Finite Group Databasebeta

← #14304785

Subgroups — #14304785

Order 1920.

460 conjugacy classes of subgroups.

The subgroup lattice has 424 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#11752446C211yesmaximal, characteristic
2C2×#91239C211yesmaximal, characteristic
3C2×C8×ΓL(1,𝔽16)C211yesmaximal, characteristic, complemented
4#11752367C211yesmaximal, characteristic, complemented
5C3×#65093C211yesmaximal, characteristic
6#11746767C211yesmaximal, characteristic, complemented
7#11746746C211yesmaximal, characteristic, complemented
8#10603987—31nomaximal, Hall
9C2×C3×#713C2211yescharacteristic
10C2×C4×ΓL(1,𝔽16)C2211yescharacteristic
11C2×#6758C2211yescharacteristic
12C2×#6758C2211yescharacteristic
13C2×D5⋊(C3⋊C8)C2211yescharacteristic
14C2×C3×C8×D5C2211yescharacteristic
15C3×#4038C2211yescharacteristic
16#91239—22no—
17#91233—21no—
18#91232—21no—
19#91232—21no—
20C4⋊(C3⋊#170)—21no—
21C8×ΓL(1,𝔽16)—22no—
22C3×C5×#103C411yescharacteristic, complemented
23C3×#3967C411yescharacteristic, complemented
24C3×#3953C411yescharacteristic, complemented
25C3×#3941C411yescharacteristic, complemented
26#70287—51nomaximal, Hall
27#65787—31no—
28C2×#4010—31no—
29C2×C8×AGL(1,𝔽5)—31no—
30#65708—31no—
31#65093D311yescharacteristic
32#64896—31no—
33#64875—31no—
34C22×ΓL(1,𝔽16)—21no—
35C2×C3×C5×Q8C2×C411yescharacteristic
36C3×#713—21no—
37C3×#713—21no—
38C3×#713—21no—
39C3×#713—21no—
40C2×C3×D20—21no—
41C2×C3×D20C2×C411yescharacteristic
42C2×C3×C4×D5—21no—
43C2×C3×C4×D5C2311yescharacteristic
44C2×C3⋊#170—21no—
45C2×C3⋊#170—21no—
46C2×C3⋊#170D411yescharacteristic
47C2×C3⋊#170D411yescharacteristic
48C4×ΓL(1,𝔽16)—21no—
49C4×ΓL(1,𝔽16)D412yes—
50#6758—21no—
51#6758—21no—
52#6758—21no—
53#6758—21no—
54D5⋊(C3⋊C8)—21no—
55D5⋊(C3⋊C8)—21no—
56C2×C3×C5×C8C2×C411yescharacteristic
57C3×C5×C4⋊C4C2×C411yescharacteristic
58C3×#685—21no—
59C3×C4⋊Dic5C2×C411yescharacteristic
60C3×C4×Dic5—21no—
61C2×C3×#168C2×C411yescharacteristic
62C3×C8×D5—21no—
63C3×C8×D5—21no—
64C3×#416—51no—
65C2×#837—51no—
66#5234—51no—
67C2×C8×Dic3—51no—
68#4999—51no—
69#4552—51no—
70#4532—51no—
71C2×#713C2×D311yescharacteristic
72C2×C4×AGL(1,𝔽5)—31no—
73C2×#700—31no—
74C2×#700—31no—
75C2×D5⋊C8—31no—
76C2×C8×D5C2×D311yescharacteristic
77#4038C2×D311yescharacteristic
78#4010—62no—
79#4004—61no—
80#4003—61no—
81#4003—61no—
82C4⋊#170—61no—
83C8×AGL(1,𝔽5)—62no—
84C5×#103Dic311yescharacteristic, complemented
85#3967Dic311yescharacteristic, complemented
86#3953Dic311yescharacteristic, complemented
87#3941Dic311yescharacteristic, complemented
88#1937—151noSylow
89C22×C3×D5—21no—
90C22×C3×D5#4111yescharacteristic
