Finite Group Databasebeta

← #44550327

Subgroups — #44550327

Order 32128.

548 conjugacy classes of subgroups; only the 97 normal subgroups are shown.

normal subgroups only

This group has 548 conjugacy classes of subgroups — too many to draw. Only its 97 normal subgroups are shown.

Dotted edges: containments with no intermediate normal subgroup — omitted non-normal subgroups may lie between.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#30632523C211yesmaximal, characteristic
2#30631703C211yesmaximal, characteristic, complemented
3#30631683C211yesmaximal, characteristic, complemented
4#30631681C211yesmaximal, characteristic, complemented
5#30631660C211yesmaximal, characteristic, complemented
6#30631655C211yesmaximal, characteristic, complemented
7#421×C251C211yesmaximal, characteristic, complemented
8#22548763—21no—
9#22548762—21no—
10#22548762C412yescomplemented
11#22548761—21no—
12#22548761C412yescomplemented
13#22548760—22no—
14#22548760—21no—
15C2×#16860810—22no—
16C2×#16860810C2211yescharacteristic
17C2×#16860810C2211yescharacteristic
18#41×D251—22no—
19#41×D251—22no—
20C22×C251⋊C8C2211yescharacteristic, complemented
21#22548674C2211yescharacteristic, complemented
22#22548641—22no—
23#22548639—21no—
24#22548638—21no—
25#22548636—21no—
26#22548635—21no—
27#22548629—22no—
28(C2×D251)⋊C8—22no—
29C2×#41×C251C2211yescharacteristic
30C2×M16×C251C2211yescharacteristic, complemented
31#117×C251C2211yescharacteristic, complemented
32#104×C251—22no—
33#16860811—22no—
34#16860811—22no—
35#16860810—24no—
36#16860810—24no—
37#16860810—22no—
38#16860810—22no—
39#16860810—21no—
40#16860810—21no—
41#16860810—21no—
42#16860810—21no—
43#16860810—21no—
44#16860810—21no—
45#16860810C2×C412yes—
46#16860810D412yes—
47#16860809—24no—
48#16860809—22no—
49#16860809—24no—
50#16860809—22no—
51C2×Dic502—22no—
52C2×Dic502C2×C411yescharacteristic
53C2×Dic502D411yescharacteristic
54Q8×D251—22no—
55Q8×D251—22no—
56C2×D1004—22no—
57C2×D1004C2×C411yescharacteristic
58C2×D1004D411yescharacteristic
59C2×C22⋊D251—24no—
60C2×C22⋊D251—21no—
61C2×C22⋊D251D412yes—
62D4×D251—24no—
63D4×D251—22no—
64D4×D251—24no—
65D4×D251—22no—
66C2×C4×D251—24no—
67C2×C4×D251—21no—
68C2×C4×D251D412yes—
69C2×C251⋊C8—21no—
70C2×C251⋊C8—21no—
71C2×C251⋊C8—21no—
72C4⋊Dic251—21no—
73C22⋊Dic251—21no—
74C4×Dic251—21no—
75#41×C251—21no—
76#41×C251—21no—
77#41×C251C2×C412yes—
78#41×C251D412yes—
79C2×Q8×C251D411yescharacteristic
80C2×D4×C251—21no—
81C2×D4×C251D411yescharacteristic
82C22×C4×C251—21no—
83C22×C4×C251C2311yescharacteristic
84M16×C251—21no—
85M16×C251—21no—
86C2×C8×C251—21no—
87C4⋊C4×C251—21no—
88C22⋊C4×C251—21no—
89C42×C251—21no—
90C22×D251—24no—
91C22×D251—21no—
92C22×D251C22⋊C412yes—
93Dic502—24no—
94Dic502—21no—
95Dic502—21no—
96Dic502—21no—
97Dic502C22⋊C412yes—
