Finite Group Databasebeta

← #14301421

Subgroups — #14301421

Order 1920.

664 conjugacy classes of subgroups; only the 110 normal subgroups are shown.

normal subgroups only

This group has 664 conjugacy classes of subgroups — too many to draw. Only its 110 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
1C3×#66260C211yesmaximal, characteristic, complemented
2#11755158C211yesmaximal, characteristic
3#11752444C211yesmaximal, characteristic, complemented
4#11751718C212yesmaximal, complemented
5#11746525C212yesmaximal, complemented
6#10604569—31nomaximal, Hall
7C2×C3×#713C2211yescharacteristic
8C2×#6772C2211yescharacteristic
9#91989C2212yes—
10C3×#4133—24no—
11C3×D5⋊M16—22no—
12C3×D5⋊M16C2212yes—
13#91833C411yescharacteristic, complemented
14C5×#947C411yescharacteristic, complemented
15#91463C411yescharacteristic, complemented
16#91440—22no—
17#91408—22no—
18C4×D5×Dic3C2211yescharacteristic
19#91364C411yescharacteristic, complemented
20#91228—22no—
21#91155—24no—
22#72021—51nomaximal, Hall
23#66260D311yescharacteristic, complemented
24#66002—31no—
25#65785—31no—
26#65769—32no—
27#64862—32no—
28C2×C3×C5×Q8C2×C411yescharacteristic
29C2×Dic30C2×C411yescharacteristic
30C2×C5×D12C2×C411yescharacteristic
31C3×#713—22no—
32C3×#713—22no—
33C2×C3×D20C2×C411yescharacteristic
34C2×C3×D20D412yes—
35C2×C3×C4×D5D412yes—
36C2×C3×C4×D5C2311yescharacteristic
37C2×#1201—22no—
38C2×#1201—21no—
39C2×D3×Dic5—21no—
40C2×D5×Dic3—21no—
41#6772—22no—
42#6772—22no—
43#6763—22no—
44C4⋊ΓL(1,𝔽16)—22no—
45C4×ΓL(1,𝔽16)—22no—
46C3×#704—22no—
47C3×#700—24no—
48C3×#700D414yescomplemented
49C3×D5⋊C8—24no—
50C3×D5⋊C8—22no—
51#6712—22no—
52C4×Dic15C2×C411yescharacteristic
53C5×(C2×D3)⋊C4—22no—
54C4×C5×Dic3C2×C411yescharacteristic
55#6665—22no—
56Dic3×Dic5—21no—
57C3×#568—51no—
58#5832—51no—
59#5240—51no—
60#5139—52no—
61#4555—52no—
62C2×#713C2×D311yescharacteristic
63C2×#711—31no—
64#4133—24no—
65#4130—32no—
66D5⋊M16—22no—
67D5⋊M16C2×D312yes—
68C5×#124—31no—
69#4090—31no—
70#4086—31no—
71#4030—62no—
72#4027—62no—
73#4021—31no—
74C42×D5—31no—
75#4008—64no—
76#3999—62no—
77#1884—151noSylow
78C22×C5×D3—21no—
79C22×C3×D5C22⋊C412yes—
80C22×C3×D5C2×D411yescharacteristic
81C2×ΓL(1,𝔽16)—42no—
82C3×C5×Q8—22no—
83C2×C3×C4×C5C22⋊C412yes—
84C2×C3×C4×C5C22×C411yescharacteristic
85C2×Dic15—21no—
86C2×Dic15—21no—
87Dic30—22no—
88C2×C5×Dic3—21no—
89C5×D12—22no—
90C2×C3×Dic5C2×D411yescharacteristic
91C3×D20—24no—
92C3×D20—22no—
93C3×C4×D5—22no—
94C3×C4×D5—22no—
95C3×C4×D5C2×D412yes—
96C3×C4×D5#4112yes—
97#1201—44no—
98#1201—42no—
99D3×Dic5—41no—
100D3×Dic5—41no—
101D5×Dic3—42no—
102C3×#170—24no—
103C2×C3×#41—51no—
104C2×#235—51no—
105C3×#131—104no—
106C2×C3×M16—52no—
107C2×C3×M16—102no—
108#947AGL(1,𝔽5)11yescharacteristic, complemented
109#944—51no—
110#940—51no—
111#939—102no—
112#936—51no—
113C4×#67—102no—
114#926—52no—
115C2×C4×Dic3—51no—
116#834—102no—
117#825—104no—
118C2×C5×Q8C4×D311yescharacteristic
