Finite Group Databasebeta

← #14301535

Subgroups — #14301535

Order 1920.

888 conjugacy classes of subgroups; only the 121 normal subgroups are shown.

normal subgroups only

This group has 888 conjugacy classes of subgroups — too many to draw. Only its 121 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#11756987C211yesmaximal, characteristic, complemented
2C3×D5×#127C211yesmaximal, characteristic
3D5⋊#833C211yesmaximal, characteristic, complemented
4#11752426C212yesmaximal, complemented
5#11746733C212yesmaximal, complemented
6#10604614—31nomaximal, Hall
7C2×C3×D4×D5—22no—
8C2×C3×D4×D5C2211yescharacteristic
9C2×C3×D4×D5C2211yescharacteristic
10C2×#6763—22no—
11D4×ΓL(1,𝔽16)—24no—
12#91989C2212yes—
13D5⋊#208C2212yes—
14C3×C5×#127C411yescharacteristic, complemented
15C3×#4086—22no—
16C3×#4086C411yescharacteristic, complemented
17C3×#4086C411yescharacteristic, complemented
18C3×#4020C411yescharacteristic, complemented
19C3×C42×D5C2211yescharacteristic
20#91242—22no—
21#91236—24no—
22#71763—51nomaximal, Hall
23#66258—31no—
24D5×#127D311yescharacteristic
25D5⋊#100—31no—
26#65767—32no—
27#64862—32no—
28C23×C3×D5—21no—
29C23×C3×D5D412yes—
30C22×ΓL(1,𝔽16)—24no—
31C2×C3×C5×D4—22no—
32C2×C3×C5×D4C2×C411yescharacteristic
33C2×C3×C5×D4C2×C411yescharacteristic, complemented
34C2×C3×C22⋊D5—22no—
35C2×C3×C22⋊D5—22no—
36C2×C3×C22⋊D5—21no—
37C2×C3×C22⋊D5D412yes—
38C3×D4×D5—22no—
39C3×D4×D5—22no—
40C3×D4×D5—22no—
41C3×D4×D5—22no—
42C3×D4×D5—48no—
43C2×C3×D20—22no—
44C2×C3×D20C2×C411yescharacteristic
45C2×C3×D20C2×C411yescharacteristic, complemented
46C2×C3×C4×D5—21no—
47C2×C3×C4×D5C2311yescharacteristic
48#6763—24no—
49#6763—22no—
50#6762—22no—
51C4⋊ΓL(1,𝔽16)D414yescomplemented
52C4×ΓL(1,𝔽16)—22no—
53#6758—22no—
54D5⋊(C3⋊C8)—22no—
55C3×C42×C5C2×C411yescharacteristic, complemented
56C3×C4×Dic5—21no—
57C3×C4×Dic5C2×C411yescharacteristic, complemented
58C2×C3×#127—51no—
59#5798—51no—
60C2×#833—51no—
61#5215—52no—
62#4523—52no—
63C2×D4×D5—22no—
64C2×D4×D5C2×D311yescharacteristic
65C2×D4×D5C2×D311yescharacteristic
66C2×C22⋊AGL(1,𝔽5)—62no—
67D4×AGL(1,𝔽5)—64no—
68#4130—32no—
69D5⋊M16—32no—
70C5×#127Dic311yescharacteristic, complemented
71#4086—22no—
72#4086Dic311yescharacteristic, complemented
73#4086Dic311yescharacteristic, complemented
74#4020Dic311yescharacteristic, complemented
75C42×D5C2×D311yescharacteristic
76#4013—62no—
77#4007—64no—
78#1887—151noSylow
79C23×C3×C5—21no—
80C23×C3×C5C22⋊C412yes—
81C22×C3×D5—22no—
82C22×C3×D5—22no—
83C22×C3×D5—21no—
84C22×C3×D5—44no—
85C22×C3×D5—44no—
86C22×C3×D5—41no—
87C22×C3×D5—41no—
88C22×C3×D5C22⋊C412yes—
89C22×C3×D5C2×D411yescharacteristic
90C2×ΓL(1,𝔽16)—24no—
91C2×ΓL(1,𝔽16)—84no—
92C3×C5×D4—22no—
93C3×C5×D4—22no—
94C3×C5×D4—44no—
95C2×C3×C4×C5—21no—
96C2×C3×C4×C5C22×C411yescharacteristic
97C3×C22⋊D5—24no—
98C3×C22⋊D5—44no—
99C3×C22⋊D5—44no—
100C3×C22⋊D5—42no—
101C2×C3×Dic5—21no—
102C2×C3×Dic5C2×D411yescharacteristic
103C3×D20—22no—
104C3×D20—22no—
105C3×D20—44no—
106C3×C4×D5—24no—
107C3×C4×D5C2×D412yes—
108C3×C4×D5#4112yes—
109C3⋊#170—42no—
110C22×C3×D4—51no—
111C22×C3×D4—51no—
112C22×C3×D4—102no—
113C3×#127—51no—
114C3×#127—51no—
115C3×#127—51no—
116C3×#127—102no—
117C3×#127AGL(1,𝔽5)11yescharacteristic, complemented
118C2×C3×C42—51no—
119C2×#217—102no—
