Finite Group Databasebeta

← #16729682

Subgroups — #16729682

Order 3328.

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#13128476C213yesmaximal, complemented
2#13124242C214yesmaximal, complemented
3#13124233C218yesmaximal, complemented
4#13124228C214yesmaximal, complemented
5C2×#11718904C218yesmaximal, complemented
6#13123297C213yesmaximal, complemented
7C13×H(7,𝔽2)C211yesmaximal, characteristic, complemented
8C13×#585—212no—
9C13×#585C2214yes—
10C2×C13×H(5,𝔽2)—212no—
11C2×C13×H(5,𝔽2)C2218yes—
12C2×C13×H(5,𝔽2)C2213yes—
13C22×#6129C2211yescharacteristic, complemented
14#11719533C2216yescomplemented
15#11719530C2216yescomplemented
16#11719529—212no—
17#11719511—212no—
18#11719509C22112yescomplemented
19#11719503C2213yescomplemented
20C2×#88189C22112yescomplemented
21#11718905C22116yescomplemented
22#11718904C22116yescomplemented
23#11718904C22132yescomplemented
24#11718901C22112yescomplemented
25C2×#88074C22124yescomplemented
26C13×2−1+4—216no—
27C13×2−1+4—212no—
28C13×H(5,𝔽2)—248no—
29C13×H(5,𝔽2)—212no—
30C13×H(5,𝔽2)—248no—
31C13×H(5,𝔽2)—224no—
32C13×H(5,𝔽2)C23116yes—
33C2×C13×#41—248no—
34C2×C13×#41—224no—
35C2×C13×#41—224no—
36C2×C13×#41D4112yescomplemented
37C2×C13×#41C2316yes—
38C22×D4×C13—216no—
39C22×D4×C13—224no—
40C22×D4×C13D4112yescomplemented
41C22×D4×C13C23112yes—
42C22×D4×C13C2311yescharacteristic
43C22×C22⋊D13C2312yescomplemented
44C23×Dic13C2312yescomplemented
45C2×#6129—212no—
46C22×C4×D13C2311yescharacteristic, complemented
47C22×Dic26C2311yescharacteristic, complemented
48#88207—212no—
49C2×C22⋊Dic13—212no—
50C2×C22⋊Dic13C23112yescomplemented
51#88200—26no—
52#88198—26no—
53#88193—224no—
54#88192—26no—
55#88191C23112yescomplemented
56#88190—26no—
57#88189—224no—
58#88189C23112yescomplemented
59#88189C23124yescomplemented
60D4×Dic13—212no—
61#88183—212no—
62C4×C22⋊D13—212no—
63#88180C2316yescomplemented
64#88178—212no—
65C2×#6101—212no—
66#88074C23148yescomplemented
67#60401—131nomaximal, Sylow
68C24×C13—28no—
69C24×C13C2×D4112yes—
70C24×C13C2412yes—
71C23×D13C2411yescharacteristic
72C13×#41—296no—
73C13×#41—248no—
74C13×#41—216no—
75C13×#41—248no—
76C13×#41—248no—
77C13×#41—224no—
78C2×Q8×C13—224no—
79C2×Q8×C13—28no—
80C2×Q8×C13C2×D416yes—
81C2×D4×C13—296no—
82C2×D4×C13—224no—
83C2×D4×C13—224no—
84C2×D4×C13—248no—
85C2×D4×C13—248no—
86C2×D4×C13—248no—
87C2×D4×C13C2×D4124yes—
88C2×D4×C13C2×D416yes—
89C2×D4×C13C2×D4112yes—
90C2×D4×C13C24112yes—
91C22×C4×C13—224no—
92C22×C4×C13—216no—
93C22×C4×C13C2×D4112yes—
94C22×C4×C13C2×D4112yes—
95C22×C4×C13C2411yescharacteristic
96C2×C22⋊D13—212no—
97C2×C22⋊D13—212no—
98C22×Dic13—212no—
99C22×Dic13—212no—
100C22×Dic13C2412yes—
101C22×Dic13C2411yescharacteristic
102#6129—416no—
