Finite Group Databasebeta

← #14307472

Subgroups — #14307472

Order 1920.

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

normal subgroups only

This group has 488 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#11752506C211yesmaximal, characteristic
2#11752345C211yesmaximal, characteristic
3C3×#65682C211yesmaximal, characteristic
4#11746524C214yesmaximal
5#10604666—31nomaximal, Hall
6C2×C4×ΓL(1,𝔽16)C2212yes—
7C2×C3×#700C2212yes—
8#91758C411yescharacteristic, complemented
9C5×#869C411yescharacteristic, complemented
10C3×C42×D5C2211yescharacteristic
11#91388C412yescomplemented
12#91360—21no—
13#91359—21no—
14#91358C2212yes—
15#91333—21no—
16Dic5×Dic6—21no—
17#91235—21no—
18#91235—21no—
19C3×#4004—21no—
20C3×#4004—21no—
21#70515—51nomaximal, Hall
22#66062—31no—
23#65686—31no—
24#65682D311yescharacteristic
25#64875—34no—
26C22×ΓL(1,𝔽16)—22no—
27C2×Dic30—21no—
28C2×C5×Dic6—21no—
29C2×C3×C4×D5D411yescharacteristic
30C2×C3×C4×D5D411yescharacteristic
31C2×C3×C4×D5C2311yescharacteristic
32C4×ΓL(1,𝔽16)—22no—
33C4×ΓL(1,𝔽16)Q814yescomplemented
34C2×C3×#170—22no—
35C3×#700—24no—
36C3×C42×C5C2×C411yescharacteristic
37C4⋊Dic15—21no—
38C4⋊Dic15C2×C412yes—
39C5×C4⋊Dic3—21no—
40C5×C4⋊Dic3C2×C412yes—
41C3×C4×Dic5C2×C411yescharacteristic
42C3×C4×Dic5D411yescharacteristic
43C3×C4×Dic5D411yescharacteristic
44Dic3⋊Dic5—22no—
45#6669—22no—
46#6665—22no—
47Dic3×Dic5—22no—
48#5470—51no—
49C3×#423—51no—
50#4932—51no—
51#4555—54no—
52C2×C4×AGL(1,𝔽5)—32no—
53C2×#700C2×D312yes—
54C5×C4⋊Q8—31no—
55#4092—61no—
56Q8×Dic5—61no—
57#4043—61no—
58#4038—32no—
59#4036—32no—
60#4033—61no—
61C42×D5C2×D311yescharacteristic
62#4015—31no—
63(C2×AGL(1,𝔽5))⋊C4—61no—
64(C2×AGL(1,𝔽5))⋊C4—61no—
65#4004—21no—
66#4004—21no—
67#1995—151noSylow
68C22×C3×D5C2×D411yescharacteristic
69C2×ΓL(1,𝔽16)—24no—
70C2×ΓL(1,𝔽16)—42no—
71C2×ΓL(1,𝔽16)—42no—
72C2×C3×C4×C5C22⋊C411yescharacteristic
73C2×C3×C4×C5C22⋊C411yescharacteristic
74C2×C3×C4×C5C22×C411yescharacteristic
75C2×Dic15—22no—
76Dic30—42no—
77C2×C5×Dic3—22no—
78C5×Dic6—42no—
79C2×C3×Dic5C22⋊C411yescharacteristic
80C2×C3×Dic5C22⋊C411yescharacteristic
81C2×C3×Dic5C2×D411yescharacteristic
82C3×C4×D5—41no—
83C3×C4×D5—41no—
84C3×C4×D5C2×Q812yes—
85C3×C4×D5#4112yes—
86C3×#170—42no—
87C2×C3×M16—52no—
88C2×C3×C42—51no—
89#926—52no—
90#924—101no—
91#924—101no—
92C2×C4×Dic3—52no—
93#870—52no—
94#869—51no—
95#869AGL(1,𝔽5)11yescharacteristic, complemented
96C4×Dic6—101no—
97C4×Dic6—101no—
98C3×C4⋊C8—101no—
99C3×C4⋊C8—101no—
