Finite Group Databasebeta

← C151×#13314 #50318239

Subgroups — C151×#13314 #50318239

Order 38656.

536 conjugacy classes of subgroups.

The subgroup lattice has 386 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C22×SD32×C151C211yesmaximal, characteristic, complemented
2C2×#473×C151C211yesmaximal, characteristic, complemented
3C2×#472×C151C211yesmaximal, characteristic
4C2×#362×C151C211yesmaximal, characteristic, complemented
5C2×#361×C151C211yesmaximal, characteristic, complemented
6#2113×C151C211yesmaximal, characteristic, complemented
7#1381×C151C211yesmaximal, characteristic, complemented
8C2×#137×C151—21no—
9C2×AGL(1,ℤ/8)×C151—21no—
10C22×Q16×C151C2211yescharacteristic
11C22×D8×C151C2211yescharacteristic
12C2×#131×C151C2211yescharacteristic
13C2×SD32×C151—22no—
14C2×SD32×C151—22no—
15C2×SD32×C151—22no—
16C2×M32×C151C2211yescharacteristic
17C22×C16×C151C2211yescharacteristic
18#473×C151—22no—
19#473×C151—22no—
20#472×C151—22no—
21#472×C151—22no—
22C2×#108×C151C2211yescharacteristic
23C2×#108×C151C2211yescharacteristic
24C2×#101×C151—21no—
25C2×#100×C151—21no—
26#362×C151—22no—
27#361×C151—22no—
28C22⋊C16×C151—22no—
29C2×#41×C151D411yescharacteristic
30C22×Q8×C151—21no—
31C22×D4×C151—21no—
32#137×C151—24no—
33#137×C151—42no—
34AGL(1,ℤ/8)×C151—24no—
35AGL(1,ℤ/8)×C151—42no—
36C2×Q16×C151—21no—
37C2×Q16×C151—21no—
38C2×Q16×C151—21no—
39C2×Q16×C151—21no—
40C2×Q16×C151—21no—
41C2×Q16×C151—41no—
42C2×Q16×C151D412yes—
43C2×SD16×C151—41no—
44C2×SD16×C151—41no—
45C2×D8×C151—21no—
46C2×D8×C151—21no—
47C2×D8×C151—21no—
48C2×D8×C151—21no—
49C2×D8×C151—21no—
50C2×D8×C151—41no—
51C2×D8×C151D412yes—
52#131×C151—21no—
53#131×C151—21no—
54#131×C151—21no—
55#131×C151—21no—
56C2×M16×C151—21no—
57C2×M16×C151—21no—
58C2×M16×C151—21no—
59C2×M16×C151D411yescharacteristic
60C22×C8×C151—21no—
61C22×C8×C151C2311yescharacteristic
62SD32×C151—28no—
63M32×C151—21no—
64M32×C151—21no—
65C2×C16×C151—21no—
66C2×C16×C151—21no—
67C2×C16×C151—21no—
68C2×C16×C151D412yes—
69#108×C151—21no—
70#108×C151—21no—
71#108×C151D412yes—
72#108×C151D412yes—
73#101×C151—24no—
74#100×C151—24no—
75C24×C151—41no—
76#41×C151—21no—
77#41×C151—21no—
78#41×C151—21no—
79#41×C151—21no—
80C2×Q8×C151—22no—
81C2×Q8×C151—41no—
82C2×Q8×C151—41no—
83C2×Q8×C151—41no—
84C2×Q8×C151—41no—
85C2×Q8×C151—41no—
86C2×Q8×C151D811yescharacteristic
87C2×D4×C151—21no—
88C2×D4×C151—22no—
89C2×D4×C151—41no—
90C2×D4×C151—41no—
91C2×D4×C151—41no—
92C2×D4×C151—41no—
93C2×D4×C151—41no—
94C2×D4×C151D811yescharacteristic
95C22×C4×C151—21no—
96C22×C4×C151—41no—
97C22×C4×C151C2×D411yescharacteristic
98Q16×C151—22no—
99Q16×C151—22no—
100Q16×C151—44no—
101Q16×C151—41no—
102Q16×C151—41no—
103SD16×C151—44no—
104SD16×C151—44no—
105D8×C151—22no—
106D8×C151—22no—
107D8×C151—44no—
108D8×C151—41no—
109D8×C151—41no—
110M16×C151—21no—
111M16×C151—22no—
112M16×C151—21no—
113M16×C151—22no—
114M16×C151—41no—
115M16×C151—41no—
116M16×C151—41no—
117M16×C151—41no—
118C2×C8×C151—21no—
119C2×C8×C151—41no—
120C2×C8×C151—41no—
121C2×C8×C151—41no—
122C2×C8×C151—41no—
123C2×C8×C151C2×D411yescharacteristic
124C2×C8×C151C2×D411yescharacteristic
125C2×C8×C151C2×D411yescharacteristic
126C2×C8×C151C2×D411yescharacteristic
