Finite Group Databasebeta

← #14513733

Subgroups — #14513733

Order 1920.

556 conjugacy classes of subgroups; only the 69 normal subgroups are shown.

normal subgroups only

This group has 556 conjugacy classes of subgroups — too many to draw. Only its 69 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×#91953C212yesmaximal, complemented
2C2×C4×C5×SL(2,𝔽3)C211yesmaximal, characteristic, complemented
3#11752114C212yesmaximal, complemented
4C4⋊(C5⋊GL(2,𝔽3))C211yesmaximal, characteristic, complemented
5C4⋊(C5⋊#226)C211yesmaximal, characteristic, complemented
6#10614172—31nomaximal, Hall
7C22×C5×SL(2,𝔽3)C2211yescharacteristic
8#91953—24no—
9C2×C5⋊GL(2,𝔽3)C2212yes—
10C2×C5⋊#226C2212yes—
11#91819—41nomaximal
12C4×C5×SL(2,𝔽3)C2211yescharacteristic
13C4×C5×SL(2,𝔽3)C2211yescharacteristic
14#85331—51nomaximal, Hall
15C2×C4×C5×Q8D311yescharacteristic, complemented
16C2×#4088—32no—
17#65811—31no—
18#65320—31no—
19#65320—31no—
20#65317—32no—
21#65317—32no—
22#65316—31no—
23#65316—31no—
24#65255—32no—
25#65213—31no—
26C22×C3×C4×C5—41no—
27C2×#1220—42no—
28C2×D60—41no—
29C2×C5×SL(2,𝔽3)D412yescomplemented
30C2×C5×SL(2,𝔽3)C2311yescharacteristic
31C5⋊GL(2,𝔽3)—22no—
32C5⋊#226—22no—
33#6714—42no—
34C4⋊Dic15—41no—
35C2×#983—52no—
36C2×C4×SL(2,𝔽3)D511yescharacteristic, complemented
37#5395—52no—
38C4⋊GL(2,𝔽3)—51no—
39C4⋊#226—51no—
40C22×C5×Q8C2×D311yescharacteristic
41C2×#709—32no—
42C4×C5×Q8—31no—
43C4×C5×Q8—31no—
44C4×C5×Q8C2×D311yescharacteristic
45C4×C5×Q8C2×D311yescharacteristic
46C2×C5×C4⋊C4—31no—
47C2×C42×C5—31no—
48C2×#689—32no—
49C2×#689—32no—
50#4088—64no—
51#4088—64no—
52C2×#688—32no—
53C2×#688—32no—
54#4076—61no—
55#4076—61no—
56C2×#681—32no—
57#4021—32no—
58#4020—31no—
59C4×D20—61no—
60C4×D20—61no—
61#4015—31no—
62#3942—32no—
63#3942—32no—
64#3941—32no—
65#3941—32no—
66C4⋊#168—31no—
67C4⋊#168—31no—
68C4⋊#168—32no—
69#3037—151noSylow
70C23×C3×C5—41no—
71C22×D15—42no—
72C2×C3×C4×C5—41no—
73C2×C3×C4×C5—41no—
74C2×C3×C4×C5—82no—
75#1220—84no—
76C2×Dic15—42no—
77D60—82no—
78C5×SL(2,𝔽3)C2×D411yescharacteristic
79C22×SL(2,𝔽3)C2×D511yescharacteristic
80#983—104no—
81C2×GL(2,𝔽3)—52no—
82C2×#226—52no—
83#930—201no—
84C4×SL(2,𝔽3)C2×D511yescharacteristic
85C4×SL(2,𝔽3)C2×D511yescharacteristic
86C2×C5×Q8—31no—
87C2×C5×Q8—32no—
88C2×C5×Q8D1212yescomplemented
89C2×C5×Q8C22×D311yescharacteristic
90C22×C4×C5—31no—
91C22×C4×C5—31no—
92C22×C4×C5S411yescharacteristic
93C2×C22⋊D5—62no—
94#709—64no—
95#709—64no—
96C2×D20—32no—
97C2×D20—61no—
98C2×D20—61no—
99C2×C4×D5—62no—
100C2×Dic10—32no—
101C5×C4⋊C4—31no—
102C5×C4⋊C4—31no—
103C5×C4⋊C4—61no—
