Finite Group Databasebeta

← #18004119

Subgroups — #18004119

Order 4736.

490 conjugacy classes of subgroups; only the 141 normal subgroups are shown.

normal subgroups only

This group has 490 conjugacy classes of subgroups — too many to draw. Only its 141 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×#10596941C211yesmaximal, characteristic, complemented
2#14558097C211yesmaximal, characteristic
3#14558011C211yesmaximal, characteristic, complemented
4#14557757C214yesmaximal, complemented
5#14557731C214yesmaximal, complemented
6#14557561C211yesmaximal, characteristic, complemented
7#14557539C211yesmaximal, characteristic, complemented
8#14557529C211yesmaximal, characteristic, complemented
9C2×#137×C37C211yesmaximal, characteristic, complemented
10C22×D148C2211yescharacteristic
11C2×#10596941—24no—
12C2×#10596941C2214yes—
13C22×#64436C2211yescharacteristic
14#11924532C2212yes—
15#11924525C2212yes—
16#11924510—22no—
17#11924504—22no—
18Q8×Dic37—22no—
19Q8×Dic37C2214yescomplemented
20#11924495—22no—
21#11924495C2211yescharacteristic
22#11924495C2211yescharacteristic
23#11924465C2214yescomplemented
24#11924460C2212yescomplemented
25#11924458C2214yes—
26#11924457C2212yescomplemented
27C2×#41×C37C2211yescharacteristic
28C22×Q8×C37C2211yescharacteristic
29#137×C37—24no—
30C2×Q16×C37C2212yescomplemented
31C2×SD16×C37C2212yescomplemented
32C2×M16×C37C2211yescharacteristic
33C23×D37—21no—
34#41×C37—22no—
35#41×C37—22no—
36C2×Q8×C37—21no—
37C2×Q8×C37—22no—
38C2×Q8×C37D414yescomplemented
39C2×Q8×C37C2311yescharacteristic
40C2×Q8×C37C2312yes—
41C2×D4×C37—21no—
42C2×D4×C37C2311yescharacteristic
43C22×C4×C37—21no—
44C22×C4×C37—21no—
45C22×C4×C37C2311yescharacteristic
46C2×C22⋊D37—24no—
47C2×D148—22no—
48C2×D148—22no—
49C2×D148—21no—
50C2×D148C2312yes—
51Q16×C37—24no—
52SD16×C37—24no—
53M16×C37—22no—
54C2×C8×C37C2312yes—
55C22⋊Dic37—21no—
56C22⋊Dic37D412yes—
57#10596941—28no—
58#10596938—24no—
59C4⋊Dic37—24no—
60C4⋊Dic37D412yes—
61C4⋊Dic37C2312yes—
62C4×Dic37—21no—
63C4×Dic37—24no—
64C4×Dic37C2312yes—
65C2×#64436—22no—
66C2×#64436C2312yes—
67C23×C37—21no—
68C23×C37C2×D411yescharacteristic
69C22×D37—22no—
70C22×D37—42no—
71C22×D37—82no—
72Q8×C37—24no—
73Q8×C37—22no—
74Q8×C37—21no—
75Q8×C37—21no—
76Q8×C37C2×D414yes—
77D4×C37—21no—
78D4×C37—21no—
79D4×C37—42no—
80C2×C4×C37—24no—
81C2×C4×C37—22no—
82C2×C4×C37—21no—
83C2×C4×C37—42no—
84C2×C4×C37C2×D411yescharacteristic
85C2×C4×C37C2×D412yes—
86C2×C4×C37#4112yes—
87C2×C4×C37C2411yescharacteristic
88C22⋊D37—84no—
89C2×Dic37—22no—
90C2×Dic37C2×D414yes—
91D148—22no—
92D148—22no—
93D148—42no—
94C8×C37—22no—
95#64436—22no—
96C22×C37—22no—
97C22×C37—41no—
98C22×C37—42no—
99C22×C37#12112yes—
100C22×C37C22×D411yescharacteristic
101C2×D37—42no—
102C2×D37—44no—
103C2×D37—84no—
104C4×C37—24no—
105C4×C37—41no—
106C4×C37#12112yes—
107C4×C37C22×D411yescharacteristic
108C4×C37C2×#4111yescharacteristic
109Dic37—24no—
110Dic37—42no—
111#3043—371nomaximal, Sylow
112C2×C37—22no—
113C2×C37—81no—
114C2×C37C2×#12111yescharacteristic
115C2×C37#57712yes—
116D37—82no—
117C2×#137D3711yescharacteristic, complemented
118C22×SD16—371no—
119#535—371no—
120#519—371no—
