Finite Group Databasebeta

← #14316636

Subgroups — #14316636

Order 1920.

708 conjugacy classes of subgroups; only the 79 normal subgroups are shown.

normal subgroups only

This group has 708 conjugacy classes of subgroups — too many to draw. Only its 79 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
1C3×C5×D42C211yesmaximal, characteristic, complemented
2#11751178C211yesmaximal, characteristic, complemented
3#11751133C212yesmaximal, complemented
4#11746366C211yesmaximal, characteristic, complemented
5#11746357C212yesmaximal, complemented
6#10602702—31nomaximal, Hall
7C22×C3×C5×D4—22no—
8#92095C2211yescharacteristic, complemented
9C3×C5×#127C2211yescharacteristic
10C3×C5×#121—22no—
11C3×C5×#120—21no—
12C3×C5×#120C2212yes—
13C3×C4×C5×D4—21no—
14#91842—22no—
15#91821—21no—
16#91118C2212yescomplemented
17#91117C2211yescharacteristic, complemented
18#91090—22no—
19#91089—21no—
20#71909—51nomaximal, Hall
21C5×D42D311yescharacteristic, complemented
22#65366—31no—
23#65321—32no—
24#64765—31no—
25#64756—32no—
26C24×C3×C5—21no—
27C24×C3×C5D412yescomplemented
28C2×C3×C5×D4—21no—
29C2×C3×C5×D4—22no—
30C2×C3×C5×D4—22no—
31C2×C3×C5×D4—22no—
32C2×C3×C5×D4—48no—
33C2×C3×C5×D4D411yescharacteristic, complemented
34C2×C3×C5×D4C2311yescharacteristic
35C22×C3×C4×C5—22no—
36C2×#1220—22no—
37C2×#1220—42no—
38#6819—41no—
39D4×D15—41no—
40#6817—21no—
41C3×C5×C4⋊C4—21no—
42C3×C5×C22⋊C4—21no—
43C3×C5×C22⋊C4—22no—
44C3×C5×C22⋊C4D412yescomplemented
45C3×C42×C5D411yescharacteristic, complemented
46#6719—22no—
47#6719—42no—
48#6716—41no—
49#6715—41no—
50#6710—21no—
51C3×D42D511yescharacteristic, complemented
52#5077—51no—
53#5032—52no—
54#4545—51no—
55#4536—52no—
56C22×C5×D4—22no—
57#4144—31no—
58C5×#127C2×D311yescharacteristic
59C5×#121—22no—
60C5×#120—21no—
61C5×#120C2×D312yes—
62C4×C5×D4—21no—
63#4099—62no—
64#4078—61no—
65#3966—32no—
66#3965—31no—
67#3938—62no—
68#3937—61no—
69#2196—151noSylow
70C23×C3×C5—22no—
71C23×C3×C5—21no—
72C23×C3×C5—42no—
73C23×C3×C5—44no—
74C23×C3×C5—44no—
75C23×C3×C5—81no—
76C23×C3×C5C2×D412yes—
77C22×D15—22no—
78C3×C5×D4—24no—
79C3×C5×D4—41no—
80C3×C5×D4—44no—
81C3×C5×D4—41no—
82C3×C5×D4—44no—
83C3×C5×D4—44no—
84C2×C3×C4×C5—21no—
85C2×C3×C4×C5—22no—
86C2×C3×C4×C5—44no—
87C2×C3×C4×C5C2×D411yescharacteristic
88#1220—42no—
89#1220—41no—
90#1220—84no—
91C2×Dic15—22no—
92C2×Dic15—42no—
93D60—21no—
94C4×D15—41no—
95Dic30—21no—
96C3⋊#168—41no—
97C22×C3×D4—22no—
98#1004—51no—
99C3×#127C2×D511yescharacteristic
100C3×#121—22no—
101C3×#120—21no—
102C3×#120C2×D512yes—
103C3×C4×D4—21no—
104#953—102no—
105#932—101no—
106#834—52no—
107#833—51no—
108#806—102no—
109#805—101no—
110C24×C5—21no—
111C24×C5#6712yescomplemented
112C2×C5×D4—21no—
113C2×C5×D4—22no—
114C2×C5×D4—22no—
115C2×C5×D4—22no—
116C2×C5×D4—48no—
117C2×C5×D4#6711yescharacteristic, complemented
118C2×C5×D4C22×D311yescharacteristic
119C22×C4×C5—22no—
120C2×C22⋊D5—62no—
121C2×C22⋊D5—122no—
122#711—121no—
123D4×D5—121no—
124#709—61no—
125C5×C4⋊C4—21no—
126C5×C22⋊C4—21no—
127C5×C22⋊C4—22no—
128C5×C22⋊C4#6712yescomplemented
129C42×C5#6711yescharacteristic, complemented
130C22⋊Dic5—62no—
131C22⋊Dic5—122no—
132#687—121no—
133#686—121no—
134#681—61no—
135D42D1511yescharacteristic, complemented
136B(4,𝔽2)—152no—
137#453—151no—
138#353—151no—
139AΣL(1,𝔽16)—152no—
140C22×C3×C5—24no—
141C22×C3×C5—22no—
142C22×C3×C5—42no—
143C22×C3×C5—42no—
144C22×C3×C5—42no—
145C22×C3×C5—44no—
146C22×C3×C5—42no—
147C22×C3×C5—41no—
148C22×C3×C5—82no—
149C22×C3×C5—84no—
150C22×C3×C5#12011yescharacteristic
151C2×D15—41no—
152C2×D15—82no—
153C3×C4×C5—22no—
154C3×C4×C5—21no—
155C3×C4×C5—42no—
156Dic15—41no—
157Dic15—82no—
158C24×C3—21no—
159C24×C3C22⋊D512yescomplemented
160C2×C3×D4—21no—
161C2×C3×D4—22no—
162C2×C3×D4—22no—
163C2×C3×D4—22no—
164C2×C3×D4—48no—
165C2×C3×D4C22⋊D511yescharacteristic, complemented
166C2×C3×D4C22×D511yescharacteristic
167C22×C3×C4—22no—
168C2×#67—102no—
169C2×#67—202no—
170#237—201no—
171D3×D4—201no—
172#235—101no—
