Finite Group Databasebeta

← C151×#24113 #50329038

Subgroups — C151×#24113 #50329038

Order 38656.

1122 conjugacy classes of subgroups.

The subgroup lattice has 408 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#3491×C151C211yesmaximal, characteristic
2#2853×C151C212yesmaximal, complemented
3#2296×C151C211yesmaximal, characteristic, complemented
4#2062×C151C212yesmaximal, complemented
5#2044×C151C212yesmaximal
6#1899×C151C212yesmaximal, complemented
7#1867×C151C214yesmaximal, complemented
8#1820×C151C211yesmaximal, characteristic, complemented
9#552×C151—24no—
10#551×C151—24no—
11#543×C151C2211yescharacteristic
12#539×C151—22no—
13#534×C151C2211yescharacteristic
14#529×C151—22no—
15C2×#123×C151C2212yes—
16#518×C151C2212yes—
17C2×#117×C151C2211yescharacteristic
18C2×#117×C151C2211yescharacteristic
19C22×C42×C151C2211yescharacteristic
20C2×#99×C151C2214yes—
21#389×C151—22no—
22#388×C151—24no—
23#387×C151—24no—
24#386×C151C2214yes—
25#385×C151C2214yes—
26#385×C151C2212yes—
27#385×C151C2212yes—
28#381×C151C2212yes—
29C4×C4⋊C4×C151—22no—
30#342×C151C2212yes—
31#342×C151C2212yes—
32#342×C151C2214yes—
33C22×D4×C151C2312yes—
34C23×C4×C151C2312yes—
35C23×C4×C151C2311yescharacteristic
36#126×C151—24no—
37#126×C151—24no—
38#125×C151—22no—
39#125×C151—24no—
40#124×C151—22no—
41#123×C151—24no—
42#123×C151—22no—
43#123×C151—22no—
44#123×C151—24no—
45#123×C151—22no—
46C22⋊Q8×C151—24no—
47C22⋊Q8×C151—22no—
48C22⋊Q8×C151—24no—
49#121×C151—22no—
50#120×C151—22no—
51C4×Q8×C151—22no—
52C4×D4×C151—22no—
53C4×D4×C151—24no—
54C4×D4×C151—24no—
55#117×C151—24no—
56#117×C151—24no—
57#117×C151—22no—
58#117×C151—24no—
59C2×C4⋊C4×C151—21no—
60C2×C4⋊C4×C151—22no—
61C2×C4⋊C4×C151—22no—
62C2×C4⋊C4×C151—24no—
63C2×C4⋊C4×C151D414yes—
64C2×C4⋊C4×C151D412yes—
65C2×C22⋊C4×C151—22no—
66C2×C22⋊C4×C151—24no—
67C2×C22⋊C4×C151D414yes—
68C2×C22⋊C4×C151C2314yes—
69C2×C22⋊C4×C151C2311yescharacteristic
70C2×C22⋊C4×C151C2311yescharacteristic
71C2×C22⋊C4×C151C2312yes—
72C2×C22⋊C4×C151C2312yes—
73C2×C42×C151—21no—
74C2×C42×C151—21no—
75C2×C42×C151—24no—
76C2×C42×C151—21no—
77C2×C42×C151D412yes—
78#99×C151—28no—
79#95×C151—22no—
80#95×C151—22no—
81#95×C151—24no—
82#95×C151—24no—
83#95×C151—24no—
84C24×C151C2×D412yes—
85C24×C151C2411yescharacteristic
86C2×Q8×C151—22no—
87C2×D4×C151—22no—
88C2×D4×C151—22no—
89C2×D4×C151—24no—
90C2×D4×C151—24no—
91C2×D4×C151—22no—
92C22×C4×C151—22no—
93C22×C4×C151—22no—
94C22×C4×C151—24no—
95C22×C4×C151—22no—
96C22×C4×C151—22no—
97C22×C4×C151—24no—
98C22×C4×C151—24no—
99C22×C4×C151—44no—
100C22×C4×C151C2×D412yes—
101C22×C4×C151C2×D414yes—
102C22×C4×C151C2×D411yescharacteristic
103C22×C4×C151C2×D411yescharacteristic
104C22×C4×C151C2×D412yes—
105C22×C4×C151C2×D412yes—
106C22×C4×C151C2×D414yes—
107C4⋊C4×C151—24no—
108C4⋊C4×C151—24no—
109C4⋊C4×C151—24no—
110C4⋊C4×C151—22no—
111C4⋊C4×C151—22no—
112C4⋊C4×C151—24no—
113C4⋊C4×C151—24no—
114C4⋊C4×C151—24no—
115C4⋊C4×C151—48no—
116C4⋊C4×C151—44no—
117C4⋊C4×C151—44no—
118C22⋊C4×C151—24no—
119C22⋊C4×C151—24no—
120C22⋊C4×C151—22no—
121C22⋊C4×C151—22no—
122C22⋊C4×C151—22no—
123C22⋊C4×C151—22no—
124C22⋊C4×C151—22no—
125C22⋊C4×C151—24no—
126C22⋊C4×C151—24no—
