Finite Group Databasebeta

← #18004069

Subgroups — #18004069

Order 4736.

484 conjugacy classes of subgroups; only the 131 normal subgroups are shown.

normal subgroups only

This group has 484 conjugacy classes of subgroups — too many to draw. Only its 131 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×#11924568C212yesmaximal, complemented
2#14558084C211yesmaximal, characteristic
3C4×Q8×D37C211yesmaximal, characteristic, complemented
4#14557758C212yesmaximal, complemented
5#14557757C212yesmaximal, complemented
6#14557740C211yesmaximal, characteristic, complemented
7#14557734C211yesmaximal, characteristic, complemented
8#14557733C211yesmaximal, characteristic, complemented
9#14557555C211yesmaximal, characteristic, complemented
10#14557533C212yesmaximal, complemented
11C37×#462C211yesmaximal, characteristic, complemented
12C2×#10596966C2212yes—
13C2×Q8×D37C2211yescharacteristic
14#11924568—28no—
15C2×#10596942C2212yes—
16C2×#10596930C2212yes—
17C2×#10596941C2212yes—
18C2×#10596929C2212yes—
19#11924533C2211yescharacteristic
20#11924525C2212yes—
21#11924511—21no—
22#11924511—21no—
23Q8×Dic37C2211yescharacteristic
24C4×Dic74—21no—
25C4×Dic74C2211yescharacteristic
26#11924501—21no—
27#11924501C2211yescharacteristic
28#11924498—21no—
29#11924498—21no—
30C4⋊C4×D37—21no—
31C4⋊C4×D37C2211yescharacteristic
32C42×D37—21no—
33C42×D37C2211yescharacteristic
34C4⋊#64436C2211yescharacteristic
35#11924465C2212yes—
36#11924460C2212yes—
37#11924458C2212yes—
38#11924456C2212yes—
39C2×Q16×C37C2212yes—
40C4⋊Q8×C37C2211yescharacteristic
41C4×Q8×C37C2211yescharacteristic
42C4⋊C8×C37C2211yescharacteristic
43#103×C37C2212yes—
44C2×Q8×C37C2312yes—
45C2×Q8×C37C2311yescharacteristic
46#10596966—24no—
47#10596966—24no—
48Q8×D37—22no—
49Q8×D37D414yescomplemented
50C2×D148—22no—
51C2×D148—21no—
52C2×D148—21no—
53C2×D148C2312yes—
54C2×C4×D37—21no—
55C2×C4×D37—21no—
56C2×C4×D37—22no—
57C2×C4×D37D411yescharacteristic
58C2×C4×D37D411yescharacteristic
59C2×C4×D37C2311yescharacteristic
60C2×Dic74—21no—
61C2×Dic74C2311yescharacteristic
62Q16×C37—24no—
63C2×C8×C37C2312yes—
64C4⋊C4×C37—21no—
65C4⋊C4×C37—21no—
66C4⋊C4×C37—21no—
67C4⋊C4×C37C2311yescharacteristic
68C42×C37—21no—
69C42×C37C2311yescharacteristic
70#10596942—24no—
71#10596941—24no—
72#10596938—22no—
73#10596938—22no—
74C4⋊Dic37—21no—
75C4⋊Dic37C2311yescharacteristic
76#10596936—21no—
77#10596936—21no—
78#10596936—21no—
79C4×Dic37—21no—
80C4×Dic37—21no—
81C4×Dic37—21no—
82C4×Dic37D411yescharacteristic
83C4×Dic37D411yescharacteristic
84C4×Dic37C2311yescharacteristic
85C2×#64436C2312yes—
86#10596930—24no—
87#10596929—24no—
88C22×D37—22no—
89C22×D37C2×D411yescharacteristic
90Q8×C37—21no—
91Q8×C37—22no—
92Q8×C37—22no—
93Q8×C37C2×D412yes—
94C2×C4×C37—21no—
95C2×C4×C37—21no—
96C2×C4×C37—22no—
97C2×C4×C37C2×D411yescharacteristic
98C2×C4×C37C2×D411yescharacteristic
99C2×C4×C37C2411yescharacteristic
100C2×Dic37—21no—
101C2×Dic37—21no—
102C2×Dic37C2×D411yescharacteristic
103C2×Dic37C2×D411yescharacteristic
104C2×Dic37C2×D411yescharacteristic
105D148—22no—
106D148—22no—
107D148—42no—
108D148—44no—
109D148—42no—
110C4×D37—24no—
111C4×D37—22no—
112C4×D37—41no—
113C4×D37—42no—
114C4×D37—44no—
115C4×D37C2×D412yes—
116C4×D37#4112yes—
117Dic74—21no—
118Dic74—24no—
119Dic74C2×D412yes—
120C8×C37—22no—
121#64436—22no—
122C22×C37C22×D411yescharacteristic
123C2×D37—21no—
124C2×D37—21no—
125C2×D37—42no—
126C2×D37—42no—
127C2×D37—42no—
128C2×D37#12112yes—
129C4×C37—22no—
130C4×C37—21no—
131C4×C37—42no—
132C4×C37—41no—
133C4×C37#13712yescomplemented
134C4×C37C22×D411yescharacteristic
135C4×C37C2×#4111yescharacteristic
136Dic37—22no—
137Dic37—21no—
138Dic37—21no—
139Dic37—41no—
140Dic37#12112yes—
