Finite Group Databasebeta

← #14306106

Subgroups — #14306106

Order 1920.

816 conjugacy classes of subgroups; only the 126 normal subgroups are shown.

normal subgroups only

This group has 816 conjugacy classes of subgroups — too many to draw. Only its 126 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
1C2×#91449C211yesmaximal, characteristic
2#11752424C211yesmaximal, characteristic, complemented
3C3×C4×C22⋊AGL(1,𝔽5)C211yesmaximal, characteristic, complemented
4C2×#91172C214yesmaximal, complemented
5#10604944—31nomaximal, Hall
6C22×C5×D12C411yescharacteristic, complemented
7C22×C3×C4×D5C2211yescharacteristic
8C22×#1201C2212yes—
9C2×#6763C2212yes—
10C2×C4⋊ΓL(1,𝔽16)—21no—
11C2×C3×C22⋊AGL(1,𝔽5)C2212yes—
12C2×C3×C4×AGL(1,𝔽5)—21no—
13C2×C4⋊Dic15C411yescharacteristic, complemented
14C2×(C2×D3)⋊Dic5C412yescomplemented
15#91449—21no—
16#91449—21no—
17#91449—21no—
18#91449—21no—
19#91235—21no—
20C3×(C2×AGL(1,𝔽5))⋊C4—21no—
21#91172—28no—
22#69456—51nomaximal, Hall
23C2×#4084—31no—
24C2×#4011—34no—
25#65765—31no—
26C4×C22⋊AGL(1,𝔽5)D311yescharacteristic, complemented
27C23×C5×D3C2×C412yes—
28C23×C3×D5C2311yescharacteristic
29C22×ΓL(1,𝔽16)—22no—
30C22×C3×AGL(1,𝔽5)—22no—
31C22×C3×C4×C5C2×C411yescharacteristic
32C22×Dic15C2×C412yes—
33C2×C5×D12—21no—
34C2×C5×D12—21no—
35C2×C5×D12—42no—
36C22×C3×Dic5C2×C411yescharacteristic
37C2×C3×C4×D5—42no—
38C2×C3×C4×D5D411yescharacteristic
39C2×C3×C4×D5D411yescharacteristic
40C2×C3×C4×D5D411yescharacteristic
41C2×C3×C4×D5D411yescharacteristic
42C2×#1201—24no—
43C2×#1201—22no—
44C2×#1201—22no—
45C2×#1201—24no—
46#6763—24no—
47C4⋊ΓL(1,𝔽16)—42no—
48C3×C22⋊AGL(1,𝔽5)—22no—
49C3×C22⋊AGL(1,𝔽5)D414yescomplemented
50C3×C4×AGL(1,𝔽5)—42no—
51C4⋊Dic15—21no—
52C4⋊Dic15—21no—
53(C2×D3)⋊Dic5—22no—
54(C2×D3)⋊Dic5—22no—
55C2×#930—51no—
56C3×C4×C22⋊C4—51no—
57#5212—51no—
58C2×#806—54no—
59C22×C5×D4—31no—
60C22×C22⋊D5—32no—
61C22×C4×D5C2×D311yescharacteristic
62C2×C22⋊AGL(1,𝔽5)—32no—
63C2×C22⋊AGL(1,𝔽5)C2×D312yes—
64C2×C4⋊AGL(1,𝔽5)—61no—
65C2×C4×AGL(1,𝔽5)—21no—
66C2×C22⋊Dic5—32no—
67#4084—61no—
68#4084—61no—
69#4084—61no—
70#4084—61no—
71C2×C4⋊Dic5—31no—
72#4011—68no—
73(C2×AGL(1,𝔽5))⋊C4—21no—
74(C2×AGL(1,𝔽5))⋊C4—61no—
75#1964—151noSylow
76C23×C3×C5C22×C411yescharacteristic
77C22×C5×D3—22no—
78C22×C5×D3—24no—
79C22×C5×D3—42no—
80C22×C5×D3—42no—
81C22×C3×D5—41no—
82C22×C3×D5—41no—
83C22×C3×D5C2×D411yescharacteristic
84C22×C3×D5C2×D411yescharacteristic
85C22×C3×D5C2×D412yes—
86C22×C3×D5#4112yes—
87C2×ΓL(1,𝔽16)—42no—
88C2×ΓL(1,𝔽16)—42no—
89C2×ΓL(1,𝔽16)—42no—
90C2×C3×AGL(1,𝔽5)—24no—
91C2×C3×AGL(1,𝔽5)—42no—
92C2×C3×AGL(1,𝔽5)—42no—
93C2×C3×C4×C5—22no—
94C2×C3×C4×C5C22⋊C411yescharacteristic
95C2×C3×C4×C5C22⋊C411yescharacteristic
96C2×Dic15—22no—
97C2×Dic15—24no—
98C5×D12—42no—
99C5×D12—42no—
