Finite Group Databasebeta

← #14298549

Subgroups — #14298549

Order 1920.

612 conjugacy classes of subgroups; only the 129 normal subgroups are shown.

normal subgroups only

This group has 612 conjugacy classes of subgroups — too many to draw. Only its 129 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×C3×#4038C211yesmaximal, characteristic
2C4×#6763C211yesmaximal, characteristic
3#11752415C211yesmaximal, characteristic
4#11746746C214yesmaximal, complemented
5#10604287—31nomaximal, Hall
6C22×C3×C4×D5C2211yescharacteristic
7C2×#6763—21no—
8C2×C4×ΓL(1,𝔽16)C2212yes—
9C2×#6758C2212yes—
10C2×C3×C5×C4⋊C4C411yescharacteristic, complemented
11C2×C3×#685—21no—
12C2×C3×C4⋊Dic5C411yescharacteristic, complemented
13C2×C3×C4×Dic5—21no—
14C3×#4038—21no—
15C3×#4038—21no—
16C3×#4038C412yescomplemented
17C3×#4038C2212yes—
18#91241—21no—
19#91235—21no—
20#91232—21no—
21#71590—51nomaximal, Hall
22C2×#4038D311yescharacteristic
23C4×C22⋊AGL(1,𝔽5)—31no—
24#65756—31no—
25#64875—34no—
26C23×C3×D5D411yescharacteristic
27C22×ΓL(1,𝔽16)—22no—
28C22×C3×C4×C5—21no—
29C22×C3×C4×C5C2×C411yescharacteristic
30C22×C3×Dic5—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—
38C2×C3⋊#170—22no—
39C4×ΓL(1,𝔽16)—24no—
40#6758—24no—
41C3×C5×C4⋊C4—21no—
42C3×C5×C4⋊C4C2×C412yes—
43C3×#685—22no—
44C3×#685—22no—
45C3×C4⋊Dic5—21no—
46C3×C4⋊Dic5C2×C412yes—
47C3×C4×Dic5—22no—
48C3×C4×Dic5—22no—
49C2×C3×#117—51no—
50C4×#217—51no—
51#5207—51no—
52#4532—54no—
53C22×C4×D5C2×D311yescharacteristic
54C2×C22⋊AGL(1,𝔽5)—61no—
55C2×C4×AGL(1,𝔽5)—32no—
56C2×#700—32no—
57C2×C5×C4⋊C4Dic311yescharacteristic, complemented
58C2×#685—21no—
59C2×C4⋊Dic5Dic311yescharacteristic, complemented
60C2×C4×Dic5—21no—
61#4038—21no—
62#4038—21no—
63#4038Dic312yescomplemented
64#4038C2×D312yes—
65C22⋊#170—61no—
66(C2×AGL(1,𝔽5))⋊C4—61no—
67#4003—61no—
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×C5—22no—
80C2×C3×C4×C5—42no—
81C2×C3×C4×C5C22⋊C411yescharacteristic
82C2×C3×C4×C5C22×C411yescharacteristic
83C2×C3×Dic5—22no—
84C2×C3×Dic5—22no—
85C2×C3×Dic5—42no—
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—
94C3×#117—52no—
95C3×#117—52no—
96C3×#117—101no—
97C3×#117—101no—
98C2×C3×C4⋊C4—51no—
99C2×C3×C4⋊C4AGL(1,𝔽5)11yescharacteristic, complemented
100C2×C3×C22⋊C4—101no—
101C2×C3×C42—101no—
102C2×#217—101no—
103C2×C4×Dic3—52no—
104C2×#208—52no—
105C3⋊#95—101no—
106#830—101no—
107#830—101no—
108C23×D5#6711yescharacteristic
109C22×AGL(1,𝔽5)—62no—
110C22×C4×C5—21no—
111C22×C4×C5C2×Dic311yescharacteristic
112C22×Dic5—21no—
113C22×Dic5#6711yescharacteristic
114C2×C4×D5—24no—
