Finite Group Databasebeta

← #15620524

Subgroups — #15620524

Order 2816.

5128 conjugacy classes of subgroups; only the 807 normal subgroups are shown.

normal subgroups only

This group has 5128 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#13082247C213yesmaximal, complemented
2#13078018C214yesmaximal, complemented
3#13078013C218yesmaximal, complemented
4#13078008C214yesmaximal, complemented
5C2×#10622857C218yesmaximal, complemented
6#13077077C213yesmaximal, complemented
7C11×H(7,𝔽2)C211yesmaximal, characteristic, complemented
8C11×#585—212no—
9C11×#585C2214yescomplemented
10C2×C11×H(5,𝔽2)—212no—
11C2×C11×H(5,𝔽2)C2213yes—
12C2×C11×H(5,𝔽2)C2218yescomplemented
13C22×D4×D11C2211yescharacteristic, complemented
14#10623361C2216yescomplemented
15#10623360—212no—
16#10623359C2216yescomplemented
17#10623335C22112yescomplemented
18D4×C22⋊D11—212no—
19C2×#66990C22112yescomplemented
20#10623329C2213yescomplemented
21#10622857C22132yescomplemented
22#10622857C22116yescomplemented
23#10622856C22116yescomplemented
24#10622853C22112yescomplemented
25C2×#66898C22124yescomplemented
26C24×D11C2312yescomplemented
27C11×2−1+4—216no—
28C11×2−1+4—212no—
29C11×H(5,𝔽2)—248no—
30C11×H(5,𝔽2)—224no—
31C11×H(5,𝔽2)—212no—
32C11×H(5,𝔽2)—248no—
33C11×H(5,𝔽2)C23116yes—
34C2×C11×#41—248no—
35C2×C11×#41—224no—
36C2×C11×#41—224no—
37C2×C11×#41D4112yescomplemented
38C2×C11×#41C2316yes—
39C22×D4×C11—216no—
40C22×D4×C11—224no—
41C22×D4×C11D4112yescomplemented
42C22×D4×C11C23112yes—
43C22×D4×C11C2311yescharacteristic
44C22×C22⋊D11—212no—
45C22×C22⋊D11C2312yescomplemented
46C2×D4×D11—212no—
47C22×D44C2311yescharacteristic, complemented
48C22×C4×D11C2311yescharacteristic, complemented
49#67006—212no—
50C2×C22⋊Dic11C23112yescomplemented
51#67000—26no—
52#66999—26no—
53#66993—26no—
54#66992—224no—
55#66991—26no—
56#66990—224no—
57#66990C23112yescomplemented
58#66990C23124yescomplemented
59#66988C23112yescomplemented
60D4×Dic11—212no—
61#66983—212no—
62#66982—212no—
63C4×C22⋊D11—212no—
64C2×(C2×D11)⋊C4—212no—
65#66979C2316yescomplemented
66#66898C23148yescomplemented
67#60399—111nomaximal, Sylow
68C24×C11—28no—
69C24×C11C2×D4112yescomplemented
70C24×C11C2412yes—
71C23×D11—212no—
72C23×D11—84no—
73C23×D11C2411yescharacteristic
74C23×D11C2412yes—
75C11×#41—296no—
76C11×#41—248no—
77C11×#41—216no—
78C11×#41—248no—
79C11×#41—248no—
80C11×#41—224no—
81C2×Q8×C11—224no—
82C2×Q8×C11—28no—
83C2×Q8×C11C2×D416yescomplemented
84C2×D4×C11—296no—
85C2×D4×C11—224no—
86C2×D4×C11—224no—
87C2×D4×C11—248no—
88C2×D4×C11—248no—
89C2×D4×C11—248no—
90C2×D4×C11C2×D4112yes—
91C2×D4×C11C2×D4124yescomplemented
92C2×D4×C11C2×D416yescomplemented
93C2×D4×C11C24112yes—
94C22×C4×C11—224no—
95C22×C4×C11—216no—
96C22×C4×C11C2×D4112yescomplemented
97C22×C4×C11C2×D4112yescomplemented
98C22×C4×C11C2411yescharacteristic
99C2×C22⋊D11—212no—
100C2×C22⋊D11—212no—
101C2×C22⋊D11—224no—
102C2×C22⋊D11—824no—
103C22×Dic11—212no—
104C22×Dic11C2411yescharacteristic
105D4×D11—416no—
106C2×D44—26no—
107C2×C4×D11—26no—
108C22⋊Dic11—212no—
109C22⋊Dic11—212no—
110C22⋊Dic11C24112yes—
111(C2×D11)⋊C4—212no—
