Finite Group Databasebeta

← #14282804

Subgroups — #14282804

Order 1920.

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

normal subgroups only

This group has 476 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#11752685C211yesmaximal, characteristic
2#11752337C211yesmaximal, characteristic, complemented
3C3×#65686C211yesmaximal, characteristic, complemented
4#11746503C212yesmaximal, complemented
5#11746497C212yesmaximal
6#10604573—31nomaximal, Hall
7C2×C3×C4×AGL(1,𝔽5)—21no—
8#91764C411yescharacteristic, complemented
9C5×#870C411yescharacteristic, complemented
10C3×C42×D5C2211yescharacteristic
11#91450C2211yescharacteristic
12#91444—21no—
13#91418C411yescharacteristic, complemented
14#91402—21no—
15#91387—21no—
16#91360C2211yescharacteristic
17#91348—21no—
18#91333C411yescharacteristic, complemented
19#91233—21no—
20#91232C2212yes—
21C4×C3⋊#170—21no—
22C3×(C2×AGL(1,𝔽5))⋊C4—21no—
23C3×(C2×AGL(1,𝔽5))⋊C4C2212yes—
24#69431—51nomaximal, Hall
25#66025—31no—
26#65686D311yescharacteristic, complemented
27#65678—31no—
28#64869—32no—
29#64868—32no—
30C22×C3×AGL(1,𝔽5)—22no—
31C2×D60C2×C411yescharacteristic
32C2×Dic30C2×C411yescharacteristic
33C2×C3×C4×D5D411yescharacteristic
34C2×C3×C4×D5D411yescharacteristic
35C2×C3×C4×D5C2311yescharacteristic
36C2×#1202—21no—
37C2×#1200—21no—
38C2×C3⋊#170—22no—
39C3×C4×AGL(1,𝔽5)—24no—
40C3×C42×C5C2×C411yescharacteristic
41#6714—21no—
42#6714—21no—
43C5×C4⋊Dic3C2×C411yescharacteristic
44C5×C4⋊Dic3C2×C411yescharacteristic
45C5×Dic3⋊C4—21no—
46C5×Dic3⋊C4—21no—
47C3×C4×Dic5C2×C411yescharacteristic
48C3×C4×Dic5D411yescharacteristic
49C3×C4×Dic5D411yescharacteristic
50Dic3⋊Dic5—21no—
51#6669—21no—
52#6667—21no—
53#6667—21no—
54#6665—21no—
55Dic3×Dic5—21no—
56#5479—51no—
57C3×#376—51no—
58#4925—51no—
59#4452—52no—
60#4451—52no—
61C2×C4×AGL(1,𝔽5)—21no—
62C5×#125—31no—
63#4044—61no—
64#4043—31no—
65#4041—31no—
66#4040—61no—
67#4039—31no—
68#4039—61no—
69#4038—61no—
70#4033—31no—
71#4021—31no—
72C42×D5C2×D311yescharacteristic
73(C2×AGL(1,𝔽5))⋊C4—21no—
74(C2×AGL(1,𝔽5))⋊C4C2×D312yes—
75#4004—61no—
76#4003—32no—
77C4×#170—61no—
78#1500—151noSylow
79C22×D15—21no—
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—
88D60—41no—
89Dic30—41no—
90C2×C5×Dic3—21no—
91C2×C5×Dic3—21no—
92C2×C3×Dic5C22⋊C411yescharacteristic
93C2×C3×Dic5C22⋊C411yescharacteristic
94C2×C3×Dic5C2×D411yescharacteristic
95C3×C4×D5—22no—
96C3×C4×D5—22no—
97C3×C4×D5—41no—
98#1202—42no—
99#1200—42no—
100C3⋊#170—42no—
101C2×C3×C42—51no—
102C2×C3×C42—101no—
103#930—51no—
104#929—101no—
105C4×#67—101no—
106#924—51no—
107#876—101no—
108#875—51no—
109C4×D12—51no—
110#872—101no—
111#870AGL(1,𝔽5)11yescharacteristic, complemented
112C4×Dic6—51no—
113C3×#95—52no—
114C3×#95—101no—
115#830—52no—
116C4⋊(C3⋊C8)—101no—
117C4×C3⋊C8—101no—
118C22×AGL(1,𝔽5)—22no—
119C2×D20—31no—
120C2×D20—61no—
121C2×C4×D5—61no—
