Finite Group Databasebeta

← #18004390

Subgroups — #18004390

Order 4736.

440 conjugacy classes of subgroups; only the 143 normal subgroups are shown.

normal subgroups only

This group has 440 conjugacy classes of subgroups — too many to draw. Only its 143 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#14558086C211yesmaximal, characteristic, complemented
2C2×D4×Dic37C211yesmaximal, characteristic
3#14557835C212yesmaximal, complemented
4#14557831C212yesmaximal, complemented
5#14557821C212yesmaximal, complemented
6#14557811C212yesmaximal, complemented
7#14557576C211yesmaximal, characteristic, complemented
8#14557575C211yesmaximal, characteristic, complemented
9#14557556C211yesmaximal, characteristic, complemented
10#14557538C211yesmaximal, characteristic
11C37×#418C211yesmaximal, characteristic
12C23×Dic37—21no—
13C2×#10596934C2211yescharacteristic
14#11924535C2211yescharacteristic, complemented
15#11924535C2211yescharacteristic, complemented
16#11924527C2211yescharacteristic, complemented
17#11924527C2211yescharacteristic, complemented
18#11924513—21no—
19#11924513—21no—
20C4×Dic74—21no—
21C4×Dic74—21no—
22D4×Dic37—21no—
23D4×Dic37—21no—
24D4×Dic37C2212yes—
25D4×Dic37C2212yes—
26#11924495C2211yescharacteristic
27C2×C4⋊Dic37C2211yescharacteristic
28C2×C22⋊Dic37—21no—
29C2×C4×Dic37C2211yescharacteristic
30#11924470C2212yescomplemented
31#11924469C2212yescomplemented
32#11924468C2212yescomplemented
33#11924467C2212yescomplemented
34#11924465C2212yescomplemented
35#11924465C2212yescomplemented
36#11924457C2212yes—
37#11924457C2212yes—
38C22×D4×C37C2211yescharacteristic
39C2×M16×C37C2211yescharacteristic
40#117×C37C2211yescharacteristic
41#102×C37C2212yes—
42#102×C37C2212yes—
43C24×C37—21no—
44C2×D4×C37—21no—
45C2×D4×C37—22no—
46C2×D4×C37Q814yescomplemented
47C2×D4×C37C2312yes—
48C22×C4×C37C2311yescharacteristic
49C22×Dic37—21no—
50C22×Dic37—22no—
51C22×Dic37—22no—
52C22×Dic37—44no—
53C22×Dic37D411yescharacteristic
54C22×Dic37D411yescharacteristic
55C2×Dic74—21no—
56C2×Dic74—21no—
57M16×C37—22no—
58C2×C8×C37C2312yes—
59C4⋊C4×C37C2311yescharacteristic
60C4⋊C4×C37C2311yescharacteristic
61C22⋊C4×C37—21no—
62C42×C37—21no—
63C22⋊Dic37—21no—
64C22⋊Dic37—22no—
65C22⋊Dic37—22no—
66C4⋊Dic37—21no—
67C4⋊Dic37—21no—
68C4⋊Dic37—21no—
69C4⋊Dic37C2312yes—
70C4⋊Dic37C2311yescharacteristic
71C4⋊Dic37C2311yescharacteristic
72#10596936—21no—
73#10596936—21no—
74#10596936—21no—
75#10596936—21no—
76C4×Dic37—21no—
77C4×Dic37D411yescharacteristic
78C4×Dic37D411yescharacteristic
79C4×Dic37C2311yescharacteristic
80C4×Dic37C2311yescharacteristic
81#10596934—22no—
82C2×#64436C2312yes—
83C23×C37—24no—
84C23×C37—22no—
85C23×C37—42no—
86C23×C37—42no—
87C23×C37C2×D411yescharacteristic
88D4×C37—24no—
89D4×C37—22no—
90D4×C37C2×Q814yes—
91C2×C4×C37—21no—
92C2×C4×C37—21no—
93C2×C4×C37C2×D411yescharacteristic
94C2×C4×C37C2×Q812yes—
95C2×C4×C37#4112yes—
96C2×C4×C37C2411yescharacteristic
97C2×Dic37—22no—
98C2×Dic37—21no—
99C2×Dic37—21no—
100C2×Dic37—22no—
101C2×Dic37—41no—
102C2×Dic37—44no—
103C2×Dic37—42no—
104C2×Dic37—44no—
105C2×Dic37C2×D411yescharacteristic
106C2×Dic37C2×D411yescharacteristic
107C2×Dic37C2×D411yescharacteristic
108C2×Dic37C2×D411yescharacteristic
109Dic74—42no—
110Dic74—42no—
111C8×C37—22no—
112#64436—22no—
113C22×C37—21no—
114C22×C37—21no—
115C22×C37—24no—
116C22×C37—48no—
117C22×C37—42no—
118C22×C37—42no—
119C22×C37C22⋊Q812yes—
120C22×C37C22×D411yescharacteristic
121C4×C37—41no—
122C4×C37—41no—
123C4×C37C22⋊Q812yes—
124C4×C37C22×Q811yescharacteristic
125C4×C37C2×#4111yescharacteristic
126Dic37—21no—
127Dic37—21no—
128Dic37—21no—
129Dic37—41no—
130Dic37—42no—
131Dic37—41no—
132Dic37AGL(1,ℤ/8)12yes—
133#3077—371nomaximal, Sylow
134C2×C37—22no—
135C2×C37—44no—
