Finite Group Databasebeta

← #1606×C251 #44546193

Subgroups — #1606×C251 #44546193

Order 32128.

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

normal subgroups only

This group has 260 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
2#451×C251C211yesmaximal, characteristic, complemented
3#448×C251C211yesmaximal, characteristic, complemented
4#393×C251C211yesmaximal, characteristic, complemented
5#328×C251C211yesmaximal, characteristic, complemented
6#327×C251C211yesmaximal, characteristic, complemented
7#323×C251C211yesmaximal, characteristic, complemented
8C22×Q8×C251C2211yescharacteristic
9C2×Q16×C251—21no—
10C2×SD16×C251—21no—
11#126×C251—21no—
12#124×C251—21no—
13#123×C251—21no—
14C22⋊Q8×C251C2211yescharacteristic
15#121×C251—21no—
16#121×C251C2211yescharacteristic
17#120×C251—21no—
18C4×D4×C251—21no—
19C2×C22⋊C4×C251—21no—
20C2×C22⋊C4×C251C2211yescharacteristic
21#103×C251—21no—
22#103×C251—21no—
23C22⋊C8×C251C2211yescharacteristic
24C22⋊C8×C251C2211yescharacteristic
25#95×C251—21no—
26#95×C251C2211yescharacteristic
27C24×C251D411yescharacteristic
28C2×Q8×C251—21no—
29C2×Q8×C251—21no—
30C2×Q8×C251—42no—
31C2×Q8×C251D411yescharacteristic
32C2×D4×C251—21no—
33C2×D4×C251—21no—
34C2×D4×C251—21no—
35C2×D4×C251D411yescharacteristic
36C22×C4×C251—21no—
37C22×C4×C251—21no—
38C22×C4×C251—21no—
39C22×C4×C251D411yescharacteristic
40C22×C4×C251C2311yescharacteristic
41Q16×C251—42no—
42SD16×C251—42no—
43C2×C8×C251—21no—
44C2×C8×C251—21no—
45C4⋊C4×C251—21no—
46C4⋊C4×C251—21no—
47C4⋊C4×C251D411yescharacteristic
48C4⋊C4×C251D411yescharacteristic
49C22⋊C4×C251—21no—
50C22⋊C4×C251—21no—
51C22⋊C4×C251—21no—
52C22⋊C4×C251—21no—
53C22⋊C4×C251—21no—
54C22⋊C4×C251—21no—
55C22⋊C4×C251—21no—
56C22⋊C4×C251—42no—
57C42×C251—21no—
58C23×C251—21no—
59C23×C251—21no—
60C23×C251—21no—
61C23×C251—21no—
62C23×C251—41no—
63C23×C251—41no—
64C23×C251C2×D411yescharacteristic
65Q8×C251—22no—
66Q8×C251—22no—
67Q8×C251—41no—
68Q8×C251—41no—
69Q8×C251—41no—
70D4×C251—41no—
71D4×C251—41no—
72D4×C251—41no—
73D4×C251—81no—
74D4×C251—81no—
75C2×C4×C251—22no—
76C2×C4×C251—21no—
77C2×C4×C251—21no—
78C2×C4×C251—21no—
79C2×C4×C251—21no—
80C2×C4×C251—41no—
81C2×C4×C251—41no—
82C2×C4×C251—41no—
83C2×C4×C251—41no—
84C2×C4×C251—41no—
85C2×C4×C251—41no—
86C2×C4×C251—41no—
87C2×C4×C251—81no—
88C2×C4×C251C2×D411yescharacteristic
89C2×C4×C251C2×D411yescharacteristic
90C8×C251—41no—
91C8×C251—41no—
92C22×C251—21no—
93C22×C251—21no—
94C22×C251—42no—
95C22×C251—41no—
96C22×C251—41no—
97C22×C251—41no—
98C22×C251—41no—
99C22×C251—41no—
100C22×C251—41no—
101C22×C251—81no—
102C22×C251#12011yescharacteristic
103C22×C251#13511yescharacteristic
104C22×C251#13711yescharacteristic
105C4×C251—21no—
106C4×C251—21no—
107C4×C251—41no—
108C4×C251—41no—
109C4×C251—41no—
110C4×C251—42no—
111C4×C251—41no—
112C4×C251—81no—
113C2×C251—21no—
114C2×C251—21no—
115C2×C251—81no—
116C2×C251—81no—
117C2×C251#45211yescharacteristic
118C2×C251#45511yescharacteristic
119C2×C251B(4,𝔽2)11yescharacteristic
120C251#160611yescharacteristic, direct product factor, Sylow
121#1606C25111yesmaximal, characteristic, direct product factor, Sylow
