Finite Group Databasebeta

← #14298566

Subgroups — #14298566

Order 1920.

612 conjugacy classes of subgroups; only the 118 normal subgroups are shown.

normal subgroups only

This group has 612 conjugacy classes of subgroups — too many to draw. Only its 118 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
1C2×#91358C211yesmaximal, characteristic
2C4×#6763C211yesmaximal, characteristic
3C3×#65756C211yesmaximal, characteristic
4#11746524C214yesmaximal, complemented
5#10604287—31nomaximal, Hall
6C22×C3×C4×D5C2211yescharacteristic
7C2×#6763—21no—
8C2×C4×ΓL(1,𝔽16)C2212yes—
9C2×C3×#700C2212yes—
10C2×C4⋊Dic15C411yescharacteristic, complemented
11C2×C5×C4⋊Dic3C411yescharacteristic, complemented
12C2×#6665—21no—
13C2×Dic3×Dic5—21no—
14#91358—21no—
15#91358—21no—
16#91358C412yescomplemented
17#91358C2212yes—
18#91235—21no—
19C3×C22⋊#170—21no—
20C3×#4003—21no—
21#69975—51nomaximal, Hall
22C2×#4038—31no—
23C4×C22⋊AGL(1,𝔽5)—31no—
24#65756D311yescharacteristic
25#64875—34no—
26C23×C3×D5D411yescharacteristic
27C22×ΓL(1,𝔽16)—22no—
28C22×C3×C4×C5C2×C411yescharacteristic
29C22×Dic15—21no—
30C22×C5×Dic3—21no—
31C22×C3×Dic5D411yescharacteristic
32C2×C3×C4×D5—24no—
33C2×C3×C4×D5C2×C411yescharacteristic
34C2×C3×C4×D5D411yescharacteristic
35C2×C3×C4×D5D411yescharacteristic
36C2×C3×C4×D5C2311yescharacteristic
37#6763—24no—
38C4×ΓL(1,𝔽16)—24no—
39C2×C3×#170—22no—
40C3×#700—24no—
41C4⋊Dic15—21no—
42C4⋊Dic15C2×C412yes—
43C5×C4⋊Dic3—21no—
44C5×C4⋊Dic3C2×C412yes—
45#6665—22no—
46#6665—22no—
47Dic3×Dic5—22no—
48Dic3×Dic5—22no—
49C2×#926—51no—
50C3×#407—51no—
51C4×#217—51no—
52#4555—54no—
53C22×C4×D5C2×D311yescharacteristic
54C2×C22⋊AGL(1,𝔽5)—61no—
55C2×C4×AGL(1,𝔽5)—32no—
56C2×#700C2×D312yes—
57C2×C5×C4⋊C4—31no—
58C2×#685—61no—
59C2×C4⋊Dic5—31no—
60C2×C4×Dic5—61no—
61#4038—32no—
62#4038—32no—
63#4038—61no—
64#4038—61no—
65C22⋊#170—21no—
66(C2×AGL(1,𝔽5))⋊C4—61no—
67#4003—21no—
68#1826—151noSylow
69C23×C3×C5C22⋊C411yescharacteristic
70C22×C3×D5—22no—
71C22×C3×D5—22no—
72C22×C3×D5—41no—
73C22×C3×D5C22⋊C411yescharacteristic
74C22×C3×D5C2×D411yescharacteristic
75C2×ΓL(1,𝔽16)—24no—
76C2×ΓL(1,𝔽16)—42no—
77C2×ΓL(1,𝔽16)—42no—
78C2×C3×C4×C5—22no—
79C2×C3×C4×C5C22⋊C411yescharacteristic
80C2×C3×C4×C5C22×C411yescharacteristic
81C2×Dic15—22no—
82C2×Dic15—42no—
83C2×C5×Dic3—22no—
84C2×C5×Dic3—42no—
85C2×C3×Dic5—22no—
86C2×C3×Dic5C22⋊C411yescharacteristic
87C2×C3×Dic5C2×D411yescharacteristic
88C3×C4×D5—22no—
89C3×C4×D5—22no—
90C3×C4×D5—22no—
91C3×C4×D5#4114yes—
92C3×#170—42no—
93C23×C3×C4—51no—
94C2×C3×M16—52no—
95C2×#217—101no—
96C2×#217—101no—
97#926—52no—
98#926—52no—
99#926—101no—
100#926—101no—
101C2×C4⋊Dic3—51no—
102C2×C4⋊Dic3AGL(1,𝔽5)11yescharacteristic, complemented
103C2×C4×Dic3—52no—
104C2×C4×Dic3—101no—
105C3×C22⋊C8—101no—
106C3×C22⋊C8—101no—
107C3⋊#95—101no—
108C23×D5D1211yescharacteristic
109C22×AGL(1,𝔽5)—62no—
110C22×C4×C5—61no—
111C22×C4×C5C4×D311yescharacteristic
112C22×Dic5—61no—
