Finite Group Databasebeta

← #16729670

Subgroups — #16729670

Order 3328.

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#13128467C213yesmaximal, complemented
2#13124238C214yesmaximal, complemented
3#13124233C218yesmaximal, complemented
4#13124228C214yesmaximal, complemented
5C2×#11718905C218yesmaximal, complemented
6#13123297C213yesmaximal, complemented
7C13×H(7,𝔽2)C211yesmaximal, characteristic, complemented
8C13×#585—212no—
9C13×#585C2214yescomplemented
10C2×C13×H(5,𝔽2)—212no—
11C2×C13×H(5,𝔽2)C2213yes—
12C2×C13×H(5,𝔽2)C2218yescomplemented
13C22×D4×D13C2211yescharacteristic, complemented
14#11719535C2216yescomplemented
15#11719534—212no—
16#11719533C2216yescomplemented
17#11719509C22112yescomplemented
18D4×C22⋊D13—212no—
19C2×#88191C22112yescomplemented
20#11719503C2213yescomplemented
21#11718905C22116yescomplemented
22#11718905C22132yescomplemented
23#11718904C22116yescomplemented
24#11718901C22112yescomplemented
25C2×#88074C22124yescomplemented
26C24×D13C2312yescomplemented
27C13×2−1+4—216no—
28C13×2−1+4—212no—
29C13×H(5,𝔽2)—248no—
30C13×H(5,𝔽2)—224no—
31C13×H(5,𝔽2)—212no—
32C13×H(5,𝔽2)—248no—
33C13×H(5,𝔽2)C23116yes—
34C2×C13×#41—248no—
35C2×C13×#41—224no—
36C2×C13×#41—224no—
37C2×C13×#41D4112yescomplemented
38C2×C13×#41C2316yes—
39C22×D4×C13—216no—
40C22×D4×C13—224no—
41C22×D4×C13D4112yescomplemented
42C22×D4×C13C23112yes—
43C22×D4×C13C2311yescharacteristic
44C22×C22⋊D13—212no—
45C22×C22⋊D13C2312yescomplemented
46C2×D4×D13—212no—
47C22×D52C2311yescharacteristic, complemented
48C22×C4×D13C2311yescharacteristic, complemented
49#88207—212no—
50C2×C22⋊Dic13C23112yescomplemented
51#88201—26no—
52#88200—26no—
53#88194—26no—
54#88193—224no—
55#88192—26no—
56#88191—224no—
57#88191C23112yescomplemented
58#88191C23124yescomplemented
59#88189C23112yescomplemented
60D4×Dic13—212no—
61#88184—212no—
62#88183—212no—
63C4×C22⋊D13—212no—
64C2×#6103—212no—
65#88180C2316yescomplemented
66#88074C23148yescomplemented
67#60399—131nomaximal, Sylow
68C24×C13—28no—
69C24×C13C2×D4112yescomplemented
70C24×C13C2412yes—
71C23×D13—212no—
72C23×D13—84no—
73C23×D13C2411yescharacteristic
74C23×D13C2412yes—
75C13×#41—296no—
76C13×#41—248no—
77C13×#41—216no—
78C13×#41—248no—
79C13×#41—248no—
80C13×#41—224no—
81C2×Q8×C13—224no—
82C2×Q8×C13—28no—
83C2×Q8×C13C2×D416yescomplemented
84C2×D4×C13—296no—
85C2×D4×C13—224no—
86C2×D4×C13—224no—
87C2×D4×C13—248no—
88C2×D4×C13—248no—
89C2×D4×C13—248no—
90C2×D4×C13C2×D4112yes—
91C2×D4×C13C2×D4124yescomplemented
92C2×D4×C13C2×D416yescomplemented
93C2×D4×C13C24112yes—
94C22×C4×C13—224no—
95C22×C4×C13—216no—
96C22×C4×C13C2×D4112yescomplemented
97C22×C4×C13C2×D4112yescomplemented
98C22×C4×C13C2411yescharacteristic
99C2×C22⋊D13—212no—
100C2×C22⋊D13—212no—
101C2×C22⋊D13—224no—
102C2×C22⋊D13—824no—
103C22×Dic13—212no—
104C22×Dic13C2411yescharacteristic
105D4×D13—416no—
106C2×D52—26no—
107C2×C4×D13—26no—
108C22⋊Dic13—212no—
109C22⋊Dic13—212no—
110C22⋊Dic13C24112yes—
111#6103—212no—
112#6103—224no—
113C4⋊Dic13—26no—
114#6101—212no—
115C4×Dic13—26no—
116H(7,𝔽2)D1311yescharacteristic, complemented
117#3484—133no—
118#3027—134no—
119#3026—138no—
120#3025—134no—
121C2×B(4,𝔽2)—138no—
122#2883—133no—
123C23×C13—216no—
124C23×C13—224no—
125C23×C13—82no—
126C23×C13—812no—
127C23×C13—168no—
128C23×C13C22×D4112yes—
129C23×C13C22×D4112yes—
130C23×C13C2511yescharacteristic
131C22×D13—23no—
132C22×D13—26no—
133C22×D13—42no—
134C22×D13—424no—
135C22×D13—48no—
136C22×D13—82no—
137C22×D13—812no—
138Q8×C13—216no—
139Q8×C13—212no—
140D4×C13—248no—
141D4×C13—248no—
142D4×C13—224no—
143D4×C13—212no—
144D4×C13#120116yes—
145C2×C4×C13—248no—
146C2×C4×C13—224no—
147C2×C4×C13—224no—
148C2×C4×C13C22×D4112yes—
149C2×C4×C13C22×D416yes—
150C22⋊D13—424no—
