Finite Group Databasebeta

← C137×#25829 #46875215

Subgroups — C137×#25829 #46875215

Order 35072.

982 conjugacy classes of subgroups.

The subgroup lattice has 514 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#3503×C137C211yesmaximal, characteristic
2#3165×C137C211yesmaximal, characteristic, complemented
3#3079×C137C211yesmaximal, characteristic
4#2862×C137C211yesmaximal, characteristic, complemented
5#2061×C137C214yesmaximal, complemented
6#1997×C137C211yesmaximal, characteristic, complemented
7#1994×C137C211yesmaximal, characteristic, complemented
8#1860×C137C214yesmaximal
9C4×#104×C137C211yesmaximal, characteristic
10Q82×C137—21no—
11#557×C137—21no—
12#554×C137—21no—
13#554×C137—22no—
14D4×Q8×C137—22no—
15D4×Q8×C137—21no—
16#544×C137—22no—
17#544×C137C2212yes—
18#533×C137—21no—
19#533×C137C2211yescharacteristic
20C2×C4⋊Q8×C137—21no—
21C2×C4⋊Q8×C137C2211yescharacteristic
22C2×C4⋊Q8×C137C2211yescharacteristic
23#520×C137C2212yes—
24C4×#41×C137C2211yescharacteristic
25#501×C137—22no—
26#500×C137—21no—
27#498×C137—21no—
28#477×C137—21no—
29#476×C137—21no—
30#475×C137—21no—
31#474×C137—21no—
32#428×C137C2212yes—
33#428×C137C2214yes—
34#423×C137C2211yescharacteristic
35C2×#104×C137C2212yes—
36#417×C137C2212yes—
37C4×M16×C137C2211yescharacteristic
38#391×C137—22no—
39#391×C137—22no—
40#382×C137C2214yes—
41C43×C137C2211yescharacteristic
42#343×C137C2214yes—
43#339×C137C2212yes—
44#339×C137C2214yes—
45C2×#41×C137C2311yescharacteristic
46C22×Q8×C137—22no—
47C22×Q8×C137—21no—
48C22×Q8×C137D412yes—
49C2×M16×C137C2312yes—
50C2×M16×C137C2312yes—
51C4⋊Q8×C137—22no—
52C4⋊Q8×C137—22no—
53C4⋊Q8×C137—24no—
54C4⋊Q8×C137—22no—
55C4⋊Q8×C137—22no—
56C4⋊Q8×C137—21no—
57C4⋊Q8×C137—21no—
58C4⋊Q8×C137—21no—
59C4⋊Q8×C137—21no—
60C4⋊Q8×C137—21no—
61C4⋊Q8×C137—21no—
62C4⋊Q8×C137—21no—
63C4⋊Q8×C137—21no—
64C4⋊Q8×C137D411yescharacteristic
65C4⋊Q8×C137D411yescharacteristic
66#125×C137—24no—
67#125×C137—22no—
68#125×C137—22no—
69#125×C137D412yes—
70C22⋊Q8×C137—22no—
71C22⋊Q8×C137—22no—
72C22⋊Q8×C137—24no—
73C22⋊Q8×C137—24no—
74C22⋊Q8×C137—21no—
75C22⋊Q8×C137—21no—
76C22⋊Q8×C137—24no—
77C4×Q8×C137—22no—
78C4×Q8×C137—22no—
79C4×Q8×C137—22no—
80C4×Q8×C137—21no—
81C4×Q8×C137—21no—
82C4×Q8×C137D411yescharacteristic
83C4×Q8×C137D411yescharacteristic
84C4×D4×C137—22no—
85C4×D4×C137—21no—
86C4×D4×C137—21no—
87C4×D4×C137—24no—
88C4×D4×C137D411yescharacteristic
89C4×D4×C137D411yescharacteristic
90#117×C137—22no—
91#117×C137—21no—
