Finite Group Databasebeta

← #14303444

Subgroups — #14303444

Order 1920.

460 conjugacy classes of subgroups.

The subgroup lattice has 432 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C4⋊C4×ΓL(1,𝔽16)C211yesmaximal, characteristic
2#11751861C211yesmaximal, characteristic
3D5×#808C211yesmaximal, characteristic
4C3×#64874C211yesmaximal, characteristic
5#11746530C211yesmaximal, characteristic
6#11746529C211yesmaximal, characteristic
7#11746517C211yesmaximal, characteristic
8#10604049—31nomaximal, Hall
9C2×C4⋊ΓL(1,𝔽16)C2211yescharacteristic
10C2×C4×ΓL(1,𝔽16)—21no—
11C2×C4×ΓL(1,𝔽16)C2211yescharacteristic
12C2×C3×C4⋊AGL(1,𝔽5)C2211yescharacteristic
13C2×C3×D5⋊C8C2211yescharacteristic
14C2×Dic3⋊AGL(1,𝔽5)—21no—
15C3×D5×C4⋊C4C2211yescharacteristic
16D5×C4⋊Dic3C2211yescharacteristic
17C2×D5×C3⋊C8C2211yescharacteristic
18#91235—21no—
19(C2×AGL(1,𝔽5))⋊Dic3—21no—
20#91092C411yescharacteristic, complemented
21C5×#808C411yescharacteristic, complemented
22#90987C411yescharacteristic, complemented
23#90986C411yescharacteristic, complemented
24#68788—51nomaximal, Hall
25#65712—31no—
26C4⋊C4×AGL(1,𝔽5)—31no—
27D5×#106—31no—
28#64893—31no—
29(C2×AGL(1,𝔽5))⋊C8—31no—
30#64874—31no—
31#64874D311yescharacteristic
32C22×ΓL(1,𝔽16)—21no—
33C22×ΓL(1,𝔽16)D411yescharacteristic
34C22×ΓL(1,𝔽16)D411yescharacteristic
35C22×C3×AGL(1,𝔽5)—21no—
36C2×C3×C4×D5—21no—
37C2×C3×C4×D5C2311yescharacteristic
38C2×D5×Dic3—21no—
39C4⋊ΓL(1,𝔽16)—21no—
40C4⋊ΓL(1,𝔽16)Q812yes—
41C4×ΓL(1,𝔽16)—21no—
42C4×ΓL(1,𝔽16)—21no—
43C4×ΓL(1,𝔽16)—42no—
44C2×C3×#170—21no—
45C3×C4⋊AGL(1,𝔽5)—21no—
46C3×C4⋊AGL(1,𝔽5)—21no—
47C3×D5⋊C8—21no—
48C3×D5⋊C8—21no—
49Dic3⋊AGL(1,𝔽5)—42no—
50C3×C5×C4⋊C4C2×C411yescharacteristic
51C4⋊Dic15C2×C411yescharacteristic
52C2×C3⋊#168C2×C411yescharacteristic
53C5×C4⋊Dic3C2×C411yescharacteristic
54C2×C5×C3⋊C8C2×C411yescharacteristic
55C3×C4⋊Dic5C2×C411yescharacteristic
56C3×#683—21no—
57#6668—21no—
58D5×C3⋊C8—21no—
59D5×C3⋊C8—21no—
60Dic3×C4⋊C4—51no—
61#4972—51no—
62C2×#808—51no—
63C3×#340—51no—
64#4551—51no—
65#4550—51no—
66#4529—51no—
67C2×C4⋊AGL(1,𝔽5)—31no—
68C2×C4⋊AGL(1,𝔽5)—61no—
69C2×C4⋊AGL(1,𝔽5)C2×D311yescharacteristic
70C2×C4×AGL(1,𝔽5)—31no—
71C2×C4×AGL(1,𝔽5)—61no—
72C2×D5⋊C8C2×D311yescharacteristic
73C2×C8×D5—31no—
74D5×C4⋊C4—31no—
75D5×C4⋊C4C2×D311yescharacteristic
76(C2×AGL(1,𝔽5))⋊C4—61no—
77(C2×AGL(1,𝔽5))⋊C4—61no—
78C5×#106—31no—
79#3949—31no—
80#3940—31no—
81#3940—31no—
82#1919—151noSylow
83C22×C3×D5C2×D411yescharacteristic
84C2×ΓL(1,𝔽16)—21no—
85C2×ΓL(1,𝔽16)—21no—
86C2×ΓL(1,𝔽16)—22no—
87C2×ΓL(1,𝔽16)—21no—
88C2×ΓL(1,𝔽16)—21no—
89C2×ΓL(1,𝔽16)—21no—
90C2×ΓL(1,𝔽16)—41no—
91C2×ΓL(1,𝔽16)—41no—
