Finite Group Databasebeta

← #14282827

Subgroups — #14282827

Order 1920.

608 conjugacy classes of subgroups.

The subgroup lattice has 452 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C3×#65858C211yesmaximal, characteristic
2C4×#6763C211yesmaximal, characteristic, complemented
3D5⋊#830C211yesmaximal, characteristic, complemented
4#11746740C212yesmaximal, complemented
5#11746739C212yesmaximal, complemented
6#10604346—31nomaximal, Hall
7C2×C3×#709C411yescharacteristic, complemented
8C22×C3×C4×D5C2211yescharacteristic
9C2×#6763—21no—
10C2×C4×ΓL(1,𝔽16)—21no—
11C2×D5⋊(C3⋊C8)—21no—
12C3×C5×#117C411yescharacteristic, complemented
13C3×#4043C411yescharacteristic, complemented
14C3×#4043C2211yescharacteristic
15C3×#4041C411yescharacteristic, complemented
16C3×#4041C2211yescharacteristic
17C3×#4032—21no—
18C3×#4028—21no—
19C3×C4×D20—21no—
20C3×C4×D20—21no—
21#91241—21no—
22#91235C2212yes—
23#91232C2212yes—
24#71707—51nomaximal, Hall
25#65858D311yescharacteristic
26C4×C22⋊AGL(1,𝔽5)—31no—
27D5⋊(C22⋊C8)—31no—
28#64869—32no—
29#64868—32no—
30C23×C3×D5D411yescharacteristic
31C22×ΓL(1,𝔽16)—22no—
32C22×C3×C4×C5C2×C411yescharacteristic
33C2×C3×C22⋊D5—21no—
34C22×C3×Dic5D411yescharacteristic
35C3×#709—42no—
36C2×C3×D20—21no—
37C2×C3×D20—21no—
38C2×C3×D20C2×C411yescharacteristic
39C2×C3×C4×D5—21no—
40C2×C3×C4×D5—42no—
41C2×C3×C4×D5C2×C411yescharacteristic
42C2×C3×C4×D5D411yescharacteristic
43C2×C3×C4×D5D411yescharacteristic
44C2×C3×C4×D5C2311yescharacteristic
45C2×C3×Dic10C2×C411yescharacteristic
46#6763—24no—
47C2×C3⋊#170—22no—
48C4×ΓL(1,𝔽16)—42no—
49D5⋊(C3⋊C8)—42no—
50C3×C5×C4⋊C4C2×C411yescharacteristic
51C3×C5×C4⋊C4C2×C411yescharacteristic
52C3×C5×C22⋊C4—21no—
53C3×C42×C5—21no—
54C3×#685—21no—
55C3×#685—21no—
56C3×#685—21no—
57C3×#685—21no—
58C3×C4⋊Dic5—21no—
59C3×C4⋊Dic5—21no—
60C3×#525—51no—
61C4×#217—51no—
62C2×#830—51no—
63#4538—52no—
64#4537—52no—
65C2×#709Dic311yescharacteristic, complemented
66C22×C4×D5C2×D311yescharacteristic
67C2×C22⋊AGL(1,𝔽5)—61no—
68C2×C4×AGL(1,𝔽5)—61no—
69C2×D5⋊C8—61no—
70C5×#117Dic311yescharacteristic, complemented
71#4043Dic311yescharacteristic, complemented
72#4043C2×D311yescharacteristic
73#4041Dic311yescharacteristic, complemented
74#4041C2×D311yescharacteristic
75#4032—21no—
76#4028—21no—
77C4×D20—21no—
78C4×D20—21no—
79C22⋊#170—61no—
80(C2×AGL(1,𝔽5))⋊C4—32no—
81#4003—32no—
82#1501—151noSylow
83C23×C3×C5C22⋊C411yescharacteristic
84C22×C3×D5—22no—
85C22×C3×D5—22no—
86C22×C3×D5—21no—
87C22×C3×D5—41no—
88C22×C3×D5C22⋊C411yescharacteristic
89C22×C3×D5C2×D411yescharacteristic
90C2×ΓL(1,𝔽16)—24no—
91C2×ΓL(1,𝔽16)—42no—
92C2×ΓL(1,𝔽16)—42no—
93C2×C3×C4×C5—22no—
94C2×C3×C4×C5—21no—
