Finite Group Databasebeta

← #14301413

Subgroups — #14301413

Order 1920.

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

normal subgroups only

This group has 664 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
1#11756985C211yesmaximal, characteristic, complemented
2#11754793C211yesmaximal, characteristic
3C3×#65771C211yesmaximal, characteristic, complemented
4#11751719C212yesmaximal, complemented
5#11746523C212yesmaximal, complemented
6#10604618—31nomaximal, Hall
7C2×C3×#711C2211yescharacteristic
8C2×#6772C2211yescharacteristic
9C2×#6762—22no—
10#91990—24no—
11D5⋊#208C2212yes—
12C3×#4130C2212yes—
13#91825C411yescharacteristic, complemented
14C5×#940C411yescharacteristic, complemented
15Dic5×#67—22no—
16#91549—22no—
17C4×D5×Dic3C2211yescharacteristic
18#91389C411yescharacteristic, complemented
19#91370C411yescharacteristic, complemented
20C3×#4011—22no—
21#91156—24no—
22#70045—51nomaximal, Hall
23#66256—31no—
24#66050—31no—
25#65771D311yescharacteristic, complemented
26#65769—32no—
27#64862—32no—
28C2×C3×C5×D4C2×C411yescharacteristic
29C2×Dic30C2×C411yescharacteristic
30C2×C5×#67—22no—
31C2×C5×D12C2×C411yescharacteristic
32C2×C3×C22⋊D5D412yes—
33C22×C3×Dic5D412yes—
34C3×#711—22no—
35C3×#711—22no—
36C2×C3×C4×D5C2311yescharacteristic
37C2×C3×Dic10C2×C411yescharacteristic
38C2×#1201—21no—
39C2×D3×Dic5—21no—
40C2×D5×Dic3—21no—
41#6772—22no—
42#6772—22no—
43#6762—24no—
44#6762—22no—
45C2×C3⋊#170—24no—
46#6758D414yescomplemented
47D5⋊(C3⋊C8)—22no—
48C3×C22⋊AGL(1,𝔽5)—22no—
49C3×C4⋊AGL(1,𝔽5)—22no—
50C3×C4×AGL(1,𝔽5)—22no—
51#6719—22no—
52C4×Dic15C2×C411yescharacteristic
53C4×C5×Dic3C2×C411yescharacteristic
54Dic3⋊Dic5—22no—
55(C2×D3)⋊Dic5—22no—
56Dic3×Dic5—21no—
57#5832—51no—
58#5819—51no—
59C3×#410—51no—
60#5005—52no—
61#4532—52no—
62C2×#711—31no—
63C2×#711C2×D311yescharacteristic
64C2×#704—62no—
65#4131—64no—
66#4130C2×D312yes—
67D5⋊M16—32no—
68C5×#127—31no—
69#4084—62no—
70#4082—31no—
71#4082—31no—
72D4×Dic5—62no—
73C42×D5—31no—
74#4015—31no—
75#4011—22no—
76#4008—64no—
77#1884—151noSylow
78C23×C3×C5C22⋊C412yes—
79C22×C5×D3—21no—
80C22×C3×D5C2×D411yescharacteristic
81C2×C3×AGL(1,𝔽5)—42no—
82C3×C5×D4—22no—
83C2×C3×C4×C5C22×C411yescharacteristic
84C2×Dic15—21no—
85C2×Dic15—21no—
86Dic30—22no—
87C5×#67—44no—
88C2×C5×Dic3—21no—
89C5×D12—22no—
90C3×C22⋊D5—24no—
91C2×C3×Dic5—22no—
92C2×C3×Dic5—22no—
93C2×C3×Dic5C22⋊C412yes—
94C2×C3×Dic5C2×D411yescharacteristic
95C3×C4×D5C2×D412yes—
96C3×C4×D5#4112yes—
97C3×Dic10—22no—
98#1201—42no—
99D3×Dic5—41no—
100D3×Dic5—41no—
101D5×Dic3—42no—
102C3⋊#170—24no—
103C2×C3×#41—51no—
104C2×#235—51no—
105C3×#117—52no—
106#948—104no—
107#947—51no—
108#944—51no—
109#940AGL(1,𝔽5)11yescharacteristic, complemented
110#939—102no—
111#936—51no—
112C4×#67—102no—
113C2×C4×Dic3—51no—
114C2×#208—52no—
115C2×#208—102no—
116C3×#99—102no—
117#805—104no—
118C2×C5×D4—31no—
