Finite Group Databasebeta

← C151×#17222 #50322147

Subgroups — C151×#17222 #50322147

Order 38656.

1054 conjugacy classes of subgroups.

The subgroup lattice has 494 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#400×C151C212yesmaximal
2#1444×C151C212yesmaximal
3#1444×C151C212yesmaximal
4#1442×C151C211yesmaximal, characteristic
5#1442×C151C211yesmaximal, characteristic
6#1442×C151C212yesmaximal
7#1442×C151C212yesmaximal
8#1442×C151C211yesmaximal, characteristic
9#1441×C151C212yesmaximal
10C22×C4⋊C4×C151C2211yescharacteristic
11C22×C4⋊C4×C151C2212yes—
12C22×C4⋊C4×C151C2212yes—
13C22×C4⋊C4×C151C2212yes—
14C22×C4⋊C4×C151C2211yescharacteristic
15C22×C4⋊C4×C151C2211yescharacteristic
16#400×C151—28no—
17C2×#95×C151C2212yes—
18C2×#95×C151C2212yes—
19C2×#95×C151C2212yes—
20C2×#95×C151C2212yes—
21C2×#95×C151C2212yes—
22C2×#95×C151C2211yescharacteristic
23C2×#95×C151C2212yes—
24C2×#95×C151C2212yes—
25C2×#95×C151C2212yes—
26C2×#95×C151C2212yes—
27C2×#95×C151C2211yescharacteristic
28C2×#95×C151C2212yes—
29C2×#95×C151C2212yes—
30C2×#95×C151C2211yescharacteristic
31C2×#95×C151C2211yescharacteristic
32C23×C4×C151C2312yes—
33C23×C4×C151C2312yes—
34C23×C4×C151C2312yes—
35C23×C4×C151C2311yescharacteristic
36C23×C4×C151C2312yes—
37C23×C4×C151C2312yes—
38C23×C4×C151C2311yescharacteristic
39C23×C4×C151C2311yescharacteristic
40C23×C4×C151C2312yes—
41C2×C4⋊C4×C151—24no—
42C2×C4⋊C4×C151—24no—
43C2×C4⋊C4×C151—24no—
44C2×C4⋊C4×C151—41no—
45C2×C4⋊C4×C151—42no—
46C2×C4⋊C4×C151—42no—
47C2×C4⋊C4×C151—42no—
48C2×C4⋊C4×C151—42no—
49C2×C4⋊C4×C151—42no—
50C2×C4⋊C4×C151—42no—
51C2×C4⋊C4×C151—42no—
52C2×C4⋊C4×C151—42no—
53C2×C4⋊C4×C151—41no—
54C2×C4⋊C4×C151—41no—
55C2×C4⋊C4×C151—42no—
56#95×C151—24no—
57#95×C151—24no—
58#95×C151—24no—
59#95×C151—24no—
60#95×C151—24no—
61#95×C151—24no—
62#95×C151—24no—
63#95×C151—22no—
64#95×C151—41no—
65#95×C151—42no—
66#95×C151—42no—
67#95×C151—42no—
68#95×C151—42no—
69#95×C151D414yes—
70#95×C151Q814yes—
71C24×C151C2411yescharacteristic
72C22×C4×C151—22no—
73C22×C4×C151—22no—
74C22×C4×C151—22no—
75C22×C4×C151—22no—
76C22×C4×C151—22no—
77C22×C4×C151—22no—
78C22×C4×C151—22no—
79C22×C4×C151—22no—
80C22×C4×C151—21no—
81C22×C4×C151—22no—
82C22×C4×C151—22no—
83C22×C4×C151—21no—
84C22×C4×C151—22no—
85C22×C4×C151—21no—
86C22×C4×C151—22no—
87C22×C4×C151—22no—
88C22×C4×C151—22no—
89C22×C4×C151—22no—
90C22×C4×C151—22no—
91C22×C4×C151—22no—
92C22×C4×C151—21no—
93C22×C4×C151—22no—
94C22×C4×C151—21no—
95C22×C4×C151—22no—
96C22×C4×C151—21no—
97C22×C4×C151—22no—
98C22×C4×C151—22no—
99C22×C4×C151—22no—
100C22×C4×C151—22no—
101C22×C4×C151—22no—
102C22×C4×C151—22no—
103C22×C4×C151—22no—
104C22×C4×C151—22no—
105C22×C4×C151—22no—
106C22×C4×C151—22no—
107C22×C4×C151—22no—
108C22×C4×C151—22no—
109C22×C4×C151—22no—
110C22×C4×C151—22no—
111C22×C4×C151—22no—
112C22×C4×C151—22no—
113C22×C4×C151—22no—
114C22×C4×C151—22no—
115C22×C4×C151—22no—
