Finite Group Databasebeta

← #14282955

Subgroups — #14282955

Order 1920.

480 conjugacy classes of subgroups; only the 118 normal subgroups are shown.

normal subgroups only

This group has 480 conjugacy classes of subgroups — too many to draw. Only its 118 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#11752506C211yesmaximal, characteristic
2#11752339C211yesmaximal, characteristic
3C3×C4×C4⋊AGL(1,𝔽5)C211yesmaximal, characteristic
4#11746502C214yesmaximal
5#10604623—31nomaximal, Hall
6C2×C3×C4⋊AGL(1,𝔽5)—21no—
7C2×C3×C4×AGL(1,𝔽5)—21no—
8#91758C411yescharacteristic, complemented
9C5×#869C411yescharacteristic, complemented
10C3×C42×D5C2211yescharacteristic
11#91388—21no—
12#91360—21no—
13#91359C2212yes—
14#91358—21no—
15#91333—21no—
16Dic5×Dic6C412yescomplemented
17#91232C2212yes—
18C4⋊(C3⋊#170)—21no—
19C4×C3⋊#170—21no—
20C3×(C2×AGL(1,𝔽5))⋊C4C2212yes—
21#69471—51nomaximal, Hall
22#66062—31no—
23C4×C4⋊AGL(1,𝔽5)D311yescharacteristic
24#65680—31no—
25#64926—34no—
26C22×C3×AGL(1,𝔽5)—22no—
27C2×Dic30—21no—
28C2×C5×Dic6C2×C412yes—
29C2×C3×C4×D5D411yescharacteristic
30C2×C3×C4×D5D411yescharacteristic
31C2×C3×C4×D5C2311yescharacteristic
32C2×C3⋊#170—22no—
33C3×C4⋊AGL(1,𝔽5)—24no—
34C3×C4×AGL(1,𝔽5)—42no—
35C3×C42×C5C2×C411yescharacteristic
36C4⋊Dic15—21no—
37C4⋊Dic15C2×C412yes—
38C5×C4⋊Dic3—21no—
39C5×C4⋊Dic3—21no—
40C3×C4×Dic5C2×C411yescharacteristic
41C3×C4×Dic5D411yescharacteristic
42C3×C4×Dic5D411yescharacteristic
43Dic3⋊Dic5—22no—
44#6669—22no—
45#6665—22no—
46Dic3×Dic5—22no—
47#5470—51no—
48C3×C4×C4⋊C4—51no—
49#4925—51no—
50#4452—54no—
51C2×C4⋊AGL(1,𝔽5)—21no—
52C2×C4×AGL(1,𝔽5)—21no—
53C5×C4⋊Q8—31no—
54#4092—32no—
55Q8×Dic5—32no—
56#4043—61no—
57#4038—61no—
58#4036—61no—
59#4033—61no—
60C42×D5C2×D311yescharacteristic
61#4015—31no—
62(C2×AGL(1,𝔽5))⋊C4C2×D312yes—
63#4003—32no—
64C4⋊#170—61no—
65C4×#170—61no—
66#1503—151noSylow
67C22×C3×D5C2×D411yescharacteristic
68C2×C3×AGL(1,𝔽5)—24no—
69C2×C3×AGL(1,𝔽5)—42no—
70C2×C3×AGL(1,𝔽5)—42no—
71C2×C3×C4×C5C22⋊C411yescharacteristic
72C2×C3×C4×C5C22⋊C411yescharacteristic
73C2×C3×C4×C5C22×C411yescharacteristic
74C2×Dic15—22no—
75Dic30—42no—
76C2×C5×Dic3—22no—
77C5×Dic6—42no—
78C2×C3×Dic5C22⋊C411yescharacteristic
79C2×C3×Dic5C22⋊C411yescharacteristic
80C2×C3×Dic5C2×D411yescharacteristic
81C3×C4×D5—22no—
82C3×C4×D5—41no—
83C3×C4×D5SD1612yes—
84C3×C4×D5Q1612yes—
85C3⋊#170—42no—
86C2×C3×C4⋊C4—101no—
87C2×C3×C42—51no—
88C2×C3×C42—101no—
89#926—101no—
90#924—52no—
91#924—101no—
92#870—101no—
93#869—51no—
94#869AGL(1,𝔽5)11yescharacteristic, complemented
95C4×Dic6—52no—
96C4×Dic6—101no—
97C3×#95—52no—
98#830—52no—
99C4⋊(C3⋊C8)—101no—
100C4×C3⋊C8—101no—
101C22×AGL(1,𝔽5)—22no—
