Finite Group Databasebeta

← #14301445

Subgroups — #14301445

Order 1920.

720 conjugacy classes of subgroups; only the 121 normal subgroups are shown.

normal subgroups only

This group has 720 conjugacy classes of subgroups — too many to draw. Only its 121 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#11756987C211yesmaximal, characteristic
2C3×D5×#124C211yesmaximal, characteristic
3#11752441C212yesmaximal, complemented
4D5⋊#833C211yesmaximal, characteristic
5#11746733C212yesmaximal, complemented
6#10604562—31nomaximal, Hall
7C2×C3×Q8×D5C2211yescharacteristic
8C2×C3×D4×D5C2211yescharacteristic
9C2×#6763—22no—
10D4×ΓL(1,𝔽16)—24no—
11#91989C2212yes—
12D5⋊#208C2212yes—
13C3×C5×#124C411yescharacteristic, complemented
14C3×#4093C411yescharacteristic, complemented
15C3×#4082C411yescharacteristic, complemented
16C3×#4031—22no—
17C3×D5×C22⋊C4—22no—
18C3×#4021C411yescharacteristic, complemented
19C3×C42×D5C2211yescharacteristic
20#91242—22no—
21#91238—24no—
22#71723—51nomaximal, Hall
23#66258—31no—
24D5×#124D311yescharacteristic
25#65782—32no—
26D5⋊#100—31no—
27#64862—32no—
28C23×C3×D5D412yes—
29C22×ΓL(1,𝔽16)—24no—
30C2×C3×C5×Q8C2×C411yescharacteristic, complemented
31C2×C3×C5×D4C2×C411yescharacteristic
32C2×C3×C22⋊D5D412yes—
33C3×Q8×D5—22no—
34C3×Q8×D5—22no—
35C3×D4×D5—22no—
36C3×D4×D5—22no—
37C2×C3×D20C2×C411yescharacteristic
38C2×C3×C4×D5—21no—
39C2×C3×C4×D5—21no—
40C2×C3×C4×D5C2311yescharacteristic
41C2×C3×Dic10—21no—
42C2×C3×Dic10C2×C411yescharacteristic, complemented
43#6763—24no—
44#6763—22no—
45#6762—22no—
46C4⋊ΓL(1,𝔽16)D414yes—
47C4×ΓL(1,𝔽16)—22no—
48#6758—22no—
49D5⋊(C3⋊C8)—22no—
50C3×C5×C22⋊C4—22no—
51C3×C42×C5C2×C411yescharacteristic, complemented
52C3×C22⋊Dic5—22no—
53C3×#685—22no—
54C3×#685—22no—
55C3×C4×Dic5—21no—
56C3×C4×Dic5C2×C411yescharacteristic, complemented
57C2×C3×#124—51no—
58#5798—51no—
59#5225—52no—
60C2×#833—51no—
61#4523—52no—
62C2×Q8×D5C2×D311yescharacteristic
63C2×D4×D5C2×D311yescharacteristic
64C2×C22⋊AGL(1,𝔽5)—62no—
65D4×AGL(1,𝔽5)—64no—
66#4130—32no—
67D5⋊M16—32no—
68C5×#124Dic311yescharacteristic, complemented
69#4093Dic311yescharacteristic, complemented
70#4082Dic311yescharacteristic, complemented
71#4031—22no—
72D5×C22⋊C4—22no—
73#4021Dic311yescharacteristic, complemented
74C42×D5C2×D311yescharacteristic
75#4013—62no—
76#4009—64no—
77#1885—151noSylow
78C23×C3×C5C22⋊C412yes—
79C22×C3×D5—22no—
80C22×C3×D5—22no—
81C22×C3×D5—44no—
82C22×C3×D5C22⋊C412yes—
83C22×C3×D5C2×D411yescharacteristic
84C2×ΓL(1,𝔽16)—24no—
85C2×ΓL(1,𝔽16)—84no—
86C3×C5×Q8—22no—
87C3×C5×D4—22no—
88C2×C3×C4×C5—21no—
89C2×C3×C4×C5—21no—
90C2×C3×C4×C5C22×C411yescharacteristic
91C3×C22⋊D5—24no—
92C2×C3×Dic5—21no—
93C2×C3×Dic5—21no—
94C2×C3×Dic5C2×D411yescharacteristic
95C3×D20—22no—
96C3×C4×D5—42no—
97C3×C4×D5—41no—
98C3×C4×D5—41no—
99C3×C4×D5C2×D412yes—
100C3×C4×D5#4112yes—
101C3×Dic10—22no—
102C3×Dic10—42no—
103C3⋊#170—42no—
104C22×C3×Q8—51no—
105C22×C3×D4—51no—
106C3×#124—51no—
107C3×#124—51no—
108C3×#124—51no—
109C3×#124—102no—