127C22⋊C4×C151—24no—
128C22⋊C4×C151—24no—
129C22⋊C4×C151—48no—
130C22⋊C4×C151#4114yes—
131C42×C151—24no—
132C42×C151—24no—
133C42×C151—22no—
134C42×C151—22no—
135C42×C151—42no—
136C42×C151—42no—
137C23×C151—22no—
138C23×C151—24no—
139C23×C151—42no—
140C23×C151—44no—
141C23×C151C22×D411yescharacteristic
142C23×C151C22×D412yes—
143C23×C151C2×#4112yes—
144C23×C1512−1+412yes—
145Q8×C151—82no—
146D4×C151—42no—
147D4×C151—42no—
148D4×C151—44no—
149D4×C151—82no—
150C2×C4×C151—21no—
151C2×C4×C151—21no—
152C2×C4×C151—21no—
153C2×C4×C151—21no—
154C2×C4×C151—28no—
155C2×C4×C151—28no—
156C2×C4×C151—24no—
157C2×C4×C151—22no—
158C2×C4×C151—24no—
159C2×C4×C151—22no—
160C2×C4×C151—22no—
161C2×C4×C151—24no—
162C2×C4×C151—24no—
163C2×C4×C151—42no—
164C2×C4×C151—42no—
165C2×C4×C151—42no—
166C2×C4×C151—44no—
167C2×C4×C151—42no—
168C2×C4×C151—42no—
169C2×C4×C151—44no—
170C2×C4×C151—42no—
171C2×C4×C151—44no—
172C2×C4×C151—44no—
173C2×C4×C151—44no—
174C2×C4×C151—84no—
175C2×C4×C151#12012yes—
176C2×C4×C151#12012yes—
177C22×C151—22no—
178C22×C151—24no—
179C22×C151—42no—
180C22×C151—42no—
181C22×C151—42no—
182C22×C151—44no—
183C22×C151—84no—
184C22×C151C2×#12011yescharacteristic
185C22×C151#52511yescharacteristic
186C22×C151#53611yescharacteristic
187C22×C151#54714yes—
188C4×C151—22no—
189C4×C151—22no—
190C4×C151—44no—
191C4×C151—42no—
192C4×C151—44no—
193C4×C151—42no—
194C4×C151—44no—
195C4×C151—42no—
196C4×C151—84no—
197C2×C151—21no—
198C2×C151—24no—
199C2×C151—21no—
200C2×C151—82no—
201C2×C151#240611yescharacteristic
202C2×C151#302512yes—
203#24113C15111yesmaximal, characteristic, direct product factor, Sylow
204C151#2411311yescharacteristic, direct product factor, Sylow
205#3491C2×C15111yescharacteristic
206#2853C2×C15112yescomplemented
207#2296C2×C15111yescharacteristic, complemented
208#2062C2×C15112yescomplemented
209#2044C2×C15112yes—
210#1899C2×C15112yescomplemented
211#1867C2×C15114yescomplemented
212#1820C2×C15111yescharacteristic, complemented
213#552—24no—
214#551—24no—
215#543C22×C15111yescharacteristic
216#539—22no—
217#534C22×C15111yescharacteristic
218#529—22no—
219C2×#123C22×C15112yes—
220#518C22×C15112yes—
221C2×#117C22×C15111yescharacteristic
222C2×#117C22×C15111yescharacteristic
223C22×C42C22×C15111yescharacteristic
224C2×#99C22×C15114yes—
225#389—22no—
226#388—24no—
227#387—24no—
228#386C22×C15114yes—
229#385C22×C15114yes—
230#385C22×C15112yes—
231#385C22×C15112yes—
232#381C22×C15112yes—
233C4×C4⋊C4—22no—
234#342C22×C15112yes—
235#342C22×C15114yes—
236#342C22×C15112yes—
237C22×D4C23×C15112yes—
238C23×C4C23×C15111yescharacteristic
239C23×C4C23×C15112yes—
240#126—24no—
241#126—24no—
242#125—22no—
243#125—24no—
244#124—22no—
245#123—24no—
246#123—22no—
247#123—22no—
248#123—24no—
249#123—22no—
250C22⋊Q8—24no—
251C22⋊Q8—22no—
252C22⋊Q8—24no—
253#121—22no—
254#120—22no—
255C4×Q8—22no—
256C4×D4—22no—
257C4×D4—24no—
258C4×D4—24no—
259#117—24no—
260#117—24no—
261#117—22no—
262#117—24no—
263C2×C4⋊C4—21no—
264C2×C4⋊C4—22no—
265C2×C4⋊C4—22no—
266C2×C4⋊C4—24no—
267C2×C4⋊C4D4×C15112yes—
268C2×C4⋊C4D4×C15114yes—
269C2×C22⋊C4—22no—
270C2×C22⋊C4—24no—
271C2×C22⋊C4D4×C15114yes—
272C2×C22⋊C4C23×C15114yes—