122#538C2×C25111yescharacteristic
123#451C2×C25111yescharacteristic, complemented
124#448C2×C25111yescharacteristic, complemented
125#393C2×C25111yescharacteristic, complemented
126#328C2×C25111yescharacteristic, complemented
127#327C2×C25111yescharacteristic, complemented
128#323C2×C25111yescharacteristic, complemented
129C22×Q8C22×C25111yescharacteristic
130C2×Q16—21no—
131C2×SD16—21no—
132#126—21no—
133#124—21no—
134#123—21no—
135C22⋊Q8C22×C25111yescharacteristic
136#121—21no—
137#121C22×C25111yescharacteristic
138#120—21no—
139C4×D4—21no—
140C2×C22⋊C4—21no—
141C2×C22⋊C4C22×C25111yescharacteristic
142#103—21no—
143#103—21no—
144C22⋊C8C22×C25111yescharacteristic
145C22⋊C8C22×C25111yescharacteristic
146#95—21no—
147#95C22×C25111yescharacteristic
148C24D4×C25111yescharacteristic
149C2×Q8—21no—
150C2×Q8—21no—
151C2×Q8—42no—
152C2×Q8D4×C25111yescharacteristic
153C2×D4—21no—
154C2×D4—21no—
155C2×D4—21no—
156C2×D4D4×C25111yescharacteristic
157C22×C4—21no—
158C22×C4—21no—
159C22×C4—21no—
160C22×C4D4×C25111yescharacteristic
161C22×C4C23×C25111yescharacteristic
162Q16—42no—
163SD16—42no—
164C2×C8—21no—
165C2×C8—21no—
166C4⋊C4—21no—
167C4⋊C4—21no—
168C4⋊C4D4×C25111yescharacteristic
169C4⋊C4D4×C25111yescharacteristic
170C22⋊C4—21no—
171C22⋊C4—21no—
172C22⋊C4—21no—
173C22⋊C4—21no—
174C22⋊C4—21no—
175C22⋊C4—21no—
176C22⋊C4—21no—
177C22⋊C4—42no—
178C42—21no—
179C23—21no—
180C23—21no—
181C23—21no—
182C23—21no—
183C23—41no—
184C23—41no—
185C23C2×D4×C25111yescharacteristic
186Q8—22no—
187Q8—22no—
188Q8—41no—
189Q8—41no—
190Q8—41no—
191D4—41no—
192D4—41no—
193D4—41no—
194D4—81no—
195D4—81no—
196C2×C4—22no—
197C2×C4—21no—
198C2×C4—21no—
199C2×C4—21no—
200C2×C4—21no—
201C2×C4—41no—
202C2×C4—41no—
203C2×C4—41no—
204C2×C4—41no—
205C2×C4—41no—
206C2×C4—41no—
207C2×C4—41no—
208C2×C4—81no—
209C2×C4C2×D4×C25111yescharacteristic
210C2×C4C2×D4×C25111yescharacteristic
211C8—41no—
212C8—41no—
213C22—21no—
214C22—21no—
215C22—42no—
216C22—41no—
217C22—41no—
218C22—41no—
219C22—41no—
220C22—41no—
221C22—41no—
222C22—81no—
223C22#120×C25111yescharacteristic
224C22#135×C25111yescharacteristic
225C22#137×C25111yescharacteristic
226C4—21no—
227C4—21no—
228C4—41no—
229C4—41no—
230C4—41no—
231C4—42no—
232C4—41no—
233C4—81no—
234C2—21no—
235C2—21no—
236C2—81no—
237C2—81no—
238C2#452×C25111yescharacteristic
239C2#455×C25111yescharacteristic
240C2B(4,𝔽2)×C25111yescharacteristic
subgroupG/H
center, socleC22×C251#120
commutator, Frattini, cosocleC22×C4C23×C251
prime radical#538×C251C2
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
2. #451×C251C2semidirect
3. #448×C251C2semidirect
4. #393×C251C2semidirect
5. #328×C251C2semidirect
6. #327×C251C2semidirect
7. #323×C251C2semidirect
120. C251#1606direct (N × H)
121. #1606C251direct (N × H)
123. #451C2×C251semidirect
124. #448C2×C251semidirect
125. #393C2×C251semidirect
126. #328C2×C251semidirect
127. #327C2×C251semidirect
128. #323C2×C251semidirect

Maximal subgroups

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

subgroupindex
#538×C2512
#451×C2512
#448×C2512
#393×C2512
#328×C2512
#327×C2512
#323×C2512
#1606251