Finite Group Databasebeta

← C137×#32922 #46882308

Subgroups — C137×#32922 #46882308

Order 35072.

1418 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
1C4×H(5,𝔽2)×C137C211yesmaximal, characteristic
2#3039×C137C212yesmaximal, complemented
3C2×#455×C137C212yesmaximal
4#2547×C137C211yesmaximal, characteristic, complemented
5#1885×C137C212yesmaximal, complemented
6#1882×C137C214yesmaximal, complemented
7#1806×C137C212yesmaximal, complemented
8#1800×C137C211yesmaximal, characteristic, complemented
9C2×H(5,𝔽2)×C137C2211yescharacteristic
10C2×#137×C137C2214yes—
11C2×C4⋊Q8×C137C2211yescharacteristic, complemented
12C2×#124×C137C2212yes—
13#520×C137—22no—
14#520×C137C2212yes—
15#518×C137—22no—
16#518×C137—22no—
17#518×C137C2211yescharacteristic
18C4×#41×C137—22no—
19C4×#41×C137C2212yes—
20C2×C4×D4×C137—22no—
21C2×C4×D4×C137—22no—
22C2×C4×D4×C137C2211yescharacteristic
23#462×C137—22no—
24#462×C137—22no—
25#461×C137—22no—
26#461×C137—22no—
27#455×C137—28no—
28#423×C137C2212yescomplemented
29C2×#104×C137C2214yes—
30#419×C137C2214yescomplemented
31C2×#100×C137C2212yes—
32#397×C137—22no—
33#391×C137C2212yescomplemented
34#380×C137—22no—
35C4×C4⋊C4×C137C2211yescharacteristic, complemented
36#343×C137C2212yescomplemented
37#339×C137C2214yescomplemented
38H(5,𝔽2)×C137—24no—
39H(5,𝔽2)×C137—22no—
40H(5,𝔽2)×C137D414yes—
41C2×#41×C137—22no—
42C2×#41×C137C2312yes—
43C22×Q8×C137C2312yes—
44C22×D4×C137—22no—
45C22×D4×C137—22no—
46C22×D4×C137C2311yescharacteristic
47C23×C4×C137—22no—
48C23×C4×C137D412yes—
49#137×C137—28no—
50#137×C137—28no—
51C2×Q16×C137—24no—
52C2×SD16×C137—24no—
53C2×M16×C137C2314yes—
54C4⋊Q8×C137—22no—
55C4⋊Q8×C137—22no—
56#124×C137—24no—
57#124×C137—24no—
58C4×Q8×C137—22no—
59C4×Q8×C137D412yes—
60C4×Q8×C137D412yes—
61C4×D4×C137—24no—
62C4×D4×C137—24no—
63C4×D4×C137—24no—
64C4×D4×C137—24no—
65C4×D4×C137—22no—
66C4×D4×C137—22no—
67C4×D4×C137—24no—
68C4×D4×C137—22no—
69C4×D4×C137—22no—
70C4×D4×C137—22no—
71C4×D4×C137—22no—
72C4×D4×C137—22no—
73C4×D4×C137D412yes—
74C4×D4×C137D412yes—
75#117×C137—24no—
76#117×C137—22no—
77#117×C137—22no—
78#117×C137C2312yes—
79C2×C4⋊C4×C137—22no—
80C2×C4⋊C4×C137—22no—
81C2×C4⋊C4×C137—21no—
82C2×C4⋊C4×C137—21no—
83C2×C4⋊C4×C137—21no—
84C2×C4⋊C4×C137C2311yescharacteristic
85C2×C22⋊C4×C137—22no—
86C2×C22⋊C4×C137—24no—
87C2×C22⋊C4×C137—22no—
88C2×C22⋊C4×C137—24no—
89C2×C22⋊C4×C137D412yes—
90C2×C42×C137—22no—
91C2×C42×C137—22no—
92C2×C42×C137C2312yes—
93C2×C42×C137C2311yescharacteristic
94C4⋊C8×C137—22no—
95C4⋊C8×C137—22no—
96#104×C137—28no—
97#103×C137—24no—
98#103×C137—24no—
99#100×C137—24no—
100#95×C137—21no—
101#95×C137—24no—
102C24×C137—22no—
103C24×C137C2×D412yes—
104#41×C137—28no—
105#41×C137—28no—
106#41×C137—22no—
107#41×C137—22no—
108#41×C137C2×D414yes—
109#41×C137#4114yes—
110C2×Q8×C137—24no—
