Finite Group Databasebeta

← C151×#22513 #50327438

Subgroups — C151×#22513 #50327438

Order 38656.

1114 conjugacy classes of subgroups.

The subgroup lattice has 528 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#554×C151C211yesmaximal, characteristic
2#2345×C151C212yesmaximal, complemented
3#1996×C151C212yesmaximal, complemented
4#1996×C151C214yesmaximal, complemented
5#1799×C151C212yesmaximal
6#1799×C151C211yesmaximal, characteristic
7C4×#99×C151C211yesmaximal, characteristic
8C4×#99×C151C212yesmaximal
9#554×C151—22no—
10#554×C151—24no—
11#554×C151—22no—
12C2×C4⋊Q8×C151C411yescharacteristic, complemented
13C2×#125×C151C412yescomplemented
14C2×C22⋊Q8×C151C412yescomplemented
15C2×C22⋊Q8×C151C411yescharacteristic, complemented
16C2×C22⋊Q8×C151C2212yes—
17C2×C22⋊Q8×C151C2211yescharacteristic
18C2×C4×Q8×C151C411yescharacteristic, complemented
19C2×C4×D4×C151C2212yes—
20C2×C4×D4×C151C2211yescharacteristic
21C22×C4⋊C4×C151C411yescharacteristic, complemented
22C22×C4⋊C4×C151C2211yescharacteristic
23C2×#99×C151C2211yescharacteristic
24C2×#99×C151C2212yes—
25C2×#99×C151C2211yescharacteristic
26#389×C151—22no—
27#387×C151—24no—
28#380×C151C2214yes—
29#380×C151C2212yes—
30C4×C4⋊C4×C151—22no—
31C4×C22⋊C4×C151C2212yes—
32C4×C22⋊C4×C151C2214yes—
33#342×C151C2214yes—
34#342×C151C2212yes—
35#342×C151C2214yes—
36#342×C151C2212yes—
37C22×Q8×C151C2×C411yescharacteristic
38C22×Q8×C151C2×C412yes—
39C22×D4×C151C2311yescharacteristic
40C23×C4×C151C2×C411yescharacteristic
41C23×C4×C151C2×C412yes—
42C23×C4×C151C2311yescharacteristic
43C23×C4×C151C2312yes—
44C4⋊Q8×C151—22no—
45C4⋊Q8×C151—22no—
46#125×C151—24no—
47#125×C151—24no—
48C22⋊Q8×C151—22no—
49C22⋊Q8×C151—22no—
50C22⋊Q8×C151—24no—
51C22⋊Q8×C151—24no—
52C22⋊Q8×C151—22no—
53C22⋊Q8×C151—22no—
54C22⋊Q8×C151—24no—
55C22⋊Q8×C151—24no—
56C4×Q8×C151—22no—
57C4×Q8×C151—21no—
58C4×Q8×C151—21no—
59C4×D4×C151—24no—
60C4×D4×C151—24no—
61C4×D4×C151—22no—
62C4×D4×C151—22no—
63C2×C4⋊C4×C151—22no—
64C2×C4⋊C4×C151—21no—
65C2×C4⋊C4×C151—21no—
66C2×C4⋊C4×C151—22no—
67C2×C4⋊C4×C151—24no—
68C2×C4⋊C4×C151C2×C412yes—
69C2×C4⋊C4×C151C2×C411yescharacteristic
70C2×C4⋊C4×C151C2×C411yescharacteristic
71C2×C4⋊C4×C151C2×C412yes—
72C2×C4⋊C4×C151C2×C412yes—
73C2×C4⋊C4×C151C2×C412yes—
74C2×C4⋊C4×C151C2×C412yes—
75C2×C4⋊C4×C151C2×C411yescharacteristic
76C2×C4⋊C4×C151C2×C412yes—
77C2×C4⋊C4×C151C2×C411yescharacteristic
78C2×C4⋊C4×C151D412yes—
79C2×C4⋊C4×C151D414yes—
80C2×C4⋊C4×C151D412yes—
81C2×C22⋊C4×C151C2×C412yes—
82C2×C22⋊C4×C151C2×C411yescharacteristic
83C2×C22⋊C4×C151C2312yes—
84C2×C22⋊C4×C151C2314yes—
85C2×C22⋊C4×C151C2312yes—
86C2×C22⋊C4×C151C2311yescharacteristic
87C2×C22⋊C4×C151C2312yes—
88C2×C42×C151—22no—
89C2×C42×C151—24no—
