Finite Group Databasebeta

← #18005354

Subgroups — #18005354

Order 4736.

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

normal subgroups only

This group has 468 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×#11924515C211yesmaximal, characteristic, complemented
2C2×#11924510C211yesmaximal, characteristic
3#14557864C212yesmaximal, complemented
4#14557852C212yesmaximal, complemented
5#14557762C214yesmaximal, complemented
6#14557686C212yesmaximal, complemented
7#14557674C212yesmaximal, complemented
8C2×C22⋊C8×C37C211yesmaximal, characteristic, complemented
9C22×Dic74C2211yescharacteristic, complemented
10C22×D148C2211yescharacteristic
11C22×C22⋊D37—21no—
12C2×#10596930C2214yescomplemented
13#11924515—21no—
14#11924515—21no—
15#11924515C2212yes—
16#11924515C2212yescomplemented
17#11924510—21no—
18#11924510—21no—
19#11924510C2212yes—
20#11924510C2212yes—
21C2×C4⋊Dic37C2211yescharacteristic
22C2×C4⋊Dic37C2211yescharacteristic, complemented
23C2×#10596936—21no—
24C2×C22⋊Dic37—21no—
25C2×#10596938—21no—
26#11924468C2214yescomplemented
27#11924461C2214yescomplemented
28#11924458C2214yescomplemented
29C23×C4×C37C2211yescharacteristic
30C22×C8×C37C2212yescomplemented
31C22⋊C8×C37C2212yes—
32C22⋊C8×C37C2212yescomplemented
33C24×C37D411yescharacteristic
34C23×D37—21no—
35C22×C4×C37—42no—
36C22×C4×C37D411yescharacteristic
37C22×C4×C37D411yescharacteristic
38C22×C4×C37D411yescharacteristic
39C22×C4×C37C2311yescharacteristic
40C22×C4×C37C2311yescharacteristic
41C22×C4×C37C2311yescharacteristic
42C2×C22⋊D37—22no—
43C2×C22⋊D37—22no—
44C2×C22⋊D37—44no—
45C22×Dic37—21no—
46C22×Dic37—21no—
47C22×Dic37—21no—
48C2×D148—21no—
49C2×D148—42no—
50C2×D148C2312yes—
51C2×Dic74—21no—
52C2×Dic74—42no—
53C2×Dic74C2312yes—
54C2×C8×C37—42no—
55C2×C8×C37C2314yes—
56C22⋊Dic37—22no—
57C22⋊Dic37—22no—
58#10596938—22no—
59#10596938—22no—
60C4⋊Dic37—21no—
61C4⋊Dic37—21no—
62C4⋊Dic37C2312yes—
63C4⋊Dic37C2312yes—
64#10596936—22no—
65#10596936—22no—
66#10596930—44no—
67C23×C37—41no—
68C23×C37—41no—
69C23×C37SD1614yescomplemented
70C23×C37C2×D411yescharacteristic
71C23×C37C2×D411yescharacteristic
72C23×C37C2×D411yescharacteristic
73C22×D37—22no—
74C22×D37—42no—
75C22×D37—42no—
76C22×D37—42no—
77C2×C4×C37—22no—
78C2×C4×C37—22no—
79C2×C4×C37—22no—
80C2×C4×C37—41no—
81C2×C4×C37—41no—
82C2×C4×C37—41no—
83C2×C4×C37C2×D411yescharacteristic
84C2×C4×C37C2×D411yescharacteristic
85C2×C4×C37C2×D411yescharacteristic
86C2×C4×C37C2411yescharacteristic
87C22⋊D37—44no—
88C22⋊D37—44no—
89C2×Dic37—22no—
90C2×Dic37—22no—
91C2×Dic37—22no—
92C2×Dic37—42no—
93C2×Dic37—42no—
94C2×Dic37—42no—
95D148—24no—
96D148—42no—
97Dic74—24no—
98Dic74—42no—
99C8×C37—24no—
100C22×C37—21no—
101C22×C37—21no—
102C22×C37—21no—
103C22×C37—21no—
104C22×C37—21no—
105C22×C37—21no—
106C22×C37—44no—
107C22×C37C2×SD1612yes—
108C22×C37C2×SD1612yes—
109C22×C37C2×SD1612yes—
110C22×C37C22×D411yescharacteristic
111C2×D37—42no—
112C2×D37—42no—
113C2×D37—42no—
114C2×D37—84no—
115C4×C37—21no—
116C4×C37—21no—
117C4×C37—21no—
118C4×C37H(5,𝔽2)12yes—
119Dic37—42no—
120Dic37—42no—
121Dic37—42no—
122#3187—371nomaximal, Sylow
123C2×C37—22no—
124C2×C37—22no—
125C2×C37—22no—
126C2×C37#53411yescharacteristic
127C2×C37C22×SD1611yescharacteristic
128C2×C37#57511yescharacteristic
129D37—82no—
130C2×C22⋊Q8—371no—
131C2×#121—371no—
132#483—372no—
133#481—372no—
134#465—374no—
135#450—372no—
136#448—372no—
137C2×C22⋊C8D3711yescharacteristic, complemented
138C37#318711yescharacteristic, complemented, Sylow
