Finite Group Databasebeta

← GL(2,𝔽3)×AGL(1,𝔽13) #72259549

Subgroups — GL(2,𝔽3)×AGL(1,𝔽13) #72259549

Order 7488.

374 conjugacy classes of subgroups; only the 43 normal subgroups are shown.

normal subgroups only

This group has 374 conjugacy classes of subgroups — too many to draw. Only its 43 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
1GL(2,𝔽3)×#261C311yesmaximal, characteristic, complemented
2#72259501C211yesmaximal, characteristic, complemented
3GL(2,𝔽3)×D13⋊C3C211yesmaximal, characteristic
4SD16×AGL(1,𝔽13)—31nomaximal
5SL(2,𝔽3)×AGL(1,𝔽13)C211yesmaximal, characteristic, complemented
6C2×D3×AGL(1,𝔽13)—41nomaximal
7SL(2,𝔽3)×D13⋊C3C2211yescharacteristic
8(C13⋊C3)⋊GL(2,𝔽3)C411yescharacteristic, complemented
9C13⋊C3×GL(2,𝔽3)C411yescharacteristic, complemented
10D13×GL(2,𝔽3)C2×C311yescharacteristic
11#261⋊SL(2,𝔽3)—21no—
12SL(2,𝔽3)×#261C2×C311yescharacteristic, complemented
13Q8×AGL(1,𝔽13)D311yescharacteristic, complemented
14D4×AGL(1,𝔽13)—31no—
15#11929041C2×C311yescharacteristic, complemented
16SD16×D13⋊C3—31no—
17#88117⋊C3—31no—
18#88115⋊C3—31no—
19#88101⋊C3—31no—
20C8×AGL(1,𝔽13)—31no—
21C2×D3×D13⋊C3—41no—
22C2×C3⋊AGL(1,𝔽13)—41no—
23C2×C3×AGL(1,𝔽13)—41no—
24D3×AGL(1,𝔽13)—44no—
25SL(2,𝔽3)×C13⋊C3C2×C411yescharacteristic, complemented
26SD16×#261—31noHall
27C2×D3×#261—41no—
28C22×AGL(1,𝔽13)—61no—
29D13⋊SL(2,𝔽3)—21no—
30SL(2,𝔽3)×D13C22×C311yescharacteristic
31Q8×D13⋊C3C2×D311yescharacteristic
32D4×D13⋊C3—31no—
33C13⋊GL(2,𝔽3)C3×C411yescharacteristic, complemented
34C13×GL(2,𝔽3)C3×C411yescharacteristic, complemented
35(C22⋊#261)⋊C3—61no—
36C4⋊AGL(1,𝔽13)—31no—
37C4⋊AGL(1,𝔽13)—31no—
38C4×AGL(1,𝔽13)—31no—
39(D13⋊C3)⋊C8—31no—
40SD16×C13⋊C3—31no—
41#6106⋊C3—31no—
42#6105⋊C3—31no—
43#6095⋊C3—31no—
44C8×D13⋊C3—31no—
45C3×C4×GL(2,𝔽3)—131nomaximal, Hall
46C2×C3⋊(D13⋊C3)—41no—
47C2×D3×C13⋊C3—41no—
48C2×C3×D13⋊C3—41no—
49D3×D13⋊C3—42no—
50D3×D13⋊C3—42no—
51C3⋊AGL(1,𝔽13)—42no—
52C3×AGL(1,𝔽13)—42no—
53Q8×#261C3×D311yescharacteristic, complemented
54D4×#261—31no—
55SD16×D13—31no—
56#88117—31no—
57#88115—31no—
58#88101—31no—
59C8×#261—31no—
60C2×D3×D13—41no—
61C2×C3⋊#261—41no—
62C2×C3×#261—41no—
63C22×D13⋊C3—61no—
64D3×#261—44no—
65C2×AGL(1,𝔽13)—31no—
66C2×AGL(1,𝔽13)—61no—
67C2×AGL(1,𝔽13)—81no—
68C2×AGL(1,𝔽13)—122no—
69C2×AGL(1,𝔽13)S411yescharacteristic
70Q8⋊(C13⋊C3)—21no—
71C13×SL(2,𝔽3)C2×C3×C411yescharacteristic, complemented
72Q8×C13⋊C3C4×D311yescharacteristic, complemented
73D4×C13⋊C3—31no—
74(C22⋊D13)⋊C3—61no—
75C4⋊(D13⋊C3)—31no—
76C4×D13⋊C3—31no—
77(C13⋊C3)⋊Q8—31no—
78#1057⋊C3—61no—
79C8×C13⋊C3—31no—
80#1055⋊C3—31no—
81C2×C3×GL(2,𝔽3)—131no—
82C3×C4×SL(2,𝔽3)—131no—
83C3×#859—131no—
84C2×C3×C13⋊C3—41no—
85C3⋊(D13⋊C3)—42no—
86D3×C13⋊C3—42no—
87C3×D13⋊C3—41no—
88C3×D13⋊C3—41no—
89C22×#261—61no—
90Q8×D13C2×C3×D311yescharacteristic
91D4×D13—31no—
92C22⋊#261—61no—
93C4⋊#261—31no—
94C4⋊#261—31no—
95C4×#261—31no—
96D13⋊C8—31no—
97C13×SD16—31no—
98#6106—31no—
99#6105—31no—
100#6095—31no—
101C8×D13—31no—
102C4×GL(2,𝔽3)—131no—
103C3×C4×SD16—391no—
104C2×D39—41no—
105C2×D3×C13—41no—
106C2×C3×D13—41no—
107C22×C13⋊C3—61no—
108D3×D13—42no—
109D3×D13—42no—
110C3⋊#261—42no—
111C3×#261—42no—
112C2×D13⋊C3—61no—
113C2×D13⋊C3—81no—
114C2×D13⋊C3—121no—
115C2×D13⋊C3—121no—
116C2×D13⋊C3C2×S411yescharacteristic
117AGL(1,𝔽13)—61no—
118AGL(1,𝔽13)—82no—
119AGL(1,𝔽13)—121no—
120AGL(1,𝔽13)GL(2,𝔽3)12yesdirect product factor
121C4×C13⋊C3—31no—
122Dic13⋊C3—31no—
123C2×C3×C4×D3—521no—
124C2×C3×SL(2,𝔽3)—131no—
125C3×GL(2,𝔽3)—131no—
126C3×GL(2,𝔽3)—131no—
127C3×C13⋊C3—41noHall
128C22×D13—61no—
129C2×#261—31no—
130C2×#261—61no—
131C2×#261—122no—
132C2×#261C3×S411yescharacteristic
133Q8×C13C3×C4×D311yescharacteristic, complemented
134D4×C13—31no—
135C22⋊D13—61no—
136D52—31no—
137C4×D13—31no—
138Dic26—31no—
