Finite Group Databasebeta

← #14301388

Subgroups — #14301388

Order 1920.

664 conjugacy classes of subgroups; only the 117 normal subgroups are shown.

normal subgroups only

This group has 664 conjugacy classes of subgroups — too many to draw. Only its 117 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
1#11756989C211yesmaximal, characteristic, complemented
2C3×#66059C211yesmaximal, characteristic
3#11752444C211yesmaximal, characteristic, complemented
4#11752443C212yesmaximal, complemented
5#11746733C212yesmaximal, complemented
6#10604676—31nomaximal, Hall
7C2×C3×#713C2211yescharacteristic
8C2×C3×#713C2211yescharacteristic
9#91992—24no—
10#91989C2212yes—
11D5⋊#208—22no—
12D5⋊#208C2212yes—
13C3×C5×C4⋊Q8C411yescharacteristic, complemented
14C3×#4093C411yescharacteristic, complemented
15C3×#4093C411yescharacteristic, complemented
16C3×#4041—22no—
17C3×#4039—22no—
18C3×#4020C411yescharacteristic, complemented
19C3×C42×D5C2211yescharacteristic
20#91239—24no—
21#91228—22no—
22#71817—51nomaximal, Hall
23#66260—31no—
24#66059D311yescharacteristic
25#65785—31no—
26#65784—32no—
27#64862—32no—
28C2×C3×C5×Q8C2×C411yescharacteristic
29C2×C3×C5×Q8C2×C411yescharacteristic
30C3×#713—22no—
31C3×#713—22no—
32C3×#713—22no—
33C3×#713—22no—
34C2×C3×D20—22no—
35C2×C3×D20—21no—
36C2×C3×D20C2×C411yescharacteristic
37C2×C3×D20C2×C411yescharacteristic
38C2×C3×D20D412yes—
39C2×C3×C4×D5—21no—
40C2×C3×C4×D5—21no—
41C2×C3×C4×D5D412yes—
42C2×C3×C4×D5C2311yescharacteristic
43#6763—22no—
44#6762—22no—
45C4⋊ΓL(1,𝔽16)—22no—
46C4×ΓL(1,𝔽16)—22no—
47#6758—24no—
48#6758D414yescomplemented
49D5⋊(C3⋊C8)—24no—
50D5⋊(C3⋊C8)—22no—
51C3×C5×C4⋊C4—22no—
52C3×C42×C5C2×C411yescharacteristic
53C3×#685—22no—
54C3×#685—22no—
55C3×C4×Dic5—21no—
56C3×C4×Dic5C2×C411yescharacteristic
57C3×#532—51no—
58#5819—51no—
59#5240—51no—
60#5236—52no—
61#4523—52no—
62C2×#713C2×D311yescharacteristic
63C2×#713C2×D311yescharacteristic
64#4133—64no—
65#4130—32no—
66D5⋊M16—32no—
67D5⋊M16—62no—
68C5×C4⋊Q8Dic311yescharacteristic, complemented
69#4093Dic311yescharacteristic, complemented
70#4093Dic311yescharacteristic, complemented
71#4041—22no—
72#4039—22no—
73#4020Dic311yescharacteristic, complemented
74C42×D5C2×D311yescharacteristic
75#4010—64no—
76#3999—62no—
77#1884—151noSylow
78C22×C3×D5—21no—
79C22×C3×D5C22⋊C412yes—
80C22×C3×D5C2×D411yescharacteristic
81C2×ΓL(1,𝔽16)—42no—
82C3×C5×Q8—22no—
83C3×C5×Q8—22no—
84C2×C3×C4×C5—21no—
85C2×C3×C4×C5—21no—
86C2×C3×C4×C5C22⋊C412yes—
87C2×C3×C4×C5C22×C411yescharacteristic
88C2×C3×Dic5—21no—
89C2×C3×Dic5C2×D411yescharacteristic
90C3×D20—24no—
91C3×D20—22no—
92C3×D20—22no—
93C3×D20—44no—
94C3×D20—42no—
95C3×C4×D5—22no—
96C3×C4×D5—22no—
97C3×C4×D5—42no—
98C3×C4×D5—41no—
99C3×C4×D5—41no—
100C3×C4×D5C2×D412yes—
101C3×C4×D5#4112yes—
102C3⋊#170—24no—
103C2×C3×#41—51no—
104C2×C3×#41—51no—
105C3×C4⋊Q8AGL(1,𝔽5)11yescharacteristic, complemented
106C3×#127—51no—
107C3×#124—51no—
108C3×#124—51no—
109C3×#121—102no—
110C3×C4×D4—102no—
111C2×C3×C42—51no—
112#948—104no—
113#926—52no—
114C2×#208—52no—
115C2×#208—102no—