100C3⋊#95—101no—
101C3⋊#95—101no—
102C22×AGL(1,𝔽5)—62no—
103C2×C5×Q8—61no—
104C2×C4×D5D1211yescharacteristic
105C2×C4×D5D1211yescharacteristic
106C2×C4×D5C22×D311yescharacteristic
107C2×Dic10—61no—
108C2×#170—22no—
109C4×AGL(1,𝔽5)—34no—
110C4×AGL(1,𝔽5)—62no—
111#700—24no—
112C5×C4⋊C4—32no—
113C5×C4⋊C4—61no—
114C42×C5C4×D311yescharacteristic
115#685—62no—
116C4⋊Dic5—32no—
117C4⋊Dic5—61no—
118C4⋊Dic5—62no—
119#683—62no—
120C4×Dic5—62no—
121C4×Dic5C4×D311yescharacteristic
122C4×Dic5#6711yescharacteristic
123C4×Dic5#6711yescharacteristic
124#533—151no—
125#423—51no—
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×C5C4×Q812yes—
136Dic15—42no—
137C3×Dic5—21no—
138C3×Dic5—21no—
139C3×Dic5#12411yescharacteristic
140C3×Dic5C4⋊Q811yescharacteristic
141C5×Dic3—42no—
142C22×C3×C4—51no—
143C22×C3×C4—51no—
144C22×C3×C4—51no—
145C22×Dic3—102no—
146C2×Dic6—21no—
147C2×Dic6—101no—
148C3×M16—104no—
149C2×C3×C8—102no—
150C3×C42—51no—
151C3×C42—51no—
152C3×C42—51no—
153C3×C42C2×AGL(1,𝔽5)11yescharacteristic
154#217—102no—
155C4⋊Dic3—21no—
156C4⋊Dic3—52no—
157C4⋊Dic3—101no—
158C4⋊Dic3C2×AGL(1,𝔽5)12yes—
159Dic3⋊C4—102no—
160Dic3⋊C4—102no—
161C4×Dic3—54no—
162C4×Dic3—102no—
163C4×Dic3—102no—
164C22×D5C2×D1211yescharacteristic
165C2×AGL(1,𝔽5)—64no—
166C2×AGL(1,𝔽5)—122no—
167C2×AGL(1,𝔽5)—122no—
168C5×Q8—122no—
169C2×C4×C5—62no—
170C2×C4×C5(C2×D3)⋊C411yescharacteristic
171C2×C4×C5(C2×D3)⋊C411yescharacteristic
172C2×C4×C5C2×C4×D311yescharacteristic
173C2×Dic5—62no—
174C2×Dic5(C2×D3)⋊C411yescharacteristic
175C2×Dic5(C2×D3)⋊C411yescharacteristic
176C2×Dic5C2×#6711yescharacteristic
177C4×D5—41no—
178C4×D5—41no—
179C4×D5#23712yes—
180C4×D5D3×Q812yes—
181Dic10—122no—
182#170—42no—
183C2×M16—52no—
184C4⋊Q8—31no—
185C4⋊Q8—151no—
186#125—152no—
187C22⋊Q8—301no—
188C22⋊Q8—301no—
189C4×Q8—301no—
190C4×Q8—301no—
191#117—152no—
192C2×C42—51no—
193C2×C42—152no—
194C4⋊C8—101no—
195C4⋊C8—101no—
196#95—301no—
197#95—301no—
198C2×C3×C5#39111yescharacteristic
199C2×C3×C5#42111yescharacteristic
200C2×C3×C5#42111yescharacteristic
201C3×D5—22no—
202C23×C3—51no—
203C2×C3×C4—51no—
204C2×C3×C4—52no—
205C2×C3×C4—52no—
206C2×C3×C4—51no—
207C2×C3×C4—51no—
208C2×C3×C4—201no—
209C2×C3×C4—201no—
210C2×C3×C4C22⋊AGL(1,𝔽5)11yescharacteristic
211C2×C3×C4C22⋊AGL(1,𝔽5)11yescharacteristic
212C2×C3×C4C22×AGL(1,𝔽5)11yescharacteristic
213C2×Dic3—22no—
214C2×Dic3—104no—
215C2×Dic3—102no—
216C2×Dic3—202no—
217C2×Dic3—202no—
218Dic6—42no—
219Dic6—202no—
220C3×C8—202no—
