Finite Group Databasebeta

← C151×#24578 #50329503

Subgroups — C151×#24578 #50329503

Order 38656.

896 conjugacy classes of subgroups.

The subgroup lattice has 466 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#481×C151C211yesmaximal, characteristic
2C22×#106×C151C211yesmaximal, characteristic, complemented
3#2501×C151C211yesmaximal, characteristic
4#2345×C151C211yesmaximal, characteristic, complemented
5#2084×C151C212yesmaximal, complemented
6#2065×C151C212yesmaximal, complemented
7#1948×C151C212yesmaximal, complemented
8#1926×C151C212yesmaximal, complemented
9#1893×C151C211yesmaximal, characteristic
10#1850×C151C211yesmaximal, characteristic, complemented
11#1823×C151C211yesmaximal, characteristic, complemented
12C23×C8×C151C2211yescharacteristic
13C2×#123×C151—21no—
14C2×#123×C151—21no—
15C2×#121×C151C2211yescharacteristic
16C22×C4⋊C4×C151C2211yescharacteristic
17C22×C4⋊C4×C151C2211yescharacteristic
18#481×C151—24no—
19C2×#106×C151—24no—
20C2×#106×C151C2211yescharacteristic
21C2×#106×C151C2212yes—
22C2×#106×C151C2212yes—
23C2×#106×C151C2211yescharacteristic
24C2×C4⋊C8×C151C2212yes—
25C2×#102×C151C2212yes—
26C2×#102×C151C2211yescharacteristic
27C2×#102×C151C2211yescharacteristic
28C2×C22⋊C8×C151C2211yescharacteristic
29#399×C151—22no—
30#390×C151C2212yes—
31#389×C151C2212yes—
32#387×C151—22no—
33#384×C151C2212yes—
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×D4×C151—21no—
44C22×D4×C151C2311yescharacteristic
45C23×C4×C151—21no—
46C23×C4×C151—21no—
47C23×C4×C151C2311yescharacteristic
48C22×C8×C151—22no—
49C22×C8×C151—24no—
50C22×C8×C151C2311yescharacteristic
51C22×C8×C151C2311yescharacteristic
52C22×C8×C151C2312yes—
53#123×C151—44no—
54#123×C151—44no—
55#121×C151—22no—
56#121×C151—22no—
57C2×C4⋊C4×C151—21no—
58C2×C4⋊C4×C151—22no—
59C2×C4⋊C4×C151—22no—
60C2×C4⋊C4×C151—22no—
61C2×C4⋊C4×C151—22no—
62C2×C4⋊C4×C151—22no—
63C2×C4⋊C4×C151—21no—
64C2×C4⋊C4×C151—22no—
65C2×C4⋊C4×C151D414yes—
66C2×C4⋊C4×C151C2312yes—
67C2×C4⋊C4×C151C2311yescharacteristic
68C2×C4⋊C4×C151C2312yes—
69C2×C4⋊C4×C151C2311yescharacteristic
70C2×C4⋊C4×C151C2311yescharacteristic
71C2×C22⋊C4×C151—22no—
72C2×C22⋊C4×C151—22no—
73C2×C22⋊C4×C151—21no—
74C2×C22⋊C4×C151—21no—
75C2×C22⋊C4×C151D411yescharacteristic
76C2×C22⋊C4×C151D411yescharacteristic
77C2×C42×C151—22no—
78C2×C42×C151C2312yes—
79#106×C151—24no—
80#106×C151—24no—
81#106×C151—22no—
82#106×C151—22no—
83C4⋊C8×C151—24no—
84#102×C151—24no—
85#102×C151—22no—
86#102×C151—22no—
87C22⋊C8×C151—22no—
88#95×C151—22no—
89#95×C151—22no—
90#95×C151—21no—
91#95×C151D412yes—
92C24×C151—21no—
93C24×C151C2×D411yescharacteristic
94C2×D4×C151—21no—
95C2×D4×C151—22no—
96C2×D4×C151—21no—
97C2×D4×C151—22no—
98C2×D4×C151—42no—
99C2×D4×C151—42no—
100C2×D4×C151—42no—
101C22×C4×C151—22no—
102C22×C4×C151—22no—
103C22×C4×C151—21no—
104C22×C4×C151—21no—
105C22×C4×C151—24no—
106C22×C4×C151—22no—
