Finite Group Databasebeta

← #14282766

Subgroups — #14282766

Order 1920.

476 conjugacy classes of subgroups; only the 105 normal subgroups are shown.

normal subgroups only

This group has 476 conjugacy classes of subgroups — too many to draw. Only its 105 normal subgroups are shown.

Dotted edges: containments with no intermediate normal subgroup — omitted non-normal subgroups may lie between.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C3×#66014C211yesmaximal, characteristic
2#11752345C211yesmaximal, characteristic, complemented
3#11752337C211yesmaximal, characteristic, complemented
4#11746797C212yesmaximal, complemented
5#11746791C212yesmaximal
6#10604537—31nomaximal, Hall
7C2×C4×ΓL(1,𝔽16)—21no—
8C3×C5×#124C411yescharacteristic, complemented
9C3×#4092C2211yescharacteristic
10C3×Q8×Dic5C411yescharacteristic, complemented
11C3×#4084C2211yescharacteristic
12C3×D4×Dic5C411yescharacteristic, complemented
13C3×#4029—21no—
14C3×#4027—21no—
15C3×#4025—21no—
16C3×#4023—21no—
17C3×C42×D5C2211yescharacteristic
18C3×#4016C411yescharacteristic, complemented
19#91235—21no—
20#91235C2212yes—
21#91233—21no—
22#91232C2212yes—
23C4×C3⋊#170—21no—
24#71754—51nomaximal, Hall
25#66014D311yescharacteristic
26#65686—31no—
27#65678—31no—
28#64926—32no—
29#64920—32no—
30C22×ΓL(1,𝔽16)—22no—
31C2×C3×C5×Q8C2×C411yescharacteristic
32C2×C3×C5×D4C2×C411yescharacteristic
33C2×C3×C22⋊D5—21no—
34C22×C3×Dic5—21no—
35C2×C3×C4×D5D411yescharacteristic
36C2×C3×C4×D5D411yescharacteristic
37C2×C3×C4×D5C2311yescharacteristic
38C2×C3⋊#170—22no—
39C4×ΓL(1,𝔽16)—24no—
40C3×C5×C22⋊C4—21no—
41C3×C5×C22⋊C4—21no—
42C3×C42×C5C2×C411yescharacteristic
43C3×C22⋊Dic5—21no—
44C3×C22⋊Dic5—21no—
45C3×#685—21no—
46C3×C4⋊Dic5—21no—
47C3×C4⋊Dic5C2×C411yescharacteristic
48C3×C4⋊Dic5C2×C411yescharacteristic
49C3×#683—21no—
50C3×#683—21no—
51C3×#683—21no—
52C3×C4×Dic5—21no—
53C3×C4×Dic5C2×C411yescharacteristic
54C3×C4×Dic5D411yescharacteristic
55C3×C4×Dic5D411yescharacteristic
56C3×#529—51no—
57#4932—51no—
58#4925—51no—
59#4538—52no—
60#4537—52no—
61C2×C4×AGL(1,𝔽5)—61no—
62C5×#124Dic311yescharacteristic, complemented
63#4092C2×D311yescharacteristic
64Q8×Dic5Dic311yescharacteristic, complemented
65#4084C2×D311yescharacteristic
66D4×Dic5Dic311yescharacteristic, complemented
67#4029—21no—
68#4027—21no—
69#4025—21no—
70#4023—21no—
71C42×D5C2×D311yescharacteristic
72#4016Dic311yescharacteristic, complemented
