Finite Group Databasebeta

← #14283008

Subgroups — #14283008

Order 1920.

668 conjugacy classes of subgroups; only the 102 normal subgroups are shown.

normal subgroups only

This group has 668 conjugacy classes of subgroups — too many to draw. Only its 102 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#11754527C211yesmaximal, characteristic
2#11752415C211yesmaximal, characteristic, complemented
3C2×C3×(C2×AGL(1,𝔽5))⋊C4C211yesmaximal, characteristic, complemented
4#11746502C212yesmaximal, complemented
5#11746497C212yesmaximal, complemented
6#10604499—31nomaximal, Hall
7C23×C3×AGL(1,𝔽5)—21no—
8C22×C3×C4×D5C2211yescharacteristic
9C2×#6773C411yescharacteristic, complemented
10C2×#6758—21no—
11#91819C411yescharacteristic, complemented
12C5×#924C411yescharacteristic, complemented
13#91564—21no—
14#91552—21no—
15#91450C2211yescharacteristic
16#91444—21no—
17#91359C2211yescharacteristic
18#91358C411yescharacteristic, complemented
19#91348—21no—
20#91241—21no—
21#91232C2212yes—
22C3×(C2×AGL(1,𝔽5))⋊C4—21no—
23C3×(C2×AGL(1,𝔽5))⋊C4C2212yes—
24#69398—51nomaximal, Hall
25#65961—31no—
26C2×(C2×AGL(1,𝔽5))⋊C4D311yescharacteristic, complemented
27#65756—31no—
28#64926—32no—
29#64868—32no—
30C23×C3×D5D411yescharacteristic
31C22×C3×AGL(1,𝔽5)—24no—
32C22×C3×AGL(1,𝔽5)—22no—
33C22×C3×AGL(1,𝔽5)—42no—
34C22×C3×AGL(1,𝔽5)—42no—
35C22×C3×C4×C5C2×C411yescharacteristic
36C2×#1220—21no—
37C2×D60C2×C411yescharacteristic, complemented
38C2×C5×Dic6C2×C411yescharacteristic, complemented
39C22×C3×Dic5D411yescharacteristic
40C2×C3×C4×D5—42no—
41C2×C3×C4×D5C2×C411yescharacteristic
42C2×C3×C4×D5D411yescharacteristic
43C2×C3×C4×D5D411yescharacteristic
44C2×C3×C4×D5C2311yescharacteristic
45C2×#1202—21no—
46C2×#1202—21no—
47C2×#1200—21no—
48#6773—42no—
49C2×C3⋊#170—22no—
50#6758—42no—
51#6714—21no—
52C4⋊Dic15C2×C411yescharacteristic, complemented
53C5×#217—21no—
54C5×C4⋊Dic3C2×C411yescharacteristic, complemented
55C5×Dic3⋊C4—21no—
56#6669—21no—
57#6667—21no—
58#6665—21no—
59#6665—21no—
60Dic3×Dic5—21no—
61#5791—51no—
62C2×C3×#95—51no—
63#5207—51no—
64#4452—52no—
65#4451—52no—
66C23×AGL(1,𝔽5)—21no—
67C2×#713—31no—
68C22×C4×D5C2×D311yescharacteristic
69C2×#700—61no—
70C5×C22⋊Q8—31no—
71#4092—31no—
72#4076—31no—
73#4041—31no—
74#4040—61no—
75#4039—61no—
76#4038—31no—
77#4028—61no—
78#4027—61no—
79C22⋊#170—61no—
80(C2×AGL(1,𝔽5))⋊C4—21no—
81(C2×AGL(1,𝔽5))⋊C4C2×D312yes—
82#4003—32no—
83#1504—151noSylow
84C23×C3×C5C22⋊C411yescharacteristic
85C22×D15—21no—
86C22×C3×D5—22no—
87C22×C3×D5—22no—
88C22×C3×D5—21no—
89C22×C3×D5—21no—
90C22×C3×D5C22⋊C411yescharacteristic
91C22×C3×D5C2×D411yescharacteristic
92C2×C3×AGL(1,𝔽5)—24no—
93C2×C3×AGL(1,𝔽5)—48no—
94C2×C3×AGL(1,𝔽5)—42no—
95C2×C3×AGL(1,𝔽5)—42no—
96C2×C3×C4×C5—41no—
97C2×C3×C4×C5C22⋊C411yescharacteristic
98C2×C3×C4×C5C22×C411yescharacteristic