139#1057—61no—
140C8×C13—31no—
141#1055—31no—
142C2×GL(2,𝔽3)—131no—
143C2×C3×SD16—391no—
144C3×C4×Q8—131no—
145C3×C4×D4—391no—
146C4×SL(2,𝔽3)—131no—
147C4×SL(2,𝔽3)—261no—
148#859—131no—
149C3×#106—391no—
150C3×#103—391no—
151C3×#102—391no—
152C3×C4×C8—391no—
153C2×C3×C13—41no—
154D39—42no—
155C3×D13—41no—
156C3×D13—41no—
157D3×C13—42no—
158C2×C13⋊C3—81no—
159C2×C13⋊C3—121no—
160C2×C13⋊C3C4×S411yescharacteristic
161D13⋊C3—81no—
162D13⋊C3—81no—
163D13⋊C3—121no—
164D13⋊C3C2×GL(2,𝔽3)11yescharacteristic
165D13⋊C3#98511yescharacteristic
166C22×C3×D3—521no—
167C2×C32×C4—521no—
168C2×C3×Dic3—521no—
169C3×C4×D3—524no—
170C3×SL(2,𝔽3)—131no—
171C4×SD16—391noSylow
172C22×C13—61no—
173C2×D13—61no—
174C2×D13—121no—
175C2×D13—121no—
176C2×D13C2×C3×S411yescharacteristic
177#261—61no—
178#261—121no—
179#261C3×GL(2,𝔽3)12yescomplemented
180C4×C13—31no—
181Dic13—31no—
182C2×C3×Q8—131no—
183C2×C3×D4—391no—
184C22×C3×C4—781no—
185C2×C4×D3—521no—
186C2×SL(2,𝔽3)—131no—
187C2×SL(2,𝔽3)—261no—
188GL(2,𝔽3)—131no—
189GL(2,𝔽3)AGL(1,𝔽13)11yescharacteristic, direct product factor
190C3×SD16—391no—
191C3×SD16—391no—
192C3×SD16—391no—
193C3×SD16—391no—
194C2×C3×C8—391no—
195C2×C3×C8—391no—
196C3×C4⋊C4—391no—
197C3×C4⋊C4—391no—
198C3×C22⋊C4—781no—
199C3×C42—391no—
200C3×C13—41no—
201C13⋊C3—81no—
202C13⋊C3C4×GL(2,𝔽3)11yescharacteristic, complemented
203C22×C32—521no—
204C2×C3×D3—521no—
205C2×C3×D3—521no—
206C2×C3×D3—522no—
207C2×C3×D3—522no—
208C32×C4—522no—
209C3×Dic3—522no—
210C2×SD16—391no—
211C4×Q8—131no—
212C4×D4—391no—
213#106—391no—
214#103—391no—
215#102—391no—
216C4×C8—391no—
217C2×C13—121no—
218C2×C13C3×C4×S411yescharacteristic
219D13—121no—
220D13C2×C3×GL(2,𝔽3)11yescharacteristic
221D13C3×#98511yescharacteristic
222C23×C3—781no—
223C22×D3—521no—
224C3×Q8—131no—
225C3×Q8—391no—
226C3×D4—391no—
227C3×D4—391no—
228C3×D4—781no—
229C2×C3×C4—131no—
230C2×C3×C4—391no—
231C2×C3×C4—391no—
232C2×C3×C4—521no—
233C2×C3×C4—781no—
234C2×C3×C4—1041no—
235C2×C3×C4—1562no—
236C2×Dic3—521no—
237C4×D3—524no—
238SL(2,𝔽3)—261no—
239SL(2,𝔽3)C2×AGL(1,𝔽13)11yescharacteristic, complemented
240C3×C8—391no—
241C3×C8—391no—
242C3×C8—781no—
243C2×C32—521no—
244C2×C32—521no—
245C2×C32—521no—
246C3×D3—522no—
247C3×D3—522no—
248C2×Q8—131no—
249C2×D4—391no—
250C22×C4—781no—
251SD16—31no—
252SD16—391no—
253SD16—391no—
254SD16—391no—
255C2×C8—391no—
256C2×C8—391no—
257C4⋊C4—391no—
258C4⋊C4—391no—
259C22⋊C4—781no—
260C42—391no—
261C13C3×C4×GL(2,𝔽3)11yescharacteristic, complemented, Sylow
262C22×C3—131no—
263C22×C3—521no—
264C22×C3—781no—
265C22×C3—781no—
266C22×C3—1041no—
267C22×C3—1561no—
268C22×C3—1561no—
269C2×D3—41no—
270C2×D3—521no—
271C2×D3—522no—
272C2×D3—522no—
273C3×C4—132no—
274C3×C4—391no—
275C3×C4—391no—
276C3×C4—522no—
277C3×C4—781no—
278C3×C4—1042no—
279C3×C4—1561no—
280Dic3—522no—
281C32—521noSylow
282C23—781no—
283Q8—391no—
284Q8D3×AGL(1,𝔽13)11yescharacteristic, complemented
285D4—31no—
286D4—391no—
287D4—781no—
288C2×C4—131no—
289C2×C4—391no—
290C2×C4—391no—
291C2×C4—781no—
292C2×C4—1562no—
293C8—31no—
294C8—391no—
295C8—781no—
296C2×C3—41no—
297C2×C3—131no—
298C2×C3—131no—
299C2×C3—131no—
300C2×C3—521no—
301C2×C3—521no—
302C2×C3—1041no—
303C2×C3—1041no—
304C2×C3—1041no—
305C2×C3—1561no—
306C2×C3—1561no—
307D3—42no—
308D3—522no—
309C22—61no—
310C22—131no—
311C22—781no—
312C22—1561no—
313C22—1561no—
314C4—31no—
315C4—132no—
316C4—391no—
317C4—781no—
318C4—1561no—
319C3—41no—
320C3—131no—
321C3—1041no—
322C2—121no—
323C2—131no—
324C2—131no—
325C2—1561no—
326C2S4×AGL(1,𝔽13)11yescharacteristic
subgroupG/H
center, Frattini, hypercenterC2S4×AGL(1,𝔽13)
commutator, hypocenterC13×SL(2,𝔽3)C2×C3×C4
FittingQ8×C13C3×C4×D3
socleC2×C13C3×C4×S4
prime radicalGL(2,𝔽3)×D13⋊C3C2
cosocleSL(2,𝔽3)×D13C22×C3
LCSG ▷ C13×SL(2,𝔽3)
UCSC1 ◁ C2

