Finite Group Databasebeta

← #14296886

Subgroups — #14296886

Order 1920.

1224 conjugacy classes of subgroups; only the 130 normal subgroups are shown.

normal subgroups only

This group has 1224 conjugacy classes of subgroups — too many to draw. Only its 130 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×#6773C211yesmaximal, characteristic
2#11752423C211yesmaximal, characteristic, complemented
3C3×#65764C211yesmaximal, characteristic, complemented
4C2×#91173C214yesmaximal, complemented
5#10604981—31nomaximal, Hall
6C22×D60C411yescharacteristic, complemented
7C22×C5×Dic6C411yescharacteristic, complemented
8C22×C3×C4×D5C2211yescharacteristic
9C22×#1202C2212yes—
10C22×#1200C412yescomplemented
11C2×#6773—22no—
12C2×#6773—22no—
13C2×#6773—28no—
14C2×#6763C2212yes—
15C2×C3×C22⋊AGL(1,𝔽5)C2212yes—
16#91235—22no—
17C3×(C2×AGL(1,𝔽5))⋊C4—22no—
18#91173—28no—
19#69537—51nomaximal, Hall
20C22×#713—31no—
21#65764—31no—
22#65764D311yescharacteristic, complemented
23C2×#3999—34no—
24C23×D15C2×C412yes—
25C23×C3×D5C2311yescharacteristic
26C22×ΓL(1,𝔽16)—22no—
27C22×C3×AGL(1,𝔽5)—22no—
28C22×C3×C4×C5C2×C411yescharacteristic
29C2×D60—24no—
30C2×D60—22no—
31C22×C5×Dic3C2×C412yes—
32C2×C5×Dic6—22no—
33C2×C5×Dic6—24no—
34C22×C3×Dic5C2×C411yescharacteristic
35C2×C3×C4×D5—24no—
36C2×C3×C4×D5D412yes—
37C2×C3×C4×D5D412yes—
38C2×#1202—22no—
39C2×#1202—24no—
40C2×#1202—22no—
41C2×#1202D418yes—
42C2×#1200—24no—
43C2×#1200—24no—
44C2×#1200—24no—
45#6773—216no—
46#6773—216no—
47#6763—24no—
48C3×C22⋊AGL(1,𝔽5)—24no—
49C22×#235—51no—
50C3×#381—51no—
51#5211—51no—
52C2×#806—54no—
53C22×C5×Q8—31no—
54C2×#713—68no—
55C2×#713—62no—
56C2×#713—62no—
57C22×D20—32no—
58C22×D20—31no—
59C22×C4×D5—32no—
60C22×C4×D5C2×D311yescharacteristic
61C2×C22⋊AGL(1,𝔽5)—32no—
62C2×C22⋊AGL(1,𝔽5)C2×D312yes—
63(C2×AGL(1,𝔽5))⋊C4—22no—
64(C2×AGL(1,𝔽5))⋊C4—62no—
65#3999—68no—
66#1782—151noSylow
67C23×C3×C5C22×C411yescharacteristic
68C22×D15—22no—
69C22×D15—22no—
70C22×D15—44no—
71C22×D15C22⋊C414yes—
72C22×C3×D5—42no—
73C22×C3×D5C2×D411yescharacteristic
74C22×C3×D5C2×D411yescharacteristic
75C22×C3×D5C2×D414yes—
76C2×ΓL(1,𝔽16)—44no—
77C2×ΓL(1,𝔽16)—42no—
78C2×C3×AGL(1,𝔽5)—44no—
79C2×C3×AGL(1,𝔽5)—42no—
80C2×C3×C4×C5—22no—
81C2×C3×C4×C5C22⋊C412yes—
82D60—28no—
83C2×C5×Dic3—22no—
84C2×C5×Dic3—22no—
85C2×C5×Dic3C22⋊C414yes—
86C5×Dic6—28no—
87C2×C3×Dic5—22no—
88C2×C3×Dic5C22⋊C412yes—
89C3×C4×D5—24no—
90C3×C4×D5—24no—
91#1202—28no—
92#1202—28no—
93#1200—216no—
94C23×C3×C4—51no—
95C22×#67—52no—
96C2×#235—108no—
97C2×#235—102no—
98C2×#235—102no—
99C22×D12—51no—
100C22×C4×D3—52no—
101C22×Dic6AGL(1,𝔽5)11yescharacteristic, complemented
102C2×C3×C22⋊C4—52no—
103C2×#217—52no—
104C3×#95—102no—
105C3⋊#95—102no—
106#806—108no—
107C23×D5—32no—
108C23×D5C22×D311yescharacteristic
109C22×AGL(1,𝔽5)—22no—
110C22×AGL(1,𝔽5)—62no—
111C2×C5×Q8—64no—
112C2×C5×Q8—62no—
113C22×C4×C5—32no—