90C2×C42×C151C2×C412yes—
91C2×C42×C151C2×C411yescharacteristic
92#99×C151—22no—
93#99×C151—24no—
94#99×C151Q814yescomplemented
95#95×C151—22no—
96#95×C151—24no—
97#95×C151—24no—
98#95×C151—22no—
99C24×C151C22×C411yescharacteristic
100C24×C151C2411yescharacteristic
101C2×Q8×C151—21no—
102C2×Q8×C151—21no—
103C2×Q8×C151—24no—
104C2×Q8×C151—42no—
105C2×Q8×C151—82no—
106C2×D4×C151—22no—
107C2×D4×C151—22no—
108C2×D4×C151C2×Q812yes—
109C2×D4×C151#4112yes—
110C22×C4×C151—22no—
111C22×C4×C151—21no—
112C22×C4×C151—21no—
113C22×C4×C151—22no—
114C22×C4×C151—22no—
115C22×C4×C151—22no—
116C22×C4×C151—24no—
117C22×C4×C151—42no—
118C22×C4×C151—42no—
119C22×C4×C151—82no—
120C22×C4×C151—82no—
121C22×C4×C151C22×C411yescharacteristic
122C22×C4×C151C22×C411yescharacteristic
123C22×C4×C151C22×C412yes—
124C22×C4×C151C22×C411yescharacteristic
125C22×C4×C151C22×C411yescharacteristic
126C22×C4×C151C22×C412yes—
127C22×C4×C151C22×C412yes—
128C22×C4×C151C22×C411yescharacteristic
129C22×C4×C151C22×C412yes—
130C22×C4×C151C2×D412yes—
131C22×C4×C151C2×D412yes—
132C22×C4×C151C2×D414yes—
133C22×C4×C151C2×D414yes—
134C4⋊C4×C151—22no—
135C4⋊C4×C151—22no—
136C4⋊C4×C151—21no—
137C4⋊C4×C151—21no—
138C4⋊C4×C151—22no—
139C4⋊C4×C151—24no—
140C4⋊C4×C151—22no—
141C4⋊C4×C151—22no—
142C4⋊C4×C151—24no—
143C4⋊C4×C151—24no—
144C4⋊C4×C151—44no—
145C4⋊C4×C151—84no—
146C4⋊C4×C151C22⋊C412yes—
147C4⋊C4×C151C22⋊C411yescharacteristic
148C4⋊C4×C151C22⋊C411yescharacteristic
149C4⋊C4×C151C22⋊C414yes—
150C4⋊C4×C151C22⋊C414yes—
151C4⋊C4×C151C22⋊C412yes—
152C4⋊C4×C151C22⋊C412yes—
153C22⋊C4×C151—22no—
154C22⋊C4×C151—24no—
155C22⋊C4×C151—22no—
156C22⋊C4×C151—22no—
157C22⋊C4×C151—24no—
158C22⋊C4×C151—22no—
159C22⋊C4×C151—24no—
160C22⋊C4×C151—22no—
161C22⋊C4×C151—24no—
162C22⋊C4×C151C2×Q814yes—
163C42×C151—22no—
164C42×C151—22no—
165C42×C151—21no—
166C42×C151—21no—
167C42×C151—44no—
168C42×C151—84no—
169C23×C151—21no—
170C23×C151—21no—
171C23×C151—41no—
172C23×C151—41no—
173C23×C151—41no—
174C23×C151—41no—
175C23×C151C4×Q812yes—
176C23×C151C23×C411yescharacteristic
177C23×C151C22×D412yes—
178C23×C151C22×Q811yescharacteristic
179C23×C151C2×#4111yescharacteristic
180C23×C1512−1+412yes—
181Q8×C151—42no—
182Q8×C151—42no—
183Q8×C151—84no—
184D4×C151—24no—
185D4×C151—24no—
186C2×C4×C151—21no—
187C2×C4×C151—21no—
188C2×C4×C151—22no—
189C2×C4×C151—24no—
190C2×C4×C151—21no—
191C2×C4×C151—22no—
192C2×C4×C151—22no—
193C2×C4×C151—21no—
194C2×C4×C151—24no—
195C2×C4×C151—42no—
196C2×C4×C151—44no—
197C2×C4×C151—44no—
198C2×C4×C151—41no—
199C2×C4×C151—42no—
200C2×C4×C151—42no—
201C2×C4×C151—42no—
202C2×C4×C151—42no—
203C2×C4×C151—42no—
204C2×C4×C151—42no—
205C2×C4×C151—42no—
206C2×C4×C151—41no—
207C2×C4×C151—42no—