107C22×C4×C151—22no—
108C22×C4×C151—21no—
109C22×C4×C151—42no—
110C22×C4×C151—42no—
111C22×C4×C151—42no—
112C22×C4×C151—82no—
113C22×C4×C151C2×D412yes—
114C22×C4×C151C2×D411yescharacteristic
115C22×C4×C151C2×D412yes—
116C22×C4×C151C2×D412yes—
117C22×C4×C151#4112yes—
118C22×C4×C151C2411yescharacteristic
119C2×C8×C151—28no—
120C2×C8×C151—22no—
121C2×C8×C151—22no—
122C2×C8×C151—21no—
123C2×C8×C151—21no—
124C2×C8×C151—22no—
125C2×C8×C151—22no—
126C2×C8×C151#4114yes—
127C4⋊C4×C151—24no—
128C4⋊C4×C151—21no—
129C4⋊C4×C151—21no—
130C4⋊C4×C151—22no—
131C4⋊C4×C151—21no—
132C4⋊C4×C151—22no—
133C4⋊C4×C151—22no—
134C4⋊C4×C151—21no—
135C4⋊C4×C151—22no—
136C4⋊C4×C151—44no—
137C4⋊C4×C151—42no—
138C4⋊C4×C151—44no—
139C4⋊C4×C151—82no—
140C4⋊C4×C151C2×D412yes—
141C4⋊C4×C151C2×D412yes—
142C22⋊C4×C151—22no—
143C22⋊C4×C151—44no—
144C22⋊C4×C151—42no—
145C22⋊C4×C151—42no—
146C22⋊C4×C151—44no—
147C22⋊C4×C151SD1614yes—
148C42×C151—22no—
149C42×C151—22no—
150C42×C151—82no—
151C23×C151—22no—
152C23×C151—22no—
153C23×C151—41no—
154C23×C151—42no—
155C23×C151—41no—
156C23×C151—41no—
157C23×C151—42no—
158C23×C151#12112yes—
159C23×C151C2×SD1612yes—
160C23×C151AGL(1,ℤ/8)12yes—
161C23×C151C22×D411yescharacteristic
162D4×C151—41no—
163D4×C151—41no—
164D4×C151—42no—
165D4×C151—84no—
166C2×C4×C151—22no—
167C2×C4×C151—22no—
168C2×C4×C151—22no—
169C2×C4×C151—22no—
170C2×C4×C151—22no—
171C2×C4×C151—24no—
172C2×C4×C151—22no—
173C2×C4×C151—22no—
174C2×C4×C151—22no—
175C2×C4×C151—22no—
176C2×C4×C151—22no—
177C2×C4×C151—42no—
178C2×C4×C151—42no—
179C2×C4×C151—41no—
180C2×C4×C151—42no—
181C2×C4×C151—42no—
182C2×C4×C151—42no—
183C2×C4×C151—41no—
184C2×C4×C151—42no—
185C2×C4×C151—42no—
186C2×C4×C151—41no—
187C2×C4×C151—42no—
188C2×C4×C151—84no—
189C2×C4×C151—84no—
190C2×C4×C151#12112yes—
191C2×C4×C151C2×SD1614yes—
192C2×C4×C151C22×D411yescharacteristic
193C2×C4×C151C2×#4112yes—
194C2×C4×C151C2×#4111yescharacteristic
195C2×C4×C1512−1+412yes—
196C8×C151—24no—
197C8×C151—41no—
198C8×C151—41no—
199C22×C151—24no—
200C22×C151—24no—
201C22×C151—22no—
202C22×C151—22no—
203C22×C151—82no—
204C22×C151—82no—
205C22×C151—81no—
206C22×C151—81no—
207C22×C151—81no—
208C22×C151#46014yes—
209C22×C151C2×#12111yescharacteristic
210C22×C151#54712yes—
211C22×C151C22×SD1611yescharacteristic
212C22×C151C2×AGL(1,ℤ/8)11yescharacteristic
213C22×C151#57712yes—
214C4×C151—24no—
215C4×C151—22no—
216C4×C151—42no—
217C4×C151—42no—
218C4×C151—42no—
219C4×C151—82no—
220C4×C151—82no—
221C4×C151—81no—
222C4×C151#52511yescharacteristic
223C4×C151#53611yescharacteristic
224C4×C151#55212yes—
225C2×C151—24no—
226C2×C151—161no—
227C2×C151#241311yescharacteristic
228C2×C151C2×#46011yescharacteristic
229C2×C151#304211yescharacteristic
230C2×C151#333512yes—
231C2×C151#334512yes—
232#24578C15111yesmaximal, characteristic, direct product factor, Sylow
233C151#2457811yescharacteristic, direct product factor, Sylow