91C2×ΓL(1,𝔽16)—22no—
92C2×ΓL(1,𝔽16)—22no—
93C2×ΓL(1,𝔽16)—22no—
94C3×C5×Q8—21no—
95C3×C5×Q8—21no—
96C2×C3×C4×C5—21no—
97C2×C3×C4×C5—21no—
98C2×C3×C4×C5C22×C411yescharacteristic
99C2×C3×Dic5—21no—
100C2×C3×Dic5C2×D411yescharacteristic
101C3×D20—21no—
102C3×D20—21no—
103C3×D20—41no—
104C3×D20—41no—
105C3×C4×D5—41no—
106C3×C4×D5—41no—
107C3×C4×D5C2×D411yescharacteristic
108C3×C4×D5#4111yescharacteristic
109C3×C4×D5#4111yescharacteristic
110C3×C4×D5#4111yescharacteristic
111C3⋊#170—21no—
112C3⋊#170—21no—
113C3⋊#170—41no—
114C3⋊#170—41no—
115C3×C5×C8—21no—
116C3×#168—21no—
117C2×C3×#41—51no—
118C22×C3×C8—51no—
119C3×#117—51no—
120C2×C4×Dic3—51no—
121C2×#208—51no—
122C2×#208—51no—
123C22×C3⋊C8—51no—
124C3×#103—51no—
125C3×#103AGL(1,𝔽5)11yescharacteristic, complemented
126C3×#102—51no—
127C3×#102—51no—
128#837—102no—
129#830—101no—
130#830—101no—
131C8×Dic3—102no—
132C4⋊(C3⋊C8)—101no—
133C4⋊(C3⋊C8)—101no—
134C22×AGL(1,𝔽5)—61no—
135C2×C5×Q8C2×Dic311yescharacteristic
136#713—21no—
137#713—21no—
138#713—21no—
139#713—21no—
140C2×D20—21no—
141C2×D20C2×Dic311yescharacteristic
142C2×C4×D5—21no—
143C2×C4×D5C22×D311yescharacteristic
144C2×#170—31no—
145C2×#170—31no—
146C2×#170—61no—
147C2×#170—61no—
148C4×AGL(1,𝔽5)—32no—
149C4×AGL(1,𝔽5)—61no—
150#700—61no—
151#700—61no—
152#700—61no—
153#700—61no—
154D5⋊C8—61no—
155D5⋊C8—61no—
156C2×C5×C8C2×Dic311yescharacteristic
157C5×C4⋊C4C2×Dic311yescharacteristic
158#685—21no—
159C4⋊Dic5C2×Dic311yescharacteristic
160C4×Dic5—21no—
161C2×#168C2×Dic311yescharacteristic
162C8×D5—21no—
163C8×D5—21no—
164#424—151no—
165C2×#104—151no—
166#416—51no—
167#408—151no—
168C2×C4×C8—151no—
169#341—151no—
170#339—151no—
171C22×C3×C5C4×D411yescharacteristic
172C2×C3×D5—21no—
173C2×C3×D5—21no—
174C2×C3×D5—41no—
175C2×C3×D5—41no—
176C2×C3×D5—41no—
177C2×C3×D5#12311yescharacteristic
178C2×C3×D5#12611yescharacteristic
179ΓL(1,𝔽16)—24no—
180C3×C4×C5—41no—
181C3×C4×C5—41no—
182C3×C4×C5#11711yescharacteristic
183C3×C4×C5C4×D411yescharacteristic
184C3×Dic5—41no—
185C3×Dic5#12111yescharacteristic
186C3×Dic5#12411yescharacteristic
187C3×#41—101no—
188C3×#41—101no—
189C3×#41—101no—
190C3×#41—101no—
191C2×C3×Q8C2×AGL(1,𝔽5)11yescharacteristic
192C2×C3×D4—51no—
193C2×C3×D4—101no—
194C22×C3×C4—51no—
195C22×C3×C4—101no—
196C22×Dic3—101no—
197C2×C3×C8—51no—
198C2×C3×C8—101no—
199C2×C3×C8—101no—
200C2×C3×C8C2×AGL(1,𝔽5)11yescharacteristic
201C3×C4⋊C4—51no—
202C3×C4⋊C4C2×AGL(1,𝔽5)11yescharacteristic
203C3×C22⋊C4—101no—
204C3×C42—101no—
205C4×Dic3—52no—
206C4×Dic3—101no—
207#208—101no—
208#208—101no—
209#208—101no—
210#208—101no—
