Finite Group Databasebeta

← #14283391

Subgroups — #14283391

Order 1920.

444 conjugacy classes of subgroups; only the 90 normal subgroups are shown.

normal subgroups only

This group has 444 conjugacy classes of subgroups — too many to draw. Only its 90 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#11752682C211yesmaximal, characteristic
2#11752343C211yesmaximal, characteristic, complemented
3C3×#65689C211yesmaximal, characteristic, complemented
4#11746503C212yesmaximal, complemented
5#11746496C212yesmaximal
6#10604392—31nomaximal, Hall
7C2×C3×C4⋊AGL(1,𝔽5)—21no—
8C4×Dic30C411yescharacteristic, complemented
9C4×C5×D12C411yescharacteristic, complemented
10C3×C42×D5C2211yescharacteristic
11#91449C2211yescharacteristic
12#91447—21no—
13C4×#1201—21no—
14#91388C411yescharacteristic, complemented
15#91373C411yescharacteristic, complemented
16#91372—21no—
17#91371—21no—
18#91360C2211yescharacteristic
19#91233—21no—
20#91232C2212yes—
21#91231—21no—
22C3×(C2×AGL(1,𝔽5))⋊C4—21no—
23C3×(C2×AGL(1,𝔽5))⋊C4C2212yes—
24#69544—51nomaximal, Hall
25#65891—31no—
26#65689D311yescharacteristic, complemented
27#65684—31no—
28#64920—32no—
29#64869—32no—
30C22×C3×AGL(1,𝔽5)—22no—
31C2×Dic30C2×C411yescharacteristic
32C2×C5×D12C2×C411yescharacteristic
33C2×C4×C5×D3—21no—
34C2×C3×C4×D5D411yescharacteristic
35C2×C3×C4×D5D411yescharacteristic
36C2×C3×C4×D5C2311yescharacteristic
37C2×#1201—21no—
38C2×C3⋊#170—22no—
39C3×C4⋊AGL(1,𝔽5)—42no—
40C3×C42×C5C2×C411yescharacteristic
41C4⋊Dic15C2×C411yescharacteristic
42#6712—21no—
43C4×Dic15—21no—
44C5×(C2×D3)⋊C4—21no—
45C5×C4⋊Dic3C2×C411yescharacteristic
46C3×C4×Dic5C2×C411yescharacteristic
47C3×C4×Dic5D411yescharacteristic
48C3×C4×Dic5D411yescharacteristic
49Dic3⋊Dic5—21no—
50Dic3⋊Dic5—21no—
51#6669—21no—
52(C2×D3)⋊Dic5—21no—
53(C2×D3)⋊Dic5—21no—
54(C2×D3)⋊Dic5—21no—
55#6665—21no—
56#5479—51no—
57C3×#382—51no—
58#4926—51no—
59#4452—52no—
60#4451—52no—
61C2×C4⋊AGL(1,𝔽5)—21no—
62C4×C5×D4—31no—
63#4084—31no—
64#4082—31no—
65C4×C22⋊D5—61no—
66#4072—61no—
67#4043—31no—
68#4036—31no—
69#4029—61no—
70#4025—61no—
71C42×D5C2×D311yescharacteristic
72C4×Dic10—31no—
73(C2×AGL(1,𝔽5))⋊C4—21no—
74(C2×AGL(1,𝔽5))⋊C4C2×D312yes—
75#4004—61no—
76#4003—32no—
77#4002—61no—
78#1512—151noSylow
79C22×C5×D3—21no—
80C22×C3×D5C2×D411yescharacteristic
81C2×C3×AGL(1,𝔽5)—42no—
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—
91C5×D12—41no—
92C4×C5×D3—42no—
93C2×C3×Dic5C22⋊C411yescharacteristic
94C2×C3×Dic5C22⋊C411yescharacteristic
95C2×C3×Dic5C2×D411yescharacteristic
96C3×C4×D5—21no—
97C3×C4×D5—21no—
98C3×C4×D5—41no—
99C3×C4×D5—41no—
100#1201—41no—
101#1201—41no—
102C3⋊#170—42no—
103C2×C3×C4⋊C4—101no—
104C2×C3×C42—51no—
105#930—51no—
106#929—101no—
107C4×#67—101no—
108#924—51no—
109#876—101no—
110#875—51no—
111C4×D12AGL(1,𝔽5)11yescharacteristic, complemented
112#872—101no—
113#870—51no—
114C4×Dic6—51no—
115C3×#95—52no—
116C3×#95—101no—
117#830—52no—
118C4⋊(C3⋊C8)—101no—
119#803—101no—
120C22×AGL(1,𝔽5)—22no—
121C2×C5×D4—31no—
122C22×C4×C5—61no—
