Finite Group Databasebeta

← #14280997

Subgroups — #14280997

Order 1920.

624 conjugacy classes of subgroups; only the 125 normal subgroups are shown.

normal subgroups only

This group has 624 conjugacy classes of subgroups — too many to draw. Only its 125 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
1C2×C3×D5×C4⋊C4C211yesmaximal, characteristic
2#11752415C212yesmaximal
3#11746785C212yesmaximal, complemented
4#11746784C212yesmaximal, complemented
5#10604292—31nomaximal, Hall
6C22×C3×C4×D5C411yescharacteristic
7C22×C3×C4×D5C411yescharacteristic
8C22×C3×C4×D5C2211yescharacteristic
9C2×#6758—22no—
10C2×C3×C5×C4⋊C4C411yescharacteristic
11C2×C3×C4⋊Dic5C2211yescharacteristic
12C2×C3×#683C411yescharacteristic
13C2×C3×#683C2211yescharacteristic
14C3×D5×C4⋊C4—22no—
15C3×D5×C4⋊C4—22no—
16#91241C2214yes—
17#91232—22no—
18#71552—51nomaximal, Hall
19C2×D5×C4⋊C4D311yescharacteristic
20#65756—32no—
21#64914—32no—
22#64913—32no—
23C23×C3×D5C2×C411yescharacteristic
24C22×C3×C4×C5C2×C411yescharacteristic
25C22×C3×C4×C5C2×C411yescharacteristic
26C22×C3×C4×C5C2×C411yescharacteristic
27C22×C3×Dic5C2×C411yescharacteristic
28C22×C3×Dic5C2×C411yescharacteristic
29C22×C3×Dic5C2311yescharacteristic
30C2×C3×C4×D5—21no—
31C2×C3×C4×D5—21no—
32C2×C3×C4×D5—22no—
33C2×C3×C4×D5—22no—
34C2×C3×C4×D5—21no—
35C2×C3×C4×D5—21no—
36C2×C3×C4×D5—42no—
37C2×C3×C4×D5—42no—
38C2×C3×C4×D5D412yes—
39C2×C3×C4×D5D412yes—
40C2×C3⋊#170—24no—
41#6758—44no—
42C3×C5×C4⋊C4—22no—
43C3×C4⋊Dic5—22no—
44C3×#683—22no—
45C3×#683—22no—
46C22×C3×C4⋊C4—51no—
47#5207—52no—
48#4528—52no—
49#4528—52no—
50C22×C4×D5Dic311yescharacteristic
51C22×C4×D5Dic311yescharacteristic
52C22×C4×D5C2×D311yescharacteristic
53C2×#700—62no—
54C2×C5×C4⋊C4Dic311yescharacteristic
55C2×C4⋊Dic5C2×D311yescharacteristic
56C2×#683Dic311yescharacteristic
57C2×#683C2×D311yescharacteristic
58D5×C4⋊C4—22no—
59D5×C4⋊C4—22no—
60C22⋊#170—34no—
61#4003—62no—
62#1463—151noSylow
63C23×C3×C5C22×C411yescharacteristic
64C22×C3×D5—21no—
65C22×C3×D5—21no—
66C22×C3×D5—22no—
67C22×C3×D5—22no—
68C22×C3×D5C22⋊C411yescharacteristic
69C22×C3×D5C22⋊C411yescharacteristic
70C2×C3×C4×C5—21no—
71C2×C3×C4×C5—21no—
72C2×C3×C4×C5—21no—
73C2×C3×C4×C5—41no—
74C2×C3×C4×C5—41no—
75C2×C3×C4×C5C22⋊C412yes—
76C2×C3×C4×C5M1612yescomplemented
77C2×C3×Dic5—21no—
78C2×C3×Dic5—21no—
