Finite Group Databasebeta

← #20253201

Subgroups — #20253201

Order 6016.

416 conjugacy classes of subgroups; only the 137 normal subgroups are shown.

normal subgroups only

This group has 416 conjugacy classes of subgroups — too many to draw. Only its 137 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×#13094023C211yesmaximal, characteristic, complemented
2#15689591C211yesmaximal, characteristic
3#15689417C212yesmaximal, complemented
4#15689412C212yesmaximal, complemented
5#15689407C212yesmaximal, complemented
6#15689402C212yesmaximal, complemented
7#15689166C211yesmaximal, characteristic, complemented
8#15689146C211yesmaximal, characteristic, complemented
9#15689125C211yesmaximal, characteristic
10#15689122C211yesmaximal, characteristic
11C47×#428C211yesmaximal, characteristic, complemented
12C2×#10625350C2211yescharacteristic
13C22×Dic94—21no—
14C2×#10625329C2211yescharacteristic
15#13094023—21no—
16#13094023—21no—
17#13094023C2212yes—
18#13094023C2212yes—
19#13093989C2211yescharacteristic
20#13093989C2211yescharacteristic
21#13093986C2211yescharacteristic
22#13093986C2211yescharacteristic
23(C22⋊D47)⋊C4—21no—
24(C22⋊D47)⋊C4—21no—
25#13093981—21no—
26C2×C4⋊Dic47C2211yescharacteristic
27C2×Dic47⋊C4—21no—
28C2×C4×Dic47C2211yescharacteristic
29C42×D47—21no—
30#13093957C2212yes—
31#13093955C2212yes—
32Dic47⋊C8C2212yes—
33Dic47⋊C8C2212yes—
34#13093949C2212yes—
35#13093947C2212yes—
36#13093946C2212yes—
37#13093944C2212yes—
38C2×M16×C47C2211yescharacteristic
39#117×C47C2211yescharacteristic
40C2×C4⋊C4×C47C2211yescharacteristic
41#107×C47C2212yes—
42#106×C47C2212yes—
43C22×C4×C47—21no—
44C22×C4×C47C2311yescharacteristic
45C2×C22⋊D47—21no—
46C22×Dic47—21no—
47C22×Dic47D411yescharacteristic
48C22×Dic47D411yescharacteristic
49#10625350—22no—
50#10625350Q814yescomplemented
51C2×D188C2311yescharacteristic
52C2×C4×D47—21no—
53C2×C4×D47—21no—
54C2×C4×D47—21no—
55C2×Dic94—22no—
56C2×Dic94—22no—
57C2×Dic94—44no—
58C2×Dic94C2311yescharacteristic
59M16×C47—22no—
60C2×C8×C47C2312yes—
61C4⋊C4×C47—21no—
62C4⋊C4×C47C2311yescharacteristic
63C4⋊C4×C47C2312yes—
64C4⋊C4×C47C2311yescharacteristic
65C22⋊C4×C47—21no—
66C42×C47—21no—
67(C2×D47)⋊C4—21no—
68(C2×D47)⋊C4—21no—
69C4⋊Dic47—21no—
70C4⋊Dic47C2312yes—
71Dic47⋊C4—22no—
72Dic47⋊C4—22no—
73Dic47⋊C4—21no—
74Dic47⋊C4—21no—
75C4×Dic47—21no—
76C4×Dic47—21no—
77C4×Dic47D411yescharacteristic
78C4×Dic47D411yescharacteristic
79C4×Dic47C2311yescharacteristic
80C4×Dic47C2311yescharacteristic
81#10625329—22no—
82C2×C47⋊C8C2312yes—
83C23×C47C2×D411yescharacteristic
84C22×D47—21no—
85C2×C4×C47—21no—
86C2×C4×C47—21no—
87C2×C4×C47—22no—
88C2×C4×C47—42no—
89C2×C4×C47C2×D411yescharacteristic
90C2×C4×C47C2×Q812yes—
91C2×C4×C47#4112yes—
92C2×C4×C47C2411yescharacteristic
93C22⋊D47—24no—
94C2×Dic47—21no—
95C2×Dic47—22no—
96C2×Dic47—22no—
97C2×Dic47—22no—
98C2×Dic47—42no—
99C2×Dic47C2×D411yescharacteristic
100C2×Dic47C2×D411yescharacteristic
101C2×Dic47C2×D411yescharacteristic
102C2×Dic47C2×D411yescharacteristic
103D188—21no—
104D188C2×Q812yescomplemented
105C4×D47—24no—
106C4×D47—42no—
107C4×D47—42no—
108Dic94—21no—
109Dic94—44no—
110Dic94—44no—
111Dic94C2×Q812yescomplemented
112C8×C47—22no—
113C47⋊C8—22no—
114C22×C47—21no—
115C22×C47—21no—
116C22×C47C22⋊Q812yes—
117C22×C47C22×D411yescharacteristic
118C2×D47—22no—
119C2×D47—41no—
120C2×D47—41no—
121C4×C47—42no—
122C4×C47—41no—
123C4×C47—41no—
124C4×C47C22⋊Q812yes—
125C4×C47C22×Q811yescharacteristic
