Finite Group Databasebeta

← C151×#23604 #50328529

Subgroups — C151×#23604 #50328529

Order 38656.

1002 conjugacy classes of subgroups.

The subgroup lattice has 544 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#430×C151C211yesmaximal, characteristic, complemented
2C2×#428×C151C211yesmaximal, characteristic, complemented
3#2905×C151C211yesmaximal, characteristic, complemented
4#2896×C151C211yesmaximal, characteristic
5C2×#417×C151C211yesmaximal, characteristic
6C2×#413×C151C211yesmaximal, characteristic
7#2064×C151C214yesmaximal, complemented
8#2063×C151C214yesmaximal, complemented
9C2×#343×C151C211yesmaximal, characteristic, complemented
10C22×#41×C151C2211yescharacteristic
11C22×M16×C151C2211yescharacteristic
12C22×M16×C151C2211yescharacteristic
13C22×M16×C151C2211yescharacteristic
14C22×M16×C151C2211yescharacteristic
15C2×#117×C151C2211yescharacteristic
16C22×C4⋊C4×C151C2211yescharacteristic
17#430×C151—24no—
18C2×#108×C151C2214yes—
19#428×C151—22no—
20#428×C151C2214yes—
21C2×#107×C151—21no—
22C2×#106×C151—21no—
23#424×C151—22no—
24#424×C151C2214yes—
25#423×C151—22no—
26#419×C151C2212yes—
27#418×C151C2212yes—
28#417×C151—22no—
29#417×C151—22no—
30#417×C151—21no—
31#417×C151—21no—
32#416×C151—22no—
33C2×#103×C151C2212yes—
34C2×#102×C151C2212yes—
35#413×C151—24no—
36C2×#101×C151C2212yes—
37C2×#100×C151C2212yes—
38#343×C151C2214yes—
39C2×#41×C151—21no—
40C2×#41×C151—21no—
41C2×#41×C151—21no—
42C2×#41×C151—21no—
43C2×#41×C151—22no—
44C2×#41×C151—22no—
45C2×#41×C151—22no—
46C2×#41×C151Q814yes—
47C22×Q8×C151C2311yescharacteristic
48C22×D4×C151—21no—
49C22×D4×C151C2311yescharacteristic
50C23×C4×C151—21no—
51C23×C4×C151—21no—
52C23×C4×C151C2311yescharacteristic
53C2×M16×C151—21no—
54C2×M16×C151—22no—
55C2×M16×C151—22no—
56C2×M16×C151—22no—
57C2×M16×C151—21no—
58C2×M16×C151—21no—
59C2×M16×C151—22no—
60C2×M16×C151—22no—
61C2×M16×C151—22no—
62C2×M16×C151—22no—
63C2×M16×C151D414yes—
64C2×M16×C151D412yes—
65C2×M16×C151D412yes—
66C2×M16×C151C2312yes—
67C2×M16×C151C2312yes—
68C2×M16×C151C2312yes—
69C22×C8×C151—21no—
70C22×C8×C151—21no—
71C22×C8×C151—21no—
72C22×C8×C151C2312yes—
73#117×C151—21no—
74#117×C151—21no—
75#117×C151D412yes—
76#117×C151C2312yes—
77C2×C4⋊C4×C151—21no—
78C2×C4⋊C4×C151—21no—
79C2×C4⋊C4×C151—21no—
80C2×C4⋊C4×C151—21no—
81C2×C4⋊C4×C151—21no—
82C2×C4⋊C4×C151D411yescharacteristic
83C2×C4⋊C4×C151D411yescharacteristic
84C2×C4⋊C4×C151C2312yes—
85C2×C22⋊C4×C151—21no—
86C2×C42×C151—21no—
87#108×C151—28no—
88#107×C151—24no—
89#106×C151—24no—
90C4⋊C8×C151—28no—
91#103×C151—24no—
92#103×C151—22no—
93#103×C151—22no—
94#102×C151—24no—
95#102×C151—22no—
