Finite Group Databasebeta

← #28384140

Subgroups — #28384140

Order 14272.

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

normal subgroups only

This group has 346 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
1C22×C22⋊D223C211yesmaximal, characteristic, complemented
2C2×D4×D223C211yesmaximal, characteristic, complemented
3#20297288C211yesmaximal, characteristic, complemented
4#20297286C212yesmaximal, complemented
5#20297285C211yesmaximal, characteristic, complemented
6#20297277C212yesmaximal, complemented
7#20297275C212yesmaximal, complemented
8#20297273C211yesmaximal, characteristic, complemented
9#20297272C212yesmaximal, complemented
10C22⋊C4×D223C211yesmaximal, characteristic, complemented
11#120×C223C211yesmaximal, characteristic, complemented
12C23×D223C2211yescharacteristic
13C23×D223C2211yescharacteristic, complemented
14C2×D892C2211yescharacteristic, complemented
15C2×C22⋊D223—24no—
16C2×C22⋊D223C2212yes—
17C2×C22⋊D223C2211yescharacteristic
18C2×C22⋊D223C2211yescharacteristic, complemented
19C2×C22⋊D223C2212yescomplemented
20C2×C22⋊D223C2212yescomplemented
21D4×D223—24no—
22C22×Dic223C2211yescharacteristic, complemented
23C2×C4×D223C2211yescharacteristic, complemented
24C2×C4×D223C2212yescomplemented
25C4⋊Dic223C2212yescomplemented
26Dic223⋊C4C2212yescomplemented
27C22⋊Dic223C2211yescharacteristic, complemented
28C22⋊Dic223C2212yescomplemented
29C22⋊Dic223C2212yescomplemented
30(C2×D223)⋊C4C2212yescomplemented
31(C2×D223)⋊C4C2212yescomplemented
32C24×C223C2211yescharacteristic, complemented
33C2×D4×C223C2211yescharacteristic
34C2×D4×C223C2212yescomplemented
35C22⋊C4×C223C2211yescharacteristic, complemented
36C22⋊C4×C223C2212yescomplemented
37C22×D223—22no—
38C22×D223—42no—
39C22×D223—42no—
40C22×D223D414yescomplemented
41C22×D223C2311yescharacteristic
42C22×D223C2311yescharacteristic
43C22×D223C2311yescharacteristic
44D892—22no—
45C22⋊D223—22no—
46C22⋊D223—22no—
47C22⋊D223—24no—
48C22⋊D223—24no—
49C22⋊D223—42no—
50C2×Dic223—22no—
51C2×Dic223C2312yes—
52C2×Dic223C2312yes—
53C2×Dic223C2311yescharacteristic
54C4×D223—22no—
55C4×D223—42no—
56C23×C223—22no—
57C23×C223—42no—
58C23×C223C2312yes—
59C23×C223C2311yescharacteristic
60C23×C223C2311yescharacteristic
61D4×C223—22no—
62D4×C223—42no—
63C2×C4×C223C2311yescharacteristic
64C2×C4×C223C2312yes—
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—22no—
75C2×D223—22no—
76C2×D223—21no—
77C2×D223—22no—
78C2×D223—44no—
79C2×D223—44no—
80C2×D223C2×D412yes—
81C2×D223C2×D412yes—
82C4×C223—21no—
83C4×C223—22no—
84Dic223—21no—
85Dic223—22no—
86Dic223—22no—
87C2×C223—22no—
88C2×C223—41no—
89C2×C223—42no—
90C2×C223C22×D411yescharacteristic
91C2×C223H(5,𝔽2)12yes—
92D223—22no—
93D223—22no—
94D223—41no—
95C223#53411yescharacteristic, complemented, Sylow
96#534—2231nomaximal, Sylow
97C22×D4—2231no—
98C22×D4—2231no—
99#123—2231no—
100#123—2232no—
101#123—2231no—
102#121—2232no—
103#121—2232no—
104#120—2231no—
105#120—2232no—
106#120D22311yescharacteristic, complemented
107C2×C22⋊C4—2231no—
108C24—2231no—
109C24—2231no—
110C24C2×D22311yescharacteristic, complemented
111C2×D4—4464no—
112C2×D4—4464no—
113C2×D4—2232no—
114C2×D4—2232no—
115C2×D4—2231no—
116C2×D4—2231no—
117C2×D4—2232no—
118C2×D4—2231no—
119C2×D4C2×D22311yescharacteristic
120C2×D4C2×D22312yescomplemented
121C22×C4—2232no—
122C22×C4—2231no—
123C22×C4—2231no—
124C4⋊C4—2232no—
125C4⋊C4—2232no—
126C22⋊C4—2231no—
127C22⋊C4—2232no—
128C22⋊C4—2232no—
129C22⋊C4—2232no—
130C22⋊C4—2232no—
131C22⋊C4C2×D22311yescharacteristic, complemented
132C22⋊C4C2×D22312yescomplemented
133C23—22no—
134C23—42no—
135C23—8922no—
136C23—8922no—
137C23—4462no—
138C23—2234no—
139C23—2231no—
140C23—2231no—
141C23—2231no—
142C23C22×D22311yescharacteristic
143C23C22×D22312yes—
144C23C22×D22311yescharacteristic
145D4—22no—
146D4—42no—
147D4—8922no—
148D4—4462no—
149D4—4462no—
150D4—4464no—
151D4—4464no—
152D4—4462no—
153C2×C4—8922no—
154C2×C4—4462no—
155C2×C4—4462no—
156C2×C4—2232no—
157C2×C4—2231no—
158C2×C4—2232no—
159C2×C4C22×D22312yes—
160C2×C4C22×D22311yescharacteristic
161C22—22no—
162C22—21no—
163C22—22no—
164C22—22no—
165C22—22no—
166C22—22no—
167C22—44no—
168C22—8924no—
169C22—8924no—
170C22—4461no—
171C22—4462no—
172C22—4462no—
173C22—4462no—
174C22—2232no—
175C22—2232no—
176C22D4×D22312yes—
177C22C23×D22311yescharacteristic
178C4—21no—
179C4—22no—
180C4—4461no—
181C4—4462no—
182C4—4462no—
183C2—22no—
184C2—42no—
185C2—41no—
186C2—8921no—
187C2—4462no—
188C2—4462no—
189C2C2×D4×D22311yescharacteristic
190C2#2029734912yes—
subgroupG/H
center, FrattiniC22C23×D223
commutator, socle, cosocleC22×C223C24
Fitting#120×C223C2
hypercenter#120D223
hypocenterC223#534
LCSG ▷ C22×C223 ▷ C223
UCSC1 ◁ C22 ◁ #120