95C2×C3×C4×C5—21no—
96C2×C3×C4×C5C22⋊C411yescharacteristic
97C2×C3×C4×C5C22×C411yescharacteristic
98C3×C22⋊D5—42no—
99C2×C3×Dic5—21no—
100C2×C3×Dic5—41no—
101C2×C3×Dic5C22⋊C411yescharacteristic
102C2×C3×Dic5C2×D411yescharacteristic
103C3×D20—41no—
104C3×D20—42no—
105C3×D20—42no—
106C3×C4×D5—22no—
107C3×C4×D5—22no—
108C3×C4×D5—41no—
109C3×C4×D5—41no—
110C3×C4×D5—42no—
111C3×Dic10—41no—
112C3⋊#170—42no—
113C2×C3×#41—51no—
114C23×C3×C4—51no—
115C3×#123—101no—
116C3×C22⋊Q8—51no—
117C3×C22⋊Q8—51no—
118C3×#121—51no—
119C3×#121—51no—
120C3×#120—101no—
121C3×C4×D4—101no—
122C3×C4×D4—101no—
123C3×#117AGL(1,𝔽5)11yescharacteristic, complemented
124C2×#217—101no—
125C2×C4×Dic3—101no—
126C22×C3⋊C8—101no—
127C3⋊#95—52no—
128#830—52no—
129#830—101no—
130C23×D5#6711yescharacteristic
131C22×AGL(1,𝔽5)—62no—
132C22×C4×C5C2×Dic311yescharacteristic
133C2×C22⋊D5—21no—
134C22×Dic5#6711yescharacteristic
135#709—42no—
136C2×D20—21no—
137C2×D20—21no—
138C2×D20C2×Dic311yescharacteristic
139C2×C4×D5—21no—
140C2×C4×D5—42no—
141C2×C4×D5C2×Dic311yescharacteristic
142C2×C4×D5#6711yescharacteristic
143C2×C4×D5#6711yescharacteristic
144C2×C4×D5C22×D311yescharacteristic
145C2×Dic10C2×Dic311yescharacteristic
146C22⋊AGL(1,𝔽5)—64no—
147C2×#170—62no—
148C4×AGL(1,𝔽5)—122no—
149D5⋊C8—122no—
150C5×C4⋊C4C2×Dic311yescharacteristic
151C5×C4⋊C4C2×Dic311yescharacteristic
152C5×C22⋊C4—21no—
153C42×C5—21no—
154#685—21no—
155#685—21no—
156#685—21no—
157#685—21no—
158C4⋊Dic5—21no—
159C4⋊Dic5—21no—
160#525—51no—
161C2×C22⋊C8—151no—
162C4×C22⋊C4—151no—
163#328—152no—
164#327—152no—
165C22×C3×C5—21no—
166C22×C3×C5—21no—
167C22×C3×C5—21no—
168C22×C3×C5C2×C22⋊C411yescharacteristic
169C2×C3×D5—21no—
170C2×C3×D5—21no—
171C2×C3×D5—21no—
172C2×C3×D5—21no—
173C2×C3×D5—44no—
174C2×C3×D5—41no—
175C2×C3×D5—41no—
176C2×C3×D5—41no—
177C2×C3×D5#10412yes—
178C2×C3×D5#13512yes—
179ΓL(1,𝔽16)—44no—
180C3×C4×C5—21no—
181C3×C4×C5—41no—
182C3×C4×C5—41no—
183C3×C4×C5#9912yes—
184C3×Dic5—21no—
185C3×Dic5—21no—
186C3×Dic5—41no—
187C24×C3—51no—
188C3×#41—202no—
189C2×C3×Q8—51no—
190C2×C3×D4—51no—
191C2×C3×D4—101no—
192C2×C3×D4—101no—
193C2×C3×D4—101no—
194C22×C3×C4—51no—
195C22×C3×C4—51no—
196C22×C3×C4—51no—
197C22×C3×C4—51no—
198C22×C3×C4—51no—
199C22×C3×C4—101no—
200C22×C3×C4—202no—
201C22×C3×C4C2×AGL(1,𝔽5)11yescharacteristic
202C22×Dic3—102no—
203C3×C4⋊C4—101no—
204C3×C4⋊C4—101no—
205C3×C4⋊C4C2×AGL(1,𝔽5)11yescharacteristic
206C3×C4⋊C4C2×AGL(1,𝔽5)11yescharacteristic
207C3×C22⋊C4—21no—
208C3×C22⋊C4—101no—
209C3×C22⋊C4—101no—
210C3×C22⋊C4—101no—
211C3×C22⋊C4—101no—
212C3×C42—21no—
213#217—104no—
214C4×Dic3—202no—
215C2×C3⋊C8—102no—
216C2×C3⋊C8—202no—
217C23×C5#21711yescharacteristic
