Finite Group Databasebeta

← #123×D223 #28384080

Subgroups — #123×D223 #28384080

Order 14272.

342 conjugacy classes of subgroups; only the 111 normal subgroups are shown.

normal subgroups only

This group has 342 conjugacy classes of subgroups — too many to draw. Only its 111 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
2C22×C4×D223C211yesmaximal, characteristic, complemented
3#20297289C212yesmaximal, complemented
4#20297288C211yesmaximal, characteristic, complemented
5#20297287C211yesmaximal, characteristic, complemented
6#20297286C212yesmaximal, complemented
7#20297285C211yesmaximal, characteristic, complemented
8C4⋊C4×D223C212yesmaximal, complemented
9C22⋊C4×D223C212yesmaximal, complemented
10C22⋊C4×D223C211yesmaximal, characteristic, complemented
11#123×C223C211yesmaximal, characteristic, complemented
12C23×D223C2211yescharacteristic
13C23×D223C2211yescharacteristic
14C2×D892C2211yescharacteristic
15C2×C22⋊D223C2211yescharacteristic
16C2×C22⋊D223C2211yescharacteristic
17D4×D223—24no—
18C22×Dic223C2211yescharacteristic, complemented
19C2×C4×D223—24no—
20C2×C4×D223C2212yes—
21C2×C4×D223C2212yes—
22C2×C4×D223C2211yescharacteristic
23C2×C4×D223C2212yescomplemented
24C4⋊Dic223C2212yescomplemented
25Dic223⋊C4C2212yescomplemented
26Dic223⋊C4C2212yescomplemented
27C22⋊Dic223C2212yescomplemented
28C22⋊Dic223C2211yescharacteristic, complemented
29(C2×D223)⋊C4C2212yescomplemented
30(C2×D223)⋊C4C2212yescomplemented
31(C2×D223)⋊C4C2212yescomplemented
32C2×D4×C223C2211yescharacteristic
33C22×C4×C223C2211yescharacteristic, complemented
34C4⋊C4×C223C2212yescomplemented
35C22⋊C4×C223C2212yescomplemented
36C22⋊C4×C223C2211yescharacteristic, complemented
37C22×D223—24no—
38C22×D223—22no—
39C22×D223—24no—
40C22×D223D414yes—
41C22×D223C2311yescharacteristic
42C22×D223C2311yescharacteristic
43C22×D223C2311yescharacteristic
44D892—22no—
45C22⋊D223—22no—
46C22⋊D223—22no—
47C2×Dic223—22no—
48C2×Dic223C2312yes—
49C2×Dic223C2312yes—
50C2×Dic223C2311yescharacteristic
51C4×D223—24no—
52C4×D223—24no—
53C4×D223—22no—
54C4×D223—24no—
55C23×C223C2311yescharacteristic
56C23×C223C2311yescharacteristic
57D4×C223—22no—
58C2×C4×C223—22no—
59C2×C4×C223C2312yes—
60C2×C4×C223C2312yes—
61C2×C4×C223C2311yescharacteristic
62C22×C223—22no—
63C22×C223—21no—
64C22×C223—22no—
65C22×C223C2×D412yes—
66C22×C223C2411yescharacteristic
67C2×D223—22no—
68C2×D223—28no—
69C2×D223—21no—
70C2×D223—22no—
71C2×D223—44no—
72C2×D223C2×D412yes—
73C2×D223C2×D412yes—
74C2×D223#4114yes—
75C4×C223—22no—
76C4×C223—21no—
77C4×C223—22no—
78Dic223—22no—
79Dic223—21no—
80Dic223—22no—
81C2×C223—22no—
82C2×C223—41no—
83C2×C223C22×D411yescharacteristic
84C2×C223C2×#4112yes—
85D223—22no—
86D223—41no—
87D223#12314yesdirect product factor
88C223C2×#12311yescharacteristic, complemented, Sylow
89C2×#123—2231nomaximal, Sylow
90C22×D4—2231no—
91C23×C4—2231no—
92#123—2231no—
93#123—2231no—
94#123—2232no—
95#123—2232no—
96#123—2231no—
97#123D22311yescharacteristic, direct product factor
98C2×C4⋊C4—2232no—
99C2×C22⋊C4—2231no—
100C2×C22⋊C4—2232no—
101C24—2231no—
102C24—2231no—
103C2×D4—4464no—
104C2×D4—2231no—
105C2×D4—2231no—
106C2×D4—2231no—
107C2×D4C2×D22311yescharacteristic
108C22×C4—4464no—
109C22×C4—2231no—
110C22×C4—2232no—
111C22×C4—2232no—
112C22×C4—2231no—
113C22×C4—2232no—
114C22×C4C2×D22311yescharacteristic, complemented
115C4⋊C4—2232no—
116C4⋊C4—2232no—
117C4⋊C4—2232no—
118C4⋊C4C2×D22312yescomplemented
119C22⋊C4—2231no—
120C22⋊C4—2232no—
121C22⋊C4—2232no—
122C22⋊C4—2232no—
123C22⋊C4—2232no—
124C22⋊C4C2×D22311yescharacteristic, complemented
125C22⋊C4C2×D22312yescomplemented
126C23—4462no—
127C23—4464no—
128C23—4464no—
129C23—2231no—
130C23—2234no—
131C23—2231no—
132C23—2231no—
133C23C22×D22311yescharacteristic
134C23C22×D22311yescharacteristic
135D4—22no—
136D4—4462no—
137D4—4462no—
138D4—4462no—
139C2×C4—22no—
140C2×C4—4464no—
141C2×C4—4464no—
142C2×C4—4462no—
143C2×C4—4464no—
144C2×C4—4462no—
145C2×C4—2232no—
146C2×C4—2232no—
147C2×C4—2231no—
148C2×C4C22×D22312yes—
149C2×C4C22×D22312yes—
150C2×C4C22×D22311yescharacteristic
151C22—22no—
152C22—21no—
153C22—22no—
154C22—8924no—
155C22—4468no—
156C22—4461no—
157C22—4462no—
158C22—4462no—
159C22—2232no—
160C22—2232no—
161C22—2234no—
162C22D4×D22312yes—
163C22C23×D22311yescharacteristic
164C4—22no—
165C4—21no—
166C4—22no—
167C4—4462no—
168C4—4461no—
169C4—4462no—
170C2—22no—
171C2—41no—
172C2—8921no—
173C2—4462no—
174C2—2234no—
175C2C2×D4×D22311yescharacteristic
176C2#41×D22312yes—
subgroupG/H
center, FrattiniC22C23×D223
commutator, socle, cosocleC22×C223C24
Fitting#123×C223C2
prime radicalC2×D4×D223C2
hypercenter#123D223
hypocenterC223C2×#123
LCSG ▷ C22×C223 ▷ C223
UCSC1 ◁ C22 ◁ #123

