Finite Group Databasebeta

← C2×#66165⋊C3 #14514216

Subgroups — C2×#66165⋊C3 #14514216

Order 1920.

850 conjugacy classes of subgroups; only the 111 normal subgroups are shown.

normal subgroups only

This group has 850 conjugacy classes of subgroups — too many to draw. Only its 111 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
1C22×C5×#41⋊C3C211yesmaximal, characteristic, complemented
2C22×D5×SL(2,𝔽3)C211yesmaximal, characteristic, complemented
3#66165⋊C3C214yesmaximal, direct product factor
4C2×Dic5×SL(2,𝔽3)C211yesmaximal, characteristic, complemented
5C2×#66165C311yesmaximal, characteristic, complemented, Hall
6C2×C5×#41⋊C3—22no—
7C2×C5×#41⋊C3C2212yescomplemented
8C22×C5×SL(2,𝔽3)C2211yescharacteristic
9C2×D5×SL(2,𝔽3)—22no—
10C2×D5×SL(2,𝔽3)C2212yescomplemented
11C2×C3×#685—41nomaximal
12Dic5×SL(2,𝔽3)C2212yescomplemented
13C2×#548⋊C3—51nomaximal, Hall
14C22×C5×#41C2×C311yescharacteristic, complemented
15C22×Q8×D5C2×C311yescharacteristic
16#66165—34no—
17#66165C2×C314yescomplemented
18C2×Q8×Dic5C2×C311yescharacteristic, complemented
19C2×#4084—31no—
20C2×#4082—31no—
21C2×C4×C22⋊D5—31no—
22C2×#4070—31no—
23C23×C3×D5—41no—
24C22×C3×C4×C5—41no—
25C22×C3×Dic5—41no—
26C5×#41⋊C3—22no—
27C2×C5×SL(2,𝔽3)D412yescomplemented
28C2×C5×SL(2,𝔽3)C2311yescharacteristic
29D5×SL(2,𝔽3)—22no—
30C3×#685—44no—
31C22×#41⋊C3D511yescharacteristic, complemented
32C23×SL(2,𝔽3)—51no—
33#548⋊C3—54no—
34C2×C4×SL(2,𝔽3)—51no—
35C2×C5×#41—22no—
36C2×C5×#41—32no—
37C2×C5×#41—62no—
38C2×C5×#41C22×C312yescomplemented
39C22×C5×Q8C22×C311yescharacteristic
40C22×C5×D4—31no—
41C23×C4×C5—31no—
42C22×C22⋊D5—31no—
43C2×Q8×D5—22no—
44C2×Q8×D5—32no—
45C2×Q8×D5—62no—
46C2×Q8×D5C22×C312yes—
47C22×C4×D5—31no—
48C22×Dic10—31no—
49C2×C22⋊Dic5—32no—
50C2×C22⋊Dic5—31no—
51Q8×Dic5—32no—
52Q8×Dic5C22×C312yescomplemented
53#4084—34no—
54#4084—34no—
55#4082—34no—
56#4082—34no—
57C4×C22⋊D5—34no—
58C4×C22⋊D5—34no—
59C2×#685A411yescharacteristic
60C2×C4⋊Dic5—31no—
61#4070—34no—
62#4070—34no—
63C2×#683—31no—
64C2×C4×Dic5—31no—
65C2×#548—51noSylow
66C23×C3×C5—41no—
67C22×C3×D5—44no—
68C22×C3×D5—42no—
69C22×C3×D5—82no—
70C22×C3×D5—82no—
71C2×C3×C4×C5—42no—
72C2×C3×C4×C5—82no—
73C2×C3×Dic5—42no—
74C2×C3×Dic5—82no—
75C5×SL(2,𝔽3)C2×D411yescharacteristic, complemented
76C2×#41⋊C3—22no—
77C2×#41⋊C3C2×D512yescomplemented
78C22×SL(2,𝔽3)—52no—
79C22×SL(2,𝔽3)—102no—
80C22×SL(2,𝔽3)C2×D511yescharacteristic
81C2×C3×C22⋊C4—201no—
82C4×SL(2,𝔽3)—52no—
83C24×C5—31no—
84C23×D5C2×A411yescharacteristic
85C5×#41—22no—
86C5×#41—64no—
87C5×#41—64no—
88C5×#41—62no—
89C2×C5×Q8—31no—
90C2×C5×Q8—32no—
91C2×C5×Q8C3×D412yes—
92C2×C5×Q8C23×C311yescharacteristic
93C2×C5×D4—32no—
94C2×C5×D4—32no—
95C2×C5×D4—62no—
96C2×C5×D4—62no—
97C22×C4×C5—31no—
98C22×C4×C5—32no—
99C22×C4×C5—32no—
