Finite Group Databasebeta

← #18006377

Subgroups — #18006377

Order 4736.

454 conjugacy classes of subgroups; only the 117 normal subgroups are shown.

normal subgroups only

This group has 454 conjugacy classes of subgroups — too many to draw. Only its 117 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#14558350C211yesmaximal, characteristic, complemented
2#14558343C212yesmaximal, complemented
3#14558334C211yesmaximal, characteristic, complemented
4#14557916C211yesmaximal, characteristic, complemented
5#14557809C211yesmaximal, characteristic, complemented
6#14557795C212yesmaximal, complemented
7#14557713C212yesmaximal, complemented
8#14557704C212yesmaximal, complemented
9#14557649C211yesmaximal, characteristic, complemented
10#14557648C211yesmaximal, characteristic, complemented
11C37×#472C211yesmaximal, characteristic, complemented
12#11924583C2212yescomplemented
13#11924581C2212yes—
14#11924569—22no—
15#11924569C2212yes—
16#11924569C2212yescomplemented
17#11924565—22no—
18#11924562C2212yes—
19#11924560—22no—
20#11924560—22no—
21#11924558—21no—
22#11924558C2211yescharacteristic
23#11924556C2211yescharacteristic
24#11924555—22no—
25#11924555—21no—
26C2×Dic148—21no—
27C2×Dic148C2211yescharacteristic, complemented
28C2×#10596940C2212yescomplemented
29SD16×D37—22no—
30D8×D37—21no—
31#11924544C2212yescomplemented
32#11924542C2211yescharacteristic
33#11924523C2212yescomplemented
34#11924471C2211yescharacteristic, complemented
35#11924463C2211yescharacteristic, complemented
36#11924462C2211yescharacteristic
37#11924462C2212yescomplemented
38#11924454C2212yescomplemented
39C8×Dic37C2211yescharacteristic, complemented
40AGL(1,ℤ/8)×C37C2212yescomplemented
41C2×D8×C37C2211yescharacteristic
42#131×C37C2211yescharacteristic, complemented
43#108×C37C2211yescharacteristic, complemented
44#100×C37C2212yescomplemented
45#41×C37C2311yescharacteristic
46C2×D4×C37C2312yes—
47C2×C22⋊D37—42no—
48Q8×D37—22no—
49#10596964—22no—
50#10596964—22no—
51#10596964—22no—
52#10596964—42no—
53D4×D37—22no—
54D4×D37—22no—
55#10596962—22no—
56#10596962C2311yescharacteristic
57#10596962C2312yes—
58C2×Dic74—22no—
59C2×Dic74C2312yes—
60SD16×C37—22no—
61D8×C37—22no—
62D8×C37—21no—
63D8×C37—21no—
64M16×C37—21no—
65M16×C37C2312yes—
66M16×C37C2311yescharacteristic
67C2×C8×C37—21no—
68C2×C8×C37C2311yescharacteristic
69C22⋊Dic37—42no—
70#10596942D411yescharacteristic
71#10596941D411yescharacteristic
72#10596940—22no—
73#10596940—22no—
74#10596940—22no—
75#10596940D411yescharacteristic
76#10596939—22no—
77#10596939D411yescharacteristic
78C4×Dic37C2311yescharacteristic
79#10596934C2311yescharacteristic
80C2×#64436C2311yescharacteristic
81Dic148—22no—
82Dic148—22no—
83Dic148—21no—
84Dic148—21no—
85Dic148D411yescharacteristic
86D296D411yescharacteristic
87#10596930—22no—
88#10596930—22no—
89#10596930D412yes—
90#10596929—21no—
91#10596929—22no—
92C8×D37—21no—
93C8×D37—22no—
94C23×C37—22no—
95C22×D37—42no—
96Q8×C37C2×D411yescharacteristic
97D4×C37—21no—
98D4×C37—22no—
99D4×C37—22no—
100D4×C37C2×D411yescharacteristic
101C2×C4×C37—21no—
102C2×C4×C37C2411yescharacteristic
103C22⋊D37—22no—
104C22⋊D37—21no—
105C22⋊D37—42no—
106C22⋊D37—42no—
107C22⋊D37—42no—
108C2×Dic37—21no—
109C2×Dic37—22no—
110C2×Dic37—42no—
111C2×Dic37—42no—
112D148C2×D411yescharacteristic
113D148C2×D412yes—
114C4×D37—21no—
115C4×D37—22no—
116C4×D37—42no—
117Dic74—22no—
118Dic74—22no—
119Dic74—42no—
120Dic74C2×D412yes—
121Dic74C2×D411yescharacteristic
122C8×C37—22no—
123C8×C37—21no—
124C8×C37C2×D411yescharacteristic
125C8×C37C2×D411yescharacteristic
126#64436—21no—
127#64436C2×D411yescharacteristic
128#64436C2×D411yescharacteristic
129C22×C37—21no—
130C22×C37—42no—
131C22×C37—82no—
132C22×C37H(5,𝔽2)11yescharacteristic
133C2×D37—22no—
134C2×D37—21no—
135C2×D37—162no—
136C4×C37—21no—
137C4×C37C22×D411yescharacteristic
138C4×C37C22×D411yescharacteristic
139Dic37—22no—
140Dic37—21no—
141Dic37—41no—
142Dic37—42no—
143#3290—371nomaximal, Sylow
144C2×C37—21no—
145C2×C37—41no—
146C2×C37—82no—
147C2×C37D4211yescharacteristic
148D37—42no—
149D37—41no—
150#578—371no—
151#577—372no—
152#576—371no—
153#495—371no—
154#473—371no—
155#472D3711yescharacteristic, complemented
156#471—372no—
157#455—372no—
158#454—372no—
159#443—371no—
160#443—371no—
161C37#329011yescharacteristic, complemented, Sylow
1622−1+4—372no—
163H(5,𝔽2)—372no—
164#137—372no—
165#137—372no—
166#137—742no—
167#137—742no—
168AGL(1,ℤ/8)—372no—
169AGL(1,ℤ/8)—742no—
170AGL(1,ℤ/8)—742no—
171AGL(1,ℤ/8)C2×D3712yescomplemented
172#135—371no—
173#135—371no—
174#135—741no—
175#135—742no—
176#135—741no—
177C2×Q16—371no—
178C2×Q16—741no—
179C2×SD16—372no—
180C2×SD16—742no—
181C2×D8—741no—
182C2×D8C2×D3711yescharacteristic
183#131—372no—
184#131—371no—
185#131C2×D3711yescharacteristic, complemented
186#124—372no—
187#108—371no—
188#108C2×D3711yescharacteristic, complemented
189#104—371no—
190#104—372no—
191#104—371no—
192#101—372no—
193#100C2×D3712yescomplemented
194C4×C8—371no—
195#41—372no—
196#41—371no—
197#41—742no—
198#41—742no—
199#41—742no—
200#41—742no—
201#41—1482no—
202#41C22×D3711yescharacteristic
203C2×Q8—372no—
204C2×Q8—742no—
205C2×Q8—742no—
206C2×D4—742no—
207C2×D4—742no—
208C2×D4—1482no—
209C2×D4C22×D3712yes—
210Q16—371no—
211Q16—371no—
212Q16—742no—
213Q16—742no—
214Q16—741no—
215Q16—741no—
216SD16—22no—
217SD16—371no—
218SD16—371no—
219SD16—372no—
220SD16—742no—
221SD16—742no—
222SD16—742no—
223SD16—742no—
224SD16—742no—
225D8—22no—
226D8—21no—
227D8—21no—
228D8—371no—
229D8—371no—
230D8—742no—
231M16—21no—
232M16—371no—
233M16—742no—
234M16—741no—
235M16C22×D3711yescharacteristic
236M16C22×D3712yes—
237C2×C8—21no—
238C2×C8—371no—
239C2×C8—742no—
240C2×C8—741no—
241C2×C8C22×D3711yescharacteristic
242C22⋊C4—1482no—
243C42—371no—
244C23—22no—
245C23—1482no—
246Q8—372no—
247Q8—371no—
248Q8—742no—
249Q8—742no—
250Q8—1482no—
251Q8D4×D3711yescharacteristic
252D4—21no—
253D4—22no—
254D4—22no—
255D4—372no—
256D4—371no—
257D4—741no—
258D4—742no—
259D4—1482no—
260D4—1482no—
261D4—1482no—
262D4D4×D3711yescharacteristic
263C2×C4—21no—
264C2×C4—741no—
265C2×C4—742no—
266C2×C4—741no—
267C2×C4—742no—
268C2×C4—1482no—
269C2×C4—1482no—
270C2×C4—1482no—
271C2×C4C23×D3711yescharacteristic
272C8—21no—
273C8—22no—
274C8—371no—
275C8—371no—
276C8—741no—
277C8D4×D3711yescharacteristic
278C8D4×D3711yescharacteristic
279C22—21no—
280C22—42no—
281C22—82no—
282C22—742no—
283C22—741no—
284C22—5922no—
285C22#1192458211yescharacteristic
286C4—21no—
287C4—741no—
288C4—742no—
289C4—1481no—
290C4—1482no—
291C4C2×D4×D3711yescharacteristic
292C4C2×D4×D3711yescharacteristic
293C2—21no—
294C2—41no—
295C2—82no—
296C2—1481no—
297C2—1482no—
298C2#1455816711yescharacteristic
subgroupG/H
centerC2#14558167
commutator, cosocleC2×C4×C37C24
FrattiniC2×C4C23×D37
FittingC37×#472C2
socleC2×C37D42
hypercenter#472D37
hypocenterC37#3290
LCSG ▷ C2×C4×C37 ▷ C2×C37 ▷ C37
UCSC1 ◁ C2 ◁ C2×C4 ◁ #472

