Finite Group Databasebeta

← #27172775

Subgroups — #27172775

Order 12928.

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

normal subgroups only

This group has 280 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#20265737C211yesmaximal, characteristic
2#100×D101C211yesmaximal, characteristic, complemented
3#99×D101C211yesmaximal, characteristic, complemented
4#20264968C211yesmaximal, characteristic, complemented
5#20264966C211yesmaximal, characteristic, complemented
6#20264964C211yesmaximal, characteristic, complemented
7AΣL(1,𝔽16)×C101C211yesmaximal, characteristic, complemented
8C22×C22⋊D101C411yescharacteristic, complemented
9C2×D4×D101C2211yescharacteristic
10M16×D101—21no—
11#15692239C411yescharacteristic, complemented
12#15692237—21no—
13#15692236C2211yescharacteristic
14#15692228—21no—
15#15692226—21no—
16#15692224C411yescharacteristic, complemented
17#15692223—21no—
18C22⋊C4×D101—21no—
19C22⋊C4×D101C411yescharacteristic, complemented
20#15692172C2211yescharacteristic
21#15692171—21no—
22#15692170C2211yescharacteristic
23#15692169—21no—
24#120×C101C2211yescharacteristic
25#100×C101C2211yescharacteristic
26#99×C101C2211yescharacteristic
27C23×D101C2×C411yescharacteristic
28C23×D101D411yescharacteristic
29C2×D404D411yescharacteristic
30C2×C22⋊D101—21no—
31C2×C22⋊D101—21no—
32C2×C22⋊D101—21no—
33C2×C22⋊D101—41no—
34C2×C22⋊D101—41no—
35C2×C22⋊D101C2×C411yescharacteristic
36C2×C22⋊D101D411yescharacteristic
37D4×D101—41no—
38D4×D101—41no—
39C22×Dic101C2×C411yescharacteristic, complemented
40C2×C4×D101—21no—
41C2×C4×D101—41no—
42C2×C4×D101D411yescharacteristic
43#13108058—21no—
44#13108057—41no—
45C8×D101—41no—
46C4⋊Dic101—21no—
47#13108053—21no—
48C22⋊Dic101—21no—
49C22⋊Dic101—21no—
50C22⋊Dic101—21no—
51C22⋊Dic101C2×C411yescharacteristic, complemented
52#13108051—21no—
53#13108051—21no—
54#13108051—41no—
55C24×C101C2×C411yescharacteristic, complemented
56C2×D4×C101—21no—
57C2×D4×C101C2311yescharacteristic
58M16×C101—21no—
59C22⋊C4×C101—21no—
60C22⋊C4×C101—21no—
61C22⋊C4×C101C2×C411yescharacteristic, complemented
62C23×C101—21no—
63C23×C101—41no—
64C23×C101—81no—
65C23×C101C22×C411yescharacteristic
66C23×C101C2×D411yescharacteristic
67C22×D101—21no—
68C22×D101—21no—
69C22×D101—21no—
70C22×D101—21no—
71C22×D101—81no—
72C22×D101—81no—
73C22×D101C22⋊C411yescharacteristic
74C22×D101C22⋊C411yescharacteristic
75C22×D101C22⋊C411yescharacteristic
76D4×C101—41no—
77D4×C101—81no—
78C2×C4×C101—21no—
79C2×C4×C101—41no—
80C2×C4×C101C2×D411yescharacteristic
81C22⋊D101—41no—
82C22⋊D101—41no—
83C22⋊D101—42no—
84C22⋊D101—41no—
85C22⋊D101—41no—
86C22⋊D101—81no—
87C2×Dic101—21no—
88C2×Dic101—21no—
89C2×Dic101—41no—
90C2×Dic101—41no—
91C2×Dic101C22⋊C411yescharacteristic
92D404—41no—
93C4×D101—21no—
94C4×D101—21no—
95C4×D101—81no—
96C4×D101—82no—
97C8×C101—41no—
98#11717579—41no—
99C22×C101—21no—
100C22×C101—21no—
101C22×C101—41no—
102C22×C101—41no—
103C22×C101—41no—
104C22×C101—41no—
105C22×C101—41no—
106C22×C101—82no—
107C22×C101C2×C22⋊C411yescharacteristic
108C2×D101—21no—
109C2×D101—21no—
110C2×D101—41no—
111C2×D101—42no—
112C2×D101—41no—
113C2×D101—41no—
114C2×D101—81no—
115C2×D101—81no—
116C2×D101—81no—
117C2×D101#9911yescharacteristic
118C2×D101#9911yescharacteristic
119C4×C101—21no—
120C4×C101—41no—
121C4×C101—81no—
122Dic101—21no—
123Dic101—41no—
124Dic101—41no—
125Dic101—81no—
126C2×C101—21no—
127C2×C101—41no—
128C2×C101—41no—
129C2×C101—81no—
130C2×C101C2×#9911yescharacteristic
131D101—21no—
132D101—21no—
133D101—41no—
