Finite Group Databasebeta

← #3257×C251 #44547844

Subgroups — #3257×C251 #44547844

Order 32128.

358 conjugacy classes of subgroups.

The subgroup lattice has 358 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#542×C251C211yesmaximal, characteristic, complemented
2#540×C251C211yesmaximal, characteristic, complemented
3#479×C251C211yesmaximal, characteristic, complemented
4#478×C251C211yesmaximal, characteristic, complemented
5#476×C251C211yesmaximal, characteristic, complemented
6#474×C251C211yesmaximal, characteristic, complemented
7#469×C251C211yesmaximal, characteristic, complemented
8#468×C251C211yesmaximal, characteristic, complemented
9#466×C251C211yesmaximal, characteristic, complemented
10#465×C251C211yesmaximal, characteristic, complemented
11#463×C251C211yesmaximal, characteristic, complemented
12#463×C251C211yesmaximal, characteristic, complemented
13#460×C251C211yesmaximal, characteristic, complemented
14#459×C251C211yesmaximal, characteristic, complemented
15#424×C251C211yesmaximal, characteristic, complemented
16C2×SD16×C251C2211yescharacteristic
17C2×SD16×C251C2211yescharacteristic
18C2×D8×C251C2211yescharacteristic
19C2×D8×C251C2211yescharacteristic
20C2×M16×C251C2211yescharacteristic
21C22×C8×C251C2211yescharacteristic
22C4⋊Q8×C251C2211yescharacteristic
23#127×C251C2211yescharacteristic
24#126×C251—21no—
25#125×C251C2211yescharacteristic
26#124×C251—21no—
27#124×C251C2211yescharacteristic
28#123×C251—21no—
29#123×C251—21no—
30#123×C251—21no—
31C22⋊Q8×C251—21no—
32C22⋊Q8×C251C2211yescharacteristic
33#121×C251—21no—
34#121×C251—21no—
35#121×C251C2211yescharacteristic
36#121×C251C2211yescharacteristic
37#121×C251C2211yescharacteristic
38C4×Q8×C251C2211yescharacteristic
39C4×D4×C251C2211yescharacteristic
40C4×D4×C251C2211yescharacteristic
41C4×D4×C251C2211yescharacteristic
42#117×C251C2211yescharacteristic
43#107×C251C2211yescharacteristic
44#107×C251C2211yescharacteristic
45#106×C251C2211yescharacteristic
46#106×C251C2211yescharacteristic
47C4⋊C8×C251C2211yescharacteristic
48C4⋊C8×C251C2211yescharacteristic
49C4⋊C8×C251C2211yescharacteristic
50C4⋊C8×C251C2211yescharacteristic
51#103×C251C2211yescharacteristic
52#103×C251C2211yescharacteristic
53#102×C251C2211yescharacteristic
54#102×C251C2211yescharacteristic
55#102×C251C2211yescharacteristic
56#102×C251C2211yescharacteristic
57#102×C251C2211yescharacteristic
58#102×C251C2211yescharacteristic
59C2×Q8×C251—21no—
60C2×Q8×C251C2311yescharacteristic
61C2×D4×C251—21no—
62C2×D4×C251—21no—
63C2×D4×C251—21no—
64C2×D4×C251—21no—
65C2×D4×C251—21no—
66C2×D4×C251C2311yescharacteristic
67C2×D4×C251C2311yescharacteristic
68C2×D4×C251C2311yescharacteristic
69C22×C4×C251—21no—
70C22×C4×C251—21no—
71C22×C4×C251—21no—
72C22×C4×C251C2311yescharacteristic
73SD16×C251—41no—
74SD16×C251—41no—
75D8×C251—41no—
76D8×C251—41no—
77M16×C251—41no—
78C2×C8×C251—41no—
79C2×C8×C251C2311yescharacteristic
80C2×C8×C251C2311yescharacteristic
81C2×C8×C251C2311yescharacteristic
82C2×C8×C251C2311yescharacteristic
83C4⋊C4×C251—21no—
84C4⋊C4×C251—21no—
85C4⋊C4×C251—21no—
86C4⋊C4×C251—21no—
87C4⋊C4×C251—21no—
88C4⋊C4×C251D411yescharacteristic
89C4⋊C4×C251D411yescharacteristic
90C4⋊C4×C251C2311yescharacteristic
91C4⋊C4×C251C2311yescharacteristic
92C4⋊C4×C251C2311yescharacteristic
