Finite Group Databasebeta

← #18007318

Subgroups — #18007318

Order 4736.

372 conjugacy classes of subgroups; only the 117 normal subgroups are shown.

normal subgroups only

This group has 372 conjugacy classes of subgroups — too many to draw. Only its 117 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#14558257C211yesmaximal, characteristic, complemented
2D4×Dic74C211yesmaximal, characteristic, complemented
3#14557894C212yesmaximal, complemented
4#14557883C211yesmaximal, characteristic, complemented
5#14557826C212yesmaximal, complemented
6#14557814C211yesmaximal, characteristic, complemented
7#14557750C212yesmaximal
8#14557660C211yesmaximal, characteristic, complemented
9#14557634C212yesmaximal, complemented
10C4×#10596939C211yesmaximal, characteristic
11C37×#461C211yesmaximal, characteristic, complemented
12C22×Dic74—21no—
13C2×#10596939C2211yescharacteristic
14#11924536C2211yescharacteristic
15#11924535C2212yes—
16#11924531—22no—
17#11924525C2212yes—
18#11924516—21no—
19#11924515—21no—
20#11924513—22no—
21Q8×Dic37C2212yes—
22C4×Dic74C2211yescharacteristic
23C4×D148C2211yescharacteristic
24D4×Dic37C2212yes—
25#11924494—21no—
26#11924470C2212yes—
27#11924469C2211yescharacteristic
28#11924465C2212yes—
29#11924460C2212yes—
30#11924460C2212yes—
31#11924458C2212yes—
32#11924457C2211yescharacteristic
33#11924456C2211yescharacteristic
34#11924446C2212yes—
35C4×#64436C2211yescharacteristic
36C2×SD16×C37C2212yes—
37C4⋊Q8×C37C2211yescharacteristic
38C4×D4×C37C2211yescharacteristic
39C4⋊C8×C37C2211yescharacteristic
40#103×C37C2212yes—
41C2×Q8×C37C2312yes—
42C2×D4×C37C2311yescharacteristic
43C22×C4×C37—21no—
44C22×Dic37—22no—
45C2×D148C2311yescharacteristic
46C2×C4×D37—21no—
47C2×Dic74—21no—
48C2×Dic74—22no—
49C2×Dic74—44no—
50C2×Dic74D411yescharacteristic
51SD16×C37—24no—
52C2×C8×C37C2312yes—
53C4⋊C4×C37—21no—
54C4⋊C4×C37—21no—
55C4⋊C4×C37C2311yescharacteristic
56C22⋊C4×C37—21no—
57C42×C37C2311yescharacteristic
58C22⋊Dic37—22no—
59#10596939—22no—
60#10596938—21no—
61#10596938—22no—
62#10596938—22no—
63C4⋊Dic37—21no—
64C4⋊Dic37—22no—
65C4⋊Dic37D411yescharacteristic
66C4⋊Dic37C2312yes—
67C4⋊Dic37C2311yescharacteristic
68#10596936—22no—
69#10596936D412yes—
70C4×Dic37—22no—
71C4×Dic37C2312yes—
72C2×#64436C2311yescharacteristic
73C2×#64436C2311yescharacteristic
74C23×C37—21no—
75C22×D37—21no—
76Q8×C37—22no—
77Q8×C37—22no—
78D4×C37—21no—
79D4×C37#4112yes—
80C2×C4×C37—22no—
81C2×C4×C37—21no—
82C2×C4×C37—42no—
83C2×C4×C37C2×D411yescharacteristic
84C2×C4×C37C2×D411yescharacteristic
85C2×C4×C37C2411yescharacteristic
86C2×Dic37—21no—
87C2×Dic37—22no—
88C2×Dic37—44no—
89C2×Dic37C2×D412yes—
90C2×Dic37C2×D412yes—
91D148—21no—
92D148—21no—
93C4×D37—42no—
94Dic74—22no—
95Dic74—44no—
96Dic74—42no—
97C8×C37—22no—
98#64436—21no—
99#64436—21no—
100C22×C37—22no—
101C22×C37—41no—
102C22×C37—41no—
103C22×C37C22×D411yescharacteristic
104C2×D37—41no—
105C2×D37—41no—
106C2×D37—41no—
107C4×C37—21no—
108C4×C37—42no—
109C4×C37—41no—
110C4×C37AGL(1,ℤ/8)12yes—
111C4×C37C2×#4111yescharacteristic
112C4×C372−1+411yescharacteristic
113Dic37—22no—
114Dic37—22no—
115Dic37—42no—
116Dic37—41no—
117#3367—371nomaximal, Sylow
118C2×C37—42no—
