Finite Group Databasebeta

← #14281188

Subgroups — #14281188

Order 1920.

560 conjugacy classes of subgroups.

The subgroup lattice has 448 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×C3×#4032C211yesmaximal, characteristic
2#11752457C211yesmaximal, characteristic, complemented
3C2×#91241C211yesmaximal, characteristic, complemented
4#11746786C212yesmaximal, complemented
5#11746784C212yesmaximal, complemented
6#10604205—31nomaximal, Hall
7C22×C3×C22⋊D5C411yescharacteristic
8C23×C3×Dic5C2211yescharacteristic
9C2×#6762—21no—
10C22×C3⋊#170—21no—
11C2×C3×C5×C22⋊C4C411yescharacteristic
12C2×C3×#685C411yescharacteristic
13C2×C3×#685C2211yescharacteristic
14C2×C3×C4⋊Dic5C411yescharacteristic
15C2×C3×C4⋊Dic5C2211yescharacteristic
16C3×#4032—21no—
17C3×#4032—21no—
18C3×#4032—21no—
19C3×#4032—21no—
20#91241—21no—
21#91241—21no—
22#91241C2212yes—
23#91241C2212yes—
24#71702—51nomaximal, Hall
25C2×#4032D311yescharacteristic
26#65798—31no—
27C2×C22⋊#170—31no—
28#64915—32no—
29#64913—32no—
30C24×C3×C5C2×C411yescharacteristic
31C23×C3×D5C2×C411yescharacteristic
32C22×C3×C4×C5C2×C411yescharacteristic
33C22×C3×C4×C5C2×C411yescharacteristic
34C2×C3×C22⋊D5—21no—
35C2×C3×C22⋊D5—21no—
36C2×C3×C22⋊D5—42no—
37C22×C3×Dic5—42no—
38C22×C3×Dic5C2×C411yescharacteristic
39C22×C3×Dic5C2×C411yescharacteristic
40C22×C3×Dic5D411yescharacteristic
41C22×C3×Dic5D411yescharacteristic
42C22×C3×Dic5D411yescharacteristic
43C22×C3×Dic5D411yescharacteristic
44C22×C3×Dic5C2311yescharacteristic
45#6762—42no—
46C2×C3⋊#170—22no—
47C2×C3⋊#170—22no—
48C2×C3⋊#170—42no—
49C3×C5×C22⋊C4—21no—
50C3×C5×C22⋊C4—21no—
51C3×#685—21no—
52C3×#685—21no—
53C3×#685—21no—
54C3×#685—21no—
55C3×C4⋊Dic5—21no—
56C3×C4⋊Dic5—21no—
57C3×C4⋊Dic5—21no—
58C3×C4⋊Dic5—21no—
59C2×C3×#123—51no—
60#5207—51no—
61C2×#830—51no—
62#4528—52no—
63#4525—52no—
64C22×C22⋊D5Dic311yescharacteristic
65C23×Dic5C2×D311yescharacteristic
66C2×#704—61no—
67C22×#170—61no—
68C2×C5×C22⋊C4Dic311yescharacteristic
69C2×#685Dic311yescharacteristic
70C2×#685C2×D311yescharacteristic
71C2×C4⋊Dic5Dic311yescharacteristic
72C2×C4⋊Dic5C2×D311yescharacteristic
73#4032—21no—
74#4032—21no—
75#4032—21no—
76#4032—21no—
77C22⋊#170—32no—
78C22⋊#170—32no—
79C22⋊#170—61no—
80C22⋊#170—61no—
81#1468—151noSylow
82C23×C3×C5—22no—
83C23×C3×C5—22no—
84C23×C3×C5—41no—
85C23×C3×C5C22⋊C411yescharacteristic
86C23×C3×C5C22⋊C411yescharacteristic
87C23×C3×C5C22×C411yescharacteristic
88C22×C3×D5—21no—
89C22×C3×D5—22no—
90C22×C3×D5—41no—
91C22×C3×D5—41no—
92C2×C3×C4×C5—21no—
93C2×C3×C4×C5—21no—
94C2×C3×C4×C5—22no—
95C2×C3×C4×C5—41no—
96C3×C22⋊D5—42no—
