Finite Group Databasebeta

← C5×#73052 #14510776

Subgroups — C5×#73052 #14510776

Order 1920.

366 conjugacy classes of subgroups; only the 64 normal subgroups are shown.

normal subgroups only

This group has 366 conjugacy classes of subgroups — too many to draw. Only its 64 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×C5×A4⋊Q8C211yesmaximal, characteristic
2C5×A4×C4⋊C4C211yesmaximal, characteristic
3C2×C5×A4⋊C8C211yesmaximal, characteristic
4C5×#1872—31nomaximal, Hall
5C2×C4×C5×A4—21no—
6C2×C4×C5×A4C2211yescharacteristic
7C2×C5×SL(2,ℤ/4)—21no—
8C5×A4⋊Q8—21no—
9C5×A4⋊Q8C412yescomplemented
10C5×A4⋊C8—21no—
11C5×#810—41nomaximal
12#73052C511yesmaximal, characteristic, direct product factor, Hall
13C2×C5×C22⋊Q8—31no—
14C22×C5×C4⋊C4D311yescharacteristic
15C2×C5×#103—31no—
16C2×C5×#103—31no—
17C2×C5×C22⋊C8—31no—
18C5×#340—31no—
19C5×#340—31no—
20C22×C5×A4D411yescharacteristic
21C2×C5×Dic6—41no—
22C4×C5×A4—41no—
23C4×C5×A4C2×C411yescharacteristic
24C4×C5×A4D411yescharacteristic
25C5×SL(2,ℤ/4)—22no—
26C3×C5×C4⋊C4—41no—
27C2×C5×C3⋊C8—41no—
28C2×A4⋊Q8C2×C511yescharacteristic
29A4×C4⋊C4C2×C511yescharacteristic
30C2×A4⋊C8C2×C511yescharacteristic
31C22×C5×Q8—31no—
32C23×C4×C5—21no—
33C23×C4×C5C2×D311yescharacteristic
34C22×C5×C8—31no—
35C22×C5×C8—31no—
36C5×C22⋊Q8—32no—
37C5×C22⋊Q8—32no—
38C5×C22⋊Q8—61no—
39C5×C22⋊Q8—61no—
40C2×C5×C4⋊C4—31no—
41C2×C5×C4⋊C4—31no—
42C2×C5×C4⋊C4—31no—
43C2×C5×C4⋊C4—61no—
44C2×C5×C4⋊C4—61no—
45C2×C5×C22⋊C4—61no—
46C5×#103—62no—
47C5×#103—61no—
48C5×#103—61no—
49C5×C22⋊C8—61no—
50C5×C22⋊C8—61no—
51#1872—31noSylow
52C2×C5×A4C22⋊C411yescharacteristic
53C2×C5×A4SD1611yescharacteristic
54C2×C5×A4Q1611yescharacteristic
55C2×C3×C4×C5—41no—
56C2×C3×C4×C5—81no—
57C2×C5×Dic3—81no—
58C5×Dic6—42no—
59C5×Dic6—81no—
60C5×C3⋊C8—81no—
61C2×C4×A4—21no—
62C2×C4×A4C22×C511yescharacteristic
63C2×SL(2,ℤ/4)—21no—
64A4⋊Q8—21no—
65A4⋊Q8C4×C512yescomplemented
66A4⋊C8—21no—
67#810—41no—
68C24×C5#6711yescharacteristic
69C2×C5×Q8—32no—
70C2×C5×Q8—61no—
71C2×C5×Q8—61no—
72C2×C5×Q8—61no—
73C2×C5×Q8—61no—
74C2×C5×Q8—61no—
75C22×C4×C5—31no—
76C22×C4×C5—31no—
77C22×C4×C5—31no—
78C22×C4×C5—31no—
79C22×C4×C5—41no—
80C22×C4×C5—61no—
81C22×C4×C5—61no—
82C22×C4×C5—61no—
83C22×C4×C5—61no—
84C22×C4×C5—121no—
85C22×C4×C5C4×D311yescharacteristic
86C22×C4×C5D1211yescharacteristic
87C2×C5×C8—61no—
88C2×C5×C8—61no—
89C2×C5×C8—61no—
90C2×C5×C8—61no—
91C2×C5×C8—61no—
92C2×C5×C8—61no—
93C5×C4⋊C4—32no—
94C5×C4⋊C4—32no—
95C5×C4⋊C4—31no—
96C5×C4⋊C4—61no—
97C5×C4⋊C4—61no—
98C5×C4⋊C4—62no—
99C5×C4⋊C4—62no—
