Finite Group Databasebeta

← #14283009

Subgroups — #14283009

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#11754526C211yesmaximal, characteristic
2C2×#91235C211yesmaximal, characteristic, complemented
3C3×#65756C211yesmaximal, characteristic, complemented
4#11746501C212yesmaximal, complemented
5#11746498C212yesmaximal, complemented
6#10604467—31nomaximal, Hall
7C23×ΓL(1,𝔽16)—21no—
8C22×C3×C4×D5C2211yescharacteristic
9C2×#6772C411yescharacteristic, complemented
10C2×C3×#700—21no—
11#91813C411yescharacteristic, complemented
12C5×#930C411yescharacteristic, complemented
13#91563—21no—
14Dic5×#67—21no—
15#91449C2211yescharacteristic
16#91447—21no—
17#91440—21no—
18#91360C2211yescharacteristic
19#91358C411yescharacteristic, complemented
20#91235—21no—
21#91235C2212yes—
22C3×C22⋊#170—21no—
23C3×#4003C2212yes—
24#69976—51nomaximal, Hall
25#65940—31no—
26C2×(C2×AGL(1,𝔽5))⋊C4—31no—
27#65756D311yescharacteristic, complemented
28#64920—32no—
29#64869—32no—
30C23×C3×D5D411yescharacteristic
31C22×ΓL(1,𝔽16)—24no—
32C22×ΓL(1,𝔽16)—22no—
33C22×ΓL(1,𝔽16)—42no—
34C22×ΓL(1,𝔽16)—42no—
35C22×C3×C4×C5C2×C411yescharacteristic
36C2×Dic30C2×C411yescharacteristic, complemented
37C2×C5×#67—21no—
38C2×C5×D12C2×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×#1201—21no—
46C2×#1201—21no—
47C2×D3×Dic5—21no—
48#6772—42no—
49C2×C3×#170—22no—
50C3×#700—42no—
51#6719—21no—
52C4⋊Dic15C2×C411yescharacteristic, complemented
53#6712—21no—
54C5×(C2×D3)⋊C4—21no—
55C5×C4⋊Dic3C2×C411yescharacteristic, complemented
56Dic3⋊Dic5—21no—
57(C2×D3)⋊Dic5—21no—
58#6665—21no—
59#6665—21no—
60Dic3×Dic5—21no—
61#5791—51no—
62C3×#407—51no—
63C2×C3⋊#95—51no—
64#4470—52no—
65#4467—52no—
66C23×AGL(1,𝔽5)—61no—
67C2×#711—31no—
68C22×C4×D5C2×D311yescharacteristic
69C2×#700—21no—
70C5×#121—31no—
71#4084—31no—
72#4083—61no—
73D4×Dic5—61no—
74#4070—31no—
75#4043—31no—
76#4038—31no—
77#4029—61no—
78#4027—61no—
79C22⋊#170—21no—
80(C2×AGL(1,𝔽5))⋊C4—32no—
81(C2×AGL(1,𝔽5))⋊C4—61no—
82#4003C2×D312yes—
83#1504—151noSylow
84C23×C3×C5C22⋊C411yescharacteristic
85C22×C5×D3—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×ΓL(1,𝔽16)—24no—
93C2×ΓL(1,𝔽16)—48no—
94C2×ΓL(1,𝔽16)—42no—
95C2×ΓL(1,𝔽16)—42no—
96C2×C3×C4×C5—41no—
97C2×C3×C4×C5C22⋊C411yescharacteristic
98C2×C3×C4×C5C22×C411yescharacteristic
99C2×Dic15—21no—
100C2×Dic15—21no—
101Dic30—41no—
102C5×#67—42no—
103C2×C5×Dic3—21no—
104C5×D12—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#1201—42no—
113#1201—41no—
114#1201—41no—
115D3×Dic5—42no—
116C3×#170—42no—
117C23×C3×C4—51no—
118C23×Dic3—101no—
119C2×#235—51no—
120C2×C3×M16—101no—
121#953—101no—
122#930—51no—
123#930AGL(1,𝔽5)11yescharacteristic, complemented
124#929—101no—
125C4×#67—101no—
126C4×#67—101no—
127#926—51no—
128#924—51no—
129#924—51no—
130C3×C22⋊C8—52no—
131C3×C22⋊C8—101no—
132C3⋊#95—52no—
133C3⋊#95—101no—
134C23×D5D1211yescharacteristic
135C22×AGL(1,𝔽5)—64no—
136C22×AGL(1,𝔽5)—62no—
137C22×AGL(1,𝔽5)—122no—
138C22×AGL(1,𝔽5)—122no—
139C2×C5×D4—31no—
140C2×C5×D4—61no—
141C22×C4×C5C4×D311yescharacteristic
142C2×C22⋊D5—61no—
143C2×C22⋊D5—61no—
144C22×Dic5—61no—
145C22×Dic5#6711yescharacteristic
146#711—122no—
147C2×C4×D5—42no—
148C2×C4×D5C4×D311yescharacteristic
