Finite Group Databasebeta

← #10617746⋊C3 #14514025

Subgroups — #10617746⋊C3 #14514025

Order 1920.

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

normal subgroups only

This group has 584 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
1C4×C5×#41⋊C3C211yesmaximal, characteristic, complemented
2#66168⋊C3C212yesmaximal, complemented
3C4⋊(C5⋊(#41⋊C3))C211yesmaximal, characteristic, complemented
4C4×C5⋊(#41⋊C3)C212yesmaximal, complemented
5SL(2,𝔽3)×Dic10C211yesmaximal, characteristic, complemented
6#10617746C311yesmaximal, characteristic, complemented, Hall
7C2×C5×#41⋊C3C2211yescharacteristic
8C2×C5⋊(#41⋊C3)C2212yes—
9C3×C4×Dic10—41nomaximal
10C4×C5×SL(2,𝔽3)C2211yescharacteristic
11C4×C5×SL(2,𝔽3)C2211yescharacteristic
12Dic5×SL(2,𝔽3)C2212yes—
13#85604—51nomaximal, Hall
14C4×C5×#41C2×C311yescharacteristic
15#66168C2×C312yes—
16#65911C2×C311yescharacteristic
17C4×#713C2×C312yes—
18Q8×Dic10C2×C311yescharacteristic
19#65897—31no—
20#65887—31no—
21#65882—32no—
22#65866—32no—
23#65811—31no—
24C4×#709—31no—
25C2×C3×Dic10—41no—
26C5×#41⋊C3—21no—
27C2×C5×SL(2,𝔽3)C2311yescharacteristic
28C5⋊(#41⋊C3)—22no—
29C3×C42×C5—41no—
30C3×C4⋊Dic5—41no—
31C3×#683—42no—
32C3×C4×Dic5—42no—
33Q8×SL(2,𝔽3)—51no—
34C4⋊(#41⋊C3)—52no—
35C4⋊(#41⋊C3)—51no—
36C4×#41⋊C3—52no—
37C4×#41⋊C3D511yescharacteristic, complemented
38C2×C5×#41C22×C311yescharacteristic
39C2×#713C22×C312yes—
40C2×#709—32no—
41C2×#709—31no—
42C4×C5×Q8C22×C311yescharacteristic
43C4×C5×Q8C22×C311yescharacteristic
44C4×C5×D4—31no—
45C4×C5×D4—31no—
46C5×#117—31no—
47C2×C42×C5—31no—
48Q8×Dic5C22×C312yes—
49#4086—32no—
50#4084—32no—
51#4076—32no—
52#4076—32no—
53C4×C22⋊D5—32no—
54C4×C22⋊D5—32no—
55#4072—31no—
56#4041—32no—
57#4039—32no—
58#4038—32no—
59#4034—32no—
60#4031—34no—
61#4030—34no—
62#4021—32no—
63#4020—31no—
64C4×D20—32no—
65C4×D20—31no—
66C4×D20—31no—
67#4018—32no—
68C42×D5—32no—
69#4015—31no—
70C4×Dic10—31no—
71C4×Dic10A411yescharacteristic
72#3483—51noSylow
73C2×C3×C4×C5—41no—
74C2×C3×C4×C5—41no—
75C2×C3×C4×C5—41no—
76C2×C3×Dic5—42no—
77C2×C3×Dic5—42no—
78C3×Dic10—44no—
79C5×SL(2,𝔽3)#4111yescharacteristic, complemented
80C2×#41⋊C3—52no—
81C2×#41⋊C3C2×D511yescharacteristic
82C3×C4×Q8—201no—
83C4×SL(2,𝔽3)—52no—
84C4×SL(2,𝔽3)C2×D511yescharacteristic
85C4×SL(2,𝔽3)C2×D511yescharacteristic
86C5×#41—21no—
87C5×#41—61no—
88C2×C5×Q8C23×C311yescharacteristic
89C2×C5×D4—31no—
90C22×C4×C5—31no—
91C22×C4×C5—31no—
92C22×C4×C5—31no—
93C2×C22⋊D5—34no—
94C2×C22⋊D5—32no—
95#713—22no—
96#713—62no—
97#709—64no—
98#709—124no—
99C2×D20—32no—
100C2×D20—31no—
101C2×D20—32no—
102C2×C4×D5—32no—
103C2×C4×D5—32no—
104C2×C4×D5—32no—
105C2×Dic10—32no—
