Finite Group Databasebeta

← #14299652

Subgroups — #14299652

Order 1920.

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

normal subgroups only

This group has 544 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
1#11756991C211yesmaximal, characteristic
2C3×(C2×AGL(1,𝔽5))⋊Q8C211yesmaximal, characteristic
3#11751727C212yesmaximal
4D5×#836C211yesmaximal, characteristic
5#11746527C212yesmaximal
6#10603931—31nomaximal, Hall
7C2×C3×Q8×D5C2211yescharacteristic
8Q8×ΓL(1,𝔽16)—22no—
9Q8×ΓL(1,𝔽16)—22no—
10#91989—22no—
11#91989C2212yes—
12C2×C3×C4⋊AGL(1,𝔽5)C2211yescharacteristic
13C2×C3×C4×AGL(1,𝔽5)C2211yescharacteristic
14D5×#208C2212yes—
15C3×(C2×AGL(1,𝔽5))⋊C4—22no—
16#91160—22no—
17#91159—22no—
18#91120C411yescharacteristic, complemented
19C5×#836C411yescharacteristic, complemented
20#90961C412yescomplemented
21#69672—51nomaximal, Hall
22#66262—31no—
23(C2×AGL(1,𝔽5))⋊Q8D311yescharacteristic
24#65727—32no—
25D5×#101—31no—
26#64900—32no—
27C22×C3×AGL(1,𝔽5)—21no—
28C22×C3×AGL(1,𝔽5)—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)D412yes—
40C4⋊ΓL(1,𝔽16)D412yes—
41C4×ΓL(1,𝔽16)—24no—
42C4×ΓL(1,𝔽16)—22no—
43C3×C4⋊AGL(1,𝔽5)—22no—
44C3×C4×AGL(1,𝔽5)—22no—
45#6710C2×C412yes—
46C5×#208C2×C412yes—
47#6649—22no—
48D5×C3⋊C8—22no—
49#5812—51no—
50C3×#391—51no—
51C2×#836—51no—
52#4964—52no—
53#4523—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)C2×D311yescharacteristic
60C2×C4×AGL(1,𝔽5)C2×D311yescharacteristic
61D5×M16—32no—
62(C2×AGL(1,𝔽5))⋊C4—22no—
63#3994—62no—
64#3993—62no—
65C5×#101—31no—
66#3968—31no—
67#3956—32no—
68#1857—151noSylow
69C22×C3×D5C2×D411yescharacteristic
70C2×ΓL(1,𝔽16)—22no—
71C2×ΓL(1,𝔽16)—22no—
72C2×C3×AGL(1,𝔽5)—22no—
73C2×C3×AGL(1,𝔽5)—41no—
74C2×C3×AGL(1,𝔽5)—41no—
75C2×C3×AGL(1,𝔽5)—42no—
76C2×C3×AGL(1,𝔽5)—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⋊#168—22no—
89C5×C3⋊C8—22no—
90C22×C3×Q8—51no—
91C2×C3×C4⋊C4—51no—
92C2×C3×C42—51no—
93Q8×Dic3—102no—
94Q8×Dic3—102no—
95#926—52no—
96#926—102no—
97C2×#208—52no—
98C3×#95—102no—
99#836—51no—
100#836—52no—
101#836AGL(1,𝔽5)11yescharacteristic, complemented
102#808—102no—
103#807—102no—
104C22×AGL(1,𝔽5)—21no—
105C22×AGL(1,𝔽5)—21no—
106C2×C5×Q8C4×D311yescharacteristic
107Q8×D5—22no—
108Q8×D5—22no—
109C2×C4×D5#6712yes—
110C2×C4×D5C22×D311yescharacteristic
111C2×Dic10Dic612yes—
112C2×Dic10C4×D311yescharacteristic
113C22⋊AGL(1,𝔽5)—62no—
114C4⋊AGL(1,𝔽5)—22no—
115C4⋊AGL(1,𝔽5)—32no—
116C4⋊AGL(1,𝔽5)—32no—
117C4⋊AGL(1,𝔽5)—62no—
118C4⋊AGL(1,𝔽5)—62no—
119C4×AGL(1,𝔽5)—22no—
120C4×AGL(1,𝔽5)—64no—
121C4×AGL(1,𝔽5)—62no—
122C5×M16—32no—
123#681—32no—
124#676—62no—
125C8×D5—62no—
126#519—151no—
127#428—152no—
128C2×#101—151no—
129#391—51no—
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—
138C3×AGL(1,𝔽5)—42no—
139C3×AGL(1,𝔽5)—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×C3×C4—101no—
152C22×C3×C4—101no—
153C3×C4⋊C4—102no—
154C3×C42—102no—
155#217—102no—
156C4⋊Dic3—52no—
157C4⋊Dic3—52no—
158C4⋊Dic3—102no—
159C4⋊Dic3—102no—
160C4×Dic3—102no—
161C4×Dic3—104no—
162#208—52no—
163#208—102no—
164#208C2×AGL(1,𝔽5)12yes—
165C2×C3⋊C8—102no—
166C22×D5C2×#6711yescharacteristic
167C2×AGL(1,𝔽5)—22no—
168C2×AGL(1,𝔽5)—41no—
