Finite Group Databasebeta

← #14305917

Subgroups — #14305917

Order 1920.

688 conjugacy classes of subgroups; only the 134 normal subgroups are shown.

normal subgroups only

This group has 688 conjugacy classes of subgroups — too many to draw. Only its 134 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
1D5×#875C211yesmaximal, characteristic
2#11752347C211yesmaximal, characteristic, complemented
3C3×#65676C211yesmaximal, characteristic, complemented
4C2×#91154C212yesmaximal, complemented
5C2×#91152C212yesmaximal
6#10604567—31nomaximal, Hall
7C2×D5×D12C2211yescharacteristic
8C2×D5×Dic6C2211yescharacteristic
9C2×C4⋊ΓL(1,𝔽16)—22no—
10C2×C4⋊ΓL(1,𝔽16)C2212yes—
11C2×C3×D5⋊C8C2212yes—
12#91764C411yescharacteristic, complemented
13C5×#875C411yescharacteristic, complemented
14C3×C42×D5C2211yescharacteristic
15D5×(C2×D3)⋊C4—22no—
16#91375C411yescharacteristic, complemented
17#91373C411yescharacteristic, complemented
18#91350—22no—
19C3×#4002—22no—
20#91154—24no—
21#91152—24no—
22#69821—51nomaximal, Hall
23D5×#124—31no—
24C2×#4009—32no—
25C2×#4007—32no—
26#65688—31no—
27#65676D311yescharacteristic, complemented
28C22×D3×D5—21no—
29C22×ΓL(1,𝔽16)—22no—
30C2×D60C2×C411yescharacteristic
31C2×Dic30C2×C411yescharacteristic
32C2×C5×D12C2×C411yescharacteristic
33C2×C5×Dic6C2×C411yescharacteristic
34C2×C3×C4×D5D412yes—
35C2×C3×C4×D5C2311yescharacteristic
36C2×#1204—21no—
37C2×#1203—21no—
38C2×D5×Dic3—21no—
39D5×D12—22no—
40D5×D12—22no—
41D5×Dic6—22no—
42D5×Dic6—22no—
43C4⋊ΓL(1,𝔽16)—22no—
44C4⋊ΓL(1,𝔽16)—22no—
45C4⋊ΓL(1,𝔽16)—44no—
46C2×C3×#170—22no—
47C3×D5⋊C8—22no—
48C3×D5⋊C8D414yes—
49C3×C42×C5C2×C411yescharacteristic
50#6714—22no—
51C5×(C2×D3)⋊C4—22no—
52C3×C4×Dic5C2×C411yescharacteristic
53C3×C4×Dic5D412yes—
54#6667—22no—
55(C2×D3)⋊Dic5—22no—
56C2×#875—51no—
57C2×C3×#97—51no—
58C2×#821—52no—
59C2×#816—52no—
60#4935—51no—
61C2×Q8×D5—31no—
62C2×D4×D5—31no—
63C2×C4⋊AGL(1,𝔽5)—32no—
64C2×C4⋊AGL(1,𝔽5)—62no—
65C2×D5⋊C8C2×D312yes—
66C5×#124—31no—
67#4093—31no—
68#4082—31no—
69#4031—62no—
70D5×C22⋊C4—62no—
71#4021—31no—
72C42×D5C2×D311yescharacteristic
73#4009—64no—
74#4007—64no—
75#4002—22no—
76#1959—151noSylow
77C22×D15—21no—
78C22×C5×D3—21no—
79C22×C3×D5C2×D411yescharacteristic
80C2×D3×D5—41no—
81C2×D3×D5—41no—
82C2×D3×D5—42no—
83C2×D3×D5—42no—
84C2×ΓL(1,𝔽16)—42no—
85C2×ΓL(1,𝔽16)—44no—
86C2×C3×C4×C5C22⋊C412yes—