93C4⋊C4×C251C2311yescharacteristic
94C22⋊C4×C251—21no—
95C22⋊C4×C251—21no—
96C22⋊C4×C251—21no—
97C22⋊C4×C251—21no—
98C22⋊C4×C251—21no—
99C22⋊C4×C251—21no—
100C22⋊C4×C251—21no—
101C22⋊C4×C251—21no—
102C22⋊C4×C251—21no—
103C22⋊C4×C251D411yescharacteristic
104C22⋊C4×C251D411yescharacteristic
105C42×C251—21no—
106C42×C251C2311yescharacteristic
107C42×C251C2311yescharacteristic
108C23×C251—21no—
109C23×C251—21no—
110C23×C251—21no—
111C23×C251C2×D411yescharacteristic
112Q8×C251—21no—
113Q8×C251—21no—
114Q8×C251—81no—
115D4×C251—21no—
116D4×C251—21no—
117D4×C251—21no—
118D4×C251—21no—
119D4×C251—21no—
120D4×C251—21no—
121D4×C251—81no—
122D4×C251—81no—
123D4×C251—81no—
124D4×C251—81no—
125D4×C251—81no—
126C2×C4×C251—21no—
127C2×C4×C251—21no—
128C2×C4×C251—21no—
129C2×C4×C251—21no—
130C2×C4×C251—21no—
131C2×C4×C251—21no—
132C2×C4×C251—21no—
133C2×C4×C251—81no—
134C2×C4×C251—81no—
135C2×C4×C251—81no—
136C2×C4×C251C2×D411yescharacteristic
137C2×C4×C251C2×D411yescharacteristic
138C2×C4×C251C2×D411yescharacteristic
139C2×C4×C251C2×D411yescharacteristic
140C2×C4×C251C2×D411yescharacteristic
141C2×C4×C251C2411yescharacteristic
142C8×C251—21no—
143C8×C251—21no—
144C8×C251—21no—
145C8×C251—21no—
146C22×C251—21no—
147C22×C251—21no—
148C22×C251—21no—
149C22×C251—41no—
150C22×C251—41no—
151C22×C251—41no—
152C22×C251—41no—
153C22×C251—41no—
154C22×C251—41no—
155C22×C251—41no—
156C22×C251—41no—
157C22×C251—41no—
158C22×C251C22×D411yescharacteristic
159C4×C251—21no—
160C4×C251—21no—
161C4×C251—21no—
162C4×C251—21no—
163C4×C251—21no—
164C4×C251—41no—
165C4×C251—41no—
166C4×C251—41no—
167C4×C251—41no—
168C4×C251—41no—
169C4×C251H(5,𝔽2)11yescharacteristic
170C4×C2512−1+411yescharacteristic
171C2×C251—41no—
172C2×C251—81no—
173C2×C251—81no—
174C2×C251—81no—
175C2×C251#53711yescharacteristic
176C2×C251#57611yescharacteristic
177C2×C251#57711yescharacteristic
178C251#325711yescharacteristic, direct product factor, Sylow
179#3257C25111yesmaximal, characteristic, direct product factor, Sylow
180#542C2×C25111yescharacteristic, complemented
181#540C2×C25111yescharacteristic, complemented
182#479C2×C25111yescharacteristic, complemented
183#478C2×C25111yescharacteristic, complemented
184#476C2×C25111yescharacteristic, complemented
185#474C2×C25111yescharacteristic, complemented
186#469C2×C25111yescharacteristic, complemented
187#468C2×C25111yescharacteristic, complemented
188#466C2×C25111yescharacteristic, complemented
189#465C2×C25111yescharacteristic, complemented
190#463C2×C25111yescharacteristic, complemented
191#463C2×C25111yescharacteristic, complemented
192#460C2×C25111yescharacteristic, complemented
193#459C2×C25111yescharacteristic, complemented
194#424C2×C25111yescharacteristic, complemented
195C2×SD16C22×C25111yescharacteristic
196C2×SD16C22×C25111yescharacteristic
197C2×D8C22×C25111yescharacteristic
198C2×D8C22×C25111yescharacteristic
199C2×M16C22×C25111yescharacteristic
200C22×C8C22×C25111yescharacteristic
201C4⋊Q8C22×C25111yescharacteristic
202#127C22×C25111yescharacteristic
203#126—21no—
204#125C22×C25111yescharacteristic
205#124—21no—
206#124C22×C25111yescharacteristic
207#123—21no—
208#123—21no—
209#123—21no—
210C22⋊Q8—21no—
211C22⋊Q8C22×C25111yescharacteristic
212#121—21no—
213#121—21no—
214#121C22×C25111yescharacteristic