100C22×C4×C5—62no—
101C22×C4×C5—62no—
102C22×C4×C5C2×A411yescharacteristic
103C2×C22⋊D5—34no—
104C2×C22⋊D5—64no—
105C22×Dic5—31no—
106C22×Dic5—31no—
107C22×Dic5C2×A411yescharacteristic
108Q8×D5—22no—
109Q8×D5—64no—
110Q8×D5—64no—
111Q8×D5—62no—
112C2×C4×D5—32no—
113C2×C4×D5—32no—
114C2×C4×D5—62no—
115C2×C4×D5—62no—
116C2×Dic10—32no—
117C2×Dic10—32no—
118C2×Dic10—62no—
119C2×Dic10—62no—
120C22⋊Dic5—38no—
121C22⋊Dic5—34no—
122#685C2×A414yes—
123C4⋊Dic5—32no—
124C4⋊Dic5—32no—
125#683—34no—
126C4×Dic5—32no—
127C4×Dic5—32no—
128C22×#41C3×D511yescharacteristic, complemented
129C23×Q8—51no—
130#548—54no—
131#548—154no—
132C2×#124—151no—
133C2×C22⋊Q8—151no—
134C2×#121—151no—
135C2×C4×Q8—51no—
136C2×C4×D4—151no—
137C22×C3×C5—42no—
138C22×C3×C5—42no—
139C22×C3×C5—42no—
140C22×C3×C5—41no—
141C2×C3×D5—44no—
142C2×C3×D5—82no—
143C2×C3×D5—82no—
144C2×C3×D5—88no—
145C3×C4×C5—82no—
146C3×Dic5—82no—
147C24×C3—201no—
148C22×C3×C4—41no—
149C22×C3×C4—201no—
150#41⋊C3—22no—
151C2×SL(2,𝔽3)—102no—
152C2×SL(2,𝔽3)C22⋊D512yescomplemented
153C2×SL(2,𝔽3)C22×D511yescharacteristic
154C3×C22⋊C4—204no—
155C23×C5—32no—
156C23×C5—62no—
157C23×C5—62no—
158C23×C5—62no—
159C23×C5C22×A411yescharacteristic
160C22×D5—22no—
161C22×D5—22no—
162C22×D5#41⋊C314yescomplemented
163C22×D5C22×A412yes—
164C5×Q8—32no—
165C5×Q8—31no—
166C5×Q8—32no—
167C5×Q8C2×C3×D411yescharacteristic
168C5×D4—62no—
169C5×D4—62no—
170C5×D4—64no—
171C2×C4×C5—22no—
172C2×C4×C5—32no—
173C2×C4×C5—32no—
174C2×C4×C5—31no—
175C2×C4×C5—31no—
176C2×C4×C5—62no—
177C2×C4×C5—62no—
178C2×C4×C5—64no—
179C2×C4×C5C22×A412yes—
180C22⋊D5—68no—
181C2×Dic5—22no—
182C2×Dic5—32no—
183C2×Dic5—32no—
184C2×Dic5—62no—
185C2×Dic5—62no—
186C2×Dic5C22×A412yes—
187C4×D5—62no—
188C4×D5—62no—
189C4×D5—64no—
190Dic10—64no—
191Dic10—62no—
192Dic10—62no—
193C2×#41—22no—
194C2×#41—32no—
195C2×#41—62no—
196C2×#41C2×C3×D512yescomplemented
197C22×Q8—52no—
198C22×Q8—102no—
199C22×Q8—151no—
200C22×Q8—152no—
201C22×Q8—302no—
202C22×Q8C2×C3×D511yescharacteristic
203C22×D4—31no—
204C22×D4—151no—
205C23×C4—31no—
206C23×C4—151no—
207#124—154no—
208#124—154no—
209C22⋊Q8—154no—
210C22⋊Q8—154no—
211#121—154no—
212#121—154no—
213C4×Q8—52no—
214C4×Q8—152no—
215C4×D4—154no—
216C4×D4—154no—
217C2×C4⋊C4—151no—
218C2×C4⋊C4—151no—
219C2×C22⋊C4—51no—
220C2×C22⋊C4—151no—
221C2×C22⋊C4—152no—
222C2×C42—151no—
223C2×C3×C5—41no—
224C2×C3×C5—41no—
225C2×C3×C5—41no—
226C2×C3×C5—44no—
227C3×D5—84no—
228C23×C3—41no—
229C23×C3—202no—
230C23×C3—204no—
231C23×C3—402no—
232C23×C3—402no—
233C2×C3×C4—42no—
234C2×C3×C4—82no—
235C2×C3×C4—202no—
236C2×C3×C4—402no—
237SL(2,𝔽3)C2×C22⋊D511yescharacteristic, complemented
238C22×C5—62no—
239C22×C5—62no—
240C22×C5—62no—
241C22×C5—124no—
242C22×C5D4×A412yes—
243C22×C5C2×#41⋊C312yes—
244C22×C52−1+4⋊C312yes—
