Finite Group Databasebeta

← #14307508

Subgroups — #14307508

Order 1920.

648 conjugacy classes of subgroups; only the 122 normal subgroups are shown.

normal subgroups only

This group has 648 conjugacy classes of subgroups — too many to draw. Only its 122 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
1C2×#91359C211yesmaximal, characteristic
2C4×#6763C211yesmaximal, characteristic
3C3×#65765C211yesmaximal, characteristic
4#11746573C214yesmaximal
5#10604982—31nomaximal, Hall
6C22×C5×Dic6C411yescharacteristic, complemented
7C22×C3×C4×D5C2211yescharacteristic
8C2×#6763C2212yes—
9C2×C4×ΓL(1,𝔽16)—21no—
10C2×C3×C22⋊AGL(1,𝔽5)C2212yes—
11C2×C3×C4⋊AGL(1,𝔽5)—21no—
12C2×C4⋊Dic15C411yescharacteristic, complemented
13C2×#6669C412yescomplemented
14C2×#6665C2212yes—
15#91359—21no—
16#91359—21no—
17#91359—21no—
18#91359—21no—
19#91235—21no—
20C3×(C2×AGL(1,𝔽5))⋊C4—21no—
21#69529—51nomaximal, Hall
22C2×#4092—31no—
23#65765D311yescharacteristic
24C4×C22⋊AGL(1,𝔽5)—31no—
25#64917—34no—
26C23×C3×D5C2311yescharacteristic
27C22×ΓL(1,𝔽16)—22no—
28C22×C3×AGL(1,𝔽5)—22no—
29C22×C3×C4×C5C2×C411yescharacteristic
30C22×Dic15C2×C412yes—
31C22×C5×Dic3C2×C412yes—
32C2×C5×Dic6—21no—
33C2×C5×Dic6—21no—
34C2×C5×Dic6—42no—
35C22×C3×Dic5C2×C411yescharacteristic
36C2×C3×C4×D5—42no—
37C2×C3×C4×D5D411yescharacteristic
38C2×C3×C4×D5D411yescharacteristic
39C2×C3×C4×D5D411yescharacteristic
40C2×C3×C4×D5D411yescharacteristic
41#6763—22no—
42#6763Q814yescomplemented
43C4×ΓL(1,𝔽16)—42no—
44C3×C22⋊AGL(1,𝔽5)—24no—
45C3×C4⋊AGL(1,𝔽5)—42no—
46C4⋊Dic15—21no—
47C4⋊Dic15—21no—
48#6669—22no—
49#6669—22no—
50#6665—22no—
51#6665—22no—
52C2×#924—51no—
53C3×#380—51no—
54C4×#217—51no—
55#4527—54no—
56C22×C5×Q8—31no—
57C22×C4×D5C2×D311yescharacteristic
58C2×C22⋊AGL(1,𝔽5)—32no—
59C2×C22⋊AGL(1,𝔽5)C2×D312yes—
60C2×C4⋊AGL(1,𝔽5)—21no—
61C2×C4×AGL(1,𝔽5)—61no—
62#4092—61no—
63#4092—61no—
64#4092—61no—
65#4092—61no—
66C2×#685—32no—
67C2×C4⋊Dic5—31no—
68C2×#683—32no—
69(C2×AGL(1,𝔽5))⋊C4—21no—
70(C2×AGL(1,𝔽5))⋊C4—61no—
71#1996—151noSylow
72C23×C3×C5C22×C411yescharacteristic
73C22×C3×D5—41no—
74C22×C3×D5—41no—
75C22×C3×D5C2×D411yescharacteristic
76C22×C3×D5C2×D411yescharacteristic
77C22×C3×D5C2×Q812yes—
78C22×C3×D5#4112yes—
79C2×ΓL(1,𝔽16)—24no—
80C2×ΓL(1,𝔽16)—42no—
81C2×ΓL(1,𝔽16)—42no—
82C2×C3×AGL(1,𝔽5)—42no—
83C2×C3×AGL(1,𝔽5)—42no—
84C2×C3×AGL(1,𝔽5)—42no—
85C2×C3×C4×C5—22no—
86C2×C3×C4×C5C22⋊C411yescharacteristic
87C2×C3×C4×C5C22⋊C411yescharacteristic
88C2×Dic15—22no—
