Finite Group Databasebeta

← #14282950

Subgroups — #14282950

Order 1920.

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

normal subgroups only

This group has 480 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
1#11752497C211yesmaximal, characteristic
2#11752337C211yesmaximal, characteristic
3C3×C4×C4⋊AGL(1,𝔽5)C211yesmaximal, characteristic
4#11746503C212yesmaximal
5#11746502C212yesmaximal
6#10604424—31nomaximal, Hall
7C2×C3×C4⋊AGL(1,𝔽5)—21no—
8C2×C3×C4×AGL(1,𝔽5)—21no—
9C4×Dic30C411yescharacteristic, complemented
10C4×C5×Dic6C411yescharacteristic, complemented
11C3×C42×D5C2211yescharacteristic
12#91468C411yescharacteristic, complemented
13C4×#1204—21no—
14#91378—21no—
15#91376C411yescharacteristic, complemented
16#91360C2211yescharacteristic
17#91359C2211yescharacteristic
18#91357—21no—
19#91349—21no—
20#91233—21no—
21#91232C2212yes—
22C4×C3⋊#170—21no—
23C3×(C2×AGL(1,𝔽5))⋊C4C2212yes—
24#69471—51nomaximal, Hall
25#65912—31no—
26C4×C4⋊AGL(1,𝔽5)D311yescharacteristic
27#65678—31no—
28#64926—32no—
29#64869—32no—
30C22×C3×AGL(1,𝔽5)—22no—
31C2×Dic30C2×C411yescharacteristic
32C2×C5×Dic6C2×C411yescharacteristic
33C2×C3×C4×D5D411yescharacteristic
34C2×C3×C4×D5D411yescharacteristic
35C2×C3×C4×D5C2311yescharacteristic
36C2×#1204—21no—
37C2×C3⋊#170—22no—
38C3×C4⋊AGL(1,𝔽5)—24no—
39C3×C4×AGL(1,𝔽5)—42no—
40C3×C42×C5C2×C411yescharacteristic
41C4⋊Dic15C2×C411yescharacteristic
42#6712—21no—
43C4×Dic15—21no—
44C5×C4⋊Dic3C2×C411yescharacteristic
45C5×Dic3⋊C4—21no—
46C4×C5×Dic3—21no—
47C3×C4×Dic5C2×C411yescharacteristic
48C3×C4×Dic5D411yescharacteristic
49C3×C4×Dic5D411yescharacteristic
50Dic3⋊Dic5—21no—
51#6669—21no—
52#6668—21no—
53#6668—21no—
54#6668—21no—
55#6665—21no—
56#6665—21no—
57#5470—51no—
58C3×C4×C4⋊C4—51no—
59#4925—51no—
60#4452—52no—
61#4452—52no—
62C2×C4⋊AGL(1,𝔽5)—21no—
63C2×C4×AGL(1,𝔽5)—21no—
64C4×C5×Q8—31no—
65#4092—31no—
66#4090—31no—
67#4043—31no—
68#4042—61no—
69#4035—61no—
70#4034—31no—
71#4018—61no—
72C42×D5C2×D311yescharacteristic
73C4×Dic10—31no—
74C4×Dic10—61no—
75(C2×AGL(1,𝔽5))⋊C4C2×D312yes—
76#4004—61no—
77#4003—32no—
78C4×#170—61no—
79#1503—151noSylow
80C22×C3×D5C2×D411yescharacteristic
81C2×C3×AGL(1,𝔽5)—24no—
82C2×C3×AGL(1,𝔽5)—42no—
83C2×C3×AGL(1,𝔽5)—42no—
84C2×C3×C4×C5C22⋊C411yescharacteristic
85C2×C3×C4×C5C22⋊C411yescharacteristic
86C2×C3×C4×C5C22×C411yescharacteristic
87C2×Dic15—21no—
88C2×Dic15—21no—
89Dic30—41no—
90C2×C5×Dic3—21no—
91C2×C5×Dic3—21no—
92C5×Dic6—41no—
93C2×C3×Dic5C22⋊C411yescharacteristic
94C2×C3×Dic5C22⋊C411yescharacteristic
95C2×C3×Dic5C2×D411yescharacteristic
96C3×C4×D5—22no—
97C3×C4×D5—41no—
98C3×C4×D5SD1612yes—
99C3×C4×D5Q1612yes—
100#1204—42no—
101C3⋊#170—42no—
102C2×C3×C4⋊C4—101no—
103C2×C3×C42—51no—
104C2×C3×C42—101no—
105#926—101no—
106#924—51no—
107#924—51no—
108#924—101no—
109#870—101no—
110#869—51no—
111#869—51no—
112C4×Dic6—51no—
113C4×Dic6—101no—
114C4×Dic6AGL(1,𝔽5)11yescharacteristic, complemented
115C3×#95—52no—
116#830—52no—
117C4⋊(C3⋊C8)—101no—
