Finite Group Databasebeta

← #14322657

Subgroups — #14322657

Order 1920.

1024 conjugacy classes of subgroups.

The subgroup lattice has 388 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×C3×D5×C4⋊C4C211yesmaximal, characteristic
2#11752424C212yesmaximal
3#11752423C212yesmaximal
4C4×#6763C212yesmaximal
5#11752370C218yesmaximal, complemented
6#10617101—31nomaximal, Hall
7C22×C3×C4×D5C2211yescharacteristic
8C22×C3×C4×D5C2211yescharacteristic
9C22×C3×C4×D5C2211yescharacteristic
10C2×#6763C2214yes—
11C2×C4⋊ΓL(1,𝔽16)C2214yes—
12C2×C4×ΓL(1,𝔽16)C2214yes—
13C2×C3×C5×C4⋊C4C411yescharacteristic, complemented
14C2×C3×C4⋊Dic5C411yescharacteristic, complemented
15C2×C3×#683C411yescharacteristic, complemented
16C2×C3×#683C411yescharacteristic, complemented
17C3×D5×C4⋊C4C414yescomplemented
18C3×D5×C4⋊C4C2214yes—
19#91235C2214yes—
20#91235C2218yes—
21#91235C2214yes—
22#82155—51nomaximal, Hall
23C2×D5×C4⋊C4D311yescharacteristic
24#65765—32no—
25#65764—32no—
26C4×C22⋊AGL(1,𝔽5)—32no—
27#65711—38no—
28C23×C3×D5C2311yescharacteristic
29C22×ΓL(1,𝔽16)C2318yes—
30C22×C3×C4×C5C2×C411yescharacteristic
31C22×C3×C4×C5C2×C411yescharacteristic
32C22×C3×C4×C5C2×C411yescharacteristic
33C22×C3×Dic5C2×C411yescharacteristic
34C22×C3×Dic5C2×C411yescharacteristic
35C22×C3×Dic5C2×C411yescharacteristic
36C2×C3×C4×D5—24no—
37C2×C3×C4×D5—42no—
38C2×C3×C4×D5—42no—
39C2×C3×C4×D5C2×C412yes—
40C2×C3×C4×D5C2×C412yes—
41C2×C3×C4×D5C2×C412yes—
42C2×C3×C4×D5C2312yes—
43C2×C3×C4×D5C2312yes—
44C2×C3×C4×D5C2312yes—
45#6763—28no—
46C4⋊ΓL(1,𝔽16)—44no—
47C4×ΓL(1,𝔽16)—44no—
48C3×C5×C4⋊C4C2×C414yes—
49C3×C4⋊Dic5C2×C414yes—
50C3×#683C2×C414yes—
51C3×#683C2×C414yes—
52C22×C3×C4⋊C4—51no—
53#5212—52no—
54#5211—52no—
55C4×#217—52no—
56#4980—58no—
57C22×C4×D5C2×D311yescharacteristic
58C22×C4×D5C2×D311yescharacteristic
59C22×C4×D5C2×D311yescharacteristic
60C2×C22⋊AGL(1,𝔽5)—34no—
61C2×C4⋊AGL(1,𝔽5)—34no—
62C2×C4×AGL(1,𝔽5)—34no—
63C2×C5×C4⋊C4Dic311yescharacteristic, complemented
64C2×C4⋊Dic5Dic311yescharacteristic, complemented
65C2×#683Dic311yescharacteristic, complemented
66C2×#683Dic311yescharacteristic, complemented
67D5×C4⋊C4Dic314yescomplemented
68D5×C4⋊C4C2×D314yes—
69(C2×AGL(1,𝔽5))⋊C4—34no—
70(C2×AGL(1,𝔽5))⋊C4—38no—
71(C2×AGL(1,𝔽5))⋊C4—34no—
72#2313—151noSylow
73C23×C3×C5C22×C411yescharacteristic
74C22×C3×D5—22no—
75C22×C3×D5—22no—
76C22×C3×D5C22×C411yescharacteristic
77C22×C3×D5#4114yes—
78C22×C3×D5C2411yescharacteristic
79C2×ΓL(1,𝔽16)—28no—
