Finite Group Databasebeta

← #30680741

Subgroups — #30680741

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
2#22554123C218yesmaximal, complemented
3#22554115C218yesmaximal, complemented
4#22554110C216yesmaximal, complemented
5C2×#16862272C216yesmaximal, complemented
6#22554079C211yesmaximal, characteristic, complemented
7C2×H(5,𝔽2)×C131C211yesmaximal, characteristic, complemented
8#16862294—21no—
9#16862293C2214yescomplemented
10#16862292C2214yescomplemented
11#16862291—23no—
12C2×#14523513C2216yescomplemented
13#41×D131—24no—
14#16862277C2214yescomplemented
15#16862276C22112yescomplemented
16#16862272C22112yescomplemented
17#16862272C22124yescomplemented
18#16862271C22112yescomplemented
19#16862266C22124yescomplemented
20C2×#14523501C22112yescomplemented
21C2×#14523498C22112yescomplemented
22#16862253C2218yescomplemented
23C2×#14523497C2216yescomplemented
24H(5,𝔽2)×C131—24no—
25H(5,𝔽2)×C131C2218yescomplemented
26C2×#41×C131—21no—
27C2×#41×C131C2211yescharacteristic
28C2×#41×C131C2213yescomplemented
29C22×D4×C131—23no—
30C22×D4×C131C2213yescomplemented
31#14523514—24no—
32#14523513—26no—
33#14523513—26no—
34#14523513C2314yes—
35#14523513C23112yescomplemented
36#14523512—212no—
37C2×Dic262C2316yescomplemented
38Q8×D131—24no—
39C2×D524C2316yescomplemented
40C2×C22⋊D131—26no—
41D4×D131—212no—
42C2×C4×D131—26no—
43#14523503C2318yescomplemented
44#14523501C23124yescomplemented
45#14523500C2318yescomplemented
46#14523498C23124yescomplemented
47#14523497C2314yes—
48#14523497C23112yescomplemented
49C2×C131⋊C8C23112yescomplemented
50C24×C131—23no—
51#41×C131—28no—
52#41×C131—26no—
53#41×C131D414yes—
54#41×C131C23112yes—
55#41×C131C2314yes—
56C2×Q8×C131D411yescharacteristic
57C2×Q8×C131C2311yescharacteristic
58C2×D4×C131—23no—
59C2×D4×C131—224no—
60C2×D4×C131—23no—
61C2×D4×C131—26no—
62C2×D4×C131D413yes—
63C2×D4×C131C23112yes—
64C2×D4×C131C2313yes—
65C22×C4×C131—23no—
66C22×C4×C131C2313yes—
67C22×D131—26no—
68Dic262—26no—
69Dic262C2414yes—
70D524—26no—
71D524C2414yes—
72C22⋊D131—212no—
73C22⋊D131—24no—
74C2×Dic131—26no—
75C4×D131—24no—
76C4×D131—212no—
77C131⋊C8C2418yes—
78C23×C131—23no—
79C23×C131—212no—
80C23×C131—86no—
81C23×C131C2×D413yes—
82Q8×C131—21no—
83Q8×C131C2×D414yes—
84Q8×C131C2412yes—
85D4×C131—212no—
86D4×C131—212no—
87D4×C131—23no—
88D4×C131C2×D4112yes—
89D4×C131C2416yes—
90C2×C4×C131—21no—
91C2×C4×C131—212no—
92C2×C4×C131C2×D413yes—
93C2×C4×C131C2×D416yes—
94C2×C4×C131C2411yescharacteristic
95C2×C4×C131C2416yes—
96C22×C131—26no—
97C22×C131—43no—
98C22×C131—46no—
99C22×C131—812no—
100C22×C131C22×D411yescharacteristic
101C22×C131C22×D416yes—
102C2×D131—24no—
103C2×D131—86no—
104C4×C131—22no—
105C4×C131C22×D416yes—
106C4×C131C22×D411yescharacteristic
107C4×C131C2511yescharacteristic
108Dic131—24no—
109C2×C131—26no—
110C2×C131—21no—
111C2×C131—46no—
112C2×C131C23×D411yescharacteristic
113D131—44no—
114C131#358511yescharacteristic, complemented, Sylow
115#3585—1311nomaximal, Sylow
116#585—1311no—
117C2×H(5,𝔽2)D13111yescharacteristic, complemented
118#577—1318no—
119#576—1318no—
120#575—1316no—
121C2×AGL(1,ℤ/8)—1316no—
122#568—1311no—
1232−1+4—2621no—
1242−1+4—1314no—
125H(5,𝔽2)—24no—
126H(5,𝔽2)—2623no—
127H(5,𝔽2)—1314no—
128H(5,𝔽2)C2×D13118yescomplemented
129C2×#41—21no—
130C2×#41—2624no—
131C2×#41—1316no—
132C2×#41C2×D13111yescharacteristic
133C2×#41C2×D13113yescomplemented
134C22×D4—23no—
135C22×D4C2×D13113yescomplemented
136#137—13112no—
137#137—1314no—
138AGL(1,ℤ/8)—13112no—
139AGL(1,ℤ/8)—13112no—
140AGL(1,ℤ/8)—13124no—
141#135—13124no—
142C2×SD16—13112no—
143C2×D8—13112no—
144#131—1318no—
145C2×M16—1316no—
146C24—23no—
147#41—28no—
148#41—26no—
149#41—2624no—
150#41—26212no—
151#41—2626no—
152#41—2626no—
153#41—1314no—
154#41—13112no—
155#41C22⋊D13114yes—
156#41C22×D131112yes—
157#41C22×D13114yes—
158C2×Q8—2624no—
159C2×Q8—1316no—
160C2×Q8C22⋊D13111yescharacteristic
161C2×Q8C22×D13111yescharacteristic
162C2×D4—23no—
163C2×D4—224no—
164C2×D4—23no—
165C2×D4—26no—
166C2×D4—2626no—
167C2×D4—26212no—
168C2×D4—1316no—
169C2×D4C22⋊D13113yes—
170C2×D4C22×D131112yes—
171C2×D4C22×D13113yes—
172C22×C4—23no—
173C22×C4—2626no—
174C22×C4C22×D13113yes—
175Q16—1318no—
176SD16—1318no—
177SD16—13124no—
178D8—13124no—
179M16—1314no—
180M16—13112no—
181C2×C8—13112no—
182C23—23no—
183C23—212no—
184C23—2626no—
185C23—86no—
186C23C2×C22⋊D13113yes—
187Q8—21no—
188Q8—2626no—
189Q8—1314no—
190Q8C2×C22⋊D13114yes—
191Q8C23×D13112yes—
192D4—212no—
193D4—212no—
194D4—23no—
195D4—2624no—
196D4—26212no—
197D4—2626no—
198D4—1314no—
199D4C2×C22⋊D131112yes—
200D4C23×D13116yes—
201C2×C4—21no—
202C2×C4—212no—
203C2×C4—2626no—
204C2×C4—2624no—
205C2×C4—26212no—
206C2×C4C2×C22⋊D13113yes—
207C2×C4C2×C22⋊D13116yes—
208C2×C4C23×D13116yes—
209C2×C4C23×D13111yescharacteristic
210C8—1318no—
211C22—26no—
212C22—43no—
213C22—46no—
214C22—2624no—
215C22—812no—
216C22—10486no—
217C22C22×C22⋊D13111yescharacteristic
218C22C22×C22⋊D13116yes—
219C4—22no—
220C4—2624no—
221C4C22×C22⋊D13116yes—
222C4C22×C22⋊D13111yescharacteristic
223C4C24×D13111yescharacteristic
224C2—21no—
225C2—26no—
226C2—46no—
227C2—5244no—
228C2C23×C22⋊D13111yescharacteristic
subgroupG/H
centerC2C23×C22⋊D131
commutator, cosocleC4×C131C25
FrattiniC4C24×D131
FittingC2×H(5,𝔽2)×C131C2
socleC2×C131C23×D4
hypercenterC2×H(5,𝔽2)D131
hypocenterC131#3585
LCSG ▷ C4×C131 ▷ C2×C131 ▷ C131
UCSC1 ◁ C2 ◁ C2×#41 ◁ C2×H(5,𝔽2)