99#1220—42no—
100C2×Dic15—21no—
101D60—41no—
102C2×C5×Dic3—21no—
103C2×C5×Dic3—21no—
104C5×Dic6—41no—
105C2×C3×Dic5—41no—
106C2×C3×Dic5C22⋊C411yescharacteristic
107C2×C3×Dic5C2×D411yescharacteristic
108C3×C4×D5—22no—
109C3×C4×D5—22no—
110C3×C4×D5—41no—
111C3×C4×D5—41no—
112#1202—42no—
113#1202—41no—
114#1202—41no—
115#1200—42no—
116C3⋊#170—42no—
117C23×C3×C4—51no—
118C23×C3×C4—101no—
119C2×#235—51no—
120#953—101no—
121#930—51no—
122#930—51no—
123#929—101no—
124C4×#67—101no—
125C4×#67—101no—
126#926—51no—
127#924—51no—
128#924AGL(1,𝔽5)11yescharacteristic, complemented
129C2×#208—101no—
130C3×#95—52no—
131C3×#95—101no—
132#830—52no—
133#830—101no—
134C23×D5#6711yescharacteristic
135C22×AGL(1,𝔽5)—24no—
136C22×AGL(1,𝔽5)—22no—
137C22×AGL(1,𝔽5)—42no—
138C22×AGL(1,𝔽5)—42no—
139C2×C5×Q8—31no—
140C22×C4×C5C4×D311yescharacteristic
141C2×C22⋊D5—61no—
142C22×Dic5D1211yescharacteristic
143#713—122no—
144C2×D20—31no—
145C2×D20—61no—
146C2×D20—61no—
147C2×C4×D5—42no—
148C2×C4×D5—61no—
149C2×C4×D5C4×D311yescharacteristic
150C2×C4×D5D1211yescharacteristic
151C2×C4×D5#6711yescharacteristic
152C2×C4×D5C22×D311yescharacteristic
153C2×#170—62no—
154#700—122no—
155C5×C4⋊C4—31no—
156C5×C4⋊C4—61no—
157C5×C22⋊C4—61no—
158#685—61no—
159#685—61no—
160#685—61no—
161#685—61no—
162C4⋊Dic5—31no—
163#683—61no—
164C4×Dic5—61no—
165#525—151no—
166#407—151no—
167C2×#95—51no—
168#328—152no—
169#327—152no—
170C22×C3×C5—21no—
171C22×C3×C5—21no—
172C22×C3×C5#10412yes—
173C22×C3×C5C2×C22⋊C411yescharacteristic
174C2×D15—41no—
175C2×D15—41no—
176C2×D15—41no—
177C2×C3×D5—24no—
178C2×C3×D5—24no—
179C2×C3×D5—21no—
180C2×C3×D5—21no—
181C2×C3×D5—21no—
182C2×C3×D5—21no—
183C2×C3×D5#9912yes—
184C2×C3×D5AGL(1,ℤ/8)11yescharacteristic
185C2×C3×D5#13711yescharacteristic
186C3×AGL(1,𝔽5)—44no—
187C3×C4×C5—21no—
188C3×C4×C5—21no—
189Dic15—41no—
190C3×Dic5—21no—
191C3×Dic5—21no—
192C5×Dic3—41no—
193C5×Dic3—41no—
194C24×C3—51no—
195C22×C3×C4—51no—
196C22×C3×C4—51no—
197C22×C3×C4—51no—
198C22×C3×C4—51no—
199C22×C3×C4—51no—
200C22×C3×C4—102no—
201C22×C3×C4—104no—
202C22×C3×C4—202no—
203C22×C3×C4—202no—
204C22×C3×C4—202no—
205C22×C3×C4C2×AGL(1,𝔽5)11yescharacteristic
206C2×#67—101no—
207C2×#67—101no—
208C2×#67—101no—
209#235—202no—
210C2×D12—51no—
211C2×C4×D3—101no—
212C2×Dic6C2×AGL(1,𝔽5)11yescharacteristic, complemented
213#217—21no—
214#217—101no—
215#217—101no—
216(C2×D3)⋊C4—101no—
217(C2×D3)⋊C4—101no—
218C4⋊Dic3—51no—
219C4⋊Dic3C2×AGL(1,𝔽5)11yescharacteristic, complemented
220Dic3⋊C4—21no—
221Dic3⋊C4—101no—
222C4×Dic3—101no—
223#208—202no—
224C2×C3⋊C8—102no—
225C23×C5(C2×D3)⋊C411yescharacteristic
226C22×D5—22no—
227C22×D5—22no—
228C22×D5—21no—
229C22×D5—21no—
230C22×D5—61no—
231C22×D5(C2×D3)⋊C411yescharacteristic
232C22×D5C2×#6711yescharacteristic
233C2×AGL(1,𝔽5)—24no—
234C2×AGL(1,𝔽5)—48no—
235C2×AGL(1,𝔽5)—42no—