113C22×Dic5#6711yescharacteristic
114C2×C4×D5—24no—
115C2×C4×D5C4×D311yescharacteristic
116C2×C4×D5D1211yescharacteristic
117C2×C4×D5#6711yescharacteristic
118C2×C4×D5C22×D311yescharacteristic
119C22⋊AGL(1,𝔽5)—64no—
120C2×#170—22no—
121C4×AGL(1,𝔽5)—64no—
122#700—24no—
123C5×C4⋊C4—32no—
124C5×C4⋊C4—61no—
125#685—62no—
126#685—62no—
127C4⋊Dic5—32no—
128C4⋊Dic5—61no—
129C4×Dic5—62no—
130C4×Dic5—62no—
131C2×#117—151no—
132#407—51no—
133C4×C22⋊C4—151no—
134#339—154no—
135C22×C3×C5—21no—
136C22×C3×C5—21no—
137C22×C3×C5#10412yes—
138C22×C3×C5C2×C22⋊C411yescharacteristic
139C2×C3×D5—21no—
140C2×C3×D5—21no—
141C2×C3×D5—21no—
142C2×C3×D5—21no—
143C2×C3×D5—21no—
144C2×C3×D5—44no—
145C2×C3×D5#12312yes—
146ΓL(1,𝔽16)—44no—
147C3×C4×C5—21no—
148C3×C4×C5#11712yes—
149Dic15—42no—
150C3×Dic5—21no—
151C3×Dic5#12312yes—
152C5×Dic3—42no—
153C24×C3—51no—
154C22×C3×C4—51no—
155C22×C3×C4—51no—
156C22×C3×C4—51no—
157C22×C3×C4—51no—
158C22×C3×C4—51no—
159C22×C3×C4—104no—
160C22×C3×C4C2×AGL(1,𝔽5)11yescharacteristic
161C22×Dic3—21no—
162C22×Dic3—102no—
163C22×Dic3—101no—
164C3×M16—104no—
165C2×C3×C8—102no—
166#217—102no—
167#217—102no—
168#217—104no—
169C4⋊Dic3—21no—
170C4⋊Dic3—52no—
171C4⋊Dic3—101no—
172C4⋊Dic3C2×AGL(1,𝔽5)12yes—
173C4×Dic3—102no—
174C4×Dic3—102no—
175C4×Dic3—104no—
176C23×C5(C2×D3)⋊C411yescharacteristic
177C22×D5—22no—
178C22×D5—22no—
179C22×D5—41no—
180C22×D5(C2×D3)⋊C411yescharacteristic
181C22×D5C2×D1211yescharacteristic
182C2×AGL(1,𝔽5)—64no—
183C2×AGL(1,𝔽5)—122no—
184C2×AGL(1,𝔽5)—122no—
185C2×C4×C5—22no—
186C2×C4×C5—62no—
187C2×C4×C5—122no—
188C2×C4×C5(C2×D3)⋊C411yescharacteristic
189C2×C4×C5C2×C4×D311yescharacteristic
190C2×Dic5—22no—
191C2×Dic5—62no—
192C2×Dic5—122no—
193C2×Dic5(C2×D3)⋊C411yescharacteristic
194C2×Dic5C2×#6711yescharacteristic
195C4×D5—22no—
196C4×D5—22no—
197C4×D5—22no—
198C4×D5#23714yes—
199#170—42no—
200C23×C4—51no—
201C2×M16—52no—
202#117—152no—
203#117—152no—
204#117—301no—
205#117—301no—
206C2×C4⋊C4—31no—
207C2×C4⋊C4—151no—
208C2×C22⋊C4—301no—
209C2×C22⋊C4—301no—
210C2×C42—152no—
211C2×C42—301no—
212C22⋊C8—101no—
213C22⋊C8—101no—
214#95—301no—
215C2×C3×C5—22no—
216C2×C3×C5#38111yescharacteristic
217C2×C3×C5C2×#10411yescharacteristic
218C2×C3×C5#42111yescharacteristic
219C3×D5—22no—
220C3×D5—41no—
221C23×C3—51no—
222C23×C3—51no—
223C23×C3—102no—
224C23×C3—102no—
225C23×C3—201no—
226C23×C3C22⋊AGL(1,𝔽5)11yescharacteristic
227C2×C3×C4—22no—
228C2×C3×C4—51no—
229C2×C3×C4—54no—
230C2×C3×C4—51no—
231C2×C3×C4—102no—
232C2×C3×C4—102no—
233C2×C3×C4—102no—
234C2×C3×C4—102no—
235C2×C3×C4C22⋊AGL(1,𝔽5)11yescharacteristic
236C2×C3×C4C22×AGL(1,𝔽5)11yescharacteristic
237C2×Dic3—22no—
238C2×Dic3—42no—
239C2×Dic3—104no—
240C2×Dic3—102no—
241C2×Dic3—202no—
242C2×Dic3—202no—
243C2×Dic3—202no—
244C3×C8—202no—
245C22×C5—21no—
246C22×C5—21no—
247C22×C5#82512yes—
248C22×C5C2×(C2×D3)⋊C411yescharacteristic
249C2×D5—21no—
250C2×D5—21no—
251C2×D5—21no—
252C2×D5—21no—
253C2×D5—21no—
254C2×D5—44no—
