Finite Group Databasebeta

← #18006394

Subgroups — #18006394

Order 4736.

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

normal subgroups only

This group has 476 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#14558338C211yesmaximal, characteristic, complemented
2#14558337C212yesmaximal, complemented
3#14558334C211yesmaximal, characteristic, complemented
4#14557908C211yesmaximal, characteristic, complemented
5#14557804C211yesmaximal, characteristic, complemented
6#14557802C212yesmaximal, complemented
7#14557703C212yesmaximal, complemented
8#14557699C212yesmaximal, complemented
9#14557649C211yesmaximal, characteristic, complemented
10#14557648C211yesmaximal, characteristic, complemented
11C37×#472C211yesmaximal, characteristic, complemented
12#11924582C2212yescomplemented
13#11924581C2212yescomplemented
14#11924564—22no—
15#11924564C2212yescomplemented
16#11924564C2212yescomplemented
17#11924563—22no—
18#11924562C2212yescomplemented
19#11924560—22no—
20#11924560—22no—
21#11924558—21no—
22#11924558C2211yescharacteristic
23#11924557—22no—
24#11924556C2211yescharacteristic
25#11924555—21no—
26C2×D296—21no—
27C2×D296C2211yescharacteristic, complemented
28C2×#10596939C2212yescomplemented
29D8×D37—22no—
30D8×D37—21no—
31#11924544C2212yescomplemented
32#11924542C2211yescharacteristic, complemented
33#11924532C2212yescomplemented
34#11924471C2211yescharacteristic, complemented
35#11924463C2211yescharacteristic, complemented
36#11924462C2212yescomplemented
37#11924462C2211yescharacteristic, complemented
38#11924452C2212yescomplemented
39C8×Dic37C2211yescharacteristic, complemented
40AGL(1,ℤ/8)×C37C2212yescomplemented
41C2×D8×C37C2211yescharacteristic, complemented
42#131×C37C2211yescharacteristic, complemented
43#108×C37C2211yescharacteristic, complemented
44#100×C37C2212yescomplemented
45#41×C37C2311yescharacteristic
46C2×D4×C37C2312yes—
47C2×C22⋊D37—42no—
48C2×C22⋊D37—42no—
49#10596966—22no—
50#10596964—22no—
51#10596964—22no—
52D4×D37—22no—
53D4×D37—22no—
54D4×D37—22no—
55D4×D37—42no—
56#10596962—22no—
57#10596962C2311yescharacteristic
58#10596962C2312yes—
59C2×D148—22no—
60C2×D148C2312yes—
61SD16×C37—22no—
62D8×C37—22no—
63D8×C37—21no—
64D8×C37—21no—
65M16×C37—21no—
66M16×C37C2312yes—
67M16×C37C2311yescharacteristic
68C2×C8×C37—21no—
69C2×C8×C37C2311yescharacteristic
70#10596942D411yescharacteristic, complemented
71#10596941D411yescharacteristic, complemented
72#10596940—22no—
73#10596940D411yescharacteristic, complemented
74#10596939—22no—
75#10596939—22no—
76#10596939—22no—
77#10596939D411yescharacteristic, complemented
78C4×Dic37C2311yescharacteristic
79#10596934C2311yescharacteristic
80C2×#64436C2311yescharacteristic
81Dic148D411yescharacteristic, complemented
82D296—22no—
83D296—22no—
84D296—21no—
85D296—21no—
86D296D411yescharacteristic, complemented
