Finite Group Databasebeta

← #44552011

Subgroups — #44552011

Order 32128.

280 conjugacy classes of subgroups; only the 73 normal subgroups are shown.

normal subgroups only

This group has 280 conjugacy classes of subgroups — too many to draw. Only its 73 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×#22548632C211yesmaximal, characteristic, complemented
2C2×#22548630C211yesmaximal, characteristic, complemented
3#30631477C212yesmaximal, complemented
4#30631474C212yesmaximal, complemented
5#534×C251C211yesmaximal, characteristic
6C2×#16860794—21no—
7C2×C22⋊Dic251—21no—
8#22548632—21no—
9#22548632C2212yes—
10#22548630—21no—
11#22548630C2212yes—
12C22×D4×C251C411yescharacteristic, complemented
13C22×D4×C251C2211yescharacteristic
14#123×C251—21no—
15#123×C251C411yescharacteristic, complemented
16#123×C251C2211yescharacteristic
17#121×C251—21no—
18#121×C251—21no—
19#120×C251—21no—
20#120×C251C411yescharacteristic, complemented
21#120×C251C2211yescharacteristic
22C2×C22⋊C4×C251C411yescharacteristic, complemented
23C22×Dic251—41no—
24#16860794—22no—
25#16860794—41no—
26C2×C251⋊C8—41no—
27C22⋊Dic251—22no—
28C22⋊Dic251—41no—
29C24×C251C2×C411yescharacteristic
30C24×C251C2×C411yescharacteristic, complemented
31C24×C251D411yescharacteristic
32C2×D4×C251—21no—
33C2×D4×C251—21no—
34C2×D4×C251—21no—
35C2×D4×C251—21no—
36C2×D4×C251—41no—
37C2×D4×C251—41no—
38C2×D4×C251—41no—
39C2×D4×C251—41no—
40C2×D4×C251C2×C411yescharacteristic
41C2×D4×C251D411yescharacteristic
42C2×D4×C251D411yescharacteristic
43C2×D4×C251C2311yescharacteristic
44C22×C4×C251—21no—
45C22×C4×C251C2×C411yescharacteristic, complemented
46C22×C4×C251D411yescharacteristic
47C4⋊C4×C251—21no—
48C4⋊C4×C251—21no—
49C22⋊C4×C251—21no—
50C22⋊C4×C251—21no—
51C22⋊C4×C251—21no—
52C22⋊C4×C251—21no—
53C22⋊C4×C251—21no—
54C22⋊C4×C251C2×C411yescharacteristic, complemented
55C22⋊C4×C251C2×C411yescharacteristic, complemented
56C2×Dic251—42no—
57C2×Dic251—82no—
58C251⋊C8—42no—
59C23×C251—21no—
60C23×C251—21no—
61C23×C251—21no—
62C23×C251—21no—
63C23×C251—21no—
64C23×C251—41no—
65C23×C251—81no—
66C23×C251—81no—
67C23×C251—81no—
68C23×C251C22⋊C411yescharacteristic
69C23×C251C22⋊C411yescharacteristic
70C23×C251C22⋊C411yescharacteristic
71C23×C251C22×C411yescharacteristic
72C23×C251C2×D411yescharacteristic
73D4×C251—41no—
74D4×C251—41no—
75D4×C251—41no—
76D4×C251—41no—
77D4×C251—42no—
78D4×C251—41no—
79D4×C251—41no—
80D4×C251—81no—
81D4×C251—81no—
82C2×C4×C251—21no—
83C2×C4×C251—21no—
84C2×C4×C251—21no—
85C2×C4×C251—21no—
86C2×C4×C251—21no—
87C2×C4×C251—41no—
88C2×C4×C251—81no—
89C2×C4×C251C22⋊C411yescharacteristic
90C2×C4×C251C2×D411yescharacteristic
91C22×C251—21no—
92C22×C251—21no—
93C22×C251—21no—
94C22×C251—21no—
95C22×C251—41no—
96C22×C251—41no—
97C22×C251—41no—
98C22×C251—41no—
99C22×C251—41no—
100C22×C251—42no—
101C22×C251—41no—
102C22×C251—41no—
103C22×C251—41no—
104C22×C251—81no—
105C22×C251—81no—
106C22×C251—81no—
107C22×C251—82no—
108C22×C251#9911yescharacteristic
109C22×C251#9911yescharacteristic
110C22×C251C2×C22⋊C411yescharacteristic
111C4×C251—21no—
112C4×C251—21no—
113C4×C251—41no—
114C4×C251—41no—
115C4×C251—41no—
116Dic251—82no—
117C2×C251—21no—
118C2×C251—21no—
119C2×C251—21no—
120C2×C251—41no—
121C2×C251—41no—
122C2×C251—41no—
123C2×C251—41no—
124C2×C251—81no—
125C2×C251C2×#9911yescharacteristic
126C251#212111yescharacteristic, complemented, Sylow
127#2121—2511nomaximal, Sylow
128#534D25111yescharacteristic
129C2×#100—2511no—
130C2×#99—2511no—
131#352—2512no—
132AΣL(1,𝔽16)—2512no—
