Finite Group Databasebeta

← #44549286

Subgroups — #44549286

Order 32128.

316 conjugacy classes of subgroups; only the 51 normal subgroups are shown.

normal subgroups only

This group has 316 conjugacy classes of subgroups — too many to draw. Only its 51 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#30632312C211yesmaximal, characteristic, complemented
2#30631838C211yesmaximal, characteristic, complemented
3#30631595C211yesmaximal, characteristic
4#30631425C211yesmaximal, characteristic, complemented
5#30631424C211yesmaximal, characteristic, complemented
6#30631410C211yesmaximal, characteristic, complemented
7#450×C251C211yesmaximal, characteristic, complemented
8C22×C22⋊D251—21no—
9C2×#16860798—21no—
10#22548695—21no—
11#22548694C2211yescharacteristic
12#22548693—21no—
13#22548693—21no—
14#22548692—21no—
15#22548691—21no—
16C2×C22⋊Dic251—21no—
17C2×(C2×D251)⋊C4C2211yescharacteristic
18C22⋊C4×D251—21no—
19#22548636—21no—
20C22⋊(C251⋊C8)C2211yescharacteristic
21C251⋊#95C2211yescharacteristic
22C22×D4×C251C2211yescharacteristic
23C2×SD16×C251—21no—
24C22⋊Q8×C251C2211yescharacteristic
25#102×C251—21no—
26C22⋊C8×C251C2211yescharacteristic
27C23×D251D411yescharacteristic
28C2×Dic502—21no—
29C2×Dic502D411yescharacteristic, complemented
30C2×C22⋊D251—42no—
31C2×C22⋊D251—42no—
32C2×C22⋊D251—42no—
33C22×Dic251—21no—
34C22×Dic251D411yescharacteristic
35C2×C4×D251—21no—
36C2×C4×D251—21no—
37#16860798—42no—
38C2×C251⋊C8—21no—
39C4⋊Dic251—21no—
40C4⋊Dic251D411yescharacteristic, complemented
41Dic251⋊C4—21no—
42Dic251⋊C4—21no—
43Dic251⋊C4—21no—
44C22⋊Dic251—21no—
45C22⋊Dic251—42no—
46(C2×D251)⋊C4—21no—
47(C2×D251)⋊C4—21no—
48(C2×D251)⋊C4—21no—
49(C2×D251)⋊C4—21no—
50C24×C251—21no—
51C2×Q8×C251D411yescharacteristic, complemented
52C2×D4×C251—21no—
53C2×D4×C251—21no—
54C2×D4×C251—42no—
55C22×C4×C251C2311yescharacteristic
56SD16×C251—42no—
57C2×C8×C251—21no—
58C4⋊C4×C251—21no—
59C4⋊C4×C251D411yescharacteristic, complemented
60C22⋊C4×C251—21no—
61C22×D251—21no—
62C22×D251—21no—
63C22×D251—41no—
64C22×D251—41no—
65C22×D251SD1612yescomplemented
66Dic502—41no—
67Dic502—81no—
68C22⋊D251—48no—
69C2×Dic251—22no—
70C2×Dic251—21no—
71C2×Dic251—21no—
72C2×Dic251—21no—
73C2×Dic251—41no—
74C2×Dic251—41no—
75C2×Dic251—41no—
76C4×D251—81no—
77C4×D251—81no—
78C251⋊C8—41no—
79C23×C251—22no—
80C23×C251—41no—
81C23×C251—41no—
82C23×C251—41no—
83C23×C251—41no—
84C23×C251—41no—
85C23×C251—41no—
86C23×C251C2×D411yescharacteristic
87Q8×C251—41no—
88D4×C251—22no—
89D4×C251—22no—
90D4×C251—41no—
91D4×C251—41no—
92C2×C4×C251—21no—
93C2×C4×C251—21no—
94C2×C4×C251—41no—
95C2×C4×C251C2×D411yescharacteristic
96C2×C4×C251C2×D411yescharacteristic
97C8×C251—41no—
98C22×C251—21no—
99C22×C251—21no—
100C22×C251—42no—
101C22×C251—42no—
102C22×C251—42no—
103C22×C251—81no—
104C22×C251—81no—
105C22×C251—81no—
106C22×C251—81no—
107C22×C251#12011yescharacteristic
108C22×C251C2×SD1611yescharacteristic
109C22×C251#13711yescharacteristic
110C2×D251—22no—
111C2×D251—41no—
112C2×D251—41no—
113C2×D251—42no—
114C2×D251—81no—
115C4×C251—21no—
116C4×C251—21no—
117C4×C251—41no—
118C4×C251—41no—
119Dic251—42no—
120Dic251—41no—
121Dic251—41no—
122Dic251—41no—
123C2×C251—21no—
124C2×C251—21no—
125C2×C251—82no—
126C2×C251#44811yescharacteristic
127C2×C251#45311yescharacteristic
128C2×C251B(4,𝔽2)11yescharacteristic
129D251—42no—
