Finite Group Databasebeta

← #14513758

Subgroups — #14513758

Order 1920.

602 conjugacy classes of subgroups; only the 65 normal subgroups are shown.

normal subgroups only

This group has 602 conjugacy classes of subgroups — too many to draw. Only its 65 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×#91953C211yesmaximal, characteristic, complemented
2C22×C5⋊GL(2,𝔽3)C211yesmaximal, characteristic, complemented
3C5×#519⋊C3C211yesmaximal, characteristic, complemented
4#11752114C212yesmaximal, complemented
5C4⋊(C5⋊GL(2,𝔽3))C212yesmaximal, complemented
6#10614282—31nomaximal, Hall
7C22×C5×SL(2,𝔽3)C2211yescharacteristic
8#91953—22no—
9C2×C5⋊GL(2,𝔽3)—22no—
10C2×C5⋊GL(2,𝔽3)C2212yes—
11C2×C5⋊GL(2,𝔽3)C2211yescharacteristic, complemented
12C2×C5⋊#226C2211yescharacteristic, complemented
13#91771—41nomaximal
14C4×C5×SL(2,𝔽3)C2212yescomplemented
15#85355—51nomaximal, Hall
16C5×#519D311yescharacteristic, complemented
17C2×#4088—31no—
18C22×#688—31no—
19#65859—31no—
20#65317—32no—
21#65317—32no—
22#65316—32no—
23#65316—32no—
24#65291—31no—
25#65290—31no—
26#65288—31no—
27C23×D15—41no—
28C2×#1220—41no—
29C2×D60—42no—
30C2×C5×SL(2,𝔽3)D412yescomplemented
31C2×C5×SL(2,𝔽3)C2311yescharacteristic
32C5⋊GL(2,𝔽3)—22no—
33C5⋊GL(2,𝔽3)—21no—
34C5⋊#226—21no—
35C3×C5×C22⋊C4—41no—
36#6714—42no—
37C2×#983—51no—
38C22×GL(2,𝔽3)—51no—
39#519⋊C3D511yescharacteristic, complemented
40#5395—52no—
41C4⋊GL(2,𝔽3)—52no—
42C22×C5×Q8C2×D311yescharacteristic
43C2×#709—31no—
44C22×D20—31no—
45C4×C5×Q8—32no—
46C4×C5×Q8C2×D312yescomplemented
47C5×#117—31no—
48C5×#117—31no—
49C2×#689—31no—
50C2×#689—31no—
51#4088—62no—
52#4088—62no—
53C2×#688—31no—
54C2×#688—32no—
55C2×#688—31no—
56C2×#688—32no—
57C2×#688—62no—
58C2×#688—62no—
59C2×#681—31no—
60C22×#168—31no—
61#4041—62no—
62#4028—61no—
63#4028—61no—
64#4021—32no—
65#4020—32no—
66#3942—32no—
67#3941—32no—
68#3941—32no—
69#3941—32no—
70C4⋊#168—32no—
71C4⋊#168—32no—
72#3043—151noSylow
73C23×C3×C5—41no—
74C22×D15—42no—
75C22×D15—41no—
76C22×D15—84no—
77C22×D15—82no—
78C2×C3×C4×C5—42no—
79#1220—82no—
80C2×Dic15—41no—
81D60—84no—
82C5×SL(2,𝔽3)C2×D411yescharacteristic
83C22×SL(2,𝔽3)C2×D511yescharacteristic
84#983—102no—
85C2×GL(2,𝔽3)—51no—
86C2×GL(2,𝔽3)—52no—
87C2×GL(2,𝔽3)—102no—
88C2×#226—51no—
89#882—201no—
90C4×SL(2,𝔽3)C2×D512yescomplemented
91C23×D5—61no—
92C2×C5×Q8—31no—
93C2×C5×Q8—32no—
94C2×C5×Q8D1212yescomplemented
95C2×C5×Q8C22×D311yescharacteristic
96C22×C4×C5—31no—
97C2×C22⋊D5—61no—
98#709—62no—
99#709—62no—
100C2×D20—32no—
101C2×D20—31no—
102C2×D20—62no—
103C2×D20—62no—
104C2×D20—61no—
105C2×D20—62no—
106C2×D20—62no—
107C2×C4×D5—61no—
108C2×Dic10—31no—
