Finite Group Databasebeta

← C151×#23755 #50328680

Subgroups — C151×#23755 #50328680

Order 38656.

880 conjugacy classes of subgroups.

The subgroup lattice has 512 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C4×#121×C151C211yesmaximal, characteristic
2#2285×C151C211yesmaximal, characteristic, complemented
3#2100×C151C211yesmaximal, characteristic, complemented
4#2094×C151C211yesmaximal, characteristic, complemented
5#2084×C151C211yesmaximal, characteristic, complemented
6#2077×C151C212yesmaximal, complemented
7#2071×C151C211yesmaximal, characteristic, complemented
8#2034×C151C212yesmaximal
9C4×#106×C151C212yesmaximal, complemented
10C4×#102×C151C212yesmaximal, complemented
11C4×C22⋊C8×C151C211yesmaximal, characteristic, complemented
12C2×#121×C151C2211yescharacteristic
13C2×C4×D4×C151—21no—
14C2×C4×D4×C151C2211yescharacteristic
15C2×#117×C151—21no—
16C22×C42×C151C2211yescharacteristic
17C2×#106×C151C2212yes—
18C2×#106×C151C2212yes—
19C2×#102×C151C2212yes—
20C2×#102×C151C2212yes—
21C2×C22⋊C8×C151C2212yes—
22C2×C4×C8×C151C2212yes—
23#390×C151C2211yescharacteristic
24#389×C151C2211yescharacteristic
25#388×C151—21no—
26#386×C151—21no—
27#384×C151C2211yescharacteristic
28#384×C151C2211yescharacteristic
29#382×C151—21no—
30#380×C151—21no—
31#380×C151C2211yescharacteristic
32#380×C151C2212yes—
33C4×C4⋊C4×C151—21no—
34C4×C4⋊C4×C151C2212yes—
35C4×C4⋊C4×C151C2211yescharacteristic
36C4×C22⋊C4×C151—21no—
37#340×C151C2212yes—
38#340×C151C2212yes—
39#340×C151C2212yes—
40#340×C151C2212yes—
41#336×C151C2211yescharacteristic
42#336×C151C2211yescharacteristic
43C22×D4×C151—21no—
44C22×D4×C151C2311yescharacteristic
45C23×C4×C151—21no—
46C23×C4×C151D411yescharacteristic
47C23×C4×C151D411yescharacteristic
48C23×C4×C151C2311yescharacteristic
49C22×C8×C151C2312yes—
50C22×C8×C151C2312yes—
51#121×C151—22no—
52#121×C151—22no—
53C4×D4×C151—24no—
54C4×D4×C151—44no—
55#117×C151—44no—
56C2×C4⋊C4×C151—21no—
57C2×C4⋊C4×C151—21no—
58C2×C4⋊C4×C151—21no—
59C2×C4⋊C4×C151—21no—
60C2×C4⋊C4×C151—21no—
61C2×C4⋊C4×C151C2311yescharacteristic
62C2×C4⋊C4×C151C2312yes—
63C2×C4⋊C4×C151C2312yes—
64C2×C4⋊C4×C151C2311yescharacteristic
65C2×C4⋊C4×C151C2311yescharacteristic
66C2×C22⋊C4×C151—21no—
67C2×C22⋊C4×C151—21no—
68C2×C22⋊C4×C151—21no—
69C2×C22⋊C4×C151—21no—
70C2×C22⋊C4×C151—22no—
71C2×C22⋊C4×C151—21no—
72C2×C42×C151—24no—
73C2×C42×C151—21no—
74C2×C42×C151—22no—
75C2×C42×C151D411yescharacteristic
76C2×C42×C151D411yescharacteristic
77C2×C42×C151C2311yescharacteristic
78C2×C42×C151C2311yescharacteristic
79#106×C151—24no—
80#106×C151—24no—
81#102×C151—24no—
82#102×C151—24no—
