Finite Group Databasebeta

← D3×#6763 #13088720

Subgroups — D3×#6763 #13088720

Order 1440.

562 conjugacy classes of subgroups; only the 106 normal subgroups are shown.

normal subgroups only

This group has 562 conjugacy classes of subgroups — too many to draw. Only its 106 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
1C22×C3×D3×D5C211yesmaximal, characteristic
2C2×D3×ΓL(1,𝔽16)C212yesmaximal, complemented
3#10624098C211yesmaximal, characteristic, complemented
4C3×#6763C211yesmaximal, characteristic, complemented
5(C2×D3)⋊ΓL(1,𝔽16)C212yesmaximal, complemented
6C2×#6763—31nomaximal
7D3×C22⋊AGL(1,𝔽5)—31nomaximal
8C22×C3×D15C411yescharacteristic, complemented
9C22×C3×C5×D3C411yescharacteristic, complemented
10C22×C32×D5C2211yescharacteristic
11C2×C3×D3×D5—24no—
12C2×C3×D3×D5C411yescharacteristic, complemented
13C2×C3×D3×D5C411yescharacteristic, complemented
14C2×C3×D3×D5C2211yescharacteristic
15C2×C3×D3×D5C2211yescharacteristic
16C2×C3⋊ΓL(1,𝔽16)C2212yescomplemented
17C2×C3×ΓL(1,𝔽16)C2212yescomplemented
18D3×ΓL(1,𝔽16)—24no—
19D3×#217—51nomaximal, Hall
20C23×C3×D5—31no—
21C22×D3×D5D311yescharacteristic
22C22×ΓL(1,𝔽16)—32no—
23C2×D3×AGL(1,𝔽5)—32no—
24#6763—31no—
25#6763—31no—
26#6763—32no—
27#6763—61no—
28#6763D311yescharacteristic, direct product factor
29C3×C22⋊AGL(1,𝔽5)—31no—
30(C2×D3)⋊AGL(1,𝔽5)—32no—
31C22×C32×C5C2×C411yescharacteristic, complemented
32C2×C3×D15—22no—
33C2×C3×D15C2×C411yescharacteristic
34C2×C3×D15C2×C411yescharacteristic, complemented
35C2×C3×C5×D3—22no—
36C2×C3×C5×D3C2×C411yescharacteristic
37C2×C3×C5×D3C2×C411yescharacteristic, complemented
38C2×C32×D5—22no—
39C2×C32×D5C2×C411yescharacteristic, complemented
40C2×C32×D5C2311yescharacteristic
41C3×D3×D5—22no—
42C3×D3×D5—22no—
43C3×D3×D5—22no—
44C3×D3×D5D414yes—
45C3⋊ΓL(1,𝔽16)—22no—
46C3×ΓL(1,𝔽16)—22no—
47C2×C22⋊AGL(1,𝔽5)—91noHall
48C23×C3×D3—51no—
49C2×D3×Dic3—52no—
50#3757—51no—
51C3×#217—51no—
52(C2×D3)⋊Dic3—52no—
53C23×C3×C5—31no—
54C22×D15Dic311yescharacteristic, complemented
55C22×C5×D3Dic311yescharacteristic, complemented
56C22×C3×D5—21no—
57C22×C3×D5—31no—
58C22×C3×D5—31no—
59C22×C3×D5—31no—
60C22×C3×D5—31no—
61C22×C3×D5—31no—
62C22×C3×D5—64no—
63C22×C3×D5C2×D311yescharacteristic
64C22×C3×D5C2×D311yescharacteristic
65C2×D3×D5—24no—
66C2×D3×D5Dic311yescharacteristic, complemented
67C2×D3×D5Dic311yescharacteristic, complemented
68C2×D3×D5C2×D311yescharacteristic
69C2×D3×D5C2×D311yescharacteristic
70C2×ΓL(1,𝔽16)—32no—
71C2×ΓL(1,𝔽16)—32no—
72C2×ΓL(1,𝔽16)—64no—
73C2×ΓL(1,𝔽16)—62no—
74C2×ΓL(1,𝔽16)C2×D312yescomplemented
75C2×C3×AGL(1,𝔽5)—32no—
76D3×AGL(1,𝔽5)—64no—
77C2×#217—151no—
78D3×C22⋊C4—151no—
79C2×C32×C5—21no—
80C2×C32×C5C22×C411yescharacteristic
81C3×D15—21no—
82C3×D15C22⋊C412yescomplemented
83C3×C5×D3—21no—
84C3×C5×D3C22⋊C412yescomplemented
85C32×D5—21no—
86C32×D5C2×D412yes—
87C23×D5—31no—
88C22×AGL(1,𝔽5)—92no—
89C22⋊AGL(1,𝔽5)—31no—
90C22⋊AGL(1,𝔽5)—92no—
91C22⋊AGL(1,𝔽5)—91no—
92C23×C32—51no—
