Finite Group Databasebeta

← #14307157

Subgroups — #14307157

Order 1920.

520 conjugacy classes of subgroups.

The subgroup lattice has 460 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#11753568C211yesmaximal, characteristic
2C4×Dic3⋊AGL(1,𝔽5)C211yesmaximal, characteristic
3Dic3×#700C211yesmaximal, characteristic
4#11746746C211yesmaximal, characteristic
5C3×#64875C211yesmaximal, characteristic
6#11746526C211yesmaximal, characteristic
7#11746524C211yesmaximal, characteristic
8#10604679—31nomaximal, Hall
9C2×C4×ΓL(1,𝔽16)C2211yescharacteristic
10C2×#6758C2211yescharacteristic
11C2×C3×C4×AGL(1,𝔽5)C2211yescharacteristic
12C2×C3×#700C2211yescharacteristic
13C2×Dic3⋊AGL(1,𝔽5)—21no—
14#91779C411yescharacteristic, complemented
15C5×#888C411yescharacteristic, complemented
16C3×#4038C2211yescharacteristic
17#91469C411yescharacteristic, complemented
18#91422—21no—
19#91421—21no—
20C4×D5×Dic3C2211yescharacteristic
21#91384C411yescharacteristic, complemented
22#91358C2211yescharacteristic
23Dic5×Dic6—21no—
24#91323—21no—
25#91169—21no—
26Dic3×#170—21no—
27(C2×AGL(1,𝔽5))⋊Dic3—21no—
28#68730—51nomaximal, Hall
29#66062—31no—
30C4×C4⋊AGL(1,𝔽5)—31no—
31C4×#700—31no—
32#64875—31no—
33#64875—31no—
34#64875—31no—
35#64875D311yescharacteristic
36C22×ΓL(1,𝔽16)—21no—
37C22×C3×AGL(1,𝔽5)—21no—
38C2×C5×Dic6—21no—
39C2×C3×C4×D5C2311yescharacteristic
40C2×#1204—21no—
41C2×D5×Dic3D411yescharacteristic
42C2×D5×Dic3D411yescharacteristic
43C2×C3⋊#170—21no—
44C4×ΓL(1,𝔽16)—21no—
45C4×ΓL(1,𝔽16)—21no—
46#6758—21no—
47#6758Q812yescomplemented
48C2×C3×#170—21no—
49C3×C4×AGL(1,𝔽5)—21no—
50C3×C4×AGL(1,𝔽5)—21no—
51C3×#700—21no—
52C3×#700Q812yescomplemented
53Dic3⋊AGL(1,𝔽5)—24no—
54C3×C5×C4⋊C4C2×C411yescharacteristic
55C4⋊Dic15—21no—
56C4⋊Dic15C2×C411yescharacteristic
57#6712—21no—
58C4×Dic15C2×C411yescharacteristic
59C5×C4⋊Dic3C2×C411yescharacteristic
60C5×Dic3⋊C4—21no—
61C4×C5×Dic3C2×C411yescharacteristic
62C3×#685—21no—
63C3×C4⋊Dic5C2×C411yescharacteristic
64C3×C4×Dic5—21no—
65Dic3⋊Dic5—21no—
66Dic3⋊Dic5—21no—
67#6668—21no—
68#6665—21no—
69Dic3×Dic5—21no—
70Dic3×Dic5D411yescharacteristic
71Dic3×Dic5D411yescharacteristic
72#5517—51no—
73Dic3×M16—51no—
74C4×Dic3⋊C4—51no—
75C3×#339—51no—
76#4555—51no—
77#4532—51no—
78#4523—51no—
79C2×C4⋊AGL(1,𝔽5)—61no—
80C2×C4×AGL(1,𝔽5)—31no—
81C2×C4×AGL(1,𝔽5)C2×D311yescharacteristic
82C2×#700—31no—
83C2×#700C2×D311yescharacteristic
84C5×C4⋊Q8—31no—
85#4092—61no—
86Q8×Dic5—61no—
87#4043—61no—
88#4038—31no—
89#4038C2×D311yescharacteristic
90#4036—31no—
91#4036—31no—
92#4033—61no—
93C42×D5—31no—
94#4015—31no—
95(C2×AGL(1,𝔽5))⋊C4—61no—
96#4002—61no—
97C4×#170—61no—
98#1986—151noSylow
99C22×C3×D5C2×D411yescharacteristic
100C2×ΓL(1,𝔽16)—22no—
101C2×ΓL(1,𝔽16)—41no—
