Finite Group Databasebeta

← #10617788⋊C3 #14514057

Subgroups — #10617788⋊C3 #14514057

Order 1920.

744 conjugacy classes of subgroups; only the 113 normal subgroups are shown.

normal subgroups only

This group has 744 conjugacy classes of subgroups — too many to draw. Only its 113 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×#4150⋊C3C211yesmaximal, characteristic
2C5×#520⋊C3C211yesmaximal, characteristic, complemented
3Dic5×#41⋊C3C211yesmaximal, characteristic, complemented
4#65906⋊C3C212yesmaximal, complemented
5C4×D5×SL(2,𝔽3)C212yesmaximal, complemented
6#10617788C311yesmaximal, characteristic, complemented, Hall
7C2×C5×#41⋊C3C2211yescharacteristic
8#4150⋊C3C414yescomplemented
9C2×D5×SL(2,𝔽3)C2212yes—
10C3×#4039—41nomaximal
11C4×C5×SL(2,𝔽3)C2212yes—
12Dic5×SL(2,𝔽3)C2211yescharacteristic
13Dic5×SL(2,𝔽3)C2211yescharacteristic
14C4×2−1+4⋊C3—51nomaximal, Hall
15C2×#4150C2×C311yescharacteristic
16C5×#520C2×C311yescharacteristic, complemented
17#41×Dic5C2×C311yescharacteristic, complemented
18#65906C2×C312yescomplemented
19C4×Q8×D5C2×C312yescomplemented
20#65880—32no—
21C4×#711—32no—
22#65856—31no—
23#65850—31no—
24#65838—31no—
25C2×#4033—31no—
26C2×C3×D20—41no—
27C2×C3×C4×D5—42no—
28C5×#41⋊C3C2×C412yescomplemented
29C2×C5×SL(2,𝔽3)C2311yescharacteristic
30D5×SL(2,𝔽3)C2×C414yescomplemented
31C3×C5×C4⋊C4—41no—
32C3×#685—42no—
33C3×C4×Dic5—41no—
34C2×2−1+4⋊C3—51no—
35#520⋊C3D511yescharacteristic, complemented
36#519⋊C3—52no—
37C4×#41⋊C3—51no—
38C2×C4×SL(2,𝔽3)—52no—
39C2×C5×#41C22×C311yescharacteristic
40#4150—34no—
41#4150C3×C414yescomplemented
42C2×Q8×D5C22×C312yes—
43C2×#711—32no—
44C2×#709—31no—
45C22×Dic10—31no—
46C4×C5×Q8C22×C312yes—
47C4×C5×D4—32no—
48C5×#117—31no—
49C2×C5×C4⋊C4—31no—
50Q8×Dic5C22×C311yescharacteristic
51Q8×Dic5C22×C311yescharacteristic
52D4×Dic5—31no—
53D4×Dic5—31no—
54C4×C22⋊D5—34no—
55#4072—31no—
56C2×#683—32no—
57C2×C4×Dic5—32no—
58C2×C4×Dic5—31no—
59#4039—31no—
60#4039A411yescharacteristic
61#4038—32no—
62D5×C4⋊C4—32no—
63#4033—32no—
64#4033—31no—
65#4033—32no—
66#4033—31no—
67#4033—32no—
68#4027—32no—
69#4027—32no—
70#4023—34no—
71#4018—32no—
72C42×D5—32no—
73C4×Dic10—32no—
74C4×Dic10—32no—
75C4×2−1+4—51noSylow
76C22×C3×D5—42no—
77C2×C3×C4×C5—41no—
78C2×C3×C4×C5—42no—
79C2×C3×Dic5—41no—
80C2×C3×Dic5—41no—
81C3×D20—44no—
82C3×C4×D5—84no—
83C5×SL(2,𝔽3)C22×C411yescharacteristic
842−1+4⋊C3—54no—
85C2×#41⋊C3C2×D511yescharacteristic
86C22×SL(2,𝔽3)—52no—
87C3×C4×D4—201no—
88C4×SL(2,𝔽3)—51no—
89C4×SL(2,𝔽3)—51no—
90C4×SL(2,𝔽3)C2×D512yes—
91C5×#41—32no—
92C5×#41C2×C3×C412yescomplemented
93C2×C5×Q8C23×C311yescharacteristic
94C2×C5×D4—31no—
95C22×C4×C5—31no—
96C22×C4×C5—32no—
97C2×C22⋊D5—32no—
98C22×Dic5—32no—
99C22×Dic5—31no—
100C22×Dic5—31no—
101Q8×D5—34no—
102Q8×D5C2×C3×C414yescomplemented
103#711—38no—
104#711—38no—
105#709—34no—
106#709—34no—
107C2×D20C2×A411yescharacteristic
108C2×C4×D5—32no—
109C2×C4×D5—32no—
110C2×C4×D5C2×A412yes—
111C2×Dic10—34no—
112C2×Dic10—31no—
113C2×Dic10—32no—
114C2×Dic10—34no—
115C5×C4⋊C4—32no—
116C5×C4⋊C4—31no—
117C5×C4⋊C4C2×A411yescharacteristic
118C5×C22⋊C4—32no—
119C42×C5—32no—
120C22⋊Dic5—32no—
121#685—32no—
122#685C2×A412yes—
123C4⋊Dic5—31no—
124C4⋊Dic5—31no—
125#683—34no—
126#683—32no—
127#683—32no—
128C4×Dic5—34no—
129C4×Dic5—32no—
130C4×Dic5—31no—
131C4×Dic5—31no—
132C4×Dic5—32no—
133C4×Dic5—31no—
134C4×Dic5C2×A411yescharacteristic
135C2×2−1+4—51no—
136#520—151no—
137#520—152no—
138#520C3×D511yescharacteristic, complemented
139#519—52no—
140#519—151no—
141C4×#41—51no—
142C4×#41—151no—
143C4×#41—152no—
144C2×C4×Q8—52no—
145C2×C4×Q8—151no—
146C22×C3×C5—41no—
147C2×C3×D5—44no—
148C2×C3×D5—82no—
149C2×C3×D5—82no—
150C3×C4×C5—42no—
151C3×C4×C5—82no—
152C3×Dic5—42no—
153C3×Dic5—81no—
154C2×C3×D4—201no—
155C22×C3×C4—202no—
156#41⋊C3C4×D512yescomplemented
157C2×SL(2,𝔽3)—54no—
158C2×SL(2,𝔽3)C22×D511yescharacteristic
159C3×C4⋊C4—41no—
160C3×C22⋊C4—202no—
161C3×C42—201no—
162C23×C5—31no—
163C22×D5C22×A412yes—