115C2×C4×D5C2×Dic311yescharacteristic
116C2×C4×D5#6711yescharacteristic
117C2×C4×D5#6711yescharacteristic
118C2×C4×D5C22×D311yescharacteristic
119C22⋊AGL(1,𝔽5)—64no—
120C2×#170—62no—
121C4×AGL(1,𝔽5)—64no—
122#700—64no—
123C5×C4⋊C4—21no—
124C5×C4⋊C4C2×Dic312yes—
125#685—22no—
126#685—22no—
127C4⋊Dic5—21no—
128C4⋊Dic5C2×Dic312yes—
129C4×Dic5—22no—
130C4×Dic5—22no—
131C2×#117—51no—
132#407—151no—
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—42no—
149C3×C4×C5#11712yes—
150C3×Dic5—21no—
151C3×Dic5—42no—
152C3×Dic5#12312yes—
153C24×C3—51no—
154C22×C3×C4—21no—
155C22×C3×C4—51no—
156C22×C3×C4—51no—
157C22×C3×C4—51no—
158C22×C3×C4—51no—
159C22×C3×C4—51no—
160C22×C3×C4—104no—
161C22×C3×C4—101no—
162C22×C3×C4C2×AGL(1,𝔽5)11yescharacteristic
163C22×Dic3—102no—
164C3×C4⋊C4—21no—
165C3×C4⋊C4—52no—
166C3×C4⋊C4—101no—
167C3×C4⋊C4C2×AGL(1,𝔽5)12yes—
168C3×C22⋊C4—102no—
169C3×C22⋊C4—102no—
170C3×C42—102no—
171C3×C42—102no—
172#217—104no—
173C4×Dic3—104no—
174#208—104no—
175C2×C3⋊C8—102no—
176C23×C5#21711yescharacteristic
177C22×D5—22no—
178C22×D5—22no—
179C22×D5—41no—
180C22×D5#21711yescharacteristic
181C22×D5C2×#6711yescharacteristic
182C2×AGL(1,𝔽5)—64no—
183C2×AGL(1,𝔽5)—122no—
184C2×AGL(1,𝔽5)—122no—
185C2×C4×C5—22no—
186C2×C4×C5—22no—
187C2×C4×C5—42no—
188C2×C4×C5#21711yescharacteristic
189C2×C4×C5C22×Dic311yescharacteristic
190C2×Dic5—22no—
191C2×Dic5—22no—
192C2×Dic5—42no—
193C2×Dic5#21711yescharacteristic
194C2×Dic5C2×#6711yescharacteristic
195C4×D5—22no—
196C4×D5—22no—
197C4×D5—22no—
198C4×D5#23514yes—
199#170—122no—
200C23×C4—51no—
201C2×M16—152no—
202#117—52no—
203#117—52no—
204#117—101no—
205#117—101no—
206C2×C4⋊C4—51no—
207C2×C4⋊C4ΓL(1,𝔽16)11yescharacteristic, complemented
208C2×C22⋊C4—101no—
209C2×C22⋊C4—301no—
210C2×C42—101no—
211C2×C42—152no—
212C22⋊C8—301no—
213C22⋊C8—301no—
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—22no—
229C2×C3×C4—42no—
230C2×C3×C4—51no—
231C2×C3×C4—54no—
232C2×C3×C4—51no—
233C2×C3×C4—102no—
234C2×C3×C4—102no—
235C2×C3×C4—102no—
236C2×C3×C4—102no—
237C2×C3×C4—102no—
238C2×C3×C4—202no—
239C2×C3×C4C22⋊AGL(1,𝔽5)11yescharacteristic
240C2×C3×C4C22×AGL(1,𝔽5)11yescharacteristic
241C2×Dic3—104no—
242C2×Dic3—202no—
243C2×Dic3—202no—
244C3⋊C8—202no—
245C22×C5—21no—
246C22×C5—21no—
247C22×C5#83712yes—
248C22×C5C2×#21711yescharacteristic
249C2×D5—21no—
250C2×D5—21no—
251C2×D5—21no—
252C2×D5—21no—
253C2×D5—21no—
254C2×D5—44no—
255C2×D5#92912yes—
256AGL(1,𝔽5)—124no—
257C4×C5—21no—
258C4×C5—42no—
259C4×C5#92612yes—
260Dic5—21no—
261Dic5—42no—
262Dic5#92912yes—
263C24—51no—
264C22×C4—21no—
265C22×C4—51no—
266C22×C4—51no—