87#10596930—22no—
88#10596930—22no—
89#10596930D412yescomplemented
90#10596929—21no—
91#10596929—22no—
92C8×D37—21no—
93C8×D37—22no—
94C23×C37—22no—
95C22×D37—22no—
96C22×D37—42no—
97C22×D37—42no—
98Q8×C37C2×D411yescharacteristic
99D4×C37—21no—
100D4×C37—22no—
101D4×C37—22no—
102D4×C37C2×D411yescharacteristic
103C2×C4×C37—21no—
104C2×C4×C37C2411yescharacteristic
105C22⋊D37—22no—
106C22⋊D37—21no—
107C22⋊D37—42no—
108C22⋊D37—42no—
109C22⋊D37—42no—
110C22⋊D37—84no—
111C2×Dic37—21no—
112C2×Dic37—42no—
113D148—22no—
114D148—22no—
115D148—42no—
116D148C2×D412yes—
117D148C2×D411yescharacteristic
118C4×D37—21no—
119C4×D37—22no—
120C4×D37—42no—
121Dic74C2×D411yescharacteristic
122Dic74C2×D412yes—
123C8×C37—22no—
124C8×C37—21no—
125C8×C37C2×D411yescharacteristic
126C8×C37C2×D411yescharacteristic
127#64436—21no—
128#64436C2×D411yescharacteristic
129#64436C2×D411yescharacteristic
130C22×C37—21no—
131C22×C37—42no—
132C22×C37—82no—
133C22×C37H(5,𝔽2)11yescharacteristic
134C2×D37—22no—
135C2×D37—21no—
136C2×D37—42no—
137C2×D37—82no—
138C2×D37—162no—
139C2×D37—162no—
140C4×C37—21no—
141C4×C37C22×D411yescharacteristic
142C4×C37C22×D411yescharacteristic
143Dic37—22no—
144Dic37—22no—
145Dic37—21no—
146#3292—371nomaximal, Sylow
147C2×C37—21no—
148C2×C37—41no—
149C2×C37—82no—
150C2×C37D4211yescharacteristic
151D37—42no—
152D37—41no—
153D37—82no—
154#576—372no—
155#576—371no—
156#576—371no—
157#493—371no—
158#472—371no—
159#472—372no—
160#472D3711yescharacteristic, complemented
161#453—372no—
162#453—372no—
163#443—371no—
164#443—371no—
165C37#329211yescharacteristic, complemented, Sylow
166H(5,𝔽2)—372no—
167H(5,𝔽2)—372no—
168AGL(1,ℤ/8)—372no—
169AGL(1,ℤ/8)—372no—
170AGL(1,ℤ/8)—372no—
171AGL(1,ℤ/8)—742no—
172AGL(1,ℤ/8)—742no—
173AGL(1,ℤ/8)—742no—
174AGL(1,ℤ/8)—742no—
175AGL(1,ℤ/8)C2×D3712yescomplemented
176#135—371no—
177#135—371no—
178#135—741no—
179#135—741no—
180#135—742no—
181C2×D8—372no—
182C2×D8—371no—
183C2×D8—741no—
184C2×D8—742no—
185C2×D8—741no—
186C2×D8C2×D3711yescharacteristic, complemented
187#131—372no—
188#131—371no—
189#131C2×D3711yescharacteristic, complemented
190#127—372no—
191#108—371no—
192#108C2×D3711yescharacteristic, complemented
193#104—371no—
194#104—372no—
195#104—371no—
196#100—372no—
197#100C2×D3712yescomplemented
198C4×C8—371no—
199#41—371no—
200#41—372no—
201#41—742no—
202#41—742no—
203#41—742no—
204#41—742no—
205#41C22×D3711yescharacteristic
206C2×D4—372no—
207C2×D4—742no—
208C2×D4—742no—
209C2×D4—742no—
210C2×D4—742no—
211C2×D4—1482no—
212C2×D4—1482no—
213C2×D4—1482no—
214C2×D4C22×D3712yes—
215Q16—371no—
216Q16—371no—
217SD16—22no—
218SD16—371no—
219SD16—371no—
220SD16—372no—
221SD16—742no—
222SD16—742no—
223SD16—742no—
224D8—22no—
225D8—21no—
226D8—21no—
227D8—371no—
228D8—371no—
229D8—742no—
230D8—742no—
231D8—742no—
232D8—742no—
233D8—742no—
234D8—741no—
235D8—741no—