106C2×Dic10C2×A411yescharacteristic
107C5×C4⋊C4—31no—
108C5×C4⋊C4—31no—
109C5×C22⋊C4—32no—
110C42×C5—31no—
111C42×C5—31no—
112C42×C5—31no—
113C42×C5C2×A411yescharacteristic
114C22⋊Dic5—32no—
115#685—34no—
116#685—32no—
117#685—32no—
118C4⋊Dic5—31no—
119C4⋊Dic5—32no—
120C4⋊Dic5C2×A411yescharacteristic
121#683—32no—
122#683C2×A412yes—
123C4×Dic5—32no—
124C4×Dic5—32no—
125C4×Dic5C2×A412yes—
126Q82—51no—
127#555—152no—
128#555—151no—
129#550—52no—
130#550—51no—
131#550—151no—
132#532—151no—
133#532—152no—
134C4×#41—52no—
135C4×#41—151no—
136C4×#41C3×D511yescharacteristic
137C22×C3×C5—41no—
138C3×C4×C5—42no—
139C3×C4×C5—42no—
140C3×C4×C5—81no—
141C3×Dic5—44no—
142C3×Dic5—82no—
143C2×C3×Q8—201no—
144#41⋊C3—21no—
145#41⋊C3—102no—
146C2×SL(2,𝔽3)C22×D511yescharacteristic
147C3×C4⋊C4—202no—
148C3×C4⋊C4—201no—
149C3×C42—41no—
150C3×C42—202no—
151C23×C5—31no—
152C22×D5—32no—
153C5×Q8—31no—
154C5×Q8C3×#4111yescharacteristic
155C5×D4—61no—
156C5×D4—61no—
157C2×C4×C5—31no—
158C2×C4×C5—31no—
159C2×C4×C5—31no—
160C2×C4×C5—61no—
161C2×C4×C5—61no—
162C2×C4×C5—62no—
163C2×C4×C5—62no—
164C2×C4×C5C22×A411yescharacteristic
165C2×C4×C5C22×A411yescharacteristic
166C2×C4×C5C22×A411yescharacteristic
167C22⋊D5—64no—
168C22⋊D5—124no—
169C2×Dic5—32no—
170C2×Dic5C22×A412yes—
171C2×Dic5C22×A412yes—
172D20—62no—
173D20—62no—
174D20—62no—
175D20—122no—
176C4×D5—62no—
177C4×D5—62no—
178C4×D5—64no—
179C4×D5—64no—
180Dic10—122no—
181Dic10#41⋊C314yescomplemented
182C2×#41—52no—
183C2×#41—151no—
184C2×#41—152no—
185C2×#41C2×C3×D511yescharacteristic
186C4⋊Q8—151no—
187C4⋊Q8—152no—
188#127—151no—
189#127—152no—
190#124—152no—
191#124—154no—
192#121—154no—
193#121—152no—
194#121—152no—
195#121—152no—
196#121—152no—
197C4×Q8—51no—
198C4×Q8—52no—
199C4×Q8—151no—
200C4×Q8C2×C3×D511yescharacteristic
201C4×Q8C2×C3×D511yescharacteristic
202C4×D4—31no—
203C4×D4—31no—
204C4×D4—151no—
205C4×D4—152no—
206C4×D4—152no—
207C4×D4—152no—
208C4×D4—152no—
209C4×D4—151no—
210#117—31no—
211#117—152no—
212#117—151no—
213#117—152no—
214C2×C42—31no—
215C2×C42—152no—
216C2×C3×C5—41no—
217C2×C3×C5—41no—
218C2×C3×C5—41no—
219C3×Q8—204no—
220C2×C3×C4—41no—
221C2×C3×C4—41no—
222C2×C3×C4—41no—
223C2×C3×C4—202no—
224C2×C3×C4—202no—
225SL(2,𝔽3)#70911yescharacteristic, complemented
226C22×C5—61no—
227C22×C5—61no—
228C22×C5—61no—
229C22×C5C23×A411yescharacteristic
230C2×D5—62no—
231C2×D5—62no—
232C2×D5—62no—
233C4×C5—21no—
234C4×C5—31no—
235C4×C5—31no—
236C4×C5—61no—
237C4×C5—61no—
238C4×C5C2×#41⋊C312yes—
239C4×C5#99412yes—
240Dic5—22no—
241Dic5—62no—
242Dic5C2×#41⋊C314yes—
243#41—21no—
244#41—61no—
245#41—102no—
246#41—304no—
247#41—302no—
248#41—604no—
