Finite Group Databasebeta

← #27186034

Subgroups — #27186034

Order 12928.

200 conjugacy classes of subgroups; only the 44 normal subgroups are shown.

normal subgroups only

This group has 200 conjugacy classes of subgroups — too many to draw. Only its 44 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#20266149C211yesmaximal, characteristic
2#20266119C211yesmaximal, characteristic
3#20265139C211yesmaximal, characteristic
4C2×#13108083C412yescomplemented
5C2×C4⋊#87938—21no—
6C2×C22⋊#87938C412yes—
7#15692312—21no—
8#15692312—21no—
9(C2×#87938)⋊C4—21no—
10C22×C4×D101C2211yescharacteristic
11C2×#13108057—21no—
12C22⋊#11717579C411yescharacteristic, complemented
13#15692167—21no—
14C22⋊C8×C101C411yescharacteristic, complemented
15C22×#87938—22no—
16#13108083—42no—
17C2×#11717581—22no—
18C4⋊#87938—42no—
19C22⋊#87938—42no—
20C23×D101C2×C411yescharacteristic
21C22×Dic101C2×C411yescharacteristic
22C2×C4×D101—42no—
23C2×C4×D101D411yescharacteristic
24C2×C4×D101D411yescharacteristic
25C2×C4×D101D411yescharacteristic
26C2×C4×D101Q811yescharacteristic
27#13108057—42no—
28C2×#11717579—21no—
29C22×C4×C101C2×C411yescharacteristic
30C2×C8×C101—21no—
31C23×C101C4211yescharacteristic
32C22×D101—21no—
33C22×D101—21no—
34C22×D101—41no—
35C22×D101—41no—
36C22×D101C22⋊C411yescharacteristic
37C22×D101C4⋊C411yescharacteristic
38C2×#87938—42no—
39C2×#87938—42no—
40C2×#87938—42no—
41C2×C4×C101—41no—
42C2×C4×C101C22⋊C411yescharacteristic
43C2×C4×C101C4⋊C411yescharacteristic
44C2×Dic101—22no—
45C2×Dic101C22⋊C411yescharacteristic
46C2×Dic101C4⋊C411yescharacteristic
47C4×D101—22no—
48C4×D101—22no—
49C4×D101—41no—
50C4×D101—41no—
51#11717581—42no—
52C8×C101—41no—
53#11717579—41no—
54C22×C101—21no—
55C22×C101—21no—
56C22×C101—21no—
57C22×C101#9511yescharacteristic
58C2×D101—21no—
59C2×D101—21no—
60C2×D101—21no—
61C2×D101—21no—
62C2×D101—21no—
63C2×D101—42no—
64C2×D101—42no—
65C2×D101#10011yescharacteristic
66C2×D101#10111yescharacteristic
67#87938—82no—
68C4×C101—21no—
69C4×C101—21no—
70Dic101—21no—
71Dic101#9912yes—
72C2×C101—41no—
73C2×C101#34211yescharacteristic
74C2×C101#34311yescharacteristic
75C2×C101#34411yescharacteristic
76D101—21no—
77D101—21no—
78D101—41no—
79#1305—1011nomaximal, Sylow
80C101#130511yescharacteristic, complemented, Sylow
81#407—1011no—
82#407—1011no—
83#380—1011no—
84C23×C4—1011no—
85C2×M16—1012no—
86C2×M16—2021no—
87C2×C4⋊C4—2021no—
88C2×C22⋊C4—1012no—
89C22⋊C8—1011no—
90C22⋊C8—2021no—
91C22⋊C8—2021no—
92C22⋊C8—2021no—
93C22⋊C8#8793811yescharacteristic, complemented
94#95—2021no—
95C24—1011no—
96C22×C4—1011no—
97C22×C4—1011no—
98C22×C4—1011no—
99C22×C4—1011no—
100C22×C4—1011no—
101C22×C4—4042no—
102C22×C4—2022no—
103C22×C4C2×#8793811yescharacteristic
104M16—4042no—
105M16—4042no—
106C2×C8—21no—
107C2×C8—2022no—
108C2×C8—2021no—
109C4⋊C4—4042no—
110C22⋊C4—4042no—
111C23—1011no—
112C23—1011no—
113C23—4041no—
114C23—4041no—
115C23—2021no—
116C23—2021no—
117C23C4×#8793811yescharacteristic
118C2×C4—41no—
119C2×C4—1011no—
120C2×C4—1011no—
121C2×C4—4041no—
122C2×C4—4041no—
123C2×C4—4042no—
124C2×C4—4042no—
125C2×C4—4042no—
126C2×C4—2022no—
127C2×C4—2022no—
128C2×C4—2022no—
129C2×C4C22⋊#8793811yescharacteristic
130C2×C4C4⋊#8793811yescharacteristic
131C8—41no—
132C8—4041no—
133C8—4042no—
134C22—21no—
135C22—21no—
136C22—21no—
137C22—1011no—
138C22—1011no—
139C22—4042no—
140C22—4042no—
141C22—2021no—
142C22—2021no—
143C22—2021no—
144C22—2021no—
145C22—2021no—
146C22(C2×#87938)⋊C411yescharacteristic
147C4—21no—
148C4—21no—
149C4—8082no—
150C4—1012no—
151C4—2021no—
152C2—41no—
153C2—4041no—
154C2—2021no—
155C2—2021no—
156C2#2026605411yescharacteristic
157C2#2026605611yescharacteristic
158C2#2026605911yescharacteristic
subgroupG/H
centerC22(C2×#87938)⋊C4
commutatorC23×C101C42
FrattiniC22×C4C2×#87938
FittingC22⋊C8×C101C4
socleC22×C101#95
prime radicalC23×D101C2×C4
cosocleC22×C4×D101C22
hypercenterC22⋊C8#87938
hypocenterC101#1305
LCSG ▷ C23×C101 ▷ C22×C101 ▷ C101
UCSC1 ◁ C22 ◁ C22×C4 ◁ C22⋊C8

Complements

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

NH ≅ G/Ntype
4. C2×#13108083C4semidirect
12. C22⋊#11717579C4semidirect
14. C22⋊C8×C101C4semidirect
80. C101#1305semidirect
93. C22⋊C8#87938semidirect

Maximal subgroups

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

subgroupindex
#202661492
#202661192
#202651392
#1305101