Finite Group Databasebeta

← #14305263

Subgroups — #14305263

Order 1920.

416 conjugacy classes of subgroups.

The subgroup lattice has 364 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C3×#65708C211yesmaximal, characteristic
2#11751822C211yesmaximal, characteristic
3C2×#91162C211yesmaximal, characteristic
4C2×ΓL(1,𝔽16)⋊C8C211yesmaximal, characteristic
5#11747432C211yesmaximal, characteristic
6#11746746C211yesmaximal, characteristic
7#11746524C211yesmaximal, characteristic
8#10604081—31nomaximal, Hall
9C2×C4×ΓL(1,𝔽16)C2211yescharacteristic
10C2×#6758C2211yescharacteristic
11C2×C3×#700C2211yescharacteristic
12C2×C3×D5⋊C8C2211yescharacteristic
13C3×#4038C2211yescharacteristic
14#91358C2211yescharacteristic
15C2×D5×C3⋊C8C2211yescharacteristic
16C3×#4004—21no—
17C3×C4⋊#170—21no—
18#91177—21no—
19Dic3⋊#170—21no—
20#91162—22no—
21ΓL(1,𝔽16)⋊C8—22no—
22#91092C411yescharacteristic, complemented
23C5×#807C411yescharacteristic, complemented
24#90990C411yescharacteristic, complemented
25#90986C411yescharacteristic, complemented
26#70537—51nomaximal, Hall
27C2×#3996—31no—
28C2×C8×AGL(1,𝔽5)—31no—
29#65708—31no—
30#65708D311yescharacteristic
31#65169—31no—
32#64875—31no—
33#64875—31no—
34C22×ΓL(1,𝔽16)—21no—
35C2×C3×C4×D5C2311yescharacteristic
36C2×C3⋊#170—21no—
37C4×ΓL(1,𝔽16)—21no—
38C4×ΓL(1,𝔽16)Q812yes—
39#6758—21no—
40#6758—21no—
41C2×C3×#170—21no—
42C2×C3×#170D411yescharacteristic
43C2×C3×#170D411yescharacteristic
44C3×#700—21no—
45C3×#700—21no—
46C3×D5⋊C8—21no—
47C3×D5⋊C8—21no—
48C3×C5×C4⋊C4C2×C411yescharacteristic
49C4⋊Dic15C2×C411yescharacteristic
50C2×C3⋊#168C2×C411yescharacteristic
51C5×C4⋊Dic3C2×C411yescharacteristic
52C2×C5×C3⋊C8C2×C411yescharacteristic
53C3×#685—21no—
54C3×C4⋊Dic5C2×C411yescharacteristic
55C3×C4×Dic5—21no—
56#6665—21no—
57Dic3×Dic5—21no—
58D5×C3⋊C8D412yes—
59D5×C3⋊C8Q812yes—
60C3×#424—51no—
61C2×#822—51no—
62#5102—51no—
63C2×C8×Dic3—51no—
64#4998—51no—
65#4555—51no—
66#4532—51no—
67C2×C4×AGL(1,𝔽5)—31no—
68C2×#700—31no—
69C2×#700C2×D311yescharacteristic
70C2×D5⋊C8C2×D311yescharacteristic
71C2×C8×D5—31no—
72#4038—31no—
73#4038C2×D311yescharacteristic
74#4004—21no—
75#4004—61no—
76C4⋊#170—21no—
77C4⋊#170—61no—
78#3996—62no—
79C8×AGL(1,𝔽5)—62no—
80C5×#107—31no—
81#3950—31no—
82#3940—31no—
83#3940—31no—
84#1945—151noSylow
85C22×C3×D5#4111yescharacteristic
86C2×ΓL(1,𝔽16)—22no—
87C2×ΓL(1,𝔽16)—22no—
88C2×ΓL(1,𝔽16)—22no—
89C2×C3×C4×C5—21no—
90C2×C3×C4×C5C22×C411yescharacteristic
91C2×Dic15—21no—
92C2×C5×Dic3—21no—
93C2×C3×Dic5—21no—
94C2×C3×Dic5C2×D411yescharacteristic
95C3×C4×D5C2×D411yescharacteristic
96C3×C4×D5C2×Q811yescharacteristic
97C3×C4×D5C2×Q811yescharacteristic
98C3×C4×D5#4111yescharacteristic
99C3⋊#170—41no—
100C3×#170—21no—
101C3×#170—21no—
102C3×#170—41no—
103C3⋊#168C4⋊C412yes—
104C5×C3⋊C8C4⋊C412yes—
105C2×C3×M16—51no—
106C22×C3×C8—51no—
107C3×#117—51no—
108#926—51no—
109C2×C4×Dic3—51no—
110C2×#208—51no—
111C22×C3⋊C8—51no—
112C3×C4⋊C8—101no—