112(C2×D11)⋊C4—224no—
113C4⋊Dic11—26no—
114Dic11⋊C4—212no—
115C4×Dic11—26no—
116H(7,𝔽2)D1111yescharacteristic, complemented
117#3484—113no—
118#3027—114no—
119#3026—118no—
120#3025—114no—
121C2×B(4,𝔽2)—118no—
122#2883—113no—
123C23×C11—216no—
124C23×C11—224no—
125C23×C11—82no—
126C23×C11—812no—
127C23×C11—168no—
128C23×C11C22×D4112yes—
129C23×C11C22×D4112yes—
130C23×C11C2511yescharacteristic
131C22×D11—23no—
132C22×D11—26no—
133C22×D11—42no—
134C22×D11—424no—
135C22×D11—48no—
136C22×D11—82no—
137C22×D11—812no—
138Q8×C11—216no—
139Q8×C11—212no—
140D4×C11—248no—
141D4×C11—248no—
142D4×C11—224no—
143D4×C11—212no—
144D4×C11#120116yes—
145C2×C4×C11—248no—
146C2×C4×C11—224no—
147C2×C4×C11—224no—
148C2×C4×C11C22×D4112yes—
149C2×C4×C11C22×D416yes—
150C22⋊D11—424no—
151C22⋊D11—48no—
152C22⋊D11—824no—
153C2×Dic11—212no—
154C2×Dic11—23no—
155C2×Dic11—812no—
156D44—44no—
157C4×D11—44no—
158#585—212no—
159#585C2×D1114yescomplemented
160C2×H(5,𝔽2)—212no—
161C2×H(5,𝔽2)C2×D1113yes—
162C2×H(5,𝔽2)C2×D1118yescomplemented
163C23×D4—111no—
164#546—2212no—
165D42—2212no—
166#535—116no—
167#534—1112no—
168#525—116no—
169C2×#120—1112no—
170#518—113no—
171#458—1116no—
172B(4,𝔽2)—1116no—
173B(4,𝔽2)—1132no—
174#410—1112no—
175C2×#99—1124no—
176C22×C11—212no—
177C22×C11—41no—
178C22×C11—412no—
179C22×C11—412no—
180C22×C11—824no—
181C22×C11—816no—
182C22×C11C2×#12018yes—
183C22×C11C23×D413yes—
184C2×D11—26no—
185C2×D11—43no—
186C2×D11—41no—
187C2×D11—412no—
188C2×D11—42no—
189C2×D11—816no—
190C4×C11—212no—
191C4×C11C2×#12014yes—
192Dic11—412no—
193Dic11—41no—
194C25—112no—
1952−1+4—216no—
1962−1+4—212no—
197H(5,𝔽2)—248no—
198H(5,𝔽2)—224no—
199H(5,𝔽2)—212no—
200H(5,𝔽2)—248no—
201H(5,𝔽2)C22×D11116yes—
202C2×#41—248no—
203C2×#41—224no—
204C2×#41—224no—
205C2×#41C22⋊D11112yescomplemented
206C2×#41C22×D1116yes—
207C22×D4—216no—
208C22×D4—224no—
209C22×D4—112no—
210C22×D4—111no—
211C22×D4—2212no—
212C22×D4—2212no—
213C22×D4C22⋊D11112yescomplemented
214C22×D4C22×D11112yes—
215C22×D4C22×D1111yescharacteristic
216C23×C4—111no—
217#127—226no—
218#124—226no—
219#123—1112no—
220#123—2212no—
221C22⋊Q8—226no—
222#121—226no—
223#121—2224no—
224#121—2212no—
225#120—1112no—
226#120—1124no—
227#120—2212no—
228#120—2224no—
229C4×D4—2212no—
230C4×D4—2212no—
231#117—116no—
232C2×C22⋊C4—1112no—
233C2×C22⋊C4—2212no—
234#99—1148no—
235C2×C11—23no—
236C2×C11—28no—
237C2×C11—412no—
238C2×C11C22×#12011yescharacteristic
239D11—42no—
240D11—82no—
241C24—28no—
242C24—111no—
243C24—112no—
244C24—2212no—
245C24—884no—
246C24C2×C22⋊D11112yescomplemented
247C24C23×D1112yes—
248#41—296no—
249#41—248no—
250#41—216no—
251#41—248no—
252#41—248no—
253#41—224no—
254C2×Q8—224no—
255C2×Q8—28no—
256C2×Q8C2×C22⋊D1116yescomplemented
257C2×D4—296no—
258C2×D4—224no—
259C2×D4—224no—
260C2×D4—248no—
261C2×D4—248no—
262C2×D4—248no—
263C2×D4—2212no—
264C2×D4—2212no—
265C2×D4—2224no—
266C2×D4—226no—
267C2×D4—4416no—
268C2×D4—8824no—
269C2×D4C2×C22⋊D11112yes—
270C2×D4C2×C22⋊D11124yescomplemented
271C2×D4C2×C22⋊D1116yescomplemented
272C2×D4C23×D11112yes—
273C22×C4—224no—
274C22×C4—216no—
