Finite Group Databasebeta

← C137×#19236 #46868622

Subgroups — C137×#19236 #46868622

Order 35072.

662 conjugacy classes of subgroups.

The subgroup lattice has 456 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#3245×C137C211yesmaximal, characteristic
2#3227×C137C211yesmaximal, characteristic, complemented
3#3141×C137C211yesmaximal, characteristic, complemented
4#2935×C137C211yesmaximal, characteristic, complemented
5#1703×C137C212yesmaximal, complemented
6#1697×C137C212yesmaximal, complemented
7#1683×C137C212yesmaximal, complemented
8#1677×C137C212yesmaximal, complemented
9#1565×C137C211yesmaximal, characteristic, complemented
10C2×#333×C137C211yesmaximal, characteristic, complemented
11#1540×C137C211yesmaximal, characteristic, complemented
12C2×#131×C137—21no—
13#532×C137C2211yescharacteristic
14#532×C137C2211yescharacteristic
15C2×C4⋊Q8×C137C2211yescharacteristic
16#498×C137—21no—
17#489×C137—22no—
18#487×C137C2212yes—
19#486×C137—21no—
20#486×C137C2212yes—
21#481×C137—22no—
22#476×C137C2212yes—
23#476×C137C2212yes—
24#474×C137—21no—
25#474×C137C2212yes—
26#466×C137—22no—
27#459×C137—21no—
28#459×C137C2212yes—
29#436×C137C2212yes—
30#435×C137—22no—
31C8×D4×C137C2212yes—
32#431×C137C2211yescharacteristic
33C2×C4⋊C8×C137C2211yescharacteristic
34C2×C4⋊C8×C137C2211yescharacteristic
35#417×C137—21no—
36#417×C137—21no—
37#408×C137—21no—
38C4×M16×C137C2211yescharacteristic
39#335×C137C2212yes—
40#334×C137C2212yes—
41#333×C137C2212yes—
42#333×C137C2212yes—
43#332×C137C2211yescharacteristic
44#332×C137C2211yescharacteristic
45#331×C137C2211yescharacteristic
46#331×C137C2211yescharacteristic
47C2×#41×C137—21no—
48C2×#41×C137—21no—
49C22×Q8×C137—21no—
50C2×D8×C137—22no—
51#131×C137—44no—
52C2×M16×C137—22no—
53C2×M16×C137D411yescharacteristic
54C2×M16×C137D411yescharacteristic
55C22×C8×C137—21no—
56C22×C8×C137—21no—
57C22×C8×C137—22no—
58C4⋊Q8×C137—21no—
59C4⋊Q8×C137—21no—
60C4⋊Q8×C137—21no—
61C4⋊Q8×C137D411yescharacteristic
62C4⋊Q8×C137C2311yescharacteristic
63C4⋊Q8×C137C2312yes—
64#127×C137D411yescharacteristic
65#127×C137C2311yescharacteristic
66#124×C137—21no—
67#124×C137—21no—
68#121×C137—21no—
69#121×C137—21no—
70#121×C137—21no—
71#121×C137D412yes—
72C4×D4×C137—21no—
73C4×D4×C137—21no—
74C4×D4×C137—21no—
75C4×D4×C137C2312yes—
76C2×C4⋊C4×C137—21no—
77C2×C4⋊C4×C137—21no—
78C2×C42×C137C2311yescharacteristic
79#107×C137—22no—
80#106×C137—22no—
81#106×C137—22no—
82C4⋊C8×C137—21no—
83C4⋊C8×C137—21no—
84C4⋊C8×C137C2311yescharacteristic
85C4⋊C8×C137C2311yescharacteristic
86C4⋊C8×C137C2312yes—
87C4⋊C8×C137C2311yescharacteristic
88C4⋊C8×C137C2311yescharacteristic
89#103×C137—22no—
90#103×C137—22no—
91#102×C137—22no—
92#102×C137—22no—
93#102×C137—22no—
94#102×C137—22no—
95#102×C137—22no—
96#102×C137—22no—
97C22⋊C8×C137—21no—
98C22⋊C8×C137—22no—
99C22⋊C8×C137—22no—
100#97×C137D412yes—
101C4×C8×C137—21no—
102C4×C8×C137C2312yes—
