Finite Group Databasebeta

← C137×#18214 #46867600

Subgroups — C137×#18214 #46867600

Order 35072.

982 conjugacy classes of subgroups.

The subgroup lattice has 560 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#3507×C137C211yesmaximal, characteristic, complemented
2#3101×C137C211yesmaximal, characteristic, complemented
3#3100×C137C211yesmaximal, characteristic
4#2536×C137C211yesmaximal, characteristic, complemented
5#1609×C137C214yesmaximal, complemented
6#1597×C137C214yesmaximal, complemented
7#1518×C137C211yesmaximal, characteristic, complemented
8#1517×C137C211yesmaximal, characteristic, complemented
9#1461×C137C211yesmaximal, characteristic, complemented
10#559×C137—21no—
11#557×C137—21no—
12#556×C137—21no—
13#556×C137—22no—
14#554×C137—21no—
15#552×C137—22no—
16#539×C137C2214yes—
17#536×C137—21no—
18#533×C137C2211yescharacteristic
19#532×C137C2211yescharacteristic
20#529×C137—22no—
21C2×C22⋊Q8×C137—21no—
22#520×C137—21no—
23C2×C4×Q8×C137C2211yescharacteristic
24C2×C4×D4×C137C2211yescharacteristic
25#483×C137C2212yes—
26#480×C137C2212yes—
27#476×C137—21no—
28#474×C137—21no—
29#461×C137—21no—
30#459×C137—21no—
31#450×C137C2212yes—
32#447×C137C2212yes—
33C4×SD16×C137—22no—
34C4×D8×C137—22no—
35#431×C137C2211yescharacteristic
36#431×C137C2211yescharacteristic
37#398×C137C2212yes—
38#394×C137C2212yes—
39#388×C137—22no—
40#382×C137C2211yescharacteristic
41#380×C137—21no—
42#328×C137C2214yes—
43#327×C137C2214yes—
44#324×C137C2214yes—
45C2×#41×C137—21no—
46C22×Q8×C137D411yescharacteristic
47C22×D4×C137C2311yescharacteristic
48C23×C4×C137—21no—
49C2×SD16×C137—22no—
50C2×D8×C137—22no—
51C4⋊Q8×C137—22no—
52C4⋊Q8×C137D411yescharacteristic
53C4⋊Q8×C137D411yescharacteristic
54C4⋊Q8×C137D411yescharacteristic
55#127×C137D411yescharacteristic
56#126×C137—24no—
57#126×C137—24no—
58#125×C137—22no—
59#125×C137—24no—
60#125×C137—21no—
61#124×C137—22no—
62#124×C137—22no—
63#124×C137—21no—
64#123×C137—24no—
65#123×C137—24no—
66C22⋊Q8×C137—22no—
67C22⋊Q8×C137—22no—
68C22⋊Q8×C137—24no—
69C22⋊Q8×C137—21no—
70C22⋊Q8×C137—44no—
71C22⋊Q8×C137C2312yes—
72#121×C137—21no—
73#121×C137C2312yes—
74C4×Q8×C137—22no—
75C4×Q8×C137—22no—
76C4×Q8×C137—21no—
77C4×Q8×C137—21no—
78C4×Q8×C137—21no—
79C4×Q8×C137—21no—
80C4×Q8×C137—21no—
81C4×Q8×C137D412yes—
82C4×D4×C137—22no—
83C4×D4×C137—22no—
84C4×D4×C137—24no—
85C4×D4×C137—21no—
86C4×D4×C137—21no—
87C4×D4×C137—21no—
88C4×D4×C137—21no—
89C4×D4×C137—21no—
90C4×D4×C137D412yes—
91#117×C137—22no—
92#117×C137—24no—
93#117×C137—21no—
94C2×C4⋊C4×C137—22no—
95C2×C4⋊C4×C137—21no—
96C2×C4⋊C4×C137—22no—
97C2×C4⋊C4×C137D411yescharacteristic
