Finite Group Databasebeta

← C151×#18210 #50323135

Subgroups — C151×#18210 #50323135

Order 38656.

1338 conjugacy classes of subgroups.

The subgroup lattice has 460 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#3496×C151C211yesmaximal, characteristic, complemented
2#3100×C151C212yesmaximal, complemented
3#2535×C151C211yesmaximal, characteristic, complemented
4#1607×C151C214yesmaximal, complemented
5#1595×C151C214yesmaximal, complemented
6#1517×C151C212yesmaximal, complemented
7#1460×C151C211yesmaximal, characteristic, complemented
8C2×H(5,𝔽2)×C151—21no—
9#547×C151—22no—
10#546×C151—24no—
11D42×C151—22no—
12#537×C151—21no—
13#534×C151C2214yes—
14#532×C151C2212yes—
15C2×#121×C151—22no—
16C2×#121×C151—21no—
17C2×C4×D4×C151C2211yescharacteristic
18C2×C4×D4×C151C2211yescharacteristic, complemented
19#480×C151C2214yescomplemented
20#474×C151—22no—
21#459×C151—22no—
22#447×C151C2214yescomplemented
23C4×D8×C151—24no—
24#431×C151C2212yescomplemented
25#397×C151C2212yescomplemented
26#392×C151C2212yescomplemented
27#386×C151—22no—
28#382×C151C2211yescharacteristic, complemented
29#380×C151—21no—
30#327×C151C2218yescomplemented
31#323×C151C2214yescomplemented
32H(5,𝔽2)×C151—44no—
33H(5,𝔽2)×C151—44no—
34C2×#41×C151—24no—
35C2×#41×C151—22no—
36C22×D4×C151—24no—
37C22×D4×C151—22no—
38C22×D4×C151—24no—
39C22×D4×C151—22no—
40C22×D4×C151D411yescharacteristic
41C22×D4×C151C2311yescharacteristic
42C23×C4×C151—21no—
43C23×C4×C151D412yes—
44C2×D8×C151—24no—
45C4⋊Q8×C151D412yescomplemented
46#127×C151D412yescomplemented
47#124×C151—22no—
48#123×C151—28no—
49C22⋊Q8×C151—24no—
50#121×C151—24no—
51#121×C151—24no—
52#121×C151—24no—
53#121×C151—22no—
54#121×C151—24no—
55#121×C151—44no—
56#121×C151C2314yes—
57#120×C151—28no—
58C4×D4×C151—22no—
59C4×D4×C151—22no—
60C4×D4×C151—22no—
61C4×D4×C151—22no—
62C4×D4×C151—22no—
63C4×D4×C151D414yescomplemented
64C2×C4⋊C4×C151—22no—
65C2×C4⋊C4×C151—21no—
66C2×C4⋊C4×C151D411yescharacteristic
67C2×C4⋊C4×C151C2311yescharacteristic
68C2×C22⋊C4×C151—24no—
69C2×C22⋊C4×C151—21no—
70C2×C22⋊C4×C151—22no—
71C2×C22⋊C4×C151—22no—
72C2×C22⋊C4×C151C2312yes—
73C2×C42×C151C2311yescharacteristic
74#107×C151—24no—
75C4⋊C8×C151—22no—
76#102×C151—24no—
77#102×C151—24no—
78C22⋊C8×C151C2314yes—
79C4×C8×C151—22no—
80#95×C151—22no—
81#95×C151—21no—
82#95×C151C2312yes—
83C24×C151—24no—
84C24×C151—21no—
85C24×C151C2×D412yes—
86#41×C151—416no—
87#41×C151—44no—
88#41×C151—44no—
89C2×Q8×C151—22no—
90C2×D4×C151—28no—
91C2×D4×C151—24no—
92C2×D4×C151—24no—
93C2×D4×C151—22no—
94C2×D4×C151—22no—
95C2×D4×C151—22no—
96C2×D4×C151—22no—
97C2×D4×C151—22no—
98C2×D4×C151—42no—
99C2×D4×C151—416no—
100C2×D4×C151—48no—
101C2×D4×C151—48no—
102C2×D4×C151—44no—
103C2×D4×C151—44no—
104C2×D4×C151—84no—
105C2×D4×C151—82no—
106C2×D4×C151D814yescomplemented
107C2×D4×C151C2×D414yes—
108C22×C4×C151—24no—
109C22×C4×C151—22no—
110C22×C4×C151—24no—
111C22×C4×C151—22no—
112C22×C4×C151—22no—
113C22×C4×C151—21no—
114C22×C4×C151—42no—
115C22×C4×C151—42no—
116C22×C4×C151—42no—
117C22×C4×C151—82no—
118C22×C4×C151C2×D412yes—
119C22×C4×C151C2×D411yescharacteristic
120C22×C4×C151C2×D411yescharacteristic
121C22×C4×C151C2411yescharacteristic
122D8×C151—48no—
123C2×C8×C151—24no—
124C4⋊C4×C151—24no—
125C4⋊C4×C151—22no—
126C4⋊C4×C151—22no—
127C4⋊C4×C151—22no—
128C4⋊C4×C151—42no—
129C4⋊C4×C151C2×D414yes—
