Finite Group Databasebeta

← #14294311

Subgroups — #14294311

Order 1920.

960 conjugacy classes of subgroups.

The subgroup lattice has 316 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C22×#6762C212yesmaximal
2C22×C3×C4×Dic5C211yesmaximal, characteristic
3#11746789C214yesmaximal, complemented
4#10604895—31nomaximal, Hall
5C23×C3×C4×C5C411yescharacteristic
6C23×C3×Dic5C411yescharacteristic
7C23×C3×Dic5C2211yescharacteristic
8C2×#6762—24no—
9C2×#6762—24no—
10C2×#6762C418yescomplemented
11C22×C3⋊#170C2214yes—
12C2×C3×C4×Dic5—24no—
13C2×C3×C4×Dic5C412yes—
14C2×C3×C4×Dic5C2212yes—
15#71527—51nomaximal, Hall
16C22×#704—32no—
17C22×C4×Dic5D311yescharacteristic
18#64918—34no—
19C24×C3×C5C2×C411yescharacteristic
20C22×C3×C4×C5—22no—
21C22×C3×C4×C5—24no—
22C22×C3×C4×C5C2×C412yes—
23C22×C3×Dic5—22no—
24C22×C3×Dic5—24no—
25C22×C3×Dic5C2×C412yes—
26C22×C3×Dic5C2×C411yescharacteristic
27C22×C3×Dic5C2×C414yes—
28C22×C3×Dic5D412yes—
29C22×C3×Dic5D414yes—
30C22×C3×Dic5Q812yes—
31C22×C3×Dic5C2311yescharacteristic
32#6762—216no—
33C2×C3⋊#170—24no—
34C2×C3⋊#170—24no—
35C2×C3⋊#170C2×C418yescomplemented
36C3×C4×Dic5—24no—
37C3×C4×Dic5—24no—
38C22×C3×C42—51no—
39C22×#208—52no—
40#4524—54no—
41C23×C4×C5Dic311yescharacteristic
42C23×Dic5Dic311yescharacteristic
43C23×Dic5C2×D311yescharacteristic
44C2×#704—38no—
45C2×#704—64no—
46C2×#704—64no—
47C22×#170—34no—
48C2×C4×Dic5—24no—
49C2×C4×Dic5Dic312yes—
50C2×C4×Dic5C2×D312yes—
51#1728—151noSylow
52C23×C3×C5—22no—
53C23×C3×C5—22no—
54C23×C3×C5C4212yes—
55C23×C3×C5C22⋊C412yes—
56C23×C3×C5C4⋊C412yes—
57C23×C3×C5C22×C411yescharacteristic
58C2×C3×C4×C5—28no—
59C2×C3×C4×C5—22no—
60C2×C3×C4×C5—22no—
61C2×C3×C4×C5M1614yescomplemented
62C2×C3×Dic5—28no—
63C2×C3×Dic5—22no—
64C2×C3×Dic5—22no—
65C2×C3×Dic5C4212yes—
66C2×C3×Dic5C22⋊C412yes—
67C2×C3×Dic5C22⋊C418yes—
68C2×C3×Dic5C4⋊C412yes—
69C2×C3×Dic5C4⋊C418yes—
70C2×C3×Dic5M1614yescomplemented
71C2×C3×Dic5C22×C412yes—
72C2×C3×Dic5C2×D411yescharacteristic
73C2×C3×Dic5C2×D412yes—
74C2×C3×Dic5C2×Q811yescharacteristic
75C3⋊#170—28no—
76C23×C3×C4—51no—
77C23×C3×C4—51no—
78C23×C3×C4AGL(1,𝔽5)11yescharacteristic
79C2×C3×C42—52no—
80C2×C3×C42—52no—
81C2×C3×C42—104no—
82C2×#208—58no—
83C2×#208—104no—
84C2×#208—104no—
85C22×C3⋊C8—54no—
86C24×C5C2×Dic311yescharacteristic
87C22×C4×C5—22no—
88C22×C4×C5—24no—
89C22×C4×C5C2×Dic312yes—
90C22×Dic5—22no—
91C22×Dic5—24no—
92C22×Dic5Dic612yes—
93C22×Dic5C4×D314yes—
94C22×Dic5D1212yes—
95C22×Dic5C2×Dic312yes—
96C22×Dic5C2×Dic311yescharacteristic
97C22×Dic5#6714yes—
98C22×Dic5C22×D311yescharacteristic
99#704—616no—
100C2×#170—38no—
101C2×#170—64no—
102C2×#170—64no—
103C4×Dic5—24no—
104C4×Dic5—24no—
105C22×M16—152no—
106C22×C42—51no—
107#336—154no—
