Finite Group Databasebeta

← #2711×C251 #44547298

Subgroups — #2711×C251 #44547298

Order 32128.

540 conjugacy classes of subgroups.

The subgroup lattice has 436 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#125×C251C211yesmaximal, characteristic, complemented
2C2×#121×C251C211yesmaximal, characteristic
3#399×C251C211yesmaximal, characteristic, complemented
4#396×C251C211yesmaximal, characteristic, complemented
5#394×C251C211yesmaximal, characteristic, complemented
6#394×C251C211yesmaximal, characteristic, complemented
7#394×C251C211yesmaximal, characteristic, complemented
8#394×C251C211yesmaximal, characteristic, complemented
9#390×C251C211yesmaximal, characteristic, complemented
10#390×C251C211yesmaximal, characteristic, complemented
11#388×C251C211yesmaximal, characteristic, complemented
12#388×C251C211yesmaximal, characteristic, complemented
13#387×C251C211yesmaximal, characteristic, complemented
14#385×C251C211yesmaximal, characteristic, complemented
15#385×C251C211yesmaximal, characteristic, complemented
16C22×D4×C251C2211yescharacteristic
17C22×D4×C251C2211yescharacteristic
18C22×D4×C251C2211yescharacteristic
19C23×C4×C251C2211yescharacteristic
20#125×C251—24no—
21#121×C251—24no—
22C2×C4⋊C4×C251C2211yescharacteristic
23C2×C4⋊C4×C251C2211yescharacteristic
24C2×C4⋊C4×C251C2211yescharacteristic
25C2×C4⋊C4×C251C2211yescharacteristic
26C2×C4⋊C4×C251C2211yescharacteristic
27C2×C4⋊C4×C251C2211yescharacteristic
28C2×C4⋊C4×C251C2211yescharacteristic
29C2×C4⋊C4×C251C2211yescharacteristic
30C2×C4⋊C4×C251C2211yescharacteristic
31C2×C22⋊C4×C251C2211yescharacteristic
32C2×C22⋊C4×C251C2211yescharacteristic
33C2×C22⋊C4×C251C2211yescharacteristic
34C2×C22⋊C4×C251C2211yescharacteristic
35C2×C22⋊C4×C251C2211yescharacteristic
36C2×C22⋊C4×C251C2211yescharacteristic
37C2×C22⋊C4×C251C2211yescharacteristic
38C2×C22⋊C4×C251C2211yescharacteristic
39C2×C22⋊C4×C251C2211yescharacteristic
40C2×C22⋊C4×C251C2211yescharacteristic
41C2×C22⋊C4×C251C2211yescharacteristic
42C2×C22⋊C4×C251C2211yescharacteristic
43C2×C22⋊C4×C251C2211yescharacteristic
44C2×C22⋊C4×C251C2211yescharacteristic
45C2×C42×C251C2211yescharacteristic
46C2×C42×C251C2211yescharacteristic
47#95×C251C2211yescharacteristic
48#95×C251C2211yescharacteristic
49#95×C251C2211yescharacteristic
50#95×C251C2211yescharacteristic
51#95×C251C2211yescharacteristic
52#95×C251C2211yescharacteristic
53C24×C251C2311yescharacteristic
54C24×C251C2311yescharacteristic
55C24×C251C2311yescharacteristic
56C2×D4×C251—22no—
57C2×D4×C251—22no—
58C2×D4×C251—22no—
59C2×D4×C251—22no—
60C2×D4×C251—22no—
61C2×D4×C251—41no—
62C2×D4×C251—41no—
63C2×D4×C251—41no—
64C2×D4×C251—41no—
65C22×C4×C251—22no—
66C22×C4×C251—41no—
67C22×C4×C251—41no—
68C22×C4×C251C2311yescharacteristic
69C22×C4×C251C2311yescharacteristic
70C22×C4×C251C2311yescharacteristic
71C22×C4×C251C2311yescharacteristic
72C22×C4×C251C2311yescharacteristic
73C22×C4×C251C2311yescharacteristic
74C22×C4×C251C2311yescharacteristic
75C22×C4×C251C2311yescharacteristic
76C22×C4×C251C2311yescharacteristic
77C22×C4×C251C2311yescharacteristic
78C22×C4×C251C2311yescharacteristic
79C22×C4×C251C2311yescharacteristic
80C4⋊C4×C251—22no—
81C4⋊C4×C251—22no—
82C4⋊C4×C251—22no—
83C4⋊C4×C251—22no—
84C4⋊C4×C251—22no—
85C4⋊C4×C251—41no—
86C4⋊C4×C251—41no—
87C4⋊C4×C251—41no—
88C4⋊C4×C251D414yes—
89C22⋊C4×C251—22no—
90C22⋊C4×C251—22no—
91C22⋊C4×C251—22no—
92C22⋊C4×C251—22no—
93C22⋊C4×C251—22no—
94C22⋊C4×C251—22no—
95C22⋊C4×C251—41no—
96C22⋊C4×C251—41no—
97C22⋊C4×C251—41no—
98C22⋊C4×C251—41no—
99C22⋊C4×C251—41no—
100C22⋊C4×C251—41no—
101C22⋊C4×C251—41no—
102C22⋊C4×C251—41no—
103C42×C251—22no—
104C42×C251—41no—
105C23×C251—21no—
106C23×C251—21no—
107C23×C251—21no—
108C23×C251—21no—
109C23×C251—21no—
110C23×C251—21no—
111C23×C251—21no—
112C23×C251—21no—
113C23×C251—21no—
