Finite Group Databasebeta

← #17995608

Subgroups — #17995608

Order 4736.

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

normal subgroups only

This group has 478 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, complemented
2C2×#11924532C211yesmaximal, characteristic, complemented
3C2×#11924463C211yesmaximal, characteristic, complemented
4#14557538C211yesmaximal, characteristic, complemented
5C2×#11924452C211yesmaximal, characteristic, complemented
6#14557294C211yesmaximal, characteristic, complemented
7C2×AGL(1,ℤ/8)×C37C211yesmaximal, characteristic, complemented
8C22×D148C2211yescharacteristic
9C22×C22⋊D37—22no—
10C2×#10596934C2211yescharacteristic
11#11924532—22no—
12#11924532—21no—
13#11924532—22no—
14#11924525—21no—
15#11924510—22no—
16#11924504—22no—
17#11924495C2211yescharacteristic
18C2×C4×Dic37C2211yescharacteristic
19#11924463—22no—
20#11924457—22no—
21#11924452—22no—
22C2×#41×C37C2211yescharacteristic
23C22×D4×C37C2211yescharacteristic
24AGL(1,ℤ/8)×C37—22no—
25AGL(1,ℤ/8)×C37—22no—
26C2×SD16×C37—21no—
27C2×D8×C37—21no—
28C2×M16×C37C2211yescharacteristic
29C24×C37—21no—
30C23×D37D412yes—
31#41×C37—21no—
32#41×C37—21no—
33#41×C37—21no—
34#41×C37—21no—
35C2×Q8×C37D411yescharacteristic, complemented
36C2×D4×C37—21no—
37C2×D4×C37—21no—
38C2×D4×C37—21no—
39C2×D4×C37—21no—
40C2×D4×C37—21no—
41C2×D4×C37—21no—
42C2×D4×C37D411yescharacteristic, complemented
43C2×D4×C37D412yescomplemented
44C22×C4×C37—21no—
45C22×C4×C37C2311yescharacteristic
46C2×C22⋊D37—28no—
47C2×C22⋊D37—24no—
48C2×C22⋊D37—48no—
49C22×Dic37—21no—
50C2×D148—21no—
51C2×D148—21no—
52C2×D148—21no—
53C2×D148—21no—
54C2×D148—21no—
55C2×D148—21no—
56SD16×C37—24no—
57D8×C37—24no—
58M16×C37—21no—
59M16×C37D412yescomplemented
60C2×C8×C37—21no—
61C22⋊Dic37—21no—
62#10596938—24no—
63C4⋊Dic37D412yescomplemented
64C4×Dic37—21no—
65C4×Dic37—21no—
66C4×Dic37D412yescomplemented
67#10596934—21no—
68#10596934—21no—
69C2×#64436—21no—
70C23×C37—21no—
71C23×C37—22no—
72C23×C37—44no—
73C23×C37—41no—
74C23×C37—41no—
75C23×C37C2×D411yescharacteristic
76C22×D37—22no—
77C22×D37—22no—
78C22×D37—22no—
79C22×D37—44no—
80Q8×C37—21no—
81Q8×C37—21no—
82D4×C37—21no—
83D4×C37—22no—
84D4×C37—22no—
85D4×C37—21no—
86D4×C37—41no—
87D4×C37—41no—
88D4×C37—41no—
89D4×C37—41no—
90C2×C4×C37—21no—
91C2×C4×C37—41no—
92C2×C4×C37—41no—
93C2×C4×C37C2×D411yescharacteristic
94C2×C4×C37C2×D411yescharacteristic
95C2×C4×C37C2×D411yescharacteristic
96C2×C4×C37C2×D411yescharacteristic
97C2×C4×C37C2×D411yescharacteristic
98C2×C4×C37#4111yescharacteristic
99C22⋊D37—416no—
100C22⋊D37—84no—
101C2×Dic37—24no—
102C2×Dic37—22no—
103C2×Dic37—22no—
104D148—41no—
105D148—41no—
106D148—41no—
107D148—41no—
108C8×C37—22no—
109#64436—41no—
110C22×C37—22no—
111C22×C37—44no—
112C22×C37—41no—
113C22×C37—42no—
114C22×C37—42no—
115C22×C37—82no—
116C22×C37—82no—
117C22×C37#12011yescharacteristic
118C22×C37#12011yescharacteristic
119C22×C37#12111yescharacteristic
120C2×D37—44no—
121C2×D37—42no—
122C2×D37—44no—
123C2×D37—82no—
124C4×C37—41no—
125C4×C37#12011yescharacteristic
126C4×C37#12111yescharacteristic
127C4×C37#12111yescharacteristic
128C4×C37#12711yescharacteristic
129Dic37—24no—
130Dic37—42no—
131#2002—371nomaximal, Sylow
132C2×C37—21no—
133C2×C37—21no—
134C2×C37—81no—
135C2×C37—82no—
136C2×C37#39211yescharacteristic
137C2×C37#45312yes—
138D37—82no—
139C2×AGL(1,ℤ/8)D3711yescharacteristic, complemented
140#535—371no—
141C2×#127—371no—
142C2×#104—371no—
143#418—371no—
144C2×#100—371no—