103C2×C4×D13—26no—
104C2×Dic26—26no—
105C22⋊Dic13—224no—
106C22⋊Dic13—212no—
107C22⋊Dic13—212no—
108C22⋊Dic13C24112yes—
109#6103—212no—
110C4⋊Dic13—26no—
111#6101—212no—
112#6101—224no—
113C4×Dic13—26no—
114H(7,𝔽2)D1311yescharacteristic, complemented
115#3485—133no—
116#3028—134no—
117#3026—138no—
118#3025—134no—
119C2×#458—138no—
120#2883—133no—
121C23×C13—216no—
122C23×C13—224no—
123C23×C13—82no—
124C23×C13—812no—
125C23×C13—168no—
126C23×C13C22×D4112yes—
127C23×C13C22×D4112yes—
128C23×C13C2511yescharacteristic
129C22×D13—23no—
130C22×D13—81no—
131Q8×C13—216no—
132Q8×C13—212no—
133D4×C13—248no—
134D4×C13—248no—
135D4×C13—224no—
136D4×C13—212no—
137D4×C13#120116yes—
138C2×C4×C13—248no—
139C2×C4×C13—224no—
140C2×C4×C13—224no—
141C2×C4×C13C22×D4112yes—
142C2×C4×C13C22×D416yes—
143C22⋊D13—48no—
144C22⋊D13—812no—
145C2×Dic13—212no—
146C2×Dic13—26no—
147C2×Dic13—23no—
148C2×Dic13—48no—
149C2×Dic13—812no—
150C4×D13—44no—
151Dic26—44no—
152#585—212no—
153#585C2×D1314yes—
154C2×H(5,𝔽2)—212no—
155C2×H(5,𝔽2)C2×D1318yes—
156C2×H(5,𝔽2)C2×D1313yes—
157C22×#41—131no—
158#547—2612no—
159#546—2612no—
160#536—136no—
161#534—1312no—
162#525—136no—
163C2×#123—1312no—
164#518—133no—
165#458—1316no—
166#458—1332no—
167B(4,𝔽2)—1316no—
168#410—1312no—
169C2×#99—1324no—
170C22×C13—212no—
171C22×C13—412no—
172C22×C13—41no—
173C22×C13—412no—
174C22×C13—824no—
175C22×C13—816no—
176C22×C13C2×#12018yes—
177C22×C13C23×D413yes—
178C2×D13—46no—
179C2×D13—41no—
180C4×C13—212no—
181C4×C13C2×#12014yes—
182Dic13—412no—
183Dic13—42no—
184Dic13—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×D13116yes—
192C2×#41—248no—
193C2×#41—224no—
194C2×#41—224no—
195C2×#41—2612no—
196C2×#41C22⋊D13112yescomplemented
197C2×#41C22×D1316yes—
198C22×Q8—131no—
199C22×D4—216no—
200C22×D4—224no—
201C22×D4—132no—
202C22×D4C22⋊D13112yescomplemented
203C22×D4C22×D13112yes—
204C22×D4C22×D1311yescharacteristic
205C23×C4—132no—
206C23×C4—131no—
207C4⋊Q8—266no—
208#124—266no—
209#123—1312no—
210#123—1324no—
211#123—2624no—
212#123—2612no—
213C22⋊Q8—266no—
214C22⋊Q8—2612no—
215#121—266no—
216#121—2624no—
217#120—1312no—
218#120—2612no—
219C4×D4—2612no—
220C4×D4—2612no—
221#117—136no—
222C2×C4⋊C4—2612no—
223C2×C22⋊C4—1312no—
224C2×C22⋊C4—2612no—
225#99—1348no—
226C2×C13—23no—
227C2×C13—28no—
228C2×C13—412no—
229C2×C13C22×#12011yescharacteristic
230D13—81no—
231C24—28no—
232C24—131no—
233C24C2×C22⋊D13112yes—
234C24C23×D1312yes—
235#41—296no—
236#41—248no—
237#41—216no—
238#41—248no—
239#41—248no—
240#41—224no—
241#41—5216no—
242C2×Q8—224no—
243C2×Q8—28no—
244C2×Q8—266no—
245C2×Q8C2×C22⋊D1316yes—
246C2×D4—296no—
247C2×D4—224no—
248C2×D4—224no—
249C2×D4—248no—
250C2×D4—248no—
251C2×D4—248no—