114C22×C4×C5C4×D311yescharacteristic
115C22×Dic5C4×D311yescharacteristic
116#713—616no—
117#713—616no—
118C2×D20—38no—
119C2×D20—64no—
120C2×D20—62no—
121C2×D20—64no—
122C2×D20—62no—
123C2×D20—62no—
124C2×C4×D5—24no—
125C2×C4×D5—64no—
126C2×C4×D5—64no—
127C2×C4×D5—64no—
128C2×C4×D5D1212yes—
129C2×C4×D5#6712yes—
130C22⋊AGL(1,𝔽5)—24no—
131C22⋊AGL(1,𝔽5)—64no—
132C22×#41—151no—
133C2×#99—154no—
134#381—51no—
135#381—151no—
136C22×C3×C5—22no—
137C22×C3×C5C2×C22⋊C412yes—
138C22×C3×C5C2×C22⋊C411yescharacteristic
139C2×D15—24no—
140C2×D15—44no—
141C2×D15—44no—
142C2×D15—44no—
143C2×C3×D5—22no—
144C2×C3×D5—22no—
145C2×C3×D5—44no—
146C2×C3×D5#12012yes—
147C2×C3×D5#12012yes—
148ΓL(1,𝔽16)—82no—
149C3×AGL(1,𝔽5)—82no—
150C3×C4×C5—22no—
151C3×Dic5—22no—
152C5×Dic3—24no—
153C24×C3—51no—
154C23×D3—52no—
155C22×C3×C4—51no—
156C22×C3×C4—52no—
157C22×C3×C4—52no—
158C22×C3×C4—104no—
159C22×C3×C4—102no—
160C22×C3×C4C2×AGL(1,𝔽5)11yescharacteristic
161C2×#67—58no—
162C2×#67—102no—
163C2×#67—102no—
164C2×#67—104no—
165C22×Dic3—102no—
166C22×Dic3C2×AGL(1,𝔽5)12yes—
167#235—1016no—
168#235—1016no—
169C2×D12—104no—
170C2×D12—102no—
171C2×C4×D3—104no—
172C2×C4×D3—104no—
173C2×C4×D3—104no—
174C2×Dic6—22no—
175C2×Dic6—24no—
176C3×C22⋊C4—104no—
177#217—104no—
178C23×C5C2×C4×D311yescharacteristic
179C22×D5—34no—
180C22×D5—42no—
181C22×D5—62no—
182C22×D5—62no—
183C22×D5—124no—
184C22×D5C2×D1211yescharacteristic
185C22×D5D3×D414yes—
186C22×D5C2×#6711yescharacteristic
187C2×AGL(1,𝔽5)—44no—
188C2×AGL(1,𝔽5)—42no—
189C2×AGL(1,𝔽5)—124no—
190C2×AGL(1,𝔽5)—122no—
191C5×Q8—68no—
192C2×C4×C5—22no—
193C2×C4×C5—34no—
194C2×C4×C5—62no—
195C2×C4×C5—62no—
196C2×C4×C5(C2×D3)⋊C412yes—
197C2×Dic5—22no—
198C2×Dic5(C2×D3)⋊C412yes—
199D20—68no—
200D20—68no—
201D20—68no—
202C4×D5—24no—
203C4×D5—24no—
204C4×D5—616no—
205C2×#41—302no—
206C2×#41—302no—
207C2×#41—308no—
208C22×Q8—31no—
209C22×D4—152no—
210C22×D4—151no—
211C23×C4—51no—
212C23×C4—152no—
213C2×C22⋊C4—52no—
214C2×C22⋊C4—152no—
215#99—308no—
216#95—102no—
217#95—302no—
218C2×C3×C5—22no—
219C2×C3×C5#37911yescharacteristic
220C2×C3×C5#41012yes—
221D15—44no—
222C3×D5—22no—
223C3×D5—22no—
224C23×C3—51no—
225C23×C3—51no—
226C23×C3—54no—
227C23×C3—202no—
228C23×C3C22×AGL(1,𝔽5)11yescharacteristic
229C22×D3—54no—
230C22×D3—102no—
231C22×D3—102no—
232C22×D3—204no—
233C2×C3×C4—22no—
234C2×C3×C4—52no—
235C2×C3×C4—104no—
236C2×C3×C4—104no—
237C2×C3×C4—102no—
238C2×C3×C4—202no—
239C2×C3×C4—204no—
240C2×C3×C4C22⋊AGL(1,𝔽5)12yes—
241#67—108no—
242#67—108no—
243C2×Dic3—22no—
244C2×Dic3—22no—
245C2×Dic3—202no—
246C2×Dic3—204no—
247C2×Dic3C22⋊AGL(1,𝔽5)14yes—
248D12—108no—
249C4×D3—1016no—
250Dic6—28no—
251C22×C5—22no—
252C22×C5D3×C22⋊C412yes—
253C22×C5C2×(C2×D3)⋊C411yescharacteristic
254C2×D5—22no—
255C2×D5—22no—
256C2×D5—44no—
257C2×D5—64no—
258C2×D5—124no—
259C2×D5—124no—
260C2×D5—124no—
261C2×D5#88212yes—
