Finite Group Databasebeta

← #44559888

Subgroups — #44559888

Order 32128.

490 conjugacy classes of subgroups; only the 141 normal subgroups are shown.

normal subgroups only

This group has 490 conjugacy classes of subgroups — too many to draw. Only its 141 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
1C22×#16860797C211yesmaximal, characteristic, complemented
2#30632251C211yesmaximal, characteristic
3#30632165C211yesmaximal, characteristic, complemented
4#30631911C214yesmaximal, complemented
5#30631885C214yesmaximal, complemented
6#30631715C211yesmaximal, characteristic, complemented
7#30631693C211yesmaximal, characteristic, complemented
8#30631683C211yesmaximal, characteristic, complemented
9C2×#137×C251C211yesmaximal, characteristic, complemented
10C22×D1004C2211yescharacteristic
11C2×#16860797—24no—
12C2×#16860797C2214yes—
13C22×C251⋊C8C2211yescharacteristic
14#22548711C2212yes—
15#22548704C2212yes—
16#22548689—22no—
17#22548683—22no—
18Q8×Dic251—22no—
19Q8×Dic251C2214yescomplemented
20#22548674—22no—
21#22548674C2211yescharacteristic
22#22548674C2211yescharacteristic
23Dic251⋊C8C2214yescomplemented
24#22548639C2212yescomplemented
25#22548637C2214yes—
26#22548636C2212yescomplemented
27C2×#41×C251C2211yescharacteristic
28C22×Q8×C251C2211yescharacteristic
29#137×C251—24no—
30C2×Q16×C251C2212yescomplemented
31C2×SD16×C251C2212yescomplemented
32C2×M16×C251C2211yescharacteristic
33C23×D251—21no—
34C2×D1004—22no—
35C2×D1004—21no—
36C2×D1004—22no—
37C2×D1004C2312yes—
38C2×C22⋊D251—24no—
39#16860797—28no—
40C2×C251⋊C8—22no—
41C2×C251⋊C8C2312yes—
42C4⋊Dic251—24no—
43C4⋊Dic251D412yes—
44C4⋊Dic251C2312yes—
45C22⋊Dic251—21no—
46C22⋊Dic251D412yes—
47(C2×D251)⋊C4—24no—
48C4×Dic251—24no—
49C4×Dic251—21no—
50C4×Dic251C2312yes—
51#41×C251—22no—
52#41×C251—22no—
53C2×Q8×C251—21no—
54C2×Q8×C251—22no—
55C2×Q8×C251D414yescomplemented
56C2×Q8×C251C2311yescharacteristic
57C2×Q8×C251C2312yes—
58C2×D4×C251—21no—
59C2×D4×C251C2311yescharacteristic
60C22×C4×C251—21no—
61C22×C4×C251—21no—
62C22×C4×C251C2311yescharacteristic
63Q16×C251—24no—
64SD16×C251—24no—
65M16×C251—22no—
66C2×C8×C251C2312yes—
67C22×D251—22no—
68C22×D251—42no—
69C22×D251—82no—
70D1004—22no—
71D1004—22no—
72D1004—42no—
73C22⋊D251—84no—
74C2×Dic251—22no—
75C2×Dic251C2×D414yes—
76C251⋊C8—22no—
77C23×C251—21no—
78C23×C251C2×D411yescharacteristic
79Q8×C251—24no—
80Q8×C251—22no—
81Q8×C251—21no—
82Q8×C251—21no—
83Q8×C251C2×D414yes—
84D4×C251—21no—
85D4×C251—21no—
86D4×C251—42no—
87C2×C4×C251—24no—
88C2×C4×C251—22no—
89C2×C4×C251—21no—
90C2×C4×C251—42no—
91C2×C4×C251C2×D411yescharacteristic
92C2×C4×C251C2×D412yes—
93C2×C4×C251#4112yes—
94C2×C4×C251C2411yescharacteristic
95C8×C251—22no—
96C22×C251—22no—
97C22×C251—41no—
98C22×C251—42no—
99C22×C251#12112yes—
100C22×C251C22×D411yescharacteristic
101C2×D251—42no—
102C2×D251—44no—
103C2×D251—84no—
104C4×C251—24no—
105C4×C251—41no—
106C4×C251#12112yes—
107C4×C251C22×D411yescharacteristic
108C4×C251C2×#4111yescharacteristic
109Dic251—24no—
110Dic251—42no—
111C2×C251—22no—
112C2×C251—81no—
113C2×C251C2×#12111yescharacteristic
114C2×C251#57712yes—
115D251—82no—
116C251#304311yescharacteristic, complemented, Sylow
117#3043—2511nomaximal, Sylow
118C2×#137D25111yescharacteristic, complemented
