Finite Group Databasebeta

← #497×D251 #44560801

Subgroups — #497×D251 #44560801

Order 32128.

564 conjugacy classes of subgroups; only the 145 normal subgroups are shown.

normal subgroups only

This group has 564 conjugacy classes of subgroups — too many to draw. Only its 145 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×D251C212yesmaximal, complemented
2C2×SD16×D251C212yesmaximal
3C4⋊Q8×D251C211yesmaximal, characteristic, complemented
4#124×D251C211yesmaximal, characteristic
5#30632082C211yesmaximal, characteristic, complemented
6#30632081C211yesmaximal, characteristic, complemented
7#30632080C212yesmaximal, complemented
8#30632079C211yesmaximal, characteristic, complemented
9#30632078C212yesmaximal, complemented
10#97×D251C211yesmaximal, characteristic, complemented
11#497×C251C211yesmaximal, characteristic, complemented
12C2×Q8×D251C2211yescharacteristic
13C2×Q8×D251C2212yes—
14C2×D4×D251C2211yescharacteristic
15C2×Dic1004C2212yescomplemented
16C2×#16860800C2212yescomplemented
17C2×#16860800C2212yescomplemented
18Q16×D251—28no—
19C2×#16860799C2212yes—
20C2×#16860798C2212yes—
21C2×#16860797C2212yes—
22SD16×D251—28no—
23C2×C8×D251C2212yes—
24#22548715C2211yescharacteristic, complemented
25#22548714—22no—
26#22548713C2212yescomplemented
27#22548704C2211yescharacteristic
28#22548703C2211yescharacteristic
29#22548702C2211yescharacteristic
30#22548701—22no—
31C4⋊C4×D251—22no—
32C22⋊C4×D251—22no—
33C42×D251C2211yescharacteristic
34#22548626C2211yescharacteristic, complemented
35#22548625C2212yescomplemented
36C2×Q16×C251C2212yescomplemented
37C2×SD16×C251C2212yes—
38C4⋊Q8×C251C2211yescharacteristic, complemented
39#124×C251C2211yescharacteristic
40#97×C251C2211yescharacteristic, complemented
41C23×D251—21no—
42C2×Dic502—21no—
43C2×Dic502—22no—
44C2×Dic502C2311yescharacteristic
45C2×Dic502C2312yes—
46Q8×D251—24no—
47Q8×D251—24no—
48Q8×D251—24no—
49C2×D1004C2311yescharacteristic
50C2×C22⋊D251—21no—
51D4×D251—24no—
52C2×C4×D251—21no—
53C2×C4×D251—22no—
54C2×C4×D251D412yes—
55C2×C4×D251C2311yescharacteristic
56Dic1004—24no—
57#16860800—24no—
58#16860800—24no—
59#16860799—24no—
60#16860798—24no—
61#16860797—24no—
62C2×C251⋊C8C2312yes—
63C8×D251D418yes—
64C4⋊Dic251—22no—
65Dic251⋊C4—24no—
66C22⋊Dic251—22no—
67(C2×D251)⋊C4—22no—
68(C2×D251)⋊C4—22no—
69C4×Dic251D412yes—
70C4×Dic251C2311yescharacteristic
71C2×Q8×C251C2311yescharacteristic
72C2×Q8×C251C2312yes—
73C2×D4×C251C2311yescharacteristic
74Q16×C251—24no—
75SD16×C251—24no—
76C2×C8×C251C2312yes—
77C4⋊C4×C251—22no—
78C22⋊C4×C251—22no—
79C42×C251C2311yescharacteristic
80C22×D251—21no—
81C22×D251—42no—
82C22×D251—44no—
83C22×D251C2×D411yescharacteristic
84Dic502—22no—
85Dic502—22no—
86Dic502—22no—
87Dic502—44no—
88Dic502—42no—
89D1004—22no—
90C22⋊D251—42no—
91C2×Dic251—21no—
92C2×Dic251—22no—
93C2×Dic251C2×D412yes—
94C2×Dic251C2×D411yescharacteristic
95C4×D251—24no—
96C4×D251—42no—
97C4×D251—44no—
98C4×D251C2×D414yes—
99C251⋊C8C2×D414yes—
100C23×C251—21no—
101Q8×C251—22no—
102Q8×C251—22no—
103Q8×C251—22no—
104D4×C251—22no—
105C2×C4×C251—21no—
106C2×C4×C251—22no—
107C2×C4×C251C2×D412yes—
108C2×C4×C251C2411yescharacteristic
109C8×C251C2×D414yes—
110C22×C251—41no—
111C22×C251—42no—
112C22×C251C22×D411yescharacteristic
113C2×D251—41no—
114C2×D251—42no—
115C2×D251—84no—
116C2×D251#12712yes—
117C2×D251#13714yes—
118C4×C251—22no—
119C4×C251—41no—
120C4×C251—42no—
121C4×C251C22×D412yes—
122Dic251—22no—
123Dic251—41no—
124Dic251—42no—
125Dic251#12712yes—
126C2×C251—81no—
127C2×C251C2×#12711yescharacteristic
128C2×C251C2×#13712yes—
129D251—81no—
130D251#49714yesdirect product factor
131C251C2×#49711yescharacteristic, complemented, Sylow
