Finite Group Databasebeta

← #14283167

Subgroups — #14283167

Order 1920.

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

normal subgroups only

This group has 588 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×#4092C211yesmaximal, characteristic
2C4×#6763C211yesmaximal, characteristic
3#11752415C211yesmaximal, characteristic
4#11746797C214yesmaximal, complemented
5#10604956—31nomaximal, Hall
6C22×C3×C5×Q8C411yescharacteristic, complemented
7C22×C3×C4×D5C2211yescharacteristic
8C2×#6763—21no—
9C2×C4×ΓL(1,𝔽16)—21no—
10C2×#6758—21no—
11C3×#4092—21no—
12C3×#4092—21no—
13C3×#4092C412yescomplemented
14C3×#4092C2212yes—
15C2×C3×#685—21no—
16C2×C3×C4⋊Dic5C411yescharacteristic, complemented
17C2×C3×#683—21no—
18#91241—21no—
19#91235C2212yes—
20#91232C2212yes—
21#71688—51nomaximal, Hall
22C2×#4092D311yescharacteristic
23C4×C22⋊AGL(1,𝔽5)—31no—
24#65756—31no—
25#64926—34no—
26C23×C3×D5D411yescharacteristic
27C22×ΓL(1,𝔽16)—22no—
28C2×C3×C5×Q8—21no—
29C2×C3×C5×Q8—42no—
30C2×C3×C5×Q8C2×C412yes—
31C22×C3×C4×C5—21no—
32C22×C3×C4×C5C2×C411yescharacteristic
33C22×C3×Dic5—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
40#6763—24no—
41C2×C3⋊#170—22no—
42C4×ΓL(1,𝔽16)—42no—
43#6758—42no—
44C3×#685—22no—
45C3×#685—22no—
46C3×C4⋊Dic5—21no—
47C3×C4⋊Dic5C2×C412yes—
48C3×#683—22no—
49C3×#683—22no—
50C2×C3×C22⋊Q8—51no—
51C4×#217—51no—
52#5207—51no—
53#4538—54no—
54C22×C5×Q8Dic311yescharacteristic, complemented
55C22×C4×D5C2×D311yescharacteristic
56C2×C22⋊AGL(1,𝔽5)—61no—
57C2×C4×AGL(1,𝔽5)—61no—
58C2×#700—61no—
59#4092—21no—
60#4092—21no—
61#4092Dic312yescomplemented
62#4092C2×D312yes—
63C2×#685—21no—
64C2×C4⋊Dic5Dic311yescharacteristic, complemented
65C2×#683—21no—
66C22⋊#170—61no—
67(C2×AGL(1,𝔽5))⋊C4—32no—
68#4003—32no—
69#1508—151noSylow
70C23×C3×C5C22⋊C411yescharacteristic
71C22×C3×D5—22no—
72C22×C3×D5—22no—
73C22×C3×D5—41no—
74C22×C3×D5C22⋊C411yescharacteristic
75C22×C3×D5C2×D411yescharacteristic
76C2×ΓL(1,𝔽16)—24no—
77C2×ΓL(1,𝔽16)—42no—
78C2×ΓL(1,𝔽16)—42no—
79C3×C5×Q8—42no—
80C3×C5×Q8—42no—
81C2×C3×C4×C5—22no—
82C2×C3×C4×C5—22no—
83C2×C3×C4×C5—42no—
84C2×C3×C4×C5C22⋊C411yescharacteristic
85C2×C3×C4×C5C22×C411yescharacteristic
86C2×C3×Dic5—22no—
87C2×C3×Dic5—41no—
88C2×C3×Dic5—42no—
89C2×C3×Dic5C22⋊C411yescharacteristic
90C2×C3×Dic5C2×D411yescharacteristic
91C3×C4×D5—22no—
92C3×C4×D5—22no—
93C3×C4×D5—41no—
94C3×C4×D5—41no—
95C3⋊#170—42no—
96C22×C3×Q8AGL(1,𝔽5)11yescharacteristic, complemented
97C23×C3×C4—51no—
98C3×C22⋊Q8—52no—
99C3×C22⋊Q8—52no—
100C3×C22⋊Q8—101no—
101C3×C22⋊Q8—101no—
102C2×C3×C4⋊C4—51no—
103C2×C3×C4⋊C4—101no—
104C2×C3×C22⋊C4—101no—
105C2×#217—101no—
106C2×C4×Dic3—101no—
107C2×#208—101no—
108C3⋊#95—52no—
109#830—52no—
110#830—101no—
111C23×D5#6711yescharacteristic
112C22×AGL(1,𝔽5)—62no—
113C2×C5×Q8—21no—
114C2×C5×Q8—42no—
115C2×C5×Q8C2×Dic312yes—
116C22×C4×C5—21no—
117C22×C4×C5C2×Dic311yescharacteristic
118C22×Dic5—21no—
119C22×Dic5#6711yescharacteristic
120C2×C4×D5—42no—
121C2×C4×D5C2×Dic311yescharacteristic
