Finite Group Databasebeta

← C151×#24579 #50329504

Subgroups — C151×#24579 #50329504

Order 38656.

798 conjugacy classes of subgroups.

The subgroup lattice has 430 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#484×C151C211yesmaximal, characteristic
2C22×#106×C151C211yesmaximal, characteristic
3#2503×C151C211yesmaximal, characteristic
4#2345×C151C211yesmaximal, characteristic
5#2085×C151C212yesmaximal, complemented
6#2066×C151C212yesmaximal, complemented
7#1949×C151C212yesmaximal, complemented
8#1927×C151C212yesmaximal, complemented
9#1892×C151C211yesmaximal, characteristic
10#1850×C151C211yesmaximal, characteristic
11#1823×C151C211yesmaximal, characteristic
12C23×C8×C151C2211yescharacteristic
13C2×C22⋊Q8×C151C2211yescharacteristic
14C22×C4⋊C4×C151C2211yescharacteristic
15C22×C4⋊C4×C151C2211yescharacteristic
16#484×C151—24no—
17C2×#107×C151C2211yescharacteristic
18C2×#107×C151C2211yescharacteristic
19C2×#106×C151—24no—
20C2×#106×C151C2212yes—
21C2×#106×C151C2212yes—
22C2×C4⋊C8×C151C2212yes—
23C2×#103×C151C2212yes—
24C2×#103×C151C2211yescharacteristic
25C2×#103×C151C2211yescharacteristic
26C2×C22⋊C8×C151C2211yescharacteristic
27#400×C151—22no—
28#391×C151C2212yes—
29#389×C151C2212yes—
30#387×C151—22no—
31#382×C151C2212yes—
32#381×C151—21no—
33#381×C151—21no—
34#380×C151—21no—
35#380×C151C2211yescharacteristic
36C4×C4⋊C4×C151C2212yes—
37C4×C22⋊C4×C151—21no—
38C4×C22⋊C4×C151C2211yescharacteristic
39#340×C151C2212yes—
40#340×C151C2212yes—
41#340×C151C2212yes—
42#336×C151C2212yes—
43C22×Q8×C151C2311yescharacteristic
44C23×C4×C151—21no—
45C23×C4×C151—21no—
46C23×C4×C151C2311yescharacteristic
47C22×C8×C151—22no—
48C22×C8×C151—24no—
49C22×C8×C151C2311yescharacteristic
50C22×C8×C151C2311yescharacteristic
51C22×C8×C151C2312yes—
52C22⋊Q8×C151—22no—
53C22⋊Q8×C151—22no—
54C2×C4⋊C4×C151—21no—
55C2×C4⋊C4×C151—22no—
56C2×C4⋊C4×C151—22no—
57C2×C4⋊C4×C151—22no—
58C2×C4⋊C4×C151—21no—
59C2×C4⋊C4×C151—21no—
60C2×C4⋊C4×C151—22no—
61C2×C4⋊C4×C151D414yes—
62C2×C4⋊C4×C151C2312yes—
63C2×C4⋊C4×C151C2311yescharacteristic
64C2×C4⋊C4×C151C2312yes—
65C2×C4⋊C4×C151C2311yescharacteristic
66C2×C4⋊C4×C151C2311yescharacteristic
67C2×C22⋊C4×C151—21no—
68C2×C22⋊C4×C151—21no—
69C2×C22⋊C4×C151D411yescharacteristic
70C2×C22⋊C4×C151D411yescharacteristic
71C2×C42×C151—22no—
72C2×C42×C151C2312yes—
73#107×C151—22no—
74#107×C151—22no—
75#106×C151—24no—
76#106×C151—24no—
77C4⋊C8×C151—24no—
78#103×C151—24no—
79#103×C151—22no—
80#103×C151—22no—
81C22⋊C8×C151—22no—
82#95×C151—22no—
83#95×C151—22no—
84#95×C151—21no—
85#95×C151—22no—
86#95×C151—22no—
87#95×C151—22no—
88#95×C151—22no—
89#95×C151D412yes—
90C24×C151C2×D411yescharacteristic
91C2×Q8×C151—21no—
92C2×Q8×C151—22no—
