Finite Group Databasebeta

← #14306001

Subgroups — #14306001

Order 1920.

720 conjugacy classes of subgroups; only the 133 normal subgroups are shown.

normal subgroups only

This group has 720 conjugacy classes of subgroups — too many to draw. Only its 133 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×#66002C211yesmaximal, characteristic
2C2×#91239C212yesmaximal, complemented
3C2×#91237C212yesmaximal, complemented
4C42×ΓL(1,𝔽16)C211yesmaximal, characteristic, complemented
5#11752341C211yesmaximal, characteristic, complemented
6#10604559—31nomaximal, Hall
7C2×C3×#713C2211yescharacteristic
8C2×C3×#711C2211yescharacteristic
9C2×C4×ΓL(1,𝔽16)—22no—
10C2×C4×ΓL(1,𝔽16)C2212yes—
11C2×#6758C2212yes—
12C3×C5×#124C411yescharacteristic, complemented
13C3×#4090C411yescharacteristic, complemented
14C3×#4086C411yescharacteristic, complemented
15C3×#4030—22no—
16C3×#4027—22no—
17C3×#4021C411yescharacteristic, complemented
18C3×C42×D5C2211yescharacteristic
19#91239—24no—
20#91237—24no—
21#91233—22no—
22#71815—51nomaximal, Hall
23#66002D311yescharacteristic
24C2×#4010—32no—
25C2×#4008—32no—
26C42×AGL(1,𝔽5)—31no—
27#65682—31no—
28C22×ΓL(1,𝔽16)—22no—
29C2×C3×C5×Q8C2×C411yescharacteristic
30C2×C3×C5×D4C2×C411yescharacteristic
31C2×C3×C22⋊D5—21no—
32C2×C3×C22⋊D5—22no—
33C22×C3×Dic5—21no—
34C3×#713—22no—
35C3×#713—22no—
36C3×#711—22no—
37C3×#711—22no—
38C2×C3×D20—21no—
39C2×C3×D20C2×C411yescharacteristic
40C2×C3×C4×D5—21no—
41C2×C3×C4×D5D412yes—
42C2×C3×C4×D5C2311yescharacteristic
43C2×C3×Dic10C2×C411yescharacteristic
44C2×C3⋊#170—22no—
45C4×ΓL(1,𝔽16)—28no—
46C4×ΓL(1,𝔽16)—22no—
47C4×ΓL(1,𝔽16)D414yescomplemented
48#6758—22no—
49#6758—22no—
50C3×C5×C22⋊C4—22no—
51C3×C42×C5C2×C411yescharacteristic
52C3×#685—22no—
53C3×#683—22no—
54C3×C4×Dic5C2×C411yescharacteristic
55C3×C4×Dic5D412yes—
56C3×#532—51no—
57C2×#837—52no—
58C2×#837—52no—
59C42×Dic3—51no—
60#4924—51no—
61C2×#713C2×D311yescharacteristic
62C2×#711C2×D311yescharacteristic
63C2×C4×AGL(1,𝔽5)—32no—
64C2×C4×AGL(1,𝔽5)—62no—
65C2×#700—32no—
66C5×#124Dic311yescharacteristic, complemented
67#4090Dic311yescharacteristic, complemented
68#4086Dic311yescharacteristic, complemented
69#4030—22no—
70#4027—22no—
71#4021Dic311yescharacteristic, complemented
72C42×D5C2×D311yescharacteristic
73#4010—64no—
74#4008—64no—
75#4004—62no—
76#1962—151noSylow
77C23×C3×C5—21no—
78C22×C3×D5—21no—
