Finite Group Databasebeta

← C137×#18717 #46868103

Subgroups — C137×#18717 #46868103

Order 35072.

984 conjugacy classes of subgroups.

The subgroup lattice has 486 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#3548×C137C211yesmaximal, characteristic
2#3230×C137C211yesmaximal, characteristic, complemented
3#3227×C137C211yesmaximal, characteristic, complemented
4#3152×C137C211yesmaximal, characteristic, complemented
5#1632×C137C214yesmaximal, complemented
6#1631×C137C214yesmaximal, complemented
7#1543×C137C211yesmaximal, characteristic, complemented
8#1485×C137C211yesmaximal, characteristic, complemented
9#1484×C137C211yesmaximal, characteristic, complemented
10C2×#137×C137—22no—
11#562×C137—22no—
12#554×C137—21no—
13#554×C137C2212yes—
14#550×C137—21no—
15#550×C137C2212yes—
16#547×C137—21no—
17#547×C137—21no—
18#540×C137—22no—
19#537×C137—22no—
20#537×C137—22no—
21#533×C137C2211yescharacteristic
22#532×C137C2211yescharacteristic
23#532×C137C2211yescharacteristic
24C2×C4⋊Q8×C137C2211yescharacteristic
25#500×C137—21no—
26#498×C137—21no—
27#497×C137—22no—
28#494×C137C2212yes—
29#492×C137C2212yes—
30#487×C137—21no—
31#486×C137—21no—
32#484×C137—22no—
33#481×C137—22no—
34#477×C137C2214yes—
35#476×C137C2214yes—
36#431×C137C2211yescharacteristic
37#431×C137C2211yescharacteristic
38C4×M16×C137C2211yescharacteristic
39#333×C137C2212yes—
40#331×C137C2212yes—
41Q8⋊C8×C137C2214yes—
42D4⋊C8×C137C2214yes—
43C2×#41×C137—22no—
44C2×#41×C137—22no—
45C2×#41×C137—21no—
46C2×#41×C137D412yes—
47C22×Q8×C137—21no—
48#137×C137—48no—
49C2×Q16×C137—24no—
50C2×SD16×C137—24no—
51C2×M16×C137—21no—
52C4⋊Q8×C137—21no—
53C4⋊Q8×C137—21no—
54C4⋊Q8×C137—21no—
55C4⋊Q8×C137D411yescharacteristic
56C4⋊Q8×C137D411yescharacteristic
57C4⋊Q8×C137D411yescharacteristic
58C4⋊Q8×C137C2312yes—
59C4⋊Q8×C137C2311yescharacteristic
60#127×C137—22no—
61#127×C137D411yescharacteristic
62#127×C137C2311yescharacteristic
63#125×C137—22no—
64#125×C137—21no—
65#124×C137—21no—
66#124×C137D412yes—
67#123×C137—24no—
68#123×C137—24no—
69C22⋊Q8×C137—22no—
70C22⋊Q8×C137—24no—
71C22⋊Q8×C137—22no—
72C22⋊Q8×C137—24no—
73C22⋊Q8×C137—21no—
74C22⋊Q8×C137D412yes—
75#121×C137—24no—
76#121×C137—24no—
77#121×C137—22no—
78#121×C137—24no—
79#121×C137—24no—
80#121×C137—21no—
81#121×C137—21no—
82#121×C137—21no—
83#121×C137—22no—
84#121×C137D412yes—
85C4×Q8×C137—21no—
86C4×Q8×C137C2312yes—
87C4×D4×C137—22no—
88C4×D4×C137—22no—
89C4×D4×C137—21no—
90C4×D4×C137—21no—
91C4×D4×C137—21no—
92C4×D4×C137C2312yes—
93#117×C137—21no—
94C2×C4⋊C4×C137—22no—
95C2×C4⋊C4×C137—22no—
96C2×C4⋊C4×C137—21no—
97C2×C4⋊C4×C137—21no—
98C2×C42×C137C2311yescharacteristic
99#107×C137—24no—
100#106×C137—24no—
101C4⋊C8×C137C2312yes—
102C4⋊C8×C137C2312yes—
103#103×C137—24no—
104#102×C137—24no—
105C22⋊C8×C137—21no—
106C22⋊C8×C137—21no—
107#97×C137—21no—
108C4×C8×C137—21no—
109C4×C8×C137—21no—
