Finite Group Databasebeta

← C151×#24122 #50329047

Subgroups — C151×#24122 #50329047

Order 38656.

1134 conjugacy classes of subgroups.

The subgroup lattice has 546 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#554×C151C211yesmaximal, characteristic
2C2×#477×C151C211yesmaximal, characteristic, complemented
3C2×C4×Q16×C151C211yesmaximal, characteristic, complemented
4#2538×C151C211yesmaximal, characteristic
5#2036×C151C212yesmaximal, complemented
6#2001×C151C212yesmaximal, complemented
7#1990×C151C211yesmaximal, characteristic, complemented
8#1954×C151C211yesmaximal, characteristic
9#1872×C151C212yesmaximal, complemented
10#1852×C151C212yesmaximal, complemented
11#1803×C151C211yesmaximal, characteristic, complemented
12C22×Q16×C151C2211yescharacteristic
13#554×C151—24no—
14#554×C151—24no—
15C2×C4⋊Q8×C151C2211yescharacteristic
16C2×#125×C151—21no—
17C2×C22⋊Q8×C151—22no—
18C2×C22⋊Q8×C151C2212yes—
19C2×C22⋊Q8×C151C2212yes—
20C2×C4×Q8×C151C2211yescharacteristic
21C2×C4×Q8×C151C2211yescharacteristic
22C2×C4×D4×C151—21no—
23C2×C4×D4×C151C2211yescharacteristic
24C22×C4⋊C4×C151C2212yes—
25#477×C151—24no—
26C4×Q16×C151—24no—
27C2×#107×C151C2211yescharacteristic
28C2×#107×C151C2212yes—
29C2×C4⋊C8×C151C2211yescharacteristic
30C2×#103×C151C2211yescharacteristic
31C2×#103×C151C2212yes—
32C2×#103×C151C2211yescharacteristic
33C2×#103×C151C2212yes—
34C2×C22⋊C8×C151C2212yes—
35C2×C22⋊C8×C151C2212yes—
36C2×C4×C8×C151C2211yescharacteristic
37#396×C151—22no—
38#390×C151—21no—
39#389×C151—21no—
40#384×C151C2211yescharacteristic
41#380×C151C2212yes—
42#340×C151C2212yes—
43#340×C151C2212yes—
44#340×C151C2212yes—
45C22×Q8×C151—21no—
46C22×Q8×C151C2311yescharacteristic
47C22×Q8×C151C2311yescharacteristic
48C22×D4×C151D411yescharacteristic
49C23×C4×C151—22no—
50C23×C4×C151C2312yes—
51C2×Q16×C151—22no—
52C2×Q16×C151—22no—
53C2×Q16×C151—22no—
54C22×C8×C151C2311yescharacteristic
55C22×C8×C151C2312yes—
56C22×C8×C151C2311yescharacteristic
57C4⋊Q8×C151—22no—
58C4⋊Q8×C151D414yes—
59#125×C151—28no—
60C22⋊Q8×C151—216no—
61C22⋊Q8×C151—24no—
62C22⋊Q8×C151—24no—
63C22⋊Q8×C151—24no—
64C22⋊Q8×C151—42no—
65C4×Q8×C151—22no—
66C4×Q8×C151—22no—
67C4×Q8×C151—22no—
68C4×Q8×C151D414yes—
69C4×D4×C151—28no—
70C4×D4×C151—22no—
71C4×D4×C151—22no—
72C2×C4⋊C4×C151—22no—
73C2×C4⋊C4×C151—21no—
74C2×C4⋊C4×C151—21no—
75C2×C4⋊C4×C151—22no—
76C2×C4⋊C4×C151—24no—
77C2×C4⋊C4×C151—24no—
78C2×C4⋊C4×C151—21no—
79C2×C4⋊C4×C151—22no—
80C2×C4⋊C4×C151—21no—
81C2×C4⋊C4×C151—21no—
82C2×C4⋊C4×C151D411yescharacteristic
83C2×C4⋊C4×C151C2312yes—
84C2×C4⋊C4×C151C2311yescharacteristic
85C2×C4⋊C4×C151C2311yescharacteristic
86C2×C4⋊C4×C151C2312yes—
87C2×C22⋊C4×C151—22no—
88C2×C22⋊C4×C151—22no—
89C2×C22⋊C4×C151—22no—
90C2×C22⋊C4×C151D412yes—
91C2×C42×C151—21no—
