Finite Group Databasebeta

← #14283124

Subgroups — #14283124

Order 1920.

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

normal subgroups only

This group has 692 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
1C2×C3×#4041C211yesmaximal, characteristic
2C4×#6763C211yesmaximal, characteristic, complemented
3#11752415C211yesmaximal, characteristic, complemented
4#11746739C214yesmaximal, complemented
5#10604258—31nomaximal, Hall
6C22×C3×D20—21no—
7C22×C3×D20C411yescharacteristic, complemented
8C22×C3×C4×D5C2211yescharacteristic
9C2×#6763—21no—
10C2×C4×ΓL(1,𝔽16)—21no—
11C2×#6758—21no—
12C2×C3×C5×C4⋊C4C411yescharacteristic, complemented
13C2×C3×#685—21no—
14C3×#4041—21no—
15C3×#4041—21no—
16C3×#4041C412yescomplemented
17C3×#4041C2212yes—
18#91241—21no—
19#91235C2212yescomplemented
20#91232C2212yescomplemented
21#71673—51nomaximal, Hall
22C2×#4041D311yescharacteristic
23C4×C22⋊AGL(1,𝔽5)—31no—
24#65756—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—44no—
36C2×C3×D20—42no—
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—24no—
44C2×C3⋊#170—22no—
45C4×ΓL(1,𝔽16)—42no—
46#6758—42no—
47C3×C5×C4⋊C4—21no—
48C3×C5×C4⋊C4C2×C412yes—
49C3×#685—22no—
50C3×#685—22no—
51C2×C3×#121—51no—
52C4×#217—51no—
53#5207—51no—
54#4537—54no—
55C22×D20—21no—
56C22×D20Dic311yescharacteristic, complemented
57C22×C4×D5C2×D311yescharacteristic
58C2×C22⋊AGL(1,𝔽5)—61no—
59C2×C4×AGL(1,𝔽5)—61no—
60C2×#700—61no—
61C2×C5×C4⋊C4Dic311yescharacteristic, complemented
62C2×#685—21no—
63#4041—21no—
64#4041—21no—
65#4041Dic312yescomplemented
66#4041C2×D312yes—
67C22⋊#170—61no—
68(C2×AGL(1,𝔽5))⋊C4—32no—
69#4003—32no—
70#1507—151noSylow
71C23×C3×C5C22⋊C411yescharacteristic
72C22×C3×D5—22no—
73C22×C3×D5—22no—
74C22×C3×D5—22no—
75C22×C3×D5—41no—
76C22×C3×D5—42no—
77C22×C3×D5—42no—
78C22×C3×D5—42no—
79C22×C3×D5C22⋊C411yescharacteristic
80C22×C3×D5C2×D411yescharacteristic
81C2×ΓL(1,𝔽16)—24no—
82C2×ΓL(1,𝔽16)—42no—
83C2×ΓL(1,𝔽16)—42no—
84C2×C3×C4×C5—22no—
85C2×C3×C4×C5—22no—
86C2×C3×C4×C5—42no—
87C2×C3×C4×C5C22⋊C411yescharacteristic
88C2×C3×C4×C5C22×C411yescharacteristic
89C2×C3×Dic5—41no—
90C2×C3×Dic5C22⋊C411yescharacteristic
91C2×C3×Dic5C2×D411yescharacteristic
92C3×D20—42no—
93C3×D20—44no—
94C3×D20—44no—
95C3×D20—42no—
96C3×C4×D5—22no—
97C3×C4×D5—22no—
98C3×C4×D5—41no—
99C3×C4×D5—41no—
100C3⋊#170—42no—
101C22×C3×D4—51no—
102C22×C3×D4—101no—
103C23×C3×C4—51no—
104C3×#121—52no—
105C3×#121—52no—
106C3×#121—101no—
107C3×#121—101no—
108C2×C3×C4⋊C4AGL(1,𝔽5)11yescharacteristic, complemented
109C2×C3×C22⋊C4—101no—
110C2×#217—101no—
111C2×C4×Dic3—101no—
112C2×#208—101no—
113C3⋊#95—52no—
114#830—52no—
115#830—101no—
116C23×D5—21no—
117C23×D5#6711yescharacteristic
118C22×AGL(1,𝔽5)—62no—
119C22×C4×C5—21no—
120C22×C4×C5C2×Dic311yescharacteristic
121C22×Dic5#6711yescharacteristic
122C2×D20—22no—
123C2×D20—22no—
124C2×D20—21no—
125C2×D20—44no—
126C2×D20—42no—
127C2×D20C2×Dic312yes—
128C2×C4×D5—42no—
129C2×C4×D5C2×Dic311yescharacteristic
130C2×C4×D5#6711yescharacteristic
131C2×C4×D5#6711yescharacteristic
132C2×C4×D5C22×D311yescharacteristic