87C2×C3×C4×C5C22×C411yescharacteristic
88C2×Dic15—21no—
89D60—22no—
90Dic30—22no—
91C2×C5×Dic3—21no—
92C5×D12—22no—
93C5×Dic6—22no—
94C2×C3×Dic5C22⋊C412yes—
95C2×C3×Dic5C2×D411yescharacteristic
96C3×C4×D5—22no—
97C3×C4×D5—22no—
98C3×C4×D5C2×D412yes—
99C3×C4×D5#4112yes—
100#1204—42no—
101#1203—42no—
102D5×Dic3—41no—
103D5×Dic3—41no—
104C3×#170—24no—
105C22×D12—51no—
106C22×Dic6—51no—
107C22×C3×C8—52no—
108C2×C3×C42—51no—
109C2×(C2×D3)⋊C4—102no—
110C2×C4⋊Dic3—52no—
111C2×C4⋊Dic3—102no—
112#875—51no—
113#875—51no—
114#875—51no—
115#875—102no—
116#875AGL(1,𝔽5)11yescharacteristic, complemented
117C3×#97—102no—
118#821—104no—
119#816—104no—
120C23×D5—61no—
121C22×AGL(1,𝔽5)—62no—
122C2×C5×Q8—31no—
123C2×C5×D4—31no—
124C2×C22⋊D5—61no—
125Q8×D5—62no—
126Q8×D5—62no—
127D4×D5—62no—
128D4×D5—62no—
129C2×D20—31no—
130C2×C4×D5—61no—
131C2×C4×D5D1212yes—
132C2×C4×D5C22×D311yescharacteristic
133C2×Dic10—31no—
134C2×Dic10—61no—
135C2×#170—22no—
136C4⋊AGL(1,𝔽5)—62no—
137C4⋊AGL(1,𝔽5)—62no—
138C4⋊AGL(1,𝔽5)—124no—
139D5⋊C8—22no—
140D5⋊C8D1214yes—
141C5×C22⋊C4—62no—
142C42×C5C4×D311yescharacteristic
143C22⋊Dic5—62no—
144#685—62no—
145#685—62no—
146C4×Dic5C4×D311yescharacteristic
147C4×Dic5#6712yes—
148C2×#124—151no—
149C2×#103—152no—
150C2×#102—152no—
151C2×#97—51no—
152#384—151no—
153C22×C3×C5C2×C22⋊C411yescharacteristic
154C2×D15—41no—
155C2×D15—42no—
156C2×C5×D3—41no—
157C2×C5×D3—42no—
158C2×C3×D5#12411yescharacteristic
159C2×C3×D5#12711yescharacteristic
160C2×C3×D5AGL(1,ℤ/8)12yes—
161C2×C3×D5#13712yes—
162D3×D5—82no—
163D3×D5—81no—
164D3×D5—81no—
165ΓL(1,𝔽16)—82no—
166C3×C4×C5—22no—
167C3×C4×C5C4×D412yes—
168Dic15—41no—
169C3×Dic5—22no—
170C3×Dic5#12112yes—
171C5×Dic3—41no—
172C23×D3—101no—
173C22×C3×C4—51no—
174C22×C3×C4—52no—
175C22×Dic3—101no—
176C22×Dic3—102no—
177C2×D12—51no—
178C2×D12—102no—
179C2×D12—102no—
180C2×D12—101no—
181C2×D12C2×AGL(1,𝔽5)11yescharacteristic
182C2×Dic6—51no—
183C2×Dic6—102no—
184C2×Dic6—102no—
185C2×Dic6—101no—
186C2×Dic6C2×AGL(1,𝔽5)11yescharacteristic
187C2×C3×C8—54no—
188C2×C3×C8—102no—
189C2×C3×C8—102no—
190C3×C42—52no—
191C3×C42—51no—
192C3×C42C2×AGL(1,𝔽5)11yescharacteristic
193(C2×D3)⋊C4—22no—
194(C2×D3)⋊C4—102no—
195(C2×D3)⋊C4—102no—
196(C2×D3)⋊C4—102no—
197C4⋊Dic3—102no—
198C4⋊Dic3—102no—
199C4⋊Dic3—204no—
200C23×C5—61no—
201C22×D5—61no—
202C22×D5—121no—
