Finite Group Databasebeta

← #14282953

Subgroups — #14282953

Order 1920.

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

normal subgroups only

This group has 480 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#11755174C211yesmaximal, characteristic
2#11751834C211yesmaximal, characteristic
3#11751762C211yesmaximal, characteristic
4#11746797C211yesmaximal, characteristic
5C3×#64926C211yesmaximal, characteristic
6#11746502C211yesmaximal, characteristic
7#11746500C211yesmaximal, characteristic
8#10604623—31nomaximal, Hall
9C2×Dic3⋊AGL(1,𝔽5)—21no—
10C2×Dic3×AGL(1,𝔽5)—21no—
11Q8×Dic15C411yescharacteristic, complemented
12C5×#944C411yescharacteristic, complemented
13C3×#4092C2211yescharacteristic
14#91469C411yescharacteristic, complemented
15#91422—21no—
16#91394—21no—
17C4×D5×Dic3C2211yescharacteristic
18#91359C2211yescharacteristic
19#91344C411yescharacteristic, complemented
20#91343—21no—
21#91325—21no—
22#91235C2211yescharacteristic
23#91232C2211yescharacteristic
24C3×(C2×AGL(1,𝔽5))⋊C4C2211yescharacteristic
25C3×#4003C2211yescharacteristic
26Dic3⋊#170—21no—
27Dic3×#170—21no—
28#68309—51nomaximal, Hall
29#66062—31no—
30C4×C4⋊AGL(1,𝔽5)—31no—
31#65680—31no—
32#64926—31no—
33#64926—31no—
34#64926—31no—
35#64926D311yescharacteristic
36C22×ΓL(1,𝔽16)—21no—
37C22×C3×AGL(1,𝔽5)—21no—
38C2×C3×C5×Q8C2×C411yescharacteristic
39C2×C5×Dic6C2×C411yescharacteristic
40C2×C3×C4×D5C2311yescharacteristic
41C2×#1204—21no—
42C2×D5×Dic3D411yescharacteristic
43C2×D5×Dic3D411yescharacteristic
44C2×C3⋊#170—21no—
45C2×C3×#170—21no—
46Dic3⋊AGL(1,𝔽5)—24no—
47Dic3×AGL(1,𝔽5)—42no—
48C4⋊Dic15—21no—
49C4⋊Dic15C2×C411yescharacteristic
50C4×Dic15—21no—
51C4×Dic15C2×C411yescharacteristic
52C5×Dic3⋊C4—21no—
53C5×Dic3⋊C4—21no—
54C4×C5×Dic3C2×C411yescharacteristic
55C3×#685—21no—
56C3×C4⋊Dic5C2×C411yescharacteristic
57C3×#683—21no—
58Dic3⋊Dic5—21no—
59Dic3⋊Dic5—21no—
60#6669—21no—
61#6665—21no—
62Dic3×Dic5—21no—
63Dic3×Dic5D411yescharacteristic
64Dic3×Dic5D411yescharacteristic
65#5622—51no—
66#4718—51no—
67#4649—51no—
68C3×#328—51no—
69#4538—51no—
70#4467—51no—
71#4452—51no—
72C2×C4⋊AGL(1,𝔽5)—61no—
73C2×C4×AGL(1,𝔽5)—61no—
74C5×C4⋊Q8—31no—
75#4092—31no—
76#4092C2×D311yescharacteristic
77Q8×Dic5—31no—
78Q8×Dic5—31no—
79#4043—61no—
80#4038—61no—
81#4036—61no—
82#4033—61no—
83C42×D5—31no—
84#4015—31no—
85(C2×AGL(1,𝔽5))⋊C4—31no—
86(C2×AGL(1,𝔽5))⋊C4C2×D311yescharacteristic
87#4003—31no—
88#4003C2×D311yescharacteristic