215#121C22×C25111yescharacteristic
216#121C22×C25111yescharacteristic
217C4×Q8C22×C25111yescharacteristic
218C4×D4C22×C25111yescharacteristic
219C4×D4C22×C25111yescharacteristic
220C4×D4C22×C25111yescharacteristic
221#117C22×C25111yescharacteristic
222#107C22×C25111yescharacteristic
223#107C22×C25111yescharacteristic
224#106C22×C25111yescharacteristic
225#106C22×C25111yescharacteristic
226C4⋊C8C22×C25111yescharacteristic
227C4⋊C8C22×C25111yescharacteristic
228C4⋊C8C22×C25111yescharacteristic
229C4⋊C8C22×C25111yescharacteristic
230#103C22×C25111yescharacteristic
231#103C22×C25111yescharacteristic
232#102C22×C25111yescharacteristic
233#102C22×C25111yescharacteristic
234#102C22×C25111yescharacteristic
235#102C22×C25111yescharacteristic
236#102C22×C25111yescharacteristic
237#102C22×C25111yescharacteristic
238C2×Q8—21no—
239C2×Q8C23×C25111yescharacteristic
240C2×D4—21no—
241C2×D4—21no—
242C2×D4—21no—
243C2×D4—21no—
244C2×D4—21no—
245C2×D4C23×C25111yescharacteristic
246C2×D4C23×C25111yescharacteristic
247C2×D4C23×C25111yescharacteristic
248C22×C4—21no—
249C22×C4—21no—
250C22×C4—21no—
251C22×C4C23×C25111yescharacteristic
252SD16—41no—
253SD16—41no—
254D8—41no—
255D8—41no—
256M16—41no—
257C2×C8—41no—
258C2×C8C23×C25111yescharacteristic
259C2×C8C23×C25111yescharacteristic
260C2×C8C23×C25111yescharacteristic
261C2×C8C23×C25111yescharacteristic
262C4⋊C4—21no—
263C4⋊C4—21no—
264C4⋊C4—21no—
265C4⋊C4—21no—
266C4⋊C4—21no—
267C4⋊C4D4×C25111yescharacteristic
268C4⋊C4D4×C25111yescharacteristic
269C4⋊C4C23×C25111yescharacteristic
270C4⋊C4C23×C25111yescharacteristic
271C4⋊C4C23×C25111yescharacteristic
272C4⋊C4C23×C25111yescharacteristic
273C22⋊C4—21no—
274C22⋊C4—21no—
275C22⋊C4—21no—
276C22⋊C4—21no—
277C22⋊C4—21no—
278C22⋊C4—21no—
279C22⋊C4—21no—
280C22⋊C4—21no—
281C22⋊C4—21no—
282C22⋊C4D4×C25111yescharacteristic
283C22⋊C4D4×C25111yescharacteristic
284C42—21no—
285C42C23×C25111yescharacteristic
286C42C23×C25111yescharacteristic
287C23—21no—
288C23—21no—
289C23—21no—
290C23C2×D4×C25111yescharacteristic
291Q8—21no—
292Q8—21no—
293Q8—81no—
294D4—21no—
295D4—21no—
296D4—21no—
297D4—21no—
298D4—21no—
299D4—21no—
300D4—81no—
301D4—81no—
302D4—81no—
303D4—81no—
304D4—81no—
305C2×C4—21no—
306C2×C4—21no—
307C2×C4—21no—
308C2×C4—21no—
309C2×C4—21no—
310C2×C4—21no—
311C2×C4—21no—
312C2×C4—81no—
313C2×C4—81no—
314C2×C4—81no—
315C2×C4C2×D4×C25111yescharacteristic
316C2×C4C2×D4×C25111yescharacteristic
317C2×C4C2×D4×C25111yescharacteristic
318C2×C4C2×D4×C25111yescharacteristic
319C2×C4C2×D4×C25111yescharacteristic
320C2×C4C24×C25111yescharacteristic
321C8—21no—
322C8—21no—
323C8—21no—
324C8—21no—
325C22—21no—
326C22—21no—
327C22—21no—
328C22—41no—
329C22—41no—
330C22—41no—
331C22—41no—
332C22—41no—
333C22—41no—
334C22—41no—
335C22—41no—
336C22—41no—
337C22C22×D4×C25111yescharacteristic
338C4—21no—
339C4—21no—
340C4—21no—
341C4—21no—
342C4—21no—
343C4—41no—
344C4—41no—
345C4—41no—
346C4—41no—
347C4—41no—
348C4H(5,𝔽2)×C25111yescharacteristic
349C42−1+4×C25111yescharacteristic
350C2—41no—
351C2—81no—
352C2—81no—
353C2—81no—
354C2#537×C25111yescharacteristic
355C2#576×C25111yescharacteristic
356C2#577×C25111yescharacteristic
subgroupG/H
center, socleC22×C251C22×D4
commutator, Frattini, cosocleC2×C4C24×C251
LCSG ▷ C2×C4 ▷ C2 ▷ C1
UCSC1 ◁ C22×C251 ◁ #117×C251 ◁ G