92C2×C42×C151—21no—
93C2×C42×C151C2311yescharacteristic
94#107×C151—22no—
95#107×C151—24no—
96C4⋊C8×C151—22no—
97#103×C151—22no—
98#103×C151—24no—
99#103×C151—22no—
100#103×C151—42no—
101C22⋊C8×C151—24no—
102C22⋊C8×C151—42no—
103C4×C8×C151—22no—
104#95×C151—22no—
105C24×C151C2×D412yes—
106C2×Q8×C151—24no—
107C2×Q8×C151—21no—
108C2×Q8×C151—21no—
109C2×Q8×C151—21no—
110C2×Q8×C151—21no—
111C2×Q8×C151—21no—
112C2×Q8×C151—21no—
113C2×Q8×C151—21no—
114C2×Q8×C151—21no—
115C2×Q8×C151—21no—
116C2×Q8×C151—21no—
117C2×Q8×C151—81no—
118C2×Q8×C151—81no—
119C2×Q8×C151C2×D412yes—
120C2×Q8×C151#4112yes—
121C2×D4×C151—22no—
122C2×D4×C151—22no—
123C2×D4×C151Q1614yescomplemented
124C22×C4×C151—21no—
125C22×C4×C151—21no—
126C22×C4×C151—21no—
127C22×C4×C151—22no—
128C22×C4×C151—28no—
129C22×C4×C151—22no—
130C22×C4×C151—22no—
131C22×C4×C151—22no—
132C22×C4×C151—22no—
133C22×C4×C151—22no—
134C22×C4×C151—22no—
135C22×C4×C151—21no—
136C22×C4×C151—22no—
137C22×C4×C151—82no—
138C22×C4×C151—82no—
139C22×C4×C151C2×D412yes—
140C22×C4×C151C2×D411yescharacteristic
141C22×C4×C151C2×D411yescharacteristic
142C22×C4×C151C2411yescharacteristic
143Q16×C151—28no—
144C2×C8×C151—21no—
145C2×C8×C151—21no—
146C2×C8×C151—21no—
147C2×C8×C151—22no—
148C2×C8×C151—22no—
149C2×C8×C151—21no—
150C2×C8×C151—22no—
151C2×C8×C151—21no—
152C2×C8×C151#4112yes—
153C4⋊C4×C151—28no—
154C4⋊C4×C151—28no—
155C4⋊C4×C151—24no—
156C4⋊C4×C151—24no—
157C4⋊C4×C151—22no—
158C4⋊C4×C151—21no—
159C4⋊C4×C151—22no—
160C4⋊C4×C151—21no—
161C4⋊C4×C151—21no—
162C4⋊C4×C151—22no—
163C4⋊C4×C151—22no—
164C4⋊C4×C151—44no—
165C4⋊C4×C151—42no—
166C4⋊C4×C151—42no—
167C4⋊C4×C151—81no—
168C4⋊C4×C151C2×D412yes—
169C4⋊C4×C151C2×D414yes—
170C22⋊C4×C151—28no—
171C22⋊C4×C151—24no—
172C22⋊C4×C151—44no—
173C22⋊C4×C151—82no—
174C42×C151—24no—
175C42×C151—42no—
176C42×C151C2×D412yes—
177C42×C151C2×D412yes—
178C23×C151—22no—
179C23×C151—22no—
180C23×C151—22no—
181C23×C151—22no—
182C23×C151—22no—
183C23×C151—22no—
184C23×C151C2×Q1614yes—
185C23×C151C22×D411yescharacteristic
186Q8×C151—22no—
187Q8×C151—22no—
188Q8×C151—22no—
189Q8×C151—22no—
190Q8×C151—41no—
191Q8×C151—41no—
192Q8×C151—41no—
193Q8×C151—41no—
194Q8×C151—84no—
195D4×C151—28no—
196C2×C4×C151—21no—
197C2×C4×C151—21no—
198C2×C4×C151—24no—
199C2×C4×C151—22no—
200C2×C4×C151—22no—
201C2×C4×C151—22no—
202C2×C4×C151—22no—
203C2×C4×C151—22no—
204C2×C4×C151—22no—
205C2×C4×C151—41no—
206C2×C4×C151—41no—
207C2×C4×C151—41no—
208C2×C4×C151—41no—
209C2×C4×C151—42no—
210C2×C4×C151—41no—
211C2×C4×C151—42no—
212C2×C4×C151—41no—
213C2×C4×C151—42no—
214C2×C4×C151—42no—
215C2×C4×C151—42no—
216C2×C4×C151—42no—
217C2×C4×C151—41no—
218C2×C4×C151—42no—
219C2×C4×C151—42no—
220C2×C4×C151—41no—
221C2×C4×C151—42no—
222C2×C4×C151—41no—
223C2×C4×C151—88no—