89C4⋊#170—61no—
90C4×#170—61no—
91#1503—151noSylow
92C22×C3×D5C2×D411yescharacteristic
93C2×ΓL(1,𝔽16)—22no—
94C2×ΓL(1,𝔽16)—41no—
95C2×ΓL(1,𝔽16)—41no—
96C2×C3×AGL(1,𝔽5)—22no—
97C2×C3×AGL(1,𝔽5)—41no—
98C2×C3×AGL(1,𝔽5)—41no—
99C3×C5×Q8—41no—
100C2×C3×C4×C5—21no—
101C2×C3×C4×C5C22×C411yescharacteristic
102C2×Dic15—21no—
103C2×Dic15C22⋊C411yescharacteristic
104C2×Dic15C22⋊C411yescharacteristic
105C2×C5×Dic3—21no—
106C2×C5×Dic3C22⋊C411yescharacteristic
107C2×C5×Dic3C22⋊C411yescharacteristic
108C5×Dic6—41no—
109C2×C3×Dic5—21no—
110C2×C3×Dic5C2×D411yescharacteristic
111C3×C4×D5—41no—
112#1204—42no—
113D5×Dic3—22no—
114D5×Dic3SD1612yes—
115D5×Dic3Q1612yes—
116C3⋊#170—41no—
117C3×#170—41no—
118C3×C22⋊Q8—51no—
119Q8×Dic3—51no—
120#944AGL(1,𝔽5)11yescharacteristic, complemented
121#924—51no—
122C2×Dic3⋊C4—101no—
123C2×C4×Dic3—51no—
124C2×C4×Dic3—101no—
125#889—101no—
126#888—51no—
127#887—51no—
128#887—101no—
129#878—101no—
130#877—101no—
131C3×C22⋊C8—51no—
132C3×#95—51no—
133C3⋊#95—51no—
134#830—51no—
135Dic3⋊C8—101no—
136C8×Dic3—101no—
137C22×AGL(1,𝔽5)—21no—
138C22×AGL(1,𝔽5)—61no—
139C2×C5×Q8—31no—
140C2×C5×Q8C4×D311yescharacteristic
141C2×C4×D5—31no—
142C2×C4×D5—31no—
143C2×C4×D5C22×D311yescharacteristic
144C2×Dic10—61no—
145C2×#170—21no—
146C2×#170—61no—
147C4⋊AGL(1,𝔽5)—64no—
148C4×AGL(1,𝔽5)—122no—
149C5×C4⋊C4—61no—
150C5×C4⋊C4—61no—
151C42×C5—31no—
152#685—21no—
153#685—61no—
154C4⋊Dic5—31no—
155C4⋊Dic5—61no—
156C4⋊Dic5—61no—
157C4⋊Dic5—61no—
158C4⋊Dic5C4×D311yescharacteristic
159#683—21no—
160#683—61no—
161C4×Dic5—31no—
162C4×Dic5—31no—
163C4×Dic5—31no—
164C4×Dic5—61no—
165C4×Dic5—61no—
166#533—151no—
167#431—151no—
168C4×C4⋊C4—151no—
169#328—51no—
170#328—151no—
171#328—151no—
172#328—151no—
173C22×C3×C5C2×C22⋊C411yescharacteristic
174C2×C3×D5—21no—
175C2×C3×D5—21no—
176C2×C3×D5C2×SD1611yescharacteristic
177C2×C3×D5C2×Q1611yescharacteristic
178ΓL(1,𝔽16)—42no—
179C3×AGL(1,𝔽5)—42no—
180C3×C4×C5—21no—
181C3×C4×C5—41no—
182Dic15—21no—
183Dic15—41no—
184Dic15#10312yes—
185C3×Dic5—21no—
186C3×Dic5—41no—
187C5×Dic3—41no—
188C5×Dic3#10312yes—
189C5×Dic3#10412yes—
190C2×C3×Q8C2×AGL(1,𝔽5)11yescharacteristic
191C22×C3×C4—51no—
192C22×C3×C4—101no—
193C22×Dic3—51no—
194C22×Dic3—51no—
195C22×Dic3—101no—
196C2×Dic6—101no—
197C2×Dic6C2×AGL(1,𝔽5)11yescharacteristic
198C2×C3×C8—101no—
199C3×C4⋊C4—51no—
200C3×C4⋊C4—101no—
201C3×C22⋊C4—101no—
202#217—101no—
203C4⋊Dic3—51no—
204C4⋊Dic3—101no—
205Dic3⋊C4—21no—