116C22×C4×C151—22no—
117C22×C4×C151—22no—
118C22×C4×C151—22no—
119C22×C4×C151—22no—
120C22×C4×C151—22no—
121C22×C4×C151—22no—
122C22×C4×C151—22no—
123C22×C4×C151C2×D412yes—
124C22×C4×C151C2×D414yes—
125C22×C4×C151C2×Q814yes—
126C22×C4×C151C2×Q812yes—
127C22×C4×C151#4112yes—
128C22×C4×C151#4114yes—
129C4⋊C4×C151—44no—
130C4⋊C4×C151—48no—
131C4⋊C4×C151—48no—
132C4⋊C4×C151—44no—
133C4⋊C4×C151—44no—
134C4⋊C4×C151—48no—
135C23×C151C22×D411yescharacteristic
136C23×C151C22×Q811yescharacteristic
137C23×C151C2×#4111yescharacteristic
138C23×C151C2×#4112yes—
139C23×C151H(5,𝔽2)12yes—
140C23×C151H(5,𝔽2)12yes—
141C23×C151H(5,𝔽2)11yescharacteristic
142C23×C1512−1+411yescharacteristic
143C23×C1512−1+411yescharacteristic
144C23×C1512−1+412yes—
145C23×C1512−1+411yescharacteristic
146C2×C4×C151—24no—
147C2×C4×C151—22no—
148C2×C4×C151—24no—
149C2×C4×C151—22no—
150C2×C4×C151—24no—
151C2×C4×C151—24no—
152C2×C4×C151—22no—
153C2×C4×C151—24no—
154C2×C4×C151—24no—
155C2×C4×C151—24no—
156C2×C4×C151—24no—
157C2×C4×C151—24no—
158C2×C4×C151—24no—
159C2×C4×C151—24no—
160C2×C4×C151—24no—
161C2×C4×C151—42no—
162C2×C4×C151—42no—
163C2×C4×C151—42no—
164C2×C4×C151—42no—
165C2×C4×C151—42no—
166C2×C4×C151—42no—
167C2×C4×C151—42no—
168C2×C4×C151—42no—
169C2×C4×C151—42no—
170C2×C4×C151—41no—
171C2×C4×C151—41no—
172C2×C4×C151—42no—
173C2×C4×C151—42no—
174C2×C4×C151—41no—
175C2×C4×C151—41no—
176C2×C4×C151—42no—
177C2×C4×C151—42no—
178C2×C4×C151—41no—
179C2×C4×C151—42no—
180C2×C4×C151—42no—
181C2×C4×C151—41no—
182C2×C4×C151—42no—
183C2×C4×C151—42no—
184C2×C4×C151—42no—
185C2×C4×C151—42no—
186C2×C4×C151—42no—
187C2×C4×C151—42no—
188C2×C4×C151—42no—
189C2×C4×C151—42no—
190C2×C4×C151—42no—
191C2×C4×C151—42no—
192C2×C4×C151—42no—
193C2×C4×C151—42no—
194C2×C4×C151—42no—
195C2×C4×C151—42no—
196C2×C4×C151—42no—
197C2×C4×C151—42no—
198C2×C4×C151—42no—
199C2×C4×C151—42no—
200C2×C4×C151—42no—
201C2×C4×C151—42no—
202C2×C4×C151—42no—
203C22×C151#53512yes—
204C22×C151#53711yescharacteristic
205C22×C151#53912yes—
206C22×C151#53911yescharacteristic
207C22×C151#53912yes—
208C22×C151#54112yes—
209C22×C151#54212yes—
210C22×C151#54412yes—
211C22×C151#54411yescharacteristic
212C22×C151#54711yescharacteristic
213C22×C151#54712yes—
214C22×C151D4×Q811yescharacteristic
215C22×C151#55111yescharacteristic
216C22×C151#55312yes—
217C22×C151#55412yes—
218C22×C151#55711yescharacteristic
219C22×C151H(3,𝔽4)11yescharacteristic
220C22×C151H(3,𝔽4)11yescharacteristic
221C22×C151#56212yes—
222C22×C151#56312yes—
223C22×C151#56311yescharacteristic
224C22×C151#56312yes—
225C22×C151#56411yescharacteristic
226C4×C151—42no—
227C4×C151—44no—
228C4×C151—44no—
229C4×C151—42no—
230C4×C151—42no—
231C4×C151—44no—
232C4×C151—82no—
233C4×C151—82no—
234C4×C151—82no—
235C2×C151#271812yes—
236C2×C151#272012yes—
237C2×C151#272211yescharacteristic
238C2×C151#272611yescharacteristic
239C2×C151#276311yescharacteristic