109C5×C4⋊C4—32no—
110C5×C4⋊C4—32no—
111C5×C22⋊C4—31no—
112C5×C22⋊C4S411yescharacteristic
113C42×C5—32no—
114C42×C5—61no—
115#689—62no—
116#689—61no—
117#689—61no—
118#688—64no—
119#688—62no—
120#688—62no—
121#688—61no—
122#688—62no—
123#688—61no—
124#685—62no—
125#685—62no—
126#681—62no—
127C2×#168—31no—
128C2×#168—31no—
129C2×#168—32no—
130C2×#168—62no—
131C2×#137—151no—
132C22×SD16—151no—
133#535—151no—
134#519D1511yescharacteristic, complemented
135#464—152no—
136#464—152no—
137#460—152no—
138#460—152no—
139#424—151no—
140#418—151no—
141#416—151no—
142C22×C3×C5—42no—
143C22×C3×C5—41no—
144C22×C3×C5—82no—
145C2×D15—44no—
146C2×D15—82no—
147C2×D15—82no—
148C2×D15—81no—
149C2×D15—82no—
150C2×D15—88no—
151C3×C4×C5—82no—
152Dic15—81no—
153C23×D3—201no—
154C2×#67—201no—
155C2×D12—202no—
156C2×SL(2,𝔽3)D2012yescomplemented
157C2×SL(2,𝔽3)C22×D511yescharacteristic
158GL(2,𝔽3)—102no—
159GL(2,𝔽3)—101no—
160#226—101no—
161C3×C22⋊C4—41no—
162(C2×D3)⋊C4—202no—
163C23×C5C2×S411yescharacteristic
164C22×D5—62no—
165C22×D5—61no—
166C22×D5—122no—
167C22×D5—242no—
168C5×Q8—32no—
169C5×Q8—31no—
170C5×Q8—61no—
171C5×Q8C2×D1211yescharacteristic
172C2×C4×C5—32no—
173C2×C4×C5—32no—
174C2×C4×C5—31no—
175C2×C4×C5—31no—
176C2×C4×C5—32no—
177C2×C4×C5C2×S412yes—
178C22⋊D5—122no—
179C2×Dic5—61no—
180D20—62no—
181D20—61no—
182D20—62no—
183D20—61no—
184D20—122no—
185D20—242no—
186D20—242no—
187C4×D5—122no—
188Dic10—61no—
189Dic10—61no—
190#168—62no—
191#168—61no—
192#168—61no—
193C2×#41—151no—
194C22×Q8C2×D1511yescharacteristic
195C22×D4—151no—
196#137—302no—
197#137—302no—
198C2×Q16—151no—
199C2×Q16—151no—
200C2×SD16—152no—
201C2×SD16—151no—
202C2×SD16—152no—
203C2×SD16—151no—
204C2×SD16—302no—
205C2×SD16—302no—
206C2×M16—151no—
207C22×C8—151no—
208#127—152no—
209#124—152no—
210#121—302no—
211#120—301no—
212#120—301no—
213C4×Q8—32no—
214C4×Q8C2×D1512yescomplemented
215#117—31no—
216#117—31no—
217C4⋊C8—152no—
218C4⋊C8—152no—
219#103—152no—
220#102—152no—
221#102—152no—
222#102—152no—
223C2×C3×C5—41no—
224C2×C3×C5—42no—
225C2×C3×C5—82no—
226D15—84no—
227D15—161no—
228C23×C3—41no—
229C22×D3—202no—
230C22×D3—201no—
231C22×D3—402no—
232C22×D3—404no—
233C2×C3×C4—42no—
234#67—402no—
235C2×Dic3—201no—
236D12—404no—
237SL(2,𝔽3)C2×D2011yescharacteristic
238C22×C5—22no—
239C22×C5C4⋊S412yes—
240C22×C5C22×S411yescharacteristic
241C2×D5—121no—
242C2×D5—122no—
243C2×D5—122no—
244C2×D5—122no—
245C2×D5—122no—
246C2×D5—244no—
247C4×C5—22no—
248C4×C5—31no—
249C4×C5—31no—
250C4×C5—32no—
251C4×C5—62no—
252Dic5—121no—
253C24—301no—
254#41—302no—
255#41—302no—
256C2×Q8—31no—
257C2×Q8—32no—
258C2×Q8—151no—