98D1004—24no—
99D1004—21no—
100D1004—21no—
101D1004—21no—
102D1004C22⋊C412yes—
103C22⋊D251—24no—
104C22⋊D251—24no—
105C22⋊D251—22no—
106C22⋊D251—22no—
107C22⋊D251—22no—
108C22⋊D251—22no—
109C2×Dic251—24no—
110C2×Dic251—21no—
111C2×Dic251—41no—
112C2×Dic251C22⋊C412yes—
113C4×D251—24no—
114C4×D251—24no—
115C4×D251—22no—
116C4×D251—22no—
117C4×D251—22no—
118C4×D251—22no—
119C251⋊C8—41no—
120C23×C251—21no—
121C23×C251C2×D411yescharacteristic
122Q8×C251—21no—
123Q8×C251C22⋊C412yes—
124D4×C251—22no—
125D4×C251—22no—
126D4×C251—21no—
127D4×C251C22⋊C412yes—
128C2×C4×C251—21no—
129C2×C4×C251—22no—
130C2×C4×C251—22no—
131C2×C4×C251—41no—
132C2×C4×C251C22×C411yescharacteristic
133C2×C4×C251C2×D411yescharacteristic
134C2×C4×C251C2×D411yescharacteristic
135C2×C4×C251C2×D411yescharacteristic
136C2×C4×C251C2×D411yescharacteristic
137C2×C4×C251C2×D411yescharacteristic
138C8×C251—41no—
139C22×C251—22no—
140C22×C251—41no—
141C22×C251—81no—
142C22×C251C2×C22⋊C411yescharacteristic
143C22×C251#12011yescharacteristic
144C22×C251#12011yescharacteristic
145C2×D251—22no—
146C2×D251—22no—
147C2×D251—42no—
148C2×D251—81no—
149C2×D251—84no—
150C4×C251—22no—
151C4×C251—81no—
152C4×C251C2×C22⋊C411yescharacteristic
153C4×C251C2×C22⋊C411yescharacteristic
154C4×C251#12011yescharacteristic
155C4×C251#12011yescharacteristic
156Dic251—22no—
157Dic251—22no—
158Dic251—81no—
159C2×C251—21no—
160C2×C251—21no—
161C2×C251—21no—
162C2×C251—42no—
163C2×C251#37911yescharacteristic
164D251—42no—
165D251—42no—
166C251#179411yescharacteristic, complemented, Sylow
167#1794—2511nomaximal, Sylow
168#585—2511no—
169#421—2511no—
170#421D25111yescharacteristic, complemented
171#416—2511no—
172#416—2511no—
173#410—2511no—
174#408—2511no—
1752−1+4—5021no—
1762−1+4—5021no—
1772−1+4—2512no—
178H(5,𝔽2)—5022no—
179H(5,𝔽2)—5021no—
180H(5,𝔽2)—5021no—
181H(5,𝔽2)—2512no—
182C2×#41—5022no—
183C2×#41—5022no—
184C2×#41—5022no—
185C2×#41—2511no—
186C2×#41—2511no—
187C2×#41C2×D25111yescharacteristic
188C2×M16C2×D25111yescharacteristic, complemented
189C22×C8—2511no—
190#117—2511no—
191#117C2×D25111yescharacteristic, complemented
192#104—22no—
193#104—5022no—
194#103—5021no—
195#103—5021no—
196#102—5021no—
197#102—5021no—
198#99—5022no—
199C22⋊C8—5022no—
200#41—21no—
201#41—21no—
202#41—5024no—
203#41—5024no—
204#41—5022no—
205#41—5022no—
206#41—5024no—
207#41—5022no—
208#41—5022no—
209#41—5022no—
210#41—5022no—
211#41—5024no—
212#41—5021no—
213#41—5021no—
214#41—5021no—
215#41—5021no—
216#41—5021no—
217#41—5021no—
218#41—2512no—
219#41—2512no—
220#41C4×D25112yes—
221#41C22⋊D25112yes—
222C2×Q8—5022no—
223C2×Q8—5022no—
224C2×Q8—5022no—
225C2×Q8—2511no—
226C2×Q8—2511no—
227C2×Q8D100411yescharacteristic
228C2×D4—21no—
229C2×D4—5022no—
230C2×D4—5024no—
231C2×D4—5022no—
232C2×D4—5024no—
233C2×D4—5024no—
