Finite Group Databasebeta

← #14306040

Subgroups — #14306040

Order 1920.

640 conjugacy classes of subgroups; only the 125 normal subgroups are shown.

normal subgroups only

This group has 640 conjugacy classes of subgroups — too many to draw. Only its 125 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
1C3×#66002C211yesmaximal, characteristic
2C2×#91239C212yesmaximal, complemented
3C2×#91237C212yesmaximal, complemented
4#11752345C211yesmaximal, characteristic, complemented
5#11752340C211yesmaximal, characteristic, complemented
6#10604560—31nomaximal, Hall
7C2×C3×#713C2211yescharacteristic
8C2×C3×#711C2211yescharacteristic
9C2×C4×ΓL(1,𝔽16)C2212yes—
10C2×#6758C2212yes—
11C3×C5×#124C411yescharacteristic, complemented
12C3×#4090C411yescharacteristic, complemented
13C3×#4086C411yescharacteristic, complemented
14C3×#4030—22no—
15C3×#4027—22no—
16C3×#4021C411yescharacteristic, complemented
17C3×C42×D5C2211yescharacteristic
18#91239—24no—
19#91237—24no—
20#91235—22no—
21C4⋊(C3⋊#170)—22no—
22#71816—51nomaximal, Hall
23#66002D311yescharacteristic
24C2×#4010—32no—
25C2×#4008—32no—
26#65686—31no—
27#65681—31no—
28C22×ΓL(1,𝔽16)—22no—
29C2×C3×C5×Q8C2×C411yescharacteristic
30C2×C3×C5×D4C2×C411yescharacteristic
31C2×C3×C22⋊D5—21no—
32C2×C3×C22⋊D5—22no—
33C22×C3×Dic5—21no—
34C3×#713—22no—
35C3×#713—22no—
36C3×#711—22no—
37C3×#711—22no—
38C2×C3×D20—21no—
39C2×C3×D20C2×C411yescharacteristic
40C2×C3×C4×D5—21no—
41C2×C3×C4×D5D412yes—
42C2×C3×C4×D5C2311yescharacteristic
43C2×C3×Dic10C2×C411yescharacteristic
44C2×C3⋊#170—22no—
45C4×ΓL(1,𝔽16)—22no—
46C4×ΓL(1,𝔽16)D414yescomplemented
47#6758—22no—
48#6758—22no—
49C3×C5×C22⋊C4—22no—
50C3×C42×C5C2×C411yescharacteristic
51C3×#685—22no—
52C3×#683—22no—
53C3×C4×Dic5C2×C411yescharacteristic
54C3×C4×Dic5D412yes—
55C3×#532—51no—
56C2×#837—52no—
57C2×#837—52no—
58#4932—51no—
59#4924—51no—
60C2×#713C2×D311yescharacteristic
61C2×#711C2×D311yescharacteristic
62C2×C4×AGL(1,𝔽5)—32no—
63C2×#700—32no—
64C5×#124Dic311yescharacteristic, complemented
65#4090Dic311yescharacteristic, complemented
66#4086Dic311yescharacteristic, complemented
67#4030—22no—
68#4027—22no—
69#4021Dic311yescharacteristic, complemented
70C42×D5C2×D311yescharacteristic
71#4010—64no—
72#4008—64no—
73(C2×AGL(1,𝔽5))⋊C4—62no—
74C4⋊#170—62no—
75#1963—151noSylow
76C23×C3×C5—21no—
77C22×C3×D5—21no—
78C22×C3×D5C2×D411yescharacteristic
79C2×ΓL(1,𝔽16)—24no—
80C2×ΓL(1,𝔽16)—44no—
81C3×C5×Q8—22no—
82C3×C5×D4—22no—
83C2×C3×C4×C5—21no—
84C2×C3×C4×C5C22⋊C412yes—
85C2×C3×C4×C5C22×C411yescharacteristic
86C3×C22⋊D5—44no—
87C3×C22⋊D5—42no—
88C2×C3×Dic5—21no—
89C2×C3×Dic5—41no—
90C2×C3×Dic5—41no—
91C2×C3×Dic5C22⋊C412yes—
92C2×C3×Dic5C2×D411yescharacteristic
93C3×D20—22no—
94C3×D20—42no—
95C3×C4×D5—42no—
96C3×C4×D5—41no—
97C3×C4×D5—41no—
98C3×C4×D5C2×D412yes—
99C3×C4×D5#4112yes—
100C3×Dic10—22no—
101C3⋊#170—42no—
102C2×C3×#41—51no—
103C2×C3×#41—51no—
104C3×C4⋊Q8—51no—
105C3×#127—51no—
106C3×#124—51no—
107C3×#124AGL(1,𝔽5)11yescharacteristic, complemented
108C3×#121—102no—
109C3×C4×D4—102no—
110C2×C3×C42—51no—
111C2×C4×Dic3—52no—
112C2×#208—52no—
