Finite Group Databasebeta

← #44549294

Subgroups — #44549294

Order 32128.

272 conjugacy classes of subgroups; only the 53 normal subgroups are shown.

normal subgroups only

This group has 272 conjugacy classes of subgroups — too many to draw. Only its 53 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#30631837C211yesmaximal, characteristic, complemented
2#30631820C211yesmaximal, characteristic
3#30631604C211yesmaximal, characteristic, complemented
4#30631424C211yesmaximal, characteristic, complemented
5#30631421C211yesmaximal, characteristic, complemented
6#30631413C211yesmaximal, characteristic, complemented
7#539×C251C211yesmaximal, characteristic, complemented
8C22×D1004C2211yescharacteristic
9C2×#16860797—21no—
10C2×#16860795—21no—
11C2×(C2×D251)⋊C4—21no—
12C2×(C2×D251)⋊C4—21no—
13#22548635—21no—
14#22548635—21no—
15C22⋊(C251⋊C8)C2211yescharacteristic
16C22⋊(C251⋊C8)C2211yescharacteristic
17C251⋊#95C2211yescharacteristic
18#126×C251—21no—
19#125×C251—21no—
20#123×C251—21no—
21#123×C251—21no—
22C22⋊Q8×C251—21no—
23C22⋊Q8×C251C2211yescharacteristic
24#121×C251C2211yescharacteristic
25C4×D4×C251—21no—
26C2×C4⋊C4×C251C2211yescharacteristic
27C23×D251—21no—
28C2×D1004—21no—
29C2×D1004—21no—
30C2×D1004—42no—
31C22×Dic251—21no—
32#16860797—42no—
33#16860795—42no—
34C2×C251⋊C8—21no—
35C2×C251⋊C8—21no—
36(C2×D251)⋊C4—42no—
37(C2×D251)⋊C4—42no—
38C2×Q8×C251D411yescharacteristic
39C2×D4×C251—21no—
40C2×D4×C251D411yescharacteristic
41C22×C4×C251—21no—
42C22×C4×C251D411yescharacteristic
43C22×C4×C251D411yescharacteristic
44C22×C4×C251C2311yescharacteristic
45C4⋊C4×C251—21no—
46C4⋊C4×C251—21no—
47C4⋊C4×C251—21no—
48C4⋊C4×C251—21no—
49C4⋊C4×C251—21no—
50C4⋊C4×C251—21no—
51C4⋊C4×C251D411yescharacteristic
52C4⋊C4×C251D411yescharacteristic
53C22⋊C4×C251—21no—
54C22⋊C4×C251—21no—
55C22⋊C4×C251—21no—
56C22⋊C4×C251—21no—
57C22⋊C4×C251—21no—
58C42×C251—21no—
59C22×D251—22no—
60C22×D251—41no—
61C22×D251—41no—
62C22×D251—41no—
63C22×D251—41no—
64C22×D251—41no—
65C22×D251—41no—
66D1004—22no—
67D1004—22no—
68D1004—41no—
69D1004—41no—
70C2×Dic251—41no—
71C2×Dic251—41no—
72C2×Dic251—41no—
73C251⋊C8—41no—
74C251⋊C8—41no—
75C23×C251—21no—
76C23×C251C2×D411yescharacteristic
77Q8×C251—41no—
78D4×C251—41no—
79D4×C251—41no—
80D4×C251—41no—
81C2×C4×C251—21no—
82C2×C4×C251—21no—
83C2×C4×C251—21no—
84C2×C4×C251—21no—
85C2×C4×C251—21no—
86C2×C4×C251—21no—
87C2×C4×C251—21no—
88C2×C4×C251—41no—
89C2×C4×C251—41no—
90C2×C4×C251—81no—
91C2×C4×C251C2×D411yescharacteristic
92C2×C4×C251C2×D411yescharacteristic
93C22×C251—21no—
94C22×C251—21no—
95C22×C251—41no—
96C22×C251—41no—
97C22×C251—41no—
98C22×C251#12011yescharacteristic
99C22×C251#13511yescharacteristic
100C22×C251AGL(1,ℤ/8)11yescharacteristic
101C2×D251—42no—
102C2×D251—42no—
103C2×D251—42no—
104C2×D251—81no—
105C2×D251—81no—
106C2×D251—81no—
107C2×D251—81no—
108C4×C251—21no—
109C4×C251—21no—
110C4×C251—41no—
111C4×C251—41no—
112C4×C251—41no—
113C4×C251—41no—
114C4×C251—41no—
115Dic251—81no—
116C2×C251—21no—
117C2×C251—21no—
118C2×C251—81no—
119C2×C251#44911yescharacteristic
120C2×C251#45411yescharacteristic
121C2×C251B(4,𝔽2)11yescharacteristic