98C2×C4⋊C4×C137C2311yescharacteristic
99C2×C4⋊C4×C137C2312yes—
100C2×C22⋊C4×C137—22no—
101C2×C22⋊C4×C137—21no—
102C2×C22⋊C4×C137—22no—
103C2×C42×C137D412yes—
104C2×C42×C137C2311yescharacteristic
105#107×C137—22no—
106#106×C137—22no—
107C4⋊C8×C137—21no—
108C4⋊C8×C137—21no—
109#103×C137—22no—
110#102×C137—22no—
111#102×C137—22no—
112#102×C137—22no—
113C22⋊C8×C137C2312yes—
114C22⋊C8×C137C2312yes—
115C4×C8×C137—21no—
116C4×C8×C137—21no—
117#95×C137—22no—
118#95×C137—21no—
119#95×C137C2312yes—
120C24×C137—21no—
121#41×C137—42no—
122#41×C137—42no—
123C2×Q8×C137—21no—
124C2×Q8×C137—21no—
125C2×Q8×C137—21no—
126C2×Q8×C137—21no—
127C2×Q8×C137—22no—
128C2×Q8×C137—22no—
129C2×Q8×C137C2×D412yes—
130C2×D4×C137—22no—
131C2×D4×C137—21no—
132C2×D4×C137—21no—
133C2×D4×C137—21no—
134C2×D4×C137—21no—
135C2×D4×C137—42no—
136C2×D4×C137C2×D412yes—
137C22×C4×C137—24no—
138C22×C4×C137—22no—
139C22×C4×C137—22no—
140C22×C4×C137—21no—
141C22×C4×C137—41no—
142C22×C4×C137—41no—
143C22×C4×C137—42no—
144C22×C4×C137—42no—
145C22×C4×C137C2×D412yes—
146C22×C4×C137C2×D412yes—
147C22×C4×C137C2×D411yescharacteristic
148C22×C4×C137C2×D411yescharacteristic
149C22×C4×C137C2411yescharacteristic
150SD16×C137—44no—
151D8×C137—44no—
152C2×C8×C137—22no—
153C2×C8×C137—22no—
154C4⋊C4×C137—24no—
155C4⋊C4×C137—24no—
156C4⋊C4×C137—24no—
157C4⋊C4×C137—21no—
158C4⋊C4×C137—21no—
159C4⋊C4×C137—21no—
160C4⋊C4×C137—21no—
161C4⋊C4×C137—22no—
162C4⋊C4×C137—21no—
163C4⋊C4×C137—21no—
164C4⋊C4×C137—22no—
165C4⋊C4×C137—22no—
166C4⋊C4×C137—22no—
167C4⋊C4×C137—21no—
168C4⋊C4×C137—21no—
169C4⋊C4×C137—44no—
170C4⋊C4×C137—42no—
171C4⋊C4×C137C2×D412yes—
172C4⋊C4×C137C2×D412yes—
173C22⋊C4×C137—24no—
174C22⋊C4×C137—24no—
175C22⋊C4×C137—24no—
176C22⋊C4×C137—22no—
177C22⋊C4×C137—22no—
178C22⋊C4×C137—22no—
179C22⋊C4×C137—44no—
180C22⋊C4×C137—44no—
181C22⋊C4×C137—41no—
182C22⋊C4×C137—41no—
183C42×C137—24no—
184C42×C137—22no—
185C42×C137—22no—
186C42×C137—22no—
187C42×C137C2×D411yescharacteristic
188C42×C137C2×D411yescharacteristic
189C42×C137C2×D411yescharacteristic
190C42×C137C2×D411yescharacteristic
191C23×C137—22no—
192C23×C137—22no—
193C23×C137—41no—
194C23×C137—41no—
195C23×C137—41no—
196C23×C137—41no—
197C23×C137—41no—
198C23×C137—41no—
199C23×C137C22×D411yescharacteristic
200Q8×C137—42no—
201Q8×C137—42no—
202Q8×C137—42no—
203Q8×C137—42no—
204Q8×C137—42no—
205D4×C137—22no—
206D4×C137—22no—
207D4×C137—42no—
208D4×C137—42no—
209D4×C137—41no—
210D4×C137—41no—
211D4×C137—42no—
212D4×C137—42no—
213D4×C137—42no—
214C2×C4×C137—22no—
215C2×C4×C137—22no—
216C2×C4×C137—22no—
217C2×C4×C137—22no—
218C2×C4×C137—22no—