122C2×C4×D5#6711yescharacteristic
123C2×C4×D5#6711yescharacteristic
124C2×C4×D5C22×D311yescharacteristic
125C22⋊AGL(1,𝔽5)—64no—
126C2×#170—62no—
127C4×AGL(1,𝔽5)—122no—
128#700—122no—
129#685—22no—
130#685—22no—
131C4⋊Dic5—21no—
132C4⋊Dic5C2×Dic312yes—
133#683—22no—
134#683—22no—
135C2×C22⋊Q8—51no—
136#407—151no—
137C4×C22⋊C4—151no—
138#328—154no—
139C22×C3×C5—21no—
140C22×C3×C5—21no—
141C22×C3×C5#10412yes—
142C22×C3×C5C2×C22⋊C411yescharacteristic
143C2×C3×D5—21no—
144C2×C3×D5—21no—
145C2×C3×D5—21no—
146C2×C3×D5—21no—
147C2×C3×D5—21no—
148C2×C3×D5—44no—
149C2×C3×D5#13712yes—
150ΓL(1,𝔽16)—44no—
151C3×C4×C5—21no—
152C3×C4×C5—42no—
153C3×C4×C5#9912yes—
154C3×Dic5—21no—
155C3×Dic5—21no—
156C3×Dic5—42no—
157C24×C3—51no—
158C2×C3×Q8—21no—
159C2×C3×Q8—42no—
160C2×C3×Q8C2×AGL(1,𝔽5)12yes—
161C22×C3×C4—21no—
162C22×C3×C4—51no—
163C22×C3×C4—51no—
164C22×C3×C4—51no—
165C22×C3×C4—51no—
166C22×C3×C4—51no—
167C22×C3×C4—101no—
168C22×C3×C4—202no—
169C22×C3×C4C2×AGL(1,𝔽5)11yescharacteristic
170C22×Dic3—102no—
171C3×C4⋊C4—52no—
172C3×C4⋊C4—102no—
173C3×C4⋊C4—102no—
174C3×C4⋊C4—101no—
175C3×C22⋊C4—102no—
176C3×C22⋊C4—102no—
177#217—104no—
178C4×Dic3—202no—
179#208—202no—
180C2×C3⋊C8—102no—
181C23×C5#21711yescharacteristic
182C22×D5—22no—
183C22×D5—22no—
184C22×D5—41no—
185C22×D5#21711yescharacteristic
186C22×D5C2×#6711yescharacteristic
187C2×AGL(1,𝔽5)—64no—
188C2×AGL(1,𝔽5)—122no—
189C2×AGL(1,𝔽5)—122no—
190C5×Q8—42no—
191C5×Q8—42no—
192C2×C4×C5—22no—
193C2×C4×C5—22no—
194C2×C4×C5—42no—
195C2×C4×C5#21711yescharacteristic
196C2×C4×C5C22×Dic311yescharacteristic
197C2×Dic5—22no—
198C2×Dic5—41no—
199C2×Dic5—42no—
200C2×Dic5#21711yescharacteristic
201C2×Dic5C2×#6711yescharacteristic
202C4×D5—22no—
203C4×D5—22no—
204C4×D5—41no—
205C4×D5—41no—
206#170—122no—
207C22×Q8ΓL(1,𝔽16)11yescharacteristic, complemented
208C23×C4—51no—
209C2×M16—301no—
210C22⋊Q8—52no—
211C22⋊Q8—52no—
212C22⋊Q8—101no—
213C22⋊Q8—101no—
214C2×C4⋊C4—51no—
215C2×C4⋊C4—101no—
216C2×C22⋊C4—101no—
217C2×C22⋊C4—301no—
218C2×C42—301no—
219C22⋊C8—152no—
220C22⋊C8—301no—
221#95—152no—
222C2×C3×C5—22no—
223C2×C3×C5C2×#9911yescharacteristic
224C2×C3×C5#41911yescharacteristic
225C2×C3×C5C2×#10411yescharacteristic
226C3×D5—22no—
227C3×D5—41no—
228C23×C3—51no—
229C23×C3—51no—
230C23×C3—102no—
231C23×C3—102no—
232C23×C3—201no—
233C23×C3C22⋊AGL(1,𝔽5)11yescharacteristic
234C3×Q8—42no—
235C3×Q8—42no—
236C2×C3×C4—22no—
237C2×C3×C4—22no—
238C2×C3×C4—42no—
239C2×C3×C4—51no—
240C2×C3×C4—51no—
241C2×C3×C4—102no—
242C2×C3×C4—102no—
243C2×C3×C4—102no—
244C2×C3×C4—201no—
245C2×C3×C4—201no—
246C2×C3×C4—201no—
247C2×C3×C4—202no—
248C2×C3×C4C22⋊AGL(1,𝔽5)11yescharacteristic
249C2×C3×C4C22×AGL(1,𝔽5)11yescharacteristic
250C2×Dic3—104no—
251C2×Dic3—202no—
252C2×Dic3—202no—
253C3⋊C8—202no—
254C22×C5—21no—
255C22×C5—21no—
256C22×C5#83712yes—
257C22×C5C2×#21711yescharacteristic
258C2×D5—21no—
259C2×D5—21no—
260C2×D5—21no—
261C2×D5—21no—
262C2×D5—21no—
263C2×D5—44no—
264C2×D5#94212yes—
265AGL(1,𝔽5)—124no—
266C4×C5—21no—
267C4×C5—42no—
