Finite Group Databasebeta

← #18005059

Subgroups — #18005059

Order 4736.

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

normal subgroups only

This group has 572 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×#11924568C212yesmaximal, complemented
2C2×#11924563C212yesmaximal, complemented
3#14558084C211yesmaximal, characteristic, complemented
4#124×D37C211yesmaximal, characteristic, complemented
5#14557925C211yesmaximal, characteristic, complemented
6#14557923C211yesmaximal, characteristic, complemented
7#14557922C211yesmaximal, characteristic, complemented
8#14557918C212yesmaximal, complemented
9#14557914C212yesmaximal, complemented
10#14557485C211yesmaximal, characteristic, complemented
11C37×#497C211yesmaximal, characteristic, complemented
12C2×#10596966C2212yes—
13C2×Q8×D37C2211yescharacteristic
14C2×D4×D37C2211yescharacteristic
15#11924568—28no—
16#11924563—28no—
17C2×#10596942C2212yescomplemented
18C2×#10596930C2212yescomplemented
19C2×#10596941C2212yes—
20C2×#10596941C2212yescomplemented
21C2×D296C2212yes—
22C2×#10596939C2212yes—
23C2×#10596929C2212yes—
24#11924533C2211yescharacteristic, complemented
25#11924525C2211yescharacteristic
26#11924525C2212yescomplemented
27#11924524C2211yescharacteristic
28#11924523C2211yescharacteristic
29#11924522—22no—
30#11924511—22no—
31#11924498—22no—
32C22⋊C4×D37—22no—
33C42×D37C2211yescharacteristic
34#11924447C2211yescharacteristic, complemented
35C8×Dic37C2212yescomplemented
36C2×Q16×C37C2212yescomplemented
37C2×SD16×C37C2212yes—
38C4⋊Q8×C37C2211yescharacteristic, complemented
39#124×C37C2211yescharacteristic
40#97×C37C2211yescharacteristic, complemented
41C23×D37—21no—
42C2×Q8×C37C2311yescharacteristic
43C2×Q8×C37C2312yes—
44C2×D4×C37C2311yescharacteristic
45C2×C22⋊D37—21no—
46#10596966—24no—
47#10596966—24no—
48Q8×D37—24no—
49D4×D37—24no—
50C2×D148—22no—
51C2×D148—22no—
52C2×D148C2311yescharacteristic
53C2×D148C2312yes—
54C2×C4×D37—21no—
55C2×C4×D37—22no—
56C2×C4×D37D412yes—
57C2×C4×D37C2311yescharacteristic
58C2×Dic74—21no—
59C2×Dic74C2311yescharacteristic
60Q16×C37—24no—
61SD16×C37—24no—
62C2×C8×C37C2312yes—
63C4⋊C4×C37—22no—
64C22⋊C4×C37—22no—
65C42×C37C2311yescharacteristic
66C22⋊Dic37—22no—
67#10596942—24no—
68#10596941—24no—
69#10596941—24no—
70#10596939—24no—
71#10596938—22no—
72#10596938—22no—
73#10596938—24no—
74C4×Dic37D412yes—
75C4×Dic37C2311yescharacteristic
76C2×#64436C2312yes—
77D296—24no—
78#10596930—24no—
79#10596929D418yescomplemented
80C23×C37—21no—
81C22×D37—21no—
82C22×D37—22no—
83C22×D37—44no—
84C22×D37—42no—
85C22×D37C2×D411yescharacteristic
86Q8×C37—22no—
87Q8×C37—22no—
88Q8×C37—22no—
89D4×C37—22no—
90C2×C4×C37—21no—
91C2×C4×C37—22no—
92C2×C4×C37C2×D412yes—
93C2×C4×C37C2411yescharacteristic
94C22⋊D37—42no—
95C2×Dic37—21no—
96C2×Dic37C2×D412yes—
97C2×Dic37C2×D411yescharacteristic
98D148—22no—
99D148—22no—
100D148—22no—
101D148—44no—
102D148—44no—
103C4×D37—42no—
104C4×D37—42no—
105C4×D37—44no—
106C4×D37C2×D414yes—
107Dic74—22no—
108Dic74—42no—
109C8×C37C2×D414yes—
110#64436C2×D414yes—
111C22×C37—41no—
112C22×C37—42no—
113C22×C37C22×D411yescharacteristic
114C2×D37—22no—
115C2×D37—42no—
116C2×D37—41no—
117C2×D37—44no—
118C2×D37—42no—
119C2×D37—84no—
120C2×D37#12712yes—
121C4×C37—22no—
122C4×C37—41no—
123C4×C37—42no—
124C4×C37C22×D412yes—
125Dic37—22no—
126Dic37—41no—
127Dic37#12712yes—
128#3153—371nomaximal, Sylow
129C2×C37—81no—
130C2×C37C2×#12711yescharacteristic
131C2×C37#57512yes—
132D37—22no—
133D37—82no—
134D37—81no—
135C2×#137—372no—
136C2×AGL(1,ℤ/8)—372no—
137#532—371no—
