Finite Group Databasebeta

← #17997768

Subgroups — #17997768

Order 4736.

520 conjugacy classes of subgroups; only the 173 normal subgroups are shown.

normal subgroups only

This group has 520 conjugacy classes of subgroups — too many to draw. Only its 173 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×#11924525C211yesmaximal, characteristic
2#14557440C211yesmaximal, characteristic, complemented
3#14557424C212yesmaximal, complemented
4#14557422C212yesmaximal, complemented
5#14557411C212yesmaximal
6#14557407C212yesmaximal, complemented
7#14557404C212yesmaximal, complemented
8#14557397C212yesmaximal, complemented
9C37×#391C211yesmaximal, characteristic, complemented
10C22×D148C2211yescharacteristic
11C22×C4×D37C2212yes—
12#11924525C418yescomplemented
13C2×C4⋊Dic37C2211yescharacteristic
14C2×#10596936C2212yes—
15C2×#10596936C2212yes—
16C2×#10596938C2212yes—
17C2×#10596938C2212yes—
18C2×#10596938C2212yes—
19C2×#10596938C2212yes—
20C2×#10596938C2212yes—
21C2×C4×Dic37C2211yescharacteristic
22C2×C4×Dic37C2211yescharacteristic
23#11924443C2212yes—
24#11924443C2212yes—
25#11924443C2212yes—
26#11924443C2212yes—
27C22×Q8×C37C2211yescharacteristic
28C2×C4⋊C4×C37C2211yescharacteristic
29C2×C42×C37C2211yescharacteristic
30#95×C37C2212yes—
31#95×C37C2212yes—
32C23×D37C2312yes—
33C2×Q8×C37—41no—
34C2×Q8×C37—41no—
35C2×Q8×C37C2×C414yescomplemented
36C22×C4×C37C2312yes—
37C22×C4×C37C2312yes—
38C22×C4×C37C2311yescharacteristic
39C22×C4×C37C2311yescharacteristic
40C22×C4×C37C2311yescharacteristic
41C22×Dic37C2312yes—
42C22×Dic37C2312yes—
43C22×Dic37C2312yes—
44C2×D148—41no—
45C2×D148—41no—
46C2×D148C2×C414yes—
47C2×C4×D37—42no—
48C2×C4×D37—42no—
49C2×C4×D37—42no—
50C4⋊C4×C37—41no—
51C42×C37—41no—
52#10596938—42no—
53#10596938—42no—
54#10596938—42no—
55#10596938C2×C418yescomplemented
56#10596938C2×C418yescomplemented
57C4⋊Dic37—41no—
58#10596936—42no—
59#10596936—42no—
60C4×Dic37—41no—
61C4×Dic37C2×C414yescomplemented
62C23×C37C2411yescharacteristic
63C22×D37—22no—
64C22×D37—22no—
65C22×D37—22no—
66C22×D37—22no—
67C22×D37—22no—
68C22×D37—22no—
69C22×D37C22×C414yes—
70Q8×C37—44no—
71C2×C4×C37—21no—
72C2×C4×C37—21no—
73C2×C4×C37—21no—
74C2×C4×C37—21no—
75C2×C4×C37—21no—
76C2×C4×C37—21no—
77C2×C4×C37—22no—
78C2×C4×C37—22no—
79C2×C4×C37—22no—
80C2×C4×C37—22no—
81C2×C4×C37—22no—
82C2×C4×C37—22no—
83C2×C4×C37C22×C412yes—
84C2×C4×C37C22×C412yes—
85C2×C4×C37C22×C412yes—
86C2×Dic37—22no—
87C2×Dic37—22no—
88C2×Dic37—22no—
89C2×Dic37—22no—
90C2×Dic37—22no—
91C2×Dic37—22no—
92C2×Dic37—22no—
93C2×Dic37—22no—
94C2×Dic37C22×C414yes—
95D148—44no—
96C4×D37—48no—
97C22×C37C23×C411yescharacteristic
98C22×C37H(5,𝔽2)11yescharacteristic
99C22×C37H(5,𝔽2)11yescharacteristic
100C22×C37H(5,𝔽2)12yes—
101C22×C372−1+412yes—
102C2×D37—24no—
103C2×D37—24no—
104C2×D37—24no—
105C2×D37—42no—
106C2×D37—42no—
107C2×D37—42no—
108C2×D37—42no—
109C4×C37—22no—
110C4×C37—22no—
111C4×C37—22no—
112C4×C37—42no—
113C4×C37—42no—
114Dic37—24no—
115Dic37—42no—
116Dic37—42no—
117#2379—371nomaximal, Sylow
118C2×C37#51811yescharacteristic
119C2×C37#52012yes—
120C2×C37H(3,𝔽4)11yescharacteristic
121C2×C37#56212yes—
122C2×C37#56311yescharacteristic
123D37—44no—
124C2×#124—371no—
