Finite Group Databasebeta

← C22×D8×D251 #44565079

Subgroups — C22×D8×D251 #44565079

Order 32128.

1976 conjugacy classes of subgroups; only the 565 normal subgroups are shown.

normal subgroups only

This group has 1976 conjugacy classes of subgroups — too many to draw. Only its 565 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×D4×D251C212yesmaximal, complemented
2C22×D2008C211yesmaximal, characteristic, complemented
3C22×#16860795C212yesmaximal, complemented
4C2×D8×D251C2124yesmaximal, direct product factor
5C22×C8×D251C211yesmaximal, characteristic, complemented
6C22×D8×C251C211yesmaximal, characteristic, complemented
7C24×D251—22no—
8C22×D1004C2212yescomplemented
9C22×C22⋊D251—22no—
10C2×D4×D251—212no—
11C2×D4×D251C22124yescomplemented
12C22×C4×D251C2211yescharacteristic
13C2×D2008C22112yescomplemented
14C2×#16860795C22124yescomplemented
15D8×D251C22164yesdirect product factor
16C22×C251⋊C8C2211yescharacteristic, complemented
17C2×C8×D251C22112yescomplemented
18C22×D4×C251C2212yescomplemented
19C2×D8×C251C22112yescomplemented
20C22×C8×C251C2211yescharacteristic, complemented
21C23×D251—224no—
22C23×D251—22no—
23C23×D251—416no—
24C23×D251D411yescharacteristic
25C2×D1004—26no—
26C2×D1004C23112yescomplemented
27C2×C22⋊D251—224no—
28D4×D251—248no—
29D4×D251C23132yescomplemented
30C22×Dic251D411yescharacteristic
31C2×C4×D251D416yes—
32C2×C4×D251C2316yes—
33D2008C23116yescomplemented
34#16860795C23132yescomplemented
35C2×C251⋊C8C2316yescomplemented
36C8×D251C23116yescomplemented
37C24×C251—22no—
38C2×D4×C251—26no—
39C2×D4×C251C23112yescomplemented
40C22×C4×C251C2311yescharacteristic
41D8×C251C23116yescomplemented
42C2×C8×C251C2316yescomplemented
43C22×D251—212no—
44C22×D251—232no—
45C22×D251—48no—
46C22×D251—496no—
47C22×D251D818yesdirect product factor
48C22×D251C2×D416yes—
49D1004—212no—
50D1004C2418yescomplemented
51C22⋊D251—232no—
52C2×Dic251C2×D416yes—
53C4×D251C2×D4112yes—
54C4×D251C2414yes—
55C251⋊C8C2414yescomplemented
56C23×C251—212no—
57C23×C251—48no—
58C23×C251C2×D411yescharacteristic
59D4×C251—212no—
60D4×C251C2418yescomplemented
61C2×C4×C251C2×D413yes—
62C2×C4×C251C2413yes—
63C8×C251C2414yescomplemented
64C22×C251—28no—
65C22×C251—424no—
66C22×C251C2×D814yescomplemented
67C22×C251C22×D413yes—
68C2×D251—28no—
69C2×D251—424no—
70C2×D251—464no—
71C2×D251C2×D8124yesdirect product factor
72C2×D251C22×D414yes—
73C4×C251C22×D413yes—
74C4×C251C2511yescharacteristic
75Dic251C22×D414yes—
76C2×C251—48no—
77C2×C251C22×D816yescomplemented
78C2×C251C23×D411yescharacteristic
79D251—48no—
80D251C22×D818yesdirect product factor
81C251C23×D811yescharacteristic, complemented, Sylow
82C23×D8—2511nomaximal, Sylow
83C23×D4—2512no—
84C22×D8—2512no—
85C22×D8—25124no—
86C22×D8—2511no—
87C22×D8D25111yescharacteristic, direct product factor
88C23×C8—2511no—
89C25—5022no—
90C22×D4—50212no—
91C22×D4—5022no—
92C22×D4—25124no—
93C22×D4—2512no—
94C22×D4C2×D25112yescomplemented
95C23×C4—2511no—
96C2×D8—25124no—
97C2×D8—25164no—
98C2×D8—25112no—
99C2×D8C2×D251112yesdirect product factor
100C22×C8—2511no—
101C22×C8—25112no—
102C22×C8C2×D25111yescharacteristic, complemented
103C24—22no—
104C24—100416no—
105C24—50224no—
106C24—5022no—
107C24—2511no—
108C2×D4—26no—
109C2×D4—50248no—
110C2×D4—50224no—
111C2×D4—5026no—
112C2×D4—25132no—
113C2×D4—25112no—
114C2×D4C22×D251112yescomplemented
115C22×C4—2511no—
116C22×C4—2516no—
117C22×C4—2516no—
118C22×C4C22×D25111yescharacteristic
119D8—25132no—
120D8—25116no—
121D8C22×D251116yesdirect product factor
122C2×C8—2516no—
123C2×C8—25116no—
124C2×C8C22×D25116yescomplemented
125C23—212no—
126C23—48no—
127C23—100496no—
128C23—10048no—
129C23—50232no—
130C23—50212no—
131C23—2516no—
132C23—2518no—
133C23D4×D25111yescharacteristic
134D4—212no—
135D4—50232no—
136D4—50212no—
137D4—2518no—
138D4C23×D25118yescomplemented
139C2×C4—2516no—
140C2×C4—25112no—
141C2×C4—2514no—
142C2×C4D4×D25113yes—
143C2×C4C23×D25113yes—
144C8—2514no—
145C8C23×D25114yescomplemented
146C22—28no—
147C22—424no—
148C22—100464no—
149C22—100424no—
150C22—5028no—
151C22—25124no—
152C22—2514no—
153C22D8×D25114yesdirect product factor
154C22C2×D4×D25113yes—
155C4—2514no—
156C4C2×D4×D25113yes—
157C4C24×D25111yescharacteristic
158C2—48no—
159C2—10048no—
160C2—2518no—
161C2C2×D8×D25116yesdirect product factor
162C2C22×D4×D25111yescharacteristic
subgroupG/H
centerC23D4×D251
commutator, cosocleC4×C251C25
FrattiniC4C24×D251
FittingC22×D8×C251C2
socleC23×C251C2×D4
hypercenterC22×D8D251
hypocenterC251C23×D8
LCSG ▷ C4×C251 ▷ C2×C251 ▷ C251
UCSC1 ◁ C23 ◁ C22×C4 ◁ C22×D8

