Finite Group Databasebeta

← #14513942

Subgroups — #14513942

Order 1920.

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

normal subgroups only

This group has 624 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×C5×2−1+4⋊C3C211yesmaximal, characteristic, complemented
2C2×#91955C211yesmaximal, characteristic, complemented
3#11752145C211yesmaximal, characteristic, complemented
4#11752140C212yesmaximal, complemented
5#11752138C212yesmaximal, complemented
6#10615845—31nomaximal, Hall
7C5×2−1+4⋊C3—22no—
8C2×C5×#41⋊C3C2211yescharacteristic
9C22×C5×SL(2,𝔽3)C2212yes—
10#91955—22no—
11C2×C5⋊GL(2,𝔽3)C2211yescharacteristic
12C2×C5⋊#226C2211yescharacteristic
13#91827—41nomaximal
14#91184C2212yes—
15#85538—51nomaximal, Hall
16C2×C5×2−1+4D311yescharacteristic, complemented
17#66167—31no—
18C2×#4096—31no—
19C2×#4088—31no—
20#65528—32no—
21#65527—32no—
22#65523—31no—
23#65521—31no—
24#65517—32no—
25#65516—32no—
26C2×#3967—31no—
27C2×C3×C5×D4—41no—
28C2×#1220—42no—
29C2×C4×D15—41no—
30C5×#41⋊C3D412yescomplemented
31C2×C5×SL(2,𝔽3)—22no—
32C2×C5×SL(2,𝔽3)C2311yescharacteristic
33C5⋊GL(2,𝔽3)—21no—
34C5⋊#226—21no—
35#6719—42no—
36C4⋊Dic15—41no—
37C2×2−1+4⋊C3D511yescharacteristic, complemented
38C2×#985—51no—
39#5426—51no—
40#5421—52no—
41#5419—52no—
42C5×2−1+4—22no—
43C5×2−1+4—62no—
44C2×C5×#41—31no—
45C2×C5×#41—32no—
46C2×C5×#41C2×D311yescharacteristic
47C22×C5×Q8—31no—
48C22×C5×Q8C2×D312yes—
49C2×#709—31no—
50#4096—62no—
51#4096—62no—
52#4092—61no—
53C2×#689—32no—
54C2×#689—31no—
55#4088—64no—
56C2×#688—32no—
57C2×#688—31no—
58#4085—62no—
59#4084—61no—
60C2×#687—31no—
61C2×#686—31no—
62#4076—32no—
63C2×C4⋊Dic5—31no—
64#4070—32no—
65C2×#681—31no—
66C22×#168—31no—
67#3967—32no—
68#3967—32no—
69#3967—32no—
70#3964—32no—
71C22⋊#168—32no—
72C22⋊#168—32no—
73#3013—151noSylow
74C23×C3×C5—42no—
75C22×D15—41no—
76C3×C5×D4—82no—
77C2×C3×C4×C5—41no—
78#1220—84no—
79C2×Dic15—41no—
80C2×Dic15—42no—
81C4×D15—82no—
82C5×SL(2,𝔽3)C2×D411yescharacteristic
832−1+4⋊C3—22no—
84C2×#41⋊C3C2×D511yescharacteristic
85C22×SL(2,𝔽3)C2×D512yes—
86#985—102no—
87C2×GL(2,𝔽3)—51no—
88C2×#226—51no—
89#938—201no—
90#859—52no—
91C5×#41—32no—
92C5×#41—62no—
93C5×#41—62no—
94C5×#41—64no—
95C5×#41—64no—
96C5×#41#6712yescomplemented
97C2×C5×Q8—22no—
98C2×C5×Q8—31no—
99C2×C5×Q8—32no—
100C2×C5×Q8—34no—
101C2×C5×Q8—62no—
102C2×C5×Q8—62no—
103C2×C5×Q8C22×D311yescharacteristic
104C2×C5×D4—31no—
105C2×C5×D4—32no—
106C2×C5×D4S411yescharacteristic
107C22×C4×C5—31no—
108C22×C4×C5—32no—
109C22×C4×C5—61no—
110C2×C22⋊D5—61no—
111C2×C22⋊D5—62no—
112C22×Dic5—61no—
113#709—62no—
114#709—62no—
115C2×D20—31no—
116C2×C4×D5—61no—
117C2×Dic10—31no—
118C22⋊Dic5—62no—
119#689—61no—
120#689—64no—