100C5×C4⋊C4S411yescharacteristic
101C5×C22⋊C4—62no—
102C5×C22⋊C4—62no—
103C2×C22⋊Q8—31no—
104C22×C4⋊C4C5×D311yescharacteristic
105C2×#103—31no—
106C2×#103—31no—
107C2×C22⋊C8—31no—
108#340—31no—
109#340—31no—
110C22×C3×C5—41no—
111C5×A4#10311yescharacteristic, complemented
112C3×C4×C5—41no—
113C3×C4×C5—41no—
114C3×C4×C5—161no—
115C5×Dic3—82no—
116C22×A4C5×D411yescharacteristic
117C2×Dic6—41no—
118C4×A4—41no—
119C4×A4C2×C4×C511yescharacteristic
120C4×A4C5×D411yescharacteristic
121SL(2,ℤ/4)—22no—
122C3×C4⋊C4—41no—
123C2×C3⋊C8—41no—
124C23×C5—31no—
125C23×C5—31no—
126C23×C5—31no—
127C23×C5—31no—
128C23×C5(C2×D3)⋊C411yescharacteristic
129C23×C5#21411yescharacteristic
130C23×C5#21611yescharacteristic
131C5×Q8—62no—
132C5×Q8—62no—
133C5×Q8—121no—
134C5×Q8—121no—
135C2×C4×C5—21no—
136C2×C4×C5—31no—
137C2×C4×C5—31no—
138C2×C4×C5—31no—
139C2×C4×C5—31no—
140C2×C4×C5—31no—
141C2×C4×C5—31no—
142C2×C4×C5—31no—
143C2×C4×C5—61no—
144C2×C4×C5—62no—
145C2×C4×C5—62no—
146C2×C4×C5—61no—
147C2×C4×C5—121no—
148C2×C4×C5—121no—
149C2×C4×C5—121no—
150C2×C4×C5—121no—
151C2×C4×C5—121no—
152C2×C4×C5—121no—
153C2×C4×C5—121no—
154C2×C4×C5—121no—
155C2×C4×C5C2×S411yescharacteristic
156C5×C8—121no—
157C5×C8—121no—
158C22×Q8—31no—
159C23×C4—21no—
160C23×C4C2×C5×D311yescharacteristic
161C22×C8—31no—
162C22×C8—31no—
163C22⋊Q8—32no—
164C22⋊Q8—32no—
165C22⋊Q8—61no—
166C22⋊Q8—61no—
167C2×C4⋊C4—31no—
168C2×C4⋊C4—31no—
169C2×C4⋊C4—31no—
170C2×C4⋊C4—61no—
171C2×C4⋊C4—61no—
172C2×C22⋊C4—61no—
173#103—62no—
174#103—61no—
175#103—61no—
176C22⋊C8—61no—
177C22⋊C8—61no—
178C2×C3×C5—41no—
179C2×C3×C5—41no—
180C2×C3×C5—41no—
181C2×A4C5×C22⋊C411yescharacteristic
182C2×A4C5×SD1611yescharacteristic
183C2×A4C5×Q1611yescharacteristic
184C2×C3×C4—41no—
185C2×C3×C4—81no—
186C2×Dic3—81no—
187Dic6—42no—
188Dic6—81no—
189C3⋊C8—81no—
190C22×C5—31no—
191C22×C5—31no—
192C22×C5—31no—
193C22×C5—31no—
194C22×C5—31no—
195C22×C5—31no—
196C22×C5—31no—
197C22×C5—31no—
198C22×C5—31no—
199C22×C5—61no—
200C22×C5#81011yescharacteristic, complemented
201C22×C5GL(2,ℤ/4)11yescharacteristic
202C4×C5—31no—
203C4×C5—31no—
204C4×C5—41no—
205C4×C5—122no—
206C4×C5—122no—
207C4×C5—121no—
208C4×C5C4×S411yescharacteristic
209C4×C5C4⋊S411yescharacteristic
210C24C5×#6711yescharacteristic
211C2×Q8—32no—
212C2×Q8—61no—
213C2×Q8—61no—
214C2×Q8—61no—
215C2×Q8—61no—
216C2×Q8—61no—
217C22×C4—31no—
218C22×C4—31no—
219C22×C4—31no—
220C22×C4—31no—
221C22×C4—41no—
222C22×C4—61no—
223C22×C4—61no—
224C22×C4—61no—
225C22×C4—61no—
226C22×C4—121no—
227C22×C4C4×C5×D311yescharacteristic