224C2×C4×C151#12012yes—
225C2×C4×C151#12012yes—
226C2×C4×C151C2×Q1612yes—
227C2×C4×C151#13712yes—
228C2×C4×C151C22×D411yescharacteristic
229C2×C4×C151C22×D411yescharacteristic
230C2×C4×C151C2×#4111yescharacteristic
231C2×C4×C151C2×#4111yescharacteristic
232C2×C4×C151H(5,𝔽2)11yescharacteristic
233C2×C4×C1512−1+411yescharacteristic
234C8×C151—22no—
235C8×C151—41no—
236C8×C151—42no—
237C22×C151—24no—
238C22×C151—24no—
239C22×C151—24no—
240C22×C151—42no—
241C22×C151—42no—
242C22×C151—42no—
243C22×C151—42no—
244C22×C151C2×#12011yescharacteristic
245C22×C151#54611yescharacteristic
246C22×C151#54711yescharacteristic
247C22×C151C22×Q1611yescharacteristic
248C22×C151C2×#13711yescharacteristic
249C22×C151#57611yescharacteristic
250C22×C151#57711yescharacteristic
251C4×C151—22no—
252C4×C151—42no—
253C4×C151—42no—
254C4×C151—42no—
255C4×C151—42no—
256C4×C151—44no—
257C4×C151—82no—
258C4×C151—81no—
259C4×C151#45114yes—
260C4×C151#52511yescharacteristic
261C4×C151#53711yescharacteristic
262C4×C151#54811yescharacteristic
263C4×C151#55011yescharacteristic
264C2×C151—44no—
265C2×C151#240711yescharacteristic
266C2×C151C2×#45111yescharacteristic
267C2×C151#301211yescharacteristic
268C2×C151#329911yescharacteristic
269C2×C151#333311yescharacteristic
270C2×C151#333811yescharacteristic
271C2×C151#335611yescharacteristic
272#24122C15111yesmaximal, characteristic, direct product factor, Sylow
273C151#2412211yescharacteristic, direct product factor, Sylow
274C2×#554C2×C15111yescharacteristic
275C2×#477C2×C15111yescharacteristic, complemented
276C2×C4×Q16C2×C15111yescharacteristic, complemented
277#2538C2×C15111yescharacteristic
278#2036C2×C15112yescomplemented
279#2001C2×C15112yescomplemented
280#1990C2×C15111yescharacteristic, complemented
281#1954C2×C15111yescharacteristic
282#1872C2×C15112yescomplemented
283#1852C2×C15112yescomplemented
284#1803C2×C15111yescharacteristic, complemented
285C22×Q16C22×C15111yescharacteristic
286#554—24no—
287#554—24no—
288C2×C4⋊Q8C22×C15111yescharacteristic
289C2×#125—21no—
290C2×C22⋊Q8—22no—
291C2×C22⋊Q8C22×C15112yes—
292C2×C22⋊Q8C22×C15112yes—
293C2×C4×Q8C22×C15111yescharacteristic
294C2×C4×Q8C22×C15111yescharacteristic
295C2×C4×D4—21no—
296C2×C4×D4C22×C15111yescharacteristic
297C22×C4⋊C4C22×C15112yes—
298#477—24no—
299C4×Q16—24no—
300C2×#107C22×C15112yes—
301C2×#107C22×C15111yescharacteristic
302C2×C4⋊C8C22×C15111yescharacteristic
303C2×#103C22×C15112yes—
304C2×#103C22×C15111yescharacteristic
305C2×#103C22×C15111yescharacteristic
306C2×#103C22×C15112yes—
307C2×C22⋊C8C22×C15112yes—
308C2×C22⋊C8C22×C15112yes—
309C2×C4×C8C22×C15111yescharacteristic
310#396—22no—
311#390—21no—
312#389—21no—
313#384C22×C15111yescharacteristic
314#380C22×C15112yes—
315#340C22×C15112yes—
316#340C22×C15112yes—
