Finite Group Databasebeta

← #14306024

Subgroups — #14306024

Order 1920.

720 conjugacy classes of subgroups; only the 126 normal subgroups are shown.

normal subgroups only

This group has 720 conjugacy classes of subgroups — too many to draw. Only its 126 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
1#11752673C211yesmaximal, characteristic
2#11752341C211yesmaximal, characteristic, complemented
3C3×C42×AGL(1,𝔽5)C211yesmaximal, characteristic, complemented
4C2×#91158C212yesmaximal, complemented
5C2×#91156C212yesmaximal, complemented
6#10604559—31nomaximal, Hall
7C2×#6773C2211yescharacteristic
8C2×#6772C2211yescharacteristic
9C2×#6758C2212yes—
10C2×C3×C4×AGL(1,𝔽5)—22no—
11C2×C3×C4×AGL(1,𝔽5)C2212yes—
12#91764C411yescharacteristic, complemented
13C5×#875C411yescharacteristic, complemented
14C3×C42×D5C2211yescharacteristic
15#91451C411yescharacteristic, complemented
16#91448—22no—
17#91405—22no—
18#91376C411yescharacteristic, complemented
19#91233—22no—
20#91158—24no—
21#91156—24no—
22#69395—51nomaximal, Hall
23#66002—31no—
24C2×#4010—32no—
25C2×#4008—32no—
26C42×AGL(1,𝔽5)D311yescharacteristic, complemented
27#65682—31no—
28C22×C3×AGL(1,𝔽5)—22no—
29C2×D60C2×C411yescharacteristic
30C2×Dic30C2×C411yescharacteristic
31C2×C5×D12C2×C411yescharacteristic
32C2×C5×Dic6C2×C411yescharacteristic
33C2×C3×C4×D5D412yes—
34C2×C3×C4×D5C2311yescharacteristic
35C2×#1203—22no—
36C2×#1202—21no—
37C2×#1201—21no—
38C2×#1200—21no—
39C2×D3×Dic5—21no—
40#6773—22no—
41#6773—22no—
42#6772—22no—
43#6772—22no—
44C2×C3⋊#170—22no—
45#6758—22no—
46#6758—22no—
47C3×C4×AGL(1,𝔽5)—28no—
48C3×C4×AGL(1,𝔽5)—22no—
49C3×C4×AGL(1,𝔽5)D414yescomplemented
50C3×C42×C5C2×C411yescharacteristic
51#6714—22no—
52C5×(C2×D3)⋊C4—22no—
53C3×C4×Dic5C2×C411yescharacteristic
54C3×C4×Dic5D412yes—
55#6668—22no—
56#5477—51no—
57C3×C43—51no—
58C2×#805—52no—
59C2×#805—52no—
60#4924—51no—
61C2×#713—31no—
62C2×#711—31no—
63C2×C4×AGL(1,𝔽5)—22no—
64C2×C4×AGL(1,𝔽5)C2×D312yes—
65C2×#700—32no—
66C5×#124—31no—
67#4090—31no—
68#4086—31no—
69#4030—62no—
70#4027—62no—
71#4021—31no—
72C42×D5C2×D311yescharacteristic
73#4010—64no—
74#4008—64no—
75#4004—62no—
76#1962—151noSylow
77C22×D15—21no—
78C22×C5×D3—21no—
79C22×C3×D5C2×D411yescharacteristic
80C2×C3×AGL(1,𝔽5)—28no—
81C2×C3×AGL(1,𝔽5)—24no—
82C2×C3×C4×C5C22⋊C412yes—
83C2×C3×C4×C5C22×C411yescharacteristic
84C2×Dic15—21no—
85D60—22no—
86Dic30—22no—
87C2×C5×Dic3—21no—
88C5×D12—22no—
89C5×Dic6—22no—
90C2×C3×Dic5C22⋊C412yes—
91C2×C3×Dic5C2×D411yescharacteristic
92C3×C4×D5—24no—
93C3×C4×D5C2×D412yes—
94C3×C4×D5#4112yes—
95#1203—44no—
96#1202—42no—
97#1201—42no—
98#1200—41no—
99#1200—41no—
100D3×Dic5—41no—
101D3×Dic5—41no—
102C3⋊#170—42no—
103C2×#235—51no—
104C2×#235—51no—
105C2×C3×C42—51no—
106C2×C3×C42—52no—
107C2×C3×C42—102no—
108#930—102no—
109C2×#208—52no—
110#875—51no—
111#875AGL(1,𝔽5)11yescharacteristic, complemented
112#874—51no—
113C4×D12—102no—
114#869—51no—
115#805—104no—
116#805—104no—
117C4⋊(C3⋊C8)—102no—
118C22×AGL(1,𝔽5)—22no—
119C2×C5×Q8—31no—
