Finite Group Databasebeta

← #28384145

Subgroups — #28384145

Order 14272.

334 conjugacy classes of subgroups; only the 103 normal subgroups are shown.

normal subgroups only

This group has 334 conjugacy classes of subgroups — too many to draw. Only its 103 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×#16813169C211yesmaximal, characteristic, complemented
2C22×C22⋊D223C211yesmaximal, characteristic, complemented
3#20297300C212yesmaximal, complemented
4#20297290C212yesmaximal, complemented
5#20297277C212yesmaximal, complemented
6#20297276C212yesmaximal, complemented
7#20297274C211yesmaximal, characteristic, complemented
8#20297273C211yesmaximal, characteristic, complemented
9#20297272C211yesmaximal, characteristic, complemented
10#20297260C211yesmaximal, characteristic, complemented
11#120×C223C211yesmaximal, characteristic, complemented
12C23×D223C2211yescharacteristic
13#16813169—24no—
14C2×Dic446C2211yescharacteristic, complemented
15C2×D892C2212yescomplemented
16C2×C22⋊D223—24no—
17C2×C22⋊D223C2211yescharacteristic
18C2×C22⋊D223C2212yes—
19C2×C22⋊D223C2212yes—
20C2×C22⋊D223C2211yescharacteristic, complemented
21C2×C22⋊D223C2212yescomplemented
22C2×C22⋊D223C2212yescomplemented
23C22×Dic223C2211yescharacteristic
24C22×Dic223C2211yescharacteristic, complemented
25C2×C4×D223C2211yescharacteristic, complemented
26Dic223⋊C4C2212yescomplemented
27C22⋊Dic223C2211yescharacteristic
28C22⋊Dic223C2212yescomplemented
29(C2×D223)⋊C4C2212yescomplemented
30(C2×D223)⋊C4C2212yescomplemented
31C4×Dic223C2212yescomplemented
32C24×C223C2211yescharacteristic
33C2×D4×C223C2211yescharacteristic
34C2×D4×C223C2212yescomplemented
35C22⋊C4×C223C2211yescharacteristic
36C22⋊C4×C223C2212yescomplemented
37C22×D223—22no—
38C22×D223—42no—
39C22×D223C2312yes—
40C22×D223C2311yescharacteristic
41Dic446—22no—
42D892—42no—
43C22⋊D223—22no—
44C22⋊D223—24no—
45C22⋊D223—24no—
46C22⋊D223—22no—
47C22⋊D223—42no—
48C22⋊D223—42no—
49C2×Dic223—22no—
50C2×Dic223D414yescomplemented
51C2×Dic223C2312yes—
52C2×Dic223C2311yescharacteristic
53C2×Dic223C2311yescharacteristic
54C2×Dic223C2311yescharacteristic
55C4×D223—22no—
56C23×C223—22no—
57C23×C223—42no—
58C23×C223C2312yes—
59C23×C223C2311yescharacteristic
60C23×C223C2311yescharacteristic
61D4×C223—22no—
62D4×C223—42no—
63C2×C4×C223C2312yes—
64C2×C4×C223C2311yescharacteristic
65C22×C223—22no—
66C22×C223—21no—
67C22×C223—22no—
68C22×C223—22no—
69C22×C223—22no—
70C22×C223—22no—
71C22×C223—44no—
72C22×C223C2×D412yes—
73C22×C223C2411yescharacteristic
74C2×D223—21no—
75C2×D223—22no—
76C2×D223—22no—
77C2×D223—22no—
78C2×D223—22no—
79C2×D223—44no—
80C4×C223—21no—
81C4×C223—22no—
82Dic223—21no—
83Dic223—22no—
84Dic223C2×D412yes—
85Dic223C2×D412yes—
86C2×C223—22no—
87C2×C223—41no—
88C2×C223—42no—
89C2×C223C22×D411yescharacteristic
90C2×C223H(5,𝔽2)12yes—
91D223—41no—
92D223—42no—
93C223#53511yescharacteristic, complemented, Sylow
94#535—2231nomaximal, Sylow
95C2×#41—2231no—
96C22×D4—2231no—
97#127—2232no—
98#124—2232no—
99#121—2232no—
100#121—2232no—
101#120—2231no—
102#120—2231no—
103#120—2231no—
104#120D22311yescharacteristic, complemented
105#117—2231no—
106C24—2231no—
107C24C2×D22311yescharacteristic
108#41—4464no—
109C2×Q8—2231no—
110C2×D4—4464no—
111C2×D4—2232no—
112C2×D4—2232no—
113C2×D4—2231no—
114C2×D4—2232no—
115C2×D4—2232no—
116C2×D4—2231no—
117C2×D4—2232no—
118C2×D4C2×D22311yescharacteristic
119C2×D4C2×D22312yescomplemented
120C22×C4—2231no—
121C22×C4—2231no—
122C22×C4—2231no—
123C4⋊C4—2232no—
124C22⋊C4—2231no—
125C22⋊C4—2232no—
126C22⋊C4—2232no—
127C22⋊C4—2232no—
128C22⋊C4C2×D22311yescharacteristic
129C22⋊C4C2×D22312yescomplemented
130C42—2232no—
131C23—22no—
132C23—42no—
133C23—8922no—
134C23—4462no—
135C23—2232no—
136C23—2231no—
137C23C22×D22312yes—
138C23C22×D22311yescharacteristic
139C23C22×D22311yescharacteristic
140Q8—4462no—
141D4—22no—
142D4—42no—
143D4—8922no—
144D4—8922no—
145D4—8922no—
146D4—4462no—
147D4—4464no—
148D4—4464no—
149D4—4462no—
150C2×C4—4462no—
151C2×C4—4462no—
152C2×C4—2232no—
153C2×C4—2231no—
154C2×C4—2234no—
155C2×C4—2231no—
156C2×C4—2231no—
157C2×C4C22×D22312yes—
158C2×C4C22×D22311yescharacteristic
159C22—22no—
160C22—21no—
161C22—22no—
162C22—22no—
163C22—22no—
164C22—22no—
165C22—44no—
166C22—8924no—
167C22—4461no—
168C22—4462no—
169C22—4462no—
170C22—4462no—
171C22—4462no—
172C22D4×D22312yes—
173C22C23×D22311yescharacteristic
174C4—21no—
175C4—22no—
176C4—4461no—
177C4—4462no—
178C4—2232no—
179C4—2232no—
180C2—22no—
181C2—42no—
182C2—41no—
183C2—8922no—
184C2—8921no—
185C2C2×D4×D22311yescharacteristic
186C2#2029734912yes—
subgroupG/H
center, FrattiniC22C23×D223
commutator, socle, cosocleC22×C223C24
Fitting#120×C223C2
hypercenter#120D223
hypocenterC223#535
LCSG ▷ C22×C223 ▷ C223
UCSC1 ◁ C22 ◁ #120

