Finite Group Databasebeta

← #14283271

Subgroups — #14283271

Order 1920.

652 conjugacy classes of subgroups; only the 109 normal subgroups are shown.

normal subgroups only

This group has 652 conjugacy classes of subgroups — too many to draw. Only its 109 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
1C2×C3×#4041C211yesmaximal, characteristic
2#11752423C211yesmaximal, characteristic, complemented
3#11752414C211yesmaximal, characteristic, complemented
4#11746739C214yesmaximal, complemented
5#10604260—31nomaximal, Hall
6C22×C3×D20—21no—
7C22×C3×D20C411yescharacteristic, complemented
8C22×C3×C4×D5C2211yescharacteristic
9C2×#6763—21no—
10C2×#6762—21no—
11C2×C3×C5×C4⋊C4C411yescharacteristic, complemented
12C2×C3×#685—21no—
13C3×#4041—21no—
14C3×#4041—21no—
15C3×#4041C412yescomplemented
16C3×#4041C2212yes—
17#91235—21no—
18#91235C2212yescomplemented
19#91232—21no—
20#91232C2212yescomplemented
21#71674—51nomaximal, Hall
22C2×#4041D311yescharacteristic
23#65764—31no—
24#65755—31no—
25#64868—34no—
26C23×C3×D5—21no—
27C23×C3×D5D411yescharacteristic
28C22×ΓL(1,𝔽16)—22no—
29C22×C3×C4×C5—21no—
30C22×C3×C4×C5C2×C411yescharacteristic
31C22×C3×Dic5D411yescharacteristic
32C2×C3×D20—22no—
33C2×C3×D20—22no—
34C2×C3×D20—21no—
35C2×C3×D20—42no—
36C2×C3×D20—44no—
37C2×C3×D20C2×C412yes—
38C2×C3×C4×D5—42no—
39C2×C3×C4×D5C2×C411yescharacteristic
40C2×C3×C4×D5D411yescharacteristic
41C2×C3×C4×D5D411yescharacteristic
42C2×C3×C4×D5C2311yescharacteristic
43#6763—42no—
44#6762—42no—
45C2×C3⋊#170—22no—
46C3×C5×C4⋊C4—21no—
47C3×C5×C4⋊C4C2×C412yes—
48C3×#685—22no—
49C3×#685—22no—
50C2×C3×#121—51no—
51#5211—51no—
52#5207—51no—
53#4537—54no—
54C22×D20—21no—
55C22×D20Dic311yescharacteristic, complemented
56C22×C4×D5C2×D311yescharacteristic
57C2×C22⋊AGL(1,𝔽5)—61no—
58C2×#704—61no—
59C2×C5×C4⋊C4Dic311yescharacteristic, complemented
60C2×#685—21no—
61#4041—21no—
62#4041—21no—
63#4041Dic312yescomplemented
64#4041C2×D312yes—
65(C2×AGL(1,𝔽5))⋊C4—32no—
66(C2×AGL(1,𝔽5))⋊C4—61no—
67#4003—32no—
68#4003—61no—
69#1510—151noSylow
70C23×C3×C5C22⋊C411yescharacteristic
71C22×C3×D5—22no—
72C22×C3×D5—22no—
73C22×C3×D5—22no—
74C22×C3×D5—41no—
75C22×C3×D5—42no—
76C22×C3×D5—42no—
77C22×C3×D5—42no—
78C22×C3×D5C22⋊C411yescharacteristic
79C22×C3×D5C2×D411yescharacteristic
80C2×ΓL(1,𝔽16)—42no—
81C2×ΓL(1,𝔽16)—42no—
82C2×ΓL(1,𝔽16)—42no—
83C2×C3×C4×C5—22no—
84C2×C3×C4×C5—22no—
85C2×C3×C4×C5—42no—
86C2×C3×C4×C5C22⋊C411yescharacteristic