92C2×ΓL(1,𝔽16)SD1612yes—
93C2×C3×AGL(1,𝔽5)—41no—
94C2×C3×AGL(1,𝔽5)—41no—
95C2×C3×AGL(1,𝔽5)—41no—
96C2×C3×C4×C5—21no—
97C2×C3×C4×C5C22×C411yescharacteristic
98C2×Dic15—21no—
99C2×C5×Dic3—21no—
100C2×C3×Dic5—21no—
101C2×C3×Dic5#4111yescharacteristic
102C3×C4×D5—41no—
103C3×C4×D5—41no—
104C3×C4×D5C2×Q811yescharacteristic
105C3×C4×D5#4111yescharacteristic
106C3×C4×D5#4111yescharacteristic
107C3×C4×D5#4111yescharacteristic
108D5×Dic3—41no—
109D5×Dic3—41no—
110C3×#170—41no—
111C3⋊#168—21no—
112C5×C3⋊C8—21no—
113C22×C3×C8—51no—
114C2×C3×C4⋊C4—51no—
115C2×C3×C4⋊C4—51no—
116C2×C4⋊Dic3—51no—
117C2×C4⋊Dic3—51no—
118C2×Dic3⋊C4—101no—
119C2×C4×Dic3—51no—
120C2×C4×Dic3—101no—
121C22×C3⋊C8—51no—
122C3⋊#95—101no—
123C3⋊#95—101no—
124#808—51no—
125#808—51no—
126#808—51no—
127#808AGL(1,𝔽5)11yescharacteristic, complemented
128C22×AGL(1,𝔽5)—21no—
129C22×AGL(1,𝔽5)—31no—
130C22×AGL(1,𝔽5)—31no—
131C22×AGL(1,𝔽5)—61no—
132C2×C4×D5—21no—
133C2×C4×D5—61no—
134C2×C4×D5C22×D311yescharacteristic
135C2×#170—21no—
136C4⋊AGL(1,𝔽5)—21no—
137C4⋊AGL(1,𝔽5)—21no—
138C4⋊AGL(1,𝔽5)—32no—
139C4⋊AGL(1,𝔽5)—61no—
140C4⋊AGL(1,𝔽5)—122no—
141C4×AGL(1,𝔽5)—61no—
142C4×AGL(1,𝔽5)—61no—
143C4×AGL(1,𝔽5)—122no—
144D5⋊C8—21no—
145D5⋊C8—21no—
146C2×C5×C8—31no—
147C5×C4⋊C4—31no—
148C5×C4⋊C4C4×D311yescharacteristic
149C4⋊Dic5—31no—
150C4⋊Dic5C4×D311yescharacteristic
151#683—21no—
152#683—61no—
153C2×#168—31no—
154C8×D5—61no—
155C8×D5—61no—
156C2×#106—151no—
157#389—151no—
158C4×C4⋊C4—151no—
159#340—51no—
160#340—151no—
161#340—151no—
162#336—151no—
163C22×C3×C5C4×D411yescharacteristic
164C2×C3×D5C22⋊Q811yescharacteristic
165C2×C3×D5#12311yescharacteristic
166C2×C3×D5C2×SD1611yescharacteristic
167C2×C3×D5#13511yescharacteristic
168C2×C3×D5AGL(1,ℤ/8)11yescharacteristic
169C2×C3×D5#13711yescharacteristic
170ΓL(1,𝔽16)—22no—
171ΓL(1,𝔽16)—42no—
172ΓL(1,𝔽16)—41no—
173C3×AGL(1,𝔽5)—81no—
174C3×C4×C5—41no—
175C3×C4×C5#11711yescharacteristic
176C3×C4×C5C4×Q811yescharacteristic
177Dic15—41no—
178C3×Dic5—41no—
179C3×Dic5#12511yescharacteristic
180C3×Dic5#12611yescharacteristic
181C5×Dic3—41no—
182C22×C3×C4—51no—
183C22×C3×C4—101no—
184C22×C3×C4—101no—
185C22×Dic3—51no—
186C22×Dic3—51no—
187C22×Dic3—101no—
188C22×Dic3—101no—
189C2×C3×C8—101no—
190C2×C3×C8—101no—
191C2×C3×C8—101no—
192C3×C4⋊C4—51no—
193C3×C4⋊C4—101no—
194C3×C4⋊C4—101no—
195C3×C4⋊C4—101no—
196C3×C4⋊C4C2×AGL(1,𝔽5)11yescharacteristic
197C4⋊Dic3—52no—
198C4⋊Dic3—51no—
199C4⋊Dic3—101no—
200C4⋊Dic3—101no—
201C4⋊Dic3C2×AGL(1,𝔽5)11yescharacteristic
202Dic3⋊C4—202no—
203C4×Dic3—101no—
204C4×Dic3—101no—
205C4×Dic3—202no—
206C2×C3⋊C8—51no—
207C2×C3⋊C8—101no—
208C2×C3⋊C8—101no—
209C2×C3⋊C8C2×AGL(1,𝔽5)11yescharacteristic
