Finite Group Databasebeta

← #17995622

Subgroups — #17995622

Order 4736.

394 conjugacy classes of subgroups; only the 75 normal subgroups are shown.

normal subgroups only

This group has 394 conjugacy classes of subgroups — too many to draw. Only its 75 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#14558097C211yesmaximal, characteristic
2C2×#11924525C211yesmaximal, characteristic, complemented
3C2×#11924463C211yesmaximal, characteristic, complemented
4#14557542C211yesmaximal, characteristic, complemented
5C2×#11924452C211yesmaximal, characteristic, complemented
6#14557294C211yesmaximal, characteristic, complemented
7C2×#137×C37C211yesmaximal, characteristic, complemented
8C22×D148C2211yescharacteristic
9C2×#10596934C2211yescharacteristic
10#11924532—21no—
11#11924525—21no—
12#11924525—22no—
13#11924525—22no—
14#11924510—22no—
15#11924504—22no—
16#11924495C2211yescharacteristic
17C2×#10596938—22no—
18C2×C4×Dic37C2211yescharacteristic
19#11924463—22no—
20#11924460—22no—
21#11924452—22no—
22C2×#41×C37C2211yescharacteristic
23C22×Q8×C37C2211yescharacteristic
24#137×C37—22no—
25#137×C37—22no—
26C2×Q16×C37—21no—
27C2×SD16×C37—21no—
28C2×M16×C37C2211yescharacteristic
29C23×D37D412yes—
30#41×C37—21no—
31#41×C37—21no—
32#41×C37—21no—
33#41×C37—21no—
34C2×Q8×C37—21no—
35C2×Q8×C37—21no—
36C2×Q8×C37—21no—
37C2×Q8×C37—21no—
38C2×Q8×C37—21no—
39C2×Q8×C37D411yescharacteristic
40C2×Q8×C37D412yescomplemented
41C2×D4×C37—21no—
42C2×D4×C37D411yescharacteristic
43C22×C4×C37—21no—
44C22×C4×C37—21no—
45C22×C4×C37C2311yescharacteristic
46C2×C22⋊D37—24no—
47C22×Dic37—21no—
48C2×D148—21no—
49C2×D148—21no—
50C2×D148—21no—
51C2×D148—21no—
52C2×D148—21no—
53C2×D148—21no—
54Q16×C37—24no—
55SD16×C37—24no—
56M16×C37—21no—
57M16×C37D412yescomplemented
58C2×C8×C37—21no—
59C22⋊Dic37—21no—
60#10596938—28no—
61#10596938—24no—
62C4⋊Dic37D412yes—
63C4×Dic37—21no—
64C4×Dic37—21no—
65C4×Dic37D412yescomplemented
66#10596934—21no—
67#10596934—21no—
68C2×#64436—21no—
69C23×C37—21no—
70C23×C37C2×D411yescharacteristic
71C22×D37—22no—
72C22×D37—22no—
73C22×D37—22no—
74C22×D37—44no—
75Q8×C37—22no—
76Q8×C37—21no—
77Q8×C37—22no—
78Q8×C37—21no—
79Q8×C37—41no—
80Q8×C37—41no—
81D4×C37—21no—
82D4×C37—21no—
83D4×C37—41no—
84D4×C37—41no—
85C2×C4×C37—21no—
86C2×C4×C37—22no—
87C2×C4×C37—41no—
88C2×C4×C37—41no—
89C2×C4×C37—41no—
90C2×C4×C37—41no—
91C2×C4×C37C2×D411yescharacteristic
92C2×C4×C37C2×D411yescharacteristic
93C2×C4×C37C2×D411yescharacteristic
94C2×C4×C37C2×D411yescharacteristic
95C2×C4×C37C2×D411yescharacteristic
96C2×C4×C37#4111yescharacteristic
97C22⋊D37—84no—
98C2×Dic37—22no—
99C2×Dic37—22no—
100C2×Dic37—42no—
101D148—41no—
102D148—41no—
103D148—41no—
104D148—41no—
105C8×C37—22no—
106#64436—41no—
107C22×C37—22no—
108C22×C37—41no—
109C22×C37—42no—
110C22×C37#12011yescharacteristic
111C22×C37#12011yescharacteristic
112C22×C37#12111yescharacteristic
113C2×D37—44no—
114C2×D37—42no—
115C2×D37—44no—
116C2×D37—82no—
117C4×C37—41no—
118C4×C37—42no—
119C4×C37#12011yescharacteristic
120C4×C37#12111yescharacteristic
121C4×C37#12111yescharacteristic
122C4×C37#12711yescharacteristic
123Dic37—42no—
124Dic37—42no—
125#2004—371nomaximal, Sylow
126C2×C37—21no—
127C2×C37—21no—
128C2×C37—81no—
129C2×C37#39211yescharacteristic
130C2×C37#45512yes—
131D37—82no—
