Finite Group Databasebeta

← #44560564

Subgroups — #44560564

Order 32128.

396 conjugacy classes of subgroups.

The subgroup lattice has 362 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#30632292C211yesmaximal, characteristic, complemented
2#30632005C211yesmaximal, characteristic, complemented
3#30632000C211yesmaximal, characteristic, complemented
4#30631994C211yesmaximal, characteristic, complemented
5#30631987C211yesmaximal, characteristic, complemented
6#30631887C211yesmaximal, characteristic, complemented
7#30631882C211yesmaximal, characteristic, complemented
8#30631832C211yesmaximal, characteristic, complemented
9#30631832C211yesmaximal, characteristic, complemented
10#30631797C211yesmaximal, characteristic, complemented
11#30631785C211yesmaximal, characteristic, complemented
12C4×#16860797C211yesmaximal, characteristic, complemented
13C4×#16860795C211yesmaximal, characteristic, complemented
14#30631748C211yesmaximal, characteristic, complemented
15#520×C251C211yesmaximal, characteristic, complemented
16C2×#16860797C2211yescharacteristic
17C2×#16860797C2211yescharacteristic
18C2×#16860795C2211yescharacteristic
19C2×#16860795C2211yescharacteristic
20#22548712C2211yescharacteristic
21#22548707C2211yescharacteristic
22#22548700—21no—
23#22548700—21no—
24#22548689C2211yescharacteristic
25#22548689C2211yescharacteristic
26#22548688—21no—
27#22548688—21no—
28C4×D1004C2211yescharacteristic
29C4×D1004C2211yescharacteristic
30#22548648C2211yescharacteristic
31#22548648C2211yescharacteristic
32#22548646C2211yescharacteristic
33#22548646C2211yescharacteristic
34C4⋊(C251⋊C8)C2211yescharacteristic
35C4⋊(C251⋊C8)C2211yescharacteristic
36#22548639C2211yescharacteristic
37#22548639C2211yescharacteristic
38#22548636C2211yescharacteristic
39#22548636C2211yescharacteristic
40#22548635C2211yescharacteristic
41#22548635C2211yescharacteristic
42#22548635C2211yescharacteristic
43#22548635C2211yescharacteristic
44C22⋊(C251⋊C8)C2211yescharacteristic
45C22⋊(C251⋊C8)C2211yescharacteristic
46#22548626C2211yescharacteristic
47C4×C251⋊C8C2211yescharacteristic
48C2×#41×C251C2211yescharacteristic
49C4×Q8×C251C2211yescharacteristic
50C4×Q8×C251C2211yescharacteristic
51C4×D4×C251—21no—
52C4×D4×C251—21no—
53C4×D4×C251C2211yescharacteristic
54C4×D4×C251C2211yescharacteristic
55#117×C251—21no—
56#117×C251C2211yescharacteristic
57C2×C4⋊C4×C251—21no—
58C2×C4⋊C4×C251C2211yescharacteristic
59C2×D1004—21no—
60C2×D1004—21no—
61C2×D1004C2311yescharacteristic
62C2×D1004C2311yescharacteristic
63C2×C22⋊D251—21no—
64C2×C22⋊D251—21no—
65C2×C4×D251—21no—
66C2×C4×D251—21no—
67#16860797—22no—
68#16860797—22no—
69#16860795—22no—
70#16860795—22no—
71C2×C251⋊C8C2311yescharacteristic
72C2×C251⋊C8C2311yescharacteristic
73C2×C251⋊C8C2311yescharacteristic
74C2×C251⋊C8C2311yescharacteristic
75C4⋊Dic251C2311yescharacteristic
76C4⋊Dic251C2311yescharacteristic
77Dic251⋊C4—21no—
78Dic251⋊C4—21no—
79(C2×D251)⋊C4—21no—
80(C2×D251)⋊C4—21no—
81(C2×D251)⋊C4—21no—
82(C2×D251)⋊C4—21no—
83#41×C251—22no—
84#41×C251—22no—
85C2×Q8×C251C2311yescharacteristic
86C2×D4×C251—21no—
87C2×D4×C251C2311yescharacteristic
88C22×C4×C251—21no—
89C22×C4×C251—21no—
90C22×C4×C251—21no—
91C22×C4×C251—21no—
