Finite Group Databasebeta

← #14299608

Subgroups — #14299608

Order 1920.

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

normal subgroups only

This group has 648 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
1C3×#66258C211yesmaximal, characteristic
2#11752439C211yesmaximal, characteristic
3#11751728C212yesmaximal, complemented
4D5×#833C211yesmaximal, characteristic
5#11746522C212yesmaximal, complemented
6#10603925—31nomaximal, Hall
7C2×C3×D4×D5C2211yescharacteristic
8C2×#6763—22no—
9C2×C4⋊ΓL(1,𝔽16)C2211yescharacteristic
10C2×C4×ΓL(1,𝔽16)C2211yescharacteristic
11C2×C3×C22⋊AGL(1,𝔽5)—22no—
12C3×D4×AGL(1,𝔽5)—22no—
13C3×D4×AGL(1,𝔽5)—22no—
14C3×#4130C2212yes—
15D5×#208C2212yes—
16#91160—22no—
17#91159—22no—
18#91117C411yescharacteristic, complemented
19C5×#833C411yescharacteristic, complemented
20#90958C412yescomplemented
21#71647—51nomaximal, Hall
22#66258D311yescharacteristic
23#65780—31no—
24#65727—32no—
25D5×#100—31no—
26#64900—32no—
27C23×C3×D5D412yes—
28C22×ΓL(1,𝔽16)—21no—
29C22×ΓL(1,𝔽16)—21no—
30C22×C3×AGL(1,𝔽5)—22no—
31C22×C3×AGL(1,𝔽5)—22no—
32C2×C3×C5×D4C2×C411yescharacteristic
33C2×C3×C22⋊D5Q812yes—
34C3×D4×D5—22no—
35C3×D4×D5—22no—
36C2×C3×D20C2×C411yescharacteristic
37C2×C3×C4×D5C2311yescharacteristic
38#6763—44no—
39C4⋊ΓL(1,𝔽16)—22no—
40C4×ΓL(1,𝔽16)—22no—
41C3×C22⋊AGL(1,𝔽5)—24no—
42C3×C22⋊AGL(1,𝔽5)—22no—
43C3×C4⋊AGL(1,𝔽5)D412yes—
44C3×C4⋊AGL(1,𝔽5)D412yes—
45C3×C4×AGL(1,𝔽5)—22no—
46#6710C2×C412yescomplemented
47C5×#208C2×C412yescomplemented
48#6649—22no—
49D5×C3⋊C8—22no—
50C3×#518—51no—
51#5221—51no—
52C2×#833—51no—
53#4995—52no—
54#4532—52no—
55C2×D4×D5C2×D311yescharacteristic
56C2×C22⋊AGL(1,𝔽5)—22no—
57C2×C22⋊AGL(1,𝔽5)—62no—
58D4×AGL(1,𝔽5)—22no—
59D4×AGL(1,𝔽5)—22no—
60#4130C2×D312yes—
61C2×C4⋊AGL(1,𝔽5)—31no—
62C2×C4×AGL(1,𝔽5)—31no—
63D5×M16—32no—
64#3994—62no—
65#3993—62no—
66C5×#100—31no—
67#3965—31no—
68#3955—32no—
69#1856—151noSylow
70C23×C3×C5C4⋊C412yes—
71C22×C3×D5—22no—
72C22×C3×D5—22no—
73C22×C3×D5—42no—
74C22×C3×D5—42no—
75C22×C3×D5C4⋊C412yes—
76C22×C3×D5C2×D411yescharacteristic
77C2×ΓL(1,𝔽16)—22no—
78C2×ΓL(1,𝔽16)—41no—
79C2×ΓL(1,𝔽16)—41no—
80C2×ΓL(1,𝔽16)—42no—
81C2×ΓL(1,𝔽16)—41no—
82C2×C3×AGL(1,𝔽5)—22no—
83C2×C3×AGL(1,𝔽5)—22no—
84C2×C3×AGL(1,𝔽5)—82no—
85C2×C3×AGL(1,𝔽5)—82no—
86C3×C5×D4—22no—
87C2×C3×C4×C5C22×C411yescharacteristic
88C3×C22⋊D5—24no—
89C2×C3×Dic5C2×Q811yescharacteristic
90C3×D20—22no—
91C3×C4×D5C2×D412yes—
92C3×C4×D5#4112yes—
93C3⋊#168—22no—
94C5×C3⋊C8—22no—
95C22×C3×D4—51no—
96C3×C4×D4—102no—
97C3×C4×D4—102no—
98C3×#117—52no—
99C2×C3×C22⋊C4—102no—
100C2×#217—102no—
101C2×C4⋊Dic3—51no—
102C2×C4×Dic3—51no—
103C2×#208—52no—
104#833—51no—
105#833—52no—
106#833AGL(1,𝔽5)11yescharacteristic, complemented
107#808—102no—
108#807—102no—
109C23×D5#6712yes—
110C22×AGL(1,𝔽5)—22no—
111C22×AGL(1,𝔽5)—22no—
112C22×AGL(1,𝔽5)—61no—
113C22×AGL(1,𝔽5)—61no—
114C2×C5×D4C4×D311yescharacteristic
115C2×C22⋊D5Dic612yes—
116D4×D5—22no—