123C2×C22⋊D5—61no—
124C2×C4×D5#6711yescharacteristic
125C2×C4×D5#6711yescharacteristic
126C2×C4×D5C22×D311yescharacteristic
127C2×Dic10—31no—
128C2×#170—62no—
129C4⋊AGL(1,𝔽5)—42no—
130C5×C4⋊C4—31no—
131C5×C22⋊C4—61no—
132C42×C5C4×D311yescharacteristic
133C22⋊Dic5—61no—
134C22⋊Dic5—61no—
135C22⋊Dic5—61no—
136#685—61no—
137C4⋊Dic5—31no—
138C4⋊Dic5—61no—
139C4⋊Dic5—61no—
140#683—61no—
141#683—61no—
142C4×Dic5—61no—
143C4×Dic5C4×D311yescharacteristic
144C4×Dic5D1211yescharacteristic
145C4×Dic5D1211yescharacteristic
146#529—151no—
147#432—151no—
148#382—51no—
149#328—152no—
150#327—152no—
151C22×C3×C5C2×C22⋊C411yescharacteristic
152C2×C5×D3—41no—
153C2×C5×D3—41no—
154C2×C5×D3—41no—
155C2×C3×D5—21no—
156C2×C3×D5—21no—
157C2×C3×D5AGL(1,ℤ/8)11yescharacteristic
158C2×C3×D5#13711yescharacteristic
159C3×AGL(1,𝔽5)—82no—
160C3×C4×C5—21no—
161C3×C4×C5—21no—
162C3×C4×C5—21no—
163Dic15—41no—
164Dic15—41no—
165C3×Dic5—21no—
166C3×Dic5—21no—
167C3×Dic5—21no—
168C5×Dic3—41no—
169C22×C3×C4—51no—
170C22×C3×C4—51no—
171C22×C3×C4—51no—
172C22×C3×C4—102no—
173C2×#67—101no—
174C2×D12C2×AGL(1,𝔽5)11yescharacteristic
175C2×C4×D3—21no—
176C2×Dic6—51no—
177C3×C4⋊C4—202no—
178C3×C42—51no—
179C3×C42—51no—
180C3×C42—51no—
181C3×C42C2×AGL(1,𝔽5)11yescharacteristic
182#217—101no—
183(C2×D3)⋊C4—21no—
184(C2×D3)⋊C4—101no—
185(C2×D3)⋊C4—101no—
186(C2×D3)⋊C4—101no—
187C4⋊Dic3—51no—
188C4⋊Dic3C2×AGL(1,𝔽5)11yescharacteristic
189Dic3⋊C4—101no—
190Dic3⋊C4—101no—
191Dic3⋊C4—101no—
192Dic3⋊C4—101no—
193C4×Dic3—101no—
194C2×C3⋊C8—102no—
195C23×C5—61no—
196C22×D5C2×#6711yescharacteristic
197C2×AGL(1,𝔽5)—42no—
198C2×AGL(1,𝔽5)—42no—
199C2×AGL(1,𝔽5)—42no—
200C5×D4—121no—
201C2×C4×C5—61no—
202C2×C4×C5—122no—
203C2×C4×C5(C2×D3)⋊C411yescharacteristic
204C2×C4×C5(C2×D3)⋊C411yescharacteristic
205C2×C4×C5C2×C4×D311yescharacteristic
206C22⋊D5—121no—
207C22⋊D5—121no—
208C2×Dic5—61no—
209C2×Dic5—61no—
210C2×Dic5(C2×D3)⋊C411yescharacteristic
211C2×Dic5(C2×D3)⋊C411yescharacteristic
212C2×Dic5C2×D1211yescharacteristic
213C4×D5—21no—
214C4×D5—21no—
215C4×D5—41no—
216C4×D5—41no—
217Dic10—121no—
218#170—122no—
219#126—301no—
220#125—151no—
221#124—151no—
222#123—301no—
223C22⋊Q8—151no—
224#121—151no—
225C4×Q8—151no—
226C4×D4—31no—
227C4×D4—301no—
228#117—301no—
229C2×C4⋊C4—101no—
230C2×C42—51no—
231C4⋊C8—301no—
232C22⋊C8—152no—
233#97—301no—
234#95—52no—
235#95—101no—
236C2×C3×C5#41011yescharacteristic
237C2×C3×C5#41711yescharacteristic
238C2×C3×C5#42111yescharacteristic
239C3×D5—21no—
240C3×D5—21no—
241C5×D3—81no—
242C23×C3—51no—
243C22×D3—21no—
244C2×C3×C4—51no—
245C2×C3×C4—51no—
246C2×C3×C4—51no—
247C2×C3×C4—101no—
248C2×C3×C4—101no—
249C2×C3×C4—201no—
250C2×C3×C4—201no—
251C2×C3×C4—202no—
252C2×C3×C4—202no—
253C2×C3×C4—202no—
254C2×C3×C4C22⋊AGL(1,𝔽5)11yescharacteristic
255C2×C3×C4C22⋊AGL(1,𝔽5)11yescharacteristic
256C2×C3×C4C22×AGL(1,𝔽5)11yescharacteristic
257#67—201no—
258#67—201no—
259C2×Dic3—21no—
260C2×Dic3—101no—
261C2×Dic3—101no—
262D12—41no—
263C4×D3—42no—
264Dic6—201no—
265C3⋊C8—202no—
266C22×C5—121no—