259C2×Q8D6012yescomplemented
260C2×Q8C22×D1511yescharacteristic
261C2×D4—152no—
262C2×D4—151no—
263C2×D4—302no—
264C2×D4—302no—
265C2×D4—301no—
266C2×D4—302no—
267C2×D4—301no—
268C2×D4—302no—
269C22×C4—31no—
270C22×C4—301no—
271Q16—302no—
272Q16—301no—
273Q16—301no—
274SD16—304no—
275SD16—302no—
276SD16—302no—
277SD16—301no—
278SD16—302no—
279SD16—301no—
280M16—302no—
281C2×C8—151no—
282C2×C8—151no—
283C2×C8—152no—
284C2×C8—302no—
285C4⋊C4—32no—
286C4⋊C4—32no—
287C22⋊C4—31no—
288C22⋊C4—302no—
289C22⋊C4—302no—
290C22⋊C4C5⋊S411yescharacteristic
291C42—32no—
292C42—61no—
293C3×C5—41noHall
294C22×C3—42no—
295C22×C3—41no—
296C22×C3—82no—
297C2×D3—204no—
298C2×D3—402no—
299C2×D3—402no—
300C2×D3—402no—
301C2×D3—401no—
302C2×D3—408no—
303C3×C4—82no—
304Dic3—401no—
305C2×C5—22no—
306C2×C5C2×C4⋊S411yescharacteristic
307C2×C5#592612yes—
308D5—242no—
309D5—241no—
310C23—302no—
311C23—301no—
312C23—602no—
313C23—1202no—
314C23C2×C5⋊S411yescharacteristic
315Q8—32no—
316Q8—31no—
317Q8—61no—
318Q8—301no—
319Q8—301no—
320Q8C2×D6011yescharacteristic
321D4—302no—
322D4—301no—
323D4—302no—
324D4—301no—
325D4—602no—
326D4—602no—
327D4—1202no—
328D4—1202no—
329C2×C4—32no—
330C2×C4—32no—
331C2×C4—31no—
332C2×C4—31no—
333C2×C4—32no—
334C2×C4—301no—
335C2×C4—602no—
336C2×C4C2×C5⋊S412yes—
337C8—302no—
338C8—301no—
339C8—301no—
340C2×C3—41no—
341C2×C3—42no—
342C2×C3—82no—
343D3—404no—
344D3—801no—
345C5#8535511yescharacteristic, complemented, Sylow
346C22—22no—
347C22—601no—
348C22—602no—
349C22—602no—
350C22—602no—
351C22—602no—
352C22—1204no—
353C22C4⋊(C5⋊S4)12yes—
354C22C22×C5⋊S411yescharacteristic
355C4—22no—
356C4—31no—
357C4—32no—
358C4—31no—
359C4—62no—
360C4—601no—
361C3—41noSylow
362C2—22no—
363C2—1202no—
364C2—1201no—
365C2C2×C4⋊(C5⋊S4)11yescharacteristic
366C2#1175691012yes—
subgroupG/H
center, FrattiniC22C22×C5⋊S4
commutator, cosocleC2×C5×SL(2,𝔽3)C23
FittingC5×#519D3
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. C22×C5⋊GL(2,𝔽3)C2semidirect
3. C5×#519⋊C3C2semidirect
4. #11752114C2semidirect
5. C4⋊(C5⋊GL(2,𝔽3))C2semidirect
11. C2×C5⋊GL(2,𝔽3)C22semidirect
12. C2×C5⋊#226C22semidirect
14. C4×C5×SL(2,𝔽3)C22semidirect
16. C5×#519D3semidirect
30. C2×C5×SL(2,𝔽3)D4semidirect
39. #519⋊C3D5semidirect
46. C4×C5×Q8C2×D3semidirect
90. C4×SL(2,𝔽3)C2×D5semidirect
94. C2×C5×Q8D12semidirect
134. #519D15semidirect
156. C2×SL(2,𝔽3)D20semidirect
214. C4×Q8C2×D15semidirect
259. C2×Q8D60semidirect
345. C5#85355semidirect

Maximal subgroups

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

subgroupindex
C2×#919532
C22×C5⋊GL(2,𝔽3)2
C5×#519⋊C32
#117521142
C4⋊(C5⋊GL(2,𝔽3))2
#106142823
#917714
#853555