110C3×#124AGL(1,𝔽5)11yescharacteristic, complemented
111C2×C3×C22⋊C4—102no—
112C2×C3×C42—51no—
113C2×#217—102no—
114D4×Dic3—104no—
115#926—52no—
116C2×#208—52no—
117#835—104no—
118#833—102no—
119C23×D5#6712yes—
120C22×AGL(1,𝔽5)—64no—
121C2×C5×Q8C2×Dic311yescharacteristic, complemented
122C2×C5×D4C2×Dic311yescharacteristic
123C2×C22⋊D5#6712yes—
124Q8×D5—22no—
125Q8×D5—22no—
126D4×D5—22no—
127D4×D5—22no—
128C2×D20C2×Dic311yescharacteristic
129C2×C4×D5—21no—
130C2×C4×D5—21no—
131C2×C4×D5C22×D311yescharacteristic
132C2×Dic10—21no—
133C2×Dic10C2×Dic311yescharacteristic, complemented
134C22⋊AGL(1,𝔽5)—64no—
135C22⋊AGL(1,𝔽5)—62no—
136#704—62no—
137C4⋊AGL(1,𝔽5)—34no—
138C4×AGL(1,𝔽5)—62no—
139#700—62no—
140D5⋊C8—62no—
141C5×C22⋊C4—22no—
142C42×C5C2×Dic311yescharacteristic, complemented
143C22⋊Dic5—22no—
144#685—22no—
145#685—22no—
146C4×Dic5—21no—
147C4×Dic5C2×Dic311yescharacteristic, complemented
148C2×#124—51no—
149#518—151no—
150#419—152no—
151C2×#100—151no—
152#339—152no—
153C22×C3×C5—22no—
154C22×C3×C5—42no—
155C22×C3×C5C2×C22⋊C411yescharacteristic
156C2×C3×D5—22no—
157C2×C3×D5—22no—
158C2×C3×D5—44no—
159C2×C3×D5—42no—
160C2×C3×D5—82no—
161C2×C3×D5#12011yescharacteristic
162C2×C3×D5#12311yescharacteristic
163ΓL(1,𝔽16)—44no—
164C3×C4×C5—41no—
165C3×C4×C5—41no—
166C3×C4×C5C4×D412yes—
167C3×Dic5—41no—
168C3×Dic5—41no—
169C3×Dic5#12112yes—
170C24×C3—52no—
171C2×C3×Q8—51no—
172C2×C3×Q8—101no—
173C2×C3×Q8—102no—
174C2×C3×Q8—102no—
175C2×C3×Q8C2×AGL(1,𝔽5)11yescharacteristic, complemented
176C2×C3×D4—52no—
177C2×C3×D4—51no—
178C2×C3×D4—102no—
179C2×C3×D4—102no—
180C2×C3×D4C2×AGL(1,𝔽5)11yescharacteristic
181C22×C3×C4—51no—
182C22×C3×C4—101no—
183C22×C3×C4—101no—
184C22×Dic3—104no—
185C3×C22⋊C4—22no—
186C3×C22⋊C4—102no—
187C3×C22⋊C4—102no—
188C3×C22⋊C4—102no—
189C3×C42—51no—
190C3×C42—101no—
191C3×C42C2×AGL(1,𝔽5)11yescharacteristic, complemented
192#217—102no—
193#217—104no—
194C4⋊Dic3—54no—
195C4×Dic3—102no—
196#208—102no—
197#208—102no—
198C2×C3⋊C8—102no—
199C23×C5#21712yes—
200C22×D5—22no—
201C22×D5—22no—
202C22×D5—44no—
203C22×D5#21712yes—
204C22×D5C2×#6711yescharacteristic
205C2×AGL(1,𝔽5)—64no—
206C2×AGL(1,𝔽5)—244no—
207C5×Q8—22no—
208C5×D4—22no—
209C2×C4×C5—21no—
210C2×C4×C5—21no—
211C2×C4×C5C22×Dic311yescharacteristic
212C22⋊D5—24no—
213C2×Dic5—21no—
214C2×Dic5—21no—
215C2×Dic5C2×#6711yescharacteristic
216D20—22no—
217C4×D5—42no—
218C4×D5—41no—
219C4×D5—41no—
220C4×D5D3×D412yes—
221C4×D5#23712yes—
222Dic10—22no—
223Dic10—42no—
224#170—122no—
225C22×Q8—51no—
226C22×D4—51no—
227C2×M16—152no—
228#124—51no—
229#124—51no—
230#124—51no—
231#124—102no—
232#124ΓL(1,𝔽16)11yescharacteristic, complemented
233C4×D4—304no—
234#117—152no—
235C2×C22⋊C4—102no—
236C2×C22⋊C4—302no—
237C2×C42—51no—
238#103—304no—
239#100—302no—
240C2×C3×C5—21no—
241C2×C3×C5—42no—
242C2×C3×C5#38611yescharacteristic
243C3×D5—21no—
244C3×D5—42no—
245C3×D5#45512yes—
246C23×C3—51no—
247C23×C3—52no—
248C23×C3—102no—
249C23×C3—102no—
250C23×C3—204no—
251C23×C3C22⋊AGL(1,𝔽5)12yes—