102C2×C5×Q8—32no—
103C2×C4×D5#6711yescharacteristic
104C2×C4×D5#6711yescharacteristic
105C2×C4×D5C22×D311yescharacteristic
106C2×Dic10—61no—
107C2×#170—62no—
108C4⋊AGL(1,𝔽5)—24no—
109C4×AGL(1,𝔽5)—42no—
110C5×C4⋊C4—61no—
111C5×C4⋊C4—61no—
112C42×C5C4×D311yescharacteristic
113#685—62no—
114C4⋊Dic5—32no—
115C4⋊Dic5—61no—
116C4⋊Dic5—62no—
117#683—62no—
118C4×Dic5—62no—
119C4×Dic5C4×D311yescharacteristic
120C4×Dic5D1211yescharacteristic
121C4×Dic5D1211yescharacteristic
122#533—151no—
123#431—151no—
124C4×C4⋊C4—51no—
125#328—154no—
126C22×C3×C5C2×C22⋊C411yescharacteristic
127C2×C3×D5—21no—
128C2×C3×D5—21no—
129C2×C3×D5C2×SD1611yescharacteristic
130C2×C3×D5C2×Q1611yescharacteristic
131C3×AGL(1,𝔽5)—44no—
132C3×C4×C5—21no—
133C3×C4×C5#10312yes—
134C3×C4×C5#10412yes—
135Dic15—42no—
136C3×Dic5—21no—
137C3×Dic5—21no—
138C3×Dic5#10312yes—
139C5×Dic3—42no—
140C22×C3×C4—51no—
141C22×C3×C4—51no—
142C22×C3×C4—51no—
143C22×C3×C4—102no—
144C2×Dic6—101no—
145C2×Dic6C2×AGL(1,𝔽5)12yes—
146C3×C4⋊C4—104no—
147C3×C42—51no—
148C3×C42—51no—
149C3×C42—51no—
150C3×C42—202no—
151C3×C42C2×AGL(1,𝔽5)11yescharacteristic
152#217—102no—
153C4⋊Dic3—21no—
154C4⋊Dic3—21no—
155C4⋊Dic3—52no—
156C4⋊Dic3—101no—
157Dic3⋊C4—102no—
158Dic3⋊C4—102no—
159C4×Dic3—102no—
160C2×C3⋊C8—102no—
161C22×D5C2×#6711yescharacteristic
162C2×AGL(1,𝔽5)—24no—
163C2×AGL(1,𝔽5)—42no—
164C2×AGL(1,𝔽5)—42no—
165C5×Q8—122no—
166C2×C4×C5—62no—
167C2×C4×C5(C2×D3)⋊C411yescharacteristic
168C2×C4×C5(C2×D3)⋊C411yescharacteristic
169C2×C4×C5C2×C4×D311yescharacteristic
170C2×Dic5—62no—
171C2×Dic5(C2×D3)⋊C411yescharacteristic
172C2×Dic5(C2×D3)⋊C411yescharacteristic
173C2×Dic5C2×D1211yescharacteristic
174C4×D5—22no—
175C4×D5—41no—
176C4×D5#21412yes—
177C4×D5#21612yes—
178Dic10—122no—
179#170—122no—
180C4⋊Q8—31no—
181C4⋊Q8—151no—
182#125—301no—
183C22⋊Q8—152no—
184C22⋊Q8—301no—
185C4×Q8—152no—
186C4×Q8—301no—
187#117—301no—
188C2×C4⋊C4—101no—
189C2×C42—51no—
190C2×C42—101no—
191C4⋊C8—301no—
192C22⋊C8—152no—
193C4×C8—301no—
194#95—52no—
195C2×C3×C5#41011yescharacteristic
196C2×C3×C5C2×#10311yescharacteristic
197C2×C3×C5C2×#10411yescharacteristic
198C3×D5—21no—
199C3×D5—21no—
200C23×C3—51no—
201C2×C3×C4—51no—
202C2×C3×C4—52no—
203C2×C3×C4—52no—
204C2×C3×C4—51no—
205C2×C3×C4—51no—
206C2×C3×C4—102no—
207C2×C3×C4—104no—
208C2×C3×C4—201no—
209C2×C3×C4—202no—
210C2×C3×C4—202no—
211C2×C3×C4C22⋊AGL(1,𝔽5)11yescharacteristic
212C2×C3×C4C22⋊AGL(1,𝔽5)11yescharacteristic
213C2×C3×C4C22×AGL(1,𝔽5)11yescharacteristic
214C2×Dic3—22no—
215C2×Dic3—102no—
216Dic6—42no—
217Dic6—202no—
218C3⋊C8—202no—
219C22×C5C2×(C2×D3)⋊C411yescharacteristic
220C2×D5—21no—
221C2×D5—21no—
222C2×D5C2×#21411yescharacteristic