111C2×Q8×C137—22no—
112C2×Q8×C137—22no—
113C2×Q8×C137—24no—
114C2×Q8×C137C2×D412yes—
115C2×D4×C137—24no—
116C2×D4×C137—28no—
117C2×D4×C137—28no—
118C2×D4×C137—28no—
119C2×D4×C137—22no—
120C2×D4×C137—28no—
121C2×D4×C137—22no—
122C2×D4×C137—22no—
123C2×D4×C137C2×D412yes—
124C2×D4×C137C2×D412yes—
125C2×D4×C137#4112yes—
126C22×C4×C137—22no—
127C22×C4×C137—22no—
128C22×C4×C137—22no—
129C22×C4×C137—24no—
130C22×C4×C137—22no—
131C22×C4×C137—22no—
132C22×C4×C137—22no—
133C22×C4×C137—22no—
134C22×C4×C137—22no—
135C22×C4×C137—22no—
136C22×C4×C137—22no—
137C22×C4×C137—44no—
138C22×C4×C137—84no—
139C22×C4×C137C2×D411yescharacteristic
140C22×C4×C137C2×D412yes—
141C22×C4×C137C2×D411yescharacteristic
142C22×C4×C137C2411yescharacteristic
143Q16×C137—48no—
144SD16×C137—48no—
145M16×C137—24no—
146M16×C137—24no—
147C2×C8×C137—24no—
148C4⋊C4×C137—24no—
149C4⋊C4×C137—22no—
150C4⋊C4×C137—22no—
151C4⋊C4×C137—22no—
152C4⋊C4×C137—42no—
153C4⋊C4×C137—42no—
154C4⋊C4×C137—41no—
155C4⋊C4×C137—41no—
156C4⋊C4×C137C2×D412yes—
157C4⋊C4×C137C2×D412yes—
158C22⋊C4×C137—24no—
159C22⋊C4×C137—24no—
160C22⋊C4×C137—22no—
161C22⋊C4×C137—22no—
162C22⋊C4×C137—22no—
163C22⋊C4×C137—24no—
164C22⋊C4×C137—48no—
165C42×C137—24no—
166C42×C137—22no—
167C42×C137—22no—
168C42×C137—22no—
169C42×C137—22no—
170C42×C137—22no—
171C42×C137C2×D412yes—
172C42×C137C2×D412yes—
173C23×C137—28no—
174C23×C137—22no—
175C23×C137—22no—
176C23×C137—22no—
177C23×C137—42no—
178C23×C137—42no—
179C23×C137—82no—
180C23×C137—82no—
181C23×C137#12114yes—
182C23×C137C22×D411yescharacteristic
183Q8×C137—22no—
184Q8×C137—44no—
185Q8×C137—44no—
186Q8×C137#12114yes—
187D4×C137—216no—
188D4×C137—24no—
189D4×C137—24no—
190D4×C137—24no—
191D4×C137—22no—
192D4×C137—24no—
193D4×C137#12114yes—
194C2×C4×C137—28no—
195C2×C4×C137—22no—
196C2×C4×C137—24no—
197C2×C4×C137—24no—
198C2×C4×C137—22no—
199C2×C4×C137—22no—
200C2×C4×C137—22no—
201C2×C4×C137—22no—
202C2×C4×C137—41no—
203C2×C4×C137—44no—
204C2×C4×C137—44no—
205C2×C4×C137—42no—
206C2×C4×C137—42no—
207C2×C4×C137—42no—
208C2×C4×C137—42no—
209C2×C4×C137—42no—
210C2×C4×C137—42no—
211C2×C4×C137—42no—
212C2×C4×C137—82no—
213C2×C4×C137—82no—
214C2×C4×C137—82no—
215C2×C4×C137—84no—
216C2×C4×C137—82no—
217C2×C4×C137#12012yes—
218C2×C4×C137#12011yescharacteristic
219C2×C4×C137#12011yescharacteristic
220C2×C4×C137C22×D411yescharacteristic
221C2×C4×C137C22×D412yes—
222C2×C4×C137C2×#4112yes—
223C2×C4×C137C2×#4111yescharacteristic
224C8×C137—44no—
225C22×C137—21no—
226C22×C137—21no—
227C22×C137—24no—
228C22×C137—24no—
229C22×C137—44no—
230C22×C137—42no—
231C22×C137—42no—
232C22×C137—42no—
233C22×C137—42no—
234C22×C137—82no—
235C22×C137—88no—
236C22×C137C2×#12011yescharacteristic
237C22×C137C2×#12111yescharacteristic
238C22×C137#52511yescharacteristic
239C4×C137—24no—
240C4×C137—21no—