252C3×Q8—22no—
253C3×Q8—102no—
254C3×Q8—202no—
255C3×D4—22no—
256C3×D4—104no—
257C3×D4—102no—
258C2×C3×C4—21no—
259C2×C3×C4—21no—
260C2×C3×C4—51no—
261C2×C3×C4—52no—
262C2×C3×C4—52no—
263C2×C3×C4—101no—
264C2×C3×C4—101no—
265C2×C3×C4—201no—
266C2×C3×C4—201no—
267C2×C3×C4—202no—
268C2×C3×C4C22×AGL(1,𝔽5)11yescharacteristic
269C2×Dic3—104no—
270C2×Dic3—404no—
271C3⋊C8—202no—
272C22×C5—22no—
273C22×C5—42no—
274C22×C5C2×#21711yescharacteristic
275C2×D5—22no—
276C2×D5—22no—
277C2×D5—44no—
278C2×D5—42no—
279C2×D5—82no—
280C2×D5#93511yescharacteristic
281C2×D5#93711yescharacteristic
282AGL(1,𝔽5)—124no—
283C4×C5—41no—
284C4×C5—41no—
285C4×C5D4×Dic312yes—
286Dic5—41no—
287Dic5—41no—
288Dic5#93912yes—
289C24—52no—
290C2×Q8—51no—
291C2×Q8—101no—
292C2×Q8—102no—
293C2×Q8—102no—
294C2×Q8C2×ΓL(1,𝔽16)11yescharacteristic, complemented
295C2×D4—52no—
296C2×D4—51no—
297C2×D4—102no—
298C2×D4—102no—
299C2×D4C2×ΓL(1,𝔽16)11yescharacteristic
300C22×C4—51no—
301C22×C4—101no—
302C22×C4—101no—
303C22×C4—304no—
304M16—302no—
305M16—302no—
306C2×C8—302no—
307C4⋊C4—154no—
308C22⋊C4—22no—
309C22⋊C4—102no—
310C22⋊C4—102no—
311C22⋊C4—102no—
312C22⋊C4—302no—
313C22⋊C4—304no—
314C42—51no—
315C42—101no—
316C42—302no—
317C42C2×ΓL(1,𝔽16)11yescharacteristic, complemented
318C3×C5#188511yescharacteristic, complemented, Hall
319C22×C3—22no—
320C22×C3—42no—
321C22×C3—51no—
322C22×C3—51no—
323C22×C3—102no—
324C22×C3—102no—
325C22×C3—202no—
326C22×C3—204no—
327C22×C3—402no—
328C22×C3C2×C22⋊AGL(1,𝔽5)11yescharacteristic
329C3×C4—41no—
330C3×C4—41no—
331C3×C4—52no—
332C3×C4—201no—
333C3×C4—201no—
334C3×C4D4×AGL(1,𝔽5)12yes—
335Dic3—204no—
336C2×C5—21no—
337C2×C5—42no—
338C2×C5#522011yescharacteristic
339D5—21no—
340D5—42no—
341D5#506112yes—
342C23—51no—
343C23—52no—
344C23—102no—
345C23—102no—
346C23—204no—
347C23#676312yes—
348Q8—22no—
349Q8—102no—
350Q8—202no—
351D4—22no—
352D4—104no—
353D4—102no—
354C2×C4—21no—
355C2×C4—21no—
356C2×C4—51no—
357C2×C4—52no—
358C2×C4—52no—
359C2×C4—101no—
360C2×C4—101no—
361C2×C4—201no—
362C2×C4—201no—
363C2×C4—202no—
364C2×C4—304no—
365C2×C4—1204no—
366C2×C4C22×ΓL(1,𝔽16)11yescharacteristic
367C8—602no—
368C2×C3—21no—
369C2×C3—42no—
370C2×C3—52no—
371C2×C3—101no—
372C2×C3—202no—
373C2×C3#6577911yescharacteristic
374C5#7172311yescharacteristic, complemented, Sylow
375C22—22no—
376C22—42no—
377C22—51no—
378C22—51no—
379C22—102no—
380C22—102no—
381C22—202no—
382C22—204no—
383C22—402no—
384C22C2×#676311yescharacteristic
385C4—41no—
386C4—41no—
387C4—52no—
388C4—201no—
389C4—201no—
390C4—604no—
391C4D4×ΓL(1,𝔽16)12yes—
392C3#1060456211yescharacteristic, complemented, Sylow
393C2—21no—
394C2—42no—
395C2—52no—
396C2—101no—
397C2—202no—
398C2#1175243811yescharacteristic
subgroupG/H
centerC2#11752438
commutatorC2×C3×C4×C5C22×C4
FrattiniC2×C4C22×ΓL(1,𝔽16)
FittingC3×C5×#124C4
socleC2×C3×C5#386
prime radicalC2×C3×D4×D5C22
cosocleC2×C3×C4×D5C23
hypercenter#124ΓL(1,𝔽16)
hypocenterC3×C5#1885
LCSG ▷ C2×C3×C4×C5 ▷ C2×C3×C5 ▷ C3×C5
UCSC1 ◁ C2 ◁ C2×C4 ◁ #124

