Finite Group Databasebeta

← #10619193⋊C3 #14514229

Subgroups — #10619193⋊C3 #14514229

Order 1920.

762 conjugacy classes of subgroups; only the 91 normal subgroups are shown.

normal subgroups only

This group has 762 conjugacy classes of subgroups — too many to draw. Only its 91 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×C5×2−1+4⋊C3C211yesmaximal, characteristic, complemented
2C2×#4150⋊C3C211yesmaximal, characteristic, complemented
3Dic5×#41⋊C3C211yesmaximal, characteristic, complemented
4SL(2,𝔽3)×C22⋊D5C214yesmaximal, complemented
5#10619193C311yesmaximal, characteristic, complemented, Hall
6C5×2−1+4⋊C3—22no—
7C2×C5×#41⋊C3C2211yescharacteristic
8C22×C5×SL(2,𝔽3)C2212yes—
9#4150⋊C3—22no—
10C2×D5×SL(2,𝔽3)C2212yes—
11C3×#4086—41nomaximal
12Dic5×SL(2,𝔽3)C2212yes—
13#3499⋊C3—51nomaximal, Hall
14C2×C5×2−1+4C2×C311yescharacteristic, complemented
15C2×#4150C2×C311yescharacteristic, complemented
16#66166—31no—
17#41×Dic5C2×C311yescharacteristic, complemented
18#66159—31no—
19#66158—34no—
20#66150—31no—
21Q8×C22⋊D5C2×C314yescomplemented
22C2×#4090—31no—
23C2×C3×C5×D4—41no—
24C2×C3×C22⋊D5—44no—
25C2×C3×D20—41no—
26C5×#41⋊C3D412yescomplemented
27C2×C5×SL(2,𝔽3)—22no—
28C2×C5×SL(2,𝔽3)C2311yescharacteristic
29D5×SL(2,𝔽3)—22no—
30C3×C4×Dic5—41no—
31C2×2−1+4⋊C3—51no—
32C2×2−1+4⋊C3D511yescharacteristic, complemented
33D4×SL(2,𝔽3)—54no—
34C4×#41⋊C3—51no—
35C5×2−1+4—22no—
36C5×2−1+4—62no—
37C2×C5×#41—31no—
38C2×C5×#41—32no—
39C2×C5×#41C22×C311yescharacteristic
40C22×C5×Q8—31no—
41C22×C5×Q8C22×C312yes—
42#4150—22no—
43#4150—62no—
44C2×Q8×D5C22×C312yes—
45C2×#711—32no—
46C2×#709—31no—
47C22×Dic10—31no—
48#4093—31no—
49#4092—34no—
50Q8×Dic5C22×C312yes—
51#4090—32no—
52#4090—34no—
53#4090—31no—
54#4086A411yescharacteristic
55#4085—34no—
56#4082—31no—
57#4081—34no—
58D4×Dic5—32no—
59#4075—34no—
60C4×C22⋊D5—34no—
61#4072—31no—
62#4070—34no—
63C2×#683—34no—
64C2×C4×Dic5—31no—
65#3499—51noSylow
66C23×C3×C5—42no—
67C22×C3×D5—42no—
68C3×C5×D4—82no—
69C2×C3×C4×C5—41no—
70C3×C22⋊D5—88no—
71C2×C3×Dic5—42no—
72C3×D20—82no—
73C5×SL(2,𝔽3)C2×D411yescharacteristic
742−1+4⋊C3—22no—
752−1+4⋊C3—102no—
76C2×#41⋊C3C2×D511yescharacteristic
77C22×SL(2,𝔽3)—52no—
78C22×SL(2,𝔽3)C2×D512yes—
79C3×#127—201no—
80C4×SL(2,𝔽3)—52no—
81C5×#41—32no—
82C5×#41—62no—
83C5×#41—62no—
84C5×#41—64no—
85C5×#41—64no—
86C5×#41C3×D412yescomplemented
87C2×C5×Q8—22no—
88C2×C5×Q8—31no—
89C2×C5×Q8—32no—
90C2×C5×Q8—62no—
91C2×C5×Q8—62no—
92C2×C5×Q8—62no—
93C2×C5×Q8C23×C311yescharacteristic
94C2×C5×D4—31no—
95C2×C5×D4—32no—
96C2×C5×D4C2×A411yescharacteristic
97C22×C4×C5—31no—
98C22×C4×C5—32no—
99C22×C4×C5—32no—
100C2×C22⋊D5—32no—
101C2×C22⋊D5C2×A414yes—
102C22×Dic5—32no—
