Finite Group Databasebeta

← #14296837

Subgroups — #14296837

Order 1920.

1760 conjugacy classes of subgroups; only the 177 normal subgroups are shown.

normal subgroups only

This group has 1760 conjugacy classes of subgroups — too many to draw. Only its 177 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
1C23×C3×D20C211yesmaximal, characteristic
2#11752459C212yesmaximal, complemented
3C2×#91228C214yesmaximal, complemented
4#10604890—31nomaximal, Hall
5C24×C3×D5C411yescharacteristic, complemented
6C24×C3×D5C2211yescharacteristic
7C23×C3×C4×C5C411yescharacteristic, complemented
8C22×C3×D20—24no—
9C22×C3×D20—28no—
10C22×C3×D20C412yescomplemented
11C22×C3×D20C2212yes—
12C2×#6763—24no—
13C2×#6763C2214yescomplemented
14#91228—28no—
15#72130—51nomaximal, Hall
16C23×D20D311yescharacteristic
17#65800—32no—
18C2×#3999—34no—
19C24×C3×C5C2×C411yescharacteristic
20C23×C3×D5—22no—
21C23×C3×D5—24no—
22C23×C3×D5—24no—
23C23×C3×D5—44no—
24C23×C3×D5—44no—
25C23×C3×D5C2×C411yescharacteristic
26C23×C3×D5C2×C412yes—
27C23×C3×D5D414yes—
28C23×C3×D5C2311yescharacteristic
29C22×ΓL(1,𝔽16)—24no—
30C22×C3×C4×C5—22no—
31C22×C3×C4×C5—24no—
32C22×C3×C4×C5C2×C412yes—
33C2×C3×D20—232no—
34C2×C3×D20—24no—
35C2×C3×D20—24no—
36C2×C3×D20—24no—
37C2×C3×D20—24no—
38C2×C3×D20—24no—
39C2×C3×D20D418yes—
40#6763—28no—
41#6763—48no—
42C23×C3×D4—51no—
43#5247—52no—
44C2×#834—54no—
45C24×D5Dic311yescharacteristic, complemented
46C24×D5C2×D311yescharacteristic
47C23×C4×C5Dic311yescharacteristic, complemented
48C22×D20—24no—
49C22×D20—28no—
50C22×D20Dic312yescomplemented
51C22×D20C2×D312yes—
52C2×C22⋊AGL(1,𝔽5)—34no—
53C2×C22⋊AGL(1,𝔽5)—64no—
54#3999—68no—
55#1781—151noSylow
56C23×C3×C5—24no—
57C23×C3×C5—22no—
58C23×C3×C5C22⋊C412yes—
59C23×C3×C5C22×C411yescharacteristic
60C22×C3×D5—28no—
61C22×C3×D5—28no—
62C22×C3×D5—22no—
63C22×C3×D5—22no—
64C22×C3×D5—22no—
65C22×C3×D5—28no—
66C22×C3×D5—42no—
67C22×C3×D5—44no—
68C22×C3×D5—42no—
69C22×C3×D5—416no—
70C22×C3×D5—416no—
71C22×C3×D5—44no—
72C22×C3×D5—44no—
73C22×C3×D5—44no—
74C22×C3×D5C22⋊C412yes—
75C22×C3×D5C22⋊C414yescomplemented
76C22×C3×D5C2×D412yes—
77C22×C3×D5C2×D414yes—
78C2×ΓL(1,𝔽16)—48no—
79C2×ΓL(1,𝔽16)—44no—
80C2×C3×C4×C5—28no—
81C2×C3×C4×C5—22no—
82C2×C3×C4×C5—22no—
83C2×C3×C4×C5C22⋊C414yescomplemented
84C3×D20—216no—
85C3×D20—216no—
86C25×C3—51no—
87C25×C3—51no—
88C22×C3×D4—52no—
89C22×C3×D4—52no—
90C22×C3×D4—108no—
91C22×C3×D4—104no—
92C23×C3×C4AGL(1,𝔽5)11yescharacteristic, complemented
93C2×#217—54no—
94C2×#217—104no—
95#834—108no—
96C24×C5C2×Dic311yescharacteristic
97C23×D5—22no—
98C23×D5—24no—
99C23×D5—24no—
100C23×D5—44no—
101C23×D5—44no—
102C23×D5C2×Dic311yescharacteristic
103C23×D5C2×Dic312yes—
104C23×D5#6714yes—
105C23×D5C22×D311yescharacteristic
106C22×AGL(1,𝔽5)—64no—
107C22×C4×C5—22no—
108C22×C4×C5—24no—
109C22×C4×C5C2×Dic312yes—
110C2×D20—232no—
111C2×D20—24no—
112C2×D20—24no—
113C2×D20—24no—
114C2×D20—24no—
115C2×D20—24no—
116C2×D20#6718yes—
117C22⋊AGL(1,𝔽5)—68no—
118C22⋊AGL(1,𝔽5)—128no—
119C23×D4—51no—
120C2×#99—154no—
121#379—152no—