275C22×C4—111no—
276C22×C4—2212no—
277C22×C4—226no—
278C22×C4C2×C22⋊D11112yescomplemented
279C22×C4C2×C22⋊D11112yescomplemented
280C22×C4C23×D1111yescharacteristic
281C4⋊C4—226no—
282C4⋊C4—2212no—
283C22⋊C4—1112no—
284C22⋊C4—2212no—
285C22⋊C4—2212no—
286C22⋊C4—2212no—
287C22⋊C4—2224no—
288C42—226no—
289C11#6039911yescharacteristic, complemented, Sylow
290C23—216no—
291C23—224no—
292C23—82no—
293C23—812no—
294C23—168no—
295C23—223no—
296C23—226no—
297C23—442no—
298C23—4424no—
299C23—448no—
300C23—882no—
301C23—8812no—
302C23C22×C22⋊D11112yes—
303C23C22×C22⋊D11112yes—
304C23C24×D1111yescharacteristic
305Q8—216no—
306Q8—212no—
307D4—248no—
308D4—248no—
309D4—224no—
310D4—212no—
311D4—4424no—
312D4—448no—
313D4—444no—
314D4—8824no—
315D4#67006116yes—
316C2×C4—248no—
317C2×C4—224no—
318C2×C4—224no—
319C2×C4—2212no—
320C2×C4—223no—
321C2×C4—444no—
322C2×C4—8812no—
323C2×C4C22×C22⋊D11112yes—
324C2×C4C22×C22⋊D1116yes—
325C22—212no—
326C22—412no—
327C22—41no—
328C22—412no—
329C22—824no—
330C22—816no—
331C22—226no—
332C22—443no—
333C22—4412no—
334C22—441no—
335C22—442no—
336C22—8816no—
337C22C2×#6700618yes—
338C22C23×C22⋊D1113yes—
339C4—212no—
340C4—4412no—
341C4—441no—
342C4C2×#6700614yes—
343C2—23no—
344C2—28no—
345C2—412no—
346C2—442no—
347C2—882no—
348C2C22×#6700611yescharacteristic
subgroupG/H
centerC2C22×#67006
commutator, cosocleC23×C11C25
FrattiniC23C24×D11
FittingC11×H(7,𝔽2)C2
socleC2×C11C22×#120
hypercenterH(7,𝔽2)D11
hypocenterC11#60399
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. #13082247C2semidirect
2. #13078018C2semidirect
3. #13078013C2semidirect
4. #13078008C2semidirect
5. C2×#10622857C2semidirect
6. #13077077C2semidirect
7. C11×H(7,𝔽2)C2semidirect
9. C11×#585C22semidirect
12. C2×C11×H(5,𝔽2)C22semidirect
13. C22×D4×D11C22semidirect
14. #10623361C22semidirect
16. #10623359C22semidirect
17. #10623335C22semidirect
19. C2×#66990C22semidirect
20. #10623329C22semidirect
21. #10622857C22semidirect
22. #10622857C22semidirect
23. #10622856C22semidirect
24. #10622853C22semidirect
25. C2×#66898C22semidirect
26. C24×D11C23semidirect
37. C2×C11×#41D4semidirect
41. C22×D4×C11D4semidirect
45. C22×C22⋊D11C23semidirect
47. C22×D44C23semidirect
48. C22×C4×D11C23semidirect
50. C2×C22⋊Dic11C23semidirect
57. #66990C23semidirect
58. #66990C23semidirect
59. #66988C23semidirect
65. #66979C23semidirect
66. #66898C23semidirect
69. C24×C11C2×D4semidirect
83. C2×Q8×C11C2×D4semidirect
91. C2×D4×C11C2×D4semidirect
92. C2×D4×C11C2×D4semidirect
96. C22×C4×C11C2×D4semidirect
97. C22×C4×C11C2×D4semidirect
116. H(7,𝔽2)D11semidirect
159. #585C2×D11semidirect
162. C2×H(5,𝔽2)C2×D11semidirect
205. C2×#41C22⋊D11semidirect
213. C22×D4C22⋊D11semidirect
246. C24C2×C22⋊D11semidirect
256. C2×Q8C2×C22⋊D11semidirect
270. C2×D4C2×C22⋊D11semidirect
271. C2×D4C2×C22⋊D11semidirect
278. C22×C4C2×C22⋊D11semidirect
279. C22×C4C2×C22⋊D11semidirect
289. C11#60399semidirect

Maximal subgroups

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

subgroupindex
#130822472
#130780182
#130780132
#130780082
C2×#106228572
#130770772
C11×H(7,𝔽2)2
#6039911