Finite Group Databasebeta

← #14299627

Subgroups — #14299627

Order 1920.

544 conjugacy classes of subgroups; only the 121 normal subgroups are shown.

normal subgroups only

This group has 544 conjugacy classes of subgroups — too many to draw. Only its 121 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#11756991C211yesmaximal, characteristic
2#11752449C211yesmaximal, characteristic
3#11752386C212yesmaximal
4C3×D5×#101C211yesmaximal, characteristic
5#11746771C212yesmaximal
6#10603931—31nomaximal, Hall
7C2×C3×Q8×D5C2211yescharacteristic
8Q8×ΓL(1,𝔽16)—22no—
9Q8×ΓL(1,𝔽16)—22no—
10#91989—22no—
11#91989C2212yes—
12C2×C4⋊ΓL(1,𝔽16)C2211yescharacteristic
13C2×C4×ΓL(1,𝔽16)C2211yescharacteristic
14C3×D5×M16C2212yes—
15#91235—22no—
16#91223—22no—
17#91222—22no—
18C3×C5×#101C411yescharacteristic, complemented
19C3×#3968C411yescharacteristic, complemented
20C3×#3956C412yescomplemented
21#70102—51nomaximal, Hall
22#66262—31no—
23(C2×AGL(1,𝔽5))⋊Q8—31no—
24#65727—32no—
25D5×#101D311yescharacteristic
26#64900—32no—
27C22×ΓL(1,𝔽16)—21no—
28C22×ΓL(1,𝔽16)—21no—
29C2×C3×C5×Q8C2×C411yescharacteristic
30C3×Q8×D5—22no—
31C3×Q8×D5—22no—
32C2×C3×C4×D5D412yes—
33C2×C3×C4×D5C2311yescharacteristic
34C2×C3×Dic10C2×C411yescharacteristic
35C2×C3×Dic10Q812yes—
36#6763—22no—
37C4⋊ΓL(1,𝔽16)—22no—
38C4⋊ΓL(1,𝔽16)—22no—
39C4⋊ΓL(1,𝔽16)—22no—
40C4⋊ΓL(1,𝔽16)D412yes—
41C4⋊ΓL(1,𝔽16)D412yes—
42C4×ΓL(1,𝔽16)—24no—
43C4×ΓL(1,𝔽16)—22no—
44C4×ΓL(1,𝔽16)—22no—
45C3×C5×M16C2×C412yes—
46C3×#681C2×C412yes—
47C3×#676—22no—
48C3×C8×D5—22no—
49#5812—51no—
50C2×C3×#101—51no—
51#5229—51no—
52#5121—52no—
53#4555—52no—
54C2×Q8×D5C2×D311yescharacteristic
55Q8×AGL(1,𝔽5)—62no—
56Q8×AGL(1,𝔽5)—62no—
57#4130—32no—
58#4130—62no—
59C2×C4⋊AGL(1,𝔽5)—31no—
60C2×C4×AGL(1,𝔽5)—31no—
61D5×M16C2×D312yes—
62(C2×AGL(1,𝔽5))⋊C4—62no—
63#3994—62no—
64#3993—62no—
65C5×#101Dic311yescharacteristic, complemented
66#3968Dic311yescharacteristic, complemented
67#3956Dic312yescomplemented
68#1857—151noSylow
69C22×C3×D5C2×D411yescharacteristic
70C2×ΓL(1,𝔽16)—22no—
71C2×ΓL(1,𝔽16)—22no—
72C2×ΓL(1,𝔽16)—22no—
73C2×ΓL(1,𝔽16)—41no—
74C2×ΓL(1,𝔽16)—41no—
75C2×ΓL(1,𝔽16)—42no—
76C2×ΓL(1,𝔽16)—41no—
77C3×C5×Q8—22no—
78C2×C3×C4×C5C4⋊C412yes—
79C2×C3×C4×C5C22×C411yescharacteristic
80C2×C3×Dic5C4⋊C412yes—
81C2×C3×Dic5C2×Q811yescharacteristic
82C3×C4×D5—22no—
83C3×C4×D5—22no—
84C3×C4×D5C2×D412yes—
85C3×C4×D5#4112yes—
86C3×Dic10—22no—
87C3×Dic10—24no—
88C3×C5×C8—22no—
89C3×#168—22no—
90C22×C3×Q8—51no—
91C2×C3×M16—52no—
92Q8×Dic3—102no—
93Q8×Dic3—102no—
94#926—52no—
95#926—102no—
96C2×C4⋊Dic3—51no—
97C2×C4×Dic3—51no—
98C3×#101—51no—
99C3×#101—52no—
100C3×#101AGL(1,𝔽5)11yescharacteristic, complemented
101C3⋊#95—102no—