122C2×C4×D5#6711yescharacteristic
123C2×C4×D5#6711yescharacteristic
124C2×C4×D5C22×D311yescharacteristic
125C2×Dic10—31no—
126C2×#170—62no—
127C4×AGL(1,𝔽5)—24no—
128C5×C4⋊C4—31no—
129C5×C4⋊C4—31no—
130C5×C4⋊C4—61no—
131C5×C4⋊C4—61no—
132C42×C5C4×D311yescharacteristic
133#685—61no—
134#685—61no—
135#685—61no—
136#685—61no—
137#685—61no—
138C4⋊Dic5—61no—
139#683—61no—
140C4×Dic5—61no—
141C4×Dic5C4×D311yescharacteristic
142C4×Dic5D1211yescharacteristic
143C4×Dic5D1211yescharacteristic
144#529—151no—
145#431—151no—
146#376—51no—
147#328—152no—
148#327—152no—
149C22×C3×C5C2×C22⋊C411yescharacteristic
150C2×D15—41no—
151C2×D15—41no—
152C2×D15—41no—
153C2×C3×D5—21no—
154C2×C3×D5—21no—
155C2×C3×D5#13512yes—
156C3×AGL(1,𝔽5)—44no—
157C3×C4×C5—21no—
158C3×C4×C5—21no—
159C3×C4×C5—21no—
160Dic15—41no—
161C3×Dic5—21no—
162C3×Dic5—21no—
163C3×Dic5#10412yes—
164C5×Dic3—41no—
165C5×Dic3—41no—
166C22×C3×C4—51no—
167C22×C3×C4—51no—
168C22×C3×C4—51no—
169C22×C3×C4—102no—
170C2×#67—101no—
171C2×D12—51no—
172C2×C4×D3—101no—
173C2×Dic6—51no—
174C3×C42—51no—
175C3×C42—51no—
176C3×C42—51no—
177C3×C42—104no—
178C3×C42C2×AGL(1,𝔽5)11yescharacteristic
179#217—101no—
180(C2×D3)⋊C4—101no—
181(C2×D3)⋊C4—101no—
182(C2×D3)⋊C4—101no—
183(C2×D3)⋊C4—101no—
184C4⋊Dic3C2×AGL(1,𝔽5)11yescharacteristic
185C4⋊Dic3C2×AGL(1,𝔽5)11yescharacteristic
186Dic3⋊C4—21no—
187Dic3⋊C4—21no—
188Dic3⋊C4—101no—
189Dic3⋊C4—101no—
190C4×Dic3—101no—
191C2×C3⋊C8—102no—
192C22×D5—61no—
193C22×D5C2×#6711yescharacteristic
194C2×AGL(1,𝔽5)—24no—
195C2×AGL(1,𝔽5)—42no—
196C2×AGL(1,𝔽5)—42no—
197C2×C4×C5—61no—
198C2×C4×C5—61no—
199C2×C4×C5(C2×D3)⋊C411yescharacteristic
200C2×C4×C5(C2×D3)⋊C411yescharacteristic
201C2×C4×C5C2×C4×D311yescharacteristic
202C2×Dic5—61no—
203C2×Dic5(C2×D3)⋊C411yescharacteristic
204C2×Dic5(C2×D3)⋊C411yescharacteristic
205C2×Dic5C2×D1211yescharacteristic
206D20—121no—
207D20—122no—
208C4×D5—22no—
209C4×D5—22no—
210C4×D5—41no—
211C4×D5—122no—
212Dic10—121no—
213#170—122no—
214#126—301no—
215#125—31no—
216#124—151no—
217#123—301no—
218C22⋊Q8—151no—
219#121—151no—
220C4×Q8—151no—
221C4×D4—151no—
222C4×D4—301no—
223#117—301no—
224C2×C42—51no—
225C2×C42—101no—
226C4⋊C8—301no—
227C22⋊C8—152no—
228C4×C8—301no—
229#95—52no—
230#95—101no—
231C2×C3×C5#41011yescharacteristic
232C2×C3×C5#41611yescharacteristic
233C2×C3×C5C2×#10411yescharacteristic
234D15—81no—
235C3×D5—22no—
236C23×C3—51no—
237C22×D3—101no—
238C2×C3×C4—51no—
239C2×C3×C4—51no—
240C2×C3×C4—51no—
241C2×C3×C4—102no—
242C2×C3×C4—102no—
243C2×C3×C4—104no—
244C2×C3×C4—201no—
245C2×C3×C4—202no—
246C2×C3×C4—202no—
247C2×C3×C4C22⋊AGL(1,𝔽5)11yescharacteristic
248C2×C3×C4C22⋊AGL(1,𝔽5)11yescharacteristic
249C2×C3×C4C22×AGL(1,𝔽5)11yescharacteristic
250#67—202no—
251C2×Dic3—21no—
252C2×Dic3—21no—
253C2×Dic3—101no—
254D12—201no—
255C4×D3—202no—
256Dic6—201no—
257C3⋊C8—202no—
258C22×C5C2×(C2×D3)⋊C411yescharacteristic