103C22×Dic5—32no—
104Q8×D5—22no—
105Q8×D5—62no—
106#711—64no—
107#711—64no—
108#709—62no—
109#709—62no—
110C2×D20C2×A411yescharacteristic
111C2×C4×D5—32no—
112C2×Dic10—31no—
113C2×Dic10—32no—
114C2×Dic10—62no—
115C2×Dic10—62no—
116C22⋊Dic5—32no—
117C22⋊Dic5—34no—
118#685—34no—
119C4⋊Dic5—32no—
120#683—38no—
121#683—34no—
122C4×Dic5—31no—
123C4×Dic5—32no—
124C4×Dic5C2×A411yescharacteristic
125C2×2−1+4—51no—
126C2×2−1+4C3×D511yescharacteristic, complemented
127D4×Q8—54no—
128#547—154no—
129#536—151no—
130#536—151no—
131#532—151no—
132C2×C4⋊Q8—151no—
133C4×#41—51no—
134C22×C3×C5—41no—
135C22×C3×C5—82no—
136C22×C3×C5—84no—
137C2×C3×D5—82no—
138C2×C3×D5—84no—
139C3×C4×C5—42no—
140C3×Dic5—44no—
141C2×C3×D4—41no—
142C2×C3×D4—201no—
143C2×C3×D4—204no—
144#41⋊C3C22⋊D512yescomplemented
145C2×SL(2,𝔽3)—22no—
146C2×SL(2,𝔽3)—102no—
147C2×SL(2,𝔽3)C22×D511yescharacteristic
148C3×C42—201no—
149C23×C5—31no—
150C23×C5C22×A412yes—
151C22×D5C22×A412yes—
152C5×Q8—31no—
153C5×Q8—62no—
154C5×Q8—62no—
155C5×Q8—62no—
156C5×Q8C2×C3×D411yescharacteristic
157C5×D4—22no—
158C5×D4—32no—
159C5×D4—32no—
160C5×D4—64no—
161C2×C4×C5—31no—
162C2×C4×C5—32no—
163C2×C4×C5—32no—
164C2×C4×C5—32no—
165C2×C4×C5—62no—
166C2×C4×C5—62no—
167C2×C4×C5—64no—
168C2×C4×C5C22×A411yescharacteristic
169C22⋊D5—28no—
170C22⋊D5—64no—
171C2×Dic5—32no—
172C2×Dic5—32no—
173C2×Dic5—64no—
174C2×Dic5—64no—
175C2×Dic5C22×A412yes—
176D20—22no—
177C4×D5—62no—
178C4×D5—62no—
179Dic10—62no—
180Dic10—62no—
181Dic10—62no—
1822−1+4—22no—
1832−1+4—62no—
1842−1+4—102no—
1852−1+4—302no—
186C2×#41—31no—
187C2×#41—32no—
188C2×#41—151no—
189C2×#41—152no—
190C2×#41C2×C3×D511yescharacteristic
191C22×Q8—31no—
192C22×Q8—52no—
193C22×Q8—151no—
194C22×Q8C2×C3×D512yes—
195C4⋊Q8—151no—
196C4⋊Q8—152no—
197C4⋊Q8—154no—
198#127—51no—
199#124—151no—
200#124—151no—
201#123—154no—
202#123—154no—
203C22⋊Q8—154no—
204C22⋊Q8—154no—
205#121—154no—
206C4×Q8—52no—
207C4×D4—154no—
208C4×D4—152no—
209#117—151no—
210C2×C4⋊C4—154no—
211C2×C42—151no—
212C2×C3×C5—41no—
213C2×C3×C5—42no—
214C2×C3×C5—162no—
215C3×D5—162no—
216C23×C3—42no—
217C23×C3—202no—
218C3×D4—82no—
219C3×D4—402no—
220C3×D4—408no—
221C2×C3×C4—41no—
222C2×C3×C4—202no—
223SL(2,𝔽3)C2×C22⋊D511yescharacteristic
224C22×C5—22no—
225C22×C5—24no—
226C22×C5—32no—
227C22×C5—62no—
228C22×C5C23×A411yescharacteristic
229C2×D5—22no—
230C2×D5—24no—
231C4×C5—31no—
232C4×C5—31no—
233C4×C5—62no—
234C4×C5D4×A412yes—
235Dic5—62no—
236Dic5—62no—
237Dic52−1+4⋊C314yescomplemented
238#41—32no—
239#41—62no—
240#41—62no—
241#41—64no—
242#41—64no—
243#41—302no—
244#41—302no—
245#41—304no—
246#41—304no—
247#41C3×C22⋊D512yescomplemented
248C2×Q8—22no—
249C2×Q8—31no—