267C22×C4—51no—
268C22×C4—51no—
269C22×C4—51no—
270C22×C4—104no—
271C22×C4—101no—
272C22×C4—302no—
273C22×C4C2×ΓL(1,𝔽16)11yescharacteristic
274M16—304no—
275C2×C8—302no—
276C4⋊C4—21no—
277C4⋊C4—52no—
278C4⋊C4—101no—
279C4⋊C4C2×ΓL(1,𝔽16)12yes—
280C22⋊C4—102no—
281C22⋊C4—102no—
282C22⋊C4—304no—
283C42—102no—
284C42—102no—
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—42no—
300C3×C4—52no—
301C3×C4—101no—
302C3×C4—202no—
303C3×C4#413012yes—
304Dic3—204no—
305C2×C5—22no—
306C2×C5#521111yescharacteristic
307C2×C5C2×#83711yescharacteristic
308C2×C5#523611yescharacteristic
309D5—22no—
310D5—41no—
311C23—51no—
312C23—51no—
313C23—102no—
314C23—102no—
315C23—201no—
316C23#676311yescharacteristic
317C2×C4—22no—
318C2×C4—22no—
319C2×C4—42no—
320C2×C4—51no—
321C2×C4—54no—
322C2×C4—51no—
323C2×C4—102no—
324C2×C4—102no—
325C2×C4—102no—
326C2×C4—102no—
327C2×C4—102no—
328C2×C4—202no—
329C2×C4—304no—
330C2×C4—602no—
331C2×C4—602no—
332C2×C4#676311yescharacteristic
333C2×C4C22×ΓL(1,𝔽16)11yescharacteristic
334C8—602no—
335C2×C3—22no—
336C2×C3—102no—
337C2×C3—201no—
338C2×C3#6576411yescharacteristic
339C2×C3#6576911yescharacteristic
340C2×C3C2×#401011yescharacteristic
341C5#7159011yescharacteristic, complemented, Sylow
342C22—21no—
343C22—21no—
344C22—52no—
345C22—101no—
346C22—101no—
347C22—101no—
348C22—101no—
349C22—101no—
350C22—204no—
351C22#9123912yes—
352C22C2×#676311yescharacteristic
353C4—21no—
354C4—42no—
355C4—52no—
356C4—101no—
357C4—202no—
358C4—604no—
359C4#9198912yes—
360C3#1060428711yescharacteristic, complemented, Sylow
361C2—22no—
362C2—102no—
363C2—201no—
364C2#1175242311yescharacteristic
365C2#1175242811yescharacteristic
366C2C2×#9123911yescharacteristic
subgroupG/H
centerC22C2×#6763
commutatorC2×C3×C4×C5C22×C4
FrattiniC2×C4C22×ΓL(1,𝔽16)
FittingC2×C3×C5×C4⋊C4C4
socleC22×C3×C5C2×C22⋊C4
prime radicalC23×C3×D5D4
cosocleC2×C3×C4×D5C23
hypercenterC2×C4⋊C4ΓL(1,𝔽16)
hypocenterC3×C5#1826
LCSG ▷ C2×C3×C4×C5 ▷ C2×C3×C5 ▷ C3×C5
UCSC1 ◁ C22 ◁ C22×C4 ◁ C2×C4⋊C4

Complements

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

NH ≅ G/Ntype
4. #11746746C2semidirect
10. C2×C3×C5×C4⋊C4C4semidirect
12. C2×C3×C4⋊Dic5C4semidirect
16. C3×#4038C4semidirect
57. C2×C5×C4⋊C4Dic3semidirect
59. C2×C4⋊Dic5Dic3semidirect
63. #4038Dic3semidirect
99. C2×C3×C4⋊C4AGL(1,𝔽5)semidirect
207. C2×C4⋊C4ΓL(1,𝔽16)semidirect
286. C3×C5#1826semidirect
341. C5#71590semidirect
360. C3#10604287semidirect

Maximal subgroups

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

subgroupindex
C2×C3×#40382
C4×#67632
#117524152
#117467462
#106042873
#715905