Finite Group Databasebeta

← #14303543

Subgroups — #14303543

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
1#11752371C211yesmaximal, characteristic
2C4⋊C4×ΓL(1,𝔽16)C211yesmaximal, characteristic
3C3×D5×#107C211yesmaximal, characteristic
4#11746764C211yesmaximal, characteristic
5#11746763C211yesmaximal, characteristic
6#11746745C211yesmaximal, characteristic
7#11746745C211yesmaximal, characteristic
8#10604074—31nomaximal, Hall
9C2×C4⋊ΓL(1,𝔽16)—21no—
10C2×C4⋊ΓL(1,𝔽16)C2211yescharacteristic
11C2×C4⋊ΓL(1,𝔽16)C2211yescharacteristic
12C2×C4×ΓL(1,𝔽16)—21no—
13C2×C4×ΓL(1,𝔽16)C2211yescharacteristic
14C2×D5⋊(C3⋊C8)C2211yescharacteristic
15C2×C3×C8×D5C2211yescharacteristic
16C3×D5×C4⋊C4C2211yescharacteristic
17C3×D5×C4⋊C4C2211yescharacteristic
18#91235—21no—
19#91235—21no—
20C3×C5×#107C411yescharacteristic, complemented
21C3×#3950C411yescharacteristic, complemented
22C3×#3939C411yescharacteristic, complemented
23C3×#3939C411yescharacteristic, complemented
24#70610—51nomaximal, Hall
25#65712—31no—
26C4⋊C4×AGL(1,𝔽5)—31no—
27D5×#107D311yescharacteristic
28#64893—31no—
29(C2×AGL(1,𝔽5))⋊C8—31no—
30#64874—31no—
31#64874—31no—
32C22×ΓL(1,𝔽16)—21no—
33C22×ΓL(1,𝔽16)—21no—
34C22×ΓL(1,𝔽16)D411yescharacteristic
35C22×ΓL(1,𝔽16)D411yescharacteristic
36C2×C3×C4×D5—21no—
37C2×C3×C4×D5—21no—
38C2×C3×C4×D5C2311yescharacteristic
39C2×C3⋊#170—21no—
40C4⋊ΓL(1,𝔽16)—21no—
41C4⋊ΓL(1,𝔽16)—21no—
42C4⋊ΓL(1,𝔽16)—21no—
43C4⋊ΓL(1,𝔽16)—42no—
44C4⋊ΓL(1,𝔽16)Q812yes—
45C4×ΓL(1,𝔽16)—21no—
46C4×ΓL(1,𝔽16)—21no—
47C4×ΓL(1,𝔽16)—42no—
48D5⋊(C3⋊C8)—21no—
49D5⋊(C3⋊C8)—21no—
50C2×C3×C5×C8C2×C411yescharacteristic
51C3×C5×C4⋊C4C2×C411yescharacteristic
52C3×C5×C4⋊C4C2×C411yescharacteristic
53C3×C4⋊Dic5C2×C411yescharacteristic
54C3×C4⋊Dic5C2×C411yescharacteristic
55C3×#683—21no—
56C3×#683—21no—
57C2×C3×#168C2×C411yescharacteristic
58C3×C8×D5—21no—
59C3×C8×D5—21no—
60C2×C3×#107—51no—
61#4984—51no—
62Dic3×C4⋊C4—51no—
63#4551—51no—
64#4550—51no—
65#4529—51no—
66#4529—51no—
67C2×C4⋊AGL(1,𝔽5)—31no—
68C2×C4⋊AGL(1,𝔽5)—31no—
69C2×C4⋊AGL(1,𝔽5)—61no—
70C2×C4×AGL(1,𝔽5)—31no—
71C2×C4×AGL(1,𝔽5)—61no—
72C2×D5⋊C8—31no—
73C2×C8×D5C2×D311yescharacteristic
74D5×C4⋊C4C2×D311yescharacteristic
75D5×C4⋊C4C2×D311yescharacteristic
76(C2×AGL(1,𝔽5))⋊C4—61no—
77(C2×AGL(1,𝔽5))⋊C4—61no—
78C5×#107Dic311yescharacteristic, complemented
79#3950Dic311yescharacteristic, complemented
80#3939Dic311yescharacteristic, complemented
81#3939Dic311yescharacteristic, complemented
82#1921—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)—41no—
93C2×ΓL(1,𝔽16)—41no—
94C2×ΓL(1,𝔽16)—41no—
95C2×ΓL(1,𝔽16)Q1612yes—
