Finite Group Databasebeta

← #14298572

Subgroups — #14298572

Order 1920.

612 conjugacy classes of subgroups; only the 114 normal subgroups are shown.

normal subgroups only

This group has 612 conjugacy classes of subgroups — too many to draw. Only its 114 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×#926C211yesmaximal, characteristic
2#11752415C211yesmaximal, characteristic
3C3×C4×C22⋊AGL(1,𝔽5)C211yesmaximal, characteristic
4#11746526C212yesmaximal, complemented
5#11746526C212yesmaximal, complemented
6#10604380—31nomaximal, Hall
7C22×C3×C4×D5C2211yescharacteristic
8C2×#6758C2212yes—
9C2×C3×C22⋊AGL(1,𝔽5)—21no—
10C2×C3×C4×AGL(1,𝔽5)C2212yes—
11#91815C411yescharacteristic, complemented
12C5×#926C411yescharacteristic, complemented
13D5×#217—21no—
14#91528—21no—
15D5×C4⋊Dic3C411yescharacteristic, complemented
16D5×C4⋊Dic3C411yescharacteristic, complemented
17C4×D5×Dic3—21no—
18#91358C2211yescharacteristic
19#91358C2211yescharacteristic
20#91357—21no—
21#91241—21no—
22#91232—21no—
23C3×(C2×AGL(1,𝔽5))⋊C4—21no—
24#69442—51nomaximal, Hall
25D5×#117—31no—
26C4×C22⋊AGL(1,𝔽5)D311yescharacteristic
27#65756—31no—
28#64875—32no—
29#64875—32no—
30C23×C3×D5D411yescharacteristic
31C22×C3×AGL(1,𝔽5)—22no—
32C22×C3×C4×C5C2×C411yescharacteristic
33C22×C3×Dic5D411yescharacteristic
34C2×C3×C4×D5—24no—
35C2×C3×C4×D5C2×C411yescharacteristic
36C2×C3×C4×D5D411yescharacteristic
37C2×C3×C4×D5D411yescharacteristic
38C2×C3×C4×D5C2311yescharacteristic
39C2×D5×Dic3—21no—
40C2×D5×Dic3—21no—
41C2×C3⋊#170—22no—
42#6758—24no—
43C3×C22⋊AGL(1,𝔽5)—24no—
44C3×C4×AGL(1,𝔽5)—24no—
45#6719—21no—
46C4⋊Dic15C2×C411yescharacteristic
47C4⋊Dic15C2×C411yescharacteristic
48C4×Dic15—21no—
49C5×#217—21no—
50C5×C4⋊Dic3C2×C411yescharacteristic
51C5×C4⋊Dic3C2×C411yescharacteristic
52C4×C5×Dic3—21no—
53#6668—21no—
54#6668—21no—
55#6665—21no—
56#6665—21no—
57Dic3×Dic5—21no—
58Dic3×Dic5—21no—
59C2×#926—51no—
60C3×C4×C22⋊C4—51no—
61#5207—51no—
62#4523—52no—
63#4523—52no—
64C22×C4×D5C2×D311yescharacteristic
65C2×C22⋊AGL(1,𝔽5)—21no—
66C2×C4×AGL(1,𝔽5)C2×D312yes—
67C2×#700—32no—
68C5×#117—31no—
69#4072—31no—
70#4038—31no—
71#4038—31no—
72D5×C4⋊C4—31no—
73D5×C4⋊C4—31no—
74D5×C22⋊C4—61no—
75#4023—61no—
76#4018—61no—
77C42×D5—61no—
78C22⋊#170—61no—
79(C2×AGL(1,𝔽5))⋊C4—21no—
80#4003—61no—
81#1826—151noSylow
82C23×C3×C5C22⋊C411yescharacteristic
83C22×C3×D5—22no—
84C22×C3×D5—22no—
85C22×C3×D5—41no—
86C22×C3×D5C22⋊C411yescharacteristic
87C22×C3×D5C2×D411yescharacteristic