110C4×C8×C137C2312yes—
111#41×C137—44no—
112#41×C137—82no—
113#41×C137—84no—
114#41×C137—84no—
115C2×Q8×C137—21no—
116C2×Q8×C137—21no—
117C2×Q8×C137—22no—
118C2×Q8×C137—44no—
119C2×Q8×C137—41no—
120C2×Q8×C137C2×D412yes—
121C2×Q8×C137C2×D412yes—
122C2×D4×C137—24no—
123C2×D4×C137—24no—
124C2×D4×C137—22no—
125C2×D4×C137—22no—
126C2×D4×C137—21no—
127C2×D4×C137—21no—
128C2×D4×C137—21no—
129C2×D4×C137—21no—
130C2×D4×C137C2×D412yes—
131C2×D4×C137C2×D412yes—
132C22×C4×C137—24no—
133C22×C4×C137—24no—
134C22×C4×C137—22no—
135C22×C4×C137—22no—
136C22×C4×C137—41no—
137C22×C4×C137C2×D411yescharacteristic
138C22×C4×C137C2×D411yescharacteristic
139C22×C4×C137C2×D411yescharacteristic
140Q16×C137—48no—
141SD16×C137—48no—
142M16×C137—24no—
143C2×C8×C137—22no—
144C2×C8×C137—22no—
145C2×C8×C137—22no—
146C4⋊C4×C137—24no—
147C4⋊C4×C137—24no—
148C4⋊C4×C137—21no—
149C4⋊C4×C137—21no—
150C4⋊C4×C137—21no—
151C4⋊C4×C137—21no—
152C4⋊C4×C137—22no—
153C4⋊C4×C137—21no—
154C4⋊C4×C137—21no—
155C4⋊C4×C137—22no—
156C4⋊C4×C137—22no—
157C4⋊C4×C137—22no—
158C4⋊C4×C137—41no—
159C4⋊C4×C137—41no—
160C4⋊C4×C137C2×D412yes—
161C4⋊C4×C137C2×D412yes—
162C22⋊C4×C137—24no—
163C22⋊C4×C137—24no—
164C22⋊C4×C137—22no—
165C22⋊C4×C137—22no—
166C22⋊C4×C137—22no—
167C22⋊C4×C137—22no—
168C22⋊C4×C137—22no—
169C42×C137—22no—
170C42×C137C2×D411yescharacteristic
171C42×C137C2×D411yescharacteristic
172C42×C137C2×D411yescharacteristic
173C42×C137C2411yescharacteristic
174C23×C137—22no—
175C23×C137—22no—
176C23×C137#12011yescharacteristic
177Q8×C137—24no—
178Q8×C137—42no—
179Q8×C137—44no—
180Q8×C137—42no—
181Q8×C137—81no—
182Q8×C137—82no—
183D4×C137—24no—
184D4×C137—42no—
185D4×C137—42no—
186D4×C137—42no—
187D4×C137—42no—
188D4×C137—81no—
189D4×C137—82no—
190D4×C137—82no—
191D4×C137—84no—
192D4×C137—84no—
193C2×C4×C137—22no—
194C2×C4×C137—22no—
195C2×C4×C137—22no—
196C2×C4×C137—22no—
197C2×C4×C137—22no—
198C2×C4×C137—22no—
199C2×C4×C137—22no—
200C2×C4×C137—44no—
201C2×C4×C137—44no—
202C2×C4×C137—42no—
203C2×C4×C137—84no—
204C2×C4×C137—84no—
205C2×C4×C137—82no—
206C2×C4×C137#12011yescharacteristic
207C2×C4×C137#12011yescharacteristic
208C2×C4×C137#12011yescharacteristic
209C2×C4×C137C22×D411yescharacteristic
210C2×C4×C137C22×D411yescharacteristic
211C2×C4×C137C22×D411yescharacteristic
212C8×C137—24no—
213C8×C137—42no—
214C8×C137—42no—
215C22×C137—21no—
216C22×C137—21no—
217C22×C137—21no—
218C22×C137—42no—
219C22×C137—42no—
220C22×C137—42no—
221C22×C137—42no—
222C22×C137—42no—
223C22×C137—42no—
224C22×C137C2×#12011yescharacteristic
225C4×C137—21no—
226C4×C137—21no—
227C4×C137—21no—
228C4×C137—42no—
229C4×C137—42no—
230C4×C137—42no—
231C4×C137—42no—
232C4×C137—82no—
233C4×C137#45312yes—
234C4×C137#57712yes—
235C4×C137#57812yes—
236C2×C137—41no—
237C2×C137—82no—
238C2×C137—82no—
239C2×C137#300911yescharacteristic
240C2×C137#301111yescharacteristic