83C22⋊C8×C151—24no—
84C4×C8×C151—24no—
85#95×C151—21no—
86#95×C151—21no—
87#95×C151—22no—
88#95×C151—22no—
89#95×C151—21no—
90C24×C151—21no—
91C24×C151C2×D411yescharacteristic
92C2×D4×C151—24no—
93C2×D4×C151—22no—
94C2×D4×C151—22no—
95C2×D4×C151—21no—
96C2×D4×C151—42no—
97C2×D4×C151—42no—
98C2×D4×C151#4112yes—
99C22×C4×C151—22no—
100C22×C4×C151—22no—
101C22×C4×C151—21no—
102C22×C4×C151—22no—
103C22×C4×C151—22no—
104C22×C4×C151—22no—
105C22×C4×C151—22no—
106C22×C4×C151—21no—
107C22×C4×C151—21no—
108C22×C4×C151—21no—
109C22×C4×C151—22no—
110C22×C4×C151—22no—
111C22×C4×C151—21no—
112C22×C4×C151—41no—
113C22×C4×C151—42no—
114C22×C4×C151—42no—
115C22×C4×C151—42no—
116C22×C4×C151SD1614yes—
117C22×C4×C151C2×D411yescharacteristic
118C22×C4×C151C2×D411yescharacteristic
119C22×C4×C151C2×D411yescharacteristic
120C22×C4×C151C2×D411yescharacteristic
121C22×C4×C151C2×D411yescharacteristic
122C22×C4×C151C2411yescharacteristic
123C2×C8×C151—22no—
124C2×C8×C151—22no—
125C2×C8×C151—22no—
126C2×C8×C151—22no—
127C2×C8×C151—22no—
128C2×C8×C151#4114yes—
129C4⋊C4×C151—21no—
130C4⋊C4×C151—22no—
131C4⋊C4×C151—22no—
132C4⋊C4×C151—22no—
133C4⋊C4×C151—22no—
134C4⋊C4×C151—22no—
135C4⋊C4×C151—22no—
136C4⋊C4×C151—42no—
137C4⋊C4×C151—42no—
138C4⋊C4×C151—42no—
139C4⋊C4×C151—42no—
140C4⋊C4×C151—42no—
141C4⋊C4×C151#4112yes—
142C22⋊C4×C151—24no—
143C22⋊C4×C151—44no—
144C22⋊C4×C151—42no—
145C22⋊C4×C151—42no—
146C22⋊C4×C151—42no—
147C22⋊C4×C151—42no—
148C42×C151—22no—
149C42×C151—22no—
150C42×C151—22no—
151C42×C151—42no—
152C42×C151—44no—
153C42×C151#4114yes—
154C23×C151—21no—
155C23×C151—21no—
156C23×C151—21no—
157C23×C151—21no—
158C23×C151—21no—
159C23×C151—21no—
160C23×C151—22no—
161C23×C151—41no—
162C23×C151—41no—
163C23×C151—41no—
164C23×C151—41no—
165C23×C151—41no—
166C23×C151—41no—
167C23×C151C2×SD1612yes—
168C23×C151C22×D411yescharacteristic
169D4×C151—24no—
170D4×C151—48no—
171D4×C151—42no—
172C2×C4×C151—21no—
173C2×C4×C151—21no—
174C2×C4×C151—21no—
175C2×C4×C151—21no—
176C2×C4×C151—21no—
177C2×C4×C151—21no—
178C2×C4×C151—28no—
179C2×C4×C151—21no—
180C2×C4×C151—21no—
181C2×C4×C151—22no—
182C2×C4×C151—21no—
183C2×C4×C151—44no—
184C2×C4×C151—41no—
185C2×C4×C151—41no—
186C2×C4×C151—42no—
187C2×C4×C151—42no—
188C2×C4×C151—41no—
189C2×C4×C151—42no—
190C2×C4×C151—41no—
191C2×C4×C151—42no—
192C2×C4×C151—41no—
193C2×C4×C151—41no—
194C2×C4×C151—41no—
195C2×C4×C151—42no—
196C2×C4×C151—42no—
197C2×C4×C151—42no—
198C2×C4×C151—41no—
199C2×C4×C151—42no—
200C2×C4×C151—84no—
201C2×C4×C151#12312yes—
202C2×C4×C151#12312yes—
203C2×C4×C151C2×SD1612yes—
204C2×C4×C151C2×SD1612yes—
205C2×C4×C151#13512yes—
