Finite Group Databasebeta

← #14296870

Subgroups — #14296870

Order 1920.

1224 conjugacy classes of subgroups; only the 161 normal subgroups are shown.

normal subgroups only

This group has 1224 conjugacy classes of subgroups — too many to draw. Only its 161 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
1C22×C3×#713C211yesmaximal, characteristic
2#11752423C212yesmaximal, complemented
3C2×#91228C214yesmaximal, complemented
4#10604981—31nomaximal, Hall
5C22×C3×C5×Q8C411yescharacteristic, complemented
6C2×C3×#713—28no—
7C2×C3×#713—24no—
8C22×C3×D20C411yescharacteristic, complemented
9C22×C3×D20C2212yes—
10C22×C3×C4×D5C412yescomplemented
11C22×C3×C4×D5C2211yescharacteristic
12C2×#6763C2214yes—
13#91235—24no—
14#91228—28no—
15#72139—51nomaximal, Hall
16C22×#713D311yescharacteristic
17#65764—32no—
18C2×#3999—34no—
19C23×C3×D5C2×C412yes—
20C23×C3×D5C2311yescharacteristic
21C22×ΓL(1,𝔽16)—24no—
22C2×C3×C5×Q8—24no—
23C2×C3×C5×Q8—22no—
24C22×C3×C4×C5C2×C412yes—
25C22×C3×C4×C5C2×C411yescharacteristic
26C22×C3×Dic5C2×C411yescharacteristic
27C3×#713—232no—
28C2×C3×D20—24no—
29C2×C3×D20—24no—
30C2×C3×D20—24no—
31C2×C3×D20—22no—
32C2×C3×D20D418yes—
33C2×C3×C4×D5—24no—
34C2×C3×C4×D5—24no—
35C2×C3×C4×D5—24no—
36C2×C3×C4×D5—24no—
37C2×C3×C4×D5D414yes—
38#6763—28no—
39C22×C3×#41—51no—
40C2×#834—54no—
41#5211—52no—
42C22×C5×Q8Dic311yescharacteristic, complemented
43C2×#713—28no—
44C2×#713—24no—
45C22×D20Dic311yescharacteristic, complemented
46C22×D20C2×D312yes—
47C22×C4×D5Dic312yescomplemented
48C22×C4×D5C2×D311yescharacteristic
49C2×C22⋊AGL(1,𝔽5)—34no—
50(C2×AGL(1,𝔽5))⋊C4—64no—
51#3999—68no—
52#1782—151noSylow
53C23×C3×C5C22×C411yescharacteristic
54C22×C3×D5—22no—
55C22×C3×D5—22no—
56C22×C3×D5—42no—
57C22×C3×D5—44no—
58C22×C3×D5C22⋊C414yes—
59C22×C3×D5C2×D412yes—
60C22×C3×D5C2×D414yes—
61C2×ΓL(1,𝔽16)—48no—
62C2×ΓL(1,𝔽16)—44no—
63C3×C5×Q8—28no—
64C2×C3×C4×C5—22no—
65C2×C3×C4×C5—22no—
66C2×C3×C4×C5—22no—
67C2×C3×C4×C5C22⋊C414yes—
68C2×C3×C4×C5C22⋊C412yes—
69C2×C3×Dic5—22no—
70C2×C3×Dic5C22⋊C412yes—
71C3×D20—28no—
72C3×D20—216no—
73C3×C4×D5—216no—
74C3×C4×D5—28no—
75C2×C3×#41—104no—
76C2×C3×#41—108no—
77C22×C3×Q8AGL(1,𝔽5)11yescharacteristic, complemented
78C22×C3×D4—52no—
79C22×C3×D4—51no—
80C23×C3×C4—51no—
81C23×C3×C4—52no—
82C2×#217—54no—
83#834—108no—
84C3⋊#95—104no—
85C23×D5C2×Dic312yes—
86C23×D5C22×D311yescharacteristic
87C22×AGL(1,𝔽5)—64no—
88C2×C5×Q8—24no—
89C2×C5×Q8—22no—
90C22×C4×C5C2×Dic312yes—
91C22×C4×C5C2×Dic311yescharacteristic