97C3×C22⋊D5—42no—
98C2×C3×Dic5—22no—
99C2×C3×Dic5—22no—
100C2×C3×Dic5—21no—
101C2×C3×Dic5—41no—
102C2×C3×Dic5—41no—
103C2×C3×Dic5—41no—
104C2×C3×Dic5—41no—
105C2×C3×Dic5—41no—
106C2×C3×Dic5C22⋊C411yescharacteristic
107C2×C3×Dic5C22⋊C411yescharacteristic
108C2×C3×Dic5C2×D411yescharacteristic
109C2×C3×Dic5C2×D411yescharacteristic
110C3⋊#170—42no—
111C3⋊#170—42no—
112C22×C3×D4—51no—
113C23×C3×C4—51no—
114C3×#123—101no—
115C3×#123—101no—
116C3×#123—101no—
117C3×#123—101no—
118C2×C3×C4⋊C4—51no—
119C2×C3×C4⋊C4—51no—
120C2×C3×C22⋊C4—51no—
121C2×C3×C22⋊C4—51no—
122C2×C3×C22⋊C4AGL(1,𝔽5)11yescharacteristic
123C2×#208—101no—
124C22×C3⋊C8—101no—
125#830—52no—
126#830—52no—
127#830—101no—
128#830—101no—
129C24×C5C2×Dic311yescharacteristic
130C23×D5C2×Dic311yescharacteristic
131C22×C4×C5C2×Dic311yescharacteristic
132C22×C4×C5C2×Dic311yescharacteristic
133C2×C22⋊D5—21no—
134C2×C22⋊D5—21no—
135C2×C22⋊D5—42no—
136C22×Dic5—42no—
137C22×Dic5C2×Dic311yescharacteristic
138C22×Dic5C2×Dic311yescharacteristic
139C22×Dic5#6711yescharacteristic
140C22×Dic5#6711yescharacteristic
141C22×Dic5#6711yescharacteristic
142C22×Dic5#6711yescharacteristic
143C22×Dic5C22×D311yescharacteristic
144#704—122no—
145C2×#170—62no—
146C2×#170—62no—
147C2×#170—122no—
148C5×C22⋊C4—21no—
149C5×C22⋊C4—21no—
150#685—21no—
151#685—21no—
152#685—21no—
153#685—21no—
154C4⋊Dic5—21no—
155C4⋊Dic5—21no—
156C4⋊Dic5—21no—
157C4⋊Dic5—21no—
158C2×#123—51no—
159#407—151no—
160C2×C22⋊C8—151no—
161#324—152no—
162#323—152no—
163C22×C3×C5—21no—
164C22×C3×C5—21no—
165C22×C3×C5—21no—
166C22×C3×C5—21no—
167C22×C3×C5—21no—
168C22×C3×C5—21no—
169C22×C3×C5—21no—
170C22×C3×C5—44no—
171C22×C3×C5#9912yes—
172C22×C3×C5C2×C22⋊C411yescharacteristic
173C22×C3×C5#13112yes—
174C2×C3×D5—41no—
175C2×C3×D5—41no—
176C2×C3×D5—41no—
177C2×C3×D5—82no—
178C3×C4×C5—41no—
179C3×C4×C5—41no—
180C3×Dic5—21no—
181C3×Dic5—21no—
182C3×Dic5—21no—
183C3×Dic5—21no—
184C3×Dic5—41no—
185C24×C3—51no—
186C24×C3C2×AGL(1,𝔽5)11yescharacteristic
187C2×C3×D4—101no—
188C2×C3×D4—101no—
189C2×C3×D4—202no—
190C22×C3×C4—51no—
191C22×C3×C4—51no—
192C22×C3×C4—51no—
193C22×C3×C4—51no—
194C22×C3×C4—51no—
195C22×C3×C4—51no—
196C22×C3×C4—51no—
197C22×C3×C4—202no—
198C22×C3×C4C2×AGL(1,𝔽5)11yescharacteristic
199C22×C3×C4C2×AGL(1,𝔽5)11yescharacteristic
200C3×C4⋊C4—101no—
201C3×C4⋊C4—101no—
202C3×C4⋊C4—101no—
203C3×C4⋊C4—101no—
204C3×C22⋊C4—21no—
205C3×C22⋊C4—21no—
206C3×C22⋊C4—101no—
207C3×C22⋊C4—101no—
208C3×C22⋊C4—101no—
209C3×C22⋊C4—101no—
210#208—202no—
211C2×C3⋊C8—102no—
212C2×C3⋊C8—102no—
213C2×C3⋊C8—202no—
214C23×C5—22no—
215C23×C5—22no—