206Dic3⋊C4—21no—
207Dic3⋊C4—101no—
208Dic3⋊C4—101no—
209Dic3⋊C4—104no—
210Dic3⋊C4—101no—
211C4×Dic3—51no—
212C4×Dic3—51no—
213C4×Dic3—51no—
214C4×Dic3—101no—
215C4×Dic3—101no—
216C4×Dic3—202no—
217C4×Dic3C2×AGL(1,𝔽5)11yescharacteristic
218C2×C3⋊C8—101no—
219C22×D5D3×D411yescharacteristic
220C2×AGL(1,𝔽5)—22no—
221C2×AGL(1,𝔽5)—41no—
222C2×AGL(1,𝔽5)—41no—
223C2×AGL(1,𝔽5)—62no—
224C2×AGL(1,𝔽5)—121no—
225C2×AGL(1,𝔽5)—121no—
226C5×Q8—41no—
227C5×Q8—121no—
228C2×C4×C5—21no—
229C2×C4×C5—31no—
230C2×C4×C5—31no—
231C2×C4×C5—61no—
232C2×C4×C5C2×C4×D311yescharacteristic
233C2×Dic5—21no—
234C2×Dic5—31no—
235C2×Dic5—31no—
236C2×Dic5—61no—
237C2×Dic5D3×D411yescharacteristic
238C4×D5—32no—
239C4×D5—32no—
240C4×D5—41no—
241C4×D5—62no—
242Dic10—122no—
243#170—41no—
244#170—121no—
245C4⋊Q8—31no—
246C4⋊Q8—151no—
247#125—301no—
248C22⋊Q8—51no—
249C22⋊Q8—151no—
250C22⋊Q8—301no—
251C4×Q8—151no—
252C4×Q8—151no—
253C4×Q8—301no—
254#117—301no—
255C2×C4⋊C4—301no—
256C2×C42—151no—
257C2×C42—301no—
258C4⋊C8—301no—
259C22⋊C8—51no—
260C22⋊C8—151no—
261C4×C8—301no—
262#95—51no—
263#95—151no—
264C2×C3×C5#41011yescharacteristic
265C2×C3×C5C2×#10311yescharacteristic
266C2×C3×C5C2×#10411yescharacteristic
267C3×D5—21no—
268C3×D5—21no—
269C23×C3—51no—
270C3×Q8—41no—
271C2×C3×C4—21no—
272C2×C3×C4—51no—
273C2×C3×C4—102no—
274C2×C3×C4—101no—
275C2×C3×C4—201no—
276C2×C3×C4—201no—
277C2×C3×C4—201no—
278C2×C3×C4C22×AGL(1,𝔽5)11yescharacteristic
279C2×Dic3—21no—
280C2×Dic3—51no—
281C2×Dic3—52no—
282C2×Dic3—52no—
283C2×Dic3—51no—
284C2×Dic3—102no—
285C2×Dic3—102no—
286C2×Dic3—101no—
287C2×Dic3—201no—
288C2×Dic3—201no—
289C2×Dic3C22⋊AGL(1,𝔽5)11yescharacteristic
290C2×Dic3C22⋊AGL(1,𝔽5)11yescharacteristic
291Dic6—41no—
292Dic6—202no—
293C3×C8—201no—
294C3⋊C8—201no—
295C22×C5D3×C22⋊C411yescharacteristic
296C2×D5—21no—
297C2×D5—21no—
298C2×D5D3×SD1611yescharacteristic
299C2×D5D3×Q1611yescharacteristic
300AGL(1,𝔽5)—42no—
301AGL(1,𝔽5)—122no—
302C4×C5—21no—
303C4×C5—32no—
304C4×C5—32no—
305C4×C5—41no—
306C4×C5—121no—
307Dic5—21no—
308Dic5—32no—
309Dic5—41no—
310Dic5—61no—
311Dic5—121no—
312C2×Q8—31no—
313C2×Q8—301no—
314C2×Q8D3×AGL(1,𝔽5)11yescharacteristic
315C22×C4—51no—
316C22×C4—101no—
317C22×C4—151no—
318C22×C4—151no—
319C22×C4—301no—
320C2×C8—101no—
321C2×C8—301no—
322C4⋊C4—51no—
323C4⋊C4—61no—
324C4⋊C4—61no—
325C4⋊C4—101no—
326C4⋊C4—151no—
327C4⋊C4—301no—
328C4⋊C4—301no—
329C4⋊C4—301no—
330C4⋊C4—301no—
331C4⋊C4—304no—
