Finite Group Databasebeta

← #14300323

Subgroups — #14300323

Order 1920.

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

normal subgroups only

This group has 748 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
2#11752438C211yesmaximal, characteristic
3#11752358C212yesmaximal, complemented
4C3×D5×#99C211yesmaximal, characteristic
5#11746737C212yesmaximal, complemented
6#10603919—31nomaximal, Hall
7C23×ΓL(1,𝔽16)C2211yescharacteristic
8C2×C3×D4×D5C2211yescharacteristic
9C2×#6763—21no—
10C2×#6763—21no—
11C2×#6763C2211yescharacteristic
12C2×#6763C2212yes—
13D4×ΓL(1,𝔽16)—22no—
14D4×ΓL(1,𝔽16)—22no—
15#91989—21no—
16C2×C4⋊ΓL(1,𝔽16)—22no—
17C3×D5×C22⋊C4C2212yes—
18#91235—22no—
19#91235—21no—
20C3×C5×#99C411yescharacteristic, complemented
21C3×#3966C411yescharacteristic, complemented
22C3×#3938C412yescomplemented
23#70021—51nomaximal, Hall
24#66258—31no—
25#65779—31no—
26#65699—32no—
27D5×#99D311yescharacteristic
28#64866—32no—
29C23×C3×D5D411yescharacteristic
30C23×C3×D5C2311yescharacteristic
31C22×ΓL(1,𝔽16)—22no—
32C22×ΓL(1,𝔽16)—21no—
33C22×ΓL(1,𝔽16)—21no—
34C22×ΓL(1,𝔽16)—21no—
35C22×ΓL(1,𝔽16)—21no—
36C22×ΓL(1,𝔽16)—42no—
37C22×ΓL(1,𝔽16)D412yes—
38C22×ΓL(1,𝔽16)D412yes—
39C2×C3×C5×D4C2×C411yescharacteristic
40C2×C3×C22⋊D5C2×C411yescharacteristic
41C2×C3×C22⋊D5Q811yescharacteristic
42C3×D4×D5—22no—
43C3×D4×D5—22no—
44C2×C3×D20Q811yescharacteristic
45C2×C3×C4×D5—22no—
46C2×C3×C4×D5D411yescharacteristic
47#6763—22no—
48#6763—22no—
49#6763—22no—
50#6763—22no—
51#6763—42no—
52C4⋊ΓL(1,𝔽16)—21no—
53C4⋊ΓL(1,𝔽16)—21no—
54C4⋊ΓL(1,𝔽16)—44no—
55C4×ΓL(1,𝔽16)—22no—
56C3×C5×C22⋊C4C2×C412yescomplemented
57C3×C22⋊Dic5C2×C412yescomplemented
58C3×#685—22no—
59#5798—51no—
60C2×C3×#99—51no—
61#5220—51no—
62#4950—52no—
63#4527—52no—
64C23×AGL(1,𝔽5)—31no—
65C2×D4×D5C2×D311yescharacteristic
66C2×C22⋊AGL(1,𝔽5)—32no—
67C2×C22⋊AGL(1,𝔽5)—31no—
68C2×C22⋊AGL(1,𝔽5)—61no—
69C2×C22⋊AGL(1,𝔽5)—61no—
70D4×AGL(1,𝔽5)—62no—
71D4×AGL(1,𝔽5)—62no—
72#4130—61no—
73C2×C4⋊AGL(1,𝔽5)—62no—
74D5×C22⋊C4C2×D312yes—
75(C2×AGL(1,𝔽5))⋊C4—62no—
76(C2×AGL(1,𝔽5))⋊C4—61no—
77C5×#99Dic311yescharacteristic, complemented
78#3966Dic311yescharacteristic, complemented
79#3938Dic312yescomplemented
80#1867—151noSylow
81C23×C3×C5C4⋊C411yescharacteristic, complemented
82C23×C3×C5C22×C411yescharacteristic
83C22×C3×D5—21no—
84C22×C3×D5—21no—
85C22×C3×D5—41no—
86C22×C3×D5—41no—
87C22×C3×D5—41no—
88C22×C3×D5—41no—
89C22×C3×D5C4⋊C411yescharacteristic, complemented
90C22×C3×D5C2×D412yes—
91C22×C3×D5C2×D411yescharacteristic
92C22×C3×D5C2×Q811yescharacteristic
93C22×C3×D5#4112yes—
94C2×ΓL(1,𝔽16)—24no—
95C2×ΓL(1,𝔽16)—22no—
96C2×ΓL(1,𝔽16)—22no—
97C2×ΓL(1,𝔽16)—22no—
98C2×ΓL(1,𝔽16)—42no—
99C2×ΓL(1,𝔽16)—41no—
100C2×ΓL(1,𝔽16)—41no—
101C2×ΓL(1,𝔽16)—42no—
102C2×ΓL(1,𝔽16)—42no—
103C2×ΓL(1,𝔽16)—41no—
104C2×ΓL(1,𝔽16)—41no—