164C5×Q8—31no—
165C5×Q8C22×C3×C411yescharacteristic
166C5×D4—32no—
167C5×D4—32no—
168C2×C4×C5—31no—
169C2×C4×C5—32no—
170C2×C4×C5—32no—
171C2×C4×C5—32no—
172C2×C4×C5—122no—
173C2×C4×C5C22×A412yes—
174C2×C4×C5C22×A411yescharacteristic
175C22⋊D5—38no—
176C2×Dic5—38no—
177C2×Dic5—32no—
178C2×Dic5—31no—
179C2×Dic5—31no—
180C2×Dic5—62no—
181C2×Dic5—121no—
182C2×Dic5C22×A411yescharacteristic
183C2×Dic5C22×A411yescharacteristic
184D20C4×A414yes—
185C4×D5—24no—
186C4×D5—34no—
187C4×D5—34no—
188C4×D5—122no—
189Dic10—34no—
190Dic10—34no—
191Dic10—34no—
1922−1+4—54no—
1932−1+4—154no—
194C2×#41—151no—
195C2×#41—152no—
196C2×#41C2×C3×D511yescharacteristic
197C22×Q8—52no—
198C22×Q8—151no—
199C4×Q8—51no—
200C4×Q8—51no—
201C4×Q8—152no—
202C4×Q8—152no—
203C4×Q8—151no—
204C4×Q8—152no—
205C4×Q8—152no—
206C4×Q8—151no—
207C4×Q8—152no—
208C4×Q8C2×C3×D512yes—
209C4×D4—32no—
210C4×D4—51no—
211C4×D4—152no—
212C4×D4—152no—
213C4×D4—151no—
214C4×D4—154no—
215C4×D4—151no—
216C4×D4—151no—
217#117—31no—
218#117—152no—
219#117—154no—
220#117—152no—
221#117—151no—
222C2×C4⋊C4—31no—
223C2×C4⋊C4—152no—
224C2×C4⋊C4—152no—
225C2×C42—152no—
226C2×C42—151no—
227C2×C42—152no—
228C2×C3×C5—41no—
229C2×C3×C5—41no—
230C2×C3×C5—41no—
231C3×D5—84no—
232C23×C3—202no—
233C3×D4—204no—
234C2×C3×C4—41no—
235C2×C3×C4—42no—
236C2×C3×C4—201no—
237C2×C3×C4—201no—
238C2×C3×C4—404no—
239SL(2,𝔽3)C2×C4×D511yescharacteristic
240C22×C5—32no—
241C22×C5—61no—
242C22×C5—61no—
243C22×C5C23×A411yescharacteristic
244C2×D5—22no—
245C2×D5—22no—
246C2×D5C2×C4×A414yes—
247C4×C5—22no—
248C4×C5—31no—
249C4×C5—31no—
250C4×C5—62no—
251C4×C5C2×C4×A412yes—
252Dic5—21no—
253Dic5—34no—
254Dic5—61no—
255Dic5—61no—
256Dic52−1+4⋊C312yescomplemented
257#41—32no—
258#41—154no—
259#41—154no—
260#41—158no—
261#41—158no—
262#41C3×C4×D512yescomplemented
263C2×Q8—54no—
264C2×Q8—154no—
265C2×Q8—154no—
266C2×Q8—152no—
267C2×Q8—151no—
268C2×Q8—154no—
269C2×Q8C22×C3×D511yescharacteristic
270C2×D4—31no—
271C2×D4—51no—
272C2×D4—152no—
273C22×C4—31no—
274C22×C4—32no—
275C22×C4—52no—
276C22×C4—152no—
277C22×C4—152no—
278C22×C4—151no—
279C22×C4—151no—
280C22×C4—152no—
281C4⋊C4—32no—
282C4⋊C4—31no—
283C4⋊C4—152no—
284C4⋊C4—151no—
285C4⋊C4—151no—
286C4⋊C4—154no—
287C4⋊C4—152no—
288C4⋊C4D5×A411yescharacteristic
289C22⋊C4—32no—
290C22⋊C4—52no—
291C22⋊C4—152no—
292C22⋊C4—152no—
293C42—32no—
294C42—51no—
295C42—151no—
296C42—152no—
297C42—151no—
298C42—154no—
299C42—152no—
300C42—151no—
301C3×C5—41noHall
302C22×C3—41no—
303C22×C3—204no—
304C22×C3—402no—
305C22×C3—402no—
306C3×C4—42no—
307C3×C4—82no—
308C3×C4—202no—
309C3×C4—401no—
310C2×C5—62no—
311C2×C5C22×C4×A411yescharacteristic
312C2×C5C2×2−1+4⋊C311yescharacteristic
313C2×C5#594511yescharacteristic
314D5—24no—
315C23—31no—
316C23—52no—
317Q8—31no—
318Q8—154no—
319Q8—154no—
320Q8—154no—
321Q8C2×C3×C4×D511yescharacteristic
322D4—32no—
323D4—32no—
324D4—54no—
325D4—158no—
326C2×C4—31no—
327C2×C4—32no—
328C2×C4—32no—
329C2×C4—32no—
330C2×C4—51no—
331C2×C4—51no—
332C2×C4—104no—
333C2×C4—122no—
334C2×C4—154no—
335C2×C4—154no—
336C2×C4—158no—
337C2×C4—152no—
338C2×C4—151no—
339C2×C4—151no—
340C2×C4—302no—
341C2×C4—602no—
342C2×C4—601no—
343C2×C4C2×D5×A412yes—
344C2×C4C2×D5×A411yescharacteristic
345C2×C3—41no—
346C2×C3—41no—
347C2×C3—41no—
348C2×C3—404no—
349C5C4×2−1+4⋊C311yescharacteristic, complemented, Sylow
350C22—32no—
351C22—54no—
352C22—61no—
353C22—61no—
354C22—102no—
355C22—102no—
356C22C22×D5×A411yescharacteristic
357C4—22no—
358C4—31no—
359C4—31no—
360C4—52no—
361C4—62no—
362C4—101no—
363C4—154no—
364C4—301no—
365C4—301no—
366C4C4×D5×A412yes—
367C3—41noSylow
368C2—62no—
369C2—104no—
370C2C2×C4×D5×A411yescharacteristic
371C2#1175694411yescharacteristic
372C2D5×2−1+4⋊C311yescharacteristic
subgroupG/H
center, FrattiniC22C22×D5×A4
commutator, hypocenterC5×Q8C22×C3×C4
FittingC5×#520C2×C3
socleC22×C5C23×A4
prime radicalC2×#4150⋊C3C2
cosocleC2×C5×Q8C23×C3
hypercenterC4⋊C4D5×A4
LCSG ▷ C5×Q8
UCSC1 ◁ C22 ◁ C4⋊C4

