Finite Group Databasebeta

← #1600×C251 #44546187

Subgroups — #1600×C251 #44546187

Order 32128.

320 conjugacy classes of subgroups; only the 64 normal subgroups are shown.

normal subgroups only

This group has 320 conjugacy classes of subgroups — too many to draw. Only its 64 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#538×C251C211yesmaximal, characteristic, complemented
2#450×C251C211yesmaximal, characteristic, complemented
3#447×C251C211yesmaximal, characteristic, complemented
4#392×C251C211yesmaximal, characteristic, complemented
5#328×C251C211yesmaximal, characteristic, complemented
6#327×C251C211yesmaximal, characteristic, complemented
7#323×C251C211yesmaximal, characteristic, complemented
8C22×D4×C251—21no—
9C22×D4×C251C2211yescharacteristic
10C2×SD16×C251—21no—
11C2×D8×C251—21no—
12#126×C251—21no—
13#124×C251—21no—
14#123×C251—21no—
15C22⋊Q8×C251C2211yescharacteristic
16#121×C251—21no—
17#121×C251C2211yescharacteristic
18#120×C251—21no—
19C4×D4×C251—21no—
20C2×C22⋊C4×C251—21no—
21C2×C22⋊C4×C251C2211yescharacteristic
22#102×C251—21no—
23#102×C251—21no—
24C22⋊C8×C251C2211yescharacteristic
25C22⋊C8×C251C2211yescharacteristic
26#95×C251C2211yescharacteristic
27C24×C251—21no—
28C24×C251D411yescharacteristic
29C2×Q8×C251D411yescharacteristic, complemented
30C2×D4×C251—21no—
31C2×D4×C251—21no—
32C2×D4×C251—21no—
33C2×D4×C251—21no—
34C2×D4×C251—21no—
35C2×D4×C251—42no—
36C2×D4×C251—42no—
37C2×D4×C251—42no—
38C2×D4×C251—42no—
39C2×D4×C251D411yescharacteristic, complemented
40C22×C4×C251—21no—
41C22×C4×C251—21no—
42C22×C4×C251D411yescharacteristic
43C22×C4×C251C2311yescharacteristic
44SD16×C251—42no—
45D8×C251—42no—
46C2×C8×C251—21no—
47C2×C8×C251—21no—
48C4⋊C4×C251—21no—
49C4⋊C4×C251—21no—
50C4⋊C4×C251D411yescharacteristic, complemented
51C4⋊C4×C251D411yescharacteristic, complemented
52C22⋊C4×C251—21no—
53C22⋊C4×C251—21no—
54C22⋊C4×C251—21no—
55C22⋊C4×C251—21no—
56C22⋊C4×C251—21no—
57C22⋊C4×C251—21no—
58C22⋊C4×C251—21no—
59C22⋊C4×C251—42no—
60C42×C251—21no—
61C23×C251—22no—
62C23×C251—21no—
63C23×C251—21no—
64C23×C251—21no—
65C23×C251—21no—
66C23×C251—41no—
67C23×C251—41no—
68C23×C251—41no—
69C23×C251—41no—
70C23×C251—41no—
71C23×C251—41no—
72C23×C251—41no—
73C23×C251—41no—
74C23×C251C2×D411yescharacteristic
75Q8×C251—41no—
76D4×C251—22no—
77D4×C251—22no—
78D4×C251—41no—
79D4×C251—41no—
80D4×C251—41no—
81D4×C251—48no—
82D4×C251—41no—
83D4×C251—41no—
84D4×C251—81no—
85D4×C251—81no—
86C2×C4×C251—22no—
87C2×C4×C251—21no—
88C2×C4×C251—21no—
89C2×C4×C251—21no—
90C2×C4×C251—21no—
91C2×C4×C251—41no—
92C2×C4×C251—41no—
93C2×C4×C251—41no—
94C2×C4×C251—41no—
95C2×C4×C251—81no—
96C2×C4×C251C2×D411yescharacteristic
97C2×C4×C251C2×D411yescharacteristic
98C8×C251—41no—
99C8×C251—41no—
100C22×C251—21no—
101C22×C251—21no—
102C22×C251—42no—
103C22×C251—42no—
104C22×C251—42no—
105C22×C251—41no—
106C22×C251—41no—
107C22×C251—41no—
108C22×C251—41no—
109C22×C251—42no—
110C22×C251—41no—
111C22×C251—41no—
112C22×C251—81no—
113C22×C251—81no—
114C22×C251—81no—
115C22×C251—81no—
116C22×C251—81no—
117C22×C251#12011yescharacteristic
118C22×C251#13511yescharacteristic
119C22×C251AGL(1,ℤ/8)11yescharacteristic
120C4×C251—21no—
121C4×C251—21no—
122C4×C251—42no—
123C4×C251—41no—
124C4×C251—41no—
125C4×C251—41no—
126C4×C251—41no—
127C2×C251—21no—
128C2×C251—21no—
129C2×C251—81no—
130C2×C251—81no—
131C2×C251—82no—
132C2×C251#44911yescharacteristic
133C2×C251#45311yescharacteristic
134C2×C251B(4,𝔽2)11yescharacteristic
135C251#160011yescharacteristic, direct product factor, Sylow
136#1600C25111yesmaximal, characteristic, direct product factor, Sylow
137#538C2×C25111yescharacteristic, complemented