105C2×ΓL(1,𝔽16)—41no—
106C2×ΓL(1,𝔽16)—81no—
107C2×ΓL(1,𝔽16)—81no—
108C3×C5×D4—22no—
109C2×C3×C4×C5—22no—
110C2×C3×C4×C5C4⋊C411yescharacteristic, complemented
111C3×C22⋊D5—22no—
112C3×C22⋊D5—22no—
113C2×C3×Dic5—22no—
114C2×C3×Dic5C4⋊C411yescharacteristic, complemented
115C3×D20—22no—
116C3×C4×D5—21no—
117C3×C4×D5—21no—
118C3×C4×D5—42no—
119C3×C4×D5—42no—
120C22×C3×D4—51no—
121C23×Dic3—51no—
122C2×C3×C22⋊C4—52no—
123C2×#217—52no—
124C2×#217—51no—
125C2×#217—101no—
126C2×#217—101no—
127D4×Dic3—102no—
128D4×Dic3—102no—
129#926—101no—
130C2×C4⋊Dic3—102no—
131C3×#99—51no—
132C3×#99—52no—
133C3×#99AGL(1,𝔽5)11yescharacteristic, complemented
134C3⋊#95—102no—
135C3⋊#95—101no—
136C23×D5D1211yescharacteristic
137C23×D5C22×D311yescharacteristic
138C22×AGL(1,𝔽5)—32no—
139C22×AGL(1,𝔽5)—32no—
140C22×AGL(1,𝔽5)—61no—
141C22×AGL(1,𝔽5)—62no—
142C22×AGL(1,𝔽5)—61no—
143C22×AGL(1,𝔽5)—61no—
144C22×AGL(1,𝔽5)—61no—
145C22×AGL(1,𝔽5)—122no—
146C2×C5×D4C2×Dic311yescharacteristic
147C2×C22⋊D5Dic611yescharacteristic
148C2×C22⋊D5C2×Dic311yescharacteristic
149D4×D5—22no—
150D4×D5—22no—
151C2×D20Dic611yescharacteristic
152C2×C4×D5—22no—
153C2×C4×D5D1211yescharacteristic
154C22⋊AGL(1,𝔽5)—62no—
155C22⋊AGL(1,𝔽5)—62no—
156C22⋊AGL(1,𝔽5)—62no—
157C22⋊AGL(1,𝔽5)—62no—
158C22⋊AGL(1,𝔽5)—122no—
159C4⋊AGL(1,𝔽5)—61no—
160C4⋊AGL(1,𝔽5)—61no—
161C4⋊AGL(1,𝔽5)—124no—
162C4×AGL(1,𝔽5)—62no—
163C5×C22⋊C4C2×Dic312yescomplemented
164C22⋊Dic5C2×Dic312yescomplemented
165#685—22no—
166#518—151no—
167C2×#99—51no—
168#386—151no—
169#385—152no—
170#342—152no—
171C22×C3×C5—21no—
172C22×C3×C5—41no—
173C22×C3×C5—41no—
174C22×C3×C5C2×C4⋊C411yescharacteristic
175C22×C3×C5C4×D412yes—
176C2×C3×D5—22no—
177C2×C3×D5—21no—
178C2×C3×D5—22no—
179C2×C3×D5—21no—
180C2×C3×D5—21no—
181C2×C3×D5—41no—
182C2×C3×D5—42no—
183C2×C3×D5—41no—
184C2×C3×D5—41no—
185C2×C3×D5—41no—
186C2×C3×D5—81no—
187C2×C3×D5#12011yescharacteristic
188C2×C3×D5C22⋊Q812yes—
189C2×C3×D5#12311yescharacteristic
190ΓL(1,𝔽16)—42no—
191ΓL(1,𝔽16)—42no—
192ΓL(1,𝔽16)—42no—
193ΓL(1,𝔽16)—81no—
194C3×C4×C5—21no—
195C3×C4×C5—42no—
196C3×Dic5—21no—
197C3×Dic5—42no—
198C24×C3—51no—
199C24×C3—51no—
200C2×C3×D4—51no—
201C2×C3×D4—51no—
202C2×C3×D4—51no—
203C2×C3×D4—102no—
204C2×C3×D4—102no—
205C2×C3×D4C2×AGL(1,𝔽5)11yescharacteristic
206C22×C3×C4—51no—
207C22×C3×C4—102no—
208C22×Dic3—52no—
209C22×Dic3—52no—
210C22×Dic3—102no—
211C22×Dic3—101no—
212C22×Dic3—101no—
213C22×Dic3—101no—
214C22×Dic3—101no—
215C22×Dic3—202no—
216C3×C22⋊C4—52no—
217C3×C22⋊C4—102no—
218C3×C22⋊C4C2×AGL(1,𝔽5)12yescomplemented
219#217—102no—
220#217—102no—
221#217—102no—
222#217—102no—
223#217—202no—
224C4⋊Dic3—101no—
225C4⋊Dic3—101no—
226C4⋊Dic3—204no—
227C4×Dic3—102no—
228C23×C5C4⋊Dic311yescharacteristic, complemented
229C23×C5C22×Dic311yescharacteristic
230C22×D5—21no—
231C22×D5—21no—