113#837—104no—
114#837—104no—
115C3⋊#95—102no—
116C4⋊(C3⋊C8)—102no—
117C22×AGL(1,𝔽5)—62no—
118C2×C5×Q8C2×Dic311yescharacteristic
119C2×C5×D4C2×Dic311yescharacteristic
120C2×C22⋊D5—21no—
121C2×C22⋊D5—22no—
122C22×Dic5—21no—
123#713—22no—
124#713—22no—
125#711—22no—
126#711—22no—
127C2×D20—21no—
128C2×D20C2×Dic311yescharacteristic
129C2×C4×D5—21no—
130C2×C4×D5#6712yes—
131C2×C4×D5C22×D311yescharacteristic
132C2×Dic10C2×Dic311yescharacteristic
133C2×#170—62no—
134C4×AGL(1,𝔽5)—34no—
135C4×AGL(1,𝔽5)—62no—
136#700—62no—
137#700—62no—
138C5×C22⋊C4—22no—
139C42×C5C2×Dic311yescharacteristic
140#685—22no—
141#683—22no—
142C4×Dic5C2×Dic311yescharacteristic
143C4×Dic5#6712yes—
144#532—51no—
145#423—151no—
146C2×#104—152no—
147C2×#104—152no—
148#376—151no—
149C22×C3×C5—41no—
150C22×C3×C5—42no—
151C22×C3×C5C2×C22⋊C411yescharacteristic
152C2×C3×D5—22no—
153C2×C3×D5—41no—
154C2×C3×D5—42no—
155C2×C3×D5#12112yes—
156ΓL(1,𝔽16)—44no—
157C3×C4×C5—22no—
158C3×C4×C5—41no—
159C3×C4×C5C4×D412yes—
160C3×Dic5—22no—
161C3×Dic5—41no—
162C3×Dic5#12411yescharacteristic
163C3×Dic5#12711yescharacteristic
164C3×#41—102no—
165C3×#41—102no—
166C3×#41—102no—
167C3×#41—102no—
168C2×C3×Q8—51no—
169C2×C3×Q8C2×AGL(1,𝔽5)11yescharacteristic
170C2×C3×D4—51no—
171C2×C3×D4—101no—
172C2×C3×D4—102no—
173C2×C3×D4—101no—
174C2×C3×D4C2×AGL(1,𝔽5)11yescharacteristic
175C22×C3×C4—51no—
176C22×C3×C4—52no—
177C22×C3×C4—101no—
178C22×C3×C4—101no—
179C22×Dic3—102no—
180C3×C4⋊C4—102no—
181C3×C22⋊C4—22no—
182C3×C22⋊C4—102no—
183C3×C42—52no—
184C3×C42—51no—
185C3×C42C2×AGL(1,𝔽5)11yescharacteristic
186C4×Dic3—54no—
187C4×Dic3—102no—
188#208—102no—
189#208—102no—
190C2×C3⋊C8—102no—
191C23×C5—21no—
192C22×D5—21no—
193C22×D5C2×#6711yescharacteristic
194C2×AGL(1,𝔽5)—64no—
195C2×AGL(1,𝔽5)—124no—
196C5×Q8—22no—
197C5×D4—22no—
198C2×C4×C5—21no—
199C2×C4×C5#21712yes—
200C2×C4×C5C22×Dic311yescharacteristic
201C22⋊D5—44no—
202C22⋊D5—42no—
203C2×Dic5—21no—
204C2×Dic5—41no—
205C2×Dic5—41no—
206C2×Dic5#21712yes—
207C2×Dic5C2×#6711yescharacteristic
208D20—22no—
209D20—42no—
210C4×D5—42no—
211C4×D5—41no—
212C4×D5—41no—
213C4×D5D3×D412yes—
214C4×D5#23712yes—
215Dic10—22no—
216#170—122no—
217C2×#41—51no—
218C2×#41—51no—
219C2×M16—152no—
220C4⋊Q8—51no—
221#127—51no—
222#124—51no—
223#124ΓL(1,𝔽16)11yescharacteristic, complemented
224#121—102no—
225C4×D4—102no—
226C2×C42—51no—
227C2×C42—152no—
228C4⋊C8—302no—
229#104—304no—
230#104—304no—
231#95—302no—
232C2×C3×C5—81no—
233C2×C3×C5#39011yescharacteristic
234C2×C3×C5#42112yes—
235C3×D5—22no—
236C3×D5—81no—
237C23×C3—21no—
238C23×C3—51no—
239C23×C3—101no—
240C3×Q8—22no—
241C3×Q8—102no—
242C3×D4—22no—
243C3×D4—102no—
244C3×D4—202no—
245C3×D4—204no—
246C3×D4—202no—
247C2×C3×C4—21no—
248C2×C3×C4—51no—
249C2×C3×C4—52no—
250C2×C3×C4—52no—
251C2×C3×C4—52no—
252C2×C3×C4—101no—
253C2×C3×C4—202no—
254C2×C3×C4—201no—
255C2×C3×C4—201no—
256C2×C3×C4—201no—
257C2×C3×C4—201no—
258C2×C3×C4C22⋊AGL(1,𝔽5)12yes—
259C2×C3×C4C22×AGL(1,𝔽5)11yescharacteristic
260C2×Dic3—104no—
261C2×Dic3—204no—
262C3⋊C8—202no—
263C22×C5—41no—
264C22×C5—42no—
265C22×C5C2×#21711yescharacteristic