79C2×C3×Dic5—21no—
80C2×C3×Dic5—41no—
81C2×C3×Dic5—41no—
82C2×C3×Dic5M1612yescomplemented
83C2×C3×Dic5C2×D412yes—
84C3×C4×D5—24no—
85C3×C4×D5—24no—
86C3×C4×D5—24no—
87C3×C4×D5—42no—
88C3×C4×D5—42no—
89C3×C4×D5—42no—
90C3⋊#170—44no—
91C23×C3×C4—51no—
92C23×C3×C4—51no—
93C23×C3×C4—51no—
94C2×C3×C4⋊C4—51no—
95C2×C3×C4⋊C4—51no—
96C2×C3×C4⋊C4—51no—
97C2×C3×C4⋊C4—102no—
98C2×C3×C4⋊C4—102no—
99C2×C3×C4⋊C4AGL(1,𝔽5)11yescharacteristic
100C2×#208—102no—
101#830—54no—
102#830—102no—
103C23×D5C2×Dic311yescharacteristic
104C22×C4×C5C2×Dic311yescharacteristic
105C22×C4×C5C2×Dic311yescharacteristic
106C22×C4×C5C2×Dic311yescharacteristic
107C22×Dic5C2×Dic311yescharacteristic
108C22×Dic5C2×Dic311yescharacteristic
109C22×Dic5C22×D311yescharacteristic
110C2×C4×D5—21no—
111C2×C4×D5—21no—
112C2×C4×D5—22no—
113C2×C4×D5—22no—
114C2×C4×D5—21no—
115C2×C4×D5—21no—
116C2×C4×D5—42no—
117C2×C4×D5—42no—
118C2×C4×D5#6712yes—
119C2×C4×D5#6712yes—
120C2×#170—64no—
121#700—124no—
122C5×C4⋊C4—22no—
123C4⋊Dic5—22no—
124#683—22no—
125#683—22no—
126C22×C4⋊C4—51no—
127#407—152no—
128#324—152no—
129#324—152no—
130C22×C3×C5—21no—
131C22×C3×C5—21no—
132C22×C3×C5C2×C22⋊C411yescharacteristic
133C22×C3×C5C2×M1611yescharacteristic
134C22×C3×C5C2×M1611yescharacteristic
135C2×C3×D5—24no—
136C2×C3×D5—24no—
137C2×C3×D5—21no—
138C2×C3×D5—21no—
139C2×C3×D5—21no—
140C2×C3×D5—21no—
141C2×C3×D5#9912yes—
142C2×C3×D5#10112yes—
143C3×C4×C5—22no—
144C3×C4×C5—22no—
145C3×C4×C5—41no—
146C3×Dic5—22no—
147C3×Dic5—22no—
148C3×Dic5—41no—
149C24×C3—51no—
150C22×C3×C4—51no—
151C22×C3×C4—51no—
152C22×C3×C4—52no—
153C22×C3×C4—52no—
154C22×C3×C4—51no—
155C22×C3×C4—102no—
156C22×C3×C4—101no—
157C22×C3×C4—101no—
158C22×C3×C4—101no—
159C22×C3×C4—101no—
160C22×C3×C4—102no—
161C22×C3×C4—202no—
162C22×C3×C4—202no—
163C22×C3×C4C2×AGL(1,𝔽5)11yescharacteristic
164C22×C3×C4C2×AGL(1,𝔽5)11yescharacteristic
165C22×C3×C4C2×AGL(1,𝔽5)11yescharacteristic
166C3×C4⋊C4—22no—
167C3×C4⋊C4—102no—
168C3×C4⋊C4—102no—
169C3×C4⋊C4—102no—
170#208—204no—
171C2×C3⋊C8—104no—
172C23×C5C22×Dic311yescharacteristic
173C22×D5—21no—
174C22×D5—21no—
175C22×D5—22no—
176C22×D5—22no—
177C22×D5#21711yescharacteristic
178C22×D5#21711yescharacteristic
179C2×C4×C5—21no—
180C2×C4×C5—21no—
181C2×C4×C5—21no—
182C2×C4×C5—41no—
183C2×C4×C5—41no—
184C2×C4×C5#20812yescomplemented
185C2×C4×C5#21712yes—