Complements

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

NH ≅ G/Ntype
2. C5×#520⋊C3C2semidirect
3. Dic5×#41⋊C3C2semidirect
4. #65906⋊C3C2semidirect
5. C4×D5×SL(2,𝔽3)C2semidirect
6. #10617788C3semidirect
8. #4150⋊C3C4semidirect
16. C5×#520C2×C3semidirect
17. #41×Dic5C2×C3semidirect
18. #65906C2×C3semidirect
19. C4×Q8×D5C2×C3semidirect
28. C5×#41⋊C3C2×C4semidirect
30. D5×SL(2,𝔽3)C2×C4semidirect
35. #520⋊C3D5semidirect
41. #4150C3×C4semidirect
92. C5×#41C2×C3×C4semidirect
102. Q8×D5C2×C3×C4semidirect
138. #520C3×D5semidirect
156. #41⋊C3C4×D5semidirect
256. Dic52−1+4⋊C3semidirect
262. #41C3×C4×D5semidirect
349. C5C4×2−1+4⋊C3semidirect

Maximal subgroups

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

subgroupindex
C2×#4150⋊C32
C5×#520⋊C32
Dic5×#41⋊C32
#65906⋊C32
C4×D5×SL(2,𝔽3)2
#106177883
C3×#40394
C4×2−1+4⋊C35