236C2×AGL(1,𝔽5)—42no—
237C5×Q8—121no—
238C2×C4×C5—41no—
239C2×C4×C5—61no—
240C2×C4×C5—61no—
241C2×C4×C5(C2×D3)⋊C411yescharacteristic
242C2×C4×C5C2×C4×D311yescharacteristic
243C22⋊D5—122no—
244C2×Dic5—41no—
245C2×Dic5—61no—
246C2×Dic5(C2×D3)⋊C411yescharacteristic
247C2×Dic5C2×D1211yescharacteristic
248D20—121no—
249D20—122no—
250D20—121no—
251D20—121no—
252C4×D5—22no—
253C4×D5—22no—
254C4×D5—41no—
255C4×D5—41no—
256C4×D5—122no—
257#170—122no—
258C2×#41—151no—
259C23×C4—51no—
260C23×C4—101no—
261C2×M16—301no—
262#123—301no—
263C22⋊Q8—31no—
264C22⋊Q8—151no—
265#121—151no—
266#121—151no—
267#120—301no—
268C4×D4—301no—
269C4×D4—301no—
270#117—151no—
271C22⋊C8—152no—
272C22⋊C8—301no—
273#95—52no—
274#95—101no—
275C2×C3×C5—22no—
276C2×C3×C5C2×#9911yescharacteristic
277C2×C3×C5#41711yescharacteristic
278C2×C3×C5C2×#10411yescharacteristic
279D15—81no—
280C3×D5—21no—
281C3×D5—22no—
282C3×D5—21no—
283C23×C3—51no—
284C23×C3—51no—
285C23×C3—101no—
286C23×C3—102no—
287C23×C3—101no—
288C23×C3—102no—
289C23×C3C22⋊AGL(1,𝔽5)11yescharacteristic
290C22×D3—101no—
291C2×C3×C4—41no—
292C2×C3×C4—51no—
293C2×C3×C4—51no—
294C2×C3×C4—102no—
295C2×C3×C4—102no—
296C2×C3×C4—104no—
297C2×C3×C4—201no—
298C2×C3×C4—201no—
299C2×C3×C4—201no—
300C2×C3×C4—202no—
301C2×C3×C4—208no—
302C2×C3×C4—202no—
303C2×C3×C4C22⋊AGL(1,𝔽5)11yescharacteristic
304C2×C3×C4C22×AGL(1,𝔽5)11yescharacteristic
305#67—202no—
306#67—201no—
307#67—201no—
308#67—202no—
309C2×Dic3—21no—
310C2×Dic3—21no—
311C2×Dic3—101no—
312D12—201no—
313C4×D3—202no—
314Dic6—41no—
315C3⋊C8—202no—
316C22×C5—21no—
317C22×C5—21no—
318C22×C5#80512yes—
319C22×C5C2×(C2×D3)⋊C411yescharacteristic
320C2×D5—24no—
321C2×D5—24no—
322C2×D5—21no—
323C2×D5—21no—
324C2×D5—21no—
325C2×D5—21no—
326C2×D5—121no—
327C2×D5—121no—
328C2×D5—121no—
329C2×D5#80612yes—
330C2×D5#93211yescharacteristic
331C2×D5#94211yescharacteristic
332AGL(1,𝔽5)—44no—
333C4×C5—21no—
334C4×C5—21no—
335C4×C5—121no—
336C4×C5—121no—
337Dic5—21no—
338Dic5—21no—
339Dic5—121no—
340C24—51no—
341#41—602no—
342C2×Q8—31no—
343C2×D4—151no—
344C2×D4—301no—
345C2×D4—301no—
346C2×D4—301no—
347C22×C4—51no—
348C22×C4—51no—
349C22×C4—51no—
350C22×C4—51no—
351C22×C4—51no—
352C22×C4—102no—
353C22×C4—104no—
354C22×C4—202no—
355C22×C4—202no—
356C22×C4—202no—
357C22×C4—301no—
358C22×C4D3×AGL(1,𝔽5)11yescharacteristic
359M16—602no—
360C2×C8—302no—
361C4⋊C4—31no—
362C4⋊C4—61no—
363C4⋊C4—151no—
364C4⋊C4—301no—
365C22⋊C4—61no—
366C22⋊C4—301no—
367C22⋊C4—301no—
368C22⋊C4—301no—
369C22⋊C4—301no—
370C42—301no—
371C3×C5#150411yescharacteristic, complemented, Hall
372C22×C3—21no—
373C22×C3—21no—
374C22×C3—51no—
375C22×C3—51no—
376C22×C3—52no—
377C22×C3—104no—
378C22×C3—104no—
379C22×C3—101no—
380C22×C3—101no—
381C22×C3—101no—
382C22×C3—101no—
383C22×C3#401012yes—
384C22×C3C2×C22⋊AGL(1,𝔽5)11yescharacteristic
385C2×D3—201no—
