Finite Group Databasebeta

← #18003953

Subgroups — #18003953

Order 4736.

762 conjugacy classes of subgroups; only the 161 normal subgroups are shown.

normal subgroups only

This group has 762 conjugacy classes of subgroups — too many to draw. Only its 161 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#14558331C212yesmaximal, complemented
2#14558100C211yesmaximal, characteristic, complemented
3#14557711C214yesmaximal, complemented
4#14557701C214yesmaximal, complemented
5#14557550C212yesmaximal, complemented
6C2×#11924453C211yesmaximal, characteristic, complemented
7C2×H(5,𝔽2)×C37C211yesmaximal, characteristic, complemented
8C2×#10596962C2211yescharacteristic, complemented
9#11924567—22no—
10#11924566C2214yescomplemented
11#11924562—22no—
12#11924561C2214yescomplemented
13#11924556—24no—
14C2×#10596934C2212yescomplemented
15#11924532C2212yescomplemented
16#11924523C2212yescomplemented
17#11924507—22no—
18#11924505—22no—
19#11924495C2211yescharacteristic, complemented
20#11924463C2218yescomplemented
21#11924453C2214yescomplemented
22H(5,𝔽2)×C37—24no—
23H(5,𝔽2)×C37—22no—
24H(5,𝔽2)×C37C2214yes—
25C2×#41×C37—22no—
26C2×#41×C37C2212yes—
27C22×D4×C37—22no—
28C22×D4×C37—22no—
29C22×D4×C37C2211yescharacteristic
30C24×C37—22no—
31C24×C37D412yes—
32#41×C37—28no—
33#41×C37—28no—
34#41×C37—22no—
35#41×C37—22no—
36#41×C37D414yescomplemented
37#41×C37C2314yes—
38C2×Q8×C37D412yescomplemented
39C2×D4×C37—24no—
40C2×D4×C37—28no—
41C2×D4×C37—28no—
42C2×D4×C37—22no—
43C2×D4×C37—28no—
44C2×D4×C37—28no—
45C2×D4×C37—22no—
46C2×D4×C37—22no—
47C2×D4×C37D412yes—
48C2×D4×C37D412yescomplemented
49C2×D4×C37C2312yes—
50C22×C4×C37—22no—
51C22×C4×C37—22no—
52C22×C4×C37C2311yescharacteristic
53C2×C22⋊D37—21no—
54C2×C22⋊D37—24no—
55#10596962—22no—
56#10596962—22no—
57C2×D148C2311yescharacteristic, complemented
58C2×C4×D37—21no—
59C2×Dic74C2311yescharacteristic, complemented
60C22⋊Dic37—21no—
61C22⋊Dic37—24no—
62#10596942—24no—
63#10596941—24no—
64#10596940—24no—
65#10596939—24no—
66C4⋊Dic37—21no—
67C4×Dic37C2312yescomplemented
68#10596934—22no—
69#10596934C2314yescomplemented
70C2×#64436—22no—
71C23×C37—28no—
72C23×C37—22no—
73C23×C37—22no—
74C23×C37—22no—
75C23×C37—82no—
76C23×C37—84no—
77C23×C37C2×D414yes—
78C23×C37C2×D411yescharacteristic
79C22×D37—21no—
80Q8×C37—22no—
81Q8×C37C2×D414yes—
82D4×C37—216no—
83D4×C37—24no—
84D4×C37—24no—
85D4×C37—24no—
86D4×C37—22no—
87D4×C37—24no—
88D4×C37C2×D414yes—
89C2×C4×C37—28no—
90C2×C4×C37—22no—
91C2×C4×C37—24no—
92C2×C4×C37—24no—
93C2×C4×C37C2×D411yescharacteristic
94C2×C4×C37C2×D412yes—
95C2×C4×C37C2×D412yes—
96C2×C4×C37C2411yescharacteristic
97C22⋊D37—41no—
98C22⋊D37—41no—
99C22⋊D37—84no—
100C2×Dic37—22no—
101C2×Dic37—21no—
102D148—22no—
103C4×D37—42no—
104Dic74—22no—
105#64436—24no—
106C22×C37—24no—
107C22×C37—24no—
108C22×C37—41no—
109C22×C37—44no—
110C22×C37—44no—
111C22×C37—44no—
112C22×C37—82no—
113C22×C37—88no—
114C22×C37#12011yescharacteristic
115C22×C37#12011yescharacteristic
116C22×C37C22×D411yescharacteristic
117C2×D37—42no—
118C2×D37—41no—
119C4×C37—24no—
120C4×C37#12012yes—
121C4×C37C22×D412yes—
122Dic37—42no—
123Dic37—41no—
124#3019—371nomaximal, Sylow
125C2×C37—21no—
126C2×C37—21no—
127C2×C37—21no—
128C2×C37—44no—
129C2×C37—44no—
130C2×C37C2×#12011yescharacteristic
131D37—81no—
132C2×H(5,𝔽2)D3711yescharacteristic, complemented
133#575—372no—
134#535—371no—
135#455—374no—
136#453—374no—
137#421—372no—
138C2×#100—371no—
139C37#301911yescharacteristic, complemented, Sylow
140H(5,𝔽2)—24no—
141H(5,𝔽2)—22no—
142H(5,𝔽2)C2×D3714yes—
143C2×#41—22no—
144C2×#41—371no—
145C2×#41C2×D3712yes—
146C22×D4—22no—
147C22×D4—22no—
148C22×D4C2×D3711yescharacteristic
149#137—374no—
150#137—742no—
151AGL(1,ℤ/8)—374no—
152AGL(1,ℤ/8)—742no—
153#135—744no—
154C2×M16—372no—
155#127—372no—
156#124—372no—
157#121—742no—
158#120—742no—
159#117—371no—
160#104—378no—
161#100—374no—
162C24—22no—
163C24C22⋊D3712yes—
164#41—28no—
165#41—28no—
166#41—22no—
167#41—22no—
168#41—742no—
169#41—742no—
170#41C22⋊D3714yescomplemented
171#41C22×D3714yes—
172C2×Q8—371no—
173C2×Q8C22⋊D3712yescomplemented
174C2×D4—24no—
175C2×D4—28no—
176C2×D4—28no—
177C2×D4—22no—
178C2×D4—28no—
179C2×D4—28no—
180C2×D4—22no—
181C2×D4—22no—
182C2×D4—371no—
183C2×D4—744no—
184C2×D4—741no—
185C2×D4C22⋊D3712yes—
186C2×D4C22⋊D3712yescomplemented
187C2×D4C22×D3712yes—
188C22×C4—22no—
189C22×C4—22no—
190C22×C4—741no—
191C22×C4C22×D3711yescharacteristic
192Q16—744no—
193SD16—744no—
194SD16—744no—
195D8—744no—
196M16—374no—
197M16—742no—
198C2×C8—742no—
199C4⋊C4—741no—
200C22⋊C4—741no—
201C22⋊C4—744no—
202C42—372no—
203C23—28no—
204C23—22no—
205C23—22no—
206C23—22no—
207C23—82no—
208C23—84no—
209C23—741no—
210C23C2×C22⋊D3714yes—
211C23C2×C22⋊D3711yescharacteristic
212Q8—22no—
213Q8—742no—
214Q8C2×C22⋊D3714yes—
215D4—216no—
216D4—24no—
217D4—24no—
218D4—24no—
219D4—22no—
220D4—24no—
221D4—2964no—
222D4—742no—
223D4—1481no—
224D4—1481no—
225D4C2×C22⋊D3714yes—
226C2×C4—28no—
227C2×C4—22no—
228C2×C4—24no—
229C2×C4—24no—
230C2×C4—742no—
231C2×C4—741no—
232C2×C4—1482no—
233C2×C4C2×C22⋊D3711yescharacteristic
234C2×C4C2×C22⋊D3712yes—
235C2×C4C2×C22⋊D3712yes—
236C2×C4C23×D3711yescharacteristic
237C8—744no—
238C22—24no—
239C22—24no—
240C22—41no—
241C22—44no—
242C22—44no—
243C22—44no—
244C22—82no—
245C22—88no—
246C22—1482no—
247C22—1481no—
248C22#1192450511yescharacteristic
249C22#1192450511yescharacteristic
250C22C22×C22⋊D3711yescharacteristic
251C4—24no—
252C4—1482no—
253C4—1481no—
254C4#1192450512yes—
255C4C22×C22⋊D3712yes—
256C2—21no—
257C2—21no—
258C2—21no—
259C2—44no—
260C2—44no—
261C2—2961no—
262C2C2×#1192450511yescharacteristic
subgroupG/H
centerC2C2×#11924505
commutator, cosocleC2×C4×C37C24
FrattiniC2×C4C23×D37
FittingC2×H(5,𝔽2)×C37C2
socleC2×C37C2×#120
hypercenterC2×H(5,𝔽2)D37
hypocenterC37#3019
LCSG ▷ C2×C4×C37 ▷ C2×C37 ▷ C37
UCSC1 ◁ C2 ◁ C22×C4 ◁ C2×H(5,𝔽2)