250C2×Q8—32no—
251C2×Q8—62no—
252C2×Q8—62no—
253C2×Q8—62no—
254C2×Q8—102no—
255C2×Q8—152no—
256C2×Q8—151no—
257C2×Q8—302no—
258C2×Q8—302no—
259C2×Q8—302no—
260C2×Q8C22×C3×D511yescharacteristic
261C2×D4—31no—
262C2×D4—32no—
263C2×D4—54no—
264C2×D4—51no—
265C2×D4—152no—
266C2×D4D5×A411yescharacteristic
267C22×C4—31no—
268C22×C4—32no—
269C22×C4—32no—
270C22×C4—152no—
271C22×C4—152no—
272C22×C4—152no—
273C4⋊C4—154no—
274C4⋊C4—152no—
275C4⋊C4—158no—
276C22⋊C4—154no—
277C22⋊C4—152no—
278C22⋊C4—154no—
279C42—51no—
280C42—152no—
281C42—151no—
282C3×C5—41noHall
283C22×C3—41no—
284C22×C3—82no—
285C22×C3—84no—
286C22×C3—404no—
287C22×C3—402no—
288C3×C4—42no—
289C3×C4—204no—
290C2×C5—42no—
291C2×C5—62no—
292C2×C5C2×D4×A411yescharacteristic
293C2×C5C2×2−1+4⋊C312yes—
294D5—42no—
295C23—31no—
296C23—52no—
297C23C2×D5×A412yes—
298Q8—31no—
299Q8—62no—
300Q8—62no—
301Q8—62no—
302Q8—302no—
303Q8—302no—
304Q8—302no—
305Q8C2×C3×C22⋊D511yescharacteristic
306D4—22no—
307D4—32no—
308D4—32no—
309D4—64no—
310D4—108no—
311D4—102no—
312D4—304no—
313C2×C4—31no—
314C2×C4—32no—
315C2×C4—32no—
316C2×C4—32no—
317C2×C4—52no—
318C2×C4—62no—
319C2×C4—62no—
320C2×C4—64no—
321C2×C4—152no—
322C2×C4—152no—
323C2×C4—302no—
324C2×C4—302no—
325C2×C4—304no—
326C2×C4—304no—
327C2×C4C2×D5×A411yescharacteristic
328C2×C3—42no—
329C2×C3—41no—
330C2×C3—162no—
331C2×C3—802no—
332C5#3499⋊C311yescharacteristic, complemented, Sylow
333C22—24no—
334C22—22no—
335C22—32no—
336C22—62no—
337C22—104no—
338C22—102no—
339C22C22×D5×A411yescharacteristic
340C4—31no—
341C4—31no—
342C4—54no—
343C4—62no—
344C4—302no—
345C4—302no—
346C4A4×C22⋊D512yes—
347C3—41noSylow
348C2—42no—
349C2—62no—
350C2—202no—
351C2D5×2−1+4⋊C312yes—
352C2C2×A4×C22⋊D511yescharacteristic
subgroupG/H
center, FrattiniC22C22×D5×A4
commutator, cosocleC2×C5×Q8C23×C3
FittingC2×C5×2−1+4C2×C3
socleC22×C5C23×A4
hypercenterC2×D4D5×A4
hypocenterC5×Q8C2×C3×D4
LCSG ▷ C2×C5×Q8 ▷ C5×Q8
UCSC1 ◁ C22 ◁ C2×D4

Complements

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

NH ≅ G/Ntype
1. C2×C5×2−1+4⋊C3C2semidirect
2. C2×#4150⋊C3C2semidirect
3. Dic5×#41⋊C3C2semidirect
4. SL(2,𝔽3)×C22⋊D5C2semidirect
5. #10619193C3semidirect
14. C2×C5×2−1+4C2×C3semidirect
15. C2×#4150C2×C3semidirect
17. #41×Dic5C2×C3semidirect
21. Q8×C22⋊D5C2×C3semidirect
26. C5×#41⋊C3D4semidirect
32. C2×2−1+4⋊C3D5semidirect
86. C5×#41C3×D4semidirect
126. C2×2−1+4C3×D5semidirect
144. #41⋊C3C22⋊D5semidirect
237. Dic52−1+4⋊C3semidirect
247. #41C3×C22⋊D5semidirect
332. C5#3499⋊C3semidirect

Maximal subgroups

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

subgroupindex
C2×C5×2−1+4⋊C32
C2×#4150⋊C32
Dic5×#41⋊C32
SL(2,𝔽3)×C22⋊D52
#106191933
C3×#40864
#3499⋊C35