Finite Group Databasebeta

← #30680742

Subgroups — #30680742

Order 16768.

1464 conjugacy classes of subgroups; only the 545 normal subgroups are shown.

normal subgroups only

This group has 1464 conjugacy classes of subgroups — too many to draw. Only its 545 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#22554148C211yesmaximal, characteristic, complemented
2C2×#16862292C211yesmaximal, characteristic, complemented
3#22554124C218yesmaximal, complemented
4#22554117C218yesmaximal, complemented
5#22554112C216yesmaximal, complemented
6C2×#16862274C216yesmaximal, complemented
7#568×C131C211yesmaximal, characteristic, complemented
8#16862294—21no—
9#16862293C2214yescomplemented
10#16862292—24no—
11#16862292C2218yescomplemented
12#16862292C2214yescomplemented
13#16862291—23no—
14C2×#14523514—21no—
15C2×#14523513C2213yescomplemented
16C2×#14523513C2216yescomplemented
17#41×D131—24no—
18C22×D524C2213yescomplemented
19C2×D4×D131—23no—
20#16862279C2214yescomplemented
21#16862279C22112yescomplemented
22#16862274C22112yescomplemented
23#16862274C22124yescomplemented
24#16862274C22112yescomplemented
25#16862268C22124yescomplemented
26C2×#14523502C22112yescomplemented
27C2×D1048C22112yescomplemented
28C2×#41×C131C2211yescharacteristic
29#131×C131C2218yescomplemented
30C2×M16×C131C2216yescomplemented
31C23×D131—23no—
32#14523514—24no—
33#14523514—28no—
34#14523513—26no—
35#14523513—26no—
36#14523513—26no—
37#14523513C23112yes—
38#14523513C2314yes—
39#14523513C23112yescomplemented
40#14523512—212no—
41C2×Dic262C2311yescharacteristic
42C2×Dic262C2316yescomplemented
43Q8×D131—24no—
44C2×D524—23no—
45C2×D524—26no—
46C2×D524C23112yes—
47C2×D524C2313yes—
48C2×D524C2316yescomplemented
49C2×C22⋊D131—23no—
50C2×C22⋊D131—26no—
51D4×D131—224no—
52D4×D131—212no—
53C2×C4×D131—23no—
54C2×C4×D131—26no—
55Dic524C2318yescomplemented
56#14523502C23124yescomplemented
57#14523502C2318yescomplemented
58D1048C23124yescomplemented
59#41×C131D414yes—
60#41×C131C2314yes—
61C2×Q8×C131D411yescharacteristic
62C2×D4×C131D413yes—
63C22×C4×C131C2313yes—
64M16×C131C2314yes—
65M16×C131C23112yescomplemented
66C2×C8×C131C23112yescomplemented
67C22×D131—23no—
68C22×D131—26no—
69C22×D131—212no—
70C22×D131—86no—
71Dic262—26no—
72Dic262—21no—
73Dic262C2414yes—
74Dic262C2412yes—
75D524—212no—
76D524—26no—
77D524—23no—
78D524C2414yes—
79D524C2416yes—
80C22⋊D131—212no—
81C22⋊D131—212no—
82C22⋊D131—24no—
83C2×Dic131—21no—
84C2×Dic131—26no—
85C4×D131—24no—
86C4×D131—212no—
87C4×D131—212no—
88C23×C131C2×D413yes—
89Q8×C131C2×D414yes—
90D4×C131C2×D4112yes—
91C2×C4×C131C2×D413yes—
92C2×C4×C131C2×D416yes—
93C2×C4×C131C2416yes—
94C2×C4×C131C2411yescharacteristic
95C8×C131C2418yes—
96C22×C131—43no—
97C22×C131C22×D411yescharacteristic
98C22×C131C22×D416yes—
99C2×D131—24no—
100C2×D131—26no—
101C2×D131—46no—
102C2×D131—86no—
103C2×D131—812no—
104C4×C131C22×D411yescharacteristic
105C4×C131C22×D416yes—
106C4×C131C2511yescharacteristic
107Dic131—22no—
108Dic131—24no—
109C2×C131—21no—
110C2×C131—26no—
111C2×C131C23×D411yescharacteristic
112D131—46no—
113D131—44no—
114C131#358511yescharacteristic, complemented, Sylow
115#3585—1311nomaximal, Sylow
116#585—1311no—
117C2×H(5,𝔽2)—1311no—
118#577—1318no—
119#576—1318no—
120#575—1316no—
121C2×AGL(1,ℤ/8)—1316no—
122#568D13111yescharacteristic, complemented
1232−1+4—2621no—
1242−1+4—1314no—
125H(5,𝔽2)—2624no—
126H(5,𝔽2)—2623no—
127H(5,𝔽2)—1314no—
128H(5,𝔽2)—1318no—
129C2×#41—2624no—
130C2×#41—2621no—
131C2×#41—1316no—
132C2×#41—1313no—
133C2×#41C2×D13111yescharacteristic
134C22×D4—2623no—
135C22×D4—1313no—
136#137—1314no—
137#137—13112no—
138AGL(1,ℤ/8)—13112no—
139AGL(1,ℤ/8)—13124no—
140AGL(1,ℤ/8)—13112no—
141#135—13124no—
142C2×SD16—13112no—
143C2×D8—13112no—
144#131C2×D13118yescomplemented
145C2×M16C2×D13116yescomplemented
146C24—2623no—
147#41—2628no—
148#41—2624no—
149#41—26212no—
150#41—2626no—
151#41—2626no—
152#41—2626no—
153#41—13112no—
154#41—1314no—
155#41—13112no—
156#41D52414yes—
157#41C22×D13114yes—
158C2×Q8—2624no—
159C2×Q8—1311no—
160C2×Q8—1316no—
161C2×Q8D52411yescharacteristic
162C2×D4—2623no—
163C2×D4—2626no—
164C2×D4—26224no—
165C2×D4—26212no—
166C2×D4—2623no—
167C2×D4—2626no—
168C2×D4—13112no—
169C2×D4—1313no—
170C2×D4—1316no—
171C2×D4D52413yes—
172C22×C4—2623no—
173C22×C4—2626no—
174C22×C4C22×D13113yes—
175Q16—1318no—
176SD16—13124no—
177SD16—1318no—
178D8—13124no—
179M16C22×D13114yes—
180M16C22×D131112yescomplemented
181C2×C8C22×D131112yescomplemented
182C23—2623no—
183C23—2626no—
184C23—26212no—
185C23—10486no—
186C23C2×D52413yes—
187Q8—2626no—
188Q8—2621no—
189Q8—1314no—
190Q8—1312no—
191Q8C2×D52414yes—
192D4—26212no—
193D4—2624no—
194D4—26212no—
195D4—2623no—
196D4—26212no—
197D4—2626no—
198D4—1314no—
199D4—1316no—
200D4C2×D524112yes—
201C2×C4—2621no—
202C2×C4—2626no—
203C2×C4—2624no—
204C2×C4—26212no—
205C2×C4—26212no—
206C2×C4C2×D52413yes—
207C2×C4C2×D52416yes—
208C2×C4C23×D13116yes—
209C2×C4C23×D13111yescharacteristic
210C8C23×D13118yes—
211C22—43no—
212C22—2624no—
213C22—2626no—
214C22—5246no—
215C22—104812no—
216C22—10486no—
217C22C22×D52411yescharacteristic
218C22C22×D52416yes—
219C4—2622no—
220C4—2624no—
221C4C22×D52411yescharacteristic
222C4C22×D52416yes—
223C4C24×D13111yescharacteristic
224C2—21no—
225C2—26no—
226C2—5246no—
227C2—5244no—
228C2C23×D52411yescharacteristic
subgroupG/H
centerC2C23×D524
commutator, cosocleC4×C131C25
FrattiniC4C24×D131
Fitting#568×C131C2
socleC2×C131C23×D4
hypercenter#568D131
hypocenterC131#3585
LCSG ▷ C4×C131 ▷ C2×C131 ▷ C131
UCSC1 ◁ C2 ◁ C2×#41 ◁ #568

