Finite Group Databasebeta

← #15620536

Subgroups — #15620536

Order 2816.

4854 conjugacy classes of subgroups; only the 807 normal subgroups are shown.

normal subgroups only

This group has 4854 conjugacy classes of subgroups — too many to draw. Only its 807 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#13082256C213yesmaximal, complemented
2#13078022C214yesmaximal, complemented
3#13078013C218yesmaximal, complemented
4#13078008C214yesmaximal, complemented
5C2×#10622856C218yesmaximal, complemented
6#13077077C213yesmaximal, complemented
7C11×H(7,𝔽2)C211yesmaximal, characteristic, complemented
8C11×#585—212no—
9C11×#585C2214yes—
10C2×C11×H(5,𝔽2)—212no—
11C2×C11×H(5,𝔽2)C2218yes—
12C2×C11×H(5,𝔽2)C2213yes—
13C22×#4341C2211yescharacteristic, complemented
14#10623359C2216yescomplemented
15#10623356C2216yescomplemented
16#10623355—212no—
17#10623337—212no—
18#10623335C22112yescomplemented
19#10623329C2213yescomplemented
20C2×#66988C22112yescomplemented
21#10622857C22116yescomplemented
22#10622856C22132yescomplemented
23#10622856C22116yescomplemented
24#10622853C22112yescomplemented
25C2×#66898C22124yescomplemented
26C11×2−1+4—216no—
27C11×2−1+4—212no—
28C11×H(5,𝔽2)—248no—
29C11×H(5,𝔽2)—212no—
30C11×H(5,𝔽2)—248no—
31C11×H(5,𝔽2)—224no—
32C11×H(5,𝔽2)C23116yes—
33C2×C11×#41—248no—
34C2×C11×#41—224no—
35C2×C11×#41—224no—
36C2×C11×#41D4112yescomplemented
37C2×C11×#41C2316yes—
38C22×D4×C11—216no—
39C22×D4×C11—224no—
40C22×D4×C11D4112yescomplemented
41C22×D4×C11C23112yes—
42C22×D4×C11C2311yescharacteristic
43C22×C22⋊D11C2312yescomplemented
44C23×Dic11C2312yescomplemented
45C2×#4341—212no—
46C22×C4×D11C2311yescharacteristic, complemented
47C22×Dic22C2311yescharacteristic, complemented
48#67006—212no—
49C2×C22⋊Dic11—212no—
50C2×C22⋊Dic11C23112yescomplemented
51#66999—26no—
52#66997—26no—
53#66992—224no—
54#66991—26no—
55#66990C23112yescomplemented
56#66989—26no—
57#66988—224no—
58#66988C23112yescomplemented
59#66988C23124yescomplemented
60D4×Dic11—212no—
61#66982—212no—
62C4×C22⋊D11—212no—
63#66979C2316yescomplemented
64#66977—212no—
65C2×Dic11⋊C4—212no—
66#66898C23148yescomplemented
67#60401—111nomaximal, Sylow
68C24×C11—28no—
69C24×C11C2×D4112yes—
70C24×C11C2412yes—
71C23×D11C2411yescharacteristic
72C11×#41—296no—
73C11×#41—248no—
74C11×#41—216no—
75C11×#41—248no—
76C11×#41—248no—
77C11×#41—224no—
78C2×Q8×C11—224no—
79C2×Q8×C11—28no—
80C2×Q8×C11C2×D416yes—
81C2×D4×C11—296no—
82C2×D4×C11—224no—
83C2×D4×C11—224no—
84C2×D4×C11—248no—
85C2×D4×C11—248no—
86C2×D4×C11—248no—
87C2×D4×C11C2×D4124yes—
88C2×D4×C11C2×D416yes—
89C2×D4×C11C2×D4112yes—
90C2×D4×C11C24112yes—
91C22×C4×C11—224no—
92C22×C4×C11—216no—
93C22×C4×C11C2×D4112yes—
94C22×C4×C11C2×D4112yes—
95C22×C4×C11C2411yescharacteristic
96C2×C22⋊D11—212no—
97C2×C22⋊D11—212no—
98C22×Dic11—212no—
99C22×Dic11—212no—
100C22×Dic11C2412yes—
101C22×Dic11C2411yescharacteristic
102#4341—416no—
103C2×C4×D11—26no—
104C2×Dic22—26no—
105C22⋊Dic11—224no—
106C22⋊Dic11—212no—
107C22⋊Dic11—212no—
108C22⋊Dic11C24112yes—
109(C2×D11)⋊C4—212no—
110C4⋊Dic11—26no—
111Dic11⋊C4—212no—
112Dic11⋊C4—224no—
113C4×Dic11—26no—
114H(7,𝔽2)D1111yescharacteristic, complemented
115#3485—113no—
116#3028—114no—
117#3026—118no—
118#3025—114no—
119C2×#458—118no—
120#2883—113no—
121C23×C11—216no—
122C23×C11—224no—
123C23×C11—82no—
124C23×C11—812no—
125C23×C11—168no—
126C23×C11C22×D4112yes—
127C23×C11C22×D4112yes—
128C23×C11C2511yescharacteristic
129C22×D11—23no—
130C22×D11—81no—
131Q8×C11—216no—
132Q8×C11—212no—
133D4×C11—248no—
134D4×C11—248no—
135D4×C11—224no—
136D4×C11—212no—
137D4×C11#120116yes—
138C2×C4×C11—248no—
139C2×C4×C11—224no—
140C2×C4×C11—224no—
