Finite Group Databasebeta

← #27170443

Subgroups — #27170443

Order 12928.

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#20265587C212yesmaximal, complemented
2#20265572C211yesmaximal, characteristic
3#20264919C212yesmaximal, complemented
4#20264917C211yesmaximal, characteristic
5#563×C101C211yesmaximal, characteristic
6C2×#13108064—21no—
7C2×#13108062—21no—
8#15692255C2211yescharacteristic
9#15692188—22no—
10#15692186—22no—
11#15692166C2211yescharacteristic
12#15692166C2212yes—
13C4⋊Q8×C101—21no—
14#126×C101—22no—
15#125×C101C2212yes—
16#124×C101C2211yescharacteristic
17#123×C101—21no—
18C22⋊Q8×C101—22no—
19C2×Dic202—21no—
20#13108064—42no—
21#13108062—42no—
22C2×#11717579—22no—
23C2×#11717579—21no—
24C4⋊Dic101—22no—
25C2×Q8×C101—21no—
26C2×Q8×C101D411yescharacteristic
27C2×D4×C101D411yescharacteristic
28C22×C4×C101—21no—
29C4⋊C4×C101—22no—
30C4⋊C4×C101—22no—
31C4⋊C4×C101—22no—
32C4⋊C4×C101D412yes—
33C4⋊C4×C101D412yes—
34C22⋊C4×C101—22no—
35C22⋊C4×C101—21no—
36C22⋊C4×C101—22no—
37C42×C101—21no—
38C42×C101C2311yescharacteristic
39C23×C101—21no—
40Q8×C101—41no—
41Q8×C101—81no—
42D4×C101—41no—
43C2×C4×C101—22no—
44C2×C4×C101—21no—
45C2×C4×C101—22no—
46C2×C4×C101—81no—
47C2×C4×C101C2×D412yes—
48C2×C4×C101C2×D411yescharacteristic
49C2×Dic101—41no—
50Dic202—41no—
51Dic202—41no—
52#11717579—24no—
53#11717579—22no—
54C22×C101—41no—
55C22×C101—42no—
56C22×C101#12011yescharacteristic
57C4×C101—21no—
58C4×C101—22no—
59C4×C101—42no—
60C4×C101—41no—
61C4×C101—42no—
62Dic101—81no—
63C2×C101—81no—
64C2×C101#45311yescharacteristic
65C2×C101#45612yes—
66#1660—1011nomaximal, Sylow
67C101#166011yescharacteristic, complemented, Sylow
68#563D10111yescharacteristic
69#501—1012no—
70#497—1011no—
71#330—1012no—
72#329—1011no—
73C2×Q16—2021no—
74C2×SD16—2021no—
75C4⋊Q8—21no—
76C4⋊Q8—1011no—
77#126—22no—
78#125C2×D10112yes—
79#124C2×D10111yescharacteristic
80#123—21no—
81C22⋊Q8—22no—
82#107—2022no—
83#106—2022no—
84#97—1012no—
85#97—1011no—
86C2×Q8—21no—
87C2×Q8—2021no—
88C2×Q8C22⋊D10111yescharacteristic
89C2×D4C22⋊D10111yescharacteristic
90C22×C4—21no—
91Q16—4042no—
92SD16—4042no—
93C2×C8—2022no—
94C2×C8—2021no—
95C4⋊C4—22no—
96C4⋊C4—22no—
97C4⋊C4—22no—
98C4⋊C4—2022no—
99C4⋊C4C22⋊D10112yes—
100C4⋊C4C22⋊D10112yes—
101C22⋊C4—22no—
102C22⋊C4—21no—
103C22⋊C4—22no—
104C42—21no—
105C42C22×D10111yescharacteristic
106C23—21no—
107Q8—41no—
108Q8—81no—
109Q8—4041no—
110Q8—4041no—
111D4—41no—
112C2×C4—22no—
113C2×C4—21no—
114C2×C4—22no—
115C2×C4—81no—
116C2×C4—4041no—
117C2×C4C2×C22⋊D10112yes—
118C2×C4C2×C22⋊D10111yescharacteristic
119C8—2022no—
120C8—2024no—
121C22—41no—
122C22—42no—
123C22#1569222411yescharacteristic
124C4—21no—
125C4—22no—
126C4—42no—
127C4—41no—
128C4—42no—
129C4—8081no—
130C2—81no—
131C2#2026534511yescharacteristic
132C2#2026536012yes—
subgroupG/H
centerC22#15692224
commutator, cosocleC42×C101C23
FrattiniC42C22×D101
Fitting#563×C101C2
socleC22×C101#120
prime radicalC2×D4×C101D4
hypercenter#563D101
hypocenterC101#1660
LCSG ▷ C42×C101 ▷ C22×C101 ▷ C101
UCSC1 ◁ C22 ◁ C42 ◁ #563

Complements

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

NH ≅ G/Ntype
1. #20265587C2semidirect
3. #20264919C2semidirect
67. C101#1660semidirect

Maximal subgroups

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

subgroupindex
#202655872
#202655722
#202649192
#202649172
#563×C1012
#1660101