93C22×C3×D3—51no—
94C22×C3×D3—51no—
95C22×C3×D3—51no—
96C22×C3×D3—51no—
97C22×C3×D3—51no—
98C22×C3×D3—104no—
99C22×C3×D3AGL(1,𝔽5)11yescharacteristic, complemented
100C2×C3⋊Dic3—52no—
101C2×C3×Dic3—52no—
102D3×Dic3—104no—
103C22×C3×C5—21no—
104C22×C3×C5—31no—
105C22×C3×C5—31no—
106C22×C3×C5—62no—
107C22×C3×C5C4×D311yescharacteristic, complemented
108C22×C3×C5C2×Dic311yescharacteristic, complemented
109C2×D15—22no—
110C2×D15C2×Dic311yescharacteristic
111C2×D15C2×Dic311yescharacteristic, complemented
112C2×C5×D3—22no—
113C2×C5×D3C2×Dic311yescharacteristic
114C2×C5×D3C2×Dic311yescharacteristic, complemented
115C2×C3×D5—22no—
116C2×C3×D5—22no—
117C2×C3×D5—21no—
118C2×C3×D5—21no—
119C2×C3×D5—31no—
120C2×C3×D5—34no—
121C2×C3×D5—31no—
122C2×C3×D5—42no—
123C2×C3×D5—62no—
124C2×C3×D5—62no—
125C2×C3×D5—62no—
126C2×C3×D5—62no—
127C2×C3×D5C4×D311yescharacteristic, complemented
128C2×C3×D5C2×Dic311yescharacteristic, complemented
129C2×C3×D5C22×D311yescharacteristic
130C2×C3×D5C22×D311yescharacteristic
131D3×D5—22no—
132D3×D5—22no—
133D3×D5—22no—
134D3×D5#6714yes—
135ΓL(1,𝔽16)—22no—
136ΓL(1,𝔽16)—62no—
137ΓL(1,𝔽16)—62no—
138ΓL(1,𝔽16)—122no—
139C3×AGL(1,𝔽5)—62no—
140C24×C3—151no—
141C23×D3—51no—
142C22×Dic3—152no—
143C2×C4×D3—152no—
144C3×C22⋊C4—151no—
145#217—51no—
146#217—152no—
147#217—151no—
148#217—151no—
149#217—301no—
150(C2×D3)⋊C4—152no—
151C32×C5C2×C22⋊C411yescharacteristic, complemented, Hall
152C23×C5—31no—
153C22×D5—31no—
154C22×D5—31no—
155C22×D5—31no—
156C22×D5—31no—
157C22×D5—31no—
158C22×D5—64no—
159C22×D5D3211yescharacteristic
160C2×AGL(1,𝔽5)—32no—
161C2×AGL(1,𝔽5)—92no—
162C2×AGL(1,𝔽5)—184no—
163C22×C32—51no—
164C22×C32—51no—
165C22×C32—102no—
166C22×C32C2×AGL(1,𝔽5)11yescharacteristic, complemented
167C2×C3×D3—22no—
168C2×C3×D3—51no—
169C2×C3×D3—54no—
170C2×C3×D3—51no—
171C2×C3×D3—102no—
172C2×C3×D3—102no—
173C2×C3×D3—102no—
174C2×C3×D3—102no—
175C2×C3×D3C2×AGL(1,𝔽5)11yescharacteristic
176C2×C3×D3C2×AGL(1,𝔽5)11yescharacteristic, complemented
177C3⋊Dic3—102no—
178C3×Dic3—102no—
179C2×C22⋊C4—451noSylow
180C2×C3×C5—21no—
181C2×C3×C5—21no—
182C2×C3×C5—21no—
183C2×C3×C5—32no—
184C2×C3×C5—41no—
185C2×C3×C5—61no—
186C2×C3×C5C2×C4×D311yescharacteristic
187C2×C3×C5C22×Dic311yescharacteristic
188D15—21no—
189D15#21712yescomplemented
190C3×D5—21no—
191C3×D5—21no—
192C3×D5—22no—
193C3×D5—32no—
194C3×D5—41no—
195C3×D5—61no—
196C3×D5D3×D412yes—
197C3×D5C2×#6712yes—
198C5×D3—21no—
199C5×D3#21712yescomplemented
200C23×C3—31no—
201C23×C3—51no—
202C23×C3—51no—
203C23×C3—101no—
204C23×C3—151no—
205C23×C3—151no—
206C23×C3—151no—
207C23×C3—151no—
208C23×C3—151no—
209C23×C3—304no—
210C22×D3—51no—
211C22×D3—51no—
212C22×D3—51no—
213C22×D3—51no—
214C22×D3—51no—
215C22×D3—104no—
216C22×D3ΓL(1,𝔽16)11yescharacteristic, complemented
217C2×C3×C4—152no—
218C2×Dic3—52no—
219C2×Dic3—152no—
220C2×Dic3—152no—
221C2×Dic3—302no—
222C2×Dic3—304no—
223C4×D3—304no—
224C22×C5—31no—
225C22×C5—31no—
226C22×C5—62no—
