Finite Group Databasebeta

← #1660×C223 #43250675

Subgroups — #1660×C223 #43250675

Order 28544.

196 conjugacy classes of subgroups; only the 60 normal subgroups are shown.

normal subgroups only

This group has 196 conjugacy classes of subgroups — too many to draw. Only its 60 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
1#563×C223C211yesmaximal, characteristic
2#501×C223C212yesmaximal, complemented
3#497×C223C211yesmaximal, characteristic
4#330×C223C212yesmaximal, complemented
5#329×C223C211yesmaximal, characteristic
6C2×Q16×C223—21no—
7C2×SD16×C223—21no—
8C4⋊Q8×C223—21no—
9C4⋊Q8×C223C2211yescharacteristic
10#126×C223—22no—
11#125×C223C2212yes—
12#124×C223C2211yescharacteristic
13#123×C223—21no—
14C22⋊Q8×C223—22no—
15#107×C223—22no—
16#106×C223—22no—
17#97×C223C2212yes—
18#97×C223C2211yescharacteristic
19C2×Q8×C223—21no—
20C2×Q8×C223—21no—
21C2×Q8×C223D411yescharacteristic
22C2×D4×C223D411yescharacteristic
23C22×C4×C223—21no—
24Q16×C223—42no—
25SD16×C223—42no—
26C2×C8×C223—22no—
27C2×C8×C223—21no—
28C4⋊C4×C223—22no—
29C4⋊C4×C223—22no—
30C4⋊C4×C223—22no—
31C4⋊C4×C223—22no—
32C4⋊C4×C223D412yes—
33C4⋊C4×C223D412yes—
34C22⋊C4×C223—22no—
35C22⋊C4×C223—21no—
36C22⋊C4×C223—22no—
37C42×C223—21no—
38C42×C223C2311yescharacteristic
39C23×C223—21no—
40Q8×C223—41no—
41Q8×C223—41no—
42Q8×C223—41no—
43Q8×C223—81no—
44D4×C223—41no—
45C2×C4×C223—22no—
46C2×C4×C223—21no—
47C2×C4×C223—22no—
48C2×C4×C223—41no—
49C2×C4×C223—81no—
50C2×C4×C223C2×D412yes—
51C2×C4×C223C2×D411yescharacteristic
52C8×C223—24no—
53C8×C223—22no—
54C22×C223—41no—
55C22×C223—42no—
56C22×C223#12011yescharacteristic
57C4×C223—21no—
58C4×C223—22no—
59C4×C223—42no—
60C4×C223—41no—
61C4×C223—42no—
62C4×C223—81no—
63C2×C223—81no—
64C2×C223#45311yescharacteristic
65C2×C223#45612yes—
66C223#166011yescharacteristic, direct product factor, Sylow
67#1660C22311yesmaximal, characteristic, direct product factor, Sylow
68#563C2×C22311yescharacteristic
69#501C2×C22312yescomplemented
70#497C2×C22311yescharacteristic
71#330C2×C22312yescomplemented
72#329C2×C22311yescharacteristic
73C2×Q16—21no—
74C2×SD16—21no—
75C4⋊Q8—21no—
76C4⋊Q8C22×C22311yescharacteristic
77#126—22no—
78#125C22×C22312yes—
79#124C22×C22311yescharacteristic
80#123—21no—
81C22⋊Q8—22no—
82#107—22no—
83#106—22no—
84#97C22×C22312yes—
85#97C22×C22311yescharacteristic
86C2×Q8—21no—
87C2×Q8—21no—
88C2×Q8D4×C22311yescharacteristic
89C2×D4D4×C22311yescharacteristic
90C22×C4—21no—
91Q16—42no—
92SD16—42no—
93C2×C8—22no—
94C2×C8—21no—
95C4⋊C4—22no—
96C4⋊C4—22no—
97C4⋊C4—22no—
98C4⋊C4—22no—
99C4⋊C4D4×C22312yes—
100C4⋊C4D4×C22312yes—
101C22⋊C4—22no—
102C22⋊C4—21no—
103C22⋊C4—22no—
104C42—21no—
105C42C23×C22311yescharacteristic
106C23—21no—
107Q8—41no—
108Q8—41no—
109Q8—41no—
110Q8—81no—
111D4—41no—
112C2×C4—22no—
113C2×C4—21no—
114C2×C4—22no—
115C2×C4—41no—
116C2×C4—81no—
117C2×C4C2×D4×C22312yes—
118C2×C4C2×D4×C22311yescharacteristic
119C8—22no—
120C8—24no—
121C22—41no—
122C22—42no—
123C22#120×C22311yescharacteristic
124C4—21no—
125C4—22no—
126C4—42no—
127C4—41no—
128C4—42no—
129C4—81no—
130C2—81no—
131C2#453×C22311yescharacteristic
132C2#456×C22312yes—
subgroupG/H
center, socleC22×C223#120
commutator, Frattini, cosocleC42C23×C223
prime radicalC2×D4×C223D4
LCSG ▷ C42 ▷ C22 ▷ C1
UCSC1 ◁ C22×C223 ◁ C42×C223 ◁ G

Complements

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

NH ≅ G/Ntype
2. #501×C223C2semidirect
4. #330×C223C2semidirect
66. C223#1660direct (N × H)
67. #1660C223direct (N × H)
69. #501C2×C223semidirect
71. #330C2×C223semidirect

Maximal subgroups

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

subgroupindex
#563×C2232
#501×C2232
#497×C2232
#330×C2232
#329×C2232
#1660223