Finite Group Databasebeta

← #14283149

Subgroups — #14283149

Order 1920.

692 conjugacy classes of subgroups; only the 106 normal subgroups are shown.

normal subgroups only

This group has 692 conjugacy classes of subgroups — too many to draw. Only its 106 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×#91450C211yesmaximal, characteristic
2C4×#6763C211yesmaximal, characteristic, complemented
3C3×#65756C211yesmaximal, characteristic, complemented
4#11746499C214yesmaximal, complemented
5#10604258—31nomaximal, Hall
6C22×D60C411yescharacteristic, complemented
7C22×C3×C4×D5C2211yescharacteristic
8C22×#1202—21no—
9C2×#6763—21no—
10C2×C4×ΓL(1,𝔽16)—21no—
11C2×C3×#700—21no—
12C2×C5×C4⋊Dic3C411yescharacteristic, complemented
13C2×#6667—21no—
14#91450—21no—
15#91450—21no—
16#91450C412yescomplemented
17#91450C2212yes—
18#91235C2212yescomplemented
19C3×C22⋊#170—21no—
20C3×#4003C2212yescomplemented
21#69977—51nomaximal, Hall
22C2×#4041—31no—
23C4×C22⋊AGL(1,𝔽5)—31no—
24#65756D311yescharacteristic, complemented
25#64868—34no—
26C23×D15—21no—
27C23×C3×D5D411yescharacteristic
28C22×ΓL(1,𝔽16)—22no—
29C22×C3×C4×C5C2×C411yescharacteristic
30C2×D60—21no—
31C2×D60—42no—
32C2×D60C2×C412yes—
33C22×C5×Dic3—21no—
34C22×C3×Dic5D411yescharacteristic
35C2×C3×C4×D5—42no—
36C2×C3×C4×D5C2×C411yescharacteristic
37C2×C3×C4×D5D411yescharacteristic
38C2×C3×C4×D5D411yescharacteristic
39C2×C3×C4×D5C2311yescharacteristic
40C2×#1202—22no—
41C2×#1202—22no—
42C2×#1202—44no—
43#6763—24no—
44C4×ΓL(1,𝔽16)—42no—
45C2×C3×#170—22no—
46C3×#700—42no—
47C5×C4⋊Dic3—21no—
48C5×C4⋊Dic3C2×C412yes—
49#6667—22no—
50#6667—22no—
51C2×#930—51no—
52C3×#407—51no—
53C4×#217—51no—
54#4470—54no—
55C22×D20—31no—
56C22×D20—61no—
57C22×C4×D5C2×D311yescharacteristic
58C2×C22⋊AGL(1,𝔽5)—61no—
59C2×C4×AGL(1,𝔽5)—61no—
60C2×#700—21no—
61C2×C5×C4⋊C4—31no—
62C2×#685—61no—
63#4041—32no—
64#4041—32no—
65#4041—61no—
66#4041—61no—
67C22⋊#170—21no—
68(C2×AGL(1,𝔽5))⋊C4—32no—
69#4003C2×D312yescomplemented
70#1507—151noSylow
71C23×C3×C5C22⋊C411yescharacteristic
72C22×D15—22no—
73C22×D15—42no—
74C22×D15—42no—
75C22×D15—42no—
76C22×C3×D5—22no—
77C22×C3×D5—22no—
78C22×C3×D5—41no—
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×C5C22⋊C411yescharacteristic
86C2×C3×C4×C5C22×C411yescharacteristic
87D60—42no—
88D60—42no—
89C2×C5×Dic3—22no—
90C2×C5×Dic3—42no—
91C2×C3×Dic5—41no—
92C2×C3×Dic5C22⋊C411yescharacteristic
93C2×C3×Dic5C2×D411yescharacteristic
94C3×C4×D5—22no—
95C3×C4×D5—22no—
96C3×C4×D5—41no—
97C3×C4×D5—41no—
98#1202—44no—
99#1202—44no—
100C3×#170—42no—
101C23×C3×C4—51no—
102C22×#67—101no—
103C22×D12—51no—
104C2×C3×M16—101no—
105C2×#217—101no—
106#930—52no—
107#930—52no—
108#930—101no—
109#930—101no—
110C2×(C2×D3)⋊C4—101no—
111C2×C4⋊Dic3AGL(1,𝔽5)11yescharacteristic, complemented
112C2×C4×Dic3—101no—
113C3×C22⋊C8—52no—
114C3×C22⋊C8—101no—
115C3⋊#95—52no—
116C23×D5—61no—
117C23×D5D1211yescharacteristic
118C22×AGL(1,𝔽5)—62no—
119C22×C4×C5—61no—
120C22×C4×C5C4×D311yescharacteristic
121C22×Dic5#6711yescharacteristic
122C2×D20—32no—
123C2×D20—62no—
124C2×D20—62no—
125C2×D20—61no—
126C2×D20—124no—
127C2×D20—122no—