227C22×C5D3×Dic311yescharacteristic, complemented
228C2×D5—22no—
229C2×D5—31no—
230C2×D5—34no—
231C2×D5—31no—
232C2×D5—62no—
233C2×D5—62no—
234C2×D5—62no—
235C2×D5—62no—
236C2×D5D3×Dic311yescharacteristic, complemented
237C2×D5C2×D3211yescharacteristic
238AGL(1,𝔽5)—62no—
239AGL(1,𝔽5)—182no—
240C2×C32—21no—
241C2×C32—52no—
242C2×C32—101no—
243C2×C32C22×AGL(1,𝔽5)11yescharacteristic
244C3×D3—21no—
245C3×D3—52no—
246C3×D3—101no—
247C3×D3C22⋊AGL(1,𝔽5)12yescomplemented
248C24—151no—
249C22×C4—452no—
250C22⋊C4—151no—
251C22⋊C4—452no—
252C22⋊C4—451no—
253C3×C5—21no—
254C3×C5D3×C22⋊C411yescharacteristic, complemented
255C3×C5C2×#21711yescharacteristic, complemented
256C22×C3—21no—
257C22×C3—31no—
258C22×C3—31no—
259C22×C3—51no—
260C22×C3—51no—
261C22×C3—51no—
262C22×C3—51no—
263C22×C3—62no—
264C22×C3—101no—
265C22×C3—101no—
266C22×C3—102no—
267C22×C3—102no—
268C22×C3—154no—
269C22×C3—151no—
270C22×C3—151no—
271C22×C3—202no—
272C22×C3—302no—
273C22×C3—302no—
274C22×C3—302no—
275C22×C3—302no—
276C22×C3D3×AGL(1,𝔽5)11yescharacteristic, complemented
277C22×C3C2×ΓL(1,𝔽16)11yescharacteristic, complemented
278C2×D3—22no—
279C2×D3—51no—
280C2×D3—54no—
281C2×D3—51no—
282C2×D3—102no—
283C2×D3—102no—
284C2×D3—102no—
285C2×D3—102no—
286C2×D3C2×ΓL(1,𝔽16)11yescharacteristic
287C2×D3C2×ΓL(1,𝔽16)11yescharacteristic, complemented
288C3×C4—302no—
289Dic3—102no—
290Dic3—302no—
291Dic3—302no—
292Dic3—602no—
293C2×C5—21no—
294C2×C5—32no—
295C2×C5—61no—
296C2×C5C2×D3×Dic311yescharacteristic
297D5—21no—
298D5—32no—
299D5—61no—
300D5D3×#6712yes—
301C32C2×C22⋊AGL(1,𝔽5)11yescharacteristic, complemented, Sylow
302C23—31no—
303C23—51no—
304C23—151no—
305C23—151no—
306C23—151no—
307C23—151no—
308C23—151no—
309C23—304no—
310C2×C4—152no—
311C2×C4—452no—
312C2×C4—904no—
313C2×C3—21no—
314C2×C3—21no—
315C2×C3—21no—
316C2×C3—32no—
317C2×C3—41no—
318C2×C3—52no—
319C2×C3—52no—
320C2×C3—61no—
321C2×C3—102no—
322C2×C3—101no—
323C2×C3—101no—
324C2×C3—152no—
325C2×C3—201no—
326C2×C3—301no—
327C2×C3C2×D3×AGL(1,𝔽5)11yescharacteristic
328C2×C3C22×ΓL(1,𝔽16)11yescharacteristic
329D3—21no—
330D3—52no—
331D3—101no—
332D3#676312yesdirect product factor
333C5D3×#21711yescharacteristic, complemented, Sylow
334C22—31no—
335C22—31no—
336C22—51no—
337C22—51no—
338C22—62no—
339C22—102no—
340C22—154no—
341C22—151no—
342C22—151no—
343C22—302no—
344C22—302no—
345C22—302no—
346C22—302no—
347C22D3×ΓL(1,𝔽16)11yescharacteristic, complemented
348C4—302no—
349C4—902no—
350C3—21no—
351C3D3×C22⋊AGL(1,𝔽5)11yescharacteristic, complemented
352C3C2×#676311yescharacteristic, complemented
353C2—21no—
354C2—32no—
355C2—52no—
356C2—61no—
357C2—101no—
358C2—152no—
359C2—301no—
360C2C2×D3×ΓL(1,𝔽16)11yescharacteristic
subgroupG/H
center, FrattiniC2C2×D3×ΓL(1,𝔽16)
commutator, socleC2×C32×C5C22×C4
FittingC22×C32×C5C2×C4
prime radicalC22×C3×D3×D5C2
cosocleC2×C32×D5C23
hypercenterC22D3×ΓL(1,𝔽16)
hypocenterC32×C5C2×C22⋊C4
LCSG ▷ C2×C32×C5 ▷ C32×C5
UCSC1 ◁ C2 ◁ C22

