Finite Group Databasebeta

← #14283213

Subgroups — #14283213

Order 1920.

484 conjugacy classes of subgroups.

The subgroup lattice has 396 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C3×#65891C211yesmaximal, characteristic
2C4×C4⋊ΓL(1,𝔽16)C211yesmaximal, characteristic, complemented
3#11752342C211yesmaximal, characteristic, complemented
4#11746791C212yesmaximal
5#11746740C212yesmaximal, complemented
6#10604391—31nomaximal, Hall
7C2×C4⋊ΓL(1,𝔽16)—21no—
8C2×C4×ΓL(1,𝔽16)—21no—
9C3×C4×C5×D4C411yescharacteristic, complemented
10C3×#4084C2211yescharacteristic
11C3×#4082C411yescharacteristic, complemented
12C3×C4×C22⋊D5—21no—
13C3×#4072—21no—
14C3×#4043C2211yescharacteristic
15C3×#4036C411yescharacteristic, complemented
16C3×#4029—21no—
17C3×#4025—21no—
18C3×C42×D5C2211yescharacteristic
19C3×C4×Dic10C411yescharacteristic, complemented
20#91235C2212yes—
21#91232C2212yes—
22#91231—21no—
23C4⋊(C3⋊#170)—21no—
24#71755—51nomaximal, Hall
25#65891D311yescharacteristic
26C4×C4⋊AGL(1,𝔽5)—31no—
27#65683—31no—
28#64920—32no—
29#64869—32no—
30C22×ΓL(1,𝔽16)—22no—
31C2×C3×C5×D4C2×C411yescharacteristic
32C22×C3×C4×C5—21no—
33C2×C3×C22⋊D5—21no—
34C2×C3×C4×D5D411yescharacteristic
35C2×C3×C4×D5D411yescharacteristic
36C2×C3×C4×D5C2311yescharacteristic
37C2×C3×Dic10C2×C411yescharacteristic
38C2×C3⋊#170—22no—
39C4⋊ΓL(1,𝔽16)—24no—
40C4×ΓL(1,𝔽16)—42no—
41C3×C5×C4⋊C4C2×C411yescharacteristic
42C3×C5×C22⋊C4—21no—
43C3×C42×C5C2×C411yescharacteristic
44C3×C22⋊Dic5—21no—
45C3×C22⋊Dic5—21no—
46C3×C22⋊Dic5—21no—
47C3×#685—21no—
48C3×C4⋊Dic5—21no—
49C3×C4⋊Dic5—21no—
50C3×C4⋊Dic5C2×C411yescharacteristic
51C3×#683—21no—
52C3×#683—21no—
53C3×C4×Dic5—21no—
54C3×C4×Dic5C2×C411yescharacteristic
55C3×C4×Dic5D411yescharacteristic
56C3×C4×Dic5D411yescharacteristic
57C3×#529—51no—
58C4×C4⋊Dic3—51no—
59#4926—51no—
60#4538—52no—
61#4537—52no—
62C2×C4⋊AGL(1,𝔽5)—61no—
63C2×C4×AGL(1,𝔽5)—61no—
64C4×C5×D4Dic311yescharacteristic, complemented
65#4084C2×D311yescharacteristic
66#4082Dic311yescharacteristic, complemented
67C4×C22⋊D5—21no—
68#4072—21no—
69#4043C2×D311yescharacteristic
70#4036Dic311yescharacteristic, complemented
71#4029—21no—
72#4025—21no—
73C42×D5C2×D311yescharacteristic
74C4×Dic10Dic311yescharacteristic, complemented
75(C2×AGL(1,𝔽5))⋊C4—32no—
76#4003—32no—
77#4002—61no—
78C4⋊#170—61no—
79#1509—151noSylow
80C23×C3×C5—21no—
81C22×C3×D5C2×D411yescharacteristic
82C2×ΓL(1,𝔽16)—24no—
83C2×ΓL(1,𝔽16)—42no—
84C2×ΓL(1,𝔽16)—42no—
85C3×C5×D4—41no—
