Finite Group Databasebeta

← C22×#4147⋊C3 #14514652

Subgroups — C22×#4147⋊C3 #14514652

Order 1920.

2574 conjugacy classes of subgroups; only the 271 normal subgroups are shown.

normal subgroups only

This group has 2574 conjugacy classes of subgroups — too many to draw. Only its 271 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
1C23×C5×SL(2,𝔽3)C211yesmaximal, characteristic, complemented
2C2×#4147⋊C3C2112yesmaximal, direct product factor
3C22×D5×SL(2,𝔽3)C211yesmaximal, characteristic, complemented
4C22×C5⋊(#41⋊C3)C211yesmaximal, characteristic, complemented
5C22×#4147C311yesmaximal, characteristic, complemented, Hall
6C22×C3×C22⋊D5—41nomaximal
7C22×C5×SL(2,𝔽3)C2211yescharacteristic
8C22×C5×SL(2,𝔽3)C2216yescomplemented
9#4147⋊C3C22116yesdirect product factor
10C2×D5×SL(2,𝔽3)C2216yescomplemented
11C2×C5⋊(#41⋊C3)C2216yescomplemented
12C22×2−1+4⋊C3—51nomaximal, Hall
13C23×C5×Q8C2×C311yescharacteristic, complemented
14C2×#4147—312no—
15C2×#4147C2×C3112yescomplemented
16C22×#713—31no—
17C22×#713C2×C311yescharacteristic, complemented
18C22×Q8×D5—31no—
19C22×Q8×D5C2×C311yescharacteristic, complemented
20C22×#709—31no—
21C22×#709—31no—
22C24×C3×C5—41no—
23C23×C3×D5—41no—
24C2×C3×C22⋊D5—412no—
25C22×C3×Dic5—41no—
26C2×C5×SL(2,𝔽3)C2313yes—
27C2×C5×SL(2,𝔽3)C2314yescomplemented
28D5×SL(2,𝔽3)C2314yescomplemented
29C5⋊(#41⋊C3)C2314yescomplemented
30C2×2−1+4⋊C3—512no—
31C22×#41⋊C3—51no—
32C23×SL(2,𝔽3)—51no—
33C23×SL(2,𝔽3)D511yescharacteristic, complemented
34C22×C5×Q8—31no—
35C22×C5×Q8—36no—
36C22×C5×Q8C22×C311yescharacteristic
37C22×C5×Q8C22×C316yescomplemented
38C23×C4×C5—31no—
39C22×C22⋊D5—31no—
40C22×C22⋊D5A411yescharacteristic
41#4147—332no—
42#4147—348no—
43#4147C22×C3116yescomplemented
44C2×#713—312no—
45C2×#713—312no—
46C2×#713—36no—
47C2×#713C22×C316yescomplemented
48C2×Q8×D5—312no—
49C2×Q8×D5—36no—
50C2×Q8×D5—312no—
51C2×Q8×D5C22×C316yescomplemented
52C2×#709—312no—
53C2×#709—312no—
54C2×#709—312no—
55C2×#709—312no—
56C22×D20—31no—
57C22×D20—31no—
58C22×C4×D5—32no—
59C22×C4×D5—31no—
60C22×C4×D5—31no—
61C22×Dic10—31no—
62C22×Dic10—31no—
63C22×2−1+4—51noSylow
64C23×C3×C5—41no—
65C23×C3×C5—46no—
66C23×C3×C5—84no—
67C22×C3×D5—46no—
68C22×C3×D5—84no—
69C3×C22⋊D5—416no—
70C2×C3×Dic5—46no—
71C5×SL(2,𝔽3)C2411yescharacteristic
72C22×C3×D4—201no—
732−1+4⋊C3—516no—
74C2×#41⋊C3—56no—
75C22×SL(2,𝔽3)—56no—
76C22×SL(2,𝔽3)C2×D511yescharacteristic
77C22×SL(2,𝔽3)C2×D516yescomplemented
78C24×C5C2×A411yescharacteristic
79C23×D5—31no—
80C23×D5C2×A411yescharacteristic
81C2×C5×Q8—36no—
82C2×C5×Q8—38no—
83C2×C5×Q8—33no—
84C2×C5×Q8—36no—
85C2×C5×Q8—312no—
86C2×C5×Q8C23×C313yes—
87C2×C5×Q8C23×C314yescomplemented
88C22×C4×C5—31no—
89C22×C4×C5—31no—
90C22×C4×C5—36no—
91C22×C4×C5—36no—
92C2×C22⋊D5—312no—
93C2×C22⋊D5C2×A4112yes—
94C22×Dic5—31no—
95C22×Dic5C2×A411yescharacteristic
96#713—348no—
97#713—38no—
98#713—316no—
99#713—312no—
100#713C23×C314yescomplemented
101Q8×D5—38no—
102Q8×D5—348no—
103Q8×D5—312no—
104Q8×D5—316no—
105Q8×D5C23×C314yescomplemented
106#709—348no—
107#709—348no—
108#709—316no—
109#709—316no—
110C2×D20—36no—
111C2×D20—36no—
112C2×D20—312no—
113C2×C4×D5—324no—
114C2×C4×D5—36no—
115C2×C4×D5—36no—
116C2×C4×D5—36no—
117C2×C4×D5—36no—
118C2×Dic10—36no—
119C2×Dic10—312no—
120C2×Dic10—36no—
121C2×2−1+4—512no—
122C2×2−1+4—1512no—
123C22×#41—51no—
124C22×#41—151no—
125C22×#41—151no—
126C22×#41—151no—
127C23×Q8—51no—
128C23×Q8—151no—
129C23×Q8C3×D511yescharacteristic, complemented
130C22×C3×C5—44no—
131C22×C3×C5—43no—
132C22×C3×C5—44no—
133C22×C3×C5—812no—
134C2×C3×D5—44no—
135C2×C3×D5—812no—
136C3×Dic5—44no—
137C24×C3—41no—
138C24×C3—201no—
139C2×C3×D4—2012no—
140C22×C3×C4—201no—
141#41⋊C3—54no—
142C2×SL(2,𝔽3)—54no—
143C2×SL(2,𝔽3)C22×D513yes—
