Finite Group Databasebeta

← #27172780

Subgroups — #27172780

Order 12928.

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

normal subgroups only

This group has 266 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#20265742C211yesmaximal, characteristic
2#20265158C211yesmaximal, characteristic, complemented
3#20265149C211yesmaximal, characteristic, complemented
4#20264965C211yesmaximal, characteristic, complemented
5#20264964C211yesmaximal, characteristic, complemented
6#20264963C211yesmaximal, characteristic, complemented
7AΣL(1,𝔽16)×C101C211yesmaximal, characteristic, complemented
8C2×#13108073C2211yescharacteristic
9C22×C22⋊D101C411yescharacteristic, complemented
10M16×D101—21no—
11#15692251—21no—
12#15692241—21no—
13#15692228—21no—
14#15692227—21no—
15#15692225C411yescharacteristic, complemented
16#15692224C411yescharacteristic, complemented
17#15692223C2211yescharacteristic
18#15692211—21no—
19#15692211C411yescharacteristic, complemented
20#15692173—21no—
21#15692172C2211yescharacteristic
22#15692170C2211yescharacteristic
23#15692169—21no—
24#120×C101C2211yescharacteristic
25#100×C101C2211yescharacteristic
26#99×C101C2211yescharacteristic
27C23×D101C2×C411yescharacteristic
28#13108073—41no—
29#13108073—41no—
30C2×Dic202D411yescharacteristic
31C2×D404—21no—
32C2×C22⋊D101—21no—
33C2×C22⋊D101—21no—
34C2×C22⋊D101—21no—
35C2×C22⋊D101—21no—
36C2×C22⋊D101—41no—
37C2×C22⋊D101—41no—
38C2×C22⋊D101C2×C411yescharacteristic
39C2×C22⋊D101D411yescharacteristic
40C22×Dic101C2×C411yescharacteristic
41C22×Dic101D411yescharacteristic
42C2×C4×D101D411yescharacteristic
43#13108058—21no—
44#13108057—41no—
45C8×D101—41no—
46#13108053—21no—
47#13108053—41no—
48C22⋊Dic101—21no—
49C22⋊Dic101—21no—
50C22⋊Dic101C2×C411yescharacteristic
51#13108051—21no—
52#13108051—21no—
53C4×Dic101—21no—
54C4×Dic101—41no—
55C24×C101C2×C411yescharacteristic
56C2×D4×C101—21no—
57C2×D4×C101C2311yescharacteristic
58M16×C101—21no—
59C22⋊C4×C101—21no—
60C22⋊C4×C101—21no—
61C22⋊C4×C101C2×C411yescharacteristic
62C23×C101—21no—
63C23×C101—41no—
64C23×C101—81no—
65C23×C101C22×C411yescharacteristic
66C23×C101C2×D411yescharacteristic
67C22×D101—21no—
68C22×D101—41no—
69C22×D101—81no—
70C22×D101C22⋊C411yescharacteristic
71D4×C101—41no—
72D4×C101—81no—
73C2×C4×C101—21no—
74C2×C4×C101—41no—
75C2×C4×C101C2×D411yescharacteristic
76C22⋊D101—41no—
77C22⋊D101—41no—
78C22⋊D101—42no—
79C22⋊D101—41no—
80C22⋊D101—41no—
81C22⋊D101—81no—
82C22⋊D101—81no—
83C2×Dic101—21no—
84C2×Dic101—21no—
85C2×Dic101—21no—
86C2×Dic101—21no—
87C2×Dic101—21no—
88C2×Dic101—41no—
89C2×Dic101C22⋊C411yescharacteristic
90C2×Dic101C22⋊C411yescharacteristic
91C2×Dic101C22⋊C411yescharacteristic
92D404—81no—
93C4×D101—21no—
94C4×D101—21no—
95Dic202—41no—
96C8×C101—41no—
97#11717579—41no—
98C22×C101—21no—
99C22×C101—21no—
100C22×C101—41no—
101C22×C101—41no—
102C22×C101—41no—
103C22×C101—41no—
104C22×C101—41no—
105C22×C101—82no—
106C22×C101C2×C22⋊C411yescharacteristic
107C2×D101—21no—
108C2×D101—41no—
109C2×D101—41no—
110C2×D101—41no—
111C2×D101—41no—
112C2×D101—82no—
113C4×C101—21no—
114C4×C101—41no—
115C4×C101—81no—
116Dic101—21no—
117Dic101—21no—
118Dic101—41no—
119Dic101—81no—
120Dic101#9911yescharacteristic
121Dic101#9911yescharacteristic
