Finite Group Databasebeta

← C137×#18383 #46867769

Subgroups — C137×#18383 #46867769

Order 35072.

986 conjugacy classes of subgroups.

The subgroup lattice has 516 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#3501×C137C211yesmaximal, characteristic
2#3189×C137C212yesmaximal, complemented
3#2564×C137C211yesmaximal, characteristic
4#1617×C137C212yesmaximal, complemented
5#1614×C137C212yesmaximal, complemented
6#1605×C137C212yesmaximal, complemented
7#1602×C137C212yesmaximal, complemented
8C2×#328×C137C212yesmaximal, complemented
9#1459×C137C211yesmaximal, characteristic
10#554×C137—24no—
11#554×C137—22no—
12D4×Q8×C137—22no—
13#544×C137C2212yes—
14#543×C137—21no—
15#543×C137C2212yes—
16#533×C137—21no—
17C2×C22⋊Q8×C137—21no—
18C2×C22⋊Q8×C137C2212yes—
19C2×C22⋊Q8×C137C2211yescharacteristic
20#518×C137—21no—
21C2×C4×D4×C137C2211yescharacteristic
22#484×C137—22no—
23#484×C137C2214yes—
24#467×C137—24no—
25#451×C137—22no—
26#451×C137C2214yes—
27C2×C22⋊C8×C137C2212yes—
28#400×C137C2211yescharacteristic
29#399×C137C2211yescharacteristic
30#395×C137C2211yescharacteristic
31#393×C137C2211yescharacteristic
32#387×C137—21no—
33#386×C137—21no—
34#385×C137—21no—
35#380×C137—21no—
36C2×#95×C137C2211yescharacteristic
37#328×C137C2214yes—
38#328×C137C2214yes—
39#324×C137C2212yes—
40#323×C137C2212yes—
41C22×Q8×C137—21no—
42C22×Q8×C137D412yes—
43C22×Q8×C137C2311yescharacteristic
44C22×D4×C137D411yescharacteristic
45C23×C4×C137—21no—
46C23×C4×C137D411yescharacteristic
47C23×C4×C137C2311yescharacteristic
48C2×Q16×C137—24no—
49C22×C8×C137—22no—
50C4⋊Q8×C137—21no—
51C4⋊Q8×C137—21no—
52C4⋊Q8×C137—24no—
53#125×C137—22no—
54#125×C137—24no—
55C22⋊Q8×C137—22no—
56C22⋊Q8×C137—22no—
57C22⋊Q8×C137—24no—
58C22⋊Q8×C137—24no—
59C22⋊Q8×C137—24no—
60C22⋊Q8×C137—22no—
61C22⋊Q8×C137—22no—
62C22⋊Q8×C137—22no—
63C22⋊Q8×C137—22no—
64C22⋊Q8×C137—24no—
65C22⋊Q8×C137—22no—
66C22⋊Q8×C137—22no—
67C22⋊Q8×C137D414yes—
68C22⋊Q8×C137C2314yes—
69C4×Q8×C137—22no—
70C4×Q8×C137—22no—
71C4×D4×C137—24no—
72C4×D4×C137—24no—
73C4×D4×C137—22no—
74C4×D4×C137—22no—
75#117×C137—22no—
76C2×C4⋊C4×C137—21no—
77C2×C4⋊C4×C137—22no—
78C2×C4⋊C4×C137—22no—
79C2×C4⋊C4×C137—21no—
80C2×C4⋊C4×C137—21no—
81C2×C4⋊C4×C137—21no—
82C2×C4⋊C4×C137D412yes—
83C2×C4⋊C4×C137C2311yescharacteristic
84C2×C4⋊C4×C137C2311yescharacteristic
85C2×C22⋊C4×C137—22no—
86C2×C22⋊C4×C137—22no—
87C2×C22⋊C4×C137—21no—
88C2×C22⋊C4×C137—21no—
