Finite Group Databasebeta

← #18004540

Subgroups — #18004540

Order 4736.

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

normal subgroups only

This group has 410 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×#11924495C211yesmaximal, characteristic
2#14557871C214yesmaximal, complemented
3#14557866C212yesmaximal, complemented
4#14557861C212yesmaximal, complemented
5#14557573C212yesmaximal, complemented
6#14557542C212yesmaximal, complemented
7#14557497C211yesmaximal, characteristic
8C2×C22⋊Q8×C37C211yesmaximal, characteristic
9C2×#10596934C2212yes—
10#11924495—21no—
11#11924495—21no—
12#11924495C2212yes—
13#11924495C2212yes—
14C2×C4⋊Dic37C2211yescharacteristic
15C2×C4⋊Dic37C2211yescharacteristic
16C2×C22⋊Dic37—21no—
17C2×C4×Dic37—21no—
18#11924469C2214yescomplemented
19#11924467C2214yescomplemented
20#11924460C2214yescomplemented
21#11924460C2214yescomplemented
22C22⋊#64436C2212yes—
23C22⋊#64436C2212yes—
24C22×Q8×C37C2211yescharacteristic
25C23×C4×C37C2211yescharacteristic
26C22⋊Q8×C37—21no—
27C22⋊Q8×C37—21no—
28C22⋊Q8×C37C2212yes—
29C22⋊Q8×C37C2212yes—
30C2×C4⋊C4×C37—21no—
31C2×C4⋊C4×C37C2211yescharacteristic
32C2×C22⋊C4×C37—21no—
33C24×C37D411yescharacteristic
34C2×Q8×C37—21no—
35C2×Q8×C37—24no—
36C2×Q8×C37C2312yes—
37C22×C4×C37—24no—
38C22×C4×C37—21no—
39C22×C4×C37—21no—
40C22×C4×C37D411yescharacteristic
41C22×C4×C37D411yescharacteristic
42C22×C4×C37D411yescharacteristic
43C22×C4×C37C2311yescharacteristic
44C22×C4×C37C2311yescharacteristic
45C22×C4×C37C2311yescharacteristic
46C22×Dic37—21no—
47C22×Dic37—21no—
48C4⋊C4×C37—22no—
49C4⋊C4×C37—22no—
50C4⋊C4×C37—21no—
51C4⋊C4×C37C2312yes—
52C22⋊C4×C37—22no—
53C22⋊C4×C37—22no—
54C22⋊Dic37—22no—
55C22⋊Dic37—22no—
56C4⋊Dic37—21no—
57C4⋊Dic37—21no—
58C4⋊Dic37C2312yes—
59C4⋊Dic37C2312yes—
60C4×Dic37—22no—
61C4×Dic37—22no—
62#10596934—24no—
63C2×#64436C2314yes—
64C23×C37—22no—
65C23×C37—22no—
66C23×C37—41no—
67C23×C37C2×D411yescharacteristic
68C23×C37C2×D411yescharacteristic
69C23×C37C2×D411yescharacteristic
70Q8×C37—24no—
71Q8×C37—42no—
72C2×C4×C37—22no—
73C2×C4×C37—22no—
74C2×C4×C37—22no—
75C2×C4×C37—22no—
76C2×C4×C37—22no—
77C2×C4×C37—22no—
78C2×C4×C37—22no—
79C2×C4×C37—42no—
80C2×C4×C37—42no—
81C2×C4×C37C2×D411yescharacteristic
82C2×C4×C37C2×D411yescharacteristic
83C2×C4×C37C2×D411yescharacteristic
84C2×C4×C37#4114yes—
85C2×C4×C37C2411yescharacteristic
86C2×Dic37—22no—
87C2×Dic37—22no—
88C2×Dic37—42no—
89C2×Dic37—42no—
90#64436—24no—
91C22×C37—21no—
92C22×C37—21no—
93C22×C37—21no—
94C22×C37—21no—
95C22×C37—21no—
96C22×C37—21no—
97C22×C37—21no—
98C22×C37—44no—
99C22×C37#12312yes—
100C22×C37#13712yes—
101C22×C37C22×D411yescharacteristic
102C4×C37—21no—
103C4×C37—21no—
104C4×C37—42no—
105C4×C37—42no—
106C4×C37#12312yes—
107C4×C37C2×#4112yes—
108Dic37—42no—
109Dic37—42no—
110#3095—371nomaximal, Sylow
111C2×C37—22no—
112C2×C37—22no—
113C2×C37—41no—
114C2×C37C2×#12311yescharacteristic
115C2×C37C2×#13711yescharacteristic
116C2×C37#57511yescharacteristic
117C2×C22⋊Q8D3711yescharacteristic
118C2×#117—371no—
119#485—374no—