228C22×C4C5×D1211yescharacteristic
229C2×C8—61no—
230C2×C8—61no—
231C2×C8—61no—
232C2×C8—61no—
233C2×C8—61no—
234C2×C8—61no—
235C4⋊C4—32no—
236C4⋊C4—32no—
237C4⋊C4—31no—
238C4⋊C4—61no—
239C4⋊C4—61no—
240C4⋊C4—62no—
241C4⋊C4—62no—
242C4⋊C4C5×S411yescharacteristic
243C22⋊C4—62no—
244C22⋊C4—62no—
245C3×C5—41noHall
246C22×C3—41no—
247A4C5×#10311yescharacteristic, complemented
248C3×C4—41no—
249C3×C4—41no—
250C3×C4—161no—
251Dic3—82no—
252C2×C5—31no—
253C2×C5—31no—
254C2×C5—31no—
255C2×C5—31no—
256C2×C5(C2×S4)⋊C411yescharacteristic
257C2×C5#541411yescharacteristic
258C2×C5#541611yescharacteristic
259C23—31no—
260C23—31no—
261C23—31no—
262C23—31no—
263C23C5×(C2×D3)⋊C411yescharacteristic
264C23C5×#21411yescharacteristic
265C23C5×#21611yescharacteristic
266Q8—62no—
267Q8—62no—
268Q8—121no—
269Q8—121no—
270C2×C4—21no—
271C2×C4—31no—
272C2×C4—31no—
273C2×C4—31no—
274C2×C4—31no—
275C2×C4—31no—
276C2×C4—31no—
277C2×C4—31no—
278C2×C4—61no—
279C2×C4—62no—
280C2×C4—62no—
281C2×C4—61no—
282C2×C4—121no—
283C2×C4—121no—
284C2×C4—121no—
285C2×C4—121no—
286C2×C4—121no—
287C2×C4—121no—
288C2×C4—121no—
289C2×C4—121no—
290C2×C4C2×C5×S411yescharacteristic
291C8—121no—
292C8—121no—
293C2×C3—41no—
294C2×C3—41no—
295C2×C3—41no—
296C5#7305211yescharacteristic, direct product factor, Sylow
297C22—31no—
298C22—31no—
299C22—31no—
300C22—31no—
301C22—31no—
302C22—31no—
303C22—31no—
304C22—31no—
305C22—31no—
306C22—61no—
307C22C5×#81011yescharacteristic, complemented
308C22C5×GL(2,ℤ/4)11yescharacteristic
309C4—31no—
310C4—31no—
311C4—41no—
312C4—122no—
313C4—122no—
314C4—121no—
315C4C4×C5×S411yescharacteristic
316C4C5×C4⋊S411yescharacteristic
317C3—41noSylow
318C2—31no—
319C2—31no—
320C2—31no—
321C2—31no—
322C2C5×(C2×S4)⋊C411yescharacteristic
323C2C5×#541411yescharacteristic
324C2C5×#541611yescharacteristic
subgroupG/H
centerC22×C5GL(2,ℤ/4)
commutatorC4×A4C2×C4×C5
FrattiniC2×C4C2×C5×S4
FittingC22×C5×C4⋊C4D3
socleC24×C5#67
prime radicalC22×C5×A4D4
cosocleC2×C4×A4C22×C5
hypercenterC5×C4⋊C4S4
hypocenterA4C5×#103
LCSG ▷ C4×A4 ▷ C2×A4 ▷ A4
UCSC1 ◁ C22×C5 ◁ C2×C4×C5 ◁ C5×C4⋊C4

Complements

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

NH ≅ G/Ntype
9. C5×A4⋊Q8C4semidirect
12. #73052C5direct (N × H)
65. A4⋊Q8C4×C5semidirect
111. C5×A4#103semidirect
200. C22×C5#810semidirect
247. A4C5×#103semidirect
296. C5#73052direct (N × H)
307. C22C5×#810semidirect

Maximal subgroups

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

subgroupindex
C2×C5×A4⋊Q82
C5×A4×C4⋊C42
C2×C5×A4⋊C82
C5×#18723
C5×#8104
#730525