Finite Group Databasebeta

← C151×#23896 #50328821

Subgroups — C151×#23896 #50328821

Order 38656.

1764 conjugacy classes of subgroups; only the 282 normal subgroups are shown.

normal subgroups only

This group has 1764 conjugacy classes of subgroups — too many to draw. Only its 282 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#3485×C151C212yesmaximal, complemented
2#3026×C151C214yesmaximal, complemented
3#2022×C151C214yesmaximal, complemented
4#1913×C151C214yesmaximal, complemented
5#1782×C151C211yesmaximal, characteristic, complemented
6C2×H(5,𝔽2)×C151C2212yescomplemented
7C22×#41×C151C2211yescharacteristic
8#547×C151—28no—
9#546×C151—28no—
10#536×C151—22no—
11#534×C151—22no—
12#534×C151C2214yes—
13#534×C151C2214yes—
14#525×C151—22no—
15C2×#123×C151—22no—
16C2×#123×C151C2214yescomplemented
17#518×C151C2212yescomplemented
18#458×C151—28no—
19B(4,𝔽2)×C151—28no—
20C2×#99×C151C2218yescomplemented
21C2×#99×C151C2214yescomplemented
22#381×C151C2212yescomplemented
23#342×C151C2214yescomplemented
24H(5,𝔽2)×C151—28no—
25H(5,𝔽2)×C151—44no—
26C2×#41×C151—28no—
27C2×#41×C151—22no—
28C2×#41×C151—24no—
29C2×#41×C151—42no—
30C22×Q8×C151D411yescharacteristic, complemented
31C22×D4×C151—22no—
32C22×D4×C151—24no—
33C22×D4×C151D411yescharacteristic, complemented
34C22×D4×C151C2314yes—
35C22×D4×C151C2312yes—
36C23×C4×C151D412yescomplemented
37C23×C4×C151C2311yescharacteristic
38C4⋊Q8×C151—24no—
39#124×C151—24no—
40#123×C151—28no—
41#123×C151—24no—
42#123×C151—24no—
43#123×C151—24no—
44#123×C151—24no—
45#123×C151—28no—
46#123×C151—28no—
47#123×C151D418yescomplemented
48C22⋊Q8×C151—24no—
49C22⋊Q8×C151—28no—
50#121×C151—24no—
51#121×C151—28no—
52#121×C151—28no—
53#121×C151—44no—
54#120×C151—24no—
55#120×C151—24no—
56#120×C151—28no—
57#120×C151D418yescomplemented
58C4×D4×C151—28no—
59C4×D4×C151—28no—
60#117×C151—22no—
61C2×C4⋊C4×C151—24no—
62C2×C4⋊C4×C151—24no—
63C2×C22⋊C4×C151—24no—
64C2×C22⋊C4×C151—22no—
65C2×C22⋊C4×C151—24no—
66C2×C22⋊C4×C151C2314yes—
67C2×C22⋊C4×C151C2314yes—
68#99×C151—28no—
69#99×C151—216no—
70#95×C151—42no—
71C24×C151—22no—
72C24×C151C2×D414yes—
73C24×C151C2×D412yes—
74C24×C151C2411yescharacteristic
75#41×C151—416no—
76#41×C151—44no—
77#41×C151—416no—
78#41×C151—44no—
79C2×Q8×C151—22no—
80C2×Q8×C151—24no—
81C2×Q8×C151—41no—
82C2×D4×C151—24no—
83C2×D4×C151—24no—
84C2×D4×C151—24no—
85C2×D4×C151—22no—
86C2×D4×C151—24no—
87C2×D4×C151—28no—
88C2×D4×C151—24no—
89C2×D4×C151—24no—
90C2×D4×C151—28no—
91C2×D4×C151—28no—
92C2×D4×C151—41no—
93C2×D4×C151—416no—
94C2×D4×C151—44no—
95C2×D4×C151—44no—
96C2×D4×C151—44no—
97C2×D4×C151C2×D418yes—
98C22×C4×C151—22no—
99C22×C4×C151—24no—
100C22×C4×C151—24no—
101C22×C4×C151—24no—
102C22×C4×C151—22no—
103C22×C4×C151—24no—
104C22×C4×C151—24no—
105C22×C4×C151—24no—
106C22×C4×C151—24no—
107C22×C4×C151—42no—
108C22×C4×C151C2×D414yes—
109C22×C4×C151C2×D411yescharacteristic
110C22×C4×C151C2×D411yescharacteristic
111C22×C4×C151C2×D412yes—
112C4⋊C4×C151—28no—
113C4⋊C4×C151—28no—
114C4⋊C4×C151—24no—
115C4⋊C4×C151—28no—
116C4⋊C4×C151—44no—
117C22⋊C4×C151—28no—
118C22⋊C4×C151—24no—
119C22⋊C4×C151—28no—
120C22⋊C4×C151—28no—
121C22⋊C4×C151—24no—
122C22⋊C4×C151—28no—
123C22⋊C4×C151—28no—
124C22⋊C4×C151—48no—
125C22⋊C4×C151C2×D418yes—
126C42×C151—24no—
127C23×C151—22no—
128C23×C151—22no—
129C23×C151—24no—
130C23×C151—28no—
131C23×C151—44no—
132C23×C151—81no—
133C23×C151—82no—
134C23×C151—84no—
135C23×C151—84no—
136C23×C151#12014yes—
137C23×C151C22×D411yescharacteristic
138C23×C151C22×D414yes—
139C23×C151H(5,𝔽2)12yes—
140Q8×C151—44no—
141Q8×C151—44no—
142D4×C151—216no—
143D4×C151—44no—
144D4×C151—44no—
145D4×C151—48no—
146D4×C151—44no—
147D4×C151—44no—
148D4×C151—48no—
149D4×C151—88no—
150D4×C151—164no—
151C2×C4×C151—22no—
152C2×C4×C151—22no—
153C2×C4×C151—22no—
154C2×C4×C151—24no—
155C2×C4×C151—28no—
156C2×C4×C151—22no—
157C2×C4×C151—21no—
158C2×C4×C151—21no—
159C2×C4×C151—28no—
160C2×C4×C151—48no—
161C2×C4×C151—48no—
162C2×C4×C151—44no—
163C2×C4×C151—44no—
164C2×C4×C151—44no—
165C2×C4×C151—84no—
166C2×C4×C151—84no—
167C2×C4×C151—84no—
168C2×C4×C151#12014yes—
169C22×C151—24no—
170C22×C151—42no—
171C22×C151—44no—
172C22×C151—41no—
173C22×C151—44no—
174C22×C151—44no—
175C22×C151—48no—