116#837—104no—
117#834—102no—
118C2×C5×Q8C2×Dic311yescharacteristic
119C2×C5×Q8C2×Dic311yescharacteristic
120#713—22no—
121#713—22no—
122#713—22no—
123#713—22no—
124C2×D20—22no—
125C2×D20—21no—
126C2×D20C2×Dic311yescharacteristic
127C2×D20C2×Dic311yescharacteristic
128C2×D20#6712yes—
129C2×C4×D5—21no—
130C2×C4×D5—21no—
131C2×C4×D5#6712yes—
132C2×C4×D5C22×D311yescharacteristic
133C22⋊AGL(1,𝔽5)—62no—
134#704—62no—
135C4⋊AGL(1,𝔽5)—62no—
136C4×AGL(1,𝔽5)—62no—
137#700—34no—
138#700—64no—
139D5⋊C8—64no—
140D5⋊C8—62no—
141C5×C4⋊C4—22no—
142C42×C5C2×Dic311yescharacteristic
143#685—22no—
144#685—22no—
145C4×Dic5—21no—
146C4×Dic5C2×Dic311yescharacteristic
147#568—151no—
148#532—51no—
149#421—152no—
150#410—151no—
151#339—152no—
152C22×C3×C5C2×C22⋊C411yescharacteristic
153C2×C3×D5—22no—
154C2×C3×D5—41no—
155C2×C3×D5—42no—
156C2×C3×D5—42no—
157C2×C3×D5—41no—
158C2×C3×D5#12112yes—
159ΓL(1,𝔽16)—82no—
160C3×C4×C5—22no—
161C3×C4×C5—22no—
162C3×C4×C5—41no—
163C3×C4×C5C4×D412yes—
164C3×Dic5—41no—
165C3×Dic5#12011yescharacteristic
166C3×Dic5#12311yescharacteristic
167C3×#41—102no—
168C3×#41—102no—
169C3×#41—102no—
170C3×#41—102no—
171C2×C3×Q8C2×AGL(1,𝔽5)11yescharacteristic
172C2×C3×Q8C2×AGL(1,𝔽5)11yescharacteristic
173C2×C3×D4—52no—
174C2×C3×D4—51no—
175C2×C3×D4—51no—
176C2×C3×D4—102no—
177C2×C3×D4—101no—
178C22×C3×C4—51no—
179C22×C3×C4—52no—
180C22×C3×C4—101no—
181C22×C3×C4—101no—
182C3×C4⋊C4—22no—
183C3×C22⋊C4—102no—
184C3×C22⋊C4—102no—
185C3×C42—51no—
186C3×C42—101no—
187C3×C42C2×AGL(1,𝔽5)11yescharacteristic
188#217—102no—
189C4⋊Dic3—102no—
190C4×Dic3—102no—
191#208—54no—
192#208—102no—
193#208—104no—
194C2×C3⋊C8—102no—
195C2×C3⋊C8—104no—
196C22×D5—21no—
197C22×D5#21712yes—
198C22×D5C2×#6711yescharacteristic
199C2×AGL(1,𝔽5)—122no—
200C5×Q8—22no—
201C5×Q8—22no—
202C2×C4×C5—21no—
203C2×C4×C5—21no—
204C2×C4×C5#21712yes—
205C2×C4×C5C22×Dic311yescharacteristic
206C2×Dic5—21no—
207C2×Dic5C2×#6711yescharacteristic
208D20—24no—
209D20—22no—
210D20—22no—
211D20—44no—
212D20—42no—
213C4×D5—22no—
214C4×D5—22no—
215C4×D5—42no—
216C4×D5—41no—
217C4×D5—41no—
218C4×D5D3×D412yes—
219C4×D5#23712yes—
220#170—64no—
221C2×#41—51no—
222C2×#41—51no—
223#131—304no—
224C2×M16—152no—
225C2×M16—302no—
226C4⋊Q8ΓL(1,𝔽16)11yescharacteristic, complemented
227#127—51no—
228#124—51no—
229#124—51no—
230#121—102no—
231C4×D4—102no—
232#117—152no—
233C2×C42—51no—
234#104—304no—
235#99—302no—
236C2×C3×C5—21no—
237C2×C3×C5#38611yescharacteristic
238C3×D5—22no—
239C3×D5—42no—
240C3×D5—81no—
241C23×C3—51no—
242C23×C3—52no—
243C23×C3—101no—
244C3×Q8—22no—
245C3×Q8—22no—
246C3×D4—104no—
247C3×D4—102no—
248C3×D4—102no—
249C3×D4—204no—
250C3×D4—202no—
251C2×C3×C4—21no—
252C2×C3×C4—21no—
253C2×C3×C4—51no—
254C2×C3×C4—52no—
255C2×C3×C4—52no—
256C2×C3×C4—102no—
257C2×C3×C4—102no—
258C2×C3×C4—101no—
259C2×C3×C4—202no—
260C2×C3×C4—201no—
261C2×C3×C4—201no—
262C2×C3×C4C22⋊AGL(1,𝔽5)12yes—
263C2×C3×C4C22×AGL(1,𝔽5)11yescharacteristic
264C2×Dic3—202no—
265C3⋊C8—104no—
266C22×C5C2×#21711yescharacteristic
267C2×D5—22no—
268C2×D5—41no—
269C2×D5—42no—