219C2×C4×C137—22no—
220C2×C4×C137—22no—
221C2×C4×C137—41no—
222C2×C4×C137—41no—
223C2×C4×C137—44no—
224C2×C4×C137—42no—
225C2×C4×C137—42no—
226C2×C4×C137—42no—
227C2×C4×C137—41no—
228C2×C4×C137—41no—
229C2×C4×C137—42no—
230C2×C4×C137—41no—
231C2×C4×C137—84no—
232C2×C4×C137—82no—
233C2×C4×C137—82no—
234C2×C4×C137#12011yescharacteristic
235C2×C4×C137#12011yescharacteristic
236C2×C4×C137#12011yescharacteristic
237C2×C4×C137#12011yescharacteristic
238C2×C4×C137#13512yes—
239C2×C4×C137AGL(1,ℤ/8)12yes—
240C2×C4×C137C22×D411yescharacteristic
241C2×C4×C137C22×D411yescharacteristic
242C8×C137—42no—
243C8×C137—42no—
244C22×C137—21no—
245C22×C137—21no—
246C22×C137—42no—
247C22×C137—42no—
248C22×C137—42no—
249C22×C137—42no—
250C22×C137—42no—
251C22×C137—42no—
252C22×C137—81no—
253C22×C137—81no—
254C22×C137—81no—
255C22×C137—81no—
256C22×C137C2×#12011yescharacteristic
257C22×C137C2×#13511yescharacteristic
258C22×C137C2×AGL(1,ℤ/8)11yescharacteristic
259C4×C137—21no—
260C4×C137—21no—
261C4×C137—21no—
262C4×C137—21no—
263C4×C137—42no—
264C4×C137—42no—
265C4×C137—42no—
266C4×C137—42no—
267C4×C137—42no—
268C4×C137—82no—
269C4×C137—81no—
270C4×C137#44912yes—
271C4×C137#44912yes—
272C2×C137—21no—
273C2×C137—21no—
274C2×C137—82no—
275C2×C137—82no—
276C2×C137C2×#44911yescharacteristic
277C2×C137#302011yescharacteristic
278C2×C137#302511yescharacteristic
279#18214C13711yesmaximal, characteristic, direct product factor, Sylow
280C137#1821411yescharacteristic, direct product factor, Sylow
281#3507C2×C13711yescharacteristic, complemented
282#3101C2×C13711yescharacteristic, complemented
283#3100C2×C13711yescharacteristic
284#2536C2×C13711yescharacteristic, complemented
285#1609C2×C13714yescomplemented
286#1597C2×C13714yescomplemented
287#1518C2×C13711yescharacteristic, complemented
288#1517C2×C13711yescharacteristic, complemented
289#1461C2×C13711yescharacteristic, complemented
290#559—21no—
291#557—21no—
292#556—21no—
293#556—22no—
294#554—21no—
295#552—22no—
296#539C22×C13714yes—
297#536—21no—
298#533C22×C13711yescharacteristic
299#532C22×C13711yescharacteristic
300#529—22no—
301C2×C22⋊Q8—21no—
302#520—21no—
303C2×C4×Q8C22×C13711yescharacteristic
304C2×C4×D4C22×C13711yescharacteristic
305#483C22×C13712yes—
306#480C22×C13712yes—
307#476—21no—
308#474—21no—
309#461—21no—
310#459—21no—
311#450C22×C13712yes—
312#447C22×C13712yes—
313C4×SD16—22no—
314C4×D8—22no—
315#431C22×C13711yescharacteristic
316#431C22×C13711yescharacteristic
317#398C22×C13712yes—
318#394C22×C13712yes—
319#388—22no—
320#382C22×C13711yescharacteristic
321#380—21no—
322#328C22×C13714yes—
323#327C22×C13714yes—