151C22⋊D13—48no—
152C22⋊D13—824no—
153C2×Dic13—212no—
154C2×Dic13—23no—
155C2×Dic13—812no—
156D52—44no—
157C4×D13—44no—
158#585—212no—
159#585C2×D1314yescomplemented
160C2×H(5,𝔽2)—212no—
161C2×H(5,𝔽2)C2×D1313yes—
162C2×H(5,𝔽2)C2×D1318yescomplemented
163C23×D4—131no—
164#546—2612no—
165D42—2612no—
166#535—136no—
167#534—1312no—
168#525—136no—
169C2×#120—1312no—
170#518—133no—
171#458—1316no—
172B(4,𝔽2)—1316no—
173B(4,𝔽2)—1332no—
174#410—1312no—
175C2×#99—1324no—
176C22×C13—212no—
177C22×C13—412no—
178C22×C13—41no—
179C22×C13—412no—
180C22×C13—824no—
181C22×C13—816no—
182C22×C13C2×#12018yes—
183C22×C13C23×D413yes—
184C2×D13—26no—
185C2×D13—43no—
186C2×D13—41no—
187C2×D13—412no—
188C2×D13—42no—
189C2×D13—816no—
190C4×C13—212no—
191C4×C13C2×#12014yes—
192Dic13—412no—
193Dic13—41no—
194C25—132no—
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×D13116yes—
202C2×#41—248no—
203C2×#41—224no—
204C2×#41—224no—
205C2×#41C22⋊D13112yescomplemented
206C2×#41C22×D1316yes—
207C22×D4—216no—
208C22×D4—224no—
209C22×D4—132no—
210C22×D4—131no—
211C22×D4—2612no—
212C22×D4—2612no—
213C22×D4C22⋊D13112yescomplemented
214C22×D4C22×D13112yes—
215C22×D4C22×D1311yescharacteristic
216C23×C4—131no—
217#127—266no—
218#124—266no—
219#123—1312no—
220#123—2612no—
221C22⋊Q8—266no—
222#121—266no—
223#121—2624no—
224#121—2612no—
225#120—1312no—
226#120—1324no—
227#120—2612no—
228#120—2624no—
229C4×D4—2612no—
230C4×D4—2612no—
231#117—136no—
232C2×C22⋊C4—1312no—
233C2×C22⋊C4—2612no—
234#99—1348no—
235C2×C13—23no—
236C2×C13—28no—
237C2×C13—412no—
238C2×C13C22×#12011yescharacteristic
239D13—42no—
240D13—82no—
241C24—28no—
242C24—131no—
243C24—132no—
244C24—2612no—
245C24—1044no—
246C24C2×C22⋊D13112yescomplemented
247C24C23×D1312yes—
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⋊D1316yescomplemented
257C2×D4—296no—
258C2×D4—224no—
259C2×D4—224no—
260C2×D4—248no—
261C2×D4—248no—
262C2×D4—248no—
263C2×D4—2612no—
264C2×D4—2612no—
265C2×D4—2624no—
266C2×D4—266no—
267C2×D4—5216no—
268C2×D4—10424no—
269C2×D4C2×C22⋊D13112yes—
270C2×D4C2×C22⋊D13124yescomplemented
271C2×D4C2×C22⋊D1316yescomplemented
272C2×D4C23×D13112yes—
273C22×C4—224no—
274C22×C4—216no—
275C22×C4—131no—
276C22×C4—2612no—
277C22×C4—266no—
278C22×C4C2×C22⋊D13112yescomplemented
279C22×C4C2×C22⋊D13112yescomplemented
280C22×C4C23×D1311yescharacteristic
281C4⋊C4—266no—
282C4⋊C4—2612no—
283C22⋊C4—1312no—
284C22⋊C4—2612no—
285C22⋊C4—2612no—
286C22⋊C4—2612no—
287C22⋊C4—2624no—
288C42—266no—
289C13#6039911yescharacteristic, complemented, Sylow
290C23—216no—
291C23—224no—
292C23—82no—
293C23—812no—
294C23—168no—
295C23—263no—
296C23—266no—
297C23—522no—
298C23—5224no—
299C23—528no—
300C23—1042no—
301C23—10412no—
302C23C22×C22⋊D13112yes—
303C23C22×C22⋊D13112yes—
304C23C24×D1311yescharacteristic
305Q8—216no—
306Q8—212no—
307D4—248no—
308D4—248no—
309D4—224no—
310D4—212no—
311D4—5224no—
312D4—528no—
313D4—524no—
314D4—10424no—
315D4#88207116yes—
316C2×C4—248no—
317C2×C4—224no—
318C2×C4—224no—
319C2×C4—2612no—
320C2×C4—263no—
321C2×C4—524no—
322C2×C4—10412no—
323C2×C4C22×C22⋊D13112yes—
324C2×C4C22×C22⋊D1316yes—
325C22—212no—
326C22—412no—
327C22—41no—
328C22—412no—
329C22—824no—
330C22—816no—
331C22—266no—
332C22—523no—
333C22—5212no—
334C22—521no—
335C22—522no—
336C22—10416no—
337C22C2×#8820718yes—
338C22C23×C22⋊D1313yes—
339C4—212no—
340C4—5212no—
341C4—521no—
342C4C2×#8820714yes—
343C2—23no—
344C2—28no—
345C2—412no—
346C2—522no—
347C2—1042no—
348C2C22×#8820711yescharacteristic
subgroupG/H
centerC2C22×#88207
commutator, cosocleC23×C13C25
FrattiniC23C24×D13
FittingC13×H(7,𝔽2)C2
socleC2×C13C22×#120
hypercenterH(7,𝔽2)D13
hypocenterC13#60399
LCSG ▷ C23×C13 ▷ C2×C13 ▷ C13
UCSC1 ◁ C2 ◁ C22×D4 ◁ H(7,𝔽2)

