Finite Group Databasebeta

← #44549676

Subgroups — #44549676

Order 32128.

196 conjugacy classes of subgroups; only the 49 normal subgroups are shown.

normal subgroups only

This group has 196 conjugacy classes of subgroups — too many to draw. Only its 49 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#30632097C212yesmaximal, complemented
2#30632082C211yesmaximal, characteristic
3#30631429C212yesmaximal, complemented
4#30631427C211yesmaximal, characteristic
5#563×C251C211yesmaximal, characteristic
6C2×#16860800—21no—
7C2×#16860798—21no—
8#22548715C2211yescharacteristic
9#22548648—22no—
10#22548646—22no—
11#22548626C2212yes—
12#22548626C2211yescharacteristic
13C4⋊Q8×C251—21no—
14#126×C251—22no—
15#125×C251C2212yes—
16#124×C251C2211yescharacteristic
17#123×C251—21no—
18C22⋊Q8×C251—22no—
19C2×Dic502—21no—
20#16860800—42no—
21#16860798—42no—
22C2×C251⋊C8—22no—
23C2×C251⋊C8—21no—
24C4⋊Dic251—22no—
25C2×Q8×C251—21no—
26C2×Q8×C251D411yescharacteristic
27C2×D4×C251D411yescharacteristic
28C22×C4×C251—21no—
29C4⋊C4×C251—22no—
30C4⋊C4×C251—22no—
31C4⋊C4×C251—22no—
32C4⋊C4×C251D412yes—
33C4⋊C4×C251D412yes—
34C22⋊C4×C251—22no—
35C22⋊C4×C251—21no—
36C22⋊C4×C251—22no—
37C42×C251—21no—
38C42×C251C2311yescharacteristic
39Dic502—41no—
40Dic502—41no—
41C2×Dic251—41no—
42C251⋊C8—24no—
43C251⋊C8—22no—
44C23×C251—21no—
45Q8×C251—41no—
46Q8×C251—81no—
47D4×C251—41no—
48C2×C4×C251—22no—
49C2×C4×C251—21no—
50C2×C4×C251—22no—
51C2×C4×C251—81no—
52C2×C4×C251C2×D412yes—
53C2×C4×C251C2×D411yescharacteristic
54C22×C251—41no—
55C22×C251—42no—
56C22×C251#12011yescharacteristic
57C4×C251—21no—
58C4×C251—22no—
59C4×C251—42no—
60C4×C251—41no—
61C4×C251—42no—
62Dic251—81no—
63C2×C251—81no—
64C2×C251#45311yescharacteristic
65C2×C251#45612yes—
66C251#166011yescharacteristic, complemented, Sylow
67#1660—2511nomaximal, Sylow
68#563D25111yescharacteristic
69#501—2512no—
70#497—2511no—
71#330—2512no—
72#329—2511no—
73C2×Q16—5021no—
74C2×SD16—5021no—
75C4⋊Q8—21no—
76C4⋊Q8—2511no—
77#126—22no—
78#125C2×D25112yes—
79#124C2×D25111yescharacteristic
80#123—21no—
81C22⋊Q8—22no—
82#107—5022no—
83#106—5022no—
84#97—2512no—
85#97—2511no—
86C2×Q8—21no—
87C2×Q8—5021no—
88C2×Q8C22⋊D25111yescharacteristic
89C2×D4C22⋊D25111yescharacteristic
90C22×C4—21no—
91Q16—10042no—
92SD16—10042no—
93C2×C8—5022no—
94C2×C8—5021no—
95C4⋊C4—22no—
96C4⋊C4—22no—
97C4⋊C4—22no—
98C4⋊C4—5022no—
99C4⋊C4C22⋊D25112yes—
100C4⋊C4C22⋊D25112yes—
101C22⋊C4—22no—
102C22⋊C4—21no—
103C22⋊C4—22no—
104C42—21no—
105C42C22×D25111yescharacteristic
106C23—21no—
107Q8—41no—
108Q8—81no—
109Q8—10041no—
110Q8—10041no—
111D4—41no—
112C2×C4—22no—
113C2×C4—21no—
114C2×C4—22no—
115C2×C4—81no—
116C2×C4—10041no—
117C2×C4C2×C22⋊D25112yes—
118C2×C4C2×C22⋊D25111yescharacteristic
119C8—5022no—
120C8—5024no—
121C22—41no—
122C22—42no—
123C22#2254868411yescharacteristic
124C4—21no—
125C4—22no—
126C4—42no—
127C4—41no—
128C4—42no—
129C4—20081no—
130C2—81no—
131C2#3063185511yescharacteristic
132C2#3063187012yes—
subgroupG/H
centerC22#22548684
commutator, cosocleC42×C251C23
FrattiniC42C22×D251
Fitting#563×C251C2
socleC22×C251#120
prime radicalC2×D4×C251D4
hypercenter#563D251
hypocenterC251#1660
LCSG ▷ C42×C251 ▷ C22×C251 ▷ C251
UCSC1 ◁ C22 ◁ C42 ◁ #563

Complements

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

NH ≅ G/Ntype
1. #30632097C2semidirect
3. #30631429C2semidirect
66. C251#1660semidirect

Maximal subgroups

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

subgroupindex
#306320972
#306320822
#306314292
#306314272
#563×C2512
#1660251