119C22×SD16—2511no—
120#535—2511no—
121#519—2511no—
122#464—2514no—
123#460—2514no—
124#424—2511no—
125#418—2511no—
126#416—2511no—
127C2×#41C2×D25111yescharacteristic
128C22×Q8C2×D25111yescharacteristic
129C22×D4—2511no—
130#137—24no—
131C2×Q16C2×D25112yescomplemented
132C2×SD16—5024no—
133C2×SD16—2514no—
134C2×SD16C2×D25112yescomplemented
135C2×M16C2×D25111yescharacteristic
136C22×C8—2511no—
137#127—2512no—
138#124—2512no—
139#121—5022no—
140#120—5022no—
141C4×Q8—5022no—
142C4×Q8—2514no—
143#117—5022no—
144#117—2511no—
145#117—2511no—
146C4⋊C8—2514no—
147#103—2512no—
148#102—2512no—
149#102—2514no—
150C24—5021no—
151#41—22no—
152#41—22no—
153C2×Q8—21no—
154C2×Q8—22no—
155C2×Q8C22⋊D25114yescomplemented
156C2×Q8C22×D25111yescharacteristic
157C2×Q8C22×D25112yes—
158C2×D4—21no—
159C2×D4—5024no—
160C2×D4—5021no—
161C2×D4—5022no—
162C2×D4—5022no—
163C2×D4—2512no—
164C2×D4C22×D25111yescharacteristic
165C22×C4—21no—
166C22×C4—21no—
167C22×C4C22×D25111yescharacteristic
168Q16—24no—
169SD16—24no—
170SD16—5028no—
171M16—22no—
172C2×C8—5022no—
173C2×C8—2512no—
174C2×C8C22×D25112yes—
175C4⋊C4—5024no—
176C4⋊C4—2512no—
177C4⋊C4—2512no—
178C22⋊C4—5024no—
179C22⋊C4—5021no—
180C22⋊C4—2512no—
181C42—5024no—
182C42—5021no—
183C42—2512no—
184C23—21no—
185C23—20082no—
186C23—10042no—
187C23—5022no—
188C23D4×D25111yescharacteristic
189Q8—22no—
190Q8—24no—
191Q8—21no—
192Q8—21no—
193Q8C2×C22⋊D25114yes—
194D4—21no—
195D4—21no—
196D4—42no—
197D4—20084no—
198D4—10042no—
199D4—5022no—
200D4—5022no—
201C2×C4—22no—
202C2×C4—24no—
203C2×C4—21no—
204C2×C4—42no—
205C2×C4—5022no—
206C2×C4—2514no—
207C2×C4D4×D25111yescharacteristic
208C2×C4C2×C22⋊D25112yes—
209C2×C4#1686080912yes—
210C2×C4C23×D25111yescharacteristic
211C8—22no—
212C8—5022no—
213C22—22no—
214C22—42no—
215C22—41no—
216C22—20084no—
217C22—10044no—
218C22—10042no—
219C22#2254868712yes—
220C22C2×D4×D25111yescharacteristic
221C4—24no—
222C4—41no—
223C4—10042no—
224C4—5024no—
225C4#2254868712yes—
226C4C22×C22⋊D25111yescharacteristic
227C4C2×#1686080911yescharacteristic
228C2—22no—
229C2—81no—
230C2—20082no—
231C2C2×#2254868711yescharacteristic
232C2#3063249612yes—
subgroupG/H
centerC22C2×D4×D251
commutator, cosocleC2×C4×C251C24
FrattiniC2×C4C23×D251
FittingC2×#137×C251C2
socleC22×C251C22×D4
prime radical#30632251C2
hypercenterC2×#137D251
hypocenterC251#3043
LCSG ▷ C2×C4×C251 ▷ C2×C251 ▷ C251
UCSC1 ◁ C22 ◁ C22×C4 ◁ C2×#137

Complements

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

NH ≅ G/Ntype
1. C22×#16860797C2semidirect
3. #30632165C2semidirect
4. #30631911C2semidirect
5. #30631885C2semidirect
6. #30631715C2semidirect
7. #30631693C2semidirect
8. #30631683C2semidirect
9. C2×#137×C251C2semidirect
19. Q8×Dic251C22semidirect
23. Dic251⋊C8C22semidirect
24. #22548639C22semidirect
26. #22548636C22semidirect
30. C2×Q16×C251C22semidirect
31. C2×SD16×C251C22semidirect
55. C2×Q8×C251D4semidirect
116. C251#3043semidirect
118. C2×#137D251semidirect
131. C2×Q16C2×D251semidirect
134. C2×SD16C2×D251semidirect
155. C2×Q8C22⋊D251semidirect

Maximal subgroups

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

subgroupindex
C22×#168607972
#306322512
#306321652
#306319112
#306318852
#306317152
#306316932
#306316832
C2×#137×C2512
#3043251