104C42×C5—31no—
105C42×C5—31no—
106C42×C5—31no—
107C42×C5—31no—
108#689—64no—
109#689—62no—
110#689—62no—
111#688—64no—
112#688—62no—
113#688—62no—
114#685—62no—
115#685—62no—
116C4⋊Dic5—61no—
117C4⋊Dic5—61no—
118#681—64no—
119C2×#168—32no—
120C2×#168—32no—
121C2×#137—152no—
122#532—151no—
123C2×C4×Q8D1511yescharacteristic, complemented
124#464—152no—
125#464—152no—
126#462—151no—
127#462—151no—
128#460—151no—
129#460—151no—
130#423—151no—
131#417—152no—
132C22×C3×C5—42no—
133C22×C3×C5—41no—
134C22×C3×C5—81no—
135C22×C3×C5—81no—
136C2×D15—82no—
137C2×D15—82no—
138C2×D15—82no—
139C3×C4×C5—42no—
140C3×C4×C5—81no—
141Dic15—82no—
142C22×C3×C4—41no—
143C2×#67—202no—
144C2×D12—201no—
145C2×SL(2,𝔽3)D2012yescomplemented
146C2×SL(2,𝔽3)C22×D511yescharacteristic
147GL(2,𝔽3)—102no—
148#226—102no—
149(C2×D3)⋊C4—202no—
150C4⋊Dic3—201no—
151C23×C5C2×S411yescharacteristic
152C22×D5—62no—
153C5×Q8—32no—
154C5×Q8—31no—
155C5×Q8—61no—
156C5×Q8C2×D1211yescharacteristic
157C2×C4×C5—22no—
158C2×C4×C5—32no—
159C2×C4×C5—32no—
160C2×C4×C5—31no—
161C2×C4×C5—31no—
162C2×C4×C5—31no—
163C2×C4×C5—31no—
164C2×C4×C5—121no—
165C2×C4×C5C2×S411yescharacteristic
166C2×C4×C5C2×S411yescharacteristic
167C22⋊D5—124no—
168C2×Dic5—62no—
169D20—62no—
170D20—62no—
171D20—122no—
172D20—122no—
173C4×D5—124no—
174Dic10—62no—
175Dic10—62no—
176#168—62no—
177#168—62no—
178C2×#41—152no—
179C22×Q8C2×D1511yescharacteristic
180#137—304no—
181#137—304no—
182C2×Q16—152no—
183C2×Q16—152no—
184C2×SD16—152no—
185C2×SD16—152no—
186C2×M16—152no—
187C4⋊Q8—151no—
188#127—151no—
189#124—152no—
190#121—301no—
191#121—301no—
192C4×Q8—31no—
193C4×Q8—31no—
194C4×Q8C2×D1511yescharacteristic
195C4×Q8C2×D1511yescharacteristic
196C4×D4—301no—
197C4×D4—301no—
198C2×C4⋊C4—31no—
199C2×C42—31no—
200C4⋊C8—151no—
201C4⋊C8—151no—
202C4⋊C8—152no—
203#103—152no—
204#103—152no—
205#102—152no—
206#102—152no—
207C2×C3×C5—41no—
208C2×C3×C5—41no—
209C2×C3×C5—41no—
210C2×C3×C5—82no—
211D15—162no—
212C23×C3—41no—
213C22×D3—202no—
214C2×C3×C4—41no—
215C2×C3×C4—41no—
216C2×C3×C4—82no—
217#67—404no—
218C2×Dic3—202no—
219D12—402no—
220SL(2,𝔽3)C2×D2011yescharacteristic
221C22×C5—21no—
222C22×C5—21no—
223C22×C5C4⋊S412yes—
224C22×C5C22×S411yescharacteristic
225C2×D5—122no—
226C2×D5—122no—
227C2×D5—122no—
228C4×C5—21no—
229C4×C5—31no—
230C4×C5—31no—
231C4×C5—32no—
232C4×C5—61no—
233C4×C5—61no—
234C4×C5#98312yescomplemented
235Dic5—122no—
236#41—304no—
237#41—304no—
238C2×Q8—31no—
239C2×Q8—32no—
240C2×Q8—152no—
241C2×Q8D6012yescomplemented
242C2×Q8C22×D1511yescharacteristic
243C2×D4—152no—
244C2×D4—301no—
245C2×D4—302no—
