Finite Group Databasebeta

← #14282912

Subgroups — #14282912

Order 1920.

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

normal subgroups only

This group has 568 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#11752674C211yesmaximal, characteristic
2#11752339C211yesmaximal, characteristic, complemented
3C3×C4×C4⋊AGL(1,𝔽5)C211yesmaximal, characteristic, complemented
4#11746497C214yesmaximal, complemented
5#10604621—31nomaximal, Hall
6C2×#6773—21no—
7C2×C3×C4⋊AGL(1,𝔽5)—21no—
8C2×C3×C4×AGL(1,𝔽5)—21no—
9#91763C411yescharacteristic, complemented
10C5×#869C411yescharacteristic, complemented
11C3×C42×D5C2211yescharacteristic
12#91450—21no—
13#91450C2212yes—
14#91418—21no—
15#91418C412yescomplemented
16#91375—21no—
17#91232C2212yes—
18C4⋊(C3⋊#170)—21no—
19C4×C3⋊#170—21no—
20C3×(C2×AGL(1,𝔽5))⋊C4C2212yes—
21#69469—51nomaximal, Hall
22#66059—31no—
23C4×C4⋊AGL(1,𝔽5)D311yescharacteristic, complemented
24#65680—31no—
25#64868—34no—
26C22×C3×AGL(1,𝔽5)—22no—
27C2×D60—21no—
28C2×D60—21no—
29C2×D60C2×C412yes—
30C2×C5×Dic6—21no—
31C2×C3×C4×D5D411yescharacteristic
32C2×C3×C4×D5D411yescharacteristic
33C2×C3×C4×D5C2311yescharacteristic
34C2×#1202—22no—
35C2×#1200—22no—
36#6773—42no—
37#6773—42no—
38C2×C3⋊#170—22no—
39C3×C4⋊AGL(1,𝔽5)—24no—
40C3×C4×AGL(1,𝔽5)—42no—
41C3×C42×C5C2×C411yescharacteristic
42C5×C4⋊Dic3—21no—
43C5×C4⋊Dic3C2×C412yes—
44C3×C4×Dic5C2×C411yescharacteristic
45C3×C4×Dic5D411yescharacteristic
46C3×C4×Dic5D411yescharacteristic
47#6667—22no—
48#6667—22no—
49#5477—51no—
50C3×C4×C4⋊C4—51no—
51#4925—51no—
52#4451—54no—
53C2×#713—61no—
54C2×C4⋊AGL(1,𝔽5)—21no—
55C2×C4×AGL(1,𝔽5)—21no—
56C5×C4⋊Q8—31no—
57#4093—61no—
58#4041—32no—
59#4041—61no—
60#4039—32no—
61#4039—61no—
62#4020—31no—
63C42×D5C2×D311yescharacteristic
64(C2×AGL(1,𝔽5))⋊C4C2×D312yes—
65#4003—32no—
66C4⋊#170—61no—
67C4×#170—61no—
68#1502—151noSylow
69C22×D15—22no—
70C22×C3×D5C2×D411yescharacteristic
71C2×C3×AGL(1,𝔽5)—24no—
72C2×C3×AGL(1,𝔽5)—42no—
73C2×C3×AGL(1,𝔽5)—42no—
74C2×C3×C4×C5C22⋊C411yescharacteristic
75C2×C3×C4×C5C22⋊C411yescharacteristic
76C2×C3×C4×C5C22×C411yescharacteristic
77D60—42no—
78D60—42no—
79D60—42no—
80C2×C5×Dic3—22no—
81C5×Dic6—42no—
82C2×C3×Dic5C22⋊C411yescharacteristic
83C2×C3×Dic5C22⋊C411yescharacteristic
84C2×C3×Dic5C2×D411yescharacteristic
85C3×C4×D5—22no—
86C3×C4×D5—41no—
87C3×C4×D5D812yes—
88C3×C4×D5SD1612yes—
89#1202—42no—
90#1202—42no—
91#1200—44no—
92C3⋊#170—42no—
93C2×#235—101no—
94C2×C3×C4⋊C4—101no—
95C2×C3×C42—51no—
96C2×C3×C42—101no—
97#930—52no—
98#930—101no—
99#875—101no—
100#874—51no—
101C4×D12—52no—
102C4×D12—101no—
103#869AGL(1,𝔽5)11yescharacteristic, complemented
104C3×#95—52no—
105#830—52no—
106C4⋊(C3⋊C8)—101no—
107C4×C3⋊C8—101no—
108C22×AGL(1,𝔽5)—22no—
109C2×C5×Q8—61no—
110#713—122no—
111#713—122no—
112C2×D20—32no—
113C2×D20—62no—
114C2×D20—61no—