93C2×Q8×C151—21no—
94C2×Q8×C151—22no—
95C22×C4×C151—22no—
96C22×C4×C151—22no—
97C22×C4×C151—21no—
98C22×C4×C151—21no—
99C22×C4×C151—24no—
100C22×C4×C151—21no—
101C22×C4×C151—21no—
102C22×C4×C151—22no—
103C22×C4×C151—22no—
104C22×C4×C151—42no—
105C22×C4×C151—42no—
106C22×C4×C151—42no—
107C22×C4×C151—82no—
108C22×C4×C151C2×D412yes—
109C22×C4×C151C2×D412yes—
110C22×C4×C151C2×D411yescharacteristic
111C22×C4×C151C2×D412yes—
112C22×C4×C151#4112yes—
113C22×C4×C151C2411yescharacteristic
114C2×C8×C151—28no—
115C2×C8×C151—22no—
116C2×C8×C151—22no—
117C2×C8×C151—21no—
118C2×C8×C151—21no—
119C2×C8×C151—22no—
120C2×C8×C151—22no—
121C2×C8×C151#4114yes—
122C4⋊C4×C151—24no—
123C4⋊C4×C151—21no—
124C4⋊C4×C151—21no—
125C4⋊C4×C151—22no—
126C4⋊C4×C151—21no—
127C4⋊C4×C151—22no—
128C4⋊C4×C151—22no—
129C4⋊C4×C151—21no—
130C4⋊C4×C151—22no—
131C4⋊C4×C151—42no—
132C4⋊C4×C151—42no—
133C4⋊C4×C151—82no—
134C4⋊C4×C151C2×D412yes—
135C4⋊C4×C151C2×D412yes—
136C22⋊C4×C151—22no—
137C22⋊C4×C151—22no—
138C22⋊C4×C151—42no—
139C22⋊C4×C151—42no—
140C42×C151—22no—
141C42×C151—22no—
142C42×C151—82no—
143C23×C151—22no—
144C23×C151—22no—
145C23×C151#12112yes—
146C23×C151#13512yes—
147C23×C151#13712yes—
148C23×C151C22×D411yescharacteristic
149Q8×C151—41no—
150Q8×C151—41no—
151Q8×C151—42no—
152C2×C4×C151—22no—
153C2×C4×C151—22no—
154C2×C4×C151—22no—
155C2×C4×C151—24no—
156C2×C4×C151—22no—
157C2×C4×C151—22no—
158C2×C4×C151—22no—
159C2×C4×C151—22no—
160C2×C4×C151—22no—
161C2×C4×C151—22no—
162C2×C4×C151—22no—
163C2×C4×C151—22no—
164C2×C4×C151—42no—
165C2×C4×C151—42no—
166C2×C4×C151—41no—
167C2×C4×C151—41no—
168C2×C4×C151—42no—
169C2×C4×C151—42no—
170C2×C4×C151—42no—
171C2×C4×C151—42no—
172C2×C4×C151—42no—
173C2×C4×C151—42no—
174C2×C4×C151—42no—
175C2×C4×C151—42no—
176C2×C4×C151—84no—
177C2×C4×C151—84no—
178C2×C4×C151#12112yes—
179C2×C4×C151C22×D411yescharacteristic
180C2×C4×C151C2×#4112yes—
181C2×C4×C151C2×#4111yescharacteristic
182C2×C4×C1512−1+412yes—
183C8×C151—24no—
184C8×C151—41no—
185C8×C151—41no—
186C22×C151—24no—
187C22×C151—24no—
188C22×C151—22no—
189C22×C151—22no—
190C22×C151#46414yes—
191C22×C151C2×#12111yescharacteristic
192C22×C151#54712yes—
193C22×C151C2×#13511yescharacteristic
194C22×C151C2×#13711yescharacteristic
195C22×C151#57712yes—
196C4×C151—22no—
197C4×C151—42no—
198C4×C151—42no—
199C4×C151—42no—
200C4×C151—42no—
201C4×C151—82no—
202C4×C151—81no—
203C4×C151—82no—
204C4×C151—81no—
205C4×C151#52511yescharacteristic
206C4×C151#53611yescharacteristic
207C4×C151#55212yes—
208C2×C151—24no—
209C2×C151#241311yescharacteristic
210C2×C151C2×#46411yescharacteristic
211C2×C151#304211yescharacteristic
212C2×C151#333712yes—
213C2×C151#334412yes—
