Finite Group Databasebeta

← #18004515

Subgroups — #18004515

Order 4736.

458 conjugacy classes of subgroups; only the 123 normal subgroups are shown.

normal subgroups only

This group has 458 conjugacy classes of subgroups — too many to draw. Only its 123 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#14558043C211yesmaximal, characteristic
2C2×C4⋊C4×D37C211yesmaximal, characteristic, complemented
3#14557870C212yesmaximal, complemented
4#14557864C211yesmaximal, characteristic, complemented
5#14557863C211yesmaximal, characteristic, complemented
6#14557850C212yesmaximal, complemented
7#14557844C211yesmaximal, characteristic, complemented
8#14557574C212yesmaximal, complemented
9#14557531C212yesmaximal, complemented
10#14557495C211yesmaximal, characteristic, complemented
11C37×#480C211yesmaximal, characteristic, complemented
12C2×#10596964C2211yescharacteristic
13C22×C4×D37—21no—
14C22×C4×D37C2211yescharacteristic
15C2×#10596929C2212yes—
16#11924518—21no—
17#11924516C2211yescharacteristic
18#11924515C2211yescharacteristic
19#11924507C2211yescharacteristic
20#11924504—21no—
21D4×Dic37—21no—
22#11924496—21no—
23#11924494C2211yescharacteristic
24C2×C4⋊Dic37C2211yescharacteristic
25C2×#10596936—21no—
26C4⋊C4×D37—21no—
27C4⋊C4×D37—21no—
28C4⋊C4×D37C2212yes—
29C4⋊C4×D37C2212yes—
30#11924469C2212yescomplemented
31#11924468C2212yescomplemented
32#11924467C2212yescomplemented
33#11924461C2212yescomplemented
34#11924459C2212yescomplemented
35#11924457C2212yescomplemented
36C22⋊#64436C2211yescharacteristic
37#11924448C2212yes—
38#121×C37C2211yescharacteristic
39C2×C4⋊C4×C37C2211yescharacteristic
40#107×C37C2212yescomplemented
41#102×C37C2212yescomplemented
42C22⋊C8×C37C2211yescharacteristic
43C23×D37D411yescharacteristic
44C2×D4×C37—21no—
45C2×D4×C37C2311yescharacteristic
46C22×C4×C37—21no—
47C22×C4×C37C2311yescharacteristic
48C2×C22⋊D37—21no—
49C2×C22⋊D37—21no—
50C22×Dic37—21no—
51C22×Dic37—21no—
52C22×Dic37D411yescharacteristic
53#10596964—24no—
54C2×C4×D37—24no—
55C2×C4×D37—22no—
56C2×C4×D37—22no—
57C2×C4×D37—44no—
58C2×C4×D37D411yescharacteristic
59C2×C4×D37D411yescharacteristic
60C2×C4×D37C2311yescharacteristic
61C2×C4×D37C2311yescharacteristic
62C2×Dic74C2311yescharacteristic
63C2×C8×C37C2312yes—
64C4⋊C4×C37—21no—
65C4⋊C4×C37C2312yes—
66C4⋊C4×C37C2311yescharacteristic
67C22⋊C4×C37—21no—
68C22⋊Dic37—21no—
69C22⋊Dic37—21no—
70#10596938—21no—
71#10596938—21no—
72C4⋊Dic37—21no—
73C4⋊Dic37—21no—
74C4⋊Dic37C2312yes—
75C4⋊Dic37C2311yescharacteristic
76#10596936—21no—
77#10596936—22no—
78#10596936—22no—
79C4×Dic37—21no—
80C2×#64436C2312yes—
81#10596929—24no—
82C23×C37—21no—
83C23×C37C2×D411yescharacteristic
84C22×D37—22no—
85C22×D37—22no—
86C22×D37—22no—
87C22×D37C2×D411yescharacteristic
88C22×D37C2×D411yescharacteristic
89D4×C37—22no—
90D4×C37—42no—
91C2×C4×C37—22no—
92C2×C4×C37—21no—
93C2×C4×C37—22no—
94C2×C4×C37—42no—
95C2×C4×C37C2×D411yescharacteristic
96C2×C4×C37C2411yescharacteristic
97C22⋊D37—42no—
98C22⋊D37—42no—
99C2×Dic37—22no—
100C2×Dic37—21no—
101C2×Dic37—22no—
102C2×Dic37—21no—
103C2×Dic37—42no—
104C2×Dic37—42no—
105C2×Dic37C2×D411yescharacteristic
106C2×Dic37C2×D411yescharacteristic
107C4×D37—22no—
108C4×D37—22no—
109C4×D37—22no—
110C4×D37—44no—
111C4×D37—44no—
112C4×D37#4114yes—
113Dic74—22no—
114C8×C37—22no—
115#64436—22no—
116C22×C37—21no—
117C22×C37—21no—
118C22×C37—41no—
119C22×C37—41no—
120C22×C37—41no—
121C22×C37AGL(1,ℤ/8)12yes—
122C22×C37C22×D411yescharacteristic
123C2×D37—28no—
124C2×D37—21no—
125C2×D37—21no—
126C2×D37—21no—
127C2×D37—21no—
128C2×D37#12312yes—
129C2×D37#13712yes—
130C4×C37—21no—
131C4×C37—42no—
132C4×C37—41no—
133C4×C37C2×#4112yes—
134Dic37—21no—
135Dic37—41no—
136Dic37—42no—
137Dic37—41no—
138Dic37#12312yes—
139#3092—371nomaximal, Sylow
140C2×C37—22no—
141C2×C37—81no—
142C2×C37C2×#12311yescharacteristic