89C2×C22⋊C4×C137—21no—
90C2×C22⋊C4×C137D411yescharacteristic
91C2×C22⋊C4×C137C2311yescharacteristic
92C2×C42×C137D411yescharacteristic
93#107×C137—24no—
94#103×C137—24no—
95#103×C137—24no—
96C22⋊C8×C137—22no—
97C22⋊C8×C137C2314yes—
98#95×C137—21no—
99#95×C137—21no—
100#95×C137—21no—
101#95×C137—21no—
102#95×C137—21no—
103#95×C137C2311yescharacteristic
104#95×C137C2311yescharacteristic
105C24×C137C2×D411yescharacteristic
106C24×C137C2×D411yescharacteristic
107C2×Q8×C137—22no—
108C2×Q8×C137—22no—
109C2×Q8×C137—22no—
110C2×Q8×C137—22no—
111C2×Q8×C137—81no—
112C2×Q8×C137—82no—
113C2×Q8×C137—82no—
114C2×Q8×C137C2×D414yes—
115C2×D4×C137—22no—
116C2×D4×C137—22no—
117C2×D4×C137Q1614yescomplemented
118C22×C4×C137—21no—
119C22×C4×C137—22no—
120C22×C4×C137—22no—
121C22×C4×C137—22no—
122C22×C4×C137—22no—
123C22×C4×C137—21no—
124C22×C4×C137—21no—
125C22×C4×C137—21no—
126C22×C4×C137—21no—
127C22×C4×C137—21no—
128C22×C4×C137—42no—
129C22×C4×C137—42no—
130C22×C4×C137—81no—
131C22×C4×C137—81no—
132C22×C4×C137—81no—
133C22×C4×C137C2×D411yescharacteristic
134C22×C4×C137C2×D411yescharacteristic
135C22×C4×C137C2×D411yescharacteristic
136C22×C4×C137C2×D411yescharacteristic
137C22×C4×C137C2×D412yes—
138C22×C4×C137C2×D412yes—
139C22×C4×C137C2411yescharacteristic
140Q16×C137—84no—
141C2×C8×C137—24no—
142C2×C8×C137—82no—
143C4⋊C4×C137—22no—
144C4⋊C4×C137—24no—
145C4⋊C4×C137—22no—
146C4⋊C4×C137—22no—
147C4⋊C4×C137—24no—
148C4⋊C4×C137—22no—
149C4⋊C4×C137—24no—
150C4⋊C4×C137—22no—
151C4⋊C4×C137—24no—
152C4⋊C4×C137—22no—
153C4⋊C4×C137—22no—
154C4⋊C4×C137—42no—
155C4⋊C4×C137—41no—
156C4⋊C4×C137—41no—
157C4⋊C4×C137—81no—
158C4⋊C4×C137C2×D414yes—
159C22⋊C4×C137—24no—
160C22⋊C4×C137—24no—
161C22⋊C4×C137—22no—
162C22⋊C4×C137—24no—
163C22⋊C4×C137—22no—
164C22⋊C4×C137—42no—
165C22⋊C4×C137—42no—
166C22⋊C4×C137—81no—
167C42×C137—24no—
168C42×C137—21no—
169C42×C137—21no—
170C23×C137—21no—
171C23×C137—21no—
172C23×C137—41no—
173C23×C137—41no—
174C23×C137—41no—
175C23×C137—41no—
176C23×C137#12012yes—
177C23×C137C2×Q1612yes—
178C23×C137C2×Q1612yes—
179C23×C137#13712yes—
180C23×C137C22×D411yescharacteristic
181Q8×C137—44no—
182Q8×C137—44no—
183Q8×C137—44no—
184Q8×C137—82no—
185Q8×C137—82no—
186D4×C137—28no—
187C2×C4×C137—21no—
188C2×C4×C137—21no—
189C2×C4×C137—22no—
190C2×C4×C137—21no—
191C2×C4×C137—24no—
192C2×C4×C137—24no—
193C2×C4×C137—21no—
194C2×C4×C137—42no—
195C2×C4×C137—41no—
196C2×C4×C137—42no—