214#24579C15111yesmaximal, characteristic, direct product factor, Sylow
215C151#2457911yescharacteristic, direct product factor, Sylow
216C2×#484C2×C15111yescharacteristic
217C22×#106C2×C15111yescharacteristic
218#2503C2×C15111yescharacteristic
219#2345C2×C15111yescharacteristic
220#2085C2×C15112yescomplemented
221#2066C2×C15112yescomplemented
222#1949C2×C15112yescomplemented
223#1927C2×C15112yescomplemented
224#1892C2×C15111yescharacteristic
225#1850C2×C15111yescharacteristic
226#1823C2×C15111yescharacteristic
227C23×C8C22×C15111yescharacteristic
228C2×C22⋊Q8C22×C15111yescharacteristic
229C22×C4⋊C4C22×C15111yescharacteristic
230C22×C4⋊C4C22×C15111yescharacteristic
231#484—24no—
232C2×#107C22×C15111yescharacteristic
233C2×#107C22×C15111yescharacteristic
234C2×#106—24no—
235C2×#106C22×C15112yes—
236C2×#106C22×C15112yes—
237C2×C4⋊C8C22×C15112yes—
238C2×#103C22×C15111yescharacteristic
239C2×#103C22×C15112yes—
240C2×#103C22×C15111yescharacteristic
241C2×C22⋊C8C22×C15111yescharacteristic
242#400—22no—
243#391C22×C15112yes—
244#389C22×C15112yes—
245#387—22no—
246#382C22×C15112yes—
247#381—21no—
248#381—21no—
249#380—21no—
250#380C22×C15111yescharacteristic
251C4×C4⋊C4C22×C15112yes—
252C4×C22⋊C4—21no—
253C4×C22⋊C4C22×C15111yescharacteristic
254#340C22×C15112yes—
255#340C22×C15112yes—
256#340C22×C15112yes—
257#336C22×C15112yes—
258C22×Q8C23×C15111yescharacteristic
259C23×C4—21no—
260C23×C4—21no—
261C23×C4C23×C15111yescharacteristic
262C22×C8—22no—
263C22×C8—24no—
264C22×C8C23×C15111yescharacteristic
265C22×C8C23×C15111yescharacteristic
266C22×C8C23×C15112yes—
267C22⋊Q8—22no—
268C22⋊Q8—22no—
269C2×C4⋊C4—22no—
270C2×C4⋊C4—21no—
271C2×C4⋊C4—22no—
272C2×C4⋊C4—22no—
273C2×C4⋊C4—21no—
274C2×C4⋊C4—21no—
275C2×C4⋊C4—22no—
276C2×C4⋊C4D4×C15114yes—
277C2×C4⋊C4C23×C15112yes—
278C2×C4⋊C4C23×C15111yescharacteristic
279C2×C4⋊C4C23×C15112yes—
280C2×C4⋊C4C23×C15111yescharacteristic
281C2×C4⋊C4C23×C15111yescharacteristic
282C2×C22⋊C4—21no—
283C2×C22⋊C4—21no—
284C2×C22⋊C4D4×C15111yescharacteristic
285C2×C22⋊C4D4×C15111yescharacteristic
286C2×C42—22no—
287C2×C42C23×C15112yes—
288#107—22no—
289#107—22no—
290#106—24no—
291#106—24no—
292C4⋊C8—24no—
293#103—24no—
294#103—22no—
295#103—22no—
296C22⋊C8—22no—
297#95—22no—
298#95—22no—
299#95—21no—
300#95—22no—
301#95—22no—
302#95—22no—
303#95—22no—
304#95D4×C15112yes—
305C24C2×D4×C15111yescharacteristic
306C2×Q8—21no—
307C2×Q8—22no—
308C2×Q8—21no—
309C2×Q8—22no—
310C22×C4—22no—
311C22×C4—22no—
312C22×C4—21no—
313C22×C4—21no—
314C22×C4—24no—
315C22×C4—21no—
316C22×C4—21no—
317C22×C4—22no—
318C22×C4—22no—
319C22×C4—42no—
320C22×C4—42no—
321C22×C4—42no—
322C22×C4—82no—
323C22×C4C2×D4×C15112yes—
324C22×C4C2×D4×C15112yes—
