Finite Group Databasebeta

← #20246998

Subgroups — #20246998

Order 6016.

914 conjugacy classes of subgroups; only the 173 normal subgroups are shown.

normal subgroups only

This group has 914 conjugacy classes of subgroups — too many to draw. Only its 173 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
1C22×#10625350C211yesmaximal, characteristic, complemented
2C2×#13094007C211yesmaximal, characteristic, complemented
3C2×#13093998C212yesmaximal, complemented
4#15689044C211yesmaximal, characteristic, complemented
5#15689041C211yesmaximal, characteristic, complemented
6#15689036C211yesmaximal, characteristic, complemented
7#15689033C211yesmaximal, characteristic, complemented
8#15689002C212yesmaximal, complemented
9#15689001C212yesmaximal, complemented
10#15688997C212yesmaximal, complemented
11C2×#95×C47C211yesmaximal, characteristic, complemented
12C2×#10625350—24no—
13C2×#10625350—28no—
14C22×Dic94C2211yescharacteristic
15C22×D188C2211yescharacteristic
16C22×D188C2212yes—
17C22×C22⋊D47C2211yescharacteristic
18C22×C22⋊D47C2212yes—
19C22×C4×D47C2212yes—
20#13094007—24no—
21#13093998—28no—
22C2×C4⋊Dic47C2212yes—
23C2×Dic47⋊C4C2211yescharacteristic
24C2×Dic47⋊C4C2211yescharacteristic
25C2×C22⋊Dic47C2211yescharacteristic
26C2×(C2×D47)⋊C4C2212yes—
27C2×(C2×D47)⋊C4C2212yes—
28C2×(C2×D47)⋊C4C2211yescharacteristic
29C2×(C2×D47)⋊C4C2212yes—
30C2×(C2×D47)⋊C4C2211yescharacteristic
31C2×(C2×D47)⋊C4C2212yes—
32C47⋊#95C2212yes—
33C47⋊#95C2212yes—
34C23×C4×C47C2212yes—
35C23×C4×C47C2211yescharacteristic
36#95×C47C2212yes—
37#95×C47C2211yescharacteristic
38#95×C47C2211yescharacteristic
39C24×C47C2311yescharacteristic
40C23×D47C2311yescharacteristic
41C23×D47C2312yes—
42C22×C4×C47—24no—
43C22×C4×C47—24no—
44C22×C4×C47—24no—
45C22×C4×C47—42no—
46C22×C4×C47D414yescomplemented
47C22×C4×C47C2312yes—
48C22×C4×C47C2311yescharacteristic
49C22×C4×C47C2311yescharacteristic
50C22×C4×C47C2312yes—
51C2×C22⋊D47—24no—
52C2×C22⋊D47—24no—
53C2×C22⋊D47—24no—
54C2×C22⋊D47—22no—
55C2×C22⋊D47D418yescomplemented
56C22×Dic47C2311yescharacteristic
57C22×Dic47C2311yescharacteristic
58C22×Dic47C2311yescharacteristic
59C22×Dic47C2312yes—
60#10625350—232no—
61C2×D188—24no—
62C2×D188—22no—
63C2×D188—24no—
64C2×D188—24no—
65C2×D188—42no—
66C2×C4×D47—24no—
67C2×C4×D47—24no—
68C2×C4×D47—24no—
69C2×Dic94—22no—
70C2×Dic94—24no—
71C22⋊Dic47—22no—
72(C2×D47)⋊C4—24no—
73(C2×D47)⋊C4—22no—
74(C2×D47)⋊C4—22no—
75(C2×D47)⋊C4—24no—
76(C2×D47)⋊C4—24no—
77(C2×D47)⋊C4—42no—
78C4⋊Dic47—24no—
79Dic47⋊C4—22no—
80Dic47⋊C4—22no—
81C23×C47—22no—
82C23×C47—21no—
83C23×C47—21no—
84C23×C47C2×D412yes—
85C23×C47C2×D414yes—