96C2×C3×C4×C5—21no—
97C2×C3×C4×C5—21no—
98C2×C3×C4×C5C22×C411yescharacteristic
99C2×C3×Dic5—21no—
100C2×C3×Dic5—21no—
101C2×C3×Dic5#4111yescharacteristic
102C3×C4×D5—41no—
103C3×C4×D5—41no—
104C3×C4×D5—41no—
105C3×C4×D5—41no—
106C3×C4×D5C2×Q811yescharacteristic
107C3×C4×D5#4111yescharacteristic
108C3×C4×D5#4111yescharacteristic
109C3×C4×D5#4111yescharacteristic
110C3⋊#170—41no—
111C3×C5×C8—21no—
112C3×#168—21no—
113C22×C3×C8—51no—
114C2×C3×C4⋊C4—51no—
115C2×C3×C4⋊C4—51no—
116C2×C4⋊Dic3—51no—
117C2×C4⋊Dic3—51no—
118C2×C4⋊Dic3—101no—
119C2×C4×Dic3—51no—
120C2×C4×Dic3—101no—
121C22×C3⋊C8—51no—
122C3×#107—51no—
123C3×#107—51no—
124C3×#107—51no—
125C3×#107AGL(1,𝔽5)11yescharacteristic, complemented
126C3⋊#95—101no—
127C3⋊#95—101no—
128C22×AGL(1,𝔽5)—31no—
129C22×AGL(1,𝔽5)—31no—
130C22×AGL(1,𝔽5)—61no—
131C22×AGL(1,𝔽5)—61no—
132C2×C4×D5—21no—
133C2×C4×D5—21no—
134C2×C4×D5C22×D311yescharacteristic
135C2×#170—61no—
136C4⋊AGL(1,𝔽5)—32no—
137C4⋊AGL(1,𝔽5)—61no—
138C4⋊AGL(1,𝔽5)—61no—
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—61no—
145D5⋊C8—61no—
146C2×C5×C8C2×Dic311yescharacteristic
147C5×C4⋊C4C2×Dic311yescharacteristic
148C5×C4⋊C4C2×Dic311yescharacteristic
149C4⋊Dic5C2×Dic311yescharacteristic
150C4⋊Dic5C2×Dic311yescharacteristic
151#683—21no—
152#683—21no—
153C2×#168C2×Dic311yescharacteristic
154C8×D5—21no—
155C8×D5—21no—
156C2×#107—51no—
157#389—151no—
158C4×C4⋊C4—151no—
159#340—151no—
160#340—151no—
161#340—151no—
162#336—151no—
163C22×C3×C5C4×D411yescharacteristic
164C2×C3×D5C22⋊Q811yescharacteristic
165C2×C3×D5#12311yescharacteristic
166C2×C3×D5C2×Q1611yescharacteristic
167C2×C3×D5#13511yescharacteristic
168C2×C3×D5AGL(1,ℤ/8)11yescharacteristic
169C2×C3×D5AGL(1,ℤ/8)11yescharacteristic
170ΓL(1,𝔽16)—22no—
171ΓL(1,𝔽16)—42no—
172ΓL(1,𝔽16)—41no—
173ΓL(1,𝔽16)—81no—
174C3×C4×C5—41no—
175C3×C4×C5—41no—
176C3×C4×C5#11711yescharacteristic
177C3×C4×C5C4×Q811yescharacteristic
178C3×Dic5—41no—
179C3×Dic5—41no—
180C3×Dic5#12511yescharacteristic
181C3×Dic5#12611yescharacteristic
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—51no—
190C2×C3×C8—101no—
191C2×C3×C8—101no—
192C2×C3×C8C2×AGL(1,𝔽5)11yescharacteristic
193C3×C4⋊C4—51no—
194C3×C4⋊C4—51no—
195C3×C4⋊C4—101no—
196C3×C4⋊C4—101no—
197C3×C4⋊C4C2×AGL(1,𝔽5)11yescharacteristic
198C3×C4⋊C4C2×AGL(1,𝔽5)11yescharacteristic
199C4⋊Dic3—52no—
200C4⋊Dic3—101no—
201C4⋊Dic3—101no—
202C4⋊Dic3—101no—
203C4⋊Dic3—202no—
204C4×Dic3—101no—
205C4×Dic3—101no—
206C4×Dic3—202no—
207C2×C3⋊C8—101no—
208C2×C3⋊C8—101no—
209C2×C3⋊C8—101no—
210C22×D5D3×D411yescharacteristic
211C2×AGL(1,𝔽5)—32no—