80C2×ΓL(1,𝔽16)—28no—
81C2×ΓL(1,𝔽16)—28no—
82C2×C3×C4×C5—22no—
83C2×C3×C4×C5—41no—
84C2×C3×C4×C5—41no—
85C2×C3×C4×C5C22×C412yes—
86C2×C3×C4×C5C22×C412yes—
87C2×C3×C4×C5C22×C412yes—
88C2×C3×Dic5—22no—
89C2×C3×Dic5—41no—
90C2×C3×Dic5—41no—
91C2×C3×Dic5C22×C412yes—
92C2×C3×Dic5C22×C412yes—
93C2×C3×Dic5C22×C412yes—
94C3×C4×D5—24no—
95C3×C4×D5—24no—
96C3×C4×D5—24no—
97C3×C4×D5—24no—
98C3×C4×D5—42no—
99C3×C4×D5—42no—
100C23×C3×C4—51no—
101C23×C3×C4—51no—
102C23×C3×C4—51no—
103C2×C3×C4⋊C4—51no—
104C2×C3×C4⋊C4—51no—
105C2×C3×C4⋊C4—51no—
106C2×C3×C4⋊C4—54no—
107C2×C3×C4⋊C4—54no—
108C2×C3×C4⋊C4AGL(1,𝔽5)11yescharacteristic, complemented
109C2×#217—54no—
110C2×C4⋊Dic3—54no—
111C2×C4×Dic3—54no—
112C3⋊#95—54no—
113C3⋊#95—58no—
114C3⋊#95—54no—
115C23×D5C22×D311yescharacteristic
116C22×AGL(1,𝔽5)—38no—
117C22×C4×C5C2×Dic311yescharacteristic
118C22×C4×C5C2×Dic311yescharacteristic
119C22×C4×C5C2×Dic311yescharacteristic
120C22×Dic5C2×Dic311yescharacteristic
121C22×Dic5C2×Dic311yescharacteristic
122C22×Dic5C2×Dic311yescharacteristic
123C2×C4×D5—24no—
124C2×C4×D5—42no—
125C2×C4×D5—42no—
126C2×C4×D5C2×Dic312yes—
127C2×C4×D5C2×Dic312yes—
128C2×C4×D5C2×Dic312yes—
129C2×C4×D5C22×D312yes—
130C2×C4×D5C22×D312yes—
131C2×C4×D5C22×D312yes—
132C22⋊AGL(1,𝔽5)—68no—
133C4⋊AGL(1,𝔽5)—124no—
134C4×AGL(1,𝔽5)—124no—
135C5×C4⋊C4C2×Dic314yes—
136C4⋊Dic5C2×Dic314yes—
137#683C2×Dic314yes—
138#683C2×Dic314yes—
139C22×C4⋊C4—51no—
140#387—158no—
141#381—152no—
142#380—152no—
143C4×C22⋊C4—152no—
144C22×C3×C5—21no—
145C22×C3×C5—21no—
146C22×C3×C5#11712yes—
147C22×C3×C5C23×C411yescharacteristic
148C2×C3×D5—21no—
149C2×C3×D5—21no—
150C2×C3×D5—28no—
151C2×C3×D5#11712yes—
152C2×C3×D5C2×#4112yes—
153C2×C3×D5H(5,𝔽2)12yes—
154C2×C3×D52−1+412yes—
155ΓL(1,𝔽16)—48no—
156C3×C4×C5—22no—
157C3×C4×C5—22no—
158C3×C4×C5—22no—
159C3×Dic5—22no—
160C3×Dic5—22no—
161C3×Dic5—22no—
162C24×C3—51no—
163C22×C3×C4—51no—
164C22×C3×C4—51no—
165C22×C3×C4—51no—
166C22×C3×C4—52no—
167C22×C3×C4—52no—
168C22×C3×C4—52no—
169C22×C3×C4—52no—
170C22×C3×C4—52no—
171C22×C3×C4—52no—
172C22×C3×C4—104no—
173C22×C3×C4—202no—
174C22×C3×C4—202no—
175C22×C3×C4C2×AGL(1,𝔽5)11yescharacteristic
176C22×C3×C4C2×AGL(1,𝔽5)11yescharacteristic
177C22×C3×C4C2×AGL(1,𝔽5)11yescharacteristic
178C22×Dic3—58no—
179C3×C4⋊C4—54no—
180C3×C4⋊C4—54no—
181C3×C4⋊C4—54no—
182C3×C4⋊C4C2×AGL(1,𝔽5)14yes—
183#217—108no—
184C4⋊Dic3—204no—
185C4×Dic3—204no—
186C23×C5C22×Dic311yescharacteristic
187C22×D5—22no—
188C22×D5—22no—
