Finite Group Databasebeta

← C151×#13323 #50318248

Subgroups — C151×#13323 #50318248

Order 38656.

534 conjugacy classes of subgroups.

The subgroup lattice has 420 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#509×C151C211yesmaximal, characteristic, complemented
2#3067×C151C211yesmaximal, characteristic, complemented
3C2×#472×C151C211yesmaximal, characteristic, complemented
4C2×#361×C151C211yesmaximal, characteristic, complemented
5#2143×C151C211yesmaximal, characteristic, complemented
6#2115×C151C211yesmaximal, characteristic, complemented
7#1388×C151C211yesmaximal, characteristic, complemented
8#575×C151—21no—
9C2×AGL(1,ℤ/8)×C151—21no—
10C2×#135×C151C2211yescharacteristic
11C22×D8×C151C2211yescharacteristic
12C2×#131×C151C2211yescharacteristic
13#509×C151—22no—
14#509×C151—22no—
15C2×SD32×C151—21no—
16C2×D16×C151—21no—
17C2×M32×C151C2211yescharacteristic
18C2×M32×C151C2211yescharacteristic
19#473×C151—21no—
20#472×C151—22no—
21#472×C151—21no—
22#472×C151—22no—
23#471×C151—22no—
24C2×#108×C151C2211yescharacteristic
25C2×#108×C151C2211yescharacteristic
26#413×C151—21no—
27C2×#100×C151—21no—
28#361×C151—22no—
29#359×C151—22no—
30#349×C151—22no—
31C2×#41×C151—21no—
32C2×#41×C151D411yescharacteristic
33C22×D4×C151—21no—
34#137×C151—22no—
35#137×C151—41no—
36AGL(1,ℤ/8)×C151—22no—
37AGL(1,ℤ/8)×C151—24no—
38AGL(1,ℤ/8)×C151—42no—
39AGL(1,ℤ/8)×C151—41no—
40#135×C151—21no—
41#135×C151—21no—
42#135×C151—21no—
43#135×C151—21no—
44#135×C151—42no—
45C2×Q16×C151D411yescharacteristic
46C2×SD16×C151—21no—
47C2×SD16×C151—41no—
48C2×D8×C151—21no—
49C2×D8×C151—21no—
50C2×D8×C151—21no—
51C2×D8×C151—21no—
52C2×D8×C151—21no—
53C2×D8×C151—41no—
54C2×D8×C151D411yescharacteristic
55C2×D8×C151D412yes—
56#131×C151—21no—
57#131×C151—21no—
58#131×C151—21no—
59#131×C151—21no—
60C2×M16×C151—21no—
61C2×M16×C151—21no—
62C2×M16×C151—21no—
63C2×M16×C151D411yescharacteristic
64C22×C8×C151—21no—
65C22×C8×C151C2311yescharacteristic
66SD32×C151—24no—
67D16×C151—24no—
68M32×C151—21no—
69M32×C151—21no—
70M32×C151—21no—
71M32×C151D412yes—
72C2×C16×C151—21no—
73C2×C16×C151—21no—
74#108×C151—21no—
75#108×C151—21no—
76#108×C151D412yes—
77#108×C151D412yes—
78#101×C151—22no—
79#100×C151—24no—
80#100×C151—22no—
81C24×C151—41no—
82#41×C151—21no—
83#41×C151—21no—
84#41×C151—21no—
85#41×C151—21no—
86#41×C151—41no—
87#41×C151—41no—
88#41×C151—41no—
89#41×C151—41no—
90C2×Q8×C151—21no—
91C2×Q8×C151D811yescharacteristic
92C2×D4×C151—21no—
93C2×D4×C151—22no—
94C2×D4×C151—21no—
95C2×D4×C151—41no—
96C2×D4×C151—41no—
97C2×D4×C151—41no—
98C2×D4×C151—41no—
99C2×D4×C151—41no—
100C2×D4×C151—41no—
101C2×D4×C151D811yescharacteristic
102C22×C4×C151—21no—
103C22×C4×C151—41no—
104C22×C4×C151C2×D411yescharacteristic
105Q16×C151—21no—
106Q16×C151—21no—
107Q16×C151—42no—
108SD16×C151—42no—
109SD16×C151—44no—
110SD16×C151—42no—
111SD16×C151—41no—
112SD16×C151—41no—
113D8×C151—21no—
114D8×C151—22no—
115D8×C151—22no—
116D8×C151—21no—
117D8×C151—44no—
118D8×C151—42no—
119D8×C151—41no—
120D8×C151—41no—
121M16×C151—21no—
122M16×C151—21no—
123M16×C151—22no—
124M16×C151—22no—
125M16×C151—41no—
126M16×C151—41no—
127M16×C151—41no—
128M16×C151—41no—
129C2×C8×C151—21no—
130C2×C8×C151—41no—
131C2×C8×C151—41no—
132C2×C8×C151—41no—
133C2×C8×C151—41no—
134C2×C8×C151C2×D411yescharacteristic
135C2×C8×C151C2×D411yescharacteristic
136C2×C8×C151C2×D411yescharacteristic
137C2×C8×C151C2×D411yescharacteristic