130C22⋊C4×C151—28no—
131C22⋊C4×C151—24no—
132C22⋊C4×C151—24no—
133C22⋊C4×C151—24no—
134C22⋊C4×C151—44no—
135C22⋊C4×C151—41no—
136C22⋊C4×C151—41no—
137C22⋊C4×C151—82no—
138C42×C151C2×D412yes—
139C42×C151C2×D412yes—
140C23×C151—22no—
141C23×C151—22no—
142C23×C151—22no—
143C23×C151—24no—
144C23×C151—42no—
145C23×C151—42no—
146C23×C151—41no—
147C23×C151—41no—
148C23×C151—41no—
149C23×C151—48no—
150C23×C151—41no—
151C23×C151—41no—
152C23×C151—41no—
153C23×C151—84no—
154C23×C151—84no—
155C23×C151C2×D814yes—
156C23×C151C22×D411yescharacteristic
157Q8×C151—44no—
158D4×C151—24no—
159D4×C151—28no—
160D4×C151—44no—
161D4×C151—42no—
162D4×C151—44no—
163D4×C151—416no—
164D4×C151—44no—
165D4×C151—44no—
166D4×C151—44no—
167D4×C151—84no—
168D4×C151—88no—
169C2×C4×C151—24no—
170C2×C4×C151—22no—
171C2×C4×C151—41no—
172C2×C4×C151—41no—
173C2×C4×C151—44no—
174C2×C4×C151—48no—
175C2×C4×C151—48no—
176C2×C4×C151—42no—
177C2×C4×C151—42no—
178C2×C4×C151—42no—
179C2×C4×C151—41no—
180C2×C4×C151—41no—
181C2×C4×C151—42no—
182C2×C4×C151—41no—
183C2×C4×C151—82no—
184C2×C4×C151—84no—
185C2×C4×C151#12011yescharacteristic
186C2×C4×C151#12011yescharacteristic
187C2×C4×C151#12012yes—
188C2×C4×C151C2×D812yes—
189C2×C4×C151AGL(1,ℤ/8)12yes—
190C2×C4×C151C22×D412yes—
191C8×C151—44no—
192C22×C151—21no—
193C22×C151—21no—
194C22×C151—24no—
195C22×C151—44no—
196C22×C151—42no—
197C22×C151—42no—
198C22×C151—42no—
199C22×C151—42no—
200C22×C151—44no—
201C22×C151—44no—
202C22×C151—42no—
203C22×C151—44no—
204C22×C151—82no—
205C22×C151—81no—
206C22×C151—81no—
207C22×C151—81no—
208C22×C151—81no—
209C22×C151—168no—
210C22×C151C2×#12011yescharacteristic
211C22×C151C22×D811yescharacteristic
212C22×C151C2×AGL(1,ℤ/8)11yescharacteristic
213C4×C151—21no—
214C4×C151—21no—
215C4×C151—22no—
216C4×C151—44no—
217C4×C151—42no—
218C4×C151—82no—
219C4×C151—81no—
220C4×C151#44714yes—
221C2×C151—21no—
222C2×C151—21no—
223C2×C151—44no—
224C2×C151—82no—
225C2×C151—84no—
226C2×C151C2×#44711yescharacteristic
227C2×C151#301911yescharacteristic
228C2×C151#302511yescharacteristic
229#18210C15111yesmaximal, characteristic, direct product factor, Sylow
230C151#1821011yescharacteristic, direct product factor, Sylow
231#3496C2×C15111yescharacteristic, complemented
232#3100C2×C15112yescomplemented
233#2535C2×C15111yescharacteristic, complemented
234#1607C2×C15114yescomplemented
235#1595C2×C15114yescomplemented
236#1517C2×C15112yescomplemented
237#1460C2×C15111yescharacteristic, complemented
238C2×H(5,𝔽2)—21no—
239#547—22no—
240#546—24no—
241D42—22no—
242#537—21no—
243#534C22×C15114yes—
244#532C22×C15112yes—
245C2×#121—22no—
246C2×#121—21no—
247C2×C4×D4C22×C15111yescharacteristic
248C2×C4×D4C22×C15111yescharacteristic, complemented
249#480C22×C15114yescomplemented
250#474—22no—
251#459—22no—
252#447C22×C15114yescomplemented
253C4×D8—24no—
254#431C22×C15112yescomplemented
255#397C22×C15112yescomplemented
256#392C22×C15112yescomplemented
257#386—22no—
258#382C22×C15111yescharacteristic, complemented
259#380—21no—
260#327C22×C15118yescomplemented
261#323C22×C15114yescomplemented
262H(5,𝔽2)—44no—
263H(5,𝔽2)—44no—
264C2×#41—24no—
265C2×#41—22no—
266C22×D4—24no—
267C22×D4—24no—
268C22×D4—22no—
269C22×D4—22no—
270C22×D4D4×C15111yescharacteristic
271C22×D4C23×C15111yescharacteristic
272C23×C4—21no—
273C23×C4D4×C15112yes—
274C2×D8—24no—
275C4⋊Q8D4×C15112yescomplemented
