Finite Group Databasebeta

← #18005058

Subgroups — #18005058

Order 4736.

572 conjugacy classes of subgroups; only the 149 normal subgroups are shown.

normal subgroups only

This group has 572 conjugacy classes of subgroups — too many to draw. Only its 149 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
1C2×#11924569C211yesmaximal, characteristic, complemented
2C2×#11924566C211yesmaximal, characteristic, complemented
3C2×#11924562C211yesmaximal, characteristic, complemented
4#14558079C211yesmaximal, characteristic, complemented
5C2×#11924523C211yesmaximal, characteristic, complemented
6#14557924C212yesmaximal, complemented
7#14557921C212yesmaximal, complemented
8#14557916C212yesmaximal, complemented
9#14557914C212yesmaximal, complemented
10M16×Dic37C211yesmaximal, characteristic, complemented
11C2×AGL(1,ℤ/8)×C37C211yesmaximal, characteristic, complemented
12C2×#10596962C2211yescharacteristic
13C22×Dic74C2211yescharacteristic
14#11924569—24no—
15#11924566—24no—
16#11924562—24no—
17C2×Dic148C2212yescomplemented
18C2×#10596942C2212yescomplemented
19C2×#10596930C2212yescomplemented
20C2×#10596940C2212yes—
21C2×#10596940C2212yescomplemented
22C2×#10596939C2212yes—
23C2×#10596934C2211yescharacteristic
24#11924534C2211yescharacteristic, complemented
25#11924532C2211yescharacteristic, complemented
26#11924525C2211yescharacteristic, complemented
27#11924523—22no—
28#11924523C2212yes—
29#11924523C2212yes—
30#11924523C2211yescharacteristic, complemented
31#11924508—22no—
32C4×C22⋊D37—22no—
33C2×C22⋊Dic37—22no—
34C2×C4×Dic37C2211yescharacteristic
35#11924446C2212yescomplemented
36C8×Dic37C2212yescomplemented
37C2×#41×C37C2211yescharacteristic
38C22×D4×C37C2211yescharacteristic
39AGL(1,ℤ/8)×C37—24no—
40C2×SD16×C37C2212yes—
41C2×D8×C37C2212yes—
42C2×M16×C37C2211yescharacteristic
43C24×C37—21no—
44#41×C37—22no—
45#41×C37—22no—
46C2×Q8×C37C2311yescharacteristic
47C2×D4×C37—21no—
48C2×D4×C37—22no—
49C2×D4×C37—22no—
50C2×D4×C37—21no—
51C2×D4×C37C2312yes—
52C2×D4×C37C2311yescharacteristic
53C22×C4×C37—21no—
54C22×C4×C37C2311yescharacteristic
55C2×C22⋊D37—21no—
56C2×C22⋊D37—22no—
57C22×Dic37—21no—
58C22×Dic37D412yes—
59#10596962—22no—
60#10596962—22no—
61C2×D148C2311yescharacteristic
62C2×C4×D37—21no—
63C2×Dic74—22no—
64C2×Dic74—22no—
65C2×Dic74—21no—
66C2×Dic74C2312yes—
67C2×Dic74C2311yescharacteristic
68SD16×C37—24no—
69D8×C37—24no—
70M16×C37D414yescomplemented
71C2×C8×C37C2312yes—
72C22⋊Dic37—24no—
73C22⋊Dic37—24no—
74C22⋊Dic37—22no—
75#10596942—24no—
76#10596940—24no—
77#10596940—24no—
78#10596939—24no—
79#10596938—22no—
80#10596936—22no—
81C4×Dic37D412yes—