268C4×C5#83412yes—
269Dic5—21no—
270Dic5—21no—
271Dic5—42no—
272C24—51no—
273C2×Q8—21no—
274C2×Q8—42no—
275C2×Q8C2×ΓL(1,𝔽16)12yes—
276C22×C4—21no—
277C22×C4—51no—
278C22×C4—51no—
279C22×C4—51no—
280C22×C4—51no—
281C22×C4—51no—
282C22×C4—101no—
283C22×C4—202no—
284C22×C4—302no—
285C22×C4C2×ΓL(1,𝔽16)11yescharacteristic
286M16—602no—
287C2×C8—302no—
288C4⋊C4—52no—
289C4⋊C4—102no—
290C4⋊C4—102no—
291C4⋊C4—101no—
292C22⋊C4—102no—
293C22⋊C4—102no—
294C22⋊C4—304no—
295C42—602no—
296C3×C5#150811yescharacteristic, complemented, Hall
297C22×C3—21no—
298C22×C3—21no—
299C22×C3—52no—
300C22×C3—101no—
301C22×C3—101no—
302C22×C3—101no—
303C22×C3—101no—
304C22×C3—101no—
305C22×C3—204no—
306C22×C3#401012yes—
307C22×C3C2×C22⋊AGL(1,𝔽5)11yescharacteristic
308C3×C4—21no—
309C3×C4—42no—
310C3×C4—101no—
311C3×C4—101no—
312C3×C4—202no—
313C3×C4#401112yes—
314Dic3—204no—
315C2×C5—22no—
316C2×C5C2×#83411yescharacteristic
317C2×C5#522511yescharacteristic
318C2×C5C2×#83711yescharacteristic
319D5—22no—
320D5—41no—
321C23—51no—
322C23—51no—
323C23—102no—
324C23—102no—
325C23—201no—
326C23#676311yescharacteristic
327Q8—42no—
328Q8—42no—
329C2×C4—22no—
330C2×C4—22no—
331C2×C4—42no—
332C2×C4—51no—
333C2×C4—51no—
334C2×C4—102no—
335C2×C4—102no—
336C2×C4—102no—
337C2×C4—201no—
338C2×C4—201no—
339C2×C4—201no—
340C2×C4—202no—
341C2×C4—304no—
342C2×C4—602no—
343C2×C4—602no—
344C2×C4#676311yescharacteristic
345C2×C4C22×ΓL(1,𝔽16)11yescharacteristic
346C8—602no—
347C2×C3—22no—
348C2×C3—102no—
349C2×C3—201no—
350C2×C3#6578211yescharacteristic
351C2×C3C2×#401011yescharacteristic
352C2×C3C2×#401111yescharacteristic
353C5#7168811yescharacteristic, complemented, Sylow
354C22—21no—
355C22—21no—
356C22—52no—
357C22—101no—
358C22—101no—
359C22—101no—
360C22—101no—
361C22—101no—
362C22—204no—
363C22#9123912yes—
364C22C2×#676311yescharacteristic
365C4—21no—
366C4—42no—
367C4—101no—
368C4—101no—
369C4—202no—
370C4—604no—
371C4#9124012yes—
372C3#1060495611yescharacteristic, complemented, Sylow
373C2—22no—
374C2—102no—
375C2—201no—
376C2#1175244111yescharacteristic
377C2C2×#9123911yescharacteristic
378C2C2×#9124011yescharacteristic
subgroupG/H
centerC22C2×#6763
commutatorC2×C3×C4×C5C22×C4
FrattiniC2×C4C22×ΓL(1,𝔽16)
FittingC22×C3×C5×Q8C4
socleC22×C3×C5C2×C22⋊C4
prime radicalC23×C3×D5D4
cosocleC2×C3×C4×D5C23
hypercenterC22×Q8ΓL(1,𝔽16)
hypocenterC3×C5#1508
LCSG ▷ C2×C3×C4×C5 ▷ C22×C3×C5 ▷ C3×C5
UCSC1 ◁ C22 ◁ C22×C4 ◁ C22×Q8

Complements

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

NH ≅ G/Ntype
4. #11746797C2semidirect
6. C22×C3×C5×Q8C4semidirect
13. C3×#4092C4semidirect
16. C2×C3×C4⋊Dic5C4semidirect
54. C22×C5×Q8Dic3semidirect
61. #4092Dic3semidirect
64. C2×C4⋊Dic5Dic3semidirect
96. C22×C3×Q8AGL(1,𝔽5)semidirect
207. C22×Q8ΓL(1,𝔽16)semidirect
296. C3×C5#1508semidirect
353. C5#71688semidirect
372. C3#10604956semidirect

Maximal subgroups

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

subgroupindex
C2×C3×#40922
C4×#67632
#117524152
#117467972
#106049563
#716885