Finite Group Databasebeta

← C22×#16862273 #30680695

Subgroups — C22×#16862273 #30680695

Order 16768.

1720 conjugacy classes of subgroups; only the 525 normal subgroups are shown.

normal subgroups only

This group has 1720 conjugacy classes of subgroups — too many to draw. Only its 525 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
1C22×#14523514C211yesmaximal, characteristic, complemented
2C22×D4×D131C211yesmaximal, characteristic, complemented
3C2×#16862273C2124yesmaximal, direct product factor
4C22×#14523500C211yesmaximal, characteristic, complemented
5C22×D1048C211yesmaximal, characteristic, complemented
6C22×#14523498C211yesmaximal, characteristic, complemented
7C22×#14523496C211yesmaximal, characteristic, complemented
8C22×SD16×C131C211yesmaximal, characteristic, complemented
9C24×D131—21no—
10C2×#14523514—26no—
11C2×#14523514C22112yescomplemented
12C22×D524—21no—
13C22×D524C2211yescharacteristic
14C22×D524C2211yescharacteristic, complemented
15C22×C22⋊D131—21no—
16C2×D4×D131—26no—
17C2×D4×D131C22112yescomplemented
18C22×C4×D131—21no—
19C22×C4×D131C2211yescharacteristic
20#16862273C22164yesdirect product factor
21C2×#14523500C22112yescomplemented
22C2×D1048C22112yescomplemented
23C2×#14523498C22112yescomplemented
24C2×#14523496C22112yescomplemented
25C22×C131⋊C8C2211yescharacteristic, complemented
26C22×Q8×C131C2211yescharacteristic, complemented
27C22×D4×C131C2211yescharacteristic
28C2×SD16×C131C22112yescomplemented
29C22×C8×C131C2211yescharacteristic, complemented
30C23×D131—212no—
31C23×D131—21no—
32C23×D131—21no—
33C23×D131—48no—
34C23×D131D411yescharacteristic
35#14523514—224no—
36#14523514C23116yescomplemented
37C2×D524—212no—
38C2×D524—23no—
39C2×D524—23no—
40C2×D524C2316yes—
41C2×D524C2316yescomplemented
42C2×C22⋊D131—212no—
43D4×D131—224no—
44D4×D131C23116yescomplemented
45C22×Dic131D411yescharacteristic
46C2×C4×D131—212no—
47C2×C4×D131D416yes—
48C2×C4×D131C2316yes—
49#14523500C23116yescomplemented
50D1048C23116yescomplemented
51#14523498C23116yescomplemented
52#14523496C23116yescomplemented
53C2×C131⋊C8C2316yescomplemented
54C24×C131—21no—
55C2×Q8×C131—23no—
56C2×Q8×C131C2316yescomplemented
57C2×D4×C131—23no—
58C2×D4×C131C2316yes—
59C22×C4×C131—21no—
60C22×C4×C131C2311yescharacteristic
61SD16×C131C23116yescomplemented
62C2×C8×C131C2316yescomplemented
63C22×D131—24no—
64C22×D131—26no—
65C22×D131—26no—
66C22×D131—216no—
67C22×D131—448no—
68C22×D131—44no—
69C22×D131—44no—
70C22×D131C2×D416yes—
71D524—216no—
72D524—26no—
73D524—26no—
74D524C2414yes—
75D524C2414yescomplemented
76C22⋊D131—216no—
77C2×Dic131C2×D416yes—
78C4×D131—216no—
79C4×D131C2×D4112yes—
80C4×D131C2414yes—
81C131⋊C8C2414yescomplemented
82C23×C131—26no—
83C23×C131—44no—
84C23×C131C2×D411yescharacteristic
85Q8×C131—26no—
86Q8×C131C2414yescomplemented
87D4×C131—26no—
88D4×C131C2414yes—
89C2×C4×C131—26no—
90C2×C4×C131C2×D413yes—
91C2×C4×C131C2413yes—
92C8×C131C2414yescomplemented
93C22×C131—24no—
94C22×C131—412no—
95C22×C131AGL(1,ℤ/8)14yescomplemented
96C22×C131C22×D413yes—
97C2×D131—212no—
98C2×D131—24no—
99C2×D131—24no—
100C2×D131—432no—
101C2×D131—412no—
102C2×D131—412no—
103C2×D131C22×D414yes—
104C4×C131—24no—
105C4×C131C22×D413yes—
106C4×C131C2511yescharacteristic
107Dic131C22×D414yes—
108C2×C131—44no—
109C2×C131C2×AGL(1,ℤ/8)16yescomplemented
110C2×C131C23×D411yescharacteristic
111D131—24no—
112D131—44no—
113D131—44no—
114C131C22×AGL(1,ℤ/8)11yescharacteristic, complemented, Sylow
115C22×AGL(1,ℤ/8)—1311nomaximal, Sylow
116C22×#41—1311no—
117C23×D4—1311no—
118C2×AGL(1,ℤ/8)—13124no—
119C22×SD16—1311no—
120C22×SD16D13111yescharacteristic, complemented
121C22×D8—1311no—
122C22×D8—1311no—
123C22×M16—1311no—
124C25—2621no—
125C2×#41—2626no—
126C2×#41—13112no—
127C22×Q8C2×D13111yescharacteristic, complemented
128C22×D4—2621no—
129C22×D4—2626no—
130C22×D4—2621no—
131C22×D4—13112no—
132C22×D4—1311no—
133C22×D4—1311no—
134C22×D4C2×D13111yescharacteristic
135C23×C4—2621no—
136C23×C4—1311no—
137AGL(1,ℤ/8)—13164no—
138C2×SD16—13112no—
139C2×SD16C2×D131112yescomplemented
140C2×D8—13112no—
141C2×D8—13112no—
142C2×M16—13112no—
143C22×C8—1311no—
144C22×C8C2×D13111yescharacteristic, complemented
145C24—21no—
146C24—26212no—
147C24—2621no—
148C24—2621no—
149C24—5248no—
150C24—1311no—
151#41—26224no—
152#41—13116no—
153C2×Q8—23no—
154C2×Q8C22×D13116yescomplemented
155C2×D4—23no—
156C2×D4—26212no—
157C2×D4—26224no—
158C2×D4—26212no—
159C2×D4—2623no—
160C2×D4—2623no—
161C2×D4—13116no—
162C2×D4—1316no—
163C2×D4—1316no—
164C2×D4C22×D13116yes—
165C22×C4—21no—
166C22×C4—26212no—
167C22×C4—1311no—
168C22×C4—1316no—
169C22×C4—1316no—
170C22×C4C22×D13111yescharacteristic
171SD16—13116no—
172SD16C22×D131116yescomplemented
173D8—13116no—
174D8—13116no—
175M16—13116no—
176C2×C8—1316no—
177C2×C8C22×D13116yescomplemented
178C23—26no—
179C23—44no—
180C23—26216no—
181C23—2624no—
182C23—2626no—
183C23—2626no—
184C23—52448no—
185C23—5244no—
186C23—5244no—
187C23—1316no—
188C23D4×D13111yescharacteristic
189Q8—26no—
190Q8C23×D13114yescomplemented
191D4—26no—
192D4—26216no—
193D4—26216no—
194D4—2626no—
195D4—2626no—
196D4—1314no—
197D4—1314no—
198D4C23×D13114yes—
199C2×C4—26no—
200C2×C4—26216no—
201C2×C4—1316no—
202C2×C4—13112no—
203C2×C4—1314no—
204C2×C4D4×D13113yes—
205C2×C4C23×D13113yes—
206C8—1314no—
207C8C23×D13114yescomplemented
208C22—24no—
209C22—412no—
210C22—26212no—
211C22—2624no—
212C22—2624no—
213C22—52432no—
214C22—52412no—
215C22—52412no—
216C22—1314no—
217C22#1686227314yesdirect product factor
218C22C2×D4×D13113yes—
219C4—24no—
220C4—1314no—
221C4C2×D4×D13113yes—
222C4C24×D13111yescharacteristic
223C2—44no—
224C2—2624no—
225C2—5244no—
226C2—5244no—
227C2C2×#1686227316yesdirect product factor
228C2C22×D4×D13111yescharacteristic
subgroupG/H
centerC23D4×D131
commutator, cosocleC4×C131C25
FrattiniC4C24×D131
FittingC22×SD16×C131C2
socleC23×C131C2×D4
hypercenterC22×SD16D131
hypocenterC131C22×AGL(1,ℤ/8)
LCSG ▷ C4×C131 ▷ C2×C131 ▷ C131
UCSC1 ◁ C23 ◁ C22×C4 ◁ C22×SD16