210C22×D5D3×D411yescharacteristic
211C2×AGL(1,𝔽5)—32no—
212C2×AGL(1,𝔽5)—41no—
213C2×AGL(1,𝔽5)—41no—
214C2×AGL(1,𝔽5)—41no—
215C2×AGL(1,𝔽5)—61no—
216C2×AGL(1,𝔽5)—61no—
217C2×AGL(1,𝔽5)—61no—
218C2×AGL(1,𝔽5)—62no—
219C2×AGL(1,𝔽5)—61no—
220C2×AGL(1,𝔽5)—61no—
221C2×AGL(1,𝔽5)—121no—
222C2×AGL(1,𝔽5)—121no—
223C2×C4×C5—21no—
224C2×C4×C5—61no—
225C2×C4×C5C2×C4×D311yescharacteristic
226C2×Dic5—21no—
227C2×Dic5—61no—
228C2×Dic5#23711yescharacteristic
229C4×D5—41no—
230C4×D5—41no—
231C4×D5—121no—
232C4×D5—121no—
233C4×D5#23511yescharacteristic
234C4×D5#23511yescharacteristic
235C4×D5#23711yescharacteristic
236C4×D5D3×Q811yescharacteristic
237#170—41no—
238C5×C8—61no—
239#168—61no—
240C22×C8—51no—
241C22×C8—151no—
242C2×C4⋊C4—51no—
243C2×C4⋊C4—51no—
244C2×C4⋊C4—151no—
245C2×C4⋊C4—151no—
246C2×C4⋊C4—301no—
247C2×C42—151no—
248C2×C42—301no—
249#106—31no—
250#106—151no—
251#106—151no—
252#106—151no—
253#95—301no—
254#95—301no—
255C2×C3×C5#38711yescharacteristic
256C2×C3×C5C4×SD1611yescharacteristic
257C2×C3×C5#44011yescharacteristic
258C3×D5#47511yescharacteristic
259C3×D5#47811yescharacteristic
260C3×D5#48111yescharacteristic
261C3×D5#48511yescharacteristic
262C23×C3—51no—
263C2×C3×C4—21no—
264C2×C3×C4—51no—
265C2×C3×C4—51no—
266C2×C3×C4—51no—
267C2×C3×C4—51no—
268C2×C3×C4—51no—
269C2×C3×C4—101no—
270C2×C3×C4—201no—
271C2×C3×C4—201no—
272C2×C3×C4—201no—
273C2×C3×C4—201no—
274C2×C3×C4—201no—
275C2×C3×C4C22×AGL(1,𝔽5)11yescharacteristic
276C2×Dic3—21no—
277C2×Dic3—52no—
278C2×Dic3—101no—
279C2×Dic3—101no—
280C2×Dic3—101no—
281C2×Dic3—102no—
282C2×Dic3—101no—
283C2×Dic3—101no—
284C2×Dic3—101no—
285C2×Dic3—201no—
286C2×Dic3—201no—
287C2×Dic3—201no—
288C2×Dic3—201no—
289C3×C8—201no—
290C3⋊C8—21no—
291C3⋊C8—101no—
292C22×C5#67⋊C411yescharacteristic
293C2×D5#88311yescharacteristic
294C2×D5#89611yescharacteristic
295C2×D5#91111yescharacteristic
296C2×D5D3×SD1611yescharacteristic
297C2×D5#91511yescharacteristic
298C2×D5#91911yescharacteristic
299AGL(1,𝔽5)—62no—
300AGL(1,𝔽5)—81no—
301AGL(1,𝔽5)—122no—
302AGL(1,𝔽5)—121no—
303C4×C5—41no—
304C4×C5—121no—
305C4×C5#87211yescharacteristic
306C4×C5#88711yescharacteristic
307Dic5—41no—
308Dic5—121no—
309Dic5#87911yescharacteristic
310Dic5#88911yescharacteristic
311C22×C4—51no—
312C22×C4—101no—
313C22×C4—101no—
314C22×C4—151no—
315C22×C4—151no—
316C22×C4—301no—
317C22×C4—301no—
318C2×C8—31no—
319C2×C8—101no—
320C2×C8—101no—
321C2×C8—101no—
322C2×C8—151no—
323C2×C8—301no—
324C2×C8—301no—
325C4⋊C4—31no—
326C4⋊C4—51no—
327C4⋊C4—101no—
328C4⋊C4—101no—
329C4⋊C4—101no—
330C4⋊C4—152no—
331C4⋊C4—151no—
332C4⋊C4—301no—
333C4⋊C4—301no—
334C4⋊C4—602no—
