Finite Group Databasebeta

← #14282885

Subgroups — #14282885

Order 1920.

568 conjugacy classes of subgroups.

The subgroup lattice has 440 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C3×#65882C211yesmaximal, characteristic
2C4×C4⋊ΓL(1,𝔽16)C211yesmaximal, characteristic, complemented
3#11752337C211yesmaximal, characteristic, complemented
4#11746791C212yesmaximal, complemented
5#11746739C212yesmaximal, complemented
6#10604389—31nomaximal, Hall
7C2×C3×#709—21no—
8C2×C4⋊ΓL(1,𝔽16)—21no—
9C2×C4×ΓL(1,𝔽16)—21no—
10C3×C4×C5×D4C411yescharacteristic, complemented
11C3×#4086C411yescharacteristic, complemented
12C3×#4084C2211yescharacteristic
13C3×C4×C22⋊D5—21no—
14C3×#4041C2211yescharacteristic
15C3×#4034C411yescharacteristic, complemented
16C3×#4031—21no—
17C3×#4030—21no—
18C3×C4×D20C411yescharacteristic, complemented
19C3×C42×D5C2211yescharacteristic
20#91235C2212yes—
21#91233—21no—
22#91232C2212yes—
23C4×C3⋊#170—21no—
24#71811—51nomaximal, Hall
25#65882D311yescharacteristic
26C4×C4⋊AGL(1,𝔽5)—31no—
27#65678—31no—
28#64920—32no—
29#64868—32no—
30C22×ΓL(1,𝔽16)—22no—
31C2×C3×C5×D4C2×C411yescharacteristic
32C22×C3×C4×C5—21no—
33C2×C3×C22⋊D5—21no—
34C2×C3×C22⋊D5—21no—
35C2×C3×C22⋊D5—21no—
36C3×#709—42no—
37C3×#709—42no—
38C2×C3×D20—21no—
39C2×C3×D20C2×C411yescharacteristic
40C2×C3×C4×D5—21no—
41C2×C3×C4×D5D411yescharacteristic
42C2×C3×C4×D5D411yescharacteristic
43C2×C3×C4×D5C2311yescharacteristic
44C2×C3×Dic10—21no—
45C2×C3⋊#170—22no—
46C4⋊ΓL(1,𝔽16)—24no—
47C4×ΓL(1,𝔽16)—42no—
48C3×C5×C4⋊C4C2×C411yescharacteristic
49C3×C5×C22⋊C4—21no—
50C3×C42×C5C2×C411yescharacteristic
51C3×C22⋊Dic5—21no—
52C3×#685—21no—
53C3×#685—21no—
54C3×C4⋊Dic5C2×C411yescharacteristic
55C3×#683—21no—
56C3×C4×Dic5C2×C411yescharacteristic
57C3×C4×Dic5D411yescharacteristic
58C3×C4×Dic5D411yescharacteristic
59C3×#532—51no—
60C4×C4⋊Dic3—51no—
61#4925—51no—
62#4537—52no—
63#4537—52no—
64C2×#709—21no—
65C2×C4⋊AGL(1,𝔽5)—61no—
66C2×C4×AGL(1,𝔽5)—61no—
67C4×C5×D4Dic311yescharacteristic, complemented
68#4086Dic311yescharacteristic, complemented
69#4084C2×D311yescharacteristic
70C4×C22⋊D5—21no—
71#4041C2×D311yescharacteristic
72#4034Dic311yescharacteristic, complemented
73#4031—21no—
74#4030—21no—
75C4×D20Dic311yescharacteristic, complemented
76C42×D5C2×D311yescharacteristic
77(C2×AGL(1,𝔽5))⋊C4—32no—
78#4004—61no—
79#4003—32no—
80C4×#170—61no—
81#1502—151noSylow
82C23×C3×C5—21no—
83C22×C3×D5—21no—
84C22×C3×D5C2×D411yescharacteristic
85C2×ΓL(1,𝔽16)—24no—
86C2×ΓL(1,𝔽16)—42no—
87C2×ΓL(1,𝔽16)—42no—
88C3×C5×D4—41no—
89C2×C3×C4×C5—21no—
90C2×C3×C4×C5—42no—
91C2×C3×C4×C5C22⋊C411yescharacteristic
92C2×C3×C4×C5C22⋊C411yescharacteristic