203C22×D5—121no—
204C22×D5—122no—
205C22×D5—122no—
206C22×D5C2×D1211yescharacteristic
207C2×AGL(1,𝔽5)—122no—
208C2×AGL(1,𝔽5)—124no—
209C5×Q8—62no—
210C5×D4—62no—
211C2×C4×C5—61no—
212C2×C4×C5(C2×D3)⋊C412yes—
213C2×C4×C5C2×C4×D311yescharacteristic
214C22⋊D5—122no—
215C2×Dic5—61no—
216C2×Dic5(C2×D3)⋊C412yes—
217C2×Dic5C2×#6711yescharacteristic
218D20—62no—
219C4×D5—22no—
220C4×D5—22no—
221C4×D5—121no—
222C4×D5—121no—
223C4×D5C2×D1212yes—
224C4×D5#23512yes—
225Dic10—62no—
226Dic10—122no—
227#170—24no—
228C22×Q8—151no—
229C22×D4—151no—
230C22×C8—52no—
231#124—31no—
232#124—151no—
233#124—151no—
234#124—151no—
235#124—302no—
236C2×C4⋊C4—152no—
237C2×C4⋊C4—302no—
238C2×C22⋊C4—302no—
239C2×C42—51no—
240#103—304no—
241#102—304no—
242#97—102no—
243C2×C3×C5#39011yescharacteristic
244C2×C3×C5#41712yes—
245D15—81no—
246C3×D5#48912yes—
247C3×D5#49611yescharacteristic
248C3×D5#49711yescharacteristic
249C5×D3—81no—
250C23×C3—51no—
251C22×D3—21no—
252C22×D3—101no—
253C22×D3—201no—
254C22×D3—201no—
255C22×D3—202no—
256C22×D3—202no—
257C2×C3×C4—51no—
258C2×C3×C4—52no—
259C2×C3×C4—52no—
260C2×C3×C4—52no—
261C2×C3×C4—102no—
262C2×C3×C4—102no—
263C2×C3×C4C22⋊AGL(1,𝔽5)12yes—
264C2×C3×C4C22×AGL(1,𝔽5)11yescharacteristic
265C2×Dic3—21no—
266C2×Dic3—101no—
267C2×Dic3—201no—
268C2×Dic3—201no—
269C2×Dic3—202no—
270C2×Dic3—204no—
271D12—22no—
272D12—102no—
273D12—202no—
274Dic6—22no—
275Dic6—102no—
276Dic6—202no—
277C3×C8—104no—
278C22×C5—122no—
279C22×C5—121no—
280C22×C5C2×(C2×D3)⋊C411yescharacteristic
281C2×D5—122no—
282C2×D5—121no—
283C2×D5—241no—
284C2×D5—242no—
285C2×D5—241no—
286C2×D5#87411yescharacteristic
287C2×D5#87511yescharacteristic
288C2×D5#90812yes—
289C2×D5#90912yes—
290AGL(1,𝔽5)—242no—
291C4×C5—22no—
292C4×C5—121no—
293C4×C5C4×D1212yes—
294Dic5—22no—
295Dic5—121no—
296Dic5#93012yes—
297C24—301no—
298C2×Q8—31no—
299C2×Q8—151no—
300C2×Q8—301no—
301C2×Q8—302no—
302C2×Q8—302no—
303C2×D4—31no—
304C2×D4—151no—
305C2×D4—302no—
306C2×D4—302no—
307C2×D4—301no—
308C22×C4—51no—
309C22×C4—52no—
310C22×C4—301no—
311C22×C4—302no—
312C2×C8—54no—
313C2×C8—102no—
314C2×C8—102no—
315C4⋊C4—302no—
316C4⋊C4—302no—
317C4⋊C4—604no—
318C22⋊C4—62no—
319C22⋊C4—302no—
320C22⋊C4—302no—
321C22⋊C4—302no—
322C42—52no—
323C42—51no—
324C42D3×AGL(1,𝔽5)11yescharacteristic
325C3×C5#195911yescharacteristic, complemented, Hall