249C2×Q8—51no—
250C2×Q8—152no—
251C2×Q8C22×C3×D511yescharacteristic
252C2×D4—31no—
253C2×D4—152no—
254C2×D4—152no—
255C2×D4—152no—
256C2×D4—154no—
257C2×D4—151no—
258C22×C4—31no—
259C22×C4—31no—
260C22×C4—31no—
261C22×C4—152no—
262C22×C4—152no—
263C22×C4—152no—
264C4⋊C4—31no—
265C4⋊C4—31no—
266C4⋊C4—52no—
267C4⋊C4—51no—
268C4⋊C4—152no—
269C4⋊C4—152no—
270C4⋊C4—151no—
271C22⋊C4—32no—
272C22⋊C4—152no—
273C22⋊C4—154no—
274C22⋊C4—152no—
275C22⋊C4—152no—
276C42—31no—
277C42—31no—
278C42—31no—
279C42—52no—
280C42—152no—
281C42—152no—
282C42D5×A411yescharacteristic
283C3×C5—41noHall
284C22×C3—41no—
285C3×C4—42no—
286C3×C4—42no—
287C3×C4—81no—
288C3×C4—204no—
289C3×C4—402no—
290C2×C5—121no—
291C2×C5C22×#41⋊C311yescharacteristic
292C2×C5A4×#4111yescharacteristic
293C2×C5C2×#99411yescharacteristic
294D5—122no—
295C23—31no—
296C23—152no—
297Q8—31no—
298Q8—54no—
299Q8—602no—
300Q8C3×#70911yescharacteristic
301D4—61no—
302D4—61no—
303D4—302no—
304D4—302no—
305D4—304no—
306D4—302no—
307D4—602no—
308D4—604no—
309C2×C4—31no—
310C2×C4—31no—
311C2×C4—31no—
312C2×C4—52no—
313C2×C4—52no—
314C2×C4—61no—
315C2×C4—61no—
316C2×C4—62no—
317C2×C4—62no—
318C2×C4—152no—
319C2×C4—302no—
320C2×C4—302no—
321C2×C4—304no—
322C2×C4—304no—
323C2×C4C2×D5×A411yescharacteristic
324C2×C4C2×D5×A411yescharacteristic
325C2×C4C2×D5×A411yescharacteristic
326C2×C3—41no—
327C2×C3—41no—
328C2×C3—41no—
329C5#8560411yescharacteristic, complemented, Sylow
330C22—61no—
331C22—61no—
332C22—61no—
333C22—302no—
334C22—302no—
335C22—302no—
336C22C22×D5×A411yescharacteristic
337C4—21no—
338C4—31no—
339C4—31no—
340C4—54no—
341C4—61no—
342C4—61no—
343C4—102no—
344C4—302no—
345C4C5⋊#99412yes—
346C4D5×#41⋊C312yes—
347C3—41noSylow
348C2—121no—
349C2—602no—
350C2A4×#70911yescharacteristic
351C2C2×C5⋊#99411yescharacteristic
352C2C2×D5×#41⋊C311yescharacteristic
subgroupG/H
centerC2×C4C2×D5×A4
commutator, cosocleC2×C5×Q8C23×C3
FrattiniC22C22×D5×A4
FittingC4×C5×#41C2×C3
socleC22×C5C23×A4
hypercenterC42D5×A4
hypocenterC5×Q8C3×#41
LCSG ▷ C2×C5×Q8 ▷ C5×Q8
UCSC1 ◁ C2×C4 ◁ C42

Complements

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

NH ≅ G/Ntype
1. C4×C5×#41⋊C3C2semidirect
2. #66168⋊C3C2semidirect
3. C4⋊(C5⋊(#41⋊C3))C2semidirect
4. C4×C5⋊(#41⋊C3)C2semidirect
5. SL(2,𝔽3)×Dic10C2semidirect
6. #10617746C3semidirect
37. C4×#41⋊C3D5semidirect
79. C5×SL(2,𝔽3)#41semidirect
181. Dic10#41⋊C3semidirect
225. SL(2,𝔽3)#709semidirect
329. C5#85604semidirect

Maximal subgroups

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

subgroupindex
C4×C5×#41⋊C32
#66168⋊C32
C4⋊(C5⋊(#41⋊C3))2
C4×C5⋊(#41⋊C3)2
SL(2,𝔽3)×Dic102
#106177463
C3×C4×Dic104
#856045