86C2×C3×C4×C5—21no—
87C2×C3×C4×C5—42no—
88C2×C3×C4×C5C22⋊C411yescharacteristic
89C2×C3×C4×C5C22⋊C411yescharacteristic
90C2×C3×C4×C5C22×C411yescharacteristic
91C3×C22⋊D5—41no—
92C3×C22⋊D5—41no—
93C2×C3×Dic5—21no—
94C2×C3×Dic5—21no—
95C2×C3×Dic5C22⋊C411yescharacteristic
96C2×C3×Dic5C22⋊C411yescharacteristic
97C2×C3×Dic5C2×D411yescharacteristic
98C3×C4×D5—21no—
99C3×C4×D5—21no—
100C3×C4×D5—22no—
101C3×C4×D5—41no—
102C3×Dic10—41no—
103C3⋊#170—42no—
104C3×#126—101no—
105C3×#125—51no—
106C3×#124—51no—
107C3×#123—101no—
108C3×C22⋊Q8—51no—
109C3×#121—51no—
110C3×C4×Q8—51no—
111C3×C4×D4—101no—
112C3×C4×D4AGL(1,𝔽5)11yescharacteristic, complemented
113C3×#117—101no—
114C2×C3×C42—51no—
115C2×C4⋊Dic3—101no—
116C2×C4×Dic3—101no—
117C3⋊#95—52no—
118#830—52no—
119C4⋊(C3⋊C8)—101no—
120#803—101no—
121C22×AGL(1,𝔽5)—62no—
122C2×C5×D4C2×Dic311yescharacteristic
123C22×C4×C5—21no—
124C2×C22⋊D5—21no—
125C2×C4×D5#6711yescharacteristic
126C2×C4×D5#6711yescharacteristic
127C2×C4×D5C22×D311yescharacteristic
128C2×Dic10C2×Dic311yescharacteristic
129C2×#170—62no—
130C4⋊AGL(1,𝔽5)—64no—
131C4×AGL(1,𝔽5)—122no—
132C5×C4⋊C4C2×Dic311yescharacteristic
133C5×C22⋊C4—21no—
134C42×C5C2×Dic311yescharacteristic
135C22⋊Dic5—21no—
136C22⋊Dic5—21no—
137C22⋊Dic5—21no—
138#685—21no—
139C4⋊Dic5—21no—
140C4⋊Dic5—21no—
141C4⋊Dic5C2×Dic311yescharacteristic
142#683—21no—
143#683—21no—
144C4×Dic5—21no—
145C4×Dic5C2×Dic311yescharacteristic
146C4×Dic5#6711yescharacteristic
147C4×Dic5#6711yescharacteristic
148#529—51no—
149#432—151no—
150C4×C4⋊C4—151no—
151#328—152no—
152#327—152no—
153C22×C3×C5—41no—
154C22×C3×C5—41no—
155C22×C3×C5—41no—
156C22×C3×C5C2×C22⋊C411yescharacteristic
157C2×C3×D5—21no—
158C2×C3×D5—21no—
159C2×C3×D5AGL(1,ℤ/8)11yescharacteristic
160C2×C3×D5#13711yescharacteristic
161ΓL(1,𝔽16)—44no—
162C3×C4×C5—21no—
163C3×C4×C5—21no—
164C3×C4×C5—41no—
165C3×C4×C5#10412yes—
166C3×Dic5—21no—
167C3×Dic5—21no—
168C3×Dic5—21no—
169C3×Dic5—41no—
170C3×Dic5—41no—
171C2×C3×Q8—51no—
172C2×C3×D4—101no—
173C2×C3×D4C2×AGL(1,𝔽5)11yescharacteristic
174C22×C3×C4—21no—
175C22×C3×C4—51no—
176C22×C3×C4—51no—
177C22×C3×C4—51no—
178C22×Dic3—102no—
179C3×C4⋊C4—51no—
180C3×C4⋊C4—101no—
181C3×C4⋊C4—101no—
182C3×C4⋊C4—101no—
183C3×C4⋊C4—101no—
184C3×C4⋊C4C2×AGL(1,𝔽5)11yescharacteristic
185C3×C22⋊C4—21no—
186C3×C22⋊C4—101no—
187C3×C22⋊C4—101no—
188C3×C22⋊C4—101no—
189C3×C22⋊C4—101no—
190C3×C42—51no—
191C3×C42—51no—
192C3×C42—51no—
193C3×C42—101no—
194C3×C42C2×AGL(1,𝔽5)11yescharacteristic