208C2×C4×C151—41no—
209C2×C4×C151—42no—
210C2×C4×C151—44no—
211C2×C4×C151—44no—
212C2×C4×C151—42no—
213C2×C4×C151—42no—
214C2×C4×C151—44no—
215C2×C4×C151—82no—
216C2×C4×C151—82no—
217C2×C4×C151C2×C22⋊C411yescharacteristic
218C2×C4×C151C2×C22⋊C414yes—
219C2×C4×C151C2×C22⋊C412yes—
220C2×C4×C151C2×C22⋊C412yes—
221C2×C4×C151C2×C22⋊C412yes—
222C2×C4×C151C2×C22⋊C411yescharacteristic
223C2×C4×C151C4×Q812yes—
224C22×C151—22no—
225C22×C151—21no—
226C22×C151—22no—
227C22×C151—21no—
228C22×C151—22no—
229C22×C151—41no—
230C22×C151—41no—
231C22×C151—42no—
232C22×C151—42no—
233C22×C151—42no—
234C22×C151—41no—
235C22×C151—41no—
236C22×C151C22×C22⋊C411yescharacteristic
237C22×C151C2×C4×Q811yescharacteristic
238C22×C151#51911yescharacteristic
239C22×C151#54812yes—
240C22×C151D4×Q812yes—
241C4×C151—22no—
242C4×C151—21no—
243C4×C151—22no—
244C4×C151—22no—
245C4×C151—22no—
246C4×C151—24no—
247C4×C151—44no—
248C4×C151—41no—
249C4×C151—42no—
250C4×C151—42no—
251C4×C151—41no—
252C4×C151—82no—
253C4×C151—84no—
254C4×C151#41012yes—
255C2×C151—22no—
256C2×C151—22no—
257C2×C151—21no—
258C2×C151—21no—
259C2×C151—42no—
260C2×C151Q8×C22⋊C411yescharacteristic
261C2×C151C2×#41011yescharacteristic
262C2×C151#288311yescharacteristic
263#22513C15111yesmaximal, characteristic, direct product factor, Sylow
264C151#2251311yescharacteristic, direct product factor, Sylow
265C2×#554C2×C15111yescharacteristic
266#2345C2×C15112yescomplemented
267#1996C2×C15114yescomplemented
268#1996C2×C15112yescomplemented
269#1799C2×C15112yes—
270#1799C2×C15111yescharacteristic
271C4×#99C2×C15112yes—
272C4×#99C2×C15111yescharacteristic
273#554—22no—
274#554—24no—
275#554—22no—
276C2×C4⋊Q8C4×C15111yescharacteristic, complemented
277C2×#125C4×C15112yescomplemented
278C2×C22⋊Q8C4×C15112yescomplemented
279C2×C22⋊Q8C4×C15111yescharacteristic, complemented
280C2×C22⋊Q8C22×C15112yes—
281C2×C22⋊Q8C22×C15111yescharacteristic
282C2×C4×Q8C4×C15111yescharacteristic, complemented
283C2×C4×D4C22×C15112yes—
284C2×C4×D4C22×C15111yescharacteristic
285C22×C4⋊C4C4×C15111yescharacteristic, complemented
286C22×C4⋊C4C22×C15111yescharacteristic
287C2×#99C22×C15111yescharacteristic
288C2×#99C22×C15112yes—
289C2×#99C22×C15111yescharacteristic
290#389—22no—
291#387—24no—
292#380C22×C15114yes—
293#380C22×C15112yes—
294C4×C4⋊C4—22no—
295C4×C22⋊C4C22×C15112yes—
296C4×C22⋊C4C22×C15114yes—
297#342C22×C15114yes—
298#342C22×C15112yes—
299#342C22×C15114yes—
300#342C22×C15112yes—
301C22×Q8C2×C4×C15111yescharacteristic
302C22×Q8C2×C4×C15112yes—
303C22×D4C23×C15111yescharacteristic
304C23×C4C2×C4×C15111yescharacteristic
305C23×C4C2×C4×C15112yes—
306C23×C4C23×C15111yescharacteristic
307C23×C4C23×C15112yes—
308C4⋊Q8—22no—
309C4⋊Q8—22no—