212C2×AGL(1,𝔽5)—61no—
213C2×AGL(1,𝔽5)—61no—
214C2×AGL(1,𝔽5)—61no—
215C2×AGL(1,𝔽5)—62no—
216C2×AGL(1,𝔽5)—61no—
217C2×AGL(1,𝔽5)—61no—
218C2×AGL(1,𝔽5)—121no—
219C2×AGL(1,𝔽5)—121no—
220C2×AGL(1,𝔽5)—121no—
221C2×AGL(1,𝔽5)—121no—
222C2×AGL(1,𝔽5)—121no—
223C2×C4×C5—21no—
224C2×C4×C5—21no—
225C2×C4×C5C22×Dic311yescharacteristic
226C2×Dic5—21no—
227C2×Dic5—21no—
228C2×Dic5#23711yescharacteristic
229C4×D5—41no—
230C4×D5—41no—
231C4×D5—41no—
232C4×D5—41no—
233C4×D5#23511yescharacteristic
234C4×D5#23511yescharacteristic
235C4×D5D3×Q811yescharacteristic
236C4×D5#23911yescharacteristic
237#170—121no—
238C5×C8—21no—
239#168—21no—
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#107—51no—
250#107—51no—
251#107—51no—
252#107ΓL(1,𝔽16)11yescharacteristic, complemented
253#95—301no—
254#95—301no—
255C2×C3×C5#38711yescharacteristic
256C2×C3×C5C4×Q1611yescharacteristic
257C2×C3×C5#44211yescharacteristic
258C3×D5#47711yescharacteristic
259C3×D5#47811yescharacteristic
260C3×D5#48211yescharacteristic
261C3×D5#48411yescharacteristic
262C23×C3—51no—
263C2×C3×C4—21no—
264C2×C3×C4—21no—
265C2×C3×C4—51no—
266C2×C3×C4—51no—
267C2×C3×C4—51no—
268C2×C3×C4—51no—
269C2×C3×C4—51no—
270C2×C3×C4—101no—
271C2×C3×C4—101no—
272C2×C3×C4—201no—
273C2×C3×C4—201no—
274C2×C3×C4—201no—
275C2×C3×C4—201no—
276C2×C3×C4C22×AGL(1,𝔽5)11yescharacteristic
277C2×Dic3—52no—
278C2×Dic3—101no—
279C2×Dic3—101no—
280C2×Dic3—101no—
281C2×Dic3—102no—
282C2×Dic3—101no—
283C2×Dic3—101no—
284C2×Dic3—201no—
285C2×Dic3—201no—
286C2×Dic3—201no—
287C2×Dic3—201no—
288C2×Dic3—201no—
289C3×C8—21no—
290C3×C8—101no—
291C3⋊C8—201no—
292C22×C5D4×Dic311yescharacteristic
293C2×D5#89411yescharacteristic
294C2×D5#89611yescharacteristic
295C2×D5#91111yescharacteristic
296C2×D5#91111yescharacteristic
297C2×D5D3×Q1611yescharacteristic
298C2×D5#91911yescharacteristic
299AGL(1,𝔽5)—62no—
300AGL(1,𝔽5)—121no—
301AGL(1,𝔽5)—122no—
302AGL(1,𝔽5)—241no—
303C4×C5—41no—
304C4×C5—41no—
305C4×C5#92611yescharacteristic
306C4×C5Q8×Dic311yescharacteristic
307Dic5—41no—
308Dic5—41no—
309Dic5#88911yescharacteristic
310Dic5#89811yescharacteristic
311C22×C4—51no—
312C22×C4—101no—
313C22×C4—101no—
314C22×C4—151no—
315C22×C4—151no—
316C22×C4—301no—
317C22×C4—301no—
318C2×C8—51no—
319C2×C8—101no—
320C2×C8—101no—
321C2×C8—301no—
322C2×C8—301no—
323C2×C8—301no—
324C2×C8C2×ΓL(1,𝔽16)11yescharacteristic
325C4⋊C4—51no—
326C4⋊C4—51no—
327C4⋊C4—101no—
328C4⋊C4—101no—
329C4⋊C4—152no—
330C4⋊C4—301no—
331C4⋊C4—301no—
332C4⋊C4—301no—
333C4⋊C4—602no—
334C4⋊C4C2×ΓL(1,𝔽16)11yescharacteristic
335C4⋊C4C2×ΓL(1,𝔽16)11yescharacteristic
