Finite Group Databasebeta

← C137×#25830 #46875216

Subgroups — C137×#25830 #46875216

Order 35072.

1054 conjugacy classes of subgroups.

The subgroup lattice has 518 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#2868×C137C211yesmaximal, characteristic, complemented
5#2060×C137C214yesmaximal, complemented
6#1997×C137C211yesmaximal, characteristic, complemented
7#1994×C137C211yesmaximal, characteristic, complemented
8#1859×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×C137—22no—
22C2×C4⋊Q8×C137C2211yescharacteristic
23C2×C4⋊Q8×C137C2211yescharacteristic
24#520×C137C2212yes—
25C4×#41×C137C2211yescharacteristic
26#501×C137—22no—
27#500×C137—21no—
28#498×C137—21no—
29#477×C137—21no—
30#476×C137—21no—
31#475×C137—21no—
32#474×C137—21no—
33#428×C137C2212yes—
34#428×C137C2214yes—
35#423×C137C2211yescharacteristic
36C2×#104×C137C2212yes—
37#417×C137C2212yes—
38C4×M16×C137C2211yescharacteristic
39#384×C137—22no—
40#384×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×C137—48no—
65C4⋊Q8×C137D411yescharacteristic
66C4⋊Q8×C137D411yescharacteristic
67#125×C137—24no—
68#125×C137—22no—
69#125×C137—22no—
70#125×C137D412yes—
71C22⋊Q8×C137—22no—
72C22⋊Q8×C137—22no—
73C22⋊Q8×C137—24no—
74C22⋊Q8×C137—24no—
75C22⋊Q8×C137—21no—
76C22⋊Q8×C137—21no—
77C22⋊Q8×C137—24no—
78C4×Q8×C137—22no—
79C4×Q8×C137—22no—
80C4×Q8×C137—22no—
81C4×Q8×C137—21no—
82C4×Q8×C137—21no—
83C4×Q8×C137D411yescharacteristic
84C4×Q8×C137D411yescharacteristic
85C4×D4×C137—22no—
86C4×D4×C137—21no—
87C4×D4×C137—21no—
88C4×D4×C137—24no—
89C4×D4×C137D411yescharacteristic
90C4×D4×C137D411yescharacteristic
91#117×C137—22no—
92#117×C137—21no—
93#117×C137C2311yescharacteristic
94#117×C137C2312yes—
95C2×C4⋊C4×C137—22no—
96C2×C4⋊C4×C137—22no—
97C2×C4⋊C4×C137—21no—
98C2×C4⋊C4×C137—28no—
99C2×C4⋊C4×C137D412yes—
100C2×C4⋊C4×C137C2312yes—
101C2×C4⋊C4×C137C2312yes—
102C2×C42×C137—21no—
103C2×C42×C137—21no—
104C2×C42×C137—21no—
105C2×C42×C137C2312yes—
106C2×C42×C137C2311yescharacteristic
107#107×C137—22no—
108#107×C137—24no—
109#106×C137—22no—
110#106×C137—24no—
111C4⋊C8×C137—21no—
112C4⋊C8×C137—21no—
113#104×C137Q818yescomplemented
114#103×C137—22no—
115#102×C137—22no—
116#97×C137—21no—
117C4×C8×C137—21no—
118#41×C137—22no—
119#41×C137C2×Q814yes—
120C2×Q8×C137—24no—
121C2×Q8×C137—21no—
122C2×Q8×C137—22no—
123C2×Q8×C137—21no—
124C2×Q8×C137—22no—
125C2×Q8×C137—44no—
126C2×Q8×C137—44no—
127C2×Q8×C137—41no—
128C2×Q8×C137—41no—
129C2×Q8×C137—84no—
130C2×Q8×C137C2×D411yescharacteristic
131C2×D4×C137—21no—
132C2×D4×C137C2×D411yescharacteristic
133C22×C4×C137—22no—
134C22×C4×C137—21no—
135C22×C4×C137—24no—
136C22×C4×C137—21no—
137C22×C4×C137—21no—
138C22×C4×C137—22no—
139C22×C4×C137C2×D411yescharacteristic
140C22×C4×C137C2×D411yescharacteristic
141C22×C4×C137C2×D414yes—
142C22×C4×C137C2411yescharacteristic
143M16×C137—22no—
144M16×C137—24no—
145M16×C137C2×Q814yescomplemented