96#102×C151—22no—
97#101×C151—24no—
98#100×C151—24no—
99C24×C151—21no—
100C24×C151#4111yescharacteristic
101#41×C151—24no—
102#41×C151—24no—
103#41×C151—24no—
104#41×C151—24no—
105#41×C151—42no—
106#41×C151—42no—
107#41×C151—42no—
108#41×C151—42no—
109C2×Q8×C151—21no—
110C2×Q8×C151—21no—
111C2×Q8×C151—21no—
112C2×Q8×C151—21no—
113C2×Q8×C151—21no—
114C2×Q8×C151C2×Q812yes—
115C2×D4×C151—24no—
116C2×D4×C151—21no—
117C2×D4×C151—21no—
118C2×D4×C151—21no—
119C2×D4×C151—21no—
120C2×D4×C151—21no—
121C2×D4×C151—41no—
122C2×D4×C151—41no—
123C2×D4×C151—42no—
124C2×D4×C151C2×Q812yes—
125C22×C4×C151—21no—
126C22×C4×C151—21no—
127C22×C4×C151—21no—
128C22×C4×C151—24no—
129C22×C4×C151—22no—
130C22×C4×C151—41no—
131C22×C4×C151—41no—
132C22×C4×C151—42no—
133C22×C4×C151—41no—
134C22×C4×C151—41no—
135C22×C4×C151—82no—
136C22×C4×C151C2×D412yes—
137C22×C4×C151C2×D411yescharacteristic
138C22×C4×C151C2×D411yescharacteristic
139C22×C4×C151C2×D411yescharacteristic
140C22×C4×C151C2×D411yescharacteristic
141C22×C4×C151C2×Q812yes—
142C22×C4×C151#4111yescharacteristic
143C22×C4×C151#4112yes—
144C22×C4×C151#4112yes—
145C22×C4×C151C2411yescharacteristic
146M16×C151—24no—
147M16×C151—24no—
148M16×C151—24no—
149M16×C151—22no—
150M16×C151—22no—
151M16×C151—22no—
152M16×C151—22no—
153M16×C151—22no—
154M16×C151—42no—
155M16×C151—42no—
156M16×C151C2×D414yes—
157C2×C8×C151—24no—
158C2×C8×C151—22no—
159C2×C8×C151—22no—
160C2×C8×C151—22no—
161C2×C8×C151—22no—
162C2×C8×C151—22no—
163C2×C8×C151—42no—
164C2×C8×C151—42no—
165C2×C8×C151C2×D414yes—
166C4⋊C4×C151—21no—
167C4⋊C4×C151—21no—
168C4⋊C4×C151—22no—
169C4⋊C4×C151—22no—
170C4⋊C4×C151—41no—
171C4⋊C4×C151—41no—
172C4⋊C4×C151C2×D412yes—
173C4⋊C4×C151C2×D412yes—
174C22⋊C4×C151—22no—
175C22⋊C4×C151—22no—
176C42×C151—22no—
177C42×C151—22no—
178C23×C151—22no—
179C23×C151—41no—
180C23×C151—41no—
181C23×C151—41no—
182C23×C151—41no—
183C23×C151—81no—
184C23×C151—81no—
185C23×C151#12111yescharacteristic
186C23×C151#12111yescharacteristic
187C23×C151C22⋊Q812yes—
188C23×C151#12612yes—
189C23×C151C2×#4111yescharacteristic
190Q8×C151—22no—
191Q8×C151—22no—
192Q8×C151—41no—
193Q8×C151—41no—
194D4×C151—22no—
195D4×C151—22no—
196D4×C151—44no—
197D4×C151—44no—
198D4×C151—41no—
199D4×C151—41no—
200C2×C4×C151—22no—
201C2×C4×C151—22no—
202C2×C4×C151—22no—
203C2×C4×C151—41no—
204C2×C4×C151—41no—
205C2×C4×C151—44no—
206C2×C4×C151—44no—
207C2×C4×C151—42no—
208C2×C4×C151—81no—
209C2×C4×C151—81no—
210C2×C4×C151—82no—
211C2×C4×C151—82no—
212C2×C4×C151—82no—
213C2×C4×C151#12112yes—
214C2×C4×C151#12112yes—
215C2×C4×C151#12112yes—
216C2×C4×C151#12112yes—
217C2×C4×C151#12111yescharacteristic
