Finite Group Databasebeta

← #20248323

Subgroups — #20248323

Order 6016.

760 conjugacy classes of subgroups; only the 157 normal subgroups are shown.

normal subgroups only

This group has 760 conjugacy classes of subgroups — too many to draw. Only its 157 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×#13094010C211yesmaximal, characteristic, complemented
2C2×#13094002C212yesmaximal, complemented
3C2×#13093997C212yesmaximal, complemented
4C2×#13093986C211yesmaximal, characteristic, complemented
5#15689058C211yesmaximal, characteristic, complemented
6#15689033C211yesmaximal, characteristic, complemented
7#15689032C211yesmaximal, characteristic, complemented
8#15689024C212yesmaximal, complemented
9#15689023C212yesmaximal, complemented
10#15688974C211yesmaximal, characteristic, complemented
11C47×#399C211yesmaximal, characteristic, complemented
12C22×Dic94C2211yescharacteristic
13C22×D188C2211yescharacteristic
14C22×D188C2212yes—
15C22×C22⋊D47C2211yescharacteristic
16C22×C22⋊D47C2212yes—
17C22×C4×D47C2212yes—
18#13094010—24no—
19#13094002—28no—
20#13093997—28no—
21#13093986—24no—
22C2×Dic47⋊C4C2212yes—
23C2×Dic47⋊C4C2212yes—
24C2×C22⋊Dic47C2211yescharacteristic
25C2×(C2×D47)⋊C4C2212yes—
26C2×(C2×D47)⋊C4C2212yes—
27C2×(C2×D47)⋊C4C2211yescharacteristic
28C2×(C2×D47)⋊C4C2212yes—
29C2×(C2×D47)⋊C4C2212yes—
30C2×(C2×D47)⋊C4C2211yescharacteristic
31C2×C4×Dic47C2212yes—
32C2×C4×Dic47C2211yescharacteristic
33C47⋊#95C2211yescharacteristic
34C2×C4⋊C4×C47C2212yes—
35C2×C4⋊C4×C47C2211yescharacteristic
36C2×C22⋊C4×C47C2212yes—
37C2×C22⋊C4×C47C2211yescharacteristic
38#95×C47C2211yescharacteristic
39C24×C47C2311yescharacteristic
40C23×D47C2311yescharacteristic
41C23×D47C2312yes—
42C22×C4×C47C2311yescharacteristic
43C22×C4×C47C2312yes—
44C22×C4×C47C2311yescharacteristic
45C22×C4×C47C2312yes—
46C2×C22⋊D47—24no—
47C2×C22⋊D47—24no—
48C2×C22⋊D47—24no—
49C2×C22⋊D47—22no—
50C2×C22⋊D47—42no—
51C22×Dic47C2311yescharacteristic
52C22×Dic47C2312yes—
53C22×Dic47C2311yescharacteristic
54C22×Dic47C2311yescharacteristic
55C2×D188—24no—
56C2×D188—22no—
57C2×D188—24no—
58C2×D188—24no—
59C2×D188—42no—
60C2×C4×D47—24no—
61C2×C4×D47—24no—
62C2×C4×D47—24no—
63C2×Dic94—24no—
64C2×Dic94—22no—
65C4⋊C4×C47—22no—
66C4⋊C4×C47—24no—
67C22⋊C4×C47—24no—
68C22⋊C4×C47—22no—
69C22⋊Dic47—22no—
70(C2×D47)⋊C4—24no—
71(C2×D47)⋊C4—22no—
72(C2×D47)⋊C4—24no—
73(C2×D47)⋊C4—24no—
74(C2×D47)⋊C4—24no—
75(C2×D47)⋊C4—22no—
76Dic47⋊C4—24no—
77Dic47⋊C4D418yescomplemented
78C4×Dic47—42no—
79C4×Dic47D414yescomplemented
80C23×C47—21no—
81C23×C47—22no—
