Finite Group Databasebeta

← #44552005

Subgroups — #44552005

Order 32128.

264 conjugacy classes of subgroups; only the 55 normal subgroups are shown.

normal subgroups only

This group has 264 conjugacy classes of subgroups — too many to draw. Only its 55 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#30632195C211yesmaximal, characteristic
2#30631668C211yesmaximal, characteristic, complemented
3#30631659C211yesmaximal, characteristic, complemented
4#30631477C211yesmaximal, characteristic, complemented
5#30631475C211yesmaximal, characteristic, complemented
6#30631473C211yesmaximal, characteristic, complemented
7#352×C251C211yesmaximal, characteristic, complemented
8C2×#16860809C2211yescharacteristic
9C22×C4×D251C411yescharacteristic, complemented
10M16×D251—21no—
11#22548698C411yescharacteristic, complemented
12#22548693—21no—
13#22548688—21no—
14#22548685C411yescharacteristic, complemented
15#22548683C2211yescharacteristic
16#22548677—21no—
17(C22⋊D251)⋊C4—21no—
18#22548671—21no—
19#22548671C411yescharacteristic, complemented
20#22548633—21no—
21#22548632C2211yescharacteristic
22#22548630C2211yescharacteristic
23#22548629—21no—
24#123×C251C2211yescharacteristic
25#100×C251C2211yescharacteristic
26#99×C251C2211yescharacteristic
27C23×D251C2×C411yescharacteristic
28#16860809—41no—
29#16860809—41no—
30C2×Dic502D411yescharacteristic
31C2×D1004—21no—
32C2×C22⋊D251—21no—
33C2×C22⋊D251C2×C411yescharacteristic
34C2×C22⋊D251D411yescharacteristic
35C22×Dic251C2×C411yescharacteristic
36C22×Dic251D411yescharacteristic
37C2×C4×D251—21no—
38C2×C4×D251—21no—
39C2×C4×D251—41no—
40C2×C4×D251—41no—
41C2×C4×D251D411yescharacteristic
42#16860794—21no—
43#16860793—41no—
44C8×D251—41no—
45Dic251⋊C4—21no—
46Dic251⋊C4—21no—
47Dic251⋊C4—41no—
48C22⋊Dic251—21no—
49C22⋊Dic251C2×C411yescharacteristic
50(C2×D251)⋊C4—21no—
51(C2×D251)⋊C4—21no—
52(C2×D251)⋊C4—21no—
53C4×Dic251—21no—
54C4×Dic251—41no—
55C2×D4×C251C2311yescharacteristic
56C22×C4×C251C2×C411yescharacteristic
57M16×C251—21no—
58C4⋊C4×C251—21no—
59C22⋊C4×C251—21no—
60C22⋊C4×C251—21no—
61C22⋊C4×C251C2×C411yescharacteristic
62C22×D251—21no—
63C22×D251—42no—
64C22×D251—41no—
65C22×D251C22⋊C411yescharacteristic
66Dic502—41no—
67D1004—42no—
68C22⋊D251—41no—
69C22⋊D251—41no—
70C22⋊D251—42no—
71C2×Dic251—21no—
72C2×Dic251—21no—
73C2×Dic251—21no—
74C2×Dic251—21no—
75C2×Dic251—21no—
76C2×Dic251—41no—
77C2×Dic251C22⋊C411yescharacteristic
78C2×Dic251C22⋊C411yescharacteristic
79C2×Dic251C22⋊C411yescharacteristic
80C4×D251—21no—
81C4×D251—21no—
82C4×D251—42no—
83C4×D251—41no—
84C4×D251—41no—
85C251⋊C8—41no—
86C23×C251C22×C411yescharacteristic
87C23×C251C2×D411yescharacteristic
88D4×C251—41no—
89C2×C4×C251—21no—
90C2×C4×C251—21no—
91C2×C4×C251—41no—
92C2×C4×C251—41no—
93C2×C4×C251C2×D411yescharacteristic
94C8×C251—41no—
95C22×C251—21no—
96C22×C251—21no—
97C22×C251—41no—
98C22×C251—41no—
99C22×C251C2×C22⋊C411yescharacteristic
100C2×D251—22no—
101C2×D251—21no—
102C2×D251—41no—
103C2×D251—42no—
104C2×D251—41no—
105C2×D251—41no—
106C2×D251—42no—
107C4×C251—21no—
108C4×C251—41no—
109C4×C251—41no—
110C4×C251—81no—
111Dic251—21no—
112Dic251—21no—
113Dic251—41no—
114Dic251—81no—
115Dic251#9911yescharacteristic
116Dic251#9911yescharacteristic