93C2×C3×C4×C5C22×C411yescharacteristic
94C3×C22⋊D5—41no—
95C3×C22⋊D5—41no—
96C3×C22⋊D5—42no—
97C3×C22⋊D5—42no—
98C2×C3×Dic5—21no—
99C2×C3×Dic5C22⋊C411yescharacteristic
100C2×C3×Dic5C22⋊C411yescharacteristic
101C2×C3×Dic5C2×D411yescharacteristic
102C3×D20—41no—
103C3×D20—41no—
104C3×D20—41no—
105C3×C4×D5—22no—
106C3×C4×D5—41no—
107C3×C4×D5—42no—
108C3×C4×D5D812yes—
109C3×C4×D5SD1612yes—
110C3×Dic10—42no—
111C3⋊#170—42no—
112C2×C3×#41—101no—
113C3×C4⋊Q8—51no—
114C3×#127—51no—
115C3×#124—101no—
116C3×#121—51no—
117C3×#121—51no—
118C3×#121—101no—
119C3×C4×D4—51no—
120C3×C4×D4—101no—
121C3×C4×D4AGL(1,𝔽5)11yescharacteristic, complemented
122C2×C3×C42—51no—
123C2×C4⋊Dic3—101no—
124C2×C4×Dic3—101no—
125C3⋊#95—52no—
126#830—52no—
127C4⋊(C3⋊C8)—101no—
128C4×C3⋊C8—101no—
129C22×AGL(1,𝔽5)—62no—
130C2×C5×D4C2×Dic311yescharacteristic
131C22×C4×C5—21no—
132C2×C22⋊D5—21no—
133C2×C22⋊D5—21no—
134C2×C22⋊D5—21no—
135#709—42no—
136#709—42no—
137C2×D20—21no—
138C2×D20C2×Dic311yescharacteristic
139C2×C4×D5—21no—
140C2×C4×D5#6711yescharacteristic
141C2×C4×D5#6711yescharacteristic
142C2×C4×D5C22×D311yescharacteristic
143C2×Dic10—21no—
144C2×#170—62no—
145C4⋊AGL(1,𝔽5)—64no—
146C4×AGL(1,𝔽5)—122no—
147C5×C4⋊C4C2×Dic311yescharacteristic
148C5×C22⋊C4—21no—
149C42×C5C2×Dic311yescharacteristic
150C22⋊Dic5—21no—
151#685—21no—
152#685—21no—
153C4⋊Dic5C2×Dic311yescharacteristic
154#683—21no—
155C4×Dic5C2×Dic311yescharacteristic
156C4×Dic5#6711yescharacteristic
157C4×Dic5#6711yescharacteristic
158#532—51no—
159#431—151no—
160C4×C4⋊C4—151no—
161#327—152no—
162#327—152no—
163C22×C3×C5—41no—
164C22×C3×C5—41no—
165C22×C3×C5—41no—
166C22×C3×C5C2×C22⋊C411yescharacteristic
167C2×C3×D5—21no—
168C2×C3×D5—21no—
169C2×C3×D5—41no—
170C2×C3×D5—41no—
171C2×C3×D5—41no—
172C2×C3×D5C2×D811yescharacteristic
173C2×C3×D5C2×SD1611yescharacteristic
174ΓL(1,𝔽16)—44no—
175C3×C4×C5—21no—
176C3×C4×C5—21no—
177C3×C4×C5—41no—
178C3×C4×C5#10212yes—
179C3×Dic5—21no—
180C3×Dic5—41no—
181C3×Dic5#10212yes—
182C3×Dic5#10412yes—
183C3×#41—202no—
184C3×#41—202no—
185C2×C3×Q8—101no—
186C2×C3×D4—51no—
187C2×C3×D4—101no—
188C2×C3×D4—101no—
189C2×C3×D4—101no—
190C2×C3×D4—101no—
191C2×C3×D4C2×AGL(1,𝔽5)11yescharacteristic
192C22×C3×C4—21no—
193C22×C3×C4—51no—
194C22×C3×C4—51no—
195C22×C3×C4—51no—
196C22×C3×C4—101no—
197C22×Dic3—102no—
198C3×C4⋊C4—51no—
199C3×C4⋊C4—101no—
200C3×C4⋊C4C2×AGL(1,𝔽5)11yescharacteristic
201C3×C22⋊C4—21no—
202C3×C22⋊C4—101no—
203C3×C22⋊C4—101no—
204C3×C22⋊C4—101no—
205C3×C42—51no—
206C3×C42—51no—
207C3×C42—51no—
208C3×C42C2×AGL(1,𝔽5)11yescharacteristic
209C4⋊Dic3—104no—
210C4×Dic3—202no—
211C2×C3⋊C8—102no—
212C23×C5—21no—
213C22×D5—21no—
214C22×D5C2×#6711yescharacteristic