276#127D4×C15112yescomplemented
277#124—22no—
278#123—28no—
279C22⋊Q8—24no—
280#121—24no—
281#121—24no—
282#121—24no—
283#121—22no—
284#121—24no—
285#121—44no—
286#121C23×C15114yes—
287#120—28no—
288C4×D4—22no—
289C4×D4—22no—
290C4×D4—22no—
291C4×D4—22no—
292C4×D4—22no—
293C4×D4D4×C15114yescomplemented
294C2×C4⋊C4—22no—
295C2×C4⋊C4—21no—
296C2×C4⋊C4D4×C15111yescharacteristic
297C2×C4⋊C4C23×C15111yescharacteristic
298C2×C22⋊C4—24no—
299C2×C22⋊C4—22no—
300C2×C22⋊C4—21no—
301C2×C22⋊C4—22no—
302C2×C22⋊C4C23×C15112yes—
303C2×C42C23×C15111yescharacteristic
304#107—24no—
305C4⋊C8—22no—
306#102—24no—
307#102—24no—
308C22⋊C8C23×C15114yes—
309C4×C8—22no—
310#95—22no—
311#95—21no—
312#95C23×C15112yes—
313C24—24no—
314C24—21no—
315C24C2×D4×C15112yes—
316#41—416no—
317#41—44no—
318#41—44no—
319C2×Q8—22no—
320C2×D4—28no—
321C2×D4—24no—
322C2×D4—24no—
323C2×D4—22no—
324C2×D4—22no—
325C2×D4—22no—
326C2×D4—22no—
327C2×D4—22no—
328C2×D4—42no—
329C2×D4—416no—
330C2×D4—48no—
331C2×D4—48no—
332C2×D4—44no—
333C2×D4—44no—
334C2×D4—84no—
335C2×D4—82no—
336C2×D4D8×C15114yescomplemented
337C2×D4C2×D4×C15114yes—
338C22×C4—24no—
339C22×C4—22no—
340C22×C4—24no—
341C22×C4—22no—
342C22×C4—22no—
343C22×C4—21no—
344C22×C4—42no—
345C22×C4—42no—
346C22×C4—42no—
347C22×C4—82no—
348C22×C4C2×D4×C15112yes—
349C22×C4C2×D4×C15111yescharacteristic
350C22×C4C2×D4×C15111yescharacteristic
351C22×C4C24×C15111yescharacteristic
352D8—48no—
353C2×C8—24no—
354C4⋊C4—24no—
355C4⋊C4—22no—
356C4⋊C4—22no—
357C4⋊C4—22no—
358C4⋊C4—42no—
359C4⋊C4C2×D4×C15114yes—
360C22⋊C4—28no—
361C22⋊C4—24no—
362C22⋊C4—24no—
363C22⋊C4—24no—
364C22⋊C4—44no—
365C22⋊C4—41no—
366C22⋊C4—41no—
367C22⋊C4—82no—
368C42C2×D4×C15112yes—
369C42C2×D4×C15112yes—
370C23—22no—
371C23—22no—
372C23—22no—
373C23—24no—
374C23—42no—
375C23—42no—
376C23—41no—
377C23—41no—
378C23—41no—
379C23—48no—
380C23—41no—
381C23—41no—
382C23—41no—
383C23—84no—
384C23—84no—
385C23C2×D8×C15114yes—
386C23C22×D4×C15111yescharacteristic
387Q8—44no—
388D4—24no—
389D4—28no—
390D4—44no—
391D4—42no—
392D4—44no—
393D4—416no—
394D4—44no—
395D4—44no—
396D4—44no—
397D4—84no—
398D4—88no—
399C2×C4—22no—
400C2×C4—24no—
401C2×C4—41no—
402C2×C4—41no—
403C2×C4—44no—
404C2×C4—48no—
405C2×C4—48no—
406C2×C4—42no—
407C2×C4—42no—
408C2×C4—42no—
409C2×C4—41no—
410C2×C4—41no—
411C2×C4—42no—
412C2×C4—41no—
413C2×C4—82no—
414C2×C4—84no—
415C2×C4#120×C15111yescharacteristic
416C2×C4#120×C15111yescharacteristic
417C2×C4#120×C15112yes—
418C2×C4C2×D8×C15112yes—
419C2×C4AGL(1,ℤ/8)×C15112yes—
420C2×C4C22×D4×C15112yes—
421C8—44no—
422C22—21no—
423C22—21no—
424C22—24no—
425C22—44no—
426C22—42no—
427C22—42no—
428C22—42no—
429C22—42no—
430C22—44no—
431C22—44no—
432C22—42no—
433C22—44no—
434C22—82no—
435C22—81no—
436C22—81no—
437C22—81no—
438C22—81no—
439C22—168no—
440C22C2×#120×C15111yescharacteristic
441C22C22×D8×C15111yescharacteristic
442C22C2×AGL(1,ℤ/8)×C15111yescharacteristic
443C4—21no—
444C4—21no—
445C4—22no—
446C4—44no—
447C4—42no—
448C4—82no—
449C4—81no—
450C4#447×C15114yes—
451C2—21no—
452C2—21no—
453C2—44no—
454C2—82no—
455C2—84no—
456C2C2×#447×C15111yescharacteristic
457C2#3019×C15111yescharacteristic
458C2#3025×C15111yescharacteristic
subgroupG/H
center, socleC22×C151C2×#120
commutator, Frattini, cosocleC22×C4C24×C151
LCSG ▷ C22×C4 ▷ C22 ▷ C1
UCSC1 ◁ C22×C151 ◁ C2×C42×C151 ◁ G