218C2×C4×C151#12111yescharacteristic
219C2×C4×C151C22⋊Q812yes—
220C2×C4×C151C22⋊Q812yes—
221C2×C4×C151C22⋊Q812yes—
222C2×C4×C151C22⋊Q812yes—
223C2×C4×C151C22⋊Q812yes—
224C2×C4×C151#12612yes—
225C2×C4×C151C22×D411yescharacteristic
226C2×C4×C151C22×D411yescharacteristic
227C2×C4×C151C22×D411yescharacteristic
228C2×C4×C151C22×Q811yescharacteristic
229C2×C4×C151C2×#4111yescharacteristic
230C2×C4×C151C2×#4111yescharacteristic
231C8×C151—24no—
232C8×C151—24no—
233C8×C151—42no—
234C8×C151—42no—
235C22×C151—21no—
236C22×C151—21no—
237C22×C151—21no—
238C22×C151—21no—
239C22×C151—21no—
240C22×C151—21no—
241C22×C151—42no—
242C22×C151—42no—
243C22×C151—42no—
244C22×C151—82no—
245C22×C151—82no—
246C22×C151—81no—
247C22×C151—81no—
248C22×C151#39612yes—
249C22×C151#39612yes—
250C22×C151C2×#12111yescharacteristic
251C22×C151C2×C22⋊Q811yescharacteristic
252C22×C151C2×#12611yescharacteristic
253C4×C151—42no—
254C4×C151—42no—
255C4×C151—42no—
256C4×C151—82no—
257C4×C151#39612yes—
258C4×C151#39612yes—
259C4×C151C2×#12111yescharacteristic
260C4×C151C2×#12111yescharacteristic
261C4×C151C2×C22⋊Q811yescharacteristic
262C4×C151C2×C22⋊Q811yescharacteristic
263C2×C151—21no—
264C2×C151—21no—
265C2×C151—22no—
266C2×C151—22no—
267C2×C151—82no—
268C2×C151C2×#39611yescharacteristic
269C2×C151#306811yescharacteristic
270C2×C151#306911yescharacteristic
271#23604C15111yesmaximal, characteristic, direct product factor, Sylow
272C151#2360411yescharacteristic, direct product factor, Sylow
273C2×#430C2×C15111yescharacteristic, complemented
274C2×#428C2×C15111yescharacteristic, complemented
275#2905C2×C15111yescharacteristic, complemented
276#2896C2×C15111yescharacteristic
277C2×#417C2×C15111yescharacteristic
278C2×#413C2×C15111yescharacteristic
279#2064C2×C15114yescomplemented
280#2063C2×C15114yescomplemented
281C2×#343C2×C15111yescharacteristic, complemented
282C22×#41C22×C15111yescharacteristic
283C22×M16C22×C15111yescharacteristic
284C22×M16C22×C15111yescharacteristic
285C22×M16C22×C15111yescharacteristic
286C22×M16C22×C15111yescharacteristic
287C2×#117C22×C15111yescharacteristic
288C22×C4⋊C4C22×C15111yescharacteristic
289#430—24no—
290C2×#108C22×C15114yes—
291#428—22no—
292#428C22×C15114yes—
293C2×#107—21no—
294C2×#106—21no—
295#424—22no—
296#424C22×C15114yes—
297#423—22no—
298#419C22×C15112yes—
299#418C22×C15112yes—
300#417—22no—
301#417—22no—
302#417—21no—
303#417—21no—
304#416—22no—
305C2×#103C22×C15112yes—
306C2×#102C22×C15112yes—
307#413—24no—
308C2×#101C22×C15112yes—
309C2×#100C22×C15112yes—
310#343C22×C15114yes—
311C2×#41—22no—
312C2×#41—22no—
313C2×#41—22no—
314C2×#41—21no—
315C2×#41—21no—
316C2×#41—21no—
