Finite Group Databasebeta

← #1663×C223 #43250678

Subgroups — #1663×C223 #43250678

Order 28544.

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

normal subgroups only

This group has 172 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#564×C223C211yesmaximal, characteristic
2#501×C223C212yesmaximal
3#491×C223C211yesmaximal, characteristic
4#330×C223C212yesmaximal
5#330×C223C211yesmaximal, characteristic
6C4⋊Q8×C223C2211yescharacteristic
7#125×C223—22no—
8#125×C223—22no—
9#125×C223—22no—
10#125×C223C2211yescharacteristic
11#125×C223C2212yes—
12#107×C223—22no—
13#106×C223—22no—
14#103×C223—22no—
15#97×C223C2211yescharacteristic
16#97×C223C2212yes—
17C2×Q8×C223—21no—
18C2×C8×C223—22no—
19C2×C8×C223—21no—
20C4⋊C4×C223—22no—
21C4⋊C4×C223—22no—
22C4⋊C4×C223—22no—
23C4⋊C4×C223—22no—
24C4⋊C4×C223—22no—
25C4⋊C4×C223—22no—
26C4⋊C4×C223—22no—
27C4⋊C4×C223D412yes—
28C4⋊C4×C223D412yes—
29C4⋊C4×C223D412yes—
30C42×C223—22no—
31C42×C223C2311yescharacteristic
32Q8×C223—42no—
33C2×C4×C223—22no—
34C2×C4×C223—22no—
35C2×C4×C223—22no—
36C2×C4×C223—41no—
37C2×C4×C223C2×D412yes—
38C2×C4×C223C2×D411yescharacteristic
39C8×C223—24no—
40C8×C223—41no—
41C22×C223#12011yescharacteristic
42C4×C223—21no—
43C4×C223—22no—
44C4×C223—42no—
45C4×C223—42no—
46C4×C223—42no—
47C4×C223—81no—
48C2×C223#45411yescharacteristic
49C2×C223#45612yes—
50C223#166311yescharacteristic, direct product factor, Sylow
51#1663C22311yesmaximal, characteristic, direct product factor, Sylow
52#564C2×C22311yescharacteristic
53#501C2×C22312yes—
54#491C2×C22311yescharacteristic
55#330C2×C22312yes—
56#330C2×C22311yescharacteristic
57C4⋊Q8C22×C22311yescharacteristic
58#125—22no—
59#125—22no—
60#125—22no—
61#125C22×C22311yescharacteristic
62#125C22×C22312yes—
63#107—22no—
64#106—22no—
65#103—22no—
66#97C22×C22311yescharacteristic
67#97C22×C22312yes—
68C2×Q8—21no—
69C2×C8—22no—
70C2×C8—21no—
71C4⋊C4—22no—
72C4⋊C4—22no—
73C4⋊C4—22no—
74C4⋊C4—22no—
75C4⋊C4—22no—
76C4⋊C4—22no—
77C4⋊C4—22no—
78C4⋊C4D4×C22312yes—
79C4⋊C4D4×C22312yes—
80C4⋊C4D4×C22312yes—
81C42—22no—
82C42C23×C22311yescharacteristic
83Q8—42no—
84C2×C4—22no—
85C2×C4—22no—
86C2×C4—22no—
87C2×C4—41no—
88C2×C4C2×D4×C22312yes—
89C2×C4C2×D4×C22311yescharacteristic
90C8—24no—
91C8—41no—
92C22#120×C22311yescharacteristic
93C4—21no—
94C4—22no—
95C4—42no—
96C4—42no—
97C4—42no—
98C4—81no—
99C2#454×C22311yescharacteristic
100C2#456×C22312yes—
subgroupG/H
center, socle, prime radicalC22×C223#120
commutator, Frattini, cosocleC42C23×C223
LCSG ▷ C42 ▷ C22 ▷ C1
UCSC1 ◁ C22×C223 ◁ C42×C223 ◁ G

Complements

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

NH ≅ G/Ntype
50. C223#1663direct (N × H)
51. #1663C223direct (N × H)

Maximal subgroups

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

subgroupindex
#564×C2232
#501×C2232
#491×C2232
#330×C2232
#1663223