87C2×C3×C4×C5C22×C411yescharacteristic
88C2×C3×Dic5—22no—
89C2×C3×Dic5C22⋊C411yescharacteristic
90C2×C3×Dic5C2×D411yescharacteristic
91C3×D20—42no—
92C3×D20—42no—
93C3×D20—44no—
94C3×D20—44no—
95C3×C4×D5—41no—
96C3×C4×D5—41no—
97C3×C4×D5—41no—
98C3×C4×D5—41no—
99C3⋊#170—42no—
100C22×C3×D4—51no—
101C22×C3×D4—101no—
102C23×C3×C4—51no—
103C3×#121—52no—
104C3×#121—52no—
105C3×#121—101no—
106C3×#121—101no—
107C2×C3×C4⋊C4AGL(1,𝔽5)11yescharacteristic, complemented
108C2×C3×C22⋊C4—101no—
109C2×#217—101no—
110C2×#208—101no—
111C3⋊#95—52no—
112C3⋊#95—101no—
113#830—52no—
114#830—101no—
115C23×D5—21no—
116C23×D5#6711yescharacteristic
117C22×AGL(1,𝔽5)—62no—
118C22×C4×C5—21no—
119C22×C4×C5C2×Dic311yescharacteristic
120C22×Dic5#6711yescharacteristic
121C2×D20—22no—
122C2×D20—22no—
123C2×D20—21no—
124C2×D20—42no—
125C2×D20—44no—
126C2×D20C2×Dic312yes—
127C2×C4×D5—42no—
128C2×C4×D5C2×Dic311yescharacteristic
129C2×C4×D5#6711yescharacteristic
130C2×C4×D5#6711yescharacteristic
131C2×C4×D5C22×D311yescharacteristic
132C22⋊AGL(1,𝔽5)—122no—
133#704—122no—
134C2×#170—62no—
135C5×C4⋊C4—21no—
136C5×C4⋊C4C2×Dic312yes—
137#685—22no—
138#685—22no—
139C2×#121—51no—
140#407—151no—
141#381—151no—
142#327—154no—
143C22×C3×C5—21no—
144C22×C3×C5—21no—
145C22×C3×C5#9912yes—
146C22×C3×C5C2×C22⋊C411yescharacteristic
147C2×C3×D5—21no—
148C2×C3×D5—21no—
149C2×C3×D5—21no—
150C2×C3×D5—21no—
151C2×C3×D5—21no—
152C2×C3×D5—44no—
153C2×C3×D5—42no—
154C2×C3×D5—42no—
155C2×C3×D5—42no—
156C2×C3×D5—84no—
157C2×C3×D5AGL(1,ℤ/8)12yes—
158ΓL(1,𝔽16)—82no—
159C3×C4×C5—21no—
160C3×C4×C5—21no—
161C3×C4×C5—42no—
162C3×Dic5—21no—
163C3×Dic5—21no—
164C24×C3—51no—
165C24×C3—101no—
166C2×C3×D4—52no—
167C2×C3×D4—102no—
168C2×C3×D4—102no—
169C2×C3×D4—101no—
170C2×C3×D4—202no—
171C2×C3×D4—204no—
172C22×C3×C4—21no—
173C22×C3×C4—51no—
174C22×C3×C4—51no—
175C22×C3×C4—51no—
176C22×C3×C4—51no—
177C22×C3×C4—51no—
178C22×C3×C4—202no—
179C22×C3×C4C2×AGL(1,𝔽5)11yescharacteristic
180C22×Dic3—102no—
181C3×C4⋊C4—21no—
182C3×C4⋊C4C2×AGL(1,𝔽5)12yes—
183C3×C22⋊C4—102no—
184C3×C22⋊C4—102no—
185#217—202no—
186#208—202no—
187C2×C3⋊C8—102no—
188C23×C5#21711yescharacteristic
189C22×D5—22no—
190C22×D5—22no—
191C22×D5—22no—
192C22×D5—41no—
193C22×D5—42no—
194C22×D5—42no—
195C22×D5—42no—
196C22×D5#21711yescharacteristic
197C22×D5C2×#6711yescharacteristic
198C2×AGL(1,𝔽5)—122no—
199C2×AGL(1,𝔽5)—122no—
200C2×AGL(1,𝔽5)—122no—
201C2×C4×C5—22no—
202C2×C4×C5—22no—
203C2×C4×C5—42no—