130C251#159611yescharacteristic, complemented, Sylow
131#1596—2511nomaximal, Sylow
132#543—2511no—
133#450—2511no—
134#450D25111yescharacteristic, complemented
135#392—2511no—
136#328—2511no—
137#328—2511no—
138#323—2511no—
139C22×D4—5021no—
140C22×D4C2×D25111yescharacteristic
141C2×SD16—21no—
142C2×SD16—5021no—
143C22⋊Q8—5021no—
144C22⋊Q8—5021no—
145C22⋊Q8—5021no—
146C22⋊Q8—5021no—
147C22⋊Q8—5021no—
148C22⋊Q8—2511no—
149C22⋊Q8C2×D25111yescharacteristic
150C2×C22⋊C4—5021no—
151C2×C22⋊C4—5021no—
152C2×C22⋊C4—2511no—
153#102—21no—
154#102—5021no—
155C22⋊C8—2511no—
156C22⋊C8C2×D25111yescharacteristic
157#95—2511no—
158C24—21no—
159C24—2511no—
160C2×Q8—5021no—
161C2×Q8—2511no—
162C2×Q8C22⋊D25111yescharacteristic, complemented
163C2×D4—21no—
164C2×D4—21no—
165C2×D4—42no—
166C2×D4—10042no—
167C2×D4—10042no—
168C2×D4—10042no—
169C22×C4—5021no—
170C22×C4—5021no—
171C22×C4—5021no—
172C22×C4—2511no—
173C22×C4C22×D25111yescharacteristic
174SD16—42no—
175SD16—10042no—
176C2×C8—21no—
177C2×C8—5021no—
178C4⋊C4—21no—
179C4⋊C4—5021no—
180C4⋊C4—5021no—
181C4⋊C4—5021no—
182C4⋊C4—5021no—
183C4⋊C4—2511no—
184C4⋊C4C22⋊D25111yescharacteristic, complemented
185C22⋊C4—21no—
186C22⋊C4—10042no—
187C22⋊C4—5021no—
188C22⋊C4—5021no—
189C22⋊C4—5021no—
190C22⋊C4—5021no—
191C22⋊C4—5021no—
192C23—22no—
193C23—41no—
194C23—41no—
195C23—41no—
196C23—41no—
197C23—41no—
198C23—41no—
199C23—10041no—
200C23—10041no—
201C23—5021no—
202C23—5021no—
203C23—2512no—
204C23D4×D25111yescharacteristic
205Q8—41no—
206Q8—20081no—
207Q8—10041no—
208D4—22no—
209D4—22no—
210D4—41no—
211D4—41no—
212D4—10048no—
213C2×C4—21no—
214C2×C4—21no—
215C2×C4—41no—
216C2×C4—20081no—
217C2×C4—20081no—
218C2×C4—10041no—
219C2×C4—10041no—
220C2×C4—10041no—
221C2×C4—5022no—
222C2×C4—5021no—
223C2×C4—5021no—
224C2×C4—5021no—
225C2×C4D4×D25111yescharacteristic
226C2×C4C2×C22⋊D25111yescharacteristic
227C8—41no—
228C8—10041no—
229C22—21no—
230C22—21no—
231C22—42no—
232C22—42no—
233C22—42no—
234C22—81no—
235C22—81no—
236C22—81no—
237C22—81no—
238C22—20081no—
239C22—10042no—
240C22—10041no—
241C22—10041no—
242C22—5022no—
243C22#2254868311yescharacteristic
244C22SD16×D25111yescharacteristic
245C22#2254874411yescharacteristic
246C4—21no—
247C4—21no—
248C4—41no—
249C4—41no—
250C4—10042no—
251C4—10041no—
252C4—10041no—
253C4—10041no—
254C2—21no—
255C2—21no—
256C2—82no—
257C2—10042no—
258C2#3063182411yescharacteristic
259C2#3063185311yescharacteristic
260C2#3063187311yescharacteristic
subgroupG/H
centerC22#22548683
commutator, cosocleC22×C4×C251C23
FrattiniC22×C4C22×D251
Fitting#450×C251C2
socleC22×C251#120
prime radical#30631595C2
hypercenter#450D251
hypocenterC251#1596
LCSG ▷ C22×C4×C251 ▷ C22×C251 ▷ C251
UCSC1 ◁ C22 ◁ C22×C4 ◁ #450

Complements

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

NH ≅ G/Ntype
1. #30632312C2semidirect
2. #30631838C2semidirect
4. #30631425C2semidirect
5. #30631424C2semidirect
6. #30631410C2semidirect
7. #450×C251C2semidirect
29. C2×Dic502D4semidirect
40. C4⋊Dic251D4semidirect
51. C2×Q8×C251D4semidirect
59. C4⋊C4×C251D4semidirect
65. C22×D251SD16semidirect
130. C251#1596semidirect
134. #450D251semidirect
162. C2×Q8C22⋊D251semidirect
184. C4⋊C4C22⋊D251semidirect

Maximal subgroups

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

subgroupindex
#306323122
#306318382
#306315952
#306314252
#306314242
#306314102
#450×C2512
#1596251