211C2×C3⋊C8—51no—
212C2×C3⋊C8—51no—
213C2×C3⋊C8—101no—
214C2×C3⋊C8—101no—
215C2×C3⋊C8—101no—
216C2×C3⋊C8—101no—
217C22×D5—21no—
218C22×D5#23711yescharacteristic
219C2×AGL(1,𝔽5)—62no—
220C2×AGL(1,𝔽5)—62no—
221C2×AGL(1,𝔽5)—62no—
222C5×Q8—21no—
223C5×Q8—21no—
224C2×C4×C5—21no—
225C2×C4×C5—21no—
226C2×C4×C5C22×Dic311yescharacteristic
227C2×Dic5—21no—
228C2×Dic5D3×D411yescharacteristic
229D20—21no—
230D20—21no—
231D20—41no—
232D20—41no—
233C4×D5—41no—
234C4×D5—41no—
235C4×D5#23511yescharacteristic
236C4×D5#23511yescharacteristic
237C4×D5D3×D411yescharacteristic
238C4×D5#23711yescharacteristic
239#170—61no—
240#170—61no—
241#170—121no—
242#170—121no—
243C5×C8—21no—
244#168—21no—
245C2×#41—51no—
246C2×M16—151no—
247C2×M16—151no—
248C22×C8—51no—
249C22×C8—151no—
250#117—51no—
251C2×C42—151no—
252C4⋊C8—301no—
253C4⋊C8—301no—
254#104—302no—
255#103—51no—
256#103ΓL(1,𝔽16)11yescharacteristic, complemented
257#102—51no—
258#102—51no—
259C22⋊C8—301no—
260C22⋊C8—301no—
261C4×C8—302no—
262C2×C3×C5#38811yescharacteristic
263C2×C3×C5#44311yescharacteristic
264C2×C3×C5#44311yescharacteristic
265C3×D5—21no—
266C3×D5—21no—
267C3×D5—81no—
268C23×C3—51no—
269C23×C3—101no—
270C3×Q8—21no—
271C3×Q8—21no—
272C3×D4—101no—
273C3×D4—101no—
274C3×D4—201no—
275C3×D4—201no—
276C2×C3×C4—21no—
277C2×C3×C4—21no—
278C2×C3×C4—51no—
279C2×C3×C4—51no—
280C2×C3×C4—51no—
281C2×C3×C4—51no—
282C2×C3×C4—51no—
283C2×C3×C4—101no—
284C2×C3×C4—201no—
285C2×C3×C4—201no—
286C2×C3×C4C22×AGL(1,𝔽5)11yescharacteristic
287C2×Dic3—102no—
288C2×Dic3—102no—
289C2×Dic3—102no—
290C3×C8—21no—
291C3×C8—101no—
292C3⋊C8—101no—
293C3⋊C8—101no—
294C3⋊C8—201no—
295C3⋊C8—201no—
296C22×C5D4×Dic311yescharacteristic
297C2×D5—21no—
298C2×D5—21no—
299C2×D5—41no—
300C2×D5—41no—
301C2×D5—41no—
302C2×D5#87911yescharacteristic
303C2×D5#88311yescharacteristic
304AGL(1,𝔽5)—64no—
305C4×C5—41no—
306C4×C5—41no—
307C4×C5#92611yescharacteristic
308C4×C5D4×Dic311yescharacteristic
309Dic5—41no—
310Dic5#88411yescharacteristic
311Dic5#88511yescharacteristic
312#41—101no—
313#41—101no—
314#41—101no—
315#41—101no—
316C2×Q8C2×ΓL(1,𝔽16)11yescharacteristic
317C2×D4—51no—
318C2×D4—101no—
319C22×C4—51no—
320C22×C4—101no—
321C22×C4—301no—
322M16—301no—
323M16—301no—
324M16—301no—
325M16—301no—
326C2×C8—51no—
327C2×C8—101no—
328C2×C8—101no—
329C2×C8—151no—
330C2×C8—151no—
331C2×C8—301no—
332C2×C8—301no—
333C2×C8—301no—
334C2×C8—301no—
335C2×C8C2×ΓL(1,𝔽16)11yescharacteristic
336C4⋊C4—51no—
337C4⋊C4C2×ΓL(1,𝔽16)11yescharacteristic
338C22⋊C4—101no—
339C42—101no—
340C42—152no—
341C42—301no—
342C3×C5#193711yescharacteristic, complemented, Hall