118C4×C3⋊C8—101no—
119C22×AGL(1,𝔽5)—22no—
120C2×C5×Q8—31no—
121C2×C4×D5#6711yescharacteristic
122C2×C4×D5#6711yescharacteristic
123C2×C4×D5C22×D311yescharacteristic
124C2×Dic10—31no—
125C2×Dic10—61no—
126C2×#170—62no—
127C4⋊AGL(1,𝔽5)—24no—
128C4×AGL(1,𝔽5)—42no—
129C5×C4⋊C4—31no—
130C5×C4⋊C4—61no—
131C42×C5—61no—
132C42×C5C4×D311yescharacteristic
133#685—61no—
134#685—61no—
135C4⋊Dic5—31no—
136C4⋊Dic5—61no—
137#683—61no—
138#683—61no—
139#683—61no—
140#683—61no—
141#683—61no—
142C4×Dic5—61no—
143C4×Dic5C4×D311yescharacteristic
144C4×Dic5D1211yescharacteristic
145C4×Dic5D1211yescharacteristic
146#533—151no—
147#431—151no—
148C4×C4⋊C4—51no—
149#328—152no—
150#328—152no—
151C22×C3×C5C2×C22⋊C411yescharacteristic
152C2×C3×D5—21no—
153C2×C3×D5—21no—
154C2×C3×D5C2×SD1611yescharacteristic
155C2×C3×D5C2×Q1611yescharacteristic
156C3×AGL(1,𝔽5)—44no—
157C3×C4×C5—21no—
158C3×C4×C5—21no—
159C3×C4×C5#10312yes—
160Dic15—41no—
161Dic15—41no—
162C3×Dic5—21no—
163C3×Dic5#10312yes—
164C3×Dic5#10412yes—
165C5×Dic3—41no—
166C5×Dic3—41no—
167C22×C3×C4—51no—
168C22×C3×C4—51no—
169C22×C3×C4—51no—
170C22×C3×C4—102no—
171C2×Dic6—51no—
172C2×Dic6—101no—
173C2×Dic6C2×AGL(1,𝔽5)11yescharacteristic
174C3×C4⋊C4—104no—
175C3×C42—51no—
176C3×C42—51no—
177C3×C42—51no—
178C3×C42—202no—
179C3×C42C2×AGL(1,𝔽5)11yescharacteristic
180#217—101no—
181#217—101no—
182C4⋊Dic3—51no—
183C4⋊Dic3—101no—
184C4⋊Dic3—101no—
185C4⋊Dic3—101no—
186C4⋊Dic3C2×AGL(1,𝔽5)11yescharacteristic
187Dic3⋊C4—21no—
188Dic3⋊C4—101no—
189Dic3⋊C4—101no—
190Dic3⋊C4—101no—
191C4×Dic3—21no—
192C4×Dic3—101no—
193C2×C3⋊C8—102no—
194C22×D5C2×#6711yescharacteristic
195C2×AGL(1,𝔽5)—24no—
196C2×AGL(1,𝔽5)—42no—
197C2×AGL(1,𝔽5)—42no—
198C5×Q8—121no—
199C2×C4×C5—61no—
200C2×C4×C5—61no—
201C2×C4×C5(C2×D3)⋊C411yescharacteristic
202C2×C4×C5(C2×D3)⋊C411yescharacteristic
203C2×C4×C5C2×C4×D311yescharacteristic
204C2×Dic5—61no—
205C2×Dic5—61no—
206C2×Dic5(C2×D3)⋊C411yescharacteristic
207C2×Dic5(C2×D3)⋊C411yescharacteristic
208C2×Dic5C2×D1211yescharacteristic
209C4×D5—22no—
210C4×D5—41no—
211C4×D5#21412yes—
212C4×D5#21612yes—
213Dic10—122no—
214Dic10—121no—
215#170—122no—
216C4⋊Q8—151no—
217C4⋊Q8—151no—
218#125—301no—
219C22⋊Q8—151no—
220C22⋊Q8—151no—
221C22⋊Q8—301no—
222C4×Q8—31no—
223C4×Q8—151no—
224C4×Q8—301no—
225#117—301no—
226C2×C4⋊C4—101no—
227C2×C42—51no—
228C2×C42—101no—
229C4⋊C8—301no—
230C22⋊C8—152no—
231C4×C8—301no—
232#95—52no—
233C2×C3×C5#41011yescharacteristic
234C2×C3×C5C2×#10311yescharacteristic
235C2×C3×C5C2×#10411yescharacteristic
236C3×D5—21no—
237C3×D5—21no—
238C23×C3—51no—
239C2×C3×C4—51no—
240C2×C3×C4—51no—
241C2×C3×C4—52no—
242C2×C3×C4—52no—
243C2×C3×C4—51no—
244C2×C3×C4—102no—
245C2×C3×C4—104no—
246C2×C3×C4—201no—
247C2×C3×C4—202no—
248C2×C3×C4—202no—
249C2×C3×C4C22⋊AGL(1,𝔽5)11yescharacteristic
250C2×C3×C4C22⋊AGL(1,𝔽5)11yescharacteristic
251C2×C3×C4C22×AGL(1,𝔽5)11yescharacteristic
252C2×Dic3—21no—
253C2×Dic3—21no—
254C2×Dic3—101no—
255C2×Dic3—101no—
256Dic6—41no—
257Dic6—202no—