89C2×Dic15—42no—
90C2×C5×Dic3—22no—
91C2×C5×Dic3—42no—
92C5×Dic6—42no—
93C5×Dic6—42no—
94C2×C3×Dic5—41no—
95C2×C3×Dic5C22⋊C411yescharacteristic
96C2×C3×Dic5C22⋊C411yescharacteristic
97C3×C4×D5—22no—
98C3×C4×D5—22no—
99C3×C4×D5—41no—
100C3×C4×D5—41no—
101C23×C3×C4—51no—
102C22×Dic6AGL(1,𝔽5)11yescharacteristic, complemented
103C2×C3×C4⋊C4—101no—
104C2×C3×C22⋊C4—52no—
105C2×#217—52no—
106C2×#217—52no—
107C2×C4⋊Dic3—51no—
108#924—101no—
109#924—101no—
110#924—101no—
111#924—101no—
112C2×Dic3⋊C4—52no—
113C2×C4×Dic3—101no—
114C3×#95—101no—
115C3⋊#95—101no—
116C23×D5C22×D311yescharacteristic
117C22×AGL(1,𝔽5)—22no—
118C22×AGL(1,𝔽5)—62no—
119C2×C5×Q8—61no—
120C2×C5×Q8—61no—
121C2×C5×Q8—122no—
122C22×C4×C5—32no—
123C22×C4×C5C4×D311yescharacteristic
124C22×Dic5—32no—
125C22×Dic5C4×D311yescharacteristic
126C2×C4×D5—42no—
127C2×C4×D5D1211yescharacteristic
128C2×C4×D5D1211yescharacteristic
129C2×C4×D5#6711yescharacteristic
130C2×C4×D5#6711yescharacteristic
131C22⋊AGL(1,𝔽5)—24no—
132C22⋊AGL(1,𝔽5)—34no—
133C22⋊AGL(1,𝔽5)—62no—
134C4⋊AGL(1,𝔽5)—42no—
135C4×AGL(1,𝔽5)—122no—
136#685—62no—
137#685—62no—
138C4⋊Dic5—61no—
139C4⋊Dic5—61no—
140#683—62no—
141#683—62no—
142C2×C22⋊Q8—151no—
143#380—51no—
144C4×C22⋊C4—151no—
145#342—154no—
146C22×C3×C5—21no—
147C22×C3×C5—21no—
148C22×C3×C5C2×C22⋊C411yescharacteristic
149C22×C3×C5C4×Q812yes—
150C2×C3×D5—21no—
151C2×C3×D5—21no—
152C2×C3×D5—21no—
153C2×C3×D5—21no—
154C2×C3×D5—42no—
155C2×C3×D5—42no—
156C2×C3×D5C22⋊Q812yes—
157C2×C3×D5#12411yescharacteristic
158C2×C3×D5C4⋊Q811yescharacteristic
159ΓL(1,𝔽16)—44no—
160C3×AGL(1,𝔽5)—82no—
161C3×C4×C5—21no—
162C3×C4×C5#9912yes—
163Dic15—42no—
164C3×Dic5—21no—
165C3×Dic5—21no—
166C5×Dic3—42no—
167C24×C3—51no—
168C22×C3×C4—51no—
169C22×C3×C4—51no—
170C22×C3×C4—51no—
171C22×C3×C4—51no—
172C22×C3×C4—51no—
173C22×C3×C4—102no—
174C22×C3×C4—202no—
175C22×C3×C4C2×AGL(1,𝔽5)11yescharacteristic
176C22×Dic3—52no—
177C22×Dic3—102no—
178C22×Dic3C2×AGL(1,𝔽5)12yes—
179C2×Dic6—21no—
180C2×Dic6—21no—
181C2×Dic6—42no—
182C3×C4⋊C4—202no—
183C3×C22⋊C4—104no—
184#217—54no—
185#217—102no—
186#217—102no—
187#217—102no—
188C4⋊Dic3—101no—
189C4⋊Dic3—101no—
190Dic3⋊C4—102no—
191Dic3⋊C4—102no—
192C4×Dic3—202no—
193C23×C5C2×C4×D311yescharacteristic
194C22×D5—41no—
195C22×D5—41no—
196C22×D5C2×D1211yescharacteristic
197C22×D5#23712yes—
198C22×D5D3×Q812yes—
199C22×D5C2×#6711yescharacteristic
200C2×AGL(1,𝔽5)—42no—
201C2×AGL(1,𝔽5)—42no—
202C2×AGL(1,𝔽5)—42no—