102C2×ΓL(1,𝔽16)—41no—
103C2×C3×AGL(1,𝔽5)—22no—
104C2×C3×AGL(1,𝔽5)—41no—
105C2×C3×AGL(1,𝔽5)—41no—
106C2×C3×C4×C5—21no—
107C2×C3×C4×C5C22×C411yescharacteristic
108C2×Dic15—21no—
109C2×Dic15C22⋊C411yescharacteristic
110C2×Dic15C22⋊C411yescharacteristic
111C2×C5×Dic3—21no—
112C2×C5×Dic3C22⋊C411yescharacteristic
113C2×C5×Dic3C22⋊C411yescharacteristic
114C5×Dic6—42no—
115C2×C3×Dic5—21no—
116C2×C3×Dic5C2×D411yescharacteristic
117C3×C4×D5C2×Q811yescharacteristic
118C3×C4×D5C2×Q811yescharacteristic
119C3×C4×D5#4111yescharacteristic
120C3×C4×D5#4111yescharacteristic
121#1204—42no—
122D5×Dic3—22no—
123D5×Dic3—41no—
124C3⋊#170—22no—
125C3×#170—22no—
126C2×C3×M16—51no—
127C3×#117—51no—
128C2×C3×C42—51no—
129#926—51no—
130C2×Dic3⋊C4—101no—
131C2×C4×Dic3—51no—
132C2×C4×Dic3—51no—
133C2×#208—51no—
134#890—51no—
135#890—51no—
136#888—51no—
137#888AGL(1,𝔽5)11yescharacteristic, complemented
138#878—101no—
139#878—101no—
140C4×Dic6—101no—
141C4×Dic6—101no—
142C3⋊#95—101no—
143#815—101no—
144C8×Dic3—101no—
145C22×AGL(1,𝔽5)—21no—
146C22×AGL(1,𝔽5)—61no—
147C2×C5×Q8—61no—
148C2×C4×D5—31no—
149C2×C4×D5—31no—
150C2×C4×D5C22×D311yescharacteristic
151C2×Dic10—61no—
152C2×#170—21no—
153C2×#170—61no—
154C4⋊AGL(1,𝔽5)—64no—
155C4×AGL(1,𝔽5)—21no—
156C4×AGL(1,𝔽5)—21no—
157C4×AGL(1,𝔽5)—61no—
158C4×AGL(1,𝔽5)—61no—
159#700—21no—
160#700—32no—
161#700—61no—
162#700Dic612yescomplemented
163C5×C4⋊C4—31no—
164C5×C4⋊C4—61no—
165C5×C4⋊C4C4×D311yescharacteristic
166C42×C5—31no—
167#685—21no—
168#685—61no—
169C4⋊Dic5—31no—
170C4⋊Dic5—61no—
171C4⋊Dic5—61no—
172C4⋊Dic5—61no—
173C4⋊Dic5C4×D311yescharacteristic
174#683—61no—
175#683—61no—
176C4×Dic5—21no—
177C4×Dic5—31no—
178C4×Dic5—31no—
179C4×Dic5—31no—
180C4×Dic5—61no—
181#533—151no—
182C4×M16—151no—
183C4×C4⋊C4—151no—
184#339—51no—
185#339—151no—
186#339—151no—
187#339—151no—
188C22×C3×C5C2×C22⋊C411yescharacteristic
189C2×C3×D5—21no—
190C2×C3×D5—21no—
191C2×C3×D5#12411yescharacteristic
192C2×C3×D5C4⋊Q811yescharacteristic
193ΓL(1,𝔽16)—42no—
194C3×AGL(1,𝔽5)—42no—
195C3×C4×C5—41no—
196C3×C4×C5C4×Q811yescharacteristic
197C3×C4×C5C4×Q811yescharacteristic
198Dic15—21no—
199Dic15—21no—
200Dic15—41no—
201C3×Dic5—41no—
202C3×Dic5C22⋊Q811yescharacteristic
203C3×Dic5C22⋊Q811yescharacteristic
204C5×Dic3—21no—
205C5×Dic3—41no—
206C5×Dic3#10412yes—
207C22×C3×C4—51no—
208C22×C3×C4—101no—
209C22×Dic3—51no—
210C22×Dic3—51no—
211C22×Dic3—101no—
212C2×Dic6—21no—
213C2×Dic6—101no—
214C3×M16—52no—
215C3×M16—101no—
216C2×C3×C8—101no—
217C3×C4⋊C4—51no—
218C3×C4⋊C4C2×AGL(1,𝔽5)11yescharacteristic
219C3×C22⋊C4—101no—
220C3×C42—101no—
221C3×C42—101no—
222C3×C42—101no—
223#217—101no—
224C4⋊Dic3—51no—
225C4⋊Dic3—101no—
226C4⋊Dic3—101no—
227C4⋊Dic3C2×AGL(1,𝔽5)11yescharacteristic