343C22×C3—51no—
344C22×C3—51no—
345C22×C3—101no—
346C22×C3—101no—
347C22×C3—201no—
348C22×C3—201no—
349C22×C3—201no—
350C22×C3D4×AGL(1,𝔽5)11yescharacteristic
351C3×C4—41no—
352C3×C4—41no—
353C3×C4—51no—
354C3×C4—51no—
355C3×C4—201no—
356C3×C4#413011yescharacteristic
357C3×C4D4×AGL(1,𝔽5)11yescharacteristic
358Dic3—104no—
359C2×C5#495111yescharacteristic
360C2×C5#519611yescharacteristic
361C2×C5#519611yescharacteristic
362D5—21no—
363D5—21no—
364D5—81no—
365C23—51no—
366C23—101no—
367Q8—21no—
368Q8—21no—
369D4—101no—
370D4—101no—
371D4—201no—
372D4—201no—
373C2×C4—21no—
374C2×C4—21no—
375C2×C4—51no—
376C2×C4—51no—
377C2×C4—51no—
378C2×C4—51no—
379C2×C4—51no—
380C2×C4—101no—
381C2×C4—201no—
382C2×C4—201no—
383C2×C4—302no—
384C2×C4—302no—
385C2×C4—302no—
386C2×C4C22×ΓL(1,𝔽16)11yescharacteristic
387C8—21no—
388C8—101no—
389C8—301no—
390C8—301no—
391C8—601no—
392C8—601no—
393C2×C3—101no—
394C2×C3—101no—
395C2×C3—401no—
396C2×C3#6570011yescharacteristic
397C2×C3#6573611yescharacteristic
398C2×C3#6574011yescharacteristic
399C5#7028711yescharacteristic, complemented, Sylow
400C22—51no—
401C22—51no—
402C22—101no—
403C22—101no—
404C22—201no—
405C22—201no—
406C22—201no—
407C22D4×ΓL(1,𝔽16)11yescharacteristic
408C4—41no—
409C4—41no—
410C4—51no—
411C4—51no—
412C4—201no—
413C4—304no—
414C4#9198911yescharacteristic
415C4D4×ΓL(1,𝔽16)11yescharacteristic
416C3#1060398711yescharacteristic, complemented, Sylow
417C2—101no—
418C2—101no—
419C2—401no—
420C2#1175235911yescharacteristic
421C2#1175239511yescharacteristic
422C2#1175239911yescharacteristic
subgroupG/H
centerC22D4×ΓL(1,𝔽16)
commutatorC2×C3×C4×C5C22×C4
FrattiniC2×C4C22×ΓL(1,𝔽16)
FittingC3×C5×#103C4
socleC22×C3×C5C4×D4
prime radicalC2×C3×#713C22
cosocleC2×C3×C4×D5C23
hypercenter#103ΓL(1,𝔽16)
hypocenterC3×C5#1937
LCSG ▷ C2×C3×C4×C5 ▷ C2×C3×C5 ▷ C3×C5
UCSC1 ◁ C22 ◁ C2×C4 ◁ #103

Complements

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

NH ≅ G/Ntype
3. C2×C8×ΓL(1,𝔽16)C2semidirect
4. #11752367C2semidirect
6. #11746767C2semidirect
7. #11746746C2semidirect
22. C3×C5×#103C4semidirect
23. C3×#3967C4semidirect
24. C3×#3953C4semidirect
25. C3×#3941C4semidirect
84. C5×#103Dic3semidirect
85. #3967Dic3semidirect
86. #3953Dic3semidirect
87. #3941Dic3semidirect
125. C3×#103AGL(1,𝔽5)semidirect
256. #103ΓL(1,𝔽16)semidirect
342. C3×C5#1937semidirect
399. C5#70287semidirect
416. C3#10603987semidirect

Maximal subgroups

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

subgroupindex
#117524462
C2×#912392
C2×C8×ΓL(1,𝔽16)2
#117523672
C3×#650932
#117467672
#117467462
#106039873
#702875