241C4×C137—22no—
242C4×C137—42no—
243C4×C137—42no—
244C4×C137—42no—
245C4×C137—82no—
246C4×C137—82no—
247C4×C137#45512yescomplemented
248C4×C137C2×#12112yes—
249C4×C137#52512yes—
250C2×C137—21no—
251C2×C137—21no—
252C2×C137—44no—
253C2×C137—44no—
254C2×C137#284711yescharacteristic
255C2×C137C2×#45511yescharacteristic
256C2×C137#302211yescharacteristic
257#32922C13711yesmaximal, characteristic, direct product factor, Sylow
258C137#3292211yescharacteristic, direct product factor, Sylow
259C4×H(5,𝔽2)C2×C13711yescharacteristic
260#3039C2×C13712yescomplemented
261C2×#455C2×C13712yes—
262#2547C2×C13711yescharacteristic, complemented
263#1885C2×C13712yescomplemented
264#1882C2×C13714yescomplemented
265#1806C2×C13712yescomplemented
266#1800C2×C13711yescharacteristic, complemented
267C2×H(5,𝔽2)C22×C13711yescharacteristic
268C2×#137C22×C13714yes—
269C2×C4⋊Q8C22×C13711yescharacteristic, complemented
270C2×#124C22×C13712yes—
271#520—22no—
272#520C22×C13712yes—
273#518—22no—
274#518—22no—
275#518C22×C13711yescharacteristic
276C4×#41—22no—
277C4×#41C22×C13712yes—
278C2×C4×D4—22no—
279C2×C4×D4—22no—
280C2×C4×D4C22×C13711yescharacteristic
281#462—22no—
282#462—22no—
283#461—22no—
284#461—22no—
285#455—28no—
286#423C22×C13712yescomplemented
287C2×#104C22×C13714yes—
288#419C22×C13714yescomplemented
289C2×#100C22×C13712yes—
290#397—22no—
291#391C22×C13712yescomplemented
292#380—22no—
293C4×C4⋊C4C22×C13711yescharacteristic, complemented
294#343C22×C13712yescomplemented
295#339C22×C13714yescomplemented
296H(5,𝔽2)—24no—
297H(5,𝔽2)—22no—
298H(5,𝔽2)D4×C13714yes—
299C2×#41—22no—
300C2×#41C23×C13712yes—
301C22×Q8C23×C13712yes—
302C22×D4—22no—
303C22×D4—22no—
304C22×D4C23×C13711yescharacteristic
305C23×C4—22no—
306C23×C4D4×C13712yes—
307#137—28no—
308#137—28no—
309C2×Q16—24no—
310C2×SD16—24no—
311C2×M16C23×C13714yes—
312C4⋊Q8—22no—
313C4⋊Q8—22no—
314#124—24no—
315#124—24no—
316C4×Q8—22no—
317C4×Q8D4×C13712yes—
318C4×Q8D4×C13712yes—
319C4×D4—24no—
320C4×D4—24no—
321C4×D4—24no—
322C4×D4—24no—
323C4×D4—22no—
324C4×D4—22no—
325C4×D4—24no—
326C4×D4—22no—
327C4×D4—22no—
328C4×D4—22no—
329C4×D4—22no—
330C4×D4—22no—
331C4×D4D4×C13712yes—
332C4×D4D4×C13712yes—
333#117—24no—
334#117—22no—
335#117—22no—
336#117C23×C13712yes—
337C2×C4⋊C4—22no—
338C2×C4⋊C4—22no—
339C2×C4⋊C4—21no—
340C2×C4⋊C4—21no—
341C2×C4⋊C4—21no—
342C2×C4⋊C4C23×C13711yescharacteristic
343C2×C22⋊C4—22no—
344C2×C22⋊C4—24no—
345C2×C22⋊C4—22no—
346C2×C22⋊C4—24no—
347C2×C22⋊C4D4×C13712yes—
348C2×C42—22no—
349C2×C42—22no—
350C2×C42C23×C13712yes—
351C2×C42C23×C13711yescharacteristic
352C4⋊C8—22no—
353C4⋊C8—22no—
354#104—28no—
355#103—24no—
356#103—24no—
357#100—24no—
358#95—21no—
359#95—24no—
360C24—22no—
361C24C2×D4×C13712yes—
362#41—28no—
363#41—28no—
364#41—22no—
365#41—22no—
366#41C2×D4×C13714yes—