127C2×C8×C151C2×D411yescharacteristic
128C2×C8×C151#4111yescharacteristic
129C16×C151—22no—
130C16×C151—41no—
131C23×C151—21no—
132C23×C151—42no—
133C23×C151—81no—
134C23×C151—81no—
135C23×C151—162no—
136C23×C151#13511yescharacteristic
137Q8×C151—21no—
138Q8×C151—21no—
139Q8×C151—42no—
140Q8×C151—42no—
141Q8×C151—81no—
142Q8×C151—81no—
143D4×C151—21no—
144D4×C151—21no—
145D4×C151—41no—
146D4×C151—41no—
147D4×C151—42no—
148D4×C151—42no—
149D4×C151—81no—
150D4×C151—81no—
151C2×C4×C151—21no—
152C2×C4×C151—41no—
153C2×C4×C151—41no—
154C2×C4×C151—42no—
155C2×C4×C151—81no—
156C2×C4×C151—81no—
157C2×C4×C151#12011yescharacteristic
158C2×C4×C151#12011yescharacteristic
159C2×C4×C151#12111yescharacteristic
160C2×C4×C151C2×D811yescharacteristic
161C2×C4×C151AGL(1,ℤ/8)11yescharacteristic
162C2×C4×C151AGL(1,ℤ/8)11yescharacteristic
163C8×C151—41no—
164C8×C151—42no—
165C8×C151—42no—
166C8×C151#12011yescharacteristic
167C8×C151#12111yescharacteristic
168C8×C151#12111yescharacteristic
169C8×C151#12711yescharacteristic
170C22×C151—22no—
171C22×C151—42no—
172C22×C151—41no—
173C22×C151—82no—
174C22×C151—162no—
175C22×C151—162no—
176C22×C151—162no—
177C22×C151#44911yescharacteristic
178C22×C151#44911yescharacteristic
179C22×C151#46611yescharacteristic
180C4×C151—41no—
181C4×C151—82no—
182C4×C151#39211yescharacteristic
183C4×C151#44711yescharacteristic
184C4×C151#44711yescharacteristic
185C4×C151#46911yescharacteristic
186C2×C151—21no—
187C2×C151—21no—
188C2×C151—81no—
189C2×C151—162no—
190C2×C151#199911yescharacteristic
191C2×C151#222312yes—
192#13314C15111yesmaximal, characteristic, direct product factor, Sylow
193C151#1331411yescharacteristic, direct product factor, Sylow
194C22×SD32C2×C15111yescharacteristic, complemented
195C2×#473C2×C15111yescharacteristic, complemented
196C2×#472C2×C15111yescharacteristic
197C2×#362C2×C15111yescharacteristic, complemented
198C2×#361C2×C15111yescharacteristic, complemented
199#2113C2×C15111yescharacteristic, complemented
200#1381C2×C15111yescharacteristic, complemented
201C2×#137—21no—
202C2×AGL(1,ℤ/8)—21no—
203C22×Q16C22×C15111yescharacteristic
204C22×D8C22×C15111yescharacteristic
205C2×#131C22×C15111yescharacteristic
206C2×SD32—22no—
207C2×SD32—22no—
208C2×SD32—22no—
209C2×M32C22×C15111yescharacteristic
210C22×C16C22×C15111yescharacteristic
211#473—22no—
212#473—22no—
213#472—22no—
214#472—22no—
215C2×#108C22×C15111yescharacteristic
216C2×#108C22×C15111yescharacteristic
217C2×#101—21no—
218C2×#100—21no—
219#362—22no—
220#361—22no—
221C22⋊C16—22no—
222C2×#41D4×C15111yescharacteristic
223C22×Q8—21no—
224C22×D4—21no—
225#137—24no—
226#137—42no—
227AGL(1,ℤ/8)—24no—
228AGL(1,ℤ/8)—42no—
229C2×Q16—21no—
230C2×Q16—21no—
231C2×Q16—21no—
232C2×Q16—21no—
233C2×Q16—21no—
234C2×Q16—41no—
235C2×Q16D4×C15112yes—
236C2×SD16—41no—
237C2×SD16—41no—
238C2×D8—21no—
239C2×D8—21no—
240C2×D8—21no—
241C2×D8—21no—
242C2×D8—21no—
243C2×D8—41no—
244C2×D8D4×C15112yes—
245#131—21no—
246#131—21no—
247#131—21no—
248#131—21no—
249C2×M16—21no—
250C2×M16—21no—
251C2×M16—21no—
252C2×M16D4×C15111yescharacteristic
253C22×C8—21no—
254C22×C8C23×C15111yescharacteristic
255SD32—28no—
256M32—21no—
257M32—21no—
258C2×C16—21no—
259C2×C16—21no—
260C2×C16—21no—
261C2×C16D4×C15112yes—