103#41×C137—42no—
104#41×C137—42no—
105#41×C137—42no—
106#41×C137—42no—
107C2×Q8×C137—21no—
108C2×Q8×C137—21no—
109C2×Q8×C137—22no—
110C2×Q8×C137—41no—
111C2×Q8×C137—82no—
112C2×D4×C137—22no—
113C2×D4×C137—21no—
114C2×D4×C137—21no—
115C2×D4×C137—21no—
116C2×D4×C137—21no—
117C2×D4×C137—21no—
118C2×D4×C137—41no—
119C2×D4×C137C2×D412yes—
120C22×C4×C137—22no—
121C22×C4×C137—41no—
122C22×C4×C137—41no—
123C22×C4×C137C2×D411yescharacteristic
124C22×C4×C137C2×D411yescharacteristic
125C22×C4×C137#4111yescharacteristic
126D8×C137—82no—
127M16×C137—22no—
128M16×C137—22no—
129M16×C137—44no—
130C2×C8×C137—21no—
131C2×C8×C137—21no—
132C2×C8×C137—21no—
133C2×C8×C137—21no—
134C2×C8×C137—22no—
135C2×C8×C137—44no—
136C2×C8×C137—42no—
137C2×C8×C137—81no—
138C2×C8×C137C2×D412yes—
139C2×C8×C137C2×D412yes—
140C4⋊C4×C137—21no—
141C4⋊C4×C137—22no—
142C4⋊C4×C137—22no—
143C4⋊C4×C137—21no—
144C4⋊C4×C137—21no—
145C4⋊C4×C137—41no—
146C4⋊C4×C137—41no—
147C4⋊C4×C137C2×D412yes—
148C22⋊C4×C137—22no—
149C22⋊C4×C137—22no—
150C22⋊C4×C137—41no—
151C22⋊C4×C137—41no—
152C42×C137C2×D411yescharacteristic
153C42×C137C2×D411yescharacteristic
154C42×C137#4111yescharacteristic
155C42×C137C2411yescharacteristic
156C23×C137—22no—
157C23×C137—41no—
158C23×C137#12111yescharacteristic
159Q8×C137—42no—
160Q8×C137—41no—
161Q8×C137—41no—
162Q8×C137—82no—
163Q8×C137—82no—
164D4×C137—22no—
165D4×C137—22no—
166D4×C137—44no—
167D4×C137—41no—
168D4×C137—41no—
169D4×C137—41no—
170D4×C137—41no—
171D4×C137—42no—
172D4×C137—81no—
173D4×C137—81no—
174D4×C137—82no—
175C2×C4×C137—21no—
176C2×C4×C137—21no—
177C2×C4×C137—22no—
178C2×C4×C137—41no—
179C2×C4×C137—42no—
180C2×C4×C137—42no—
181C2×C4×C137—41no—
182C2×C4×C137—42no—
183C2×C4×C137—82no—
184C2×C4×C137—82no—
185C2×C4×C137#12111yescharacteristic
186C2×C4×C137#12111yescharacteristic
187C2×C4×C137#12111yescharacteristic
188C2×C4×C137#13512yes—
189C2×C4×C137AGL(1,ℤ/8)12yes—
190C2×C4×C137C22×D411yescharacteristic
191C2×C4×C137C22×D411yescharacteristic
192C2×C4×C137C2×#4111yescharacteristic
193C8×C137—22no—
194C8×C137—22no—
195C8×C137—41no—
196C8×C137—41no—
197C8×C137—41no—
198C8×C137—42no—
199C8×C137—41no—
200C22×C137—21no—
201C22×C137—21no—
202C22×C137—42no—
203C22×C137—42no—
204C22×C137—42no—
205C22×C137—81no—
206C22×C137—81no—
207C22×C137—81no—
208C22×C137#46312yes—
209C22×C137C2×#12111yescharacteristic
210C4×C137—21no—
211C4×C137—21no—
212C4×C137—21no—
213C4×C137—42no—
214C4×C137—81no—
215C4×C137—82no—
216C4×C137#46312yes—
217C4×C137C2×#13511yescharacteristic
218C4×C137C2×AGL(1,ℤ/8)11yescharacteristic
219C4×C137#57711yescharacteristic
220C4×C137#57811yescharacteristic
221C2×C137—22no—
222C2×C137—82no—
223C2×C137—161no—
224C2×C137C2×#46311yescharacteristic
225C2×C137#306311yescharacteristic
226C2×C137#306811yescharacteristic
227#19236C13711yesmaximal, characteristic, direct product factor, Sylow