206C2×C4×C151AGL(1,ℤ/8)12yes—
207C2×C4×C151C2×#4112yes—
208C2×C4×C151C2×#4111yescharacteristic
209C2×C4×C151C2×#4111yescharacteristic
210C2×C4×C151C2×#4112yes—
211C8×C151—24no—
212C8×C151—42no—
213C22×C151—22no—
214C22×C151—22no—
215C22×C151—22no—
216C22×C151—41no—
217C22×C151—41no—
218C22×C151—41no—
219C22×C151—41no—
220C22×C151—42no—
221C22×C151—42no—
222C22×C151—42no—
223C22×C151—81no—
224C22×C151—81no—
225C22×C151—81no—
226C22×C151—81no—
227C22×C151C2×#12311yescharacteristic
228C22×C151#52511yescharacteristic
229C22×C151#53211yescharacteristic
230C22×C151C22×SD1611yescharacteristic
231C22×C151C2×#13511yescharacteristic
232C22×C151C2×AGL(1,ℤ/8)11yescharacteristic
233C22×C151#57511yescharacteristic
234C4×C151—22no—
235C4×C151—22no—
236C4×C151—22no—
237C4×C151—42no—
238C4×C151—41no—
239C4×C151—81no—
240C4×C151—82no—
241C4×C151—81no—
242C4×C151—82no—
243C4×C151#48114yes—
244C4×C151#52912yes—
245C4×C151#52912yes—
246C2×C151—42no—
247C2×C151—82no—
248C2×C151#239511yescharacteristic
249C2×C151C2×#48111yescharacteristic
250C2×C151#309111yescharacteristic
251C2×C151#310111yescharacteristic
252C2×C151#310611yescharacteristic
253C2×C151#316711yescharacteristic
254C2×C151#317111yescharacteristic
255#23755C15111yesmaximal, characteristic, direct product factor, Sylow
256C151#2375511yescharacteristic, direct product factor, Sylow
257C4×#121C2×C15111yescharacteristic
258#2285C2×C15111yescharacteristic, complemented
259#2100C2×C15111yescharacteristic, complemented
260#2094C2×C15111yescharacteristic, complemented
261#2084C2×C15111yescharacteristic, complemented
262#2077C2×C15112yescomplemented
263#2071C2×C15111yescharacteristic, complemented
264#2034C2×C15112yes—
265C4×#106C2×C15112yescomplemented
266C4×#102C2×C15112yescomplemented
267C4×C22⋊C8C2×C15111yescharacteristic, complemented
268C2×#121C22×C15111yescharacteristic
269C2×C4×D4—21no—
270C2×C4×D4C22×C15111yescharacteristic
271C2×#117—21no—
272C22×C42C22×C15111yescharacteristic
273C2×#106C22×C15112yes—
274C2×#106C22×C15112yes—
275C2×#102C22×C15112yes—
276C2×#102C22×C15112yes—
277C2×C22⋊C8C22×C15112yes—
278C2×C4×C8C22×C15112yes—
279#390C22×C15111yescharacteristic
280#389C22×C15111yescharacteristic
281#388—21no—
282#386—21no—
283#384C22×C15111yescharacteristic
284#384C22×C15111yescharacteristic
285#382—21no—
286#380—21no—
287#380C22×C15111yescharacteristic
288#380C22×C15112yes—
289C4×C4⋊C4—21no—
290C4×C4⋊C4C22×C15112yes—
291C4×C4⋊C4C22×C15111yescharacteristic
292C4×C22⋊C4—21no—
293#340C22×C15112yes—
294#340C22×C15112yes—
295#340C22×C15112yes—
296#340C22×C15112yes—
297#336C22×C15111yescharacteristic