141C2×C4×C11C22×D4112yes—
142C2×C4×C11C22×D416yes—
143C22⋊D11—48no—
144C22⋊D11—812no—
145C2×Dic11—212no—
146C2×Dic11—26no—
147C2×Dic11—23no—
148C2×Dic11—48no—
149C2×Dic11—812no—
150C4×D11—44no—
151Dic22—44no—
152#585—212no—
153#585C2×D1114yes—
154C2×H(5,𝔽2)—212no—
155C2×H(5,𝔽2)C2×D1118yes—
156C2×H(5,𝔽2)C2×D1113yes—
157C22×#41—111no—
158#547—2212no—
159#546—2212no—
160#536—116no—
161#534—1112no—
162#525—116no—
163C2×#123—1112no—
164#518—113no—
165#458—1116no—
166#458—1132no—
167B(4,𝔽2)—1116no—
168#410—1112no—
169C2×#99—1124no—
170C22×C11—212no—
171C22×C11—41no—
172C22×C11—412no—
173C22×C11—412no—
174C22×C11—824no—
175C22×C11—816no—
176C22×C11C2×#12018yes—
177C22×C11C23×D413yes—
178C2×D11—46no—
179C2×D11—41no—
180C4×C11—212no—
181C4×C11C2×#12014yes—
182Dic11—412no—
183Dic11—42no—
184Dic11—41no—
1852−1+4—216no—
1862−1+4—212no—
187H(5,𝔽2)—248no—
188H(5,𝔽2)—212no—
189H(5,𝔽2)—248no—
190H(5,𝔽2)—224no—
191H(5,𝔽2)C22×D11116yes—
192C2×#41—248no—
193C2×#41—224no—
194C2×#41—224no—
195C2×#41—2212no—
196C2×#41C22⋊D11112yescomplemented
197C2×#41C22×D1116yes—
198C22×Q8—111no—
199C22×D4—216no—
200C22×D4—224no—
201C22×D4—112no—
202C22×D4C22⋊D11112yescomplemented
203C22×D4C22×D11112yes—
204C22×D4C22×D1111yescharacteristic
205C23×C4—112no—
206C23×C4—111no—
207C4⋊Q8—226no—
208#124—226no—
209#123—1112no—
210#123—1124no—
211#123—2224no—
212#123—2212no—
213C22⋊Q8—226no—
214C22⋊Q8—2212no—
215#121—226no—
216#121—2224no—
217#120—1112no—
218#120—2212no—
219C4×D4—2212no—
220C4×D4—2212no—
221#117—116no—
222C2×C4⋊C4—2212no—
223C2×C22⋊C4—1112no—
224C2×C22⋊C4—2212no—
225#99—1148no—
226C2×C11—23no—
227C2×C11—28no—
228C2×C11—412no—
229C2×C11C22×#12011yescharacteristic
230D11—81no—
231C24—28no—
232C24—111no—
233C24C2×C22⋊D11112yes—
234C24C23×D1112yes—
235#41—296no—
236#41—248no—
237#41—216no—
238#41—248no—
239#41—248no—
240#41—224no—
241#41—4416no—
242C2×Q8—224no—
243C2×Q8—28no—
244C2×Q8—226no—
245C2×Q8C2×C22⋊D1116yes—
246C2×D4—296no—
247C2×D4—224no—
248C2×D4—224no—
249C2×D4—248no—
250C2×D4—248no—
251C2×D4—248no—
252C2×D4—2212no—
253C2×D4—2212no—
254C2×D4C2×C22⋊D11124yes—
255C2×D4C2×C22⋊D1116yes—
256C2×D4C2×C22⋊D11112yes—
257C2×D4C23×D11112yes—
258C22×C4—224no—
259C22×C4—216no—
260C22×C4—111no—
261C22×C4—112no—
262C22×C4—2212no—
263C22×C4—2212no—
264C22×C4—226no—
265C22×C4C2×C22⋊D11112yes—
266C22×C4C2×C22⋊D11112yes—
267C22×C4C23×D1111yescharacteristic
268C4⋊C4—2212no—
269C4⋊C4—2224no—
270C4⋊C4—226no—
271C22⋊C4—1112no—
272C22⋊C4—2212no—
273C22⋊C4—2212no—
274C22⋊C4—2224no—
275C22⋊C4—2212no—
276C42—226no—
277C11#6040111yescharacteristic, complemented, Sylow
278C23—216no—
279C23—224no—
280C23—82no—
281C23—812no—
282C23—168no—
283C23—223no—
284C23—881no—
285C23C22×C22⋊D11112yes—
286C23C22×C22⋊D11112yes—
287C23C24×D1111yescharacteristic
288Q8—216no—
289Q8—212no—
290Q8—444no—
291D4—248no—
292D4—248no—
293D4—224no—
294D4—212no—
295D4—448no—
296D4—8812no—
297D4#67006116yes—
298C2×C4—248no—
299C2×C4—224no—
300C2×C4—224no—
301C2×C4—2212no—
302C2×C4—223no—
303C2×C4—226no—
304C2×C4—448no—
305C2×C4—444no—
306C2×C4—8812no—
307C2×C4C22×C22⋊D11112yes—
308C2×C4C22×C22⋊D1116yes—
309C22—212no—
310C22—412no—
311C22—41no—
312C22—412no—
313C22—824no—
314C22—816no—
315C22—446no—
316C22—441no—
317C22C2×#6700618yes—
318C22C23×C22⋊D1113yes—
319C4—212no—
320C4—4412no—
321C4—441no—
322C4—442no—
323C4C2×#6700614yes—
324C2—23no—
325C2—28no—
326C2—412no—
327C2—881no—
328C2C22×#6700611yescharacteristic
subgroupG/H
centerC2C22×#67006
commutator, cosocleC23×C11C25
FrattiniC23C24×D11
FittingC11×H(7,𝔽2)C2
socleC2×C11C22×#120
hypercenterH(7,𝔽2)D11
hypocenterC11#60401
LCSG ▷ C23×C11 ▷ C2×C11 ▷ C11
UCSC1 ◁ C2 ◁ C22×D4 ◁ H(7,𝔽2)