100C2×C3×Dic5—41no—
101C2×C3×Dic5C22⋊C411yescharacteristic
102C2×C3×Dic5C22⋊C411yescharacteristic
103C3×C4×D5—22no—
104C3×C4×D5—22no—
105C3×C4×D5—41no—
106C3×C4×D5—41no—
107#1201—44no—
108#1201—42no—
109#1201—42no—
110C23×C3×C4—51no—
111C22×#67—52no—
112C22×D12AGL(1,𝔽5)11yescharacteristic, complemented
113C2×C3×C22⋊C4—52no—
114C2×C3×C42—101no—
115C2×#217—52no—
116#930—101no—
117#930—101no—
118#930—101no—
119#930—101no—
120C2×(C2×D3)⋊C4—52no—
121C2×C4⋊Dic3—51no—
122C2×C4⋊Dic3—101no—
123C3×#95—101no—
124C3⋊#95—101no—
125#806—108no—
126C24×C5—32no—
127C23×D5C22×D311yescharacteristic
128C22×AGL(1,𝔽5)—22no—
129C22×AGL(1,𝔽5)—62no—
130C2×C5×D4—61no—
131C2×C5×D4—61no—
132C2×C5×D4—122no—
133C22×C4×C5C4×D311yescharacteristic
134C2×C22⋊D5—64no—
135C2×C22⋊D5—62no—
136C2×C22⋊D5—64no—
137C2×C22⋊D5—62no—
138C22×Dic5—32no—
139C22×Dic5C4×D311yescharacteristic
140C2×C4×D5—42no—
141C2×C4×D5D1211yescharacteristic
142C2×C4×D5D1211yescharacteristic
143C2×C4×D5#6711yescharacteristic
144C2×C4×D5#6711yescharacteristic
145C22⋊AGL(1,𝔽5)—22no—
146C22⋊AGL(1,𝔽5)—64no—
147C22⋊AGL(1,𝔽5)D1214yescomplemented
148C4⋊AGL(1,𝔽5)—122no—
149C4×AGL(1,𝔽5)—42no—
150C22⋊Dic5—62no—
151C22⋊Dic5—62no—
152C4⋊Dic5—61no—
153C4⋊Dic5—61no—
154C2×#121—151no—
155C2×#99—154no—
156#380—151no—
157C4×C22⋊C4—51no—
158C22×C3×C5—21no—
159C22×C3×C5—21no—
160C22×C3×C5C2×C22⋊C411yescharacteristic
161C22×C3×C5C4×D412yes—
162C2×C5×D3—42no—
163C2×C5×D3—42no—
164C2×C5×D3—42no—
165C2×C5×D3—84no—
166C2×C3×D5—21no—
167C2×C3×D5—21no—
168C2×C3×D5—21no—
169C2×C3×D5—21no—
170C2×C3×D5—42no—
171C2×C3×D5—42no—
172C2×C3×D5#12112yes—
173C2×C3×D5#12411yescharacteristic
174C2×C3×D5#12711yescharacteristic
175ΓL(1,𝔽16)—82no—
176C3×AGL(1,𝔽5)—44no—
177C3×C4×C5—21no—
178C3×C4×C5#9912yescomplemented
179Dic15—42no—
180C3×Dic5—21no—
181C3×Dic5—21no—
182C24×C3—51no—
183C23×D3C2×AGL(1,𝔽5)12yes—
184C22×C3×C4—51no—
185C22×C3×C4—51no—
186C22×C3×C4—51no—
187C22×C3×C4—51no—
188C22×C3×C4—51no—
189C22×C3×C4—102no—
190C22×C3×C4—202no—
191C22×C3×C4C2×AGL(1,𝔽5)11yescharacteristic
192C2×#67—104no—
193C2×#67—102no—
194C2×#67—102no—
195C2×#67—104no—
196C22×Dic3—52no—
197C22×Dic3—102no—
198C2×D12—21no—
199C2×D12—21no—
200C2×D12—42no—
201C3×C22⋊C4—54no—
202C3×C22⋊C4—102no—
203C3×C42—202no—
204#217—104no—
205(C2×D3)⋊C4—102no—
206(C2×D3)⋊C4—102no—
207C4⋊Dic3—101no—
208C4⋊Dic3—101no—
209C4⋊Dic3—202no—
210C23×C5—62no—
211C23×C5—64no—
212C23×C5—122no—
213C23×C5—122no—
214C23×C5C2×C4×D311yescharacteristic
215C22×D5—41no—
216C22×D5—41no—
217C22×D5C2×D1211yescharacteristic
218C22×D5C2×D1212yes—
219C22×D5#23512yes—
220C22×D5C2×#6711yescharacteristic
221C2×AGL(1,𝔽5)—24no—
222C2×AGL(1,𝔽5)—42no—
223C2×AGL(1,𝔽5)—42no—
224C2×AGL(1,𝔽5)—122no—
225C2×AGL(1,𝔽5)—122no—
226C2×AGL(1,𝔽5)—122no—
227C5×D4—122no—
228C5×D4—122no—
