Finite Group Databasebeta

← #121×D223 #28384068

Subgroups — #121×D223 #28384068

Order 14272.

426 conjugacy classes of subgroups; only the 121 normal subgroups are shown.

normal subgroups only

This group has 426 conjugacy classes of subgroups — too many to draw. Only its 121 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
1C2×D4×D223C211yesmaximal, characteristic, complemented
2C2×D4×D223C212yesmaximal, complemented
3C22×C4×D223C211yesmaximal, characteristic, complemented
4#20297279C211yesmaximal, characteristic, complemented
5#20297278C211yesmaximal, characteristic, complemented
6#20297277C212yesmaximal, complemented
7#20297276C212yesmaximal, complemented
8#20297275C211yesmaximal, characteristic, complemented
9C4⋊C4×D223C211yesmaximal, characteristic, complemented
10C22⋊C4×D223C212yesmaximal, complemented
11#121×C223C211yesmaximal, characteristic, complemented
12C23×D223C2212yes—
13C23×D223C2211yescharacteristic
14C2×D892C2212yescomplemented
15C2×D892C2211yescharacteristic, complemented
16C2×C22⋊D223C2212yescomplemented
17C2×C22⋊D223C2212yescomplemented
18C2×C22⋊D223C2212yescomplemented
19D4×D223—28no—
20D4×D223—24no—
21C22×Dic223C2211yescharacteristic, complemented
22C2×C4×D223—24no—
23C2×C4×D223C2211yescharacteristic
24C2×C4×D223C2212yes—
25C2×C4×D223C2211yescharacteristic
26C2×C4×D223C2211yescharacteristic, complemented
27C2×C4×D223C2211yescharacteristic, complemented
28C4⋊Dic223C2211yescharacteristic, complemented
29Dic223⋊C4C2212yescomplemented
30C22⋊Dic223C2212yescomplemented
31(C2×D223)⋊C4C2212yescomplemented
32(C2×D223)⋊C4C2212yescomplemented
33C2×D4×C223C2211yescharacteristic, complemented
34C2×D4×C223C2212yescomplemented
35C22×C4×C223C2211yescharacteristic, complemented
36C4⋊C4×C223C2211yescharacteristic, complemented
37C22⋊C4×C223C2212yescomplemented
38C22×D223—22no—
39C22×D223—22no—
40C22×D223—24no—
41C22×D223—24no—
42C22×D223—24no—
43C22×D223D414yescomplemented
44C22×D223C2312yes—
45C22×D223C2311yescharacteristic
46C22×D223C2311yescharacteristic
47D892—24no—
48D892—22no—
49C22⋊D223—24no—
50C22⋊D223—24no—
51C22⋊D223—24no—
52C2×Dic223—22no—
53C2×Dic223C2311yescharacteristic
54C2×Dic223C2312yes—
55C2×Dic223C2311yescharacteristic
56C4×D223—22no—
57C4×D223—24no—
58C4×D223—22no—
59C4×D223—22no—
60C4×D223D414yescomplemented
61C23×C223C2312yes—
62C23×C223C2311yescharacteristic
63D4×C223—24no—
64D4×C223—22no—
65C2×C4×C223—22no—
66C2×C4×C223C2311yescharacteristic
67C2×C4×C223C2312yes—
68C2×C4×C223C2311yescharacteristic
69C22×C223—21no—
70C22×C223—21no—
71C22×C223—22no—
72C22×C223—22no—
73C22×C223—22no—
74C22×C223C2×D412yescomplemented
75C22×C223C2411yescharacteristic
76C2×D223—21no—
77C2×D223—21no—
78C2×D223—28no—
79C2×D223—22no—
80C2×D223—22no—
81C2×D223—22no—
82C2×D223—48no—
83C2×D223C2×D412yes—
84C2×D223C2×D412yes—
85C2×D223C2×D412yescomplemented
86C2×D223#4112yes—
87C4×C223—21no—
88C4×C223—22no—
89C4×C223C2×D412yescomplemented
90Dic223—21no—
91Dic223—22no—
92Dic223C2×D412yescomplemented
93C2×C223—22no—
94C2×C223—42no—
95C2×C223C22×D411yescharacteristic
96C2×C223C22×D411yescharacteristic
97C2×C223C2×#4111yescharacteristic
98D223—22no—
99D223—42no—
100D223#12114yesdirect product factor
101C223C2×#12111yescharacteristic, complemented, Sylow
102C2×#121—2231nomaximal, Sylow
103C22×D4—2232no—
104C22×D4—2231no—
105C23×C4—2231no—
106#121—2231no—
107#121—2232no—
108#121—2231no—
109#121—2231no—
110#121—2232no—
111#121D22311yescharacteristic, direct product factor
112C2×C4⋊C4—2231no—
113C2×C22⋊C4—2232no—
114C24—2232no—
115C24—2231no—
116C2×D4—4468no—
117C2×D4—4464no—
118C2×D4—2232no—
119C2×D4—2232no—
120C2×D4—2232no—
121C2×D4—2232no—
122C2×D4—2231no—
123C2×D4C2×D22312yescomplemented
124C2×D4C2×D22311yescharacteristic, complemented
125C22×C4—4464no—
126C22×C4—2231no—
127C22×C4—2231no—
128C22×C4—2232no—
129C22×C4—2231no—
130C22×C4—2231no—
131C22×C4—2231no—
132C22×C4C2×D22311yescharacteristic, complemented
133C4⋊C4—2232no—
134C4⋊C4—2231no—
135C4⋊C4C2×D22311yescharacteristic, complemented
136C22⋊C4—2232no—
137C22⋊C4—2232no—
138C22⋊C4—2232no—
139C22⋊C4C2×D22312yescomplemented
140C23—4464no—
141C23—4464no—
142C23—4464no—
143C23—4462no—
144C23—4462no—
145C23—2232no—
146C23—2234no—
147C23—2231no—
148C23—2231no—
149C23C22×D22312yes—
150C23C22×D22311yescharacteristic
151D4—24no—
152D4—22no—
153D4—4464no—
154D4—4464no—
155D4—4464no—
156D4—4464no—
157D4—4462no—
158C2×C4—22no—
159C2×C4—4462no—
160C2×C4—4464no—
161C2×C4—4462no—
162C2×C4—4462no—
163C2×C4—4462no—
164C2×C4—2231no—
165C2×C4—2232no—
166C2×C4—2231no—
167C2×C4—2234no—
168C2×C4C22×D22311yescharacteristic
169C2×C4C22×D22312yes—
170C2×C4C22×D22311yescharacteristic
171C22—21no—
172C22—21no—
173C22—22no—
174C22—22no—
175C22—22no—
176C22—8928no—
177C22—4468no—
178C22—4462no—
179C22—4462no—
180C22—4462no—
181C22—4461no—
182C22—4461no—
183C22—2232no—
184C22—2232no—
185C22—2232no—
186C22—2232no—
187C22D4×D22312yescomplemented
188C22C23×D22311yescharacteristic
189C4—21no—
190C4—22no—
191C4—4461no—
192C4—4462no—
193C4—2232no—
194C4D4×D22312yescomplemented
195C2—22no—
196C2—42no—
197C2—8922no—
198C2—4462no—
199C2—2234no—
200C2C2×D4×D22311yescharacteristic
201C2C2×D4×D22311yescharacteristic
202C2#41×D22311yescharacteristic
subgroupG/H
center, FrattiniC22C23×D223
commutator, socle, cosocleC22×C223C24
Fitting#121×C223C2
hypercenter#121D223
hypocenterC223C2×#121
LCSG ▷ C22×C223 ▷ C223
UCSC1 ◁ C22 ◁ #121

