Finite Group Databasebeta

← #14513354

Subgroups — #14513354

Order 1920.

694 conjugacy classes of subgroups; only the 63 normal subgroups are shown.

normal subgroups only

This group has 694 conjugacy classes of subgroups — too many to draw. Only its 63 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×D5×SL(2,𝔽3)C211yesmaximal, characteristic, complemented
2C2×C5×#983C211yesmaximal, characteristic, complemented
3#11752139C211yesmaximal, characteristic, complemented
4#11751613C212yesmaximal, complemented
5#11751606C212yesmaximal, complemented
6#10615710—31nomaximal, Hall
7C22×C5×SL(2,𝔽3)C2211yescharacteristic
8C2×D5×SL(2,𝔽3)—22no—
9C2×D5×SL(2,𝔽3)C2212yes—
10C5×#983—22no—
11C2×C5×GL(2,𝔽3)C2211yescharacteristic, complemented
12C2×C5×#226C2211yescharacteristic, complemented
13#91563—41nomaximal
14#91184C2212yescomplemented
15#85507—51nomaximal, Hall
16C22×Q8×D5D311yescharacteristic, complemented
17C2×C5×#137—31no—
18#66162—31no—
19C2×#4097—31no—
20#65514—31no—
21#65475—32no—
22#65475—32no—
23#65459—32no—
24#65459—32no—
25#65415—31no—
26#65409—31no—
27C23×C3×D5—41no—
28C2×C5×#67—41no—
29C2×C5×SL(2,𝔽3)D412yescomplemented
30C2×C5×SL(2,𝔽3)C2311yescharacteristic
31C2×#1201—42no—
32D5×SL(2,𝔽3)—22no—
33C5×GL(2,𝔽3)—21no—
34C5×#226—21no—
35#6719—41no—
36#6665—42no—
37C23×SL(2,𝔽3)—51no—
38C2×#983D511yescharacteristic, complemented
39#5421—52no—
40#5420—51no—
41#5419—52no—
42C2×C5×#41—31no—
43C22×C5×Q8C2×D311yescharacteristic
44C2×Q8×D5—22no—
45C2×Q8×D5—32no—
46C2×Q8×D5—62no—
47C2×Q8×D5C2×D312yes—
48C22×C4×D5—31no—
49C22×Dic10—31no—
50C5×#137—62no—
51C5×#137—62no—
52C2×C5×Q16—31no—
53C2×C5×Q16—31no—
54C2×C5×SD16—31no—
55C2×C5×SD16—31no—
56C2×C5×M16—31no—
57#4097—64no—
58#4092—32no—
59C2×#689—32no—
60#4084—32no—
61#4083—61no—
62#4081—61no—
63C2×#687—32no—
64C4×C22⋊D5—62no—
65#4072—31no—
66C2×#681—31no—
67#3967—32no—
68#3967—32no—
69#3952—32no—
70#3952—32no—
71#3948—32no—
72#3948—32no—
73#3004—151noSylow
74C23×C3×C5—41no—
75C22×C5×D3—41no—
76C22×C3×D5—42no—
77C22×C3×D5—84no—
78C22×C3×D5—82no—
79C2×Dic15—42no—
80C5×#67—82no—
81C2×C5×Dic3—41no—
82C5×SL(2,𝔽3)C2×D411yescharacteristic
83#1201—84no—
84C22×SL(2,𝔽3)—52no—
85C22×SL(2,𝔽3)—102no—
86C22×SL(2,𝔽3)C2×D511yescharacteristic
87#983—22no—
88C2×GL(2,𝔽3)C2×D511yescharacteristic, complemented
89C2×#226C2×D511yescharacteristic, complemented
90#953—201no—
91#859—52no—
92C23×D5S411yescharacteristic
93C5×#41—62no—
94C5×#41—62no—
95C2×C5×Q8—31no—
96C2×C5×Q8—32no—
97C2×C5×Q8—31no—
98C2×C5×Q8#6712yescomplemented
99C2×C5×Q8C22×D311yescharacteristic
100C2×C5×D4—31no—
101C2×C5×D4—61no—
102C22×C4×C5—31no—
103C22×C4×C5—61no—
104C2×C22⋊D5—62no—
105C22×Dic5—31no—
106Q8×D5—22no—
107Q8×D5—62no—
108Q8×D5—64no—
109Q8×D5—64no—
110C2×C4×D5—32no—
111C2×C4×D5—32no—
112C2×C4×D5—62no—
113C2×C4×D5—62no—
