Finite Group Databasebeta

← #44558453

Subgroups — #44558453

Order 32128.

522 conjugacy classes of subgroups.

The subgroup lattice has 464 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C251⋊#401C211yesmaximal, characteristic, complemented
2#30631616C211yesmaximal, characteristic, complemented
3#30631615C211yesmaximal, characteristic, complemented
4#30631612C211yesmaximal, characteristic, complemented
5#30631610C211yesmaximal, characteristic, complemented
6#30631603C211yesmaximal, characteristic, complemented
7#30631602C211yesmaximal, characteristic, complemented
8#30631601C211yesmaximal, characteristic, complemented
9#30631595C211yesmaximal, characteristic, complemented
10#30631595C211yesmaximal, characteristic, complemented
11#30631595C211yesmaximal, characteristic, complemented
12#30631567C211yesmaximal, characteristic, complemented
13#30631567C211yesmaximal, characteristic, complemented
14#30631564C211yesmaximal, characteristic, complemented
15#392×C251C211yesmaximal, characteristic, complemented
16C22×C22⋊D251C2211yescharacteristic
17C22×C22⋊D251C2211yescharacteristic
18C22×C22⋊D251C2211yescharacteristic
19C23×Dic251C2211yescharacteristic
20C2×C4⋊Dic251C2211yescharacteristic
21C2×Dic251⋊C4C2211yescharacteristic
22C2×Dic251⋊C4C2211yescharacteristic
23C2×C22⋊Dic251C2211yescharacteristic
24C2×C22⋊Dic251C2211yescharacteristic
25C2×C22⋊Dic251C2211yescharacteristic
26C2×C22⋊Dic251C2211yescharacteristic
27C2×C22⋊Dic251C2211yescharacteristic
28C2×C22⋊Dic251C2211yescharacteristic
29C2×C22⋊Dic251C2211yescharacteristic
30C2×C22⋊Dic251C2211yescharacteristic
31C2×(C2×D251)⋊C4C2211yescharacteristic
32C2×(C2×D251)⋊C4C2211yescharacteristic
33C2×(C2×D251)⋊C4C2211yescharacteristic
34C2×(C2×D251)⋊C4C2211yescharacteristic
35C251⋊#95C2211yescharacteristic
36C251⋊#95C2211yescharacteristic
37C251⋊#95C2211yescharacteristic
38C251⋊#95C2211yescharacteristic
39C251⋊#95C2211yescharacteristic
40C251⋊#95C2211yescharacteristic
41C251⋊#95C2211yescharacteristic
42C251⋊#95C2211yescharacteristic
43C251⋊#95C2211yescharacteristic
44C22×D4×C251C2211yescharacteristic
45C22×D4×C251C2211yescharacteristic
46C22×D4×C251C2211yescharacteristic
47C2×C22⋊C4×C251C2211yescharacteristic
48C2×C22⋊C4×C251C2211yescharacteristic
49C2×C22⋊C4×C251C2211yescharacteristic
50#95×C251C2211yescharacteristic
51C23×D251C2311yescharacteristic
52C2×C22⋊D251—22no—
53C2×C22⋊D251—41no—
54C2×C22⋊D251—41no—
55C2×C22⋊D251—41no—
56C2×C22⋊D251—41no—
57C2×C22⋊D251—41no—
58C2×C22⋊D251—41no—
59C2×C22⋊D251—41no—
60C2×C22⋊D251—41no—
61C22×Dic251—41no—
62C22×Dic251—41no—
63C22×Dic251—41no—
64C22×Dic251C2311yescharacteristic
65C22×Dic251C2311yescharacteristic
66C22×Dic251C2311yescharacteristic
67C22×Dic251C2311yescharacteristic
68C22×Dic251C2311yescharacteristic
69C22×Dic251C2311yescharacteristic