92C22×Dic5C2×Dic311yescharacteristic
93#713—232no—
94C2×D20—24no—
95C2×D20—24no—
96C2×D20—24no—
97C2×D20—22no—
98C2×D20#6718yes—
99C2×C4×D5—24no—
100C2×C4×D5—24no—
101C2×C4×D5—24no—
102C2×C4×D5—24no—
103C2×C4×D5#6714yes—
104C22⋊AGL(1,𝔽5)—68no—
105C22×#41—51no—
106C2×#99—154no—
107#381—152no—
108C22×C3×C5—22no—
109C22×C3×C5C2×C22⋊C412yes—
110C22×C3×C5C2×C22⋊C411yescharacteristic
111C2×C3×D5—24no—
112C2×C3×D5—24no—
113C2×C3×D5—44no—
114C2×C3×D5—44no—
115C2×C3×D5—44no—
116C2×C3×D5—44no—
117C2×C3×D5#12014yes—
118ΓL(1,𝔽16)—84no—
119C3×C4×C5—24no—
120C3×C4×C5—22no—
121C3×Dic5—22no—
122C24×C3—51no—
123C24×C3—52no—
124C3×#41—1032no—
125C2×C3×Q8—22no—
126C2×C3×Q8—24no—
127C2×C3×D4—58no—
128C2×C3×D4—104no—
129C2×C3×D4—104no—
130C2×C3×D4—104no—
131C2×C3×D4—102no—
132C22×C3×C4—51no—
133C22×C3×C4—54no—
134C22×C3×C4—104no—
135C22×C3×C4—104no—
136C22×C3×C4—104no—
137C22×C3×C4—104no—
138C22×C3×C4C2×AGL(1,𝔽5)12yes—
139C22×C3×C4C2×AGL(1,𝔽5)11yescharacteristic
140C22×Dic3—104no—
141#217—108no—
142C23×C5C22×Dic311yescharacteristic
143C22×D5—22no—
144C22×D5—22no—
145C22×D5—42no—
146C22×D5—44no—
147C22×D5#21714yes—
148C22×D5C2×#6712yes—
149C22×D5C2×#6714yes—
150C2×AGL(1,𝔽5)—128no—
151C2×AGL(1,𝔽5)—124no—
152C5×Q8—28no—
153C2×C4×C5—22no—
154C2×C4×C5—22no—
155C2×C4×C5—22no—
156C2×C4×C5#21714yes—
157C2×C4×C5#21712yes—
158C2×Dic5—22no—
159C2×Dic5#21712yes—
160D20—28no—
161D20—216no—
162C4×D5—216no—
163C4×D5—28no—
164C2×#41—104no—
165C2×#41—108no—
166C22×Q8ΓL(1,𝔽16)11yescharacteristic, complemented
167C22×D4—52no—
168C22×D4—51no—
169C23×C4—51no—
170C23×C4—52no—
171C2×C22⋊C4—154no—
172#99—308no—
173#95—304no—
174C2×C3×C5—22no—
175C2×C3×C5#37911yescharacteristic
176C2×C3×C5#41012yes—
177C3×D5—24no—
178C3×D5—44no—
179C23×C3—52no—
180C23×C3—54no—
181C23×C3—54no—
182C23×C3—102no—
183C23×C3—102no—
184C23×C3—202no—
185C23×C3—204no—
186C23×C3C22×AGL(1,𝔽5)11yescharacteristic
187C3×Q8—28no—
188C3×D4—1016no—
189C3×D4—108no—
190C2×C3×C4—22no—
191C2×C3×C4—22no—
192C2×C3×C4—22no—
193C2×C3×C4—52no—
194C2×C3×C4—108no—
195C2×C3×C4—1016no—
196C2×C3×C4—102no—
197C2×C3×C4C22⋊AGL(1,𝔽5)12yes—
198C2×C3×C4C22⋊AGL(1,𝔽5)14yes—
199C2×Dic3—204no—
200C2×Dic3—208no—
201C22×C5—22no—
202C22×C5C2×#21712yes—
203C22×C5C2×#21711yescharacteristic
204C2×D5—24no—
205C2×D5—24no—
206C2×D5—44no—
207C2×D5—44no—
208C2×D5—44no—
209C2×D5—44no—
210C2×D5#95314yes—
211AGL(1,𝔽5)—244no—
212C4×C5—24no—
213C4×C5—22no—
214Dic5—22no—
215C24—51no—
216C24—52no—
217#41—1032no—
218C2×Q8—24no—
