Finite Group Databasebeta

← #27172770

Subgroups — #27172770

Order 12928.

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#20265683C211yesmaximal, characteristic
2#20265158C211yesmaximal, characteristic, complemented
3#20265149C211yesmaximal, characteristic, complemented
4#20264968C211yesmaximal, characteristic, complemented
5#20264966C211yesmaximal, characteristic, complemented
6#20264964C211yesmaximal, characteristic, complemented
7AΣL(1,𝔽16)×C101C211yesmaximal, characteristic, complemented
8C2×#13108073C2211yescharacteristic
9C23×Dic101C411yescharacteristic, complemented
10M16×D101—21no—
11#15692239C411yescharacteristic, complemented
12#15692236C2211yescharacteristic
13#15692232—21no—
14#15692227—21no—
15#15692224C411yescharacteristic, complemented
16D4×Dic101—21no—
17#15692215—21no—
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
27#13108073—41no—
28#13108073—41no—
29C2×Dic202D411yescharacteristic
30C2×C22⋊D101—21no—
31C2×C22⋊D101C2×C411yescharacteristic
32C2×C22⋊D101D411yescharacteristic
33C22×Dic101—21no—
34C22×Dic101—21no—
35C22×Dic101—41no—
36C22×Dic101—41no—
37C22×Dic101C2×C411yescharacteristic
38C22×Dic101C2×C411yescharacteristic
39C22×Dic101D411yescharacteristic
40C2×C4×D101D411yescharacteristic
41#13108058—21no—
42#13108057—41no—
43C8×D101—41no—
44C4⋊Dic101—21no—
45#13108053—21no—
46#13108053—21no—
47#13108053—41no—
48C22⋊Dic101—21no—
49C22⋊Dic101—21no—
50C22⋊Dic101—21no—
51C22⋊Dic101C2×C411yescharacteristic
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—42no—
64C23×C101—41no—
65C23×C101C22×C411yescharacteristic
66C23×C101C2×D411yescharacteristic
67C22×D101C22⋊C411yescharacteristic
68D4×C101—41no—
69D4×C101—42no—
70C2×C4×C101—21no—
71C2×C4×C101—41no—
72C2×C4×C101C2×D411yescharacteristic
73C22⋊D101—41no—
74C22⋊D101—41no—
75C22⋊D101—42no—
76C2×Dic101—21no—
77C2×Dic101—21no—
78C2×Dic101—21no—
79C2×Dic101—21no—
80C2×Dic101—21no—
81C2×Dic101—21no—
82C2×Dic101—42no—
83C2×Dic101—41no—
84C2×Dic101—41no—
85C2×Dic101—41no—
86C2×Dic101—41no—
87C2×Dic101C22⋊C411yescharacteristic
88C2×Dic101C22⋊C411yescharacteristic
89C2×Dic101C22⋊C411yescharacteristic
90C4×D101—21no—
91C4×D101—21no—
92Dic202—41no—
93C8×C101—41no—
94#11717579—41no—
95C22×C101—22no—
96C22×C101—21no—
97C22×C101—21no—
98C22×C101—41no—
99C22×C101—41no—
100C22×C101—41no—
101C22×C101—42no—
102C22×C101—42no—
103C22×C101—41no—
104C22×C101C2×C22⋊C411yescharacteristic
105C2×D101—21no—
106C2×D101—41no—
107C4×C101—21no—
108C4×C101—41no—
109C4×C101—81no—
110Dic101—21no—
111Dic101—21no—
112Dic101—41no—
113Dic101—41no—
114Dic101—81no—
115Dic101#9911yescharacteristic
116Dic101#9911yescharacteristic
117C2×C101—21no—
118C2×C101—41no—
119C2×C101—42no—
120C2×C101—41no—
121C2×C101C2×#9911yescharacteristic
122D101—41no—
123#2120—1011nomaximal, Sylow
124C101#212011yescharacteristic, complemented, Sylow
125#525—1011no—
