Finite Group Databasebeta

← #44548701

Subgroups — #44548701

Order 32128.

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

normal subgroups only

This group has 316 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
1C2×C4×C22⋊D251C211yesmaximal, characteristic
2#30631651C211yesmaximal, characteristic, complemented
3#30631412C212yesmaximal, complemented
4#30631410C212yesmaximal, complemented
5#407×C251C211yesmaximal, characteristic, complemented
6C22×C22⋊D251C411yescharacteristic
7C22×C4×D251C411yescharacteristic
8C2×#16860794—21no—
9C4×C22⋊D251—21no—
10C4×C22⋊D251—21no—
11C4×C22⋊D251—21no—
12C4×C22⋊D251—21no—
13C2×Dic251⋊C4C2211yescharacteristic
14C2×C22⋊Dic251C411yescharacteristic
15C2×(C2×D251)⋊C4C2211yescharacteristic
16C2×C4×Dic251C411yescharacteristic
17C22⋊(C251⋊C8)—21no—
18C22⋊(C251⋊C8)C2212yes—
19C23×C4×C251C2211yescharacteristic
20C2×M16×C251—21no—
21C22⋊C8×C251—21no—
22C22⋊C8×C251C2212yes—
23C23×D251C2×C411yescharacteristic
24C2×C22⋊D251—22no—
25C2×C22⋊D251—22no—
26C2×C22⋊D251—21no—
27C2×C22⋊D251—21no—
28C22×Dic251C2×C411yescharacteristic
29C22×Dic251C2×C411yescharacteristic
30C22×Dic251C2×C411yescharacteristic
31C2×C4×D251—21no—
32C2×C4×D251—21no—
33C2×C4×D251—42no—
34#16860794—42no—
35C2×C251⋊C8—22no—
36Dic251⋊C4—21no—
37Dic251⋊C4—21no—
38C22⋊Dic251—21no—
39C22⋊Dic251—21no—
40(C2×D251)⋊C4—21no—
41(C2×D251)⋊C4—21no—
42C4×Dic251—21no—
43C4×Dic251—21no—
44C24×C251C2×C411yescharacteristic
45C22×C4×C251—42no—
46C22×C4×C251C2×C411yescharacteristic
47C22×C4×C251D411yescharacteristic
48C22×C4×C251D411yescharacteristic
49C22×C4×C251D411yescharacteristic
50C22×C4×C251D411yescharacteristic
51C22×C4×C251C2311yescharacteristic
52M16×C251—42no—
53C2×C8×C251—22no—
54C22×D251—21no—
55C22×D251—21no—
56C22×D251—41no—
57C22×D251—41no—
58C22×D251M1612yescomplemented
59C22⋊D251—24no—
60C22⋊D251—24no—
61C2×Dic251—21no—
62C2×Dic251—21no—
63C2×Dic251—21no—
64C2×Dic251—21no—
65C2×Dic251—41no—
66C2×Dic251—41no—
67C2×Dic251M1612yescomplemented
68C4×D251—42no—
69C4×D251—42no—
70C251⋊C8—42no—
71C23×C251—21no—
72C23×C251—21no—
73C23×C251—41no—
74C23×C251—41no—
75C23×C251C22⋊C411yescharacteristic
76C23×C251C22⋊C411yescharacteristic
77C23×C251C22×C411yescharacteristic
78C2×C4×C251—22no—
79C2×C4×C251—22no—
80C2×C4×C251—22no—
81C2×C4×C251—41no—
82C2×C4×C251—41no—
83C2×C4×C251—41no—
84C2×C4×C251C22⋊C411yescharacteristic
85C2×C4×C251C22⋊C411yescharacteristic
86C2×C4×C251C2×D411yescharacteristic
87C2×C4×C251C2×D411yescharacteristic
88C8×C251—42no—
89C22×C251—21no—
90C22×C251—21no—
91C22×C251—21no—
92C22×C251—21no—
93C22×C251—21no—
94C22×C251—21no—
95C22×C251—21no—
96C22×C251—21no—
97C22×C251—42no—
98C22×C251—42no—
99C22×C251C2×C22⋊C411yescharacteristic
100C22×C251C2×M1611yescharacteristic
101C22×C251C2×M1611yescharacteristic
102C2×D251—22no—
103C2×D251—42no—
104C2×D251—41no—
105C2×D251—41no—
106C2×D251—42no—
107C4×C251—21no—
108C4×C251—21no—
109C4×C251—21no—
110C4×C251#9912yes—
111Dic251—22no—
112Dic251—41no—
113Dic251—41no—
114C2×C251—21no—
115C2×C251—21no—
116C2×C251—41no—
117C2×C251—41no—
118C2×C251#40711yescharacteristic
119C2×C251C2×#9911yescharacteristic
120C2×C251#41311yescharacteristic
121D251—42no—