136C2×C37C2×C22⋊Q811yescharacteristic
137C2×C37C2×AGL(1,ℤ/8)11yescharacteristic
138C2×C37#57511yescharacteristic
139#533—371no—
140C2×C4×D4—371no—
141#478—372no—
142#478—372no—
143#476—372no—
144#474—372no—
145#428—371no—
146#428—371no—
147#423—371no—
148#418—371no—
149#418D3711yescharacteristic
150C37#307711yescharacteristic, complemented, Sylow
151C22×D4C2×D3711yescharacteristic
152C23×C4—741no—
153C2×M16—371no—
154C2×M16C2×D3711yescharacteristic
155C4⋊Q8—371no—
156C4⋊Q8—371no—
157#125—371no—
158#125—371no—
159C22⋊Q8—741no—
160C22⋊Q8—741no—
161C4×Q8—741no—
162C4×Q8—741no—
163C4×D4—372no—
164C4×D4—372no—
165C4×D4—741no—
166C4×D4—741no—
167#117—371no—
168#117C2×D3711yescharacteristic
169C2×C4⋊C4—371no—
170C2×C22⋊C4—741no—
171C2×C42—371no—
172#107—372no—
173#107—372no—
174#106—372no—
175#106—372no—
176C4⋊C8—372no—
177C4⋊C8—372no—
178#102—372no—
179#102—372no—
180#102C2×D3712yes—
181#102C2×D3712yes—
182C24—21no—
183C2×Q8—741no—
184C2×Q8—741no—
185C2×D4—21no—
186C2×D4—22no—
187C2×D4Dic7414yescomplemented
188C2×D4C22×D3712yes—
189C22×C4—371no—
190C22×C4—371no—
191C22×C4—742no—
192C22×C4—742no—
193C22×C4—741no—
194C22×C4—1484no—
195C22×C4C22×D3711yescharacteristic
196M16—22no—
197M16—742no—
198C2×C8—372no—
199C2×C8C22×D3712yes—
200C4⋊C4—371no—
201C4⋊C4—371no—
202C4⋊C4—372no—
203C4⋊C4—741no—
204C4⋊C4—741no—
205C4⋊C4—741no—
206C4⋊C4—741no—
207C4⋊C4—741no—
208C4⋊C4—741no—
209C4⋊C4—741no—
210C4⋊C4C22×D3711yescharacteristic
211C4⋊C4C22×D3711yescharacteristic
212C22⋊C4—21no—
213C22⋊C4—742no—
214C22⋊C4—742no—
215C22⋊C4—741no—
216C42—21no—
217C42—371no—
218C42—371no—
219C42—371no—
220C42—371no—
221C42—741no—
222C23—24no—
223C23—22no—
224C23—42no—
225C23—42no—
226C23D4×D3711yescharacteristic
227Q8—1482no—
228Q8—1482no—
229D4—24no—
230D4—22no—
231D4C2×Dic7414yes—
232C2×C4—21no—
233C2×C4—21no—
234C2×C4—371no—
235C2×C4—371no—
236C2×C4—371no—
237C2×C4—371no—
238C2×C4—742no—
239C2×C4—741no—
240C2×C4—741no—
241C2×C4—742no—
242C2×C4—1484no—
243C2×C4—1481no—
244C2×C4—1484no—
245C2×C4—1482no—
246C2×C4C2×Dic7412yes—
247C2×C4D4×D3711yescharacteristic
248C2×C4#1059696412yes—
249C2×C4C23×D3711yescharacteristic
250C8—22no—
251C8—742no—
252C22—21no—
253C22—21no—
254C22—24no—
255C22—48no—
256C22—42no—
257C22—42no—
258C22#1192451312yes—
259C22C2×D4×D3711yescharacteristic
260C4—41no—
261C4—41no—
262C4—372no—
263C4—741no—
264C4—741no—
265C4—741no—
266C4—1482no—
267C4—1481no—
268C4—1481no—
269C4#1192451312yes—
270C4C22×Dic7411yescharacteristic
271C4C2×#1059696411yescharacteristic
272C2—22no—
273C2—44no—
274C2C2×#1192451311yescharacteristic
275C2AGL(1,ℤ/8)×D3711yescharacteristic
276C2#1455832911yescharacteristic
subgroupG/H
centerC22C2×D4×D37
commutator, cosocleC2×C4×C37C24
FrattiniC2×C4C23×D37
FittingC37×#418C2
socleC22×C37C22×D4
prime radicalC22×D4×C37C22
hypercenter#418D37
hypocenterC37#3077
LCSG ▷ C2×C4×C37 ▷ C2×C37 ▷ C37
UCSC1 ◁ C22 ◁ C22×C4 ◁ #418

Complements

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

NH ≅ G/Ntype
1. #14558086C2semidirect
3. #14557835C2semidirect
4. #14557831C2semidirect
5. #14557821C2semidirect
6. #14557811C2semidirect
7. #14557576C2semidirect
8. #14557575C2semidirect
9. #14557556C2semidirect
14. #11924535C22semidirect
15. #11924535C22semidirect
16. #11924527C22semidirect
17. #11924527C22semidirect
30. #11924470C22semidirect
31. #11924469C22semidirect
32. #11924468C22semidirect
33. #11924467C22semidirect
34. #11924465C22semidirect
35. #11924465C22semidirect
46. C2×D4×C37Q8semidirect
150. C37#3077semidirect
187. C2×D4Dic74semidirect

Maximal subgroups

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

subgroupindex
#145580862
C2×D4×Dic372
#145578352
#145578312
#145578212
#145578112
#145575762
#145575752
#145575562
#145575382
C37×#4182
#307737