176C22×C151—44no—
177C22×C151—82no—
178C22×C151—88no—
179C22×C151—164no—
180C22×C151C2×#12012yes—
181C22×C151#53411yescharacteristic
182C22×C151D4214yes—
183C4×C151—24no—
184C4×C151—48no—
185C4×C151—41no—
186C4×C151—44no—
187C4×C151—41no—
188C4×C151—84no—
189C2×C151—22no—
190C2×C151—24no—
191C2×C151—21no—
192C2×C151—44no—
193C2×C151—84no—
194C2×C151#240311yescharacteristic
195#23896C15111yesmaximal, characteristic, direct product factor, Sylow
196C151#2389611yescharacteristic, direct product factor, Sylow
197#3485C2×C15112yescomplemented
198#3026C2×C15114yescomplemented
199#2022C2×C15114yescomplemented
200#1913C2×C15114yescomplemented
201#1782C2×C15111yescharacteristic, complemented
202C2×H(5,𝔽2)C22×C15112yescomplemented
203C22×#41C22×C15111yescharacteristic
204#547—28no—
205#546—28no—
206#536—22no—
207#534—22no—
208#534C22×C15114yes—
209#534C22×C15114yes—
210#525—22no—
211C2×#123—22no—
212C2×#123C22×C15114yescomplemented
213#518C22×C15112yescomplemented
214#458—28no—
215B(4,𝔽2)—28no—
216C2×#99C22×C15118yescomplemented
217C2×#99C22×C15114yescomplemented
218#381C22×C15112yescomplemented
219#342C22×C15114yescomplemented
220H(5,𝔽2)—28no—
221H(5,𝔽2)—44no—
222C2×#41—28no—
223C2×#41—22no—
224C2×#41—24no—
225C2×#41—42no—
226C22×Q8D4×C15111yescharacteristic, complemented
227C22×D4—22no—
228C22×D4—24no—
229C22×D4D4×C15111yescharacteristic, complemented
230C22×D4C23×C15114yes—
231C22×D4C23×C15112yes—
232C23×C4D4×C15112yescomplemented
233C23×C4C23×C15111yescharacteristic
234C4⋊Q8—24no—
235#124—24no—
236#123—28no—
237#123—24no—
238#123—24no—
239#123—24no—
240#123—24no—
241#123—28no—
242#123—28no—
243#123D4×C15118yescomplemented
244C22⋊Q8—24no—
245C22⋊Q8—28no—
246#121—24no—
247#121—28no—
248#121—28no—
249#121—44no—
250#120—24no—
251#120—24no—
252#120—28no—
253#120D4×C15118yescomplemented
254C4×D4—28no—
255C4×D4—28no—
256#117—22no—
257C2×C4⋊C4—24no—
258C2×C4⋊C4—24no—
259C2×C22⋊C4—24no—
260C2×C22⋊C4—22no—
261C2×C22⋊C4—24no—
262C2×C22⋊C4C23×C15114yes—
263C2×C22⋊C4C23×C15114yes—
264#99—28no—
265#99—216no—
266#95—42no—
267C24—22no—
268C24C2×D4×C15114yes—
269C24C2×D4×C15112yes—
270C24C24×C15111yescharacteristic
271#41—416no—
272#41—44no—
273#41—416no—
274#41—44no—
275C2×Q8—22no—
276C2×Q8—24no—
277C2×Q8—41no—
278C2×D4—24no—
279C2×D4—24no—
280C2×D4—24no—
281C2×D4—22no—
282C2×D4—24no—
283C2×D4—28no—
284C2×D4—24no—
285C2×D4—24no—
286C2×D4—28no—
287C2×D4—28no—
288C2×D4—41no—
289C2×D4—416no—
290C2×D4—44no—
291C2×D4—44no—
292C2×D4—44no—
293C2×D4C2×D4×C15118yes—
294C22×C4—22no—
295C22×C4—24no—
296C22×C4—24no—
297C22×C4—24no—
298C22×C4—22no—
299C22×C4—24no—
300C22×C4—24no—
301C22×C4—24no—
302C22×C4—24no—
303C22×C4—42no—
304C22×C4C2×D4×C15114yes—
305C22×C4C2×D4×C15111yescharacteristic
306C22×C4C2×D4×C15111yescharacteristic
307C22×C4C2×D4×C15112yes—
308C4⋊C4—28no—
309C4⋊C4—28no—
310C4⋊C4—24no—
311C4⋊C4—28no—
312C4⋊C4—44no—
313C22⋊C4—28no—
314C22⋊C4—24no—
315C22⋊C4—28no—
316C22⋊C4—28no—
317C22⋊C4—24no—
318C22⋊C4—28no—
319C22⋊C4—28no—
320C22⋊C4—48no—
321C22⋊C4C2×D4×C15118yes—
322C42—24no—
323C23—22no—
324C23—22no—
325C23—24no—
326C23—28no—
327C23—44no—
328C23—81no—
329C23—82no—
330C23—84no—
331C23—84no—
332C23#120×C15114yes—
333C23C22×D4×C15111yescharacteristic
334C23C22×D4×C15114yes—
335C23H(5,𝔽2)×C15112yes—
336Q8—44no—
337Q8—44no—
338D4—216no—
339D4—44no—
340D4—44no—
341D4—48no—
342D4—44no—
343D4—44no—
344D4—48no—
345D4—88no—
346D4—164no—
347C2×C4—22no—
348C2×C4—22no—
349C2×C4—28no—
350C2×C4—22no—
351C2×C4—22no—
352C2×C4—24no—
353C2×C4—21no—
354C2×C4—21no—
355C2×C4—28no—
356C2×C4—48no—
357C2×C4—48no—
358C2×C4—44no—
359C2×C4—44no—
360C2×C4—44no—
361C2×C4—84no—
362C2×C4—84no—
363C2×C4—84no—
364C2×C4#120×C15114yes—
365C22—24no—
366C22—42no—
367C22—44no—
368C22—41no—
369C22—44no—
370C22—44no—
371C22—48no—
372C22—44no—
373C22—82no—
374C22—88no—
375C22—164no—
376C22C2×#120×C15112yes—
377C22#534×C15111yescharacteristic
378C22D42×C15114yes—
379C4—24no—
380C4—48no—
381C4—41no—
382C4—44no—
383C4—41no—
384C4—84no—
385C2—22no—
386C2—24no—
387C2—21no—
388C2—44no—
389C2—84no—
390C2#2403×C15111yescharacteristic
subgroupG/H
center, socleC2×C151#2403
commutator, Frattini, cosocleC24C24×C151
LCSG ▷ C24 ▷ C2 ▷ C1
UCSC1 ◁ C2×C151 ◁ C24×C151 ◁ G