195C4⋊Dic3—104no—
196C4×Dic3—202no—
197C2×C3⋊C8—102no—
198C23×C5—21no—
199C22×D5C2×#6711yescharacteristic
200C2×AGL(1,𝔽5)—64no—
201C2×AGL(1,𝔽5)—122no—
202C2×AGL(1,𝔽5)—122no—
203C5×D4—41no—
204C2×C4×C5—21no—
205C2×C4×C5—42no—
206C2×C4×C5#21711yescharacteristic
207C2×C4×C5#21711yescharacteristic
208C2×C4×C5C22×Dic311yescharacteristic
209C22⋊D5—41no—
210C22⋊D5—41no—
211C2×Dic5—21no—
212C2×Dic5—21no—
213C2×Dic5#21711yescharacteristic
214C2×Dic5#21711yescharacteristic
215C2×Dic5C2×#6711yescharacteristic
216C4×D5—21no—
217C4×D5—21no—
218C4×D5—22no—
219C4×D5—41no—
220Dic10—41no—
221#170—122no—
222#126—101no—
223#125—51no—
224#124—51no—
225#123—101no—
226C22⋊Q8—51no—
227#121—51no—
228C4×Q8—51no—
229C4×D4—101no—
230C4×D4ΓL(1,𝔽16)11yescharacteristic, complemented
231#117—101no—
232C2×C4⋊C4—301no—
233C2×C42—51no—
234C2×C42—301no—
235C4⋊C8—301no—
236C22⋊C8—152no—
237#97—301no—
238#95—152no—
239C2×C3×C5—81no—
240C2×C3×C5#41011yescharacteristic
241C2×C3×C5#41711yescharacteristic
242C2×C3×C5C2×#10411yescharacteristic
243C3×D5—21no—
244C3×D5—21no—
245C23×C3—21no—
246C23×C3—51no—
247C3×Q8—201no—
248C3×D4—41no—
249C3×D4—201no—
250C3×D4—201no—
251C2×C3×C4—21no—
252C2×C3×C4—42no—
253C2×C3×C4—51no—
254C2×C3×C4—51no—
255C2×C3×C4—51no—
256C2×C3×C4—101no—
257C2×C3×C4—101no—
258C2×C3×C4—102no—
259C2×C3×C4—101no—
260C2×C3×C4—101no—
261C2×C3×C4—201no—
262C2×C3×C4C22⋊AGL(1,𝔽5)11yescharacteristic
263C2×C3×C4C22⋊AGL(1,𝔽5)11yescharacteristic
264C2×C3×C4C22×AGL(1,𝔽5)11yescharacteristic
265C2×Dic3—104no—
266C2×Dic3—202no—
267C2×Dic3—202no—
268C3⋊C8—202no—
269C22×C5—41no—
270C22×C5—41no—
271C22×C5—41no—
272C22×C5C2×#21711yescharacteristic
273C2×D5—21no—
274C2×D5—21no—
275C2×D5#94911yescharacteristic
276C2×D5#95111yescharacteristic
277AGL(1,𝔽5)—124no—
278C4×C5—21no—
279C4×C5—21no—
280C4×C5—41no—
281C4×C5#83712yes—
282Dic5—21no—
283Dic5—21no—
284Dic5—21no—
285Dic5—41no—
286Dic5—41no—
287C2×Q8—51no—
288C2×D4—101no—
289C2×D4C2×ΓL(1,𝔽16)11yescharacteristic
290C22×C4—21no—
291C22×C4—51no—
292C22×C4—51no—
293C22×C4—51no—
294C22×C4—302no—
295C2×C8—302no—
296C4⋊C4—51no—
297C4⋊C4—101no—
298C4⋊C4—101no—
299C4⋊C4—101no—
300C4⋊C4—101no—
301C4⋊C4—304no—
302C4⋊C4C2×ΓL(1,𝔽16)11yescharacteristic
303C22⋊C4—21no—
304C22⋊C4—101no—
305C22⋊C4—101no—
306C22⋊C4—101no—
307C22⋊C4—101no—
308C42—51no—
309C42—51no—
310C42—51no—
311C42—101no—
312C42—602no—
313C42C2×ΓL(1,𝔽16)11yescharacteristic
314C3×C5#150911yescharacteristic, complemented, Hall
315C22×C3—41no—
316C22×C3—41no—
317C22×C3—41no—