145#339—371no—
146C37#200211yescharacteristic, complemented, Sylow
147C2×#41C2×D3711yescharacteristic
148C22×D4—371no—
149C22×D4—742no—
150C22×D4C2×D3711yescharacteristic
151AGL(1,ℤ/8)—22no—
152AGL(1,ℤ/8)—22no—
153C2×SD16—21no—
154C2×D8—21no—
155C2×M16—371no—
156C2×M16C2×D3711yescharacteristic
157#127—742no—
158#127—742no—
159#127—741no—
160#124—741no—
161#121—742no—
162#120—742no—
163#117—371no—
164C2×C42—371no—
165#104—742no—
166#102—742no—
167#100—742no—
168C24—21no—
169C24—372no—
170#41—21no—
171#41—21no—
172#41—21no—
173#41—21no—
174C2×Q8C22⋊D3711yescharacteristic, complemented
175C2×D4—21no—
176C2×D4—21no—
177C2×D4—21no—
178C2×D4—21no—
179C2×D4—21no—
180C2×D4—21no—
181C2×D4—744no—
182C2×D4—748no—
183C2×D4—741no—
184C2×D4—741no—
185C2×D4—741no—
186C2×D4—741no—
187C2×D4—741no—
188C2×D4—741no—
189C2×D4—1488no—
190C2×D4C22⋊D3711yescharacteristic, complemented
191C2×D4C22⋊D3712yescomplemented
192C22×C4—21no—
193C22×C4—741no—
194C22×C4C22×D3711yescharacteristic
195SD16—24no—
196D8—24no—
197M16—21no—
198M16—741no—
199M16—741no—
200M16C22⋊D3712yescomplemented
201C2×C8—21no—
202C2×C8—741no—
203C4⋊C4—372no—
204C22⋊C4—744no—
205C22⋊C4—741no—
206C42—372no—
207C42—741no—
208C42—741no—
209C23—22no—
210C23—21no—
211C23—44no—
212C23—41no—
213C23—41no—
214C23—742no—
215C23—742no—
216C23—742no—
217C23—1484no—
218C23D4×D3711yescharacteristic
219Q8—21no—
220Q8—21no—
221D4—21no—
222D4—22no—
223D4—22no—
224D4—21no—
225D4—41no—
226D4—41no—
227D4—41no—
228D4—41no—
229D4—2964no—
230D4—14816no—
231D4—1481no—
232D4—1481no—
233D4—1481no—
234D4—1481no—
235C2×C4—21no—
236C2×C4—41no—
237C2×C4—41no—
238C2×C4—744no—
239C2×C4—742no—
240C2×C4—742no—
241C2×C4D4×D3711yescharacteristic
242C2×C4D4×D3711yescharacteristic
243C2×C4#1059696411yescharacteristic
244C2×C4C2×C22⋊D3711yescharacteristic
245C2×C4C2×C22⋊D3711yescharacteristic
246C2×C4C2×C22⋊D3711yescharacteristic
247C8—22no—
248C8—1481no—
249C22—22no—
250C22—44no—
251C22—41no—
252C22—42no—
253C22—42no—
254C22—82no—
255C22—82no—
256C22—2962no—
257C22—1484no—
258C22—1482no—
259C22—1484no—
260C22#1192450411yescharacteristic
261C22#1192450411yescharacteristic
262C22#1192450711yescharacteristic
263C4—41no—
264C4—744no—
265C4—1482no—
266C4#1192450511yescharacteristic
267C4#1192450811yescharacteristic
268C4#1192450811yescharacteristic
269C4#1192453211yescharacteristic
270C2—21no—
271C2—21no—
272C2—82no—
273C2—81no—
274C2—2962no—
275C2#1455744111yescharacteristic
276C2#1455769912yes—
subgroupG/H
centerC22#11924504
commutator, cosocleC22×C4×C37C23
FrattiniC22×C4C22×D37
FittingC2×AGL(1,ℤ/8)×C37C2
socleC22×C37#120
hypercenterC2×AGL(1,ℤ/8)D37
hypocenterC37#2002
LCSG ▷ C22×C4×C37 ▷ C2×C37 ▷ C37
UCSC1 ◁ C22 ◁ C22×C4 ◁ C2×AGL(1,ℤ/8)

Complements

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

NH ≅ G/Ntype
1. #14558097C2semidirect
2. C2×#11924532C2semidirect
3. C2×#11924463C2semidirect
4. #14557538C2semidirect
5. C2×#11924452C2semidirect
6. #14557294C2semidirect
7. C2×AGL(1,ℤ/8)×C37C2semidirect
35. C2×Q8×C37D4semidirect
42. C2×D4×C37D4semidirect
43. C2×D4×C37D4semidirect
59. M16×C37D4semidirect
63. C4⋊Dic37D4semidirect
66. C4×Dic37D4semidirect
139. C2×AGL(1,ℤ/8)D37semidirect
146. C37#2002semidirect
174. C2×Q8C22⋊D37semidirect
190. C2×D4C22⋊D37semidirect
191. C2×D4C22⋊D37semidirect
200. M16C22⋊D37semidirect

Maximal subgroups

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

subgroupindex
#145580972
C2×#119245322
C2×#119244632
#145575382
C2×#119244522
#145572942
C2×AGL(1,ℤ/8)×C372
#200237