234C2×D4—5021no—
235C2×D4—5022no—
236C2×D4—2512no—
237C2×D4—2511no—
238C2×D4—2511no—
239C2×D4D100411yescharacteristic
240C22×C4—21no—
241C22×C4—5024no—
242C22×C4—5021no—
243C22×C4—2512no—
244C22×C4C22×D25111yescharacteristic
245M16—21no—
246M16—21no—
247C2×C8—21no—
248C2×C8—5021no—
249C2×C8—5021no—
250C2×C8—5021no—
251C4⋊C4—21no—
252C4⋊C4—5021no—
253C22⋊C4—21no—
254C22⋊C4—5021no—
255C42—21no—
256C42—5021no—
257C23—21no—
258C23—5024no—
259C23—5021no—
260C23—2512no—
261C23D4×D25111yescharacteristic
262Q8—21no—
263Q8—5024no—
264Q8—5021no—
265Q8—5021no—
266Q8—5021no—
267Q8—2512no—
268Q8(C2×D251)⋊C412yes—
269D4—22no—
270D4—22no—
271D4—21no—
272D4—5024no—
273D4—5024no—
274D4—5024no—
275D4—5022no—
276D4—5022no—
277D4—5022no—
278D4—5022no—
279D4—5021no—
280D4—5021no—
281D4—5021no—
282D4—2512no—
283D4(C2×D251)⋊C412yes—
284C2×C4—21no—
285C2×C4—22no—
286C2×C4—22no—
287C2×C4—41no—
288C2×C4—10041no—
289C2×C4—5024no—
290C2×C4—5024no—
291C2×C4—5024no—
292C2×C4—5021no—
293C2×C4—5022no—
294C2×C4—5022no—
295C2×C4—5022no—
296C2×C4—5022no—
297C2×C4—2512no—
298C2×C4C2×C4×D25111yescharacteristic
299C2×C4D4×D25111yescharacteristic
300C2×C4D4×D25111yescharacteristic
301C2×C4D4×D25111yescharacteristic
302C2×C4C2×C22⋊D25111yescharacteristic
303C2×C4C2×D100411yescharacteristic
304C8—41no—
305C8—10041no—
306C22—22no—
307C22—41no—
308C22—81no—
309C22—20081no—
310C22—20084no—
311C22—10042no—
312C22—5022no—
313C22—5022no—
314C22C22⋊C4×D25111yescharacteristic
315C22#2254868311yescharacteristic
316C22#2254868511yescharacteristic
317C4—22no—
318C4—81no—
319C4—20081no—
320C4—5022no—
321C4—5022no—
322C4C22⋊C4×D25111yescharacteristic
323C4C2×(C2×D251)⋊C411yescharacteristic
324C4#2254868311yescharacteristic
325C4#2254868511yescharacteristic
326C2—21no—
327C2—21no—
328C2—21no—
329C2—42no—
330C2—10042no—
331C2—10042no—
332C2#3063154211yescharacteristic
subgroupG/H
centerC4#22548683
commutatorC2×C4×C251C22×C4
FrattiniC22×C4C22×D251
Fitting#421×C251C2
socleC2×C251#379
prime radical#30632523C2
cosocleC22×C4×C251C23
hypercenter#421D251
hypocenterC251#1794
LCSG ▷ C2×C4×C251 ▷ C2×C251 ▷ C251
UCSC1 ◁ C4 ◁ C22×C4 ◁ #421

Complements

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

NH ≅ G/Ntype
2. #30631703C2semidirect
3. #30631683C2semidirect
4. #30631681C2semidirect
5. #30631660C2semidirect
6. #30631655C2semidirect
7. #421×C251C2semidirect
10. #22548762C4semidirect
12. #22548761C4semidirect
20. C22×C251⋊C8C22semidirect
21. #22548674C22semidirect
30. C2×M16×C251C22semidirect
31. #117×C251C22semidirect
166. C251#1794semidirect
170. #421D251semidirect
188. C2×M16C2×D251semidirect
191. #117C2×D251semidirect

Maximal subgroups

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

subgroupindex
#306325232
#306317032
#306316832
#306316812
#306316602
#306316552
#421×C2512
#1794251