113C3×C4⋊C8—101no—
114#822—102no—
115Dic3⋊C8—101no—
116Dic3⋊C8—101no—
117C8×Dic3—102no—
118#808—51no—
119#808—51no—
120#807—51no—
121#807AGL(1,𝔽5)11yescharacteristic, complemented
122C22×AGL(1,𝔽5)—61no—
123C2×C4×D5C22×D311yescharacteristic
124C2×#170—21no—
125C2×#170—61no—
126C2×#170#6711yescharacteristic
127C2×#170#6711yescharacteristic
128C4×AGL(1,𝔽5)—32no—
129C4×AGL(1,𝔽5)—61no—
130#700—21no—
131#700—21no—
132#700—61no—
133#700—61no—
134D5⋊C8—21no—
135D5⋊C8—21no—
136C2×C5×C8—31no—
137C5×C4⋊C4—31no—
138C5×C4⋊C4C4×D311yescharacteristic
139#685—21no—
140#685—61no—
141C4⋊Dic5—31no—
142C4⋊Dic5C4×D311yescharacteristic
143C4×Dic5—21no—
144C4×Dic5—61no—
145C2×#168—31no—
146C8×D5—32no—
147C8×D5—32no—
148C2×#108—151no—
149#427—151no—
150#424—51no—
151#424—151no—
152C2×C4×C8—151no—
153#339—151no—
154#339—151no—
155C22×C3×C5C4×D411yescharacteristic
156C2×C3×D5—21no—
157C2×C3×D5—21no—
158C2×C3×D5C22⋊Q811yescharacteristic
159C2×C3×D5#12511yescharacteristic
160ΓL(1,𝔽16)—24no—
161C3×C4×C5—41no—
162C3×C4×C5C2×C4⋊C411yescharacteristic
163C3×C4×C5C4×Q811yescharacteristic
164Dic15—41no—
165C3×Dic5—41no—
166C3×Dic5#12111yescharacteristic
167C3×Dic5C4⋊Q811yescharacteristic
168C5×Dic3—41no—
169C22×C3×C4—51no—
170C22×Dic3—101no—
171C3×M16—101no—
172C3×M16—101no—
173C2×C3×C8—51no—
174C2×C3×C8—51no—
175C2×C3×C8—101no—
176C2×C3×C8—101no—
177C2×C3×C8—101no—
178C3×C4⋊C4—51no—
179C3×C4⋊C4C2×AGL(1,𝔽5)11yescharacteristic
180C3×C22⋊C4—101no—
181C3×C42—101no—
182#217—101no—
183C4⋊Dic3—51no—
184C4⋊Dic3C2×AGL(1,𝔽5)11yescharacteristic
185C4×Dic3—52no—
186C4×Dic3—101no—
187C4×Dic3—101no—
188#208—101no—
189#208—101no—
190C2×C3⋊C8—51no—
191C2×C3⋊C8—52no—
192C2×C3⋊C8—52no—
193C2×C3⋊C8—101no—
194C2×C3⋊C8C2×AGL(1,𝔽5)11yescharacteristic
195C22×D5#23511yescharacteristic
196C2×AGL(1,𝔽5)—62no—
197C2×AGL(1,𝔽5)—62no—
198C2×AGL(1,𝔽5)—62no—
199C2×C4×C5—21no—
200C2×C4×C5—61no—
201C2×C4×C5C2×C4×D311yescharacteristic
202C2×Dic5—21no—
203C2×Dic5—61no—
204C2×Dic5C2×#6711yescharacteristic
205C4×D5D3×D411yescharacteristic
206C4×D5#23711yescharacteristic
207C4×D5D3×Q811yescharacteristic
208C4×D5D3×Q811yescharacteristic
209#170—21no—
210#170—21no—
211#170—41no—
212#170—121no—
213C5×C8—32no—
214#168—32no—
215C2×M16—51no—
216C2×M16—151no—
217C22×C8—51no—
218C22×C8—151no—
219#117—51no—
220#117—151no—
221C2×C42—151no—
222#108—302no—
223#107—31no—
224#107—151no—
225#106—151no—
226#106—151no—
227C4⋊C8—101no—
228C4⋊C8—101no—
229C4⋊C8—301no—
230C4⋊C8—301no—
231C4×C8—302no—
232C2×C3×C5#38911yescharacteristic
233C2×C3×C5#44311yescharacteristic
234C2×C3×C5#44311yescharacteristic
235C3×D5—21no—
236C3×D5—21no—
237C23×C3—51no—
238C2×C3×C4—21no—
239C2×C3×C4—51no—
240C2×C3×C4—51no—
241C2×C3×C4—51no—
242C2×C3×C4—51no—
243C2×C3×C4—51no—
244C2×C3×C4—101no—
245C2×C3×C4C22×AGL(1,𝔽5)11yescharacteristic
246C2×Dic3—21no—
247C2×Dic3—102no—
248C2×Dic3—102no—
249C2×Dic3—102no—
250C2×Dic3—101no—
251C3×C8—101no—
252C3×C8—101no—
253C3×C8—201no—