122C22×C3×C5—28no—
123C22×C3×C5—22no—
124C22×C3×C5—22no—
125C22×C3×C5—22no—
126C22×C3×C5#9914yescomplemented
127C22×C3×C5C2×C22⋊C412yes—
128C22×C3×C5C2×C22⋊C411yescharacteristic
129C2×C3×D5—24no—
130C2×C3×D5—22no—
131C2×C3×D5—24no—
132C2×C3×D5—24no—
133C2×C3×D5—44no—
134C2×C3×D5—416no—
135C2×C3×D5—44no—
136C2×C3×D5—416no—
137C2×C3×D5—44no—
138C2×C3×D5—44no—
139C2×C3×D5—44no—
140C2×C3×D5#12014yes—
141ΓL(1,𝔽16)—84no—
142C3×C4×C5—24no—
143C24×C3—51no—
144C24×C3—54no—
145C24×C3—51no—
146C24×C3—52no—
147C24×C3—104no—
148C24×C3—104no—
149C24×C3—102no—
150C24×C3—204no—
151C24×C3—204no—
152C24×C3C2×AGL(1,𝔽5)11yescharacteristic
153C2×C3×D4—58no—
154C2×C3×D4—104no—
155C2×C3×D4—104no—
156C2×C3×D4—104no—
157C2×C3×D4—104no—
158C2×C3×D4—1032no—
159C2×C3×D4—104no—
160C22×C3×C4—22no—
161C22×C3×C4—24no—
162C22×C3×C4C2×AGL(1,𝔽5)12yes—
163C22×Dic3—104no—
164#217—108no—
165#217—208no—
166C23×C5—24no—
167C23×C5—22no—
168C23×C5#21712yes—
169C23×C5C22×Dic311yescharacteristic
170C22×D5—28no—
171C22×D5—28no—
172C22×D5—22no—
173C22×D5—22no—
174C22×D5—22no—
175C22×D5—28no—
176C22×D5—42no—
177C22×D5—44no—
178C22×D5—42no—
179C22×D5—416no—
180C22×D5—416no—
181C22×D5—44no—
182C22×D5—44no—
183C22×D5—44no—
184C22×D5#21712yes—
185C22×D5#21714yescomplemented
186C22×D5C2×#6712yes—
187C22×D5C2×#6714yes—
188C2×AGL(1,𝔽5)—128no—
189C2×AGL(1,𝔽5)—124no—
190C2×C4×C5—28no—
191C2×C4×C5—22no—
192C2×C4×C5—22no—
193C2×C4×C5#21714yescomplemented
194D20—216no—
195D20—216no—
196C25—51no—
197C25—51no—
198C22×D4—52no—
199C22×D4—52no—
200C22×D4—108no—
201C22×D4—104no—
202C23×C4ΓL(1,𝔽16)11yescharacteristic, complemented
203C2×C22⋊C4—154no—
204C2×C22⋊C4—304no—
205#99—308no—
206C2×C3×C5—24no—
207C2×C3×C5—22no—
208C2×C3×C5#37911yescharacteristic
209C2×C3×C5C2×#9912yes—
210C3×D5—24no—
211C3×D5—42no—
212C3×D5—44no—
213C23×C3—22no—
214C23×C3—24no—
215C23×C3—52no—
216C23×C3—54no—
217C23×C3—52no—
218C23×C3—54no—
219C23×C3—108no—
220C23×C3—108no—
221C23×C3—102no—
222C23×C3—102no—
223C23×C3—102no—
224C23×C3—108no—
225C23×C3—204no—
226C23×C3—202no—
227C23×C3—202no—
228C23×C3—204no—
229C23×C3—2016no—
230C23×C3—2016no—
231C23×C3—204no—
232C23×C3—204no—
233C23×C3C22⋊AGL(1,𝔽5)12yes—
234C23×C3C22×AGL(1,𝔽5)11yescharacteristic
235C3×D4—1016no—
236C3×D4—1016no—
237C2×C3×C4—28no—
238C2×C3×C4—22no—
239C2×C3×C4—22no—
240C2×C3×C4C22⋊AGL(1,𝔽5)14yescomplemented
241C2×Dic3—204no—
242C2×Dic3—208no—
243C22×C5—28no—
244C22×C5—22no—
245C22×C5—22no—
246C22×C5—22no—
247C22×C5#83414yescomplemented
248C22×C5C2×#21712yes—
249C22×C5C2×#21711yescharacteristic
250C2×D5—24no—
251C2×D5—24no—
252C2×D5—22no—
253C2×D5—24no—
254C2×D5—44no—
255C2×D5—416no—
256C2×D5—44no—
257C2×D5—416no—
258C2×D5—44no—
259C2×D5—44no—
260C2×D5—44no—
261C2×D5#95314yes—
262AGL(1,𝔽5)—244no—
263C4×C5—24no—
264C24—51no—
265C24—54no—
266C24—51no—
267C24—52no—
268C24—104no—
269C24—104no—
270C24—102no—
271C24—204no—
272C24—204no—
273C24C2×ΓL(1,𝔽16)11yescharacteristic
274C2×D4—58no—
275C2×D4—104no—
276C2×D4—104no—
277C2×D4—104no—
278C2×D4—104no—
279C2×D4—1032no—
280C2×D4—104no—