92#117×C137C2311yescharacteristic
93#117×C137C2312yes—
94C2×C4⋊C4×C137—22no—
95C2×C4⋊C4×C137—22no—
96C2×C4⋊C4×C137—21no—
97C2×C4⋊C4×C137D412yes—
98C2×C4⋊C4×C137C2312yes—
99C2×C4⋊C4×C137C2312yes—
100C2×C42×C137—21no—
101C2×C42×C137—21no—
102C2×C42×C137—21no—
103C2×C42×C137C2312yes—
104C2×C42×C137C2311yescharacteristic
105#107×C137—22no—
106#107×C137—24no—
107#106×C137—22no—
108#106×C137—24no—
109C4⋊C8×C137—21no—
110C4⋊C8×C137—21no—
111#104×C137Q818yescomplemented
112#103×C137—22no—
113#102×C137—22no—
114#97×C137—21no—
115C4×C8×C137—21no—
116#95×C137—28no—
117#41×C137—22no—
118#41×C137C2×Q814yes—
119C2×Q8×C137—24no—
120C2×Q8×C137—21no—
121C2×Q8×C137—22no—
122C2×Q8×C137—21no—
123C2×Q8×C137—22no—
124C2×Q8×C137—44no—
125C2×Q8×C137—44no—
126C2×Q8×C137—41no—
127C2×Q8×C137—41no—
128C2×Q8×C137—84no—
129C2×Q8×C137C2×D411yescharacteristic
130C2×D4×C137—21no—
131C2×D4×C137C2×D411yescharacteristic
132C22×C4×C137—21no—
133C22×C4×C137—22no—
134C22×C4×C137—24no—
135C22×C4×C137—21no—
136C22×C4×C137—21no—
137C22×C4×C137—22no—
138C22×C4×C137C2×D411yescharacteristic
139C22×C4×C137C2×D411yescharacteristic
140C22×C4×C137C2×D414yes—
141C22×C4×C137C2411yescharacteristic
142M16×C137—22no—
143M16×C137—24no—
144M16×C137C2×Q814yescomplemented
145C2×C8×C137—22no—
146C2×C8×C137—22no—
147C4⋊C4×C137—24no—
148C4⋊C4×C137—24no—
149C4⋊C4×C137—24no—
150C4⋊C4×C137—24no—
151C4⋊C4×C137—21no—
152C4⋊C4×C137—22no—
153C4⋊C4×C137—22no—
154C4⋊C4×C137—21no—
155C4⋊C4×C137—21no—
156C4⋊C4×C137—21no—
157C4⋊C4×C137—22no—
158C4⋊C4×C137—22no—
159C4⋊C4×C137—22no—
160C4⋊C4×C137—22no—
161C4⋊C4×C137—22no—
162C4⋊C4×C137—22no—
163C4⋊C4×C137—41no—
164C4⋊C4×C137—41no—
165C4⋊C4×C137C2×D412yes—
166C4⋊C4×C137C2×D411yescharacteristic
167C4⋊C4×C137C2×D412yes—
168C4⋊C4×C137C2×D411yescharacteristic
169C22⋊C4×C137—24no—
170C22⋊C4×C137—22no—
171C22⋊C4×C137—21no—
172C22⋊C4×C137—21no—
173C22⋊C4×C137—21no—
174C42×C137—24no—
175C42×C137—21no—
176C42×C137—21no—
177C42×C137—21no—
178C42×C137—22no—
179C42×C137—22no—
180C42×C137—42no—
181C42×C137—42no—
182C42×C137C2×D411yescharacteristic
183C42×C137C2×D411yescharacteristic
184C42×C137C2×D411yescharacteristic
185C42×C137C2×D411yescharacteristic
186C42×C137C2×Q814yescomplemented
187C23×C137—21no—
188C23×C137C22×D411yescharacteristic
189Q8×C137—21no—
190Q8×C137—42no—
191Q8×C137—44no—
192Q8×C137—42no—
193Q8×C137—82no—
194Q8×C137—82no—
195Q8×C137—84no—
196Q8×C137—82no—
197Q8×C137C4⋊Q812yes—
198D4×C137—24no—
199D4×C137—21no—
200D4×C137C4⋊Q812yes—
201C2×C4×C137—21no—