255C2×D5#88612yes—
256AGL(1,𝔽5)—124no—
257C4×C5—21no—
258C4×C5—122no—
259C4×C5#87712yes—
260Dic5—21no—
261Dic5—122no—
262Dic5#93512yes—
263C24—51no—
264C22×C4—51no—
265C22×C4—51no—
266C22×C4—51no—
267C22×C4—51no—
268C22×C4—51no—
269C22×C4—61no—
270C22×C4—104no—
271C22×C4—302no—
272C22×C4—301no—
273C22×C4D3×AGL(1,𝔽5)11yescharacteristic
274M16—104no—
275C2×C8—102no—
276C4⋊C4—32no—
277C4⋊C4—61no—
278C4⋊C4—152no—
279C4⋊C4—301no—
280C22⋊C4—302no—
281C22⋊C4—302no—
282C22⋊C4—304no—
283C42—302no—
284C42—302no—
285C42—304no—
286C3×C5#182611yescharacteristic, complemented, Hall
287C22×C3—21no—
288C22×C3—21no—
289C22×C3—52no—
290C22×C3—101no—
291C22×C3—101no—
292C22×C3—101no—
293C22×C3—101no—
294C22×C3—101no—
295C22×C3—204no—
296C22×C3#401012yes—
297C22×C3C2×C22⋊AGL(1,𝔽5)11yescharacteristic
298C3×C4—21no—
299C3×C4—52no—
300C3×C4—101no—
301C3×C4#413012yes—
302Dic3—42no—
303Dic3—204no—
304Dic3—202no—
305C2×C5—22no—
306C2×C5#494311yescharacteristic
307C2×C5C2×#82511yescharacteristic
308C2×C5#513911yescharacteristic
309D5—22no—
310D5—41no—
311C23—51no—
312C23—51no—
313C23—102no—
314C23—102no—
315C23—201no—
316C23(C2×D3)⋊AGL(1,𝔽5)11yescharacteristic
317C2×C4—22no—
318C2×C4—51no—
319C2×C4—54no—
320C2×C4—51no—
321C2×C4—62no—
322C2×C4—102no—
323C2×C4—102no—
324C2×C4—102no—
325C2×C4—102no—
326C2×C4—122no—
327C2×C4—304no—
328C2×C4—302no—
329C2×C4—602no—
330C2×C4—602no—
331C2×C4—602no—
332C2×C4(C2×D3)⋊AGL(1,𝔽5)11yescharacteristic
333C2×C4C2×D3×AGL(1,𝔽5)11yescharacteristic
334C8—202no—
335C2×C3—22no—
336C2×C3—102no—
337C2×C3—201no—
338C2×C3#6576411yescharacteristic
339C2×C3#6576911yescharacteristic
340C2×C3C2×#401011yescharacteristic
341C5#6997511yescharacteristic, complemented, Sylow
342C22—21no—
343C22—21no—
344C22—52no—
345C22—101no—
346C22—101no—
347C22—101no—
348C22—101no—
349C22—101no—
350C22—204no—
351C22#9115712yes—
352C22C2×(C2×D3)⋊AGL(1,𝔽5)11yescharacteristic
353C4—21no—
354C4—52no—
355C4—101no—
356C4—122no—
357C4—604no—
358C4—602no—
359C4#9192312yes—
360C3#1060428711yescharacteristic, complemented, Sylow
361C2—22no—
362C2—102no—
363C2—201no—
364C2#1175171811yescharacteristic
365C2C2×#9115711yescharacteristic
366C2#1175198411yescharacteristic
subgroupG/H
centerC22C2×(C2×D3)⋊AGL(1,𝔽5)
commutatorC2×C3×C4×C5C22×C4
FrattiniC2×C4C2×D3×AGL(1,𝔽5)
FittingC22×C3×C4×C5C2×C4
socleC22×C3×C5C2×C22⋊C4
prime radicalC23×C3×D5D4
cosocleC2×C3×C4×D5C23
hypercenterC22×C4D3×AGL(1,𝔽5)
hypocenterC3×C5#1826
LCSG ▷ C2×C3×C4×C5 ▷ C2×C3×C5 ▷ C3×C5
UCSC1 ◁ C22 ◁ C22×C4

Complements

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

NH ≅ G/Ntype
4. #11746524C2semidirect
10. C2×C4⋊Dic15C4semidirect
11. C2×C5×C4⋊Dic3C4semidirect
16. #91358C4semidirect
102. C2×C4⋊Dic3AGL(1,𝔽5)semidirect
286. C3×C5#1826semidirect
341. C5#69975semidirect
360. C3#10604287semidirect

Maximal subgroups

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

subgroupindex
C2×#913582
C4×#67632
C3×#657562
#117465242
#106042873
#699755