Finite Group Databasebeta

← C2×C5×#5406 #14511187

Subgroups — C2×C5×#5406 #14511187

Order 1920.

514 conjugacy classes of subgroups; only the 76 normal subgroups are shown.

normal subgroups only

This group has 514 conjugacy classes of subgroups — too many to draw. Only its 76 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×#986C211yesmaximal, characteristic
2C2×C5×#984C211yesmaximal, characteristic, complemented
3C2×C5×#131⋊C3C211yesmaximal, characteristic, complemented
4C5×#5406C214yesmaximal, direct product factor
5C2×C5×#471—31nomaximal, Hall
6C2×C5×#41⋊C3C2211yescharacteristic
7C5×#986—21no—
8C5×#986C2212yes—
9C5×#984—21no—
10C5×#984C2212yescomplemented
11C2×C5×GL(2,𝔽3)—21no—
12C2×C5×#226—21no—
13C2×C5×#204—41nomaximal
14C5×#131⋊C3C2212yescomplemented
15C2×#5406C511yesmaximal, characteristic, direct product factor, Hall
16C2×C5×#137—31no—
17C2×C5×AGL(1,ℤ/8)—31no—
18C22×C5×SD16—31no—
19C2×C5×#131D311yescharacteristic, complemented
20C5×#471—34no—
21C5×#471—34no—
22C2×C5×#108—31no—
23C2×C5×#101—31no—
24C2×C5×#100—31no—
25C2×C5×D12—41no—
26C2×C5×Dic6—41no—
27C5×#41⋊C3D411yescharacteristic
28C5×#41⋊C3C2311yescharacteristic
29C2×C5×SL(2,𝔽3)D411yescharacteristic
30C5×GL(2,𝔽3)—22no—
31C5×#226—22no—
32C2×C3×C5×C8—41no—
33C5×#204—44no—
34C2×#986C2×C511yescharacteristic
35C2×#984C2×C511yescharacteristic, complemented
36C2×#131⋊C3C2×C511yescharacteristic, complemented
37#5406C2×C514yesdirect product factor
38C2×C5×#41C2×D311yescharacteristic
39C22×C5×Q8—31no—
40C22×C5×D4—31no—
41C5×#137—32no—
42C5×#137—32no—
43C5×#137—61no—
44C5×#137—61no—
45C5×AGL(1,ℤ/8)—32no—
46C5×AGL(1,ℤ/8)—32no—
47C5×AGL(1,ℤ/8)—61no—
48C5×AGL(1,ℤ/8)—61no—
49C2×C5×Q16—61no—
50C2×C5×SD16—34no—
51C2×C5×SD16—61no—
52C2×C5×SD16—62no—
53C2×C5×SD16—61no—
54C2×C5×SD16—61no—
55C2×C5×SD16—61no—
56C2×C5×D8—61no—
57C5×#131—32no—
58C5×#131C2×D312yescomplemented
59C2×C5×M16—31no—
60C2×C5×M16—31no—
61C2×C5×M16—31no—
62C22×C5×C8—31no—
63C5×#108—34no—
64C5×#101—34no—
65C5×#100—34no—
66C2×#471—31noSylow
67C22×C5×D3—81no—
68C2×C3×C4×C5—41no—
69C2×C5×Dic3—81no—
70C5×D12—42no—
71C5×D12—81no—
72C5×Dic6—42no—
73C5×Dic6—81no—
74C5×SL(2,𝔽3)C2×D411yescharacteristic
75C3×C5×C8—42no—
76C2×#41⋊C3C22×C511yescharacteristic
77#986—21no—
78#986C22×C512yes—
79#984—21no—
80#984C22×C512yescomplemented
81C2×GL(2,𝔽3)—21no—
82C2×#226—21no—
83C2×#204—41no—
84#131⋊C3C22×C512yescomplemented
85C24×C5—61no—
86C5×#41—31no—
87C5×#41—31no—
88C5×#41D1211yescharacteristic
89C5×#41C22×D311yescharacteristic
90C2×C5×Q8—32no—
91C2×C5×Q8—61no—
92C2×C5×Q8—61no—
93C2×C5×Q8—61no—
94C2×C5×Q8—61no—
95C2×C5×Q8—61no—
96C2×C5×Q8D1211yescharacteristic
97C2×C5×D4—31no—
98C2×C5×D4—32no—
99C2×C5×D4—61no—
100C2×C5×D4—61no—