245C22×C5C23×A411yescharacteristic
246C2×D5—22no—
247C2×D5—22no—
248C2×D5—28no—
249C2×D5C2×#41⋊C314yescomplemented
250C4×C5—22no—
251C4×C5—32no—
252C4×C5—31no—
253C4×C5—31no—
254Dic5—22no—
255Dic5—62no—
256Dic5—62no—
257C24—31no—
258C24—51no—
259#41—22no—
260#41—64no—
261#41—64no—
262#41—62no—
263C2×Q8—31no—
264C2×Q8—32no—
265C2×Q8—102no—
266C2×Q8—152no—
267C2×Q8—152no—
268C2×Q8—302no—
269C2×Q8—302no—
270C2×Q8—304no—
271C2×Q8—302no—
272C2×Q8—304no—
273C2×Q8C3×C22⋊D512yes—
274C2×Q8C22×C3×D511yescharacteristic
275C2×D4—32no—
276C2×D4—32no—
277C2×D4—62no—
278C2×D4—62no—
279C2×D4—154no—
280C2×D4—304no—
281C22×C4—31no—
282C22×C4—32no—
283C22×C4—32no—
284C22×C4—51no—
285C22×C4—62no—
286C22×C4—62no—
287C22×C4—152no—
288C22×C4—152no—
289C22×C4—151no—
290C22×C4—151no—
291C22×C4—302no—
292C22×C4—302no—
293C22×C4D5×A411yescharacteristic
294C4⋊C4—152no—
295C4⋊C4—152no—
296C4⋊C4—154no—
297C22⋊C4—54no—
298C22⋊C4—154no—
299C22⋊C4—158no—
300C42—152no—
301C42—152no—
302C3×C5—41noHall
303C22×C3—42no—
304C22×C3—42no—
305C22×C3—42no—
306C22×C3—41no—
307C22×C3—204no—
308C22×C3—408no—
309C22×C3—402no—
310C22×C3—402no—
311C3×C4—82no—
312C3×C4—402no—
313C2×C5—122no—
314C2×C5#548⋊C314yescomplemented
315C2×C5C2×D4×A411yescharacteristic
316C2×C5C22×#41⋊C311yescharacteristic
317C2×C5C2×2−1+4⋊C311yescharacteristic
318D5—24no—
319C23—32no—
320C23—54no—
321C23—52no—
322C23—62no—
323C23—62no—
324C23—62no—
325C23—102no—
326C23—102no—
327C23C2×D5×A411yescharacteristic
328Q8—32no—
329Q8—31no—
330Q8—32no—
331Q8—302no—
332Q8—302no—
333Q8—304no—
334Q8C2×C3×C22⋊D511yescharacteristic
335D4—62no—
336D4—62no—
337D4—64no—
338D4—308no—
339C2×C4—22no—
340C2×C4—32no—
341C2×C4—32no—
342C2×C4—31no—
343C2×C4—31no—
344C2×C4—52no—
345C2×C4—62no—
346C2×C4—62no—
347C2×C4—64no—
348C2×C4—102no—
349C2×C4—152no—
350C2×C4—152no—
351C2×C4—302no—
352C2×C4—302no—
353C2×C4—304no—
354C2×C4—302no—
355C2×C4—302no—
356C2×C4C2×D5×A412yes—
357C2×C3—44no—
358C2×C3—41no—
359C2×C3—41no—
360C2×C3—41no—
361C2×C3—404no—
362C5C2×#548⋊C311yescharacteristic, complemented, Sylow
363C22—54no—
364C22—62no—
365C22—62no—
366C22—62no—
367C22—108no—
368C22—102no—
369C22—102no—
370C22—124no—
371C22D5×#41⋊C312yes—
372C22#4150⋊C312yes—
373C22A4×C22⋊D512yes—
374C22C22×D5×A411yescharacteristic
375C4—22no—
376C4—31no—
377C4—32no—
378C4—31no—
379C4—102no—
380C4—302no—
381C4—302no—
382C3—41noSylow
383C2—104no—
384C2—122no—
385C2#66165⋊C314yesdirect product factor
386C2C2×D5×#41⋊C311yescharacteristic
387C2C2×#4150⋊C311yescharacteristic
388C2C2×A4×C22⋊D511yescharacteristic
subgroupG/H
centerC23C2×D5×A4
commutator, cosocleC2×C5×Q8C23×C3
FrattiniC22C22×D5×A4
FittingC22×C5×#41C2×C3
socleC23×C5C22×A4
hypercenterC22×C4D5×A4
hypocenterC5×Q8C2×C3×D4
LCSG ▷ C2×C5×Q8 ▷ C5×Q8
UCSC1 ◁ C23 ◁ C22×C4

