Finite Group Databasebeta

← #27173023

Subgroups — #27173023

Order 12928.

142 conjugacy classes of subgroups; only the 53 normal subgroups are shown.

normal subgroups only

This group has 142 conjugacy classes of subgroups — too many to draw. Only its 53 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#20265617C211yesmaximal, characteristic
2#20265285C211yesmaximal, characteristic, complemented
3#20265001C211yesmaximal, characteristic, complemented
4#20264992C211yesmaximal, characteristic, complemented
5#20264987C211yesmaximal, characteristic
6#20264955C211yesmaximal, characteristic, complemented
7#444×C101C211yesmaximal, characteristic
8C2×Dic404C2211yescharacteristic
9C4×Dic202—21no—
10#15692201C412yescomplemented
11#15692199C412yescomplemented
12#15692195C2211yescharacteristic
13#15692195C2211yescharacteristic
14#15692187C2211yescharacteristic
15#15692180—21no—
16#135×C101C2211yescharacteristic
17#131×C101—21no—
18#108×C101C2211yescharacteristic
19#104×C101—21no—
20#97×C101C2211yescharacteristic
21C2×Dic202—21no—
22Dic404—21no—
23Dic404C2×C412yes—
24C4⋊Dic101—21no—
25#13108053—41no—
26C4×Dic101—41no—
27#13108049—21no—
28#13108049C2×C412yes—
29#41×C101—21no—
30Q16×C101C2×C411yescharacteristic
31SD16×C101—21no—
32D8×C101C2×C411yescharacteristic
33M16×C101—21no—
34M16×C101—41no—
35C2×C8×C101—41no—
36C2×C8×C101D411yescharacteristic
37C2×C8×C101C2311yescharacteristic
38C42×C101D411yescharacteristic
39Q8×C101—21no—
40D4×C101—21no—
41D4×C101—41no—
42C2×C4×C101—21no—
43C2×C4×C101—41no—
44C2×C4×C101C2×D411yescharacteristic
45C2×Dic101—41no—
46C2×Dic101—41no—
47Dic202—22no—
48Dic202—41no—
49C8×C101—21no—
50C8×C101—41no—
51C8×C101C22×C411yescharacteristic
52C8×C101#4111yescharacteristic
53C22×C101—41no—
54C22×C101AGL(1,ℤ/8)11yescharacteristic
55C4×C101—41no—
56C4×C101—41no—
57C4×C101C4×D411yescharacteristic
58C4×C101AGL(1,ℤ/8)11yescharacteristic
59Dic101—42no—
60Dic101—81no—
61C2×C101—21no—
62C2×C101—81no—
63C2×C101#44211yescharacteristic
64#2180—1011nomaximal, Sylow
65C101#218011yescharacteristic, complemented, Sylow
66#510—1011no—
67#444D10111yescharacteristic
68#441—1011no—
69#365—1011no—
70#362—1011no—
71#360—1011no—
72#347—1011no—
73#135C2×D10111yescharacteristic
74C2×Q16—1011no—
75#131—21no—
76C4×Q8—2021no—
77Q32—1012no—
78SD32—1012no—
79M32—1011no—
80M32—1011no—
81#108C2×D10111yescharacteristic
82#106—1011no—
83#104—21no—
84#103—2021no—
85#97C2×D10111yescharacteristic
86#41—21no—
87C2×Q8—2021no—
88Q16—1012no—
89Q16—2021no—
90Q16C4×D10111yescharacteristic
91SD16—21no—
92D8C4×D10111yescharacteristic
93M16—21no—
94M16—41no—
95C2×C8—41no—
96C2×C8C22⋊D10111yescharacteristic
97C2×C8C22×D10111yescharacteristic
98C4⋊C4—4041no—
99C4⋊C4—2021no—
100C42—4041no—
101C42C22⋊D10111yescharacteristic
102C16—1012no—
103C16—2021no—
104Q8—21no—
105Q8—4041no—
106Q8—2022no—
107D4—21no—
108D4—41no—
109C2×C4—21no—
110C2×C4—41no—
111C2×C4—4041no—
112C2×C4—4041no—
113C2×C4C2×C22⋊D10111yescharacteristic
114C8—21no—
115C8—41no—
116C8C2×C4×D10111yescharacteristic
117C8#1310807411yescharacteristic
118C22—41no—
119C22#1569228011yescharacteristic
120C4—41no—
121C4—41no—
122C4—8081no—
123C4—4042no—
124C4C4×C22⋊D10111yescharacteristic
125C4#1569228111yescharacteristic
126C2—21no—
127C2—81no—
128C2#2026528711yescharacteristic
subgroupG/H
centerC2#20265287
commutatorC8×C101C22×C4
FrattiniC2×C8C22×D101
Fitting#444×C101C2
socleC2×C101#442
prime radical#135×C101C22
cosocleC2×C8×C101C23
hypercenter#444D101
hypocenterC101#2180
LCSG ▷ C8×C101 ▷ C4×C101 ▷ C2×C101 ▷ C101
UCSC1 ◁ C2 ◁ C2×C4 ◁ #97 ◁ #444

Complements

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

NH ≅ G/Ntype
2. #20265285C2semidirect
3. #20265001C2semidirect
4. #20264992C2semidirect
6. #20264955C2semidirect
10. #15692201C4semidirect
11. #15692199C4semidirect
65. C101#2180semidirect

Maximal subgroups

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

subgroupindex
#202656172
#202652852
#202650012
#202649922
#202649872
#202649552
#444×C1012
#2180101