Finite Group Databasebeta

← #14305965

Subgroups — #14305965

Order 1920.

824 conjugacy classes of subgroups; only the 141 normal subgroups are shown.

normal subgroups only

This group has 824 conjugacy classes of subgroups — too many to draw. Only its 141 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×D5×#127C211yesmaximal, characteristic
2C2×#91236C214yesmaximal, complemented
3#11752348C211yesmaximal, characteristic, complemented
4#11752335C211yesmaximal, characteristic, complemented
5#10604617—31nomaximal, Hall
6C2×C3×D4×D5—22no—
7C2×C3×D4×D5C2212yes—
8C2×C4⋊ΓL(1,𝔽16)C2212yes—
9C2×D5⋊(C3⋊C8)C2212yes—
10C3×C5×#127C411yescharacteristic, complemented
11C3×#4086—22no—
12C3×#4086C412yescomplemented
13C3×#4020C411yescharacteristic, complemented
14C3×C42×D5C2211yescharacteristic
15#91236—28no—
16#91235—22no—
17#91231—22no—
18#71765—51nomaximal, Hall
19D5×#127D311yescharacteristic
20C2×#4007—34no—
21#65689—31no—
22#65676—31no—
23C23×C3×D5—22no—
24C22×ΓL(1,𝔽16)—22no—
25C2×C3×C5×D4—22no—
26C2×C3×C5×D4C2×C412yes—
27C2×C3×C22⋊D5—24no—
28C2×C3×C22⋊D5—22no—
29C3×D4×D5—24no—
30C3×D4×D5—24no—
31C3×D4×D5—48no—
32C2×C3×D20—22no—
33C2×C3×D20C2×C412yes—
34C2×C3×C4×D5D412yes—
35C2×C3×C4×D5C2311yescharacteristic
36C2×C3⋊#170—22no—
37C4⋊ΓL(1,𝔽16)—24no—
38D5⋊(C3⋊C8)—22no—
39D5⋊(C3⋊C8)D414yescomplemented
40C3×C42×C5C2×C411yescharacteristic
41C3×C4×Dic5C2×C411yescharacteristic
42C3×C4×Dic5D412yes—
43C2×C3×#127—51no—
44C2×#832—54no—
45#4936—51no—
46C2×#803—51no—
47C2×D4×D5—22no—
48C2×D4×D5C2×D312yes—
49C2×C4⋊AGL(1,𝔽5)—32no—
50C2×D5⋊C8—32no—
51C5×#127Dic311yescharacteristic, complemented
52#4086—22no—
53#4086Dic312yescomplemented
54#4020Dic311yescharacteristic, complemented
55C42×D5C2×D311yescharacteristic
56#4007—68no—
57(C2×AGL(1,𝔽5))⋊C4—62no—
58#4002—62no—
59#1961—151noSylow
60C23×C3×C5—22no—
61C22×C3×D5—22no—
62C22×C3×D5—42no—
63C22×C3×D5—42no—
64C22×C3×D5—48no—
65C22×C3×D5C2×D411yescharacteristic
66C2×ΓL(1,𝔽16)—42no—
67C2×ΓL(1,𝔽16)—44no—
68C3×C5×D4—24no—
69C3×C5×D4—44no—
70C2×C3×C4×C5C22⋊C412yes—
71C2×C3×C4×C5C22×C411yescharacteristic
72C3×C22⋊D5—48no—
73C3×C22⋊D5—44no—
74C2×C3×Dic5C22⋊C412yes—
75C2×C3×Dic5C2×D411yescharacteristic
76C3×D20—24no—
77C3×D20—44no—
78C3×C4×D5—24no—
79C3×C4×D5C2×D412yes—
80C3×C4×D5#4112yes—
81C3⋊#170—24no—
82C22×C3×D4—52no—
83C22×C3×D4—102no—
84C3×#127—52no—
85C3×#127—51no—
86C3×#127—102no—
87C3×#127AGL(1,𝔽5)11yescharacteristic, complemented
88C2×C3×C42—51no—
89C2×C4⋊Dic3—52no—
90C22×C3⋊C8—52no—
91#832—108no—
92C3⋊#95—102no—
93#803—102no—
94C23×D5—22no—
95C22×AGL(1,𝔽5)—62no—
96C2×C5×D4—22no—
97C2×C5×D4C2×Dic312yes—
98C2×C22⋊D5—24no—
99C2×C22⋊D5—22no—
