Finite Group Databasebeta

← #44558038

Subgroups — #44558038

Order 32128.

448 conjugacy classes of subgroups.

The subgroup lattice has 388 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#30631629C211yesmaximal, characteristic, complemented
2#30631628C211yesmaximal, characteristic, complemented
3#30631628C211yesmaximal, characteristic, complemented
4#30631615C211yesmaximal, characteristic, complemented
5#30631612C211yesmaximal, characteristic, complemented
6#30631601C211yesmaximal, characteristic
7#30631581C211yesmaximal, characteristic, complemented
8#30631580C211yesmaximal, characteristic, complemented
9#30631567C211yesmaximal, characteristic
10#30631560C211yesmaximal, characteristic, complemented
11#30631560C211yesmaximal, characteristic, complemented
12#30631559C211yesmaximal, characteristic, complemented
13#30631559C211yesmaximal, characteristic, complemented
14#30631551C211yesmaximal, characteristic, complemented
15#386×C251C211yesmaximal, characteristic
16C23×Dic251C2211yescharacteristic
17C2×C4⋊Dic251C2211yescharacteristic
18C2×Dic251⋊C4C2211yescharacteristic
19C2×Dic251⋊C4C2211yescharacteristic
20C2×Dic251⋊C4C2211yescharacteristic
21C2×Dic251⋊C4C2211yescharacteristic
22C2×Dic251⋊C4C2211yescharacteristic
23C2×Dic251⋊C4C2211yescharacteristic
24C2×C22⋊Dic251C2211yescharacteristic
25C2×C22⋊Dic251C2211yescharacteristic
26C2×C22⋊Dic251C2211yescharacteristic
27C2×C22⋊Dic251C2211yescharacteristic
28C2×C22⋊Dic251C2211yescharacteristic
29C2×C22⋊Dic251C2211yescharacteristic
30C2×C22⋊Dic251C2211yescharacteristic
31C2×C4×Dic251C2211yescharacteristic
32C251⋊#95C2211yescharacteristic
33C251⋊#95C2211yescharacteristic
34C251⋊#95C2211yescharacteristic
35C251⋊#95C2211yescharacteristic
36C251⋊#95C2211yescharacteristic
37C251⋊#95C2211yescharacteristic
38C251⋊#95C2211yescharacteristic
39C251⋊#95C2211yescharacteristic
40C251⋊#95C2211yescharacteristic
41C251⋊#95C2211yescharacteristic
42C251⋊#95C2211yescharacteristic
43C251⋊#95C2211yescharacteristic
44C22×D4×C251C2211yescharacteristic
45C23×C4×C251C2211yescharacteristic
46C2×C22⋊C4×C251C2211yescharacteristic
47C2×C22⋊C4×C251C2211yescharacteristic
48C2×C22⋊C4×C251C2211yescharacteristic
49#95×C251C2211yescharacteristic
50#95×C251C2211yescharacteristic
51C22×Dic251—41no—
52C22×Dic251—41no—
53C22×Dic251—41no—
54C22×Dic251C2311yescharacteristic
55C22×Dic251C2311yescharacteristic
56C22×Dic251C2311yescharacteristic
57C22×Dic251C2311yescharacteristic
58C22×Dic251C2311yescharacteristic
59C22×Dic251C2311yescharacteristic
60C22×Dic251C2311yescharacteristic
61C22×Dic251C2311yescharacteristic
62C4⋊Dic251—41no—
63Dic251⋊C4—41no—
64Dic251⋊C4—41no—
65Dic251⋊C4—41no—
66Dic251⋊C4—41no—
67Dic251⋊C4—41no—
68Dic251⋊C4—41no—
69C22⋊Dic251—22no—
70C22⋊Dic251—22no—
71C22⋊Dic251—22no—
72C22⋊Dic251—22no—
73C22⋊Dic251—22no—
74C22⋊Dic251—41no—
75C22⋊Dic251—41no—
76C4×Dic251—22no—
77C24×C251C2311yescharacteristic
78C24×C251C2311yescharacteristic
79C2×D4×C251—22no—
80C2×D4×C251—22no—
81C2×D4×C251—41no—
82C22×C4×C251—41no—
83C22×C4×C251—41no—
84C22×C4×C251—41no—
85C22×C4×C251C2311yescharacteristic
86C22×C4×C251C2311yescharacteristic
87C22×C4×C251C2311yescharacteristic
88C22×C4×C251C2311yescharacteristic
89C22×C4×C251C2311yescharacteristic
90C22⋊C4×C251—22no—
