Finite Group Databasebeta

← C151×#13578 #50318503

Subgroups — C151×#13578 #50318503

Order 38656.

420 conjugacy classes of subgroups; only the 90 normal subgroups are shown.

normal subgroups only

This group has 420 conjugacy classes of subgroups — too many to draw. Only its 90 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#3153×C151C211yesmaximal, characteristic
2C2×#355×C151C211yesmaximal, characteristic, complemented
3C2×#354×C151C211yesmaximal, characteristic, complemented
4#1407×C151C212yesmaximal, complemented
5#1406×C151C212yesmaximal, complemented
6C2×#137×C151—21no—
7C2×AGL(1,ℤ/8)×C151—21no—
8#532×C151C411yescharacteristic, complemented
9C2×#124×C151C2211yescharacteristic
10#497×C151C411yescharacteristic, complemented
11#497×C151C2211yescharacteristic
12#496×C151C411yescharacteristic, complemented
13#496×C151C2211yescharacteristic
14#495×C151—21no—
15#495×C151—21no—
16C2×#101×C151—21no—
17C2×#99×C151—21no—
18C4×M16×C151C411yescharacteristic, complemented
19#355×C151—21no—
20#355×C151C2212yes—
21#354×C151—21no—
22#354×C151C2212yes—
23C2×#41×C151—21no—
24C22×Q8×C151D411yescharacteristic
25C22×D4×C151D411yescharacteristic
26#137×C151—42no—
27#137×C151—42no—
28AGL(1,ℤ/8)×C151—42no—
29AGL(1,ℤ/8)×C151—42no—
30C2×Q16×C151—21no—
31C2×Q16×C151—21no—
32C2×SD16×C151—21no—
33C2×SD16×C151—21no—
34C2×SD16×C151—21no—
35C2×SD16×C151—21no—
36C2×D8×C151—21no—
37C2×D8×C151—21no—
38C2×M16×C151—21no—
39C2×M16×C151—41no—
40C4⋊Q8×C151C2×C411yescharacteristic, complemented
41#127×C151C2×C411yescharacteristic, complemented
42#124×C151—21no—
43#124×C151—41no—
44#124×C151C2×C411yescharacteristic
45#124×C151D411yescharacteristic
46#124×C151D411yescharacteristic
47#124×C151C2311yescharacteristic
48#121×C151—41no—
49C4×D4×C151—41no—
50C2×C22⋊C4×C151—41no—
51C2×C22⋊C4×C151—41no—
52C2×C42×C151C2×C411yescharacteristic
53#101×C151—22no—
54#101×C151—41no—
55#99×C151—22no—
56#99×C151—41no—
57#97×C151C2×C411yescharacteristic, complemented
58#97×C151C2×C411yescharacteristic, complemented
59C4×C8×C151—21no—
60C24×C151—21no—
61#41×C151—41no—
62#41×C151—41no—
63#41×C151—41no—
64#41×C151—41no—
65C2×Q8×C151—21no—
66C2×Q8×C151—21no—
67C2×Q8×C151—41no—
68C2×Q8×C151—41no—
69C2×Q8×C151C22⋊C411yescharacteristic
70C2×Q8×C151C2×D411yescharacteristic
71C2×D4×C151—21no—
72C2×D4×C151—21no—
73C2×D4×C151—41no—
74C2×D4×C151—41no—
75C2×D4×C151—41no—
76C2×D4×C151—41no—
77C2×D4×C151C22⋊C411yescharacteristic
78C2×D4×C151C2×D411yescharacteristic
79C22×C4×C151—21no—
80C22×C4×C151—21no—
81C22×C4×C151—41no—
82C22×C4×C151—81no—
83C22×C4×C151C22⋊C411yescharacteristic
84Q16×C151—42no—
85Q16×C151—42no—
86SD16×C151—42no—
87SD16×C151—42no—
88SD16×C151—42no—
89SD16×C151—42no—
90D8×C151—42no—
91D8×C151—42no—
92M16×C151—22no—
93M16×C151—22no—
94M16×C151—42no—
95M16×C151—81no—
96C2×C8×C151—21no—
97C2×C8×C151—21no—
98C2×C8×C151—81no—
99C4⋊C4×C151—41no—
100C22⋊C4×C151—41no—
101C22⋊C4×C151—41no—
102C22⋊C4×C151—41no—
103C22⋊C4×C151—41no—
104C22⋊C4×C151—42no—
105C22⋊C4×C151—42no—
106C22⋊C4×C151—81no—
107C42×C151—21no—
108C42×C151C22⋊C411yescharacteristic