202C2×C4×C137—24no—
203C2×C4×C137—22no—
204C2×C4×C137—22no—
205C2×C4×C137—22no—
206C2×C4×C137—21no—
207C2×C4×C137—21no—
208C2×C4×C137—24no—
209C2×C4×C137—22no—
210C2×C4×C137—24no—
211C2×C4×C137—42no—
212C2×C4×C137—42no—
213C2×C4×C137—42no—
214C2×C4×C137—42no—
215C2×C4×C137—44no—
216C2×C4×C137—44no—
217C2×C4×C137—42no—
218C2×C4×C137—84no—
219C2×C4×C137#12011yescharacteristic
220C2×C4×C137#12011yescharacteristic
221C2×C4×C137#12011yescharacteristic
222C2×C4×C137#12011yescharacteristic
223C2×C4×C137C22×D411yescharacteristic
224C2×C4×C137C22×D411yescharacteristic
225C2×C4×C137C22×Q812yes—
226C2×C4×C1372−1+412yes—
227C8×C137—24no—
228C8×C137—42no—
229C22×C137—21no—
230C22×C137—21no—
231C22×C137—22no—
232C22×C137—41no—
233C22×C137—41no—
234C22×C137C2×#12011yescharacteristic
235C22×C137D4×Q812yes—
236C4×C137—22no—
237C4×C137—21no—
238C4×C137—21no—
239C4×C137—44no—
240C4×C137—41no—
241C4×C137—44no—
242C4×C137—42no—
243C4×C137—41no—
244C4×C137—42no—
245C4×C137—82no—
246C4×C137#45412yes—
247C4×C137#45512yes—
248C4×C137C2×C4⋊Q811yescharacteristic
249C4×C137#53611yescharacteristic
250C4×C137D4×Q812yes—
251C2×C137—22no—
252C2×C137—42no—
253C2×C137#243411yescharacteristic
254C2×C137C2×#45511yescharacteristic
255C2×C137C2×#45411yescharacteristic
256#25829C13711yesmaximal, characteristic, direct product factor, Sylow
257C137#2582911yescharacteristic, direct product factor, Sylow
258#3503C2×C13711yescharacteristic
259#3165C2×C13711yescharacteristic, complemented
260#3079C2×C13711yescharacteristic
261#2862C2×C13711yescharacteristic, complemented
262#2061C2×C13714yescomplemented
263#1997C2×C13711yescharacteristic, complemented
264#1994C2×C13711yescharacteristic, complemented
265#1860C2×C13714yes—
266C4×#104C2×C13711yescharacteristic
267Q82—21no—
268#557—21no—
269#554—21no—
270#554—22no—
271D4×Q8—22no—
272D4×Q8—21no—
273#544—22no—
274#544C22×C13712yes—
275#533—21no—
276#533C22×C13711yescharacteristic
277C2×C4⋊Q8—21no—
278C2×C4⋊Q8C22×C13711yescharacteristic
279C2×C4⋊Q8C22×C13711yescharacteristic
280#520C22×C13712yes—
281C4×#41C22×C13711yescharacteristic
282#501—22no—
283#500—21no—
284#498—21no—
285#477—21no—
286#476—21no—
287#475—21no—
288#474—21no—
289#428C22×C13712yes—
290#428C22×C13714yes—
291#423C22×C13711yescharacteristic
292C2×#104C22×C13712yes—
293#417C22×C13712yes—
294C4×M16C22×C13711yescharacteristic
295#391—22no—
296#391—22no—
297#382C22×C13714yes—
298C43C22×C13711yescharacteristic
299#343C22×C13714yes—
300#339C22×C13712yes—