91C22⋊C4×C251—41no—
92C22⋊C4×C251—41no—
93C2×Dic251—21no—
94C2×Dic251—21no—
95C2×Dic251—21no—
96C2×Dic251—21no—
97C2×Dic251—21no—
98C2×Dic251—21no—
99C2×Dic251—21no—
100C2×Dic251—21no—
101C2×Dic251—21no—
102C2×Dic251—21no—
103C2×Dic251—21no—
104C2×Dic251—21no—
105C2×Dic251—21no—
106C2×Dic251—21no—
107C2×Dic251—21no—
108C2×Dic251—21no—
109C2×Dic251—21no—
110C2×Dic251—21no—
111C2×Dic251—21no—
112C2×Dic251—21no—
113C2×Dic251—21no—
114C2×Dic251—21no—
115C2×Dic251—21no—
116C2×Dic251—44no—
117C2×Dic251#4112yes—
118C23×C251—21no—
119C23×C251—21no—
120C23×C251—21no—
121C23×C251—21no—
122C23×C251—21no—
123C23×C251—21no—
124C23×C251—21no—
125C23×C251—21no—
126C23×C251—21no—
127C23×C251—21no—
128C23×C251—21no—
129C23×C251—21no—
130C23×C251#4112yes—
131C23×C251#4112yes—
132C23×C251C2411yescharacteristic
133D4×C251—44no—
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—44no—
150C22×C251—22no—
151C22×C251—22no—
152C22×C251—22no—
153C22×C251—22no—
154C22×C251—22no—
155C22×C251—22no—
156C22×C251—41no—
157C22×C251—41no—
158C22×C251—41no—
159C22×C251—41no—
160C22×C251—41no—
161C22×C251—41no—
162C22×C251—41no—
163C22×C251—41no—
164C22×C251C2×#4111yescharacteristic
165C22×C251C2×#4111yescharacteristic
166C22×C251C2×#4111yescharacteristic
167C22×C251H(5,𝔽2)11yescharacteristic
168C22×C251H(5,𝔽2)11yescharacteristic
169C22×C251H(5,𝔽2)11yescharacteristic
170C22×C2512−1+411yescharacteristic
171C4×C251—41no—
172C4×C251—41no—
173C4×C251—41no—
174C4×C251—41no—
175C4×C251—41no—
176Dic251—22no—
177Dic251—41no—
178Dic251—41no—
179Dic251—41no—
180Dic251—41no—
181Dic251—41no—
182Dic251—41no—
183Dic251—41no—
184C2×C251—42no—
185C2×C251—42no—
186C2×C251#53911yescharacteristic
187C2×C251#53911yescharacteristic
188C2×C251#53911yescharacteristic
189C2×C251#55111yescharacteristic
190C2×C251#55311yescharacteristic
191C2×C251#55311yescharacteristic
192C2×C251H(3,𝔽4)11yescharacteristic
193C251#278211yescharacteristic, complemented, Sylow
194#2782—2511nomaximal, Sylow
195#400—2511no—
196#400—2511no—
197#400—2511no—
198#397—2511no—
199#397—2511no—
200#394—2511no—
201#388—2511no—
202#388—2511no—
203#386—2511no—
204#386D25111yescharacteristic
205#385—2511no—
206#385—2511no—
207#385—2511no—
208#385—2511no—
209#382—2511no—
210C22×D4C2×D25111yescharacteristic
211C23×C4—2511no—
212C23×C4C2×D25111yescharacteristic
213C2×C4⋊C4—2511no—
214C2×C4⋊C4—2511no—
215C2×C4⋊C4—2511no—
216C2×C4⋊C4—2511no—
217C2×C4⋊C4—2511no—
218C2×C4⋊C4—2511no—
219C2×C4⋊C4—2511no—
220C2×C22⋊C4—2511no—
221C2×C22⋊C4—2511no—
222C2×C22⋊C4—2511no—
223C2×C22⋊C4—2511no—
224C2×C22⋊C4—2511no—
225C2×C22⋊C4—2511no—
226C2×C22⋊C4—2511no—
227C2×C22⋊C4C2×D25111yescharacteristic
228C2×C22⋊C4C2×D25111yescharacteristic
229C2×C22⋊C4C2×D25111yescharacteristic
230C2×C42—2511no—
231#95—2511no—
232#95—2511no—
233#95—2511no—
234#95—2511no—
235#95—2511no—
236#95—2511no—
237#95—2511no—
238#95—2511no—
239#95—2511no—
240#95—2511no—
241#95—2511no—
242#95—2511no—
243#95C2×D25111yescharacteristic
244#95C2×D25111yescharacteristic
245C24C22×D25111yescharacteristic
246C24C22×D25111yescharacteristic
247C2×D4—22no—
248C2×D4—22no—
249C2×D4—41no—
250C22×C4—41no—
251C22×C4—41no—
252C22×C4—41no—
253C22×C4—10041no—
254C22×C4—10041no—
255C22×C4—10041no—
256C22×C4—2511no—
257C22×C4—2511no—