204C2×C4×C5#21711yescharacteristic
205C2×C4×C5C22×Dic311yescharacteristic
206C2×Dic5—22no—
207C2×Dic5#21711yescharacteristic
208C2×Dic5C2×#6711yescharacteristic
209D20—42no—
210D20—42no—
211D20—44no—
212D20—44no—
213C4×D5—41no—
214C4×D5—41no—
215C4×D5—41no—
216C4×D5—41no—
217#170—122no—
218C22×D4—51no—
219C22×D4—101no—
220C23×C4—51no—
221C2×M16—301no—
222#121—52no—
223#121—52no—
224#121—101no—
225#121—101no—
226C2×C4⋊C4ΓL(1,𝔽16)11yescharacteristic, complemented
227C2×C22⋊C4—101no—
228C2×C22⋊C4—301no—
229C22⋊C8—152no—
230C22⋊C8—301no—
231#95—152no—
232#95—301no—
233C2×C3×C5—22no—
234C2×C3×C5C2×#9911yescharacteristic
235C2×C3×C5#41811yescharacteristic
236C2×C3×C5#42111yescharacteristic
237C3×D5—22no—
238C3×D5—41no—
239C3×D5—82no—
240C23×C3—51no—
241C23×C3—51no—
242C23×C3—102no—
243C23×C3—102no—
244C23×C3—102no—
245C23×C3—201no—
246C23×C3—202no—
247C23×C3—202no—
248C23×C3—202no—
249C23×C3C22⋊AGL(1,𝔽5)11yescharacteristic
250C3×D4—202no—
251C3×D4—202no—
252C3×D4—204no—
253C3×D4—204no—
254C2×C3×C4—22no—
255C2×C3×C4—22no—
256C2×C3×C4—42no—
257C2×C3×C4—51no—
258C2×C3×C4—51no—
259C2×C3×C4—102no—
260C2×C3×C4—201no—
261C2×C3×C4—201no—
262C2×C3×C4—201no—
263C2×C3×C4—201no—
264C2×C3×C4C22⋊AGL(1,𝔽5)11yescharacteristic
265C2×C3×C4C22×AGL(1,𝔽5)11yescharacteristic
266C2×Dic3—202no—
267C2×Dic3—202no—
268C2×Dic3—202no—
269C3⋊C8—202no—
270C22×C5—21no—
271C22×C5—21no—
272C22×C5#83412yes—
273C22×C5C2×#21711yescharacteristic
274C2×D5—21no—
275C2×D5—21no—
276C2×D5—21no—
277C2×D5—21no—
278C2×D5—21no—
279C2×D5—44no—
280C2×D5—42no—
281C2×D5—42no—
282C2×D5—42no—
283C2×D5—84no—
284C2×D5#93212yes—
285AGL(1,𝔽5)—242no—
286C4×C5—21no—
287C4×C5—21no—
288C4×C5—42no—
289Dic5—21no—
290Dic5—21no—
291C24—51no—
292C24—101no—
293C2×D4—52no—
294C2×D4—102no—
295C2×D4—102no—
296C2×D4—101no—
297C2×D4—202no—
298C2×D4—204no—
299C22×C4—21no—
300C22×C4—51no—
301C22×C4—51no—
302C22×C4—51no—
303C22×C4—51no—
304C22×C4—51no—
305C22×C4—202no—
306C22×C4—302no—
307C22×C4C2×ΓL(1,𝔽16)11yescharacteristic
308M16—602no—
309C2×C8—302no—
310C4⋊C4—21no—
311C4⋊C4C2×ΓL(1,𝔽16)12yes—
312C22⋊C4—102no—
313C22⋊C4—102no—
314C22⋊C4—602no—
315C3×C5#151011yescharacteristic, complemented, Hall
316C22×C3—21no—
317C22×C3—21no—
318C22×C3—52no—
319C22×C3—101no—
320C22×C3—101no—
321C22×C3—101no—
322C22×C3—101no—
323C22×C3—101no—
324C22×C3—204no—
325C22×C3—202no—
326C22×C3—202no—
327C22×C3—202no—
328C22×C3—404no—
329C22×C3#399912yes—
330C22×C3C2×C22⋊AGL(1,𝔽5)11yescharacteristic
331C3×C4—21no—