108C22×C3×C5—24no—
109C22×C3×C5—22no—
110C22×C3×C5—22no—
111C22×C3×C5—24no—
112C22×C3×C5#9514yes—
113C22×C3×C5C2×C4211yescharacteristic
114C22×C3×C5C2×C22⋊C411yescharacteristic
115C22×C3×C5C2×C4⋊C411yescharacteristic
116C22×C3×C5C2×M1612yes—
117C22×C3×C5C2×M1612yes—
118C3×C4×C5—24no—
119C3×Dic5—24no—
120C3×Dic5#9514yes—
121C3×Dic5C2×C22⋊C412yes—
122C3×Dic5C2×C4⋊C412yes—
123C24×C3C2×AGL(1,𝔽5)11yescharacteristic
124C22×C3×C4—22no—
125C22×C3×C4—24no—
126C22×C3×C4—54no—
127C22×C3×C4—52no—
128C22×C3×C4—52no—
129C22×C3×C4—54no—
130C22×C3×C4—51no—
131C22×C3×C4—51no—
132C22×C3×C4—52no—
133C22×C3×C4—104no—
134C22×C3×C4—102no—
135C22×C3×C4C2×AGL(1,𝔽5)12yes—
136C3×C42—104no—
137C3×C42—104no—
138#208—1016no—
139C2×C3⋊C8—58no—
140C2×C3⋊C8—104no—
141C2×C3⋊C8—104no—
142C23×C5—22no—
143C23×C5—22no—
144C23×C5C4×Dic312yes—
145C23×C5C4⋊Dic312yes—
146C23×C5#21712yes—
147C23×C5C22×Dic311yescharacteristic
148C2×C4×C5—28no—
149C2×C4×C5—22no—
150C2×C4×C5—22no—
151C2×C4×C5#20814yescomplemented
152C2×Dic5—28no—
153C2×Dic5—22no—
154C2×Dic5—22no—
155C2×Dic5#20814yescomplemented
156C2×Dic5C4×Dic312yes—
157C2×Dic5Dic3⋊C418yes—
158C2×Dic5C4⋊Dic312yes—
159C2×Dic5(C2×D3)⋊C418yes—
160C2×Dic5#21712yes—
161C2×Dic5C2×Dic611yescharacteristic
162C2×Dic5C2×C4×D312yes—
163C2×Dic5C2×D1211yescharacteristic
164C2×Dic5C2×#6712yes—
165#170—68no—
166C23×C4—51no—
167C23×C4—51no—
168C23×C4ΓL(1,𝔽16)11yescharacteristic
169C2×M16—158no—
170C2×M16—304no—
171C2×M16—304no—
172C22×C8—154no—
173C2×C42—52no—
174C2×C42—52no—
175C2×C42—104no—
176C2×C3×C5—24no—
177C2×C3×C5C2×#9511yescharacteristic
178C2×C3×C5C4×M1612yes—
179C2×C3×C5#40711yescharacteristic
180C2×C3×C5#40711yescharacteristic
181C2×C3×C5#42312yes—
182C23×C3—22no—
183C23×C3—22no—
184C23×C3C4×AGL(1,𝔽5)12yes—
185C23×C3C4⋊AGL(1,𝔽5)12yes—
186C23×C3C22⋊AGL(1,𝔽5)12yes—
187C23×C3C22×AGL(1,𝔽5)11yescharacteristic
188C2×C3×C4—28no—
189C2×C3×C4—22no—
190C2×C3×C4—22no—
191C2×C3×C4—52no—
192C2×C3×C4—52no—
193C2×C3×C4—52no—
194C2×C3×C4—51no—
195C2×C3×C4—51no—
196C2×C3×C4—52no—
197C2×C3×C4—52no—
198C2×C3×C4—58no—
199C2×C3×C4—58no—
200C2×C3×C4—54no—
201C2×C3×C4—108no—
202C2×C3×C4—102no—
203C2×C3×C4—102no—
204C2×C3×C4#70414yescomplemented
205C3⋊C8—108no—
206C22×C5—24no—
207C22×C5—24no—
208C22×C5—22no—
209C22×C5—22no—
210C22×C5C3⋊#9514yes—
211C22×C5C2×#20812yes—
212C22×C5C2×#20812yes—
213C22×C5C2×C4×Dic311yescharacteristic
214C22×C5C2×C4⋊Dic311yescharacteristic
215C22×C5C2×#21711yescharacteristic
216C4×C5—24no—
217Dic5—24no—
218Dic5C3⋊#9514yes—
219Dic5C2×Dic3⋊C412yes—
220Dic5C2×(C2×D3)⋊C412yes—
221C24C2×ΓL(1,𝔽16)11yescharacteristic
222C22×C4—22no—
223C22×C4—24no—
224C22×C4—54no—
225C22×C4—52no—
226C22×C4—52no—
227C22×C4—54no—
228C22×C4—51no—
229C22×C4—51no—