281C22×C4—22no—
282C22×C4—24no—
283C22×C4—304no—
284C22×C4C2×ΓL(1,𝔽16)12yes—
285C22⋊C4—308no—
286C22⋊C4—608no—
287C3×C5#178111yescharacteristic, complemented, Hall
288C22×C3—28no—
289C22×C3—22no—
290C22×C3—22no—
291C22×C3—22no—
292C22×C3—54no—
293C22×C3—104no—
294C22×C3—102no—
295C22×C3—104no—
296C22×C3—104no—
297C22×C3—204no—
298C22×C3—2016no—
299C22×C3—204no—
300C22×C3—204no—
301C22×C3—204no—
302C22×C3—2016no—
303C22×C3—204no—
304C22×C3#399914yescomplemented
305C22×C3C2×C22⋊AGL(1,𝔽5)12yes—
306C22×C3C2×C22⋊AGL(1,𝔽5)11yescharacteristic
307C3×C4—24no—
308Dic3—404no—
309C2×C5—24no—
310C2×C5—22no—
311C2×C5C2×#83412yes—
312C2×C5#524711yescharacteristic
313D5—24no—
314D5—42no—
315D5—44no—
316C23—24no—
317C23—22no—
318C23—52no—
319C23—54no—
320C23—52no—
321C23—54no—
322C23—108no—
323C23—108no—
324C23—108no—
325C23—102no—
326C23—102no—
327C23—102no—
328C23—204no—
329C23—202no—
330C23—202no—
331C23—204no—
332C23—2016no—
333C23—2016no—
334C23—204no—
335C23—204no—
336C23#676312yes—
337C23C22×ΓL(1,𝔽16)11yescharacteristic
338D4—1016no—
339D4—1016no—
340C2×C4—28no—
341C2×C4—22no—
342C2×C4—22no—
343C2×C4—604no—
344C2×C4—608no—
345C2×C4#676314yescomplemented
346C2×C3—22no—
347C2×C3—24no—
348C2×C3—104no—
349C2×C3—202no—
350C2×C3—204no—
351C2×C3C2×#399912yes—
352C2×C3#6580011yescharacteristic
353C5#7213011yescharacteristic, complemented, Sylow
354C22—28no—
355C22—22no—
356C22—22no—
357C22—22no—
358C22—54no—
359C22—104no—
360C22—102no—
361C22—104no—
362C22—104no—
363C22—204no—
364C22—2016no—
365C22—204no—
366C22—2016no—
367C22—204no—
368C22—204no—
369C22—204no—
370C22#9122814yescomplemented
371C22C2×#676312yes—
372C22C2×#676311yescharacteristic
373C4—24no—
374C4—1204no—
375C3#1060489011yescharacteristic, complemented, Sylow
376C2—24no—
377C2—22no—
378C2—104no—
379C2—202no—
380C2—204no—
381C2C2×#9122812yes—
382C2#1175245911yescharacteristic
subgroupG/H
centerC22C2×#6763
commutatorC23×C3×C5C22×C4
FrattiniC23C22×ΓL(1,𝔽16)
FittingC23×C3×C4×C5C4
socleC22×C3×C5C2×C22⋊C4
prime radicalC23×C3×D20C2
cosocleC23×C3×D5C23
hypercenterC23×C4ΓL(1,𝔽16)
hypocenterC3×C5#1781
LCSG ▷ C23×C3×C5 ▷ C2×C3×C5 ▷ C3×C5
UCSC1 ◁ C22 ◁ C24 ◁ C23×C4

Complements

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

NH ≅ G/Ntype
2. #11752459C2semidirect
3. C2×#91228C2semidirect
5. C24×C3×D5C4semidirect
7. C23×C3×C4×C5C4semidirect
10. C22×C3×D20C4semidirect
13. C2×#6763C22semidirect
45. C24×D5Dic3semidirect
47. C23×C4×C5Dic3semidirect
50. C22×D20Dic3semidirect
75. C22×C3×D5C22⋊C4semidirect
83. C2×C3×C4×C5C22⋊C4semidirect
92. C23×C3×C4AGL(1,𝔽5)semidirect
126. C22×C3×C5#99semidirect
185. C22×D5#217semidirect
193. C2×C4×C5#217semidirect
202. C23×C4ΓL(1,𝔽16)semidirect
240. C2×C3×C4C22⋊AGL(1,𝔽5)semidirect
247. C22×C5#834semidirect
287. C3×C5#1781semidirect
304. C22×C3#3999semidirect
345. C2×C4#6763semidirect
353. C5#72130semidirect
370. C22#91228semidirect
375. C3#10604890semidirect

Maximal subgroups

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

subgroupindex
C23×C3×D202
#117524592
C2×#912282
#106048903
#721305