Finite Group Databasebeta

← #27172777

Subgroups — #27172777

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#20265738C211yesmaximal, characteristic
2#100×D101C211yesmaximal, characteristic, complemented
3#99×D101C211yesmaximal, characteristic, complemented
4#20264967C211yesmaximal, characteristic, complemented
5#20264965C211yesmaximal, characteristic, complemented
6#20264963C211yesmaximal, characteristic, complemented
7#352×C101C211yesmaximal, characteristic, complemented
8C22×D404C411yescharacteristic, complemented
9C2×D4×D101C2211yescharacteristic
10M16×D101—21no—
11#15692240—21no—
12#15692238C411yescharacteristic, complemented
13#15692230—21no—
14#15692228—21no—
15#15692225—21no—
16#15692225C411yescharacteristic, complemented
17#15692223C2211yescharacteristic
18C22⋊C4×D101—21no—
19C22⋊C4×D101C411yescharacteristic, complemented
20#15692172C2211yescharacteristic
21#15692171—21no—
22#15692170C2211yescharacteristic
23#15692169—21no—
24#123×C101C2211yescharacteristic
25#100×C101C2211yescharacteristic
26#99×C101C2211yescharacteristic
27C23×D101C2×C411yescharacteristic
28C23×D101C2×C411yescharacteristic, complemented
29C23×D101D411yescharacteristic
30C2×D404—21no—
31C2×D404—21no—
32C2×D404—21no—
33C2×D404—41no—
34C2×D404—41no—
35C2×D404D411yescharacteristic
36C2×C22⋊D101—21no—
37C2×C22⋊D101C2×C411yescharacteristic
38C2×C22⋊D101D411yescharacteristic
39D4×D101—41no—
40D4×D101—41no—
41C2×C4×D101—21no—
42C2×C4×D101—41no—
43C2×C4×D101D411yescharacteristic
44#13108058—21no—
45#13108057—41no—
46C8×D101—41no—
47#13108053—21no—
48C22⋊Dic101—21no—
49C22⋊Dic101C2×C411yescharacteristic, complemented
50#13108051—21no—
51#13108051—21no—
52#13108051—21no—
53#13108051—21no—
54#13108051—41no—
55C2×D4×C101C2311yescharacteristic
56C22×C4×C101C2×C411yescharacteristic, complemented
57M16×C101—21no—
58C4⋊C4×C101—21no—
59C22⋊C4×C101—21no—
60C22⋊C4×C101—21no—
61C22⋊C4×C101C2×C411yescharacteristic, complemented
62C23×C101C22×C411yescharacteristic
63C23×C101C2×D411yescharacteristic
64C22×D101—21no—
65C22×D101—21no—
66C22×D101—21no—
67C22×D101—21no—
68C22×D101—21no—
69C22×D101—41no—
70C22×D101—81no—
71C22×D101—81no—
72C22×D101—81no—
73C22×D101C22⋊C411yescharacteristic
74C22×D101C22⋊C411yescharacteristic
75C22×D101C22⋊C411yescharacteristic
76D4×C101—41no—
77C2×C4×C101—21no—
78C2×C4×C101—21no—
79C2×C4×C101—41no—
80C2×C4×C101—41no—
81C2×C4×C101C2×D411yescharacteristic
82C22⋊D101—41no—
83C22⋊D101—41no—
84C22⋊D101—81no—
85C2×Dic101—21no—
86C2×Dic101—41no—
87C2×Dic101C22⋊C411yescharacteristic
88D404—41no—
89D404—42no—
90D404—41no—
91D404—41no—
92D404—81no—
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×C101C2×C22⋊C411yescharacteristic
104C2×D101—21no—
105C2×D101—21no—
106C2×D101—41no—
107C2×D101—42no—
108C2×D101—41no—
109C2×D101—41no—
110C2×D101—41no—
111C2×D101—41no—
112C2×D101—41no—
113C2×D101—81no—
114C2×D101—81no—
115C2×D101—81no—
116C2×D101—82no—
117C2×D101#9911yescharacteristic
118C2×D101#9911yescharacteristic
119C4×C101—21no—
120C4×C101—41no—
121C4×C101—41no—
122C4×C101—81no—
123Dic101—21no—
124Dic101—41no—
125Dic101—81no—
126C2×C101—21no—