Complements

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

NH ≅ G/Ntype
1. C22×C22⋊D223C2semidirect
2. C2×D4×D223C2semidirect
3. #20297288C2semidirect
4. #20297286C2semidirect
5. #20297285C2semidirect
6. #20297277C2semidirect
7. #20297275C2semidirect
8. #20297273C2semidirect
9. #20297272C2semidirect
10. C22⋊C4×D223C2semidirect
11. #120×C223C2semidirect
13. C23×D223C22semidirect
14. C2×D892C22semidirect
18. C2×C22⋊D223C22semidirect
19. C2×C22⋊D223C22semidirect
20. C2×C22⋊D223C22semidirect
22. C22×Dic223C22semidirect
23. C2×C4×D223C22semidirect
24. C2×C4×D223C22semidirect
25. C4⋊Dic223C22semidirect
26. Dic223⋊C4C22semidirect
27. C22⋊Dic223C22semidirect
28. C22⋊Dic223C22semidirect
29. C22⋊Dic223C22semidirect
30. (C2×D223)⋊C4C22semidirect
31. (C2×D223)⋊C4C22semidirect
32. C24×C223C22semidirect
34. C2×D4×C223C22semidirect
35. C22⋊C4×C223C22semidirect
36. C22⋊C4×C223C22semidirect
40. C22×D223D4semidirect
95. C223#534semidirect
106. #120D223semidirect
110. C24C2×D223semidirect
120. C2×D4C2×D223semidirect
131. C22⋊C4C2×D223semidirect
132. C22⋊C4C2×D223semidirect

Maximal subgroups

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

subgroupindex
C22×C22⋊D2232
C2×D4×D2232
#202972882
#202972862
#202972852
#202972772
#202972752
#202972732
#202972722
C22⋊C4×D2232
#120×C2232
#534223