144C2×SL(2,𝔽3)C22×D514yescomplemented
145C23×C5—24no—
146C23×C5C22×A416yes—
147C23×C5C22×A411yescharacteristic
148C22×D5—24no—
149C22×D5—36no—
150C22×D5—64no—
151C22×D5C22×A416yes—
152C5×Q8—33no—
153C5×Q8—34no—
154C5×Q8—312no—
155C5×Q8—32no—
156C5×Q8C24×C311yescharacteristic
157C2×C4×C5—33no—
158C2×C4×C5—36no—
159C2×C4×C5—33no—
160C2×C4×C5—312no—
161C2×C4×C5—34no—
162C22⋊D5—316no—
163C22⋊D5C22×A4116yes—
164C2×Dic5—36no—
165C2×Dic5C22×A416yes—
166D20—34no—
167D20—312no—
168D20—316no—
169C4×D5—332no—
170C4×D5—34no—
171C4×D5—312no—
172C4×D5—312no—
173C4×D5—34no—
174Dic10—312no—
175Dic10—316no—
176Dic10—34no—
1772−1+4—516no—
1782−1+4—1548no—
1792−1+4—1532no—
180C2×#41—56no—
181C2×#41—1512no—
182C2×#41—1512no—
183C2×#41—1512no—
184C2×#41—1512no—
185C2×#41—1512no—
186C2×#41—1512no—
187C2×#41—156no—
188C22×Q8—31no—
189C22×Q8—36no—
190C22×Q8—56no—
191C22×Q8—151no—
192C22×Q8—151no—
193C22×Q8—1512no—
194C22×Q8—1512no—
195C22×Q8—156no—
196C22×Q8C2×C3×D511yescharacteristic
197C22×Q8C2×C3×D516yescomplemented
198C22×D4—51no—
199C22×D4—151no—
200C22×D4—151no—
201C22×D4—151no—
202C23×C4—31no—
203C23×C4—151no—
204C23×C4—151no—
205C23×C4—152no—
206C2×C3×C5—46no—
207C2×C3×C5—41no—
208C2×C3×C5—84no—
209C3×D5—84no—
210C23×C3—46no—
211C23×C3—41no—
212C23×C3—84no—
213C23×C3—206no—
214C23×C3—404no—
215C3×D4—2016no—
216C2×C3×C4—206no—
217SL(2,𝔽3)C23×D511yescharacteristic
218C22×C5—212no—
219C22×C52−1+4⋊C314yescomplemented
220C22×C5C23×A414yes—
221C22×C5C23×A413yes—
222C2×D5—212no—
223C2×D5—34no—
224C2×D5—612no—
225C2×D5C23×A414yes—
226C4×C5—33no—
227C4×C5—34no—
228C4×C5—31no—
229Dic5—34no—
230Dic5C23×A414yes—
231C24—51no—
232C24—151no—
233C24D5×A411yescharacteristic
234#41—54no—
235#41—1516no—
236#41—1548no—
237#41—1548no—
238#41—1516no—
239#41—158no—
240#41—1548no—
241#41—1516no—
242#41—1512no—
243C2×Q8—36no—
244C2×Q8—38no—
245C2×Q8—33no—
246C2×Q8—36no—
247C2×Q8—312no—
248C2×Q8—54no—
249C2×Q8—1512no—
250C2×Q8—156no—
251C2×Q8—156no—
252C2×Q8—1516no—
253C2×Q8—1548no—
254C2×Q8—158no—
255C2×Q8—1512no—
256C2×Q8C22×C3×D513yes—
257C2×Q8C22×C3×D514yescomplemented
258C2×D4—512no—
259C2×D4—1512no—
260C2×D4—156no—
261C2×D4—156no—
262C2×D4—1512no—
263C22×C4—36no—
264C22×C4—36no—
265C22×C4—31no—
266C22×C4—31no—
267C22×C4—51no—
268C22×C4—156no—
269C22×C4—156no—
270C22×C4—156no—
271C22×C4—156no—
272C22×C4—1524no—
273C22×C4—151no—
274C3×C5—41noHall
275C22×C3—44no—
276C22×C3—43no—
277C22×C3—44no—
278C22×C3—812no—
279C22×C3—204no—
280C22×C3—4012no—
281C3×C4—204no—
282C2×C5—24no—
283C2×C5C2×2−1+4⋊C316yescomplemented
284C2×C5C24×A411yescharacteristic
285D5—24no—
286D5—64no—
287C23—24no—
288C23—56no—
289C23—104no—
290C23—156no—
291C23—304no—
292C23C2×D5×A416yes—
293C23C2×D5×A411yescharacteristic
294Q8—33no—
295Q8—34no—
296Q8—312no—
297Q8—32no—
298Q8—1516no—
299Q8—154no—
300Q8—1512no—
301Q8C23×C3×D511yescharacteristic
302D4—516no—
303D4—1516no—
304D4—154no—
305D4—1512no—
306D4—1516no—
307C2×C4—33no—
308C2×C4—36no—
309C2×C4—33no—
310C2×C4—34no—
311C2×C4—312no—
312C2×C4—56no—
313C2×C4—154no—
314C2×C4—1512no—
315C2×C4—154no—
316C2×C4—1512no—
317C2×C4—1532no—
318C2×C4—156no—
319C2×C3—41no—
320C2×C3—46no—
321C2×C3—84no—
322C2×C3—404no—
323C5C22×2−1+4⋊C311yescharacteristic, complemented, Sylow
324C22—212no—
325C22—54no—
326C22—1012no—
327C22—154no—
328C22—3012no—
329C22#4147⋊C314yesdirect product factor
330C22C22×D5×A414yes—
331C22C22×D5×A413yes—
332C4—31no—
333C4—33no—
334C4—34no—
335C4—54no—
336C4—154no—
337C3—41noSylow
338C2—24no—
339C2—104no—
340C2—304no—
341C2C2×#4147⋊C316yesdirect product factor
342C2C23×D5×A411yescharacteristic
subgroupG/H
centerC23C2×D5×A4
commutator, cosocle, hypocenterC5×Q8C24×C3
FrattiniC2C23×D5×A4
FittingC23×C5×Q8C2×C3
socleC23×C5C22×A4
hypercenterC24D5×A4
LCSG ▷ C5×Q8
UCSC1 ◁ C23 ◁ C24

