Finite Group Databasebeta

← #10617705⋊C3 #14514022

Subgroups — #10617705⋊C3 #14514022

Order 1920.

554 conjugacy classes of subgroups; only the 83 normal subgroups are shown.

normal subgroups only

This group has 554 conjugacy classes of subgroups — too many to draw. Only its 83 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×C4×C5×SL(2,𝔽3)C211yesmaximal, characteristic, complemented
2SL(2,𝔽3)×C22⋊D5C212yesmaximal, complemented
3#65906⋊C3C212yesmaximal, complemented
4SL(2,𝔽3)×D20C211yesmaximal, characteristic, complemented
5#65904⋊C3C211yesmaximal, characteristic, complemented
6#10617705C311yesmaximal, characteristic, complemented, Hall
7C22×C5×SL(2,𝔽3)C2211yescharacteristic
8C2×D5×SL(2,𝔽3)C2212yes—
9C3×#4076—41nomaximal
10C4×C5×SL(2,𝔽3)C2211yescharacteristic
11C4×C5×SL(2,𝔽3)C2211yescharacteristic
12Dic5×SL(2,𝔽3)C2212yes—
13#3523⋊C3—51nomaximal, Hall
14C2×C4×C5×Q8C2×C311yescharacteristic, complemented
15Q8×C22⋊D5C2×C312yescomplemented
16#65918—31no—
17#65917—32no—
18Q8×D20C2×C311yescharacteristic, complemented
19#65906C2×C312yescomplemented
20#65904C2×C311yescharacteristic, complemented
21#65901—31no—
22#65847—32no—
23#65813—31no—
24#65804—31no—
25C22×C3×C4×C5—41no—
26C2×C3×C22⋊D5—42no—
27C2×C3×D20—41no—
28C2×C5×SL(2,𝔽3)—21no—
29C2×C5×SL(2,𝔽3)C2311yescharacteristic
30D5×SL(2,𝔽3)—22no—
31C3×#685—42no—
32C3×C4⋊Dic5—41no—
33#557⋊C3—51no—
34D4×SL(2,𝔽3)—52no—
35D4×SL(2,𝔽3)—51no—
36#519⋊C3—52no—
37C2×C4×SL(2,𝔽3)D511yescharacteristic, complemented
38C22×C5×Q8C22×C311yescharacteristic
39C2×Q8×D5C22×C312yes—
40C4×C5×Q8—31no—
41C4×C5×Q8—31no—
42C4×C5×Q8C22×C311yescharacteristic
43C4×C5×Q8C22×C311yescharacteristic
44C2×C5×C4⋊C4—31no—
45C2×C42×C5—31no—
46#4092—32no—
47Q8×Dic5C22×C312yes—
48#4090—32no—
49#4076A411yescharacteristic
50#4075—32no—
51C4×C22⋊D5—32no—
52#4072—32no—
53#4070—31no—
54#4070—32no—
55#4044—34no—
56#4043—32no—
57#4038—32no—
58#4036—32no—
59#4035—34no—
60#4033—32no—
61#4022—32no—
62#4021—31no—
63C4×D20—31no—
64#4018—32no—
65#4016—32no—
66#4016—31no—
67#4015—31no—
68#4015—31no—
69C4×Dic10—32no—
70C4×Dic10—31no—
71C4×Dic10—32no—
72#3523—51noSylow
73C23×C3×C5—41no—
74C22×C3×D5—42no—
75C2×C3×C4×C5—41no—
76C2×C3×C4×C5—41no—
77C2×C3×C4×C5—82no—
78C3×C22⋊D5—84no—
79C2×C3×Dic5—42no—
80C3×D20—82no—
81C5×SL(2,𝔽3)#4111yescharacteristic
82C22×SL(2,𝔽3)—52no—
83C22×SL(2,𝔽3)C2×D511yescharacteristic
84C3×#121—201no—
85C4×SL(2,𝔽3)—52no—
86C4×SL(2,𝔽3)C2×D511yescharacteristic
87C4×SL(2,𝔽3)C2×D511yescharacteristic
88C2×C5×Q8—21no—
89C2×C5×Q8—31no—
90C2×C5×Q8—61no—
91C2×C5×Q8C23×C311yescharacteristic
92C22×C4×C5—31no—
93C22×C4×C5—31no—
94C22×C4×C5C2×A411yescharacteristic
95C2×C22⋊D5C2×A412yes—
96Q8×D5—22no—
97Q8×D5—62no—
98C2×D20C2×A411yescharacteristic
99C2×C4×D5—32no—
100C2×Dic10—31no—