Complements

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

NH ≅ G/Ntype
1. #13128467C2semidirect
2. #13124238C2semidirect
3. #13124233C2semidirect
4. #13124228C2semidirect
5. C2×#11718905C2semidirect
6. #13123297C2semidirect
7. C13×H(7,𝔽2)C2semidirect
9. C13×#585C22semidirect
12. C2×C13×H(5,𝔽2)C22semidirect
13. C22×D4×D13C22semidirect
14. #11719535C22semidirect
16. #11719533C22semidirect
17. #11719509C22semidirect
19. C2×#88191C22semidirect
20. #11719503C22semidirect
21. #11718905C22semidirect
22. #11718905C22semidirect
23. #11718904C22semidirect
24. #11718901C22semidirect
25. C2×#88074C22semidirect
26. C24×D13C23semidirect
37. C2×C13×#41D4semidirect
41. C22×D4×C13D4semidirect
45. C22×C22⋊D13C23semidirect
47. C22×D52C23semidirect
48. C22×C4×D13C23semidirect
50. C2×C22⋊Dic13C23semidirect
57. #88191C23semidirect
58. #88191C23semidirect
59. #88189C23semidirect
65. #88180C23semidirect
66. #88074C23semidirect
69. C24×C13C2×D4semidirect
83. C2×Q8×C13C2×D4semidirect
91. C2×D4×C13C2×D4semidirect
92. C2×D4×C13C2×D4semidirect
96. C22×C4×C13C2×D4semidirect
97. C22×C4×C13C2×D4semidirect
116. H(7,𝔽2)D13semidirect
159. #585C2×D13semidirect
162. C2×H(5,𝔽2)C2×D13semidirect
205. C2×#41C22⋊D13semidirect
213. C22×D4C22⋊D13semidirect
246. C24C2×C22⋊D13semidirect
256. C2×Q8C2×C22⋊D13semidirect
270. C2×D4C2×C22⋊D13semidirect
271. C2×D4C2×C22⋊D13semidirect
278. C22×C4C2×C22⋊D13semidirect
279. C22×C4C2×C22⋊D13semidirect
289. C13#60399semidirect

Maximal subgroups

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

subgroupindex
#131284672
#131242382
#131242332
#131242282
C2×#117189052
#131232972
C13×H(7,𝔽2)2
#6039913