73(C2×AGL(1,𝔽5))⋊C4—32no—
74(C2×AGL(1,𝔽5))⋊C4—61no—
75#4004—61no—
76#4003—32no—
77C4×#170—61no—
78#1500—151noSylow
79C23×C3×C5—21no—
80C22×C3×D5C2×D411yescharacteristic
81C2×ΓL(1,𝔽16)—24no—
82C2×ΓL(1,𝔽16)—42no—
83C2×ΓL(1,𝔽16)—42no—
84C3×C5×Q8—41no—
85C3×C5×D4—41no—
86C2×C3×C4×C5—21no—
87C2×C3×C4×C5C22⋊C411yescharacteristic
88C2×C3×C4×C5C22⋊C411yescharacteristic
89C2×C3×C4×C5C22×C411yescharacteristic
90C3×C22⋊D5—42no—
91C2×C3×Dic5—21no—
92C2×C3×Dic5—21no—
93C2×C3×Dic5—42no—
94C2×C3×Dic5C22⋊C411yescharacteristic
95C2×C3×Dic5C22⋊C411yescharacteristic
96C2×C3×Dic5C2×D411yescharacteristic
97C3×C4×D5—22no—
98C3×C4×D5—22no—
99C3×C4×D5—41no—
100C3⋊#170—42no—
101C3×#126—101no—
102C3×#125—51no—
103C3×#124AGL(1,𝔽5)11yescharacteristic, complemented
104C3×#123—101no—
105C3×C22⋊Q8—51no—
106C3×#121—51no—
107C3×C4×Q8—51no—
108C3×C4×D4—51no—
109C3×C4×D4—101no—
110C3×#117—101no—
111C2×C3×C42—51no—
112C2×C4×Dic3—101no—
113C3⋊#95—52no—
114C3⋊#95—101no—
115#830—52no—
116C4⋊(C3⋊C8)—101no—
117C4×C3⋊C8—101no—
118C22×AGL(1,𝔽5)—62no—
119C2×C5×Q8C2×Dic311yescharacteristic
120C2×C5×D4C2×Dic311yescharacteristic
121C2×C22⋊D5—21no—
122C22×Dic5—21no—
123C2×C4×D5#6711yescharacteristic
124C2×C4×D5#6711yescharacteristic
125C2×C4×D5C22×D311yescharacteristic
126C2×#170—62no—
127C4×AGL(1,𝔽5)—64no—
128C5×C22⋊C4—21no—
129C5×C22⋊C4—21no—
130C42×C5C2×Dic311yescharacteristic
131C22⋊Dic5—21no—
132C22⋊Dic5—21no—
133#685—21no—
134C4⋊Dic5—21no—
135C4⋊Dic5C2×Dic311yescharacteristic
136C4⋊Dic5C2×Dic311yescharacteristic
137#683—21no—
138#683—21no—
139#683—21no—
140C4×Dic5—21no—
141C4×Dic5C2×Dic311yescharacteristic
142C4×Dic5#6711yescharacteristic
143C4×Dic5#6711yescharacteristic
144#529—51no—
145#431—151no—
146#376—151no—
147#328—152no—
148#327—152no—
149C22×C3×C5—41no—
150C22×C3×C5—41no—
151C22×C3×C5—41no—
152C22×C3×C5C2×C22⋊C411yescharacteristic
153C2×C3×D5—21no—
154C2×C3×D5—21no—
155C2×C3×D5#13512yes—
156ΓL(1,𝔽16)—44no—
157C3×C4×C5—21no—
158C3×C4×C5—21no—
159C3×C4×C5—21no—
160C3×C4×C5—41no—
161C3×Dic5—21no—
162C3×Dic5—21no—
163C3×Dic5—41no—
164C3×Dic5—41no—
165C3×Dic5#10412yes—
166C2×C3×Q8C2×AGL(1,𝔽5)11yescharacteristic
167C2×C3×D4—101no—
168C2×C3×D4C2×AGL(1,𝔽5)11yescharacteristic
169C22×C3×C4—51no—
170C22×C3×C4—51no—
171C22×C3×C4—51no—
172C22×C3×C4—101no—
173C22×Dic3—102no—
174C3×C4⋊C4—51no—
175C3×C4⋊C4—51no—
176C3×C4⋊C4—101no—
177C3×C4⋊C4—101no—
178C3×C4⋊C4—101no—
179C3×C4⋊C4—101no—
180C3×C22⋊C4—21no—