120#484—372no—
121#483—372no—
122#428—372no—
123#419—372no—
124#407—371no—
125C37#309511yescharacteristic, complemented, Sylow
126C22×Q8C2×D3711yescharacteristic
127C23×C4C2×D3711yescharacteristic
128C2×M16—372no—
129C22⋊Q8—21no—
130C22⋊Q8—21no—
131C22⋊Q8C2×D3712yes—
132C22⋊Q8C2×D3712yes—
133#117—372no—
134#117—372no—
135#117—741no—
136#117—741no—
137C2×C4⋊C4—21no—
138C2×C4⋊C4—371no—
139C2×C4⋊C4—371no—
140C2×C4⋊C4C2×D3711yescharacteristic
141C2×C22⋊C4—21no—
142C2×C22⋊C4—741no—
143C2×C42—741no—
144#107—374no—
145#106—374no—
146#103—374no—
147#103—374no—
148C22⋊C8—372no—
149C22⋊C8—372no—
150C24C22⋊D3711yescharacteristic
151C2×Q8—21no—
152C2×Q8—24no—
153C2×Q8C22×D3712yes—
154C22×C4—24no—
155C22×C4—21no—
156C22×C4—21no—
157C22×C4—741no—
158C22×C4—741no—
159C22×C4C22⋊D3711yescharacteristic
160C22×C4C22⋊D3711yescharacteristic
161C22×C4C22⋊D3711yescharacteristic
162C22×C4C22×D3711yescharacteristic
163C22×C4C22×D3711yescharacteristic
164C22×C4C22×D3711yescharacteristic
165M16—744no—
166C2×C8—374no—
167C4⋊C4—22no—
168C4⋊C4—22no—
169C4⋊C4—21no—
170C4⋊C4—372no—
171C4⋊C4—372no—
172C4⋊C4—741no—
173C4⋊C4—741no—
174C4⋊C4C22×D3712yes—
175C22⋊C4—22no—
176C22⋊C4—22no—
177C22⋊C4—742no—
178C22⋊C4—742no—
179C42—742no—
180C42—742no—
181C23—22no—
182C23—22no—
183C23—41no—
184C23C2×C22⋊D3711yescharacteristic
185C23C2×C22⋊D3711yescharacteristic
186C23C2×C22⋊D3711yescharacteristic
187Q8—24no—
188Q8—42no—
189C2×C4—22no—
190C2×C4—22no—
191C2×C4—22no—
192C2×C4—22no—
193C2×C4—22no—
194C2×C4—22no—
195C2×C4—22no—
196C2×C4—42no—
197C2×C4—42no—
198C2×C4—742no—
199C2×C4—742no—
200C2×C4—1482no—
201C2×C4—1482no—
202C2×C4#1059696414yes—
203C2×C4C2×C22⋊D3711yescharacteristic
204C2×C4C2×C22⋊D3711yescharacteristic
205C2×C4C2×C22⋊D3711yescharacteristic
206C2×C4C23×D3711yescharacteristic
207C8—744no—
208C22—21no—
209C22—21no—
210C22—21no—
211C22—21no—
212C22—21no—
213C22—21no—
214C22—21no—
215C22—44no—
216C22#1192451712yes—
217C22#1192456712yes—
218C22C22×C22⋊D3711yescharacteristic
219C4—21no—
220C4—21no—
221C4—42no—
222C4—42no—
223C4—1482no—
224C4—1482no—
225C4#1192451712yes—
226C4C2×#1059696412yes—
227C2—22no—
228C2—22no—
229C2—41no—
230C2C2×#1192451711yescharacteristic
231C2C2×#1192456711yescharacteristic
232C2#1455833111yescharacteristic
subgroupG/H
centerC22C22×C22⋊D37
commutator, cosocleC2×C4×C37C24
FrattiniC2×C4C23×D37
FittingC2×C22⋊Q8×C37C2
socleC22×C37C22×D4
prime radicalC24×C37D4
hypercenterC2×C22⋊Q8D37
hypocenterC37#3095
LCSG ▷ C2×C4×C37 ▷ C2×C37 ▷ C37
UCSC1 ◁ C22 ◁ C23×C4 ◁ C2×C22⋊Q8

Complements

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

NH ≅ G/Ntype
2. #14557871C2semidirect
3. #14557866C2semidirect
4. #14557861C2semidirect
5. #14557573C2semidirect
6. #14557542C2semidirect
18. #11924469C22semidirect
19. #11924467C22semidirect
20. #11924460C22semidirect
21. #11924460C22semidirect
125. C37#3095semidirect

Maximal subgroups

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

subgroupindex
C2×#119244952
#145578712
#145578662
#145578612
#145575732
#145575422
#145574972
C2×C22⋊Q8×C372
#309537