133C22⋊AGL(1,𝔽5)—64no—
134C2×#170—62no—
135C4×AGL(1,𝔽5)—122no—
136#700—122no—
137C5×C4⋊C4—21no—
138C5×C4⋊C4C2×Dic312yes—
139#685—22no—
140#685—22no—
141C2×#121—51no—
142#407—151no—
143C4×C22⋊C4—151no—
144#327—154no—
145C22×C3×C5—21no—
146C22×C3×C5—21no—
147C22×C3×C5#10412yes—
148C22×C3×C5C2×C22⋊C411yescharacteristic
149C2×C3×D5—21no—
150C2×C3×D5—21no—
151C2×C3×D5—21no—
152C2×C3×D5—21no—
153C2×C3×D5—21no—
154C2×C3×D5—44no—
155C2×C3×D5—42no—
156C2×C3×D5—42no—
157C2×C3×D5—42no—
158C2×C3×D5—84no—
159C2×C3×D5AGL(1,ℤ/8)12yes—
160ΓL(1,𝔽16)—44no—
161C3×C4×C5—21no—
162C3×C4×C5—42no—
163C3×C4×C5#9912yes—
164C3×Dic5—21no—
165C3×Dic5—21no—
166C24×C3—51no—
167C24×C3—101no—
168C2×C3×D4—52no—
169C2×C3×D4—102no—
170C2×C3×D4—102no—
171C2×C3×D4—101no—
172C2×C3×D4—202no—
173C2×C3×D4—204no—
174C22×C3×C4—21no—
175C22×C3×C4—51no—
176C22×C3×C4—51no—
177C22×C3×C4—51no—
178C22×C3×C4—51no—
179C22×C3×C4—51no—
180C22×C3×C4—202no—
181C22×C3×C4C2×AGL(1,𝔽5)11yescharacteristic
182C22×Dic3—102no—
183C3×C4⋊C4—21no—
184C3×C4⋊C4C2×AGL(1,𝔽5)12yes—
185C3×C22⋊C4—102no—
186C3×C22⋊C4—102no—
187#217—104no—
188C4×Dic3—202no—
189#208—202no—
190C2×C3⋊C8—102no—
191C23×C5#21711yescharacteristic
192C22×D5—22no—
193C22×D5—22no—
194C22×D5—22no—
195C22×D5—41no—
196C22×D5—42no—
197C22×D5—42no—
198C22×D5—42no—
199C22×D5#21711yescharacteristic
200C22×D5C2×#6711yescharacteristic
201C2×AGL(1,𝔽5)—64no—
202C2×AGL(1,𝔽5)—122no—
203C2×AGL(1,𝔽5)—122no—
204C2×C4×C5—22no—
205C2×C4×C5—22no—
206C2×C4×C5—42no—
207C2×C4×C5#21711yescharacteristic
208C2×C4×C5C22×Dic311yescharacteristic
209C2×Dic5—41no—
210C2×Dic5#21711yescharacteristic
211C2×Dic5C2×#6711yescharacteristic
212D20—42no—
213D20—44no—
214D20—44no—
215D20—42no—
216C4×D5—22no—
217C4×D5—22no—
218C4×D5—41no—
219C4×D5—41no—
220#170—122no—
221C22×D4—51no—
222C22×D4—101no—
223C23×C4—51no—
224C2×M16—301no—
225#121—52no—
226#121—52no—
227#121—101no—
228#121—101no—
229C2×C4⋊C4ΓL(1,𝔽16)11yescharacteristic, complemented
230C2×C22⋊C4—101no—
231C2×C22⋊C4—301no—
232C2×C42—301no—
233C22⋊C8—152no—
234C22⋊C8—301no—
235#95—152no—
236C2×C3×C5—22no—
237C2×C3×C5C2×#9911yescharacteristic
238C2×C3×C5#41811yescharacteristic
239C2×C3×C5C2×#10411yescharacteristic
240C3×D5—22no—
241C3×D5—41no—
242C3×D5—82no—
243C23×C3—51no—
244C23×C3—51no—
245C23×C3—102no—
246C23×C3—102no—
247C23×C3—102no—
248C23×C3—201no—
249C23×C3—202no—
250C23×C3—202no—
251C23×C3—202no—
252C23×C3C22⋊AGL(1,𝔽5)11yescharacteristic
253C3×D4—202no—
254C3×D4—204no—
255C3×D4—204no—
256C3×D4—202no—
257C2×C3×C4—22no—
258C2×C3×C4—22no—
259C2×C3×C4—42no—
260C2×C3×C4—51no—
261C2×C3×C4—51no—
262C2×C3×C4—102no—
263C2×C3×C4—102no—
264C2×C3×C4—201no—
265C2×C3×C4—201no—
266C2×C3×C4—201no—
267C2×C3×C4C22⋊AGL(1,𝔽5)11yescharacteristic
268C2×C3×C4C22×AGL(1,𝔽5)11yescharacteristic
269C2×Dic3—104no—
270C2×Dic3—202no—
271C2×Dic3—202no—
272C3⋊C8—202no—
273C22×C5—21no—
274C22×C5—21no—
275C22×C5#83712yes—
276C22×C5C2×#21711yescharacteristic
277C2×D5—21no—
278C2×D5—21no—
279C2×D5—21no—
280C2×D5—21no—
281C2×D5—21no—
282C2×D5—44no—
283C2×D5—42no—
284C2×D5—42no—
285C2×D5—42no—
286C2×D5—84no—