181C3×C22⋊C4—21no—
182C3×C22⋊C4—101no—
183C3×C22⋊C4—101no—
184C3×C22⋊C4—101no—
185C3×C42—51no—
186C3×C42—51no—
187C3×C42—51no—
188C3×C42—101no—
189C3×C42C2×AGL(1,𝔽5)11yescharacteristic
190C4×Dic3—104no—
191C2×C3⋊C8—102no—
192C23×C5—21no—
193C22×D5C2×#6711yescharacteristic
194C2×AGL(1,𝔽5)—64no—
195C2×AGL(1,𝔽5)—122no—
196C2×AGL(1,𝔽5)—122no—
197C5×Q8—41no—
198C5×D4—41no—
199C2×C4×C5—21no—
200C2×C4×C5#21711yescharacteristic
201C2×C4×C5#21711yescharacteristic
202C2×C4×C5C22×Dic311yescharacteristic
203C22⋊D5—42no—
204C2×Dic5—21no—
205C2×Dic5—21no—
206C2×Dic5—42no—
207C2×Dic5#21711yescharacteristic
208C2×Dic5#21711yescharacteristic
209C2×Dic5C2×#6711yescharacteristic
210C4×D5—22no—
211C4×D5—22no—
212C4×D5—41no—
213#170—122no—
214#126—101no—
215#125—51no—
216#124ΓL(1,𝔽16)11yescharacteristic, complemented
217#123—101no—
218C22⋊Q8—51no—
219#121—51no—
220C4×Q8—51no—
221C4×D4—51no—
222C4×D4—101no—
223#117—101no—
224C2×C42—51no—
225C2×C42—301no—
226C4⋊C8—301no—
227C22⋊C8—152no—
228C4×C8—301no—
229#95—152no—
230#95—301no—
231C2×C3×C5—81no—
232C2×C3×C5#41011yescharacteristic
233C2×C3×C5#41611yescharacteristic
234C2×C3×C5C2×#10411yescharacteristic
235C3×D5—22no—
236C23×C3—21no—
237C23×C3—51no—
238C3×Q8—41no—
239C3×D4—41no—
240C3×D4—202no—
241C2×C3×C4—21no—
242C2×C3×C4—51no—
243C2×C3×C4—51no—
244C2×C3×C4—51no—
245C2×C3×C4—102no—
246C2×C3×C4—102no—
247C2×C3×C4—101no—
248C2×C3×C4—101no—
249C2×C3×C4—201no—
250C2×C3×C4—202no—
251C2×C3×C4C22⋊AGL(1,𝔽5)11yescharacteristic
252C2×C3×C4C22⋊AGL(1,𝔽5)11yescharacteristic
253C2×C3×C4C22×AGL(1,𝔽5)11yescharacteristic
254C2×Dic3—104no—
255C2×Dic3—202no—
256C2×Dic3—202no—
257C3⋊C8—202no—
258C22×C5—41no—
259C22×C5—41no—
260C22×C5—41no—
261C22×C5C2×#21711yescharacteristic
262C2×D5—21no—
263C2×D5—21no—
264C2×D5#95012yes—
265AGL(1,𝔽5)—124no—
266C4×C5—21no—
267C4×C5—21no—
268C4×C5—21no—
269C4×C5—41no—
270Dic5—21no—
271Dic5—21no—
272Dic5—41no—
273Dic5—41no—
274Dic5#83712yes—
275C2×Q8C2×ΓL(1,𝔽16)11yescharacteristic
276C2×D4—101no—
277C2×D4C2×ΓL(1,𝔽16)11yescharacteristic
278C22×C4—51no—
279C22×C4—51no—
280C22×C4—51no—
281C22×C4—101no—
282C22×C4—302no—
283C2×C8—302no—
284C4⋊C4—51no—
285C4⋊C4—51no—
286C4⋊C4—101no—
287C4⋊C4—101no—
288C4⋊C4—101no—
289C4⋊C4—101no—
290C22⋊C4—21no—
291C22⋊C4—21no—
292C22⋊C4—101no—
293C22⋊C4—101no—
294C22⋊C4—101no—
295C42—51no—
296C42—51no—
297C42—51no—
298C42—101no—
299C42—304no—
300C42C2×ΓL(1,𝔽16)11yescharacteristic