Complements

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

NH ≅ G/Ntype
1. #13082256C2semidirect
2. #13078022C2semidirect
3. #13078013C2semidirect
4. #13078008C2semidirect
5. C2×#10622856C2semidirect
6. #13077077C2semidirect
7. C11×H(7,𝔽2)C2semidirect
13. C22×#4341C22semidirect
14. #10623359C22semidirect
15. #10623356C22semidirect
18. #10623335C22semidirect
19. #10623329C22semidirect
20. C2×#66988C22semidirect
21. #10622857C22semidirect
22. #10622856C22semidirect
23. #10622856C22semidirect
24. #10622853C22semidirect
25. C2×#66898C22semidirect
36. C2×C11×#41D4semidirect
40. C22×D4×C11D4semidirect
43. C22×C22⋊D11C23semidirect
44. C23×Dic11C23semidirect
46. C22×C4×D11C23semidirect
47. C22×Dic22C23semidirect
50. C2×C22⋊Dic11C23semidirect
55. #66990C23semidirect
58. #66988C23semidirect
59. #66988C23semidirect
63. #66979C23semidirect
66. #66898C23semidirect
114. H(7,𝔽2)D11semidirect
196. C2×#41C22⋊D11semidirect
202. C22×D4C22⋊D11semidirect
277. C11#60401semidirect

Maximal subgroups

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

subgroupindex
#130822562
#130780222
#130780132
#130780082
C2×#106228562
#130770772
C11×H(7,𝔽2)2
#6040111