139C22×Q8—371no—
140C22×D4—371no—
141C22×D4—741no—
142C23×C4C2×D3711yescharacteristic
143C2×SD16—374no—
144C22×C8C2×D3712yescomplemented
145C22⋊Q8—372no—
146C22⋊Q8—372no—
147C22⋊Q8—741no—
148C22⋊Q8—741no—
149#121—372no—
150#121—372no—
151#121—741no—
152#121—741no—
153C2×C4⋊C4—371no—
154C2×C4⋊C4—371no—
155C2×C4⋊C4—741no—
156C2×C22⋊C4—741no—
157C2×C22⋊C4—741no—
158#106—374no—
159#103—374no—
160#102—374no—
161C22⋊C8C2×D3712yes—
162C22⋊C8C2×D3712yescomplemented
163C24—741no—
164C24D14811yescharacteristic
165C2×Q8—372no—
166C2×Q8—741no—
167C2×Q8—1482no—
168C2×D4—372no—
169C2×D4—742no—
170C2×D4—742no—
171C2×D4—741no—
172C2×D4—1482no—
173C2×D4—1484no—
174C22×C4—42no—
175C22×C4—741no—
176C22×C4—741no—
177C22×C4—741no—
178C22×C4D14811yescharacteristic
179C22×C4D14811yescharacteristic
180C22×C4D14811yescharacteristic
181C22×C4C22×D3711yescharacteristic
182C22×C4C22×D3711yescharacteristic
183C22×C4C22×D3711yescharacteristic
184SD16—1484no—
185C2×C8—42no—
186C2×C8C22×D3714yes—
187C4⋊C4—372no—
188C4⋊C4—372no—
189C4⋊C4—742no—
190C4⋊C4—742no—
191C4⋊C4—741no—
192C4⋊C4—741no—
193C22⋊C4—742no—
194C22⋊C4—742no—
195C22⋊C4—742no—
196C22⋊C4—742no—
197C23—41no—
198C23—41no—
199C23—742no—
200C23—1482no—
201C23—1482no—
202C23—1482no—
203C23#1059693014yescomplemented
204C23C2×D14811yescharacteristic
205C23C2×D14811yescharacteristic
206C23C2×D14811yescharacteristic
207Q8—744no—
208Q8—1482no—
209D4—744no—
210D4—1482no—
211D4—1484no—
212D4—1484no—
213C2×C4—22no—
214C2×C4—22no—
215C2×C4—22no—
216C2×C4—41no—
217C2×C4—41no—
218C2×C4—41no—
219C2×C4—742no—
220C2×C4—742no—
221C2×C4—742no—
222C2×C4—1482no—
223C2×C4—1482no—
224C2×C4—1482no—
225C2×C4C2×D14811yescharacteristic
226C2×C4C2×D14811yescharacteristic
227C2×C4C2×D14811yescharacteristic
228C2×C4C23×D3711yescharacteristic
229C8—24no—
230C22—21no—
231C22—21no—
232C22—21no—
233C22—21no—
234C22—21no—
235C22—21no—
236C22—44no—
237C22—2964no—
238C22—1482no—
239C22—1482no—
240C22—1482no—
241C22C2×#1059693012yes—
242C22C2×#1059693012yes—
243C22C2×#1059693012yes—
244C22C22×D14811yescharacteristic
245C4—21no—
246C4—21no—
247C4—21no—
248C4—1482no—
249C4—1482no—
250C4—1482no—
251C4#1192458112yes—
252C2—22no—
253C2—22no—
254C2—22no—
255C2—2962no—
256C2#1455809511yescharacteristic
257C2C22×#1059693011yescharacteristic
258C2#1455833311yescharacteristic
subgroupG/H
centerC22C22×D148
commutator, cosocleC2×C4×C37C24
FrattiniC2×C4C23×D37
FittingC2×C22⋊C8×C37C2
socleC22×C37C22×D4
prime radicalC2×#11924510C2
hypercenterC2×C22⋊C8D37
hypocenterC37#3187
LCSG ▷ C2×C4×C37 ▷ C2×C37 ▷ C37
UCSC1 ◁ C22 ◁ C23×C4 ◁ C2×C22⋊C8

Complements

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

NH ≅ G/Ntype
1. C2×#11924515C2semidirect
3. #14557864C2semidirect
4. #14557852C2semidirect
5. #14557762C2semidirect
6. #14557686C2semidirect
7. #14557674C2semidirect
8. C2×C22⋊C8×C37C2semidirect
9. C22×Dic74C22semidirect
12. C2×#10596930C22semidirect
16. #11924515C22semidirect
22. C2×C4⋊Dic37C22semidirect
26. #11924468C22semidirect
27. #11924461C22semidirect
28. #11924458C22semidirect
30. C22×C8×C37C22semidirect
32. C22⋊C8×C37C22semidirect
69. C23×C37SD16semidirect
137. C2×C22⋊C8D37semidirect
138. C37#3187semidirect
144. C22×C8C2×D37semidirect
162. C22⋊C8C2×D37semidirect
203. C23#10596930semidirect

Maximal subgroups

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

subgroupindex
C2×#119245152
C2×#119245102
#145578642
#145578522
#145577622
#145576862
#145576742
C2×C22⋊C8×C372
#318737