117D4×D5—22no—
118C2×D20C4×D311yescharacteristic
119C2×C4×D5C22×D311yescharacteristic
120C22⋊AGL(1,𝔽5)—24no—
121C22⋊AGL(1,𝔽5)—22no—
122C22⋊AGL(1,𝔽5)—124no—
123C4⋊AGL(1,𝔽5)—62no—
124C4⋊AGL(1,𝔽5)#6712yes—
125C4⋊AGL(1,𝔽5)#6712yes—
126C4×AGL(1,𝔽5)—22no—
127C4×AGL(1,𝔽5)—62no—
128C5×M16—32no—
129#681—32no—
130#676—62no—
131C8×D5—62no—
132#518—51no—
133#428—152no—
134C2×#100—151no—
135#390—151no—
136#339—152no—
137C22×C3×C5—22no—
138C22×C3×C5—42no—
139C22×C3×C5C2×C4⋊C411yescharacteristic
140C2×C3×D5—21no—
141C2×C3×D5—21no—
142C2×C3×D5—22no—
143C2×C3×D5—42no—
144C2×C3×D5—42no—
145C2×C3×D5—42no—
146C2×C3×D5—82no—
147C2×C3×D5#12011yescharacteristic
148C2×C3×D5#12311yescharacteristic
149ΓL(1,𝔽16)—42no—
150ΓL(1,𝔽16)—81no—
151C3×AGL(1,𝔽5)—42no—
152C3×AGL(1,𝔽5)—42no—
153C3×C4×C5C4×D412yes—
154C3×Dic5C22⋊Q812yes—
155C24×C3—52no—
156C2×C3×D4—52no—
157C2×C3×D4—51no—
158C2×C3×D4—102no—
159C2×C3×D4—102no—
160C2×C3×D4C2×AGL(1,𝔽5)11yescharacteristic
161C22×C3×C4—51no—
162C22×C3×C4—102no—
163C22×C3×C4—102no—
164C22×Dic3—101no—
165C22×Dic3—101no—
166C3×C4⋊C4—52no—
167C3×C4⋊C4—52no—
168C3×C22⋊C4—102no—
169C3×C22⋊C4—104no—
170C3×C42—102no—
171#217—204no—
172C4⋊Dic3—102no—
173C4×Dic3—102no—
174#208—52no—
175#208—102no—
176#208C2×AGL(1,𝔽5)12yescomplemented
177C2×C3⋊C8—102no—
178C23×C5Dic3⋊C412yes—
179C22×D5—22no—
180C22×D5—22no—
181C22×D5—42no—
182C22×D5—42no—
183C22×D5Dic3⋊C412yes—
184C22×D5C2×#6711yescharacteristic
185C2×AGL(1,𝔽5)—22no—
186C2×AGL(1,𝔽5)—22no—
187C2×AGL(1,𝔽5)—62no—
188C2×AGL(1,𝔽5)—82no—
189C2×AGL(1,𝔽5)—82no—
190C2×AGL(1,𝔽5)—121no—
191C2×AGL(1,𝔽5)—121no—
192C2×AGL(1,𝔽5)—122no—
193C2×AGL(1,𝔽5)—121no—
194C5×D4—22no—
195C2×C4×C5C2×C4×D311yescharacteristic
196C22⋊D5—24no—
197C2×Dic5C2×Dic611yescharacteristic
198D20—22no—
199C4×D5#23512yes—
200C4×D5C2×#6712yes—
201C5×C8—62no—
202#168—62no—
203C22×D4—51no—
204C2×M16—152no—
205C4×D4—102no—
206C4×D4—102no—
207#117—52no—
208C2×C4⋊C4—151no—
209C2×C22⋊C4—102no—
210C2×C22⋊C4—302no—
211C2×C42—151no—
212#107—302no—
213#106—302no—
214#100—31no—
215#100—152no—
216#100—151no—
217C2×C3×C5—21no—
218C2×C3×C5—42no—
219C2×C3×C5#38511yescharacteristic
220C3×D5—21no—
221C3×D5—42no—
222C3×D5#45311yescharacteristic
223C3×D5#45511yescharacteristic
224C23×C3—51no—
225C23×C3—52no—
226C23×C3—102no—
227C23×C3—102no—
228C23×C3—202no—
229C23×C3—202no—
230C23×C3C4⋊AGL(1,𝔽5)12yes—
231C3×D4—22no—
232C3×D4—104no—
233C3×D4—102no—
234C2×C3×C4—51no—
235C2×C3×C4—52no—
236C2×C3×C4—52no—
237C2×C3×C4—102no—
238C2×C3×C4—102no—
239C2×C3×C4—402no—
240C2×C3×C4—402no—
241C2×C3×C4C22×AGL(1,𝔽5)11yescharacteristic
242C2×Dic3—102no—
243C2×Dic3—201no—
244C2×Dic3—201no—
245C2×Dic3—201no—
246C2×Dic3—202no—
247C3⋊C8—22no—
248C3⋊C8—102no—
249C22×C5—22no—
250C22×C5—42no—
251C22×C5C2×Dic3⋊C411yescharacteristic
252C2×D5—21no—
253C2×D5—21no—
254C2×D5—22no—
255C2×D5—42no—
256C2×D5—42no—
257C2×D5—42no—
258C2×D5—82no—
259C2×D5#92911yescharacteristic
260C2×D5#95311yescharacteristic