386C2×D3—201no—
387C2×D3—201no—
388C3×C4—21no—
389C3×C4—21no—
390C3×C4—101no—
391C3×C4—101no—
392C3×C4—204no—
393Dic3—41no—
394Dic3—41no—
395Dic3—201no—
396C2×C5—22no—
397C2×C5C2×#80511yescharacteristic
398C2×C5C2×#80611yescharacteristic
399C2×C5#496611yescharacteristic
400D5—21no—
401D5—22no—
402D5—21no—
403D5—241no—
404C23—51no—
405C23—51no—
406C23—101no—
407C23—102no—
408C23—101no—
409C23—102no—
410C23—301no—
411C23(C2×D3)⋊AGL(1,𝔽5)11yescharacteristic
412Q8—121no—
413D4—601no—
414D4—602no—
415D4—601no—
416D4—601no—
417D4—602no—
418C2×C4—41no—
419C2×C4—51no—
420C2×C4—51no—
421C2×C4—61no—
422C2×C4—61no—
423C2×C4—102no—
424C2×C4—102no—
425C2×C4—104no—
426C2×C4—201no—
427C2×C4—201no—
428C2×C4—201no—
429C2×C4—202no—
430C2×C4—208no—
431C2×C4—202no—
432C2×C4—301no—
433C2×C4—602no—
434C2×C4(C2×D3)⋊AGL(1,𝔽5)11yescharacteristic
435C2×C4C2×D3×AGL(1,𝔽5)11yescharacteristic
436C8—602no—
437C2×C3—22no—
438C2×C3—101no—
439C2×C3—101no—
440C2×C3—102no—
441C2×C3D5⋊#9911yescharacteristic
442C2×C3C2×#401011yescharacteristic
443C2×C3#6579311yescharacteristic
444D3—401no—
445C5#6939811yescharacteristic, complemented, Sylow
446C22—21no—
447C22—21no—
448C22—51no—
449C22—51no—
450C22—52no—
451C22—104no—
452C22—104no—
453C22—101no—
454C22—101no—
455C22—101no—
456C22—101no—
457C22—601no—
458C22—601no—
459C22—601no—
460C22#9115812yes—
461C22C2×(C2×D3)⋊AGL(1,𝔽5)11yescharacteristic
462C4—21no—
463C4—21no—
464C4—101no—
465C4—101no—
466C4—121no—
467C4—121no—
468C4—204no—
469C4—601no—
470C3#1060449911yescharacteristic, complemented, Sylow
471C2—22no—
472C2—101no—
473C2—101no—
474C2—102no—
475C2—1201no—
476C2#1175170511yescharacteristic
477C2C2×#9115811yescharacteristic
478C2D5⋊#80611yescharacteristic
subgroupG/H
centerC22C2×(C2×D3)⋊AGL(1,𝔽5)
commutatorC2×C3×C4×C5C22×C4
FrattiniC2×C4C2×D3×AGL(1,𝔽5)
FittingC22×C3×C4×C5C2×C4
socleC22×C3×C5C2×C22⋊C4
prime radical#11754527C2
cosocleC2×C3×C4×D5C23
hypercenterC22×C4D3×AGL(1,𝔽5)
hypocenterC3×C5#1504
LCSG ▷ C2×C3×C4×C5 ▷ C22×C3×C5 ▷ C3×C5
UCSC1 ◁ C22 ◁ C22×C4

Complements

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

NH ≅ G/Ntype
2. #11752415C2semidirect
3. C2×C3×(C2×AGL(1,𝔽5))⋊C4C2semidirect
4. #11746502C2semidirect
5. #11746497C2semidirect
9. C2×#6773C4semidirect
11. #91819C4semidirect
12. C5×#924C4semidirect
18. #91358C4semidirect
26. C2×(C2×AGL(1,𝔽5))⋊C4D3semidirect
37. C2×D60C2×C4semidirect
38. C2×C5×Dic6C2×C4semidirect
52. C4⋊Dic15C2×C4semidirect
54. C5×C4⋊Dic3C2×C4semidirect
128. #924AGL(1,𝔽5)semidirect
212. C2×Dic6C2×AGL(1,𝔽5)semidirect
219. C4⋊Dic3C2×AGL(1,𝔽5)semidirect
371. C3×C5#1504semidirect
445. C5#69398semidirect
470. C3#10604499semidirect

Maximal subgroups

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

subgroupindex
#117545272
#117524152
C2×C3×(C2×AGL(1,𝔽5))⋊C42
#117465022
#117464972
#106044993
#693985