82C4×Dic37C2311yescharacteristic
83C4×Dic37C2311yescharacteristic
84#10596934D414yescomplemented
85C2×#64436C2312yes—
86Dic148—24no—
87#10596930—24no—
88C23×C37—22no—
89C23×C37—21no—
90C23×C37—44no—
91C23×C37—42no—
92C23×C37C2×D411yescharacteristic
93C22×D37—21no—
94Q8×C37—21no—
95Q8×C37—21no—
96D4×C37—22no—
97D4×C37—21no—
98D4×C37—21no—
99D4×C37—22no—
100D4×C37—42no—
101D4×C37—42no—
102C2×C4×C37—21no—
103C2×C4×C37—42no—
104C2×C4×C37C2×D411yescharacteristic
105C2×C4×C37C2×D412yes—
106C2×C4×C37C2×D412yes—
107C2×C4×C37C2411yescharacteristic
108C22⋊D37—44no—
109C22⋊D37—42no—
110C2×Dic37—21no—
111C2×Dic37—22no—
112C2×Dic37—42no—
113C2×Dic37—42no—
114C2×Dic37C2×D412yes—
115C2×Dic37C2×D412yes—
116D148—21no—
117D148—21no—
118C4×D37—42no—
119Dic74—22no—
120Dic74—21no—
121Dic74—21no—
122Dic74—22no—
123Dic74—42no—
124C8×C37C2×D414yes—
125#64436C2×D414yes—
126C22×C37—22no—
127C22×C37—41no—
128C22×C37—42no—
129C22×C37—42no—
130C22×C37—44no—
131C22×C37—84no—
132C22×C37#12712yes—
133C22×C37C22×D411yescharacteristic
134C2×D37—41no—
135C2×D37—42no—
136C4×C37—41no—
137C4×C37#12712yes—
138C4×C37C22×D411yescharacteristic
139C4×C37C22×D411yescharacteristic
140Dic37—22no—
141Dic37—22no—
142Dic37—42no—
143Dic37—41no—
144#3153—371nomaximal, Sylow
145C2×C37—22no—
146C2×C37—81no—
147C2×C37—82no—
148C2×C37C2×#12711yescharacteristic
149C2×C37#57512yes—
150D37—81no—
151C2×#137—371no—
152C2×#137—371no—
153C2×AGL(1,ℤ/8)—371no—
154C2×AGL(1,ℤ/8)D3711yescharacteristic, complemented
155#532—371no—
156C2×#124—371no—
157#497—372no—
158#496—372no—
159#495—372no—
160#495—372no—
161C4×M16—371no—
162C37#315311yescharacteristic, complemented, Sylow
163C2×#41—371no—
164C2×#41C2×D3711yescharacteristic
165C22×Q8—371no—
166C22×D4C2×D3711yescharacteristic
167#137—744no—
168#137—744no—
169AGL(1,ℤ/8)—24no—
170AGL(1,ℤ/8)—744no—
171C2×Q16—372no—
172C2×Q16—372no—
173C2×SD16—372no—
174C2×SD16—372no—
175C2×SD16—372no—
176C2×SD16C2×D3712yes—
177C2×D8—372no—
178C2×D8C2×D3712yes—
179C2×M16—371no—
180C2×M16C2×D3711yescharacteristic
181C4⋊Q8—371no—
182#127—371no—
183#124—371no—
184#124—372no—
185#124—372no—
186#124—371no—
187#124—742no—
188#121—742no—
189C4×D4—742no—
190C2×C22⋊C4—742no—
191C2×C42—371no—
192#97—372no—
193C4×C8—372no—
194C24—21no—
195#41—22no—
196#41—22no—
197#41—742no—
198#41—742no—
199C2×Q8—372no—
200C2×Q8—371no—
201C2×Q8—741no—
202C2×Q8—742no—
203C2×Q8—742no—
204C2×Q8C22×D3711yescharacteristic
205C2×D4—21no—
206C2×D4—22no—
207C2×D4—21no—
208C2×D4—22no—
209C2×D4—371no—
210C2×D4—742no—
211C2×D4—741no—
212C2×D4C22×D3712yes—
213C2×D4C22×D3711yescharacteristic
214C22×C4—21no—
215C22×C4—372no—
216C22×C4—741no—