Complements

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

NH ≅ G/Ntype
1. #14558331C2semidirect
2. #14558100C2semidirect
3. #14557711C2semidirect
4. #14557701C2semidirect
5. #14557550C2semidirect
6. C2×#11924453C2semidirect
7. C2×H(5,𝔽2)×C37C2semidirect
8. C2×#10596962C22semidirect
10. #11924566C22semidirect
12. #11924561C22semidirect
14. C2×#10596934C22semidirect
15. #11924532C22semidirect
16. #11924523C22semidirect
19. #11924495C22semidirect
20. #11924463C22semidirect
21. #11924453C22semidirect
36. #41×C37D4semidirect
38. C2×Q8×C37D4semidirect
48. C2×D4×C37D4semidirect
57. C2×D148C23semidirect
59. C2×Dic74C23semidirect
67. C4×Dic37C23semidirect
69. #10596934C23semidirect
132. C2×H(5,𝔽2)D37semidirect
139. C37#3019semidirect
170. #41C22⋊D37semidirect
173. C2×Q8C22⋊D37semidirect
186. C2×D4C22⋊D37semidirect

Maximal subgroups

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

subgroupindex
#145583312
#145581002
#145577112
#145577012
#145575502
C2×#119244532
C2×H(5,𝔽2)×C372
#301937