258Dic6—201no—
259C3⋊C8—202no—
260C22×C5C2×(C2×D3)⋊C411yescharacteristic
261C2×D5—21no—
262C2×D5—21no—
263C2×D5C2×#21411yescharacteristic
264C2×D5C2×#21611yescharacteristic
265AGL(1,𝔽5)—44no—
266C4×C5—21no—
267C4×C5—21no—
268C4×C5—121no—
269C4×C5—121no—
270C4×C5#81012yes—
271Dic5—21no—
272Dic5—121no—
273Dic5—121no—
274Dic5#80512yes—
275Dic5#81012yes—
276C2×Q8—31no—
277C2×Q8—151no—
278C2×Q8—301no—
279C22×C4—51no—
280C22×C4—51no—
281C22×C4—51no—
282C22×C4—102no—
283C2×C8—302no—
284C4⋊C4—31no—
285C4⋊C4—61no—
286C4⋊C4—104no—
287C4⋊C4—151no—
288C4⋊C4—301no—
289C4⋊C4—301no—
290C4⋊C4—301no—
291C4⋊C4—301no—
292C4⋊C4—301no—
293C4⋊C4—301no—
294C22⋊C4—301no—
295C22⋊C4—301no—
296C42—51no—
297C42—51no—
298C42—51no—
299C42—61no—
300C42—202no—
301C42—301no—
302C42D3×AGL(1,𝔽5)11yescharacteristic
303C3×C5#150311yescharacteristic, complemented, Hall
304C22×C3—51no—
305C22×C3—51no—
306C22×C3—101no—
307C22×C3—101no—
308C22×C3C2×C22⋊AGL(1,𝔽5)11yescharacteristic
309C3×C4—21no—
310C3×C4—21no—
311C3×C4—52no—
312C3×C4—52no—
313C3×C4—101no—
314C3×C4—204no—
315C3×C4#400912yes—
316Dic3—41no—
317Dic3—41no—
318Dic3—201no—
319Dic3—201no—
320C2×C5C2×#80511yescharacteristic
321C2×C5C2×#81011yescharacteristic
322C2×C5#500611yescharacteristic
323D5—21no—
324D5—21no—
325C23—51no—
326Q8—121no—
327Q8—601no—
328Q8—602no—
329C2×C4—51no—
330C2×C4—51no—
331C2×C4—52no—
332C2×C4—52no—
333C2×C4—51no—
334C2×C4—61no—
335C2×C4—61no—
336C2×C4—102no—
337C2×C4—104no—
338C2×C4—201no—
339C2×C4—202no—
340C2×C4—202no—
341C2×C4—301no—
342C2×C4—301no—
343C2×C4(C2×D3)⋊AGL(1,𝔽5)11yescharacteristic
344C2×C4(C2×D3)⋊AGL(1,𝔽5)11yescharacteristic
345C2×C4C2×D3×AGL(1,𝔽5)11yescharacteristic
346C8—602no—
347C2×C3—101no—
348C2×C3—101no—
349C2×C3#6574511yescharacteristic
350C2×C3C2×#400911yescharacteristic
351C2×C3D5⋊#10411yescharacteristic
352C5#6947111yescharacteristic, complemented, Sylow
353C22—51no—
354C22—51no—
355C22—101no—
356C22—101no—
357C22C2×(C2×D3)⋊AGL(1,𝔽5)11yescharacteristic
358C4—21no—
359C4—21no—
360C4—52no—
361C4—52no—
362C4—101no—
363C4—121no—
364C4—121no—
365C4—204no—
366C4—601no—
367C4—601no—
368C4#9115312yes—
369C3#1060442411yescharacteristic, complemented, Sylow
370C2—101no—
371C2—101no—
372C2C2×#9115311yescharacteristic
373C2D5⋊#80511yescharacteristic
374C2#1175181511yescharacteristic
subgroupG/H
centerC2×C4(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 radicalC22×C3×D5C2×D4
cosocleC2×C3×C4×D5C23
hypercenterC42D3×AGL(1,𝔽5)
hypocenterC3×C5#1503
LCSG ▷ C2×C3×C4×C5 ▷ C22×C3×C5 ▷ C3×C5
UCSC1 ◁ C2×C4 ◁ C42

Complements

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

NH ≅ G/Ntype
9. C4×Dic30C4semidirect
10. C4×C5×Dic6C4semidirect
12. #91468C4semidirect
15. #91376C4semidirect
114. C4×Dic6AGL(1,𝔽5)semidirect
303. C3×C5#1503semidirect
352. C5#69471semidirect
369. C3#10604424semidirect

Maximal subgroups

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

subgroupindex
#117524972
#117523372
C3×C4×C4⋊AGL(1,𝔽5)2
#117465032
#117465022
#106044243
#694715