Complements

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

NH ≅ G/Ntype
2. C2×D3×ΓL(1,𝔽16)C2semidirect
3. #10624098C2semidirect
4. C3×#6763C2semidirect
5. (C2×D3)⋊ΓL(1,𝔽16)C2semidirect
8. C22×C3×D15C4semidirect
9. C22×C3×C5×D3C4semidirect
12. C2×C3×D3×D5C4semidirect
13. C2×C3×D3×D5C4semidirect
16. C2×C3⋊ΓL(1,𝔽16)C22semidirect
17. C2×C3×ΓL(1,𝔽16)C22semidirect
28. #6763D3direct (N × H)
31. C22×C32×C5C2×C4semidirect
34. C2×C3×D15C2×C4semidirect
37. C2×C3×C5×D3C2×C4semidirect
39. C2×C32×D5C2×C4semidirect
54. C22×D15Dic3semidirect
55. C22×C5×D3Dic3semidirect
66. C2×D3×D5Dic3semidirect
67. C2×D3×D5Dic3semidirect
74. C2×ΓL(1,𝔽16)C2×D3semidirect
82. C3×D15C22⋊C4semidirect
84. C3×C5×D3C22⋊C4semidirect
99. C22×C3×D3AGL(1,𝔽5)semidirect
107. C22×C3×C5C4×D3semidirect
108. C22×C3×C5C2×Dic3semidirect
111. C2×D15C2×Dic3semidirect
114. C2×C5×D3C2×Dic3semidirect
127. C2×C3×D5C4×D3semidirect
128. C2×C3×D5C2×Dic3semidirect
151. C32×C5C2×C22⋊C4semidirect
166. C22×C32C2×AGL(1,𝔽5)semidirect
176. C2×C3×D3C2×AGL(1,𝔽5)semidirect
189. D15#217semidirect
199. C5×D3#217semidirect
216. C22×D3ΓL(1,𝔽16)semidirect
227. C22×C5D3×Dic3semidirect
236. C2×D5D3×Dic3semidirect
247. C3×D3C22⋊AGL(1,𝔽5)semidirect
254. C3×C5D3×C22⋊C4semidirect
255. C3×C5C2×#217semidirect
276. C22×C3D3×AGL(1,𝔽5)semidirect
277. C22×C3C2×ΓL(1,𝔽16)semidirect
287. C2×D3C2×ΓL(1,𝔽16)semidirect
301. C32C2×C22⋊AGL(1,𝔽5)semidirect
332. D3#6763direct (N × H)
333. C5D3×#217semidirect
347. C22D3×ΓL(1,𝔽16)semidirect
351. C3D3×C22⋊AGL(1,𝔽5)semidirect
352. C3C2×#6763semidirect

Maximal subgroups

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

subgroupindex
C22×C3×D3×D52
C2×D3×ΓL(1,𝔽16)2
#106240982
C3×#67632
(C2×D3)⋊ΓL(1,𝔽16)2
C2×#67633
D3×C22⋊AGL(1,𝔽5)3
D3×#2175