236M16—21no—
237M16—371no—
238M16—742no—
239M16—741no—
240M16C22×D3712yes—
241M16C22×D3711yescharacteristic
242C2×C8—21no—
243C2×C8—371no—
244C2×C8—742no—
245C2×C8—741no—
246C2×C8C22×D3711yescharacteristic
247C42—371no—
248C23—22no—
249C23—742no—
250C23—1482no—
251C23—1482no—
252Q8—371no—
253Q8—372no—
254Q8D4×D3711yescharacteristic
255D4—21no—
256D4—22no—
257D4—22no—
258D4—371no—
259D4—372no—
260D4—2964no—
261D4—742no—
262D4—741no—
263D4—742no—
264D4—742no—
265D4—1482no—
266D4—1482no—
267D4—1482no—
268D4—1482no—
269D4D4×D3711yescharacteristic
270C2×C4—21no—
271C2×C4—741no—
272C2×C4—742no—
273C2×C4—741no—
274C2×C4—1482no—
275C2×C4—1482no—
276C2×C4C23×D3711yescharacteristic
277C8—21no—
278C8—22no—
279C8—371no—
280C8—371no—
281C8—741no—
282C8D4×D3711yescharacteristic
283C8D4×D3711yescharacteristic
284C22—21no—
285C22—42no—
286C22—82no—
287C22—2962no—
288C22—742no—
289C22—741no—
290C22—5922no—
291C22—5922no—
292C22—1482no—
293C22#1192458211yescharacteristic
294C4—21no—
295C4—742no—
296C4—742no—
297C4—741no—
298C4C2×D4×D3711yescharacteristic
299C4C2×D4×D3711yescharacteristic
300C2—21no—
301C2—41no—
302C2—82no—
303C2—2962no—
304C2—1482no—
305C2—1481no—
306C2#1455816711yescharacteristic
subgroupG/H
centerC2#14558167
commutator, cosocleC2×C4×C37C24
FrattiniC2×C4C23×D37
FittingC37×#472C2
socleC2×C37D42
hypercenter#472D37
hypocenterC37#3292
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. #14558338C2semidirect
2. #14558337C2semidirect
3. #14558334C2semidirect
4. #14557908C2semidirect
5. #14557804C2semidirect
6. #14557802C2semidirect
7. #14557703C2semidirect
8. #14557699C2semidirect
9. #14557649C2semidirect
10. #14557648C2semidirect
11. C37×#472C2semidirect
12. #11924582C22semidirect
13. #11924581C22semidirect
15. #11924564C22semidirect
16. #11924564C22semidirect
18. #11924562C22semidirect
27. C2×D296C22semidirect
28. C2×#10596939C22semidirect
31. #11924544C22semidirect
32. #11924542C22semidirect
33. #11924532C22semidirect
34. #11924471C22semidirect
35. #11924463C22semidirect
36. #11924462C22semidirect
37. #11924462C22semidirect
38. #11924452C22semidirect
39. C8×Dic37C22semidirect
40. AGL(1,ℤ/8)×C37C22semidirect
41. C2×D8×C37C22semidirect
42. #131×C37C22semidirect
43. #108×C37C22semidirect
44. #100×C37C22semidirect
70. #10596942D4semidirect
71. #10596941D4semidirect
73. #10596940D4semidirect
77. #10596939D4semidirect
81. Dic148D4semidirect
86. D296D4semidirect
89. #10596930D4semidirect
160. #472D37semidirect
165. C37#3292semidirect
175. AGL(1,ℤ/8)C2×D37semidirect
186. C2×D8C2×D37semidirect
189. #131C2×D37semidirect
192. #108C2×D37semidirect
197. #100C2×D37semidirect

Maximal subgroups

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

subgroupindex
#145583382
#145583372
#145583342
#145579082
#145578042
#145578022
#145577032
#145576992
#145576492
#145576482
C37×#4722
#329237