262#108—21no—
263#108—21no—
264#108D4×C15112yes—
265#108D4×C15112yes—
266#101—24no—
267#100—24no—
268C24—41no—
269#41—21no—
270#41—21no—
271#41—21no—
272#41—21no—
273C2×Q8—22no—
274C2×Q8—41no—
275C2×Q8—41no—
276C2×Q8—41no—
277C2×Q8—41no—
278C2×Q8—41no—
279C2×Q8D8×C15111yescharacteristic
280C2×D4—21no—
281C2×D4—22no—
282C2×D4—41no—
283C2×D4—41no—
284C2×D4—41no—
285C2×D4—41no—
286C2×D4—41no—
287C2×D4D8×C15111yescharacteristic
288C22×C4—21no—
289C22×C4—41no—
290C22×C4C2×D4×C15111yescharacteristic
291Q16—22no—
292Q16—22no—
293Q16—44no—
294Q16—41no—
295Q16—41no—
296SD16—44no—
297SD16—44no—
298D8—22no—
299D8—22no—
300D8—44no—
301D8—41no—
302D8—41no—
303M16—21no—
304M16—22no—
305M16—21no—
306M16—22no—
307M16—41no—
308M16—41no—
309M16—41no—
310M16—41no—
311C2×C8—21no—
312C2×C8—41no—
313C2×C8—41no—
314C2×C8—41no—
315C2×C8—41no—
316C2×C8C2×D4×C15111yescharacteristic
317C2×C8C2×D4×C15111yescharacteristic
318C2×C8C2×D4×C15111yescharacteristic
319C2×C8C2×D4×C15111yescharacteristic
320C2×C8C2×D4×C15111yescharacteristic
321C2×C8#41×C15111yescharacteristic
322C16—22no—
323C16—41no—
324C23—21no—
325C23—42no—
326C23—81no—
327C23—81no—
328C23—162no—
329C23#135×C15111yescharacteristic
330Q8—21no—
331Q8—21no—
332Q8—42no—
333Q8—42no—
334Q8—81no—
335Q8—81no—
336D4—21no—
337D4—21no—
338D4—41no—
339D4—41no—
340D4—42no—
341D4—42no—
342D4—81no—
343D4—81no—
344C2×C4—21no—
345C2×C4—41no—
346C2×C4—41no—
347C2×C4—42no—
348C2×C4—81no—
349C2×C4—81no—
350C2×C4#120×C15111yescharacteristic
351C2×C4#120×C15111yescharacteristic
352C2×C4#121×C15111yescharacteristic
353C2×C4C2×D8×C15111yescharacteristic
354C2×C4AGL(1,ℤ/8)×C15111yescharacteristic
355C2×C4AGL(1,ℤ/8)×C15111yescharacteristic
356C8—41no—
357C8—42no—
358C8—42no—
359C8#120×C15111yescharacteristic
360C8#121×C15111yescharacteristic
361C8#121×C15111yescharacteristic
362C8#127×C15111yescharacteristic
363C22—22no—
364C22—42no—
365C22—41no—
366C22—82no—
367C22—162no—
368C22—162no—
369C22—162no—
370C22#449×C15111yescharacteristic
371C22#449×C15111yescharacteristic
372C22#466×C15111yescharacteristic
373C4—41no—
374C4—82no—
375C4#392×C15111yescharacteristic
376C4#447×C15111yescharacteristic
377C4#447×C15111yescharacteristic
378C4#469×C15111yescharacteristic
379C2—21no—
380C2—21no—
381C2—81no—
382C2—162no—
383C2#1999×C15111yescharacteristic
384C2#2223×C15112yes—
subgroupG/H
center, socleC22×C151#466
commutator, Frattini, cosocleC22×C8C23×C151
prime radicalC2×#472×C151C2
LCSG ▷ C22×C8 ▷ C4 ▷ C2 ▷ C1
UCSC1 ◁ C22×C151 ◁ C22×C4×C151 ◁ C2×#131×C151 ◁ G

Complements

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

NH ≅ G/Ntype
1. C22×SD32×C151C2semidirect
2. C2×#473×C151C2semidirect
4. C2×#362×C151C2semidirect
5. C2×#361×C151C2semidirect
6. #2113×C151C2semidirect
7. #1381×C151C2semidirect
192. #13314C151direct (N × H)
193. C151#13314direct (N × H)
194. C22×SD32C2×C151semidirect
195. C2×#473C2×C151semidirect
197. C2×#362C2×C151semidirect
198. C2×#361C2×C151semidirect
199. #2113C2×C151semidirect
200. #1381C2×C151semidirect

Maximal subgroups

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

subgroupindex
C22×SD32×C1512
C2×#473×C1512
C2×#472×C1512
C2×#362×C1512
C2×#361×C1512
#2113×C1512
#1381×C1512
#13314151