114C23×C251—21no—
115C23×C251—21no—
116C23×C251—21no—
117C23×C251—21no—
118C23×C251—21no—
119C23×C251—21no—
120C23×C251—21no—
121C23×C251—21no—
122C23×C251—21no—
123C23×C251—21no—
124C23×C251—21no—
125C23×C251#4112yes—
126C23×C251C2411yescharacteristic
127D4×C251—44no—
128D4×C251—44no—
129D4×C251—44no—
130C2×C4×C251—21no—
131C2×C4×C251—21no—
132C2×C4×C251—21no—
133C2×C4×C251—21no—
134C2×C4×C251—21no—
135C2×C4×C251—21no—
136C2×C4×C251—21no—
137C2×C4×C251—21no—
138C2×C4×C251—21no—
139C2×C4×C251—21no—
140C2×C4×C251—21no—
141C2×C4×C251—21no—
142C2×C4×C251—21no—
143C2×C4×C251—21no—
144C2×C4×C251—21no—
145C2×C4×C251—21no—
146C2×C4×C251—21no—
147C2×C4×C251—21no—
148C2×C4×C251—21no—
149C2×C4×C251—21no—
150C2×C4×C251—21no—
151C2×C4×C251—21no—
152C2×C4×C251—21no—
153C2×C4×C251—21no—
154C2×C4×C251—21no—
155C2×C4×C251—21no—
156C2×C4×C251—21no—
157C2×C4×C251—21no—
158C2×C4×C251—21no—
159C2×C4×C251—21no—
160C2×C4×C251—21no—
161C2×C4×C251—21no—
162C2×C4×C251—44no—
163C2×C4×C251C2×D412yes—
164C2×C4×C251C2×D412yes—
165C2×C4×C251C2×D412yes—
166C2×C4×C251#4112yes—
167C22×C251—22no—
168C22×C251—22no—
169C22×C251—22no—
170C22×C251—41no—
171C22×C251—41no—
172C22×C251—41no—
173C22×C251—41no—
174C22×C251—41no—
175C22×C251—41no—
176C22×C251—41no—
177C22×C251—41no—
178C22×C251—41no—
179C22×C251—41no—
180C22×C251—41no—
181C22×C251—41no—
182C22×C251—41no—
183C22×C251—41no—
184C22×C251—41no—
185C22×C251—41no—
186C22×C251—41no—
187C22×C251—41no—
188C22×C251C22×D411yescharacteristic
189C22×C251C2×#4111yescharacteristic
190C22×C251C2×#4111yescharacteristic
191C22×C251H(5,𝔽2)11yescharacteristic
192C22×C251H(5,𝔽2)11yescharacteristic
193C22×C251H(5,𝔽2)11yescharacteristic
194C22×C2512−1+411yescharacteristic
195C4×C251—22no—
196C4×C251—22no—
197C4×C251—22no—
198C4×C251—22no—
199C4×C251—41no—
200C4×C251—41no—
201C4×C251—41no—
202C4×C251—41no—
203C4×C251—41no—
204C4×C251—41no—
205C4×C251—41no—
206C4×C251—41no—
207C2×C251—42no—
208C2×C251—81no—
209C2×C251—81no—
210C2×C251#53511yescharacteristic
211C2×C251#53911yescharacteristic
212C2×C251#54011yescharacteristic
213C2×C251#54611yescharacteristic
214C2×C251#54811yescharacteristic
215C2×C251#55311yescharacteristic
216C2×C251#56211yescharacteristic
217C251#271111yescharacteristic, direct product factor, Sylow
218#2711C25111yesmaximal, characteristic, direct product factor, Sylow
219C2×#125C2×C25111yescharacteristic, complemented
220C2×#121C2×C25111yescharacteristic
221#399C2×C25111yescharacteristic, complemented
222#396C2×C25111yescharacteristic, complemented
223#394C2×C25111yescharacteristic, complemented
224#394C2×C25111yescharacteristic, complemented
225#394C2×C25111yescharacteristic, complemented
226#394C2×C25111yescharacteristic, complemented
227#390C2×C25111yescharacteristic, complemented
228#390C2×C25111yescharacteristic, complemented
229#388C2×C25111yescharacteristic, complemented
230#388C2×C25111yescharacteristic, complemented
231#387C2×C25111yescharacteristic, complemented
232#385C2×C25111yescharacteristic, complemented
233#385C2×C25111yescharacteristic, complemented
234C22×D4C22×C25111yescharacteristic
235C22×D4C22×C25111yescharacteristic
236C22×D4C22×C25111yescharacteristic
237C23×C4C22×C25111yescharacteristic
238#125—24no—
239#121—24no—
240C2×C4⋊C4C22×C25111yescharacteristic
241C2×C4⋊C4C22×C25111yescharacteristic
242C2×C4⋊C4C22×C25111yescharacteristic
243C2×C4⋊C4C22×C25111yescharacteristic
244C2×C4⋊C4C22×C25111yescharacteristic
245C2×C4⋊C4C22×C25111yescharacteristic
246C2×C4⋊C4C22×C25111yescharacteristic
247C2×C4⋊C4C22×C25111yescharacteristic
248C2×C4⋊C4C22×C25111yescharacteristic
249C2×C22⋊C4C22×C25111yescharacteristic
250C2×C22⋊C4C22×C25111yescharacteristic
251C2×C22⋊C4C22×C25111yescharacteristic
252C2×C22⋊C4C22×C25111yescharacteristic
253C2×C22⋊C4C22×C25111yescharacteristic