120D4×Dic3—104no—
121#926—52no—
122C2×#208—52no—
123#833—102no—
124#832—104no—
125C23×D5—21no—
126C23×D5#6712yes—
127C22×AGL(1,𝔽5)—64no—
128C2×C5×D4—22no—
129C2×C5×D4C2×Dic311yescharacteristic
130C2×C5×D4C2×Dic311yescharacteristic, complemented
131C2×C22⋊D5—22no—
132C2×C22⋊D5—22no—
133C2×C22⋊D5—21no—
134C2×C22⋊D5#6712yes—
135D4×D5—22no—
136D4×D5—22no—
137D4×D5—22no—
138D4×D5—22no—
139D4×D5—48no—
140C2×D20—22no—
141C2×D20C2×Dic311yescharacteristic
142C2×D20C2×Dic311yescharacteristic, complemented
143C2×C4×D5—21no—
144C2×C4×D5C22×D311yescharacteristic
145C22⋊AGL(1,𝔽5)—64no—
146C22⋊AGL(1,𝔽5)—62no—
147#704—62no—
148C4⋊AGL(1,𝔽5)—34no—
149C4×AGL(1,𝔽5)—62no—
150#700—62no—
151D5⋊C8—62no—
152C42×C5C2×Dic311yescharacteristic, complemented
153C4×Dic5—21no—
154C4×Dic5C2×Dic311yescharacteristic, complemented
155C2×#127—51no—
156#518—151no—
157#418—152no—
158C2×#100—151no—
159#339—152no—
160C22×C3×C5—22no—
161C22×C3×C5—42no—
162C22×C3×C5—42no—
163C22×C3×C5—41no—
164C22×C3×C5C2×C22⋊C411yescharacteristic
165C2×C3×D5—22no—
166C2×C3×D5—22no—
167C2×C3×D5—44no—
168C2×C3×D5—42no—
169C2×C3×D5—42no—
170C2×C3×D5—41no—
171C2×C3×D5—82no—
172C2×C3×D5—82no—
173C2×C3×D5—82no—
174C2×C3×D5#12011yescharacteristic
175C2×C3×D5#12311yescharacteristic
176ΓL(1,𝔽16)—44no—
177C3×C4×C5—22no—
178C3×C4×C5C4×D412yes—
179C3×Dic5—22no—
180C3×Dic5#12112yes—
181C24×C3—52no—
182C24×C3—101no—
183C2×C3×D4—22no—
184C2×C3×D4—52no—
185C2×C3×D4—51no—
186C2×C3×D4—51no—
187C2×C3×D4—102no—
188C2×C3×D4—102no—
189C2×C3×D4—102no—
190C2×C3×D4—102no—
191C2×C3×D4—102no—
192C2×C3×D4—101no—
193C2×C3×D4—102no—
194C2×C3×D4—102no—
195C2×C3×D4—208no—
196C2×C3×D4C2×AGL(1,𝔽5)11yescharacteristic
197C2×C3×D4C2×AGL(1,𝔽5)11yescharacteristic, complemented
198C22×C3×C4—51no—
199C22×C3×C4—101no—
200C22×Dic3—104no—
201C3×C42—51no—
202C3×C42—101no—
203C3×C42C2×AGL(1,𝔽5)11yescharacteristic, complemented
204#217—102no—
205#217—104no—
206C4⋊Dic3—54no—
207C4×Dic3—102no—
208#208—102no—
209#208—102no—
210C2×C3⋊C8—102no—
211C23×C5—21no—
212C23×C5#21712yes—
213C22×D5—22no—
214C22×D5—22no—
215C22×D5—21no—
216C22×D5—44no—
217C22×D5—44no—
218C22×D5—41no—
219C22×D5—41no—
220C22×D5#21712yes—
221C22×D5C2×#6711yescharacteristic
222C2×AGL(1,𝔽5)—64no—
223C2×AGL(1,𝔽5)—244no—
224C5×D4—22no—
225C5×D4—22no—
226C5×D4—44no—
227C2×C4×C5—21no—
228C2×C4×C5C22×Dic311yescharacteristic
229C22⋊D5—24no—
230C22⋊D5—44no—
231C22⋊D5—44no—
232C22⋊D5—42no—
233C2×Dic5—21no—
234C2×Dic5C2×#6711yescharacteristic
235D20—22no—
236D20—22no—
237D20—44no—
238C4×D5—24no—
239C4×D5D3×D412yes—
240C4×D5#23712yes—
241#170—122no—
242C22×D4—51no—
243C22×D4—51no—
244C22×D4—102no—
245C2×M16—152no—
246#127—51no—
247#127—51no—
248#127—51no—
249#127—102no—
250#127ΓL(1,𝔽16)11yescharacteristic, complemented
251C4×D4—304no—
252#117—152no—
253C2×C22⋊C4—302no—
254C2×C42—51no—
255#102—304no—
256#100—302no—
257C2×C3×C5—21no—
258C2×C3×C5—42no—
259C2×C3×C5—81no—
260C2×C3×C5#38611yescharacteristic
261C3×D5—21no—
262C3×D5—42no—
263C3×D5—81no—
264C3×D5#45312yes—
265C23×C3—21no—
266C23×C3—51no—
267C23×C3—52no—
268C23×C3—102no—
269C23×C3—102no—
270C23×C3—101no—
271C23×C3—201no—
272C23×C3—201no—
273C23×C3—204no—
274C23×C3—204no—
275C23×C3C22⋊AGL(1,𝔽5)12yes—
276C3×D4—22no—
277C3×D4—22no—
278C3×D4—44no—
279C3×D4—104no—