141#3037—371nomaximal, Sylow
142C2×C37C2×#12111yescharacteristic
143C2×C37C2×#13711yescharacteristic
144C2×C37#57511yescharacteristic
145D37—22no—
146D37—82no—
147C2×#137—372no—
148#532—371no—
149C2×C4×Q8—371no—
150#464—372no—
151#464—372no—
152#462—371no—
153#462D3711yescharacteristic, complemented
154#460—371no—
155#460—371no—
156#423—371no—
157#417—372no—
158C37#303711yescharacteristic, complemented, Sylow
159C2×#41—372no—
160C22×Q8—371no—
161#137—748no—
162C2×Q16—372no—
163C2×Q16C2×D3712yes—
164C2×SD16—372no—
165C2×SD16—372no—
166C2×M16—372no—
167C4⋊Q8C2×D3711yescharacteristic
168#127—371no—
169#124—372no—
170#121—741no—
171#121—741no—
172C4×Q8—371no—
173C4×Q8—371no—
174C4×Q8—371no—
175C4×Q8—741no—
176C4×Q8—741no—
177C4×Q8C2×D3711yescharacteristic
178C4×D4—741no—
179C4×D4—741no—
180C2×C4⋊C4—371no—
181C2×C4⋊C4—741no—
182C2×C42—371no—
183C2×C42—741no—
184C4⋊C8—372no—
185C4⋊C8—371no—
186C4⋊C8C2×D3711yescharacteristic
187#103—372no—
188#103C2×D3712yes—
189#102—372no—
190#102—372no—
191#41—744no—
192#41—744no—
193C2×Q8—374no—
194C2×Q8—371no—
195C2×Q8—741no—
196C2×Q8—742no—
197C2×Q8C22×D3712yes—
198C2×Q8C22×D3711yescharacteristic
199C2×D4—372no—
200C2×D4—742no—
201C2×D4—741no—
202C2×D4—741no—
203C22×C4—371no—
204C22×C4—371no—
205C22×C4—371no—
206C22×C4—742no—
207C22×C4—741no—
208C22×C4—741no—
209Q16—24no—
210Q16—744no—
211SD16—744no—
212SD16—744no—
213M16—744no—
214C2×C8—372no—
215C2×C8C22×D3712yes—
216C4⋊C4—21no—
217C4⋊C4—21no—
218C4⋊C4—21no—
219C4⋊C4—371no—
220C4⋊C4—741no—
221C4⋊C4—741no—
222C4⋊C4—741no—
223C4⋊C4—741no—
224C4⋊C4C22×D3711yescharacteristic
225C22⋊C4—742no—
226C22⋊C4—742no—
227C42—21no—
228C42—371no—
229C42—371no—
230C42—371no—
231C42—741no—
232C42—741no—
233C42—741no—
234C42C22×D3711yescharacteristic
235C23—371no—
236C23—742no—
237Q8—21no—
238Q8—22no—
239Q8—22no—
240Q8—372no—
241Q8—744no—
242Q8—741no—
243Q8D4×D3712yes—
244D4—742no—
245D4—742no—
246D4—1482no—
247D4—1484no—
248D4—1482no—
249C2×C4—21no—
250C2×C4—21no—
251C2×C4—22no—
252C2×C4—371no—
253C2×C4—372no—
254C2×C4—372no—
255C2×C4—371no—
256C2×C4—371no—
257C2×C4—744no—
258C2×C4—741no—
259C2×C4—742no—
260C2×C4—741no—
261C2×C4—1481no—
262C2×C4—1484no—
263C2×C4—1482no—
264C2×C4D4×D3711yescharacteristic
265C2×C4D4×D3711yescharacteristic
266C2×C4C23×D3711yescharacteristic
267C8—22no—
268C8—742no—
269C22—372no—
270C22—741no—
271C22—741no—
272C22—1482no—
273C22—1482no—
274C22—1482no—
275C22C2×D4×D3711yescharacteristic
276C4—22no—
277C4—21no—
278C4—42no—
279C4—41no—
280C4—372no—
281C4—742no—
282C4—741no—
283C4—741no—
284C4—1481no—
285C4#1192456812yescomplemented
286C4C2×D4×D3711yescharacteristic
287C4#41×D3711yescharacteristic
288C2—2962no—
289C2—742no—
290C2#121×D3711yescharacteristic
291C2C2×#1192456811yescharacteristic
292C2#1455833211yescharacteristic
subgroupG/H
centerC22C2×D4×D37
commutator, cosocleC2×C4×C37C24
FrattiniC2×C4C23×D37
FittingC37×#462C2
socleC22×C37C22×D4
prime radical#14558084C2
hypercenter#462D37
hypocenterC37#3037
LCSG ▷ C2×C4×C37 ▷ C2×C37 ▷ C37
UCSC1 ◁ C22 ◁ C42 ◁ #462

Complements

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

NH ≅ G/Ntype
1. C2×#11924568C2semidirect
3. C4×Q8×D37C2semidirect
4. #14557758C2semidirect
5. #14557757C2semidirect
6. #14557740C2semidirect
7. #14557734C2semidirect
8. #14557733C2semidirect
9. #14557555C2semidirect
10. #14557533C2semidirect
11. C37×#462C2semidirect
49. Q8×D37D4semidirect
133. C4×C37#137semidirect
153. #462D37semidirect
158. C37#3037semidirect
285. C4#11924568semidirect

Maximal subgroups

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

subgroupindex
C2×#119245682
#145580842
C4×Q8×D372
#145577582
#145577572
#145577402
#145577342
#145577332
#145575552
#145575332
C37×#4622
#303737