121#689—61no—
122#688—61no—
123#688—64no—
124#688—61no—
125#687—62no—
126#686—62no—
127#685—62no—
128C4⋊Dic5—32no—
129C4⋊Dic5—61no—
130#683—62no—
131#681—62no—
132C2×#168—32no—
133C2×#168—31no—
134C2×#168—31no—
135C2×#168—62no—
136C2×2−1+4D1511yescharacteristic, complemented
137C2×#137—151no—
138C2×#135—151no—
139#537—151no—
140#452—152no—
141#451—152no—
142#449—152no—
143#448—152no—
144#417—151no—
145C2×#103—151no—
146#408—151no—
147C22×C3×C5—41no—
148C22×C3×C5—82no—
149C22×C3×C5—82no—
150C22×C3×C5—82no—
151C2×D15—42no—
152C2×D15—81no—
153C2×D15—81no—
154C3×C4×C5—42no—
155Dic15—81no—
156Dic15—82no—
157C2×C3×D4—41no—
158C2×#67—202no—
159C2×C4×D3—201no—
160#41⋊C3C22⋊D512yescomplemented
161C2×SL(2,𝔽3)—22no—
162C2×SL(2,𝔽3)C22×D511yescharacteristic
163GL(2,𝔽3)—101no—
164#226—101no—
165#217—202no—
166C4⋊Dic3—201no—
167C23×C5—31no—
168C23×C5C2×S412yes—
169C22×D5—61no—
170C5×Q8—31no—
171C5×Q8—34no—
172C5×Q8—62no—
173C5×Q8—62no—
174C5×Q8C2×#6711yescharacteristic
175C5×D4—22no—
176C5×D4—32no—
177C5×D4—61no—
178C5×D4—64no—
179C2×C4×C5—31no—
180C2×C4×C5—32no—
181C2×C4×C5—32no—
182C2×C4×C5—32no—
183C2×C4×C5—62no—
184C2×C4×C5—62no—
185C2×C4×C5—64no—
186C2×C4×C5C2×S411yescharacteristic
187C22⋊D5—122no—
188C22⋊D5—242no—
189C2×Dic5—61no—
190C2×Dic5—62no—
191C2×Dic5—241no—
192D20—61no—
193D20—61no—
194C4×D5—122no—
195Dic10—61no—
196Dic10—61no—
197#168—61no—
198#168—61no—
199#168—62no—
2002−1+4—22no—
2012−1+4—62no—
202C2×#41—31no—
203C2×#41—32no—
204C2×#41—151no—
205C2×#41C2×D1511yescharacteristic
206C22×Q8—31no—
207C22×Q8C2×D1512yes—
208#137—304no—
209#135—302no—
210#135—302no—
211C2×Q16—152no—
212C2×Q16—151no—
213C2×SD16—151no—
214C2×SD16—152no—
215C2×SD16—151no—
216C2×D8—151no—
217C2×M16—151no—
218C22×C8—151no—
219C22⋊Q8—152no—
220C22⋊Q8—301no—
221#121—152no—
222#121—302no—
223#121—301no—
224C2×C4⋊C4—151no—
225#103—152no—
226#103—152no—
227#103—152no—
228#102—152no—
229C22⋊C8—152no—
230C22⋊C8—152no—
231C2×C3×C5—41no—
232C2×C3×C5—41no—
233C2×C3×C5—41no—
234C2×C3×C5—162no—
235D15—82no—
236C23×C3—42no—
237C22×D3—201no—
238C3×D4—82no—
239C2×C3×C4—41no—
240#67—404no—
241C2×Dic3—202no—
242C2×Dic3—201no—
243C4×D3—402no—
244SL(2,𝔽3)C2×C22⋊D511yescharacteristic
245C22×C5—22no—
246C22×C5—22no—
247C22×C5—22no—
248C22×C5—32no—
249C22×C5—61no—
250C22×C5—61no—
251C22×C5C22×S411yescharacteristic
252C2×D5—121no—
253C2×D5—121no—
254C2×D5—121no—
255C4×C5—31no—
256C4×C5—31no—
257C4×C5—62no—
258C4×C5GL(2,ℤ/4)12yes—
259Dic5—122no—
260Dic5—121no—
261#41—32no—
262#41—62no—
263#41—62no—
264#41—64no—
265#41—64no—
266#41—302no—
267#41—302no—
268#41#122012yescomplemented
269C2×Q8—22no—
270C2×Q8—31no—
271C2×Q8—32no—
272C2×Q8—34no—
273C2×Q8—62no—
274C2×Q8—62no—