109C42×C151C22×C411yescharacteristic
110C23×C151—21no—
111C23×C151—21no—
112C23×C151—41no—
113C23×C151—41no—
114C23×C151—41no—
115C23×C151—82no—
116C23×C151#9911yescharacteristic
117Q8×C151—41no—
118Q8×C151—41no—
119Q8×C151—41no—
120Q8×C151—41no—
121Q8×C151—81no—
122D4×C151—41no—
123D4×C151—41no—
124D4×C151—41no—
125D4×C151—41no—
126D4×C151—81no—
127D4×C151—82no—
128D4×C151—81no—
129D4×C151—81no—
130C2×C4×C151—21no—
131C2×C4×C151—21no—
132C2×C4×C151—21no—
133C2×C4×C151—21no—
134C2×C4×C151—21no—
135C2×C4×C151—21no—
136C2×C4×C151—41no—
137C2×C4×C151—41no—
138C2×C4×C151—41no—
139C2×C4×C151—81no—
140C2×C4×C151—81no—
141C2×C4×C151—81no—
142C2×C4×C151—82no—
143C2×C4×C151—162no—
144C2×C4×C151#9911yescharacteristic
145C2×C4×C151C2×C22⋊C411yescharacteristic
146C8×C151—22no—
147C8×C151—22no—
148C8×C151—82no—
149C22×C151—41no—
150C22×C151—41no—
151C22×C151—41no—
152C22×C151—81no—
153C22×C151—81no—
154C22×C151—82no—
155C22×C151—82no—
156C22×C151—81no—
157C22×C151—161no—
158C22×C151#35311yescharacteristic
159C22×C151#35311yescharacteristic
160C22×C151C2×#9911yescharacteristic
161C4×C151—21no—
162C4×C151—21no—
163C4×C151—41no—
164C4×C151—41no—
165C4×C151—41no—
166C4×C151—41no—
167C4×C151—81no—
168C4×C151—162no—
169C2×C151—21no—
170C2×C151—21no—
171C2×C151—21no—
172C2×C151—81no—
173C2×C151—81no—
174C2×C151—161no—
175C2×C151C2×#35311yescharacteristic
176#13578C15111yesmaximal, characteristic, direct product factor, Sylow
177C151#1357811yescharacteristic, direct product factor, Sylow
178#3153C2×C15111yescharacteristic
179C2×#355C2×C15111yescharacteristic, complemented
180C2×#354C2×C15111yescharacteristic, complemented
181#1407C2×C15112yescomplemented
182#1406C2×C15112yescomplemented
183C2×#137—21no—
184C2×AGL(1,ℤ/8)—21no—
185#532C4×C15111yescharacteristic, complemented
186C2×#124C22×C15111yescharacteristic
187#497C4×C15111yescharacteristic, complemented
188#497C22×C15111yescharacteristic
189#496C4×C15111yescharacteristic, complemented
190#496C22×C15111yescharacteristic
191#495—21no—
192#495—21no—
193C2×#101—21no—
194C2×#99—21no—
195C4×M16C4×C15111yescharacteristic, complemented
196#355—21no—
197#355C22×C15112yes—
198#354—21no—
199#354C22×C15112yes—
200C2×#41—21no—
201C22×Q8D4×C15111yescharacteristic
202C22×D4D4×C15111yescharacteristic
203#137—42no—
204#137—42no—
205AGL(1,ℤ/8)—42no—
206AGL(1,ℤ/8)—42no—
207C2×Q16—21no—
208C2×Q16—21no—
209C2×SD16—21no—
210C2×SD16—21no—
211C2×SD16—21no—
212C2×SD16—21no—
213C2×D8—21no—
214C2×D8—21no—
215C2×M16—21no—
216C2×M16—41no—
217C4⋊Q8C2×C4×C15111yescharacteristic, complemented
218#127C2×C4×C15111yescharacteristic, complemented
219#124—21no—
220#124—41no—
221#124C2×C4×C15111yescharacteristic
222#124D4×C15111yescharacteristic
223#124D4×C15111yescharacteristic
224#124C23×C15111yescharacteristic
225#121—41no—
226C4×D4—41no—
227C2×C22⋊C4—41no—
228C2×C22⋊C4—41no—
229C2×C42C2×C4×C15111yescharacteristic
230#101—22no—
231#101—41no—
232#99—22no—
233#99—41no—
234#97C2×C4×C15111yescharacteristic, complemented
235#97C2×C4×C15111yescharacteristic, complemented
236C4×C8—21no—
237C24—21no—
238#41—41no—
239#41—41no—