298#336C22×C15111yescharacteristic
299C22×D4—21no—
300C22×D4C23×C15111yescharacteristic
301C23×C4—21no—
302C23×C4D4×C15111yescharacteristic
303C23×C4D4×C15111yescharacteristic
304C23×C4C23×C15111yescharacteristic
305C22×C8C23×C15112yes—
306C22×C8C23×C15112yes—
307#121—22no—
308#121—22no—
309C4×D4—24no—
310C4×D4—44no—
311#117—44no—
312C2×C4⋊C4—21no—
313C2×C4⋊C4—21no—
314C2×C4⋊C4—21no—
315C2×C4⋊C4—21no—
316C2×C4⋊C4—21no—
317C2×C4⋊C4C23×C15111yescharacteristic
318C2×C4⋊C4C23×C15112yes—
319C2×C4⋊C4C23×C15112yes—
320C2×C4⋊C4C23×C15111yescharacteristic
321C2×C4⋊C4C23×C15111yescharacteristic
322C2×C22⋊C4—21no—
323C2×C22⋊C4—22no—
324C2×C22⋊C4—21no—
325C2×C22⋊C4—21no—
326C2×C22⋊C4—21no—
327C2×C22⋊C4—21no—
328C2×C42—24no—
329C2×C42—22no—
330C2×C42—21no—
331C2×C42D4×C15111yescharacteristic
332C2×C42D4×C15111yescharacteristic
333C2×C42C23×C15111yescharacteristic
334C2×C42C23×C15111yescharacteristic
335#106—24no—
336#106—24no—
337#102—24no—
338#102—24no—
339C22⋊C8—24no—
340C4×C8—24no—
341#95—21no—
342#95—21no—
343#95—22no—
344#95—22no—
345#95—21no—
346C24—21no—
347C24C2×D4×C15111yescharacteristic
348C2×D4—24no—
349C2×D4—22no—
350C2×D4—22no—
351C2×D4—21no—
352C2×D4—42no—
353C2×D4—42no—
354C2×D4#41×C15112yes—
355C22×C4—22no—
356C22×C4—22no—
357C22×C4—21no—
358C22×C4—22no—
359C22×C4—22no—
360C22×C4—22no—
361C22×C4—22no—
362C22×C4—21no—
363C22×C4—21no—
364C22×C4—21no—
365C22×C4—22no—
366C22×C4—22no—
367C22×C4—21no—
368C22×C4—41no—
369C22×C4—42no—
370C22×C4—42no—
371C22×C4—42no—
372C22×C4SD16×C15114yes—
373C22×C4C2×D4×C15111yescharacteristic
374C22×C4C2×D4×C15111yescharacteristic
375C22×C4C2×D4×C15111yescharacteristic
376C22×C4C2×D4×C15111yescharacteristic
377C22×C4C2×D4×C15111yescharacteristic
378C22×C4C24×C15111yescharacteristic
379C2×C8—22no—
380C2×C8—22no—
381C2×C8—22no—
382C2×C8—22no—
383C2×C8—22no—
384C2×C8#41×C15114yes—
385C4⋊C4—21no—
386C4⋊C4—22no—
387C4⋊C4—22no—
388C4⋊C4—22no—
389C4⋊C4—22no—
390C4⋊C4—22no—
391C4⋊C4—22no—
392C4⋊C4—42no—
393C4⋊C4—42no—
394C4⋊C4—42no—
395C4⋊C4—42no—
396C4⋊C4—42no—
397C4⋊C4#41×C15112yes—
398C22⋊C4—24no—
399C22⋊C4—44no—
400C22⋊C4—42no—
401C22⋊C4—42no—
402C22⋊C4—42no—
403C22⋊C4—42no—
404C42—22no—
405C42—22no—
406C42—22no—
407C42—42no—
408C42—44no—
409C42#41×C15114yes—
410C23—21no—
411C23—21no—
412C23—21no—
413C23—21no—
414C23—21no—
415C23—21no—
416C23—22no—
417C23—41no—
418C23—41no—
419C23—41no—
420C23—41no—
421C23—41no—
422C23—41no—
423C23C2×SD16×C15112yes—
424C23C22×D4×C15111yescharacteristic
425D4—24no—
426D4—48no—
427D4—42no—
428C2×C4—21no—
429C2×C4—21no—
430C2×C4—21no—
431C2×C4—21no—
432C2×C4—21no—
433C2×C4—21no—
434C2×C4—28no—
435C2×C4—21no—