101C2×Dic10—32no—
102C5×C4⋊C4—31no—
103C5×C4⋊C4—31no—
104C5×C4⋊C4—32no—
105C42×C5—31no—
106C42×C5—31no—
107C42×C5—31no—
108C42×C5—31no—
109C22⋊Dic5—32no—
110#685—32no—
111#685—32no—
112#685—32no—
113#685C2×A412yes—
114C4⋊Dic5—32no—
115C4⋊Dic5—32no—
116C4⋊Dic5—31no—
117C4⋊Dic5—32no—
118C4⋊Dic5C2×A411yescharacteristic
119#683—34no—
120#683—32no—
121#683—32no—
122#683—32no—
123#683—32no—
124C4×Dic5—34no—
125C4×Dic5—32no—
126#557—51no—
127#557—151no—
128#556—151no—
129#552—152no—
130D4×Q8—52no—
131D4×Q8—51no—
132#541—152no—
133#533—151no—
134#529—151no—
135#519—52no—
136C2×C4×Q8C3×D511yescharacteristic, complemented
137C22×C3×C5—42no—
138C22×C3×C5—41no—
139C22×C3×C5—81no—
140C22×C3×C5—81no—
141C2×C3×D5—82no—
142C2×C3×D5—82no—
143C2×C3×D5—82no—
144C3×C4×C5—42no—
145C3×C4×C5—81no—
146C3×Dic5—82no—
147C2×C3×D4—201no—
148C2×C3×D4—202no—
149C22×C3×C4—41no—
150C2×SL(2,𝔽3)—21no—
151C2×SL(2,𝔽3)—102no—
152C2×SL(2,𝔽3)C22×D511yescharacteristic
153C3×C4⋊C4—201no—
154C3×C22⋊C4—202no—
155C23×C5C22×A411yescharacteristic
156C22×D5C22×A412yes—
157C5×Q8—31no—
158C5×Q8—61no—
159C5×Q8—61no—
160C5×Q8C3×#4111yescharacteristic
161C2×C4×C5—22no—
162C2×C4×C5—31no—
163C2×C4×C5—31no—
164C2×C4×C5—31no—
165C2×C4×C5—31no—
166C2×C4×C5—61no—
167C2×C4×C5—61no—
168C2×C4×C5—62no—
169C2×C4×C5C22×A411yescharacteristic
170C2×C4×C5C22×A411yescharacteristic
171C22⋊D5—24no—
172C2×Dic5—32no—
173C2×Dic5—32no—
174C2×Dic5C22×A412yes—
175D20—22no—
176C4×D5—62no—
177C4×D5—62no—
178Dic10—62no—
179Dic10—62no—
180Dic10—62no—
181C22×Q8—52no—
182C22×Q8C2×C3×D511yescharacteristic
183C4⋊Q8—151no—
184C4⋊Q8—151no—
185C4⋊Q8—152no—
186#126—152no—
187#126—154no—
188#125—151no—
189#125—152no—
190#125—152no—
191#125—154no—
192#124—151no—
193#123—152no—
194C22⋊Q8—151no—
195C22⋊Q8—152no—
196C22⋊Q8—152no—
197C22⋊Q8—152no—
198#121—51no—
199C4×Q8—31no—
200C4×Q8—31no—
201C4×Q8—52no—
202C4×Q8—151no—
203C4×Q8—152no—
204C4×Q8—152no—
205C4×Q8—152no—
206C4×Q8C2×C3×D511yescharacteristic
207C4×Q8C2×C3×D511yescharacteristic
208C4×D4—151no—
209C4×D4—152no—
210#117—152no—
211#117—152no—
212#117—152no—
213C2×C4⋊C4—31no—
214C2×C42—31no—
215C2×C3×C5—41no—
216C2×C3×C5—41no—
217C2×C3×C5—41no—
218C2×C3×C5—82no—
219C3×D5—162no—
220C23×C3—41no—
221C23×C3—202no—
222C3×D4—402no—
223C3×D4—404no—
224C2×C3×C4—41no—
225C2×C3×C4—41no—
226C2×C3×C4—82no—
227C2×C3×C4—202no—
228SL(2,𝔽3)#70911yescharacteristic
229C22×C5—21no—
230C22×C5—21no—
231C22×C52−1+4⋊C312yes—
232C22×C5C23×A411yescharacteristic
233C2×D5—22no—
234C2×D5—22no—
235C2×D5—22no—
236C4×C5—21no—
237C4×C5—31no—
238C4×C5—31no—
239C4×C5—61no—
240C4×C5—61no—
241C4×C5—61no—
242C4×C52−1+4⋊C312yes—
243Dic5—22no—
244Dic5—62no—
245Dic5—62no—
246C2×Q8—21no—
247C2×Q8—31no—
248C2×Q8—61no—
249C2×Q8—102no—
250C2×Q8—152no—