133C22×D4Dic25111yescharacteristic, complemented
134C22×D4C2×D25111yescharacteristic
135C2×M16—5021no—
136#123—21no—
137#123Dic25111yescharacteristic, complemented
138#123C2×D25111yescharacteristic
139#121—21no—
140#121—21no—
141#120—21no—
142#120Dic25111yescharacteristic, complemented
143#120C2×D25111yescharacteristic
144C2×C22⋊C4—5021no—
145C2×C22⋊C4Dic25111yescharacteristic, complemented
146#100—5021no—
147#100—2512no—
148#99—5021no—
149#99—2512no—
150C24C2×Dic25111yescharacteristic
151C24C2×Dic25111yescharacteristic, complemented
152C24C22⋊D25111yescharacteristic
153C2×D4—21no—
154C2×D4—21no—
155C2×D4—21no—
156C2×D4—21no—
157C2×D4—41no—
158C2×D4—41no—
159C2×D4—41no—
160C2×D4—41no—
161C2×D4C2×Dic25111yescharacteristic
162C2×D4C22⋊D25111yescharacteristic
163C2×D4C22⋊D25111yescharacteristic
164C2×D4C22×D25111yescharacteristic
165C22×C4—21no—
166C22×C4—10041no—
167C22×C4C2×Dic25111yescharacteristic, complemented
168C22×C4C22⋊D25111yescharacteristic
169M16—10041no—
170M16—5022no—
171C2×C8—10041no—
172C4⋊C4—21no—
173C4⋊C4—21no—
174C22⋊C4—21no—
175C22⋊C4—21no—
176C22⋊C4—21no—
177C22⋊C4—21no—
178C22⋊C4—21no—
179C22⋊C4—10041no—
180C22⋊C4—5022no—
181C22⋊C4C2×Dic25111yescharacteristic, complemented
182C22⋊C4C2×Dic25111yescharacteristic, complemented
183C23—21no—
184C23—21no—
185C23—21no—
186C23—21no—
187C23—21no—
188C23—41no—
189C23—81no—
190C23—81no—
191C23—81no—
192C23C22⋊Dic25111yescharacteristic
193C23C22⋊Dic25111yescharacteristic
194C23C22⋊Dic25111yescharacteristic
195C23C22×Dic25111yescharacteristic
196C23C2×C22⋊D25111yescharacteristic
197D4—41no—
198D4—41no—
199D4—41no—
200D4—41no—
201D4—42no—
202D4—41no—
203D4—41no—
204D4—81no—
205D4—81no—
206C2×C4—21no—
207C2×C4—21no—
208C2×C4—21no—
209C2×C4—21no—
210C2×C4—21no—
211C2×C4—41no—
212C2×C4—81no—
213C2×C4—20082no—
214C2×C4—10042no—
215C2×C4C22⋊Dic25111yescharacteristic
216C2×C4C2×C22⋊D25111yescharacteristic
217C8—10042no—
218C22—21no—
219C22—21no—
220C22—21no—
221C22—21no—
222C22—41no—
223C22—41no—
224C22—41no—
225C22—41no—
226C22—41no—
227C22—42no—
228C22—41no—
229C22—41no—
230C22—41no—
231C22—81no—
232C22—81no—
233C22—81no—
234C22—82no—
235C22#2254863011yescharacteristic
236C22#2254863011yescharacteristic
237C22C2×C22⋊Dic25111yescharacteristic
238C4—21no—
239C4—21no—
240C4—41no—
241C4—41no—
242C4—41no—
243C4—20082no—
244C2—21no—
245C2—21no—
246C2—21no—
247C2—41no—
248C2—41no—
249C2—41no—
250C2—41no—
251C2—81no—
252C2C2×#2254863011yescharacteristic
subgroupG/H
centerC2C2×#22548630
commutatorC23×C251C22×C4
FrattiniC2×D4C22×D251
Fitting, prime radical#534×C251C2
socleC2×C251C2×#99
cosocleC2×D4×C251C23
hypercenter#534D251
hypocenterC251#2121
LCSG ▷ C23×C251 ▷ C22×C251 ▷ C2×C251 ▷ C251
UCSC1 ◁ C2 ◁ C23 ◁ C22×D4 ◁ #534

Complements

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

NH ≅ G/Ntype
1. C2×#22548632C2semidirect
2. C2×#22548630C2semidirect
3. #30631477C2semidirect
4. #30631474C2semidirect
12. C22×D4×C251C4semidirect
15. #123×C251C4semidirect
20. #120×C251C4semidirect
22. C2×C22⋊C4×C251C4semidirect
30. C24×C251C2×C4semidirect
45. C22×C4×C251C2×C4semidirect
54. C22⋊C4×C251C2×C4semidirect
55. C22⋊C4×C251C2×C4semidirect
126. C251#2121semidirect
133. C22×D4Dic251semidirect
137. #123Dic251semidirect
142. #120Dic251semidirect
145. C2×C22⋊C4Dic251semidirect
151. C24C2×Dic251semidirect
167. C22×C4C2×Dic251semidirect
181. C22⋊C4C2×Dic251semidirect
182. C22⋊C4C2×Dic251semidirect

Maximal subgroups

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

subgroupindex
C2×#225486322
C2×#225486302
#306314772
#306314742
#534×C2512
#2121251