Finite Group Databasebeta

← #18010501

Subgroups — #18010501

Order 4736.

300 conjugacy classes of subgroups; only the 74 normal subgroups are shown.

normal subgroups only

This group has 300 conjugacy classes of subgroups — too many to draw. Only its 74 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
1D37⋊(C22⋊C8)C211yesmaximal, characteristic
2#14558475C211yesmaximal, characteristic
3#14558406C212yesmaximal, complemented
4#14558406C212yesmaximal, complemented
5#117×D37C211yesmaximal, characteristic
6C2×D37⋊C8C2212yes—
7C2×C4⋊#3866C2212yes—
8C2×C22⋊#3866—21no—
9#11924593—21no—
10C22⋊#64438—21no—
11(C2×#3866)⋊C4—21no—
12C22×C4×D37C2211yescharacteristic
13#11924495C411yescharacteristic, complemented
14#11924494C411yescharacteristic, complemented
15#11924494C411yescharacteristic, complemented
16#11924493—21no—
17#11924492—21no—
18C4⋊C4×D37C2211yescharacteristic
19C4⋊C4×D37C2211yescharacteristic
20C22⋊C4×D37—21no—
21C42×D37—21no—
22#117×C37C411yescharacteristic, complemented
23C23×D37D411yescharacteristic
24C22×#3866—22no—
25C22×C4×C37C2×C411yescharacteristic
26C22×Dic37D411yescharacteristic
27C2×C4×D37—24no—
28C2×C4×D37—21no—
29C2×C4×D37—21no—
30C2×C4×D37C2×C411yescharacteristic
31C2×C4×D37D411yescharacteristic
32C2×C4×D37D411yescharacteristic
33C2×C4×D37C2311yescharacteristic
34C22⋊#3866—42no—
35C2×#64438—22no—
36C4⋊#3866—24no—
37D37⋊C8—24no—
38C4⋊C4×C37C2×C411yescharacteristic
39C4⋊C4×C37C2×C411yescharacteristic
40C22⋊C4×C37—21no—
41C42×C37—21no—
42C22⋊Dic37—21no—
43#10596938—21no—
44#10596938—21no—
45C4⋊Dic37C2×C411yescharacteristic
46C4⋊Dic37C2×C411yescharacteristic
47#10596936—21no—
48#10596936—21no—
49C4×Dic37—21no—
50C4×Dic37—21no—
51C4×Dic37—21no—
52C23×C37C22⋊C411yescharacteristic
53C22×D37—22no—
54C22×D37—22no—
55C22×D37—21no—
56C22×D37—21no—
57C22×D37C22⋊C411yescharacteristic
58C22×D37C2×D411yescharacteristic
59C2×#3866—42no—
60C2×#3866—42no—
61C2×#3866—42no—
62C2×C4×C37—22no—
63C2×C4×C37—21no—
64C2×C4×C37—21no—
65C2×C4×C37C22⋊C411yescharacteristic
66C2×C4×C37C22×C411yescharacteristic
67C2×Dic37—22no—
68C2×Dic37—21no—
69C2×Dic37—21no—
70C2×Dic37C22⋊C411yescharacteristic
71C2×Dic37C2×D411yescharacteristic
72C4×D37—22no—
73C4×D37—22no—
74C4×D37—22no—
75C4×D37—42no—
76C4×D37—41no—
77C4×D37—41no—
78C4×D37#4114yes—
79#64438—42no—
80C22×C37—21no—
81C22×C37—21no—
82C22×C37—21no—
83C22×C37C2×C22⋊C411yescharacteristic
84C2×D37—21no—
85C2×D37—21no—
86C2×D37—21no—
87C2×D37—42no—
88C2×D37—42no—
89C2×D37#12312yes—
90C2×D37#13512yes—
91C2×D37AGL(1,ℤ/8)11yescharacteristic
92C2×D37#13711yescharacteristic
93#3866—82no—
94C4×C37—21no—
95C4×C37—41no—
96C4×C37—41no—
97C4×C37#11712yes—
98Dic37—21no—
99Dic37—41no—
100Dic37—41no—
101Dic37#12312yes—
102#1825—371nomaximal, Sylow
103C2×C37—41no—
104C2×C37#38111yescharacteristic