70C22×Dic251C2311yescharacteristic
71C4⋊Dic251—41no—
72Dic251⋊C4—41no—
73Dic251⋊C4—41no—
74C22⋊Dic251—41no—
75C22⋊Dic251—41no—
76C22⋊Dic251—41no—
77C22⋊Dic251—41no—
78C22⋊Dic251—41no—
79C22⋊Dic251—41no—
80C22⋊Dic251—41no—
81C22⋊Dic251—41no—
82(C2×D251)⋊C4—41no—
83(C2×D251)⋊C4—41no—
84(C2×D251)⋊C4—41no—
85(C2×D251)⋊C4—41no—
86C24×C251C2311yescharacteristic
87C24×C251C2311yescharacteristic
88C24×C251C2311yescharacteristic
89C2×D4×C251—22no—
90C2×D4×C251—22no—
91C2×D4×C251—41no—
92C2×D4×C251—41no—
93C2×D4×C251—41no—
94C2×D4×C251—41no—
95C2×D4×C251—41no—
96C2×D4×C251—41no—
97C2×D4×C251—41no—
98C22×C4×C251C2311yescharacteristic
99C22×C4×C251C2311yescharacteristic
100C22×C4×C251C2311yescharacteristic
101C22×C4×C251C2311yescharacteristic
102C22⋊C4×C251—41no—
103C22⋊C4×C251—41no—
104C22⋊C4×C251—41no—
105C22×D251—21no—
106C22×D251—21no—
107C22×D251—21no—
108C22×D251—21no—
109C22×D251—21no—
110C22×D251—21no—
111C22×D251—21no—
112C22⋊D251—44no—
113C22⋊D251—44no—
114C22⋊D251—44no—
115C2×Dic251—21no—
116C2×Dic251—21no—
117C2×Dic251—21no—
118C2×Dic251—21no—
119C2×Dic251—21no—
120C2×Dic251—21no—
121C2×Dic251—21no—
122C2×Dic251—21no—
123C2×Dic251—21no—
124C2×Dic251—21no—
125C2×Dic251—21no—
126C2×Dic251—21no—
127C2×Dic251—21no—
128C2×Dic251—21no—
129C2×Dic251—21no—
130C2×Dic251—21no—
131C2×Dic251—21no—
132C2×Dic251—21no—
133C2×Dic251—21no—
134C2×Dic251—21no—
135C2×Dic251—21no—
136C2×Dic251—44no—
137C23×C251—21no—
138C23×C251—21no—
139C23×C251—21no—
140C23×C251—21no—
141C23×C251—21no—
142C23×C251—21no—
143C23×C251—21no—
144C23×C251—21no—
145C23×C251—21no—
146C23×C251—21no—
147C23×C251—21no—
148C23×C251—21no—
149C23×C251—21no—
150C23×C251—21no—
151C23×C251—21no—
152C23×C251—21no—
153C23×C251—21no—
154C23×C251—21no—
155C23×C251—21no—
156C23×C251—21no—
157C23×C251#4112yes—
158C23×C251C2411yescharacteristic
159D4×C251—44no—
160D4×C251—44no—
161D4×C251—44no—
162C2×C4×C251—21no—
163C2×C4×C251—21no—
164C2×C4×C251—21no—
165C2×C4×C251—21no—
166C2×C4×C251—21no—
167C2×C4×C251—21no—
168C2×C4×C251—21no—
169C2×C4×C251—21no—
170C2×C4×C251—21no—
171C2×C4×C251—21no—
172C2×C4×C251—21no—
173C2×C4×C251—21no—
174C22×C251—22no—
175C22×C251—22no—
176C22×C251—22no—
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×C251—41no—
189C22×C251—41no—
190C22×C251—41no—
191C22×C251—41no—
192C22×C251—41no—
193C22×C251—41no—
194C22×C251—41no—
195C22×C251C2×#4111yescharacteristic
196C22×C251H(5,𝔽2)11yescharacteristic
197C22×C251H(5,𝔽2)11yescharacteristic
198C22×C251H(5,𝔽2)11yescharacteristic
199C22×C251H(5,𝔽2)11yescharacteristic
200C22×C251H(5,𝔽2)11yescharacteristic
201C22×C251H(5,𝔽2)11yescharacteristic
202C2×D251—41no—
203C2×D251—41no—
204C2×D251—41no—
205C2×D251—41no—
206C2×D251—41no—
207C2×D251—41no—
208C2×D251—41no—
209C4×C251—41no—
210C4×C251—41no—
211C4×C251—41no—
212C4×C251—41no—
213Dic251—41no—