Complements

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

NH ≅ G/Ntype
1. C2×D4×D223C2semidirect
2. C2×D4×D223C2semidirect
3. C22×C4×D223C2semidirect
4. #20297279C2semidirect
5. #20297278C2semidirect
6. #20297277C2semidirect
7. #20297276C2semidirect
8. #20297275C2semidirect
9. C4⋊C4×D223C2semidirect
10. C22⋊C4×D223C2semidirect
11. #121×C223C2semidirect
14. C2×D892C22semidirect
15. C2×D892C22semidirect
16. C2×C22⋊D223C22semidirect
17. C2×C22⋊D223C22semidirect
18. C2×C22⋊D223C22semidirect
21. C22×Dic223C22semidirect
26. C2×C4×D223C22semidirect
27. C2×C4×D223C22semidirect
28. C4⋊Dic223C22semidirect
29. Dic223⋊C4C22semidirect
30. C22⋊Dic223C22semidirect
31. (C2×D223)⋊C4C22semidirect
32. (C2×D223)⋊C4C22semidirect
33. C2×D4×C223C22semidirect
34. C2×D4×C223C22semidirect
35. C22×C4×C223C22semidirect
36. C4⋊C4×C223C22semidirect
37. C22⋊C4×C223C22semidirect
43. C22×D223D4semidirect
60. C4×D223D4semidirect
74. C22×C223C2×D4semidirect
85. C2×D223C2×D4semidirect
89. C4×C223C2×D4semidirect
92. Dic223C2×D4semidirect
100. D223#121direct (N × H)
101. C223C2×#121semidirect
111. #121D223direct (N × H)
123. C2×D4C2×D223semidirect
124. C2×D4C2×D223semidirect
132. C22×C4C2×D223semidirect
135. C4⋊C4C2×D223semidirect
139. C22⋊C4C2×D223semidirect
187. C22D4×D223semidirect
194. C4D4×D223semidirect

Maximal subgroups

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

subgroupindex
C2×D4×D2232
C22×C4×D2232
#202972792
#202972782
#202972772
#202972762
#202972752
C4⋊C4×D2232
C22⋊C4×D2232
#121×C2232
C2×#121223