219C2×Q8—22no—
220C2×D4—58no—
221C2×D4—104no—
222C2×D4—104no—
223C2×D4—104no—
224C2×D4—102no—
225C22×C4—51no—
226C22×C4—54no—
227C22×C4—104no—
228C22×C4—104no—
229C22×C4—104no—
230C22×C4—104no—
231C22×C4—304no—
232C22×C4C2×ΓL(1,𝔽16)12yes—
233C22×C4C2×ΓL(1,𝔽16)11yescharacteristic
234C22⋊C4—308no—
235C3×C5#178211yescharacteristic, complemented, Hall
236C22×C3—22no—
237C22×C3—54no—
238C22×C3—104no—
239C22×C3—104no—
240C22×C3—204no—
241C22×C3—204no—
242C22×C3—204no—
243C22×C3—204no—
244C22×C3C2×C22⋊AGL(1,𝔽5)12yes—
245C22×C3C2×C22⋊AGL(1,𝔽5)11yescharacteristic
246C3×C4—22no—
247C3×C4—24no—
248C3×C4—102no—
249Dic3—404no—
250C2×C5—22no—
251C2×C5#524012yes—
252C2×C5#524711yescharacteristic
253D5—24no—
254D5—44no—
255C23—52no—
256C23—54no—
257C23—54no—
258C23—102no—
259C23—102no—
260C23—202no—
261C23—204no—
262C23C22×ΓL(1,𝔽16)11yescharacteristic
263Q8—28no—
264D4—1016no—
265D4—108no—
266C2×C4—22no—
267C2×C4—22no—
268C2×C4—22no—
269C2×C4—52no—
270C2×C4—108no—
271C2×C4—1016no—
272C2×C4—102no—
273C2×C4—604no—
274C2×C4—608no—
275C2×C4#676314yes—
276C2×C4#676312yes—
277C2×C3—22no—
278C2×C3—104no—
279C2×C3—204no—
280C2×C3#6578512yes—
281C2×C3#6580011yescharacteristic
282C5#7213911yescharacteristic, complemented, Sylow
283C22—22no—
284C22—54no—
285C22—104no—
286C22—104no—
287C22—204no—
288C22—204no—
289C22—204no—
290C22—204no—
291C22C2×#676312yes—
292C22C2×#676311yescharacteristic
293C4—24no—
294C4—22no—
295C4—102no—
296C4—1204no—
297C3#1060498111yescharacteristic, complemented, Sylow
298C2—22no—
299C2—104no—
300C2—204no—
301C2#1175244412yes—
302C2#1175245911yescharacteristic
subgroupG/H
centerC22C2×#6763
commutatorC23×C3×C5C22×C4
FrattiniC23C22×ΓL(1,𝔽16)
FittingC22×C3×C5×Q8C4
socleC22×C3×C5C2×C22⋊C4
prime radicalC22×C3×#713C2
cosocleC23×C3×D5C23
hypercenterC22×Q8ΓL(1,𝔽16)
hypocenterC3×C5#1782
LCSG ▷ C23×C3×C5 ▷ C2×C3×C5 ▷ C3×C5
UCSC1 ◁ C22 ◁ C22×C4 ◁ C22×Q8

Complements

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

NH ≅ G/Ntype
2. #11752423C2semidirect
3. C2×#91228C2semidirect
5. C22×C3×C5×Q8C4semidirect
8. C22×C3×D20C4semidirect
10. C22×C3×C4×D5C4semidirect
42. C22×C5×Q8Dic3semidirect
45. C22×D20Dic3semidirect
47. C22×C4×D5Dic3semidirect
77. C22×C3×Q8AGL(1,𝔽5)semidirect
166. C22×Q8ΓL(1,𝔽16)semidirect
235. C3×C5#1782semidirect
282. C5#72139semidirect
297. C3#10604981semidirect

Maximal subgroups

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

subgroupindex
C22×C3×#7132
#117524232
C2×#912282
#106049813
#721395