229C2×C4×C5—22no—
230C2×C4×C5(C2×D3)⋊C411yescharacteristic
231C2×C4×C5(C2×D3)⋊C411yescharacteristic
232C22⋊D5—122no—
233C22⋊D5—122no—
234C22⋊D5—124no—
235C2×Dic5—41no—
236C2×Dic5—64no—
237C2×Dic5—62no—
238C2×Dic5(C2×D3)⋊C411yescharacteristic
239C2×Dic5(C2×D3)⋊C411yescharacteristic
240C4×D5—22no—
241C4×D5—22no—
242C4×D5—41no—
243C4×D5—41no—
244C22×D4—31no—
245C22×D4—152no—
246C23×C4—51no—
247#121—301no—
248#121—301no—
249#121—301no—
250#121—301no—
251C2×C4⋊C4—151no—
252C2×C4⋊C4—301no—
253C2×C22⋊C4—52no—
254C2×C22⋊C4—152no—
255C2×C22⋊C4—152no—
256C2×C42—101no—
257#99—308no—
258#95—101no—
259#95—301no—
260C2×C3×C5—22no—
261C2×C3×C5#39011yescharacteristic
262C2×C3×C5C2×#9911yescharacteristic
263C2×C3×C5#41011yescharacteristic
264C3×D5—22no—
265C3×D5—21no—
266C3×D5—21no—
267C5×D3—82no—
268C23×C3—51no—
269C23×C3—51no—
270C23×C3—52no—
271C23×C3—52no—
272C23×C3—201no—
273C23×C3—201no—
274C23×C3C22×AGL(1,𝔽5)11yescharacteristic
275C22×D3—22no—
276C22×D3—24no—
277C22×D3—42no—
278C22×D3—42no—
279C2×C3×C4—22no—
280C2×C3×C4—51no—
281C2×C3×C4—51no—
282C2×C3×C4—102no—
283C2×C3×C4—102no—
284C2×C3×C4—104no—
285C2×C3×C4—201no—
286C2×C3×C4—201no—
287C2×C3×C4—201no—
288C2×C3×C4—202no—
289C2×C3×C4—202no—
290C2×C3×C4C22⋊AGL(1,𝔽5)11yescharacteristic
291C2×C3×C4C22⋊AGL(1,𝔽5)11yescharacteristic
292#67—204no—
293#67—202no—
294#67—202no—
295C2×Dic3—104no—
296C2×Dic3—102no—
297C2×Dic3—202no—
298C2×Dic3—202no—
299C2×Dic3—202no—
300D12—42no—
301D12—42no—
302C22×C5—21no—
303C22×C5—21no—
304C22×C5—122no—
305C22×C5—122no—
306C22×C5—122no—
307C22×C5—244no—
308C22×C5C4×D1212yes—
309C22×C5C2×(C2×D3)⋊C411yescharacteristic
310C2×D5—21no—
311C2×D5—21no—
312C2×D5—21no—
313C2×D5—21no—
314C2×D5—42no—
315C2×D5—42no—
316C2×D5#87411yescharacteristic
317C2×D5#87511yescharacteristic
318C2×D5#93012yes—
319AGL(1,𝔽5)—44no—
320AGL(1,𝔽5)—242no—
321C4×C5—21no—
322C4×C5#80612yescomplemented
323Dic5—21no—
324Dic5—21no—
325Dic5—122no—
326C24—32no—
327C24—51no—
328C2×D4—61no—
329C2×D4—61no—
330C2×D4—122no—
331C2×D4—304no—
332C2×D4—302no—
333C2×D4—302no—
334C2×D4—304no—
335C22×C4—51no—
336C22×C4—51no—
337C22×C4—51no—
338C22×C4—51no—
339C22×C4—51no—
340C22×C4—102no—
341C22×C4—152no—
342C22×C4—202no—
343C22×C4—302no—
344C22×C4D3×AGL(1,𝔽5)11yescharacteristic
345C4⋊C4—301no—
346C4⋊C4—301no—
347C4⋊C4—602no—
348C22⋊C4—54no—
349C22⋊C4—102no—
350C22⋊C4—302no—
351C22⋊C4—302no—
352C22⋊C4—304no—
353C42—202no—
354C3×C5#196411yescharacteristic, complemented, Hall
355C22×C3—21no—
356C22×C3—21no—
357C22×C3—52no—
358C22×C3—51no—
359C22×C3—51no—
360C22×C3—101no—
361C22×C3—101no—
362C22×C3—101no—
363C22×C3—101no—
364C22×C3—202no—
365C22×C3—202no—
366C22×C3D4×AGL(1,𝔽5)12yes—
367C22×C3C2×C22⋊AGL(1,𝔽5)11yescharacteristic
368C2×D3—42no—
369C2×D3—42no—
370C2×D3—42no—
