Finite Group Databasebeta

← #14283063

Subgroups — #14283063

Order 1920.

544 conjugacy classes of subgroups; only the 94 normal subgroups are shown.

normal subgroups only

This group has 544 conjugacy classes of subgroups — too many to draw. Only its 94 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#11752671C211yesmaximal, characteristic
2#11752345C211yesmaximal, characteristic, complemented
3C3×#65684C211yesmaximal, characteristic, complemented
4#11746499C212yesmaximal, complemented
5#11746498C212yesmaximal, complemented
6#10604390—31nomaximal, Hall
7C2×#6771—21no—
8C2×C4×ΓL(1,𝔽16)—21no—
9C4×D60C411yescharacteristic, complemented
10C4×C5×D12C411yescharacteristic, complemented
11C3×C42×D5C2211yescharacteristic
12#91468C411yescharacteristic, complemented
13#91451C411yescharacteristic, complemented
14#91450C2211yescharacteristic
15#91449C2211yescharacteristic
16#91448—21no—
17C4×#1203—21no—
18#91350—21no—
19#91235—21no—
20#91235C2212yes—
21C3×#4004—21no—
22C3×#4003C2212yes—
23C3×#4002—21no—
24#70832—51nomaximal, Hall
25#65882—31no—
26#65686—31no—
27#65684D311yescharacteristic, complemented
28#64920—32no—
29#64868—32no—
30C22×ΓL(1,𝔽16)—22no—
31C2×D60C2×C411yescharacteristic
32C2×C4×D15—21no—
33C2×C5×D12C2×C411yescharacteristic
34C2×C4×C5×D3—21no—
35C2×C3×C4×D5D411yescharacteristic
36C2×C3×C4×D5D411yescharacteristic
37C2×C3×C4×D5C2311yescharacteristic
38C2×#1204—21no—
39C2×#1203—21no—
40C2×#1203—21no—
41C2×#1202—21no—
42C2×#1201—21no—
43#6771—44no—
44C4×ΓL(1,𝔽16)—24no—
45C2×C3×#170—22no—
46C3×C42×C5C2×C411yescharacteristic
47#6714—21no—
48C4⋊Dic15C2×C411yescharacteristic
49C5×(C2×D3)⋊C4—21no—
50C5×C4⋊Dic3C2×C411yescharacteristic
51C3×C4×Dic5C2×C411yescharacteristic
52C3×C4×Dic5D411yescharacteristic
53C3×C4×Dic5D411yescharacteristic
54#6668—21no—
55#6667—21no—
56(C2×D3)⋊Dic5—21no—
57#5477—51no—
58C3×#432—51no—
59#4932—51no—
60#4470—52no—
61#4470—52no—
62C2×#709—61no—
63C2×C4×AGL(1,𝔽5)—61no—
64C4×C5×D4—31no—
65#4086—31no—
66#4084—31no—
67C4×C22⋊D5—61no—
68#4041—31no—
69#4034—31no—
70#4031—61no—
71#4030—61no—
72C4×D20—31no—
73C42×D5C2×D311yescharacteristic
74(C2×AGL(1,𝔽5))⋊C4—32no—
75(C2×AGL(1,𝔽5))⋊C4—61no—
76#4004—21no—
77#4003C2×D312yes—
78#4002—21no—
79#1505—151noSylow
80C22×D15—21no—
81C22×C5×D3—21no—
82C22×C3×D5C2×D411yescharacteristic
83C2×ΓL(1,𝔽16)—24no—
84C2×ΓL(1,𝔽16)—42no—
85C2×ΓL(1,𝔽16)—42no—
86C2×C3×C4×C5C22⋊C411yescharacteristic
87C2×C3×C4×C5C22⋊C411yescharacteristic
88C2×C3×C4×C5C22×C411yescharacteristic
89C2×Dic15—21no—
90D60—41no—
91C4×D15—42no—
92C2×C5×Dic3—21no—
93C5×D12—41no—
94C4×C5×D3—42no—
95C2×C3×Dic5C22⋊C411yescharacteristic
96C2×C3×Dic5C22⋊C411yescharacteristic
97C2×C3×Dic5C2×D411yescharacteristic
98C3×C4×D5—22no—
99C3×C4×D5—22no—
100C3×C4×D5—41no—
101#1204—42no—
102#1203—42no—
103#1203—42no—
104#1202—42no—
105#1201—42no—
106C3×#170—42no—
107C2×#235—101no—
108C2×C3×C42—51no—
109#930—51no—
110#930—51no—
111#930—101no—
112C2×C4×Dic3—101no—
113#875—101no—
114#874—51no—
115C4×D12—51no—
116C4×D12—101no—
117C4×D12AGL(1,𝔽5)11yescharacteristic, complemented
118#869—51no—
119C3×C4⋊C8—101no—
120C3×C22⋊C8—52no—
121C3×#97—101no—