310#125—24no—
311#125—24no—
312C22⋊Q8—24no—
313C22⋊Q8—24no—
314C22⋊Q8—22no—
315C22⋊Q8—22no—
316C22⋊Q8—24no—
317C22⋊Q8—24no—
318C22⋊Q8—22no—
319C22⋊Q8—22no—
320C4×Q8—22no—
321C4×Q8—21no—
322C4×Q8—21no—
323C4×D4—24no—
324C4×D4—24no—
325C4×D4—22no—
326C4×D4—22no—
327C2×C4⋊C4—22no—
328C2×C4⋊C4—21no—
329C2×C4⋊C4—21no—
330C2×C4⋊C4—22no—
331C2×C4⋊C4—24no—
332C2×C4⋊C4C2×C4×C15112yes—
333C2×C4⋊C4C2×C4×C15111yescharacteristic
334C2×C4⋊C4C2×C4×C15111yescharacteristic
335C2×C4⋊C4C2×C4×C15112yes—
336C2×C4⋊C4C2×C4×C15112yes—
337C2×C4⋊C4C2×C4×C15112yes—
338C2×C4⋊C4C2×C4×C15112yes—
339C2×C4⋊C4C2×C4×C15111yescharacteristic
340C2×C4⋊C4C2×C4×C15112yes—
341C2×C4⋊C4C2×C4×C15111yescharacteristic
342C2×C4⋊C4D4×C15112yes—
343C2×C4⋊C4D4×C15114yes—
344C2×C4⋊C4D4×C15112yes—
345C2×C22⋊C4C2×C4×C15112yes—
346C2×C22⋊C4C2×C4×C15111yescharacteristic
347C2×C22⋊C4C23×C15112yes—
348C2×C22⋊C4C23×C15114yes—
349C2×C22⋊C4C23×C15112yes—
350C2×C22⋊C4C23×C15111yescharacteristic
351C2×C22⋊C4C23×C15112yes—
352C2×C42—22no—
353C2×C42—24no—
354C2×C42C2×C4×C15112yes—
355C2×C42C2×C4×C15111yescharacteristic
356#99—22no—
357#99—24no—
358#99Q8×C15114yescomplemented
359#95—22no—
360#95—24no—
361#95—24no—
362#95—22no—
363C24C22×C4×C15111yescharacteristic
364C24C24×C15111yescharacteristic
365C2×Q8—21no—
366C2×Q8—21no—
367C2×Q8—24no—
368C2×Q8—42no—
369C2×Q8—82no—
370C2×D4—22no—
371C2×D4—22no—
372C2×D4C2×Q8×C15112yes—
373C2×D4#41×C15112yes—
374C22×C4—22no—
375C22×C4—21no—
376C22×C4—21no—
377C22×C4—22no—
378C22×C4—22no—
379C22×C4—22no—
380C22×C4—24no—
381C22×C4—42no—
382C22×C4—42no—
383C22×C4—82no—
384C22×C4—82no—
385C22×C4C22×C4×C15111yescharacteristic
386C22×C4C22×C4×C15111yescharacteristic
387C22×C4C22×C4×C15112yes—
388C22×C4C22×C4×C15111yescharacteristic
389C22×C4C22×C4×C15111yescharacteristic
390C22×C4C22×C4×C15112yes—
391C22×C4C22×C4×C15112yes—
392C22×C4C22×C4×C15111yescharacteristic
393C22×C4C22×C4×C15112yes—
394C22×C4C2×D4×C15112yes—
395C22×C4C2×D4×C15112yes—
396C22×C4C2×D4×C15114yes—
397C22×C4C2×D4×C15114yes—
398C4⋊C4—22no—
399C4⋊C4—22no—
400C4⋊C4—21no—
401C4⋊C4—21no—
402C4⋊C4—22no—
403C4⋊C4—24no—
404C4⋊C4—22no—
405C4⋊C4—22no—
406C4⋊C4—24no—
407C4⋊C4—24no—
408C4⋊C4—44no—
409C4⋊C4—84no—
410C4⋊C4C22⋊C4×C15112yes—
411C4⋊C4C22⋊C4×C15111yescharacteristic
412C4⋊C4C22⋊C4×C15111yescharacteristic
413C4⋊C4C22⋊C4×C15114yes—
414C4⋊C4C22⋊C4×C15114yes—
415C4⋊C4C22⋊C4×C15112yes—
416C4⋊C4C22⋊C4×C15112yes—
417C22⋊C4—24no—
418C22⋊C4—22no—
419C22⋊C4—22no—
420C22⋊C4—24no—
421C22⋊C4—22no—
422C22⋊C4—22no—
423C22⋊C4—24no—
424C22⋊C4—22no—
425C22⋊C4—24no—
426C22⋊C4C2×Q8×C15114yes—
427C42—22no—
428C42—22no—
429C42—21no—
430C42—21no—
431C42—44no—
432C42—84no—
433C23—21no—
434C23—21no—
435C23—41no—
436C23—41no—