Complements

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

NH ≅ G/Ntype
1. GL(2,𝔽3)×#261C3semidirect
2. #72259501C2semidirect
5. SL(2,𝔽3)×AGL(1,𝔽13)C2semidirect
8. (C13⋊C3)⋊GL(2,𝔽3)C4semidirect
9. C13⋊C3×GL(2,𝔽3)C4semidirect
12. SL(2,𝔽3)×#261C2×C3semidirect
13. Q8×AGL(1,𝔽13)D3semidirect
15. #11929041C2×C3semidirect
25. SL(2,𝔽3)×C13⋊C3C2×C4semidirect
33. C13⋊GL(2,𝔽3)C3×C4semidirect
34. C13×GL(2,𝔽3)C3×C4semidirect
53. Q8×#261C3×D3semidirect
71. C13×SL(2,𝔽3)C2×C3×C4semidirect
72. Q8×C13⋊C3C4×D3semidirect
120. AGL(1,𝔽13)GL(2,𝔽3)direct (N × H)
133. Q8×C13C3×C4×D3semidirect
179. #261C3×GL(2,𝔽3)semidirect
189. GL(2,𝔽3)AGL(1,𝔽13)direct (N × H)
202. C13⋊C3C4×GL(2,𝔽3)semidirect
239. SL(2,𝔽3)C2×AGL(1,𝔽13)semidirect
261. C13C3×C4×GL(2,𝔽3)semidirect
284. Q8D3×AGL(1,𝔽13)semidirect

Maximal subgroups

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

subgroupindex
GL(2,𝔽3)×#2613
#722595012
GL(2,𝔽3)×D13⋊C32
SD16×AGL(1,𝔽13)3
SL(2,𝔽3)×AGL(1,𝔽13)2
C2×D3×AGL(1,𝔽13)4
C3×C4×GL(2,𝔽3)13