Finite Group Databasebeta

← #18004234

Subgroups — #18004234

Order 4736.

492 conjugacy classes of subgroups; only the 133 normal subgroups are shown.

normal subgroups only

This group has 492 conjugacy classes of subgroups — too many to draw. Only its 133 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×#10596940C211yesmaximal, characteristic, complemented
2C2×#11924544C211yesmaximal, characteristic, complemented
3#14558109C211yesmaximal, characteristic, complemented
4#14558100C211yesmaximal, characteristic
5#14557778C214yesmaximal, complemented
6#14557761C214yesmaximal, complemented
7#14557569C211yesmaximal, characteristic, complemented
8#14557543C211yesmaximal, characteristic, complemented
9C37×#418C211yesmaximal, characteristic, complemented
10C2×#10596962C2211yescharacteristic
11C22×Dic74C2211yescharacteristic
12C2×#10596940—24no—
13C2×#10596940C2212yes—
14C2×#10596940C2212yes—
15#11924544—24no—
16C2×#10596929C2212yescomplemented
17C22×#64436C2211yescharacteristic
18C2×C8×D37C2212yescomplemented
19#11924536—21no—
20#11924532—21no—
21#11924524—21no—
22#11924523—21no—
23#11924520—22no—
24#11924516C2214yescomplemented
25#11924507C2214yes—
26#11924505—22no—
27#11924495C2211yescharacteristic
28#11924469C2212yescomplemented
29#11924467C2212yescomplemented
30#11924461C2214yescomplemented
31C22×D4×C37C2211yescharacteristic
32C2×M16×C37C2211yescharacteristic
33#117×C37C2211yescharacteristic
34#102×C37C2214yes—
35C24×C37—21no—
36C2×D4×C37—22no—
37C2×D4×C37—21no—
38C2×D4×C37—22no—
39C2×D4×C37C2312yes—
40C22×C4×C37C2311yescharacteristic
41C2×C22⋊D37—24no—
42C2×C22⋊D37D412yes—
43C22×Dic37—21no—
44#10596962—22no—
45#10596962—22no—
46C2×D148D411yescharacteristic
47C2×C4×D37C2312yes—
48C2×Dic74—22no—
49C2×Dic74—22no—
50C2×Dic74—21no—
51C2×Dic74D411yescharacteristic
52C2×Dic74C2312yes—
53M16×C37—22no—
54C2×C8×C37C2312yes—
55C4⋊C4×C37C2312yes—
56C22⋊C4×C37—21no—
57C42×C37—21no—
58C22⋊Dic37—21no—
59C22⋊Dic37—24no—
60#10596940—24no—
61#10596940—24no—
62#10596938—24no—
63C4⋊Dic37—24no—
64C4⋊Dic37C2312yes—
65C4×Dic37—21no—
66C2×#64436D414yescomplemented
67C2×#64436C2311yescharacteristic
68C2×#64436C2311yescharacteristic
69#10596929—24no—
70C8×D37—24no—
71C23×C37—22no—
72C23×C37—42no—
73C23×C37—82no—
74C23×C37C2×D411yescharacteristic
75C22×D37C2×D412yes—
76D4×C37—22no—
77D4×C37—22no—
78D4×C37—42no—
79C2×C4×C37—22no—
80C2×C4×C37C2×D411yescharacteristic
81C2×C4×C37C2×D412yes—
82C2×C4×C37#4112yes—
83C2×C4×C37C2411yescharacteristic
84C22⋊D37—24no—
85C22⋊D37—84no—
86C2×Dic37—22no—
87C2×Dic37—22no—
88C2×Dic37—42no—
89C2×Dic37C2×D412yes—
90D148—22no—
91C4×D37—22no—
92C4×D37—22no—
93Dic74—22no—
94Dic74—22no—
95Dic74—22no—
96Dic74—42no—
97C8×C37—22no—
98#64436C2×D412yes—
99#64436C2×D412yes—
100C22×C37—22no—
101C22×C37—42no—
102C22×C37—44no—
103C22×C37—84no—
104C22×C37#12112yes—
105C22×C37C22×D411yescharacteristic
106C2×D37—22no—
107C2×D37—24no—
108C4×C37—42no—
109C4×C37#12112yes—
110C4×C37C22×D411yescharacteristic
111C4×C37C2×#4111yescharacteristic
112Dic37—22no—
113Dic37—42no—
114Dic37—42no—
115#3057—371nomaximal, Sylow
116C2×C37—22no—
117C2×C37—82no—
118C2×C37C2×#12111yescharacteristic