146C2×C8×C137—22no—
147C2×C8×C137—22no—
148C4⋊C4×C137—24no—
149C4⋊C4×C137—24no—
150C4⋊C4×C137—24no—
151C4⋊C4×C137—24no—
152C4⋊C4×C137—21no—
153C4⋊C4×C137—22no—
154C4⋊C4×C137—22no—
155C4⋊C4×C137—21no—
156C4⋊C4×C137—21no—
157C4⋊C4×C137—21no—
158C4⋊C4×C137—22no—
159C4⋊C4×C137—22no—
160C4⋊C4×C137—22no—
161C4⋊C4×C137—22no—
162C4⋊C4×C137—22no—
163C4⋊C4×C137—22no—
164C4⋊C4×C137—416no—
165C4⋊C4×C137—41no—
166C4⋊C4×C137—41no—
167C4⋊C4×C137C2×D412yes—
168C4⋊C4×C137C2×D411yescharacteristic
169C4⋊C4×C137C2×D412yes—
170C4⋊C4×C137C2×D411yescharacteristic
171C22⋊C4×C137—24no—
172C22⋊C4×C137—22no—
173C22⋊C4×C137—21no—
174C22⋊C4×C137—21no—
175C22⋊C4×C137—21no—
176C42×C137—24no—
177C42×C137—24no—
178C42×C137—24no—
179C42×C137—21no—
180C42×C137—21no—
181C42×C137—21no—
182C42×C137—22no—
183C42×C137—22no—
184C42×C137C2×D411yescharacteristic
185C42×C137C2×D411yescharacteristic
186C42×C137C2×D411yescharacteristic
187C42×C137C2×D411yescharacteristic
188C42×C137C2×Q814yescomplemented
189C23×C137—21no—
190C23×C137C22×D411yescharacteristic
191Q8×C137—21no—
192Q8×C137—44no—
193Q8×C137—42no—
194Q8×C137—42no—
195Q8×C137—82no—
196Q8×C137—82no—
197Q8×C137—84no—
198Q8×C137—82no—
199Q8×C137C4⋊Q812yescomplemented
200D4×C137—24no—
201D4×C137—21no—
202D4×C137C4⋊Q812yescomplemented
203C2×C4×C137—24no—
204C2×C4×C137—24no—
205C2×C4×C137—21no—
206C2×C4×C137—24no—
207C2×C4×C137—24no—
208C2×C4×C137—22no—
209C2×C4×C137—22no—
210C2×C4×C137—22no—
211C2×C4×C137—21no—
212C2×C4×C137—21no—
213C2×C4×C137—24no—
214C2×C4×C137—22no—
215C2×C4×C137—42no—
216C2×C4×C137—42no—
217C2×C4×C137—44no—
218C2×C4×C137—44no—
219C2×C4×C137—42no—
220C2×C4×C137—84no—
221C2×C4×C137#12011yescharacteristic
222C2×C4×C137#12011yescharacteristic
223C2×C4×C137#12011yescharacteristic
224C2×C4×C137#12011yescharacteristic
225C2×C4×C137C22×D411yescharacteristic
226C2×C4×C137C22×D411yescharacteristic
227C2×C4×C137C22×Q812yes—
228C2×C4×C1372−1+412yes—
229C8×C137—24no—
230C8×C137—42no—
231C22×C137—21no—
232C22×C137—21no—
233C22×C137—22no—
234C22×C137—41no—
235C22×C137—41no—
236C22×C137C2×#12011yescharacteristic
237C22×C137D4×Q812yes—
238C4×C137—28no—
239C4×C137—22no—
240C4×C137—21no—
241C4×C137—21no—
242C4×C137—41no—
243C4×C137—44no—
244C4×C137—42no—
245C4×C137—41no—
246C4×C137—42no—
247C4×C137—82no—
248C4×C137#45312yes—
249C4×C137#45612yes—
250C4×C137C2×C4⋊Q811yescharacteristic
251C4×C137#53611yescharacteristic
252C4×C137D4×Q812yes—
253C2×C137—22no—
254C2×C137—42no—
255C2×C137#243411yescharacteristic
256C2×C137C2×#45311yescharacteristic
257C2×C137C2×#45611yescharacteristic
258#25830C13711yesmaximal, characteristic, direct product factor, Sylow
259C137#2583011yescharacteristic, direct product factor, Sylow
260#3503C2×C13711yescharacteristic
261#3165C2×C13711yescharacteristic, complemented
262#3079C2×C13711yescharacteristic
263#2868C2×C13711yescharacteristic, complemented
264#2060C2×C13714yescomplemented