301#339C22×C13714yes—
302C2×#41C23×C13711yescharacteristic
303C22×Q8—22no—
304C22×Q8—21no—
305C22×Q8D4×C13712yes—
306C2×M16C23×C13712yes—
307C2×M16C23×C13712yes—
308C4⋊Q8—22no—
309C4⋊Q8—22no—
310C4⋊Q8—24no—
311C4⋊Q8—22no—
312C4⋊Q8—22no—
313C4⋊Q8—21no—
314C4⋊Q8—21no—
315C4⋊Q8—21no—
316C4⋊Q8—21no—
317C4⋊Q8—21no—
318C4⋊Q8—21no—
319C4⋊Q8—21no—
320C4⋊Q8—21no—
321C4⋊Q8D4×C13711yescharacteristic
322C4⋊Q8D4×C13711yescharacteristic
323#125—24no—
324#125—22no—
325#125—22no—
326#125D4×C13712yes—
327C22⋊Q8—22no—
328C22⋊Q8—22no—
329C22⋊Q8—24no—
330C22⋊Q8—24no—
331C22⋊Q8—21no—
332C22⋊Q8—21no—
333C22⋊Q8—24no—
334C4×Q8—22no—
335C4×Q8—22no—
336C4×Q8—22no—
337C4×Q8—21no—
338C4×Q8—21no—
339C4×Q8D4×C13711yescharacteristic
340C4×Q8D4×C13711yescharacteristic
341C4×D4—22no—
342C4×D4—21no—
343C4×D4—21no—
344C4×D4—24no—
345C4×D4D4×C13711yescharacteristic
346C4×D4D4×C13711yescharacteristic
347#117—22no—
348#117—21no—
349#117C23×C13711yescharacteristic
350#117C23×C13712yes—
351C2×C4⋊C4—22no—
352C2×C4⋊C4—22no—
353C2×C4⋊C4—21no—
354C2×C4⋊C4D4×C13712yes—
355C2×C4⋊C4C23×C13712yes—
356C2×C4⋊C4C23×C13712yes—
357C2×C42—21no—
358C2×C42—21no—
359C2×C42—21no—
360C2×C42C23×C13712yes—
361C2×C42C23×C13711yescharacteristic
362#107—22no—
363#107—24no—
364#106—22no—
365#106—24no—
366C4⋊C8—21no—
367C4⋊C8—21no—
368#104Q8×C13718yescomplemented
369#103—22no—
370#102—22no—
371#97—21no—
372C4×C8—21no—
373#95—28no—
374#41—22no—
375#41C2×Q8×C13714yes—
376C2×Q8—24no—
377C2×Q8—21no—
378C2×Q8—22no—
379C2×Q8—21no—
380C2×Q8—22no—
381C2×Q8—44no—
382C2×Q8—44no—
383C2×Q8—41no—
384C2×Q8—41no—
385C2×Q8—84no—
386C2×Q8C2×D4×C13711yescharacteristic
387C2×D4—21no—
388C2×D4C2×D4×C13711yescharacteristic
389C22×C4—21no—
390C22×C4—22no—
391C22×C4—24no—
392C22×C4—21no—
393C22×C4—21no—
394C22×C4—22no—
395C22×C4C2×D4×C13711yescharacteristic
396C22×C4C2×D4×C13711yescharacteristic
397C22×C4C2×D4×C13714yes—
398C22×C4C24×C13711yescharacteristic
399M16—22no—
400M16—24no—
401M16C2×Q8×C13714yescomplemented
402C2×C8—22no—
403C2×C8—22no—
404C4⋊C4—24no—
405C4⋊C4—24no—
406C4⋊C4—24no—
407C4⋊C4—24no—
408C4⋊C4—21no—
409C4⋊C4—22no—
410C4⋊C4—22no—
411C4⋊C4—21no—
412C4⋊C4—21no—
413C4⋊C4—21no—
414C4⋊C4—22no—
415C4⋊C4—22no—
416C4⋊C4—22no—
417C4⋊C4—22no—
418C4⋊C4—22no—
419C4⋊C4—22no—
420C4⋊C4—41no—
421C4⋊C4—41no—
422C4⋊C4C2×D4×C13712yes—
423C4⋊C4C2×D4×C13711yescharacteristic
424C4⋊C4C2×D4×C13712yes—
425C4⋊C4C2×D4×C13711yescharacteristic
426C22⋊C4—24no—
427C22⋊C4—22no—
428C22⋊C4—21no—
429C22⋊C4—21no—
430C22⋊C4—21no—
