Finite Group Databasebeta

← #27187413

Subgroups — #27187413

Order 12928.

270 conjugacy classes of subgroups; only the 54 normal subgroups are shown.

normal subgroups only

This group has 270 conjugacy classes of subgroups — too many to draw. Only its 54 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#20266156C212yesmaximal, complemented
2#20266082C214yesmaximal, complemented
3#20265746C211yesmaximal, characteristic
4#15692334—22no—
5#15692314C2214yescomplemented
6#15692313—22no—
7C2×#13108074C2211yescharacteristic
8C22×D404C411yescharacteristic, complemented
9#15692252C2212yes—
10#15692243C412yescomplemented
11#15692230—22no—
12#15692225—22no—
13#117×C101C411yescharacteristic, complemented
14C4⋊#87938—42no—
15C22⋊#87938—24no—
16C4×#87938—42no—
17C23×D101—21no—
18#13108074—42no—
19C2×Dic202C2×C411yescharacteristic
20C2×D404—21no—
21C2×D404—24no—
22C2×D404—42no—
23C2×D404C2×C412yes—
24C2×D404C2311yescharacteristic
25C2×C22⋊D101D412yes—
26C2×C4×D101D412yes—
27#13108051—24no—
28C22×C4×C101C2×C411yescharacteristic
29C4⋊C4×C101—21no—
30C22⋊C4×C101—21no—
31C42×C101C2×C412yes—
32C23×C101C22⋊C411yescharacteristic
33C22×D101—22no—
34C22×D101—42no—
35C22×D101—82no—
36C22×D101C2×D412yes—
37C2×#87938—44no—
38C2×C4×C101—22no—
39C2×C4×C101—22no—
40C2×C4×C101C22⋊C411yescharacteristic
41C2×C4×C101C22×C411yescharacteristic
42C22⋊D101—42no—
43C2×Dic101C22⋊C412yes—
44D404—41no—
45D404—42no—
46D404—42no—
47D404—84no—
48C4×D101—22no—
49C4×D101—22no—
50Dic202—41no—
51C22×C101—21no—
52C22×C101—41no—
53C22×C101C2×C22⋊C411yescharacteristic
54C2×D101—22no—
55C2×D101—42no—
56C2×D101—44no—
57C2×D101—42no—
58C2×D101—84no—
59#87938—84no—
60C4×C101—21no—
61C4×C101—42no—
62C4×C101#9912yes—
63Dic101—22no—
64C2×C101—21no—
65C2×C101—41no—
66C2×C101C2×#9911yescharacteristic
67D101—42no—
68D101—82no—
69#2128—1011nomaximal, Sylow
70C101#212811yescharacteristic, complemented, Sylow
71#535—1011no—
72#410—1012no—
73#353—1014no—
74C2×#41—1011no—
75C22×D4—1011no—
76#127—1012no—
77#124—1012no—
78#121—2022no—
79#120—2022no—
80#117—2022no—
81#117#8793811yescharacteristic, complemented
82#99—1014no—
83#99—2022no—
84C24—2021no—
85#41—4042no—
86C2×Q8—1011no—
87C2×D4—1012no—
88C2×D4—1011no—
89C2×D4—1012no—
90C2×D4—4042no—
91C2×D4—2024no—
92C2×D4—2021no—
93C22×C4—1012no—
94C22×C4C2×#8793811yescharacteristic
95C4⋊C4—21no—
96C4⋊C4—4042no—
97C22⋊C4—21no—
98C22⋊C4—2024no—
99C22⋊C4—2024no—
100C42—4042no—
101C42C2×#8793812yes—
102C23—8082no—
103C23—1012no—
104C23—4042no—
105C23—2022no—
106C23C22⋊#8793811yescharacteristic
107Q8—4041no—
108D4—8084no—
109D4—4042no—
110D4—4041no—
111D4—4042no—
112D4—4042no—
113C2×C4—22no—
114C2×C4—22no—
115C2×C4—1012no—
116C2×C4—4044no—
117C2×C4—2022no—
118C2×C4—2022no—
119C2×C4C22⋊#8793811yescharacteristic
120C2×C4C22×#8793811yescharacteristic
121C22—21no—
122C22—41no—
123C22—8084no—
124C22—4044no—
125C22—4042no—
126C22—4042no—
127C22—2022no—
128C22C2×C22⋊#8793811yescharacteristic
129C4—21no—
130C4—42no—
131C4—8084no—
132C4—2022no—
133C4#1569231412yes—
134C2—21no—
135C2—41no—
136C2—8082no—
137C2—4042no—
138C2C2×#1569231411yescharacteristic
subgroupG/H
centerC2C2×#15692314
commutatorC2×C4×C101C22×C4
FrattiniC2×C4C22×#87938
Fitting#117×C101C4
socleC2×C101C2×#99
prime radical#20265746C2
cosocleC2×D404C23
hypercenter#117#87938
hypocenterC101#2128
LCSG ▷ C2×C4×C101 ▷ C22×C101 ▷ C2×C101 ▷ C101
UCSC1 ◁ C2 ◁ C2×C4 ◁ C22×C4 ◁ #117

Complements

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

NH ≅ G/Ntype
1. #20266156C2semidirect
2. #20266082C2semidirect
5. #15692314C22semidirect
8. C22×D404C4semidirect
10. #15692243C4semidirect
13. #117×C101C4semidirect
70. C101#2128semidirect
81. #117#87938semidirect

Maximal subgroups

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

subgroupindex
#202661562
#202660822
#202657462
#2128101