Finite Group Databasebeta

← #14283059

Subgroups — #14283059

Order 1920.

544 conjugacy classes of subgroups; only the 98 normal subgroups are shown.

normal subgroups only

This group has 544 conjugacy classes of subgroups — too many to draw. Only its 98 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#11752345C211yesmaximal, characteristic, complemented
3C3×#65683C211yesmaximal, characteristic, complemented
4#11746499C214yesmaximal, complemented
5#10604624—31nomaximal, Hall
6C2×#6773—21no—
7C2×C4×ΓL(1,𝔽16)—21no—
8#91763C411yescharacteristic, complemented
9C5×#869C411yescharacteristic, complemented
10C3×C42×D5C2211yescharacteristic
11#91450—21no—
12#91450C2212yes—
13#91418—21no—
14#91418C412yescomplemented
15#91375—21no—
16#91235—21no—
17#91235C2212yes—
18C3×#4003C2212yes—
19C3×#4002—21no—
20C3×C4⋊#170—21no—
21#70832—51nomaximal, Hall
22#66059—31no—
23#65686—31no—
24#65683D311yescharacteristic, complemented
25#64868—34no—
26C22×ΓL(1,𝔽16)—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—44no—
37C4×ΓL(1,𝔽16)—24no—
38C2×C3×#170—22no—
39C3×C42×C5C2×C411yescharacteristic
40C5×C4⋊Dic3—21no—
41C5×C4⋊Dic3C2×C412yes—
42C3×C4×Dic5C2×C411yescharacteristic
43C3×C4×Dic5D411yescharacteristic
44C3×C4×Dic5D411yescharacteristic
45#6667—22no—
46#6667—22no—
47#5477—51no—
48C3×#432—51no—
49#4932—51no—
50#4470—54no—
51C2×#713—61no—
52C2×C4×AGL(1,𝔽5)—61no—
53C5×C4⋊Q8—31no—
54#4093—61no—
55#4041—32no—
56#4041—61no—
57#4039—32no—
58#4039—61no—
59#4020—31no—
60C42×D5C2×D311yescharacteristic
61(C2×AGL(1,𝔽5))⋊C4—32no—
62(C2×AGL(1,𝔽5))⋊C4—61no—
63#4003C2×D312yes—
64#4002—21no—
65C4⋊#170—21no—
66#1505—151noSylow
67C22×D15—22no—
68C22×C3×D5C2×D411yescharacteristic
69C2×ΓL(1,𝔽16)—24no—
70C2×ΓL(1,𝔽16)—42no—
71C2×ΓL(1,𝔽16)—42no—
72C2×C3×C4×C5C22⋊C411yescharacteristic
73C2×C3×C4×C5C22⋊C411yescharacteristic
74C2×C3×C4×C5C22×C411yescharacteristic
75D60—42no—
76D60—42no—
77D60—42no—
78C2×C5×Dic3—22no—
79C5×Dic6—42no—
80C2×C3×Dic5C22⋊C411yescharacteristic
81C2×C3×Dic5C22⋊C411yescharacteristic
82C2×C3×Dic5C2×D411yescharacteristic
83C3×C4×D5—22no—
84C3×C4×D5—22no—
85C3×C4×D5—41no—
86#1202—44no—
87#1200—44no—
88C3×#170—42no—
89C2×#235—101no—
90C2×C3×C42—51no—
91#930—52no—
92#930—101no—
93C2×C4×Dic3—101no—
94#875—101no—
95#874—51no—
96C4×D12—52no—
97C4×D12—101no—
98#869AGL(1,𝔽5)11yescharacteristic, complemented
99C3×C4⋊C8—101no—
100C3×C22⋊C8—52no—
101C3×#97—101no—
102C3⋊#95—52no—
103C3⋊#95—101no—
104C22×AGL(1,𝔽5)—62no—
105C2×C5×Q8—61no—
106#713—124no—
107C2×D20—32no—
108C2×D20—62no—
109C2×D20—61no—
110C2×D20—61no—
