Finite Group Databasebeta

← #470×D251 #44559963

Subgroups — #470×D251 #44559963

Order 32128.

492 conjugacy classes of subgroups; only the 135 normal subgroups are shown.

normal subgroups only

This group has 492 conjugacy classes of subgroups — too many to draw. Only its 135 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
1C2×Q16×D251C211yesmaximal, characteristic, complemented
2C2×M16×D251C211yesmaximal, characteristic
3C22⋊Q8×D251C212yesmaximal
4#30631946C211yesmaximal, characteristic, complemented
5#30631945C211yesmaximal, characteristic, complemented
6#30631944C212yesmaximal, complemented
7#30631943C212yesmaximal, complemented
8#30631942C211yesmaximal, characteristic, complemented
9#106×D251C211yesmaximal, characteristic, complemented
10#103×D251C212yesmaximal, complemented
11#470×C251C211yesmaximal, characteristic, complemented
12C2×Q8×D251C2212yes—
13C22×C4×D251C2211yescharacteristic
14C2×Dic1004C2211yescharacteristic, complemented
15C2×#16860800C2212yescomplemented
16Q16×D251—24no—
17C2×#16860794C2211yescharacteristic
18C2×#16860793C2211yescharacteristic
19C2×#16860793C2211yescharacteristic
20M16×D251—24no—
21C2×C8×D251C2211yescharacteristic
22C2×C8×D251C2211yescharacteristic
23#22548695C2212yes—
24#22548694C2212yes—
25#22548693—22no—
26#22548692—22no—
27#22548691C2212yes—
28C4⋊C4×D251—22no—
29C4⋊C4×D251C2212yes—
30C22⋊C4×D251—22no—
31#22548647C2211yescharacteristic, complemented
32#22548646C2212yescomplemented
33#22548640C2212yescomplemented
34#22548639C2212yescomplemented
35#22548638C2212yescomplemented
36C2×Q16×C251C2211yescharacteristic, complemented
37C2×M16×C251C2211yescharacteristic
38C22⋊Q8×C251C2212yes—
39#106×C251C2211yescharacteristic, complemented
40#103×C251C2212yescomplemented
41C23×D251D411yescharacteristic
42C2×Dic502—22no—
43C2×Dic502C2312yes—
44Q8×D251—24no—
45Q8×D251—24no—
46C22×Dic251D411yescharacteristic
47C2×C4×D251—22no—
48C2×C4×D251—22no—
49C2×C4×D251—22no—
50C2×C4×D251—22no—
51C2×C4×D251D411yescharacteristic
52C2×C4×D251D411yescharacteristic
53C2×C4×D251C2311yescharacteristic
54C2×C4×D251C2311yescharacteristic
55Dic1004—22no—
56#16860800—24no—
57#16860794—22no—
58#16860793—22no—
59#16860793—22no—
60C2×C251⋊C8C2311yescharacteristic
61C2×C251⋊C8C2311yescharacteristic
62C8×D251—22no—
63C8×D251D414yes—
64C4⋊Dic251—22no—
65C4⋊Dic251C2312yes—
66Dic251⋊C4—22no—
67Dic251⋊C4—22no—
68Dic251⋊C4—22no—
69C22⋊Dic251—22no—
70(C2×D251)⋊C4—22no—
71(C2×D251)⋊C4—22no—
72C2×Q8×C251C2312yes—
73C22×C4×C251C2311yescharacteristic
74Q16×C251—22no—
75M16×C251—22no—
76C2×C8×C251C2311yescharacteristic
77C2×C8×C251C2311yescharacteristic
78C4⋊C4×C251—22no—
79C4⋊C4×C251C2312yes—
80C22⋊C4×C251—22no—
81C22×D251—22no—
82C22×D251—24no—
83C22×D251C2×D411yescharacteristic
84C22×D251C2×D411yescharacteristic
85Dic502—22no—
86Dic502—22no—
87Dic502—44no—
88C2×Dic251—22no—
89C2×Dic251—22no—
90C2×Dic251—22no—
91C2×Dic251C2×D411yescharacteristic
92C2×Dic251C2×D411yescharacteristic
93C4×D251—22no—
94C4×D251—21no—
95C4×D251—21no—
96C4×D251—22no—
97C4×D251—44no—
98C4×D251—44no—
99C4×D251C2×D412yes—
100C4×D251#4112yes—
101C251⋊C8—21no—
102C251⋊C8C2×D412yes—
103C23×C251C2×D411yescharacteristic
104Q8×C251—22no—
105Q8×C251—22no—
106C2×C4×C251—21no—
107C2×C4×C251—21no—
108C2×C4×C251—22no—
109C2×C4×C251—22no—
110C2×C4×C251C2×D411yescharacteristic
111C2×C4×C251C2411yescharacteristic
112C8×C251—21no—
113C8×C251C2×D412yes—
114C22×C251—21no—
115C22×C251—22no—
116C22×C251C22×D411yescharacteristic
117C2×D251—21no—
118C2×D251—22no—
119C2×D251—44no—
120C2×D251#12112yes—
121C2×D251#13714yes—
122C4×C251—21no—
123C4×C251—42no—
124C4×C251—42no—
125C4×C251C22×D411yescharacteristic
126C4×C251C2×#4111yescharacteristic
127Dic251—21no—
128Dic251—42no—
129Dic251—42no—
130Dic251#12112yes—
131C2×C251—41no—
132C2×C251C2×#12111yescharacteristic
133C2×C251C2×#13712yes—
134D251—41no—
135D251#47014yesdirect product factor