317#340C22×C15112yes—
318C22×Q8—21no—
319C22×Q8C23×C15111yescharacteristic
320C22×Q8C23×C15111yescharacteristic
321C22×D4D4×C15111yescharacteristic
322C23×C4—22no—
323C23×C4C23×C15112yes—
324C2×Q16—22no—
325C2×Q16—22no—
326C2×Q16—22no—
327C22×C8C23×C15111yescharacteristic
328C22×C8C23×C15112yes—
329C22×C8C23×C15111yescharacteristic
330C4⋊Q8—22no—
331C4⋊Q8D4×C15114yes—
332#125—28no—
333C22⋊Q8—216no—
334C22⋊Q8—24no—
335C22⋊Q8—24no—
336C22⋊Q8—24no—
337C22⋊Q8—42no—
338C4×Q8—22no—
339C4×Q8—22no—
340C4×Q8—22no—
341C4×Q8D4×C15114yes—
342C4×D4—28no—
343C4×D4—22no—
344C4×D4—22no—
345C2×C4⋊C4—22no—
346C2×C4⋊C4—21no—
347C2×C4⋊C4—21no—
348C2×C4⋊C4—22no—
349C2×C4⋊C4—24no—
350C2×C4⋊C4—24no—
351C2×C4⋊C4—21no—
352C2×C4⋊C4—22no—
353C2×C4⋊C4—21no—
354C2×C4⋊C4—21no—
355C2×C4⋊C4D4×C15111yescharacteristic
356C2×C4⋊C4C23×C15111yescharacteristic
357C2×C4⋊C4C23×C15112yes—
358C2×C4⋊C4C23×C15111yescharacteristic
359C2×C4⋊C4C23×C15112yes—
360C2×C22⋊C4—22no—
361C2×C22⋊C4—22no—
362C2×C22⋊C4—22no—
363C2×C22⋊C4D4×C15112yes—
364C2×C42—21no—
365C2×C42—21no—
366C2×C42C23×C15111yescharacteristic
367#107—22no—
368#107—24no—
369C4⋊C8—22no—
370#103—22no—
371#103—22no—
372#103—24no—
373#103—42no—
374C22⋊C8—24no—
375C22⋊C8—42no—
376C4×C8—22no—
377#95—22no—
378C24C2×D4×C15112yes—
379C2×Q8—24no—
380C2×Q8—21no—
381C2×Q8—21no—
382C2×Q8—21no—
383C2×Q8—21no—
384C2×Q8—21no—
385C2×Q8—21no—
386C2×Q8—21no—
387C2×Q8—21no—
388C2×Q8—21no—
389C2×Q8—21no—
390C2×Q8—81no—
391C2×Q8—81no—
392C2×Q8C2×D4×C15112yes—
393C2×Q8#41×C15112yes—
394C2×D4—22no—
395C2×D4—22no—
396C2×D4Q16×C15114yescomplemented
397C22×C4—21no—
398C22×C4—21no—
399C22×C4—21no—
400C22×C4—22no—
401C22×C4—28no—
402C22×C4—22no—
403C22×C4—22no—
404C22×C4—22no—
405C22×C4—22no—
406C22×C4—22no—
407C22×C4—22no—
408C22×C4—21no—
409C22×C4—22no—
410C22×C4—82no—
411C22×C4—82no—
412C22×C4C2×D4×C15112yes—
413C22×C4C2×D4×C15111yescharacteristic
414C22×C4C2×D4×C15111yescharacteristic
415C22×C4C24×C15111yescharacteristic
416Q16—28no—
417C2×C8—21no—
418C2×C8—21no—
419C2×C8—21no—
420C2×C8—22no—
421C2×C8—22no—
422C2×C8—21no—
423C2×C8—22no—
424C2×C8—21no—
425C2×C8#41×C15112yes—
426C4⋊C4—28no—
427C4⋊C4—28no—
428C4⋊C4—24no—
429C4⋊C4—24no—
430C4⋊C4—22no—
431C4⋊C4—21no—
432C4⋊C4—22no—
433C4⋊C4—21no—
434C4⋊C4—21no—
435C4⋊C4—22no—
436C4⋊C4—22no—
437C4⋊C4—44no—
438C4⋊C4—42no—
439C4⋊C4—42no—
440C4⋊C4—81no—
441C4⋊C4C2×D4×C15112yes—
442C4⋊C4C2×D4×C15114yes—
443C22⋊C4—28no—
444C22⋊C4—24no—
445C22⋊C4—44no—
446C22⋊C4—82no—
447C42—24no—
448C42—42no—
449C42C2×D4×C15112yes—
450C42C2×D4×C15112yes—
451C23—22no—
452C23—22no—
453C23—22no—
454C23—22no—
455C23—22no—
456C23—22no—
457C23C2×Q16×C15114yes—
458C23C22×D4×C15111yescharacteristic
459Q8—22no—
460Q8—22no—
461Q8—22no—
462Q8—22no—
463Q8—41no—
464Q8—41no—
465Q8—41no—