114C2×Dic10—32no—
115C2×Dic10—34no—
116C2×Dic10—61no—
117C2×Dic10—62no—
118C5×Q16—62no—
119C5×Q16—61no—
120C5×Q16—61no—
121C5×SD16—62no—
122C5×SD16—61no—
123C5×SD16—61no—
124C5×M16—62no—
125C2×C5×C8—31no—
126C2×C5×C8—31no—
127C22⋊Dic5—62no—
128C22⋊Dic5—61no—
129#689—64no—
130#687—64no—
131#685—62no—
132C4⋊Dic5—32no—
133#683—62no—
134C4×Dic5—61no—
135#681—62no—
136C2×#168—32no—
137C23×Q8—51no—
138C2×#137—31no—
139C2×#137—151no—
140#525—151no—
141#451—152no—
142#451—152no—
143#448—152no—
144#448—152no—
145#419—151no—
146#419—151no—
147#407—151no—
148C22×C3×C5—42no—
149C22×C3×C5—41no—
150C22×C3×C5—82no—
151C2×C5×D3—81no—
152C2×C5×D3—82no—
153C2×C3×D5—44no—
154C2×C3×D5—82no—
155C2×C3×D5—82no—
156C2×C3×D5—88no—
157Dic15—82no—
158C5×Dic3—81no—
159C24×C3—201no—
160C2×#67—41no—
161C2×#67—202no—
162C2×SL(2,𝔽3)—102no—
163C2×SL(2,𝔽3)C22⋊D512yescomplemented
164C2×SL(2,𝔽3)C22×D511yescharacteristic
165GL(2,𝔽3)—21no—
166#226—21no—
167#217—202no—
168#217—201no—
169C23×C5—61no—
170C23×C5C2×S411yescharacteristic
171C22×D5—22no—
172C22×D5—24no—
173C22×D5C2×S412yes—
174C5×Q8—32no—
175C5×Q8—31no—
176C5×Q8—61no—
177C5×Q8—61no—
178C5×Q8—61no—
179C5×Q8C2×#6711yescharacteristic
180C5×D4—61no—
181C5×D4—61no—
182C5×D4—122no—
183C2×C4×C5—32no—
184C2×C4×C5—32no—
185C2×C4×C5—31no—
186C2×C4×C5—31no—
187C2×C4×C5—61no—
188C2×C4×C5—122no—
189C22⋊D5—124no—
190C2×Dic5—32no—
191C2×Dic5—62no—
192C2×Dic5—62no—
193C4×D5—62no—
194C4×D5—62no—
195C4×D5—64no—
196Dic10—34no—
197Dic10—64no—
198Dic10—62no—
199C5×C8—61no—
200C5×C8—61no—
201#168—62no—
202C2×#41—31no—
203C22×Q8—52no—
204C22×Q8—102no—
205C22×Q8—151no—
206C22×Q8—152no—
207C22×Q8—302no—
208C22×Q8D3×D511yescharacteristic
209C23×C4—151no—
210#137—62no—
211#137—62no—
212#137—304no—
213C2×Q16—31no—
214C2×Q16—31no—
215C2×Q16—152no—
216C2×SD16—31no—
217C2×SD16—31no—
218C2×SD16—152no—
219C2×M16—31no—
220C2×M16—151no—
221#123—301no—
222C22⋊Q8—152no—
223#121—152no—
224#120—301no—
225C4×D4—302no—
226#117—151no—
227#103—152no—
228#103—152no—
229#103—152no—
230#103—152no—
231C22⋊C8—152no—
232C22⋊C8—152no—
233C2×C3×C5—42no—
234C2×C3×C5—41no—
235C2×C3×C5—82no—
236C3×D5—84no—
237C5×D3—161no—
238C23×C3—41no—
239C23×C3—202no—
240C23×C3—402no—
241C23×C3—404no—
242C22×D3—41no—
243#67—82no—
244#67—404no—
245C2×Dic3—41no—
246C2×Dic3—202no—
247SL(2,𝔽3)C2×C22⋊D511yescharacteristic
248C22×C5—22no—
249C22×C5—122no—
250C22×C5—121no—
251C22×C5GL(2,ℤ/4)12yes—
252C22×C5C22×S411yescharacteristic
253C2×D5—28no—
254C2×D5—22no—
255C2×D5—22no—
256C2×D5#98314yescomplemented
257C4×C5—31no—
258C4×C5—31no—
259C4×C5—32no—
260C4×C5—121no—
261Dic5—62no—
262Dic5—122no—
263C24—51no—
264#41—62no—
265#41—62no—
266C2×Q8—31no—
267C2×Q8—32no—
268C2×Q8—31no—
269C2×Q8—102no—
270C2×Q8—152no—
271C2×Q8—154no—
272C2×Q8—301no—
273C2×Q8—302no—