234C2×#481C2×C15111yescharacteristic
235C22×#106C2×C15111yescharacteristic, complemented
236#2501C2×C15111yescharacteristic
237#2345C2×C15111yescharacteristic, complemented
238#2084C2×C15112yescomplemented
239#2065C2×C15112yescomplemented
240#1948C2×C15112yescomplemented
241#1926C2×C15112yescomplemented
242#1893C2×C15111yescharacteristic
243#1850C2×C15111yescharacteristic, complemented
244#1823C2×C15111yescharacteristic, complemented
245C23×C8C22×C15111yescharacteristic
246C2×#123—21no—
247C2×#123—21no—
248C2×#121C22×C15111yescharacteristic
249C22×C4⋊C4C22×C15111yescharacteristic
250C22×C4⋊C4C22×C15111yescharacteristic
251#481—24no—
252C2×#106—24no—
253C2×#106C22×C15112yes—
254C2×#106C22×C15111yescharacteristic
255C2×#106C22×C15111yescharacteristic
256C2×#106C22×C15112yes—
257C2×C4⋊C8C22×C15112yes—
258C2×#102C22×C15111yescharacteristic
259C2×#102C22×C15112yes—
260C2×#102C22×C15111yescharacteristic
261C2×C22⋊C8C22×C15111yescharacteristic
262#399—22no—
263#390C22×C15112yes—
264#389C22×C15112yes—
265#387—22no—
266#384C22×C15112yes—
267#380—21no—
268#380C22×C15111yescharacteristic
269C4×C4⋊C4C22×C15112yes—
270C4×C22⋊C4—21no—
271C4×C22⋊C4C22×C15111yescharacteristic
272#340C22×C15112yes—
273#340C22×C15112yes—
274#340C22×C15112yes—
275#336C22×C15112yes—
276C22×D4—21no—
277C22×D4C23×C15111yescharacteristic
278C23×C4—21no—
279C23×C4—21no—
280C23×C4C23×C15111yescharacteristic
281C22×C8—22no—
282C22×C8—24no—
283C22×C8C23×C15111yescharacteristic
284C22×C8C23×C15111yescharacteristic
285C22×C8C23×C15112yes—
286#123—44no—
287#123—44no—
288#121—22no—
289#121—22no—
290C2×C4⋊C4—22no—
291C2×C4⋊C4—21no—
292C2×C4⋊C4—22no—
293C2×C4⋊C4—22no—
294C2×C4⋊C4—21no—
295C2×C4⋊C4—22no—
296C2×C4⋊C4—22no—
297C2×C4⋊C4—22no—
298C2×C4⋊C4D4×C15114yes—
299C2×C4⋊C4C23×C15112yes—
300C2×C4⋊C4C23×C15111yescharacteristic
301C2×C4⋊C4C23×C15112yes—
302C2×C4⋊C4C23×C15111yescharacteristic
303C2×C4⋊C4C23×C15111yescharacteristic
304C2×C22⋊C4—21no—
305C2×C22⋊C4—21no—
306C2×C22⋊C4—22no—
307C2×C22⋊C4—22no—
308C2×C22⋊C4D4×C15111yescharacteristic
309C2×C22⋊C4D4×C15111yescharacteristic
310C2×C42—22no—
311C2×C42C23×C15112yes—
312#106—24no—
313#106—24no—
314#106—22no—
315#106—22no—
316C4⋊C8—24no—
317#102—24no—
318#102—22no—
319#102—22no—
320C22⋊C8—22no—
321#95—22no—
322#95—22no—
323#95—21no—
324#95D4×C15112yes—
325C24—21no—
326C24C2×D4×C15111yescharacteristic
327C2×D4—21no—
328C2×D4—22no—
329C2×D4—21no—
330C2×D4—22no—
331C2×D4—42no—
332C2×D4—42no—
333C2×D4—42no—
334C22×C4—22no—
335C22×C4—22no—
336C22×C4—21no—
337C22×C4—21no—
338C22×C4—24no—
339C22×C4—22no—
340C22×C4—22no—
341C22×C4—21no—
342C22×C4—42no—
343C22×C4—42no—
344C22×C4—42no—
345C22×C4—82no—
346C22×C4C2×D4×C15112yes—
347C22×C4C2×D4×C15111yescharacteristic
348C22×C4C2×D4×C15112yes—
349C22×C4C2×D4×C15112yes—
350C22×C4#41×C15112yes—
351C22×C4C24×C15111yescharacteristic
352C2×C8—28no—
353C2×C8—22no—
354C2×C8—22no—
