Finite Group Databasebeta

← #20247027

Subgroups — #20247027

Order 6016.

800 conjugacy classes of subgroups; only the 181 normal subgroups are shown.

normal subgroups only

This group has 800 conjugacy classes of subgroups — too many to draw. Only its 181 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×Q8×D47C211yesmaximal, characteristic
2C2×#13094023C212yesmaximal, complemented
3C2×#13094002C211yesmaximal, characteristic
4#15689044C211yesmaximal, characteristic, complemented
5#15689043C211yesmaximal, characteristic, complemented
6#15689042C211yesmaximal, characteristic, complemented
7#15689027C212yesmaximal, complemented
8#15688997C212yesmaximal
9#15688994C212yesmaximal
10#15688970C211yesmaximal, characteristic
11C47×#395C211yesmaximal, characteristic, complemented
12C22×Dic94C2211yescharacteristic
13C22×Dic94C2211yescharacteristic
14C22×Dic94C2212yes—
15C2×Q8×D47—24no—
16C2×Q8×D47—28no—
17C22×C4×D47C2212yes—
18C22×C4×D47C2211yescharacteristic
19#13094023—28no—
20#13094002—24no—
21C2×C4⋊Dic47C2212yes—
22C2×Dic47⋊C4C2212yes—
23C2×Dic47⋊C4C2212yes—
24C2×Dic47⋊C4C2211yescharacteristic
25C2×Dic47⋊C4C2211yescharacteristic
26C2×(C2×D47)⋊C4C2211yescharacteristic
27C2×(C2×D47)⋊C4C2211yescharacteristic
28C2×(C2×D47)⋊C4C2212yes—
29C2×C4×Dic47C2212yes—
30C47⋊#95C2211yescharacteristic
31C47⋊#95C2212yes—
32C47⋊#95C2212yes—
33C47⋊#95C2212yes—
34C22×Q8×C47C2211yescharacteristic
35C2×C4⋊C4×C47C2212yes—
36C2×C4⋊C4×C47C2211yescharacteristic
37#95×C47C2211yescharacteristic
38#95×C47C2212yes—
39C23×D47C2311yescharacteristic
40C2×Q8×C47—22no—
41C2×Q8×C47—24no—
42C22×C4×C47C2311yescharacteristic
43C22×C4×C47C2311yescharacteristic
44C22×C4×C47C2311yescharacteristic
45C22×C4×C47C2312yes—
46C22×C4×C47C2312yes—
47C22×Dic47C2311yescharacteristic
48C22×Dic47C2311yescharacteristic
49C22×Dic47C2312yes—
50C22×Dic47C2312yes—
51C22×Dic47C2311yescharacteristic
52Q8×D47—232no—
53C2×C4×D47—24no—
54C2×C4×D47—24no—
55C2×C4×D47—24no—
56C2×C4×D47—24no—
57C2×C4×D47D414yes—
58C2×Dic94—22no—
59C2×Dic94—24no—
60C2×Dic94—24no—
61C2×Dic94—24no—
62C2×Dic94—22no—
63C2×Dic94—24no—
64C2×Dic94D418yes—
65C4⋊C4×C47—22no—
66C4⋊C4×C47—24no—
67(C2×D47)⋊C4—22no—
68(C2×D47)⋊C4—22no—
69(C2×D47)⋊C4Q818yescomplemented
70C4⋊Dic47—24no—
71Dic47⋊C4—22no—
72Dic47⋊C4—24no—
73Dic47⋊C4—22no—
74Dic47⋊C4—24no—
75C4×Dic47—24no—
76C23×C47C2411yescharacteristic
77C22×D47—22no—
78C22×D47—21no—
79C22×D47—21no—
80C22×D47C2×D412yes—
81C22×D47C2×Q814yes—
82Q8×C47—28no—
83C2×C4×C47—22no—
84C2×C4×C47—22no—
85C2×C4×C47—22no—
86C2×C4×C47—22no—
87C2×C4×C47—22no—
88C2×C4×C47—21no—
89C2×C4×C47—22no—
90C2×C4×C47—21no—
91C2×C4×C47—21no—
92C2×C4×C47—21no—
93C2×C4×C47C2×D412yes—
94C2×C4×C47C2×D414yes—
95C2×C4×C47C2×Q814yescomplemented
96C2×Dic47—22no—
97C2×Dic47—22no—
98C2×Dic47—22no—
99C2×Dic47—22no—
100C2×Dic47—21no—
101C2×Dic47—22no—
102C2×Dic47—21no—
103C2×Dic47—21no—
104C2×Dic47—21no—
105C2×Dic47C2×D414yes—
106C2×Dic47C2×D414yes—
107C2×Dic47C2×D412yes—
108C2×Dic47C2×Q814yescomplemented
109C4×D47—216no—
110C4×D47—28no—
111Dic94—28no—
112Dic94—216no—
113Dic94—44no—
114C22×C47C22×D411yescharacteristic
115C22×C47C22×D412yes—
116C22×C47C22×Q812yes—
117C22×C472−1+411yescharacteristic
118C22×C472−1+411yescharacteristic
119C2×D47—22no—
120C2×D47—22no—
121C2×D47—24no—
122C2×D47—24no—
123C2×D47C4⋊Q814yescomplemented
124C4×C47—22no—
125C4×C47—24no—
126C4×C47—24no—
127C4×C47—41no—
128C4×C47—41no—
129Dic47—24no—
130Dic47—24no—