228Dic3⋊C4—21no—
229Dic3⋊C4—101no—
230Dic3⋊C4—101no—
231Dic3⋊C4—104no—
232Dic3⋊C4—101no—
233C4×Dic3—51no—
234C4×Dic3—51no—
235C4×Dic3—51no—
236C4×Dic3—101no—
237C4×Dic3—101no—
238C4×Dic3—101no—
239C4×Dic3C2×AGL(1,𝔽5)11yescharacteristic
240#208—52no—
241#208—101no—
242C2×C3⋊C8—101no—
243C22×D5D3×D411yescharacteristic
244C2×AGL(1,𝔽5)—22no—
245C2×AGL(1,𝔽5)—41no—
246C2×AGL(1,𝔽5)—41no—
247C2×AGL(1,𝔽5)—62no—
248C2×AGL(1,𝔽5)—121no—
249C2×AGL(1,𝔽5)—121no—
250C5×Q8—122no—
251C2×C4×C5—21no—
252C2×C4×C5—31no—
253C2×C4×C5—31no—
254C2×C4×C5—61no—
255C2×C4×C5C2×C4×D311yescharacteristic
256C2×Dic5—21no—
257C2×Dic5—31no—
258C2×Dic5—31no—
259C2×Dic5—61no—
260C2×Dic5D3×D411yescharacteristic
261C4×D5—62no—
262C4×D5—121no—
263C4×D5C2×Dic611yescharacteristic
264C4×D5#23511yescharacteristic
265C4×D5#23711yescharacteristic
266C4×D5D3×Q811yescharacteristic
267Dic10—122no—
268#170—22no—
269#170—62no—
270C2×M16—51no—
271C2×M16—151no—
272C4⋊Q8—31no—
273C4⋊Q8—151no—
274#125—151no—
275#125—151no—
276C22⋊Q8—301no—
277C22⋊Q8—301no—
278C4×Q8—301no—
279C4×Q8—301no—
280#117—51no—
281#117—151no—
282C2×C4⋊C4—301no—
283C2×C42—51no—
284C2×C42—151no—
285C2×C42—151no—
286#97—301no—
287C4×C8—301no—
288#95—301no—
289C2×C3×C5#39111yescharacteristic
290C2×C3×C5C2×#10411yescharacteristic
291C2×C3×C5#42111yescharacteristic
292C3×D5—21no—
293C3×D5—21no—
294C23×C3—51no—
295C2×C3×C4—21no—
296C2×C3×C4—51no—
297C2×C3×C4—51no—
298C2×C3×C4—51no—
299C2×C3×C4—51no—
300C2×C3×C4—51no—
301C2×C3×C4—102no—
302C2×C3×C4—101no—
303C2×C3×C4—201no—
304C2×C3×C4—201no—
305C2×C3×C4C22×AGL(1,𝔽5)11yescharacteristic
306C2×Dic3—21no—
307C2×Dic3—51no—
308C2×Dic3—51no—
309C2×Dic3—102no—
310C2×Dic3—102no—
311C2×Dic3—101no—
312C2×Dic3—201no—
313C2×Dic3—201no—
314C2×Dic3—201no—
315C2×Dic3C22⋊AGL(1,𝔽5)11yescharacteristic
316C2×Dic3C22⋊AGL(1,𝔽5)11yescharacteristic
317Dic6—42no—
318Dic6—202no—
319C3×C8—102no—
320C3⋊C8—102no—
321C22×C5D3×C22⋊C411yescharacteristic
322C2×D5—21no—
323C2×D5—21no—
324C2×D5#88511yescharacteristic
325C2×D5#88811yescharacteristic
326AGL(1,𝔽5)—42no—
327AGL(1,𝔽5)—122no—
328C4×C5—32no—
329C4×C5—41no—
330C4×C5—61no—
331C4×C5—121no—
332C4×C5C4×Dic611yescharacteristic
333C4×C5#88711yescharacteristic
334Dic5—41no—
335Dic5—61no—
336Dic5—61no—
337Dic5—121no—
338Dic5#87811yescharacteristic
339Dic5#89611yescharacteristic
340C2×Q8—61no—
341C2×Q8—301no—
342C22×C4—51no—
343C22×C4—101no—
344C22×C4—151no—
345C22×C4—151no—
346C22×C4—301no—
347M16—52no—
348M16—101no—
349M16—152no—
350M16—301no—
351C2×C8—101no—
352C2×C8—301no—
353C4⋊C4—31no—
354C4⋊C4—51no—
355C4⋊C4—61no—
356C4⋊C4—151no—
357C4⋊C4—301no—
358C4⋊C4—301no—
359C4⋊C4—301no—
360C4⋊C4—301no—
361C4⋊C4—301no—
362C4⋊C4—304no—
363C4⋊C4D3×AGL(1,𝔽5)11yescharacteristic
364C22⋊C4—101no—