138C2×C8×C151C2×D411yescharacteristic
139C2×C8×C151#4111yescharacteristic
140C16×C151—22no—
141C16×C151—41no—
142C23×C151—21no—
143C23×C151—42no—
144C23×C151—41no—
145C23×C151—81no—
146C23×C151—81no—
147C23×C151—162no—
148C23×C151#13511yescharacteristic
149Q8×C151—21no—
150Q8×C151—21no—
151Q8×C151—41no—
152Q8×C151—41no—
153D4×C151—21no—
154D4×C151—21no—
155D4×C151—41no—
156D4×C151—41no—
157D4×C151—41no—
158D4×C151—42no—
159D4×C151—41no—
160D4×C151—42no—
161D4×C151—81no—
162D4×C151—81no—
163D4×C151—81no—
164D4×C151—81no—
165C2×C4×C151—21no—
166C2×C4×C151—41no—
167C2×C4×C151—41no—
168C2×C4×C151—41no—
169C2×C4×C151—81no—
170C2×C4×C151—81no—
171C2×C4×C151#12011yescharacteristic
172C2×C4×C151#12011yescharacteristic
173C2×C4×C151#12111yescharacteristic
174C2×C4×C151C2×D811yescharacteristic
175C2×C4×C151AGL(1,ℤ/8)11yescharacteristic
176C2×C4×C151AGL(1,ℤ/8)11yescharacteristic
177C8×C151—41no—
178C8×C151—42no—
179C8×C151—42no—
180C8×C151#12011yescharacteristic
181C8×C151#12111yescharacteristic
182C8×C151#12111yescharacteristic
183C8×C151#12711yescharacteristic
184C22×C151—41no—
185C22×C151—41no—
186C22×C151—81no—
187C22×C151—81no—
188C22×C151—82no—
189C22×C151—161no—
190C22×C151—162no—
191C22×C151—162no—
192C22×C151—162no—
193C22×C151#44911yescharacteristic
194C22×C151#44911yescharacteristic
195C22×C151#46611yescharacteristic
196C4×C151—41no—
197C4×C151—81no—
198C4×C151#39211yescharacteristic
199C4×C151#44711yescharacteristic
200C4×C151#44711yescharacteristic
201C4×C151#46911yescharacteristic
202C2×C151—21no—
203C2×C151—21no—
204C2×C151—21no—
205C2×C151—81no—
206C2×C151—161no—
207C2×C151—162no—
208C2×C151#199911yescharacteristic
209#13323C15111yesmaximal, characteristic, direct product factor, Sylow
210C151#1332311yescharacteristic, direct product factor, Sylow
211C2×#509C2×C15111yescharacteristic, complemented
212#3067C2×C15111yescharacteristic, complemented
213C2×#472C2×C15111yescharacteristic, complemented
214C2×#361C2×C15111yescharacteristic, complemented
215#2143C2×C15111yescharacteristic, complemented
216#2115C2×C15111yescharacteristic, complemented
217#1388C2×C15111yescharacteristic, complemented
218#575—21no—
219C2×AGL(1,ℤ/8)—21no—
220C2×#135C22×C15111yescharacteristic
221C22×D8C22×C15111yescharacteristic
222C2×#131C22×C15111yescharacteristic
223#509—22no—
224#509—22no—
225C2×SD32—21no—
226C2×D16—21no—
227C2×M32C22×C15111yescharacteristic
228C2×M32C22×C15111yescharacteristic
229#473—21no—
230#472—22no—
231#472—21no—
232#472—22no—
233#471—22no—
234C2×#108C22×C15111yescharacteristic
235C2×#108C22×C15111yescharacteristic
236#413—21no—
237C2×#100—21no—
238#361—22no—
239#359—22no—
240#349—22no—
241C2×#41—21no—
242C2×#41D4×C15111yescharacteristic
243C22×D4—21no—
244#137—22no—
245#137—41no—
246AGL(1,ℤ/8)—24no—
247AGL(1,ℤ/8)—22no—
248AGL(1,ℤ/8)—42no—
249AGL(1,ℤ/8)—41no—
250#135—21no—
251#135—21no—
252#135—21no—
253#135—21no—
254#135—42no—
255C2×Q16D4×C15111yescharacteristic
256C2×SD16—21no—
257C2×SD16—41no—
258C2×D8—21no—
259C2×D8—21no—
260C2×D8—21no—
261C2×D8—21no—
262C2×D8—21no—
263C2×D8—41no—
264C2×D8D4×C15112yes—
265C2×D8D4×C15111yescharacteristic
266#131—21no—
267#131—21no—
268#131—21no—
269#131—21no—
270C2×M16—21no—
271C2×M16—21no—
272C2×M16—21no—
273C2×M16D4×C15111yescharacteristic
274C22×C8—21no—
275C22×C8C23×C15111yescharacteristic
276SD32—24no—
277D16—24no—
278M32—21no—
279M32—21no—
280M32—21no—