325C22×C4C2×D4×C15111yescharacteristic
326C22×C4C2×D4×C15112yes—
327C22×C4#41×C15112yes—
328C22×C4C24×C15111yescharacteristic
329C2×C8—28no—
330C2×C8—22no—
331C2×C8—22no—
332C2×C8—21no—
333C2×C8—21no—
334C2×C8—22no—
335C2×C8—22no—
336C2×C8#41×C15114yes—
337C4⋊C4—24no—
338C4⋊C4—21no—
339C4⋊C4—21no—
340C4⋊C4—22no—
341C4⋊C4—21no—
342C4⋊C4—22no—
343C4⋊C4—22no—
344C4⋊C4—21no—
345C4⋊C4—22no—
346C4⋊C4—42no—
347C4⋊C4—42no—
348C4⋊C4—82no—
349C4⋊C4C2×D4×C15112yes—
350C4⋊C4C2×D4×C15112yes—
351C22⋊C4—22no—
352C22⋊C4—22no—
353C22⋊C4—42no—
354C22⋊C4—42no—
355C42—22no—
356C42—22no—
357C42—82no—
358C23—22no—
359C23—22no—
360C23#121×C15112yes—
361C23#135×C15112yes—
362C23#137×C15112yes—
363C23C22×D4×C15111yescharacteristic
364Q8—41no—
365Q8—41no—
366Q8—42no—
367C2×C4—22no—
368C2×C4—22no—
369C2×C4—22no—
370C2×C4—24no—
371C2×C4—22no—
372C2×C4—22no—
373C2×C4—22no—
374C2×C4—22no—
375C2×C4—22no—
376C2×C4—22no—
377C2×C4—22no—
378C2×C4—22no—
379C2×C4—42no—
380C2×C4—42no—
381C2×C4—41no—
382C2×C4—41no—
383C2×C4—42no—
384C2×C4—42no—
385C2×C4—42no—
386C2×C4—42no—
387C2×C4—42no—
388C2×C4—42no—
389C2×C4—42no—
390C2×C4—42no—
391C2×C4—84no—
392C2×C4—84no—
393C2×C4#121×C15112yes—
394C2×C4C22×D4×C15111yescharacteristic
395C2×C4C2×#41×C15112yes—
396C2×C4C2×#41×C15111yescharacteristic
397C2×C42−1+4×C15112yes—
398C8—24no—
399C8—41no—
400C8—41no—
401C22—22no—
402C22—22no—
403C22—24no—
404C22—24no—
405C22#464×C15114yes—
406C22C2×#121×C15111yescharacteristic
407C22#547×C15112yes—
408C22C2×#135×C15111yescharacteristic
409C22C2×#137×C15111yescharacteristic
410C22#577×C15112yes—
411C4—22no—
412C4—42no—
413C4—42no—
414C4—42no—
415C4—42no—
416C4—81no—
417C4—82no—
418C4—81no—
419C4—82no—
420C4#525×C15111yescharacteristic
421C4#536×C15111yescharacteristic
422C4#552×C15112yes—
423C2—24no—
424C2#2413×C15111yescharacteristic
425C2C2×#464×C15111yescharacteristic
426C2#3042×C15111yescharacteristic
427C2#3337×C15112yes—
428C2#3344×C15112yes—
subgroupG/H
center, socleC23×C151C22×D4
commutator, Frattini, cosocleC22×C4C24×C151
prime radicalC24×C151C2×D4
LCSG ▷ C22×C4 ▷ C2 ▷ C1
UCSC1 ◁ C23×C151 ◁ C4×C22⋊C4×C151 ◁ G

Complements

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

NH ≅ G/Ntype
5. #2085×C151C2semidirect
6. #2066×C151C2semidirect
7. #1949×C151C2semidirect
8. #1927×C151C2semidirect
214. #24579C151direct (N × H)
215. C151#24579direct (N × H)
220. #2085C2×C151semidirect
221. #2066C2×C151semidirect
222. #1949C2×C151semidirect
223. #1927C2×C151semidirect

Maximal subgroups

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

subgroupindex
C2×#484×C1512
C22×#106×C1512
#2503×C1512
#2345×C1512
#2085×C1512
#2066×C1512
#1949×C1512
#1927×C1512
#1892×C1512
#1850×C1512
#1823×C1512
#24579151