119C2×C37#57712yes—
120D37—42no—
121C22×SD16—371no—
122C2×#131—371no—
123#536—371no—
124#535—371no—
125#468—374no—
126#465—374no—
127#427—371no—
128#419—371no—
129#418D3711yescharacteristic, complemented
130C37#305711yescharacteristic, complemented, Sylow
131C2×#41—371no—
132C22×Q8—371no—
133C22×D4C2×D3711yescharacteristic
134C2×SD16—372no—
135C2×SD16—372no—
136C2×SD16—744no—
137#131—744no—
138C2×M16—372no—
139C2×M16C2×D3711yescharacteristic
140C22×C8—371no—
141C22×C8—372no—
142C4⋊Q8—741no—
143#127—741no—
144#124—741no—
145#124—741no—
146#123—742no—
147C22⋊Q8—374no—
148#121—374no—
149#120—742no—
150#117—371no—
151#117C2×D3711yescharacteristic
152#107—372no—
153#106—372no—
154#103—374no—
155#102C2×D3714yes—
156C24—21no—
157#41—742no—
158#41—742no—
159C2×Q8—371no—
160C2×Q8—372no—
161C2×Q8—741no—
162C2×Q8—742no—
163C2×Q8—742no—
164C2×D4—22no—
165C2×D4—22no—
166C2×D4—21no—
167C2×D4—372no—
168C2×D4—371no—
169C2×D4—744no—
170C2×D4C22×D3712yes—
171C22×C4—372no—
172C22×C4—741no—
173C22×C4C22×D3711yescharacteristic
174SD16—744no—
175SD16—744no—
176M16—22no—
177M16—744no—
178C2×C8—374no—
179C2×C8—371no—
180C2×C8—371no—
181C2×C8—744no—
182C2×C8C22×D3712yes—
183C4⋊C4—372no—
184C4⋊C4—744no—
185C4⋊C4C22×D3712yes—
186C22⋊C4—21no—
187C22⋊C4—744no—
188C22⋊C4—744no—
189C22⋊C4—741no—
190C42—21no—
191C42—741no—
192C23—22no—
193C23—42no—
194C23—82no—
195C23—372no—
196C23D4×D3711yescharacteristic
197Q8—742no—
198Q8—742no—
199Q8—742no—
200Q8—1482no—
201D4—22no—
202D4—22no—
203D4—42no—
204D4—2964no—
205D4—744no—
206D4—742no—
207C2×C4—22no—
208C2×C4—372no—
209C2×C4—742no—
210C2×C4—742no—
211C2×C4—742no—
212C2×C4—742no—
213C2×C4—1482no—
214C2×C4#1059696212yes—
215C2×C4D4×D3711yescharacteristic
216C2×C4D4×D3712yes—
217C2×C4C23×D3711yescharacteristic
218C8—22no—
219C8—372no—
220C8—372no—
221C22—22no—
222C22—42no—
223C22—44no—
224C22—84no—
225C22—742no—
226C22—744no—
227C22#1192450912yes—
228C22C2×D4×D3711yescharacteristic
229C4—42no—
230C4—742no—
231C4—1482no—
232C4—1482no—
233C4#1192450912yes—
234C4C2×D4×D3711yescharacteristic
235C4C2×#1059696211yescharacteristic
236C2—22no—
237C2—82no—
238C2—1482no—
239C2C2×#1192450911yescharacteristic
240C2#1455834312yes—
subgroupG/H
centerC22C2×D4×D37
commutator, cosocleC2×C4×C37C24
FrattiniC2×C4C23×D37
FittingC37×#418C2
socleC22×C37C22×D4
prime radical#14558100C2
hypercenter#418D37
hypocenterC37#3057
LCSG ▷ C2×C4×C37 ▷ C2×C37 ▷ C37
UCSC1 ◁ C22 ◁ C22×C4 ◁ #418

Complements

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

NH ≅ G/Ntype
1. C22×#10596940C2semidirect
2. C2×#11924544C2semidirect
3. #14558109C2semidirect
5. #14557778C2semidirect
6. #14557761C2semidirect
7. #14557569C2semidirect
8. #14557543C2semidirect
9. C37×#418C2semidirect
16. C2×#10596929C22semidirect
18. C2×C8×D37C22semidirect
24. #11924516C22semidirect
28. #11924469C22semidirect
29. #11924467C22semidirect
30. #11924461C22semidirect
66. C2×#64436D4semidirect
129. #418D37semidirect
130. C37#3057semidirect

Maximal subgroups

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

subgroupindex
C22×#105969402
C2×#119245442
#145581092
#145581002
#145577782
#145577612
#145575692
#145575432
C37×#4182
#305737