230C22×C4—52no—
231C22×C4—104no—
232C22×C4—102no—
233C22×C4C2×ΓL(1,𝔽16)12yes—
234M16—3016no—
235C2×C8—158no—
236C2×C8—304no—
237C2×C8—304no—
238C42—104no—
239C42—104no—
240C3×C5#172811yescharacteristic, complemented, Hall
241C22×C3—22no—
242C22×C3—24no—
243C22×C3—22no—
244C22×C3—24no—
245C22×C3(C2×AGL(1,𝔽5))⋊C414yes—
246C22×C3D5⋊M1612yes—
247C22×C3C2×C4×AGL(1,𝔽5)11yescharacteristic
248C22×C3C2×C4⋊AGL(1,𝔽5)11yescharacteristic
249C22×C3C2×#70412yes—
250C22×C3C2×C22⋊AGL(1,𝔽5)11yescharacteristic
251C3×C4—24no—
252C3×C4—54no—
253C3×C4—52no—
254C3×C4—52no—
255C3×C4—104no—
256C2×C5—24no—
257C2×C5C4×#20812yes—
258C2×C5#492412yes—
259C2×C5#520711yescharacteristic
260C2×C5#520711yescharacteristic
261C2×C5C2×C3⋊#9511yescharacteristic
262C23—22no—
263C23—22no—
264C23C4×ΓL(1,𝔽16)12yes—
265C23C4⋊ΓL(1,𝔽16)12yes—
266C23#676312yes—
267C23C22×ΓL(1,𝔽16)11yescharacteristic
268C2×C4—28no—
269C2×C4—22no—
270C2×C4—22no—
271C2×C4—52no—
272C2×C4—52no—
273C2×C4—51no—
274C2×C4—52no—
275C2×C4—51no—
276C2×C4—52no—
277C2×C4—52no—
278C2×C4—58no—
279C2×C4—58no—
280C2×C4—54no—
281C2×C4—108no—
282C2×C4—102no—
283C2×C4—102no—
284C2×C4#676214yescomplemented
285C8—308no—
286C2×C3—24no—
287C2×C3C4×#70412yes—
288C2×C3#6575511yescharacteristic
289C2×C3#6575812yes—
290C2×C3C2×(C2×AGL(1,𝔽5))⋊C411yescharacteristic
291C2×C3#6579811yescharacteristic
292C5#7152711yescharacteristic, complemented, Sylow
293C22—22no—
294C22—24no—
295C22—22no—
296C22—24no—
297C22#9123514yes—
298C22D5⋊#20812yes—
299C22C2×C4×ΓL(1,𝔽16)11yescharacteristic
300C22C2×C4⋊ΓL(1,𝔽16)11yescharacteristic
301C22C2×#676212yes—
302C22C2×#676311yescharacteristic
303C4—24no—
304C4—52no—
305C4—54no—
306C4—52no—
307C4—104no—
308C3#1060489511yescharacteristic, complemented, Sylow
309C2—24no—
310C2C4×#676212yes—
311C2#1175241411yescharacteristic
312C2#1175241712yes—
313C2C2×#9123511yescharacteristic
314C2#1175245711yescharacteristic
subgroupG/H
center, FrattiniC23C22×ΓL(1,𝔽16)
commutatorC22×C3×C5C2×C42
FittingC23×C3×C4×C5C4
socleC23×C3×C5C22×C4
prime radicalC24×C3×C5C2×C4
cosocleC22×C3×Dic5C23
hypercenterC23×C4ΓL(1,𝔽16)
hypocenterC3×C5#1728
LCSG ▷ C22×C3×C5 ▷ C3×C5
UCSC1 ◁ C23 ◁ C23×C4

Complements

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

NH ≅ G/Ntype
3. #11746789C2semidirect
10. C2×#6762C4semidirect
35. C2×C3⋊#170C2×C4semidirect
61. C2×C3×C4×C5M16semidirect
70. C2×C3×Dic5M16semidirect
151. C2×C4×C5#208semidirect
155. C2×Dic5#208semidirect
204. C2×C3×C4#704semidirect
240. C3×C5#1728semidirect
284. C2×C4#6762semidirect
292. C5#71527semidirect
308. C3#10604895semidirect

Maximal subgroups

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

subgroupindex
C22×#67622
C22×C3×C4×Dic52
#117467892
#106048953
#715275