82C23×C47—22no—
83C23×C47—21no—
84C23×C47—21no—
85C23×C47C2411yescharacteristic
86C22×D47—22no—
87C22×D47—22no—
88C22×D47—22no—
89C22×D47—22no—
90C22×D47—22no—
91C22×D47—22no—
92C22×D47—21no—
93C22×D47—22no—
94C22×D47—22no—
95C22×D47—21no—
96C22×D47—21no—
97C22×D47#4114yes—
98C2×C4×C47—22no—
99C2×C4×C47—22no—
100C2×C4×C47—22no—
101C2×C4×C47—22no—
102C2×C4×C47—21no—
103C2×C4×C47—22no—
104C2×C4×C47—22no—
105C2×C4×C47—22no—
106C2×C4×C47C2×D412yes—
107C2×C4×C47C2×D414yes—
108C22⋊D47—48no—
109C22⋊D47—44no—
110C2×Dic47—22no—
111C2×Dic47—21no—
112C2×Dic47—22no—
113C2×Dic47—21no—
114C2×Dic47—22no—
115C2×Dic47—21no—
116C2×Dic47C2×D412yes—
117C2×Dic47C2×D414yes—
118C2×Dic47C2×D412yes—
119C2×Dic47C2×D414yes—
120D188—28no—
121D188—48no—
122C4×D47—48no—
123Dic94—44no—
124C22×C47—41no—
125C22×C47—41no—
126C22×C47—41no—
127C22×C47—42no—
128C22×C47—42no—
129C22×C47C22×D411yescharacteristic
130C22×C47C22×D412yes—
131C22×C47C2×#4112yes—
132C22×C47H(5,𝔽2)11yescharacteristic
133C22×C472−1+411yescharacteristic
134C2×D47—24no—
135C2×D47—24no—
136C2×D47—24no—
137C2×D47—42no—
138C2×D47—42no—
139C2×D47—42no—
140C2×D47—42no—
141C2×D47—41no—
142C2×D47—41no—
143C2×D47—41no—
144C2×D47—42no—
145C2×D47—42no—
146C4×C47—22no—
147C4×C47—24no—
148C4×C47—41no—
149C4×C47—42no—
150Dic47—22no—
151Dic47—24no—
152Dic47—41no—
153Dic47#12714yes—
154#2540—471nomaximal, Sylow
155C2×C47—81no—
156C2×C47#52511yescharacteristic
157C2×C47C2×#12711yescharacteristic
158C2×C47#53711yescharacteristic
159C2×C47#54612yes—
160C2×C47#54812yes—
161D47—44no—
162D47—81no—
163C2×#124—471no—
164C2×C22⋊Q8—472no—
165C2×#121—472no—
166C2×C4×D4—471no—
167#399—471no—
168#399D4711yescharacteristic, complemented
169#392—471no—
170#392—471no—
171#390—472no—
172#390—472no—
173C4×C4⋊C4—471no—
174C47#254011yescharacteristic, complemented, Sylow
175C22×Q8—471no—
176C22×D4—471no—
177C22×D4—472no—
178C22×D4—471no—
179C22×D4—472no—
180C23×C4—472no—
181#124—944no—
182C22⋊Q8—948no—
183#121—948no—
184C4×D4—944no—
185C2×C4⋊C4—472no—
186C2×C4⋊C4—472no—
187C2×C4⋊C4C2×D4711yescharacteristic
188C2×C4⋊C4C2×D4712yes—
189C2×C22⋊C4—471no—
190C2×C22⋊C4—472no—
191C2×C22⋊C4—472no—
192C2×C22⋊C4—472no—
193C2×C22⋊C4—472no—
194C2×C22⋊C4—471no—
195C2×C22⋊C4—471no—
196C2×C22⋊C4C2×D4712yes—
197C2×C22⋊C4C2×D4711yescharacteristic
198C2×C42—471no—
199C2×C42—472no—
200#95—471no—
201#95C2×D4711yescharacteristic
202C24—472no—
203C24—471no—
204C24C22×D4711yescharacteristic
205C2×Q8—944no—
206C2×Q8—942no—
207C2×D4—944no—
208C2×D4—944no—
209C2×D4—944no—
210C2×D4—944no—
211C2×D4—942no—