173C3×C4⋊C4—21no—
174C3×C22⋊C4—21no—
175C3×C22⋊C4—22no—
176C3×C22⋊C4C22⋊D512yescomplemented
177C3×C42C22⋊D511yescharacteristic, complemented
178#217—102no—
179#217—202no—
180#214—201no—
181#213—201no—
182#208—101no—
183C23×C5—22no—
184C23×C5—21no—
185C23×C5—42no—
186C23×C5—44no—
187C23×C5—44no—
188C23×C5—81no—
189C23×C5C2×#6712yes—
190C22×D5—62no—
191C5×D4—24no—
192C5×D4—41no—
193C5×D4—41no—
194C5×D4—44no—
195C5×D4—44no—
196C5×D4—44no—
197C2×C4×C5—21no—
198C2×C4×C5—22no—
199C2×C4×C5—44no—
200C2×C4×C5C2×#6711yescharacteristic
201C22⋊D5—122no—
202C22⋊D5—121no—
203C22⋊D5—244no—
204C2×Dic5—62no—
205C2×Dic5—122no—
206D20—61no—
207C4×D5—121no—
208Dic10—61no—
209#168—121no—
210H(5,𝔽2)—151no—
211C22×D4—22no—
212AGL(1,ℤ/8)—301no—
213#127C2×D1511yescharacteristic
214#121—22no—
215#120—21no—
216#120—302no—
217#120C2×D1512yes—
218C4×D4—21no—
219#104—301no—
220#100—151no—
221#99—152no—
222#99—302no—
223C2×C3×C5—21no—
224C2×C3×C5—42no—
225C2×C3×C5—44no—
226C2×C3×C5—81no—
227C2×C3×C5B(4,𝔽2)11yescharacteristic
228D15—81no—
229C23×C3—22no—
230C23×C3—21no—
231C23×C3—42no—
232C23×C3—44no—
233C23×C3—44no—
234C23×C3—81no—
235C23×C3C2×C22⋊D512yes—
236C22×D3—102no—
237C3×D4—24no—
238C3×D4—41no—
239C3×D4—41no—
240C3×D4—44no—
241C3×D4—44no—
242C3×D4—44no—
243C2×C3×C4—21no—
244C2×C3×C4—22no—
245C2×C3×C4—44no—
246C2×C3×C4C2×C22⋊D511yescharacteristic
247#67—202no—
248#67—201no—
249#67—404no—
250C2×Dic3—102no—
251C2×Dic3—202no—
252D12—101no—
253C4×D3—201no—
254Dic6—101no—
255C3⋊C8—201no—
256C22×C5—24no—
257C22×C5—22no—
258C22×C5—42no—
259C22×C5—42no—
260C22×C5—42no—
261C22×C5—44no—
262C22×C5—42no—
263C22×C5—41no—
264C22×C5—82no—
265C22×C5—84no—
266C22×C5#95311yescharacteristic
267C2×D5—121no—
268C2×D5—242no—
269C4×C5—22no—
270C4×C5—21no—
271C4×C5—42no—
272Dic5—121no—
273Dic5—242no—
274C24—21no—
275C24#122012yescomplemented
276#41—301no—
277#41—601no—
278C2×D4—21no—
279C2×D4—22no—
280C2×D4—22no—
281C2×D4—22no—
282C2×D4—48no—
283C2×D4—302no—
284C2×D4—601no—
285C2×D4—602no—
286C2×D4#122011yescharacteristic, complemented
287C2×D4C22×D1511yescharacteristic
288C22×C4—22no—
289SD16—601no—
290D8—601no—
291M16—301no—
292C4⋊C4—21no—
293C22⋊C4—21no—
294C22⋊C4—22no—
295C22⋊C4—302no—
296C22⋊C4—602no—
297C22⋊C4#122012yescomplemented
298C42#122011yescharacteristic, complemented
299C3×C5#219611yescharacteristic, complemented, Hall
300C22×C3—24no—
301C22×C3—22no—
302C22×C3—42no—
303C22×C3—42no—
304C22×C3—42no—
305C22×C3—44no—
306C22×C3—42no—
307C22×C3—41no—
308C22×C3—82no—
309C22×C3—84no—
310C22×C3#409911yescharacteristic
311C2×D3—201no—
312C2×D3—402no—
313C3×C4—22no—
314C3×C4—21no—
315C3×C4—42no—
316Dic3—201no—
317Dic3—402no—
318C2×C5—21no—
319C2×C5—42no—
320C2×C5—44no—
321C2×C5—81no—
322C2×C5#524411yescharacteristic
323D5—241no—
324C23—22no—
325C23—21no—
326C23—42no—
327C23—44no—
328C23—44no—
329C23—81no—
330C23—302no—
331C23C2×#122012yes—
332Q8—301no—
333D4—24no—
334D4—41no—
335D4—41no—
336D4—44no—
337D4—44no—
338D4—44no—
339D4—301no—
340D4—602no—
341D4—601no—
342D4—1204no—
343C2×C4—21no—
344C2×C4—22no—
345C2×C4—44no—
346C2×C4—302no—
347C2×C4—601no—
348C2×C4—602no—
349C2×C4C2×#122011yescharacteristic
350C8—601no—
351C2×C3—21no—
352C2×C3—42no—
353C2×C3—44no—
354C2×C3—81no—
355C2×C3#6553311yescharacteristic
356D3—401no—
357C5#7190911yescharacteristic, complemented, Sylow
358C22—24no—
359C22—22no—
360C22—42no—
361C22—42no—
362C22—42no—
363C22—44no—
364C22—42no—
365C22—41no—
366C22—82no—
367C22—84no—
368C22—601no—
369C22—1202no—
370C22#9184211yescharacteristic
371C4—22no—
372C4—21no—
373C4—42no—
374C4—601no—
375C4—1202no—
376C3#1060270211yescharacteristic, complemented, Sylow
377C2—21no—
378C2—42no—
379C2—44no—
380C2—81no—
381C2—1201no—
382C2#1175134511yescharacteristic
subgroupG/H
centerC2#11751345
commutator, cosocleC2×C3×C5×D4C23
FrattiniC2×D4C22×D15
FittingC3×C5×D42C2
socleC2×C3×C5B(4,𝔽2)
hypercenterD42D15
hypocenterC3×C5#2196
LCSG ▷ C2×C3×C5×D4 ▷ C22×C3×C5 ▷ C2×C3×C5 ▷ C3×C5
UCSC1 ◁ C2 ◁ C22 ◁ C2×D4 ◁ D42