240#41—41no—
241#41—41no—
242C2×Q8—21no—
243C2×Q8—21no—
244C2×Q8—41no—
245C2×Q8—41no—
246C2×Q8C22⋊C4×C15111yescharacteristic
247C2×Q8C2×D4×C15111yescharacteristic
248C2×D4—21no—
249C2×D4—21no—
250C2×D4—41no—
251C2×D4—41no—
252C2×D4—41no—
253C2×D4—41no—
254C2×D4C22⋊C4×C15111yescharacteristic
255C2×D4C2×D4×C15111yescharacteristic
256C22×C4—21no—
257C22×C4—21no—
258C22×C4—41no—
259C22×C4—81no—
260C22×C4C22⋊C4×C15111yescharacteristic
261Q16—42no—
262Q16—42no—
263SD16—42no—
264SD16—42no—
265SD16—42no—
266SD16—42no—
267D8—42no—
268D8—42no—
269M16—22no—
270M16—22no—
271M16—42no—
272M16—81no—
273C2×C8—21no—
274C2×C8—21no—
275C2×C8—81no—
276C4⋊C4—41no—
277C22⋊C4—41no—
278C22⋊C4—41no—
279C22⋊C4—41no—
280C22⋊C4—41no—
281C22⋊C4—42no—
282C22⋊C4—42no—
283C22⋊C4—81no—
284C42—21no—
285C42C22⋊C4×C15111yescharacteristic
286C42C22×C4×C15111yescharacteristic
287C23—21no—
288C23—21no—
289C23—41no—
290C23—41no—
291C23—41no—
292C23—82no—
293C23#99×C15111yescharacteristic
294Q8—41no—
295Q8—41no—
296Q8—41no—
297Q8—41no—
298Q8—81no—
299D4—41no—
300D4—41no—
301D4—41no—
302D4—41no—
303D4—81no—
304D4—82no—
305D4—81no—
306D4—81no—
307C2×C4—21no—
308C2×C4—21no—
309C2×C4—21no—
310C2×C4—21no—
311C2×C4—21no—
312C2×C4—21no—
313C2×C4—41no—
314C2×C4—41no—
315C2×C4—41no—
316C2×C4—81no—
317C2×C4—81no—
318C2×C4—81no—
319C2×C4—82no—
320C2×C4—162no—
321C2×C4#99×C15111yescharacteristic
322C2×C4C2×C22⋊C4×C15111yescharacteristic
323C8—22no—
324C8—22no—
325C8—82no—
326C22—41no—
327C22—41no—
328C22—41no—
329C22—81no—
330C22—81no—
331C22—82no—
332C22—82no—
333C22—81no—
334C22—161no—
335C22#353×C15111yescharacteristic
336C22#353×C15111yescharacteristic
337C22C2×#99×C15111yescharacteristic
338C4—21no—
339C4—21no—
340C4—41no—
341C4—41no—
342C4—41no—
343C4—41no—
344C4—81no—
345C4—162no—
346C2—21no—
347C2—21no—
348C2—21no—
349C2—81no—
350C2—81no—
351C2—161no—
352C2C2×#353×C15111yescharacteristic
subgroupG/H
center, socleC2×C151C2×#353
commutatorC42C22×C4×C151
Frattini, cosocle#124C23×C151
prime radical#3153×C151C2
LCSG ▷ C42 ▷ C2×C4 ▷ C22 ▷ C2 ▷ C1
UCSC1 ◁ C2×C151 ◁ C23×C151 ◁ C22×C4×C151 ◁ C2×#124×C151 ◁ G

Complements

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

NH ≅ G/Ntype
2. C2×#355×C151C2semidirect
3. C2×#354×C151C2semidirect
4. #1407×C151C2semidirect
5. #1406×C151C2semidirect
8. #532×C151C4semidirect
10. #497×C151C4semidirect
12. #496×C151C4semidirect
18. C4×M16×C151C4semidirect
40. C4⋊Q8×C151C2×C4semidirect
41. #127×C151C2×C4semidirect
57. #97×C151C2×C4semidirect
58. #97×C151C2×C4semidirect
176. #13578C151direct (N × H)
177. C151#13578direct (N × H)
179. C2×#355C2×C151semidirect
180. C2×#354C2×C151semidirect
181. #1407C2×C151semidirect
182. #1406C2×C151semidirect
185. #532C4×C151semidirect
187. #497C4×C151semidirect
189. #496C4×C151semidirect
195. C4×M16C4×C151semidirect
217. C4⋊Q8C2×C4×C151semidirect
218. #127C2×C4×C151semidirect
234. #97C2×C4×C151semidirect
235. #97C2×C4×C151semidirect

Maximal subgroups

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

subgroupindex
#3153×C1512
C2×#355×C1512
C2×#354×C1512
#1407×C1512
#1406×C1512
#13578151