169C2×AGL(1,𝔽5)—41no—
170C2×AGL(1,𝔽5)—42no—
171C2×AGL(1,𝔽5)—41no—
172C2×AGL(1,𝔽5)—62no—
173C2×AGL(1,𝔽5)—62no—
174C5×Q8—22no—
175C2×C4×C5Dic3⋊C412yes—
176C2×C4×C5C2×C4×D311yescharacteristic
177C2×Dic5Dic3⋊C412yes—
178C2×Dic5C2×Dic611yescharacteristic
179C4×D5—22no—
180C4×D5—22no—
181C4×D5D3×D412yes—
182C4×D5#23712yes—
183Dic10—22no—
184Dic10—24no—
185C5×C8—62no—
186#168—62no—
187C22×Q8—51no—
188C2×M16—152no—
189C4×Q8—302no—
190C4×Q8—302no—
191#117—152no—
192#117—302no—
193C2×C4⋊C4—51no—
194C2×C42—51no—
195#107—302no—
196#106—302no—
197#101—31no—
198#101—152no—
199#101—151no—
200#95—102no—
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×C4—102no—
217C2×C3×C4—201no—
218C2×C3×C4—201no—
219C2×C3×C4—201no—
220C2×C3×C4—202no—
221C2×C3×C4C4⋊AGL(1,𝔽5)12yes—
222C2×C3×C4C22×AGL(1,𝔽5)11yescharacteristic
223C2×Dic3—102no—
224C2×Dic3—102no—
225C3⋊C8—22no—
226C3⋊C8—102no—
227C22×C5C2×Dic3⋊C411yescharacteristic
228C2×D5—21no—
229C2×D5—21no—
230C2×D5#93511yescharacteristic
231C2×D5#93711yescharacteristic
232AGL(1,𝔽5)—42no—
233AGL(1,𝔽5)—81no—
234AGL(1,𝔽5)—122no—
235AGL(1,𝔽5)—122no—
236C4×C5—22no—
237C4×C5#67⋊C412yes—
238Dic5—22no—
239Dic5#87812yes—
240C2×Q8—52no—
241C2×Q8—51no—
242C2×Q8—102no—
243C2×Q8—102no—
244C2×Q8D3×AGL(1,𝔽5)11yescharacteristic
245C22×C4—51no—
246C22×C4—52no—
247C22×C4—101no—
248C22×C4—101no—
249M16—32no—
250M16—152no—
251M16—302no—
252C2×C8—302no—
253C4⋊C4—102no—
254C4⋊C4—152no—
255C4⋊C4—152no—
256C4⋊C4—302no—
257C4⋊C4—302no—
258C22⋊C4—302no—
259C42—102no—
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×C4—202no—
272C3×C4—401no—
273C3×C4D4×AGL(1,𝔽5)12yes—
274Dic3—202no—
275Dic3—202no—
276C2×C5—21no—
277C2×C5#494611yescharacteristic
278D5—21no—
279D5#506211yescharacteristic
280D5#509511yescharacteristic
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—102no—
292C2×C4—201no—
293C2×C4—201no—
294C2×C4—201no—
295C2×C4—202no—
296C2×C4—302no—
297C2×C4—302no—
298C2×C4Dic3⋊AGL(1,𝔽5)12yes—
299C2×C4C2×D3×AGL(1,𝔽5)11yescharacteristic
300C8—62no—
301C8—302no—
302C2×C3—21no—
303C2×C3—51no—
304C2×C3—51no—
305C2×C3—101no—
306C2×C3#6569911yescharacteristic
307C5#6967211yescharacteristic, complemented, Sylow
308C22—51no—
309C22—51no—
310C22—101no—
311C22—101no—
312C22C2×Dic3⋊AGL(1,𝔽5)11yescharacteristic
313C4—22no—
314C4—52no—
315C4—102no—
316C4—202no—
317C4—401no—
318C4—602no—
319C4—602no—
320C4#67⋊AGL(1,𝔽5)12yes—
321C3#1060393111yescharacteristic, complemented, Sylow
322C2—21no—
323C2—51no—
324C2—51no—
325C2—101no—
326C2#1175202911yescharacteristic
subgroupG/H
centerC2#11752029
commutatorC2×C3×C4×C5C22×C4
FrattiniC2×C4C2×D3×AGL(1,𝔽5)
FittingC2×C3×C5×Q8C2×C4
socleC2×C3×C5#385
prime radicalC22×C3×D5C2×D4
cosocleC2×C3×C4×D5C23
hypercenterC2×Q8D3×AGL(1,𝔽5)
hypocenterC3×C5#1857
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
18. #91120C4semidirect
19. C5×#836C4semidirect
20. #90961C4semidirect
101. #836AGL(1,𝔽5)semidirect
262. C3×C5#1857semidirect
307. C5#69672semidirect
321. C3#10603931semidirect

Maximal subgroups

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

subgroupindex
#117569912
C3×(C2×AGL(1,𝔽5))⋊Q82
#117517272
D5×#8362
#117465272
#106039313
#696725