132C2×#497—2511nomaximal, Sylow
133C22×Q16—2512no—
134C22×SD16—2512no—
135C2×C4⋊Q8—2511no—
136C2×#124—2511no—
137#497—2512no—
138#497—2512no—
139#497—2511no—
140#497—2511no—
141#497—2511no—
142#497D25111yescharacteristic, direct product factor
143C2×#97—2511no—
144C22×Q8—2511no—
145C22×Q8—2512no—
146C22×D4—2511no—
147C2×Q16—5028no—
148C2×Q16—2512no—
149C2×Q16—2512no—
150C2×Q16—2512no—
151C2×Q16C2×D25112yescomplemented
152C2×SD16—5028no—
153C2×SD16—2512no—
154C2×SD16—2512no—
155C2×SD16—2512no—
156C2×SD16C2×D25112yes—
157C22×C8—2512no—
158C4⋊Q8—5022no—
159C4⋊Q8—2512no—
160C4⋊Q8—2511no—
161C4⋊Q8C2×D25111yescharacteristic, complemented
162#124—5022no—
163#124—2511no—
164#124—2511no—
165#124—2511no—
166#124C2×D25111yescharacteristic
167C2×C4⋊C4—5022no—
168C2×C22⋊C4—5022no—
169C2×C42—2511no—
170#97—2512no—
171#97—2511no—
172#97C2×D25111yescharacteristic, complemented
173C24—5021no—
174C2×Q8—5021no—
175C2×Q8—5022no—
176C2×Q8—5024no—
177C2×Q8—5024no—
178C2×Q8—5024no—
179C2×Q8—2511no—
180C2×Q8—2512no—
181C2×Q8C22×D25111yescharacteristic
182C2×Q8C22×D25112yes—
183C2×D4—5024no—
184C2×D4—5021no—
185C2×D4—2511no—
186C2×D4C22×D25111yescharacteristic
187C22×C4—5021no—
188C22×C4—5022no—
189C22×C4—2512no—
190C22×C4—2511no—
191Q16—24no—
192Q16—5024no—
193Q16—5024no—
194Q16—5024no—
195SD16—24no—
196SD16—5024no—
197SD16—5024no—
198SD16—5024no—
199C2×C8—2512no—
200C2×C8—2518no—
201C2×C8C22×D25112yes—
202C4⋊C4—22no—
203C4⋊C4—5024no—
204C4⋊C4—5022no—
205C22⋊C4—22no—
206C22⋊C4—5022no—
207C22⋊C4—5022no—
208C22⋊C4—5022no—
209C42—2512no—
210C42—2511no—
211C42C22×D25111yescharacteristic
212C23—21no—
213C23—10042no—
214C23—10044no—
215C23—5021no—
216C23—2511no—
217Q8—22no—
218Q8—22no—
219Q8—22no—
220Q8—10042no—
221Q8—10044no—
222Q8—5022no—
223Q8—5022no—
224Q8—5022no—
225D4—22no—
226D4—10042no—
227D4—5022no—
228C2×C4—21no—
229C2×C4—22no—
230C2×C4—10042no—
231C2×C4—10044no—
232C2×C4—5024no—
233C2×C4—5021no—
234C2×C4—5022no—
235C2×C4—2512no—
236C2×C4—2511no—
237C2×C4—2514no—
238C2×C4D4×D25112yes—
239C2×C4C23×D25111yescharacteristic
240C8—2514no—
241C8D4×D25114yes—
242C22—42no—
243C22—41no—
244C22—20084no—
245C22—10042no—
246C22—10041no—
247C22—2514no—
248C22—2512no—
249C22C2×D4×D25111yescharacteristic
250C4—22no—
251C4—41no—
252C4—42no—
253C4—10041no—
254C4—10042no—
255C4—5022no—
256C4—2512no—
257C4C2×D4×D25112yes—
258C2—81no—
259C2—20081no—
260C2—2514no—
261C2#127×D25111yescharacteristic
262C2#137×D25112yes—
subgroupG/H
centerC22C2×D4×D251
commutator, cosocleC2×C4×C251C24
FrattiniC2×C4C23×D251
Fitting#497×C251C2
socleC22×C251C22×D4
prime radicalC2×D4×D251C22
hypercenter#497D251
hypocenterC251C2×#497
LCSG ▷ C2×C4×C251 ▷ C2×C251 ▷ C251
UCSC1 ◁ C22 ◁ C42 ◁ #497

Complements

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

NH ≅ G/Ntype
1. C2×Q16×D251C2semidirect
3. C4⋊Q8×D251C2semidirect
5. #30632082C2semidirect
6. #30632081C2semidirect
7. #30632080C2semidirect
8. #30632079C2semidirect
9. #30632078C2semidirect
10. #97×D251C2semidirect
11. #497×C251C2semidirect
15. C2×Dic1004C22semidirect
16. C2×#16860800C22semidirect
17. C2×#16860800C22semidirect
24. #22548715C22semidirect
26. #22548713C22semidirect
34. #22548626C22semidirect
35. #22548625C22semidirect
36. C2×Q16×C251C22semidirect
38. C4⋊Q8×C251C22semidirect
40. #97×C251C22semidirect
130. D251#497direct (N × H)
131. C251C2×#497semidirect
142. #497D251direct (N × H)
151. C2×Q16C2×D251semidirect
161. C4⋊Q8C2×D251semidirect
172. #97C2×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×SD16×D2512
C4⋊Q8×D2512
#124×D2512
#306320822
#306320812
#306320802
#306320792
#306320782
#97×D2512
#497×C2512
C2×#497251