Complements

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

NH ≅ G/Ntype
1. C3×C5×D42C2semidirect
2. #11751178C2semidirect
3. #11751133C2semidirect
4. #11746366C2semidirect
5. #11746357C2semidirect
8. #92095C22semidirect
16. #91118C22semidirect
17. #91117C22semidirect
21. C5×D42D3semidirect
27. C24×C3×C5D4semidirect
33. C2×C3×C5×D4D4semidirect
44. C3×C5×C22⋊C4D4semidirect
45. C3×C42×C5D4semidirect
51. C3×D42D5semidirect
111. C24×C5#67semidirect
117. C2×C5×D4#67semidirect
128. C5×C22⋊C4#67semidirect
129. C42×C5#67semidirect
135. D42D15semidirect
159. C24×C3C22⋊D5semidirect
165. C2×C3×D4C22⋊D5semidirect
176. C3×C22⋊C4C22⋊D5semidirect
177. C3×C42C22⋊D5semidirect
275. C24#1220semidirect
286. C2×D4#1220semidirect
297. C22⋊C4#1220semidirect
298. C42#1220semidirect
299. C3×C5#2196semidirect
357. C5#71909semidirect
376. C3#10602702semidirect

Maximal subgroups

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

subgroupindex
C3×C5×D422
#117511782
#117511332
#117463662
#117463572
#106027023
#719095