318C22×C3—51no—
319C22×C3—51no—
320C22×C3—101no—
321C22×C3—101no—
322C22×C3C2×C22⋊AGL(1,𝔽5)11yescharacteristic
323C3×C4—21no—
324C3×C4—21no—
325C3×C4—41no—
326C3×C4—101no—
327C3×C4—101no—
328C3×C4—101no—
329C3×C4—201no—
330C3×C4—201no—
331C3×C4#400812yes—
332Dic3—204no—
333C2×C5—81no—
334C2×C5#523211yescharacteristic
335C2×C5C2×#83711yescharacteristic
336C2×C5#524011yescharacteristic
337D5—21no—
338D5—21no—
339C23—21no—
340C23—51no—
341Q8—201no—
342D4—41no—
343D4—201no—
344D4—201no—
345C2×C4—21no—
346C2×C4—42no—
347C2×C4—51no—
348C2×C4—51no—
349C2×C4—51no—
350C2×C4—101no—
351C2×C4—101no—
352C2×C4—102no—
353C2×C4—101no—
354C2×C4—101no—
355C2×C4—201no—
356C2×C4—304no—
357C2×C4—602no—
358C2×C4—602no—
359C2×C4#676311yescharacteristic
360C2×C4#676311yescharacteristic
361C2×C4C22×ΓL(1,𝔽16)11yescharacteristic
362C8—602no—
363C2×C3—81no—
364C2×C3—101no—
365C2×C3—101no—
366C2×C3#6574511yescharacteristic
367C2×C3C2×#400811yescharacteristic
368C2×C3#6579311yescharacteristic
369C5#7175511yescharacteristic, complemented, Sylow
370C22—41no—
371C22—41no—
372C22—41no—
373C22—51no—
374C22—51no—
375C22—101no—
376C22—101no—
377C22C2×#676311yescharacteristic
378C4—21no—
379C4—21no—
380C4—41no—
381C4—101no—
382C4—101no—
383C4—101no—
384C4—201no—
385C4—201no—
386C4—604no—
387C4#9123712yes—
388C3#1060439111yescharacteristic, complemented, Sylow
389C2—81no—
390C2—101no—
391C2—101no—
392C2#1175240411yescharacteristic
393C2C2×#9123711yescharacteristic
394C2#1175245211yescharacteristic
subgroupG/H
centerC22C2×#6763
commutatorC2×C3×C4×C5C22×C4
FrattiniC2×C4C22×ΓL(1,𝔽16)
FittingC3×C4×C5×D4C4
socleC22×C3×C5C2×C22⋊C4
prime radicalC3×#4084C22
cosocleC2×C3×C4×D5C23
hypercenterC4×D4ΓL(1,𝔽16)
hypocenterC3×C5#1509
LCSG ▷ C2×C3×C4×C5 ▷ C22×C3×C5 ▷ C3×C5
UCSC1 ◁ C22 ◁ C42 ◁ C4×D4

Complements

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

NH ≅ G/Ntype
2. C4×C4⋊ΓL(1,𝔽16)C2semidirect
3. #11752342C2semidirect
5. #11746740C2semidirect
9. C3×C4×C5×D4C4semidirect
11. C3×#4082C4semidirect
15. C3×#4036C4semidirect
19. C3×C4×Dic10C4semidirect
64. C4×C5×D4Dic3semidirect
66. #4082Dic3semidirect
70. #4036Dic3semidirect
74. C4×Dic10Dic3semidirect
112. C3×C4×D4AGL(1,𝔽5)semidirect
230. C4×D4ΓL(1,𝔽16)semidirect
314. C3×C5#1509semidirect
369. C5#71755semidirect
388. C3#10604391semidirect

Maximal subgroups

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

subgroupindex
C3×#658912
C4×C4⋊ΓL(1,𝔽16)2
#117523422
#117467912
#117467402
#106043913
#717555