324#324C22×C13714yes—
325C2×#41—21no—
326C22×Q8D4×C13711yescharacteristic
327C22×D4C23×C13711yescharacteristic
328C23×C4—21no—
329C2×SD16—22no—
330C2×D8—22no—
331C4⋊Q8—22no—
332C4⋊Q8D4×C13711yescharacteristic
333C4⋊Q8D4×C13711yescharacteristic
334C4⋊Q8D4×C13711yescharacteristic
335#127D4×C13711yescharacteristic
336#126—24no—
337#126—24no—
338#125—22no—
339#125—24no—
340#125—21no—
341#124—22no—
342#124—22no—
343#124—21no—
344#123—24no—
345#123—24no—
346C22⋊Q8—22no—
347C22⋊Q8—22no—
348C22⋊Q8—24no—
349C22⋊Q8—21no—
350C22⋊Q8—44no—
351C22⋊Q8C23×C13712yes—
352#121—21no—
353#121C23×C13712yes—
354C4×Q8—22no—
355C4×Q8—22no—
356C4×Q8—21no—
357C4×Q8—21no—
358C4×Q8—21no—
359C4×Q8—21no—
360C4×Q8—21no—
361C4×Q8D4×C13712yes—
362C4×D4—22no—
363C4×D4—22no—
364C4×D4—24no—
365C4×D4—21no—
366C4×D4—21no—
367C4×D4—21no—
368C4×D4—21no—
369C4×D4—21no—
370C4×D4D4×C13712yes—
371#117—22no—
372#117—24no—
373#117—21no—
374C2×C4⋊C4—22no—
375C2×C4⋊C4—21no—
376C2×C4⋊C4—22no—
377C2×C4⋊C4D4×C13711yescharacteristic
378C2×C4⋊C4C23×C13711yescharacteristic
379C2×C4⋊C4C23×C13712yes—
380C2×C22⋊C4—21no—
381C2×C22⋊C4—22no—
382C2×C22⋊C4—22no—
383C2×C42D4×C13712yes—
384C2×C42C23×C13711yescharacteristic
385#107—22no—
386#106—22no—
387C4⋊C8—21no—
388C4⋊C8—21no—
389#103—22no—
390#102—22no—
391#102—22no—
392#102—22no—
393C22⋊C8C23×C13712yes—
394C22⋊C8C23×C13712yes—
395C4×C8—21no—
396C4×C8—21no—
397#95—22no—
398#95—21no—
399#95C23×C13712yes—
400C24—21no—
401#41—42no—
402#41—42no—
403C2×Q8—21no—
404C2×Q8—21no—
405C2×Q8—21no—
406C2×Q8—21no—
407C2×Q8—22no—
408C2×Q8—22no—
409C2×Q8C2×D4×C13712yes—
410C2×D4—22no—
411C2×D4—21no—
412C2×D4—21no—
413C2×D4—21no—
414C2×D4—21no—
415C2×D4—42no—
416C2×D4C2×D4×C13712yes—
417C22×C4—24no—
418C22×C4—22no—
419C22×C4—22no—
420C22×C4—21no—
421C22×C4—41no—
422C22×C4—41no—
423C22×C4—42no—
424C22×C4—42no—
425C22×C4C2×D4×C13712yes—
426C22×C4C2×D4×C13712yes—
427C22×C4C2×D4×C13711yescharacteristic
428C22×C4C2×D4×C13711yescharacteristic
429C22×C4C24×C13711yescharacteristic
430SD16—44no—
431D8—44no—
432C2×C8—22no—
433C2×C8—22no—
434C4⋊C4—24no—
435C4⋊C4—24no—
436C4⋊C4—24no—
437C4⋊C4—21no—
438C4⋊C4—21no—
439C4⋊C4—21no—
440C4⋊C4—21no—
441C4⋊C4—22no—
442C4⋊C4—21no—
443C4⋊C4—21no—
444C4⋊C4—22no—
445C4⋊C4—22no—
446C4⋊C4—22no—
447C4⋊C4—21no—
448C4⋊C4—21no—
449C4⋊C4—44no—
450C4⋊C4—42no—
451C4⋊C4C2×D4×C13712yes—
452C4⋊C4C2×D4×C13712yes—
453C22⋊C4—24no—
454C22⋊C4—24no—
455C22⋊C4—24no—
456C22⋊C4—22no—
457C22⋊C4—22no—
458C22⋊C4—22no—
459C22⋊C4—44no—
460C22⋊C4—44no—
461C22⋊C4—41no—
462C22⋊C4—41no—
463C42—24no—
464C42—22no—
465C42—22no—
466C42—22no—
467C42C2×D4×C13711yescharacteristic