126#413—1011no—
127#410—1011no—
128#352—1011no—
129#352—1011no—
130AΣL(1,𝔽16)—1011no—
131AΣL(1,𝔽16)D10111yescharacteristic, complemented
132C2×#41—1011no—
133C23×C4—1011no—
134C2×M16—2021no—
135#123—1011no—
136#123—1011no—
137C22⋊Q8—2021no—
138#121—2021no—
139#120—1011no—
140#120C2×D10111yescharacteristic
141C4×D4—2021no—
142C4×D4—2021no—
143#117—1011no—
144#117—2021no—
145#101—2021no—
146#100—1011no—
147#100C2×D10111yescharacteristic
148#99—1011no—
149#99—2021no—
150#99C2×D10111yescharacteristic
151C24C4×D10111yescharacteristic
152#41—4041no—
153#41—4041no—
154C2×Q8—1011no—
155C2×D4—21no—
156C2×D4—1011no—
157C2×D4—1011no—
158C2×D4—2021no—
159C2×D4C22×D10111yescharacteristic
160C22×C4—1011no—
161C22×C4—1011no—
162C22×C4—1011no—
163C22×C4—1011no—
164C22×C4—4041no—
165C22×C4—4041no—
166C22×C4—2021no—
167C22×C4—2021no—
168M16—21no—
169M16—4041no—
170M16—2021no—
171C2×C8—4041no—
172C4⋊C4—4041no—
173C4⋊C4—2021no—
174C4⋊C4—2021no—
175C4⋊C4—2021no—
176C22⋊C4—21no—
177C22⋊C4—21no—
178C22⋊C4—1011no—
179C22⋊C4—2021no—
180C22⋊C4—2021no—
181C22⋊C4—2021no—
182C22⋊C4—2021no—
183C22⋊C4C4×D10111yescharacteristic
184C42—4041no—
185C42—2021no—
186C23—21no—
187C23—41no—
188C23—42no—
189C23—1011no—
190C23C2×C4×D10111yescharacteristic
191C23D4×D10111yescharacteristic
192Q8—4041no—
193D4—41no—
194D4—42no—
195D4—4042no—
196D4—4041no—
197D4—4041no—
198C2×C4—21no—
199C2×C4—41no—
200C2×C4—1011no—
201C2×C4—1011no—
202C2×C4—1011no—
203C2×C4—4041no—
204C2×C4—4041no—
205C2×C4—4041no—
206C2×C4—4042no—
207C2×C4—4041no—
208C2×C4—2021no—
209C2×C4—2021no—
210C2×C4—2021no—
211C2×C4—2021no—
212C2×C4—2021no—
213C2×C4—2021no—
214C2×C4—2021no—
215C2×C4—2021no—
216C2×C4D4×D10111yescharacteristic
217C8—41no—
218C8—4041no—
219C22—22no—
220C22—21no—
221C22—21no—
222C22—41no—
223C22—41no—
224C22—41no—
225C22—42no—
226C22—42no—
227C22—41no—
228C22—4041no—
229C22—2021no—
230C22C22⋊C4×D10111yescharacteristic
231C4—21no—
232C4—41no—
233C4—81no—
234C4—8081no—
235C4—1011no—
236C4—1011no—
237C4—4041no—
238C4—4041no—
239C4—2021no—
240C4—2021no—
241C2—21no—
242C2—41no—
243C2—42no—
244C2—41no—
245C2—4041no—
246C2#99×D10111yescharacteristic
subgroupG/H
centerC2#99×D101
commutatorC23×C101C22×C4
FrattiniC2×D4C22×D101
FittingAΣL(1,𝔽16)×C101C2
socleC2×C101C2×#99
prime radical#20265683C2
cosocleC2×D4×C101C23
hypercenterAΣL(1,𝔽16)D101
hypocenterC101#2120
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. #20264968C2semidirect
5. #20264966C2semidirect
6. #20264964C2semidirect
7. AΣL(1,𝔽16)×C101C2semidirect
9. C23×Dic101C4semidirect
11. #15692239C4semidirect
15. #15692224C4semidirect
19. #15692211C4semidirect
124. C101#2120semidirect
131. AΣL(1,𝔽16)D101semidirect

Maximal subgroups

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

subgroupindex
#202656832
#202651582
#202651492
#202649682
#202649662
#202649642
AΣL(1,𝔽16)×C1012
#2120101