203C2×AGL(1,𝔽5)—64no—
204C2×AGL(1,𝔽5)—122no—
205C2×AGL(1,𝔽5)—122no—
206C5×Q8—122no—
207C5×Q8—122no—
208C2×C4×C5—22no—
209C2×C4×C5—62no—
210C2×C4×C5—122no—
211C2×C4×C5(C2×D3)⋊C411yescharacteristic
212C2×C4×C5(C2×D3)⋊C411yescharacteristic
213C2×Dic5—41no—
214C2×Dic5—62no—
215C2×Dic5—122no—
216C2×Dic5(C2×D3)⋊C411yescharacteristic
217C2×Dic5(C2×D3)⋊C411yescharacteristic
218C4×D5—22no—
219C4×D5—22no—
220C4×D5—41no—
221C4×D5—41no—
222C22×Q8—31no—
223C23×C4—51no—
224C22⋊Q8—301no—
225C22⋊Q8—301no—
226C22⋊Q8—301no—
227C22⋊Q8—301no—
228C2×C4⋊C4—101no—
229C2×C4⋊C4—152no—
230C2×C4⋊C4—151no—
231C2×C22⋊C4—52no—
232C2×C22⋊C4—152no—
233C2×C22⋊C4—152no—
234C2×C42—301no—
235#95—101no—
236#95—301no—
237C2×C3×C5—22no—
238C2×C3×C5#39111yescharacteristic
239C2×C3×C5C2×#9911yescharacteristic
240C2×C3×C5#41011yescharacteristic
241C3×D5—21no—
242C3×D5—22no—
243C3×D5—21no—
244C23×C3—51no—
245C23×C3—51no—
246C23×C3—52no—
247C23×C3—52no—
248C23×C3—201no—
249C23×C3—201no—
250C23×C3C22×AGL(1,𝔽5)11yescharacteristic
251C2×C3×C4—22no—
252C2×C3×C4—51no—
253C2×C3×C4—51no—
254C2×C3×C4—102no—
255C2×C3×C4—102no—
256C2×C3×C4—201no—
257C2×C3×C4—201no—
258C2×C3×C4—201no—
259C2×C3×C4—202no—
260C2×C3×C4—202no—
261C2×C3×C4—202no—
262C2×C3×C4C22⋊AGL(1,𝔽5)11yescharacteristic
263C2×C3×C4C22⋊AGL(1,𝔽5)11yescharacteristic
264C2×Dic3—22no—
265C2×Dic3—42no—
266C2×Dic3—104no—
267C2×Dic3—102no—
268C2×Dic3—202no—
269C2×Dic3—202no—
270C2×Dic3—202no—
271Dic6—42no—
272Dic6—42no—
273C22×C5—21no—
274C22×C5—21no—
275C22×C5#88712yes—
276C22×C5C2×(C2×D3)⋊C411yescharacteristic
277C2×D5—21no—
278C2×D5—21no—
279C2×D5—21no—
280C2×D5—21no—
281C2×D5—42no—
282C2×D5—42no—
283C2×D5#89712yes—
284C2×D5#93611yescharacteristic
285C2×D5#94411yescharacteristic
286AGL(1,𝔽5)—82no—
287AGL(1,𝔽5)—124no—
288C4×C5—21no—
289C4×C5—122no—
290C4×C5#80612yes—
291Dic5—21no—
292Dic5—21no—
293Dic5—122no—
294C24—51no—
295C2×Q8—61no—
296C2×Q8—61no—
297C2×Q8—122no—
298C22×C4—32no—
299C22×C4—51no—
300C22×C4—51no—
301C22×C4—51no—
302C22×C4—51no—
303C22×C4—51no—
304C22×C4—102no—
305C22×C4—152no—
306C22×C4—202no—
307C22×C4—302no—
308C22×C4D3×AGL(1,𝔽5)11yescharacteristic
309C4⋊C4—202no—
310C4⋊C4—302no—
311C4⋊C4—302no—
312C4⋊C4—301no—
313C4⋊C4—301no—
314C22⋊C4—104no—
315C22⋊C4—154no—
316C22⋊C4—302no—
317C22⋊C4—302no—
318C22⋊C4—302no—
319C42—602no—
320C3×C5#199611yescharacteristic, complemented, Hall
321C22×C3—21no—
322C22×C3—21no—
323C22×C3—52no—
324C22×C3—51no—
325C22×C3—51no—
326C22×C3—101no—
327C22×C3—101no—