189C22×D5#23514yes—
190C22×D5C22×Dic311yescharacteristic
191C22×D5C23×D311yescharacteristic
192C2×AGL(1,𝔽5)—68no—
193C2×AGL(1,𝔽5)—68no—
194C2×AGL(1,𝔽5)—68no—
195C2×C4×C5—22no—
196C2×C4×C5—41no—
197C2×C4×C5—41no—
198C2×C4×C5C22×Dic312yes—
199C2×C4×C5C22×Dic312yes—
200C2×C4×C5C22×Dic312yes—
201C2×Dic5—22no—
202C2×Dic5—41no—
203C2×Dic5—41no—
204C2×Dic5C22×Dic312yes—
205C2×Dic5C22×Dic312yes—
206C2×Dic5C22×Dic312yes—
207C4×D5—24no—
208C4×D5—24no—
209C4×D5—24no—
210C4×D5—24no—
211C4×D5—42no—
212C4×D5—42no—
213C23×C4—51no—
214C23×C4—51no—
215C23×C4—51no—
216C2×C4⋊C4—51no—
217C2×C4⋊C4—51no—
218C2×C4⋊C4—51no—
219C2×C4⋊C4—54no—
220C2×C4⋊C4—54no—
221C2×C4⋊C4—154no—
222C2×C4⋊C4ΓL(1,𝔽16)11yescharacteristic, complemented
223C2×C22⋊C4—154no—
224C2×C42—154no—
225#95—154no—
226#95—154no—
227#95—158no—
228C2×C3×C5—22no—
229C2×C3×C5C2×#11711yescharacteristic
230C2×C3×C5#51811yescharacteristic
231C2×C3×C5#51911yescharacteristic
232C3×D5—22no—
233C3×D5#53914yes—
234C23×C3—51no—
235C23×C3—54no—
236C23×C3—51no—
237C23×C3—102no—
238C23×C3—102no—
239C23×C3C22×AGL(1,𝔽5)11yescharacteristic
240C2×C3×C4—22no—
241C2×C3×C4—41no—
242C2×C3×C4—41no—
243C2×C3×C4—52no—
244C2×C3×C4—52no—
245C2×C3×C4—52no—
246C2×C3×C4—104no—
247C2×C3×C4—104no—
248C2×C3×C4—104no—
249C2×C3×C4—104no—
250C2×C3×C4—102no—
251C2×C3×C4—202no—
252C2×C3×C4—202no—
253C2×C3×C4—201no—
254C2×C3×C4—201no—
255C2×C3×C4C22×AGL(1,𝔽5)12yes—
256C2×C3×C4C22×AGL(1,𝔽5)12yes—
257C2×C3×C4C22×AGL(1,𝔽5)12yes—
258C2×Dic3—108no—
259C2×Dic3—108no—
260C2×Dic3—108no—
261C22×C5—21no—
262C22×C5—21no—
263C22×C5#92612yes—
264C22×C5C23×Dic311yescharacteristic
265C2×D5—21no—
266C2×D5—21no—
267C2×D5—28no—
268C2×D5#92612yes—
269C2×D5C2×#23512yes—
270C2×D5#100412yes—
271C2×D5#100712yes—
272AGL(1,𝔽5)—128no—
273C4×C5—22no—
274C4×C5—22no—
275C4×C5—22no—
276Dic5—22no—
277Dic5—22no—
278Dic5—22no—
279C24—51no—
280C22×C4—51no—
281C22×C4—51no—
282C22×C4—51no—
283C22×C4—52no—
284C22×C4—52no—
285C22×C4—52no—
286C22×C4—52no—
287C22×C4—52no—
288C22×C4—52no—
289C22×C4—104no—
290C22×C4—158no—
291C22×C4—202no—
292C22×C4—202no—
293C22×C4C2×ΓL(1,𝔽16)11yescharacteristic
294C22×C4C2×ΓL(1,𝔽16)11yescharacteristic
295C22×C4C2×ΓL(1,𝔽16)11yescharacteristic
296C4⋊C4—54no—
297C4⋊C4—54no—
298C4⋊C4—54no—
299C4⋊C4—604no—
300C4⋊C4C2×ΓL(1,𝔽16)14yes—
301C22⋊C4—308no—
302C42—604no—
303C3×C5#231311yescharacteristic, complemented, Hall
304C22×C3—21no—
305C22×C3—21no—
306C22×C3—52no—
307C22×C3—52no—
308C22×C3—52no—
309C22×C3—52no—