143C2×C37C2×AGL(1,ℤ/8)11yescharacteristic
144C2×C37C2×#13711yescharacteristic
145D37—22no—
146D37—22no—
147#525—371no—
148C22×C4⋊C4—371no—
149#484—372no—
150#483—371no—
151#483—371no—
152#481—372no—
153#480—371no—
154#480D3711yescharacteristic, complemented
155#428—372no—
156#417—372no—
157#407—371no—
158C37#309211yescharacteristic, complemented, Sylow
159C2×#41—371no—
160C23×C4—371no—
161C23×C4—741no—
162C2×M16—372no—
163#123—741no—
164C22⋊Q8—371no—
165C22⋊Q8—371no—
166#121—371no—
167#121C2×D3711yescharacteristic
168#120—741no—
169C4×D4—741no—
170C4×D4—741no—
171#117—371no—
172C2×C4⋊C4—371no—
173C2×C4⋊C4—372no—
174C2×C4⋊C4—372no—
175C2×C4⋊C4—741no—
176C2×C4⋊C4—741no—
177C2×C4⋊C4—741no—
178C2×C4⋊C4C2×D3711yescharacteristic
179#107—372no—
180#107C2×D3712yescomplemented
181#106—372no—
182#106—372no—
183#103—372no—
184#103—372no—
185#102—372no—
186#102C2×D3712yescomplemented
187C22⋊C8—371no—
188C22⋊C8—372no—
189C22⋊C8C2×D3711yescharacteristic
190C24—371no—
191#41—744no—
192C2×Q8—371no—
193C2×D4—21no—
194C2×D4—741no—
195C2×D4—741no—
196C2×D4C22×D3711yescharacteristic
197C22×C4—21no—
198C22×C4—371no—
199C22×C4—371no—
200C22×C4—371no—
201C22×C4—371no—
202C22×C4—371no—
203C22×C4—744no—
204C22×C4—742no—
205C22×C4—742no—
206C22×C4—741no—
207C22×C4—741no—
208C22×C4—1484no—
209C22×C4C22×D3711yescharacteristic
210M16—744no—
211C2×C8—372no—
212C2×C8C22×D3712yes—
213C4⋊C4—21no—
214C4⋊C4—372no—
215C4⋊C4—371no—
216C4⋊C4—742no—
217C4⋊C4—742no—
218C4⋊C4—741no—
219C4⋊C4—741no—
220C4⋊C4—741no—
221C4⋊C4C22×D3712yes—
222C4⋊C4C22×D3711yescharacteristic
223C22⋊C4—21no—
224C22⋊C4—741no—
225C22⋊C4—741no—
226C22⋊C4—741no—
227C22⋊C4—741no—
228C42—741no—
229C23—21no—
230C23—371no—
231C23—371no—
232C23—742no—
233C23—742no—
234C23—742no—
235C23D4×D3711yescharacteristic
236Q8—742no—
237D4—22no—
238D4—42no—
239D4—1482no—
240D4—1482no—
241C2×C4—22no—
242C2×C4—21no—
243C2×C4—22no—
244C2×C4—42no—
245C2×C4—371no—
246C2×C4—374no—
247C2×C4—371no—
248C2×C4—742no—
249C2×C4—742no—
250C2×C4—742no—
251C2×C4—742no—
252C2×C4—741no—
253C2×C4—741no—
254C2×C4—742no—
255C2×C4—1484no—
256C2×C4—1484no—
257C2×C4—1482no—
258C2×C4—1482no—
259C2×C4D4×D3711yescharacteristic
260C2×C4C23×D3711yescharacteristic
261C8—22no—
262C8—742no—
263C22—21no—
264C22—21no—
265C22—41no—
266C22—41no—
267C22—41no—
268C22—372no—
269C22—372no—
270C22—748no—
271C22—741no—
272C22—741no—
273C22—741no—
274C22—741no—
275C22#1192456012yes—
276C22C2×D4×D3711yescharacteristic
277C4—21no—
278C4—42no—
279C4—41no—
280C4—372no—
281C4—741no—
282C4—1481no—
283C4—1482no—
284C4—1481no—
285C4#41×D3712yes—
286C2—22no—
287C2—81no—
288C2—742no—
289C2—742no—
290C2#123×D3711yescharacteristic
291C2C2×#1192456011yescharacteristic
292C2#137×D3711yescharacteristic
subgroupG/H
centerC22C2×D4×D37
commutator, cosocleC2×C4×C37C24
FrattiniC2×C4C23×D37
FittingC37×#480C2
socleC22×C37C22×D4
prime radical#14558043C2
hypercenter#480D37
hypocenterC37#3092
LCSG ▷ C2×C4×C37 ▷ C2×C37 ▷ C37
UCSC1 ◁ C22 ◁ C22×C4 ◁ #480

Complements

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

NH ≅ G/Ntype
2. C2×C4⋊C4×D37C2semidirect
3. #14557870C2semidirect
4. #14557864C2semidirect
5. #14557863C2semidirect
6. #14557850C2semidirect
7. #14557844C2semidirect
8. #14557574C2semidirect
9. #14557531C2semidirect
10. #14557495C2semidirect
11. C37×#480C2semidirect
30. #11924469C22semidirect
31. #11924468C22semidirect
32. #11924467C22semidirect
33. #11924461C22semidirect
34. #11924459C22semidirect
35. #11924457C22semidirect
40. #107×C37C22semidirect
41. #102×C37C22semidirect
154. #480D37semidirect
158. C37#3092semidirect
180. #107C2×D37semidirect
186. #102C2×D37semidirect

Maximal subgroups

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

subgroupindex
#145580432
C2×C4⋊C4×D372
#145578702
#145578642
#145578632
#145578502
#145578442
#145575742
#145575312
#145574952
C37×#4802
#309237