121#464—374no—
122#460—374no—
123#424—371no—
124#418—371no—
125#416—371no—
126C37#304311yescharacteristic, complemented, Sylow
127C2×#41C2×D3711yescharacteristic
128C22×Q8C2×D3711yescharacteristic
129C22×D4—371no—
130#137—24no—
131C2×Q16C2×D3712yescomplemented
132C2×SD16—374no—
133C2×SD16—744no—
134C2×SD16C2×D3712yescomplemented
135C2×M16C2×D3711yescharacteristic
136C22×C8—371no—
137#127—372no—
138#124—372no—
139#121—742no—
140#120—742no—
141C4×Q8—374no—
142C4×Q8—742no—
143#117—371no—
144#117—371no—
145#117—742no—
146C4⋊C8—374no—
147#103—372no—
148#102—372no—
149#102—374no—
150C24—741no—
151#41—22no—
152#41—22no—
153C2×Q8—21no—
154C2×Q8—22no—
155C2×Q8C22⋊D3714yescomplemented
156C2×Q8C22×D3711yescharacteristic
157C2×Q8C22×D3712yes—
158C2×D4—21no—
159C2×D4—372no—
160C2×D4—744no—
161C2×D4—741no—
162C2×D4—742no—
163C2×D4—742no—
164C2×D4C22×D3711yescharacteristic
165C22×C4—21no—
166C22×C4—21no—
167C22×C4C22×D3711yescharacteristic
168Q16—24no—
169SD16—24no—
170SD16—748no—
171M16—22no—
172C2×C8—372no—
173C2×C8—742no—
174C2×C8C22×D3712yes—
175C4⋊C4—372no—
176C4⋊C4—372no—
177C4⋊C4—744no—
178C22⋊C4—372no—
179C22⋊C4—744no—
180C22⋊C4—741no—
181C42—372no—
182C42—744no—
183C42—741no—
184C23—21no—
185C23—2962no—
186C23—742no—
187C23—1482no—
188C23D4×D3711yescharacteristic
189Q8—24no—
190Q8—22no—
191Q8—21no—
192Q8—21no—
193Q8C2×C22⋊D3714yes—
194D4—21no—
195D4—21no—
196D4—42no—
197D4—2964no—
198D4—742no—
199D4—742no—
200D4—1482no—
201C2×C4—24no—
202C2×C4—22no—
203C2×C4—21no—
204C2×C4—42no—
205C2×C4—374no—
206C2×C4—742no—
207C2×C4D4×D3711yescharacteristic
208C2×C4#1059696412yes—
209C2×C4C2×C22⋊D3712yes—
210C2×C4C23×D3711yescharacteristic
211C8—22no—
212C8—742no—
213C22—22no—
214C22—41no—
215C22—42no—
216C22—2964no—
217C22—1482no—
218C22—1484no—
219C22#1192450812yes—
220C22C2×D4×D3711yescharacteristic
221C4—24no—
222C4—41no—
223C4—744no—
224C4—1482no—
225C4#1192450812yes—
226C4C22×C22⋊D3711yescharacteristic
227C4C2×#1059696411yescharacteristic
228C2—22no—
229C2—81no—
230C2—2962no—
231C2C2×#1192450811yescharacteristic
232C2#1455834212yes—
subgroupG/H
centerC22C2×D4×D37
commutator, cosocleC2×C4×C37C24
FrattiniC2×C4C23×D37
FittingC2×#137×C37C2
socleC22×C37C22×D4
prime radical#14558097C2
hypercenterC2×#137D37
hypocenterC37#3043
LCSG ▷ C2×C4×C37 ▷ C2×C37 ▷ C37
UCSC1 ◁ C22 ◁ C22×C4 ◁ C2×#137

Complements

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

NH ≅ G/Ntype
1. C22×#10596941C2semidirect
3. #14558011C2semidirect
4. #14557757C2semidirect
5. #14557731C2semidirect
6. #14557561C2semidirect
7. #14557539C2semidirect
8. #14557529C2semidirect
9. C2×#137×C37C2semidirect
19. Q8×Dic37C22semidirect
23. #11924465C22semidirect
24. #11924460C22semidirect
26. #11924457C22semidirect
30. C2×Q16×C37C22semidirect
31. C2×SD16×C37C22semidirect
38. C2×Q8×C37D4semidirect
117. C2×#137D37semidirect
126. C37#3043semidirect
131. C2×Q16C2×D37semidirect
134. C2×SD16C2×D37semidirect
155. C2×Q8C22⋊D37semidirect

Maximal subgroups

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

subgroupindex
C22×#105969412
#145580972
#145580112
#145577572
#145577312
#145575612
#145575392
#145575292
C2×#137×C372
#304337