317C2×#41—21no—
318C2×#41Q8×C15114yes—
319C22×Q8C23×C15111yescharacteristic
320C22×D4—21no—
321C22×D4C23×C15111yescharacteristic
322C23×C4—21no—
323C23×C4—21no—
324C23×C4C23×C15111yescharacteristic
325C2×M16—22no—
326C2×M16—22no—
327C2×M16—21no—
328C2×M16—22no—
329C2×M16—21no—
330C2×M16—21no—
331C2×M16—22no—
332C2×M16—22no—
333C2×M16—22no—
334C2×M16—22no—
335C2×M16D4×C15114yes—
336C2×M16D4×C15112yes—
337C2×M16D4×C15112yes—
338C2×M16C23×C15112yes—
339C2×M16C23×C15112yes—
340C2×M16C23×C15112yes—
341C22×C8—21no—
342C22×C8—21no—
343C22×C8—21no—
344C22×C8C23×C15112yes—
345#117—21no—
346#117—21no—
347#117D4×C15112yes—
348#117C23×C15112yes—
349C2×C4⋊C4—21no—
350C2×C4⋊C4—21no—
351C2×C4⋊C4—21no—
352C2×C4⋊C4—21no—
353C2×C4⋊C4—21no—
354C2×C4⋊C4D4×C15111yescharacteristic
355C2×C4⋊C4D4×C15111yescharacteristic
356C2×C4⋊C4C23×C15112yes—
357C2×C22⋊C4—21no—
358C2×C42—21no—
359#108—28no—
360#107—24no—
361#106—24no—
362C4⋊C8—28no—
363#103—24no—
364#103—22no—
365#103—22no—
366#102—24no—
367#102—22no—
368#102—22no—
369#101—24no—
370#100—24no—
371C24—21no—
372C24#41×C15111yescharacteristic
373#41—24no—
374#41—24no—
375#41—24no—
376#41—24no—
377#41—42no—
378#41—42no—
379#41—42no—
380#41—42no—
381C2×Q8—21no—
382C2×Q8—21no—
383C2×Q8—21no—
384C2×Q8—21no—
385C2×Q8—21no—
386C2×Q8C2×Q8×C15112yes—
387C2×D4—24no—
388C2×D4—21no—
389C2×D4—21no—
390C2×D4—21no—
391C2×D4—21no—
392C2×D4—21no—
393C2×D4—41no—
394C2×D4—41no—
395C2×D4—42no—
396C2×D4C2×Q8×C15112yes—
397C22×C4—21no—
398C22×C4—21no—
399C22×C4—21no—
400C22×C4—24no—
401C22×C4—22no—
402C22×C4—41no—
403C22×C4—41no—
404C22×C4—42no—
405C22×C4—41no—
406C22×C4—41no—
407C22×C4—82no—
408C22×C4C2×D4×C15112yes—
409C22×C4C2×D4×C15111yescharacteristic
410C22×C4C2×D4×C15111yescharacteristic
411C22×C4C2×D4×C15111yescharacteristic
412C22×C4C2×D4×C15111yescharacteristic
413C22×C4C2×Q8×C15112yes—
414C22×C4#41×C15111yescharacteristic
415C22×C4#41×C15112yes—
416C22×C4#41×C15112yes—
417C22×C4C24×C15111yescharacteristic
418M16—24no—
419M16—24no—
420M16—24no—
421M16—22no—
422M16—22no—
423M16—22no—
424M16—22no—
425M16—22no—
426M16—42no—
427M16—42no—
428M16C2×D4×C15114yes—
429C2×C8—24no—
430C2×C8—22no—
431C2×C8—22no—
432C2×C8—22no—
433C2×C8—22no—
434C2×C8—22no—
435C2×C8—42no—
436C2×C8—42no—
437C2×C8C2×D4×C15114yes—
438C4⋊C4—21no—
439C4⋊C4—21no—
440C4⋊C4—22no—
441C4⋊C4—22no—
442C4⋊C4—41no—
443C4⋊C4—41no—
444C4⋊C4C2×D4×C15112yes—
445C4⋊C4C2×D4×C15112yes—
446C22⋊C4—22no—
447C22⋊C4—22no—
448C42—22no—
449C42—22no—
450C23—22no—
451C23—41no—
452C23—41no—
453C23—41no—
454C23—41no—
455C23—81no—
456C23—81no—
457C23#121×C15111yescharacteristic
458C23#121×C15111yescharacteristic