216C23×C5—41no—
217C23×C5#21711yescharacteristic
218C23×C5#21711yescharacteristic
219C23×C5C22×Dic311yescharacteristic
220C22×D5—21no—
221C22×D5—22no—
222C22×D5—41no—
223C22×D5—41no—
224C2×C4×C5—22no—
225C2×C4×C5—21no—
226C2×C4×C5—21no—
227C2×C4×C5—41no—
228C22⋊D5—42no—
229C22⋊D5—42no—
230C2×Dic5—22no—
231C2×Dic5—22no—
232C2×Dic5—21no—
233C2×Dic5—41no—
234C2×Dic5—41no—
235C2×Dic5—41no—
236C2×Dic5—41no—
237C2×Dic5—41no—
238C2×Dic5#21711yescharacteristic
239C2×Dic5#21711yescharacteristic
240C2×Dic5C2×#6711yescharacteristic
241C2×Dic5C2×#6711yescharacteristic
242#170—122no—
243#170—122no—
244C22×D4—51no—
245C23×C4—51no—
246C2×M16—301no—
247C22×C8—301no—
248#123—101no—
249#123—101no—
250#123—101no—
251#123—101no—
252C2×C4⋊C4—51no—
253C2×C4⋊C4—51no—
254C2×C22⋊C4—51no—
255C2×C22⋊C4—51no—
256C2×C22⋊C4ΓL(1,𝔽16)11yescharacteristic
257C22⋊C8—152no—
258C22⋊C8—152no—
259C22⋊C8—301no—
260C22⋊C8—301no—
261C2×C3×C5—22no—
262C2×C3×C5—22no—
263C2×C3×C5—41no—
264C2×C3×C5#40811yescharacteristic
265C2×C3×C5C2×#9911yescharacteristic
266C2×C3×C5#41311yescharacteristic
267C3×D5—81no—
268C23×C3—22no—
269C23×C3—22no—
270C23×C3—41no—
271C23×C3—101no—
272C23×C3—102no—
273C23×C3—201no—
274C23×C3—201no—
275C23×C3C22⋊AGL(1,𝔽5)11yescharacteristic
276C23×C3C22⋊AGL(1,𝔽5)11yescharacteristic
277C23×C3C22×AGL(1,𝔽5)11yescharacteristic
278C3×D4—202no—
279C3×D4—202no—
280C2×C3×C4—22no—
281C2×C3×C4—21no—
282C2×C3×C4—21no—
283C2×C3×C4—41no—
284C2×C3×C4—51no—
285C2×C3×C4—51no—
286C2×C3×C4—51no—
287C2×C3×C4—51no—
288C2×C3×C4—102no—
289C2×C3×C4—102no—
290C2×C3×C4—101no—
291C2×C3×C4—201no—
292C2×C3×C4—201no—
293C2×C3×C4—201no—
294C2×C3×C4—201no—
295C2×C3×C4—201no—
296C3⋊C8—202no—
297C3⋊C8—202no—
298C22×C5—21no—
299C22×C5—21no—
300C22×C5—21no—
301C22×C5—21no—
302C22×C5—21no—
303C22×C5—21no—
304C22×C5—21no—
305C22×C5—44no—
306C22×C5#83412yes—
307C22×C5#94812yes—
308C22×C5C2×#21711yescharacteristic
309C2×D5—41no—
310C2×D5—41no—
311C2×D5—41no—
312C2×D5—82no—
313C4×C5—41no—
314C4×C5—41no—
315Dic5—21no—
316Dic5—21no—
317Dic5—21no—
318Dic5—21no—
319Dic5—41no—
320C24—51no—
321C24C2×ΓL(1,𝔽16)11yescharacteristic
322C2×D4—101no—
323C2×D4—101no—
324C2×D4—202no—
325C22×C4—51no—
326C22×C4—51no—
327C22×C4—51no—
328C22×C4—51no—
329C22×C4—51no—
330C22×C4—51no—
331C22×C4—51no—
332C22×C4—202no—
333C22×C4C2×ΓL(1,𝔽16)11yescharacteristic
334C22×C4C2×ΓL(1,𝔽16)11yescharacteristic
335M16—602no—
336C2×C8—302no—
337C2×C8—302no—
338C2×C8—602no—
339C4⋊C4—101no—
340C4⋊C4—101no—
341C4⋊C4—101no—
342C4⋊C4—101no—
343C22⋊C4—21no—
344C22⋊C4—21no—
345C22⋊C4—101no—
346C22⋊C4—101no—
347C22⋊C4—101no—
348C22⋊C4—101no—