105C2×C37#41611yescharacteristic
106C2×C37#41711yescharacteristic
107D37—41no—
108D37#48212yes—
109D37#48512yes—
110C2×#117—371no—
111C2×C22⋊C8—371no—
112#380—371no—
113#340—372no—
114#340—372no—
115C37#182511yescharacteristic, complemented, Sylow
116C23×C4—371no—
117C22×C8—372no—
118#117—371no—
119#117—371no—
120#117—371no—
121#117—741no—
122#117—741no—
123#117#386611yescharacteristic, complemented
124C2×C4⋊C4—371no—
125C2×C4⋊C4—371no—
126C2×C4⋊C4—372no—
127C2×C22⋊C4—741no—
128C2×C22⋊C4—741no—
129C2×C42—741no—
130C22⋊C8—741no—
131C22⋊C8—741no—
132#95—741no—
133C24—371no—
134C22×C4—371no—
135C22×C4—371no—
136C22×C4—371no—
137C22×C4—371no—
138C22×C4—371no—
139C22×C4—744no—
140C22×C4—741no—
141C22×C4—741no—
142C22×C4—742no—
143C22×C4C2×#386611yescharacteristic
144C2×C8—744no—
145C2×C8—742no—
146C4⋊C4—371no—
147C4⋊C4—371no—
148C4⋊C4—741no—
149C4⋊C4—741no—
150C4⋊C4—744no—
151C4⋊C4C2×#386611yescharacteristic
152C4⋊C4C2×#386611yescharacteristic
153C22⋊C4—21no—
154C22⋊C4—741no—
155C22⋊C4—741no—
156C22⋊C4—741no—
157C22⋊C4—1482no—
158C42—21no—
159C42—741no—
160C42—741no—
161C42—741no—
162C23—371no—
163C23—371no—
164C23—741no—
165C23—742no—
166C23—742no—
167C23—741no—
168C23C22⋊#386611yescharacteristic
169C2×C4—22no—
170C2×C4—21no—
171C2×C4—21no—
172C2×C4—371no—
173C2×C4—374no—
174C2×C4—371no—
175C2×C4—742no—
176C2×C4—742no—
177C2×C4—742no—
178C2×C4—742no—
179C2×C4—741no—
180C2×C4—741no—
181C2×C4—1482no—
182C2×C4—1481no—
183C2×C4—1481no—
184C2×C4—1482no—
185C2×C4—1482no—
186C2×C4—1482no—
187C2×C4C22⋊#386611yescharacteristic
188C2×C4C22×#386611yescharacteristic
189C8—1482no—
190C22—21no—
191C22—21no—
192C22—21no—
193C22—372no—
194C22—372no—
195C22—371no—
196C22—371no—
197C22—741no—
198C22—741no—
199C22—741no—
200C22—1482no—
201C22—1482no—
202C22C2×C22⋊#386611yescharacteristic
203C4—21no—
204C4—41no—
205C4—41no—
206C4—372no—
207C4—2962no—
208C4—741no—
209C4—1481no—
210C4—1481no—
211C4#1192461512yes—
212C2—41no—
213C2—372no—
214C2—372no—
215C2—1481no—
216C2#1455847611yescharacteristic
217C2#1455852311yescharacteristic
218C2#1455852411yescharacteristic
subgroupG/H
centerC22C2×C22⋊#3866
commutatorC2×C4×C37C22×C4
FrattiniC2×C4C22×#3866
Fitting#117×C37C4
socleC22×C37C2×C22⋊C4
prime radicalC23×D37D4
cosocleC2×C4×D37C23
hypercenter#117#3866
hypocenterC37#1825
LCSG ▷ C2×C4×C37 ▷ C2×C37 ▷ C37
UCSC1 ◁ C22 ◁ C22×C4 ◁ #117

Complements

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

NH ≅ G/Ntype
3. #14558406C2semidirect
4. #14558406C2semidirect
13. #11924495C4semidirect
14. #11924494C4semidirect
15. #11924494C4semidirect
22. #117×C37C4semidirect
115. C37#1825semidirect
123. #117#3866semidirect

Maximal subgroups

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

subgroupindex
D37⋊(C22⋊C8)2
#145584752
#145584062
#117×D372
#182537