259C2×D5—21no—
260C2×D5—21no—
261C2×D5—121no—
262C2×D5—121no—
263C2×D5—121no—
264C2×D5#95012yes—
265AGL(1,𝔽5)—44no—
266C4×C5—21no—
267C4×C5—21no—
268C4×C5—21no—
269C4×C5—121no—
270C4×C5—121no—
271Dic5—21no—
272Dic5—21no—
273Dic5—121no—
274Dic5#80512yes—
275C2×Q8—151no—
276C2×D4—151no—
277C2×D4—301no—
278C22×C4—51no—
279C22×C4—51no—
280C22×C4—51no—
281C22×C4—102no—
282C22×C4—301no—
283C2×C8—302no—
284C4⋊C4—31no—
285C4⋊C4—31no—
286C4⋊C4—61no—
287C4⋊C4—61no—
288C4⋊C4—301no—
289C4⋊C4—301no—
290C22⋊C4—301no—
291C22⋊C4—301no—
292C22⋊C4—301no—
293C22⋊C4—301no—
294C22⋊C4—301no—
295C42—51no—
296C42—51no—
297C42—51no—
298C42—104no—
299C42—301no—
300C42D3×AGL(1,𝔽5)11yescharacteristic
301C3×C5#150011yescharacteristic, complemented, Hall
302C22×C3—52no—
303C22×C3—101no—
304C22×C3—101no—
305C22×C3C2×C22⋊AGL(1,𝔽5)11yescharacteristic
306C2×D3—201no—
307C2×D3—201no—
308C2×D3—201no—
309C3×C4—21no—
310C3×C4—21no—
311C3×C4—21no—
312C3×C4—52no—
313C3×C4—101no—
314C3×C4—101no—
315C3×C4—204no—
316Dic3—41no—
317Dic3—41no—
318Dic3—201no—
319C2×C5C2×#80511yescharacteristic
320C2×C5#500211yescharacteristic
321C2×C5#500611yescharacteristic
322D5—22no—
323D5—241no—
324C23—51no—
325C23—301no—
326Q8—601no—
327D4—601no—
328D4—602no—
329C2×C4—51no—
330C2×C4—51no—
331C2×C4—51no—
332C2×C4—61no—
333C2×C4—61no—
334C2×C4—102no—
335C2×C4—102no—
336C2×C4—104no—
337C2×C4—201no—
338C2×C4—202no—
339C2×C4—202no—
340C2×C4—301no—
341C2×C4—602no—
342C2×C4(C2×D3)⋊AGL(1,𝔽5)11yescharacteristic
343C2×C4(C2×D3)⋊AGL(1,𝔽5)11yescharacteristic
344C2×C4C2×D3×AGL(1,𝔽5)11yescharacteristic
345C8—602no—
346C2×C3—102no—
347C2×C3#6578511yescharacteristic
348C2×C3D5⋊#10411yescharacteristic
349C2×C3#6579411yescharacteristic
350D3—401no—
351C5#6943111yescharacteristic, complemented, Sylow
352C22—52no—
353C22—101no—
354C22—101no—
355C22—601no—
356C22—601no—
357C22—601no—
358C22C2×(C2×D3)⋊AGL(1,𝔽5)11yescharacteristic
359C4—21no—
360C4—21no—
361C4—21no—
362C4—52no—
363C4—101no—
364C4—101no—
365C4—121no—
366C4—121no—
367C4—204no—
368C4—601no—
369C3#1060457311yescharacteristic, complemented, Sylow
370C2—102no—
371C2—1201no—
372C2#1175170311yescharacteristic
373C2D5⋊#80511yescharacteristic
374C2#1175181711yescharacteristic
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#91450C22
cosocleC2×C3×C4×D5C23
hypercenterC42D3×AGL(1,𝔽5)
hypocenterC3×C5#1500
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. #11752337C2semidirect
3. C3×#65686C2semidirect
4. #11746503C2semidirect
8. #91764C4semidirect
9. C5×#870C4semidirect
13. #91418C4semidirect
18. #91333C4semidirect
26. #65686D3semidirect
111. #870AGL(1,𝔽5)semidirect
301. C3×C5#1500semidirect
351. C5#69431semidirect
369. C3#10604573semidirect

Maximal subgroups

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

subgroupindex
#117526852
#117523372
C3×#656862
#117465032
#117464972
#106045733
#694315