92C22×C4×C251C2311yescharacteristic
93C4⋊C4×C251—21no—
94C4⋊C4×C251—21no—
95C4⋊C4×C251—21no—
96C4⋊C4×C251D411yescharacteristic
97C4⋊C4×C251D411yescharacteristic
98C4⋊C4×C251C2311yescharacteristic
99C4⋊C4×C251C2311yescharacteristic
100C22⋊C4×C251—21no—
101C22⋊C4×C251—21no—
102C22⋊C4×C251D411yescharacteristic
103C22⋊C4×C251D411yescharacteristic
104C42×C251—21no—
105C42×C251—21no—
106C42×C251C2311yescharacteristic
107C42×C251C2311yescharacteristic
108C22×D251—21no—
109C22×D251—21no—
110D1004—21no—
111D1004—21no—
112D1004—21no—
113D1004—21no—
114D1004—81no—
115D1004—81no—
116C22⋊D251—81no—
117C22⋊D251—81no—
118C2×Dic251—21no—
119C2×Dic251—21no—
120C4×D251—81no—
121C4×D251—81no—
122C251⋊C8—21no—
123C251⋊C8—21no—
124C251⋊C8—21no—
125C251⋊C8—21no—
126C23×C251—21no—
127C23×C251C2×D411yescharacteristic
128Q8×C251—21no—
129Q8×C251#4112yes—
130D4×C251—24no—
131D4×C251—21no—
132D4×C251#4112yes—
133C2×C4×C251—24no—
134C2×C4×C251—21no—
135C2×C4×C251—21no—
136C2×C4×C251—21no—
137C2×C4×C251—21no—
138C2×C4×C251—21no—
139C2×C4×C251—81no—
140C2×C4×C251—81no—
141C2×C4×C251—81no—
142C2×C4×C251C2×D411yescharacteristic
143C2×C4×C251C2×D411yescharacteristic
144C2×C4×C251C2×D411yescharacteristic
145C2×C4×C251C2×D411yescharacteristic
146C2×C4×C251C2×D411yescharacteristic
147C2×C4×C251C2411yescharacteristic
148C22×C251—21no—
149C22×C251—22no—
150C22×C251—21no—
151C22×C251—21no—
152C22×C251—41no—
153C22×C251—41no—
154C22×C251C22×D411yescharacteristic
155C2×D251—41no—
156C2×D251—41no—
157C2×D251—41no—
158C2×D251—41no—
159C2×D251—41no—
160C2×D251—41no—
161C4×C251—21no—
162C4×C251—22no—
163C4×C251—21no—
164C4×C251—21no—
165C4×C251—21no—
166C4×C251—21no—
167C4×C251—41no—
168C4×C251—41no—
169C4×C251C2×#4111yescharacteristic
170C4×C251C2×#4111yescharacteristic
171Dic251—41no—
172Dic251—41no—
173C2×C251—41no—
174C2×C251—42no—
175C2×C251#52511yescharacteristic
176C2×C251#57611yescharacteristic
177C2×C251#57711yescharacteristic
178D251—81no—
179D251—81no—
180C251#311911yescharacteristic, complemented, Sylow
181#3119—2511nomaximal, Sylow
182#540—2511no—
183#520D25111yescharacteristic, complemented
184#481—2511no—
185#480—2511no—
186#479—2511no—
187#478—2511no—
188#460—2511no—
189#459—2511no—
190#449—2511no—
191#449—2511no—
192#442—2511no—
193#440—2511no—
194C4×SD16—2511no—
195C4×D8—2511no—
196#433—2511no—
197C2×#41C2×D25111yescharacteristic
198C2×SD16—2511no—
199C2×SD16—2511no—
200C2×D8—2511no—
201C2×D8—2511no—
202#127—2511no—
203#125—2511no—
204#123—5021no—
205#123—5021no—
206#121—5021no—
207#121—5021no—
208#121—2511no—
209#121—2511no—
210C4×Q8C2×D25111yescharacteristic
211C4×Q8C2×D25111yescharacteristic
212C4×D4—21no—
213C4×D4—21no—
214C4×D4—2511no—
215C4×D4—2511no—
216C4×D4C2×D25111yescharacteristic
217C4×D4C2×D25111yescharacteristic
218#117—21no—
219#117C2×D25111yescharacteristic
220C2×C4⋊C4—21no—
221C2×C4⋊C4C2×D25111yescharacteristic
222#107—2511no—
223#107—2511no—
224#106—2511no—
225#106—2511no—
226C4⋊C8—2511no—
227C4⋊C8—2511no—
228#103—2511no—
229#103—2511no—
230#102—2511no—
231#102—2511no—
232#102—2511no—
233#102—2511no—
234#102—2511no—
235#102—2511no—
236C22⋊C8—2511no—
237C22⋊C8—2511no—
238#97—2511no—
239C4×C8—2511no—
240#41—22no—
241#41—22no—
242C2×Q8C22×D25111yescharacteristic
243C2×D4—21no—