Complements

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

NH ≅ G/Ntype
1. C22×C5×#41⋊C3C2semidirect
2. C22×D5×SL(2,𝔽3)C2semidirect
3. #66165⋊C3C2direct (N × H)
4. C2×Dic5×SL(2,𝔽3)C2semidirect
5. C2×#66165C3semidirect
7. C2×C5×#41⋊C3C22semidirect
10. C2×D5×SL(2,𝔽3)C22semidirect
12. Dic5×SL(2,𝔽3)C22semidirect
14. C22×C5×#41C2×C3semidirect
17. #66165C2×C3semidirect
18. C2×Q8×Dic5C2×C3semidirect
27. C2×C5×SL(2,𝔽3)D4semidirect
31. C22×#41⋊C3D5semidirect
38. C2×C5×#41C22×C3semidirect
52. Q8×Dic5C22×C3semidirect
75. C5×SL(2,𝔽3)C2×D4semidirect
77. C2×#41⋊C3C2×D5semidirect
128. C22×#41C3×D5semidirect
152. C2×SL(2,𝔽3)C22⋊D5semidirect
162. C22×D5#41⋊C3semidirect
196. C2×#41C2×C3×D5semidirect
237. SL(2,𝔽3)C2×C22⋊D5semidirect
249. C2×D5C2×#41⋊C3semidirect
314. C2×C5#548⋊C3semidirect
362. C5C2×#548⋊C3semidirect
385. C2#66165⋊C3direct (N × H)

Maximal subgroups

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

subgroupindex
C22×C5×#41⋊C32
C22×D5×SL(2,𝔽3)2
#66165⋊C32
C2×Dic5×SL(2,𝔽3)2
C2×#661653
C2×C3×#6854
C2×#548⋊C35