214Dic251—41no—
215Dic251—41no—
216Dic251—41no—
217Dic251—41no—
218Dic251—41no—
219Dic251—41no—
220C2×C251—42no—
221C2×C251—81no—
222C2×C251—81no—
223C2×C251#53811yescharacteristic
224C2×C251#53811yescharacteristic
225C2×C251#53811yescharacteristic
226C2×C251#56011yescharacteristic
227C2×C251#56011yescharacteristic
228C2×C251#56011yescharacteristic
229C2×C251#56011yescharacteristic
230D251—81no—
231C251#282511yescharacteristic, complemented, Sylow
232#2825—2511nomaximal, Sylow
233#401—2511no—
234#397—2511no—
235#397—2511no—
236#397—2511no—
237#396—2511no—
238#394—2511no—
239#394—2511no—
240#394—2511no—
241#392—2511no—
242#392—2511no—
243#392—2511no—
244#392D25111yescharacteristic, complemented
245#386—2511no—
246#386—2511no—
247#386—2511no—
248C22×D4—2511no—
249C22×D4—2511no—
250C22×D4—2511no—
251C22×D4C2×D25111yescharacteristic
252C22×D4C2×D25111yescharacteristic
253C22×D4C2×D25111yescharacteristic
254C23×C4—2511no—
255C2×C4⋊C4—2511no—
256C2×C4⋊C4—2511no—
257C2×C4⋊C4—2511no—
258C2×C22⋊C4—2511no—
259C2×C22⋊C4—2511no—
260C2×C22⋊C4—2511no—
261C2×C22⋊C4—2511no—
262C2×C22⋊C4—2511no—
263C2×C22⋊C4—2511no—
264C2×C22⋊C4—2511no—
265C2×C22⋊C4—2511no—
266C2×C22⋊C4—2511no—
267C2×C22⋊C4—2511no—
268C2×C22⋊C4—2511no—
269C2×C22⋊C4—2511no—
270C2×C22⋊C4C2×D25111yescharacteristic
271C2×C22⋊C4C2×D25111yescharacteristic
272C2×C22⋊C4C2×D25111yescharacteristic
273#95—2511no—
274#95—2511no—
275#95—2511no—
276#95—2511no—
277#95—2511no—
278#95—2511no—
279#95—2511no—
280#95—2511no—
281#95—2511no—
282#95C2×D25111yescharacteristic
283C24—2511no—
284C24C22×D25111yescharacteristic
285C24C22×D25111yescharacteristic
286C24C22×D25111yescharacteristic
287C2×D4—22no—
288C2×D4—22no—
289C2×D4—41no—
290C2×D4—41no—
291C2×D4—41no—
292C2×D4—41no—
293C2×D4—41no—
294C2×D4—41no—
295C2×D4—41no—
296C2×D4—10041no—
297C2×D4—10041no—
298C2×D4—10041no—
299C2×D4—10041no—
300C2×D4—10041no—
301C2×D4—10041no—
302C2×D4—10041no—
303C2×D4—10041no—
304C2×D4—5022no—
305C22×C4—10041no—
306C22×C4—10041no—
307C22×C4—10041no—
308C22×C4—2511no—
309C22×C4—2511no—
310C22×C4—2511no—
311C22×C4—2511no—
312C22×C4—2511no—
313C22×C4—2511no—
314C22×C4—2511no—
315C22×C4C22×D25111yescharacteristic
316C22×C4C22×D25111yescharacteristic
317C22×C4C22×D25111yescharacteristic
318C22×C4C22×D25111yescharacteristic
319C4⋊C4—10041no—
320C4⋊C4—10041no—
321C4⋊C4—10041no—
322C22⋊C4—41no—
323C22⋊C4—41no—
324C22⋊C4—41no—
325C22⋊C4—10041no—
326C22⋊C4—10041no—
327C22⋊C4—10041no—
328C22⋊C4—10041no—
329C22⋊C4—10041no—
330C22⋊C4—10041no—
331C22⋊C4—10041no—
332C22⋊C4—10041no—
333C22⋊C4—10041no—
334C22⋊C4—10041no—
335C22⋊C4—10041no—
336C22⋊C4—10041no—
337C23—21no—
338C23—21no—
339C23—21no—
340C23—21no—
341C23—21no—
342C23—21no—
343C23—21no—
344C23—21no—
345C23—21no—
346C23—21no—
347C23—21no—
348C23—21no—
349C23—21no—
350C23—21no—