132C2×#137D3711yescharacteristic, complemented
133#535—371no—
134C2×#124—371no—
135C2×#104—371no—
136#419—371no—
137C2×#100—371no—
138#339—371no—
139C37#200411yescharacteristic, complemented, Sylow
140C2×#41C2×D3711yescharacteristic
141C22×Q8C2×D3711yescharacteristic
142C22×D4—371no—
143#137—22no—
144#137—22no—
145C2×Q16—21no—
146C2×SD16—21no—
147C2×M16—371no—
148C2×M16C2×D3711yescharacteristic
149#127—741no—
150#124—742no—
151#124—741no—
152#124—742no—
153#121—742no—
154#120—742no—
155#117—371no—
156C2×C22⋊C4—742no—
157C2×C42—371no—
158#104—742no—
159#103—742no—
160#100—742no—
161C24—372no—
162#41—21no—
163#41—21no—
164#41—21no—
165#41—21no—
166C2×Q8—21no—
167C2×Q8—21no—
168C2×Q8—21no—
169C2×Q8—21no—
170C2×Q8—21no—
171C2×Q8C22⋊D3711yescharacteristic
172C2×Q8C22⋊D3712yescomplemented
173C2×D4—21no—
174C2×D4—744no—
175C2×D4—741no—
176C2×D4—741no—
177C2×D4—741no—
178C2×D4—741no—
179C2×D4—741no—
180C2×D4—741no—
181C2×D4C22⋊D3711yescharacteristic
182C22×C4—21no—
183C22×C4—21no—
184C22×C4—741no—
185C22×C4C22×D3711yescharacteristic
186Q16—24no—
187SD16—24no—
188M16—21no—
189M16—741no—
190M16—741no—
191M16C22⋊D3712yescomplemented
192C2×C8—21no—
193C2×C8—741no—
194C4⋊C4—372no—
195C22⋊C4—748no—
196C22⋊C4—744no—
197C22⋊C4—741no—
198C42—372no—
199C42—741no—
200C42—741no—
201C23—21no—
202C23—742no—
203C23—742no—
204C23—742no—
205C23—1484no—
206C23D4×D3711yescharacteristic
207Q8—22no—
208Q8—21no—
209Q8—22no—
210Q8—21no—
211Q8—41no—
212Q8—41no—
213D4—21no—
214D4—21no—
215D4—41no—
216D4—41no—
217D4—2964no—
218D4—1481no—
219D4—1481no—
220D4—1481no—
221D4—1481no—
222C2×C4—22no—
223C2×C4—21no—
224C2×C4—41no—
225C2×C4—41no—
226C2×C4—41no—
227C2×C4—41no—
228C2×C4—742no—
229C2×C4—742no—
230C2×C4—1482no—
231C2×C4D4×D3711yescharacteristic
232C2×C4D4×D3711yescharacteristic
233C2×C4#1059696411yescharacteristic
234C2×C4C2×C22⋊D3711yescharacteristic
235C2×C4C2×C22⋊D3711yescharacteristic
236C2×C4C2×C22⋊D3711yescharacteristic
237C8—22no—
238C8—1481no—
239C22—22no—
240C22—41no—
241C22—42no—
242C22—2962no—
243C22—1484no—
244C22—1482no—
245C22—1484no—
246C22#1192450411yescharacteristic
247C22#1192450411yescharacteristic
248C22#1192450711yescharacteristic
249C4—41no—
250C4—42no—
251C4—1482no—
252C4—1482no—
253C4#1192450511yescharacteristic
254C4#1192450811yescharacteristic
255C4#1192450811yescharacteristic
256C4#1192453211yescharacteristic
257C2—21no—
258C2—21no—
259C2—81no—
260C2—2962no—
261C2#1455744111yescharacteristic
262C2#1455770912yes—
subgroupG/H
centerC22#11924504
commutator, cosocleC22×C4×C37C23
FrattiniC22×C4C22×D37
FittingC2×#137×C37C2
socleC22×C37#120
prime radical#14558097C2
hypercenterC2×#137D37
hypocenterC37#2004
LCSG ▷ C22×C4×C37 ▷ C2×C37 ▷ C37
UCSC1 ◁ C22 ◁ C22×C4 ◁ C2×#137

Complements

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

NH ≅ G/Ntype
2. C2×#11924525C2semidirect
3. C2×#11924463C2semidirect
4. #14557542C2semidirect
5. C2×#11924452C2semidirect
6. #14557294C2semidirect
7. C2×#137×C37C2semidirect
40. C2×Q8×C37D4semidirect
57. M16×C37D4semidirect
65. C4×Dic37D4semidirect
132. C2×#137D37semidirect
139. C37#2004semidirect
172. C2×Q8C22⋊D37semidirect
191. M16C22⋊D37semidirect

Maximal subgroups

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

subgroupindex
#145580972
C2×#119245252
C2×#119244632
#145575422
C2×#119244522
#145572942
C2×#137×C372
#200437