436C2×C4—21no—
437C2×C4—22no—
438C2×C4—21no—
439C2×C4—44no—
440C2×C4—41no—
441C2×C4—41no—
442C2×C4—42no—
443C2×C4—42no—
444C2×C4—41no—
445C2×C4—42no—
446C2×C4—41no—
447C2×C4—42no—
448C2×C4—41no—
449C2×C4—41no—
450C2×C4—41no—
451C2×C4—42no—
452C2×C4—42no—
453C2×C4—42no—
454C2×C4—41no—
455C2×C4—42no—
456C2×C4—84no—
457C2×C4#123×C15112yes—
458C2×C4#123×C15112yes—
459C2×C4C2×SD16×C15112yes—
460C2×C4C2×SD16×C15112yes—
461C2×C4#135×C15112yes—
462C2×C4AGL(1,ℤ/8)×C15112yes—
463C2×C4C2×#41×C15112yes—
464C2×C4C2×#41×C15111yescharacteristic
465C2×C4C2×#41×C15111yescharacteristic
466C2×C4C2×#41×C15112yes—
467C8—24no—
468C8—42no—
469C22—22no—
470C22—22no—
471C22—22no—
472C22—41no—
473C22—41no—
474C22—41no—
475C22—41no—
476C22—42no—
477C22—42no—
478C22—42no—
479C22—81no—
480C22—81no—
481C22—81no—
482C22—81no—
483C22C2×#123×C15111yescharacteristic
484C22#525×C15111yescharacteristic
485C22#532×C15111yescharacteristic
486C22C22×SD16×C15111yescharacteristic
487C22C2×#135×C15111yescharacteristic
488C22C2×AGL(1,ℤ/8)×C15111yescharacteristic
489C22#575×C15111yescharacteristic
490C4—22no—
491C4—22no—
492C4—22no—
493C4—42no—
494C4—41no—
495C4—82no—
496C4—81no—
497C4—82no—
498C4—81no—
499C4#481×C15114yes—
500C4#529×C15112yes—
501C4#529×C15112yes—
502C2—42no—
503C2—82no—
504C2#2395×C15111yescharacteristic
505C2C2×#481×C15111yescharacteristic
506C2#3091×C15111yescharacteristic
507C2#3101×C15111yescharacteristic
508C2#3106×C15111yescharacteristic
509C2#3167×C15111yescharacteristic
510C2#3171×C15111yescharacteristic
subgroupG/H
centerC22×C4×C151C2×D4
commutator, Frattini, cosocleC22×C4C24×C151
socleC23×C151C22×D4
prime radicalC2×#121×C151C22
LCSG ▷ C22×C4 ▷ C2 ▷ C1
UCSC1 ◁ C22×C4×C151 ◁ C22×C42×C151 ◁ G

Complements

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

NH ≅ G/Ntype
2. #2285×C151C2semidirect
3. #2100×C151C2semidirect
4. #2094×C151C2semidirect
5. #2084×C151C2semidirect
6. #2077×C151C2semidirect
7. #2071×C151C2semidirect
9. C4×#106×C151C2semidirect
10. C4×#102×C151C2semidirect
11. C4×C22⋊C8×C151C2semidirect
255. #23755C151direct (N × H)
256. C151#23755direct (N × H)
258. #2285C2×C151semidirect
259. #2100C2×C151semidirect
260. #2094C2×C151semidirect
261. #2084C2×C151semidirect
262. #2077C2×C151semidirect
263. #2071C2×C151semidirect
265. C4×#106C2×C151semidirect
266. C4×#102C2×C151semidirect
267. C4×C22⋊C8C2×C151semidirect

Maximal subgroups

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

subgroupindex
C4×#121×C1512
#2285×C1512
#2100×C1512
#2094×C1512
#2084×C1512
#2077×C1512
#2071×C1512
#2034×C1512
C4×#106×C1512
C4×#102×C1512
C4×C22⋊C8×C1512
#23755151