367#41#41×C13714yes—
368C2×Q8—24no—
369C2×Q8—22no—
370C2×Q8—22no—
371C2×Q8—24no—
372C2×Q8C2×D4×C13712yes—
373C2×D4—24no—
374C2×D4—28no—
375C2×D4—28no—
376C2×D4—28no—
377C2×D4—22no—
378C2×D4—28no—
379C2×D4—22no—
380C2×D4—22no—
381C2×D4C2×D4×C13712yes—
382C2×D4C2×D4×C13712yes—
383C2×D4#41×C13712yes—
384C22×C4—22no—
385C22×C4—22no—
386C22×C4—22no—
387C22×C4—24no—
388C22×C4—22no—
389C22×C4—22no—
390C22×C4—22no—
391C22×C4—22no—
392C22×C4—22no—
393C22×C4—22no—
394C22×C4—22no—
395C22×C4—44no—
396C22×C4—84no—
397C22×C4C2×D4×C13711yescharacteristic
398C22×C4C2×D4×C13712yes—
399C22×C4C2×D4×C13711yescharacteristic
400C22×C4C24×C13711yescharacteristic
401Q16—48no—
402SD16—48no—
403M16—24no—
404M16—24no—
405C2×C8—24no—
406C4⋊C4—24no—
407C4⋊C4—22no—
408C4⋊C4—22no—
409C4⋊C4—22no—
410C4⋊C4—42no—
411C4⋊C4—42no—
412C4⋊C4—41no—
413C4⋊C4—41no—
414C4⋊C4C2×D4×C13712yes—
415C4⋊C4C2×D4×C13712yes—
416C22⋊C4—24no—
417C22⋊C4—24no—
418C22⋊C4—22no—
419C22⋊C4—22no—
420C22⋊C4—22no—
421C22⋊C4—24no—
422C22⋊C4—48no—
423C42—24no—
424C42—22no—
425C42—22no—
426C42—22no—
427C42—22no—
428C42—22no—
429C42C2×D4×C13712yes—
430C42C2×D4×C13712yes—
431C23—28no—
432C23—22no—
433C23—22no—
434C23—22no—
435C23—42no—
436C23—42no—
437C23—82no—
438C23—82no—
439C23#121×C13714yes—
440C23C22×D4×C13711yescharacteristic
441Q8—22no—
442Q8—44no—
443Q8—44no—
444Q8#121×C13714yes—
445D4—216no—
446D4—24no—
447D4—24no—
448D4—24no—
449D4—22no—
450D4—24no—
451D4#121×C13714yes—
452C2×C4—28no—
453C2×C4—22no—
454C2×C4—22no—
455C2×C4—24no—
456C2×C4—24no—
457C2×C4—22no—
458C2×C4—22no—
459C2×C4—22no—
460C2×C4—41no—
461C2×C4—44no—
462C2×C4—44no—
463C2×C4—42no—
464C2×C4—42no—
465C2×C4—42no—
466C2×C4—42no—
467C2×C4—42no—
468C2×C4—42no—
469C2×C4—42no—
470C2×C4—82no—
471C2×C4—82no—
472C2×C4—84no—
473C2×C4—82no—
474C2×C4—82no—
475C2×C4#120×C13712yes—
476C2×C4#120×C13711yescharacteristic
477C2×C4#120×C13711yescharacteristic
478C2×C4C22×D4×C13711yescharacteristic
479C2×C4C22×D4×C13712yes—
480C2×C4C2×#41×C13712yes—
481C2×C4C2×#41×C13711yescharacteristic
482C8—44no—
483C22—21no—
484C22—21no—
485C22—24no—
486C22—24no—
487C22—44no—
488C22—42no—
489C22—42no—
490C22—42no—
491C22—42no—
492C22—82no—
493C22—88no—
494C22C2×#120×C13711yescharacteristic
495C22C2×#121×C13711yescharacteristic
496C22#525×C13711yescharacteristic
497C4—24no—
498C4—21no—
499C4—22no—
500C4—42no—
501C4—42no—
502C4—42no—
503C4—82no—
504C4—82no—
505C4#455×C13712yescomplemented
506C4C2×#121×C13712yes—
507C4#525×C13712yes—
508C2—21no—
509C2—21no—
510C2—44no—
511C2—44no—
512C2#2847×C13711yescharacteristic
513C2C2×#455×C13711yescharacteristic
514C2#3022×C13711yescharacteristic
subgroupG/H
center, socleC22×C137C2×#120
commutator, Frattini, cosocleC22×C4C24×C137
prime radicalC2×H(5,𝔽2)×C137C22
LCSG ▷ C22×C4 ▷ C2 ▷ C1
UCSC1 ◁ C22×C137 ◁ C2×C42×C137 ◁ G