125#391—371no—
126#391D3711yescharacteristic, complemented
127#388—372no—
128#388—372no—
129#386—372no—
130#385—372no—
131#385—372no—
132#382—372no—
133C37#237911yescharacteristic, complemented, Sylow
134C22×Q8C2×D3711yescharacteristic
135C22×D4—371no—
136C23×C4—372no—
137#124—378no—
138C2×C4⋊C4—372no—
139C2×C4⋊C4—372no—
140C2×C4⋊C4—371no—
141C2×C4⋊C4C2×D3711yescharacteristic
142C2×C22⋊C4—372no—
143C2×C22⋊C4—372no—
144C2×C22⋊C4—372no—
145C2×C22⋊C4—372no—
146C2×C22⋊C4—372no—
147C2×C42—371no—
148C2×C42—371no—
149C2×C42C2×D3711yescharacteristic
150#95—372no—
151#95—372no—
152#95—372no—
153#95—372no—
154#95C2×D3712yes—
155#95C2×D3712yes—
156C24—372no—
157C2×Q8—41no—
158C2×Q8—41no—
159C2×Q8C4×D3714yescomplemented
160C2×D4—374no—
161C2×D4—1481no—
162C2×D4—1481no—
163C22×C4—372no—
164C22×C4—372no—
165C22×C4—372no—
166C22×C4—1482no—
167C22×C4—1482no—
168C22×C4—1482no—
169C22×C4C22×D3712yes—
170C22×C4C22×D3712yes—
171C22×C4C22×D3711yescharacteristic
172C22×C4C22×D3711yescharacteristic
173C22×C4C22×D3711yescharacteristic
174C4⋊C4—41no—
175C4⋊C4—1482no—
176C4⋊C4—1482no—
177C4⋊C4—1481no—
178C22⋊C4—378no—
179C22⋊C4—378no—
180C22⋊C4—1482no—
181C22⋊C4—1482no—
182C22⋊C4—1482no—
183C42—41no—
184C42—374no—
185C42—1481no—
186C23—374no—
187C23—742no—
188C23—742no—
189C23—742no—
190C23—742no—
191C23—742no—
192C23—742no—
193C23C23×D3711yescharacteristic
194Q8—44no—
195D4—1484no—
196C2×C4—21no—
197C2×C4—21no—
198C2×C4—21no—
199C2×C4—21no—
200C2×C4—21no—
201C2×C4—21no—
202C2×C4—22no—
203C2×C4—22no—
204C2×C4—22no—
205C2×C4—22no—
206C2×C4—22no—
207C2×C4—22no—
208C2×C4—374no—
209C2×C4—742no—
210C2×C4—742no—
211C2×C4—742no—
212C2×C4—742no—
213C2×C4—742no—
214C2×C4—742no—
215C2×C4—742no—
216C2×C4—742no—
217C2×C4—1488no—
218C2×C4C2×C4×D3712yes—
219C2×C4C2×C4×D3712yes—
220C2×C4C2×C4×D3712yes—
221C22—744no—
222C22—744no—
223C22—744no—
224C22—1482no—
225C22—1482no—
226C22—1482no—
227C22—1482no—
228C22C22×C4×D3711yescharacteristic
229C22#1192458111yescharacteristic
230C22#1192458111yescharacteristic
231C22#1192458212yes—
232C22#1192458412yes—
233C4—22no—
234C4—22no—
235C4—22no—
236C4—42no—
237C4—42no—
238C4—744no—
239C4—1482no—
240C4—1482no—
241C2—1484no—
242C2#1455800611yescharacteristic
243C2#1455801512yes—
244C2#1455827511yescharacteristic
245C2#1455827912yes—
246C2#1455828211yescharacteristic
subgroupG/H
center, FrattiniC23C23×D37
commutatorC22×C37C23×C4
FittingC37×#391C2
socle, cosocleC23×C37C24
prime radicalC22×D148C22
hypercenter#391D37
hypocenterC37#2379
LCSG ▷ C22×C37 ▷ C37
UCSC1 ◁ C23 ◁ #391

Complements

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

NH ≅ G/Ntype
2. #14557440C2semidirect
3. #14557424C2semidirect
4. #14557422C2semidirect
6. #14557407C2semidirect
7. #14557404C2semidirect
8. #14557397C2semidirect
9. C37×#391C2semidirect
12. #11924525C4semidirect
35. C2×Q8×C37C2×C4semidirect
55. #10596938C2×C4semidirect
56. #10596938C2×C4semidirect
61. C4×Dic37C2×C4semidirect
126. #391D37semidirect
133. C37#2379semidirect
159. C2×Q8C4×D37semidirect

Maximal subgroups

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

subgroupindex
C2×#119245252
#145574402
#145574242
#145574222
#145574112
#145574072
#145574042
#145573972
C37×#3912
#237937