119C2×C5×D4—62no—
120C2×C5×D4C4×D311yescharacteristic
121C2×C22⋊D5—61no—
122C2×C22⋊D5#6712yes—
123C22×Dic5—61no—
124C22×Dic5D1212yes—
125#711—22no—
126#711—22no—
127#711—62no—
128#711—62no—
129C2×C4×D5—61no—
130C2×C4×D5C22×D311yescharacteristic
131C2×Dic10—31no—
132C2×Dic10C4×D311yescharacteristic
133C22⋊AGL(1,𝔽5)—22no—
134#704—64no—
135#704—62no—
136C2×#170—64no—
137C4⋊AGL(1,𝔽5)—22no—
138C4×AGL(1,𝔽5)—22no—
139#700—34no—
140D5⋊C8—62no—
141C42×C5—31no—
142C22⋊Dic5—62no—
143C22⋊Dic5—62no—
144C4⋊Dic5—62no—
145C4×Dic5—31no—
146C4×Dic5—61no—
147#568—151no—
148#532—151no—
149#421—152no—
150#410—51no—
151#339—152no—
152C22×C3×C5—22no—
153C22×C3×C5—42no—
154C22×C3×C5C2×C22⋊C411yescharacteristic
155C2×C5×D3—42no—
156C2×C5×D3—41no—
157C2×C3×D5—41no—
158C2×C3×D5#12112yes—
159C3×AGL(1,𝔽5)—82no—
160C3×C4×C5C4×D412yes—
161Dic15—41no—
162Dic15—41no—
163C3×Dic5—22no—
164C3×Dic5#12011yescharacteristic
165C3×Dic5#12311yescharacteristic
166C5×Dic3—22no—
167C3×#41—102no—
168C3×#41—102no—
169C2×C3×Q8—51no—
170C2×C3×D4—52no—
171C2×C3×D4C2×AGL(1,𝔽5)11yescharacteristic
172C22×C3×C4—51no—
173C22×C3×C4—52no—
174C2×#67—22no—
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×C4⋊C4—102no—
183C3×C22⋊C4—102no—
184C3×C42—102no—
185#217—102no—
186(C2×D3)⋊C4—102no—
187Dic3⋊C4—102no—
188C4×Dic3—51no—
189C4×Dic3—101no—
190C4×Dic3C2×AGL(1,𝔽5)11yescharacteristic
191#208—54no—
192#208—102no—
193#208—104no—
194C2×C3⋊C8—102no—
195C2×C3⋊C8—104no—
196C23×C5—61no—
197C23×C5(C2×D3)⋊C412yes—
198C22×D5C2×#6711yescharacteristic
199C2×AGL(1,𝔽5)—42no—
200C5×D4—22no—
201C5×D4—62no—
202C5×D4—124no—
203C2×C4×C5—61no—
204C2×C4×C5C2×C4×D311yescharacteristic
205C22⋊D5—24no—
206C22⋊D5—122no—
207C2×Dic5—22no—
208C2×Dic5—22no—
209C2×Dic5—61no—
210C2×Dic5—61no—
211C2×Dic5—121no—
212C2×Dic5—121no—
213C2×Dic5(C2×D3)⋊C412yes—
214C2×Dic5C2×D1211yescharacteristic
215C4×D5—122no—
216C4×D5D3×D412yes—
217C4×D5#23712yes—
218Dic10—22no—
219Dic10—62no—
220#170—64no—
221C2×#41—51no—
222C2×#41—151no—
223#131—304no—
224C2×M16—152no—
225C2×M16—302no—
226C4⋊Q8—151no—
227#127—31no—
228#124—151no—
229#124—151no—
230#121—302no—
231C4×D4—302no—
232#117—52no—
233C2×C42—151no—
234#104—304no—
235#99—102no—
236C2×C3×C5—21no—
237C2×C3×C5—42no—
238C2×C3×C5#38611yescharacteristic
239C3×D5—22no—
240C5×D3—81no—
241C23×C3—51no—
242C23×C3C22⋊AGL(1,𝔽5)12yes—
243C22×D3—21no—
244C3×Q8—102no—
245C3×D4—22no—
246C3×D4—104no—
247C2×C3×C4—51no—
248C2×C3×C4—52no—
249C2×C3×C4—52no—
250C2×C3×C4—52no—
251C2×C3×C4—102no—
252C2×C3×C4—102no—
253C2×C3×C4—202no—
254C2×C3×C4C22×AGL(1,𝔽5)11yescharacteristic
255#67—44no—
256#67—202no—
257C2×Dic3—21no—
258C2×Dic3—101no—
259C2×Dic3—101no—
260C2×Dic3—202no—
261D12—22no—
262C4×D3—201no—
263C4×D3—201no—
264Dic6—102no—
265C3⋊C8—104no—
266C22×C5—22no—
267C22×C5—42no—
268C22×C5—122no—
269C22×C5—121no—
270C22×C5C2×(C2×D3)⋊C411yescharacteristic
271C2×D5—41no—
272C2×D5#93912yes—