149C2×C4×D5D1211yescharacteristic
150C2×C4×D5#6711yescharacteristic
151C2×C4×D5C22×D311yescharacteristic
152C2×Dic10—31no—
153C2×#170—22no—
154#700—42no—
155C5×C4⋊C4—31no—
156C5×C22⋊C4—61no—
157C22⋊Dic5—61no—
158C22⋊Dic5—61no—
159#685—61no—
160#685—61no—
161C4⋊Dic5—31no—
162C4⋊Dic5—61no—
163#683—61no—
164C4×Dic5—61no—
165#525—151no—
166#407—51no—
167C2×#95—151no—
168#328—152no—
169#327—152no—
170C22×C3×C5—21no—
171C22×C3×C5—21no—
172C22×C3×C5#10412yes—
173C22×C3×C5C2×C22⋊C411yescharacteristic
174C2×C5×D3—41no—
175C2×C5×D3—41no—
176C2×C5×D3—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
186ΓL(1,𝔽16)—44no—
187C3×C4×C5—21no—
188C3×C4×C5—21no—
189Dic15—41no—
190Dic15—41no—
191C3×Dic5—21no—
192C3×Dic5—21no—
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—202no—
201C22×C3×C4C2×AGL(1,𝔽5)11yescharacteristic
202C2×#67—21no—
203C2×#67—101no—
204C2×#67—101no—
205C22×Dic3—102no—
206C22×Dic3—104no—
207C22×Dic3—202no—
208C22×Dic3—202no—
209#235—202no—
210C2×D12C2×AGL(1,𝔽5)11yescharacteristic, complemented
211C2×C4×D3—101no—
212C2×Dic6—51no—
213C3×M16—202no—
214C2×C3×C8—102no—
215#217—101no—
216#217—101no—
217#217—101no—
218(C2×D3)⋊C4—21no—
219(C2×D3)⋊C4—101no—
220C4⋊Dic3—51no—
221C4⋊Dic3C2×AGL(1,𝔽5)11yescharacteristic, complemented
222Dic3⋊C4—101no—
223Dic3⋊C4—101no—
224C4×Dic3—101no—
225C23×C5—61no—
226C23×C5(C2×D3)⋊C411yescharacteristic
227C22×D5—22no—
228C22×D5—22no—
229C22×D5—21no—
230C22×D5—21no—
231C22×D5(C2×D3)⋊C411yescharacteristic
232C22×D5C2×D1211yescharacteristic
233C2×AGL(1,𝔽5)—64no—
234C2×AGL(1,𝔽5)—128no—
235C2×AGL(1,𝔽5)—122no—
236C2×AGL(1,𝔽5)—122no—
237C5×D4—121no—
238C5×D4—122no—
239C2×C4×C5—41no—
240C2×C4×C5—61no—
241C2×C4×C5(C2×D3)⋊C411yescharacteristic
242C2×C4×C5C2×C4×D311yescharacteristic
243C22⋊D5—122no—
244C22⋊D5—121no—
245C22⋊D5—121no—
246C2×Dic5—41no—
247C2×Dic5—61no—
248C2×Dic5—61no—
249C2×Dic5—122no—
250C2×Dic5(C2×D3)⋊C411yescharacteristic
251C2×Dic5C2×#6711yescharacteristic
252C4×D5—22no—
253C4×D5—22no—
254C4×D5—41no—
255C4×D5—41no—
256Dic10—121no—
257#170—42no—
258C2×#41—151no—
259C23×C4—51no—
260C23×C4—301no—
261C2×M16—101no—
262#123—301no—
263C22⋊Q8—151no—
264C22⋊Q8—151no—
265#121—31no—
266#121—151no—
267#120—301no—
268C4×D4—301no—
269C4×D4—301no—
270#117—151no—
271C22⋊C8—52no—
272C22⋊C8—101no—
273#95—152no—
274#95—301no—
275C2×C3×C5—22no—
276C2×C3×C5C2×#9911yescharacteristic
277C2×C3×C5#41711yescharacteristic
278C2×C3×C5C2×#10411yescharacteristic
279C3×D5—21no—
280C3×D5—22no—
281C3×D5—21no—
282C5×D3—81no—
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—21no—
291C2×C3×C4—41no—
292C2×C3×C4—51no—
293C2×C3×C4—51no—
294C2×C3×C4—102no—
295C2×C3×C4—102no—
296C2×C3×C4—201no—
297C2×C3×C4—201no—
298C2×C3×C4—201no—
299C2×C3×C4C22⋊AGL(1,𝔽5)11yescharacteristic
300C2×C3×C4C22×AGL(1,𝔽5)11yescharacteristic
301#67—42no—
302#67—202no—
303#67—201no—
304#67—201no—
305C2×Dic3—21no—
306C2×Dic3—104no—
307C2×Dic3—101no—
308C2×Dic3—101no—
309C2×Dic3—202no—
310C2×Dic3—208no—
311C2×Dic3—202no—
312D12—41no—
313C4×D3—202no—
314Dic6—201no—
315C3×C8—202no—
316C22×C5—21no—
317C22×C5—21no—
318C22×C5—121no—
319C22×C5—121no—
320C22×C5—121no—
321C22×C5#82512yes—
322C22×C5C2×(C2×D3)⋊C411yescharacteristic
