect
46. C2×M16×C137C2×C4semidirect
53. C2×C4⋊C4×C137C2×C4semidirect
165. D4×C137#99semidirect
238. #9365C137direct (N × H)
239. C137#9365direct (N × H)
241. #2339C2×C137semidirect
242. #1964C2×C137semidirect
243. #1789C2×C137semidirect
244. #1507C2×C137semidirect
245. #1305C2×C137semidirect
246. #1297C2×C137semidirect
248. C2×AGL(1,ℤ/8)C4×C137semidirect
255. #417C4×C137semidirect
276. C22×D4C2×C4×C137semidirect
285. C2×M16C2×C4×C137semidirect
292. C2×C4⋊C4C2×C4×C137semidirect
404. D4#99×C137semidirect

Maximal subgroups

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

subgroupindex
#3006×C1372
#2339×C1372
#1964×C1372
#1789×C1372
#1507×C1372
#1305×C1372
#1297×C1372
#9365137