252C2×D4—2612no—
253C2×D4—2612no—
254C2×D4C2×C22⋊D13124yes—
255C2×D4C2×C22⋊D1316yes—
256C2×D4C2×C22⋊D13112yes—
257C2×D4C23×D13112yes—
258C22×C4—224no—
259C22×C4—216no—
260C22×C4—131no—
261C22×C4—132no—
262C22×C4—2612no—
263C22×C4—2612no—
264C22×C4—266no—
265C22×C4C2×C22⋊D13112yes—
266C22×C4C2×C22⋊D13112yes—
267C22×C4C23×D1311yescharacteristic
268C4⋊C4—2612no—
269C4⋊C4—2624no—
270C4⋊C4—266no—
271C22⋊C4—1312no—
272C22⋊C4—2612no—
273C22⋊C4—2612no—
274C22⋊C4—2624no—
275C22⋊C4—2612no—
276C42—266no—
277C13#6040111yescharacteristic, complemented, Sylow
278C23—216no—
279C23—224no—
280C23—82no—
281C23—812no—
282C23—168no—
283C23—263no—
284C23—1041no—
285C23C22×C22⋊D13112yes—
286C23C22×C22⋊D13112yes—
287C23C24×D1311yescharacteristic
288Q8—216no—
289Q8—212no—
290Q8—524no—
291D4—248no—
292D4—248no—
293D4—224no—
294D4—212no—
295D4—528no—
296D4—10412no—
297D4#88207116yes—
298C2×C4—248no—
299C2×C4—224no—
300C2×C4—224no—
301C2×C4—2612no—
302C2×C4—263no—
303C2×C4—266no—
304C2×C4—528no—
305C2×C4—524no—
306C2×C4—10412no—
307C2×C4C22×C22⋊D13112yes—
308C2×C4C22×C22⋊D1316yes—
309C22—212no—
310C22—412no—
311C22—41no—
312C22—412no—
313C22—824no—
314C22—816no—
315C22—526no—
316C22—521no—
317C22C2×#8820718yes—
318C22C23×C22⋊D1313yes—
319C4—212no—
320C4—5212no—
321C4—521no—
322C4—522no—
323C4C2×#8820714yes—
324C2—23no—
325C2—28no—
326C2—412no—
327C2—1041no—
328C2C22×#8820711yescharacteristic
subgroupG/H
centerC2C22×#88207
commutator, cosocleC23×C13C25
FrattiniC23C24×D13
FittingC13×H(7,𝔽2)C2
socleC2×C13C22×#120
hypercenterH(7,𝔽2)D13
hypocenterC13#60401
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. #13128476C2semidirect
2. #13124242C2semidirect
3. #13124233C2semidirect
4. #13124228C2semidirect
5. C2×#11718904C2semidirect
6. #13123297C2semidirect
7. C13×H(7,𝔽2)C2semidirect
13. C22×#6129C22semidirect
14. #11719533C22semidirect
15. #11719530C22semidirect
18. #11719509C22semidirect
19. #11719503C22semidirect
20. C2×#88189C22semidirect
21. #11718905C22semidirect
22. #11718904C22semidirect
23. #11718904C22semidirect
24. #11718901C22semidirect
25. C2×#88074C22semidirect
36. C2×C13×#41D4semidirect
40. C22×D4×C13D4semidirect
43. C22×C22⋊D13C23semidirect
44. C23×Dic13C23semidirect
46. C22×C4×D13C23semidirect
47. C22×Dic26C23semidirect
50. C2×C22⋊Dic13C23semidirect
55. #88191C23semidirect
58. #88189C23semidirect
59. #88189C23semidirect
63. #88180C23semidirect
66. #88074C23semidirect
114. H(7,𝔽2)D13semidirect
196. C2×#41C22⋊D13semidirect
202. C22×D4C22⋊D13semidirect
277. C13#60401semidirect

Maximal subgroups

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

subgroupindex
#131284762
#131242422
#131242332
#131242282
C2×#117189042
#131232972
C13×H(7,𝔽2)2
#6040113