215C2×AGL(1,𝔽5)—64no—
216C2×AGL(1,𝔽5)—122no—
217C2×AGL(1,𝔽5)—122no—
218C5×D4—41no—
219C2×C4×C5—21no—
220C2×C4×C5—42no—
221C2×C4×C5#21711yescharacteristic
222C2×C4×C5#21711yescharacteristic
223C2×C4×C5C22×Dic311yescharacteristic
224C22⋊D5—41no—
225C22⋊D5—41no—
226C22⋊D5—42no—
227C22⋊D5—42no—
228C2×Dic5—21no—
229C2×Dic5#21711yescharacteristic
230C2×Dic5#21711yescharacteristic
231C2×Dic5C2×#6711yescharacteristic
232D20—41no—
233D20—41no—
234D20—41no—
235C4×D5—22no—
236C4×D5—41no—
237C4×D5—42no—
238C4×D5#21312yes—
239C4×D5#21412yes—
240Dic10—42no—
241#170—122no—
242C2×#41—101no—
243C4⋊Q8—51no—
244#127—51no—
245#124—101no—
246#121—51no—
247#121—51no—
248#121—101no—
249C4×D4—51no—
250C4×D4—101no—
251C4×D4ΓL(1,𝔽16)11yescharacteristic, complemented
252C2×C4⋊C4—301no—
253C2×C42—51no—
254C2×C42—301no—
255C4⋊C8—301no—
256C22⋊C8—152no—
257C4×C8—301no—
258#95—152no—
259C2×C3×C5—81no—
260C2×C3×C5#41011yescharacteristic
261C2×C3×C5C2×#10211yescharacteristic
262C2×C3×C5C2×#10411yescharacteristic
263C3×D5—21no—
264C3×D5—21no—
265C3×D5—81no—
266C23×C3—21no—
267C23×C3—51no—
268C23×C3—101no—
269C3×Q8—202no—
270C3×D4—41no—
271C3×D4—201no—
272C3×D4—201no—
273C3×D4—201no—
274C3×D4—202no—
275C3×D4—202no—
276C3×D4—201no—
277C3×D4—201no—
278C2×C3×C4—21no—
279C2×C3×C4—42no—
280C2×C3×C4—51no—
281C2×C3×C4—51no—
282C2×C3×C4—52no—
283C2×C3×C4—52no—
284C2×C3×C4—51no—
285C2×C3×C4—102no—
286C2×C3×C4—101no—
287C2×C3×C4—201no—
288C2×C3×C4—202no—
289C2×C3×C4C22⋊AGL(1,𝔽5)11yescharacteristic
290C2×C3×C4C22⋊AGL(1,𝔽5)11yescharacteristic
291C2×C3×C4C22×AGL(1,𝔽5)11yescharacteristic
292C2×Dic3—104no—
293C2×Dic3—202no—
294C2×Dic3—202no—
295C3⋊C8—202no—
296C22×C5—41no—
297C22×C5—41no—
298C22×C5—41no—
299C22×C5C2×#21711yescharacteristic
300C2×D5—21no—
301C2×D5—21no—
302C2×D5—41no—
303C2×D5—41no—
304C2×D5—41no—
305C2×D5C2×#21311yescharacteristic
306C2×D5C2×#21411yescharacteristic
307AGL(1,𝔽5)—124no—
308C4×C5—21no—
309C4×C5—21no—
310C4×C5—41no—
311C4×C5#83212yes—
312Dic5—21no—
313Dic5—41no—
314Dic5#83212yes—
315Dic5#83712yes—
316#41—202no—
317#41—202no—
318C2×Q8—101no—
319C2×D4—51no—
320C2×D4—101no—
321C2×D4—101no—
322C2×D4—101no—
323C2×D4—101no—
324C2×D4C2×ΓL(1,𝔽16)11yescharacteristic
325C22×C4—21no—
326C22×C4—51no—
327C22×C4—51no—
328C22×C4—51no—
329C22×C4—101no—
330C22×C4—302no—
331C2×C8—302no—
332C4⋊C4—51no—
333C4⋊C4—101no—
334C4⋊C4—304no—
335C4⋊C4C2×ΓL(1,𝔽16)11yescharacteristic
336C22⋊C4—21no—
337C22⋊C4—101no—
338C22⋊C4—101no—
339C22⋊C4—101no—
340C42—51no—
341C42—51no—
342C42—51no—
343C42—602no—
344C42C2×ΓL(1,𝔽16)11yescharacteristic
345C3×C5#150211yescharacteristic, complemented, Hall
346C22×C3—41no—
347C22×C3—41no—
348C22×C3—41no—