119C2×C5×D4—31no—
120C2×C22⋊D5—62no—
121C2×C22⋊D5—61no—
122C22×Dic5—61no—
123#713—22no—
124#713—22no—
125#711—62no—
126#711—62no—
127C2×D20C4×D311yescharacteristic
128C2×D20D1212yes—
129C2×C4×D5—61no—
130C2×C4×D5#6712yes—
131C2×C4×D5C22×D311yescharacteristic
132C2×Dic10—31no—
133C22⋊AGL(1,𝔽5)—62no—
134#704—22no—
135C4⋊AGL(1,𝔽5)—62no—
136C4×AGL(1,𝔽5)—62no—
137#700—24no—
138#700#6714yescomplemented
139D5⋊C8—24no—
140D5⋊C8—22no—
141C5×C22⋊C4—62no—
142C42×C5—31no—
143#685—62no—
144#683—62no—
145C4×Dic5—31no—
146C4×Dic5—61no—
147#568—51no—
148#532—151no—
149#421—152no—
150#410—151no—
151#339—152no—
152C22×C3×C5C2×C22⋊C411yescharacteristic
153C2×C5×D3—42no—
154C2×C5×D3—41no—
155C2×C3×D5—22no—
156C2×C3×D5—41no—
157C2×C3×D5—42no—
158C2×C3×D5#12112yes—
159ΓL(1,𝔽16)—82no—
160C3×C4×C5—22no—
161C3×C4×C5C4×D412yes—
162Dic15—22no—
163Dic15—41no—
164C3×Dic5#12011yescharacteristic
165C3×Dic5#12311yescharacteristic
166C5×Dic3—41no—
167C3×#41—102no—
168C3×#41—102no—
169C2×C3×Q8C2×AGL(1,𝔽5)11yescharacteristic
170C2×C3×D4—52no—
171C2×C3×D4—51no—
172C22×C3×C4—51no—
173C22×C3×C4—52no—
174C2×#67—102no—
175C2×#67—101no—
176C22×Dic3—101no—
177#235—102no—
178#235—102no—
179C2×D12C2×AGL(1,𝔽5)11yescharacteristic
180C2×C4×D3—101no—
181C2×Dic6—51no—
182C3×M16—54no—
183C3×M16—102no—
184C3×M16—104no—
185C2×C3×C8—102no—
186C2×C3×C8—104no—
187#217—102no—
188#217—102no—
189(C2×D3)⋊C4—22no—
190C4⋊Dic3—102no—
191Dic3⋊C4—102no—
192C4×Dic3—51no—
193C4×Dic3—101no—
194C4×Dic3—102no—
195C4×Dic3C2×AGL(1,𝔽5)11yescharacteristic
196C23×C5—61no—
197C22×D5(C2×D3)⋊C412yes—
198C22×D5C2×D1211yescharacteristic
199C2×AGL(1,𝔽5)—122no—
200C5×Q8—22no—
201C5×D4—62no—
202C2×C4×C5—61no—
203C2×C4×C5(C2×D3)⋊C412yes—
204C2×C4×C5C2×C4×D311yescharacteristic
205C22⋊D5—124no—
206C22⋊D5—122no—
207C2×Dic5—61no—
208C2×Dic5—61no—
209C2×Dic5—121no—
210C2×Dic5—121no—
211C2×Dic5C2×#6711yescharacteristic
212D20—24no—
213D20—22no—
214C4×D5—22no—
215C4×D5—22no—
216C4×D5—122no—
217C4×D5#23512yes—
218C4×D5C2×#6712yes—
219Dic10—62no—
220#170—24no—
221C2×#41—51no—
222C2×#41—151no—
223#131—104no—
224C2×M16—52no—
225C2×M16—102no—
226C4⋊Q8—151no—
227#127—151no—
228#124—31no—
229#124—151no—
230#121—302no—
231C4×D4—302no—
232#117—152no—
233C2×C42—151no—
234#104—304no—
235#99—302no—
236C2×C3×C5—21no—
237C2×C3×C5#38611yescharacteristic
238C3×D5—22no—
239C3×D5—42no—
240C5×D3—81no—
241C23×C3—51no—
242C23×C3—52no—
243C22×D3—21no—
244C3×Q8—22no—
245C3×D4—104no—
246C3×D4—102no—
247C2×C3×C4—51no—
248C2×C3×C4—52no—
249C2×C3×C4—52no—
250C2×C3×C4—102no—
251C2×C3×C4—102no—
252C2×C3×C4C22⋊AGL(1,𝔽5)12yes—
253C2×C3×C4C22×AGL(1,𝔽5)11yescharacteristic
254#67—204no—
255#67—202no—
256C2×Dic3—21no—
257C2×Dic3—101no—
258C2×Dic3—101no—
259C2×Dic3—202no—
260C2×Dic3—202no—
261D12—22no—
262C4×D3—201no—
263C4×D3—201no—
264Dic6—102no—
265C3×C8—104no—
266C22×C5—122no—
267C22×C5—121no—
268C22×C5C2×(C2×D3)⋊C411yescharacteristic
269C2×D5—22no—
270C2×D5—41no—