280C3×D4—102no—
281C3×D4—102no—
282C3×D4—204no—
283C3×D4—202no—
284C3×D4—204no—
285C3×D4—204no—
286C2×C3×C4—21no—
287C2×C3×C4—51no—
288C2×C3×C4—52no—
289C2×C3×C4—52no—
290C2×C3×C4—104no—
291C2×C3×C4—101no—
292C2×C3×C4C22×AGL(1,𝔽5)11yescharacteristic
293C2×Dic3—104no—
294C2×Dic3—404no—
295C3⋊C8—202no—
296C22×C5—22no—
297C22×C5—42no—
298C22×C5—42no—
299C22×C5—41no—
300C22×C5C2×#21711yescharacteristic
301C2×D5—22no—
302C2×D5—22no—
303C2×D5—44no—
304C2×D5—42no—
305C2×D5—42no—
306C2×D5—41no—
307C2×D5—82no—
308C2×D5—82no—
309C2×D5—82no—
310C2×D5#93511yescharacteristic
311C2×D5#93711yescharacteristic
312AGL(1,𝔽5)—124no—
313C4×C5—22no—
314C4×C5D4×Dic312yes—
315Dic5—22no—
316Dic5#93912yes—
317C24—52no—
318C24—101no—
319C2×D4—22no—
320C2×D4—52no—
321C2×D4—51no—
322C2×D4—51no—
323C2×D4—102no—
324C2×D4—102no—
325C2×D4—102no—
326C2×D4—102no—
327C2×D4—102no—
328C2×D4—101no—
329C2×D4—102no—
330C2×D4—102no—
331C2×D4—208no—
332C2×D4C2×ΓL(1,𝔽16)11yescharacteristic
333C2×D4C2×ΓL(1,𝔽16)11yescharacteristic, complemented
334C22×C4—51no—
335C22×C4—101no—
336C22×C4—304no—
337M16—302no—
338M16—302no—
339C2×C8—302no—
340C4⋊C4—154no—
341C22⋊C4—302no—
342C22⋊C4—304no—
343C42—51no—
344C42—101no—
345C42—302no—
346C42C2×ΓL(1,𝔽16)11yescharacteristic, complemented
347C3×C5#188711yescharacteristic, complemented, Hall
348C22×C3—22no—
349C22×C3—42no—
350C22×C3—42no—
351C22×C3—41no—
352C22×C3—51no—
353C22×C3—51no—
354C22×C3—102no—
355C22×C3—102no—
356C22×C3—202no—
357C22×C3—202no—
358C22×C3—204no—
359C22×C3—201no—
360C22×C3—402no—
361C22×C3—402no—
362C22×C3—402no—
363C22×C3C2×C22⋊AGL(1,𝔽5)11yescharacteristic
364C3×C4—22no—
365C3×C4—52no—
366C3×C4—102no—
367C3×C4D4×AGL(1,𝔽5)12yes—
368Dic3—204no—
369C2×C5—21no—
370C2×C5—42no—
371C2×C5—81no—
372C2×C5#522011yescharacteristic
373D5—21no—
374D5—42no—
375D5—81no—
376D5#507712yes—
377C23—21no—
378C23—51no—
379C23—52no—
380C23—102no—
381C23—102no—
382C23—101no—
383C23—201no—
384C23—201no—
385C23—204no—
386C23—204no—
387C23#676312yes—
388D4—22no—
389D4—22no—
390D4—44no—
391D4—104no—
392D4—102no—
393D4—102no—
394D4—204no—
395D4—202no—
396D4—204no—
397D4—204no—
398C2×C4—21no—
399C2×C4—51no—
400C2×C4—52no—
401C2×C4—52no—
402C2×C4—104no—
403C2×C4—101no—
404C2×C4—304no—
405C2×C4—1204no—
406C2×C4C22×ΓL(1,𝔽16)11yescharacteristic
407C8—602no—
408C2×C3—21no—
409C2×C3—42no—
410C2×C3—52no—
411C2×C3—81no—
412C2×C3—101no—
413C2×C3—202no—
414C2×C3—401no—
415C2×C3#6577911yescharacteristic
416C5#7176311yescharacteristic, complemented, Sylow
417C22—22no—
418C22—42no—
419C22—42no—
420C22—41no—
421C22—51no—
422C22—51no—
423C22—102no—
424C22—102no—
425C22—202no—
426C22—202no—
427C22—204no—
428C22—201no—
429C22—402no—
430C22—402no—
431C22—402no—
432C22C2×#676311yescharacteristic
433C4—22no—
434C4—52no—
435C4—102no—
436C4—604no—
437C4D4×ΓL(1,𝔽16)12yes—
438C3#1060461411yescharacteristic, complemented, Sylow
439C2—21no—
440C2—42no—
441C2—52no—
442C2—81no—
443C2—101no—
444C2—202no—
445C2—401no—
446C2#1175243811yescharacteristic
subgroupG/H
centerC2#11752438
commutatorC2×C3×C4×C5C22×C4
FrattiniC2×C4C22×ΓL(1,𝔽16)
FittingC3×C5×#127C4
socleC2×C3×C5#386
prime radicalC3×D5×#127C2
cosocleC2×C3×C4×D5C23
hypercenter#127ΓL(1,𝔽16)
hypocenterC3×C5#1887
LCSG ▷ C2×C3×C4×C5 ▷ C2×C3×C5 ▷ C3×C5
UCSC1 ◁ C2 ◁ C2×C4 ◁ #127