275C2×Q8—151no—
276C2×Q8C22×D1511yescharacteristic
277C2×D4—31no—
278C2×D4—32no—
279C2×D4—151no—
280C2×D4—301no—
281C2×D4—302no—
282C2×D4C5⋊S411yescharacteristic
283C22×C4—31no—
284C22×C4—32no—
285C22×C4—61no—
286C22×C4—301no—
287C22×C4—301no—
288Q16—301no—
289Q16—304no—
290Q16—301no—
291SD16—302no—
292SD16—301no—
293SD16—304no—
294SD16—301no—
295D8—302no—
296M16—302no—
297C2×C8—152no—
298C2×C8—151no—
299C2×C8—151no—
300C2×C8—302no—
301C4⋊C4—152no—
302C4⋊C4—302no—
303C4⋊C4—301no—
304C22⋊C4—302no—
305C22⋊C4—302no—
306C3×C5—41noHall
307C22×C3—41no—
308C22×C3—82no—
309C22×C3—82no—
310C22×C3—82no—
311C2×D3—202no—
312C2×D3—401no—
313C2×D3—401no—
314C3×C4—42no—
315Dic3—401no—
316Dic3—402no—
317C2×C5—42no—
318C2×C5—62no—
319C2×C5#592611yescharacteristic
320C2×C5#592711yescharacteristic
321C2×C5C2×GL(2,ℤ/4)11yescharacteristic
322D5—241no—
323C23—31no—
324C23—301no—
325C23C2×C5⋊S412yes—
326Q8—31no—
327Q8—34no—
328Q8—62no—
329Q8—62no—
330Q8—301no—
331Q8—301no—
332Q8C2×#122011yescharacteristic
333D4—22no—
334D4—32no—
335D4—61no—
336D4—64no—
337D4—301no—
338D4—301no—
339D4—602no—
340D4—1202no—
341C2×C4—31no—
342C2×C4—32no—
343C2×C4—32no—
344C2×C4—32no—
345C2×C4—62no—
346C2×C4—62no—
347C2×C4—64no—
348C2×C4—302no—
349C2×C4—301no—
350C2×C4—602no—
351C2×C4—1201no—
352C2×C4C2×C5⋊S411yescharacteristic
353C8—301no—
354C8—301no—
355C8—302no—
356C2×C3—41no—
357C2×C3—41no—
358C2×C3—41no—
359C2×C3—162no—
360D3—402no—
361C5#8553811yescharacteristic, complemented, Sylow
362C22—22no—
363C22—22no—
364C22—22no—
365C22—32no—
366C22—61no—
367C22—61no—
368C22—601no—
369C22—601no—
370C22—601no—
371C22C22×C5⋊S411yescharacteristic
372C4—31no—
373C4—31no—
374C4—62no—
375C4—602no—
376C4—601no—
377C4#9195812yes—
378C3—41noSylow
379C2—42no—
380C2—62no—
381C2—1201no—
382C2#1175691011yescharacteristic
383C2#1175691111yescharacteristic
384C2C2×#9195811yescharacteristic
subgroupG/H
center, FrattiniC22C22×C5⋊S4
commutator, cosocleC2×C5×SL(2,𝔽3)C23
FittingC2×C5×2−1+4D3
socleC22×C5C22×S4
hypercenterC2×D4C5⋊S4
hypocenterC5×SL(2,𝔽3)C2×D4
LCSG ▷ C2×C5×SL(2,𝔽3) ▷ C5×SL(2,𝔽3)
UCSC1 ◁ C22 ◁ C2×D4

Complements

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

NH ≅ G/Ntype
1. C2×C5×2−1+4⋊C3C2semidirect
2. C2×#91955C2semidirect
3. #11752145C2semidirect
4. #11752140C2semidirect
5. #11752138C2semidirect
16. C2×C5×2−1+4D3semidirect
30. C5×#41⋊C3D4semidirect
37. C2×2−1+4⋊C3D5semidirect
96. C5×#41#67semidirect
136. C2×2−1+4D15semidirect
160. #41⋊C3C22⋊D5semidirect
268. #41#1220semidirect
361. C5#85538semidirect

Maximal subgroups

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

subgroupindex
C2×C5×2−1+4⋊C32
C2×#919552
#117521452
#117521402
#117521382
#106158453
#918274
#855385