119C2×C37#54811yescharacteristic
120C2×C37C2×AGL(1,ℤ/8)11yescharacteristic
121C2×C37#57811yescharacteristic
122D37—81no—
123#556—371no—
124D4×Q8—371no—
125#489—372no—
126#487—371no—
127#477—372no—
128#474—371no—
129#463—372no—
130#461D3711yescharacteristic, complemented
131#446—371no—
132#440—372no—
133C4×D8—371no—
134C37#336711yescharacteristic, complemented, Sylow
135C22×Q8—741no—
136C2×SD16C2×D3712yes—
137C2×D8—371no—
138C4⋊Q8—372no—
139C4⋊Q8—371no—
140C4⋊Q8C2×D3711yescharacteristic
141#126—742no—
142#124—372no—
143C22⋊Q8—742no—
144C22⋊Q8—741no—
145C22⋊Q8—741no—
146C4×Q8—371no—
147C4×Q8—372no—
148C4×D4—372no—
149C4×D4—371no—
150C4×D4C2×D3711yescharacteristic
151#117—741no—
152#107—372no—
153#107—371no—
154C4⋊C8—372no—
155C4⋊C8C2×D3711yescharacteristic
156#103—372no—
157#103—372no—
158#103C2×D3712yes—
159#102—371no—
160#102—372no—
161#102—371no—
162#97—372no—
163C4×C8—371no—
164C2×Q8—371no—
165C2×Q8—742no—
166C2×Q8—741no—
167C2×Q8—1484no—
168C2×Q8C22×D3712yes—
169C2×D4—371no—
170C2×D4C22×D3711yescharacteristic
171C22×C4—21no—
172C22×C4—742no—
173C22×C4—741no—
174SD16—24no—
175D8—742no—
176C2×C8—371no—
177C2×C8—371no—
178C2×C8C22×D3712yes—
179C4⋊C4—21no—
180C4⋊C4—21no—
181C4⋊C4—372no—
182C4⋊C4—371no—
183C4⋊C4—372no—
184C4⋊C4—371no—
185C4⋊C4—742no—
186C4⋊C4—742no—
187C4⋊C4—741no—
188C4⋊C4C22×D3711yescharacteristic
189C22⋊C4—21no—
190C22⋊C4—742no—
191C22⋊C4—742no—
192C22⋊C4—742no—
193C22⋊C4—741no—
194C42—372no—
195C42—742no—
196C42C22×D3711yescharacteristic
197C23—21no—
198C23—741no—
199Q8—22no—
200Q8—22no—
201Q8—742no—
202Q8—1484no—
203Q8—1482no—
204D4—21no—
205D4—741no—
206D4—741no—
207D4#1059696412yes—
208C2×C4—22no—
209C2×C4—21no—
210C2×C4—42no—
211C2×C4—372no—
212C2×C4—372no—
213C2×C4—741no—
214C2×C4—742no—
215C2×C4—1484no—
216C2×C4—1482no—
217C2×C4D4×D3711yescharacteristic
218C2×C4D4×D3711yescharacteristic
219C2×C4C23×D3711yescharacteristic
220C8—22no—
221C8—741no—
222C8—741no—
223C22—22no—
224C22—41no—
225C22—41no—
226C22—1481no—
227C22—1481no—
228C22—1481no—
229C22C2×D4×D3711yescharacteristic
230C4—21no—
231C4—41no—
232C4—42no—
233C4—742no—
234C4—742no—
235C4—1482no—
236C4—1481no—
237C4#1192456312yes—
238C4C2×#1059696411yescharacteristic
239C4#1192458311yescharacteristic
240C2—42no—
241C2—2961no—
242C2#1455818911yescharacteristic
243C2C2×#1192456311yescharacteristic
244C2#1455834911yescharacteristic
subgroupG/H
centerC22C2×D4×D37
commutator, cosocleC2×C4×C37C24
FrattiniC2×C4C23×D37
FittingC37×#461C2
socleC22×C37C22×D4
prime radicalC2×#10596939C22
hypercenter#461D37
hypocenterC37#3367
LCSG ▷ C2×C4×C37 ▷ C2×C37 ▷ C37
UCSC1 ◁ C22 ◁ C42 ◁ #461

Complements

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

NH ≅ G/Ntype
1. #14558257C2semidirect
2. D4×Dic74C2semidirect
3. #14557894C2semidirect
4. #14557883C2semidirect
5. #14557826C2semidirect
6. #14557814C2semidirect
8. #14557660C2semidirect
9. #14557634C2semidirect
11. C37×#461C2semidirect
130. #461D37semidirect
134. C37#3367semidirect

Maximal subgroups

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

subgroupindex
#145582572
D4×Dic742
#145578942
#145578832
#145578262
#145578142
#145577502
#145576602
#145576342
C4×#105969392
C37×#4612
#336737