223C2×D5C2×#21611yescharacteristic
224AGL(1,𝔽5)—44no—
225C4×C5—21no—
226C4×C5—122no—
227C4×C5#80512yes—
228C4×C5#81012yes—
229Dic5—21no—
230Dic5—21no—
231Dic5—122no—
232Dic5#81012yes—
233C2×Q8—32no—
234C2×Q8—301no—
235C22×C4—51no—
236C22×C4—51no—
237C22×C4—51no—
238C22×C4—102no—
239C2×C8—302no—
240C4⋊C4—61no—
241C4⋊C4—61no—
242C4⋊C4—104no—
243C4⋊C4—152no—
244C4⋊C4—302no—
245C4⋊C4—302no—
246C4⋊C4—301no—
247C22⋊C4—302no—
248C42—51no—
249C42—51no—
250C42—51no—
251C42—202no—
252C42—302no—
253C42D3×AGL(1,𝔽5)11yescharacteristic
254C3×C5#150311yescharacteristic, complemented, Hall
255C22×C3—51no—
256C22×C3—51no—
257C22×C3—101no—
258C22×C3—101no—
259C22×C3C2×C22⋊AGL(1,𝔽5)11yescharacteristic
260C3×C4—21no—
261C3×C4—52no—
262C3×C4—101no—
263C3×C4—101no—
264C3×C4—204no—
265C3×C4#400912yes—
266C3×C4#401012yes—
267Dic3—42no—
268Dic3—202no—
269C2×C5C2×#80511yescharacteristic
270C2×C5C2×#81011yescharacteristic
271C2×C5#500611yescharacteristic
272D5—21no—
273D5—21no—
274C23—51no—
275Q8—122no—
276Q8—602no—
277C2×C4—51no—
278C2×C4—52no—
279C2×C4—52no—
280C2×C4—51no—
281C2×C4—51no—
282C2×C4—62no—
283C2×C4—102no—
284C2×C4—104no—
285C2×C4—201no—
286C2×C4—202no—
287C2×C4—202no—
288C2×C4—302no—
289C2×C4(C2×D3)⋊AGL(1,𝔽5)11yescharacteristic
290C2×C4(C2×D3)⋊AGL(1,𝔽5)11yescharacteristic
291C2×C4C2×D3×AGL(1,𝔽5)11yescharacteristic
292C8—602no—
293C2×C3—101no—
294C2×C3—101no—
295C2×C3#6577111yescharacteristic
296C2×C3C2×#400911yescharacteristic
297C2×C3C2×#401011yescharacteristic
298C5#6947111yescharacteristic, complemented, Sylow
299C22—51no—
300C22—51no—
301C22—101no—
302C22—101no—
303C22C2×(C2×D3)⋊AGL(1,𝔽5)11yescharacteristic
304C4—21no—
305C4—52no—
306C4—101no—
307C4—101no—
308C4—122no—
309C4—204no—
310C4—602no—
311C4#9115312yes—
312C4#9115812yes—
313C3#1060462311yescharacteristic, complemented, Sylow
314C2—101no—
315C2—101no—
316C2C2×#9115311yescharacteristic
317C2C2×#9115811yescharacteristic
318C2#1175181611yescharacteristic
subgroupG/H
centerC22C2×(C2×D3)⋊AGL(1,𝔽5)
commutatorC2×C3×C4×C5C22×C4
FrattiniC2×C4C2×D3×AGL(1,𝔽5)
FittingC3×C42×C5C2×C4
socleC22×C3×C5C2×C22⋊C4
prime radicalC22×C3×D5C2×D4
cosocleC2×C3×C4×D5C23
hypercenterC42D3×AGL(1,𝔽5)
hypocenterC3×C5#1503
LCSG ▷ C2×C3×C4×C5 ▷ C22×C3×C5 ▷ C3×C5
UCSC1 ◁ C22 ◁ C42

Complements

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

NH ≅ G/Ntype
8. #91758C4semidirect
9. C5×#869C4semidirect
16. Dic5×Dic6C4semidirect
94. #869AGL(1,𝔽5)semidirect
254. C3×C5#1503semidirect
298. C5#69471semidirect
313. C3#10604623semidirect

Maximal subgroups

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

subgroupindex
#117525062
#117523392
C3×C4×C4⋊AGL(1,𝔽5)2
#117465022
#106046233
#694715