355C2×C8—21no—
356C2×C8—21no—
357C2×C8—22no—
358C2×C8—22no—
359C2×C8#41×C15114yes—
360C4⋊C4—24no—
361C4⋊C4—21no—
362C4⋊C4—21no—
363C4⋊C4—22no—
364C4⋊C4—21no—
365C4⋊C4—22no—
366C4⋊C4—22no—
367C4⋊C4—21no—
368C4⋊C4—22no—
369C4⋊C4—44no—
370C4⋊C4—42no—
371C4⋊C4—44no—
372C4⋊C4—82no—
373C4⋊C4C2×D4×C15112yes—
374C4⋊C4C2×D4×C15112yes—
375C22⋊C4—22no—
376C22⋊C4—44no—
377C22⋊C4—42no—
378C22⋊C4—42no—
379C22⋊C4—44no—
380C22⋊C4SD16×C15114yes—
381C42—22no—
382C42—22no—
383C42—82no—
384C23—22no—
385C23—22no—
386C23—41no—
387C23—42no—
388C23—41no—
389C23—41no—
390C23—42no—
391C23#121×C15112yes—
392C23C2×SD16×C15112yes—
393C23AGL(1,ℤ/8)×C15112yes—
394C23C22×D4×C15111yescharacteristic
395D4—41no—
396D4—41no—
397D4—42no—
398D4—84no—
399C2×C4—22no—
400C2×C4—22no—
401C2×C4—22no—
402C2×C4—22no—
403C2×C4—22no—
404C2×C4—24no—
405C2×C4—22no—
406C2×C4—22no—
407C2×C4—22no—
408C2×C4—22no—
409C2×C4—22no—
410C2×C4—42no—
411C2×C4—42no—
412C2×C4—41no—
413C2×C4—42no—
414C2×C4—42no—
415C2×C4—42no—
416C2×C4—41no—
417C2×C4—42no—
418C2×C4—42no—
419C2×C4—41no—
420C2×C4—42no—
421C2×C4—84no—
422C2×C4—84no—
423C2×C4#121×C15112yes—
424C2×C4C2×SD16×C15114yes—
425C2×C4C22×D4×C15111yescharacteristic
426C2×C4C2×#41×C15112yes—
427C2×C4C2×#41×C15111yescharacteristic
428C2×C42−1+4×C15112yes—
429C8—24no—
430C8—41no—
431C8—41no—
432C22—22no—
433C22—22no—
434C22—24no—
435C22—24no—
436C22—81no—
437C22—81no—
438C22—81no—
439C22—82no—
440C22—82no—
441C22#460×C15114yes—
442C22C2×#121×C15111yescharacteristic
443C22#547×C15112yes—
444C22C22×SD16×C15111yescharacteristic
445C22C2×AGL(1,ℤ/8)×C15111yescharacteristic
446C22#577×C15112yes—
447C4—22no—
448C4—24no—
449C4—42no—
450C4—42no—
451C4—42no—
452C4—82no—
453C4—81no—
454C4—82no—
455C4#525×C15111yescharacteristic
456C4#536×C15111yescharacteristic
457C4#552×C15112yes—
458C2—24no—
459C2—161no—
460C2#2413×C15111yescharacteristic
461C2C2×#460×C15111yescharacteristic
462C2#3042×C15111yescharacteristic
463C2#3335×C15112yes—
464C2#3345×C15112yes—
subgroupG/H
center, socleC23×C151C22×D4
commutator, Frattini, cosocleC22×C4C24×C151
prime radicalC2×#121×C151C22
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
2. C22×#106×C151C2semidirect
4. #2345×C151C2semidirect
5. #2084×C151C2semidirect
6. #2065×C151C2semidirect
7. #1948×C151C2semidirect
8. #1926×C151C2semidirect
10. #1850×C151C2semidirect
11. #1823×C151C2semidirect
232. #24578C151direct (N × H)
233. C151#24578direct (N × H)
235. C22×#106C2×C151semidirect
237. #2345C2×C151semidirect
238. #2084C2×C151semidirect
239. #2065C2×C151semidirect
240. #1948C2×C151semidirect
241. #1926C2×C151semidirect
243. #1850C2×C151semidirect
244. #1823C2×C151semidirect

Maximal subgroups

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

subgroupindex
C2×#481×C1512
C22×#106×C1512
#2501×C1512
#2345×C1512
#2084×C1512
#2065×C1512
#1948×C1512
#1926×C1512
#1893×C1512
#1850×C1512
#1823×C1512
#24578151