102#818—102no—
103#817—102no—
104C22×AGL(1,𝔽5)—61no—
105C22×AGL(1,𝔽5)—61no—
106C2×C5×Q8C2×Dic311yescharacteristic
107Q8×D5—22no—
108Q8×D5—22no—
109C2×C4×D5D1212yes—
110C2×C4×D5C22×D311yescharacteristic
111C2×Dic10Dic612yes—
112C2×Dic10C2×Dic311yescharacteristic
113C22⋊AGL(1,𝔽5)—62no—
114C4⋊AGL(1,𝔽5)—32no—
115C4⋊AGL(1,𝔽5)—32no—
116C4⋊AGL(1,𝔽5)—62no—
117C4⋊AGL(1,𝔽5)—62no—
118C4⋊AGL(1,𝔽5)—62no—
119C4×AGL(1,𝔽5)—64no—
120C4×AGL(1,𝔽5)—62no—
121C4×AGL(1,𝔽5)—62no—
122C5×M16C2×Dic312yes—
123#681C2×Dic312yes—
124#676—22no—
125C8×D5—22no—
126#519—151no—
127#428—152no—
128C2×#101—51no—
129#391—151no—
130#339—152no—
131C22×C3×C5C2×C4⋊C411yescharacteristic
132C2×C3×D5—21no—
133C2×C3×D5—21no—
134C2×C3×D5#12011yescharacteristic
135C2×C3×D5#12311yescharacteristic
136ΓL(1,𝔽16)—42no—
137ΓL(1,𝔽16)—42no—
138ΓL(1,𝔽16)—42no—
139ΓL(1,𝔽16)—81no—
140C3×C4×C5—22no—
141C3×C4×C5C4×D412yes—
142C3×Dic5—22no—
143C3×Dic5C22⋊Q812yes—
144C2×C3×Q8—52no—
145C2×C3×Q8—51no—
146C2×C3×Q8—102no—
147C2×C3×Q8—102no—
148C2×C3×Q8C2×AGL(1,𝔽5)11yescharacteristic
149C22×C3×C4—51no—
150C22×C3×C4—52no—
151C22×Dic3—101no—
152C22×Dic3—101no—
153C3×M16—52no—
154C3×M16—102no—
155C3×M16C2×AGL(1,𝔽5)12yes—
156C2×C3×C8—102no—
157#217—102no—
158C4⋊Dic3—52no—
159C4⋊Dic3—52no—
160C4⋊Dic3—102no—
161C4⋊Dic3—102no—
162C4⋊Dic3—102no—
163C4×Dic3—102no—
164C4×Dic3—104no—
165C4×Dic3—102no—
166C22×D5C2×D1211yescharacteristic
167C2×AGL(1,𝔽5)—62no—
168C2×AGL(1,𝔽5)—62no—
169C2×AGL(1,𝔽5)—62no—
170C2×AGL(1,𝔽5)—121no—
171C2×AGL(1,𝔽5)—121no—
172C2×AGL(1,𝔽5)—122no—
173C2×AGL(1,𝔽5)—121no—
174C5×Q8—22no—
175C2×C4×C5C4⋊Dic312yes—
176C2×C4×C5C22×Dic311yescharacteristic
177C2×Dic5C4⋊Dic312yes—
178C2×Dic5C2×Dic611yescharacteristic
179C4×D5—22no—
180C4×D5—22no—
181C4×D5D3×D412yes—
182C4×D5#23712yes—
183Dic10—22no—
184Dic10—24no—
185C5×C8—22no—
186#168—22no—
187C22×Q8—51no—
188C2×M16—52no—
189C4×Q8—302no—
190C4×Q8—302no—
191#117—152no—
192#117—302no—
193C2×C4⋊C4—151no—
194C2×C42—151no—
195#107—302no—
196#106—302no—
197#101—51no—
198#101—52no—
199#101ΓL(1,𝔽16)11yescharacteristic, complemented
200#95—302no—
201C2×C3×C5—21no—
202C2×C3×C5#38511yescharacteristic
203C3×D5—21no—
204C3×D5#45411yescharacteristic
205C3×D5#45611yescharacteristic
206C23×C3—51no—
207C3×Q8—22no—
208C3×Q8—104no—
209C3×Q8—102no—
210C2×C3×C4—51no—
211C2×C3×C4—52no—
212C2×C3×C4—52no—
213C2×C3×C4—52no—
214C2×C3×C4—102no—
215C2×C3×C4—102no—
216C2×C3×C4C4⋊AGL(1,𝔽5)12yes—
217C2×C3×C4C22×AGL(1,𝔽5)11yescharacteristic
218C2×Dic3—102no—
219C2×Dic3—102no—
220C2×Dic3—102no—
221C2×Dic3—201no—
222C2×Dic3—201no—
223C2×Dic3—201no—
224C2×Dic3—202no—
225C3×C8—22no—
226C3×C8—102no—
227C22×C5C2×C4⋊Dic311yescharacteristic
228C2×D5—21no—
229C2×D5—21no—