100D4×D5—24no—
101D4×D5—24no—
102D4×D5—48no—
103C2×D20—22no—
104C2×D20C2×Dic312yes—
105C2×C4×D5#6712yes—
106C2×C4×D5C22×D311yescharacteristic
107C2×#170—62no—
108C4⋊AGL(1,𝔽5)—64no—
109D5⋊C8—34no—
110D5⋊C8—62no—
111C42×C5C2×Dic311yescharacteristic
112C4×Dic5C2×Dic311yescharacteristic
113C4×Dic5#6712yes—
114C2×#127—51no—
115C2×#102—154no—
116C2×#97—151no—
117#382—151no—
118C22×C3×C5—42no—
119C22×C3×C5—44no—
120C22×C3×C5C2×C22⋊C411yescharacteristic
121C2×C3×D5—42no—
122C2×C3×D5—44no—
123C2×C3×D5—84no—
124C2×C3×D5—84no—
125C2×C3×D5#12411yescharacteristic
126C2×C3×D5#12711yescharacteristic
127C2×C3×D5AGL(1,ℤ/8)14yes—
128ΓL(1,𝔽16)—82no—
129C3×C4×C5—22no—
130C3×C4×C5C4×D412yes—
131C3×Dic5—22no—
132C3×Dic5#12112yes—
133C24×C3—102no—
134C2×C3×D4—22no—
135C2×C3×D4—52no—
136C2×C3×D4—104no—
137C2×C3×D4—104no—
138C2×C3×D4—104no—
139C2×C3×D4—102no—
140C2×C3×D4—102no—
141C2×C3×D4—208no—
142C2×C3×D4C2×AGL(1,𝔽5)12yes—
143C22×C3×C4—51no—
144C22×C3×C4—52no—
145C22×Dic3—102no—
146C3×C42—52no—
147C3×C42—51no—
148C3×C42C2×AGL(1,𝔽5)11yescharacteristic
149C4⋊Dic3—104no—
150C2×C3⋊C8—54no—
151C2×C3⋊C8—102no—
152C2×C3⋊C8—102no—
153C23×C5—22no—
154C22×D5—22no—
155C22×D5—48no—
156C22×D5—42no—
157C22×D5—42no—
158C22×D5C2×#6711yescharacteristic
159C2×AGL(1,𝔽5)—124no—
160C2×AGL(1,𝔽5)—122no—
161C5×D4—24no—
162C5×D4—44no—
163C2×C4×C5#21712yes—
164C2×C4×C5C22×Dic311yescharacteristic
165C22⋊D5—48no—
166C22⋊D5—44no—
167C2×Dic5#21712yes—
168C2×Dic5C2×#6711yescharacteristic
169D20—24no—
170D20—44no—
171C4×D5—24no—
172C4×D5D3×D412yes—
173C4×D5#23712yes—
174#170—64no—
175C22×D4—52no—
176C22×D4—102no—
177C22×C8—152no—
178#127—52no—
179#127—51no—
180#127—102no—
181#127ΓL(1,𝔽16)11yescharacteristic, complemented
182C2×C4⋊C4—152no—
183C2×C42—51no—
184#102—308no—
185#97—302no—
186#95—302no—
187C2×C3×C5—82no—
188C2×C3×C5#39011yescharacteristic
189C2×C3×C5#41812yes—
190C3×D5—82no—
191C3×D5#49012yes—
192C3×D5#49612yes—
193C23×C3—22no—
194C23×C3—51no—
195C23×C3—102no—
196C23×C3—202no—
197C23×C3—202no—
198C23×C3—208no—
199C3×D4—24no—
200C3×D4—44no—
201C3×D4—104no—
202C3×D4—204no—
203C3×D4—204no—
204C3×D4—208no—
205C2×C3×C4—51no—
206C2×C3×C4—52no—
207C2×C3×C4—52no—
208C2×C3×C4—52no—
209C2×C3×C4—104no—
210C2×C3×C4C22⋊AGL(1,𝔽5)12yes—
211C2×C3×C4C22×AGL(1,𝔽5)11yescharacteristic
212C2×Dic3—202no—
213C2×Dic3—204no—
214C3⋊C8—104no—
215C22×C5—44no—
216C22×C5—42no—
217C22×C5C2×#21711yescharacteristic
218C2×D5—44no—
219C2×D5—42no—
220C2×D5—84no—
221C2×D5—84no—
222C2×D5#93214yes—
223C2×D5#93611yescharacteristic
224C2×D5#94011yescharacteristic
225AGL(1,𝔽5)—242no—
226C4×C5—22no—
227C4×C5D4×Dic312yes—
228Dic5—22no—
