Finite Group Databasebeta

← #14307452

Subgroups — #14307452

Order 1920.

488 conjugacy classes of subgroups; only the 125 normal subgroups are shown.

normal subgroups only

This group has 488 conjugacy classes of subgroups — too many to draw. Only its 125 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×#66062C211yesmaximal, characteristic
2#11752345C211yesmaximal, characteristic
3#11752341C211yesmaximal, characteristic
4#11746746C214yesmaximal
5#10604666—31nomaximal, Hall
6C2×C4×ΓL(1,𝔽16)C2212yes—
7C2×#6758C2212yes—
8C3×C5×C4⋊Q8C411yescharacteristic, complemented
9C3×#4092—21no—
10C3×Q8×Dic5—21no—
11C3×#4043—21no—
12C3×#4038C2212yes—
13C3×#4036C412yescomplemented
14C3×#4033—21no—
15C3×C42×D5C2211yescharacteristic
16C3×#4015C411yescharacteristic, complemented
17#91235—21no—
18#91235—21no—
19#91233—21no—
20#91233—21no—
21#71840—51nomaximal, Hall
22#66062D311yescharacteristic
23#65686—31no—
24#65682—31no—
25#64875—34no—
26C22×ΓL(1,𝔽16)—22no—
27C2×C3×C5×Q8—21no—
28C2×C3×C4×D5D411yescharacteristic
29C2×C3×C4×D5D411yescharacteristic
30C2×C3×C4×D5C2311yescharacteristic
31C2×C3×Dic10—21no—
32C2×C3⋊#170—22no—
33C4×ΓL(1,𝔽16)—22no—
34C4×ΓL(1,𝔽16)Q814yescomplemented
35#6758—24no—
36C3×C5×C4⋊C4—21no—
37C3×C5×C4⋊C4C2×C412yes—
38C3×C42×C5C2×C411yescharacteristic
39C3×#685—22no—
40C3×C4⋊Dic5—21no—
41C3×C4⋊Dic5—22no—
42C3×C4⋊Dic5C2×C412yes—
43C3×#683—22no—
44C3×C4×Dic5—22no—
45C3×C4×Dic5C2×C411yescharacteristic
46C3×C4×Dic5D411yescharacteristic
47C3×C4×Dic5D411yescharacteristic
48C3×#533—51no—
49#4932—51no—
50#4924—51no—
51#4532—54no—
52C2×C4×AGL(1,𝔽5)—32no—
53C2×#700—32no—
54C5×C4⋊Q8Dic311yescharacteristic, complemented
55#4092—21no—
56Q8×Dic5—21no—
57#4043—21no—
58#4038C2×D312yes—
59#4036Dic312yescomplemented
60#4033—21no—
61C42×D5C2×D311yescharacteristic
62#4015Dic311yescharacteristic, complemented
63(C2×AGL(1,𝔽5))⋊C4—61no—
64(C2×AGL(1,𝔽5))⋊C4—61no—
65#4004—61no—
66#4004—61no—
67#1995—151noSylow
68C22×C3×D5C2×D411yescharacteristic
69C2×ΓL(1,𝔽16)—24no—
70C2×ΓL(1,𝔽16)—42no—
71C2×ΓL(1,𝔽16)—42no—
72C3×C5×Q8—42no—
73C2×C3×C4×C5—22no—
74C2×C3×C4×C5C22⋊C411yescharacteristic
75C2×C3×C4×C5C22⋊C411yescharacteristic
76C2×C3×C4×C5C22×C411yescharacteristic
77C2×C3×Dic5—22no—
78C2×C3×Dic5C22⋊C411yescharacteristic
79C2×C3×Dic5C22⋊C411yescharacteristic
80C2×C3×Dic5C2×D411yescharacteristic
81C3×C4×D5—41no—
82C3×C4×D5—41no—
83C3×C4×D5C2×Q812yes—
84C3×C4×D5#4112yes—
85C3×Dic10—42no—
86C3⋊#170—42no—
87C3×C4⋊Q8—51no—
88C3×C4⋊Q8AGL(1,𝔽5)11yescharacteristic, complemented
89C3×#125—52no—
90C3×C22⋊Q8—101no—
91C3×C22⋊Q8—101no—
92C3×C4×Q8—101no—
93C3×C4×Q8—101no—
94C3×#117—52no—
95C2×C3×C42—51no—
96C2×C4×Dic3—52no—
97C2×#208—52no—
98C3⋊#95—101no—
99C3⋊#95—101no—
100C4⋊(C3⋊C8)—101no—
101C4⋊(C3⋊C8)—101no—