265#1997C2×C13711yescharacteristic, complemented
266#1994C2×C13711yescharacteristic, complemented
267#1859C2×C13714yes—
268C4×#104C2×C13711yescharacteristic
269Q82—21no—
270#557—21no—
271#554—21no—
272#554—22no—
273D4×Q8—22no—
274D4×Q8—21no—
275#544—22no—
276#544C22×C13712yes—
277#533—21no—
278#533C22×C13711yescharacteristic
279C2×C4⋊Q8—21no—
280C2×C4⋊Q8—22no—
281C2×C4⋊Q8C22×C13711yescharacteristic
282C2×C4⋊Q8C22×C13711yescharacteristic
283#520C22×C13712yes—
284C4×#41C22×C13711yescharacteristic
285#501—22no—
286#500—21no—
287#498—21no—
288#477—21no—
289#476—21no—
290#475—21no—
291#474—21no—
292#428C22×C13712yes—
293#428C22×C13714yes—
294#423C22×C13711yescharacteristic
295C2×#104C22×C13712yes—
296#417C22×C13712yes—
297C4×M16C22×C13711yescharacteristic
298#384—22no—
299#384C22×C13714yes—
300C43C22×C13711yescharacteristic
301#343C22×C13714yes—
302#339C22×C13712yes—
303#339C22×C13714yes—
304C2×#41C23×C13711yescharacteristic
305C22×Q8—22no—
306C22×Q8—21no—
307C22×Q8D4×C13712yes—
308C2×M16C23×C13712yes—
309C2×M16C23×C13712yes—
310C4⋊Q8—22no—
311C4⋊Q8—22no—
312C4⋊Q8—24no—
313C4⋊Q8—22no—
314C4⋊Q8—22no—
315C4⋊Q8—21no—
316C4⋊Q8—21no—
317C4⋊Q8—21no—
318C4⋊Q8—21no—
319C4⋊Q8—21no—
320C4⋊Q8—21no—
321C4⋊Q8—21no—
322C4⋊Q8—21no—
323C4⋊Q8—48no—
324C4⋊Q8D4×C13711yescharacteristic
325C4⋊Q8D4×C13711yescharacteristic
326#125—24no—
327#125—22no—
328#125—22no—
329#125D4×C13712yes—
330C22⋊Q8—22no—
331C22⋊Q8—22no—
332C22⋊Q8—24no—
333C22⋊Q8—24no—
334C22⋊Q8—21no—
335C22⋊Q8—21no—
336C22⋊Q8—24no—
337C4×Q8—22no—
338C4×Q8—22no—
339C4×Q8—22no—
340C4×Q8—21no—
341C4×Q8—21no—
342C4×Q8D4×C13711yescharacteristic
343C4×Q8D4×C13711yescharacteristic
344C4×D4—22no—
345C4×D4—21no—
346C4×D4—21no—
347C4×D4—24no—
348C4×D4D4×C13711yescharacteristic
349C4×D4D4×C13711yescharacteristic
350#117—22no—
351#117—21no—
352#117C23×C13711yescharacteristic
353#117C23×C13712yes—
354C2×C4⋊C4—22no—
355C2×C4⋊C4—22no—
356C2×C4⋊C4—28no—
357C2×C4⋊C4—21no—
358C2×C4⋊C4D4×C13712yes—
359C2×C4⋊C4C23×C13712yes—
360C2×C4⋊C4C23×C13712yes—
361C2×C42—21no—
362C2×C42—21no—
363C2×C42—21no—
364C2×C42C23×C13712yes—
365C2×C42C23×C13711yescharacteristic
366#107—22no—
367#107—24no—
368#106—22no—
369#106—24no—
370C4⋊C8—21no—
371C4⋊C8—21no—
372#104Q8×C13718yescomplemented
373#103—22no—
374#102—22no—
375#97—21no—
376C4×C8—21no—
377#41—22no—
378#41C2×Q8×C13714yes—
379C2×Q8—24no—
380C2×Q8—21no—
381C2×Q8—22no—
382C2×Q8—21no—
383C2×Q8—22no—
384C2×Q8—44no—
385C2×Q8—44no—
386C2×Q8—41no—
387C2×Q8—41no—
388C2×Q8—84no—
389C2×Q8C2×D4×C13711yescharacteristic
390C2×D4—21no—
391C2×D4C2×D4×C13711yescharacteristic
392C22×C4—22no—
393C22×C4—21no—
394C22×C4—24no—
395C22×C4—21no—
396C22×C4—21no—
397C22×C4—22no—
398C22×C4C2×D4×C13711yescharacteristic
399C22×C4C2×D4×C13711yescharacteristic
400C22×C4C2×D4×C13714yes—
401C22×C4C24×C13711yescharacteristic
402M16—22no—
403M16—24no—
404M16C2×Q8×C13714yescomplemented