101C2×C5×D4—61no—
102C2×C5×D4—61no—
103C2×C5×D4—61no—
104C22×C4×C5—31no—
105C22×C4×C5—61no—
106C5×Q16—62no—
107C5×Q16—62no—
108C5×SD16—64no—
109C5×SD16—62no—
110C5×SD16—62no—
111C5×SD16—62no—
112C5×SD16—62no—
113C5×SD16—64no—
114C5×D8—62no—
115C5×D8—62no—
116C5×M16—32no—
117C5×M16—32no—
118C5×M16—32no—
119C5×M16—61no—
120C5×M16—61no—
121C5×M16—61no—
122C2×C5×C8—31no—
123C2×C5×C8—32no—
124C2×C5×C8—32no—
125C2×C5×C8—61no—
126C2×C5×C8—61no—
127C2×C5×C8S411yescharacteristic
128C2×#137—31no—
129C2×AGL(1,ℤ/8)—31no—
130C22×SD16—31no—
131C2×#131C5×D311yescharacteristic, complemented
132#471—34no—
133#471—34no—
134C2×#108—31no—
135C2×#101—31no—
136C2×#100—31no—
137C22×C3×C5—41no—
138C2×C5×D3—82no—
139C2×C5×D3—162no—
140C3×C4×C5—41no—
141C3×C4×C5—41no—
142C5×Dic3—82no—
143C2×D12—41no—
144C2×Dic6—41no—
145#41⋊C3C5×D411yescharacteristic
146#41⋊C3C23×C511yescharacteristic
147C2×SL(2,𝔽3)C5×D411yescharacteristic
148GL(2,𝔽3)—22no—
149#226—22no—
150C2×C3×C8—41no—
151#204—44no—
152C23×C5—31no—
153C23×C5—62no—
154C23×C5—121no—
155C23×C5—121no—
156C23×C5—242no—
157C5×Q8—31no—
158C5×Q8—62no—
159C5×Q8—62no—
160C5×Q8—121no—
161C5×Q8—121no—
162C5×Q8C2×D1211yescharacteristic
163C5×D4—31no—
164C5×D4—31no—
165C5×D4—61no—
166C5×D4—62no—
167C5×D4—62no—
168C5×D4—121no—
169C5×D4—121no—
170C2×C4×C5—31no—
171C2×C4×C5—31no—
172C2×C4×C5—31no—
173C2×C4×C5—31no—
174C2×C4×C5—31no—
175C2×C4×C5—62no—
176C2×C4×C5—121no—
177C2×C4×C5—121no—
178C2×C4×C5C2×S411yescharacteristic
179C5×C8—32no—
180C5×C8—62no—
181C5×C8—62no—
182C5×C8C2×S412yes—
183C2×#41C2×C5×D311yescharacteristic
184C22×Q8—31no—
185C22×D4—31no—
186#137—32no—
187#137—32no—
188#137—61no—
189#137—61no—
190AGL(1,ℤ/8)—32no—
191AGL(1,ℤ/8)—32no—
192AGL(1,ℤ/8)—61no—
193AGL(1,ℤ/8)—61no—
194C2×Q16—61no—
195C2×SD16—34no—
196C2×SD16—61no—
197C2×SD16—62no—
198C2×SD16—61no—
199C2×SD16—61no—
200C2×SD16—61no—
201C2×D8—61no—
202#131—32no—
203#131C2×C5×D312yescomplemented
204C2×M16—31no—
205C2×M16—31no—
206C2×M16—31no—
207C22×C8—31no—
208#108—34no—
209#101—34no—
210#100—34no—
211C2×C3×C5—41no—
212C2×C3×C5—42no—
213C5×D3—162no—
214C22×D3—81no—
215C2×C3×C4—41no—
216C2×Dic3—81no—
217D12—42no—
218D12—81no—
219Dic6—42no—
220Dic6—81no—
221SL(2,𝔽3)C2×C5×D411yescharacteristic
222C3×C8—42no—
223C22×C5—31no—
224C22×C5—31no—
225C22×C5—62no—
226C22×C5—122no—
227C22×C5—242no—
228C22×C5—242no—
229C22×C5—242no—
230C22×C5C4⋊S411yescharacteristic
231C4×C5—31no—
232C4×C5—31no—
233C4×C5—122no—
234C4×C5C4⋊S411yescharacteristic
235C4×C5C22×S411yescharacteristic
236C24—61no—
237#41—31no—
238#41—31no—
239#41C5×D1211yescharacteristic
240#41C22×C5×D311yescharacteristic
241C2×Q8—32no—
242C2×Q8—61no—
243C2×Q8—61no—
244C2×Q8—61no—
245C2×Q8—61no—