244C2×D4—5021no—
245C2×D4—5021no—
246C2×D4—5021no—
247C2×D4—5021no—
248C2×D4—2511no—
249C2×D4—2511no—
250C2×D4C22×D25111yescharacteristic
251C22×C4—21no—
252C22×C4—21no—
253C22×C4—21no—
254C22×C4—21no—
255C22×C4—5021no—
256C22×C4—5021no—
257C22×C4C22×D25111yescharacteristic
258SD16—5022no—
259SD16—5022no—
260D8—5022no—
261D8—5022no—
262C2×C8—2511no—
263C2×C8—2511no—
264C2×C8—2511no—
265C2×C8—2511no—
266C4⋊C4—21no—
267C4⋊C4—21no—
268C4⋊C4—21no—
269C4⋊C4—5021no—
270C4⋊C4—5021no—
271C4⋊C4—2511no—
272C4⋊C4—2511no—
273C4⋊C4C22⋊D25111yescharacteristic
274C4⋊C4C22⋊D25111yescharacteristic
275C4⋊C4C22×D25111yescharacteristic
276C4⋊C4C22×D25111yescharacteristic
277C22⋊C4—21no—
278C22⋊C4—21no—
279C22⋊C4—5021no—
280C22⋊C4—5021no—
281C22⋊C4—5021no—
282C22⋊C4—5021no—
283C22⋊C4C22⋊D25111yescharacteristic
284C22⋊C4C22⋊D25111yescharacteristic
285C42—21no—
286C42—21no—
287C42C22×D25111yescharacteristic
288C42C22×D25111yescharacteristic
289C23—21no—
290C23—5021no—
291C23—5021no—
292C23C2×C22⋊D25111yescharacteristic
293Q8—21no—
294Q8#1686081012yes—
295D4—24no—
296D4—21no—
297D4—20081no—
298D4—20081no—
299D4—20081no—
300D4—20081no—
301D4—5021no—
302D4—5021no—
303D4—5021no—
304D4—5021no—
305D4#1686081012yes—
306C2×C4—24no—
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×C4—20081no—
316C2×C4—20081no—
317C2×C4—5021no—
318C2×C4—5021no—
319C2×C4C2×C22⋊D25111yescharacteristic
320C2×C4C2×C22⋊D25111yescharacteristic
321C2×C4C2×C22⋊D25111yescharacteristic
322C2×C4C2×C22⋊D25111yescharacteristic
323C2×C4C2×C22⋊D25111yescharacteristic
324C2×C4C23×D25111yescharacteristic
325C8—5021no—
326C8—5021no—
327C8—5021no—
328C8—5021no—
329C22—21no—
330C22—22no—
331C22—21no—
332C22—21no—
333C22—41no—
334C22—41no—
335C22—10041no—
336C22—10041no—
337C22—10041no—
338C22—10041no—
339C22—10041no—
340C22—10041no—
341C22C22×C22⋊D25111yescharacteristic
342C4—21no—
343C4—22no—
344C4—21no—
345C4—21no—
346C4—21no—
347C4—21no—
348C4—41no—
349C4—41no—
350C4—10041no—
351C4—10041no—
352C4C2×#1686081011yescharacteristic
353C4C2×#1686081011yescharacteristic
354C2—41no—
355C2—42no—
356C2—20081no—
357C2—20081no—
358C2#3063219811yescharacteristic
359C2#3063249011yescharacteristic
360C2#3063249511yescharacteristic
subgroupG/H
centerC22C22×C22⋊D251
commutator, cosocleC2×C4×C251C24
FrattiniC2×C4C23×D251
Fitting#520×C251C2
socleC22×C251C22×D4
hypercenter#520D251
hypocenterC251#3119
LCSG ▷ C2×C4×C251 ▷ C2×C251 ▷ C251
UCSC1 ◁ C22 ◁ #117 ◁ #520

Complements

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

NH ≅ G/Ntype
1. #30632292C2semidirect
2. #30632005C2semidirect
3. #30632000C2semidirect
4. #30631994C2semidirect
5. #30631987C2semidirect
6. #30631887C2semidirect
7. #30631882C2semidirect
8. #30631832C2semidirect
9. #30631832C2semidirect
10. #30631797C2semidirect
11. #30631785C2semidirect
12. C4×#16860797C2semidirect
13. C4×#16860795C2semidirect
14. #30631748C2semidirect
15. #520×C251C2semidirect
180. C251#3119semidirect
183. #520D251semidirect

Maximal subgroups

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

subgroupindex
#306322922
#306320052
#306320002
#306319942
#306319872
#306318872
#306318822
#306318322
#306317972
#306317852
C4×#168607972
C4×#168607952
#306317482
#520×C2512
#3119251