Complements

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

NH ≅ G/Ntype
1. C22×D4×D251C2semidirect
2. C22×D2008C2semidirect
3. C22×#16860795C2semidirect
4. C2×D8×D251C2direct (N × H)
5. C22×C8×D251C2semidirect
6. C22×D8×C251C2semidirect
8. C22×D1004C22semidirect
11. C2×D4×D251C22semidirect
13. C2×D2008C22semidirect
14. C2×#16860795C22semidirect
15. D8×D251C22direct (N × H)
16. C22×C251⋊C8C22semidirect
17. C2×C8×D251C22semidirect
18. C22×D4×C251C22semidirect
19. C2×D8×C251C22semidirect
20. C22×C8×C251C22semidirect
26. C2×D1004C23semidirect
29. D4×D251C23semidirect
33. D2008C23semidirect
34. #16860795C23semidirect
35. C2×C251⋊C8C23semidirect
36. C8×D251C23semidirect
39. C2×D4×C251C23semidirect
41. D8×C251C23semidirect
42. C2×C8×C251C23semidirect
47. C22×D251D8direct (N × H)
50. D1004C24semidirect
55. C251⋊C8C24semidirect
60. D4×C251C24semidirect
63. C8×C251C24semidirect
66. C22×C251C2×D8semidirect
71. C2×D251C2×D8direct (N × H)
77. C2×C251C22×D8semidirect
80. D251C22×D8direct (N × H)
81. C251C23×D8semidirect
87. C22×D8D251direct (N × H)
94. C22×D4C2×D251semidirect
99. C2×D8C2×D251direct (N × H)
102. C22×C8C2×D251semidirect
114. C2×D4C22×D251semidirect
121. D8C22×D251direct (N × H)
124. C2×C8C22×D251semidirect
138. D4C23×D251semidirect
145. C8C23×D251semidirect
153. C22D8×D251direct (N × H)
161. C2C2×D8×D251direct (N × H)

Maximal subgroups

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

subgroupindex
C22×D4×D2512
C22×D20082
C22×#168607952
C2×D8×D2512
C22×C8×D2512
C22×D8×C2512
C23×D8251