102C22×AGL(1,𝔽5)—62no—
103C2×C5×Q8—21no—
104C2×C4×D5#6711yescharacteristic
105C2×C4×D5#6711yescharacteristic
106C2×C4×D5C22×D311yescharacteristic
107C2×Dic10—21no—
108C2×#170—62no—
109C4×AGL(1,𝔽5)—34no—
110C4×AGL(1,𝔽5)—62no—
111#700—64no—
112C5×C4⋊C4—21no—
113C5×C4⋊C4C2×Dic312yes—
114C42×C5C2×Dic311yescharacteristic
115#685—22no—
116C4⋊Dic5—21no—
117C4⋊Dic5—22no—
118C4⋊Dic5C2×Dic312yes—
119#683—22no—
120C4×Dic5—22no—
121C4×Dic5C2×Dic311yescharacteristic
122C4×Dic5#6711yescharacteristic
123C4×Dic5#6711yescharacteristic
124#533—51no—
125#423—151no—
126#376—151no—
127#339—154no—
128C22×C3×C5C2×C22⋊C411yescharacteristic
129C2×C3×D5—21no—
130C2×C3×D5—21no—
131C2×C3×D5C22⋊Q812yes—
132ΓL(1,𝔽16)—44no—
133C3×C4×C5—21no—
134C3×C4×C5—21no—
135C3×C4×C5—42no—
136C3×C4×C5C4×Q812yes—
137C3×Dic5—21no—
138C3×Dic5—21no—
139C3×Dic5—42no—
140C3×Dic5#12411yescharacteristic
141C3×Dic5C4⋊Q811yescharacteristic
142C2×C3×Q8—21no—
143C2×C3×Q8—101no—
144C22×C3×C4—51no—
145C22×C3×C4—51no—
146C22×C3×C4—51no—
147C22×Dic3—102no—
148C3×C4⋊C4—21no—
149C3×C4⋊C4—52no—
150C3×C4⋊C4—102no—
151C3×C4⋊C4—102no—
152C3×C4⋊C4—101no—
153C3×C4⋊C4C2×AGL(1,𝔽5)12yes—
154C3×C22⋊C4—102no—
155C3×C42—51no—
156C3×C42—51no—
157C3×C42—51no—
158C3×C42—102no—
159C3×C42C2×AGL(1,𝔽5)11yescharacteristic
160C4×Dic3—54no—
161C4×Dic3—102no—
162#208—104no—
163C2×C3⋊C8—102no—
164C22×D5C2×#6711yescharacteristic
165C2×AGL(1,𝔽5)—64no—
166C2×AGL(1,𝔽5)—122no—
167C2×AGL(1,𝔽5)—122no—
168C5×Q8—42no—
169C2×C4×C5—22no—
170C2×C4×C5#21711yescharacteristic
171C2×C4×C5#21711yescharacteristic
172C2×C4×C5C22×Dic311yescharacteristic
173C2×Dic5—22no—
174C2×Dic5#21711yescharacteristic
175C2×Dic5#21711yescharacteristic
176C2×Dic5C2×#6711yescharacteristic
177C4×D5—41no—
178C4×D5—41no—
179C4×D5D3×Q812yes—
180C4×D5#23912yes—
181Dic10—42no—
182#170—122no—
183C2×M16—152no—
184C4⋊Q8—51no—
185C4⋊Q8ΓL(1,𝔽16)11yescharacteristic, complemented
186#125—52no—
187C22⋊Q8—101no—
188C22⋊Q8—101no—
189C4×Q8—101no—
190C4×Q8—101no—
191#117—52no—
192C2×C42—51no—
193C2×C42—152no—
194C4⋊C8—301no—
195C4⋊C8—301no—
196#95—301no—
197#95—301no—
198C2×C3×C5#39111yescharacteristic
199C2×C3×C5#42111yescharacteristic
200C2×C3×C5#42111yescharacteristic
201C3×D5—22no—
202C23×C3—51no—
203C3×Q8—42no—
204C3×Q8—202no—
205C2×C3×C4—22no—
206C2×C3×C4—51no—
207C2×C3×C4—52no—
208C2×C3×C4—52no—
209C2×C3×C4—51no—
210C2×C3×C4—51no—
211C2×C3×C4—102no—
212C2×C3×C4—201no—
213C2×C3×C4—201no—
214C2×C3×C4C22⋊AGL(1,𝔽5)11yescharacteristic
215C2×C3×C4C22⋊AGL(1,𝔽5)11yescharacteristic
216C2×C3×C4C22×AGL(1,𝔽5)11yescharacteristic
217C2×Dic3—104no—
218C2×Dic3—202no—
219C2×Dic3—202no—
220C3⋊C8—202no—
221C22×C5C2×#21711yescharacteristic
222C2×D5—21no—
223C2×D5—21no—
224C2×D5#94612yes—
225AGL(1,𝔽5)—124no—