88C2×C3×AGL(1,𝔽5)—24no—
89C2×C3×AGL(1,𝔽5)—42no—
90C2×C3×AGL(1,𝔽5)—42no—
91C2×C3×C4×C5—22no—
92C2×C3×C4×C5C22⋊C411yescharacteristic
93C2×C3×C4×C5C22×C411yescharacteristic
94C2×Dic15—21no—
95C2×Dic15—21no—
96C2×C5×Dic3—21no—
97C2×C5×Dic3—21no—
98C2×C3×Dic5—22no—
99C2×C3×Dic5C22⋊C411yescharacteristic
100C2×C3×Dic5C2×D411yescharacteristic
101C3×C4×D5—22no—
102C3×C4×D5—22no—
103C3×C4×D5—22no—
104C3×C4×D5#4114yes—
105D5×Dic3—42no—
106D5×Dic3—42no—
107C3⋊#170—42no—
108C23×C3×C4—51no—
109C2×C3×C22⋊C4—101no—
110C2×C3×C42—52no—
111C2×#217—101no—
112#926—51no—
113#926—51no—
114#926—51no—
115#926—101no—
116#926—101no—
117#926AGL(1,𝔽5)11yescharacteristic, complemented
118C2×C4⋊Dic3—51no—
119C2×C4⋊Dic3—51no—
120C2×C4×Dic3—101no—
121C2×#208—52no—
122C3×#95—101no—
123#830—101no—
124#830—101no—
125C23×D5#6711yescharacteristic
126C22×AGL(1,𝔽5)—22no—
127C22×C4×C5C4×D311yescharacteristic
128C22×Dic5D1211yescharacteristic
129C2×C4×D5—24no—
130C2×C4×D5—61no—
131C2×C4×D5—61no—
132C2×C4×D5C4×D311yescharacteristic
133C2×C4×D5D1211yescharacteristic
134C2×C4×D5#6711yescharacteristic
135C2×C4×D5C22×D311yescharacteristic
136C22⋊AGL(1,𝔽5)—24no—
137C2×#170—62no—
138C4×AGL(1,𝔽5)—24no—
139#700—64no—
140C5×C4⋊C4—31no—
141C5×C4⋊C4—31no—
142C5×C22⋊C4—61no—
143C42×C5—61no—
144C22⋊Dic5—61no—
145#685—61no—
146#685—61no—
147C4⋊Dic5—31no—
148C4⋊Dic5—31no—
149#683—61no—
150#683—61no—
151C4×Dic5—61no—
152C4×Dic5—61no—
153C4×Dic5—61no—
154C2×#117—151no—
155#407—151no—
156C4×C22⋊C4—51no—
157#339—152no—
158#339—152no—
159C22×C3×C5—21no—
160C22×C3×C5—21no—
161C22×C3×C5—21no—
162C22×C3×C5C2×C22⋊C411yescharacteristic
163C2×C3×D5—21no—
164C2×C3×D5—21no—
165C2×C3×D5—21no—
166C2×C3×D5—21no—
167C2×C3×D5—44no—
168C2×C3×D5#10412yes—
169C2×C3×D5#12312yes—
170C3×AGL(1,𝔽5)—44no—
171C3×C4×C5—21no—
172C3×C4×C5#11712yes—
173Dic15—41no—
174Dic15—41no—
175C3×Dic5—21no—
176C3×Dic5#12312yes—
177C5×Dic3—41no—
178C5×Dic3—41no—
179C24×C3—51no—
180C22×C3×C4—51no—
181C22×C3×C4—51no—
182C22×C3×C4—51no—
183C22×C3×C4—51no—
184C22×C3×C4—51no—
185C22×C3×C4—104no—
186C22×C3×C4—102no—
187C22×C3×C4C2×AGL(1,𝔽5)11yescharacteristic
188C22×Dic3—101no—
189C22×Dic3—101no—
190C3×C22⋊C4—104no—
191C3×C42—104no—
192#217—21no—
193#217—101no—
194#217—101no—
195#217—101no—
196C4⋊Dic3—51no—
197C4⋊Dic3—51no—
198C4⋊Dic3—101no—
199C4⋊Dic3—101no—
200C4⋊Dic3C2×AGL(1,𝔽5)11yescharacteristic
201C4⋊Dic3C2×AGL(1,𝔽5)11yescharacteristic
202C4×Dic3—21no—
203C4×Dic3—101no—
204C4×Dic3—101no—
205C4×Dic3—101no—
206#208—104no—
207C2×C3⋊C8—102no—