138C2×#124—371no—
139#497—371no—
140#497D3711yescharacteristic, complemented
141#496—371no—
142#496—371no—
143#495—372no—
144#495—372no—
145C4×M16—371no—
146C37#315311yescharacteristic, complemented, Sylow
147C2×#41—372no—
148C22×Q8—371no—
149C22×D4—371no—
150#137—748no—
151AGL(1,ℤ/8)—748no—
152C2×Q16—372no—
153C2×Q16C2×D3712yescomplemented
154C2×SD16—372no—
155C2×SD16—372no—
156C2×SD16—372no—
157C2×SD16C2×D3712yes—
158C2×D8—372no—
159C2×D8—372no—
160C2×M16—372no—
161C4⋊Q8C2×D3711yescharacteristic, complemented
162#127—371no—
163#124—371no—
164#124—372no—
165#124—371no—
166#124—371no—
167#124—742no—
168#124C2×D3711yescharacteristic
169#121—742no—
170C4×D4—742no—
171C2×C22⋊C4—742no—
172C2×C42—371no—
173#97—371no—
174#97C2×D3711yescharacteristic, complemented
175C4×C8—372no—
176C24—741no—
177#41—744no—
178#41—744no—
179C2×Q8—371no—
180C2×Q8—741no—
181C2×Q8—744no—
182C2×Q8C22×D3711yescharacteristic
183C2×Q8C22×D3712yes—
184C2×D4—371no—
185C2×D4—372no—
186C2×D4—742no—
187C2×D4—744no—
188C2×D4—741no—
189C2×D4—742no—
190C2×D4C22×D3711yescharacteristic
191C22×C4—371no—
192C22×C4—372no—
193C22×C4—742no—
194C22×C4—741no—
195Q16—24no—
196Q16—744no—
197SD16—24no—
198SD16—744no—
199SD16—744no—
200SD16—744no—
201D8—744no—
202D8—744no—
203M16—378no—
204C2×C8—372no—
205C2×C8C22×D3712yes—
206C4⋊C4—22no—
207C22⋊C4—22no—
208C22⋊C4—742no—
209C22⋊C4—744no—
210C22⋊C4—742no—
211C22⋊C4—742no—
212C42—372no—
213C42—371no—
214C42C22×D3711yescharacteristic
215C23—21no—
216C23—371no—
217C23—741no—
218C23—742no—
219C23—1482no—
220C23—1484no—
221Q8—22no—
222Q8—22no—
223Q8—22no—
224Q8—742no—
225Q8—1482no—
226D4—22no—
227D4—742no—
228D4—742no—
229D4—742no—
230D4—1484no—
231D4—1484no—
232D4—1482no—
233C2×C4—21no—
234C2×C4—22no—
235C2×C4—371no—
236C2×C4—374no—
237C2×C4—372no—
238C2×C4—741no—
239C2×C4—1482no—
240C2×C4—1484no—
241C2×C4—1482no—
242C2×C4D4×D3712yes—
243C2×C4C23×D3711yescharacteristic
244C8—374no—
245C8D4×D3714yes—
246C22—41no—
247C22—42no—
248C22—372no—
249C22—2964no—
250C22—742no—
251C22—1482no—
252C22—1481no—
253C22—1484no—
254C22—1482no—
255C22C2×D4×D3711yescharacteristic
256C4—22no—
257C4—42no—
258C4—41no—
259C4—372no—
260C4—742no—
261C4—1481no—
262C4C2×D4×D3712yes—
263C2—81no—
264C2—2961no—
265C2—2962no—
266C2—742no—
267C2#127×D3711yescharacteristic
268C2#1455833212yes—
subgroupG/H
centerC22C2×D4×D37
commutator, cosocleC2×C4×C37C24
FrattiniC2×C4C23×D37
FittingC37×#497C2
socleC22×C37C22×D4
hypercenter#497D37
hypocenterC37#3153
LCSG ▷ C2×C4×C37 ▷ C2×C37 ▷ C37
UCSC1 ◁ C22 ◁ C42 ◁ #497

Complements

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

NH ≅ G/Ntype
1. C2×#11924568C2semidirect
2. C2×#11924563C2semidirect
3. #14558084C2semidirect
4. #124×D37C2semidirect
5. #14557925C2semidirect
6. #14557923C2semidirect
7. #14557922C2semidirect
8. #14557918C2semidirect
9. #14557914C2semidirect
10. #14557485C2semidirect
11. C37×#497C2semidirect
17. C2×#10596942C22semidirect
18. C2×#10596930C22semidirect
20. C2×#10596941C22semidirect
24. #11924533C22semidirect
26. #11924525C22semidirect
34. #11924447C22semidirect
35. C8×Dic37C22semidirect
36. C2×Q16×C37C22semidirect
38. C4⋊Q8×C37C22semidirect
40. #97×C37C22semidirect
79. #10596929D4semidirect
140. #497D37semidirect
146. C37#3153semidirect
153. C2×Q16C2×D37semidirect
161. C4⋊Q8C2×D37semidirect
174. #97C2×D37semidirect

Maximal subgroups

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

subgroupindex
C2×#119245682
C2×#119245632
#145580842
#124×D372
#145579252
#145579232
#145579222
#145579182
#145579142
#145574852
C37×#4972
#315337