326C22×C3—52no—
327C22×C3—52no—
328C22×C3—51no—
329C22×C3—51no—
330C22×C3C2×C22⋊AGL(1,𝔽5)11yescharacteristic
331C2×D3—42no—
332C2×D3—41no—
333C2×D3—202no—
334C2×D3—201no—
335C2×D3—401no—
336C2×D3—402no—
337C2×D3—401no—
338C3×C4—22no—
339C3×C4—52no—
340C3×C4—102no—
341C3×C4D4×AGL(1,𝔽5)12yes—
342Dic3—41no—
343Dic3—201no—
344Dic3—402no—
345C2×C5—241no—
346C2×C5#493911yescharacteristic
347C2×C5#513312yes—
348D5—241no—
349D5#471211yescharacteristic
350D5#471412yes—
351D5#471511yescharacteristic
352C23—51no—
353C23—61no—
354C23—301no—
355C23—601no—
356C23—601no—
357C23—602no—
358C23—602no—
359Q8—62no—
360Q8—302no—
361Q8—602no—
362D4—62no—
363D4—302no—
364D4—602no—
365C2×C4—51no—
366C2×C4—52no—
367C2×C4—52no—
368C2×C4—52no—
369C2×C4—61no—
370C2×C4—102no—
371C2×C4—102no—
372C2×C4—301no—
373C2×C4—601no—
374C2×C4—601no—
375C2×C4—602no—
376C2×C4—604no—
377C2×C4(C2×D3)⋊AGL(1,𝔽5)12yes—
378C2×C4C2×D3×AGL(1,𝔽5)11yescharacteristic
379C8—104no—
380C2×C3—51no—
381C2×C3—52no—
382C2×C3—51no—
383C2×C3#6578011yescharacteristic
384C2×C3#6579312yes—
385D3—81no—
386D3—401no—
387C5#6982111yescharacteristic, complemented, Sylow
388C22—52no—
389C22—52no—
390C22—51no—
391C22—51no—
392C22—122no—
393C22—121no—
394C22—602no—
395C22—601no—
396C22—1201no—
397C22—1202no—
398C22—1201no—
399C22C2×(C2×D3)⋊AGL(1,𝔽5)11yescharacteristic
400C4—22no—
401C4—52no—
402C4—102no—
403C4—121no—
404C4—601no—
405C4—1202no—
406C4AGL(1,𝔽5)×D1212yes—
407C3#1060456711yescharacteristic, complemented, Sylow
408C2—51no—
409C2—52no—
410C2—51no—
411C2—241no—
412C2—1201no—
413C2#1175170812yes—
414C2#1175185011yescharacteristic
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 radicalC2×D5×D12C22
cosocleC2×C3×C4×D5C23
hypercenterC42D3×AGL(1,𝔽5)
hypocenterC3×C5#1959
LCSG ▷ C2×C3×C4×C5 ▷ C2×C3×C5 ▷ C3×C5
UCSC1 ◁ C22 ◁ C42

Complements

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

NH ≅ G/Ntype
2. #11752347C2semidirect
3. C3×#65676C2semidirect
4. C2×#91154C2semidirect
12. #91764C4semidirect
13. C5×#875C4semidirect
16. #91375C4semidirect
17. #91373C4semidirect
27. #65676D3semidirect
116. #875AGL(1,𝔽5)semidirect
325. C3×C5#1959semidirect
387. C5#69821semidirect
407. C3#10604567semidirect

Maximal subgroups

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

subgroupindex
D5×#8752
#117523472
C3×#656762
C2×#911542
C2×#911522
#106045673
#698215