232C22×D5—41no—
233C22×D5—41no—
234C22×D5—41no—
235C22×D5—41no—
236C22×D5C4⋊Dic311yescharacteristic, complemented
237C22×D5C2×Dic611yescharacteristic
238C22×D5C2×D1211yescharacteristic
239C22×D5D3×D412yes—
240C22×D5#23712yes—
241C2×AGL(1,𝔽5)—64no—
242C2×AGL(1,𝔽5)—62no—
243C2×AGL(1,𝔽5)—62no—
244C2×AGL(1,𝔽5)—62no—
245C2×AGL(1,𝔽5)—122no—
246C2×AGL(1,𝔽5)—121no—
247C2×AGL(1,𝔽5)—121no—
248C2×AGL(1,𝔽5)—122no—
249C2×AGL(1,𝔽5)—122no—
250C2×AGL(1,𝔽5)—121no—
251C2×AGL(1,𝔽5)—121no—
252C2×AGL(1,𝔽5)—121no—
253C2×AGL(1,𝔽5)—241no—
254C2×AGL(1,𝔽5)—241no—
255C5×D4—22no—
256C2×C4×C5—22no—
257C2×C4×C5C4⋊Dic311yescharacteristic, complemented
258C22⋊D5—22no—
259C22⋊D5—22no—
260C2×Dic5—22no—
261C2×Dic5C4⋊Dic311yescharacteristic, complemented
262D20—22no—
263C4×D5—21no—
264C4×D5—21no—
265C4×D5—42no—
266C4×D5—42no—
267C22×D4—51no—
268C23×C4—151no—
269C4×D4—302no—
270C4×D4—302no—
271#117—301no—
272C2×C4⋊C4—302no—
273C2×C22⋊C4—52no—
274C2×C22⋊C4—152no—
275C2×C22⋊C4—151no—
276C2×C22⋊C4—301no—
277C2×C22⋊C4—301no—
278#99—51no—
279#99—52no—
280#99ΓL(1,𝔽16)11yescharacteristic, complemented
281#95—301no—
282#95—302no—
283C2×C3×C5—22no—
284C2×C3×C5—21no—
285C2×C3×C5—41no—
286C2×C3×C5#38511yescharacteristic
287C3×D5—21no—
288C3×D5—22no—
289C3×D5—41no—
290C3×D5B(4,𝔽2)11yescharacteristic
291C3×D5#45811yescharacteristic
292C23×C3—51no—
293C23×C3—52no—
294C23×C3—51no—
295C23×C3—52no—
296C23×C3—51no—
297C23×C3—101no—
298C23×C3—101no—
299C23×C3—201no—
300C23×C3—201no—
301C23×C3—201no—
302C23×C3—201no—
303C23×C3C4⋊AGL(1,𝔽5)11yescharacteristic, complemented
304C23×C3C22×AGL(1,𝔽5)11yescharacteristic
305C3×D4—22no—
306C3×D4—102no—
307C3×D4—102no—
308C3×D4—102no—
309C2×C3×C4—22no—
310C2×C3×C4—51no—
311C2×C3×C4—101no—
312C2×C3×C4—101no—
313C2×C3×C4—102no—
314C2×C3×C4—202no—
315C2×C3×C4—202no—
316C2×C3×C4C4⋊AGL(1,𝔽5)11yescharacteristic, complemented
317C2×Dic3—102no—
318C2×Dic3—102no—
319C2×Dic3—104no—
320C2×Dic3—102no—
321C2×Dic3—201no—
322C2×Dic3—202no—
323C2×Dic3—201no—
324C2×Dic3—202no—
325C2×Dic3—202no—
326C2×Dic3—201no—
327C2×Dic3—201no—
328C2×Dic3—201no—
329C2×Dic3—401no—
330C2×Dic3—401no—
331C22×C5—21no—
332C22×C5—41no—
333C22×C5—41no—
334C22×C5C2×C4⋊Dic311yescharacteristic
335C22×C5D4×Dic312yes—
336C2×D5—22no—
337C2×D5—21no—
338C2×D5—22no—
339C2×D5—21no—
340C2×D5—21no—
341C2×D5—41no—
342C2×D5—42no—
343C2×D5—41no—
344C2×D5—41no—
345C2×D5—41no—
346C2×D5—81no—
347C2×D5#87812yes—
348C2×D5#88211yescharacteristic
349C2×D5#88611yescharacteristic
350AGL(1,𝔽5)—122no—
351AGL(1,𝔽5)—122no—
352AGL(1,𝔽5)—122no—
353AGL(1,𝔽5)—241no—
354C4×C5—21no—
355C4×C5—42no—
356Dic5—21no—
357Dic5—42no—
358C24—51no—
359C24—51no—
360C2×D4—51no—
361C2×D4—51no—
362C2×D4—51no—
363C2×D4—102no—
364C2×D4—102no—
365C2×D4C2×ΓL(1,𝔽16)11yescharacteristic
366C22×C4—51no—
367C22×C4—102no—
368C22×C4—152no—