273C2×C22⋊C4C23×C15111yescharacteristic
274C2×C22⋊C4C23×C15111yescharacteristic
275C2×C22⋊C4C23×C15112yes—
276C2×C22⋊C4C23×C15112yes—
277C2×C42—21no—
278C2×C42—21no—
279C2×C42—24no—
280C2×C42—21no—
281C2×C42D4×C15112yes—
282#99—28no—
283#95—22no—
284#95—22no—
285#95—24no—
286#95—24no—
287#95—24no—
288C24C2×D4×C15112yes—
289C24C24×C15111yescharacteristic
290C2×Q8—22no—
291C2×D4—22no—
292C2×D4—22no—
293C2×D4—24no—
294C2×D4—24no—
295C2×D4—22no—
296C22×C4—22no—
297C22×C4—22no—
298C22×C4—24no—
299C22×C4—22no—
300C22×C4—22no—
301C22×C4—24no—
302C22×C4—24no—
303C22×C4—44no—
304C22×C4C2×D4×C15112yes—
305C22×C4C2×D4×C15114yes—
306C22×C4C2×D4×C15111yescharacteristic
307C22×C4C2×D4×C15111yescharacteristic
308C22×C4C2×D4×C15112yes—
309C22×C4C2×D4×C15112yes—
310C22×C4C2×D4×C15114yes—
311C4⋊C4—24no—
312C4⋊C4—24no—
313C4⋊C4—24no—
314C4⋊C4—22no—
315C4⋊C4—22no—
316C4⋊C4—24no—
317C4⋊C4—24no—
318C4⋊C4—24no—
319C4⋊C4—48no—
320C4⋊C4—44no—
321C4⋊C4—44no—
322C22⋊C4—24no—
323C22⋊C4—24no—
324C22⋊C4—22no—
325C22⋊C4—22no—
326C22⋊C4—22no—
327C22⋊C4—22no—
328C22⋊C4—22no—
329C22⋊C4—24no—
330C22⋊C4—24no—
331C22⋊C4—24no—
332C22⋊C4—24no—
333C22⋊C4—48no—
334C22⋊C4#41×C15114yes—
335C42—24no—
336C42—24no—
337C42—22no—
338C42—22no—
339C42—42no—
340C42—42no—
341C23—22no—
342C23—24no—
343C23—42no—
344C23—44no—
345C23C22×D4×C15111yescharacteristic
346C23C22×D4×C15112yes—
347C23C2×#41×C15112yes—
348C232−1+4×C15112yes—
349Q8—82no—
350D4—42no—
351D4—42no—
352D4—44no—
353D4—82no—
354C2×C4—21no—
355C2×C4—21no—
356C2×C4—21no—
357C2×C4—21no—
358C2×C4—28no—
359C2×C4—28no—
360C2×C4—24no—
361C2×C4—22no—
362C2×C4—22no—
363C2×C4—22no—
364C2×C4—24no—
365C2×C4—24no—
366C2×C4—24no—
367C2×C4—42no—
368C2×C4—42no—
369C2×C4—42no—
370C2×C4—44no—
371C2×C4—42no—
372C2×C4—42no—
373C2×C4—44no—
374C2×C4—42no—
375C2×C4—44no—
376C2×C4—44no—
377C2×C4—44no—
378C2×C4—84no—
379C2×C4#120×C15112yes—
380C2×C4#120×C15112yes—
381C22—22no—
382C22—24no—
383C22—42no—
384C22—42no—
385C22—42no—
386C22—44no—
387C22—84no—
388C22C2×#120×C15111yescharacteristic
389C22#525×C15111yescharacteristic
390C22#536×C15111yescharacteristic
391C22#547×C15114yes—
392C4—22no—
393C4—22no—
394C4—44no—
395C4—42no—
396C4—44no—
397C4—42no—
398C4—44no—
399C4—42no—
400C4—84no—
401C2—21no—
402C2—24no—
403C2—21no—
404C2—82no—
405C2#2406×C15111yescharacteristic
406C2#3025×C15112yes—
subgroupG/H
center, socleC22×C151C2×#120
commutator, Frattini, cosocleC24C24×C151
prime radical#534×C151C22
LCSG ▷ C24 ▷ C2 ▷ C1
UCSC1 ◁ C22×C151 ◁ C23×C4×C151 ◁ G

Complements

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

NH ≅ G/Ntype
2. #2853×C151C2semidirect
3. #2296×C151C2semidirect
4. #2062×C151C2semidirect
6. #1899×C151C2semidirect
7. #1867×C151C2semidirect
8. #1820×C151C2semidirect
203. #24113C151direct (N × H)
204. C151#24113direct (N × H)
206. #2853C2×C151semidirect
207. #2296C2×C151semidirect
208. #2062C2×C151semidirect
210. #1899C2×C151semidirect
211. #1867C2×C151semidirect
212. #1820C2×C151semidirect

Maximal subgroups

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

subgroupindex
#3491×C1512
#2853×C1512
#2296×C1512
#2062×C1512
#2044×C1512
#1899×C1512
#1867×C1512
#1820×C1512
#24113151