240C2×C151#278211yescharacteristic
241C2×C151#280212yes—
242C2×C151#281911yescharacteristic
243C2×C151#283511yescharacteristic
244C2×C151#283912yes—
245C2×C151#284411yescharacteristic
246#17222C15111yesmaximal, characteristic, direct product factor, Sylow
247C151#1722211yescharacteristic, direct product factor, Sylow
248C2×#400C2×C15112yes—
249#1444C2×C15112yes—
250#1444C2×C15112yes—
251#1442C2×C15111yescharacteristic
252#1442C2×C15112yes—
253#1442C2×C15112yes—
254#1442C2×C15111yescharacteristic
255#1442C2×C15111yescharacteristic
256#1441C2×C15112yes—
257C22×C4⋊C4C22×C15111yescharacteristic
258C22×C4⋊C4C22×C15112yes—
259C22×C4⋊C4C22×C15112yes—
260C22×C4⋊C4C22×C15112yes—
261C22×C4⋊C4C22×C15111yescharacteristic
262C22×C4⋊C4C22×C15111yescharacteristic
263#400—28no—
264C2×#95C22×C15112yes—
265C2×#95C22×C15112yes—
266C2×#95C22×C15112yes—
267C2×#95C22×C15112yes—
268C2×#95C22×C15112yes—
269C2×#95C22×C15111yescharacteristic
270C2×#95C22×C15112yes—
271C2×#95C22×C15112yes—
272C2×#95C22×C15112yes—
273C2×#95C22×C15112yes—
274C2×#95C22×C15111yescharacteristic
275C2×#95C22×C15112yes—
276C2×#95C22×C15112yes—
277C2×#95C22×C15111yescharacteristic
278C2×#95C22×C15111yescharacteristic
279C23×C4C23×C15112yes—
280C23×C4C23×C15112yes—
281C23×C4C23×C15112yes—
282C23×C4C23×C15112yes—
283C23×C4C23×C15111yescharacteristic
284C23×C4C23×C15112yes—
285C23×C4C23×C15111yescharacteristic
286C23×C4C23×C15112yes—
287C23×C4C23×C15111yescharacteristic
288C2×C4⋊C4—24no—
289C2×C4⋊C4—24no—
290C2×C4⋊C4—24no—
291C2×C4⋊C4—41no—
292C2×C4⋊C4—42no—
293C2×C4⋊C4—42no—
294C2×C4⋊C4—42no—
295C2×C4⋊C4—42no—
296C2×C4⋊C4—42no—
297C2×C4⋊C4—42no—
298C2×C4⋊C4—42no—
299C2×C4⋊C4—42no—
300C2×C4⋊C4—41no—
301C2×C4⋊C4—41no—
302C2×C4⋊C4—42no—
303#95—24no—
304#95—24no—
305#95—24no—
306#95—22no—
307#95—24no—
308#95—24no—
309#95—24no—
310#95—24no—
311#95—41no—
312#95—42no—
313#95—42no—
314#95—42no—
315#95—42no—
316#95D4×C15114yes—
317#95Q8×C15114yes—
318C24C24×C15111yescharacteristic
319C22×C4—22no—
320C22×C4—22no—
321C22×C4—22no—
322C22×C4—22no—
323C22×C4—22no—
324C22×C4—22no—
325C22×C4—22no—
326C22×C4—22no—
327C22×C4—21no—
328C22×C4—22no—
329C22×C4—22no—
330C22×C4—21no—
331C22×C4—22no—
332C22×C4—21no—
333C22×C4—22no—
334C22×C4—22no—
335C22×C4—22no—
336C22×C4—22no—
337C22×C4—22no—
338C22×C4—22no—
339C22×C4—21no—
340C22×C4—22no—
341C22×C4—21no—
342C22×C4—22no—
343C22×C4—21no—
344C22×C4—22no—
345C22×C4—22no—
346C22×C4—22no—
347C22×C4—22no—
348C22×C4—22no—
349C22×C4—22no—
350C22×C4—22no—
351C22×C4—22no—
352C22×C4—22no—
353C22×C4—22no—
354C22×C4—22no—
355C22×C4—22no—
356C22×C4—22no—
357C22×C4—22no—
358C22×C4—22no—
359C22×C4—22no—
360C22×C4—22no—
361C22×C4—22no—
362C22×C4—22no—
363C22×C4—22no—
364C22×C4—22no—
365C22×C4—22no—
366C22×C4—22no—
367C22×C4—22no—
368C22×C4—22no—
369C22×C4—22no—
370C22×C4C2×D4×C15112yes—
371C22×C4C2×D4×C15114yes—
372C22×C4C2×Q8×C15114yes—
373C22×C4C2×Q8×C15112yes—