369C22×C4—152no—
370C22×C4—301no—
371C22×C4—302no—
372C22×C4—301no—
373C22×C4—301no—
374C22×C4—301no—
375C22×C4—602no—
376C4⋊C4—301no—
377C4⋊C4—301no—
378C4⋊C4—604no—
379C22⋊C4—52no—
380C22⋊C4—102no—
381C22⋊C4—302no—
382C22⋊C4—302no—
383C22⋊C4—302no—
384C22⋊C4—302no—
385C22⋊C4—602no—
386C22⋊C4C2×ΓL(1,𝔽16)12yescomplemented
387C42—302no—
388C3×C5#186711yescharacteristic, complemented, Hall
389C22×C3—21no—
390C22×C3—41no—
391C22×C3—41no—
392C22×C3—51no—
393C22×C3—52no—
394C22×C3—51no—
395C22×C3—102no—
396C22×C3—101no—
397C22×C3—102no—
398C22×C3—101no—
399C22×C3—101no—
400C22×C3—202no—
401C22×C3—201no—
402C22×C3—201no—
403C22×C3—201no—
404C22×C3—201no—
405C22×C3—401no—
406C22×C3C2×C4⋊AGL(1,𝔽5)11yescharacteristic
407C22×C3D4×AGL(1,𝔽5)12yes—
408C3×C4—21no—
409C3×C4—42no—
410C3×C4—101no—
411C3×C4—202no—
412Dic3—202no—
413Dic3—202no—
414Dic3—202no—
415Dic3—401no—
416C2×C5—22no—
417C2×C5—21no—
418C2×C5—41no—
419C2×C5#495011yescharacteristic
420D5—21no—
421D5—22no—
422D5—41no—
423D5#474111yescharacteristic
424D5#474211yescharacteristic
425C23—51no—
426C23—52no—
427C23—51no—
428C23—52no—
429C23—51no—
430C23—101no—
431C23—101no—
432C23—201no—
433C23—201no—
434C23—201no—
435C23—201no—
436C23C4⋊ΓL(1,𝔽16)11yescharacteristic, complemented
437C23C22×ΓL(1,𝔽16)11yescharacteristic
438D4—22no—
439D4—102no—
440D4—102no—
441D4—102no—
442C2×C4—22no—
443C2×C4—51no—
444C2×C4—101no—
445C2×C4—101no—
446C2×C4—102no—
447C2×C4—202no—
448C2×C4—202no—
449C2×C4—302no—
450C2×C4—302no—
451C2×C4—302no—
452C2×C4—304no—
453C2×C4—601no—
454C2×C4—602no—
455C2×C4—601no—
456C2×C4—602no—
457C2×C4—602no—
458C2×C4—601no—
459C2×C4—601no—
460C2×C4—601no—
461C2×C4—1201no—
462C2×C4—1201no—
463C2×C4C4⋊ΓL(1,𝔽16)11yescharacteristic, complemented
464C2×C3—22no—
465C2×C3—21no—
466C2×C3—41no—
467C2×C3—51no—
468C2×C3—51no—
469C2×C3—101no—
470C2×C3—102no—
471C2×C3—201no—
472C2×C3#6569911yescharacteristic
473C5#7002111yescharacteristic, complemented, Sylow
474C22—21no—
475C22—41no—
476C22—41no—
477C22—51no—
478C22—51no—
479C22—52no—
480C22—102no—
481C22—102no—
482C22—101no—
483C22—101no—
484C22—101no—
485C22—202no—
486C22—201no—
487C22—201no—
488C22—201no—
489C22—201no—
490C22—401no—
491C22C2×C4⋊ΓL(1,𝔽16)11yescharacteristic
492C22D4×ΓL(1,𝔽16)12yes—
493C4—21no—
494C4—42no—
495C4—101no—
496C4—202no—
497C4—602no—
498C4—602no—
499C4—602no—
500C4—1201no—
501C3#1060391911yescharacteristic, complemented, Sylow
502C2—22no—
503C2—21no—
504C2—41no—
505C2—51no—
506C2—51no—
507C2—101no—
508C2—102no—
509C2—201no—
510C2#1175235811yescharacteristic
subgroupG/H
centerC2#11752358
commutatorC23×C3×C5C22×C4
FrattiniC23C22×ΓL(1,𝔽16)
FittingC3×C5×#99C4
socleC2×C3×C5#385
prime radicalC2×C3×D4×D5C22
cosocleC23×C3×D5C23
hypercenter#99ΓL(1,𝔽16)
hypocenterC3×C5#1867
LCSG ▷ C23×C3×C5 ▷ C2×C3×C5 ▷ C3×C5
UCSC1 ◁ C2 ◁ C23 ◁ #99