Complements

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

NH ≅ G/Ntype
1. #14558350C2semidirect
2. #14558343C2semidirect
3. #14558334C2semidirect
4. #14557916C2semidirect
5. #14557809C2semidirect
6. #14557795C2semidirect
7. #14557713C2semidirect
8. #14557704C2semidirect
9. #14557649C2semidirect
10. #14557648C2semidirect
11. C37×#472C2semidirect
12. #11924583C22semidirect
16. #11924569C22semidirect
27. C2×Dic148C22semidirect
28. C2×#10596940C22semidirect
31. #11924544C22semidirect
33. #11924523C22semidirect
34. #11924471C22semidirect
35. #11924463C22semidirect
37. #11924462C22semidirect
38. #11924454C22semidirect
39. C8×Dic37C22semidirect
40. AGL(1,ℤ/8)×C37C22semidirect
42. #131×C37C22semidirect
43. #108×C37C22semidirect
44. #100×C37C22semidirect
155. #472D37semidirect
161. C37#3290semidirect
171. AGL(1,ℤ/8)C2×D37semidirect
185. #131C2×D37semidirect
188. #108C2×D37semidirect
193. #100C2×D37semidirect

Maximal subgroups

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

subgroupindex
#145583502
#145583432
#145583342
#145579162
#145578092
#145577952
#145577132
#145577042
#145576492
#145576482
C37×#4722
#329037