228C137#1923611yescharacteristic, direct product factor, Sylow
229#3245C2×C13711yescharacteristic
230#3227C2×C13711yescharacteristic, complemented
231#3141C2×C13711yescharacteristic, complemented
232#2935C2×C13711yescharacteristic, complemented
233#1703C2×C13712yescomplemented
234#1697C2×C13712yescomplemented
235#1683C2×C13712yescomplemented
236#1677C2×C13712yescomplemented
237#1565C2×C13711yescharacteristic, complemented
238C2×#333C2×C13711yescharacteristic, complemented
239#1540C2×C13711yescharacteristic, complemented
240C2×#131—21no—
241#532C22×C13711yescharacteristic
242#532C22×C13711yescharacteristic
243C2×C4⋊Q8C22×C13711yescharacteristic
244#498—21no—
245#489—22no—
246#487C22×C13712yes—
247#486—21no—
248#486C22×C13712yes—
249#481—22no—
250#476C22×C13712yes—
251#476C22×C13712yes—
252#474—21no—
253#474C22×C13712yes—
254#466—22no—
255#459—21no—
256#459C22×C13712yes—
257#436C22×C13712yes—
258#435—22no—
259C8×D4C22×C13712yes—
260#431C22×C13711yescharacteristic
261C2×C4⋊C8C22×C13711yescharacteristic
262C2×C4⋊C8C22×C13711yescharacteristic
263#417—21no—
264#417—21no—
265#408—21no—
266C4×M16C22×C13711yescharacteristic
267#335C22×C13712yes—
268#334C22×C13712yes—
269#333C22×C13712yes—
270#333C22×C13712yes—
271#332C22×C13711yescharacteristic
272#332C22×C13711yescharacteristic
273#331C22×C13711yescharacteristic
274#331C22×C13711yescharacteristic
275C2×#41—21no—
276C2×#41—21no—
277C22×Q8—21no—
278C2×D8—22no—
279#131—44no—
280C2×M16—22no—
281C2×M16D4×C13711yescharacteristic
282C2×M16D4×C13711yescharacteristic
283C22×C8—21no—
284C22×C8—21no—
285C22×C8—22no—
286C4⋊Q8—21no—
287C4⋊Q8—21no—
288C4⋊Q8—21no—
289C4⋊Q8D4×C13711yescharacteristic
290C4⋊Q8C23×C13712yes—
291C4⋊Q8C23×C13711yescharacteristic
292#127D4×C13711yescharacteristic
293#127C23×C13711yescharacteristic
294#124—21no—
295#124—21no—
296#121—21no—
297#121—21no—
298#121—21no—
299#121D4×C13712yes—
300C4×D4—21no—
301C4×D4—21no—
302C4×D4—21no—
303C4×D4C23×C13712yes—
304C2×C4⋊C4—21no—
305C2×C4⋊C4—21no—
306C2×C42C23×C13711yescharacteristic
307#107—22no—
308#106—22no—
309#106—22no—
310C4⋊C8—21no—
311C4⋊C8—21no—
312C4⋊C8C23×C13711yescharacteristic
313C4⋊C8C23×C13711yescharacteristic
314C4⋊C8C23×C13711yescharacteristic
315C4⋊C8C23×C13711yescharacteristic
316C4⋊C8C23×C13712yes—
317#103—22no—
318#103—22no—
319#102—22no—
320#102—22no—
321#102—22no—
322#102—22no—
323#102—22no—
324#102—22no—
325C22⋊C8—21no—
326C22⋊C8—22no—
327C22⋊C8—22no—
328#97D4×C13712yes—
329C4×C8—21no—
330C4×C8C23×C13712yes—
331#41—42no—
332#41—42no—
333#41—42no—
334#41—42no—
335C2×Q8—21no—
336C2×Q8—21no—
337C2×Q8—22no—
338C2×Q8—41no—
339C2×Q8—82no—
340C2×D4—22no—
341C2×D4—21no—
342C2×D4—21no—
343C2×D4—21no—
344C2×D4—21no—
345C2×D4—21no—
346C2×D4—41no—
347C2×D4C2×D4×C13712yes—
348C22×C4—22no—
349C22×C4—41no—
350C22×C4—41no—
351C22×C4C2×D4×C13711yescharacteristic
352C22×C4C2×D4×C13711yescharacteristic
353C22×C4#41×C13711yescharacteristic