122C251#146511yescharacteristic, complemented, Sylow
123#1465—2511nomaximal, Sylow
124C2×C4×D4—2511no—
125#407—2511no—
126#407D25111yescharacteristic, complemented
127#324—2512no—
128#323—2512no—
129C22×D4—2511no—
130C23×C4—2511no—
131C23×C4C2×D25111yescharacteristic
132C2×M16—21no—
133C2×M16—5021no—
134C4×D4—5021no—
135C4×D4—5021no—
136C4×D4—5021no—
137C4×D4—5021no—
138C2×C4⋊C4—2511no—
139C2×C22⋊C4—2511no—
140C2×C22⋊C4—2511no—
141C2×C42—2511no—
142C22⋊C8—21no—
143C22⋊C8—5021no—
144C22⋊C8—2512no—
145C22⋊C8C2×D25112yes—
146C24—2511no—
147C24C4×D25111yescharacteristic
148C2×D4—5022no—
149C2×D4—5021no—
150C2×D4—5021no—
151C2×D4—5022no—
152C22×C4—42no—
153C22×C4—10042no—
154C22×C4—5021no—
155C22×C4—5021no—
156C22×C4—2511no—
157C22×C4—2511no—
158C22×C4—2511no—
159C22×C4C4×D25111yescharacteristic
160C22×C4C22⋊D25111yescharacteristic
161C22×C4C22⋊D25111yescharacteristic
162C22×C4D100411yescharacteristic
163C22×C4D100411yescharacteristic
164C22×C4C22×D25111yescharacteristic
165M16—42no—
166M16—10042no—
167C2×C8—22no—
168C2×C8—5022no—
169C4⋊C4—5021no—
170C4⋊C4—5021no—
171C22⋊C4—5021no—
172C22⋊C4—5021no—
173C22⋊C4—5021no—
174C22⋊C4—5021no—
175C42—5021no—
176C42—5021no—
177C23—21no—
178C23—21no—
179C23—41no—
180C23—41no—
181C23—10041no—
182C23—10041no—
183C23—5021no—
184C23—5021no—
185C23—2512no—
186C23(C2×D251)⋊C411yescharacteristic
187C23(C2×D251)⋊C411yescharacteristic
188C23C2×C4×D25111yescharacteristic
189D4—5024no—
190D4—5024no—
191C2×C4—22no—
192C2×C4—22no—
193C2×C4—22no—
194C2×C4—41no—
195C2×C4—41no—
196C2×C4—41no—
197C2×C4—10042no—
198C2×C4—10042no—
199C2×C4—10041no—
200C2×C4—10041no—
201C2×C4—5021no—
202C2×C4—5021no—
203C2×C4—5021no—
204C2×C4—5021no—
205C2×C4—2512no—
206C2×C4(C2×D251)⋊C411yescharacteristic
207C2×C4(C2×D251)⋊C411yescharacteristic
208C2×C4C2×C22⋊D25111yescharacteristic
209C2×C4C2×D100411yescharacteristic
210C8—42no—
211C8—10042no—
212C22—21no—
213C22—21no—
214C22—21no—
215C22—21no—
216C22—21no—
217C22—21no—
218C22—21no—
219C22—21no—
220C22—42no—
221C22—42no—
222C22—10042no—
223C22—10042no—
224C22—10041no—
225C22—10041no—
226C22—5022no—
227C22C2×(C2×D251)⋊C411yescharacteristic
228C22M16×D25111yescharacteristic
229C22M16×D25111yescharacteristic
230C4—21no—
231C4—21no—
232C4—21no—
233C4—10041no—
234C4—10041no—
235C4—5022no—
236C4#2254862912yes—
237C2—21no—
238C2—21no—
239C2—41no—
240C2—41no—
241C2—10042no—
242C2#3063165011yescharacteristic
243C2C2×#2254862911yescharacteristic
244C2#3063166911yescharacteristic
subgroupG/H
centerC22C2×(C2×D251)⋊C4
commutatorC23×C251C22×C4
FrattiniC22×C4C22×D251
Fitting#407×C251C2
socleC22×C251C2×C22⋊C4
prime radicalC22×C22⋊D251C4
cosocleC22×C4×C251C23
hypercenter#407D251
hypocenterC251#1465
LCSG ▷ C23×C251 ▷ C2×C251 ▷ C251
UCSC1 ◁ C22 ◁ C23×C4 ◁ #407

Complements

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

NH ≅ G/Ntype
2. #30631651C2semidirect
3. #30631412C2semidirect
4. #30631410C2semidirect
5. #407×C251C2semidirect
58. C22×D251M16semidirect
67. C2×Dic251M16semidirect
122. C251#1465semidirect
126. #407D251semidirect

Maximal subgroups

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

subgroupindex
C2×C4×C22⋊D2512
#306316512
#306314122
#306314102
#407×C2512
#1465251