117C2×C251—21no—
118C2×C251—41no—
119C2×C251—41no—
120C2×C251C2×#9911yescharacteristic
121D251—41no—
122D251—42no—
123C251#212011yescharacteristic, complemented, Sylow
124#2120—2511nomaximal, Sylow
125#525—2511no—
126#413—2511no—
127#410—2511no—
128#352—2511no—
129#352D25111yescharacteristic, complemented
130AΣL(1,𝔽16)—2511no—
131AΣL(1,𝔽16)—2511no—
132C2×#41—2511no—
133C23×C4—2511no—
134C2×M16—5021no—
135#123—2511no—
136#123C2×D25111yescharacteristic
137C22⋊Q8—5021no—
138#121—5021no—
139#120—2511no—
140#120—2511no—
141C4×D4—5021no—
142C4×D4—5021no—
143#117—5021no—
144#117—2511no—
145#101—5021no—
146#100—2511no—
147#100C2×D25111yescharacteristic
148#99—5021no—
149#99—2511no—
150#99C2×D25111yescharacteristic
151C24—2511no—
152#41—10041no—
153#41—10041no—
154C2×Q8—2511no—
155C2×D4—5021no—
156C2×D4—5021no—
157C2×D4—2511no—
158C2×D4—2511no—
159C2×D4C22×D25111yescharacteristic
160C22×C4—10041no—
161C22×C4—10041no—
162C22×C4—5021no—
163C22×C4—5021no—
164C22×C4—2511no—
165C22×C4—2511no—
166C22×C4—2511no—
167C22×C4C4×D25111yescharacteristic
168M16—21no—
169M16—10041no—
170M16—5021no—
171C2×C8—10041no—
172C4⋊C4—21no—
173C4⋊C4—10041no—
174C4⋊C4—5021no—
175C4⋊C4—5021no—
176C22⋊C4—21no—
177C22⋊C4—21no—
178C22⋊C4—5021no—
179C22⋊C4—5021no—
180C22⋊C4—5021no—
181C22⋊C4—5021no—
182C22⋊C4—2511no—
183C22⋊C4C4×D25111yescharacteristic
184C42—10041no—
185C42—5021no—
186C23—10041no—
187C23—10042no—
188C23—5021no—
189C23—2511no—
190C23C2×C4×D25111yescharacteristic
191C23D4×D25111yescharacteristic
192Q8—10041no—
193D4—41no—
194D4—10042no—
195D4—10042no—
196D4—10041no—
197D4—10041no—
198C2×C4—21no—
199C2×C4—21no—
200C2×C4—41no—
201C2×C4—41no—
202C2×C4—10042no—
203C2×C4—10041no—
204C2×C4—10041no—
205C2×C4—10041no—
206C2×C4—5021no—
207C2×C4—5021no—
208C2×C4—5021no—
209C2×C4—5021no—
210C2×C4—5021no—
211C2×C4—5021no—
212C2×C4—5021no—
213C2×C4—2511no—
214C2×C4—2511no—
215C2×C4—2511no—
216C2×C4D4×D25111yescharacteristic
217C8—41no—
218C8—10041no—
219C22—21no—
220C22—21no—
221C22—41no—
222C22—41no—
223C22—10041no—
224C22—10041no—
225C22—10042no—
226C22—10042no—
227C22—10041no—
228C22—5022no—
229C22—5021no—
230C22C22⋊C4×D25111yescharacteristic
231C4—21no—
232C4—41no—
233C4—41no—
234C4—81no—
235C4—20081no—
236C4—10041no—
237C4—5021no—
238C4—5021no—
239C4—2511no—
240C4—2511no—
241C2—21no—
242C2—41no—
243C2—41no—
244C2—10042no—
245C2—10041no—
246C2#99×D25111yescharacteristic
subgroupG/H
centerC2#99×D251
commutatorC23×C251C22×C4
FrattiniC2×D4C22×D251
Fitting#352×C251C2
socleC2×C251C2×#99
prime radical#30632195C2
cosocleC2×D4×C251C23
hypercenter#352D251
hypocenterC251#2120
LCSG ▷ C23×C251 ▷ C22×C251 ▷ C2×C251 ▷ C251
UCSC1 ◁ C2 ◁ C22 ◁ C2×D4 ◁ #352

Complements

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

NH ≅ G/Ntype
2. #30631668C2semidirect
3. #30631659C2semidirect
4. #30631477C2semidirect
5. #30631475C2semidirect
6. #30631473C2semidirect
7. #352×C251C2semidirect
9. C22×C4×D251C4semidirect
11. #22548698C4semidirect
14. #22548685C4semidirect
19. #22548671C4semidirect
123. C251#2120semidirect
129. #352D251semidirect

Maximal subgroups

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

subgroupindex
#306321952
#306316682
#306316592
#306314772
#306314752
#306314732
#352×C2512
#2120251