120C2×C5×D4—31no—
121C2×C22⋊D5—61no—
122C2×C22⋊D5—62no—
123C22×Dic5—61no—
124#713—62no—
125#713—62no—
126#711—62no—
127#711—62no—
128C2×D20—31no—
129C2×D20—61no—
130C2×C4×D5—61no—
131C2×C4×D5#6712yes—
132C2×C4×D5C22×D311yescharacteristic
133C2×Dic10—31no—
134C2×#170—62no—
135C4×AGL(1,𝔽5)—28no—
136C4×AGL(1,𝔽5)—22no—
137C4×AGL(1,𝔽5)D1214yescomplemented
138#700—62no—
139#700—62no—
140C5×C22⋊C4—62no—
141C42×C5C4×D311yescharacteristic
142#685—62no—
143#683—62no—
144C4×Dic5C4×D311yescharacteristic
145C4×Dic5D1212yes—
146#532—151no—
147#423—151no—
148C2×#104—152no—
149C2×#104—152no—
150C43—51no—
151C22×C3×C5C2×C22⋊C411yescharacteristic
152C2×D15—41no—
153C2×D15—42no—
154C2×C5×D3—41no—
155C2×C5×D3—42no—
156C2×C3×D5—22no—
157C2×C3×D5#12112yes—
158C3×AGL(1,𝔽5)—28no—
159C3×C4×C5—22no—
160C3×C4×C5C4×D412yes—
161Dic15—41no—
162C3×Dic5#10414yescomplemented
163C3×Dic5#12411yescharacteristic
164C3×Dic5#12711yescharacteristic
165C5×Dic3—41no—
166C22×C3×C4—51no—
167C22×C3×C4—52no—
168C22×C3×C4—102no—
169C2×#67—101no—
170C2×#67—101no—
171#235—102no—
172#235—102no—
173#235—102no—
174#235—102no—
175C2×D12—51no—
176C2×D12—102no—
177C2×D12C2×AGL(1,𝔽5)11yescharacteristic
178C2×C4×D3—101no—
179C2×C4×D3—101no—
180C2×Dic6—51no—
181C2×Dic6C2×AGL(1,𝔽5)11yescharacteristic
182C3×C42—52no—
183C3×C42—51no—
184C3×C42—54no—
185C3×C42—102no—
186C3×C42—108no—
187C3×C42C2×AGL(1,𝔽5)11yescharacteristic
188(C2×D3)⋊C4—22no—
189(C2×D3)⋊C4—102no—
190C4⋊Dic3—102no—
191#208—102no—
192#208—102no—
193C2×C3⋊C8—102no—
194C23×C5—61no—
195C22×D5—61no—
196C22×D5C2×#6711yescharacteristic
197C2×AGL(1,𝔽5)—28no—
198C2×AGL(1,𝔽5)—24no—
199C5×Q8—62no—
200C5×D4—62no—
201C2×C4×C5—61no—
202C2×C4×C5(C2×D3)⋊C412yes—
203C2×C4×C5C2×C4×D311yescharacteristic
204C22⋊D5—122no—
205C22⋊D5—124no—
206C2×Dic5—61no—
207C2×Dic5—121no—
208C2×Dic5—121no—
209C2×Dic5(C2×D3)⋊C412yes—
210C2×Dic5C2×D1211yescharacteristic
211D20—62no—
212D20—122no—
213C4×D5—24no—
214C4×D5—121no—
215C4×D5—121no—
216C4×D5C2×D1212yes—
217C4×D5#23512yes—
218Dic10—62no—
219#170—122no—
220C2×#41—151no—
221C2×#41—151no—
222C2×M16—152no—
223C4⋊Q8—151no—
224#127—151no—
225#124—31no—
226#124—151no—
227#121—302no—
228C4×D4—302no—
229C2×C42—51no—
230C2×C42—52no—
231C2×C42—102no—
232C4⋊C8—302no—
233#104—304no—
234#104—304no—
235C2×C3×C5#39011yescharacteristic
236C2×C3×C5C2×#10412yes—
237D15—81no—
238C3×D5—22no—
239C5×D3—81no—
240C23×C3—51no—
241C22×D3—21no—
242C22×D3—101no—
243C2×C3×C4—52no—
244C2×C3×C4—51no—
245C2×C3×C4—52no—
246C2×C3×C4—52no—
247C2×C3×C4—104no—
248C2×C3×C4—104no—
249C2×C3×C4—108no—
250C2×C3×C4C22⋊AGL(1,𝔽5)12yes—
251C2×C3×C4C22×AGL(1,𝔽5)11yescharacteristic
252#67—202no—
253#67—202no—
254C2×Dic3—21no—
255C2×Dic3—101no—
256D12—22no—
257D12—102no—
258D12—204no—
259C4×D3—201no—
260C4×D3—201no—
261C4×D3—201no—
262C4×D3—201no—
263Dic6—22no—
264Dic6—102no—
265C3⋊C8—202no—
266C22×C5—122no—
267C22×C5—121no—
268C22×C5C2×(C2×D3)⋊C411yescharacteristic
269C2×D5—22no—
270C2×D5—122no—
271C2×D5—121no—
272C2×D5#93012yes—
273AGL(1,𝔽5)—28no—
