Finite Group Databasebeta

← C151×#11757 #50316682

Subgroups — C151×#11757 #50316682

Order 38656.

700 conjugacy classes of subgroups; only the 106 normal subgroups are shown.

normal subgroups only

This group has 700 conjugacy classes of subgroups — too many to draw. Only its 106 normal subgroups are shown.

Dotted edges: containments with no intermediate normal subgroup — omitted non-normal subgroups may lie between.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#2539×C151C211yesmaximal, characteristic
2C2×#342×C151C211yesmaximal, characteristic, complemented
3#1462×C151C211yesmaximal, characteristic, complemented
4#1352×C151C212yesmaximal, complemented
5#1343×C151C212yesmaximal, complemented
6C2×#123×C151—21no—
7C2×#121×C151C411yescharacteristic, complemented
8C2×#120×C151—21no—
9C22×C22⋊C4×C151—21no—
10C22×C22⋊C4×C151C2211yescharacteristic
11#407×C151—21no—
12#399×C151C411yescharacteristic, complemented
13#399×C151C2211yescharacteristic
14#392×C151C411yescharacteristic, complemented
15#392×C151C2211yescharacteristic
16#390×C151—21no—
17#390×C151—21no—
18C4×C22⋊C4×C151C411yescharacteristic, complemented
19#342×C151—21no—
20#342×C151C2212yes—
21#323×C151—21no—
22#323×C151C2212yes—
23C25×C151D411yescharacteristic
24C22×D4×C151—21no—
25C22×D4×C151—21no—
26C22×D4×C151—21no—
27C22×D4×C151C2×C411yescharacteristic, complemented
28C23×C4×C151—41no—
29C23×C4×C151C2×C411yescharacteristic
30C23×C4×C151D411yescharacteristic
31C2×M16×C151—41no—
32#123×C151—44no—
33#121×C151—41no—
34#121×C151—41no—
35#120×C151—44no—
36C2×C4⋊C4×C151—21no—
37C2×C4⋊C4×C151—21no—
38C2×C4⋊C4×C151C2×C411yescharacteristic, complemented
39C2×C22⋊C4×C151—24no—
40C2×C22⋊C4×C151—21no—
41C2×C22⋊C4×C151—21no—
42C2×C22⋊C4×C151—21no—
43C2×C22⋊C4×C151—21no—
44C2×C22⋊C4×C151—21no—
45C2×C22⋊C4×C151—22no—
46C2×C22⋊C4×C151—21no—
47C2×C22⋊C4×C151—41no—
48C2×C22⋊C4×C151—41no—
49C2×C22⋊C4×C151—41no—
50C2×C22⋊C4×C151—42no—
51C2×C22⋊C4×C151C2×C411yescharacteristic
52C2×C22⋊C4×C151D411yescharacteristic
53C2×C22⋊C4×C151D411yescharacteristic
54C2×C22⋊C4×C151C2311yescharacteristic
55C2×C42×C151—21no—
56C22⋊C8×C151—22no—
57C22⋊C8×C151—41no—
58#95×C151C2×C411yescharacteristic, complemented
59#95×C151C2×C411yescharacteristic, complemented
60C24×C151—21no—
61C24×C151—21no—
62C24×C151—22no—
63C24×C151—21no—
64C24×C151—21no—
65C24×C151—21no—
66C24×C151—42no—
67C24×C151—81no—
68C24×C151C22⋊C411yescharacteristic
69C24×C151C22⋊C411yescharacteristic
70C24×C151C2×D411yescharacteristic
71C2×D4×C151—41no—
72C2×D4×C151—42no—
73C2×D4×C151—42no—
74C2×D4×C151—42no—
75C2×D4×C151—42no—
76C2×D4×C151—41no—
77C2×D4×C151—41no—
78C2×D4×C151—42no—
79C2×D4×C151—41no—
80C2×D4×C151—41no—
81C2×D4×C151—81no—
82C2×D4×C151—81no—
83C2×D4×C151—81no—
84C22×C4×C151—21no—
85C22×C4×C151—21no—
86C22×C4×C151—21no—
87C22×C4×C151—21no—
88C22×C4×C151—21no—
89C22×C4×C151—21no—
90C22×C4×C151—21no—
91C22×C4×C151—42no—
92C22×C4×C151—44no—
93C22×C4×C151—42no—
94C22×C4×C151—81no—
95C22×C4×C151—82no—
96C22×C4×C151—82no—
97C22×C4×C151C22⋊C411yescharacteristic
98C22×C4×C151C22⋊C411yescharacteristic
99C22×C4×C151C22×C411yescharacteristic
100C22×C4×C151C2×D411yescharacteristic
101M16×C151—82no—
102C2×C8×C151—42no—
103C4⋊C4×C151—41no—
104C4⋊C4×C151—42no—
105C4⋊C4×C151—42no—
106C22⋊C4×C151—24no—
107C22⋊C4×C151—24no—
108C22⋊C4×C151—41no—
109C22⋊C4×C151—41no—
110C22⋊C4×C151—41no—
111C22⋊C4×C151—41no—
112C22⋊C4×C151—42no—
113C22⋊C4×C151—42no—
114C22⋊C4×C151—42no—
115C22⋊C4×C151—42no—
116C22⋊C4×C151—42no—
117C22⋊C4×C151—44no—
118C22⋊C4×C151—42no—
119C42×C151—81no—
120C23×C151—22no—
121C23×C151—22no—
122C23×C151—22no—
123C23×C151—22no—
124C23×C151—22no—
125C23×C151—21no—
