Finite Group Databasebeta

← #18004070

Subgroups — #18004070

Order 4736.

484 conjugacy classes of subgroups; only the 145 normal subgroups are shown.

normal subgroups only

This group has 484 conjugacy classes of subgroups — too many to draw. Only its 145 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×#11924569C212yesmaximal, complemented
2#14558083C211yesmaximal, characteristic
3C2×C4×Dic74C211yesmaximal, characteristic, complemented
4#14557756C214yesmaximal, complemented
5#14557744C212yesmaximal, complemented
6#14557734C212yesmaximal, complemented
7#14557535C212yesmaximal, complemented
8C37×#423C211yesmaximal, characteristic, complemented
9C2×#10596962C2212yes—
10C22×Dic74C2211yescharacteristic
11#11924569—28no—
12C2×Dic148C2214yes—
13C2×#10596930C2214yes—
14#11924536C2211yescharacteristic
15#11924533C2211yescharacteristic
16#11924524C2212yes—
17#11924510—21no—
18#11924510—21no—
19C4×Dic74—21no—
20C4×Dic74—21no—
21C4×Dic74C2212yes—
22C4×Dic74C2212yes—
23C4×D148—21no—
24C4×D148—21no—
25C2×C4⋊Dic37C2211yescharacteristic
26C2×#10596936—21no—
27C2×C4×Dic37—21no—
28#11924461C2214yes—
29#11924458C2214yes—
30C2×M16×C37C2212yes—
31C2×C42×C37C2211yescharacteristic
32C4⋊C8×C37C2212yes—
33C4⋊C8×C37C2212yes—
34C22×C4×C37D411yescharacteristic
35C22×C4×C37D411yescharacteristic
36C22×C4×C37C2311yescharacteristic
37C2×C22⋊D37—22no—
38C22×Dic37—21no—
39C22×Dic37—21no—
40#10596962—24no—
41#10596962—24no—
42C2×D148—21no—
43C2×D148—21no—
44C2×D148C2312yes—
45C2×C4×D37—22no—
46C2×Dic74—21no—
47C2×Dic74—22no—
48C2×Dic74D414yescomplemented
49C2×Dic74C2312yes—
50C2×Dic74C2312yes—
51M16×C37—24no—
52C2×C8×C37C2314yes—
53C42×C37D411yescharacteristic
54C42×C37D411yescharacteristic
55C42×C37C2311yescharacteristic
56C42×C37C2311yescharacteristic
57#10596938—22no—
58#10596938—22no—
59C4⋊Dic37—21no—
60C4⋊Dic37—21no—
61C4⋊Dic37—21no—
62C4⋊Dic37C2312yes—
63#10596936—22no—
64#10596936—22no—
65C4×Dic37—22no—
66C4×Dic37—22no—
67Dic148—28no—
68#10596930—28no—
69C23×C37C2×D411yescharacteristic
70C22×D37—22no—
71C2×C4×C37—22no—
72C2×C4×C37—41no—
73C2×C4×C37C2×D411yescharacteristic
74C2×C4×C37C2×D411yescharacteristic
75C2×C4×C37C2×D411yescharacteristic
76C2×C4×C37C2×D411yescharacteristic
77C2×C4×C37C2×D411yescharacteristic
78C2×C4×C37C2×D412yes—
79C2×C4×C37#4112yes—
80C2×C4×C37C2411yescharacteristic
81C22⋊D37—44no—
82C2×Dic37—24no—
83C2×Dic37—22no—
84C2×Dic37—22no—
85C2×Dic37—22no—
86C2×Dic37—42no—
87D148—22no—
88D148—22no—
89D148—42no—
90D148—42no—
91C4×D37—44no—
92Dic74—24no—
93Dic74—22no—
94Dic74—22no—
95Dic74—22no—
96Dic74C2×D414yes—
97C8×C37—24no—
98C22×C37—21no—
99C22×C37—21no—
100C22×C37#12112yes—
101C22×C37C22×D411yescharacteristic
102C2×D37—42no—
103C2×D37—42no—
104C2×D37—42no—
105C4×C37—21no—
106C4×C37—21no—
107C4×C37—21no—
108C4×C37#12112yes—
109C4×C37#13712yescomplemented
110C4×C37C22×D411yescharacteristic
111C4×C37C2×#4111yescharacteristic
112Dic37—24no—
113Dic37—42no—
114Dic37—42no—
115#3037—371nomaximal, Sylow
116C2×C37—22no—
117C2×C37C2×#12111yescharacteristic
118C2×C37C2×#13711yescharacteristic
119C2×C37#57511yescharacteristic
120D37—82no—
121C2×#137—372no—
122#532—371no—
123C2×C4×Q8—371no—