351C23—21no—
352C23—21no—
353C23—21no—
354C23—21no—
355C23—21no—
356C23—21no—
357C23—5021no—
358C23—5021no—
359C23—5021no—
360C23—5021no—
361C23—5021no—
362C23—5021no—
363C23—5021no—
364C23#1686080912yes—
365C23C23×D25111yescharacteristic
366D4—44no—
367D4—44no—
368D4—44no—
369D4—10044no—
370D4—10044no—
371D4—10044no—
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—21no—
381C2×C4—21no—
382C2×C4—21no—
383C2×C4—21no—
384C2×C4—10044no—
385C2×C4—5021no—
386C2×C4—5021no—
387C2×C4—5021no—
388C2×C4—5021no—
389C2×C4—5021no—
390C2×C4—5021no—
391C2×C4—5021no—
392C2×C4—5021no—
393C2×C4—5021no—
394C2×C4—5021no—
395C2×C4—5021no—
396C2×C4—5021no—
397C2×C4—5021no—
398C2×C4—5021no—
399C2×C4—5021no—
400C2×C4—5021no—
401C2×C4—5021no—
402C2×C4—5021no—
403C2×C4—5021no—
404C2×C4—5021no—
405C2×C4—5021no—
406C22—22no—
407C22—22no—
408C22—22no—
409C22—41no—
410C22—41no—
411C22—41no—
412C22—41no—
413C22—41no—
414C22—41no—
415C22—41no—
416C22—41no—
417C22—41no—
418C22—41no—
419C22—41no—
420C22—41no—
421C22—41no—
422C22—41no—
423C22—41no—
424C22—41no—
425C22—41no—
426C22—41no—
427C22—10041no—
428C22—10041no—
429C22—10041no—
430C22—10041no—
431C22—10041no—
432C22—10041no—
433C22—10041no—
434C22C2×#1686080911yescharacteristic
435C22#2254876011yescharacteristic
436C22#2254876011yescharacteristic
437C22#2254876011yescharacteristic
438C22#2254876011yescharacteristic
439C22#2254876011yescharacteristic
440C22#2254876111yescharacteristic
441C4—41no—
442C4—41no—
443C4—41no—
444C4—41no—
445C4—10041no—
446C4—10041no—
447C4—10041no—
448C4—10041no—
449C4—10041no—
450C4—10041no—
451C4—10041no—
452C2—42no—
453C2—81no—
454C2—81no—
455C2—20081no—
456C2#3063227211yescharacteristic
457C2#3063227211yescharacteristic
458C2#3063227411yescharacteristic
459C2#3063242411yescharacteristic
460C2#3063242511yescharacteristic
461C2#3063242511yescharacteristic
462C2#3063242611yescharacteristic
subgroupG/H
center, FrattiniC23C23×D251
commutator, socle, cosocleC23×C251C24
Fitting#392×C251C2
hypercenter#392D251
hypocenterC251#2825
LCSG ▷ C23×C251 ▷ C251
UCSC1 ◁ C23 ◁ #392

Complements

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

NH ≅ G/Ntype
1. C251⋊#401C2semidirect
2. #30631616C2semidirect
3. #30631615C2semidirect
4. #30631612C2semidirect
5. #30631610C2semidirect
6. #30631603C2semidirect
7. #30631602C2semidirect
8. #30631601C2semidirect
9. #30631595C2semidirect
10. #30631595C2semidirect
11. #30631595C2semidirect
12. #30631567C2semidirect
13. #30631567C2semidirect
14. #30631564C2semidirect
15. #392×C251C2semidirect
231. C251#2825semidirect
244. #392D251semidirect

Maximal subgroups

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

subgroupindex
C251⋊#4012
#306316162
#306316152
#306316122
#306316102
#306316032
#306316022
#306316012
#306315952
#306315672
#306315642
#392×C2512
#2825251