Complements

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

NH ≅ G/Ntype
2. C22×C4×D223C2semidirect
3. #20297289C2semidirect
4. #20297288C2semidirect
5. #20297287C2semidirect
6. #20297286C2semidirect
7. #20297285C2semidirect
8. C4⋊C4×D223C2semidirect
9. C22⋊C4×D223C2semidirect
10. C22⋊C4×D223C2semidirect
11. #123×C223C2semidirect
18. C22×Dic223C22semidirect
23. C2×C4×D223C22semidirect
24. C4⋊Dic223C22semidirect
25. Dic223⋊C4C22semidirect
26. Dic223⋊C4C22semidirect
27. C22⋊Dic223C22semidirect
28. C22⋊Dic223C22semidirect
29. (C2×D223)⋊C4C22semidirect
30. (C2×D223)⋊C4C22semidirect
31. (C2×D223)⋊C4C22semidirect
33. C22×C4×C223C22semidirect
34. C4⋊C4×C223C22semidirect
35. C22⋊C4×C223C22semidirect
36. C22⋊C4×C223C22semidirect
87. D223#123direct (N × H)
88. C223C2×#123semidirect
97. #123D223direct (N × H)
114. C22×C4C2×D223semidirect
118. C4⋊C4C2×D223semidirect
124. C22⋊C4C2×D223semidirect
125. C22⋊C4C2×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
#202972892
#202972882
#202972872
#202972862
#202972852
C4⋊C4×D2232
C22⋊C4×D2232
#123×C2232
C2×#123223