226C4×C5—21no—
227C4×C5—21no—
228C4×C5—42no—
229C4×C5Q8×Dic312yes—
230Dic5—21no—
231Dic5—21no—
232Dic5—42no—
233Dic5#94411yescharacteristic
234Dic5#94711yescharacteristic
235C2×Q8—21no—
236C2×Q8—101no—
237C22×C4—51no—
238C22×C4—51no—
239C22×C4—51no—
240C22×C4—302no—
241M16—304no—
242C2×C8—302no—
243C4⋊C4—21no—
244C4⋊C4—52no—
245C4⋊C4—102no—
246C4⋊C4—102no—
247C4⋊C4—101no—
248C4⋊C4C2×ΓL(1,𝔽16)12yes—
249C22⋊C4—102no—
250C42—51no—
251C42—51no—
252C42—51no—
253C42—102no—
254C42—154no—
255C42—302no—
256C42C2×ΓL(1,𝔽16)11yescharacteristic
257C3×C5#199511yescharacteristic, complemented, Hall
258C22×C3—52no—
259C22×C3—101no—
260C22×C3—101no—
261C22×C3C2×C22⋊AGL(1,𝔽5)11yescharacteristic
262C3×C4—21no—
263C3×C4—21no—
264C3×C4—42no—
265C3×C4—51no—
266C3×C4—51no—
267C3×C4—101no—
268C3×C4—101no—
269C3×C4—202no—
270C3×C4Q8×AGL(1,𝔽5)12yes—
271Dic3—204no—
272C2×C5#522911yescharacteristic
273C2×C5#523611yescharacteristic
274C2×C5#523611yescharacteristic
275D5—22no—
276C23—51no—
277Q8—42no—
278Q8—202no—
279C2×C4—22no—
280C2×C4—51no—
281C2×C4—52no—
282C2×C4—52no—
283C2×C4—51no—
284C2×C4—51no—
285C2×C4—102no—
286C2×C4—201no—
287C2×C4—201no—
288C2×C4—304no—
289C2×C4—602no—
290C2×C4—602no—
291C2×C4#676311yescharacteristic
292C2×C4#676311yescharacteristic
293C2×C4C22×ΓL(1,𝔽16)11yescharacteristic
294C8—602no—
295C2×C3—102no—
296C2×C3#6576911yescharacteristic
297C2×C3#6578411yescharacteristic
298C2×C3(C2×AGL(1,𝔽5))⋊Q811yescharacteristic
299C5#7184011yescharacteristic, complemented, Sylow
300C22—52no—
301C22—101no—
302C22—101no—
303C22C2×#676311yescharacteristic
304C4—21no—
305C4—21no—
306C4—42no—
307C4—51no—
308C4—51no—
309C4—101no—
310C4—101no—
311C4—202no—
312C4—604no—
313C4Q8×ΓL(1,𝔽16)12yes—
314C3#1060466611yescharacteristic, complemented, Sylow
315C2—102no—
316C2#1175242811yescharacteristic
317C2#1175244311yescharacteristic
318C2#1175244911yescharacteristic
subgroupG/H
centerC22C2×#6763
commutatorC2×C3×C4×C5C22×C4
FrattiniC2×C4C22×ΓL(1,𝔽16)
FittingC3×C5×C4⋊Q8C4
socleC22×C3×C5C2×C22⋊C4
prime radicalC22×C3×D5C2×D4
cosocleC2×C3×C4×D5C23
hypercenterC4⋊Q8ΓL(1,𝔽16)
hypocenterC3×C5#1995
LCSG ▷ C2×C3×C4×C5 ▷ C2×C3×C5 ▷ C3×C5
UCSC1 ◁ C22 ◁ C42 ◁ C4⋊Q8

Complements

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

NH ≅ G/Ntype
8. C3×C5×C4⋊Q8C4semidirect
13. C3×#4036C4semidirect
16. C3×#4015C4semidirect
34. C4×ΓL(1,𝔽16)Q8semidirect
54. C5×C4⋊Q8Dic3semidirect
59. #4036Dic3semidirect
62. #4015Dic3semidirect
88. C3×C4⋊Q8AGL(1,𝔽5)semidirect
185. C4⋊Q8ΓL(1,𝔽16)semidirect
257. C3×C5#1995semidirect
299. C5#71840semidirect
314. C3#10604666semidirect

Maximal subgroups

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

subgroupindex
C3×#660622
#117523452
#117523412
#117467462
#106046663
#718405