Finite Group Databasebeta

← #44548870

Subgroups — #44548870

Order 32128.

334 conjugacy classes of subgroups; only the 77 normal subgroups are shown.

normal subgroups only

This group has 334 conjugacy classes of subgroups — too many to draw. Only its 77 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#30632198C211yesmaximal, characteristic
2C2×C251⋊#95C211yesmaximal, characteristic, complemented
3#30631425C212yesmaximal, complemented
4#30631422C212yesmaximal, complemented
5#407×C251C211yesmaximal, characteristic, complemented
6C2×#16860810C411yescharacteristic, complemented
7C23×Dic251—21no—
8#22548698—21no—
9#22548694C411yescharacteristic, complemented
10#22548694C2211yescharacteristic
11#22548689C411yescharacteristic, complemented
12#22548689C2211yescharacteristic
13#22548684—21no—
14C4×C22⋊D251—21no—
15C4×C22⋊D251—21no—
16#22548674C411yescharacteristic, complemented
17C251⋊#95—21no—
18C251⋊#95C2212yes—
19C23×C4×C251C2211yescharacteristic
20C2×M16×C251—21no—
21C22⋊C8×C251—21no—
22C22⋊C8×C251C2212yes—
23#16860810—42no—
24C2×Dic502C2×C411yescharacteristic, complemented
25C2×D1004C2×C411yescharacteristic, complemented
26C2×C22⋊D251—21no—
27C2×C22⋊D251—21no—
28C2×C22⋊D251—21no—
29C22×Dic251—24no—
30C22×Dic251—22no—
31C22×Dic251—42no—
32C22×Dic251—42no—
33C2×C4×D251—21no—
34C4⋊Dic251C2×C411yescharacteristic, complemented
35C4⋊Dic251C2×C411yescharacteristic, complemented
36Dic251⋊C4—21no—
37Dic251⋊C4—21no—
38C22⋊Dic251—21no—
39C22⋊Dic251—21no—
40C22⋊Dic251—21no—
41(C2×D251)⋊C4—21no—
42(C2×D251)⋊C4—21no—
43C4×Dic251—21no—
44C24×C251D411yescharacteristic
45C22×C4×C251—42no—
46C22×C4×C251C2×C411yescharacteristic
47C22×C4×C251C2×C411yescharacteristic
48C22×C4×C251D411yescharacteristic
49C22×C4×C251D411yescharacteristic
50C22×C4×C251D411yescharacteristic
51C22×C4×C251C2311yescharacteristic
52M16×C251—42no—
53C2×C8×C251—22no—
54C22×D251—21no—
55Dic502—41no—
56D1004—41no—
57C22⋊D251—42no—
58C22⋊D251—42no—
59C22⋊D251—41no—
60C22⋊D251—41no—
61C2×Dic251—24no—
62C2×Dic251—21no—
63C2×Dic251—21no—
64C2×Dic251—21no—
65C2×Dic251—48no—
66C2×Dic251—42no—
67C2×Dic251—42no—
68C4×D251—42no—
69C23×C251—22no—
70C23×C251—22no—
71C23×C251—21no—
72C23×C251—21no—
73C23×C251C22⋊C411yescharacteristic
74C23×C251C22⋊C411yescharacteristic
75C23×C251C2×D411yescharacteristic
76C2×C4×C251—22no—
77C2×C4×C251—22no—
78C2×C4×C251—41no—
79C2×C4×C251—41no—
80C2×C4×C251—41no—
81C2×C4×C251—41no—
82C2×C4×C251C22⋊C411yescharacteristic
83C2×C4×C251C22⋊C411yescharacteristic
84C2×C4×C251C22×C411yescharacteristic
85C2×C4×C251C2×D411yescharacteristic
86C8×C251—42no—
87C22×C251—24no—
88C22×C251—24no—
89C22×C251—21no—
90C22×C251—21no—
91C22×C251—21no—
92C22×C251—21no—
93C22×C251—21no—
94C22×C251—21no—
95C22×C251#9912yes—
96C22×C251#10412yes—
97C22×C251C2×C22⋊C411yescharacteristic
98C22×C251AGL(1,ℤ/8)11yescharacteristic
99C22×C251#13711yescharacteristic
100C2×D251—41no—
101C2×D251—41no—
102C2×D251—41no—
103C4×C251—21no—
104C4×C251—21no—
105C4×C251—21no—
106C4×C251—21no—
107Dic251—44no—
108Dic251—41no—
109Dic251—41no—
110Dic251—41no—
111C2×C251—21no—
112C2×C251—22no—
113C2×C251—22no—
114C2×C251—21no—
115C2×C251C2×#9911yescharacteristic
116C2×C251#41711yescharacteristic
117C2×C251C2×#10411yescharacteristic
118D251—81no—
119C251#150411yescharacteristic, complemented, Sylow
