Finite Group Databasebeta

← #27170453

Subgroups — #27170453

Order 12928.

196 conjugacy classes of subgroups; only the 45 normal subgroups are shown.

normal subgroups only

This group has 196 conjugacy classes of subgroups — too many to draw. Only its 45 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#20265925C211yesmaximal, characteristic
2#20265539C212yesmaximal, complemented
3#20264919C211yesmaximal, characteristic, complemented
4#20264916C212yesmaximal, complemented
5#491×C101C211yesmaximal, characteristic, complemented
6#15692243C2212yes—
7#15692240—22no—
8#15692233—22no—
9#15692230—22no—
10#15692178—22no—
11#15692175—22no—
12#15692166C2212yes—
13C4⋊Q8×C101C2211yescharacteristic
14#125×C101C2211yescharacteristic
15#103×C101—22no—
16#97×C101C2211yescharacteristic
17C2×Dic202D412yes—
18C2×D404—22no—
19C2×D404D412yes—
20C2×C4×D101—22no—
21C2×#11717579—22no—
22#13108053—22no—
23#13108051—22no—
24#13108051—22no—
25#13108051—22no—
26C2×Q8×C101—21no—
27C2×C8×C101—21no—
28C4⋊C4×C101—22no—
29C4⋊C4×C101—22no—
30C4⋊C4×C101D412yes—
31C42×C101C2311yescharacteristic
32C22×D101—22no—
33Q8×C101—42no—
34C2×C4×C101—22no—
35C2×C4×C101—41no—
36C2×C4×C101C2×D412yes—
37C2×C4×C101C2×D411yescharacteristic
38C2×Dic101—22no—
39D404—42no—
40D404—82no—
41C4×D101—82no—
42Dic202—42no—
43C8×C101—41no—
44#11717579—42no—
45C22×C101#12011yescharacteristic
46C2×D101—42no—
47C2×D101—42no—
48C2×D101—42no—
49C4×C101—21no—
50C4×C101—22no—
51C4×C101—42no—
52C4×C101—81no—
53Dic101—42no—
54C2×C101#45411yescharacteristic
55C2×C101#45512yes—
56D101—82no—
57#1662—1011nomaximal, Sylow
58C101#166211yescharacteristic, complemented, Sylow
59#562—1011no—
60#491D10111yescharacteristic, complemented
61#489—1012no—
62#330—1011no—
63#329—1012no—
64C4⋊Q8C2×D10111yescharacteristic
65#125C2×D10111yescharacteristic
66#124—1012no—
67#123—2022no—
68C22⋊Q8—2022no—
69#121—2022no—
70#103—22no—
71#103—2022no—
72#102—2022no—
73#97—1012no—
74#97C2×D10111yescharacteristic
75C2×Q8—21no—
76C2×Q8—1012no—
77C2×D4—1012no—
78C2×D4—2022no—
79C22×C4—2022no—
80C2×C8—21no—
81C2×C8—2022no—
82C4⋊C4—22no—
83C4⋊C4—22no—
84C4⋊C4—2022no—
85C4⋊C4C22⋊D10112yes—
86C22⋊C4—2022no—
87C22⋊C4—2022no—
88C22⋊C4—2022no—
89C42C22×D10111yescharacteristic
90C23—2022no—
91Q8—42no—
92Q8—4042no—
93D4—8082no—
94D4—4042no—
95C2×C4—22no—
96C2×C4—41no—
97C2×C4—8082no—
98C2×C4—2022no—
99C2×C4D4×D10112yes—
100C2×C4C2×C22⋊D10111yescharacteristic
101C8—41no—
102C8—4042no—
103C22—4042no—
104C22—4042no—
105C22—4042no—
106C22#1569222311yescharacteristic
107C4—21no—
108C4—22no—
109C4—42no—
110C4—81no—
111C4—4042no—
112C2—8082no—
113C2#2026534911yescharacteristic
114C2#2026535312yes—
subgroupG/H
centerC22#15692223
commutator, cosocleC42×C101C23
FrattiniC42C22×D101
Fitting#491×C101C2
socleC22×C101#120
prime radical#20265925C2
hypercenter#491D101
hypocenterC101#1662
LCSG ▷ C42×C101 ▷ C22×C101 ▷ C101
UCSC1 ◁ C22 ◁ C42 ◁ #491

Complements

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

NH ≅ G/Ntype
2. #20265539C2semidirect
3. #20264919C2semidirect
4. #20264916C2semidirect
5. #491×C101C2semidirect
58. C101#1662semidirect
60. #491D101semidirect

Maximal subgroups

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

subgroupindex
#202659252
#202655392
#202649192
#202649162
#491×C1012
#1662101