79C22×C3×D5C2×D411yescharacteristic
80C2×ΓL(1,𝔽16)—28no—
81C2×ΓL(1,𝔽16)—24no—
82C3×C5×Q8—22no—
83C3×C5×D4—22no—
84C2×C3×C4×C5—21no—
85C2×C3×C4×C5C22⋊C412yes—
86C2×C3×C4×C5C22×C411yescharacteristic
87C3×C22⋊D5—42no—
88C3×C22⋊D5—44no—
89C2×C3×Dic5—21no—
90C2×C3×Dic5—41no—
91C2×C3×Dic5—41no—
92C2×C3×Dic5C22⋊C412yes—
93C2×C3×Dic5C2×D411yescharacteristic
94C3×D20—22no—
95C3×D20—42no—
96C3×C4×D5—24no—
97C3×C4×D5—41no—
98C3×C4×D5—41no—
99C3×C4×D5C2×D412yes—
100C3×C4×D5#4112yes—
101C3×Dic10—22no—
102C3⋊#170—42no—
103C2×C3×#41—51no—
104C2×C3×#41—51no—
105C3×C4⋊Q8—51no—
106C3×#127—51no—
107C3×#124—51no—
108C3×#124AGL(1,𝔽5)11yescharacteristic, complemented
109C3×#121—102no—
110C3×C4×D4—102no—
111C2×C3×C42—51no—
112C2×C4×Dic3—52no—
113C2×C4×Dic3—102no—
114C2×#208—52no—
115#837—104no—
116#837—104no—
117C4⋊(C3⋊C8)—102no—
118C22×AGL(1,𝔽5)—62no—
119C2×C5×Q8C2×Dic311yescharacteristic
120C2×C5×D4C2×Dic311yescharacteristic
121C2×C22⋊D5—21no—
122C2×C22⋊D5—22no—
123C22×Dic5—21no—
124#713—22no—
125#713—22no—
126#711—22no—
127#711—22no—
128C2×D20—21no—
129C2×D20C2×Dic311yescharacteristic
130C2×C4×D5—21no—
131C2×C4×D5#6712yes—
132C2×C4×D5C22×D311yescharacteristic
133C2×Dic10C2×Dic311yescharacteristic
134C2×#170—62no—
135C4×AGL(1,𝔽5)—34no—
136C4×AGL(1,𝔽5)—68no—
137C4×AGL(1,𝔽5)—62no—
138#700—62no—
139#700—62no—
140C5×C22⋊C4—22no—
141C42×C5C2×Dic311yescharacteristic
142#685—22no—
143#683—22no—
144C4×Dic5C2×Dic311yescharacteristic
145C4×Dic5#6712yes—
146#532—51no—
147#423—151no—
148C2×#104—152no—
149C2×#104—152no—
150C43—151no—
151C22×C3×C5—41no—
152C22×C3×C5—42no—
153C22×C3×C5C2×C22⋊C411yescharacteristic
154C2×C3×D5—22no—
155C2×C3×D5—41no—
156C2×C3×D5—42no—
157C2×C3×D5#12112yes—
158ΓL(1,𝔽16)—28no—
159C3×C4×C5—22no—
160C3×C4×C5—41no—
161C3×C4×C5C4×D412yes—
162C3×Dic5—41no—
163C3×Dic5#10414yescomplemented
164C3×Dic5#12411yescharacteristic
165C3×Dic5#12711yescharacteristic
166C3×#41—102no—
167C3×#41—102no—
168C3×#41—102no—
169C3×#41—102no—
170C2×C3×Q8—51no—
171C2×C3×Q8C2×AGL(1,𝔽5)11yescharacteristic
172C2×C3×D4—51no—
173C2×C3×D4—101no—
174C2×C3×D4—102no—
175C2×C3×D4—101no—
176C2×C3×D4C2×AGL(1,𝔽5)11yescharacteristic
177C22×C3×C4—51no—
178C22×C3×C4—52no—
179C22×C3×C4—101no—
180C22×C3×C4—101no—
181C22×Dic3—102no—
182C3×C4⋊C4—102no—
183C3×C22⋊C4—22no—
184C3×C22⋊C4—102no—
185C3×C42—52no—
186C3×C42—51no—
187C3×C42C2×AGL(1,𝔽5)11yescharacteristic