186C2×Dic5—21no—
187C2×Dic5—21no—
188C2×Dic5—21no—
189C2×Dic5—41no—
190C2×Dic5—41no—
191C2×Dic5#20812yescomplemented
192C2×Dic5C2×#6712yes—
193C4×D5—24no—
194C4×D5—24no—
195C4×D5—24no—
196C4×D5—42no—
197C4×D5—42no—
198C4×D5—42no—
199#170—124no—
200C23×C4—51no—
201C23×C4—51no—
202C23×C4—51no—
203C2×M16—302no—
204C2×C4⋊C4—51no—
205C2×C4⋊C4—51no—
206C2×C4⋊C4—51no—
207C2×C4⋊C4—102no—
208C2×C4⋊C4—102no—
209C2×C4⋊C4ΓL(1,𝔽16)11yescharacteristic
210C22⋊C8—154no—
211C22⋊C8—302no—
212C2×C3×C5—21no—
213C2×C3×C5—21no—
214C2×C3×C5#40711yescharacteristic
215C2×C3×C5C2×#9911yescharacteristic
216C2×C3×C5C2×#10111yescharacteristic
217C3×D5—22no—
218C3×D5—22no—
219C23×C3—51no—
220C23×C3—51no—
221C23×C3—102no—
222C23×C3—102no—
223C23×C3—101no—
224C23×C3—101no—
225C23×C3C22×AGL(1,𝔽5)11yescharacteristic
226C2×C3×C4—21no—
227C2×C3×C4—21no—
228C2×C3×C4—21no—
229C2×C3×C4—41no—
230C2×C3×C4—41no—
231C2×C3×C4—52no—
232C2×C3×C4—52no—
233C2×C3×C4—104no—
234C2×C3×C4—104no—
235C2×C3×C4—104no—
236C2×C3×C4—101no—
237C2×C3×C4—101no—
238C2×C3×C4—101no—
239C2×C3×C4—202no—
240C2×C3×C4—202no—
241C2×C3×C4—202no—
242C2×C3×C4—201no—
243C2×C3×C4—201no—
244C2×C3×C4#70012yescomplemented
245C2×C3×C4C22⋊AGL(1,𝔽5)12yes—
246C3⋊C8—204no—
247C22×C5—21no—
248C22×C5—21no—
249C22×C5C2×#20811yescharacteristic
250C22×C5C2×#20811yescharacteristic
251C22×C5C2×#21711yescharacteristic
252C2×D5—24no—
253C2×D5—24no—
254C2×D5—21no—
255C2×D5—21no—
256C2×D5—21no—
257C2×D5—21no—
258C2×D5#83412yes—
259C2×D5#83612yes—
260C4×C5—22no—
261C4×C5—22no—
262C4×C5—41no—
263Dic5—22no—
264Dic5—22no—
265Dic5—41no—
266C24—51no—
267C22×C4—51no—
268C22×C4—51no—
269C22×C4—52no—
270C22×C4—52no—
271C22×C4—51no—
272C22×C4—102no—
273C22×C4—101no—
274C22×C4—101no—
275C22×C4—101no—
276C22×C4—101no—
277C22×C4—102no—
278C22×C4—202no—
279C22×C4—202no—
280C22×C4C2×ΓL(1,𝔽16)11yescharacteristic
281C22×C4C2×ΓL(1,𝔽16)11yescharacteristic
282C22×C4C2×ΓL(1,𝔽16)11yescharacteristic
283M16—604no—
284C2×C8—304no—
285C4⋊C4—22no—
286C4⋊C4—102no—
287C4⋊C4—102no—
288C4⋊C4—102no—
289C3×C5#146311yescharacteristic, complemented, Hall
290C22×C3—21no—
291C22×C3—21no—
292C22×C3—52no—
293C22×C3—52no—
294C22×C3—104no—
295C22×C3—104no—
296C22×C3—101no—
297C22×C3—101no—
298C22×C3—101no—
299C22×C3—101no—
300C22×C3C2×#70011yescharacteristic
301C22×C3D5⋊M1611yescharacteristic