127C2×C101—41no—
128C2×C101—41no—
129C2×C101C2×#9911yescharacteristic
130D101—21no—
131D101—21no—
132D101—41no—
133D101—41no—
134D101—81no—
135#2121—1011nomaximal, Sylow
136C101#212111yescharacteristic, complemented, Sylow
137#534—1011no—
138C2×#100—1011no—
139C2×#99—1011no—
140#352—1011no—
141#352D10111yescharacteristic, complemented
142AΣL(1,𝔽16)—1011no—
143AΣL(1,𝔽16)—1011no—
144C22×D4—1011no—
145C22×D4—1011no—
146C2×M16—2021no—
147#123—1011no—
148#123—2021no—
149#123C2×D10111yescharacteristic
150#121—2021no—
151#121—2021no—
152#120—1011no—
153#120—1011no—
154#120—2021no—
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—
165C24—1011no—
166C2×D4—1011no—
167C2×D4—1011no—
168C2×D4—1011no—
169C2×D4—4041no—
170C2×D4—4041no—
171C2×D4—4041no—
172C2×D4—4041no—
173C2×D4—2021no—
174C2×D4—2021no—
175C2×D4—2021no—
176C2×D4—2021no—
177C2×D4C22×D10111yescharacteristic
178C22×C4—1011no—
179C22×C4—4041no—
180C22×C4—2021no—
181C22×C4C4×D10111yescharacteristic, complemented
182M16—21no—
183M16—4041no—
184M16—2021no—
185C2×C8—4041no—
186C4⋊C4—21no—
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—8081no—
199C23—8081no—
200C23—8081no—
201C23—1011no—
202C23—1011no—
203C23—1011no—
204C23—4041no—
205C23—2021no—
206C23—2021no—
207C23—2021no—
208C23—2021no—
209C23—2021no—
210C23C2×C4×D10111yescharacteristic
211C23D4×D10111yescharacteristic
212D4—41no—
213D4—8081no—
214D4—8081no—
215D4—4041no—
216D4—4041no—
217D4—4041no—
218D4—4042no—
219D4—4041no—
220D4—4041no—
221C2×C4—21no—
222C2×C4—21no—
223C2×C4—41no—
224C2×C4—41no—
225C2×C4—8081no—
226C2×C4—8082no—
227C2×C4—1011no—
228C2×C4—4041no—
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—8082no—
240C22—8081no—
241C22—8081no—
242C22—8081no—
243C22—1011no—
244C22—1011no—
245C22—4041no—
246C22—4041no—
247C22—4041no—
248C22—4041no—
249C22—4041no—
250C22—4042no—
251C22—4041no—
252C22—2021no—
253C22—2021no—
254C22C22⋊C4×D10111yescharacteristic
255C4—21no—
256C4—41no—
257C4—41no—
258C4—81no—
259C4—8081no—
260C4—4041no—
261C4—2021no—
262C2—21no—
263C2—41no—
264C2—41no—
265C2—8081no—
266C2—4041no—
267C2—4041no—
268C2—2021no—
269C2—2021no—
270C2#99×D10111yescharacteristic
subgroupG/H
centerC2#99×D101
commutatorC23×C101C22×C4
FrattiniC2×D4C22×D101
Fitting#352×C101C2
socleC2×C101C2×#99
prime radical#20265738C2
cosocleC2×D4×C101C23
hypercenter#352D101
hypocenterC101#2121
LCSG ▷ C23×C101 ▷ C22×C101 ▷ C2×C101 ▷ C101
UCSC1 ◁ C2 ◁ C22 ◁ C2×D4 ◁ #352

Complements

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

NH ≅ G/Ntype
2. #100×D101C2semidirect
3. #99×D101C2semidirect
4. #20264967C2semidirect
5. #20264965C2semidirect
6. #20264963C2semidirect
7. #352×C101C2semidirect
8. C22×D404C4semidirect
12. #15692238C4semidirect
16. #15692225C4semidirect
19. C22⋊C4×D101C4semidirect
28. C23×D101C2×C4semidirect
49. C22⋊Dic101C2×C4semidirect
56. C22×C4×C101C2×C4semidirect
61. C22⋊C4×C101C2×C4semidirect
136. C101#2121semidirect
141. #352D101semidirect
181. C22×C4C4×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
#202657382
#100×D1012
#99×D1012
#202649672
#202649652
#202649632
#352×C1012
#2121101