128C2×C4×D5—42no—
129C2×C4×D5C4×D311yescharacteristic
130C2×C4×D5D1211yescharacteristic
131C2×C4×D5#6711yescharacteristic
132C2×C4×D5C22×D311yescharacteristic
133C22⋊AGL(1,𝔽5)—64no—
134C2×#170—22no—
135C4×AGL(1,𝔽5)—122no—
136#700—42no—
137C5×C4⋊C4—32no—
138C5×C4⋊C4—61no—
139#685—62no—
140#685—62no—
141C2×#121—151no—
142#407—51no—
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×D15—42no—
150C2×D15—42no—
151C2×D15—42no—
152C2×D15—84no—
153C2×C3×D5—21no—
154C2×C3×D5—21no—
155C2×C3×D5—21no—
156C2×C3×D5—21no—
157C2×C3×D5—21no—
158C2×C3×D5—44no—
159C2×C3×D5AGL(1,ℤ/8)12yes—
160ΓL(1,𝔽16)—44no—
161C3×C4×C5—21no—
162C3×C4×C5#9912yes—
163C3×Dic5—21no—
164C3×Dic5—21no—
165C5×Dic3—42no—
166C24×C3—51no—
167C23×D3—101no—
168C22×C3×C4—51no—
169C22×C3×C4—51no—
170C22×C3×C4—51no—
171C22×C3×C4—51no—
172C22×C3×C4—51no—
173C22×C3×C4—202no—
174C22×C3×C4C2×AGL(1,𝔽5)11yescharacteristic
175C2×#67—102no—
176C2×#67—102no—
177C2×#67—204no—
178C22×Dic3—21no—
179C22×Dic3—102no—
180C2×D12—52no—
181C2×D12—101no—
182C2×D12—202no—
183C3×M16—202no—
184C2×C3×C8—102no—
185#217—104no—
186(C2×D3)⋊C4—102no—
187(C2×D3)⋊C4—102no—
188C4⋊Dic3—21no—
189C4⋊Dic3C2×AGL(1,𝔽5)12yes—
190C4×Dic3—202no—
191C23×C5(C2×D3)⋊C411yescharacteristic
192C22×D5—22no—
193C22×D5—22no—
194C22×D5—41no—
195C22×D5—62no—
196C22×D5—122no—
197C22×D5—122no—
198C22×D5—122no—
199C22×D5(C2×D3)⋊C411yescharacteristic
200C22×D5C2×D1211yescharacteristic
201C2×AGL(1,𝔽5)—64no—
202C2×AGL(1,𝔽5)—122no—
203C2×AGL(1,𝔽5)—122no—
204C2×C4×C5—22no—
205C2×C4×C5—62no—
206C2×C4×C5—122no—
207C2×C4×C5(C2×D3)⋊C411yescharacteristic
208C2×C4×C5C2×C4×D311yescharacteristic
209C2×Dic5—41no—
210C2×Dic5(C2×D3)⋊C411yescharacteristic
211C2×Dic5C2×#6711yescharacteristic
212D20—122no—
213D20—124no—
214D20—124no—
215D20—122no—
216C4×D5—22no—
217C4×D5—22no—
218C4×D5—41no—
219C4×D5—41no—
220#170—42no—
221C22×D4—151no—
222C22×D4—301no—
223C23×C4—51no—
224C2×M16—101no—
225#121—152no—
226#121—152no—
227#121—301no—
228#121—301no—
229C2×C4⋊C4—31no—
230C2×C22⋊C4—301no—
231C2×C22⋊C4—301no—
232C2×C42—301no—
233C22⋊C8—52no—
234C22⋊C8—101no—
235#95—152no—
236C2×C3×C5—22no—
237C2×C3×C5C2×#9911yescharacteristic
238C2×C3×C5#41811yescharacteristic
239C2×C3×C5C2×#10411yescharacteristic
240D15—82no—
241C3×D5—22no—
242C3×D5—41no—
243C23×C3—51no—
244C23×C3—51no—
245C23×C3—102no—
246C23×C3—102no—
247C23×C3—201no—
248C23×C3C22⋊AGL(1,𝔽5)11yescharacteristic
249C22×D3—102no—
250C22×D3—202no—
251C22×D3—202no—
252C22×D3—202no—
253C2×C3×C4—22no—
254C2×C3×C4—51no—
255C2×C3×C4—51no—
256C2×C3×C4—102no—
257C2×C3×C4—102no—
258C2×C3×C4—201no—
259C2×C3×C4—201no—
260C2×C3×C4—201no—
261C2×C3×C4C22⋊AGL(1,𝔽5)11yescharacteristic
262C2×C3×C4C22×AGL(1,𝔽5)11yescharacteristic
263#67—204no—
264#67—204no—
265C2×Dic3—22no—
266C2×Dic3—42no—
267C2×Dic3—104no—
268C2×Dic3—202no—
269C2×Dic3—202no—
270D12—202no—
271D12—202no—
272C3×C8—202no—
273C22×C5—21no—
274C22×C5—21no—
275C22×C5#82512yes—
276C22×C5C2×(C2×D3)⋊C411yescharacteristic
277C2×D5—21no—
278C2×D5—21no—
279C2×D5—21no—
280C2×D5—21no—
281C2×D5—21no—
282C2×D5—44no—
283C2×D5—122no—
284C2×D5—122no—
285C2×D5—122no—
286C2×D5—244no—