197C2×C4×C137—41no—
198C2×C4×C137—42no—
199C2×C4×C137—41no—
200C2×C4×C137—41no—
201C2×C4×C137—41no—
202C2×C4×C137—41no—
203C2×C4×C137—41no—
204C2×C4×C137—41no—
205C2×C4×C137—41no—
206C2×C4×C137—41no—
207C2×C4×C137—41no—
208C2×C4×C137—41no—
209C2×C4×C137—41no—
210C2×C4×C137—41no—
211C2×C4×C137—41no—
212C2×C4×C137—81no—
213C2×C4×C137—82no—
214C2×C4×C137—82no—
215C2×C4×C137—82no—
216C2×C4×C137—82no—
217C2×C4×C137—84no—
218C2×C4×C137#12011yescharacteristic
219C2×C4×C137#12011yescharacteristic
220C2×C4×C137C2×Q1612yes—
221C2×C4×C137C22×D412yes—
222C8×C137—44no—
223C22×C137—21no—
224C22×C137—22no—
225C22×C137—21no—
226C22×C137—22no—
227C22×C137—22no—
228C22×C137—42no—
229C22×C137—42no—
230C22×C137—42no—
231C22×C137—41no—
232C22×C137—41no—
233C22×C137—81no—
234C22×C137#45114yes—
235C22×C137C2×#12011yescharacteristic
236C22×C137C22×Q1611yescharacteristic
237C22×C137C2×#13711yescharacteristic
238C4×C137—22no—
239C4×C137—22no—
240C4×C137—21no—
241C4×C137—21no—
242C4×C137—42no—
243C4×C137—41no—
244C4×C137—44no—
245C4×C137—41no—
246C4×C137—44no—
247C4×C137—81no—
248C4×C137—81no—
249C4×C137—81no—
250C2×C137—21no—
251C2×C137—21no—
252C2×C137—24no—
253C2×C137—42no—
254C2×C137C2×#45111yescharacteristic
255C2×C137#301811yescharacteristic
256C2×C137#302611yescharacteristic
257#18383C13711yesmaximal, characteristic, direct product factor, Sylow
258C137#1838311yescharacteristic, direct product factor, Sylow
259#3501C2×C13711yescharacteristic
260#3189C2×C13712yescomplemented
261#2564C2×C13711yescharacteristic
262#1617C2×C13712yescomplemented
263#1614C2×C13712yescomplemented
264#1605C2×C13712yescomplemented
265#1602C2×C13712yescomplemented
266C2×#328C2×C13712yescomplemented
267#1459C2×C13711yescharacteristic
268#554—24no—
269#554—22no—
270D4×Q8—22no—
271#544C22×C13712yes—
272#543—21no—
273#543C22×C13712yes—
274#533—21no—
275C2×C22⋊Q8—21no—
276C2×C22⋊Q8C22×C13712yes—
277C2×C22⋊Q8C22×C13711yescharacteristic
278#518—21no—
279C2×C4×D4C22×C13711yescharacteristic
280#484—22no—
281#484C22×C13714yes—
282#467—24no—
283#451—22no—
284#451C22×C13714yes—
285C2×C22⋊C8C22×C13712yes—
286#400C22×C13711yescharacteristic
287#399C22×C13711yescharacteristic
288#395C22×C13711yescharacteristic
289#393C22×C13711yescharacteristic
290#387—21no—
291#386—21no—
292#385—21no—
293#380—21no—
294C2×#95C22×C13711yescharacteristic
295#328C22×C13714yes—
296#328C22×C13714yes—
297#324C22×C13712yes—