Complements

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

NH ≅ G/Ntype
1. #11756987C2semidirect
3. D5⋊#833C2semidirect
4. #11752426C2semidirect
5. #11746733C2semidirect
14. C3×C5×#127C4semidirect
16. C3×#4086C4semidirect
17. C3×#4086C4semidirect
18. C3×#4020C4semidirect
33. C2×C3×C5×D4C2×C4semidirect
45. C2×C3×D20C2×C4semidirect
51. C4⋊ΓL(1,𝔽16)D4semidirect
55. C3×C42×C5C2×C4semidirect
57. C3×C4×Dic5C2×C4semidirect
70. C5×#127Dic3semidirect
72. #4086Dic3semidirect
73. #4086Dic3semidirect
74. #4020Dic3semidirect
117. C3×#127AGL(1,𝔽5)semidirect
130. C2×C5×D4C2×Dic3semidirect
142. C2×D20C2×Dic3semidirect
152. C42×C5C2×Dic3semidirect
154. C4×Dic5C2×Dic3semidirect
197. C2×C3×D4C2×AGL(1,𝔽5)semidirect
203. C3×C42C2×AGL(1,𝔽5)semidirect
250. #127ΓL(1,𝔽16)semidirect
333. C2×D4C2×ΓL(1,𝔽16)semidirect
346. C42C2×ΓL(1,𝔽16)semidirect
347. C3×C5#1887semidirect
416. C5#71763semidirect
438. C3#10604614semidirect

Maximal subgroups

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

subgroupindex
#117569872
C3×D5×#1272
D5⋊#8332
#117524262
#117467332
#106046143
#717635