Finite Group Databasebeta

← #18004392

Subgroups — #18004392

Order 4736.

440 conjugacy classes of subgroups; only the 135 normal subgroups are shown.

normal subgroups only

This group has 440 conjugacy classes of subgroups — too many to draw. Only its 135 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
1#14558086C211yesmaximal, characteristic, complemented
2C2×#11924498C211yesmaximal, characteristic
3#14557836C212yesmaximal, complemented
4#14557832C212yesmaximal, complemented
5#14557822C212yesmaximal, complemented
6#14557812C212yesmaximal, complemented
7#14557573C211yesmaximal, characteristic, complemented
8#14557556C211yesmaximal, characteristic, complemented
9#14557539C211yesmaximal, characteristic
10#14557537C211yesmaximal, characteristic
11C37×#428C211yesmaximal, characteristic, complemented
12C22×D148C2211yescharacteristic
13C22×C4×D37—21no—
14C2×#10596934C2211yescharacteristic
15#11924535C2211yescharacteristic, complemented
16#11924535C2211yescharacteristic, complemented
17#11924527C2211yescharacteristic, complemented
18#11924527C2211yescharacteristic, complemented
19#11924513—21no—
20#11924513—21no—
21C4×Dic74—21no—
22C4×Dic74—21no—
23#11924498—21no—
24#11924498—21no—
25#11924498C2212yes—
26#11924498C2212yes—
27#11924495C2211yescharacteristic
28C2×#10596938—21no—
29C2×C4×Dic37C2211yescharacteristic
30#11924469C2212yescomplemented
31#11924467C2212yescomplemented
32#11924465C2212yescomplemented
33#11924465C2212yescomplemented
34#11924458C2212yes—
35#11924458C2212yes—
36#11924456C2212yes—
37#11924456C2212yes—
38C2×M16×C37C2211yescharacteristic
39#117×C37C2211yescharacteristic
40C2×C4⋊C4×C37C2211yescharacteristic
41#107×C37C2212yescomplemented
42#106×C37C2212yescomplemented
43C23×D37—21no—
44C22×C4×C37—21no—
45C22×C4×C37C2311yescharacteristic
46C22×Dic37D411yescharacteristic
47C22×Dic37D411yescharacteristic
48C2×D148—21no—
49C2×D148—22no—
50C2×D148Q814yescomplemented
51C2×D148C2312yes—
52C2×C4×D37—22no—
53C2×C4×D37—22no—
54C2×C4×D37—44no—
55C2×Dic74—21no—
56C2×Dic74—21no—
57M16×C37—22no—
58C2×C8×C37C2312yes—
59C4⋊C4×C37—21no—
60C4⋊C4×C37C2312yes—
61C4⋊C4×C37C2311yescharacteristic
62C4⋊C4×C37C2311yescharacteristic
63C22⋊C4×C37—21no—
64C42×C37—21no—
65C22⋊Dic37—21no—
66#10596938—22no—
67#10596938—22no—
68C4⋊Dic37—21no—
69C4⋊Dic37—21no—
70C4⋊Dic37C2311yescharacteristic
71C4⋊Dic37C2311yescharacteristic
72#10596936—21no—
73#10596936—21no—
74#10596936—21no—
75#10596936—21no—
76C4×Dic37—21no—
77C4×Dic37D411yescharacteristic
78C4×Dic37D411yescharacteristic
79C4×Dic37C2311yescharacteristic
80C4×Dic37C2311yescharacteristic
81#10596934—22no—
82C2×#64436C2312yes—
83C23×C37C2×D411yescharacteristic
84C22×D37—24no—
85C22×D37—22no—
86C22×D37—42no—
87C22×D37—42no—
88C2×C4×C37—21no—
89C2×C4×C37—21no—
90C2×C4×C37—22no—
91C2×C4×C37—42no—
92C2×C4×C37C2×D411yescharacteristic
93C2×C4×C37C2×Q812yes—
94C2×C4×C37#4112yes—
95C2×C4×C37C2411yescharacteristic
96C2×Dic37—22no—
97C2×Dic37—21no—
98C2×Dic37—21no—
99C2×Dic37—41no—
100C2×Dic37C2×D411yescharacteristic
101C2×Dic37C2×D411yescharacteristic
102C2×Dic37C2×D411yescharacteristic
103C2×Dic37C2×D411yescharacteristic
104D148—24no—
105D148—22no—
106D148C2×Q814yes—
107C4×D37—44no—
108C4×D37—44no—
109Dic74—42no—
110Dic74—42no—
111C8×C37—22no—
112#64436—22no—
113C22×C37—21no—
114C22×C37—21no—
115C22×C37C22⋊Q812yes—
116C22×C37C22×D411yescharacteristic
117C2×D37—24no—
118C2×D37—48no—
119C2×D37—42no—
120C2×D37—42no—
121C4×C37—41no—
122C4×C37—42no—
123C4×C37—41no—
124C4×C37C22⋊Q812yes—
125C4×C37C22×Q811yescharacteristic
126C4×C37C2×#4111yescharacteristic
127Dic37—21no—
128Dic37—21no—
129Dic37—21no—
130Dic37—41no—
131Dic37—41no—
132Dic37AGL(1,ℤ/8)12yes—
133#3077—371nomaximal, Sylow
134C2×C37—22no—
135C2×C37C2×C22⋊Q811yescharacteristic