254C2×C22⋊C4C22×C25111yescharacteristic
255C2×C22⋊C4C22×C25111yescharacteristic
256C2×C22⋊C4C22×C25111yescharacteristic
257C2×C22⋊C4C22×C25111yescharacteristic
258C2×C22⋊C4C22×C25111yescharacteristic
259C2×C22⋊C4C22×C25111yescharacteristic
260C2×C22⋊C4C22×C25111yescharacteristic
261C2×C22⋊C4C22×C25111yescharacteristic
262C2×C22⋊C4C22×C25111yescharacteristic
263C2×C42C22×C25111yescharacteristic
264C2×C42C22×C25111yescharacteristic
265#95C22×C25111yescharacteristic
266#95C22×C25111yescharacteristic
267#95C22×C25111yescharacteristic
268#95C22×C25111yescharacteristic
269#95C22×C25111yescharacteristic
270#95C22×C25111yescharacteristic
271C24C23×C25111yescharacteristic
272C24C23×C25111yescharacteristic
273C24C23×C25111yescharacteristic
274C2×D4—22no—
275C2×D4—22no—
276C2×D4—22no—
277C2×D4—22no—
278C2×D4—22no—
279C2×D4—41no—
280C2×D4—41no—
281C2×D4—41no—
282C2×D4—41no—
283C22×C4—22no—
284C22×C4—41no—
285C22×C4—41no—
286C22×C4C23×C25111yescharacteristic
287C22×C4C23×C25111yescharacteristic
288C22×C4C23×C25111yescharacteristic
289C22×C4C23×C25111yescharacteristic
290C22×C4C23×C25111yescharacteristic
291C22×C4C23×C25111yescharacteristic
292C22×C4C23×C25111yescharacteristic
293C22×C4C23×C25111yescharacteristic
294C22×C4C23×C25111yescharacteristic
295C22×C4C23×C25111yescharacteristic
296C22×C4C23×C25111yescharacteristic
297C22×C4C23×C25111yescharacteristic
298C4⋊C4—22no—
299C4⋊C4—22no—
300C4⋊C4—22no—
301C4⋊C4—22no—
302C4⋊C4—22no—
303C4⋊C4—41no—
304C4⋊C4—41no—
305C4⋊C4—41no—
306C4⋊C4D4×C25114yes—
307C22⋊C4—22no—
308C22⋊C4—22no—
309C22⋊C4—22no—
310C22⋊C4—22no—
311C22⋊C4—22no—
312C22⋊C4—22no—
313C22⋊C4—41no—
314C22⋊C4—41no—
315C22⋊C4—41no—
316C22⋊C4—41no—
317C22⋊C4—41no—
318C22⋊C4—41no—
319C22⋊C4—41no—
320C22⋊C4—41no—
321C42—22no—
322C42—41no—
323C23—21no—
324C23—21no—
325C23—21no—
326C23—21no—
327C23—21no—
328C23—21no—
329C23—21no—
330C23—21no—
331C23—21no—
332C23—21no—
333C23—21no—
334C23—21no—
335C23—21no—
336C23—21no—
337C23—21no—
338C23—21no—
339C23—21no—
340C23—21no—
341C23—21no—
342C23—21no—
343C23#41×C25112yes—
344C23C24×C25111yescharacteristic
345D4—44no—
346D4—44no—
347D4—44no—
348C2×C4—21no—
349C2×C4—21no—
350C2×C4—21no—
351C2×C4—21no—
352C2×C4—21no—
353C2×C4—21no—
354C2×C4—21no—
355C2×C4—21no—
356C2×C4—21no—
357C2×C4—21no—
358C2×C4—21no—
359C2×C4—21no—
360C2×C4—21no—
361C2×C4—21no—
362C2×C4—21no—
363C2×C4—21no—
364C2×C4—21no—
365C2×C4—21no—
366C2×C4—21no—
367C2×C4—21no—
368C2×C4—21no—
369C2×C4—21no—
370C2×C4—21no—
371C2×C4—21no—
372C2×C4—21no—
373C2×C4—21no—
374C2×C4—21no—
375C2×C4—21no—
376C2×C4—21no—
377C2×C4—21no—
378C2×C4—21no—
379C2×C4—21no—
380C2×C4—44no—
381C2×C4C2×D4×C25112yes—
382C2×C4C2×D4×C25112yes—
383C2×C4C2×D4×C25112yes—
384C2×C4#41×C25112yes—
385C22—22no—
386C22—22no—
387C22—22no—
388C22—41no—
389C22—41no—
390C22—41no—
391C22—41no—
392C22—41no—
393C22—41no—
394C22—41no—
395C22—41no—
396C22—41no—
397C22—41no—
398C22—41no—
399C22—41no—
400C22—41no—
401C22—41no—
402C22—41no—
403C22—41no—
404C22—41no—
405C22—41no—
406C22C22×D4×C25111yescharacteristic
407C22C2×#41×C25111yescharacteristic
408C22C2×#41×C25111yescharacteristic
409C22H(5,𝔽2)×C25111yescharacteristic
410C22H(5,𝔽2)×C25111yescharacteristic
411C22H(5,𝔽2)×C25111yescharacteristic
412C222−1+4×C25111yescharacteristic
413C4—22no—
414C4—22no—
415C4—22no—
416C4—22no—
417C4—41no—
418C4—41no—
419C4—41no—
420C4—41no—
421C4—41no—
422C4—41no—
423C4—41no—
424C4—41no—
425C2—42no—
426C2—81no—
427C2—81no—
428C2#535×C25111yescharacteristic
429C2#539×C25111yescharacteristic
430C2#540×C25111yescharacteristic
431C2#546×C25111yescharacteristic
432C2#548×C25111yescharacteristic
433C2#553×C25111yescharacteristic
434C2#562×C25111yescharacteristic
subgroupG/H
center, socleC23×C251C24
commutator, Frattini, cosocleC23C24×C251
prime radicalC2×#121×C251C2
LCSG ▷ C23 ▷ C1
UCSC1 ◁ C23×C251 ◁ G