Complements

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

NH ≅ G/Ntype
1. #22554148C2semidirect
2. #22554123C2semidirect
3. #22554115C2semidirect
4. #22554110C2semidirect
5. C2×#16862272C2semidirect
6. #22554079C2semidirect
7. C2×H(5,𝔽2)×C131C2semidirect
9. #16862293C22semidirect
10. #16862292C22semidirect
12. C2×#14523513C22semidirect
14. #16862277C22semidirect
15. #16862276C22semidirect
16. #16862272C22semidirect
17. #16862272C22semidirect
18. #16862271C22semidirect
19. #16862266C22semidirect
20. C2×#14523501C22semidirect
21. C2×#14523498C22semidirect
22. #16862253C22semidirect
23. C2×#14523497C22semidirect
25. H(5,𝔽2)×C131C22semidirect
28. C2×#41×C131C22semidirect
30. C22×D4×C131C22semidirect
35. #14523513C23semidirect
37. C2×Dic262C23semidirect
39. C2×D524C23semidirect
43. #14523503C23semidirect
44. #14523501C23semidirect
45. #14523500C23semidirect
46. #14523498C23semidirect
48. #14523497C23semidirect
49. C2×C131⋊C8C23semidirect
114. C131#3585semidirect
117. C2×H(5,𝔽2)D131semidirect
128. H(5,𝔽2)C2×D131semidirect
133. C2×#41C2×D131semidirect
135. C22×D4C2×D131semidirect

Maximal subgroups

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

subgroupindex
#225541482
#225541232
#225541152
#225541102
C2×#168622722
#225540792
C2×H(5,𝔽2)×C1312
#3585131