120#1504—2511nomaximal, Sylow
121#525—2511no—
122#407D25111yescharacteristic, complemented
123C2×#95—2511no—
124#328—2512no—
125#327—2512no—
126C2×#41—2511no—
127C23×C4—5021no—
128C23×C4C2×D25111yescharacteristic
129C2×M16—21no—
130#123—5021no—
131C22⋊Q8—2511no—
132C22⋊Q8—2511no—
133#121—2511no—
134#121—2511no—
135#120—5021no—
136C4×D4—5021no—
137C4×D4—5021no—
138#117—2511no—
139C22⋊C8—21no—
140C22⋊C8C2×D25112yes—
141#95—5021no—
142#95—2512no—
143C24D100411yescharacteristic
144#41—10042no—
145C2×Q8—2511no—
146C2×D4—5021no—
147C2×D4—5021no—
148C2×D4—5021no—
149C2×D4—2511no—
150C22×C4—42no—
151C22×C4—10042no—
152C22×C4—10042no—
153C22×C4—5024no—
154C22×C4—5022no—
155C22×C4—5021no—
156C22×C4C4×D25111yescharacteristic
157C22×C4C4×D25111yescharacteristic
158C22×C4C22⋊D25111yescharacteristic
159C22×C4C22⋊D25111yescharacteristic
160C22×C4D100411yescharacteristic
161C22×C4C22×D25111yescharacteristic
162M16—42no—
163C2×C8—22no—
164C4⋊C4—5021no—
165C4⋊C4—5021no—
166C4⋊C4—2511no—
167C4⋊C4—2511no—
168C22⋊C4—5021no—
169C22⋊C4—5021no—
170C22⋊C4—5021no—
171C22⋊C4—5021no—
172C22⋊C4—5021no—
173C42—5021no—
174C23—21no—
175C23—21no—
176C23—22no—
177C23—22no—
178C23—5021no—
179C23(C2×D251)⋊C411yescharacteristic
180C23(C2×D251)⋊C411yescharacteristic
181C23C2×D100411yescharacteristic
182Q8—10041no—
183D4—10041no—
184D4—10041no—
185D4—10041no—
186D4—10042no—
187D4—10042no—
188C2×C4—22no—
189C2×C4—22no—
190C2×C4—41no—
191C2×C4—41no—
192C2×C4—41no—
193C2×C4—41no—
194C2×C4—10042no—
195C2×C4—10048no—
196C2×C4—10042no—
197C2×C4—10042no—
198C2×C4—5024no—
199C2×C4—5021no—
200C2×C4—5021no—
201C2×C4—5021no—
202C2×C4(C2×D251)⋊C411yescharacteristic
203C2×C4(C2×D251)⋊C411yescharacteristic
204C2×C4C2×C4×D25111yescharacteristic
205C2×C4C2×C22⋊D25111yescharacteristic
206C8—42no—
207C22—24no—
208C22—24no—
209C22—21no—
210C22—21no—
211C22—21no—
212C22—21no—
213C22—21no—
214C22—21no—
215C22—10041no—
216C22—10041no—
217C22—10041no—
218C22#2254862912yes—
219C22#2254864112yes—
220C22C2×(C2×D251)⋊C411yescharacteristic
221C22#2254874311yescharacteristic
222C22#2254874811yescharacteristic
223C4—21no—
224C4—21no—
225C4—21no—
226C4—21no—
227C4—10044no—
228C4—10041no—
229C4—10041no—
230C4—10041no—
231C2—21no—
232C2—21no—
233C2—22no—
234C2—22no—
235C2—20081no—
236C2C2×#2254862911yescharacteristic
237C2#3063168911yescharacteristic
238C2C2×#2254864111yescharacteristic
subgroupG/H
centerC22C2×(C2×D251)⋊C4
commutatorC2×C4×C251C22×C4
FrattiniC22×C4C22×D251
Fitting#407×C251C2
socleC22×C251C2×C22⋊C4
prime radical#30632198C2
cosocleC22×C4×C251C23
hypercenter#407D251
hypocenterC251#1504
LCSG ▷ C2×C4×C251 ▷ C22×C251 ▷ C251
UCSC1 ◁ C22 ◁ C23×C4 ◁ #407

Complements

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

NH ≅ G/Ntype
2. C2×C251⋊#95C2semidirect
3. #30631425C2semidirect
4. #30631422C2semidirect
5. #407×C251C2semidirect
6. C2×#16860810C4semidirect
9. #22548694C4semidirect
11. #22548689C4semidirect
16. #22548674C4semidirect
24. C2×Dic502C2×C4semidirect
25. C2×D1004C2×C4semidirect
34. C4⋊Dic251C2×C4semidirect
35. C4⋊Dic251C2×C4semidirect
119. C251#1504semidirect
122. #407D251semidirect

Maximal subgroups

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

subgroupindex
#306321982
C2×C251⋊#952
#306314252
#306314222
#407×C2512
#1504251