86C23×C47C2411yescharacteristic
87C22×D47—22no—
88C22×D47—22no—
89C22×D47—22no—
90C22×D47—22no—
91C22×D47—22no—
92C22×D47—22no—
93C22×D47—21no—
94C22×D47—22no—
95C22×D47—22no—
96C22×D47—21no—
97C22×D47—21no—
98C22×D47C2×D414yes—
99C2×C4×C47—28no—
100C2×C4×C47—22no—
101C2×C4×C47—22no—
102C2×C4×C47—22no—
103C2×C4×C47—22no—
104C2×C4×C47—22no—
105C2×C4×C47—22no—
106C2×C4×C47—22no—
107C2×C4×C47—48no—
108C2×C4×C47C2×D412yes—
109C2×C4×C47C2×D412yes—
110C2×C4×C47#4114yescomplemented
111C22⋊D47—216no—
112C22⋊D47—44no—
113C2×Dic47—22no—
114C2×Dic47—22no—
115C2×Dic47—21no—
116C2×Dic47—21no—
117C2×Dic47—21no—
118C2×Dic47—22no—
119C2×Dic47—22no—
120C2×Dic47—21no—
121C2×Dic47—21no—
122C2×Dic47C2×D414yes—
123D188—28no—
124D188—48no—
125C4×D47—216no—
126Dic94—28no—
127C22×C47—22no—
128C22×C47—22no—
129C22×C47—24no—
130C22×C47—24no—
131C22×C47#12014yes—
132C22×C47C22×D412yes—
133C22×C47C22×D411yescharacteristic
134C22×C47C2×#4112yes—
135C22×C47H(5,𝔽2)11yescharacteristic
136C22×C472−1+411yescharacteristic
137C2×D47—24no—
138C2×D47—24no—
139C2×D47—24no—
140C2×D47—42no—
141C2×D47—42no—
142C2×D47—42no—
143C2×D47—42no—
144C2×D47—41no—
145C2×D47—41no—
146C2×D47—41no—
147C2×D47—42no—
148C2×D47—42no—
149C4×C47—22no—
150C4×C47—22no—
151C4×C47—24no—
152C4×C47—42no—
153Dic47—24no—
154Dic47—41no—
155Dic47—41no—
156Dic47—41no—
157#2430—471nomaximal, Sylow
158C2×C47—24no—
159C2×C47C2×#12011yescharacteristic
160C2×C47#53211yescharacteristic
161C2×C47#53711yescharacteristic
162C2×C47#54612yes—
163C2×C47#54712yes—
164D47—44no—
165D47—81no—
166C22×#41—471no—
167C2×#123—471no—
168C2×#121—472no—
169#395—471no—
170#394—471no—
171#393—471no—
172#392—471no—
173#386—472no—
174#386—472no—
175#385—472no—
176C2×#95D4711yescharacteristic, complemented
177C47#243011yescharacteristic, complemented, Sylow
178C2×#41—948no—
179C2×#41—944no—
180C22×Q8—471no—
181C22×D4—472no—
182C22×D4—471no—
183C22×D4—471no—
184C22×D4—472no—
185C23×C4—472no—
186C23×C4C2×D4712yes—
187C23×C4C2×D4711yescharacteristic
188#123—944no—
189#121—948no—
190C2×C4⋊C4—472no—
191C2×C4⋊C4—471no—
192C2×C4⋊C4—471no—
193C2×C22⋊C4—471no—
194C2×C22⋊C4—472no—
195C2×C22⋊C4—472no—
196C2×C22⋊C4—471no—
197C2×C22⋊C4—471no—
198C2×C22⋊C4—472no—
199C2×C22⋊C4—472no—
200#95—472no—
201#95—472no—
202#95C2×D4712yes—
203#95C2×D4711yescharacteristic
204#95C2×D4711yescharacteristic
205C24—472no—
206C24—471no—
207C24C22×D4711yescharacteristic
208#41—9432no—
209C2×Q8—944no—
210C2×Q8—942no—
211C2×D4—478no—
212C2×D4—944no—
213C2×D4—944no—
214C2×D4—944no—
215C2×D4—944no—
216C2×D4—942no—
217C2×D4—944no—
218C2×D4—942no—
219C2×D4—944no—
220C2×D4—1882no—
221C22×C4—24no—
222C22×C4—24no—