332C3×C4—21no—
333C3×C4—42no—
334C3×C4—101no—
335C3×C4—101no—
336Dic3—402no—
337C2×C5—22no—
338C2×C5#521511yescharacteristic
339C2×C5C2×#83411yescharacteristic
340C2×C5#523611yescharacteristic
341D5—22no—
342D5—41no—
343D5—82no—
344C23—51no—
345C23—51no—
346C23—102no—
347C23—102no—
348C23—102no—
349C23—201no—
350C23—202no—
351C23—202no—
352C23—202no—
353C23#676311yescharacteristic
354D4—202no—
355D4—202no—
356D4—204no—
357D4—204no—
358C2×C4—22no—
359C2×C4—22no—
360C2×C4—42no—
361C2×C4—51no—
362C2×C4—51no—
363C2×C4—102no—
364C2×C4—201no—
365C2×C4—201no—
366C2×C4—201no—
367C2×C4—201no—
368C2×C4—602no—
369C2×C4—602no—
370C2×C4—602no—
371C2×C4#676311yescharacteristic
372C2×C4C22×ΓL(1,𝔽16)11yescharacteristic
373C8—602no—
374C2×C3—22no—
375C2×C3—102no—
376C2×C3—201no—
377C2×C3—402no—
378C2×C3C2×#399911yescharacteristic
379C2×C3#6576711yescharacteristic
380C2×C3#6578411yescharacteristic
381C5#7167411yescharacteristic, complemented, Sylow
382C22—21no—
383C22—21no—
384C22—52no—
385C22—101no—
386C22—101no—
387C22—101no—
388C22—101no—
389C22—101no—
390C22—204no—
391C22—202no—
392C22—202no—
393C22—202no—
394C22—404no—
395C22#9122812yes—
396C22C2×#676311yescharacteristic
397C4—21no—
398C4—21no—
399C4—42no—
400C4—101no—
401C4—101no—
402C4—1202no—
403C3#1060426011yescharacteristic, complemented, Sylow
404C2—22no—
405C2—102no—
406C2—201no—
407C2—402no—
408C2C2×#9122811yescharacteristic
409C2#1175242611yescharacteristic
410C2#1175244311yescharacteristic
subgroupG/H
centerC22C2×#6763
commutatorC2×C3×C4×C5C22×C4
FrattiniC2×C4C22×ΓL(1,𝔽16)
FittingC2×C3×C5×C4⋊C4C4
socleC22×C3×C5C2×C22⋊C4
prime radicalC2×C3×#4041C2
cosocleC2×C3×C4×D5C23
hypercenterC2×C4⋊C4ΓL(1,𝔽16)
hypocenterC3×C5#1510
LCSG ▷ C2×C3×C4×C5 ▷ C22×C3×C5 ▷ C3×C5
UCSC1 ◁ C22 ◁ C22×C4 ◁ C2×C4⋊C4

Complements

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

NH ≅ G/Ntype
2. #11752423C2semidirect
3. #11752414C2semidirect
4. #11746739C2semidirect
7. C22×C3×D20C4semidirect
11. C2×C3×C5×C4⋊C4C4semidirect
15. C3×#4041C4semidirect
18. #91235C22semidirect
20. #91232C22semidirect
55. C22×D20Dic3semidirect
59. C2×C5×C4⋊C4Dic3semidirect
63. #4041Dic3semidirect
107. C2×C3×C4⋊C4AGL(1,𝔽5)semidirect
226. C2×C4⋊C4ΓL(1,𝔽16)semidirect
315. C3×C5#1510semidirect
381. C5#71674semidirect
403. C3#10604260semidirect

Maximal subgroups

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

subgroupindex
C2×C3×#40412
#117524232
#117524142
#117467392
#106042603
#716745