246C2×Q8—61no—
247C2×Q8C5×D1211yescharacteristic
248C2×D4—31no—
249C2×D4—32no—
250C2×D4—61no—
251C2×D4—61no—
252C2×D4—61no—
253C2×D4—61no—
254C2×D4—61no—
255C22×C4—31no—
256C22×C4—61no—
257Q16—62no—
258Q16—62no—
259SD16—64no—
260SD16—62no—
261SD16—62no—
262SD16—62no—
263SD16—62no—
264SD16—64no—
265D8—62no—
266D8—62no—
267M16—32no—
268M16—32no—
269M16—32no—
270M16—61no—
271M16—61no—
272M16—61no—
273C2×C8—31no—
274C2×C8—32no—
275C2×C8—32no—
276C2×C8—61no—
277C2×C8—61no—
278C2×C8C5×S411yescharacteristic
279C3×C5—41noHall
280C22×C3—41no—
281C2×D3—82no—
282C2×D3—162no—
283C3×C4—41no—
284C3×C4—41no—
285Dic3—82no—
286C2×C5—61no—
287C2×C5—61no—
288C2×C5—242no—
289C2×C5#540612yesdirect product factor
290C2×C5C2×C4⋊S411yescharacteristic
291C23—31no—
292C23—62no—
293C23—121no—
294C23—121no—
295C23—242no—
296Q8—31no—
297Q8—62no—
298Q8—62no—
299Q8—121no—
300Q8—121no—
301Q8C2×C5×D1211yescharacteristic
302D4—31no—
303D4—31no—
304D4—61no—
305D4—62no—
306D4—62no—
307D4—121no—
308D4—121no—
309C2×C4—31no—
310C2×C4—31no—
311C2×C4—31no—
312C2×C4—31no—
313C2×C4—31no—
314C2×C4—62no—
315C2×C4—121no—
316C2×C4—121no—
317C2×C4C2×C5×S411yescharacteristic
318C8—32no—
319C8—62no—
320C8—62no—
321C8C2×C5×S412yes—
322C2×C3—41no—
323C2×C3—42no—
324D3—162no—
325C5C2×#540611yescharacteristic, direct product factor, Sylow
326C22—31no—
327C22—31no—
328C22—62no—
329C22—122no—
330C22—242no—
331C22—242no—
332C22—242no—
333C22C5×C4⋊S411yescharacteristic
334C4—31no—
335C4—31no—
336C4—122no—
337C4C5×C4⋊S411yescharacteristic
338C4C22×C5×S411yescharacteristic
339C3—41noSylow
340C2—61no—
341C2—61no—
342C2—242no—
343C2C5×#540612yesdirect product factor
344C2C2×C5×C4⋊S411yescharacteristic
subgroupG/H
center, socleC22×C5C4⋊S4
commutator, cosocle#41⋊C3C23×C5
FrattiniC4C22×C5×S4
FittingC2×C5×#131D3
prime radicalC2×C5×#986C2
hypercenterC2×C5×C8S4
hypocenterSL(2,𝔽3)C2×C5×D4
LCSG ▷ #41⋊C3 ▷ SL(2,𝔽3)
UCSC1 ◁ C22×C5 ◁ C2×C4×C5 ◁ C2×C5×C8

Complements

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

NH ≅ G/Ntype
2. C2×C5×#984C2semidirect
3. C2×C5×#131⋊C3C2semidirect
4. C5×#5406C2direct (N × H)
10. C5×#984C22semidirect
14. C5×#131⋊C3C22semidirect
15. C2×#5406C5direct (N × H)
19. C2×C5×#131D3semidirect
35. C2×#984C2×C5semidirect
36. C2×#131⋊C3C2×C5semidirect
37. #5406C2×C5direct (N × H)
58. C5×#131C2×D3semidirect
80. #984C22×C5semidirect
84. #131⋊C3C22×C5semidirect
131. C2×#131C5×D3semidirect
203. #131C2×C5×D3semidirect
289. C2×C5#5406direct (N × H)
325. C5C2×#5406direct (N × H)
343. C2C5×#5406direct (N × H)

Maximal subgroups

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

subgroupindex
C2×C5×#9862
C2×C5×#9842
C2×C5×#131⋊C32
C5×#54062
C2×C5×#4713
C2×C5×#2044
C2×#54065