126C4×C47C2×#4111yescharacteristic
127Dic47—22no—
128Dic47—21no—
129Dic47—21no—
130Dic47—42no—
131Dic47AGL(1,ℤ/8)12yes—
132Dic47#13712yes—
133#3079—471nomaximal, Sylow
134C2×C47—22no—
135C2×C47C2×C22⋊Q811yescharacteristic
136C2×C47C2×AGL(1,ℤ/8)11yescharacteristic
137C2×C47C2×#13711yescharacteristic
138D47—42no—
139C2×C4⋊Q8—471no—
140C4×#41—471no—
141#477—472no—
142#476—472no—
143#475—472no—
144#474—472no—
145#428—471no—
146#428D4711yescharacteristic, complemented
147#423—471no—
148#417—471no—
149#417—471no—
150C47#307911yescharacteristic, complemented, Sylow
151C2×#41—471no—
152C22×Q8—941no—
153C2×M16—471no—
154C2×M16C2×D4711yescharacteristic
155C4⋊Q8—472no—
156C4⋊Q8—472no—
157C4⋊Q8—941no—
158C4⋊Q8—941no—
159C4×Q8—471no—
160C4×Q8—471no—
161C4×D4—471no—
162C4×D4—471no—
163C4×D4—941no—
164C4×D4—941no—
165#117—941no—
166#117C2×D4711yescharacteristic
167C2×C4⋊C4—471no—
168C2×C4⋊C4—941no—
169C2×C4⋊C4C2×D4711yescharacteristic
170C2×C42—471no—
171C2×C42—941no—
172#107—472no—
173#107C2×D4712yes—
174#106—472no—
175#106C2×D4712yes—
176C4⋊C8—472no—
177C4⋊C8—472no—
178#103—472no—
179#103—472no—
180#102—472no—
181#102—472no—
182#41—474no—
183#41—942no—
184C2×Q8—471no—
185C2×Q8—942no—
186C2×Q8—942no—
187C2×Q8—1884no—
188C2×D4—471no—
189C2×D4—941no—
190C22×C4—21no—
191C22×C4—471no—
192C22×C4—471no—
193C22×C4—941no—
194C22×C4—941no—
195C22×C4—941no—
196C22×C4—941no—
197C22×C4C22×D4711yescharacteristic
198M16—22no—
199M16—942no—
200C2×C8—472no—
201C2×C8C22×D4712yes—
202C4⋊C4—21no—
203C4⋊C4—472no—
204C4⋊C4—942no—
205C4⋊C4—942no—
206C4⋊C4—941no—
207C4⋊C4—941no—
208C4⋊C4—941no—
209C4⋊C4C22×D4712yes—
210C4⋊C4C22×D4711yescharacteristic
211C4⋊C4C22×D4711yescharacteristic
212C22⋊C4—21no—
213C22⋊C4—941no—
214C22⋊C4—941no—
215C42—21no—
216C42—471no—
217C42—471no—
218C42—471no—
219C42—471no—
220C42—941no—
221C42—941no—
222C23—941no—
223C23D4×D4711yescharacteristic
224Q8—472no—
225Q8—941no—
226Q8—1884no—
227Q8—1884no—
228D4—472no—
229D4—944no—
230D4—941no—
231C2×C4—21no—
232C2×C4—21no—
233C2×C4—22no—
234C2×C4—42no—
235C2×C4—471no—
236C2×C4—471no—
237C2×C4—471no—
238C2×C4—471no—
239C2×C4—942no—
240C2×C4—942no—
241C2×C4—941no—
242C2×C4—944no—
243C2×C4—942no—
244C2×C4—1882no—
245C2×C4—1882no—
246C2×C4—1882no—
247C2×C4#1062535012yes—
248C2×C4D4×D4711yescharacteristic
249C2×C4Q8×D4712yes—
250C2×C4C23×D4711yescharacteristic
251C8—22no—
252C8—942no—
253C22—21no—
254C22—21no—
255C22—942no—
256C22—1881no—
257C22—1881no—
258C22#1309400212yes—
259C22C2×D4×D4711yescharacteristic
260C4—42no—
261C4—41no—
262C4—41no—
263C4—472no—
264C4—472no—
265C4—942no—
266C4—941no—
267C4—941no—
268C4—1882no—
269C4#1309400212yes—
270C4C2×Q8×D4711yescharacteristic
271C4C2×#1062535011yescharacteristic
272C2—22no—
273C2—1882no—
274C2C2×#1309400211yescharacteristic
275C2AGL(1,ℤ/8)×D4711yescharacteristic
276C2#137×D4711yescharacteristic
subgroupG/H
centerC22C2×D4×D47
commutatorC2×C4×C47C24
FrattiniC2×C4C23×D47
FittingC47×#428C2
socleC22×C47C22×D4
hypercenter#428D47
hypocenterC47#3079
LCSG ▷ C2×C4×C47 ▷ C2×C47 ▷ C47
UCSC1 ◁ C22 ◁ C22×C4 ◁ #428

Complements

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

NH ≅ G/Ntype
1. C2×#13094023C2semidirect
3. #15689417C2semidirect
4. #15689412C2semidirect
5. #15689407C2semidirect
6. #15689402C2semidirect
7. #15689166C2semidirect
8. #15689146C2semidirect
11. C47×#428C2semidirect
50. #10625350Q8semidirect
104. D188C2×Q8semidirect
111. Dic94C2×Q8semidirect
146. #428D47semidirect
150. C47#3079semidirect