Finite Group Databasebeta

← #14313652

Subgroups — #14313652

Order 1920.

500 conjugacy classes of subgroups; only the 96 normal subgroups are shown.

normal subgroups only

This group has 500 conjugacy classes of subgroups — too many to draw. Only its 96 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
1C3×#66059C211yesmaximal, characteristic
2#11749369C212yesmaximal, complemented
3#11746369C211yesmaximal, characteristic, complemented
4C5×#4548C211yesmaximal, characteristic, complemented
5#11746012C212yesmaximal, complemented
6#10599965—31nomaximal, Hall
7C2×C3×#713C411yescharacteristic
8C2×C3×#713C2211yescharacteristic
9C3×C5×C4⋊Q8C2211yescharacteristic
10C3×#4093C411yescharacteristic
11C3×#4093C2211yescharacteristic
12C3×#4041—22no—
13C3×#4039—22no—
14C3×#4020C411yescharacteristic
15C3×C42×D5C411yescharacteristic
16D5×#208—22no—
17#91120C2212yes—
18C5×#836C2212yes—
19#90958—22no—
20#71820—51nomaximal, Hall
21#66059D311yescharacteristic
22#65040—32no—
23C5×#356—31no—
24#64768—31no—
25#64697—32no—
26C2×C3×C5×Q8C2×C411yescharacteristic
27C2×C3×C5×Q8C2311yescharacteristic
28C3×#713—41no—
29C3×#713—41no—
30C3×#713—41no—
31C3×#713—41no—
32C2×C3×D20—21no—
33C2×C3×D20—22no—
34C2×C3×D20C2×C411yescharacteristic
35C2×C3×D20C2×C411yescharacteristic
36C2×C3×D20D412yes—
37C2×C3×C4×D5—21no—
38C2×C3×C4×D5—21no—
39C2×C3×C4×D5C2×C411yescharacteristic
40C2×C3×C4×D5D412yes—
41C3×C5×C4⋊C4—22no—
42C3×C42×C5C2×C411yescharacteristic
43#6710—22no—
44C5×#208—22no—
45C3×#685—22no—
46C3×#685—22no—
47C3×C4×Dic5—21no—
48C3×C4×Dic5C2×C411yescharacteristic
49#6649—42no—
50D5×C3⋊C8—42no—
51C3×#532—51no—
52#5237—52no—
53#4548—51no—
54#4548D511yescharacteristic, complemented
55#4542—52no—
56C2×#713Dic311yescharacteristic
57C2×#713C2×D311yescharacteristic
58C5×C4⋊Q8C2×D311yescharacteristic
59#4093Dic311yescharacteristic
60#4093C2×D311yescharacteristic
61D5×M16—62no—
62#4041—22no—
63#4039—22no—
64#4020Dic311yescharacteristic
65C42×D5Dic311yescharacteristic
66C5×#101—32no—
67#3968—32no—
68#3955—62no—
69#2132—151noSylow
70C22×C3×D5—21no—
71C22×C3×D5C22⋊C412yes—
72C22×C3×D5C22⋊C411yescharacteristic
73C3×C5×Q8—41no—
74C3×C5×Q8—41no—
75C2×C3×C4×C5—21no—
76C2×C3×C4×C5—21no—
77C2×C3×C4×C5C22×C411yescharacteristic
78C2×C3×C4×C5C2×D412yes—
79C2×C3×Dic5—21no—
80C2×C3×Dic5C22⋊C411yescharacteristic
81C3×D20—41no—
82C3×D20—42no—
83C3×D20—44no—
84C3×D20—41no—
85C3×D20—42no—
86C3×C4×D5—21no—
87C3×C4×D5—21no—
88C3×C4×D5—22no—
89C3×C4×D5—22no—
90C3×C4×D5—42no—
91C3×C4×D5—41no—
92C3×C4×D5—41no—
93C3⋊#168—42no—
94C5×C3⋊C8—42no—
95C2×C3×#41—51no—
96C2×C3×#41—51no—
97C3×C4⋊Q8C2×D511yescharacteristic
98C3×#127—51no—
99C3×#124—51no—
100C3×#124—51no—
101C3×#121—102no—
102C3×C4×D4—102no—
103C2×C3×C42—51no—