241C2×C137C2×#45311yescharacteristic
242#18717C13711yesmaximal, characteristic, direct product factor, Sylow
243C137#1871711yescharacteristic, direct product factor, Sylow
244#3548C2×C13711yescharacteristic
245#3230C2×C13711yescharacteristic, complemented
246#3227C2×C13711yescharacteristic, complemented
247#3152C2×C13711yescharacteristic, complemented
248#1632C2×C13714yescomplemented
249#1631C2×C13714yescomplemented
250#1543C2×C13711yescharacteristic, complemented
251#1485C2×C13711yescharacteristic, complemented
252#1484C2×C13711yescharacteristic, complemented
253C2×#137—22no—
254#562—22no—
255#554—21no—
256#554C22×C13712yes—
257#550—21no—
258#550C22×C13712yes—
259#547—21no—
260#547—21no—
261#540—22no—
262#537—22no—
263#537—22no—
264#533C22×C13711yescharacteristic
265#532C22×C13711yescharacteristic
266#532C22×C13711yescharacteristic
267C2×C4⋊Q8C22×C13711yescharacteristic
268#500—21no—
269#498—21no—
270#497—22no—
271#494C22×C13712yes—
272#492C22×C13712yes—
273#487—21no—
274#486—21no—
275#484—22no—
276#481—22no—
277#477C22×C13714yes—
278#476C22×C13714yes—
279#431C22×C13711yescharacteristic
280#431C22×C13711yescharacteristic
281C4×M16C22×C13711yescharacteristic
282#333C22×C13712yes—
283#331C22×C13712yes—
284Q8⋊C8C22×C13714yes—
285D4⋊C8C22×C13714yes—
286C2×#41—22no—
287C2×#41—22no—
288C2×#41—21no—
289C2×#41D4×C13712yes—
290C22×Q8—21no—
291#137—48no—
292C2×Q16—24no—
293C2×SD16—24no—
294C2×M16—21no—
295C4⋊Q8—21no—
296C4⋊Q8—21no—
297C4⋊Q8—21no—
298C4⋊Q8D4×C13711yescharacteristic
299C4⋊Q8D4×C13711yescharacteristic
300C4⋊Q8D4×C13711yescharacteristic
301C4⋊Q8C23×C13712yes—
302C4⋊Q8C23×C13711yescharacteristic
303#127—22no—
304#127D4×C13711yescharacteristic
305#127C23×C13711yescharacteristic
306#125—22no—
307#125—21no—
308#124—21no—
309#124D4×C13712yes—
310#123—24no—
311#123—24no—
312C22⋊Q8—22no—
313C22⋊Q8—24no—
314C22⋊Q8—22no—
315C22⋊Q8—24no—
316C22⋊Q8—21no—
317C22⋊Q8D4×C13712yes—
318#121—24no—
319#121—24no—
320#121—22no—
321#121—24no—
322#121—24no—
323#121—21no—
324#121—21no—
325#121—21no—
326#121—22no—
327#121D4×C13712yes—
328C4×Q8—21no—
329C4×Q8C23×C13712yes—
330C4×D4—22no—
331C4×D4—22no—
332C4×D4—21no—
333C4×D4—21no—
334C4×D4—21no—
335C4×D4C23×C13712yes—
336#117—21no—
337C2×C4⋊C4—22no—
338C2×C4⋊C4—22no—
339C2×C4⋊C4—21no—
340C2×C4⋊C4—21no—
341C2×C42C23×C13711yescharacteristic
342#107—24no—
343#106—24no—
344C4⋊C8C23×C13712yes—
345C4⋊C8C23×C13712yes—
346#103—24no—
347#102—24no—
348C22⋊C8—21no—
349C22⋊C8—21no—
350#97—21no—
351C4×C8—21no—
352C4×C8—21no—
353C4×C8C23×C13712yes—
354#41—44no—
355#41—82no—
356#41—84no—
357#41—84no—
358C2×Q8—21no—
359C2×Q8—21no—
360C2×Q8—22no—
361C2×Q8—44no—
362C2×Q8—41no—
363C2×Q8C2×D4×C13712yes—
364C2×Q8C2×D4×C13712yes—
365C2×D4—24no—
366C2×D4—24no—
367C2×D4—22no—
368C2×D4—22no—
369C2×D4—21no—
370C2×D4—21no—
371C2×D4—21no—
372C2×D4—21no—
373C2×D4C2×D4×C13712yes—
374C2×D4C2×D4×C13712yes—