267C22×C5—121no—
268C22×C5—121no—
269C22×C5C2×(C2×D3)⋊C411yescharacteristic
270C2×D5—21no—
271C2×D5—21no—
272C2×D5#93211yescharacteristic
273C2×D5#94211yescharacteristic
274AGL(1,𝔽5)—82no—
275C4×C5—21no—
276C4×C5—21no—
277C4×C5—21no—
278C4×C5—121no—
279Dic5—21no—
280Dic5—21no—
281Dic5—21no—
282Dic5—121no—
283Dic5—121no—
284C2×Q8—151no—
285C2×D4—31no—
286C2×D4—301no—
287C22×C4—51no—
288C22×C4—51no—
289C22×C4—51no—
290C22×C4—61no—
291C22×C4—102no—
292C2×C8—302no—
293C4⋊C4—31no—
294C4⋊C4—151no—
295C4⋊C4—202no—
296C4⋊C4—301no—
297C4⋊C4—301no—
298C4⋊C4—301no—
299C4⋊C4—301no—
300C22⋊C4—61no—
301C22⋊C4—301no—
302C22⋊C4—301no—
303C22⋊C4—301no—
304C22⋊C4—301no—
305C42—51no—
306C42—51no—
307C42—51no—
308C42—301no—
309C42D3×AGL(1,𝔽5)11yescharacteristic
310C3×C5#151211yescharacteristic, complemented, Hall
311C22×C3—51no—
312C22×C3—51no—
313C22×C3—101no—
314C22×C3—101no—
315C22×C3C2×C22⋊AGL(1,𝔽5)11yescharacteristic
316C2×D3—41no—
317C2×D3—41no—
318C2×D3—41no—
319C3×C4—21no—
320C3×C4—21no—
321C3×C4—21no—
322C3×C4—101no—
323C3×C4—101no—
324C3×C4—101no—
325C3×C4—402no—
326Dic3—41no—
327Dic3—201no—
328Dic3—201no—
329C2×C5—241no—
330C2×C5#496611yescharacteristic
331C2×C5#500511yescharacteristic
332C2×C5#500611yescharacteristic
333D5—21no—
334D5—21no—
335C23—51no—
336C23—61no—
337Q8—601no—
338D4—121no—
339D4—601no—
340D4—601no—
341C2×C4—51no—
342C2×C4—51no—
343C2×C4—51no—
344C2×C4—61no—
345C2×C4—101no—
346C2×C4—101no—
347C2×C4—122no—
348C2×C4—201no—
349C2×C4—201no—
350C2×C4—202no—
351C2×C4—202no—
352C2×C4—202no—
353C2×C4—301no—
354C2×C4—301no—
355C2×C4(C2×D3)⋊AGL(1,𝔽5)11yescharacteristic
356C2×C4(C2×D3)⋊AGL(1,𝔽5)11yescharacteristic
357C2×C4C2×D3×AGL(1,𝔽5)11yescharacteristic
358C8—602no—
359C2×C3—101no—
360C2×C3—101no—
361C2×C3#6574511yescharacteristic
362C2×C3#6576911yescharacteristic
363C2×C3#6579311yescharacteristic
364D3—81no—
365C5#6954411yescharacteristic, complemented, Sylow
366C22—51no—
367C22—51no—
368C22—101no—
369C22—101no—
370C22—121no—
371C22—121no—
372C22—121no—
373C22C2×(C2×D3)⋊AGL(1,𝔽5)11yescharacteristic
374C4—21no—
375C4—21no—
376C4—21no—
377C4—101no—
378C4—101no—
379C4—101no—
380C4—121no—
381C4—402no—
382C4—601no—
383C4—601no—
384C3#1060439211yescharacteristic, complemented, Sylow
385C2—101no—
386C2—101no—
387C2—241no—
388C2#1175170511yescharacteristic
389C2#1175171911yescharacteristic
390C2#1175181511yescharacteristic
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#91449C22
cosocleC2×C3×C4×D5C23
hypercenterC42D3×AGL(1,𝔽5)
hypocenterC3×C5#1512
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. #11752343C2semidirect
3. C3×#65689C2semidirect
4. #11746503C2semidirect
8. C4×Dic30C4semidirect
9. C4×C5×D12C4semidirect
14. #91388C4semidirect
15. #91373C4semidirect
26. #65689D3semidirect
111. C4×D12AGL(1,𝔽5)semidirect
310. C3×C5#1512semidirect
365. C5#69544semidirect
384. C3#10604392semidirect

Maximal subgroups

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

subgroupindex
#117526822
#117523432
C3×#656892
#117465032
#117464962
#106043923
#695445