374C22×C4#41×C15112yes—
375C22×C4#41×C15114yes—
376C4⋊C4—44no—
377C4⋊C4—48no—
378C4⋊C4—48no—
379C4⋊C4—44no—
380C4⋊C4—48no—
381C4⋊C4—44no—
382C23C22×D4×C15111yescharacteristic
383C23C22×Q8×C15111yescharacteristic
384C23C2×#41×C15111yescharacteristic
385C23C2×#41×C15112yes—
386C23H(5,𝔽2)×C15112yes—
387C23H(5,𝔽2)×C15112yes—
388C23H(5,𝔽2)×C15111yescharacteristic
389C232−1+4×C15111yescharacteristic
390C232−1+4×C15111yescharacteristic
391C232−1+4×C15112yes—
392C232−1+4×C15111yescharacteristic
393C2×C4—24no—
394C2×C4—22no—
395C2×C4—24no—
396C2×C4—24no—
397C2×C4—22no—
398C2×C4—24no—
399C2×C4—22no—
400C2×C4—24no—
401C2×C4—24no—
402C2×C4—24no—
403C2×C4—24no—
404C2×C4—24no—
405C2×C4—24no—
406C2×C4—24no—
407C2×C4—24no—
408C2×C4—42no—
409C2×C4—42no—
410C2×C4—42no—
411C2×C4—42no—
412C2×C4—42no—
413C2×C4—42no—
414C2×C4—42no—
415C2×C4—42no—
416C2×C4—42no—
417C2×C4—41no—
418C2×C4—41no—
419C2×C4—42no—
420C2×C4—42no—
421C2×C4—41no—
422C2×C4—42no—
423C2×C4—41no—
424C2×C4—42no—
425C2×C4—42no—
426C2×C4—41no—
427C2×C4—41no—
428C2×C4—42no—
429C2×C4—42no—
430C2×C4—42no—
431C2×C4—42no—
432C2×C4—42no—
433C2×C4—42no—
434C2×C4—42no—
435C2×C4—42no—
436C2×C4—42no—
437C2×C4—42no—
438C2×C4—42no—
439C2×C4—42no—
440C2×C4—42no—
441C2×C4—42no—
442C2×C4—42no—
443C2×C4—42no—
444C2×C4—42no—
445C2×C4—42no—
446C2×C4—42no—
447C2×C4—42no—
448C2×C4—42no—
449C2×C4—42no—
450C22#535×C15112yes—
451C22#537×C15111yescharacteristic
452C22#539×C15112yes—
453C22#539×C15112yes—
454C22#539×C15111yescharacteristic
455C22#541×C15112yes—
456C22#542×C15112yes—
457C22#544×C15112yes—
458C22#544×C15111yescharacteristic
459C22#547×C15111yescharacteristic
460C22#547×C15112yes—
461C22D4×Q8×C15111yescharacteristic
462C22#551×C15111yescharacteristic
463C22#553×C15112yes—
464C22#554×C15112yes—
465C22#557×C15111yescharacteristic
466C22H(3,𝔽4)×C15111yescharacteristic
467C22H(3,𝔽4)×C15111yescharacteristic
468C22#562×C15112yes—
469C22#563×C15112yes—
470C22#563×C15111yescharacteristic
471C22#563×C15112yes—
472C22#564×C15111yescharacteristic
473C4—44no—
474C4—44no—
475C4—42no—
476C4—42no—
477C4—44no—
478C4—42no—
479C4—82no—
480C4—82no—
481C4—82no—
482C2#2718×C15112yes—
483C2#2720×C15112yes—
484C2#2722×C15111yescharacteristic
485C2#2726×C15111yescharacteristic
486C2#2763×C15111yescharacteristic
487C2#2782×C15111yescharacteristic
488C2#2802×C15112yes—
489C2#2819×C15111yescharacteristic
490C2#2835×C15111yescharacteristic
491C2#2839×C15112yes—
492C2#2844×C15111yescharacteristic
subgroupG/H
center, socle, prime radicalC24×C151C24
commutator, Frattini, cosocleC24C24×C151
LCSG ▷ C24 ▷ C1
UCSC1 ◁ C24×C151 ◁ G

Complements

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

NH ≅ G/Ntype
246. #17222C151direct (N × H)
247. C151#17222direct (N × H)

Maximal subgroups

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

subgroupindex
C2×#400×C1512
#1444×C1512
#1442×C1512
#1441×C1512
#17222151