287C2×D5#93212yes—
288AGL(1,𝔽5)—124no—
289C4×C5—21no—
290C4×C5—42no—
291C4×C5#83412yes—
292Dic5—21no—
293Dic5—21no—
294C24—51no—
295C24—101no—
296C2×D4—52no—
297C2×D4—102no—
298C2×D4—102no—
299C2×D4—101no—
300C2×D4—202no—
301C2×D4—204no—
302C22×C4—21no—
303C22×C4—51no—
304C22×C4—51no—
305C22×C4—51no—
306C22×C4—51no—
307C22×C4—51no—
308C22×C4—202no—
309C22×C4—302no—
310C22×C4C2×ΓL(1,𝔽16)11yescharacteristic
311M16—602no—
312C2×C8—302no—
313C4⋊C4—21no—
314C4⋊C4C2×ΓL(1,𝔽16)12yes—
315C22⋊C4—102no—
316C22⋊C4—102no—
317C22⋊C4—304no—
318C42—602no—
319C3×C5#150711yescharacteristic, complemented, Hall
320C22×C3—21no—
321C22×C3—21no—
322C22×C3—52no—
323C22×C3—101no—
324C22×C3—101no—
325C22×C3—101no—
326C22×C3—101no—
327C22×C3—101no—
328C22×C3—204no—
329C22×C3—202no—
330C22×C3—202no—
331C22×C3—202no—
332C22×C3—404no—
333C22×C3#401012yes—
334C22×C3C2×C22⋊AGL(1,𝔽5)11yescharacteristic
335C3×C4—21no—
336C3×C4—42no—
337C3×C4—101no—
338C3×C4—101no—
339C3×C4#399912yes—
340Dic3—204no—
341C2×C5—22no—
342C2×C5#521511yescharacteristic
343C2×C5C2×#83411yescharacteristic
344C2×C5C2×#83711yescharacteristic
345D5—22no—
346D5—41no—
347D5—82no—
348C23—51no—
349C23—51no—
350C23—102no—
351C23—102no—
352C23—102no—
353C23—201no—
354C23—202no—
355C23—202no—
356C23—202no—
357C23#676311yescharacteristic
358D4—202no—
359D4—204no—
360D4—204no—
361D4—202no—
362C2×C4—22no—
363C2×C4—22no—
364C2×C4—42no—
365C2×C4—51no—
366C2×C4—51no—
367C2×C4—102no—
368C2×C4—102no—
369C2×C4—201no—
370C2×C4—201no—
371C2×C4—201no—
372C2×C4—304no—
373C2×C4—602no—
374C2×C4—602no—
375C2×C4#676311yescharacteristic
376C2×C4C22×ΓL(1,𝔽16)11yescharacteristic
377C8—602no—
378C2×C3—22no—
379C2×C3—102no—
380C2×C3—201no—
381C2×C3—402no—
382C2×C3C2×#399911yescharacteristic
383C2×C3#6576711yescharacteristic
384C2×C3C2×#401011yescharacteristic
385C5#7167311yescharacteristic, complemented, Sylow
386C22—21no—
387C22—21no—
388C22—52no—
389C22—101no—
390C22—101no—
391C22—101no—
392C22—101no—
393C22—101no—
394C22—204no—
395C22—202no—
396C22—202no—
397C22—202no—
398C22—404no—
399C22#9123912yes—
400C22C2×#676311yescharacteristic
401C4—21no—
402C4—42no—
403C4—101no—
404C4—101no—
405C4—604no—
406C4#9122812yes—
407C3#1060425811yescharacteristic, complemented, Sylow
408C2—22no—
409C2—102no—
410C2—201no—
411C2—402no—
412C2C2×#9122811yescharacteristic
413C2#1175242611yescharacteristic
414C2C2×#9123911yescharacteristic
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#1507
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. C4×#6763C2semidirect
3. #11752415C2semidirect
4. #11746739C2semidirect
7. C22×C3×D20C4semidirect
12. C2×C3×C5×C4⋊C4C4semidirect
16. C3×#4041C4semidirect
19. #91235C22semidirect
20. #91232C22semidirect
56. C22×D20Dic3semidirect
61. C2×C5×C4⋊C4Dic3semidirect
65. #4041Dic3semidirect
108. C2×C3×C4⋊C4AGL(1,𝔽5)semidirect
229. C2×C4⋊C4ΓL(1,𝔽16)semidirect
319. C3×C5#1507semidirect
385. C5#71673semidirect
407. C3#10604258semidirect

Maximal subgroups

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

subgroupindex
C2×C3×#40412
C4×#67632
#117524152
#117467392
#106042583
#716735