Finite Group Databasebeta

← #14283252

Subgroups — #14283252

Order 1920.

484 conjugacy classes of subgroups; only the 98 normal subgroups are shown.

normal subgroups only

This group has 484 conjugacy classes of subgroups — too many to draw. Only its 98 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#11752682C211yesmaximal, characteristic
2C4×C4⋊ΓL(1,𝔽16)C211yesmaximal, characteristic, complemented
3C3×#65683C211yesmaximal, characteristic, complemented
4#11746501C212yesmaximal, complemented
5#11746498C212yesmaximal
6#10604391—31nomaximal, Hall
7C2×C4⋊ΓL(1,𝔽16)—21no—
8C2×C4×ΓL(1,𝔽16)—21no—
9C4×Dic30C411yescharacteristic, complemented
10C4×C5×D12C411yescharacteristic, complemented
11C3×C42×D5C2211yescharacteristic
12#91449C2211yescharacteristic
13#91447—21no—
14C4×#1201—21no—
15#91388C411yescharacteristic, complemented
16#91373C411yescharacteristic, complemented
17#91372—21no—
18#91371—21no—
19#91360C2211yescharacteristic
20#91235C2212yes—
21C3×#4003C2212yes—
22C3×#4002—21no—
23C3×C4⋊#170—21no—
24#70833—51nomaximal, Hall
25#65891—31no—
26C4×C4⋊AGL(1,𝔽5)—31no—
27#65683D311yescharacteristic, complemented
28#64920—32no—
29#64869—32no—
30C22×ΓL(1,𝔽16)—22no—
31C2×Dic30C2×C411yescharacteristic
32C2×C5×D12C2×C411yescharacteristic
33C2×C4×C5×D3—21no—
34C2×C3×C4×D5D411yescharacteristic
35C2×C3×C4×D5D411yescharacteristic
36C2×C3×C4×D5C2311yescharacteristic
37C2×#1201—21no—
38C4⋊ΓL(1,𝔽16)—24no—
39C4×ΓL(1,𝔽16)—42no—
40C2×C3×#170—22no—
41C3×C42×C5C2×C411yescharacteristic
42C4⋊Dic15C2×C411yescharacteristic
43#6712—21no—
44C4×Dic15—21no—
45C5×(C2×D3)⋊C4—21no—
46C5×C4⋊Dic3C2×C411yescharacteristic
47C3×C4×Dic5C2×C411yescharacteristic
48C3×C4×Dic5D411yescharacteristic
49C3×C4×Dic5D411yescharacteristic
50Dic3⋊Dic5—21no—
51Dic3⋊Dic5—21no—
52#6669—21no—
53(C2×D3)⋊Dic5—21no—
54(C2×D3)⋊Dic5—21no—
55(C2×D3)⋊Dic5—21no—
56#6665—21no—
57#5479—51no—
58C3×#432—51no—
59C4×C4⋊Dic3—51no—
60#4470—52no—
61#4467—52no—
62C2×C4⋊AGL(1,𝔽5)—61no—
63C2×C4×AGL(1,𝔽5)—61no—
64C4×C5×D4—31no—
65#4084—31no—
66#4082—31no—
67C4×C22⋊D5—61no—
68#4072—61no—
69#4043—31no—
70#4036—31no—
71#4029—61no—
72#4025—61no—
73C42×D5C2×D311yescharacteristic
74C4×Dic10—31no—
75(C2×AGL(1,𝔽5))⋊C4—32no—
76#4003C2×D312yes—
77#4002—21no—
78C4⋊#170—21no—
79#1509—151noSylow
80C22×C5×D3—21no—
81C22×C3×D5C2×D411yescharacteristic
82C2×ΓL(1,𝔽16)—24no—
83C2×ΓL(1,𝔽16)—42no—
84C2×ΓL(1,𝔽16)—42no—
85C2×C3×C4×C5C22⋊C411yescharacteristic
86C2×C3×C4×C5C22⋊C411yescharacteristic
87C2×C3×C4×C5C22×C411yescharacteristic
88C2×Dic15—21no—
89C2×Dic15—21no—
90Dic30—41no—
91C2×C5×Dic3—21no—
92C5×D12—41no—
93C4×C5×D3—42no—
94C2×C3×Dic5C22⋊C411yescharacteristic
95C2×C3×Dic5C22⋊C411yescharacteristic
96C2×C3×Dic5C2×D411yescharacteristic
97C3×C4×D5—21no—
98C3×C4×D5—21no—
99C3×C4×D5—22no—
100C3×C4×D5—41no—
101#1201—41no—
102#1201—41no—
103C3×#170—42no—
104C2×C3×C42—51no—
105#930—51no—
106#929—101no—
107C4×#67—101no—
108C2×C4⋊Dic3—101no—
109#924—51no—
110C2×C4×Dic3—101no—
111#876—101no—
112#875—51no—
113C4×D12AGL(1,𝔽5)11yescharacteristic, complemented
114#872—101no—
115#870—51no—
116C4×Dic6—51no—
117C3×C4⋊C8—101no—