Complements

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

NH ≅ G/Ntype
3. #11752358C2semidirect
5. #11746737C2semidirect
20. C3×C5×#99C4semidirect
21. C3×#3966C4semidirect
22. C3×#3938C4semidirect
56. C3×C5×C22⋊C4C2×C4semidirect
57. C3×C22⋊Dic5C2×C4semidirect
77. C5×#99Dic3semidirect
78. #3966Dic3semidirect
79. #3938Dic3semidirect
81. C23×C3×C5C4⋊C4semidirect
89. C22×C3×D5C4⋊C4semidirect
110. C2×C3×C4×C5C4⋊C4semidirect
114. C2×C3×Dic5C4⋊C4semidirect
133. C3×#99AGL(1,𝔽5)semidirect
163. C5×C22⋊C4C2×Dic3semidirect
164. C22⋊Dic5C2×Dic3semidirect
218. C3×C22⋊C4C2×AGL(1,𝔽5)semidirect
228. C23×C5C4⋊Dic3semidirect
236. C22×D5C4⋊Dic3semidirect
257. C2×C4×C5C4⋊Dic3semidirect
261. C2×Dic5C4⋊Dic3semidirect
280. #99ΓL(1,𝔽16)semidirect
303. C23×C3C4⋊AGL(1,𝔽5)semidirect
316. C2×C3×C4C4⋊AGL(1,𝔽5)semidirect
386. C22⋊C4C2×ΓL(1,𝔽16)semidirect
388. C3×C5#1867semidirect
436. C23C4⋊ΓL(1,𝔽16)semidirect
463. C2×C4C4⋊ΓL(1,𝔽16)semidirect
473. C5#70021semidirect
501. C3#10603919semidirect

Maximal subgroups

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

subgroupindex
#117569872
#117524382
#117523582
C3×D5×#992
#117467372
#106039193
#700215