371C2×D3—84no—
372C3×C4—21no—
373C3×C4—101no—
374C3×C4—101no—
375C3×C4—204no—
376C3×C4#401112yescomplemented
377Dic3—202no—
378Dic3—402no—
379C2×C5—22no—
380C2×C5—242no—
381C2×C5#493911yescharacteristic
382C2×C5C2×#80611yescharacteristic
383C2×C5#500611yescharacteristic
384D5—22no—
385D5—21no—
386D5—21no—
387C23—51no—
388C23—51no—
389C23—52no—
390C23—52no—
391C23—62no—
392C23—64no—
393C23—122no—
394C23—122no—
395C23—201no—
396C23—201no—
397C23C2×D3×AGL(1,𝔽5)11yescharacteristic
398D4—122no—
399D4—122no—
400D4—604no—
401D4—602no—
402D4—602no—
403C2×C4—22no—
404C2×C4—51no—
405C2×C4—51no—
406C2×C4—102no—
407C2×C4—102no—
408C2×C4—104no—
409C2×C4—201no—
410C2×C4—201no—
411C2×C4—201no—
412C2×C4—202no—
413C2×C4—202no—
414C2×C4—304no—
415C2×C4—302no—
416C2×C4—602no—
417C2×C4—602no—
418C2×C4—602no—
419C2×C4(C2×D3)⋊AGL(1,𝔽5)11yescharacteristic
420C2×C4(C2×D3)⋊AGL(1,𝔽5)11yescharacteristic
421C2×C3—22no—
422C2×C3—101no—
423C2×C3—102no—
424C2×C3—101no—
425C2×C3#6577111yescharacteristic
426C2×C3#6578011yescharacteristic
427C2×C3C2×#401111yescharacteristic
428D3—82no—
429C5#6945611yescharacteristic, complemented, Sylow
430C22—21no—
431C22—21no—
432C22—52no—
433C22—51no—
434C22—51no—
435C22—101no—
436C22—101no—
437C22—101no—
438C22—101no—
439C22—122no—
440C22—122no—
441C22—122no—
442C22—202no—
443C22—202no—
444C22—244no—
445C22AGL(1,𝔽5)×D1212yes—
446C22C2×(C2×D3)⋊AGL(1,𝔽5)11yescharacteristic
447C4—21no—
448C4—101no—
449C4—101no—
450C4—204no—
451C4—602no—
452C4—1202no—
453C4#9117212yescomplemented
454C3#1060494411yescharacteristic, complemented, Sylow
455C2—22no—
456C2—102no—
457C2—101no—
458C2—101no—
459C2—242no—
460C2#1175181611yescharacteristic
461C2#1175185011yescharacteristic
462C2C2×#9117211yescharacteristic
subgroupG/H
centerC22C2×(C2×D3)⋊AGL(1,𝔽5)
commutatorC23×C3×C5C22×C4
FrattiniC23C2×D3×AGL(1,𝔽5)
FittingC22×C3×C4×C5C2×C4
socleC22×C3×C5C2×C22⋊C4
prime radicalC2×#91449C2
cosocleC23×C3×D5C23
hypercenterC22×C4D3×AGL(1,𝔽5)
hypocenterC3×C5#1964
LCSG ▷ C23×C3×C5 ▷ C2×C3×C5 ▷ C3×C5
UCSC1 ◁ C22 ◁ C22×C4

Complements

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

NH ≅ G/Ntype
2. #11752424C2semidirect
3. C3×C4×C22⋊AGL(1,𝔽5)C2semidirect
4. C2×#91172C2semidirect
6. C22×C5×D12C4semidirect
13. C2×C4⋊Dic15C4semidirect
14. C2×(C2×D3)⋊Dic5C4semidirect
26. C4×C22⋊AGL(1,𝔽5)D3semidirect
49. C3×C22⋊AGL(1,𝔽5)D4semidirect
112. C22×D12AGL(1,𝔽5)semidirect
147. C22⋊AGL(1,𝔽5)D12semidirect
178. C3×C4×C5#99semidirect
322. C4×C5#806semidirect
354. C3×C5#1964semidirect
376. C3×C4#4011semidirect
429. C5#69456semidirect
453. C4#91172semidirect
454. C3#10604944semidirect

Maximal subgroups

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

subgroupindex
C2×#914492
#117524242
C3×C4×C22⋊AGL(1,𝔽5)2
C2×#911722
#106049443
#694565