122C2×C101—21no—
123C2×C101—41no—
124C2×C101—41no—
125C2×C101—81no—
126C2×C101C2×#9911yescharacteristic
127D101—41no—
128D101—81no—
129#2122—1011nomaximal, Sylow
130C101#212211yescharacteristic, complemented, Sylow
131#535—1011no—
132#413—1011no—
133#410—1011no—
134AΣL(1,𝔽16)—1011no—
135AΣL(1,𝔽16)—1011no—
136AΣL(1,𝔽16)—1011no—
137AΣL(1,𝔽16)D10111yescharacteristic, complemented
138C2×#41—1011no—
139C22×D4—1011no—
140C2×M16—2021no—
141#127—2021no—
142#124—2021no—
143#121—2021no—
144#121—2021no—
145#120—1011no—
146#120—1011no—
147#120—1011no—
148#120C2×D10111yescharacteristic
149#117—1011no—
150#117—2021no—
151#101—2021no—
152#100—1011no—
153#100C2×D10111yescharacteristic
154#99—1011no—
155#99—2021no—
156#99C2×D10111yescharacteristic
157C24—1011no—
158C24C4×D10111yescharacteristic
159#41—4041no—
160#41—4041no—
161C2×Q8—1011no—
162C2×D4—21no—
163C2×D4—1011no—
164C2×D4—1011no—
165C2×D4—4041no—
166C2×D4—4041no—
167C2×D4—2021no—
168C2×D4—2021no—
169C2×D4—2021no—
170C2×D4—2021no—
171C2×D4—2021no—
172C2×D4C22×D10111yescharacteristic
173C22×C4—1011no—
174C22×C4—1011no—
175C22×C4—1011no—
176M16—21no—
177M16—4041no—
178M16—2021no—
179C2×C8—4041no—
180C4⋊C4—4041no—
181C4⋊C4—2021no—
182C22⋊C4—21no—
183C22⋊C4—21no—
184C22⋊C4—1011no—
185C22⋊C4—2021no—
186C22⋊C4—2021no—
187C22⋊C4—2021no—
188C22⋊C4—2021no—
189C22⋊C4C4×D10111yescharacteristic
190C42—4041no—
191C42—2021no—
192C23—21no—
193C23—41no—
194C23—81no—
195C23—8081no—
196C23—1011no—
197C23—4041no—
198C23—2021no—
199C23C2×C4×D10111yescharacteristic
200C23D4×D10111yescharacteristic
201Q8—4041no—
202D4—41no—
203D4—81no—
204D4—8081no—
205D4—8081no—
206D4—8081no—
207D4—4041no—
208D4—4041no—
209D4—4041no—
210D4—4041no—
211D4—4042no—
212C2×C4—21no—
213C2×C4—41no—
214C2×C4—1011no—
215C2×C4—1011no—
216C2×C4—1011no—
217C2×C4—4041no—
218C2×C4—2021no—
219C2×C4—2021no—
220C2×C4—2021no—
221C2×C4—2021no—
222C2×C4—2021no—
223C2×C4—2021no—
224C2×C4—2021no—
225C2×C4D4×D10111yescharacteristic
226C8—41no—
227C8—4041no—
228C22—21no—
229C22—21no—
230C22—41no—
231C22—41no—
232C22—41no—
233C22—41no—
234C22—41no—
235C22—82no—
236C22—8082no—
237C22—4041no—
238C22—4041no—
239C22—4041no—
240C22—4041no—
241C22—2021no—
242C22C22⋊C4×D10111yescharacteristic
243C4—21no—
244C4—41no—
245C4—81no—
246C4—8081no—
247C4—1011no—
248C4—1011no—
249C4—4041no—
250C4—2021no—
251C4—2021no—
252C2—21no—
253C2—41no—
254C2—41no—
255C2—81no—
256C2—8081no—
257C2—4041no—
258C2#99×D10111yescharacteristic
subgroupG/H
centerC2#99×D101
commutatorC23×C101C22×C4
FrattiniC2×D4C22×D101
FittingAΣL(1,𝔽16)×C101C2
socleC2×C101C2×#99
prime radical#20265742C2
cosocleC2×D4×C101C23
hypercenterAΣL(1,𝔽16)D101
hypocenterC101#2122
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. #20265158C2semidirect
3. #20265149C2semidirect
4. #20264965C2semidirect
5. #20264964C2semidirect
6. #20264963C2semidirect
7. AΣL(1,𝔽16)×C101C2semidirect
9. C22×C22⋊D101C4semidirect
15. #15692225C4semidirect
16. #15692224C4semidirect
19. #15692211C4semidirect
130. C101#2122semidirect
137. AΣL(1,𝔽16)D101semidirect

Maximal subgroups

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

subgroupindex
#202657422
#202651582
#202651492
#202649652
#202649642
#202649632
AΣL(1,𝔽16)×C1012
#2122101