301C3×C5#150011yescharacteristic, complemented, Hall
302C22×C3—41no—
303C22×C3—41no—
304C22×C3—41no—
305C22×C3—52no—
306C22×C3—101no—
307C22×C3—101no—
308C22×C3C2×C22⋊AGL(1,𝔽5)11yescharacteristic
309C3×C4—21no—
310C3×C4—21no—
311C3×C4—21no—
312C3×C4—41no—
313C3×C4—52no—
314C3×C4—101no—
315C3×C4—101no—
316C3×C4—201no—
317C3×C4—201no—
318Dic3—204no—
319C2×C5—81no—
320C2×C5#523311yescharacteristic
321C2×C5C2×#83711yescharacteristic
322C2×C5#524011yescharacteristic
323D5—22no—
324C23—21no—
325C23—51no—
326Q8—41no—
327D4—41no—
328D4—202no—
329C2×C4—21no—
330C2×C4—51no—
331C2×C4—51no—
332C2×C4—51no—
333C2×C4—102no—
334C2×C4—102no—
335C2×C4—101no—
336C2×C4—101no—
337C2×C4—201no—
338C2×C4—202no—
339C2×C4—304no—
340C2×C4—602no—
341C2×C4—602no—
342C2×C4#676311yescharacteristic
343C2×C4#676311yescharacteristic
344C2×C4C22×ΓL(1,𝔽16)11yescharacteristic
345C8—602no—
346C2×C3—81no—
347C2×C3—102no—
348C2×C3#6577111yescharacteristic
349C2×C3D5⋊#10411yescharacteristic
350C2×C3#6579411yescharacteristic
351C5#7175411yescharacteristic, complemented, Sylow
352C22—41no—
353C22—41no—
354C22—41no—
355C22—52no—
356C22—101no—
357C22—101no—
358C22C2×#676311yescharacteristic
359C4—21no—
360C4—21no—
361C4—21no—
362C4—41no—
363C4—52no—
364C4—101no—
365C4—101no—
366C4—201no—
367C4—201no—
368C4—604no—
369C3#1060453711yescharacteristic, complemented, Sylow
370C2—81no—
371C2—102no—
372C2#1175243011yescharacteristic
373C2D5⋊#83711yescharacteristic
374C2#1175245311yescharacteristic
subgroupG/H
centerC22C2×#6763
commutatorC2×C3×C4×C5C22×C4
FrattiniC2×C4C22×ΓL(1,𝔽16)
FittingC3×C5×#124C4
socleC22×C3×C5C2×C22⋊C4
prime radicalC3×#4084C22
cosocleC2×C3×C4×D5C23
hypercenter#124ΓL(1,𝔽16)
hypocenterC3×C5#1500
LCSG ▷ C2×C3×C4×C5 ▷ C22×C3×C5 ▷ C3×C5
UCSC1 ◁ C22 ◁ C42 ◁ #124

Complements

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

NH ≅ G/Ntype
2. #11752345C2semidirect
3. #11752337C2semidirect
4. #11746797C2semidirect
8. C3×C5×#124C4semidirect
10. C3×Q8×Dic5C4semidirect
12. C3×D4×Dic5C4semidirect
18. C3×#4016C4semidirect
62. C5×#124Dic3semidirect
64. Q8×Dic5Dic3semidirect
66. D4×Dic5Dic3semidirect
72. #4016Dic3semidirect
103. C3×#124AGL(1,𝔽5)semidirect
216. #124ΓL(1,𝔽16)semidirect
301. C3×C5#1500semidirect
351. C5#71754semidirect
369. C3#10604537semidirect

Maximal subgroups

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

subgroupindex
C3×#660142
#117523452
#117523372
#117467972
#117467912
#106045373
#717545