122C3⋊#95—52no—
123C3⋊#95—101no—
124C22×AGL(1,𝔽5)—62no—
125C2×C5×D4—31no—
126C22×C4×C5—61no—
127C2×C22⋊D5—61no—
128C2×C22⋊D5—61no—
129C2×C22⋊D5—61no—
130#709—124no—
131C2×D20—31no—
132C2×D20—61no—
133C2×C4×D5—61no—
134C2×C4×D5D1211yescharacteristic
135C2×C4×D5D1211yescharacteristic
136C2×C4×D5C22×D311yescharacteristic
137C2×Dic10—61no—
138C2×#170—22no—
139C4×AGL(1,𝔽5)—64no—
140C5×C4⋊C4—31no—
141C5×C22⋊C4—61no—
142C42×C5C4×D311yescharacteristic
143C22⋊Dic5—61no—
144#685—61no—
145#685—61no—
146C4⋊Dic5—31no—
147#683—61no—
148C4×Dic5C4×D311yescharacteristic
149C4×Dic5#6711yescharacteristic
150C4×Dic5#6711yescharacteristic
151#532—151no—
152#432—51no—
153#376—151no—
154#327—152no—
155#327—152no—
156C22×C3×C5C2×C22⋊C411yescharacteristic
157C2×D15—41no—
158C2×D15—41no—
159C2×D15—41no—
160C2×C5×D3—41no—
161C2×C5×D3—41no—
162C2×C5×D3—41no—
163C2×C3×D5—21no—
164C2×C3×D5—21no—
165C2×C3×D5AGL(1,ℤ/8)12yes—
166ΓL(1,𝔽16)—44no—
167C3×C4×C5—21no—
168C3×C4×C5—21no—
169C3×C4×C5—21no—
170Dic15—41no—
171C3×Dic5—21no—
172C3×Dic5—21no—
173C3×Dic5#10412yes—
174C5×Dic3—41no—
175C22×C3×C4—51no—
176C22×C3×C4—51no—
177C22×C3×C4—51no—
178C2×#67—101no—
179C2×#67—101no—
180C22×Dic3—102no—
181#235—204no—
182C2×D12—51no—
183C2×D12—101no—
184C2×D12—101no—
185C2×D12C2×AGL(1,𝔽5)11yescharacteristic
186C2×C4×D3—21no—
187C2×C4×D3—101no—
188C2×Dic6—101no—
189C2×C3×C8—102no—
190C3×C42—51no—
191C3×C42—51no—
192C3×C42—51no—
193C3×C42C2×AGL(1,𝔽5)11yescharacteristic
194(C2×D3)⋊C4—21no—
195(C2×D3)⋊C4—101no—
196(C2×D3)⋊C4—101no—
197(C2×D3)⋊C4—101no—
198C4⋊Dic3—51no—
199C4⋊Dic3—101no—
200C4⋊Dic3C2×AGL(1,𝔽5)11yescharacteristic
201C4×Dic3—104no—
202C23×C5—61no—
203C22×D5—61no—
204C22×D5C2×D1211yescharacteristic
205C2×AGL(1,𝔽5)—64no—
206C2×AGL(1,𝔽5)—122no—
207C2×AGL(1,𝔽5)—122no—
208C5×D4—121no—
209C2×C4×C5—61no—
210C2×C4×C5—122no—
211C2×C4×C5(C2×D3)⋊C411yescharacteristic
212C2×C4×C5(C2×D3)⋊C411yescharacteristic
213C2×C4×C5C2×C4×D311yescharacteristic
214C22⋊D5—122no—
215C22⋊D5—122no—
216C22⋊D5—122no—
217C2×Dic5—61no—
218C2×Dic5(C2×D3)⋊C411yescharacteristic
219C2×Dic5(C2×D3)⋊C411yescharacteristic
220C2×Dic5C2×#6711yescharacteristic
221D20—121no—
222D20—122no—
223C4×D5—22no—
224C4×D5—22no—
225C4×D5—41no—
226C4×D5—122no—
227Dic10—122no—
228#170—42no—
229C2×#41—301no—
230C4⋊Q8—151no—
231#127—151no—
232#124—301no—
233#121—151no—
234#121—151no—
235#121—301no—
236C4×D4—31no—
237C4×D4—151no—
238C4×D4—301no—
239C2×C42—51no—
240C2×C42—301no—
241C4⋊C8—101no—
242C22⋊C8—52no—
243#97—101no—
244#95—152no—
245#95—301no—
246C2×C3×C5#41011yescharacteristic
247C2×C3×C5#41811yescharacteristic
248C2×C3×C5C2×#10411yescharacteristic
249D15—81no—
250C3×D5—22no—
251C5×D3—81no—
252C23×C3—51no—
253C22×D3—21no—
254C22×D3—101no—
255C2×C3×C4—51no—
256C2×C3×C4—51no—
257C2×C3×C4—51no—
258C2×C3×C4—102no—
259C2×C3×C4—102no—
260C2×C3×C4—201no—
261C2×C3×C4C22⋊AGL(1,𝔽5)11yescharacteristic
262C2×C3×C4C22⋊AGL(1,𝔽5)11yescharacteristic
263C2×C3×C4C22×AGL(1,𝔽5)11yescharacteristic
264#67—202no—
265#67—202no—
266C2×Dic3—21no—
267C2×Dic3—104no—
268C2×Dic3—101no—
269C2×Dic3—202no—
270C2×Dic3—202no—
271D12—41no—
272D12—201no—
273D12—202no—