332C22⋊C4—101no—
333C22⋊C4—301no—
334C42—31no—
335C42—151no—
336C42—151no—
337C42—151no—
338C42—301no—
339C42—301no—
340C42—602no—
341C3×C5#150311yescharacteristic, complemented, Hall
342C22×C3—51no—
343C22×C3—51no—
344C22×C3—101no—
345C22×C3—101no—
346C22×C3C2×C22⋊AGL(1,𝔽5)11yescharacteristic
347C3×C4—21no—
348C3×C4—41no—
349C3×C4—101no—
350C3×C4—202no—
351C3×C4—201no—
352Dic3—41no—
353Dic3—52no—
354Dic3—101no—
355Dic3—202no—
356Dic3—201no—
357Dic3#400912yes—
358Dic3#401012yes—
359C2×C5#474011yescharacteristic
360C2×C5D3×#10311yescharacteristic
361C2×C5D3×#10411yescharacteristic
362D5—21no—
363D5—21no—
364C23—51no—
365Q8—41no—
366Q8—121no—
367Q8—602no—
368C2×C4—21no—
369C2×C4—31no—
370C2×C4—31no—
371C2×C4—51no—
372C2×C4—61no—
373C2×C4—102no—
374C2×C4—101no—
375C2×C4—152no—
376C2×C4—152no—
377C2×C4—151no—
378C2×C4—151no—
379C2×C4—201no—
380C2×C4—201no—
381C2×C4—201no—
382C2×C4—302no—
383C2×C4—302no—
384C2×C4—301no—
385C2×C4—601no—
386C2×C4—601no—
387C2×C4C2×D3×AGL(1,𝔽5)11yescharacteristic
388C8—201no—
389C8—601no—
390C2×C3—101no—
391C2×C3—101no—
392C2×C3#6577111yescharacteristic
393C2×C3C2×#400911yescharacteristic
394C2×C3C2×#401011yescharacteristic
395C5#6830911yescharacteristic, complemented, Sylow
396C22—51no—
397C22—51no—
398C22—101no—
399C22—101no—
400C22D3×C22⋊AGL(1,𝔽5)11yescharacteristic
401C4—21no—
402C4—32no—
403C4—32no—
404C4—41no—
405C4—101no—
406C4—121no—
407C4—152no—
408C4—202no—
409C4—201no—
410C4—301no—
411C4—602no—
412C4—601no—
413C3#1060462311yescharacteristic, complemented, Sylow
414C2—101no—
415C2—101no—
416C2D3×#400911yescharacteristic
417C2D3×#401011yescharacteristic
418C2#1175203111yescharacteristic
subgroupG/H
centerC22D3×C22⋊AGL(1,𝔽5)
commutatorC2×C3×C4×C5C22×C4
FrattiniC2×C4C2×D3×AGL(1,𝔽5)
FittingC2×C3×C5×Q8C2×C4
socleC22×C3×C5C2×C22⋊C4
prime radicalC22×C3×D5C2×D4
cosocleC2×C3×C4×D5C23
hypercenterC2×Q8D3×AGL(1,𝔽5)
hypocenterC3×C5#1503
LCSG ▷ C2×C3×C4×C5 ▷ C22×C3×C5 ▷ C3×C5
UCSC1 ◁ C22 ◁ C2×C4 ◁ C2×Q8

Complements

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

NH ≅ G/Ntype
11. Q8×Dic15C4semidirect
12. C5×#944C4semidirect
14. #91469C4semidirect
19. #91344C4semidirect
120. #944AGL(1,𝔽5)semidirect
341. C3×C5#1503semidirect
395. C5#68309semidirect
413. C3#10604623semidirect

Maximal subgroups

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

subgroupindex
#117551742
#117518342
#117517622
#117467972
C3×#649262
#117465022
#117465002
#106046233
#683095