Complements

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

NH ≅ G/Ntype
1. #3485×C151C2semidirect
2. #3026×C151C2semidirect
3. #2022×C151C2semidirect
4. #1913×C151C2semidirect
5. #1782×C151C2semidirect
6. C2×H(5,𝔽2)×C151C22semidirect
16. C2×#123×C151C22semidirect
17. #518×C151C22semidirect
20. C2×#99×C151C22semidirect
21. C2×#99×C151C22semidirect
22. #381×C151C22semidirect
23. #342×C151C22semidirect
30. C22×Q8×C151D4semidirect
33. C22×D4×C151D4semidirect
36. C23×C4×C151D4semidirect
47. #123×C151D4semidirect
57. #120×C151D4semidirect
195. #23896C151direct (N × H)
196. C151#23896direct (N × H)
197. #3485C2×C151semidirect
198. #3026C2×C151semidirect
199. #2022C2×C151semidirect
200. #1913C2×C151semidirect
201. #1782C2×C151semidirect
202. C2×H(5,𝔽2)C22×C151semidirect
212. C2×#123C22×C151semidirect
213. #518C22×C151semidirect
216. C2×#99C22×C151semidirect
217. C2×#99C22×C151semidirect
218. #381C22×C151semidirect
219. #342C22×C151semidirect
226. C22×Q8D4×C151semidirect
229. C22×D4D4×C151semidirect
232. C23×C4D4×C151semidirect
243. #123D4×C151semidirect
253. #120D4×C151semidirect

Maximal subgroups

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

subgroupindex
#3485×C1512
#3026×C1512
#2022×C1512
#1913×C1512
#1782×C1512
#23896151