Complements

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

NH ≅ G/Ntype
1. C22×#14523514C2semidirect
2. C22×D4×D131C2semidirect
3. C2×#16862273C2direct (N × H)
4. C22×#14523500C2semidirect
5. C22×D1048C2semidirect
6. C22×#14523498C2semidirect
7. C22×#14523496C2semidirect
8. C22×SD16×C131C2semidirect
11. C2×#14523514C22semidirect
14. C22×D524C22semidirect
17. C2×D4×D131C22semidirect
20. #16862273C22direct (N × H)
21. C2×#14523500C22semidirect
22. C2×D1048C22semidirect
23. C2×#14523498C22semidirect
24. C2×#14523496C22semidirect
25. C22×C131⋊C8C22semidirect
26. C22×Q8×C131C22semidirect
28. C2×SD16×C131C22semidirect
29. C22×C8×C131C22semidirect
36. #14523514C23semidirect
41. C2×D524C23semidirect
44. D4×D131C23semidirect
49. #14523500C23semidirect
50. D1048C23semidirect
51. #14523498C23semidirect
52. #14523496C23semidirect
53. C2×C131⋊C8C23semidirect
56. C2×Q8×C131C23semidirect
61. SD16×C131C23semidirect
62. C2×C8×C131C23semidirect
75. D524C24semidirect
81. C131⋊C8C24semidirect
86. Q8×C131C24semidirect
92. C8×C131C24semidirect
95. C22×C131AGL(1,ℤ/8)semidirect
109. C2×C131C2×AGL(1,ℤ/8)semidirect
114. C131C22×AGL(1,ℤ/8)semidirect
120. C22×SD16D131semidirect
127. C22×Q8C2×D131semidirect
139. C2×SD16C2×D131semidirect
144. C22×C8C2×D131semidirect
154. C2×Q8C22×D131semidirect
172. SD16C22×D131semidirect
177. C2×C8C22×D131semidirect
190. Q8C23×D131semidirect
207. C8C23×D131semidirect
217. C22#16862273direct (N × H)
227. C2C2×#16862273direct (N × H)

Maximal subgroups

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

subgroupindex
C22×#145235142
C22×D4×D1312
C2×#168622732
C22×#145235002
C22×D10482
C22×#145234982
C22×#145234962
C22×SD16×C1312
C22×AGL(1,ℤ/8)131