287C2×D5#90812yes—
288AGL(1,𝔽5)—124no—
289C4×C5—21no—
290C4×C5—122no—
291C4×C5#80612yes—
292Dic5—21no—
293Dic5—21no—
294C24—51no—
295C24—301no—
296C2×D4—152no—
297C2×D4—302no—
298C2×D4—302no—
299C2×D4—301no—
300C2×D4—602no—
301C2×D4—604no—
302C22×C4—51no—
303C22×C4—51no—
304C22×C4—51no—
305C22×C4—51no—
306C22×C4—51no—
307C22×C4—61no—
308C22×C4—202no—
309C22×C4—302no—
310C22×C4D3×AGL(1,𝔽5)11yescharacteristic
311M16—202no—
312C2×C8—102no—
313C4⋊C4—32no—
314C4⋊C4—61no—
315C22⋊C4—302no—
316C22⋊C4—302no—
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#401012yes—
330C22×C3C2×C22⋊AGL(1,𝔽5)11yescharacteristic
331C2×D3—202no—
332C2×D3—202no—
333C2×D3—202no—
334C2×D3—404no—
335C3×C4—21no—
336C3×C4—101no—
337C3×C4—101no—
338C3×C4#399912yes—
339Dic3—42no—
340Dic3—204no—
341C2×C5—22no—
342C2×C5C2×#80611yescharacteristic
343C2×C5#513111yescharacteristic
344C2×C5C2×#82511yescharacteristic
345D5—22no—
346D5—41no—
347D5—242no—
348C23—51no—
349C23—51no—
350C23—102no—
351C23—102no—
352C23—201no—
353C23—302no—
354C23—602no—
355C23—602no—
356C23—602no—
357C23(C2×D3)⋊AGL(1,𝔽5)11yescharacteristic
358D4—602no—
359D4—604no—
360D4—604no—
361D4—602no—
362C2×C4—22no—
363C2×C4—51no—
364C2×C4—51no—
365C2×C4—62no—
366C2×C4—102no—
367C2×C4—102no—
368C2×C4—122no—
369C2×C4—201no—
370C2×C4—201no—
371C2×C4—201no—
372C2×C4—304no—
373C2×C4—602no—
374C2×C4—602no—
375C2×C4(C2×D3)⋊AGL(1,𝔽5)11yescharacteristic
376C2×C4C2×D3×AGL(1,𝔽5)11yescharacteristic
377C8—202no—
378C2×C3—22no—
379C2×C3—102no—
380C2×C3—201no—
381C2×C3C2×#399911yescharacteristic
382C2×C3#6576711yescharacteristic
383C2×C3C2×#401011yescharacteristic
384D3—402no—
385C5#6997711yescharacteristic, complemented, Sylow
386C22—21no—
387C22—21no—
388C22—52no—
389C22—101no—
390C22—101no—
391C22—101no—
392C22—101no—
393C22—101no—
394C22—204no—
395C22—602no—
396C22—602no—
397C22—602no—
398C22—1204no—
399C22#9115712yes—
400C22C2×(C2×D3)⋊AGL(1,𝔽5)11yescharacteristic
401C4—21no—
402C4—101no—
403C4—101no—
404C4—122no—
405C4—604no—
406C4#9117312yes—
407C3#1060425811yescharacteristic, complemented, Sylow
408C2—22no—
409C2—102no—
410C2—201no—
411C2—1202no—
412C2#1175170711yescharacteristic
413C2C2×#9115711yescharacteristic
414C2C2×#9117311yescharacteristic
subgroupG/H
centerC22C2×(C2×D3)⋊AGL(1,𝔽5)
commutatorC2×C3×C4×C5C22×C4
FrattiniC2×C4C2×D3×AGL(1,𝔽5)
FittingC22×C3×C4×C5C2×C4
socleC22×C3×C5C2×C22⋊C4
prime radicalC2×#91450C2
cosocleC2×C3×C4×D5C23
hypercenterC22×C4D3×AGL(1,𝔽5)
hypocenterC3×C5#1507
LCSG ▷ C2×C3×C4×C5 ▷ C22×C3×C5 ▷ C3×C5
UCSC1 ◁ C22 ◁ C22×C4

Complements

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

NH ≅ G/Ntype
2. C4×#6763C2semidirect
3. C3×#65756C2semidirect
4. #11746499C2semidirect
6. C22×D60C4semidirect
12. C2×C5×C4⋊Dic3C4semidirect
16. #91450C4semidirect
18. #91235C22semidirect
20. C3×#4003C22semidirect
24. #65756D3semidirect
69. #4003C2×D3semidirect
111. C2×C4⋊Dic3AGL(1,𝔽5)semidirect
319. C3×C5#1507semidirect
385. C5#69977semidirect
407. C3#10604258semidirect

Maximal subgroups

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

subgroupindex
C2×#914502
C4×#67632
C3×#657562
#117464992
#106042583
#699775