136C251C2×#47011yescharacteristic, complemented, Sylow
137C2×#470—2511nomaximal, Sylow
138C22×Q16—2511no—
139C22×M16—2511no—
140C2×C22⋊Q8—2512no—
141#470—2511no—
142#470—2511no—
143#470—2512no—
144#470—2511no—
145#470—2512no—
146#470D25111yescharacteristic, direct product factor
147C2×#106—2511no—
148C2×#103—2512no—
149C22×Q8—2512no—
150C23×C4—2511no—
151C2×Q16—5024no—
152C2×Q16—2512no—
153C2×Q16—2511no—
154C2×Q16C2×D25111yescharacteristic, complemented
155C2×M16—5024no—
156C2×M16—2511no—
157C2×M16—2511no—
158C2×M16—2511no—
159C2×M16C2×D25111yescharacteristic
160C22×C8—2511no—
161C22×C8—2511no—
162C22⋊Q8—5022no—
163C22⋊Q8—5022no—
164C22⋊Q8—2512no—
165C22⋊Q8—2512no—
166C22⋊Q8—2512no—
167C22⋊Q8C2×D25112yes—
168C2×C4⋊C4—5022no—
169C2×C4⋊C4—2512no—
170C2×C22⋊C4—5022no—
171#106—2511no—
172#106—2512no—
173#106C2×D25111yescharacteristic, complemented
174#103—2512no—
175#103—2512no—
176#103—2512no—
177#103C2×D25112yescomplemented
178C24—2511no—
179C2×Q8—5022no—
180C2×Q8—5024no—
181C2×Q8—5024no—
182C2×Q8—2512no—
183C2×Q8C22×D25112yes—
184C22×C4—5022no—
185C22×C4—5022no—
186C22×C4—5022no—
187C22×C4—5022no—
188C22×C4—2511no—
189C22×C4—2511no—
190C22×C4—2511no—
191C22×C4—2511no—
192C22×C4—2511no—
193C22×C4C22×D25111yescharacteristic
194Q16—22no—
195Q16—5024no—
196Q16—5022no—
197M16—22no—
198M16—5022no—
199M16—5022no—
200M16—5022no—
201C2×C8—5022no—
202C2×C8—2511no—
203C2×C8—2511no—
204C2×C8—2514no—
205C2×C8C22×D25111yescharacteristic
206C2×C8C22×D25111yescharacteristic
207C4⋊C4—22no—
208C4⋊C4—5022no—
209C4⋊C4—5022no—
210C4⋊C4—5022no—
211C4⋊C4—5022no—
212C4⋊C4—2512no—
213C4⋊C4C22×D25112yes—
214C22⋊C4—22no—
215C22⋊C4—5022no—
216C22⋊C4—5022no—
217C22⋊C4—5022no—
218C23—5022no—
219C23—5024no—
220C23—2511no—
221C23—2511no—
222C23D4×D25111yescharacteristic
223Q8—22no—
224Q8—22no—
225Q8—10044no—
226Q8—5022no—
227Q8—5022no—
228C2×C4—21no—
229C2×C4—21no—
230C2×C4—22no—
231C2×C4—22no—
232C2×C4—10044no—
233C2×C4—10044no—
234C2×C4—5022no—
235C2×C4—5022no—
236C2×C4—5022no—
237C2×C4—5022no—
238C2×C4—5022no—
239C2×C4—5021no—
240C2×C4—5021no—
241C2×C4—2511no—
242C2×C4—2511no—
243C2×C4—2512no—
244C2×C4—2512no—
245C2×C4D4×D25111yescharacteristic
246C2×C4C23×D25111yescharacteristic
247C8—21no—
248C8—5021no—
249C8—2512no—
250C8D4×D25112yes—
251C22—22no—
252C22—21no—
253C22—10044no—
254C22—5022no—
255C22—5021no—
256C22—2514no—
257C22—2512no—
258C22C2×D4×D25111yescharacteristic
259C4—21no—
260C4—42no—
261C4—42no—
262C4—10042no—
263C4—10042no—
264C4—5021no—
265C4—2512no—
266C4C2×D4×D25111yescharacteristic
267C4#41×D25111yescharacteristic
268C2—41no—
269C2—10041no—
270C2—2514no—
271C2#121×D25111yescharacteristic
272C2#137×D25112yes—
subgroupG/H
centerC22C2×D4×D251
commutator, cosocleC2×C4×C251C24
FrattiniC2×C4C23×D251
Fitting#470×C251C2
socleC22×C251C22×D4
prime radicalC23×D251D4
hypercenter#470D251
hypocenterC251C2×#470
LCSG ▷ C2×C4×C251 ▷ C2×C251 ▷ C251
UCSC1 ◁ C22 ◁ C22×C4 ◁ #470

Complements

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

NH ≅ G/Ntype
1. C2×Q16×D251C2semidirect
4. #30631946C2semidirect
5. #30631945C2semidirect
6. #30631944C2semidirect
7. #30631943C2semidirect
8. #30631942C2semidirect
9. #106×D251C2semidirect
10. #103×D251C2semidirect
11. #470×C251C2semidirect
14. C2×Dic1004C22semidirect
15. C2×#16860800C22semidirect
31. #22548647C22semidirect
32. #22548646C22semidirect
33. #22548640C22semidirect
34. #22548639C22semidirect
35. #22548638C22semidirect
36. C2×Q16×C251C22semidirect
39. #106×C251C22semidirect
40. #103×C251C22semidirect
135. D251#470direct (N × H)
136. C251C2×#470semidirect
146. #470D251direct (N × H)
154. C2×Q16C2×D251semidirect
173. #106C2×D251semidirect
177. #103C2×D251semidirect

Maximal subgroups

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

subgroupindex
C2×Q16×D2512
C2×M16×D2512
C22⋊Q8×D2512
#306319462
#306319452
#306319442
#306319432
#306319422
#106×D2512
#103×D2512
#470×C2512
C2×#470251