218C22×D5—22no—
219C22×D5—22no—
220C22×D5—21no—
221C22×D5—41no—
222C22×D5#21711yescharacteristic
223C22×D5C2×#6711yescharacteristic
224C2×AGL(1,𝔽5)—64no—
225C2×AGL(1,𝔽5)—122no—
226C2×AGL(1,𝔽5)—122no—
227C2×C4×C5—22no—
228C2×C4×C5—21no—
229C2×C4×C5—21no—
230C2×C4×C5#21711yescharacteristic
231C2×C4×C5C22×Dic311yescharacteristic
232C22⋊D5—42no—
233C2×Dic5—21no—
234C2×Dic5—41no—
235C2×Dic5#21711yescharacteristic
236C2×Dic5C2×#6711yescharacteristic
237D20—41no—
238D20—42no—
239D20—42no—
240C4×D5—22no—
241C4×D5—22no—
242C4×D5—41no—
243C4×D5—41no—
244C4×D5—42no—
245Dic10—41no—
246#170—122no—
247C2×#41—51no—
248C23×C4—51no—
249C22×C8—301no—
250#123—101no—
251C22⋊Q8—51no—
252C22⋊Q8—51no—
253#121—51no—
254#121—51no—
255#120—101no—
256C4×D4—101no—
257C4×D4—101no—
258#117ΓL(1,𝔽16)11yescharacteristic, complemented
259C2×C22⋊C4—301no—
260C2×C42—301no—
261C22⋊C8—152no—
262C22⋊C8—301no—
263#95—152no—
264C2×C3×C5—41no—
265C2×C3×C5C2×#9911yescharacteristic
266C2×C3×C5#41611yescharacteristic
267C2×C3×C5C2×#10411yescharacteristic
268C3×D5—22no—
269C3×D5—22no—
270C3×D5—81no—
271C23×C3—51no—
272C23×C3—51no—
273C23×C3—102no—
274C23×C3—102no—
275C23×C3—101no—
276C23×C3—201no—
277C23×C3C22⋊AGL(1,𝔽5)11yescharacteristic
278C3×Q8—201no—
279C3×D4—201no—
280C3×D4—202no—
281C3×D4—202no—
282C3×D4—202no—
283C2×C3×C4—22no—
284C2×C3×C4—21no—
285C2×C3×C4—21no—
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—202no—
295C2×C3×C4C22⋊AGL(1,𝔽5)11yescharacteristic
296C2×C3×C4C22×AGL(1,𝔽5)11yescharacteristic
297C2×Dic3—104no—
298C2×Dic3—202no—
299C2×Dic3—202no—
300C3⋊C8—202no—
301C22×C5—21no—
302C22×C5—21no—
303C22×C5—21no—
304C22×C5C2×#21711yescharacteristic
305C2×D5—21no—
306C2×D5—21no—
307C2×D5—21no—
308C2×D5—21no—
309C2×D5—44no—
310C2×D5—41no—
311C2×D5—41no—
312C2×D5—41no—
313C2×D5#83712yes—
314C2×D5#95012yes—
315AGL(1,𝔽5)—124no—
316C4×C5—21no—
317C4×C5—41no—
318C4×C5—41no—
319C4×C5#83412yes—
320Dic5—21no—
321Dic5—21no—
322Dic5—41no—
323C24—51no—
324#41—202no—
325C2×Q8—51no—
326C2×D4—51no—
327C2×D4—101no—
328C2×D4—101no—
329C2×D4—101no—
330C22×C4—51no—
331C22×C4—51no—
332C22×C4—51no—
333C22×C4—51no—
334C22×C4—51no—
335C22×C4—101no—
336C22×C4—202no—
337C22×C4—302no—
338C22×C4C2×ΓL(1,𝔽16)11yescharacteristic
339C2×C8—302no—
340C2×C8—602no—
341C4⋊C4—101no—
342C4⋊C4—101no—
343C4⋊C4C2×ΓL(1,𝔽16)11yescharacteristic
344C4⋊C4C2×ΓL(1,𝔽16)11yescharacteristic
345C22⋊C4—21no—
346C22⋊C4—101no—
347C22⋊C4—101no—
348C22⋊C4—101no—
349C22⋊C4—101no—
350C22⋊C4—304no—
351C42—21no—
352C42—602no—
353C3×C5#150111yescharacteristic, complemented, Hall
354C22×C3—21no—
355C22×C3—21no—
356C22×C3—21no—
357C22×C3—52no—
358C22×C3—52no—
359C22×C3—101no—
360C22×C3—101no—