223C22×C4—24no—
224C22×C4—42no—
225C22×C4—472no—
226C22×C4—471no—
227C22×C4—471no—
228C22×C4—471no—
229C22×C4—944no—
230C22×C4—944no—
231C22×C4—944no—
232C22×C4D18814yescomplemented
233C22×C4C22×D4712yes—
234C22×C4C22×D4711yescharacteristic
235C22×C4C22×D4711yescharacteristic
236C22×C4C22×D4712yes—
237C4⋊C4—944no—
238C4⋊C4—942no—
239C4⋊C4—942no—
240C22⋊C4—944no—
241C22⋊C4—944no—
242C22⋊C4—942no—
243C22⋊C4—942no—
244C22⋊C4—942no—
245C22⋊C4—944no—
246C22⋊C4—1882no—
247C23—22no—
248C23—21no—
249C23—21no—
250C23—474no—
251C23—942no—
252C23—942no—
253C23—942no—
254C23—942no—
255C23—942no—
256C23—942no—
257C23—942no—
258C23—941no—
259C23—942no—
260C23—941no—
261C23—941no—
262C23C2×D18812yes—
263C23D4×D4714yes—
264C23C23×D4711yescharacteristic
265Q8—948no—
266D4—948no—
267D4—9416no—
268D4—1888no—
269D4—1884no—
270C2×C4—28no—
271C2×C4—22no—
272C2×C4—22no—
273C2×C4—22no—
274C2×C4—22no—
275C2×C4—22no—
276C2×C4—22no—
277C2×C4—22no—
278C2×C4—48no—
279C2×C4—474no—
280C2×C4—9416no—
281C2×C4—942no—
282C2×C4—942no—
283C2×C4—942no—
284C2×C4—941no—
285C2×C4—941no—
286C2×C4—941no—
287C2×C4—942no—
288C2×C4—941no—
289C2×C4—941no—
290C2×C4C2×D18812yes—
291C2×C4C2×D18812yes—
292C2×C4#1062535014yescomplemented
293C22—24no—
294C22—24no—
295C22—22no—
296C22—22no—
297C22—944no—
298C22—944no—
299C22—944no—
300C22—1882no—
301C22—1882no—
302C22—1881no—
303C22—1881no—
304C22—1881no—
305C22—1882no—
306C22—1882no—
307C22—1882no—
308C22—1882no—
309C22#1309399414yes—
310C22C2×D4×D4712yes—
311C22C22×D18811yescharacteristic
312C22C2×#1062535012yes—
313C22#1309406911yescharacteristic
314C22#1309407211yescharacteristic
315C4—24no—
316C4—22no—
317C4—22no—
318C4—42no—
319C4—944no—
320C4—1881no—
321C4—1881no—
322C4—1881no—
323C2—24no—
324C2—3761no—
325C2—1884no—
326C2C2×#1309399411yescharacteristic
327C2#1568967311yescharacteristic
328C2#1568970411yescharacteristic
329C2#1568976312yes—
330C2#1568977412yes—
subgroupG/H
center, FrattiniC23C23×D47
commutator, socleC23×C47C24
FittingC2×#95×C47C2
hypercenterC2×#95D47
hypocenterC47#2430
LCSG ▷ C23×C47 ▷ C47
UCSC1 ◁ C23 ◁ C2×#95

Complements

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

NH ≅ G/Ntype
1. C22×#10625350C2semidirect
2. C2×#13094007C2semidirect
3. C2×#13093998C2semidirect
4. #15689044C2semidirect
5. #15689041C2semidirect
6. #15689036C2semidirect
7. #15689033C2semidirect
8. #15689002C2semidirect
9. #15689001C2semidirect
10. #15688997C2semidirect
11. C2×#95×C47C2semidirect
46. C22×C4×C47D4semidirect
55. C2×C22⋊D47D4semidirect
110. C2×C4×C47#41semidirect
176. C2×#95D47semidirect
177. C47#2430semidirect
232. C22×C4D188semidirect
292. C2×C4#10625350semidirect