468C42C2×D4×C13711yescharacteristic
469C42C2×D4×C13711yescharacteristic
470C42C2×D4×C13711yescharacteristic
471C23—22no—
472C23—22no—
473C23—41no—
474C23—41no—
475C23—41no—
476C23—41no—
477C23—41no—
478C23—41no—
479C23C22×D4×C13711yescharacteristic
480Q8—42no—
481Q8—42no—
482Q8—42no—
483Q8—42no—
484Q8—42no—
485D4—22no—
486D4—22no—
487D4—42no—
488D4—41no—
489D4—41no—
490D4—42no—
491D4—42no—
492D4—42no—
493D4—42no—
494C2×C4—22no—
495C2×C4—22no—
496C2×C4—22no—
497C2×C4—22no—
498C2×C4—22no—
499C2×C4—22no—
500C2×C4—22no—
501C2×C4—41no—
502C2×C4—41no—
503C2×C4—44no—
504C2×C4—42no—
505C2×C4—42no—
506C2×C4—42no—
507C2×C4—41no—
508C2×C4—41no—
509C2×C4—42no—
510C2×C4—41no—
511C2×C4—84no—
512C2×C4—82no—
513C2×C4—82no—
514C2×C4#120×C13711yescharacteristic
515C2×C4#120×C13711yescharacteristic
516C2×C4#120×C13711yescharacteristic
517C2×C4#120×C13711yescharacteristic
518C2×C4#135×C13712yes—
519C2×C4AGL(1,ℤ/8)×C13712yes—
520C2×C4C22×D4×C13711yescharacteristic
521C2×C4C22×D4×C13711yescharacteristic
522C8—42no—
523C8—42no—
524C22—21no—
525C22—21no—
526C22—42no—
527C22—42no—
528C22—42no—
529C22—42no—
530C22—42no—
531C22—42no—
532C22—81no—
533C22—81no—
534C22—81no—
535C22—81no—
536C22C2×#120×C13711yescharacteristic
537C22C2×#135×C13711yescharacteristic
538C22C2×AGL(1,ℤ/8)×C13711yescharacteristic
539C4—21no—
540C4—21no—
541C4—21no—
542C4—21no—
543C4—42no—
544C4—42no—
545C4—42no—
546C4—42no—
547C4—42no—
548C4—82no—
549C4—81no—
550C4#449×C13712yes—
551C4#449×C13712yes—
552C2—21no—
553C2—21no—
554C2—82no—
555C2—82no—
556C2C2×#449×C13711yescharacteristic
557C2#3020×C13711yescharacteristic
558C2#3025×C13711yescharacteristic
subgroupG/H
center, socleC22×C137C2×#120
commutator, Frattini, cosocleC22×C4C24×C137
prime radical#3100×C137C2
LCSG ▷ C22×C4 ▷ C22 ▷ C1
UCSC1 ◁ C22×C137 ◁ C2×C42×C137 ◁ G

Complements

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

NH ≅ G/Ntype
1. #3507×C137C2semidirect
2. #3101×C137C2semidirect
4. #2536×C137C2semidirect
5. #1609×C137C2semidirect
6. #1597×C137C2semidirect
7. #1518×C137C2semidirect
8. #1517×C137C2semidirect
9. #1461×C137C2semidirect
279. #18214C137direct (N × H)
280. C137#18214direct (N × H)
281. #3507C2×C137semidirect
282. #3101C2×C137semidirect
284. #2536C2×C137semidirect
285. #1609C2×C137semidirect
286. #1597C2×C137semidirect
287. #1518C2×C137semidirect
288. #1517C2×C137semidirect
289. #1461C2×C137semidirect

Maximal subgroups

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

subgroupindex
#3507×C1372
#3101×C1372
#3100×C1372
#2536×C1372
#1609×C1372
#1597×C1372
#1518×C1372
#1517×C1372
#1461×C1372
#18214137