336C42—301no—
337C42—301no—
338C42—602no—
339C3×C5#192111yescharacteristic, 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—41no—
349C3×C4—51no—
350C3×C4—51no—
351C3×C4—201no—
352C3×C4—201no—
353C3×C4#413011yescharacteristic
354C3×C4Q8×AGL(1,𝔽5)11yescharacteristic
355Dic3—102no—
356Dic3—201no—
357Dic3—202no—
358Dic3—401no—
359C2×C5#498011yescharacteristic
360C2×C5#515211yescharacteristic
361C2×C5Dic3×Q1611yescharacteristic
362D5#487511yescharacteristic
363D5#488411yescharacteristic
364D5#488511yescharacteristic
365D5#489111yescharacteristic
366C23—51no—
367C2×C4—21no—
368C2×C4—21no—
369C2×C4—51no—
370C2×C4—51no—
371C2×C4—51no—
372C2×C4—51no—
373C2×C4—51no—
374C2×C4—101no—
375C2×C4—101no—
376C2×C4—152no—
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—601no—
388C2×C4—601no—
389C2×C4—601no—
390C2×C4—601no—
391C2×C4—601no—
392C2×C4C22×ΓL(1,𝔽16)11yescharacteristic
393C8—21no—
394C8—101no—
395C8—601no—
396C2×C3—51no—
397C2×C3—51no—
398C2×C3—51no—
399C2×C3—51no—
400C2×C3#6571111yescharacteristic
401C2×C3#6573111yescharacteristic
402C2×C3Q16×AGL(1,𝔽5)11yescharacteristic
403C5#7061011yescharacteristic, complemented, Sylow
404C22—51no—
405C22—51no—
406C22—51no—
407C22—51no—
408C22—51no—
409C22—51no—
410C22D4×ΓL(1,𝔽16)11yescharacteristic
411C4—41no—
412C4—41no—
413C4—51no—
414C4—51no—
415C4—201no—
416C4—201no—
417C4—302no—
418C4—601no—
419C4—602no—
420C4—1201no—
421C4#9198911yescharacteristic
422C4Q8×ΓL(1,𝔽16)11yescharacteristic
423C3#1060407411yescharacteristic, complemented, Sylow
424C2—51no—
425C2—51no—
426C2—51no—
427C2—51no—
428C2#1175237011yescharacteristic
429C2#1175239011yescharacteristic
430C2Q16×ΓL(1,𝔽16)11yescharacteristic
subgroupG/H
centerC22D4×ΓL(1,𝔽16)
commutatorC2×C3×C4×C5C22×C4
FrattiniC2×C4C22×ΓL(1,𝔽16)
FittingC3×C5×#107C4
socleC22×C3×C5C4×D4
prime radicalC22×C3×D5C2×D4
cosocleC2×C3×C4×D5C23
hypercenter#107ΓL(1,𝔽16)
hypocenterC3×C5#1921
LCSG ▷ C2×C3×C4×C5 ▷ C2×C3×C5 ▷ C3×C5
UCSC1 ◁ C22 ◁ C2×C4 ◁ #107

Complements

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

NH ≅ G/Ntype
20. C3×C5×#107C4semidirect
21. C3×#3950C4semidirect
22. C3×#3939C4semidirect
23. C3×#3939C4semidirect
78. C5×#107Dic3semidirect
79. #3950Dic3semidirect
80. #3939Dic3semidirect
81. #3939Dic3semidirect
125. C3×#107AGL(1,𝔽5)semidirect
252. #107ΓL(1,𝔽16)semidirect
339. C3×C5#1921semidirect
403. C5#70610semidirect
423. C3#10604074semidirect

Maximal subgroups

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

subgroupindex
#117523712
C4⋊C4×ΓL(1,𝔽16)2
C3×D5×#1072
#117467642
#117467632
#117467452
#106040743
#706105