134D101—41no—
135#2121—1011nomaximal, Sylow
136C101#212111yescharacteristic, complemented, Sylow
137#534—1011no—
138C2×#100—1011no—
139C2×#99—1011no—
140#352—1011no—
141#352—1011no—
142AΣL(1,𝔽16)—1011no—
143AΣL(1,𝔽16)D10111yescharacteristic, complemented
144C22×D4—1011no—
145C22×D4—1011no—
146C2×M16—2021no—
147#123—1011no—
148#123—1011no—
149#123—2021no—
150#121—2021no—
151#121—2021no—
152#120—1011no—
153#120—2021no—
154#120C2×D10111yescharacteristic
155C2×C22⋊C4—1011no—
156C2×C22⋊C4—2021no—
157#100—1011no—
158#100—2021no—
159#100C2×D10111yescharacteristic
160#99—1011no—
161#99—2021no—
162#99C2×D10111yescharacteristic
163C24—1011no—
164C24—1011no—
165C24C4×D10111yescharacteristic, complemented
166C2×D4—21no—
167C2×D4—1011no—
168C2×D4—1011no—
169C2×D4—1011no—
170C2×D4—4041no—
171C2×D4—4041no—
172C2×D4—4041no—
173C2×D4—4041no—
174C2×D4—2021no—
175C2×D4—2021no—
176C2×D4—2021no—
177C2×D4C22×D10111yescharacteristic
178C22×C4—1011no—
179C22×C4—1011no—
180C22×C4—4041no—
181C22×C4—2021no—
182M16—21no—
183M16—4041no—
184M16—2021no—
185C2×C8—4041no—
186C4⋊C4—2021no—
187C4⋊C4—2021no—
188C22⋊C4—21no—
189C22⋊C4—21no—
190C22⋊C4—1011no—
191C22⋊C4—4041no—
192C22⋊C4—2021no—
193C22⋊C4—2021no—
194C22⋊C4—2021no—
195C22⋊C4—2021no—
196C22⋊C4—2021no—
197C22⋊C4C4×D10111yescharacteristic, complemented
198C23—21no—
199C23—41no—
200C23—81no—
201C23—8081no—
202C23—8081no—
203C23—1011no—
204C23—1011no—
205C23—1011no—
206C23—2021no—
207C23—2021no—
208C23—2021no—
209C23—2021no—
210C23C2×C4×D10111yescharacteristic
211C23D4×D10111yescharacteristic
212D4—41no—
213D4—81no—
214D4—8081no—
215D4—4041no—
216D4—4041no—
217D4—4041no—
218D4—4042no—
219D4—4041no—
220D4—4041no—
221C2×C4—21no—
222C2×C4—41no—
223C2×C4—8081no—
224C2×C4—8082no—
225C2×C4—1011no—
226C2×C4—4041no—
227C2×C4—4041no—
228C2×C4—2021no—
229C2×C4—2021no—
230C2×C4—2021no—
231C2×C4—2021no—
232C2×C4D4×D10111yescharacteristic
233C8—41no—
234C8—4041no—
235C22—21no—
236C22—21no—
237C22—41no—
238C22—41no—
239C22—41no—
240C22—41no—
241C22—41no—
242C22—82no—
243C22—8081no—
244C22—8081no—
245C22—8081no—
246C22—1011no—
247C22—1011no—
248C22—4041no—
249C22—4041no—
250C22—4041no—
251C22—4042no—
252C22—2021no—
253C22—2021no—
254C22C22⋊C4×D10111yescharacteristic
255C4—21no—
256C4—41no—
257C4—81no—
258C4—8081no—
259C4—4041no—
260C4—4041no—
261C4—2021no—
262C2—21no—
263C2—41no—
264C2—41no—
265C2—81no—
266C2—4041no—
267C2—4041no—
268C2—2021no—
269C2—2021no—
270C2#99×D10111yescharacteristic
subgroupG/H
centerC2#99×D101
commutatorC23×C101C22×C4
FrattiniC2×D4C22×D101
FittingAΣL(1,𝔽16)×C101C2
socleC2×C101C2×#99
prime radical#20265737C2
cosocleC2×D4×C101C23
hypercenterAΣL(1,𝔽16)D101
hypocenterC101#2121
LCSG ▷ C23×C101 ▷ C22×C101 ▷ C2×C101 ▷ C101
UCSC1 ◁ C2 ◁ C22 ◁ C2×D4 ◁ AΣL(1,𝔽16)

Complements

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

NH ≅ G/Ntype
2. #100×D101C2semidirect
3. #99×D101C2semidirect
4. #20264968C2semidirect
5. #20264966C2semidirect
6. #20264964C2semidirect
7. AΣL(1,𝔽16)×C101C2semidirect
8. C22×C22⋊D101C4semidirect
11. #15692239C4semidirect
16. #15692224C4semidirect
19. C22⋊C4×D101C4semidirect
39. C22×Dic101C2×C4semidirect
51. C22⋊Dic101C2×C4semidirect
55. C24×C101C2×C4semidirect
61. C22⋊C4×C101C2×C4semidirect
136. C101#2121semidirect
143. AΣL(1,𝔽16)D101semidirect
165. C24C4×D101semidirect
197. C22⋊C4C4×D101semidirect

Maximal subgroups

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

subgroupindex
#202657372
#100×D1012
#99×D1012
#202649682
#202649662
#202649642
AΣL(1,𝔽16)×C1012
#2121101