188C4×Dic3—54no—
189C4×Dic3—102no—
190C4×Dic3—108no—
191#208—102no—
192#208—102no—
193C2×C3⋊C8—102no—
194C23×C5—21no—
195C22×D5—21no—
196C22×D5C2×#6711yescharacteristic
197C2×AGL(1,𝔽5)—68no—
198C2×AGL(1,𝔽5)—64no—
199C5×Q8—22no—
200C5×D4—22no—
201C2×C4×C5—21no—
202C2×C4×C5#21712yes—
203C2×C4×C5C22×Dic311yescharacteristic
204C22⋊D5—42no—
205C22⋊D5—44no—
206C2×Dic5—21no—
207C2×Dic5—41no—
208C2×Dic5—41no—
209C2×Dic5#21712yes—
210C2×Dic5C2×#6711yescharacteristic
211D20—22no—
212D20—42no—
213C4×D5—24no—
214C4×D5—41no—
215C4×D5—41no—
216C4×D5D3×D412yes—
217C4×D5#23712yes—
218Dic10—22no—
219#170—122no—
220C2×#41—51no—
221C2×#41—51no—
222C2×M16—152no—
223C4⋊Q8—51no—
224#127—51no—
225#124—51no—
226#124ΓL(1,𝔽16)11yescharacteristic, complemented
227#121—102no—
228C4×D4—102no—
229C2×C42—51no—
230C2×C42—152no—
231C2×C42—302no—
232C4⋊C8—302no—
233#104—304no—
234#104—304no—
235C2×C3×C5—81no—
236C2×C3×C5#39011yescharacteristic
237C2×C3×C5C2×#10412yes—
238C3×D5—22no—
239C3×D5—81no—
240C23×C3—21no—
241C23×C3—51no—
242C23×C3—101no—
243C3×Q8—22no—
244C3×Q8—102no—
245C3×D4—22no—
246C3×D4—102no—
247C3×D4—202no—
248C3×D4—204no—
249C3×D4—202no—
250C2×C3×C4—21no—
251C2×C3×C4—52no—
252C2×C3×C4—51no—
253C2×C3×C4—52no—
254C2×C3×C4—52no—
255C2×C3×C4—104no—
256C2×C3×C4—101no—
257C2×C3×C4—201no—
258C2×C3×C4—201no—
259C2×C3×C4—201no—
260C2×C3×C4—201no—
261C2×C3×C4C22⋊AGL(1,𝔽5)12yes—
262C2×C3×C4C22×AGL(1,𝔽5)11yescharacteristic
263C2×Dic3—104no—
264C2×Dic3—108no—
265C3⋊C8—202no—
266C22×C5—41no—
267C22×C5—42no—
268C22×C5C2×#21711yescharacteristic
269C2×D5—22no—
270C2×D5—41no—
271C2×D5—42no—
272C2×D5#93812yes—
273AGL(1,𝔽5)—68no—
274C4×C5—22no—
275C4×C5—41no—
276C4×C5D4×Dic312yes—
277Dic5—41no—
278Dic5#83714yescomplemented
279Dic5#93611yescharacteristic
280Dic5#94011yescharacteristic
281#41—102no—
282#41—102no—
283#41—102no—
284#41—102no—
285C2×Q8—51no—
286C2×Q8C2×ΓL(1,𝔽16)11yescharacteristic
287C2×D4—51no—
288C2×D4—101no—
289C2×D4—102no—
290C2×D4—101no—
291C2×D4C2×ΓL(1,𝔽16)11yescharacteristic
292C22×C4—51no—
293C22×C4—52no—
294C22×C4—101no—
295C22×C4—101no—
296C22×C4—302no—
297M16—302no—
298M16—302no—
299C2×C8—302no—
300C4⋊C4—102no—
301C22⋊C4—22no—
302C22⋊C4—102no—
303C42—52no—
304C42—51no—
305C42—154no—
306C42—302no—
307C42—308no—
308C42C2×ΓL(1,𝔽16)11yescharacteristic
309C3×C5#196211yescharacteristic, complemented, Hall
310C22×C3—42no—