328C22×C3—101no—
329C22×C3—101no—
330C22×C3—202no—
331C22×C3—202no—
332C22×C3Q8×AGL(1,𝔽5)12yes—
333C22×C3C2×C22⋊AGL(1,𝔽5)11yescharacteristic
334C3×C4—21no—
335C3×C4—101no—
336C3×C4—101no—
337C3×C4—402no—
338C3×C4#401112yes—
339Dic3—42no—
340Dic3—204no—
341Dic3—202no—
342C2×C5—22no—
343C2×C5C2×#80611yescharacteristic
344C2×C5#497311yescharacteristic
345C2×C5#500611yescharacteristic
346D5—21no—
347D5—22no—
348D5—21no—
349C23—51no—
350C23—51no—
351C23—52no—
352C23—52no—
353C23—201no—
354C23—201no—
355C23C2×D3×AGL(1,𝔽5)11yescharacteristic
356Q8—122no—
357Q8—122no—
358C2×C4—22no—
359C2×C4—51no—
360C2×C4—51no—
361C2×C4—62no—
362C2×C4—102no—
363C2×C4—102no—
364C2×C4—122no—
365C2×C4—201no—
366C2×C4—201no—
367C2×C4—201no—
368C2×C4—202no—
369C2×C4—202no—
370C2×C4—202no—
371C2×C4—304no—
372C2×C4—302no—
373C2×C4—602no—
374C2×C4—602no—
375C2×C4—602no—
376C2×C4(C2×D3)⋊AGL(1,𝔽5)11yescharacteristic
377C2×C4(C2×D3)⋊AGL(1,𝔽5)11yescharacteristic
378C2×C3—22no—
379C2×C3—102no—
380C2×C3—101no—
381C2×C3—101no—
382C2×C3#6578511yescharacteristic
383C2×C3(C2×AGL(1,𝔽5))⋊Q811yescharacteristic
384C2×C3C2×#401111yescharacteristic
385C5#6952911yescharacteristic, complemented, Sylow
386C22—21no—
387C22—21no—
388C22—51no—
389C22—52no—
390C22—51no—
391C22—101no—
392C22—101no—
393C22—101no—
394C22—101no—
395C22—202no—
396C22—202no—
397C22ΓL(1,𝔽16)⋊Q812yes—
398C22C2×(C2×D3)⋊AGL(1,𝔽5)11yescharacteristic
399C4—21no—
400C4—101no—
401C4—101no—
402C4—122no—
403C4—402no—
404C4—604no—
405C4—602no—
406C4#9117212yes—
407C3#1060498211yescharacteristic, complemented, Sylow
408C2—22no—
409C2—101no—
410C2—102no—
411C2—101no—
412C2#1175175211yescharacteristic
413C2#1175181711yescharacteristic
414C2C2×#9117211yescharacteristic
subgroupG/H
centerC22C2×(C2×D3)⋊AGL(1,𝔽5)
commutatorC23×C3×C5C22×C4
FrattiniC23C2×D3×AGL(1,𝔽5)
FittingC22×C3×C4×C5C2×C4
socleC22×C3×C5C2×C22⋊C4
prime radical, cosocleC23×C3×D5C23
hypercenterC22×C4D3×AGL(1,𝔽5)
hypocenterC3×C5#1996
LCSG ▷ C23×C3×C5 ▷ C2×C3×C5 ▷ C3×C5
UCSC1 ◁ C22 ◁ C22×C4

Complements

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

NH ≅ G/Ntype
6. C22×C5×Dic6C4semidirect
12. C2×C4⋊Dic15C4semidirect
13. C2×#6669C4semidirect
42. #6763Q8semidirect
102. C22×Dic6AGL(1,𝔽5)semidirect
320. C3×C5#1996semidirect
385. C5#69529semidirect
407. C3#10604982semidirect

Maximal subgroups

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

subgroupindex
C2×#913592
C4×#67632
C3×#657652
#117465732
#106049823
#695295