323C2×D5—24no—
324C2×D5—24no—
325C2×D5—21no—
326C2×D5—21no—
327C2×D5—21no—
328C2×D5—21no—
329C2×D5#80612yes—
330C2×D5#90811yescharacteristic
331C2×D5#90911yescharacteristic
332AGL(1,𝔽5)—124no—
333C4×C5—21no—
334C4×C5—21no—
335C4×C5—121no—
336Dic5—21no—
337Dic5—21no—
338Dic5—121no—
339Dic5—121no—
340C24—51no—
341#41—602no—
342C2×Q8—151no—
343C2×D4—31no—
344C2×D4—61no—
345C2×D4—301no—
346C2×D4—301no—
347C22×C4—51no—
348C22×C4—51no—
349C22×C4—51no—
350C22×C4—51no—
351C22×C4—51no—
352C22×C4—202no—
353C22×C4—301no—
354C22×C4—302no—
355C22×C4—304no—
356C22×C4—602no—
357C22×C4—602no—
358C22×C4D3×AGL(1,𝔽5)11yescharacteristic
359M16—202no—
360C2×C8—102no—
361C4⋊C4—31no—
362C4⋊C4—151no—
363C4⋊C4—301no—
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#400812yes—
384C22×C3C2×C22⋊AGL(1,𝔽5)11yescharacteristic
385C2×D3—41no—
386C2×D3—41no—
387C2×D3—41no—
388C3×C4—21no—
389C3×C4—21no—
390C3×C4—101no—
391C3×C4—101no—
392Dic3—41no—
393Dic3—204no—
394Dic3—201no—
395Dic3—201no—
396C2×C5—22no—
397C2×C5—241no—
398C2×C5C2×#80611yescharacteristic
399C2×C5#513311yescharacteristic
400C2×C5C2×#82511yescharacteristic
401D5—21no—
402D5—22no—
403D5—21no—
404C23—51no—
405C23—51no—
406C23—61no—
407C23—102no—
408C23—101no—
409C23—101no—
410C23—102no—
411C23(C2×D3)⋊AGL(1,𝔽5)11yescharacteristic
412Q8—601no—
413D4—121no—
414D4—122no—
415D4—602no—
416D4—601no—
417D4—601no—
418C2×C4—41no—
419C2×C4—51no—
420C2×C4—51no—
421C2×C4—61no—
422C2×C4—102no—
423C2×C4—102no—
424C2×C4—201no—
425C2×C4—201no—
426C2×C4—201no—
427C2×C4—304no—
428C2×C4—301no—
429C2×C4—301no—
430C2×C4—602no—
431C2×C4—602no—
432C2×C4—608no—
433C2×C4—602no—
434C2×C4(C2×D3)⋊AGL(1,𝔽5)11yescharacteristic
435C2×C4C2×D3×AGL(1,𝔽5)11yescharacteristic
436C8—202no—
437C2×C3—22no—
438C2×C3—101no—
439C2×C3—101no—
440C2×C3—102no—
441C2×C3C2×#400811yescharacteristic
442C2×C3D5⋊#9911yescharacteristic
443C2×C3#6579311yescharacteristic
444D3—81no—
445C5#6997611yescharacteristic, 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—121no—
458C22—121no—
459C22—121no—
460C22#9115512yes—
461C22C2×(C2×D3)⋊AGL(1,𝔽5)11yescharacteristic
462C4—21no—
463C4—21no—
464C4—101no—
465C4—101no—
466C4—121no—
467C4—604no—
468C4—601no—
469C4—601no—
470C3#1060446711yescharacteristic, complemented, Sylow
471C2—22no—
472C2—101no—
473C2—101no—
474C2—102no—
475C2—241no—
476C2#1175170811yescharacteristic
477C2C2×#9115511yescharacteristic
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#11754526C2
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. C2×#91235C2semidirect
3. C3×#65756C2semidirect
4. #11746501C2semidirect
5. #11746498C2semidirect
9. C2×#6772C4semidirect
11. #91813C4semidirect
12. C5×#930C4semidirect
19. #91358C4semidirect
27. #65756D3semidirect
36. C2×Dic30C2×C4semidirect
38. C2×C5×D12C2×C4semidirect
52. C4⋊Dic15C2×C4semidirect
55. C5×C4⋊Dic3C2×C4semidirect
123. #930AGL(1,𝔽5)semidirect
210. C2×D12C2×AGL(1,𝔽5)semidirect
221. C4⋊Dic3C2×AGL(1,𝔽5)semidirect
371. C3×C5#1504semidirect
445. C5#69976semidirect
470. C3#10604467semidirect

Maximal subgroups

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

subgroupindex
#117545262
C2×#912352
C3×#657562
#117465012
#117464982
#106044673
#699765