217C22×C4—741no—
218C22×C4C22×D3711yescharacteristic
219Q16—744no—
220Q16—744no—
221SD16—24no—
222SD16—744no—
223SD16—744no—
224SD16—744no—
225D8—24no—
226D8—744no—
227M16—374no—
228M16C22⋊D3714yescomplemented
229C2×C8—372no—
230C2×C8C22×D3712yes—
231C4⋊C4—742no—
232C22⋊C4—742no—
233C22⋊C4—744no—
234C22⋊C4—744no—
235C22⋊C4—742no—
236C42—372no—
237C42—371no—
238C42—371no—
239C23—22no—
240C23—21no—
241C23—44no—
242C23—42no—
243C23—741no—
244C23D4×D3711yescharacteristic
245Q8—21no—
246Q8—21no—
247Q8—741no—
248Q8—742no—
249Q8—742no—
250Q8—741no—
251Q8—1482no—
252D4—22no—
253D4—21no—
254D4—21no—
255D4—22no—
256D4—42no—
257D4—42no—
258D4—741no—
259D4—741no—
260D4—1484no—
261D4—1482no—
262C2×C4—21no—
263C2×C4—42no—
264C2×C4—372no—
265C2×C4—372no—
266C2×C4—742no—
267C2×C4—741no—
268C2×C4—1482no—
269C2×C4—1482no—
270C2×C4—1482no—
271C2×C4D4×D3711yescharacteristic
272C2×C4D4×D3712yes—
273C2×C4C2×C22⋊D3712yes—
274C2×C4C23×D3711yescharacteristic
275C8—374no—
276C8C2×C22⋊D3714yes—
277C22—22no—
278C22—41no—
279C22—42no—
280C22—42no—
281C22—44no—
282C22—84no—
283C22—1481no—
284C22—1482no—
285C22#1192453212yes—
286C22C2×D4×D3711yescharacteristic
287C4—41no—
288C4—742no—
289C4—742no—
290C4—1481no—
291C4—1482no—
292C4#1192453212yes—
293C4C2×D4×D3711yescharacteristic
294C4C22×C22⋊D3711yescharacteristic
295C2—22no—
296C2—82no—
297C2—81no—
298C2—2961no—
299C2C2×#1192453211yescharacteristic
300C2#1455832912yes—
subgroupG/H
centerC22C2×D4×D37
commutator, cosocleC2×C4×C37C24
FrattiniC2×C4C23×D37
FittingC2×AGL(1,ℤ/8)×C37C2
socleC22×C37C22×D4
hypercenterC2×AGL(1,ℤ/8)D37
hypocenterC37#3153
LCSG ▷ C2×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. C2×#11924569C2semidirect
2. C2×#11924566C2semidirect
3. C2×#11924562C2semidirect
4. #14558079C2semidirect
5. C2×#11924523C2semidirect
6. #14557924C2semidirect
7. #14557921C2semidirect
8. #14557916C2semidirect
9. #14557914C2semidirect
10. M16×Dic37C2semidirect
11. C2×AGL(1,ℤ/8)×C37C2semidirect
17. C2×Dic148C22semidirect
18. C2×#10596942C22semidirect
19. C2×#10596930C22semidirect
21. C2×#10596940C22semidirect
24. #11924534C22semidirect
25. #11924532C22semidirect
26. #11924525C22semidirect
30. #11924523C22semidirect
35. #11924446C22semidirect
36. C8×Dic37C22semidirect
70. M16×C37D4semidirect
84. #10596934D4semidirect
154. C2×AGL(1,ℤ/8)D37semidirect
162. C37#3153semidirect
228. M16C22⋊D37semidirect

Maximal subgroups

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

subgroupindex
C2×#119245692
C2×#119245662
C2×#119245622
#145580792
C2×#119245232
#145579242
#145579212
#145579162
#145579142
M16×Dic372
C2×AGL(1,ℤ/8)×C372
#315337