136C2×C37C2×AGL(1,ℤ/8)11yescharacteristic
137C2×C37#57511yescharacteristic
138D37—44no—
139#533—371no—
140C2×C4×D4—371no—
141#478—372no—
142#478—372no—
143#476—372no—
144#474—372no—
145#428—371no—
146#428D3711yescharacteristic, complemented
147#423—371no—
148#418—371no—
149#418—371no—
150C37#307711yescharacteristic, complemented, Sylow
151C22×D4—371no—
152C23×C4—741no—
153C2×M16—371no—
154C2×M16C2×D3711yescharacteristic
155C4⋊Q8—371no—
156C4⋊Q8—371no—
157#125—371no—
158#125—371no—
159C22⋊Q8—741no—
160C22⋊Q8—741no—
161C4×Q8—741no—
162C4×Q8—741no—
163C4×D4—372no—
164C4×D4—372no—
165C4×D4—741no—
166C4×D4—741no—
167#117—371no—
168#117C2×D3711yescharacteristic
169C2×C4⋊C4C2×D3711yescharacteristic
170C2×C22⋊C4—741no—
171C2×C42—371no—
172#107—372no—
173#107C2×D3712yescomplemented
174#106—372no—
175#106C2×D3712yescomplemented
176C4⋊C8—372no—
177C4⋊C8—372no—
178#102—372no—
179#102—372no—
180#102—372no—
181#102—372no—
182C24—741no—
183C2×Q8—741no—
184C2×Q8—741no—
185C2×D4—374no—
186C2×D4—372no—
187C2×D4—742no—
188C2×D4—741no—
189C22×C4—21no—
190C22×C4—371no—
191C22×C4—371no—
192C22×C4—742no—
193C22×C4—742no—
194C22×C4—1484no—
195C22×C4C22×D3711yescharacteristic
196M16—22no—
197M16—742no—
198C2×C8—372no—
199C2×C8C22×D3712yes—
200C4⋊C4—21no—
201C4⋊C4—371no—
202C4⋊C4—371no—
203C4⋊C4—741no—
204C4⋊C4—741no—
205C4⋊C4—741no—
206C4⋊C4—741no—
207C4⋊C4—741no—
208C4⋊C4—741no—
209C4⋊C4C22×D3711yescharacteristic
210C4⋊C4C22×D3711yescharacteristic
211C4⋊C4C22×D3712yes—
212C22⋊C4—21no—
213C22⋊C4—742no—
214C22⋊C4—742no—
215C22⋊C4—741no—
216C42—21no—
217C42—371no—
218C42—371no—
219C42—371no—
220C42—371no—
221C42—741no—
222C23—744no—
223C23—742no—
224C23—1482no—
225C23—1482no—
226C23D4×D3711yescharacteristic
227Q8—1482no—
228Q8—1482no—
229D4—374no—
230D4—744no—
231D4—742no—
232C2×C4—21no—
233C2×C4—21no—
234C2×C4—22no—
235C2×C4—42no—
236C2×C4—371no—
237C2×C4—371no—
238C2×C4—371no—
239C2×C4—371no—
240C2×C4—742no—
241C2×C4—741no—
242C2×C4—741no—
243C2×C4—1484no—
244C2×C4—1484no—
245C2×C4—1481no—
246C2×C4#1059696212yes—
247C2×C4D4×D3711yescharacteristic
248C2×C4Q8×D3712yes—
249C2×C4C23×D3711yescharacteristic
250C8—22no—
251C8—742no—
252C22—21no—
253C22—21no—
254C22—744no—
255C22—1488no—
256C22—1482no—
257C22—1482no—
258C22#1192451412yes—
259C22C2×D4×D3711yescharacteristic
260C4—42no—
261C4—41no—
262C4—41no—
263C4—372no—
264C4—741no—
265C4—741no—
266C4—741no—
267C4—1481no—
268C4—1481no—
269C4#1192451412yes—
270C4C2×Q8×D3711yescharacteristic
271C4C2×#1059696211yescharacteristic
272C2—22no—
273C2—1484no—
274C2C2×#1192451411yescharacteristic
275C2AGL(1,ℤ/8)×D3711yescharacteristic
276C2#1455833211yescharacteristic
subgroupG/H
centerC22C2×D4×D37
commutator, cosocleC2×C4×C37C24
FrattiniC2×C4C23×D37
FittingC37×#428C2
socleC22×C37C22×D4
prime radicalC22×D148C22
hypercenter#428D37
hypocenterC37#3077
LCSG ▷ C2×C4×C37 ▷ C2×C37 ▷ C37
UCSC1 ◁ C22 ◁ C22×C4 ◁ #428

Complements

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

NH ≅ G/Ntype
1. #14558086C2semidirect
3. #14557836C2semidirect
4. #14557832C2semidirect
5. #14557822C2semidirect
6. #14557812C2semidirect
7. #14557573C2semidirect
8. #14557556C2semidirect
11. C37×#428C2semidirect
15. #11924535C22semidirect
16. #11924535C22semidirect
17. #11924527C22semidirect
18. #11924527C22semidirect
30. #11924469C22semidirect
31. #11924467C22semidirect
32. #11924465C22semidirect
33. #11924465C22semidirect
41. #107×C37C22semidirect
42. #106×C37C22semidirect
50. C2×D148Q8semidirect
146. #428D37semidirect
150. C37#3077semidirect
173. #107C2×D37semidirect
175. #106C2×D37semidirect

Maximal subgroups

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

subgroupindex
#145580862
C2×#119244982
#145578362
#145578322
#145578222
#145578122
#145575732
#145575562
#145575392
#145575372
C37×#4282
#307737