311C22×C3—41no—
312C22×C3—52no—
313C22×C3—102no—
314C22×C3—202no—
315C22×C3—201no—
316C22×C3C2×C22⋊AGL(1,𝔽5)11yescharacteristic
317C3×C4—22no—
318C3×C4—41no—
319C3×C4—54no—
320C3×C4—51no—
321C3×C4—51no—
322C3×C4—201no—
323C3×C4D4×AGL(1,𝔽5)12yes—
324Dic3—108no—
325C2×C5—81no—
326C2×C5#522111yescharacteristic
327C2×C5C2×#83712yes—
328D5—22no—
329D5—81no—
330C23—21no—
331C23—51no—
332C23—101no—
333Q8—22no—
334Q8—102no—
335D4—22no—
336D4—102no—
337D4—202no—
338D4—204no—
339D4—202no—
340C2×C4—21no—
341C2×C4—52no—
342C2×C4—51no—
343C2×C4—52no—
344C2×C4—52no—
345C2×C4—104no—
346C2×C4—101no—
347C2×C4—201no—
348C2×C4—201no—
349C2×C4—201no—
350C2×C4—201no—
351C2×C4—304no—
352C2×C4—308no—
353C2×C4#676312yes—
354C2×C4C22×ΓL(1,𝔽16)11yescharacteristic
355C8—602no—
356C2×C3—81no—
357C2×C3—102no—
358C2×C3—401no—
359C2×C3#6578011yescharacteristic
360C2×C3D5⋊#10412yes—
361C5#7181511yescharacteristic, complemented, Sylow
362C22—42no—
363C22—41no—
364C22—52no—
365C22—102no—
366C22—202no—
367C22—201no—
368C22C2×#676311yescharacteristic
369C4—22no—
370C4—41no—
371C4—54no—
372C4—51no—
373C4—51no—
374C4—201no—
375C4—308no—
376C4D4×ΓL(1,𝔽16)12yes—
377C3#1060455911yescharacteristic, complemented, Sylow
378C2—81no—
379C2—102no—
380C2—401no—
381C2#1175243911yescharacteristic
382C2D5⋊#83712yes—
subgroupG/H
centerC22C2×#6763
commutatorC2×C3×C4×C5C22×C4
FrattiniC2×C4C22×ΓL(1,𝔽16)
FittingC3×C5×#124C4
socleC22×C3×C5C2×C22⋊C4
prime radicalC3×#66002C2
cosocleC2×C3×C4×D5C23
hypercenter#124ΓL(1,𝔽16)
hypocenterC3×C5#1962
LCSG ▷ C2×C3×C4×C5 ▷ C2×C3×C5 ▷ C3×C5
UCSC1 ◁ C22 ◁ C42 ◁ #124

Complements

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

NH ≅ G/Ntype
2. C2×#91239C2semidirect
3. C2×#91237C2semidirect
4. C42×ΓL(1,𝔽16)C2semidirect
5. #11752341C2semidirect
12. C3×C5×#124C4semidirect
13. C3×#4090C4semidirect
14. C3×#4086C4semidirect
17. C3×#4021C4semidirect
47. C4×ΓL(1,𝔽16)D4semidirect
66. C5×#124Dic3semidirect
67. #4090Dic3semidirect
68. #4086Dic3semidirect
71. #4021Dic3semidirect
108. C3×#124AGL(1,𝔽5)semidirect
163. C3×Dic5#104semidirect
226. #124ΓL(1,𝔽16)semidirect
278. Dic5#837semidirect
309. C3×C5#1962semidirect
361. C5#71815semidirect
377. C3#10604559semidirect

Maximal subgroups

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

subgroupindex
C3×#660022
C2×#912392
C2×#912372
C42×ΓL(1,𝔽16)2
#117523412
#106045593
#718155