126C23×C151—21no—
127C23×C151—21no—
128C23×C151—21no—
129C23×C151—21no—
130C23×C151—21no—
131C23×C151—41no—
132C23×C151—42no—
133C23×C151—41no—
134C23×C151—41no—
135C23×C151—41no—
136C23×C151—41no—
137C23×C151—41no—
138C23×C151—41no—
139C23×C151—41no—
140C23×C151—41no—
141C23×C151—42no—
142C23×C151—41no—
143C23×C151—41no—
144C23×C151—41no—
145C23×C151—81no—
146C23×C151—81no—
147C23×C151—84no—
148C23×C151—82no—
149C23×C151—82no—
150C23×C151#9911yescharacteristic
151C23×C151#9911yescharacteristic
152C23×C151#10412yes—
153C23×C151C2×C22⋊C411yescharacteristic
154C23×C151AGL(1,ℤ/8)11yescharacteristic
155C23×C151#13711yescharacteristic
156D4×C151—84no—
157D4×C151—82no—
158D4×C151—82no—
159D4×C151—82no—
160D4×C151—82no—
161D4×C151—82no—
162C2×C4×C151—22no—
163C2×C4×C151—22no—
164C2×C4×C151—21no—
165C2×C4×C151—21no—
166C2×C4×C151—21no—
167C2×C4×C151—21no—
168C2×C4×C151—41no—
169C2×C4×C151—41no—
170C2×C4×C151—42no—
171C2×C4×C151—41no—
172C2×C4×C151—41no—
173C2×C4×C151—41no—
174C2×C4×C151—41no—
175C2×C4×C151—41no—
176C2×C4×C151—41no—
177C2×C4×C151—44no—
178C2×C4×C151—41no—
179C2×C4×C151—41no—
180C2×C4×C151—41no—
181C2×C4×C151—81no—
182C2×C4×C151—81no—
183C2×C4×C151—81no—
184C2×C4×C151—88no—
185C2×C4×C151—82no—
186C2×C4×C151—82no—
187C8×C151—82no—
188C22×C151—21no—
189C22×C151—21no—
190C22×C151—21no—
191C22×C151—21no—
192C22×C151—21no—
193C22×C151—21no—
194C22×C151—21no—
195C22×C151—21no—
196C22×C151—22no—
197C22×C151—21no—
198C22×C151—42no—
199C22×C151—42no—
200C22×C151—42no—
201C22×C151—41no—
202C22×C151—41no—
203C22×C151—44no—
204C22×C151—44no—
205C22×C151—41no—
206C22×C151—41no—
207C22×C151—41no—
208C22×C151—41no—
209C22×C151—81no—
210C22×C151—81no—
211C22×C151—81no—
212C22×C151—81no—
213C22×C151—81no—
214C22×C151—81no—
215C22×C151—82no—
216C22×C151—82no—
217C22×C151—81no—
218C22×C151—81no—
219C22×C151—81no—
220C22×C151#35312yes—
221C22×C151C2×#9911yescharacteristic
222C22×C151#41711yescharacteristic
223C22×C151C2×#10411yescharacteristic
224C4×C151—42no—
225C4×C151—42no—
226C4×C151—41no—
227C4×C151—41no—
228C4×C151—41no—
229C4×C151—41no—
230C4×C151—84no—
231C4×C151—81no—
232C2×C151—21no—
233C2×C151—21no—
234C2×C151—22no—
235C2×C151—41no—
236C2×C151—41no—
237C2×C151—42no—
238C2×C151—41no—
239C2×C151—161no—
240C2×C151#150411yescharacteristic
241C2×C151#212111yescharacteristic
242C2×C151C2×#35311yescharacteristic
243#11757C15111yesmaximal, characteristic, direct product factor, Sylow
244C151#1175711yescharacteristic, direct product factor, Sylow
245#2539C2×C15111yescharacteristic
246C2×#342C2×C15111yescharacteristic, complemented
247#1462C2×C15111yescharacteristic, complemented
248#1352C2×C15112yescomplemented
249#1343C2×C15112yescomplemented
250C2×#123—21no—
251C2×#121C4×C15111yescharacteristic, complemented
252C2×#120—21no—
253C22×C22⋊C4—21no—
254C22×C22⋊C4C22×C15111yescharacteristic
255#407—21no—
256#399C4×C15111yescharacteristic, complemented
257#399C22×C15111yescharacteristic
258#392C4×C15111yescharacteristic, complemented
259#392C22×C15111yescharacteristic
260#390—21no—
261#390—21no—
262C4×C22⋊C4C4×C15111yescharacteristic, complemented
263#342—21no—
264#342C22×C15112yes—
265#323—21no—
266#323C22×C15112yes—
267C25D4×C15111yescharacteristic
268C22×D4—21no—
269C22×D4—21no—
270C22×D4—21no—
271C22×D4C2×C4×C15111yescharacteristic, complemented
272C23×C4—41no—
273C23×C4C2×C4×C15111yescharacteristic
274C23×C4D4×C15111yescharacteristic
275C2×M16—41no—
276#123—44no—
277#121—41no—
278#121—41no—
279#120—44no—
280C2×C4⋊C4—21no—
281C2×C4⋊C4—21no—
282C2×C4⋊C4C2×C4×C15111yescharacteristic, complemented
283C2×C22⋊C4—21no—
284C2×C22⋊C4—24no—
285C2×C22⋊C4—22no—
286C2×C22⋊C4—21no—
287C2×C22⋊C4—21no—