349C3×C5#146811yescharacteristic, complemented, Hall
350C22×C3—21no—
351C22×C3—21no—
352C22×C3—21no—
353C22×C3—21no—
354C22×C3—21no—
355C22×C3—21no—
356C22×C3—21no—
357C22×C3—44no—
358C22×C3—201no—
359C22×C3—201no—
360C22×C3—201no—
361C22×C3—402no—
362C22×C3#399912yes—
363C22×C3#413112yes—
364C22×C3C2×C22⋊AGL(1,𝔽5)11yescharacteristic
365C3×C4—41no—
366C3×C4—41no—
367C3×C4—101no—
368C3×C4—101no—
369C3×C4—101no—
370C3×C4—101no—
371C3×C4—201no—
372C2×C5—22no—
373C2×C5—22no—
374C2×C5—41no—
375C2×C5C2×#83411yescharacteristic
376C2×C5#523411yescharacteristic
377C2×C5#523711yescharacteristic
378D5—81no—
379C23—22no—
380C23—22no—
381C23—41no—
382C23—101no—
383C23—102no—
384C23—201no—
385C23—201no—
386C23#676311yescharacteristic
387C23#676311yescharacteristic
388C23C22×ΓL(1,𝔽16)11yescharacteristic
389D4—202no—
390D4—202no—
391C2×C4—22no—
392C2×C4—21no—
393C2×C4—21no—
394C2×C4—41no—
395C2×C4—51no—
396C2×C4—51no—
397C2×C4—51no—
398C2×C4—51no—
399C2×C4—102no—
400C2×C4—102no—
401C2×C4—101no—
402C2×C4—201no—
403C2×C4—201no—
404C2×C4—201no—
405C2×C4—201no—
406C2×C4—201no—
407C8—602no—
408C8—602no—
409C2×C3—22no—
410C2×C3—22no—
411C2×C3—41no—
412C2×C3—401no—
413C2×C3C2×#399911yescharacteristic
414C2×C3#6577511yescharacteristic
415C2×C3#6577711yescharacteristic
416C5#7170211yescharacteristic, complemented, Sylow
417C22—21no—
418C22—21no—
419C22—21no—
420C22—21no—
421C22—21no—
422C22—21no—
423C22—21no—
424C22—44no—
425C22—201no—
426C22—201no—
427C22—201no—
428C22—402no—
429C22#9122812yes—
430C22#9199012yes—
431C22C2×#676311yescharacteristic
432C4—41no—
433C4—41no—
434C4—101no—
435C4—101no—
436C4—101no—
437C4—101no—
438C4—201no—
439C3#1060420511yescharacteristic, complemented, Sylow
440C2—22no—
441C2—22no—
442C2—41no—
443C2—401no—
444C2C2×#9122811yescharacteristic
445C2#1175243411yescharacteristic
446C2#1175243611yescharacteristic
subgroupG/H
centerC22C2×#6763
commutatorC23×C3×C5C22×C4
FrattiniC23C22×ΓL(1,𝔽16)
FittingC2×C3×C5×C22⋊C4C4
socleC22×C3×C5C2×C22⋊C4
prime radicalC22×C3×C22⋊D5C4
cosocleC22×C3×Dic5C23
hypercenterC2×C22⋊C4ΓL(1,𝔽16)
hypocenterC3×C5#1468
LCSG ▷ C23×C3×C5 ▷ C2×C3×C5 ▷ C3×C5
UCSC1 ◁ C22 ◁ C24 ◁ C2×C22⋊C4

Complements

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

NH ≅ G/Ntype
2. #11752457C2semidirect
3. C2×#91241C2semidirect
4. #11746786C2semidirect
5. #11746784C2semidirect
349. C3×C5#1468semidirect
416. C5#71702semidirect
439. C3#10604205semidirect

Maximal subgroups

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

subgroupindex
C2×C3×#40322
#117524572
C2×#912412
#117467862
#117467842
#106042053
#717025