437C23—41no—
438C23—41no—
439C23C4×Q8×C15112yes—
440C23C23×C4×C15111yescharacteristic
441C23C22×D4×C15112yes—
442C23C22×Q8×C15111yescharacteristic
443C23C2×#41×C15111yescharacteristic
444C232−1+4×C15112yes—
445Q8—42no—
446Q8—42no—
447Q8—84no—
448D4—24no—
449D4—24no—
450C2×C4—21no—
451C2×C4—21no—
452C2×C4—24no—
453C2×C4—22no—
454C2×C4—22no—
455C2×C4—21no—
456C2×C4—21no—
457C2×C4—22no—
458C2×C4—24no—
459C2×C4—44no—
460C2×C4—44no—
461C2×C4—41no—
462C2×C4—42no—
463C2×C4—42no—
464C2×C4—42no—
465C2×C4—42no—
466C2×C4—42no—
467C2×C4—42no—
468C2×C4—42no—
469C2×C4—41no—
470C2×C4—42no—
471C2×C4—42no—
472C2×C4—41no—
473C2×C4—42no—
474C2×C4—44no—
475C2×C4—44no—
476C2×C4—42no—
477C2×C4—42no—
478C2×C4—44no—
479C2×C4—82no—
480C2×C4—82no—
481C2×C4C2×C22⋊C4×C15112yes—
482C2×C4C2×C22⋊C4×C15112yes—
483C2×C4C2×C22⋊C4×C15111yescharacteristic
484C2×C4C2×C22⋊C4×C15112yes—
485C2×C4C2×C22⋊C4×C15114yes—
486C2×C4C2×C22⋊C4×C15111yescharacteristic
487C2×C4C4×Q8×C15112yes—
488C22—22no—
489C22—21no—
490C22—22no—
491C22—21no—
492C22—22no—
493C22—41no—
494C22—41no—
495C22—42no—
496C22—42no—
497C22—42no—
498C22—41no—
499C22—41no—
500C22C22×C22⋊C4×C15111yescharacteristic
501C22C2×C4×Q8×C15111yescharacteristic
502C22#519×C15111yescharacteristic
503C22#548×C15112yes—
504C22D4×Q8×C15112yes—
505C4—22no—
506C4—21no—
507C4—22no—
508C4—22no—
509C4—22no—
510C4—24no—
511C4—44no—
512C4—41no—
513C4—42no—
514C4—42no—
515C4—41no—
516C4—82no—
517C4—84no—
518C4#410×C15112yes—
519C2—22no—
520C2—22no—
521C2—21no—
522C2—21no—
523C2—42no—
524C2Q8×C22⋊C4×C15111yescharacteristic
525C2C2×#410×C15111yescharacteristic
526C2#2883×C15111yescharacteristic
subgroupG/H
center, socleC22×C151C22×C22⋊C4
commutatorC23C23×C4×C151
Frattini, cosocleC24C24×C151
prime radicalC22×D4×C151C23
LCSG ▷ C23 ▷ C2 ▷ C1
UCSC1 ◁ C22×C151 ◁ C22×C4⋊C4×C151 ◁ G

Complements

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

NH ≅ G/Ntype
2. #2345×C151C2semidirect
3. #1996×C151C2semidirect
4. #1996×C151C2semidirect
12. C2×C4⋊Q8×C151C4semidirect
13. C2×#125×C151C4semidirect
14. C2×C22⋊Q8×C151C4semidirect
15. C2×C22⋊Q8×C151C4semidirect
18. C2×C4×Q8×C151C4semidirect
21. C22×C4⋊C4×C151C4semidirect
94. #99×C151Q8semidirect
263. #22513C151direct (N × H)
264. C151#22513direct (N × H)
266. #2345C2×C151semidirect
267. #1996C2×C151semidirect
268. #1996C2×C151semidirect
276. C2×C4⋊Q8C4×C151semidirect
277. C2×#125C4×C151semidirect
278. C2×C22⋊Q8C4×C151semidirect
279. C2×C22⋊Q8C4×C151semidirect
282. C2×C4×Q8C4×C151semidirect
285. C22×C4⋊C4C4×C151semidirect
358. #99Q8×C151semidirect

Maximal subgroups

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

subgroupindex
C2×#554×C1512
#2345×C1512
#1996×C1512
#1799×C1512
C4×#99×C1512
#22513151