335C4⋊C4D3×AGL(1,𝔽5)11yescharacteristic
336C42—301no—
337C42—301no—
338C42—602no—
339C3×C5#191911yescharacteristic, complemented, Hall
340C22×C3—51no—
341C22×C3—51no—
342C22×C3—51no—
343C22×C3—51no—
344C22×C3—51no—
345C22×C3—51no—
346C22×C3D4×AGL(1,𝔽5)11yescharacteristic
347C3×C4—41no—
348C3×C4—51no—
349C3×C4—51no—
350C3×C4—201no—
351C3×C4—401no—
352C3×C4#413011yescharacteristic
353C3×C4Q8×AGL(1,𝔽5)11yescharacteristic
354Dic3—41no—
355Dic3—102no—
356Dic3—201no—
357Dic3—202no—
358Dic3—201no—
359C2×C5#465211yescharacteristic
360C2×C5#475711yescharacteristic
361C2×C5#478811yescharacteristic
362D5#477711yescharacteristic
363D5#481411yescharacteristic
364D5#485411yescharacteristic
365D5#489111yescharacteristic
366C23—51no—
367C2×C4—21no—
368C2×C4—51no—
369C2×C4—51no—
370C2×C4—51no—
371C2×C4—51no—
372C2×C4—51no—
373C2×C4—61no—
374C2×C4—101no—
375C2×C4—152no—
376C2×C4—201no—
377C2×C4—201no—
378C2×C4—201no—
379C2×C4—201no—
380C2×C4—201no—
381C2×C4—301no—
382C2×C4—301no—
383C2×C4—301no—
384C2×C4—302no—
385C2×C4—301no—
386C2×C4—301no—
387C2×C4—301no—
388C2×C4—601no—
389C2×C4—601no—
390C2×C4—601no—
391C2×C4—601no—
392C2×C4C2×D3×AGL(1,𝔽5)11yescharacteristic
393C8—61no—
394C8—201no—
395C8—301no—
396C2×C3—51no—
397C2×C3—51no—
398C2×C3—51no—
399C2×C3—51no—
400C2×C3#6571111yescharacteristic
401C2×C3SD16×AGL(1,𝔽5)11yescharacteristic
402C2×C3#6573511yescharacteristic
403C5#6878811yescharacteristic, complemented, Sylow
404C22—51no—
405C22—51no—
406C22—51no—
407C22—51no—
408C22—51no—
409C22—51no—
410C22#67⋊AGL(1,𝔽5)11yescharacteristic
411C4—41no—
412C4—51no—
413C4—51no—
414C4—121no—
415C4—201no—
416C4—302no—
417C4—401no—
418C4—601no—
419C4—602no—
420C4—601no—
421C4ΓL(1,𝔽16)⋊Q811yescharacteristic
422C4#9190911yescharacteristic
423C3#1060404911yescharacteristic, complemented, Sylow
424C2—51no—
425C2—51no—
426C2—51no—
427C2—51no—
428C2#1175174711yescharacteristic
429C2#1175194611yescharacteristic
430C2#1175196511yescharacteristic
subgroupG/H
centerC22#67⋊AGL(1,𝔽5)
commutatorC2×C3×C4×C5C22×C4
FrattiniC2×C4C2×D3×AGL(1,𝔽5)
FittingC3×C5×C4⋊C4C2×C4
socleC22×C3×C5C4×D4
prime radicalC22×C3×D5C2×D4
cosocleC2×C3×C4×D5C23
hypercenterC4⋊C4D3×AGL(1,𝔽5)
hypocenterC3×C5#1919
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
20. #91092C4semidirect
21. C5×#808C4semidirect
22. #90987C4semidirect
23. #90986C4semidirect
127. #808AGL(1,𝔽5)semidirect
339. C3×C5#1919semidirect
403. C5#68788semidirect
423. C3#10604049semidirect

Maximal subgroups

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

subgroupindex
C4⋊C4×ΓL(1,𝔽16)2
#117518612
D5×#8082
C3×#648742
#117465302
#117465292
#117465172
#106040493
#687885