298#323C22×C13712yes—
299C22×Q8—21no—
300C22×Q8D4×C13712yes—
301C22×Q8C23×C13711yescharacteristic
302C22×D4D4×C13711yescharacteristic
303C23×C4—21no—
304C23×C4D4×C13711yescharacteristic
305C23×C4C23×C13711yescharacteristic
306C2×Q16—24no—
307C22×C8—22no—
308C4⋊Q8—21no—
309C4⋊Q8—21no—
310C4⋊Q8—24no—
311#125—22no—
312#125—24no—
313C22⋊Q8—22no—
314C22⋊Q8—22no—
315C22⋊Q8—24no—
316C22⋊Q8—24no—
317C22⋊Q8—24no—
318C22⋊Q8—22no—
319C22⋊Q8—22no—
320C22⋊Q8—22no—
321C22⋊Q8—22no—
322C22⋊Q8—24no—
323C22⋊Q8—22no—
324C22⋊Q8—22no—
325C22⋊Q8D4×C13714yes—
326C22⋊Q8C23×C13714yes—
327C4×Q8—22no—
328C4×Q8—22no—
329C4×D4—24no—
330C4×D4—24no—
331C4×D4—22no—
332C4×D4—22no—
333#117—22no—
334C2×C4⋊C4—21no—
335C2×C4⋊C4—22no—
336C2×C4⋊C4—22no—
337C2×C4⋊C4—21no—
338C2×C4⋊C4—21no—
339C2×C4⋊C4—21no—
340C2×C4⋊C4D4×C13712yes—
341C2×C4⋊C4C23×C13711yescharacteristic
342C2×C4⋊C4C23×C13711yescharacteristic
343C2×C22⋊C4—22no—
344C2×C22⋊C4—22no—
345C2×C22⋊C4—21no—
346C2×C22⋊C4—21no—
347C2×C22⋊C4—21no—
348C2×C22⋊C4D4×C13711yescharacteristic
349C2×C22⋊C4C23×C13711yescharacteristic
350C2×C42D4×C13711yescharacteristic
351#107—24no—
352#103—24no—
353#103—24no—
354C22⋊C8—22no—
355C22⋊C8C23×C13714yes—
356#95—21no—
357#95—21no—
358#95—21no—
359#95—21no—
360#95—21no—
361#95C23×C13711yescharacteristic
362#95C23×C13711yescharacteristic
363C24C2×D4×C13711yescharacteristic
364C24C2×D4×C13711yescharacteristic
365C2×Q8—22no—
366C2×Q8—22no—
367C2×Q8—22no—
368C2×Q8—22no—
369C2×Q8—81no—
370C2×Q8—82no—
371C2×Q8—82no—
372C2×Q8C2×D4×C13714yes—
373C2×D4—22no—
374C2×D4—22no—
375C2×D4Q16×C13714yescomplemented
376C22×C4—21no—
377C22×C4—22no—
378C22×C4—22no—
379C22×C4—22no—
380C22×C4—22no—
381C22×C4—21no—
382C22×C4—21no—
383C22×C4—21no—
384C22×C4—21no—
385C22×C4—21no—
386C22×C4—42no—
387C22×C4—42no—
388C22×C4—81no—
389C22×C4—81no—
390C22×C4—81no—
391C22×C4C2×D4×C13711yescharacteristic
392C22×C4C2×D4×C13711yescharacteristic
393C22×C4C2×D4×C13711yescharacteristic
394C22×C4C2×D4×C13711yescharacteristic
395C22×C4C2×D4×C13712yes—
396C22×C4C2×D4×C13712yes—
397C22×C4C24×C13711yescharacteristic
398Q16—84no—
399C2×C8—24no—
400C2×C8—82no—
401C4⋊C4—22no—
402C4⋊C4—24no—
403C4⋊C4—22no—
404C4⋊C4—22no—
405C4⋊C4—24no—
406C4⋊C4—22no—
407C4⋊C4—24no—
408C4⋊C4—22no—
409C4⋊C4—24no—
410C4⋊C4—22no—
411C4⋊C4—22no—
412C4⋊C4—42no—
413C4⋊C4—41no—
414C4⋊C4—41no—
415C4⋊C4—81no—
416C4⋊C4C2×D4×C13714yes—
417C22⋊C4—24no—
418C22⋊C4—24no—
419C22⋊C4—22no—