111C2×C4×D5—62no—
112C2×C4×D5D1211yescharacteristic
113C2×C4×D5D1211yescharacteristic
114C2×C4×D5C22×D311yescharacteristic
115C2×#170—22no—
116C4×AGL(1,𝔽5)—64no—
117C5×C4⋊C4—32no—
118C5×C4⋊C4—61no—
119C42×C5C4×D311yescharacteristic
120#685—62no—
121#685—62no—
122C4×Dic5C4×D311yescharacteristic
123C4×Dic5#6711yescharacteristic
124C4×Dic5#6711yescharacteristic
125#532—151no—
126#432—51no—
127#376—151no—
128#327—154no—
129C22×C3×C5C2×C22⋊C411yescharacteristic
130C2×D15—42no—
131C2×D15—42no—
132C2×D15—42no—
133C2×C3×D5—21no—
134C2×C3×D5—21no—
135C2×C3×D5AGL(1,ℤ/8)12yes—
136ΓL(1,𝔽16)—44no—
137C3×C4×C5—21no—
138C3×C4×C5—21no—
139C3×C4×C5#10412yes—
140C3×Dic5—21no—
141C3×Dic5—21no—
142C3×Dic5—21no—
143C5×Dic3—42no—
144C22×C3×C4—51no—
145C22×C3×C4—51no—
146C22×C3×C4—51no—
147C2×#67—102no—
148C22×Dic3—102no—
149#235—204no—
150C2×D12—52no—
151C2×D12—101no—
152C2×D12—101no—
153C2×C4×D3—102no—
154C2×Dic6—21no—
155C2×C3×C8—102no—
156C3×C42—51no—
157C3×C42—51no—
158C3×C42—51no—
159C3×C42C2×AGL(1,𝔽5)11yescharacteristic
160(C2×D3)⋊C4—102no—
161(C2×D3)⋊C4—102no—
162C4⋊Dic3—21no—
163C4⋊Dic3C2×AGL(1,𝔽5)12yes—
164C4×Dic3—104no—
165C22×D5—62no—
166C22×D5C2×D1211yescharacteristic
167C2×AGL(1,𝔽5)—64no—
168C2×AGL(1,𝔽5)—122no—
169C2×AGL(1,𝔽5)—122no—
170C5×Q8—122no—
171C2×C4×C5—62no—
172C2×C4×C5(C2×D3)⋊C411yescharacteristic
173C2×C4×C5(C2×D3)⋊C411yescharacteristic
174C2×C4×C5C2×C4×D311yescharacteristic
175C2×Dic5(C2×D3)⋊C411yescharacteristic
176C2×Dic5(C2×D3)⋊C411yescharacteristic
177C2×Dic5C2×#6711yescharacteristic
178D20—122no—
179D20—122no—
180D20—124no—
181D20—122no—
182C4×D5—22no—
183C4×D5—22no—
184C4×D5—41no—
185C4×D5—124no—
186#170—42no—
187C2×#41—301no—
188C4⋊Q8—31no—
189#127—151no—
190#124—301no—
191#121—152no—
192#121—301no—
193C4×D4—152no—
194C4×D4—301no—
195C2×C42—51no—
196C2×C42—301no—
197C4⋊C8—101no—
198C22⋊C8—52no—
199#97—101no—
200#95—152no—
201#95—301no—
202C2×C3×C5#41011yescharacteristic
203C2×C3×C5#41811yescharacteristic
204C2×C3×C5C2×#10411yescharacteristic
205D15—82no—
206C3×D5—22no—
207C23×C3—51no—
208C22×D3—102no—
209C2×C3×C4—51no—
210C2×C3×C4—51no—
211C2×C3×C4—51no—
212C2×C3×C4—102no—
213C2×C3×C4—102no—
214C2×C3×C4—201no—
215C2×C3×C4C22⋊AGL(1,𝔽5)11yescharacteristic
216C2×C3×C4C22⋊AGL(1,𝔽5)11yescharacteristic
217C2×C3×C4C22×AGL(1,𝔽5)11yescharacteristic
218#67—204no—
219C2×Dic3—22no—
220C2×Dic3—104no—
221C2×Dic3—202no—
222C2×Dic3—202no—
223D12—202no—
224D12—202no—
225D12—202no—
226C4×D3—204no—
227Dic6—42no—
228C3×C8—202no—
229C22×C5C2×(C2×D3)⋊C411yescharacteristic
230C2×D5—21no—