431C42—24no—
432C42—21no—
433C42—21no—
434C42—21no—
435C42—22no—
436C42—22no—
437C42—42no—
438C42—42no—
439C42C2×D4×C13711yescharacteristic
440C42C2×D4×C13711yescharacteristic
441C42C2×D4×C13711yescharacteristic
442C42C2×D4×C13711yescharacteristic
443C42C2×Q8×C13714yescomplemented
444C23—21no—
445C23C22×D4×C13711yescharacteristic
446Q8—21no—
447Q8—42no—
448Q8—44no—
449Q8—42no—
450Q8—82no—
451Q8—82no—
452Q8—84no—
453Q8—82no—
454Q8C4⋊Q8×C13712yes—
455D4—24no—
456D4—21no—
457D4C4⋊Q8×C13712yes—
458C2×C4—21no—
459C2×C4—22no—
460C2×C4—22no—
461C2×C4—24no—
462C2×C4—22no—
463C2×C4—21no—
464C2×C4—21no—
465C2×C4—24no—
466C2×C4—22no—
467C2×C4—24no—
468C2×C4—42no—
469C2×C4—42no—
470C2×C4—42no—
471C2×C4—42no—
472C2×C4—44no—
473C2×C4—44no—
474C2×C4—42no—
475C2×C4—84no—
476C2×C4#120×C13711yescharacteristic
477C2×C4#120×C13711yescharacteristic
478C2×C4#120×C13711yescharacteristic
479C2×C4#120×C13711yescharacteristic
480C2×C4C22×D4×C13711yescharacteristic
481C2×C4C22×D4×C13711yescharacteristic
482C2×C4C22×Q8×C13712yes—
483C2×C42−1+4×C13712yes—
484C8—24no—
485C8—42no—
486C22—21no—
487C22—21no—
488C22—22no—
489C22—41no—
490C22—41no—
491C22C2×#120×C13711yescharacteristic
492C22D4×Q8×C13712yes—
493C4—22no—
494C4—21no—
495C4—21no—
496C4—44no—
497C4—41no—
498C4—44no—
499C4—42no—
500C4—41no—
501C4—42no—
502C4—82no—
503C4#454×C13712yes—
504C4#455×C13712yes—
505C4C2×C4⋊Q8×C13711yescharacteristic
506C4#536×C13711yescharacteristic
507C4D4×Q8×C13712yes—
508C2—22no—
509C2—42no—
510C2#2434×C13711yescharacteristic
511C2C2×#455×C13711yescharacteristic
512C2C2×#454×C13711yescharacteristic
subgroupG/H
center, socleC22×C137C2×#120
commutator, Frattini, cosocleC22×C4C24×C137
prime radicalC2×#41×C137C23
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. #3165×C137C2semidirect
4. #2862×C137C2semidirect
5. #2061×C137C2semidirect
6. #1997×C137C2semidirect
7. #1994×C137C2semidirect
111. #104×C137Q8semidirect
144. M16×C137C2×Q8semidirect
186. C42×C137C2×Q8semidirect
256. #25829C137direct (N × H)
257. C137#25829direct (N × H)
259. #3165C2×C137semidirect
261. #2862C2×C137semidirect
262. #2061C2×C137semidirect
263. #1997C2×C137semidirect
264. #1994C2×C137semidirect
368. #104Q8×C137semidirect
401. M16C2×Q8×C137semidirect
443. C42C2×Q8×C137semidirect

Maximal subgroups

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

subgroupindex
#3503×C1372
#3165×C1372
#3079×C1372
#2862×C1372
#2061×C1372
#1997×C1372
#1994×C1372
#1860×C1372
C4×#104×C1372
#25829137