266C2×D5—22no—
267C2×D5—41no—
268C2×D5—42no—
269C2×D5#93812yes—
270AGL(1,𝔽5)—124no—
271C4×C5—22no—
272C4×C5—41no—
273C4×C5D4×Dic312yes—
274Dic5—22no—
275Dic5—41no—
276Dic5#93611yescharacteristic
277Dic5#94011yescharacteristic
278#41—102no—
279#41—102no—
280#41—102no—
281#41—102no—
282C2×Q8—51no—
283C2×Q8C2×ΓL(1,𝔽16)11yescharacteristic
284C2×D4—51no—
285C2×D4—101no—
286C2×D4—102no—
287C2×D4—101no—
288C2×D4C2×ΓL(1,𝔽16)11yescharacteristic
289C22×C4—51no—
290C22×C4—52no—
291C22×C4—101no—
292C22×C4—101no—
293C22×C4—302no—
294M16—302no—
295M16—302no—
296C2×C8—302no—
297C4⋊C4—102no—
298C22⋊C4—22no—
299C22⋊C4—102no—
300C42—52no—
301C42—51no—
302C42—154no—
303C42—302no—
304C42C2×ΓL(1,𝔽16)11yescharacteristic
305C3×C5#196311yescharacteristic, complemented, Hall
306C22×C3—42no—
307C22×C3—41no—
308C22×C3—52no—
309C22×C3—102no—
310C22×C3—202no—
311C22×C3—201no—
312C22×C3C2×C22⋊AGL(1,𝔽5)11yescharacteristic
313C3×C4—22no—
314C3×C4—41no—
315C3×C4—51no—
316C3×C4—51no—
317C3×C4—102no—
318C3×C4—201no—
319C3×C4D4×AGL(1,𝔽5)12yes—
320Dic3—204no—
321C2×C5—81no—
322C2×C5#522111yescharacteristic
323C2×C5#523612yes—
324D5—22no—
325D5—81no—
326C23—21no—
327C23—51no—
328C23—101no—
329Q8—22no—
330Q8—102no—
331D4—22no—
332D4—102no—
333D4—202no—
334D4—204no—
335D4—202no—
336C2×C4—21no—
337C2×C4—51no—
338C2×C4—52no—
339C2×C4—52no—
340C2×C4—52no—
341C2×C4—101no—
342C2×C4—202no—
343C2×C4—201no—
344C2×C4—201no—
345C2×C4—201no—
346C2×C4—201no—
347C2×C4—304no—
348C2×C4—604no—
349C2×C4#676312yes—
350C2×C4C22×ΓL(1,𝔽16)11yescharacteristic
351C8—602no—
352C2×C3—81no—
353C2×C3—102no—
354C2×C3—401no—
355C2×C3#6578011yescharacteristic
356C2×C3#6579512yes—
357C5#7181611yescharacteristic, complemented, Sylow
358C22—42no—
359C22—41no—
360C22—52no—
361C22—102no—
362C22—202no—
363C22—201no—
364C22C2×#676311yescharacteristic
365C4—22no—
366C4—41no—
367C4—51no—
368C4—51no—
369C4—102no—
370C4—201no—
371C4—604no—
372C4D4×ΓL(1,𝔽16)12yes—
373C3#1060456011yescharacteristic, complemented, Sylow
374C2—81no—
375C2—102no—
376C2—401no—
377C2#1175243911yescharacteristic
378C2#1175245412yes—
subgroupG/H
centerC22C2×#6763
commutatorC2×C3×C4×C5C22×C4
FrattiniC2×C4C22×ΓL(1,𝔽16)
FittingC3×C5×#124C4
socleC22×C3×C5C2×C22⋊C4
prime radicalC3×#66002C2
cosocleC2×C3×C4×D5C23
hypercenter#124ΓL(1,𝔽16)
hypocenterC3×C5#1963
LCSG ▷ C2×C3×C4×C5 ▷ C2×C3×C5 ▷ C3×C5
UCSC1 ◁ C22 ◁ C42 ◁ #124

Complements

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

NH ≅ G/Ntype
2. C2×#91239C2semidirect
3. C2×#91237C2semidirect
4. #11752345C2semidirect
5. #11752340C2semidirect
11. C3×C5×#124C4semidirect
12. C3×#4090C4semidirect
13. C3×#4086C4semidirect
16. C3×#4021C4semidirect
46. C4×ΓL(1,𝔽16)D4semidirect
64. C5×#124Dic3semidirect
65. #4090Dic3semidirect
66. #4086Dic3semidirect
69. #4021Dic3semidirect
107. C3×#124AGL(1,𝔽5)semidirect
223. #124ΓL(1,𝔽16)semidirect
305. C3×C5#1963semidirect
357. C5#71816semidirect
373. C3#10604560semidirect

Maximal subgroups

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

subgroupindex
C3×#660022
C2×#912392
C2×#912372
#117523452
#117523402
#106045603
#718165