Finite Group Databasebeta

← #27171679

Subgroups — #27171679

Order 12928.

340 conjugacy classes of subgroups; only the 89 normal subgroups are shown.

normal subgroups only

This group has 340 conjugacy classes of subgroups — too many to draw. Only its 89 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#20265741C211yesmaximal, characteristic
2#20265161C211yesmaximal, characteristic, complemented
3C2×#15692171C212yesmaximal, complemented
4#20264949C212yesmaximal, complemented
5#568×C101C211yesmaximal, characteristic, complemented
6C22×D404C2211yescharacteristic
7C2×#13108058C2212yes—
8#15692251—21no—
9#15692244—21no—
10#15692229—22no—
11#15692223C414yes—
12#15692214C2211yescharacteristic
13#15692174—21no—
14#15692172—21no—
15#15692171—24no—
16C2×#41×C101C2211yescharacteristic
17#131×C101—24no—
18C2×M16×C101—22no—
19C2×M16×C101C2212yes—
20C23×D101C2×C412yes—
21C2×D404—22no—
22C2×D404—21no—
23C2×D404—21no—
24C2×D404—22no—
25C2×C22⋊D101—24no—
26#13108058—22no—
27#13108058—22no—
28C2×#11717579—22no—
29C4⋊Dic101—21no—
30C22⋊Dic101C2×C412yes—
31#13108051—24no—
32C4×Dic101—21no—
33#41×C101—22no—
34#41×C101—22no—
35C2×Q8×C101D411yescharacteristic
36C2×D4×C101C2×C412yes—
37C2×D4×C101D411yescharacteristic
38C22×C4×C101D412yes—
39C22×C4×C101C2311yescharacteristic
40M16×C101—24no—
41M16×C101—22no—
42M16×C101D414yescomplemented
43C2×C8×C101—24no—
44C2×C8×C101—22no—
45C23×C101C22⋊C412yes—
46C23×C101C22×C411yescharacteristic
47C22×D101—22no—
48C22×D101—24no—
49C22×D101—82no—
50Q8×C101—22no—
51D4×C101—24no—
52D4×C101—22no—
53C2×C4×C101—22no—
54C2×C4×C101—22no—
55C2×C4×C101C22⋊C412yes—
56C2×C4×C101C2×D412yes—
57C2×C4×C101C2×D411yescharacteristic
58C2×C4×C101C2×D411yescharacteristic
59C2×C4×C101#4112yes—
60C22⋊D101—84no—
61C2×Dic101—22no—
62D404—41no—
63D404—42no—
64D404—41no—
65C8×C101—24no—
66#11717579—42no—
67C22×C101—22no—
68C22×C101—41no—
69C22×C101—42no—
70C22×C101C2×C22⋊C411yescharacteristic
71C22×C101C4×D412yes—
72C2×D101—44no—
73C2×D101—42no—
74C2×D101—84no—
75C4×C101—22no—
76C4×C101#12011yescharacteristic
77C4×C101#12112yes—
78C4×C101#12311yescharacteristic
79Dic101—42no—
80C2×C101—21no—
81C2×C101—22no—
82C2×C101—42no—
83C2×C101#38611yescharacteristic
84D101—82no—
85#1914—1011nomaximal, Sylow
86C101#191411yescharacteristic, complemented, Sylow
87#568D10111yescharacteristic, complemented
88#535—1011no—
89#413—1011no—
90C2×#100—1012no—
91#344—1012no—
92C2×#41C2×D10111yescharacteristic
93C22×D4—1011no—
94#131—24no—
95C2×M16—22no—
96C2×M16—1012no—
97C2×M16C2×D10112yes—
98#127—2021no—
99#124—2021no—
100#121—2022no—
101#120—1014no—
102#117—1011no—
103#101—2021no—
104#100—2021no—
105#100—2024no—
106C24—1012no—
107#41—22no—
108#41—22no—
109C2×Q8D40411yescharacteristic
110C2×D4—2024no—
111C2×D4—2022no—
112C2×D4—2021no—
113C2×D4—2021no—
114C2×D4—2022no—
115C2×D4C4×D10112yes—
116C2×D4D40411yescharacteristic
117C22×C4C22⋊D10112yes—
118C22×C4C22×D10111yescharacteristic
119M16—24no—
120M16—22no—
121M16—2022no—
122M16—2022no—
123M16C22⋊D10114yescomplemented
124C2×C8—24no—
125C2×C8—22no—
126C2×C8—2022no—
127C4⋊C4—2021no—
128C22⋊C4—1012no—
129C22⋊C4—2024no—
130C42—2021no—
131C23—8082no—
132C23—2024no—
133C23—2022no—
134C23#1310805112yes—
135C23C2×C4×D10111yescharacteristic
136Q8—22no—
137D4—24no—
138D4—22no—
139D4—8084no—
140D4—4041no—
141D4—4042no—
142D4—4041no—
143C2×C4—22no—
144C2×C4—22no—
145C2×C4—2022no—
146C2×C4#1310805112yes—
147C2×C4C2×C22⋊D10112yes—
148C2×C4C2×C22⋊D10111yescharacteristic
149C2×C4C2×D40411yescharacteristic
150C2×C4#1310807412yes—
151C8—24no—
152C8—4042no—
153C22—22no—
154C22—41no—
155C22—42no—
156C22—8084no—
157C22—4044no—
158C22—4042no—
159C22C2×#1310805111yescharacteristic
160C22C4×C22⋊D10112yes—
161C4—22no—
162C4—4042no—
163C4#1569222411yescharacteristic
164C4#1569222912yes—
165C4#1569223811yescharacteristic
166C2—21no—
167C2—22no—
168C2—42no—
169C2—8082no—
170C2#2026505611yescharacteristic
subgroupG/H
centerC2#20265056
commutatorC23×C101C22×C4
FrattiniC22×C4C22×D101
Fitting#568×C101C2
socleC2×C101#386
prime radical#20265741C2
cosocleC22×C4×C101C23
hypercenter#568D101
hypocenterC101#1914
LCSG ▷ C23×C101 ▷ C2×C101 ▷ C101
UCSC1 ◁ C2 ◁ C22×C4 ◁ #568

Complements

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

NH ≅ G/Ntype
2. #20265161C2semidirect
3. C2×#15692171C2semidirect
4. #20264949C2semidirect
5. #568×C101C2semidirect
42. M16×C101D4semidirect
86. C101#1914semidirect
87. #568D101semidirect
123. M16C22⋊D101semidirect

Maximal subgroups

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

subgroupindex
#202657412
#202651612
C2×#156921712
#202649492
#568×C1012
#1914101