246C2×D4—301no—
247C22×C4—31no—
248C22×C4—31no—
249C22×C4—302no—
250C22×C4C5⋊S411yescharacteristic
251Q16—304no—
252Q16—302no—
253Q16—302no—
254SD16—304no—
255SD16—302no—
256SD16—302no—
257M16—304no—
258C2×C8—152no—
259C2×C8—152no—
260C4⋊C4—31no—
261C4⋊C4—31no—
262C4⋊C4—61no—
263C4⋊C4—301no—
264C4⋊C4—301no—
265C22⋊C4—302no—
266C22⋊C4—302no—
267C42—31no—
268C42—31no—
269C42—31no—
270C42—31no—
271C3×C5—41noHall
272C22×C3—42no—
273C22×C3—41no—
274C22×C3—81no—
275C22×C3—81no—
276C2×D3—402no—
277C2×D3—402no—
278C2×D3—402no—
279C3×C4—42no—
280C3×C4—81no—
281Dic3—402no—
282C2×C5—22no—
283C2×C5C2×C4⋊S411yescharacteristic
284C2×C5C2×#98311yescharacteristic
285C2×C5#592311yescharacteristic
286D5—242no—
287C23—302no—
288C23C2×C5⋊S411yescharacteristic
289Q8—32no—
290Q8—31no—
291Q8—61no—
292Q8—302no—
293Q8—302no—
294Q8C2×D6011yescharacteristic
295D4—302no—
296D4—302no—
297D4—602no—
298D4—604no—
299D4—602no—
300C2×C4—22no—
301C2×C4—32no—
302C2×C4—32no—
303C2×C4—31no—
304C2×C4—31no—
305C2×C4—31no—
306C2×C4—31no—
307C2×C4—121no—
308C2×C4—302no—
309C2×C4—604no—
310C2×C4C2×C5⋊S411yescharacteristic
311C2×C4C2×C5⋊S411yescharacteristic
312C8—302no—
313C8—302no—
314C2×C3—41no—
315C2×C3—41no—
316C2×C3—41no—
317C2×C3—82no—
318D3—802no—
319C5#8533111yescharacteristic, complemented, Sylow
320C22—21no—
321C22—21no—
322C22—602no—
323C22—602no—
324C22—602no—
325C22C4⋊(C5⋊S4)12yes—
326C22C22×C5⋊S411yescharacteristic
327C4—21no—
328C4—31no—
329C4—32no—
330C4—31no—
331C4—61no—
332C4—61no—
333C4—602no—
334C4#9195312yescomplemented
335C3—41noSylow
336C2—22no—
337C2—1202no—
338C2C2×C4⋊(C5⋊S4)11yescharacteristic
339C2C2×#9195311yescharacteristic
340C2#1175690711yescharacteristic
subgroupG/H
center, FrattiniC22C22×C5⋊S4
commutator, cosocleC2×C5×SL(2,𝔽3)C23
FittingC2×C4×C5×Q8D3
socleC22×C5C22×S4
hypercenterC22×C4C5⋊S4
hypocenterC5×SL(2,𝔽3)C2×D4
LCSG ▷ C2×C5×SL(2,𝔽3) ▷ C5×SL(2,𝔽3)
UCSC1 ◁ C22 ◁ C22×C4

Complements

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

NH ≅ G/Ntype
1. C2×#91953C2semidirect
2. C2×C4×C5×SL(2,𝔽3)C2semidirect
3. #11752114C2semidirect
4. C4⋊(C5⋊GL(2,𝔽3))C2semidirect
5. C4⋊(C5⋊#226)C2semidirect
15. C2×C4×C5×Q8D3semidirect
29. C2×C5×SL(2,𝔽3)D4semidirect
36. C2×C4×SL(2,𝔽3)D5semidirect
88. C2×C5×Q8D12semidirect
123. C2×C4×Q8D15semidirect
145. C2×SL(2,𝔽3)D20semidirect
234. C4×C5#983semidirect
241. C2×Q8D60semidirect
319. C5#85331semidirect
334. C4#91953semidirect

Maximal subgroups

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

subgroupindex
C2×#919532
C2×C4×C5×SL(2,𝔽3)2
#117521142
C4⋊(C5⋊GL(2,𝔽3))2
C4⋊(C5⋊#226)2
#106141723
#918194
#853315