361C22×C3—101no—
362C22×C3—101no—
363C22×C3—204no—
364C22×C3—201no—
365C22×C3—201no—
366C22×C3—201no—
367C22×C3C2×C22⋊AGL(1,𝔽5)11yescharacteristic
368C3×C4—21no—
369C3×C4—41no—
370C3×C4—41no—
371C3×C4—101no—
372C3×C4—101no—
373C3×C4—201no—
374C3×C4#399912yes—
375Dic3—204no—
376C2×C5—41no—
377C2×C5C2×#83411yescharacteristic
378C2×C5#523311yescharacteristic
379C2×C5C2×#83711yescharacteristic
380D5—22no—
381D5—22no—
382D5—81no—
383C23—51no—
384C23—51no—
385C23—102no—
386C23—102no—
387C23—101no—
388C23—201no—
389C23#676311yescharacteristic
390Q8—201no—
391D4—201no—
392D4—202no—
393D4—202no—
394D4—202no—
395C2×C4—22no—
396C2×C4—21no—
397C2×C4—21no—
398C2×C4—51no—
399C2×C4—51no—
400C2×C4—102no—
401C2×C4—102no—
402C2×C4—101no—
403C2×C4—201no—
404C2×C4—201no—
405C2×C4—201no—
406C2×C4—202no—
407C2×C4—304no—
408C2×C4—602no—
409C2×C4—602no—
410C2×C4#676311yescharacteristic
411C2×C4C22×ΓL(1,𝔽16)11yescharacteristic
412C8—602no—
413C2×C3—41no—
414C2×C3—102no—
415C2×C3—102no—
416C2×C3—401no—
417C2×C3C2×#399911yescharacteristic
418C2×C3D5⋊#10411yescharacteristic
419C2×C3#6579411yescharacteristic
420C5#7170711yescharacteristic, complemented, Sylow
421C22—21no—
422C22—21no—
423C22—21no—
424C22—52no—
425C22—52no—
426C22—101no—
427C22—101no—
428C22—101no—
429C22—101no—
430C22—204no—
431C22—201no—
432C22—201no—
433C22—201no—
434C22C2×#676311yescharacteristic
435C4—21no—
436C4—41no—
437C4—41no—
438C4—101no—
439C4—101no—
440C4—201no—
441C4—604no—
442C4#9122812yes—
443C3#1060434611yescharacteristic, complemented, Sylow
444C2—41no—
445C2—102no—
446C2—102no—
447C2—401no—
448C2C2×#9122811yescharacteristic
449C2D5⋊#83711yescharacteristic
450C2#1175245311yescharacteristic
subgroupG/H
centerC2×C4#6763
commutatorC2×C3×C4×C5C22×C4
FrattiniC2×C4C22×ΓL(1,𝔽16)
FittingC3×C5×#117C4
socleC22×C3×C5C2×C22⋊C4
prime radicalC3×#65858C2
cosocleC2×C3×C4×D5C23
hypercenter#117ΓL(1,𝔽16)
hypocenterC3×C5#1501
LCSG ▷ C2×C3×C4×C5 ▷ C22×C3×C5 ▷ C3×C5
UCSC1 ◁ C2×C4 ◁ C22×C4 ◁ #117

Complements

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

NH ≅ G/Ntype
2. C4×#6763C2semidirect
3. D5⋊#830C2semidirect
4. #11746740C2semidirect
5. #11746739C2semidirect
7. C2×C3×#709C4semidirect
12. C3×C5×#117C4semidirect
13. C3×#4043C4semidirect
15. C3×#4041C4semidirect
65. C2×#709Dic3semidirect
70. C5×#117Dic3semidirect
71. #4043Dic3semidirect
73. #4041Dic3semidirect
123. C3×#117AGL(1,𝔽5)semidirect
258. #117ΓL(1,𝔽16)semidirect
353. C3×C5#1501semidirect
420. C5#71707semidirect
443. C3#10604346semidirect

Maximal subgroups

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

subgroupindex
C3×#658582
C4×#67632
D5⋊#8302
#117467402
#117467392
#106043463
#717075