262C2×D5#93712yes—
263AGL(1,𝔽5)—82no—
264AGL(1,𝔽5)—242no—
265C4×C5—22no—
266C4×C5—64no—
267Dic5—22no—
268C24—51no—
269C24—152no—
270#41—3016no—
271#41—3016no—
272C2×Q8—64no—
273C2×Q8—62no—
274C2×D4—158no—
275C2×D4—304no—
276C2×D4—302no—
277C2×D4—302no—
278C2×D4—304no—
279C2×D4—302no—
280C22×C4—32no—
281C22×C4—51no—
282C22×C4—52no—
283C22×C4—52no—
284C22×C4—104no—
285C22×C4—102no—
286C22×C4—304no—
287C22×C4—304no—
288C22×C4—304no—
289C22×C4—302no—
290C22×C4D3×AGL(1,𝔽5)11yescharacteristic
291C22⋊C4—104no—
292C22⋊C4—304no—
293C3×C5#178211yescharacteristic, complemented, Hall
294C22×C3—22no—
295C22×C3—52no—
296C22×C3—52no—
297C22×C3—102no—
298C22×C3—102no—
299C22×C3—204no—
300C22×C3C2×C22⋊AGL(1,𝔽5)11yescharacteristic
301C22×C3C2×C22⋊AGL(1,𝔽5)12yes—
302C2×D3—104no—
303C2×D3—204no—
304C2×D3—204no—
305C2×D3—204no—
306C3×C4—22no—
307C3×C4—102no—
308C3×C4—402no—
309Dic3—24no—
310Dic3—402no—
311C2×C5—22no—
312C2×C5#495511yescharacteristic
313C2×C5#500612yes—
314D5—22no—
315D5—22no—
316D5—124no—
317C23—51no—
318C23—51no—
319C23—54no—
320C23—154no—
321C23—202no—
322C23—302no—
323C23—302no—
324C23—604no—
325C23C2×D3×AGL(1,𝔽5)11yescharacteristic
326Q8—68no—
327D4—308no—
328D4—308no—
329D4—308no—
330C2×C4—22no—
331C2×C4—34no—
332C2×C4—52no—
333C2×C4—62no—
334C2×C4—62no—
335C2×C4—104no—
336C2×C4—104no—
337C2×C4—102no—
338C2×C4—202no—
339C2×C4—204no—
340C2×C4—3016no—
341C2×C4—602no—
342C2×C4—604no—
343C2×C4(C2×D3)⋊AGL(1,𝔽5)12yes—
344C2×C3—22no—
345C2×C3—102no—
346C2×C3—102no—
347C2×C3#6578512yes—
348C2×C3#6580011yescharacteristic
349D3—204no—
350C5#6953711yescharacteristic, complemented, Sylow
351C22—22no—
352C22—52no—
353C22—52no—
354C22—102no—
355C22—102no—
356C22—204no—
357C22—304no—
358C22—604no—
359C22—604no—
360C22—604no—
361C22C2×(C2×D3)⋊AGL(1,𝔽5)11yescharacteristic
362C22D3×C22⋊AGL(1,𝔽5)12yes—
363C4—22no—
364C4—64no—
365C4—102no—
366C4—402no—
367C4—1202no—
368C3#1060498111yescharacteristic, complemented, Sylow
369C2—22no—
370C2—102no—
371C2—102no—
372C2—604no—
373C2#1175181712yes—
374C2#1175202511yescharacteristic
subgroupG/H
centerC22C2×(C2×D3)⋊AGL(1,𝔽5)
commutatorC23×C3×C5C22×C4
FrattiniC23C2×D3×AGL(1,𝔽5)
FittingC22×C3×C4×C5C2×C4
socleC22×C3×C5C2×C22⋊C4
prime radicalC22×#6773C2
cosocleC23×C3×D5C23
hypercenterC22×C4D3×AGL(1,𝔽5)
hypocenterC3×C5#1782
LCSG ▷ C23×C3×C5 ▷ C2×C3×C5 ▷ C3×C5
UCSC1 ◁ C22 ◁ C22×C4

Complements

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

NH ≅ G/Ntype
2. #11752423C2semidirect
3. C3×#65764C2semidirect
4. C2×#91173C2semidirect
6. C22×D60C4semidirect
7. C22×C5×Dic6C4semidirect
10. C22×#1200C4semidirect
22. #65764D3semidirect
101. C22×Dic6AGL(1,𝔽5)semidirect
293. C3×C5#1782semidirect
350. C5#69537semidirect
368. C3#10604981semidirect

Maximal subgroups

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

subgroupindex
C22×#67732
#117524232
C3×#657642
C2×#911732
#106049813
#695375