118C3×C22⋊C8—52no—
119C3×#97—101no—
120C3⋊#95—52no—
121C22×AGL(1,𝔽5)—62no—
122C2×C5×D4—31no—
123C22×C4×C5—61no—
124C2×C22⋊D5—61no—
125C2×C4×D5D1211yescharacteristic
126C2×C4×D5D1211yescharacteristic
127C2×C4×D5C22×D311yescharacteristic
128C2×Dic10—31no—
129C2×#170—22no—
130C4⋊AGL(1,𝔽5)—64no—
131C4×AGL(1,𝔽5)—122no—
132C5×C4⋊C4—31no—
133C5×C22⋊C4—61no—
134C42×C5C4×D311yescharacteristic
135C22⋊Dic5—61no—
136C22⋊Dic5—61no—
137C22⋊Dic5—61no—
138#685—61no—
139C4⋊Dic5—31no—
140C4⋊Dic5—61no—
141C4⋊Dic5—61no—
142#683—61no—
143#683—61no—
144C4×Dic5—61no—
145C4×Dic5C4×D311yescharacteristic
146C4×Dic5#6711yescharacteristic
147C4×Dic5#6711yescharacteristic
148#529—151no—
149#432—51no—
150C4×C4⋊C4—151no—
151#328—152no—
152#327—152no—
153C22×C3×C5C2×C22⋊C411yescharacteristic
154C2×C5×D3—41no—
155C2×C5×D3—41no—
156C2×C5×D3—41no—
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#10412yes—
165Dic15—41no—
166Dic15—41no—
167C3×Dic5—21no—
168C3×Dic5—21no—
169C3×Dic5—21no—
170C5×Dic3—41no—
171C22×C3×C4—51no—
172C22×C3×C4—51no—
173C22×C3×C4—51no—
174C2×#67—101no—
175C22×Dic3—102no—
176C2×D12C2×AGL(1,𝔽5)11yescharacteristic
177C2×C4×D3—21no—
178C2×Dic6—51no—
179C2×C3×C8—102no—
180C3×C42—51no—
181C3×C42—51no—
182C3×C42—51no—
183C3×C42C2×AGL(1,𝔽5)11yescharacteristic
184#217—101no—
185(C2×D3)⋊C4—21no—
186(C2×D3)⋊C4—101no—
187(C2×D3)⋊C4—101no—
188(C2×D3)⋊C4—101no—
189C4⋊Dic3—51no—
190C4⋊Dic3—104no—
191C4⋊Dic3C2×AGL(1,𝔽5)11yescharacteristic
192Dic3⋊C4—101no—
193Dic3⋊C4—101no—
194Dic3⋊C4—101no—
195Dic3⋊C4—101no—
196C4×Dic3—101no—
197C4×Dic3—202no—
198C23×C5—61no—
199C22×D5C2×D1211yescharacteristic
200C2×AGL(1,𝔽5)—64no—
201C2×AGL(1,𝔽5)—122no—
202C2×AGL(1,𝔽5)—122no—
203C5×D4—121no—
204C2×C4×C5—61no—
205C2×C4×C5—122no—
206C2×C4×C5(C2×D3)⋊C411yescharacteristic
207C2×C4×C5(C2×D3)⋊C411yescharacteristic
208C2×C4×C5C2×C4×D311yescharacteristic
209C22⋊D5—121no—
210C22⋊D5—121no—
211C2×Dic5—61no—
212C2×Dic5—61no—
213C2×Dic5(C2×D3)⋊C411yescharacteristic
214C2×Dic5(C2×D3)⋊C411yescharacteristic
215C2×Dic5C2×#6711yescharacteristic
216C4×D5—21no—
217C4×D5—21no—
218C4×D5—22no—
219C4×D5—41no—
220Dic10—121no—
221#170—42no—
222#126—301no—
223#125—151no—
224#124—151no—
225#123—301no—
226C22⋊Q8—151no—
227#121—151no—
228C4×Q8—151no—
229C4×D4—31no—
230C4×D4—301no—
231#117—301no—
232C2×C4⋊C4—301no—
233C2×C42—51no—
234C2×C42—301no—
235C4⋊C8—101no—
236C22⋊C8—52no—
237#97—101no—
238#95—152no—
239C2×C3×C5#41011yescharacteristic
240C2×C3×C5#41711yescharacteristic
241C2×C3×C5C2×#10411yescharacteristic
242C3×D5—21no—
243C3×D5—21no—
244C5×D3—81no—
245C23×C3—51no—
246C22×D3—21no—
247C2×C3×C4—51no—
248C2×C3×C4—51no—
249C2×C3×C4—51no—
250C2×C3×C4—101no—
251C2×C3×C4—101no—
252C2×C3×C4—102no—
253C2×C3×C4—201no—
254C2×C3×C4C22⋊AGL(1,𝔽5)11yescharacteristic
255C2×C3×C4C22⋊AGL(1,𝔽5)11yescharacteristic
256C2×C3×C4C22×AGL(1,𝔽5)11yescharacteristic
257#67—201no—
258#67—201no—
259C2×Dic3—21no—
260C2×Dic3—104no—
261C2×Dic3—101no—
262C2×Dic3—101no—
263C2×Dic3—202no—
264C2×Dic3—202no—
265D12—41no—
266C4×D3—42no—
267Dic6—201no—
268C3×C8—202no—
269C22×C5—121no—
270C22×C5—121no—
271C22×C5—121no—