274C2×Q8—304no—
275C2×Q8—302no—
276C2×Q8—304no—
277C2×Q8#120112yescomplemented
278C2×Q8C2×D3×D511yescharacteristic
279C2×D4—31no—
280C2×D4—61no—
281C2×D4—302no—
282C22×C4—31no—
283C22×C4—61no—
284C22×C4—152no—
285C22×C4—152no—
286C22×C4—151no—
287C22×C4—302no—
288C22×C4—302no—
289Q16—62no—
290Q16—61no—
291Q16—61no—
292Q16—304no—
293SD16—62no—
294SD16—61no—
295SD16—61no—
296SD16—304no—
297M16—62no—
298M16—302no—
299C2×C8—31no—
300C2×C8—31no—
301C2×C8—152no—
302C4⋊C4—152no—
303C4⋊C4—302no—
304C22⋊C4—302no—
305C22⋊C4—302no—
306C22⋊C4—301no—
307C42—301no—
308C3×C5—41noHall
309C22×C3—42no—
310C22×C3—41no—
311C22×C3—82no—
312C22×C3—204no—
313C22×C3—402no—
314C22×C3—402no—
315C22×C3—408no—
316C2×D3—81no—
317C2×D3—82no—
318Dic3—81no—
319Dic3—402no—
320C2×C5—22no—
321C2×C5—241no—
322C2×C5C2×#98312yes—
323C2×C5C2×GL(2,ℤ/4)11yescharacteristic
324D5—24no—
325C23—52no—
326C23—61no—
327C23—104no—
328C23—102no—
329C23D5×S411yescharacteristic
330Q8—32no—
331Q8—31no—
332Q8—61no—
333Q8—61no—
334Q8—61no—
335Q8—154no—
336Q8—302no—
337Q8—304no—
338Q8C2×#120111yescharacteristic
339D4—61no—
340D4—61no—
341D4—122no—
342D4—604no—
343C2×C4—32no—
344C2×C4—32no—
345C2×C4—31no—
346C2×C4—31no—
347C2×C4—61no—
348C2×C4—122no—
349C2×C4—152no—
350C2×C4—302no—
351C2×C4—302no—
352C2×C4—304no—
353C2×C4—302no—
354C2×C4—302no—
355C8—61no—
356C8—61no—
357C8—302no—
358C2×C3—42no—
359C2×C3—41no—
360C2×C3—82no—
361C2×C3—404no—
362D3—161no—
363C5#8550711yescharacteristic, complemented, Sylow
364C22—22no—
365C22—54no—
366C22—108no—
367C22—102no—
368C22—102no—
369C22—122no—
370C22—121no—
371C22#9190412yes—
372C22C2×D5×S411yescharacteristic
373C4—31no—
374C4—31no—
375C4—32no—
376C4—121no—
377C4—302no—
378C4—602no—
379C3—41noSylow
380C2—22no—
381C2—104no—
382C2—241no—
383C2D5×#98312yes—
384C2C2×#9190411yescharacteristic
subgroupG/H
center, FrattiniC22C2×D5×S4
commutator, cosocleC2×C5×SL(2,𝔽3)C23
FittingC22×C5×Q8C2×D3
socleC22×C5C22×S4
hypercenterC23D5×S4
hypocenterC5×SL(2,𝔽3)C2×D4
LCSG ▷ C2×C5×SL(2,𝔽3) ▷ C5×SL(2,𝔽3)
UCSC1 ◁ C22 ◁ C23

Complements

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

NH ≅ G/Ntype
1. C22×D5×SL(2,𝔽3)C2semidirect
2. C2×C5×#983C2semidirect
3. #11752139C2semidirect
4. #11751613C2semidirect
5. #11751606C2semidirect
11. C2×C5×GL(2,𝔽3)C22semidirect
12. C2×C5×#226C22semidirect
14. #91184C22semidirect
16. C22×Q8×D5D3semidirect
29. C2×C5×SL(2,𝔽3)D4semidirect
38. C2×#983D5semidirect
88. C2×GL(2,𝔽3)C2×D5semidirect
89. C2×#226C2×D5semidirect
98. C2×C5×Q8#67semidirect
163. C2×SL(2,𝔽3)C22⋊D5semidirect
256. C2×D5#983semidirect
277. C2×Q8#1201semidirect
363. C5#85507semidirect

Maximal subgroups

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

subgroupindex
C22×D5×SL(2,𝔽3)2
C2×C5×#9832
#117521392
#117516132
#117516062
#106157103
#915634
#855075