405C2×C8—22no—
406C2×C8—22no—
407C4⋊C4—24no—
408C4⋊C4—24no—
409C4⋊C4—24no—
410C4⋊C4—24no—
411C4⋊C4—21no—
412C4⋊C4—22no—
413C4⋊C4—22no—
414C4⋊C4—21no—
415C4⋊C4—21no—
416C4⋊C4—21no—
417C4⋊C4—22no—
418C4⋊C4—22no—
419C4⋊C4—22no—
420C4⋊C4—22no—
421C4⋊C4—22no—
422C4⋊C4—22no—
423C4⋊C4—416no—
424C4⋊C4—41no—
425C4⋊C4—41no—
426C4⋊C4C2×D4×C13712yes—
427C4⋊C4C2×D4×C13711yescharacteristic
428C4⋊C4C2×D4×C13712yes—
429C4⋊C4C2×D4×C13711yescharacteristic
430C22⋊C4—24no—
431C22⋊C4—22no—
432C22⋊C4—21no—
433C22⋊C4—21no—
434C22⋊C4—21no—
435C42—24no—
436C42—24no—
437C42—24no—
438C42—21no—
439C42—21no—
440C42—21no—
441C42—22no—
442C42—22no—
443C42C2×D4×C13711yescharacteristic
444C42C2×D4×C13711yescharacteristic
445C42C2×D4×C13711yescharacteristic
446C42C2×D4×C13711yescharacteristic
447C42C2×Q8×C13714yescomplemented
448C23—21no—
449C23C22×D4×C13711yescharacteristic
450Q8—21no—
451Q8—44no—
452Q8—42no—
453Q8—42no—
454Q8—82no—
455Q8—82no—
456Q8—84no—
457Q8—82no—
458Q8C4⋊Q8×C13712yescomplemented
459D4—24no—
460D4—21no—
461D4C4⋊Q8×C13712yescomplemented
462C2×C4—24no—
463C2×C4—24no—
464C2×C4—21no—
465C2×C4—24no—
466C2×C4—22no—
467C2×C4—22no—
468C2×C4—24no—
469C2×C4—22no—
470C2×C4—21no—
471C2×C4—21no—
472C2×C4—24no—
473C2×C4—22no—
474C2×C4—42no—
475C2×C4—42no—
476C2×C4—44no—
477C2×C4—44no—
478C2×C4—42no—
479C2×C4—84no—
480C2×C4#120×C13711yescharacteristic
481C2×C4#120×C13711yescharacteristic
482C2×C4#120×C13711yescharacteristic
483C2×C4#120×C13711yescharacteristic
484C2×C4C22×D4×C13711yescharacteristic
485C2×C4C22×D4×C13711yescharacteristic
486C2×C4C22×Q8×C13712yes—
487C2×C42−1+4×C13712yes—
488C8—24no—
489C8—42no—
490C22—21no—
491C22—21no—
492C22—22no—
493C22—41no—
494C22—41no—
495C22C2×#120×C13711yescharacteristic
496C22D4×Q8×C13712yes—
497C4—28no—
498C4—22no—
499C4—21no—
500C4—21no—
501C4—41no—
502C4—44no—
503C4—42no—
504C4—41no—
505C4—42no—
506C4—82no—
507C4#453×C13712yes—
508C4#456×C13712yes—
509C4C2×C4⋊Q8×C13711yescharacteristic
510C4#536×C13711yescharacteristic
511C4D4×Q8×C13712yes—
512C2—22no—
513C2—42no—
514C2#2434×C13711yescharacteristic
515C2C2×#453×C13711yescharacteristic
516C2C2×#456×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. #2868×C137C2semidirect
5. #2060×C137C2semidirect
6. #1997×C137C2semidirect
7. #1994×C137C2semidirect
113. #104×C137Q8semidirect
145. M16×C137C2×Q8semidirect
188. C42×C137C2×Q8semidirect
199. Q8×C137C4⋊Q8semidirect
202. D4×C137C4⋊Q8semidirect
258. #25830C137direct (N × H)
259. C137#25830direct (N × H)
261. #3165C2×C137semidirect
263. #2868C2×C137semidirect
264. #2060C2×C137semidirect
265. #1997C2×C137semidirect
266. #1994C2×C137semidirect
372. #104Q8×C137semidirect
404. M16C2×Q8×C137semidirect
447. C42C2×Q8×C137semidirect
458. Q8C4⋊Q8×C137semidirect
461. D4C4⋊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
#2868×C1372
#2060×C1372
#1997×C1372
#1994×C1372
#1859×C1372
C4×#104×C1372
#25830137