Complements

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

NH ≅ G/Ntype
1. C23×C5×SL(2,𝔽3)C2semidirect
2. C2×#4147⋊C3C2direct (N × H)
3. C22×D5×SL(2,𝔽3)C2semidirect
4. C22×C5⋊(#41⋊C3)C2semidirect
5. C22×#4147C3semidirect
8. C22×C5×SL(2,𝔽3)C22semidirect
9. #4147⋊C3C22direct (N × H)
10. C2×D5×SL(2,𝔽3)C22semidirect
11. C2×C5⋊(#41⋊C3)C22semidirect
13. C23×C5×Q8C2×C3semidirect
15. C2×#4147C2×C3semidirect
17. C22×#713C2×C3semidirect
19. C22×Q8×D5C2×C3semidirect
27. C2×C5×SL(2,𝔽3)C23semidirect
28. D5×SL(2,𝔽3)C23semidirect
29. C5⋊(#41⋊C3)C23semidirect
33. C23×SL(2,𝔽3)D5semidirect
37. C22×C5×Q8C22×C3semidirect
43. #4147C22×C3semidirect
47. C2×#713C22×C3semidirect
51. C2×Q8×D5C22×C3semidirect
77. C22×SL(2,𝔽3)C2×D5semidirect
87. C2×C5×Q8C23×C3semidirect
100. #713C23×C3semidirect
105. Q8×D5C23×C3semidirect
129. C23×Q8C3×D5semidirect
144. C2×SL(2,𝔽3)C22×D5semidirect
197. C22×Q8C2×C3×D5semidirect
219. C22×C52−1+4⋊C3semidirect
257. C2×Q8C22×C3×D5semidirect
283. C2×C5C2×2−1+4⋊C3semidirect
323. C5C22×2−1+4⋊C3semidirect
329. C22#4147⋊C3direct (N × H)
341. C2C2×#4147⋊C3direct (N × H)

Maximal subgroups

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

subgroupindex
C23×C5×SL(2,𝔽3)2
C2×#4147⋊C32
C22×D5×SL(2,𝔽3)2
C22×C5⋊(#41⋊C3)2
C22×#41473
C22×C3×C22⋊D54
C22×2−1+4⋊C35