288C2×C22⋊C4—21no—
289C2×C22⋊C4—21no—
290C2×C22⋊C4—21no—
291C2×C22⋊C4—41no—
292C2×C22⋊C4—41no—
293C2×C22⋊C4—41no—
294C2×C22⋊C4—42no—
295C2×C22⋊C4C2×C4×C15111yescharacteristic
296C2×C22⋊C4D4×C15111yescharacteristic
297C2×C22⋊C4D4×C15111yescharacteristic
298C2×C22⋊C4C23×C15111yescharacteristic
299C2×C42—21no—
300C22⋊C8—22no—
301C22⋊C8—41no—
302#95C2×C4×C15111yescharacteristic, complemented
303#95C2×C4×C15111yescharacteristic, complemented
304C24—21no—
305C24—21no—
306C24—22no—
307C24—21no—
308C24—21no—
309C24—21no—
310C24—42no—
311C24—81no—
312C24C22⋊C4×C15111yescharacteristic
313C24C22⋊C4×C15111yescharacteristic
314C24C2×D4×C15111yescharacteristic
315C2×D4—41no—
316C2×D4—42no—
317C2×D4—42no—
318C2×D4—42no—
319C2×D4—42no—
320C2×D4—41no—
321C2×D4—41no—
322C2×D4—42no—
323C2×D4—41no—
324C2×D4—41no—
325C2×D4—81no—
326C2×D4—81no—
327C2×D4—81no—
328C22×C4—21no—
329C22×C4—21no—
330C22×C4—21no—
331C22×C4—21no—
332C22×C4—21no—
333C22×C4—21no—
334C22×C4—21no—
335C22×C4—42no—
336C22×C4—44no—
337C22×C4—42no—
338C22×C4—81no—
339C22×C4—82no—
340C22×C4—82no—
341C22×C4C22⋊C4×C15111yescharacteristic
342C22×C4C22⋊C4×C15111yescharacteristic
343C22×C4C22×C4×C15111yescharacteristic
344C22×C4C2×D4×C15111yescharacteristic
345M16—82no—
346C2×C8—42no—
347C4⋊C4—41no—
348C4⋊C4—42no—
349C4⋊C4—42no—
350C22⋊C4—24no—
351C22⋊C4—24no—
352C22⋊C4—41no—
353C22⋊C4—41no—
354C22⋊C4—41no—
355C22⋊C4—41no—
356C22⋊C4—42no—
357C22⋊C4—42no—
358C22⋊C4—42no—
359C22⋊C4—42no—
360C22⋊C4—42no—
361C22⋊C4—44no—
362C22⋊C4—42no—
363C42—81no—
364C23—22no—
365C23—22no—
366C23—22no—
367C23—22no—
368C23—22no—
369C23—21no—
370C23—21no—
371C23—21no—
372C23—21no—
373C23—21no—
374C23—21no—
375C23—41no—
376C23—42no—
377C23—41no—
378C23—41no—
379C23—41no—
380C23—41no—
381C23—41no—
382C23—41no—
383C23—41no—
384C23—41no—
385C23—42no—
386C23—41no—
387C23—41no—
388C23—41no—
389C23—81no—
390C23—81no—
391C23—84no—
392C23—82no—
393C23—82no—
394C23#99×C15111yescharacteristic
395C23#99×C15111yescharacteristic
396C23#104×C15112yes—
397C23C2×C22⋊C4×C15111yescharacteristic
398C23AGL(1,ℤ/8)×C15111yescharacteristic
399C23#137×C15111yescharacteristic
400D4—84no—
401D4—82no—
402D4—82no—
403D4—82no—
404D4—82no—
405D4—82no—
406C2×C4—22no—
407C2×C4—22no—
408C2×C4—21no—
409C2×C4—21no—
410C2×C4—21no—
411C2×C4—21no—
412C2×C4—41no—
413C2×C4—41no—
414C2×C4—42no—
415C2×C4—41no—
416C2×C4—41no—
417C2×C4—41no—
418C2×C4—41no—
419C2×C4—41no—
420C2×C4—41no—
421C2×C4—44no—
422C2×C4—41no—
423C2×C4—41no—
424C2×C4—41no—
425C2×C4—81no—
426C2×C4—81no—
427C2×C4—81no—
428C2×C4—88no—
429C2×C4—82no—
430C2×C4—82no—
431C8—82no—
432C22—21no—
433C22—21no—
434C22—21no—
435C22—21no—
436C22—21no—
437C22—21no—
438C22—21no—
439C22—21no—
440C22—22no—
441C22—21no—
442C22—42no—
443C22—42no—
444C22—42no—
445C22—41no—
446C22—41no—
447C22—44no—
448C22—44no—
449C22—41no—
450C22—41no—
451C22—41no—
452C22—41no—
453C22—81no—
454C22—81no—
455C22—81no—
456C22—81no—
457C22—81no—
458C22—81no—
459C22—82no—
460C22—82no—
461C22—81no—
462C22—81no—
463C22—81no—
464C22#353×C15112yes—
465C22C2×#99×C15111yescharacteristic
466C22#417×C15111yescharacteristic
467C22C2×#104×C15111yescharacteristic
468C4—42no—
469C4—42no—
470C4—41no—
471C4—41no—
472C4—41no—
473C4—41no—
474C4—84no—
475C4—81no—
476C2—21no—
477C2—21no—
478C2—22no—
479C2—41no—
480C2—41no—
481C2—42no—
482C2—41no—
483C2—161no—
484C2#1504×C15111yescharacteristic
485C2#2121×C15111yescharacteristic
486C2C2×#353×C15111yescharacteristic
subgroupG/H
center, socleC22×C151C2×#99
commutatorC22×C4C22×C4×C151
Frattini, cosocleC2×C22⋊C4C23×C151
prime radical#2539×C151C2
LCSG ▷ C22×C4 ▷ C23 ▷ C2 ▷ C1
UCSC1 ◁ C22×C151 ◁ C24×C151 ◁ C22×C22⋊C4×C151 ◁ G