131Dic47—41no—
132Dic47—41no—
133Dic47#12014yes—
134#2434—471nomaximal, Sylow
135C2×C47C2×#12011yescharacteristic
136C2×C47C2×C4⋊Q811yescharacteristic
137C2×C47#53611yescharacteristic
138C2×C47D4×Q812yes—
139C2×C47D4×Q812yes—
140D47—24no—
141C23×Q8—471no—
142C2×C4⋊Q8—472no—
143C2×C22⋊Q8—471no—
144#395—471no—
145#395—471no—
146#395—471no—
147#395D4711yescharacteristic, complemented
148#391—472no—
149#385—472no—
150#385—472no—
151C4×C22⋊C4—471no—
152C47#243411yescharacteristic, complemented, Sylow
153C22×Q8—472no—
154C22×Q8—471no—
155C22×Q8—471no—
156C22×Q8—944no—
157C22×Q8—948no—
158C22×Q8C2×D4711yescharacteristic
159C23×C4—471no—
160C23×C4—472no—
161C4⋊Q8—948no—
162C22⋊Q8—944no—
163C2×C4⋊C4—471no—
164C2×C4⋊C4—471no—
165C2×C4⋊C4—472no—
166C2×C4⋊C4—472no—
167C2×C4⋊C4—472no—
168C2×C4⋊C4C2×D4711yescharacteristic
169C2×C4⋊C4C2×D4712yes—
170C2×C22⋊C4—472no—
171C2×C22⋊C4—471no—
172C2×C22⋊C4—471no—
173C2×C42—472no—
174#95—471no—
175#95—472no—
176#95—472no—
177#95—472no—
178#95C2×D4711yescharacteristic
179#95C2×D4712yes—
180C24—471no—
181C2×Q8—22no—
182C2×Q8—24no—
183C2×Q8—478no—
184C2×Q8—944no—
185C2×Q8—944no—
186C2×Q8—944no—
187C2×Q8—942no—
188C2×Q8—944no—
189C2×Q8—9432no—
190C2×Q8—942no—
191C22×C4—471no—
192C22×C4—472no—
193C22×C4—472no—
194C22×C4—474no—
195C22×C4—471no—
196C22×C4—471no—
197C22×C4—944no—
198C22×C4—944no—
199C22×C4—944no—
200C22×C4—944no—
201C22×C4C22×D4711yescharacteristic
202C22×C4C22×D4711yescharacteristic
203C22×C4C22×D4711yescharacteristic
204C22×C4C22×D4712yes—
205C22×C4C22×D4712yes—
206C4⋊C4—22no—
207C4⋊C4—24no—
208C4⋊C4—942no—
209C4⋊C4—944no—
210C4⋊C4—944no—
211C4⋊C4—942no—
212C4⋊C4—944no—
213C22⋊C4—478no—
214C22⋊C4—942no—
215C22⋊C4—942no—
216C42—944no—
217C23—472no—
218C23—474no—
219C23—942no—
220C23—941no—
221C23—941no—
222C23C23×D4711yescharacteristic
223Q8—28no—
224Q8—9416no—
225Q8—948no—
226Q8—1884no—
227C2×C4—22no—
228C2×C4—22no—
229C2×C4—22no—
230C2×C4—22no—
231C2×C4—22no—
232C2×C4—21no—
233C2×C4—21no—
234C2×C4—22no—
235C2×C4—21no—
236C2×C4—21no—
237C2×C4—474no—
238C2×C4—472no—
239C2×C4—474no—
240C2×C4—474no—
241C2×C4—948no—
242C2×C4—9416no—
243C2×C4—942no—
244C2×C4—942no—
245C2×C4—942no—
246C2×C4—942no—
247C2×C4—941no—
248C2×C4—941no—
249C2×C4—942no—
250C2×C4—941no—
251C2×C4—941no—
252C2×C4D4×D4712yes—
253C2×C4D4×D4714yes—
254C2×C4Q8×D4714yescomplemented
255C22—474no—
256C22—944no—
257C22—944no—
258C22—942no—
259C22—942no—
260C22C2×D4×D4712yes—
261C22C2×D4×D4711yescharacteristic
262C22C2×Q8×D4712yes—
263C22#1309407111yescharacteristic
264C22#1309407211yescharacteristic
265C4—24no—
266C4—22no—
267C4—24no—
268C4—41no—
269C4—41no—
270C4—474no—
271C4—944no—
272C4—944no—
273C4—1881no—
274C4—1881no—
275C2—944no—
276C2#120×D4711yescharacteristic
277C2C4⋊Q8×D4711yescharacteristic
278C2#1568969611yescharacteristic
279C2#1568978612yes—
280C2#1568978812yes—
subgroupG/H
center, FrattiniC23C23×D47
commutator, socleC23×C47C24
FittingC47×#395C2
hypercenter#395D47
hypocenterC47#2434
LCSG ▷ C23×C47 ▷ C47
UCSC1 ◁ C23 ◁ #395

Complements

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

NH ≅ G/Ntype
2. C2×#13094023C2semidirect
4. #15689044C2semidirect
5. #15689043C2semidirect
6. #15689042C2semidirect
7. #15689027C2semidirect
11. C47×#395C2semidirect
69. (C2×D47)⋊C4Q8semidirect
95. C2×C4×C47C2×Q8semidirect
108. C2×Dic47C2×Q8semidirect
123. C2×D47C4⋊Q8semidirect
147. #395D47semidirect
152. C47#2434semidirect
254. C2×C4Q8×D47semidirect