208C23×C5(C2×D3)⋊C411yescharacteristic
209C22×D5—22no—
210C22×D5—22no—
211C22×D5—41no—
212C22×D5(C2×D3)⋊C411yescharacteristic
213C22×D5C2×#6711yescharacteristic
214C2×AGL(1,𝔽5)—24no—
215C2×AGL(1,𝔽5)—42no—
216C2×AGL(1,𝔽5)—42no—
217C2×C4×C5—22no—
218C2×C4×C5—61no—
219C2×C4×C5—61no—
220C2×C4×C5(C2×D3)⋊C411yescharacteristic
221C2×C4×C5C2×C4×D311yescharacteristic
222C2×Dic5—22no—
223C2×Dic5—61no—
224C2×Dic5—61no—
225C2×Dic5(C2×D3)⋊C411yescharacteristic
226C2×Dic5C2×D1211yescharacteristic
227C4×D5—22no—
228C4×D5—22no—
229C4×D5—22no—
230C4×D5—122no—
231C4×D5—122no—
232C4×D5#23714yes—
233#170—122no—
234C23×C4—51no—
235C2×M16—152no—
236#117—31no—
237#117—151no—
238#117—151no—
239#117—151no—
240#117—301no—
241#117—301no—
242C2×C4⋊C4—151no—
243C2×C4⋊C4—151no—
244C2×C22⋊C4—101no—
245C2×C22⋊C4—301no—
246C2×C42—52no—
247C2×C42—301no—
248C22⋊C8—301no—
249C22⋊C8—301no—
250#95—101no—
251C2×C3×C5—41no—
252C2×C3×C5#38111yescharacteristic
253C2×C3×C5C2×#10411yescharacteristic
254C2×C3×C5#42111yescharacteristic
255C3×D5—22no—
256C3×D5—22no—
257C23×C3—51no—
258C23×C3—51no—
259C23×C3—102no—
260C23×C3—102no—
261C23×C3—201no—
262C23×C3C22⋊AGL(1,𝔽5)11yescharacteristic
263C2×C3×C4—22no—
264C2×C3×C4—51no—
265C2×C3×C4—54no—
266C2×C3×C4—51no—
267C2×C3×C4—102no—
268C2×C3×C4—102no—
269C2×C3×C4—102no—
270C2×C3×C4—102no—
271C2×C3×C4—104no—
272C2×C3×C4—202no—
273C2×C3×C4—202no—
274C2×C3×C4C22⋊AGL(1,𝔽5)11yescharacteristic
275C2×C3×C4C22×AGL(1,𝔽5)11yescharacteristic
276C2×Dic3—21no—
277C2×Dic3—21no—
278C2×Dic3—101no—
279C2×Dic3—101no—
280C2×Dic3—202no—
281C2×Dic3—202no—
282C3⋊C8—202no—
283C22×C5—21no—
284C22×C5—21no—
285C22×C5—21no—
286C22×C5C2×(C2×D3)⋊C411yescharacteristic
287C2×D5—21no—
288C2×D5—21no—
289C2×D5—21no—
290C2×D5—21no—
291C2×D5—44no—
292C2×D5#80512yes—
293C2×D5#93512yes—
294AGL(1,𝔽5)—44no—
295C4×C5—21no—
296C4×C5—121no—
297C4×C5—121no—
298C4×C5#87712yes—
299Dic5—21no—
300Dic5—121no—
301Dic5—121no—
302Dic5#88612yes—
303C24—51no—
304C22×C4—51no—
305C22×C4—51no—
306C22×C4—51no—
307C22×C4—51no—
308C22×C4—51no—
309C22×C4—104no—
310C22×C4—102no—
311C22×C4—301no—
312C22×C4—301no—
313C22×C4D3×AGL(1,𝔽5)11yescharacteristic
314M16—304no—
315C2×C8—302no—
316C4⋊C4—31no—
317C4⋊C4—31no—
318C4⋊C4—151no—
319C4⋊C4—151no—
320C4⋊C4—301no—
321C4⋊C4—301no—
322C22⋊C4—61no—
323C22⋊C4—104no—
324C22⋊C4—301no—
325C22⋊C4—301no—
326C22⋊C4—301no—
327C42—61no—
328C42—104no—
329C42—301no—
330C42—301no—
331C42—301no—
332C3×C5#182611yescharacteristic, complemented, Hall