261AGL(1,𝔽5)—42no—
262AGL(1,𝔽5)—42no—
263AGL(1,𝔽5)—122no—
264AGL(1,𝔽5)—241no—
265C4×C5C4×#6712yes—
266Dic5#92412yes—
267C24—52no—
268C2×D4—52no—
269C2×D4—51no—
270C2×D4—102no—
271C2×D4—102no—
272C2×D4D3×AGL(1,𝔽5)11yescharacteristic
273C22×C4—51no—
274C22×C4—102no—
275C22×C4—102no—
276C22×C4—301no—
277C22×C4—301no—
278M16—32no—
279M16—152no—
280M16—302no—
281C2×C8—302no—
282C4⋊C4—52no—
283C4⋊C4—52no—
284C4⋊C4—302no—
285C22⋊C4—102no—
286C22⋊C4—104no—
287C22⋊C4—604no—
288C42—102no—
289C42—302no—
290C3×C5#185611yescharacteristic, complemented, Hall
291C22×C3—22no—
292C22×C3—42no—
293C22×C3—51no—
294C22×C3—51no—
295C22×C3—101no—
296C22×C3—101no—
297C22×C3—102no—
298C22×C3—202no—
299C22×C3—202no—
300C22×C3—202no—
301C22×C3—402no—
302C22×C3C2×C4⋊AGL(1,𝔽5)11yescharacteristic
303C3×C4—52no—
304C3×C4—202no—
305C3×C4—202no—
306C3×C4D4×AGL(1,𝔽5)12yes—
307Dic3—202no—
308Dic3—401no—
309C2×C5—21no—
310C2×C5—42no—
311C2×C5#521011yescharacteristic
312D5—21no—
313D5—42no—
314D5#524111yescharacteristic
315D5#524211yescharacteristic
316C23—51no—
317C23—52no—
318C23—102no—
319C23—102no—
320C23—202no—
321C23—202no—
322C23Dic3⋊AGL(1,𝔽5)12yes—
323D4—22no—
324D4—104no—
325D4—102no—
326C2×C4—51no—
327C2×C4—52no—
328C2×C4—52no—
329C2×C4—102no—
330C2×C4—102no—
331C2×C4—302no—
332C2×C4—402no—
333C2×C4—402no—
334C2×C4—601no—
335C2×C4—601no—
336C2×C4—601no—
337C2×C4—602no—
338C2×C4C2×D3×AGL(1,𝔽5)11yescharacteristic
339C8—62no—
340C8—302no—
341C2×C3—21no—
342C2×C3—42no—
343C2×C3—51no—
344C2×C3—51no—
345C2×C3—101no—
346C2×C3—202no—
347C2×C3#6569911yescharacteristic
348C5#7164711yescharacteristic, complemented, Sylow
349C22—22no—
350C22—42no—
351C22—51no—
352C22—51no—
353C22—101no—
354C22—101no—
355C22—102no—
356C22—202no—
357C22—202no—
358C22—202no—
359C22—402no—
360C22C2×Dic3⋊AGL(1,𝔽5)11yescharacteristic
361C4—52no—
362C4—202no—
363C4—202no—
364C4—602no—
365C4—1201no—
366C4AGL(1,𝔽5)×#6712yes—
367C3#1060392511yescharacteristic, complemented, Sylow
368C2—21no—
369C2—42no—
370C2—51no—
371C2—51no—
372C2—101no—
373C2—202no—
374C2#1175202711yescharacteristic
subgroupG/H
centerC2#11752027
commutatorC2×C3×C4×C5C22×C4
FrattiniC2×C4C2×D3×AGL(1,𝔽5)
FittingC2×C3×C5×D4C2×C4
socleC2×C3×C5#385
prime radicalC2×C3×D4×D5C22
cosocleC2×C3×C4×D5C23
hypercenterC2×D4D3×AGL(1,𝔽5)
hypocenterC3×C5#1856
LCSG ▷ C2×C3×C4×C5 ▷ C2×C3×C5 ▷ C3×C5
UCSC1 ◁ C2 ◁ C2×C4 ◁ C2×D4

Complements

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

NH ≅ G/Ntype
3. #11751728C2semidirect
5. #11746522C2semidirect
18. #91117C4semidirect
19. C5×#833C4semidirect
20. #90958C4semidirect
46. #6710C2×C4semidirect
47. C5×#208C2×C4semidirect
106. #833AGL(1,𝔽5)semidirect
176. #208C2×AGL(1,𝔽5)semidirect
290. C3×C5#1856semidirect
348. C5#71647semidirect
367. C3#10603925semidirect

Maximal subgroups

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

subgroupindex
C3×#662582
#117524392
#117517282
D5×#8332
#117465222
#106039253
#716475