310C22×C3—108no—
311C22×C3—101no—
312C22×C3—101no—
313C22×C3#413012yes—
314C22×C3C23×AGL(1,𝔽5)11yescharacteristic
315C3×C4—22no—
316C3×C4—22no—
317C3×C4—22no—
318C3×C4—102no—
319C3×C4—102no—
320C3×C4—102no—
321Dic3—208no—
322C2×C5—22no—
323C2×C5C2×#92611yescharacteristic
324C2×C5#579811yescharacteristic
325C2×C5#581211yescharacteristic
326D5—22no—
327D5#551514yes—
328C23—51no—
329C23—54no—
330C23—51no—
331C23—102no—
332C23—102no—
333C23C22×ΓL(1,𝔽16)11yescharacteristic
334C2×C4—22no—
335C2×C4—41no—
336C2×C4—41no—
337C2×C4—52no—
338C2×C4—52no—
339C2×C4—52no—
340C2×C4—104no—
341C2×C4—104no—
342C2×C4—104no—
343C2×C4—104no—
344C2×C4—102no—
345C2×C4—202no—
346C2×C4—202no—
347C2×C4—201no—
348C2×C4—201no—
349C2×C4—308no—
350C2×C4—308no—
351C2×C4—308no—
352C2×C4C22×ΓL(1,𝔽16)12yes—
353C2×C4C22×ΓL(1,𝔽16)12yes—
354C2×C4C22×ΓL(1,𝔽16)12yes—
355C2×C3—22no—
356C2×C3—54no—
357C2×C3—102no—
358C2×C3C2×#413011yescharacteristic
359C2×C3#6625811yescharacteristic
360C2×C3#6626211yescharacteristic
361C5#8215511yescharacteristic, complemented, Sylow
362C22—21no—
363C22—21no—
364C22—52no—
365C22—52no—
366C22—52no—
367C22—52no—
368C22—108no—
369C22—101no—
370C22—101no—
371C22#9198912yes—
372C22C23×ΓL(1,𝔽16)11yescharacteristic
373C4—22no—
374C4—22no—
375C4—22no—
376C4—102no—
377C4—102no—
378C4—102no—
379C4—608no—
380C3#1061710111yescharacteristic, complemented, Sylow
381C2—22no—
382C2—54no—
383C2—102no—
384C2C2×#9198911yescharacteristic
385C2#1175698711yescharacteristic
386C2#1175699111yescharacteristic
subgroupG/H
center, FrattiniC22C23×ΓL(1,𝔽16)
commutator, socleC22×C3×C5C23×C4
FittingC2×C3×C5×C4⋊C4C4
prime radicalC23×C3×D5C23
cosocleC22×C3×D5C24
hypercenterC2×C4⋊C4ΓL(1,𝔽16)
hypocenterC3×C5#2313
LCSG ▷ C22×C3×C5 ▷ C3×C5
UCSC1 ◁ C22 ◁ C2×C4⋊C4

Complements

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

NH ≅ G/Ntype
5. #11752370C2semidirect
13. C2×C3×C5×C4⋊C4C4semidirect
14. C2×C3×C4⋊Dic5C4semidirect
15. C2×C3×#683C4semidirect
16. C2×C3×#683C4semidirect
17. C3×D5×C4⋊C4C4semidirect
63. C2×C5×C4⋊C4Dic3semidirect
64. C2×C4⋊Dic5Dic3semidirect
65. C2×#683Dic3semidirect
66. C2×#683Dic3semidirect
67. D5×C4⋊C4Dic3semidirect
108. C2×C3×C4⋊C4AGL(1,𝔽5)semidirect
222. C2×C4⋊C4ΓL(1,𝔽16)semidirect
303. C3×C5#2313semidirect
361. C5#82155semidirect
380. C3#10617101semidirect

Maximal subgroups

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

subgroupindex
C2×C3×D5×C4⋊C42
#117524242
#117524232
C4×#67632
#117523702
#106171013
#821555