122D251—82no—
123C251#159711yescharacteristic, complemented, Sylow
124#1597—2511nomaximal, Sylow
125#539D25111yescharacteristic, complemented
126#450—2511no—
127#447—2511no—
128#394—2511no—
129#328—2511no—
130#327—2511no—
131#324—2511no—
132C22×D4—2511no—
133C2×SD16—5021no—
134C2×D8—5021no—
135#126—21no—
136#125—21no—
137#123—21no—
138#123—21no—
139C22⋊Q8—21no—
140C22⋊Q8C2×D25111yescharacteristic
141#121C2×D25111yescharacteristic
142C4×D4—21no—
143C2×C4⋊C4C2×D25111yescharacteristic
144C2×C22⋊C4—5021no—
145C2×C22⋊C4—5021no—
146#102—5021no—
147#102—5021no—
148C22⋊C8—2511no—
149C22⋊C8—2511no—
150#95—2511no—
151C24—5021no—
152C2×Q8C22⋊D25111yescharacteristic
153C2×D4—21no—
154C2×D4—10042no—
155C2×D4—5021no—
156C2×D4—5021no—
157C2×D4C22⋊D25111yescharacteristic
158C22×C4—21no—
159C22×C4—5021no—
160C22×C4C22⋊D25111yescharacteristic
161C22×C4C22⋊D25111yescharacteristic
162C22×C4C22×D25111yescharacteristic
163SD16—10042no—
164D8—10042no—
165C2×C8—5021no—
166C2×C8—5021no—
167C4⋊C4—21no—
168C4⋊C4—21no—
169C4⋊C4—21no—
170C4⋊C4—21no—
171C4⋊C4—21no—
172C4⋊C4—21no—
173C4⋊C4C22⋊D25111yescharacteristic
174C4⋊C4C22⋊D25111yescharacteristic
175C22⋊C4—21no—
176C22⋊C4—21no—
177C22⋊C4—21no—
178C22⋊C4—21no—
179C22⋊C4—21no—
180C22⋊C4—10042no—
181C22⋊C4—10042no—
182C42—21no—
183C23—21no—
184C23—10041no—
185C23—10041no—
186C23—10041no—
187C23—10041no—
188C23—10041no—
189C23—10041no—
190C23—5022no—
191C23C2×C22⋊D25111yescharacteristic
192Q8—41no—
193D4—41no—
194D4—41no—
195D4—41no—
196D4—10041no—
197D4—10041no—
198D4—5022no—
199D4—5022no—
200C2×C4—21no—
201C2×C4—21no—
202C2×C4—21no—
203C2×C4—21no—
204C2×C4—21no—
205C2×C4—21no—
206C2×C4—21no—
207C2×C4—41no—
208C2×C4—41no—
209C2×C4—81no—
210C2×C4—10041no—
211C2×C4—10041no—
212C2×C4—10041no—
213C2×C4C2×C22⋊D25111yescharacteristic
214C2×C4C2×C22⋊D25111yescharacteristic
215C8—10041no—
216C8—10041no—
217C22—21no—
218C22—21no—
219C22—41no—
220C22—41no—
221C22—41no—
222C22—20081no—
223C22—20081no—
224C22—20081no—
225C22—20081no—
226C22—10042no—
227C22—10042no—
228C22—10042no—
229C22#2254868411yescharacteristic
230C22#2254873511yescharacteristic
231C22#2254874011yescharacteristic
232C4—21no—
233C4—21no—
234C4—41no—
235C4—41no—
236C4—41no—
237C4—41no—
238C4—41no—
239C4—20081no—
240C2—21no—
241C2—21no—
242C2—81no—
243C2—20082no—
244C2#3063183211yescharacteristic
245C2#3063186011yescharacteristic
246C2#3063187411yescharacteristic
subgroupG/H
centerC22#22548684
commutator, cosocleC22×C4×C251C23
FrattiniC22×C4C22×D251
Fitting#539×C251C2
socleC22×C251#120
prime radical#30631820C2
hypercenter#539D251
hypocenterC251#1597
LCSG ▷ C22×C4×C251 ▷ C22×C251 ▷ C251
UCSC1 ◁ C22 ◁ C22×C4 ◁ #539

Complements

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

NH ≅ G/Ntype
1. #30631837C2semidirect
3. #30631604C2semidirect
4. #30631424C2semidirect
5. #30631421C2semidirect
6. #30631413C2semidirect
7. #539×C251C2semidirect
123. C251#1597semidirect
125. #539D251semidirect

Maximal subgroups

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

subgroupindex
#306318372
#306318202
#306316042
#306314242
#306314212
#306314132
#539×C2512
#1597251