420C22⋊C4—24no—
421C22⋊C4—22no—
422C22⋊C4—42no—
423C22⋊C4—42no—
424C22⋊C4—81no—
425C42—24no—
426C42—21no—
427C42—21no—
428C23—21no—
429C23—21no—
430C23—41no—
431C23—41no—
432C23—41no—
433C23—41no—
434C23#120×C13712yes—
435C23C2×Q16×C13712yes—
436C23C2×Q16×C13712yes—
437C23#137×C13712yes—
438C23C22×D4×C13711yescharacteristic
439Q8—44no—
440Q8—44no—
441Q8—44no—
442Q8—82no—
443Q8—82no—
444D4—28no—
445C2×C4—21no—
446C2×C4—21no—
447C2×C4—22no—
448C2×C4—21no—
449C2×C4—24no—
450C2×C4—24no—
451C2×C4—21no—
452C2×C4—42no—
453C2×C4—41no—
454C2×C4—42no—
455C2×C4—41no—
456C2×C4—42no—
457C2×C4—41no—
458C2×C4—41no—
459C2×C4—41no—
460C2×C4—41no—
461C2×C4—41no—
462C2×C4—41no—
463C2×C4—41no—
464C2×C4—41no—
465C2×C4—41no—
466C2×C4—41no—
467C2×C4—41no—
468C2×C4—41no—
469C2×C4—41no—
470C2×C4—81no—
471C2×C4—82no—
472C2×C4—82no—
473C2×C4—82no—
474C2×C4—82no—
475C2×C4—84no—
476C2×C4#120×C13711yescharacteristic
477C2×C4#120×C13711yescharacteristic
478C2×C4C2×Q16×C13712yes—
479C2×C4C22×D4×C13712yes—
480C8—44no—
481C22—21no—
482C22—22no—
483C22—21no—
484C22—22no—
485C22—22no—
486C22—42no—
487C22—42no—
488C22—42no—
489C22—41no—
490C22—41no—
491C22—81no—
492C22#451×C13714yes—
493C22C2×#120×C13711yescharacteristic
494C22C22×Q16×C13711yescharacteristic
495C22C2×#137×C13711yescharacteristic
496C4—22no—
497C4—22no—
498C4—21no—
499C4—21no—
500C4—42no—
501C4—41no—
502C4—44no—
503C4—41no—
504C4—44no—
505C4—81no—
506C4—81no—
507C4—81no—
508C2—21no—
509C2—21no—
510C2—24no—
511C2—42no—
512C2C2×#451×C13711yescharacteristic
513C2#3018×C13711yescharacteristic
514C2#3026×C13711yescharacteristic
subgroupG/H
center, socleC22×C137C2×#120
commutator, Frattini, cosocleC22×C4C24×C137
prime radicalC22×D4×C137D4
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. #3189×C137C2semidirect
4. #1617×C137C2semidirect
5. #1614×C137C2semidirect
6. #1605×C137C2semidirect
7. #1602×C137C2semidirect
8. C2×#328×C137C2semidirect
117. C2×D4×C137Q16semidirect
257. #18383C137direct (N × H)
258. C137#18383direct (N × H)
260. #3189C2×C137semidirect
262. #1617C2×C137semidirect
263. #1614C2×C137semidirect
264. #1605C2×C137semidirect
265. #1602C2×C137semidirect
266. C2×#328C2×C137semidirect
375. C2×D4Q16×C137semidirect

Maximal subgroups

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

subgroupindex
#3501×C1372
#3189×C1372
#2564×C1372
#1617×C1372
#1614×C1372
#1605×C1372
#1602×C1372
C2×#328×C1372
#1459×C1372
#18383137