254C3⋊C8—52no—
255C3⋊C8—201no—
256C3⋊C8C4⋊AGL(1,𝔽5)12yes—
257C22×C5C4×#6711yescharacteristic
258C2×D5—21no—
259C2×D5—21no—
260C2×D5#88911yescharacteristic
261C2×D5#89611yescharacteristic
262AGL(1,𝔽5)—64no—
263C4×C5—41no—
264C4×C5—121no—
265C4×C5#88711yescharacteristic
266C4×C5D3×C4⋊C411yescharacteristic
267Dic5—41no—
268Dic5—121no—
269Dic5#93911yescharacteristic
270Dic5#94411yescharacteristic
271C22×C4—51no—
272C22×C4—301no—
273M16—101no—
274M16—101no—
275M16—301no—
276M16—301no—
277C2×C8—31no—
278C2×C8—51no—
279C2×C8—51no—
280C2×C8—101no—
281C2×C8—101no—
282C2×C8—101no—
283C2×C8—152no—
284C2×C8—152no—
285C2×C8—151no—
286C2×C8—301no—
287C4⋊C4—31no—
288C4⋊C4—51no—
289C4⋊C4—151no—
290C4⋊C4D3×AGL(1,𝔽5)11yescharacteristic
291C22⋊C4—101no—
292C22⋊C4—301no—
293C42—101no—
294C42—152no—
295C42—301no—
296C42—301no—
297C3×C5#194511yescharacteristic, complemented, Hall
298C22×C3—51no—
299C22×C3—51no—
300C22×C3—101no—
301C22×C3—101no—
302C22×C3D4×AGL(1,𝔽5)11yescharacteristic
303C3×C4—41no—
304C3×C4—51no—
305C3×C4—51no—
306C3×C4—201no—
307C3×C4C2×C4⋊AGL(1,𝔽5)11yescharacteristic
308C3×C4Q8×AGL(1,𝔽5)11yescharacteristic
309Dic3—41no—
310Dic3—104no—
311Dic3—201no—
312C2×C5#497611yescharacteristic
313C2×C5#514411yescharacteristic
314C2×C5#514411yescharacteristic
315D5—21no—
316D5—21no—
317C23—51no—
318C2×C4—21no—
319C2×C4—51no—
320C2×C4—51no—
321C2×C4—51no—
322C2×C4—51no—
323C2×C4—51no—
324C2×C4—61no—
325C2×C4—101no—
326C2×C4—302no—
327C2×C4—302no—
328C2×C4—302no—
329C2×C4—301no—
330C2×C4C2×D3×AGL(1,𝔽5)11yescharacteristic
331C8—32no—
332C8—101no—
333C8—101no—
334C8—152no—
335C8—201no—
336C8—601no—
337C2×C3—101no—
338C2×C3—101no—
339C2×C3#6571211yescharacteristic
340C2×C3#6573211yescharacteristic
341C2×C3#6574011yescharacteristic
342C5#7053711yescharacteristic, complemented, Sylow
343C22—51no—
344C22—51no—
345C22—101no—
346C22—101no—
347C22AGL(1,𝔽5)×#6711yescharacteristic
348C4—41no—
349C4—51no—
350C4—51no—
351C4—121no—
352C4—201no—
353C4—304no—
354C4—601no—
355C4ΓL(1,𝔽16)⋊Q811yescharacteristic
356C4D3×C4⋊AGL(1,𝔽5)11yescharacteristic
357C3#1060408111yescharacteristic, complemented, Sylow
358C2—101no—
359C2—101no—
360C2#1175183711yescharacteristic
361C2#1175195111yescharacteristic
362C2#1175197711yescharacteristic
subgroupG/H
centerC22AGL(1,𝔽5)×#67
commutatorC2×C3×C4×C5C22×C4
FrattiniC2×C4C2×D3×AGL(1,𝔽5)
FittingC3×C5×C4⋊C4C2×C4
socleC22×C3×C5C4×D4
prime radicalC22×C3×D5#41
cosocleC2×C3×C4×D5C23
hypercenterC4⋊C4D3×AGL(1,𝔽5)
hypocenterC3×C5#1945
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
22. #91092C4semidirect
23. C5×#807C4semidirect
24. #90990C4semidirect
25. #90986C4semidirect
121. #807AGL(1,𝔽5)semidirect
297. C3×C5#1945semidirect
342. C5#70537semidirect
357. C3#10604081semidirect

Maximal subgroups

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

subgroupindex
C3×#657082
#117518222
C2×#911622
C2×ΓL(1,𝔽16)⋊C82
#117474322
#117467462
#117465242
#106040813
#705375