365C22⋊C4—301no—
366C42—31no—
367C42—101no—
368C42—101no—
369C42—101no—
370C42—151no—
371C42—151no—
372C42—151no—
373C42—301no—
374C42—301no—
375C42—301no—
376C3×C5#198611yescharacteristic, complemented, Hall
377C22×C3—51no—
378C22×C3—51no—
379C22×C3—101no—
380C22×C3—101no—
381C22×C3C2×C22⋊AGL(1,𝔽5)11yescharacteristic
382C3×C4—41no—
383C3×C4—51no—
384C3×C4—51no—
385C3×C4—202no—
386C3×C4—201no—
387C3×C4Q8×AGL(1,𝔽5)11yescharacteristic
388C3×C4Q8×AGL(1,𝔽5)11yescharacteristic
389Dic3—21no—
390Dic3—41no—
391Dic3—101no—
392Dic3—101no—
393Dic3—202no—
394Dic3—201no—
395Dic3#400812yes—
396C2×C5#464611yescharacteristic
397C2×C5D3×#10411yescharacteristic
398C2×C5#482111yescharacteristic
399D5—21no—
400D5—21no—
401C23—51no—
402Q8—122no—
403Q8—602no—
404C2×C4—21no—
405C2×C4—31no—
406C2×C4—31no—
407C2×C4—51no—
408C2×C4—51no—
409C2×C4—51no—
410C2×C4—51no—
411C2×C4—51no—
412C2×C4—61no—
413C2×C4—102no—
414C2×C4—101no—
415C2×C4—151no—
416C2×C4—151no—
417C2×C4—201no—
418C2×C4—201no—
419C2×C4—302no—
420C2×C4—302no—
421C2×C4—301no—
422C2×C4—601no—
423C2×C4—601no—
424C2×C4—601no—
425C2×C4C2×D3×AGL(1,𝔽5)11yescharacteristic
426C8—102no—
427C8—302no—
428C2×C3—101no—
429C2×C3—101no—
430C2×C3C2×#400811yescharacteristic
431C2×C3#6578411yescharacteristic
432C2×C3(C2×AGL(1,𝔽5))⋊Q811yescharacteristic
433C5#6873011yescharacteristic, complemented, Sylow
434C22—51no—
435C22—51no—
436C22—101no—
437C22—101no—
438C22D3×C22⋊AGL(1,𝔽5)11yescharacteristic
439C4—32no—
440C4—41no—
441C4—51no—
442C4—51no—
443C4—61no—
444C4—121no—
445C4—202no—
446C4—201no—
447C4—301no—
448C4—301no—
449C4—602no—
450C4—601no—
451C4AGL(1,𝔽5)×Dic611yescharacteristic
452C4ΓL(1,𝔽16)⋊Q811yescharacteristic
453C3#1060467911yescharacteristic, complemented, Sylow
454C2—101no—
455C2—101no—
456C2#1175184211yescharacteristic
457C2D3×#400811yescharacteristic
458C2#1175197411yescharacteristic
subgroupG/H
centerC22D3×C22⋊AGL(1,𝔽5)
commutatorC2×C3×C4×C5C22×C4
FrattiniC2×C4C2×D3×AGL(1,𝔽5)
FittingC3×C5×C4⋊C4C2×C4
socleC22×C3×C5C2×C22⋊C4
prime radicalC22×C3×D5C2×D4
cosocleC2×C3×C4×D5C23
hypercenterC4⋊C4D3×AGL(1,𝔽5)
hypocenterC3×C5#1986
LCSG ▷ C2×C3×C4×C5 ▷ C2×C3×C5 ▷ C3×C5
UCSC1 ◁ C22 ◁ C2×C4 ◁ C4⋊C4

Complements

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

NH ≅ G/Ntype
14. #91779C4semidirect
15. C5×#888C4semidirect
17. #91469C4semidirect
21. #91384C4semidirect
47. #6758Q8semidirect
52. C3×#700Q8semidirect
137. #888AGL(1,𝔽5)semidirect
162. #700Dic6semidirect
376. C3×C5#1986semidirect
433. C5#68730semidirect
453. C3#10604679semidirect

Maximal subgroups

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

subgroupindex
#117535682
C4×Dic3⋊AGL(1,𝔽5)2
Dic3×#7002
#117467462
C3×#648752
#117465262
#117465242
#106046793
#687305