Complements

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

NH ≅ G/Ntype
1. #22554148C2semidirect
2. C2×#16862292C2semidirect
3. #22554124C2semidirect
4. #22554117C2semidirect
5. #22554112C2semidirect
6. C2×#16862274C2semidirect
7. #568×C131C2semidirect
9. #16862293C22semidirect
11. #16862292C22semidirect
12. #16862292C22semidirect
15. C2×#14523513C22semidirect
16. C2×#14523513C22semidirect
18. C22×D524C22semidirect
20. #16862279C22semidirect
21. #16862279C22semidirect
22. #16862274C22semidirect
23. #16862274C22semidirect
24. #16862274C22semidirect
25. #16862268C22semidirect
26. C2×#14523502C22semidirect
27. C2×D1048C22semidirect
29. #131×C131C22semidirect
30. C2×M16×C131C22semidirect
39. #14523513C23semidirect
42. C2×Dic262C23semidirect
48. C2×D524C23semidirect
55. Dic524C23semidirect
56. #14523502C23semidirect
57. #14523502C23semidirect
58. D1048C23semidirect
65. M16×C131C23semidirect
66. C2×C8×C131C23semidirect
114. C131#3585semidirect
122. #568D131semidirect
144. #131C2×D131semidirect
145. C2×M16C2×D131semidirect
180. M16C22×D131semidirect
181. C2×C8C22×D131semidirect

Maximal subgroups

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

subgroupindex
#225541482
C2×#168622922
#225541242
#225541172
#225541122
C2×#168622742
#568×C1312
#3585131