375C22×C4—24no—
376C22×C4—24no—
377C22×C4—22no—
378C22×C4—22no—
379C22×C4—41no—
380C22×C4C2×D4×C13711yescharacteristic
381C22×C4C2×D4×C13711yescharacteristic
382C22×C4C2×D4×C13711yescharacteristic
383Q16—48no—
384SD16—48no—
385M16—24no—
386C2×C8—22no—
387C2×C8—22no—
388C2×C8—22no—
389C4⋊C4—24no—
390C4⋊C4—24no—
391C4⋊C4—21no—
392C4⋊C4—21no—
393C4⋊C4—21no—
394C4⋊C4—21no—
395C4⋊C4—22no—
396C4⋊C4—21no—
397C4⋊C4—21no—
398C4⋊C4—22no—
399C4⋊C4—22no—
400C4⋊C4—22no—
401C4⋊C4—41no—
402C4⋊C4—41no—
403C4⋊C4C2×D4×C13712yes—
404C4⋊C4C2×D4×C13712yes—
405C22⋊C4—24no—
406C22⋊C4—24no—
407C22⋊C4—22no—
408C22⋊C4—22no—
409C22⋊C4—22no—
410C22⋊C4—22no—
411C22⋊C4—22no—
412C42—22no—
413C42C2×D4×C13711yescharacteristic
414C42C2×D4×C13711yescharacteristic
415C42C2×D4×C13711yescharacteristic
416C42C24×C13711yescharacteristic
417C23—22no—
418C23—22no—
419C23#120×C13711yescharacteristic
420Q8—24no—
421Q8—42no—
422Q8—44no—
423Q8—42no—
424Q8—81no—
425Q8—82no—
426D4—24no—
427D4—42no—
428D4—42no—
429D4—42no—
430D4—42no—
431D4—81no—
432D4—82no—
433D4—82no—
434D4—84no—
435D4—84no—
436C2×C4—22no—
437C2×C4—22no—
438C2×C4—22no—
439C2×C4—22no—
440C2×C4—22no—
441C2×C4—22no—
442C2×C4—22no—
443C2×C4—44no—
444C2×C4—44no—
445C2×C4—42no—
446C2×C4—84no—
447C2×C4—84no—
448C2×C4—82no—
449C2×C4#120×C13711yescharacteristic
450C2×C4#120×C13711yescharacteristic
451C2×C4#120×C13711yescharacteristic
452C2×C4C22×D4×C13711yescharacteristic
453C2×C4C22×D4×C13711yescharacteristic
454C2×C4C22×D4×C13711yescharacteristic
455C8—24no—
456C8—42no—
457C8—42no—
458C22—21no—
459C22—21no—
460C22—21no—
461C22—42no—
462C22—42no—
463C22—42no—
464C22—42no—
465C22—42no—
466C22—42no—
467C22C2×#120×C13711yescharacteristic
468C4—21no—
469C4—21no—
470C4—21no—
471C4—42no—
472C4—42no—
473C4—42no—
474C4—42no—
475C4—82no—
476C4#453×C13712yes—
477C4#577×C13712yes—
478C4#578×C13712yes—
479C2—41no—
480C2—82no—
481C2—82no—
482C2#3009×C13711yescharacteristic
483C2#3011×C13711yescharacteristic
484C2C2×#453×C13711yescharacteristic
subgroupG/H
center, socleC22×C137C2×#120
commutator, Frattini, cosocleC42C24×C137
prime radical#3548×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. #3230×C137C2semidirect
3. #3227×C137C2semidirect
4. #3152×C137C2semidirect
5. #1632×C137C2semidirect
6. #1631×C137C2semidirect
7. #1543×C137C2semidirect
8. #1485×C137C2semidirect
9. #1484×C137C2semidirect
242. #18717C137direct (N × H)
243. C137#18717direct (N × H)
245. #3230C2×C137semidirect
246. #3227C2×C137semidirect
247. #3152C2×C137semidirect
248. #1632C2×C137semidirect
249. #1631C2×C137semidirect
250. #1543C2×C137semidirect
251. #1485C2×C137semidirect
252. #1484C2×C137semidirect

Maximal subgroups

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

subgroupindex
#3548×C1372
#3230×C1372
#3227×C1372
#3152×C1372
#1632×C1372
#1631×C1372
#1543×C1372
#1485×C1372
#1484×C1372
#18717137