349C22×C3—51no—
350C22×C3—51no—
351C22×C3—101no—
352C22×C3—101no—
353C22×C3—201no—
354C22×C3—201no—
355C22×C3—201no—
356C22×C3C2×C22⋊AGL(1,𝔽5)11yescharacteristic
357C3×C4—21no—
358C3×C4—21no—
359C3×C4—41no—
360C3×C4—52no—
361C3×C4—52no—
362C3×C4—101no—
363C3×C4—201no—
364C3×C4#400712yes—
365Dic3—204no—
366C2×C5—81no—
367C2×C5C2×#83211yescharacteristic
368C2×C5C2×#83711yescharacteristic
369C2×C5#524011yescharacteristic
370D5—21no—
371D5—21no—
372D5—81no—
373C23—21no—
374C23—51no—
375C23—101no—
376Q8—202no—
377D4—41no—
378D4—201no—
379D4—201no—
380D4—201no—
381D4—202no—
382D4—202no—
383D4—201no—
384D4—201no—
385C2×C4—21no—
386C2×C4—42no—
387C2×C4—51no—
388C2×C4—51no—
389C2×C4—52no—
390C2×C4—52no—
391C2×C4—51no—
392C2×C4—102no—
393C2×C4—101no—
394C2×C4—201no—
395C2×C4—202no—
396C2×C4—304no—
397C2×C4—602no—
398C2×C4—602no—
399C2×C4#676311yescharacteristic
400C2×C4#676311yescharacteristic
401C2×C4C22×ΓL(1,𝔽16)11yescharacteristic
402C8—602no—
403C2×C3—81no—
404C2×C3—101no—
405C2×C3—101no—
406C2×C3—401no—
407C2×C3#6574511yescharacteristic
408C2×C3C2×#400711yescharacteristic
409C2×C3D5⋊#10411yescharacteristic
410C5#7181111yescharacteristic, complemented, Sylow
411C22—41no—
412C22—41no—
413C22—41no—
414C22—51no—
415C22—51no—
416C22—101no—
417C22—101no—
418C22—201no—
419C22—201no—
420C22—201no—
421C22C2×#676311yescharacteristic
422C4—21no—
423C4—21no—
424C4—41no—
425C4—52no—
426C4—52no—
427C4—101no—
428C4—201no—
429C4—604no—
430C4#9123612yes—
431C3#1060438911yescharacteristic, complemented, Sylow
432C2—81no—
433C2—101no—
434C2—101no—
435C2—401no—
436C2#1175240411yescharacteristic
437C2C2×#9123611yescharacteristic
438C2D5⋊#83711yescharacteristic
subgroupG/H
centerC2×C4#6763
commutatorC2×C3×C4×C5C22×C4
FrattiniC2×C4C22×ΓL(1,𝔽16)
FittingC3×C4×C5×D4C4
socleC22×C3×C5C2×C22⋊C4
prime radicalC3×#65882C2
cosocleC2×C3×C4×D5C23
hypercenterC4×D4ΓL(1,𝔽16)
hypocenterC3×C5#1502
LCSG ▷ C2×C3×C4×C5 ▷ C22×C3×C5 ▷ C3×C5
UCSC1 ◁ C2×C4 ◁ C42 ◁ C4×D4

Complements

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

NH ≅ G/Ntype
2. C4×C4⋊ΓL(1,𝔽16)C2semidirect
3. #11752337C2semidirect
4. #11746791C2semidirect
5. #11746739C2semidirect
10. C3×C4×C5×D4C4semidirect
11. C3×#4086C4semidirect
15. C3×#4034C4semidirect
18. C3×C4×D20C4semidirect
67. C4×C5×D4Dic3semidirect
68. #4086Dic3semidirect
72. #4034Dic3semidirect
75. C4×D20Dic3semidirect
121. C3×C4×D4AGL(1,𝔽5)semidirect
251. C4×D4ΓL(1,𝔽16)semidirect
345. C3×C5#1502semidirect
410. C5#71811semidirect
431. C3#10604389semidirect

Maximal subgroups

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

subgroupindex
C3×#658822
C4×C4⋊ΓL(1,𝔽16)2
#117523372
#117467912
#117467392
#106043893
#718115