274C4×C5—22no—
275C4×C5—121no—
276C4×C5C4×D1212yes—
277Dic5—121no—
278Dic5#80514yescomplemented
279Dic5#87411yescharacteristic
280Dic5#87511yescharacteristic
281#41—302no—
282#41—302no—
283#41—302no—
284#41—302no—
285C2×Q8—31no—
286C2×Q8—151no—
287C2×D4—31no—
288C2×D4—151no—
289C2×D4—302no—
290C2×D4—301no—
291C2×D4—301no—
292C22×C4—51no—
293C22×C4—52no—
294C22×C4—102no—
295C22×C4—301no—
296C22×C4—301no—
297M16—302no—
298M16—302no—
299C2×C8—302no—
300C4⋊C4—302no—
301C22⋊C4—62no—
302C22⋊C4—302no—
303C42—52no—
304C42—51no—
305C42—54no—
306C42—102no—
307C42—108no—
308C42D3×AGL(1,𝔽5)11yescharacteristic
309C3×C5#196211yescharacteristic, complemented, Hall
310C22×C3—52no—
311C22×C3—102no—
312C22×C3C2×C22⋊AGL(1,𝔽5)11yescharacteristic
313C2×D3—42no—
314C2×D3—41no—
315C2×D3—202no—
316C2×D3—201no—
317C3×C4—22no—
318C3×C4—54no—
319C3×C4—51no—
320C3×C4—51no—
321C3×C4—108no—
322C3×C4D4×AGL(1,𝔽5)12yes—
323Dic3—41no—
324Dic3—201no—
325C2×C5—241no—
326C2×C5C2×#80512yes—
327C2×C5#493911yescharacteristic
328D5—22no—
329D5—241no—
330C23—51no—
331C23—61no—
332C23—301no—
333Q8—62no—
334Q8—302no—
335D4—62no—
336D4—302no—
337D4—604no—
338D4—602no—
339D4—602no—
340C2×C4—52no—
341C2×C4—51no—
342C2×C4—52no—
343C2×C4—52no—
344C2×C4—61no—
345C2×C4—104no—
346C2×C4—104no—
347C2×C4—108no—
348C2×C4—301no—
349C2×C4—601no—
350C2×C4—601no—
351C2×C4—601no—
352C2×C4—601no—
353C2×C4(C2×D3)⋊AGL(1,𝔽5)12yes—
354C2×C4C2×D3×AGL(1,𝔽5)11yescharacteristic
355C8—602no—
356C2×C3—102no—
357C2×C3#6578011yescharacteristic
358C2×C3D5⋊#10412yes—
359D3—81no—
360D3—401no—
361C5#6939511yescharacteristic, complemented, Sylow
362C22—52no—
363C22—102no—
364C22—122no—
365C22—121no—
366C22—602no—
367C22—601no—
368C22C2×(C2×D3)⋊AGL(1,𝔽5)11yescharacteristic
369C4—22no—
370C4—54no—
371C4—51no—
372C4—51no—
373C4—108no—
374C4—121no—
375C4—601no—
376C4AGL(1,𝔽5)×D1212yes—
377C3#1060455911yescharacteristic, complemented, Sylow
378C2—102no—
379C2—241no—
380C2—1201no—
381C2D5⋊#80512yes—
382C2#1175185011yescharacteristic
subgroupG/H
centerC22C2×(C2×D3)⋊AGL(1,𝔽5)
commutatorC2×C3×C4×C5C22×C4
FrattiniC2×C4C2×D3×AGL(1,𝔽5)
FittingC3×C42×C5C2×C4
socleC22×C3×C5C2×C22⋊C4
prime radical#11752673C2
cosocleC2×C3×C4×D5C23
hypercenterC42D3×AGL(1,𝔽5)
hypocenterC3×C5#1962
LCSG ▷ C2×C3×C4×C5 ▷ C2×C3×C5 ▷ C3×C5
UCSC1 ◁ C22 ◁ C42

Complements

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

NH ≅ G/Ntype
2. #11752341C2semidirect
3. C3×C42×AGL(1,𝔽5)C2semidirect
4. C2×#91158C2semidirect
5. C2×#91156C2semidirect
12. #91764C4semidirect
13. C5×#875C4semidirect
15. #91451C4semidirect
18. #91376C4semidirect
26. C42×AGL(1,𝔽5)D3semidirect
49. C3×C4×AGL(1,𝔽5)D4semidirect
111. #875AGL(1,𝔽5)semidirect
137. C4×AGL(1,𝔽5)D12semidirect
162. C3×Dic5#104semidirect
278. Dic5#805semidirect
309. C3×C5#1962semidirect
361. C5#69395semidirect
377. C3#10604559semidirect

Maximal subgroups

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

subgroupindex
#117526732
#117523412
C3×C42×AGL(1,𝔽5)2
C2×#911582
C2×#911562
#106045593
#693955