138#450C2×C25111yescharacteristic, complemented
139#447C2×C25111yescharacteristic, complemented
140#392C2×C25111yescharacteristic, complemented
141#328C2×C25111yescharacteristic, complemented
142#327C2×C25111yescharacteristic, complemented
143#323C2×C25111yescharacteristic, complemented
144C22×D4—21no—
145C22×D4C22×C25111yescharacteristic
146C2×SD16—21no—
147C2×D8—21no—
148#126—21no—
149#124—21no—
150#123—21no—
151C22⋊Q8C22×C25111yescharacteristic
152#121—21no—
153#121C22×C25111yescharacteristic
154#120—21no—
155C4×D4—21no—
156C2×C22⋊C4—21no—
157C2×C22⋊C4C22×C25111yescharacteristic
158#102—21no—
159#102—21no—
160C22⋊C8C22×C25111yescharacteristic
161C22⋊C8C22×C25111yescharacteristic
162#95C22×C25111yescharacteristic
163C24—21no—
164C24D4×C25111yescharacteristic
165C2×Q8D4×C25111yescharacteristic, complemented
166C2×D4—21no—
167C2×D4—21no—
168C2×D4—21no—
169C2×D4—21no—
170C2×D4—21no—
171C2×D4—42no—
172C2×D4—42no—
173C2×D4—42no—
174C2×D4—42no—
175C2×D4D4×C25111yescharacteristic, complemented
176C22×C4—21no—
177C22×C4—21no—
178C22×C4D4×C25111yescharacteristic
179C22×C4C23×C25111yescharacteristic
180SD16—42no—
181D8—42no—
182C2×C8—21no—
183C2×C8—21no—
184C4⋊C4—21no—
185C4⋊C4—21no—
186C4⋊C4D4×C25111yescharacteristic, complemented
187C4⋊C4D4×C25111yescharacteristic, complemented
188C22⋊C4—21no—
189C22⋊C4—21no—
190C22⋊C4—21no—
191C22⋊C4—21no—
192C22⋊C4—21no—
193C22⋊C4—21no—
194C22⋊C4—21no—
195C22⋊C4—42no—
196C42—21no—
197C23—22no—
198C23—21no—
199C23—21no—
200C23—21no—
201C23—21no—
202C23—41no—
203C23—41no—
204C23—41no—
205C23—41no—
206C23—41no—
207C23—41no—
208C23—41no—
209C23—41no—
210C23C2×D4×C25111yescharacteristic
211Q8—41no—
212D4—22no—
213D4—22no—
214D4—41no—
215D4—41no—
216D4—41no—
217D4—48no—
218D4—41no—
219D4—41no—
220D4—81no—
221D4—81no—
222C2×C4—22no—
223C2×C4—21no—
224C2×C4—21no—
225C2×C4—21no—
226C2×C4—21no—
227C2×C4—41no—
228C2×C4—41no—
229C2×C4—41no—
230C2×C4—41no—
231C2×C4—81no—
232C2×C4C2×D4×C25111yescharacteristic
233C2×C4C2×D4×C25111yescharacteristic
234C8—41no—
235C8—41no—
236C22—21no—
237C22—21no—
238C22—42no—
239C22—42no—
240C22—42no—
241C22—41no—
242C22—41no—
243C22—41no—
244C22—41no—
245C22—42no—
246C22—41no—
247C22—41no—
248C22—81no—
249C22—81no—
250C22—81no—
251C22—81no—
252C22—81no—
253C22#120×C25111yescharacteristic
254C22#135×C25111yescharacteristic
255C22AGL(1,ℤ/8)×C25111yescharacteristic
256C4—21no—
257C4—21no—
258C4—42no—
259C4—41no—
260C4—41no—
261C4—41no—
262C4—41no—
263C2—21no—
264C2—21no—
265C2—81no—
266C2—82no—
267C2—81no—
268C2#449×C25111yescharacteristic
269C2#453×C25111yescharacteristic
270C2B(4,𝔽2)×C25111yescharacteristic
subgroupG/H
center, socleC22×C251#120
commutator, Frattini, cosocleC22×C4C23×C251
LCSG ▷ C22×C4 ▷ C22 ▷ C1
UCSC1 ◁ C22×C251 ◁ C22×C4×C251 ◁ G

Complements

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

NH ≅ G/Ntype
1. #538×C251C2semidirect
2. #450×C251C2semidirect
3. #447×C251C2semidirect
4. #392×C251C2semidirect
5. #328×C251C2semidirect
6. #327×C251C2semidirect
7. #323×C251C2semidirect
29. C2×Q8×C251D4semidirect
39. C2×D4×C251D4semidirect
50. C4⋊C4×C251D4semidirect
51. C4⋊C4×C251D4semidirect
135. C251#1600direct (N × H)
136. #1600C251direct (N × H)
137. #538C2×C251semidirect
138. #450C2×C251semidirect
139. #447C2×C251semidirect
140. #392C2×C251semidirect
141. #328C2×C251semidirect
142. #327C2×C251semidirect
143. #323C2×C251semidirect
165. C2×Q8D4×C251semidirect
175. C2×D4D4×C251semidirect
186. C4⋊C4D4×C251semidirect
187. C4⋊C4D4×C251semidirect

Maximal subgroups

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

subgroupindex
#538×C2512
#450×C2512
#447×C2512
#392×C2512
#328×C2512
#327×C2512
#323×C2512
#1600251