104C2×#208—102no—
105#836—52no—
106#836C2×D512yes—
107#833—102no—
108C2×C5×Q8C2×Dic311yescharacteristic
109C2×C5×Q8C22×D311yescharacteristic
110#713—41no—
111#713—41no—
112#713—41no—
113#713—41no—
114C2×D20—21no—
115C2×D20—22no—
116C2×D20C2×Dic311yescharacteristic
117C2×D20C2×Dic311yescharacteristic
118C2×D20#6712yes—
119C2×C4×D5—21no—
120C2×C4×D5—21no—
121C2×C4×D5C2×Dic311yescharacteristic
122C2×C4×D5#6712yes—
123C5×M16—62no—
124C5×C4⋊C4—22no—
125C42×C5C2×Dic311yescharacteristic
126#685—22no—
127#685—22no—
128C4×Dic5—21no—
129C4×Dic5C2×Dic311yescharacteristic
130#681—62no—
131#676—122no—
132C8×D5—122no—
133#532—51no—
134#413—152no—
135#356—31no—
136#356—151no—
137#355—152no—
138C22×C3×C5C2×C22⋊C411yescharacteristic
139C2×C3×D5—21no—
140C2×C3×D5—22no—
141C2×C3×D5—41no—
142C2×C3×D5—42no—
143C2×C3×D5—42no—
144C2×C3×D5—41no—
145C3×C4×C5—22no—
146C3×C4×C5—22no—
147C3×C4×C5—21no—
148C3×C4×C5—41no—
149C3×Dic5—41no—
150C3×Dic5#9911yescharacteristic
151C3×Dic5#9911yescharacteristic
152C3×#41—201no—
153C3×#41—201no—
154C3×#41—201no—
155C3×#41—201no—
156C2×C3×Q8C4×D511yescharacteristic
157C2×C3×Q8C22×D511yescharacteristic
158C2×C3×D4—51no—
159C2×C3×D4—51no—
160C2×C3×D4—52no—
161C2×C3×D4—102no—
162C2×C3×D4—101no—
163C22×C3×C4—51no—
164C22×C3×C4—52no—
165C22×C3×C4—101no—
166C22×C3×C4—101no—
167C3×C4⋊C4—22no—
168C3×C22⋊C4—102no—
169C3×C22⋊C4—102no—
170C3×C42—51no—
171C3×C42—101no—
172C3×C42C4×D511yescharacteristic
173#208—22no—
174#208—102no—
175#208—202no—
176C2×C3⋊C8—202no—
177C22×D5—21no—
178C22×D5#21712yes—
179C22×D5#21711yescharacteristic
180C5×Q8—41no—
181C5×Q8—41no—
182C2×C4×C5—21no—
183C2×C4×C5—21no—
184C2×C4×C5C22×Dic311yescharacteristic
185C2×C4×C5C2×#6712yes—
186C2×Dic5—21no—
187C2×Dic5#21711yescharacteristic
188D20—41no—
189D20—42no—
190D20—41no—
191D20—44no—
192D20—42no—
193C4×D5—22no—
194C4×D5—22no—
195C4×D5—21no—
196C4×D5—21no—
197C4×D5—42no—
198C4×D5—41no—
199C4×D5—41no—
200C5×C8—122no—
201#168—122no—
202C2×#41—51no—
203C2×#41—51no—
204C2×M16—302no—
205C4⋊Q8D3×D511yescharacteristic
206#127—51no—
207#124—51no—
208#124—51no—
209#121—102no—
210C4×D4—102no—
211C2×C42—51no—
212#101—32no—
213#101—152no—
214#100—302no—
215C2×C3×C5—21no—
216C2×C3×C5C2×#9911yescharacteristic
217C3×D5—41no—
218C3×D5—42no—
219C3×D5—81no—
220C23×C3—52no—
221C23×C3—51no—
222C23×C3—101no—
223C3×Q8—41no—
224C3×Q8—41no—
225C3×D4—202no—
226C3×D4—201no—
227C3×D4—201no—
228C3×D4—204no—
229C3×D4—202no—
230C2×C3×C4—21no—
231C2×C3×C4—21no—
232C2×C3×C4—51no—
233C2×C3×C4—102no—
234C2×C3×C4—102no—
235C2×C3×C4—101no—
236C2×C3×C4—101no—
237C2×C3×C4—101no—
238C2×C3×C4—202no—
239C2×C3×C4—201no—
240C2×C3×C4—201no—
241C2×C3×C4C2×C4×D511yescharacteristic
242C2×C3×C4D4×D512yes—
243C3⋊C8—42no—
