Finite Group Databasebeta

← #44549663

Subgroups — #44549663

Order 32128.

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

normal subgroups only

This group has 206 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#30632077C211yesmaximal, characteristic
2#30632053C212yesmaximal, complemented
3#30631429C212yesmaximal, complemented
4#30631427C211yesmaximal, characteristic, complemented
5#563×C251C211yesmaximal, characteristic, complemented
6C2×#16860797—21no—
7C2×#16860795—21no—
8#22548712C2211yescharacteristic
9#22548635—22no—
10#22548635—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×D1004—22no—
20C2×D1004—21no—
21#16860797—42no—
22#16860795—42no—
23C2×C251⋊C8—21no—
24C2×C251⋊C8—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
39C22×D251—41no—
40D1004—41no—
41D1004—41no—
42D1004—44no—
43C251⋊C8—22no—
44C251⋊C8—42no—
45C23×C251—21no—
46Q8×C251—41no—
47Q8×C251—81no—
48D4×C251—41no—
49C2×C4×C251—22no—
50C2×C4×C251—21no—
51C2×C4×C251—22no—
52C2×C4×C251—81no—
53C2×C4×C251C2×D412yes—
54C2×C4×C251C2×D411yescharacteristic
55C22×C251—41no—
56C22×C251—42no—
57C22×C251#12011yescharacteristic
58C2×D251—81no—
59C2×D251—82no—
60C4×C251—21no—
61C4×C251—22no—
62C4×C251—42no—
63C4×C251—41no—
64C4×C251—42no—
65C2×C251—81no—
66C2×C251#45311yescharacteristic
67C2×C251#45412yes—
68D251—161no—
69C251#165711yescharacteristic, complemented, Sylow
70#1657—2511nomaximal, Sylow
71#563D25111yescharacteristic, complemented
72#496—2511no—
73#490—2512no—
74#330—2512no—
75#329—2511no—
76C2×SD16—5021no—
77C2×D8—5021no—
78C4⋊Q8—21no—
79#127—2511no—
80#126—22no—
81#125C2×D25112yes—
82#124C2×D25111yescharacteristic
83#123—21no—
84C22⋊Q8—22no—
85#102—5022no—
86#102—5022no—
87#97—2512no—
88#97—2511no—
89C2×Q8—21no—
90C2×Q8C22⋊D25111yescharacteristic
91C2×D4—5022no—
92C2×D4—5021no—
93C2×D4C22⋊D25111yescharacteristic
94C22×C4—21no—
95SD16—10042no—
96D8—10042no—
97C2×C8—5022no—
98C2×C8—5021no—
99C4⋊C4—22no—
100C4⋊C4—22no—
101C4⋊C4—22no—
102C4⋊C4C22⋊D25112yes—
103C4⋊C4C22⋊D25112yes—
104C22⋊C4—22no—
105C22⋊C4—21no—
106C22⋊C4—22no—
107C42—21no—
108C42C22×D25111yescharacteristic
109C23—21no—
110C23—10041no—
111Q8—41no—
112Q8—81no—
113D4—41no—
114D4—10041no—
115D4—10041no—
116D4—10044no—
117C2×C4—22no—
118C2×C4—21no—
119C2×C4—22no—
120C2×C4—81no—
121C2×C4C2×C22⋊D25112yes—
122C2×C4C2×C22⋊D25111yescharacteristic
123C8—10042no—
124C8—5022no—
125C22—41no—
126C22—42no—
127C22—20081no—
128C22—20082no—
129C22#2254868411yescharacteristic
130C4—21no—
131C4—22no—
132C4—42no—
133C4—41no—
134C4—42no—
135C2—81no—
136C2—40161no—
137C2#3063185511yescharacteristic
138C2#3063186012yes—
subgroupG/H
centerC22#22548684
commutator, cosocleC42×C251C23
FrattiniC42C22×D251
Fitting#563×C251C2
socleC22×C251#120
prime radical#30632077C2
hypercenter#563D251
hypocenterC251#1657
LCSG ▷ C42×C251 ▷ C22×C251 ▷ C251
UCSC1 ◁ C22 ◁ C42 ◁ #563

Complements

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

NH ≅ G/Ntype
2. #30632053C2semidirect
3. #30631429C2semidirect
4. #30631427C2semidirect
5. #563×C251C2semidirect
69. C251#1657semidirect
71. #563D251semidirect

Maximal subgroups

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

subgroupindex
#306320772
#306320532
#306314292
#306314272
#563×C2512
#1657251