466Q8—41no—
467Q8—84no—
468D4—28no—
469C2×C4—21no—
470C2×C4—21no—
471C2×C4—22no—
472C2×C4—22no—
473C2×C4—22no—
474C2×C4—24no—
475C2×C4—22no—
476C2×C4—22no—
477C2×C4—22no—
478C2×C4—41no—
479C2×C4—41no—
480C2×C4—41no—
481C2×C4—41no—
482C2×C4—41no—
483C2×C4—42no—
484C2×C4—41no—
485C2×C4—42no—
486C2×C4—42no—
487C2×C4—42no—
488C2×C4—42no—
489C2×C4—42no—
490C2×C4—41no—
491C2×C4—42no—
492C2×C4—42no—
493C2×C4—41no—
494C2×C4—42no—
495C2×C4—41no—
496C2×C4—88no—
497C2×C4#120×C15112yes—
498C2×C4#120×C15112yes—
499C2×C4C2×Q16×C15112yes—
500C2×C4#137×C15112yes—
501C2×C4C22×D4×C15111yescharacteristic
502C2×C4C22×D4×C15111yescharacteristic
503C2×C4C2×#41×C15111yescharacteristic
504C2×C4C2×#41×C15111yescharacteristic
505C2×C4H(5,𝔽2)×C15111yescharacteristic
506C2×C42−1+4×C15111yescharacteristic
507C8—22no—
508C8—41no—
509C8—42no—
510C22—24no—
511C22—24no—
512C22—24no—
513C22—42no—
514C22—42no—
515C22—42no—
516C22—42no—
517C22C2×#120×C15111yescharacteristic
518C22#546×C15111yescharacteristic
519C22#547×C15111yescharacteristic
520C22C22×Q16×C15111yescharacteristic
521C22C2×#137×C15111yescharacteristic
522C22#576×C15111yescharacteristic
523C22#577×C15111yescharacteristic
524C4—22no—
525C4—42no—
526C4—42no—
527C4—42no—
528C4—42no—
529C4—44no—
530C4—81no—
531C4—82no—
532C4#451×C15114yes—
533C4#525×C15111yescharacteristic
534C4#537×C15111yescharacteristic
535C4#548×C15111yescharacteristic
536C4#550×C15111yescharacteristic
537C2—44no—
538C2#2407×C15111yescharacteristic
539C2C2×#451×C15111yescharacteristic
540C2#3012×C15111yescharacteristic
541C2#3299×C15111yescharacteristic
542C2#3333×C15111yescharacteristic
543C2#3338×C15111yescharacteristic
544C2#3356×C15111yescharacteristic
subgroupG/H
center, socleC23×C151C22×D4
commutator, Frattini, cosocleC22×C4C24×C151
prime radicalC22×D4×C151D4
LCSG ▷ C22×C4 ▷ C2 ▷ C1
UCSC1 ◁ C23×C151 ◁ C2×C4×D4×C151 ◁ G

Complements

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

NH ≅ G/Ntype
2. C2×#477×C151C2semidirect
3. C2×C4×Q16×C151C2semidirect
5. #2036×C151C2semidirect
6. #2001×C151C2semidirect
7. #1990×C151C2semidirect
9. #1872×C151C2semidirect
10. #1852×C151C2semidirect
11. #1803×C151C2semidirect
123. C2×D4×C151Q16semidirect
272. #24122C151direct (N × H)
273. C151#24122direct (N × H)
275. C2×#477C2×C151semidirect
276. C2×C4×Q16C2×C151semidirect
278. #2036C2×C151semidirect
279. #2001C2×C151semidirect
280. #1990C2×C151semidirect
282. #1872C2×C151semidirect
283. #1852C2×C151semidirect
284. #1803C2×C151semidirect
396. C2×D4Q16×C151semidirect

Maximal subgroups

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

subgroupindex
C2×#554×C1512
C2×#477×C1512
C2×C4×Q16×C1512
#2538×C1512
#2036×C1512
#2001×C1512
#1990×C1512
#1954×C1512
#1872×C1512
#1852×C1512
#1803×C1512
#24122151