302C22×C3C2×C22⋊AGL(1,𝔽5)11yescharacteristic
303C3×C4—22no—
304C3×C4—22no—
305C3×C4—41no—
306C3×C4—102no—
307C3×C4—102no—
308C3×C4—201no—
309C2×C5—21no—
310C2×C5—21no—
311C2×C5#520711yescharacteristic
312C2×C5C2×#83411yescharacteristic
313C2×C5C2×#83611yescharacteristic
314D5—22no—
315D5—22no—
316C23—51no—
317C23—51no—
318C23—102no—
319C23—102no—
320C23—101no—
321C23—101no—
322C23C22×ΓL(1,𝔽16)11yescharacteristic
323C2×C4—21no—
324C2×C4—21no—
325C2×C4—21no—
326C2×C4—41no—
327C2×C4—41no—
328C2×C4—52no—
329C2×C4—52no—
330C2×C4—104no—
331C2×C4—104no—
332C2×C4—104no—
333C2×C4—101no—
334C2×C4—101no—
335C2×C4—101no—
336C2×C4—202no—
337C2×C4—202no—
338C2×C4—202no—
339C2×C4—201no—
340C2×C4—201no—
341C2×C4#675812yescomplemented
342C2×C4#676312yes—
343C8—604no—
344C2×C3—21no—
345C2×C3—21no—
346C2×C3—102no—
347C2×C3—102no—
348C2×C3#6575611yescharacteristic
349C2×C3D5⋊#9911yescharacteristic
350C2×C3D5⋊#10111yescharacteristic
351C5#7155211yescharacteristic, complemented, Sylow
352C22—21no—
353C22—21no—
354C22—52no—
355C22—52no—
356C22—104no—
357C22—104no—
358C22—101no—
359C22—101no—
360C22—101no—
361C22—101no—
362C22C2×#675811yescharacteristic
363C22D5⋊#20811yescharacteristic
364C22C2×#676311yescharacteristic
365C4—22no—
366C4—22no—
367C4—41no—
368C4—102no—
369C4—102no—
370C4—201no—
371C3#1060429211yescharacteristic, complemented, Sylow
372C2—21no—
373C2—21no—
374C2—102no—
375C2—102no—
376C2#1175241511yescharacteristic
377C2D5⋊#83411yescharacteristic
378C2D5⋊#83611yescharacteristic
subgroupG/H
centerC22C2×#6763
commutatorC23×C3×C5C22×C4
FrattiniC23C22×ΓL(1,𝔽16)
FittingC2×C3×C5×C4⋊C4C4
socleC22×C3×C5C2×C22⋊C4
prime radicalC23×C3×D5C2×C4
cosocleC22×C3×Dic5C23
hypercenterC2×C4⋊C4ΓL(1,𝔽16)
hypocenterC3×C5#1463
LCSG ▷ C23×C3×C5 ▷ C2×C3×C5 ▷ C3×C5
UCSC1 ◁ C22 ◁ C22×C4 ◁ C2×C4⋊C4

Complements

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

NH ≅ G/Ntype
3. #11746785C2semidirect
4. #11746784C2semidirect
76. C2×C3×C4×C5M16semidirect
82. C2×C3×Dic5M16semidirect
184. C2×C4×C5#208semidirect
191. C2×Dic5#208semidirect
244. C2×C3×C4#700semidirect
289. C3×C5#1463semidirect
341. C2×C4#6758semidirect
351. C5#71552semidirect
371. C3#10604292semidirect

Maximal subgroups

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

subgroupindex
C2×C3×D5×C4⋊C42
#117524152
#117467852
#117467842
#106042923
#715525