Complements

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

NH ≅ G/Ntype
3. #11752441C2semidirect
5. #11746733C2semidirect
13. C3×C5×#124C4semidirect
14. C3×#4093C4semidirect
15. C3×#4082C4semidirect
18. C3×#4021C4semidirect
30. C2×C3×C5×Q8C2×C4semidirect
42. C2×C3×Dic10C2×C4semidirect
51. C3×C42×C5C2×C4semidirect
56. C3×C4×Dic5C2×C4semidirect
68. C5×#124Dic3semidirect
69. #4093Dic3semidirect
70. #4082Dic3semidirect
73. #4021Dic3semidirect
110. C3×#124AGL(1,𝔽5)semidirect
121. C2×C5×Q8C2×Dic3semidirect
133. C2×Dic10C2×Dic3semidirect
142. C42×C5C2×Dic3semidirect
147. C4×Dic5C2×Dic3semidirect
175. C2×C3×Q8C2×AGL(1,𝔽5)semidirect
191. C3×C42C2×AGL(1,𝔽5)semidirect
232. #124ΓL(1,𝔽16)semidirect
294. C2×Q8C2×ΓL(1,𝔽16)semidirect
317. C42C2×ΓL(1,𝔽16)semidirect
318. C3×C5#1885semidirect
374. C5#71723semidirect
392. C3#10604562semidirect

Maximal subgroups

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

subgroupindex
#117569872
C3×D5×#1242
#117524412
D5⋊#8332
#117467332
#106045623
#717235