251C2×Q8—151no—
252C2×Q8—302no—
253C2×Q8C22×C3×D511yescharacteristic
254C2×D4—52no—
255C2×D4—51no—
256C22×C4—31no—
257C22×C4—31no—
258C22×C4—152no—
259C22×C4D5×A411yescharacteristic
260C4⋊C4—31no—
261C4⋊C4—31no—
262C4⋊C4—32no—
263C4⋊C4—51no—
264C4⋊C4—152no—
265C4⋊C4—152no—
266C4⋊C4—152no—
267C4⋊C4—152no—
268C4⋊C4—152no—
269C4⋊C4—152no—
270C4⋊C4—154no—
271C4⋊C4—152no—
272C4⋊C4—151no—
273C22⋊C4—52no—
274C22⋊C4—152no—
275C22⋊C4—152no—
276C22⋊C4—152no—
277C22⋊C4—152no—
278C42—31no—
279C42—31no—
280C42—31no—
281C42—31no—
282C42—152no—
283C42—154no—
284C3×C5—41noHall
285C22×C3—42no—
286C22×C3—41no—
287C22×C3—81no—
288C22×C3—81no—
289C22×C3—402no—
290C22×C3—402no—
291C22×C3—402no—
292C3×C4—42no—
293C3×C4—81no—
294C3×C4—402no—
295C2×C5—22no—
296C2×C5A4×#4111yescharacteristic
297C2×C5C2×2−1+4⋊C311yescharacteristic
298C2×C5C2×2−1+4⋊C311yescharacteristic
299D5—42no—
300C23—52no—
301C23C2×D5×A411yescharacteristic
302Q8—31no—
303Q8—61no—
304Q8—61no—
305Q8—302no—
306Q8—302no—
307Q8—302no—
308Q8C3×#70911yescharacteristic
309D4—104no—
310D4—102no—
311C2×C4—22no—
312C2×C4—31no—
313C2×C4—31no—
314C2×C4—31no—
315C2×C4—31no—
316C2×C4—52no—
317C2×C4—61no—
318C2×C4—61no—
319C2×C4—62no—
320C2×C4—152no—
321C2×C4—152no—
322C2×C4—302no—
323C2×C4—302no—
324C2×C4C2×D5×A411yescharacteristic
325C2×C4C2×D5×A411yescharacteristic
326C2×C3—41no—
327C2×C3—41no—
328C2×C3—41no—
329C2×C3—82no—
330C2×C3—802no—
331C5#3523⋊C311yescharacteristic, complemented, Sylow
332C22—21no—
333C22—21no—
334C22—102no—
335C22—102no—
336C22—102no—
337C22#4150⋊C312yes—
338C22C22×D5×A411yescharacteristic
339C4—21no—
340C4—31no—
341C4—31no—
342C4—61no—
343C4—61no—
344C4—61no—
345C4—102no—
346C4—302no—
347C4—302no—
348C4#4147⋊C312yes—
349C3—41noSylow
350C2—22no—
351C2—202no—
352C2A4×#70911yescharacteristic
353C2C2×#4147⋊C311yescharacteristic
354C2C2×#4150⋊C311yescharacteristic
subgroupG/H
center, FrattiniC22C22×D5×A4
commutator, cosocleC2×C5×Q8C23×C3
FittingC2×C4×C5×Q8C2×C3
socleC22×C5C23×A4
hypercenterC22×C4D5×A4
hypocenterC5×Q8C3×#41
LCSG ▷ C2×C5×Q8 ▷ C5×Q8
UCSC1 ◁ C22 ◁ C22×C4

Complements

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

NH ≅ G/Ntype
1. C2×C4×C5×SL(2,𝔽3)C2semidirect
2. SL(2,𝔽3)×C22⋊D5C2semidirect
3. #65906⋊C3C2semidirect
4. SL(2,𝔽3)×D20C2semidirect
5. #65904⋊C3C2semidirect
6. #10617705C3semidirect
14. C2×C4×C5×Q8C2×C3semidirect
15. Q8×C22⋊D5C2×C3semidirect
18. Q8×D20C2×C3semidirect
19. #65906C2×C3semidirect
20. #65904C2×C3semidirect
37. C2×C4×SL(2,𝔽3)D5semidirect
136. C2×C4×Q8C3×D5semidirect
331. C5#3523⋊C3semidirect

Maximal subgroups

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

subgroupindex
C2×C4×C5×SL(2,𝔽3)2
SL(2,𝔽3)×C22⋊D52
#65906⋊C32
SL(2,𝔽3)×D202
#65904⋊C32
#106177053
C3×#40764
#3523⋊C35