270C2×D5—42no—
271C2×D5—41no—
272C2×D5#93912yes—
273AGL(1,𝔽5)—242no—
274C4×C5—22no—
275C4×C5—22no—
276C4×C5—41no—
277C4×C5D4×Dic312yes—
278Dic5—41no—
279Dic5#93511yescharacteristic
280Dic5#93711yescharacteristic
281#41—102no—
282#41—102no—
283#41—102no—
284#41—102no—
285C2×Q8C2×ΓL(1,𝔽16)11yescharacteristic
286C2×Q8C2×ΓL(1,𝔽16)11yescharacteristic
287C2×D4—52no—
288C2×D4—51no—
289C2×D4—51no—
290C2×D4—102no—
291C2×D4—101no—
292C22×C4—51no—
293C22×C4—52no—
294C22×C4—101no—
295C22×C4—101no—
296M16—154no—
297M16—302no—
298M16—304no—
299C2×C8—302no—
300C2×C8—304no—
301C4⋊C4—22no—
302C4⋊C4—302no—
303C22⋊C4—102no—
304C22⋊C4—102no—
305C22⋊C4—302no—
306C42—51no—
307C42—101no—
308C42—302no—
309C42C2×ΓL(1,𝔽16)11yescharacteristic
310C3×C5#188411yescharacteristic, complemented, Hall
311C22×C3—52no—
312C22×C3—102no—
313C22×C3—201no—
314C22×C3—202no—
315C22×C3—202no—
316C22×C3—201no—
317C22×C3C2×C22⋊AGL(1,𝔽5)11yescharacteristic
318C3×C4—22no—
319C3×C4—22no—
320C3×C4—41no—
321C3×C4—51no—
322C3×C4—51no—
323C3×C4—201no—
324C3×C4D4×AGL(1,𝔽5)12yes—
325Dic3—402no—
326C2×C5—21no—
327C2×C5#522011yescharacteristic
328D5—22no—
329D5—42no—
330D5—81no—
331C23—51no—
332C23—52no—
333C23—101no—
334Q8—22no—
335Q8—22no—
336D4—104no—
337D4—102no—
338D4—102no—
339D4—204no—
340D4—202no—
341C2×C4—21no—
342C2×C4—21no—
343C2×C4—51no—
344C2×C4—52no—
345C2×C4—52no—
346C2×C4—102no—
347C2×C4—102no—
348C2×C4—101no—
349C2×C4—202no—
350C2×C4—201no—
351C2×C4—201no—
352C2×C4—602no—
353C2×C4#676312yes—
354C2×C4C22×ΓL(1,𝔽16)11yescharacteristic
355C8—304no—
356C2×C3—21no—
357C2×C3—102no—
358C2×C3—202no—
359C2×C3—401no—
360C2×C3#6577911yescharacteristic
361C5#7181711yescharacteristic, complemented, Sylow
362C22—52no—
363C22—102no—
364C22—201no—
365C22—202no—
366C22—202no—
367C22—201no—
368C22C2×#676311yescharacteristic
369C4—22no—
370C4—22no—
371C4—41no—
372C4—51no—
373C4—51no—
374C4—201no—
375C4—1202no—
376C4D4×ΓL(1,𝔽16)12yes—
377C3#1060467611yescharacteristic, complemented, Sylow
378C2—21no—
379C2—102no—
380C2—202no—
381C2—401no—
382C2#1175243811yescharacteristic
subgroupG/H
centerC2#11752438
commutatorC2×C3×C4×C5C22×C4
FrattiniC2×C4C22×ΓL(1,𝔽16)
FittingC3×C5×C4⋊Q8C4
socleC2×C3×C5#386
prime radicalC3×#66059C2
cosocleC2×C3×C4×D5C23
hypercenterC4⋊Q8ΓL(1,𝔽16)
hypocenterC3×C5#1884
LCSG ▷ C2×C3×C4×C5 ▷ C2×C3×C5 ▷ C3×C5
UCSC1 ◁ C2 ◁ C2×C4 ◁ C4⋊Q8

Complements

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

NH ≅ G/Ntype
1. #11756989C2semidirect
3. #11752444C2semidirect
4. #11752443C2semidirect
5. #11746733C2semidirect
13. C3×C5×C4⋊Q8C4semidirect
14. C3×#4093C4semidirect
15. C3×#4093C4semidirect
18. C3×#4020C4semidirect
48. #6758D4semidirect
68. C5×C4⋊Q8Dic3semidirect
69. #4093Dic3semidirect
70. #4093Dic3semidirect
73. #4020Dic3semidirect
105. C3×C4⋊Q8AGL(1,𝔽5)semidirect
226. C4⋊Q8ΓL(1,𝔽16)semidirect
310. C3×C5#1884semidirect
361. C5#71817semidirect
377. C3#10604676semidirect

Maximal subgroups

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

subgroupindex
#117569892
C3×#660592
#117524442
#117524432
#117467332
#106046763
#718175