231C2×D5—21no—
232C2×D5—122no—
233C2×D5—122no—
234C2×D5—122no—
235C2×D5#90812yes—
236AGL(1,𝔽5)—124no—
237C4×C5—21no—
238C4×C5—21no—
239C4×C5—122no—
240C4×C5#82512yes—
241Dic5—21no—
242Dic5—21no—
243Dic5—21no—
244#41—604no—
245C2×Q8—61no—
246C2×D4—152no—
247C2×D4—302no—
248C2×D4—301no—
249C2×D4—301no—
250C22×C4—51no—
251C22×C4—51no—
252C22×C4—51no—
253C22×C4—302no—
254C22×C4—302no—
255C2×C8—102no—
256C4⋊C4—32no—
257C4⋊C4—61no—
258C22⋊C4—302no—
259C22⋊C4—302no—
260C42—51no—
261C42—51no—
262C42—51no—
263C42—304no—
264C42D3×AGL(1,𝔽5)11yescharacteristic
265C3×C5#150511yescharacteristic, complemented, Hall
266C22×C3—52no—
267C22×C3—101no—
268C22×C3—101no—
269C22×C3C2×C22⋊AGL(1,𝔽5)11yescharacteristic
270C2×D3—202no—
271C2×D3—202no—
272C2×D3—202no—
273C3×C4—21no—
274C3×C4—21no—
275C3×C4—101no—
276C3×C4—101no—
277C3×C4—101no—
278C3×C4#401012yes—
279Dic3—42no—
280Dic3—204no—
281C2×C5#500611yescharacteristic
282C2×C5#513111yescharacteristic
283C2×C5C2×#82511yescharacteristic
284D5—22no—
285D5—242no—
286C23—51no—
287C23—302no—
288Q8—122no—
289D4—602no—
290D4—602no—
291D4—604no—
292D4—602no—
293C2×C4—51no—
294C2×C4—51no—
295C2×C4—51no—
296C2×C4—62no—
297C2×C4—102no—
298C2×C4—102no—
299C2×C4—201no—
300C2×C4—304no—
301C2×C4—604no—
302C2×C4—602no—
303C2×C4—602no—
304C2×C4(C2×D3)⋊AGL(1,𝔽5)11yescharacteristic
305C2×C4(C2×D3)⋊AGL(1,𝔽5)11yescharacteristic
306C2×C4C2×D3×AGL(1,𝔽5)11yescharacteristic
307C8—202no—
308C2×C3—102no—
309C2×C3#6576711yescharacteristic
310C2×C3C2×#401011yescharacteristic
311C2×C3#6578511yescharacteristic
312D3—402no—
313C5#7083211yescharacteristic, complemented, Sylow
314C22—52no—
315C22—101no—
316C22—101no—
317C22—602no—
318C22—602no—
319C22—602no—
320C22C2×(C2×D3)⋊AGL(1,𝔽5)11yescharacteristic
321C4—21no—
322C4—21no—
323C4—101no—
324C4—101no—
325C4—101no—
326C4—122no—
327C4—604no—
328C4#9115712yes—
329C3#1060462411yescharacteristic, complemented, Sylow
330C2—102no—
331C2—1202no—
332C2#1175170711yescharacteristic
333C2C2×#9115711yescharacteristic
334C2#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#1505
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. #11752345C2semidirect
3. C3×#65683C2semidirect
4. #11746499C2semidirect
8. #91763C4semidirect
9. C5×#869C4semidirect
14. #91418C4semidirect
24. #65683D3semidirect
98. #869AGL(1,𝔽5)semidirect
265. C3×C5#1505semidirect
313. C5#70832semidirect
329. C3#10604624semidirect

Maximal subgroups

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

subgroupindex
#117526742
#117523452
C3×#656832
#117464992
#106046243
#708325