281M32D4×C15112yes—
282C2×C16—21no—
283C2×C16—21no—
284#108—21no—
285#108—21no—
286#108D4×C15112yes—
287#108D4×C15112yes—
288#101—22no—
289#100—24no—
290#100—22no—
291C24—41no—
292#41—21no—
293#41—21no—
294#41—21no—
295#41—21no—
296#41—41no—
297#41—41no—
298#41—41no—
299#41—41no—
300C2×Q8—21no—
301C2×Q8D8×C15111yescharacteristic
302C2×D4—21no—
303C2×D4—22no—
304C2×D4—21no—
305C2×D4—41no—
306C2×D4—41no—
307C2×D4—41no—
308C2×D4—41no—
309C2×D4—41no—
310C2×D4—41no—
311C2×D4D8×C15111yescharacteristic
312C22×C4—21no—
313C22×C4—41no—
314C22×C4C2×D4×C15111yescharacteristic
315Q16—21no—
316Q16—21no—
317Q16—42no—
318SD16—42no—
319SD16—44no—
320SD16—42no—
321SD16—41no—
322SD16—41no—
323D8—21no—
324D8—22no—
325D8—22no—
326D8—21no—
327D8—44no—
328D8—42no—
329D8—41no—
330D8—41no—
331M16—21no—
332M16—21no—
333M16—22no—
334M16—22no—
335M16—41no—
336M16—41no—
337M16—41no—
338M16—41no—
339C2×C8—21no—
340C2×C8—41no—
341C2×C8—41no—
342C2×C8—41no—
343C2×C8—41no—
344C2×C8C2×D4×C15111yescharacteristic
345C2×C8C2×D4×C15111yescharacteristic
346C2×C8C2×D4×C15111yescharacteristic
347C2×C8C2×D4×C15111yescharacteristic
348C2×C8C2×D4×C15111yescharacteristic
349C2×C8#41×C15111yescharacteristic
350C16—22no—
351C16—41no—
352C23—21no—
353C23—41no—
354C23—42no—
355C23—81no—
356C23—81no—
357C23—162no—
358C23#135×C15111yescharacteristic
359Q8—21no—
360Q8—21no—
361Q8—41no—
362Q8—41no—
363D4—21no—
364D4—21no—
365D4—41no—
366D4—41no—
367D4—42no—
368D4—41no—
369D4—42no—
370D4—41no—
371D4—81no—
372D4—81no—
373D4—81no—
374D4—81no—
375C2×C4—21no—
376C2×C4—41no—
377C2×C4—41no—
378C2×C4—41no—
379C2×C4—81no—
380C2×C4—81no—
381C2×C4#120×C15111yescharacteristic
382C2×C4#120×C15111yescharacteristic
383C2×C4#121×C15111yescharacteristic
384C2×C4C2×D8×C15111yescharacteristic
385C2×C4AGL(1,ℤ/8)×C15111yescharacteristic
386C2×C4AGL(1,ℤ/8)×C15111yescharacteristic
387C8—41no—
388C8—42no—
389C8—42no—
390C8#120×C15111yescharacteristic
391C8#121×C15111yescharacteristic
392C8#121×C15111yescharacteristic
393C8#127×C15111yescharacteristic
394C22—41no—
395C22—41no—
396C22—81no—
397C22—81no—
398C22—82no—
399C22—162no—
400C22—161no—
401C22—162no—
402C22—162no—
403C22#449×C15111yescharacteristic
404C22#449×C15111yescharacteristic
405C22#466×C15111yescharacteristic
406C4—41no—
407C4—81no—
408C4#392×C15111yescharacteristic
409C4#447×C15111yescharacteristic
410C4#447×C15111yescharacteristic
411C4#469×C15111yescharacteristic
412C2—21no—
413C2—21no—
414C2—21no—
415C2—81no—
416C2—162no—
417C2—161no—
418C2#1999×C15111yescharacteristic
subgroupG/H
center, socleC2×C151#1999
commutator, Frattini, cosocleC22×C8C23×C151
LCSG ▷ C22×C8 ▷ C4 ▷ C2 ▷ C1
UCSC1 ◁ C2×C151 ◁ C22×C4×C151 ◁ C2×#131×C151 ◁ G

Complements

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

NH ≅ G/Ntype
1. C2×#509×C151C2semidirect
2. #3067×C151C2semidirect
3. C2×#472×C151C2semidirect
4. C2×#361×C151C2semidirect
5. #2143×C151C2semidirect
6. #2115×C151C2semidirect
7. #1388×C151C2semidirect
209. #13323C151direct (N × H)
210. C151#13323direct (N × H)
211. C2×#509C2×C151semidirect
212. #3067C2×C151semidirect
213. C2×#472C2×C151semidirect
214. C2×#361C2×C151semidirect
215. #2143C2×C151semidirect
216. #2115C2×C151semidirect
217. #1388C2×C151semidirect

Maximal subgroups

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

subgroupindex
C2×#509×C1512
#3067×C1512
C2×#472×C1512
C2×#361×C1512
#2143×C1512
#2115×C1512
#1388×C1512
#13323151