230C2×D5#88211yescharacteristic
231C2×D5#88611yescharacteristic
232AGL(1,𝔽5)—122no—
233AGL(1,𝔽5)—122no—
234AGL(1,𝔽5)—122no—
235AGL(1,𝔽5)—241no—
236C4×C5—22no—
237C4×C5D4×Dic312yes—
238Dic5—22no—
239Dic5#87812yes—
240C2×Q8—52no—
241C2×Q8—51no—
242C2×Q8—102no—
243C2×Q8—102no—
244C2×Q8C2×ΓL(1,𝔽16)11yescharacteristic
245C22×C4—51no—
246C22×C4—52no—
247C22×C4—301no—
248C22×C4—301no—
249M16—52no—
250M16—102no—
251M16C2×ΓL(1,𝔽16)12yes—
252C2×C8—102no—
253C4⋊C4—152no—
254C4⋊C4—152no—
255C4⋊C4—302no—
256C4⋊C4—302no—
257C4⋊C4—302no—
258C22⋊C4—302no—
259C42—302no—
260C42—302no—
261C42—304no—
262C3×C5#185711yescharacteristic, complemented, Hall
263C22×C3—51no—
264C22×C3—51no—
265C22×C3—101no—
266C22×C3—101no—
267C22×C3C2×C4⋊AGL(1,𝔽5)11yescharacteristic
268C3×C4—22no—
269C3×C4—52no—
270C3×C4—102no—
271C3×C4D4×AGL(1,𝔽5)12yes—
272Dic3—202no—
273Dic3—202no—
274Dic3—202no—
275Dic3—401no—
276C2×C5—21no—
277C2×C5#495011yescharacteristic
278D5—21no—
279D5#475211yescharacteristic
280D5#475311yescharacteristic
281C23—51no—
282Q8—22no—
283Q8—104no—
284Q8—102no—
285C2×C4—51no—
286C2×C4—52no—
287C2×C4—52no—
288C2×C4—52no—
289C2×C4—102no—
290C2×C4—102no—
291C2×C4—302no—
292C2×C4—302no—
293C2×C4—302no—
294C2×C4—601no—
295C2×C4—601no—
296C2×C4—601no—
297C2×C4—602no—
298C2×C4C4⋊ΓL(1,𝔽16)12yes—
299C2×C4C22×ΓL(1,𝔽16)11yescharacteristic
300C8—22no—
301C8—102no—
302C2×C3—21no—
303C2×C3—51no—
304C2×C3—51no—
305C2×C3—101no—
306C2×C3#6569911yescharacteristic
307C5#7010211yescharacteristic, complemented, Sylow
308C22—51no—
309C22—51no—
310C22—101no—
311C22—101no—
312C22C2×C4⋊ΓL(1,𝔽16)11yescharacteristic
313C4—22no—
314C4—52no—
315C4—102no—
316C4—602no—
317C4—602no—
318C4—602no—
319C4—1201no—
320C4D4×ΓL(1,𝔽16)12yes—
321C3#1060393111yescharacteristic, complemented, Sylow
322C2—21no—
323C2—51no—
324C2—51no—
325C2—101no—
326C2#1175235811yescharacteristic
subgroupG/H
centerC2#11752358
commutatorC2×C3×C4×C5C22×C4
FrattiniC2×C4C22×ΓL(1,𝔽16)
FittingC3×C5×#101C4
socleC2×C3×C5#385
prime radicalC22×C3×D5C2×D4
cosocleC2×C3×C4×D5C23
hypercenter#101ΓL(1,𝔽16)
hypocenterC3×C5#1857
LCSG ▷ C2×C3×C4×C5 ▷ C2×C3×C5 ▷ C3×C5
UCSC1 ◁ C2 ◁ C2×C4 ◁ #101

Complements

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

NH ≅ G/Ntype
18. C3×C5×#101C4semidirect
19. C3×#3968C4semidirect
20. C3×#3956C4semidirect
65. C5×#101Dic3semidirect
66. #3968Dic3semidirect
67. #3956Dic3semidirect
100. C3×#101AGL(1,𝔽5)semidirect
199. #101ΓL(1,𝔽16)semidirect
262. C3×C5#1857semidirect
307. C5#70102semidirect
321. C3#10603931semidirect

Maximal subgroups

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

subgroupindex
#117569912
#117524492
#117523862
C3×D5×#1012
#117467712
#106039313
#701025