244C3⋊C8—202no—
245C22×C5C2×#21711yescharacteristic
246C2×D5—21no—
247C2×D5—22no—
248C2×D5—41no—
249C2×D5—42no—
250C2×D5—42no—
251C2×D5—41no—
252C4×C5—22no—
253C4×C5—22no—
254C4×C5—21no—
255C4×C5—41no—
256Dic5—41no—
257Dic5#83411yescharacteristic
258Dic5#83411yescharacteristic
259#41—201no—
260#41—201no—
261#41—201no—
262#41—201no—
263C2×Q8D5×Dic311yescharacteristic
264C2×Q8C2×D3×D511yescharacteristic
265C2×D4—51no—
266C2×D4—51no—
267C2×D4—52no—
268C2×D4—102no—
269C2×D4—101no—
270C22×C4—51no—
271C22×C4—52no—
272C22×C4—101no—
273C22×C4—101no—
274M16—62no—
275M16—302no—
276M16—602no—
277C2×C8—602no—
278C4⋊C4—22no—
279C22⋊C4—102no—
280C22⋊C4—102no—
281C42—51no—
282C42—101no—
283C42D5×Dic311yescharacteristic
284C3×C5#213211yescharacteristic, complemented, Hall
285C22×C3—101no—
286C22×C3—102no—
287C22×C3—201no—
288C22×C3—202no—
289C22×C3—202no—
290C22×C3—201no—
291C22×C3D5×C22⋊C411yescharacteristic
292C3×C4—22no—
293C3×C4—22no—
294C3×C4—21no—
295C3×C4—41no—
296C3×C4—51no—
297C3×C4—51no—
298C3×C4—201no—
299C2×C5—21no—
300C2×C5C2×#83411yescharacteristic
301D5—41no—
302D5—42no—
303D5—81no—
304C23—52no—
305C23—51no—
306C23—101no—
307Q8—41no—
308Q8—41no—
309D4—202no—
310D4—201no—
311D4—201no—
312D4—204no—
313D4—202no—
314C2×C4—21no—
315C2×C4—21no—
316C2×C4—51no—
317C2×C4—102no—
318C2×C4—102no—
319C2×C4—101no—
320C2×C4—101no—
321C2×C4—101no—
322C2×C4—202no—
323C2×C4—201no—
324C2×C4—201no—
325C2×C4C2×D5×Dic311yescharacteristic
326C2×C4D5×#6712yes—
327C8—122no—
328C8—602no—
329C2×C3—21no—
330C2×C3—201no—
331C2×C3—202no—
332C2×C3—401no—
333C2×C3D5×#9911yescharacteristic
334C5#7182011yescharacteristic, complemented, Sylow
335C22—101no—
336C22—102no—
337C22—201no—
338C22—202no—
339C22—202no—
340C22—201no—
341C22D5×#21711yescharacteristic
342C4—22no—
343C4—22no—
344C4—21no—
345C4—41no—
346C4—51no—
347C4—51no—
348C4—201no—
349C3#1059996511yescharacteristic, complemented, Sylow
350C2—21no—
351C2—201no—
352C2—202no—
353C2—401no—
354C2D5×#83411yescharacteristic
subgroupG/H
centerC2D5×#834
commutatorC2×C3×C4×C5C22×C4
FrattiniC2×Q8C2×D3×D5
FittingC3×C5×C4⋊Q8C22
socleC2×C3×C5C2×#99
prime radicalC3×#66059C2
cosocleC2×C3×C5×Q8C23
hypercenterC4⋊Q8D3×D5
hypocenterC3×C5#2132
LCSG ▷ C2×C3×C4×C5 ▷ C22×C3×C5 ▷ C2×C3×C5 ▷ C3×C5
UCSC1 ◁ C2 ◁ C22 ◁ C2×Q8 ◁ C4⋊Q8

Complements

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

NH ≅ G/Ntype
2. #11749369C2semidirect
3. #11746369C2semidirect
4. C5×#4548C2semidirect
5. #11746012C2semidirect
54. #4548D5semidirect
284. C3×C5#2132semidirect
334. C5#71820semidirect
349. C3#10599965semidirect

Maximal subgroups

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

subgroupindex
C3×#660592
#117493692
#117463692
C5×#45482
#117460122
#105999653
#718205