124#464—374no—
125#462—372no—
126#460—372no—
127#423D3711yescharacteristic, complemented
128#417—372no—
129C37#303711yescharacteristic, complemented, Sylow
130C2×#41—372no—
131C22×Q8—371no—
132#137—748no—
133C2×Q16—374no—
134C2×SD16—374no—
135C2×M16C2×D3712yes—
136C4⋊Q8—371no—
137#127—371no—
138#124—372no—
139#121—741no—
140#121—741no—
141C4×Q8—372no—
142C4×Q8—372no—
143C4×Q8—741no—
144C4×Q8—741no—
145C4×D4—741no—
146C4×D4—741no—
147C2×C4⋊C4—371no—
148C2×C4⋊C4—741no—
149C2×C42—741no—
150C2×C42C2×D3711yescharacteristic
151C4⋊C8C2×D3712yes—
152C4⋊C8C2×D3712yes—
153#103—374no—
154#102—374no—
155#41—744no—
156#41—744no—
157C2×Q8—374no—
158C2×Q8—372no—
159C2×Q8—372no—
160C2×Q8—741no—
161C2×Q8—742no—
162C2×D4—372no—
163C2×D4—742no—
164C2×D4—741no—
165C2×D4—741no—
166C22×C4—741no—
167C22×C4—741no—
168C22×C4—742no—
169C22×C4D14811yescharacteristic
170C22×C4D14811yescharacteristic
171C22×C4C22×D3711yescharacteristic
172Q16—748no—
173SD16—748no—
174M16—24no—
175C2×C8C22×D3714yes—
176C4⋊C4—372no—
177C4⋊C4—742no—
178C4⋊C4—742no—
179C4⋊C4—741no—
180C4⋊C4—741no—
181C4⋊C4—741no—
182C22⋊C4—742no—
183C22⋊C4—742no—
184C42—742no—
185C42—742no—
186C42D14811yescharacteristic
187C42D14811yescharacteristic
188C42C22×D3711yescharacteristic
189C42C22×D3711yescharacteristic
190C23—742no—
191C23C2×D14811yescharacteristic
192Q8—374no—
193Q8—744no—
194Q8—742no—
195Q8—742no—
196Q8—742no—
197D4—742no—
198D4—742no—
199D4—1482no—
200D4—1484no—
201D4—1482no—
202C2×C4—22no—
203C2×C4—41no—
204C2×C4—744no—
205C2×C4—742no—
206C2×C4—742no—
207C2×C4—742no—
208C2×C4—1482no—
209C2×C4—1484no—
210C2×C4C2×D14811yescharacteristic
211C2×C4C2×D14811yescharacteristic
212C2×C4C2×D14811yescharacteristic
213C2×C4C2×D14811yescharacteristic
214C2×C4C2×D14811yescharacteristic
215C2×C4D4×D3712yes—
216C2×C4#1059696612yes—
217C2×C4C23×D3711yescharacteristic
218C8—24no—
219C22—21no—
220C22—21no—
221C22—1482no—
222C22—1482no—
223C22—1482no—
224C22#1192451112yes—
225C22C22×D14811yescharacteristic
226C4—21no—
227C4—21no—
228C4—21no—
229C4—744no—
230C4—1482no—
231C4—1482no—
232C4#1192451112yes—
233C4#1192456912yescomplemented
234C4C2×D4×D3711yescharacteristic
235C4C2×#1059696611yescharacteristic
236C2—22no—
237C2—2962no—
238C2C2×#1192451111yescharacteristic
239C2C2×#1192456911yescharacteristic
240C2#1455833311yescharacteristic
subgroupG/H
centerC22C22×D148
commutator, cosocleC2×C4×C37C24
FrattiniC2×C4C23×D37
FittingC37×#423C2
socleC22×C37C22×D4
prime radical#14558083C2
hypercenter#423D37
hypocenterC37#3037
LCSG ▷ C2×C4×C37 ▷ C2×C37 ▷ C37
UCSC1 ◁ C22 ◁ C2×C42 ◁ #423

Complements

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

NH ≅ G/Ntype
1. C2×#11924569C2semidirect
3. C2×C4×Dic74C2semidirect
4. #14557756C2semidirect
5. #14557744C2semidirect
6. #14557734C2semidirect
7. #14557535C2semidirect
8. C37×#423C2semidirect
48. C2×Dic74D4semidirect
109. C4×C37#137semidirect
127. #423D37semidirect
129. C37#3037semidirect
233. C4#11924569semidirect

Maximal subgroups

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

subgroupindex
C2×#119245692
#145580832
C2×C4×Dic742
#145577562
#145577442
#145577342
#145575352
C37×#4232
#303737