212C2×D4—944no—
213C2×D4—942no—
214C2×D4—944no—
215C2×D4—1882no—
216C2×D4—1882no—
217C22×C4—471no—
218C22×C4—472no—
219C22×C4—471no—
220C22×C4—471no—
221C22×C4—944no—
222C22×C4—944no—
223C22×C4—944no—
224C22×C4C22×D4711yescharacteristic
225C22×C4C22×D4712yes—
226C22×C4C22×D4711yescharacteristic
227C22×C4C22×D4712yes—
228C4⋊C4—22no—
229C4⋊C4—24no—
230C4⋊C4—478no—
231C4⋊C4—944no—
232C22⋊C4—24no—
233C22⋊C4—22no—
234C22⋊C4—944no—
235C22⋊C4—944no—
236C22⋊C4—944no—
237C22⋊C4—942no—
238C22⋊C4—942no—
239C22⋊C4—944no—
240C22⋊C4—942no—
241C42—474no—
242C42—1882no—
243C23—21no—
244C23—22no—
245C23—22no—
246C23—21no—
247C23—21no—
248C23—474no—
249C23—942no—
250C23—942no—
251C23—942no—
252C23—942no—
253C23—942no—
254C23—942no—
255C23—941no—
256C23—942no—
257C23—942no—
258C23—941no—
259C23—941no—
260C23C23×D4711yescharacteristic
261Q8—1884no—
262D4—948no—
263D4—1888no—
264D4—1884no—
265D4—1888no—
266C2×C4—22no—
267C2×C4—22no—
268C2×C4—22no—
269C2×C4—22no—
270C2×C4—21no—
271C2×C4—22no—
272C2×C4—22no—
273C2×C4—22no—
274C2×C4—472no—
275C2×C4—474no—
276C2×C4—472no—
277C2×C4—474no—
278C2×C4—942no—
279C2×C4—941no—
280C2×C4—942no—
281C2×C4—941no—
282C2×C4—941no—
283C2×C4—942no—
284C2×C4—1888no—
285C2×C4D4×D4712yes—
286C2×C4D4×D4714yes—
287C22—42no—
288C22—42no—
289C22—41no—
290C22—41no—
291C22—41no—
292C22—944no—
293C22—944no—
294C22—944no—
295C22—1882no—
296C22—1882no—
297C22—1881no—
298C22—1881no—
299C22—1881no—
300C22—1882no—
301C22—1882no—
302C22—1882no—
303C22—1882no—
304C22C2×D4×D4712yes—
305C22C2×D4×D4711yescharacteristic
306C22#41×D4712yes—
307C22#1309407011yescharacteristic
308C22#1309407211yescharacteristic
309C4—24no—
310C4—22no—
311C4—42no—
312C4—41no—
313C4—474no—
314C4—944no—
315C4—942no—
316C4—1881no—
317C2—81no—
318C2—3761no—
319C2—1884no—
320C2#1568963111yescharacteristic
321C2#127×D4711yescharacteristic
322C2#1568970311yescharacteristic
323C2#1568976512yes—
324C2#1568978312yes—
subgroupG/H
center, FrattiniC23C23×D47
commutator, socleC23×C47C24
FittingC47×#399C2
hypercenter#399D47
hypocenterC47#2540
LCSG ▷ C23×C47 ▷ C47
UCSC1 ◁ C23 ◁ #399

Complements

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

NH ≅ G/Ntype
1. C2×#13094010C2semidirect
2. C2×#13094002C2semidirect
3. C2×#13093997C2semidirect
4. C2×#13093986C2semidirect
5. #15689058C2semidirect
6. #15689033C2semidirect
7. #15689032C2semidirect
8. #15689024C2semidirect
9. #15689023C2semidirect
10. #15688974C2semidirect
11. C47×#399C2semidirect
77. Dic47⋊C4D4semidirect
79. C4×Dic47D4semidirect
168. #399D47semidirect
174. C47#2540semidirect