333C22×C3—21no—
334C22×C3—21no—
335C22×C3—21no—
336C22×C3—52no—
337C22×C3—52no—
338C22×C3—101no—
339C22×C3—101no—
340C22×C3—101no—
341C22×C3—101no—
342C22×C3—204no—
343C22×C3C2×C22⋊AGL(1,𝔽5)11yescharacteristic
344C3×C4—21no—
345C3×C4—52no—
346C3×C4—101no—
347C3×C4—204no—
348C3×C4#413012yes—
349Dic3—41no—
350Dic3—41no—
351Dic3—201no—
352Dic3—201no—
353C2×C5—41no—
354C2×C5C2×#80511yescharacteristic
355C2×C5#494311yescharacteristic
356C2×C5#500511yescharacteristic
357D5—22no—
358D5—22no—
359C23—51no—
360C23—51no—
361C23—102no—
362C23—102no—
363C23—201no—
364C23(C2×D3)⋊AGL(1,𝔽5)11yescharacteristic
365C2×C4—22no—
366C2×C4—51no—
367C2×C4—54no—
368C2×C4—51no—
369C2×C4—61no—
370C2×C4—61no—
371C2×C4—102no—
372C2×C4—102no—
373C2×C4—102no—
374C2×C4—102no—
375C2×C4—104no—
376C2×C4—202no—
377C2×C4—202no—
378C2×C4—301no—
379C2×C4—301no—
380C2×C4—602no—
381C2×C4—602no—
382C2×C4(C2×D3)⋊AGL(1,𝔽5)11yescharacteristic
383C2×C4C2×D3×AGL(1,𝔽5)11yescharacteristic
384C8—602no—
385C2×C3—41no—
386C2×C3—102no—
387C2×C3—102no—
388C2×C3#6576411yescharacteristic
389C2×C3D5⋊#10411yescharacteristic
390C2×C3#6579511yescharacteristic
391C5#6944211yescharacteristic, complemented, Sylow
392C22—21no—
393C22—21no—
394C22—21no—
395C22—52no—
396C22—52no—
397C22—101no—
398C22—101no—
399C22—101no—
400C22—101no—
401C22—204no—
402C22C2×(C2×D3)⋊AGL(1,𝔽5)11yescharacteristic
403C4—21no—
404C4—52no—
405C4—101no—
406C4—121no—
407C4—121no—
408C4—204no—
409C4—601no—
410C4—601no—
411C4#9192312yes—
412C3#1060438011yescharacteristic, complemented, Sylow
413C2—41no—
414C2—102no—
415C2—102no—
416C2D5⋊#80511yescharacteristic
417C2#1175172011yescharacteristic
418C2#1175198411yescharacteristic
subgroupG/H
centerC22C2×(C2×D3)⋊AGL(1,𝔽5)
commutatorC2×C3×C4×C5C22×C4
FrattiniC2×C4C2×D3×AGL(1,𝔽5)
FittingC22×C3×C4×C5C2×C4
socleC22×C3×C5C2×C22⋊C4
prime radicalC23×C3×D5D4
cosocleC2×C3×C4×D5C23
hypercenterC22×C4D3×AGL(1,𝔽5)
hypocenterC3×C5#1826
LCSG ▷ C2×C3×C4×C5 ▷ C2×C3×C5 ▷ C3×C5
UCSC1 ◁ C22 ◁ C22×C4

Complements

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

NH ≅ G/Ntype
4. #11746526C2semidirect
5. #11746526C2semidirect
11. #91815C4semidirect
12. C5×#926C4semidirect
15. D5×C4⋊Dic3C4semidirect
16. D5×C4⋊Dic3C4semidirect
117. #926AGL(1,𝔽5)semidirect
332. C3×C5#1826semidirect
391. C5#69442semidirect
412. C3#10604380semidirect

Maximal subgroups

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

subgroupindex
D5×#9262
#117524152
C3×C4×C22⋊AGL(1,𝔽5)2
#117465262
#106043803
#694425