Complements

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

NH ≅ G/Ntype
2. C2×#342×C151C2semidirect
3. #1462×C151C2semidirect
4. #1352×C151C2semidirect
5. #1343×C151C2semidirect
7. C2×#121×C151C4semidirect
12. #399×C151C4semidirect
14. #392×C151C4semidirect
18. C4×C22⋊C4×C151C4semidirect
27. C22×D4×C151C2×C4semidirect
38. C2×C4⋊C4×C151C2×C4semidirect
58. #95×C151C2×C4semidirect
59. #95×C151C2×C4semidirect
243. #11757C151direct (N × H)
244. C151#11757direct (N × H)
246. C2×#342C2×C151semidirect
247. #1462C2×C151semidirect
248. #1352C2×C151semidirect
249. #1343C2×C151semidirect
251. C2×#121C4×C151semidirect
256. #399C4×C151semidirect
258. #392C4×C151semidirect
262. C4×C22⋊C4C4×C151semidirect
271. C22×D4C2×C4×C151semidirect
282. C2×C4⋊C4C2×C4×C151semidirect
302. #95C2×C4×C151semidirect
303. #95C2×C4×C151semidirect

Maximal subgroups

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

subgroupindex
#2539×C1512
C2×#342×C1512
#1462×C1512
#1352×C1512
#1343×C1512
#11757151