Complements

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

NH ≅ G/Ntype
1. C2×#125×C251C2semidirect
3. #399×C251C2semidirect
4. #396×C251C2semidirect
5. #394×C251C2semidirect
6. #394×C251C2semidirect
7. #394×C251C2semidirect
8. #394×C251C2semidirect
9. #390×C251C2semidirect
10. #390×C251C2semidirect
11. #388×C251C2semidirect
12. #388×C251C2semidirect
13. #387×C251C2semidirect
14. #385×C251C2semidirect
15. #385×C251C2semidirect
217. C251#2711direct (N × H)
218. #2711C251direct (N × H)
219. C2×#125C2×C251semidirect
221. #399C2×C251semidirect
222. #396C2×C251semidirect
223. #394C2×C251semidirect
224. #394C2×C251semidirect
225. #394C2×C251semidirect
226. #394C2×C251semidirect
227. #390C2×C251semidirect
228. #390C2×C251semidirect
229. #388C2×C251semidirect
230. #388C2×C251semidirect
231. #387C2×C251semidirect
232. #385C2×C251semidirect
233. #385C2×C251semidirect

Maximal subgroups

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

subgroupindex
C2×#125×C2512
C2×#121×C2512
#399×C2512
#396×C2512
#394×C2512
#390×C2512
#388×C2512
#387×C2512
#385×C2512
#2711251