Complements

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

NH ≅ G/Ntype
1. #542×C251C2semidirect
2. #540×C251C2semidirect
3. #479×C251C2semidirect
4. #478×C251C2semidirect
5. #476×C251C2semidirect
6. #474×C251C2semidirect
7. #469×C251C2semidirect
8. #468×C251C2semidirect
9. #466×C251C2semidirect
10. #465×C251C2semidirect
11. #463×C251C2semidirect
12. #463×C251C2semidirect
13. #460×C251C2semidirect
14. #459×C251C2semidirect
15. #424×C251C2semidirect
178. C251#3257direct (N × H)
179. #3257C251direct (N × H)
180. #542C2×C251semidirect
181. #540C2×C251semidirect
182. #479C2×C251semidirect
183. #478C2×C251semidirect
184. #476C2×C251semidirect
185. #474C2×C251semidirect
186. #469C2×C251semidirect
187. #468C2×C251semidirect
188. #466C2×C251semidirect
189. #465C2×C251semidirect
190. #463C2×C251semidirect
191. #463C2×C251semidirect
192. #460C2×C251semidirect
193. #459C2×C251semidirect
194. #424C2×C251semidirect

Maximal subgroups

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

subgroupindex
#542×C2512
#540×C2512
#479×C2512
#478×C2512
#476×C2512
#474×C2512
#469×C2512
#468×C2512
#466×C2512
#465×C2512
#463×C2512
#460×C2512
#459×C2512
#424×C2512
#3257251