459C23C22⋊Q8×C15112yes—
460C23#126×C15112yes—
461C23C2×#41×C15111yescharacteristic
462Q8—22no—
463Q8—22no—
464Q8—41no—
465Q8—41no—
466D4—22no—
467D4—22no—
468D4—44no—
469D4—44no—
470D4—41no—
471D4—41no—
472C2×C4—22no—
473C2×C4—22no—
474C2×C4—22no—
475C2×C4—41no—
476C2×C4—41no—
477C2×C4—44no—
478C2×C4—44no—
479C2×C4—42no—
480C2×C4—81no—
481C2×C4—81no—
482C2×C4—82no—
483C2×C4—82no—
484C2×C4—82no—
485C2×C4#121×C15111yescharacteristic
486C2×C4#121×C15111yescharacteristic
487C2×C4#121×C15112yes—
488C2×C4#121×C15112yes—
489C2×C4#121×C15112yes—
490C2×C4#121×C15112yes—
491C2×C4C22⋊Q8×C15112yes—
492C2×C4C22⋊Q8×C15112yes—
493C2×C4C22⋊Q8×C15112yes—
494C2×C4C22⋊Q8×C15112yes—
495C2×C4C22⋊Q8×C15112yes—
496C2×C4#126×C15112yes—
497C2×C4C22×D4×C15111yescharacteristic
498C2×C4C22×D4×C15111yescharacteristic
499C2×C4C22×D4×C15111yescharacteristic
500C2×C4C22×Q8×C15111yescharacteristic
501C2×C4C2×#41×C15111yescharacteristic
502C2×C4C2×#41×C15111yescharacteristic
503C8—24no—
504C8—24no—
505C8—42no—
506C8—42no—
507C22—21no—
508C22—21no—
509C22—21no—
510C22—21no—
511C22—21no—
512C22—21no—
513C22—42no—
514C22—42no—
515C22—42no—
516C22—82no—
517C22—82no—
518C22—81no—
519C22—81no—
520C22#396×C15112yes—
521C22#396×C15112yes—
522C22C2×#121×C15111yescharacteristic
523C22C2×C22⋊Q8×C15111yescharacteristic
524C22C2×#126×C15111yescharacteristic
525C4—42no—
526C4—42no—
527C4—42no—
528C4—82no—
529C4#396×C15112yes—
530C4#396×C15112yes—
531C4C2×#121×C15111yescharacteristic
532C4C2×#121×C15111yescharacteristic
533C4C2×C22⋊Q8×C15111yescharacteristic
534C4C2×C22⋊Q8×C15111yescharacteristic
535C2—21no—
536C2—21no—
537C2—22no—
538C2—22no—
539C2—82no—
540C2C2×#396×C15111yescharacteristic
541C2#3068×C15111yescharacteristic
542C2#3069×C15111yescharacteristic
subgroupG/H
center, socleC22×C151C2×#121
commutator, Frattini, cosocleC22×C4C24×C151
prime radicalC22×#41×C151C22
LCSG ▷ C22×C4 ▷ C2 ▷ C1
UCSC1 ◁ C22×C151 ◁ C23×C4×C151 ◁ G

Complements

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

NH ≅ G/Ntype
1. C2×#430×C151C2semidirect
2. C2×#428×C151C2semidirect
3. #2905×C151C2semidirect
7. #2064×C151C2semidirect
8. #2063×C151C2semidirect
9. C2×#343×C151C2semidirect
271. #23604C151direct (N × H)
272. C151#23604direct (N × H)
273. C2×#430C2×C151semidirect
274. C2×#428C2×C151semidirect
275. #2905C2×C151semidirect
279. #2064C2×C151semidirect
280. #2063C2×C151semidirect
281. C2×#343C2×C151semidirect

Maximal subgroups

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

subgroupindex
C2×#430×C1512
C2×#428×C1512
#2905×C1512
#2896×C1512
C2×#417×C1512
C2×#413×C1512
#2064×C1512
#2063×C1512
C2×#343×C1512
#23604151