273AGL(1,𝔽5)—82no—
274C4×C5—62no—
275C4×C5#67⋊C412yes—
276Dic5—22no—
277Dic5—121no—
278Dic5—121no—
279Dic5#88211yescharacteristic
280Dic5#88611yescharacteristic
281#41—102no—
282#41—102no—
283#41—302no—
284#41—302no—
285C2×Q8—51no—
286C2×Q8—151no—
287C2×D4—31no—
288C2×D4—52no—
289C2×D4—62no—
290C2×D4—301no—
291C2×D4D3×AGL(1,𝔽5)11yescharacteristic
292C22×C4—51no—
293C22×C4—52no—
294C22×C4—301no—
295C22×C4—301no—
296M16—154no—
297M16—302no—
298M16—304no—
299C2×C8—302no—
300C2×C8—304no—
301C4⋊C4—102no—
302C4⋊C4—302no—
303C22⋊C4—102no—
304C22⋊C4—302no—
305C22⋊C4—302no—
306C42—31no—
307C42—102no—
308C42—151no—
309C42—301no—
310C3×C5#188411yescharacteristic, complemented, Hall
311C22×C3—22no—
312C22×C3—42no—
313C22×C3—52no—
314C22×C3—201no—
315C22×C3C2×C22⋊AGL(1,𝔽5)11yescharacteristic
316C2×D3—42no—
317C2×D3—41no—
318C3×C4—51no—
319C3×C4—51no—
320C3×C4—102no—
321C3×C4—402no—
322C3×C4D4×AGL(1,𝔽5)12yes—
323Dic3—22no—
324Dic3—201no—
325Dic3—201no—
326C2×C5—21no—
327C2×C5—42no—
328C2×C5—241no—
329C2×C5#495811yescharacteristic
330D5—22no—
331C23—51no—
332C23—61no—
333C23(C2×D3)⋊AGL(1,𝔽5)12yes—
334Q8—102no—
335Q8—302no—
336D4—22no—
337D4—62no—
338D4—104no—
339D4—124no—
340D4—602no—
341C2×C4—51no—
342C2×C4—52no—
343C2×C4—52no—
344C2×C4—52no—
345C2×C4—61no—
346C2×C4—102no—
347C2×C4—102no—
348C2×C4—202no—
349C2×C4—301no—
350C2×C4—301no—
351C2×C4—601no—
352C2×C4—601no—
353C2×C4—602no—
354C2×C4C2×D3×AGL(1,𝔽5)11yescharacteristic
355C8—304no—
356C2×C3—21no—
357C2×C3—42no—
358C2×C3—102no—
359C2×C3#6577911yescharacteristic
360D3—81no—
361C5#7004511yescharacteristic, complemented, Sylow
362C22—22no—
363C22—42no—
364C22—52no—
365C22—122no—
366C22—121no—
367C22—201no—
368C22C2×(C2×D3)⋊AGL(1,𝔽5)11yescharacteristic
369C4—51no—
370C4—51no—
371C4—62no—
372C4—102no—
373C4—402no—
374C4—601no—
375C4—601no—
376C4#67⋊AGL(1,𝔽5)12yes—
377C3#1060461811yescharacteristic, complemented, Sylow
378C2—21no—
379C2—42no—
380C2—102no—
381C2—241no—
382C2#1175202611yescharacteristic
subgroupG/H
centerC2#11752026
commutatorC2×C3×C4×C5C22×C4
FrattiniC2×C4C2×D3×AGL(1,𝔽5)
FittingC2×C3×C5×D4C2×C4
socleC2×C3×C5#386
prime radical#11754793C2
cosocleC2×C3×C4×D5C23
hypercenterC2×D4D3×AGL(1,𝔽5)
hypocenterC3×C5#1884
LCSG ▷ C2×C3×C4×C5 ▷ C2×C3×C5 ▷ C3×C5
UCSC1 ◁ C2 ◁ C2×C4 ◁ C2×D4

Complements

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

NH ≅ G/Ntype
1. #11756985C2semidirect
3. C3×#65771C2semidirect
4. #11751719C2semidirect
5. #11746523C2semidirect
13. #91825C4semidirect
14. C5×#940C4semidirect
18. #91389C4semidirect
19. #91370C4semidirect
25. #65771D3semidirect
46. #6758D4semidirect
109. #940AGL(1,𝔽5)semidirect
310. C3×C5#1884semidirect
361. C5#70045semidirect
377. C3#10604618semidirect

Maximal subgroups

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

subgroupindex
#117569852
#117547932
C3×#657712
#117517192
#117465232
#106046183
#700455