258C22×C4—2511no—
259C22×C4—2511no—
260C22×C4—2511no—
261C22×C4—2511no—
262C22×C4—2511no—
263C22×C4—2511no—
264C22×C4C22×D25111yescharacteristic
265C22×C4C22×D25111yescharacteristic
266C22×C4C22×D25111yescharacteristic
267C22×C4C22×D25111yescharacteristic
268C22×C4C22×D25111yescharacteristic
269C4⋊C4—10041no—
270C4⋊C4—10041no—
271C4⋊C4—10041no—
272C4⋊C4—10041no—
273C4⋊C4—10041no—
274C4⋊C4—10041no—
275C4⋊C4—10041no—
276C22⋊C4—22no—
277C22⋊C4—41no—
278C22⋊C4—41no—
279C22⋊C4—10041no—
280C22⋊C4—10041no—
281C22⋊C4—5022no—
282C22⋊C4—5022no—
283C22⋊C4—5022no—
284C22⋊C4—5022no—
285C22⋊C4—5022no—
286C42—5022no—
287C23—21no—
288C23—21no—
289C23—21no—
290C23—21no—
291C23—21no—
292C23—21no—
293C23—21no—
294C23—21no—
295C23—21no—
296C23—21no—
297C23—21no—
298C23—21no—
299C23#1686080912yes—
300C23#1686081012yes—
301C23C23×D25111yescharacteristic
302D4—44no—
303C2×C4—21no—
304C2×C4—21no—
305C2×C4—21no—
306C2×C4—21no—
307C2×C4—21no—
308C2×C4—21no—
309C2×C4—21no—
310C2×C4—21no—
311C2×C4—21no—
312C2×C4—21no—
313C2×C4—21no—
314C2×C4—21no—
315C2×C4—21no—
316C2×C4—21no—
317C2×C4—21no—
318C2×C4—44no—
319C2×C4—10044no—
320C2×C4—5021no—
321C2×C4—5021no—
322C2×C4—5021no—
323C2×C4—5021no—
324C2×C4—5021no—
325C2×C4—5021no—
326C2×C4—5021no—
327C2×C4—5021no—
328C2×C4—5021no—
329C2×C4—5021no—
330C2×C4—5021no—
331C2×C4—5021no—
332C2×C4—5021no—
333C2×C4—5021no—
334C2×C4—5021no—
335C2×C4—5021no—
336C2×C4—5021no—
337C2×C4—5021no—
338C2×C4—5021no—
339C2×C4—5021no—
340C2×C4—5021no—
341C2×C4—5021no—
342C2×C4—5021no—
343C2×C4—2512no—
344C22—22no—
345C22—22no—
346C22—22no—
347C22—22no—
348C22—22no—
349C22—22no—
350C22—41no—
351C22—41no—
352C22—41no—
353C22—41no—
354C22—41no—
355C22—41no—
356C22—41no—
357C22—41no—
358C22#41×D25111yescharacteristic
359C22C2×#1686080911yescharacteristic
360C22C2×#1686081011yescharacteristic
361C22#2254876011yescharacteristic
362C22#2254876011yescharacteristic
363C22#2254876011yescharacteristic
364C22#2254876211yescharacteristic
365C4—41no—
366C4—41no—
367C4—41no—
368C4—41no—
369C4—41no—
370C4—10041no—
371C4—10041no—
372C4—10041no—
373C4—10041no—
374C4—10041no—
375C4—10041no—
376C4—10041no—
377C4—5022no—
378C2—42no—
379C2—42no—
380C2#3063227911yescharacteristic
381C2#3063228011yescharacteristic
382C2#3063228211yescharacteristic
383C2#3063236311yescharacteristic
384C2#3063238011yescharacteristic
385C2#3063238611yescharacteristic
386C2#3063243011yescharacteristic
subgroupG/H
center, FrattiniC23C23×D251
commutator, socle, cosocleC23×C251C24
Fitting#386×C251C2
prime radicalC22×D4×C251C22
hypercenter#386D251
hypocenterC251#2782
LCSG ▷ C23×C251 ▷ C251
UCSC1 ◁ C23 ◁ #386

Complements

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

NH ≅ G/Ntype
1. #30631629C2semidirect
2. #30631628C2semidirect
3. #30631628C2semidirect
4. #30631615C2semidirect
5. #30631612C2semidirect
7. #30631581C2semidirect
8. #30631580C2semidirect
10. #30631560C2semidirect
11. #30631560C2semidirect
12. #30631559C2semidirect
13. #30631559C2semidirect
14. #30631551C2semidirect
193. C251#2782semidirect

Maximal subgroups

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

subgroupindex
#306316292
#306316282
#306316152
#306316122
#306316012
#306315812
#306315802
#306315672
#306315602
#306315592
#306315512
#386×C2512
#2782251