271C2×D5—42no—
272C2×D5#93012yes—
273AGL(1,𝔽5)—242no—
274C4×C5—22no—
275C4×C5—121no—
276C4×C5C4×#6712yes—
277Dic5—62no—
278Dic5—121no—
279Dic5#92911yescharacteristic
280Dic5#95311yescharacteristic
281#41—102no—
282#41—102no—
283#41—302no—
284#41—302no—
285C2×Q8—151no—
286C2×Q8D3×AGL(1,𝔽5)11yescharacteristic
287C2×D4—31no—
288C2×D4—52no—
289C2×D4—51no—
290C2×D4—302no—
291C2×D4—301no—
292C22×C4—51no—
293C22×C4—52no—
294C22×C4—301no—
295C22×C4—301no—
296M16—54no—
297M16—102no—
298M16—104no—
299C2×C8—102no—
300C2×C8—104no—
301C4⋊C4—302no—
302C4⋊C4—302no—
303C22⋊C4—62no—
304C22⋊C4—302no—
305C22⋊C4—302no—
306C42—31no—
307C42—151no—
308C42—301no—
309C42—302no—
310C3×C5#188411yescharacteristic, complemented, Hall
311C22×C3—52no—
312C22×C3—102no—
313C22×C3—201no—
314C22×C3—202no—
315C22×C3C2×C22⋊AGL(1,𝔽5)11yescharacteristic
316C2×D3—42no—
317C2×D3—41no—
318C3×C4—22no—
319C3×C4—51no—
320C3×C4—51no—
321C3×C4D4×AGL(1,𝔽5)12yes—
322Dic3—41no—
323Dic3—102no—
324Dic3—201no—
325Dic3—402no—
326C2×C5—21no—
327C2×C5—241no—
328C2×C5#521311yescharacteristic
329D5—22no—
330D5—42no—
331C23—51no—
332C23—52no—
333C23—61no—
334Q8—22no—
335Q8—302no—
336D4—62no—
337D4—104no—
338D4—102no—
339D4—604no—
340D4—602no—
341C2×C4—51no—
342C2×C4—52no—
343C2×C4—52no—
344C2×C4—61no—
345C2×C4—102no—
346C2×C4—102no—
347C2×C4—301no—
348C2×C4—301no—
349C2×C4—601no—
350C2×C4—601no—
351C2×C4—602no—
352C2×C4—602no—
353C2×C4(C2×D3)⋊AGL(1,𝔽5)12yes—
354C2×C4C2×D3×AGL(1,𝔽5)11yescharacteristic
355C8—104no—
356C2×C3—21no—
357C2×C3—102no—
358C2×C3—202no—
359C2×C3#6577911yescharacteristic
360D3—81no—
361C5#7202111yescharacteristic, complemented, Sylow
362C22—52no—
363C22—102no—
364C22—122no—
365C22—121no—
366C22—201no—
367C22—202no—
368C22C2×(C2×D3)⋊AGL(1,𝔽5)11yescharacteristic
369C4—22no—
370C4—51no—
371C4—51no—
372C4—121no—
373C4—302no—
374C4—601no—
375C4—1202no—
376C4AGL(1,𝔽5)×#6712yes—
377C3#1060456911yescharacteristic, complemented, Sylow
378C2—21no—
379C2—102no—
380C2—202no—
381C2—241no—
382C2#1175202211yescharacteristic
subgroupG/H
centerC2#11752022
commutatorC2×C3×C4×C5C22×C4
FrattiniC2×C4C2×D3×AGL(1,𝔽5)
FittingC2×C3×C5×Q8C2×C4
socleC2×C3×C5#386
prime radical#11755158C2
cosocleC2×C3×C4×D5C23
hypercenterC2×Q8D3×AGL(1,𝔽5)
hypocenterC3×C5#1884
LCSG ▷ C2×C3×C4×C5 ▷ C2×C3×C5 ▷ C3×C5
UCSC1 ◁ C2 ◁ C2×C4 ◁ C2×Q8

Complements

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

NH ≅ G/Ntype
1. C3×#66260C2semidirect
3. #11752444C2semidirect
4. #11751718C2semidirect
5. #11746525C2semidirect
13. #91833C4semidirect
14. C5×#947C4semidirect
15. #91463C4semidirect
19. #91364C4semidirect
23. #66260D3semidirect
48. C3×#700D4semidirect
108. #947AGL(1,𝔽5)semidirect
138. #700#67semidirect
310. C3×C5#1884semidirect
361. C5#72021semidirect
377. C3#10604569semidirect

Maximal subgroups

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

subgroupindex
C3×#662602
#117551582
#117524442
#117517182
#117465252
#106045693
#720215