Complements

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

NH ≅ G/Ntype
1. C2×#16813169C2semidirect
2. C22×C22⋊D223C2semidirect
3. #20297300C2semidirect
4. #20297290C2semidirect
5. #20297277C2semidirect
6. #20297276C2semidirect
7. #20297274C2semidirect
8. #20297273C2semidirect
9. #20297272C2semidirect
10. #20297260C2semidirect
11. #120×C223C2semidirect
14. C2×Dic446C22semidirect
15. C2×D892C22semidirect
20. C2×C22⋊D223C22semidirect
21. C2×C22⋊D223C22semidirect
22. C2×C22⋊D223C22semidirect
24. C22×Dic223C22semidirect
25. C2×C4×D223C22semidirect
26. Dic223⋊C4C22semidirect
28. C22⋊Dic223C22semidirect
29. (C2×D223)⋊C4C22semidirect
30. (C2×D223)⋊C4C22semidirect
31. C4×Dic223C22semidirect
34. C2×D4×C223C22semidirect
36. C22⋊C4×C223C22semidirect
50. C2×Dic223D4semidirect
93. C223#535semidirect
104. #120D223semidirect
119. C2×D4C2×D223semidirect
129. C22⋊C4C2×D223semidirect

Maximal subgroups

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

subgroupindex
C2×#168131692
C22×C22⋊D2232
#202973002
#202972902
#202972772
#202972762
#202972742
#202972732
#202972722
#202972602
#120×C2232
#535223