272C22×C5C2×(C2×D3)⋊C411yescharacteristic
273C2×D5—21no—
274C2×D5—21no—
275C2×D5#90811yescharacteristic
276C2×D5#90911yescharacteristic
277AGL(1,𝔽5)—124no—
278C4×C5—21no—
279C4×C5—21no—
280C4×C5—121no—
281C4×C5#82512yes—
282Dic5—21no—
283Dic5—21no—
284Dic5—21no—
285Dic5—121no—
286Dic5—121no—
287C2×Q8—151no—
288C2×D4—31no—
289C2×D4—301no—
290C22×C4—51no—
291C22×C4—51no—
292C22×C4—51no—
293C22×C4—61no—
294C22×C4—302no—
295C2×C8—102no—
296C4⋊C4—31no—
297C4⋊C4—151no—
298C4⋊C4—301no—
299C4⋊C4—301no—
300C4⋊C4—301no—
301C4⋊C4—304no—
302C4⋊C4—301no—
303C22⋊C4—61no—
304C22⋊C4—301no—
305C22⋊C4—301no—
306C22⋊C4—301no—
307C22⋊C4—301no—
308C42—51no—
309C42—51no—
310C42—51no—
311C42—301no—
312C42—602no—
313C42D3×AGL(1,𝔽5)11yescharacteristic
314C3×C5#150911yescharacteristic, complemented, Hall
315C22×C3—51no—
316C22×C3—51no—
317C22×C3—101no—
318C22×C3—101no—
319C22×C3C2×C22⋊AGL(1,𝔽5)11yescharacteristic
320C2×D3—41no—
321C2×D3—41no—
322C2×D3—41no—
323C3×C4—21no—
324C3×C4—21no—
325C3×C4—101no—
326C3×C4—101no—
327C3×C4—101no—
328C3×C4#400812yes—
329Dic3—41no—
330Dic3—204no—
331Dic3—201no—
332Dic3—201no—
333C2×C5—241no—
334C2×C5#500611yescharacteristic
335C2×C5#513311yescharacteristic
336C2×C5C2×#82511yescharacteristic
337D5—21no—
338D5—21no—
339C23—51no—
340C23—61no—
341Q8—601no—
342D4—121no—
343D4—601no—
344D4—601no—
345C2×C4—51no—
346C2×C4—51no—
347C2×C4—51no—
348C2×C4—61no—
349C2×C4—101no—
350C2×C4—101no—
351C2×C4—102no—
352C2×C4—122no—
353C2×C4—201no—
354C2×C4—304no—
355C2×C4—301no—
356C2×C4—301no—
357C2×C4—602no—
358C2×C4—602no—
359C2×C4(C2×D3)⋊AGL(1,𝔽5)11yescharacteristic
360C2×C4(C2×D3)⋊AGL(1,𝔽5)11yescharacteristic
361C2×C4C2×D3×AGL(1,𝔽5)11yescharacteristic
362C8—202no—
363C2×C3—101no—
364C2×C3—101no—
365C2×C3#6574511yescharacteristic
366C2×C3C2×#400811yescharacteristic
367C2×C3#6579311yescharacteristic
368D3—81no—
369C5#7083311yescharacteristic, complemented, Sylow
370C22—51no—
371C22—51no—
372C22—101no—
373C22—101no—
374C22—121no—
375C22—121no—
376C22—121no—
377C22C2×(C2×D3)⋊AGL(1,𝔽5)11yescharacteristic
378C4—21no—
379C4—21no—
380C4—101no—
381C4—101no—
382C4—101no—
383C4—121no—
384C4—604no—
385C4—601no—
386C4—601no—
387C4#9115512yes—
388C3#1060439111yescharacteristic, complemented, Sylow
389C2—101no—
390C2—101no—
391C2—241no—
392C2#1175170811yescharacteristic
393C2C2×#9115511yescharacteristic
394C2#1175181511yescharacteristic
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#91449C22
cosocleC2×C3×C4×D5C23
hypercenterC42D3×AGL(1,𝔽5)
hypocenterC3×C5#1509
LCSG ▷ C2×C3×C4×C5 ▷ C22×C3×C5 ▷ C3×C5
UCSC1 ◁ C22 ◁ C42

Complements

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

NH ≅ G/Ntype
2. C4×C4⋊ΓL(1,𝔽16)C2semidirect
3. C3×#65683C2semidirect
4. #11746501C2semidirect
9. C4×Dic30C4semidirect
10. C4×C5×D12C4semidirect
15. #91388C4semidirect
16. #91373C4semidirect
27. #65683D3semidirect
113. C4×D12AGL(1,𝔽5)semidirect
314. C3×C5#1509semidirect
369. C5#70833semidirect
388. C3#10604391semidirect

Maximal subgroups

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

subgroupindex
#117526822
C4×C4⋊ΓL(1,𝔽16)2
C3×#656832
#117465012
#117464982
#106043913
#708335