229Dic5#93812yes—
230C24—102no—
231C2×D4—22no—
232C2×D4—52no—
233C2×D4—104no—
234C2×D4—104no—
235C2×D4—104no—
236C2×D4—102no—
237C2×D4—102no—
238C2×D4—208no—
239C2×D4C2×ΓL(1,𝔽16)12yes—
240C22×C4—51no—
241C22×C4—52no—
242C22×C4—302no—
243C2×C8—154no—
244C2×C8—302no—
245C2×C8—302no—
246C4⋊C4—304no—
247C42—52no—
248C42—51no—
249C42C2×ΓL(1,𝔽16)11yescharacteristic
250C3×C5#196111yescharacteristic, complemented, Hall
251C22×C3—44no—
252C22×C3—42no—
253C22×C3—54no—
254C22×C3—51no—
255C22×C3—51no—
256C22×C3—204no—
257C22×C3—202no—
258C22×C3—404no—
259C22×C3—404no—
260C22×C3C2×C22⋊AGL(1,𝔽5)11yescharacteristic
261C3×C4—22no—
262C3×C4—52no—
263C3×C4—102no—
264C3×C4D4×AGL(1,𝔽5)12yes—
265Dic3—402no—
266C2×C5—82no—
267C2×C5#521512yes—
268C2×C5#522111yescharacteristic
269D5—82no—
270D5#507112yes—
271D5#507412yes—
272C23—22no—
273C23—51no—
274C23—102no—
275C23—202no—
276C23—202no—
277C23—208no—
278D4—24no—
279D4—44no—
280D4—104no—
281D4—204no—
282D4—204no—
283D4—208no—
284C2×C4—51no—
285C2×C4—52no—
286C2×C4—52no—
287C2×C4—52no—
288C2×C4—104no—
289C2×C4—602no—
290C2×C4—604no—
291C2×C4#676312yes—
292C2×C4C22×ΓL(1,𝔽16)11yescharacteristic
293C8—304no—
294C2×C3—52no—
295C2×C3—52no—
296C2×C3—82no—
297C2×C3—402no—
298C2×C3#6576712yes—
299C2×C3#6578011yescharacteristic
300C5#7176511yescharacteristic, complemented, Sylow
301C22—44no—
302C22—42no—
303C22—54no—
304C22—51no—
305C22—51no—
306C22—204no—
307C22—202no—
308C22—404no—
309C22—404no—
310C22C2×#676311yescharacteristic
311C4—22no—
312C4—52no—
313C4—102no—
314C4—1202no—
315C4D4×ΓL(1,𝔽16)12yes—
316C3#1060461711yescharacteristic, complemented, Sylow
317C2—52no—
318C2—52no—
319C2—82no—
320C2—402no—
321C2#1175242612yes—
322C2#1175243911yescharacteristic
subgroupG/H
centerC22C2×#6763
commutatorC2×C3×C4×C5C22×C4
FrattiniC2×C4C22×ΓL(1,𝔽16)
FittingC3×C5×#127C4
socleC22×C3×C5C2×C22⋊C4
prime radicalC3×D5×#127C2
cosocleC2×C3×C4×D5C23
hypercenter#127ΓL(1,𝔽16)
hypocenterC3×C5#1961
LCSG ▷ C2×C3×C4×C5 ▷ C2×C3×C5 ▷ C3×C5
UCSC1 ◁ C22 ◁ C42 ◁ #127

Complements

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

NH ≅ G/Ntype
2. C2×#91236C2semidirect
3. #11752348C2semidirect
4. #11752335C2semidirect
10. C3×C5×#127C4semidirect
12. C3×#4086C4semidirect
13. C3×#4020C4semidirect
39. D5⋊(C3⋊C8)D4semidirect
51. C5×#127Dic3semidirect
53. #4086Dic3semidirect
54. #4020Dic3semidirect
87. C3×#127AGL(1,𝔽5)semidirect
181. #127ΓL(1,𝔽16)semidirect
250. C3×C5#1961semidirect
300. C5#71765semidirect
316. C3#10604617semidirect

Maximal subgroups

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

subgroupindex
C3×D5×#1272
C2×#912362
#117523482
#117523352
#106046173
#717655