115C2×D20—61no—
116C2×C4×D5—62no—
117C2×C4×D5#6711yescharacteristic
118C2×C4×D5#6711yescharacteristic
119C2×C4×D5C22×D311yescharacteristic
120C2×#170—62no—
121C4⋊AGL(1,𝔽5)—24no—
122C4×AGL(1,𝔽5)—42no—
123C5×C4⋊C4—32no—
124C5×C4⋊C4—61no—
125C42×C5C4×D311yescharacteristic
126#685—62no—
127#685—62no—
128C4×Dic5C4×D311yescharacteristic
129C4×Dic5D1211yescharacteristic
130C4×Dic5D1211yescharacteristic
131#532—151no—
132#431—151no—
133C4×C4⋊C4—51no—
134#327—154no—
135C22×C3×C5C2×C22⋊C411yescharacteristic
136C2×D15—42no—
137C2×D15—42no—
138C2×D15—42no—
139C2×C3×D5—21no—
140C2×C3×D5—21no—
141C2×C3×D5C2×D811yescharacteristic
142C2×C3×D5C2×SD1611yescharacteristic
143C3×AGL(1,𝔽5)—44no—
144C3×C4×C5—21no—
145C3×C4×C5#10212yes—
146C3×C4×C5#10412yes—
147C3×Dic5—21no—
148C3×Dic5—21no—
149C3×Dic5#10212yes—
150C5×Dic3—42no—
151C22×C3×C4—51no—
152C22×C3×C4—51no—
153C22×C3×C4—51no—
154C22×C3×C4—102no—
155C2×#67—102no—
156#235—202no—
157#235—202no—
158C2×D12—52no—
159C2×D12—101no—
160C2×D12—101no—
161C2×C4×D3—102no—
162C2×Dic6—21no—
163C3×C4⋊C4—104no—
164C3×C42—51no—
165C3×C42—51no—
166C3×C42—51no—
167C3×C42—202no—
168C3×C42C2×AGL(1,𝔽5)11yescharacteristic
169(C2×D3)⋊C4—102no—
170(C2×D3)⋊C4—102no—
171C4⋊Dic3—21no—
172C4⋊Dic3C2×AGL(1,𝔽5)12yes—
173C2×C3⋊C8—102no—
174C22×D5—62no—
175C22×D5C2×#6711yescharacteristic
176C2×AGL(1,𝔽5)—24no—
177C2×AGL(1,𝔽5)—42no—
178C2×AGL(1,𝔽5)—42no—
179C5×Q8—122no—
180C2×C4×C5—62no—
181C2×C4×C5(C2×D3)⋊C411yescharacteristic
182C2×C4×C5(C2×D3)⋊C411yescharacteristic
183C2×C4×C5C2×C4×D311yescharacteristic
184C2×Dic5(C2×D3)⋊C411yescharacteristic
185C2×Dic5(C2×D3)⋊C411yescharacteristic
186C2×Dic5C2×D1211yescharacteristic
187D20—122no—
188D20—122no—
189D20—122no—
190D20—122no—
191D20—122no—
192C4×D5—22no—
193C4×D5—41no—
194C4×D5—124no—
195C4×D5#21312yes—
196C4×D5#21512yes—
197#170—122no—
198C2×#41—301no—
199C4⋊Q8—31no—
200#127—151no—
201#124—301no—
202#121—152no—
203#121—301no—
204C4×D4—152no—
205C4×D4—301no—
206C2×C4⋊C4—101no—
207C2×C42—51no—
208C2×C42—101no—
209C4⋊C8—301no—
210C22⋊C8—152no—
211C4×C8—301no—
212#95—52no—
213C2×C3×C5#41011yescharacteristic
214C2×C3×C5C2×#10211yescharacteristic
215C2×C3×C5C2×#10411yescharacteristic
216D15—82no—
217C3×D5—21no—
218C3×D5—21no—
219C23×C3—51no—
220C22×D3—102no—
221C2×C3×C4—51no—
222C2×C3×C4—52no—
223C2×C3×C4—52no—
224C2×C3×C4—51no—
225C2×C3×C4—51no—
226C2×C3×C4—102no—
227C2×C3×C4—104no—
228C2×C3×C4—201no—
229C2×C3×C4—202no—
230C2×C3×C4—202no—
231C2×C3×C4C22⋊AGL(1,𝔽5)11yescharacteristic
232C2×C3×C4C22⋊AGL(1,𝔽5)11yescharacteristic
233C2×C3×C4C22×AGL(1,𝔽5)11yescharacteristic
234#67—202no—
235#67—202no—
236C2×Dic3—22no—
237D12—202no—
238D12—202no—
239D12—202no—
240C4×D3—204no—
241Dic6—42no—
242C3⋊C8—202no—
243C22×C5C2×(C2×D3)⋊C411yescharacteristic
244C2×D5—21no—
245C2×D5—21no—
246C2×D5—122no—
247C2×D5—122no—
248C2×D5—122no—