274D12—202no—
275C4×D3—42no—
276C4×D3—202no—
277Dic6—202no—
278C3×C8—202no—
279C22×C5—121no—
280C22×C5—121no—
281C22×C5—121no—
282C22×C5C2×(C2×D3)⋊C411yescharacteristic
283C2×D5—21no—
284C2×D5—21no—
285C2×D5—121no—
286C2×D5—121no—
287C2×D5—121no—
288C2×D5#90812yes—
289AGL(1,𝔽5)—124no—
290C4×C5—21no—
291C4×C5—21no—
292C4×C5—21no—
293C4×C5—121no—
294Dic5—21no—
295Dic5—21no—
296Dic5—121no—
297Dic5#82512yes—
298#41—604no—
299C2×Q8—301no—
300C2×D4—31no—
301C2×D4—151no—
302C2×D4—301no—
303C2×D4—301no—
304C2×D4—301no—
305C2×D4—301no—
306C22×C4—51no—
307C22×C4—51no—
308C22×C4—51no—
309C22×C4—61no—
310C22×C4—302no—
311C22×C4—301no—
312C2×C8—102no—
313C4⋊C4—31no—
314C4⋊C4—151no—
315C4⋊C4—301no—
316C22⋊C4—61no—
317C22⋊C4—301no—
318C22⋊C4—301no—
319C22⋊C4—301no—
320C42—51no—
321C42—51no—
322C42—51no—
323C42—304no—
324C42D3×AGL(1,𝔽5)11yescharacteristic
325C3×C5#150511yescharacteristic, complemented, Hall
326C22×C3—52no—
327C22×C3—101no—
328C22×C3—101no—
329C22×C3C2×C22⋊AGL(1,𝔽5)11yescharacteristic
330C2×D3—41no—
331C2×D3—41no—
332C2×D3—41no—
333C2×D3—201no—
334C2×D3—201no—
335C2×D3—201no—
336C3×C4—21no—
337C3×C4—21no—
338C3×C4—21no—
339C3×C4—52no—
340C3×C4—101no—
341C3×C4—101no—
342Dic3—41no—
343Dic3—204no—
344Dic3—201no—
345C2×C5—241no—
346C2×C5#500611yescharacteristic
347C2×C5#513111yescharacteristic
348C2×C5C2×#82511yescharacteristic
349D5—22no—
350D5—241no—
351C23—51no—
352C23—61no—
353C23—301no—
354Q8—602no—
355D4—121no—
356D4—601no—
357D4—602no—
358D4—602no—
359D4—602no—
360D4—602no—
361C2×C4—51no—
362C2×C4—51no—
363C2×C4—51no—
364C2×C4—61no—
365C2×C4—102no—
366C2×C4—102no—
367C2×C4—122no—
368C2×C4—201no—
369C2×C4—304no—
370C2×C4—301no—
371C2×C4—602no—
372C2×C4—602no—
373C2×C4—602no—
374C2×C4(C2×D3)⋊AGL(1,𝔽5)11yescharacteristic
375C2×C4(C2×D3)⋊AGL(1,𝔽5)11yescharacteristic
376C2×C4C2×D3×AGL(1,𝔽5)11yescharacteristic
377C8—202no—
378C2×C3—102no—
379C2×C3#6574511yescharacteristic
380C2×C3#6576711yescharacteristic
381C2×C3D5⋊#10411yescharacteristic
382D3—81no—
383D3—401no—
384C5#7083211yescharacteristic, complemented, Sylow
385C22—52no—
386C22—101no—
387C22—101no—
388C22—121no—
389C22—121no—
390C22—121no—
391C22—601no—
392C22—601no—
393C22—601no—
394C22C2×(C2×D3)⋊AGL(1,𝔽5)11yescharacteristic
395C4—21no—
396C4—21no—
397C4—21no—
398C4—52no—
399C4—101no—
400C4—101no—
401C4—121no—
402C4—604no—
403C4—601no—
404C3#1060439011yescharacteristic, complemented, Sylow
405C2—102no—
406C2—241no—
407C2—1201no—
408C2#1175170711yescharacteristic
409C2D5⋊#82511yescharacteristic
410C2#1175181511yescharacteristic
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#11752671C2
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×#65684C2semidirect
4. #11746499C2semidirect
5. #11746498C2semidirect
9. C4×D60C4semidirect
10. C4×C5×D12C4semidirect
12. #91468C4semidirect
13. #91451C4semidirect
27. #65684D3semidirect
117. C4×D12AGL(1,𝔽5)semidirect
325. C3×C5#1505semidirect
384. C5#70832semidirect
404. C3#10604390semidirect

Maximal subgroups

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

subgroupindex
#117526712
#117523452
C3×#656842
#117464992
#117464982
#106043903
#708325