221C22×C5C2×(C2×D3)⋊C411yescharacteristic
222C2×D5—21no—
223C2×D5—21no—
224C2×D5#89712yes—
225AGL(1,𝔽5)—124no—
226C4×C5—21no—
227C4×C5—21no—
228C4×C5—122no—
229C4×C5#88712yes—
230Dic5—21no—
231Dic5—21no—
232Dic5—122no—
233Dic5#93611yescharacteristic
234Dic5#94411yescharacteristic
235C2×Q8—61no—
236C2×Q8—301no—
237C22×C4—51no—
238C22×C4—51no—
239C22×C4—51no—
240C22×C4—302no—
241M16—104no—
242C2×C8—102no—
243C4⋊C4—32no—
244C4⋊C4—61no—
245C4⋊C4—152no—
246C4⋊C4—302no—
247C4⋊C4—302no—
248C4⋊C4—301no—
249C22⋊C4—302no—
250C42—51no—
251C42—51no—
252C42—51no—
253C42—154no—
254C42—302no—
255C42—302no—
256C42D3×AGL(1,𝔽5)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—51no—
265C3×C4—51no—
266C3×C4—101no—
267C3×C4—101no—
268C3×C4Q8×AGL(1,𝔽5)12yes—
269Dic3—42no—
270Dic3—204no—
271Dic3—202no—
272C2×C5#497311yescharacteristic
273C2×C5#513911yescharacteristic
274C2×C5#513911yescharacteristic
275D5—22no—
276C23—51no—
277Q8—122no—
278Q8—602no—
279C2×C4—51no—
280C2×C4—52no—
281C2×C4—52no—
282C2×C4—51no—
283C2×C4—51no—
284C2×C4—62no—
285C2×C4—201no—
286C2×C4—201no—
287C2×C4—304no—
288C2×C4—302no—
289C2×C4—602no—
290C2×C4—602no—
291C2×C4(C2×D3)⋊AGL(1,𝔽5)11yescharacteristic
292C2×C4(C2×D3)⋊AGL(1,𝔽5)11yescharacteristic
293C2×C4C2×D3×AGL(1,𝔽5)11yescharacteristic
294C8—202no—
295C2×C3—102no—
296C2×C3#6576911yescharacteristic
297C2×C3#6578411yescharacteristic
298C2×C3(C2×AGL(1,𝔽5))⋊Q811yescharacteristic
299C5#7051511yescharacteristic, complemented, Sylow
300C22—52no—
301C22—101no—
302C22—101no—
303C22C2×(C2×D3)⋊AGL(1,𝔽5)11yescharacteristic
304C4—21no—
305C4—21no—
306C4—51no—
307C4—51no—
308C4—101no—
309C4—101no—
310C4—122no—
311C4—604no—
312C4—602no—
313C4ΓL(1,𝔽16)⋊Q812yes—
314C3#1060466611yescharacteristic, complemented, Sylow
315C2—102no—
316C2#1175171811yescharacteristic
317C2#1175172311yescharacteristic
318C2#1175175211yescharacteristic
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 radicalC22×C3×D5C2×D4
cosocleC2×C3×C4×D5C23
hypercenterC42D3×AGL(1,𝔽5)
hypocenterC3×C5#1995
LCSG ▷ C2×C3×C4×C5 ▷ C2×C3×C5 ▷ C3×C5
UCSC1 ◁ C22 ◁ C42

Complements

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

NH ≅ G/Ntype
8. #91758C4semidirect
9. C5×#869C4semidirect
11. #91388C4semidirect
33. C4×ΓL(1,𝔽16)Q8semidirect
95. #869AGL(1,𝔽5)semidirect
257. C3×C5#1995semidirect
299. C5#70515semidirect
314. C3#10604666semidirect

Maximal subgroups

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

subgroupindex
#117525062
#117523452
C3×#656822
#117465242
#106046663
#705155