Complements

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

NH ≅ G/Ntype
2. #3039×C137C2semidirect
4. #2547×C137C2semidirect
5. #1885×C137C2semidirect
6. #1882×C137C2semidirect
7. #1806×C137C2semidirect
8. #1800×C137C2semidirect
11. C2×C4⋊Q8×C137C22semidirect
28. #423×C137C22semidirect
30. #419×C137C22semidirect
33. #391×C137C22semidirect
35. C4×C4⋊C4×C137C22semidirect
36. #343×C137C22semidirect
37. #339×C137C22semidirect
247. C4×C137#455semidirect
257. #32922C137direct (N × H)
258. C137#32922direct (N × H)
260. #3039C2×C137semidirect
262. #2547C2×C137semidirect
263. #1885C2×C137semidirect
264. #1882C2×C137semidirect
265. #1806C2×C137semidirect
266. #1800C2×C137semidirect
269. C2×C4⋊Q8C22×C137semidirect
286. #423C22×C137semidirect
288. #419C22×C137semidirect
291. #391C22×C137semidirect
293. C4×C4⋊C4C22×C137semidirect
294. #343C22×C137semidirect
295. #339C22×C137semidirect
505. C4#455×C137semidirect

Maximal subgroups

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

subgroupindex
C4×H(5,𝔽2)×C1372
#3039×C1372
C2×#455×C1372
#2547×C1372
#1885×C1372
#1882×C1372
#1806×C1372
#1800×C1372
#32922137