Complements

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

NH ≅ G/Ntype
1. #3496×C151C2semidirect
2. #3100×C151C2semidirect
3. #2535×C151C2semidirect
4. #1607×C151C2semidirect
5. #1595×C151C2semidirect
6. #1517×C151C2semidirect
7. #1460×C151C2semidirect
18. C2×C4×D4×C151C22semidirect
19. #480×C151C22semidirect
22. #447×C151C22semidirect
24. #431×C151C22semidirect
25. #397×C151C22semidirect
26. #392×C151C22semidirect
28. #382×C151C22semidirect
30. #327×C151C22semidirect
31. #323×C151C22semidirect
45. C4⋊Q8×C151D4semidirect
46. #127×C151D4semidirect
63. C4×D4×C151D4semidirect
106. C2×D4×C151D8semidirect
229. #18210C151direct (N × H)
230. C151#18210direct (N × H)
231. #3496C2×C151semidirect
232. #3100C2×C151semidirect
233. #2535C2×C151semidirect
234. #1607C2×C151semidirect
235. #1595C2×C151semidirect
236. #1517C2×C151semidirect
237. #1460C2×C151semidirect
248. C2×C4×D4C22×C151semidirect
249. #480C22×C151semidirect
252. #447C22×C151semidirect
254. #431C22×C151semidirect
255. #397C22×C151semidirect
256. #392C22×C151semidirect
258. #382C22×C151semidirect
260. #327C22×C151semidirect
261. #323C22×C151semidirect
275. C4⋊Q8D4×C151semidirect
276. #127D4×C151semidirect
293. C4×D4D4×C151semidirect
336. C2×D4D8×C151semidirect

Maximal subgroups

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

subgroupindex
#3496×C1512
#3100×C1512
#2535×C1512
#1607×C1512
#1595×C1512
#1517×C1512
#1460×C1512
#18210151