249C2×D5C2×#21311yescharacteristic
250C2×D5C2×#21511yescharacteristic
251AGL(1,𝔽5)—44no—
252C4×C5—21no—
253C4×C5—122no—
254C4×C5#80512yes—
255C4×C5#80912yes—
256Dic5—21no—
257Dic5—21no—
258Dic5#80912yes—
259#41—602no—
260#41—602no—
261C2×Q8—61no—
262C2×D4—152no—
263C2×D4—302no—
264C2×D4—301no—
265C2×D4—301no—
266C22×C4—51no—
267C22×C4—51no—
268C22×C4—51no—
269C22×C4—102no—
270C22×C4—302no—
271C2×C8—302no—
272C4⋊C4—32no—
273C4⋊C4—61no—
274C4⋊C4—104no—
275C22⋊C4—302no—
276C22⋊C4—302no—
277C42—51no—
278C42—51no—
279C42—51no—
280C42—202no—
281C42D3×AGL(1,𝔽5)11yescharacteristic
282C3×C5#150211yescharacteristic, complemented, Hall
283C22×C3—51no—
284C22×C3—51no—
285C22×C3—101no—
286C22×C3—101no—
287C22×C3C2×C22⋊AGL(1,𝔽5)11yescharacteristic
288C2×D3—202no—
289C2×D3—202no—
290C2×D3—202no—
291C3×C4—21no—
292C3×C4—52no—
293C3×C4—101no—
294C3×C4—101no—
295C3×C4—204no—
296C3×C4#400712yes—
297C3×C4#401012yes—
298Dic3—42no—
299C2×C5C2×#80511yescharacteristic
300C2×C5C2×#80911yescharacteristic
301C2×C5#500611yescharacteristic
302D5—21no—
303D5—21no—
304D5—242no—
305C23—51no—
306C23—302no—
307Q8—122no—
308D4—602no—
309D4—602no—
310D4—602no—
311D4—602no—
312D4—602no—
313C2×C4—51no—
314C2×C4—52no—
315C2×C4—52no—
316C2×C4—51no—
317C2×C4—51no—
318C2×C4—62no—
319C2×C4—102no—
320C2×C4—104no—
321C2×C4—201no—
322C2×C4—202no—
323C2×C4—202no—
324C2×C4—604no—
325C2×C4(C2×D3)⋊AGL(1,𝔽5)11yescharacteristic
326C2×C4(C2×D3)⋊AGL(1,𝔽5)11yescharacteristic
327C2×C4C2×D3×AGL(1,𝔽5)11yescharacteristic
328C8—602no—
329C2×C3—101no—
330C2×C3—101no—
331C2×C3C2×#400711yescharacteristic
332C2×C3C2×#401011yescharacteristic
333C2×C3#6578511yescharacteristic
334D3—402no—
335C5#6946911yescharacteristic, complemented, Sylow
336C22—51no—
337C22—51no—
338C22—101no—
339C22—101no—
340C22—602no—
341C22—602no—
342C22—602no—
343C22C2×(C2×D3)⋊AGL(1,𝔽5)11yescharacteristic
344C4—21no—
345C4—52no—
346C4—101no—
347C4—101no—
348C4—122no—
349C4—204no—
350C4#9115112yes—
351C4#9115812yes—
352C3#1060462111yescharacteristic, complemented, Sylow
353C2—101no—
354C2—101no—
355C2—1202no—
356C2C2×#9115111yescharacteristic
357C2C2×#9115811yescharacteristic
358C2#1175181711yescharacteristic
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 radical#11752674C2
cosocleC2×C3×C4×D5C23
hypercenterC42D3×AGL(1,𝔽5)
hypocenterC3×C5#1502
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
2. #11752339C2semidirect
3. C3×C4×C4⋊AGL(1,𝔽5)C2semidirect
4. #11746497C2semidirect
9. #91763C4semidirect
10. C5×#869C4semidirect
15. #91418C4semidirect
23. C4×C4⋊AGL(1,𝔽5)D3semidirect
103. #869AGL(1,𝔽5)semidirect
282. C3×C5#1502semidirect
335. C5#69469semidirect
352. C3#10604621semidirect

Maximal subgroups

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

subgroupindex
#117526742
#117523392
C3×C4×C4⋊AGL(1,𝔽5)2
#117464972
#106046213
#694695