354D8—82no—
355M16—22no—
356M16—22no—
357M16—44no—
358C2×C8—21no—
359C2×C8—21no—
360C2×C8—21no—
361C2×C8—21no—
362C2×C8—22no—
363C2×C8—44no—
364C2×C8—42no—
365C2×C8—81no—
366C2×C8C2×D4×C13712yes—
367C2×C8C2×D4×C13712yes—
368C4⋊C4—21no—
369C4⋊C4—22no—
370C4⋊C4—22no—
371C4⋊C4—21no—
372C4⋊C4—21no—
373C4⋊C4—41no—
374C4⋊C4—41no—
375C4⋊C4C2×D4×C13712yes—
376C22⋊C4—22no—
377C22⋊C4—22no—
378C22⋊C4—41no—
379C22⋊C4—41no—
380C42C2×D4×C13711yescharacteristic
381C42C2×D4×C13711yescharacteristic
382C42#41×C13711yescharacteristic
383C42C24×C13711yescharacteristic
384C23—22no—
385C23—41no—
386C23#121×C13711yescharacteristic
387Q8—42no—
388Q8—41no—
389Q8—41no—
390Q8—82no—
391Q8—82no—
392D4—22no—
393D4—22no—
394D4—44no—
395D4—42no—
396D4—41no—
397D4—41no—
398D4—41no—
399D4—41no—
400D4—81no—
401D4—81no—
402D4—82no—
403C2×C4—21no—
404C2×C4—21no—
405C2×C4—22no—
406C2×C4—41no—
407C2×C4—42no—
408C2×C4—42no—
409C2×C4—41no—
410C2×C4—42no—
411C2×C4—82no—
412C2×C4—82no—
413C2×C4#121×C13711yescharacteristic
414C2×C4#121×C13711yescharacteristic
415C2×C4#121×C13711yescharacteristic
416C2×C4#135×C13712yes—
417C2×C4AGL(1,ℤ/8)×C13712yes—
418C2×C4C22×D4×C13711yescharacteristic
419C2×C4C22×D4×C13711yescharacteristic
420C2×C4C2×#41×C13711yescharacteristic
421C8—22no—
422C8—22no—
423C8—41no—
424C8—41no—
425C8—42no—
426C8—41no—
427C8—41no—
428C22—21no—
429C22—21no—
430C22—42no—
431C22—42no—
432C22—42no—
433C22—81no—
434C22—81no—
435C22—81no—
436C22#463×C13712yes—
437C22C2×#121×C13711yescharacteristic
438C4—21no—
439C4—21no—
440C4—21no—
441C4—42no—
442C4—82no—
443C4—81no—
444C4#463×C13712yes—
445C4C2×#135×C13711yescharacteristic
446C4C2×AGL(1,ℤ/8)×C13711yescharacteristic
447C4#577×C13711yescharacteristic
448C4#578×C13711yescharacteristic
449C2—22no—
450C2—82no—
451C2—161no—
452C2C2×#463×C13711yescharacteristic
453C2#3063×C13711yescharacteristic
454C2#3068×C13711yescharacteristic
subgroupG/H
center, socleC22×C137C2×#121
commutator, Frattini, cosocleC42C24×C137
prime radical#3245×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
2. #3227×C137C2semidirect
3. #3141×C137C2semidirect
4. #2935×C137C2semidirect
5. #1703×C137C2semidirect
6. #1697×C137C2semidirect
7. #1683×C137C2semidirect
8. #1677×C137C2semidirect
9. #1565×C137C2semidirect
10. C2×#333×C137C2semidirect
11. #1540×C137C2semidirect
227. #19236C137direct (N × H)
228. C137#19236direct (N × H)
230. #3227C2×C137semidirect
231. #3141C2×C137semidirect
232. #2935C2×C137semidirect
233. #1703C2×C137semidirect
234. #1697C2×C137semidirect
235. #1683C2×C137semidirect
236. #1677C2×C137semidirect
237. #1565C2×C137semidirect
238. C2×#333C2×C137semidirect
239. #1540C2×C137semidirect

Maximal subgroups

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

subgroupindex
#3245×C1372
#3227×C1372
#3141×C1372
#2935×C1372
#1703×C1372
#1697×C1372
#1683×C1372
#1677×C1372
#1565×C1372
C2×#333×C1372
#1540×C1372
#19236137