Finite Group Databasebeta

← #27171680

Subgroups — #27171680

Order 12928.

340 conjugacy classes of subgroups; only the 93 normal subgroups are shown.

normal subgroups only

This group has 340 conjugacy classes of subgroups — too many to draw. Only its 93 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#20265944C211yesmaximal, characteristic, complemented
2#20265161C211yesmaximal, characteristic, complemented
3C2×#15692172C212yesmaximal, complemented
4#20264947C212yesmaximal, complemented
5#535×C101C211yesmaximal, characteristic
6#15692262—24no—
7C2×#13108058—22no—
8C2×#13108058C2212yes—
9C2×#13108058C2212yes—
10#15692174—21no—
11#15692172—24no—
12#15692172—21no—
13C2×#41×C101C2211yescharacteristic
14C22×D4×C101C2211yescharacteristic
15#127×C101—21no—
16#124×C101—21no—
17#121×C101—22no—
18#120×C101C414yes—
19#117×C101C2211yescharacteristic
20#13108058—24no—
21#13108058—22no—
22#13108058—22no—
23#13108058—22no—
24#13108058D414yescomplemented
25C2×#11717579—24no—
26C2×#11717579—22no—
27C2×#11717579—22no—
28C24×C101C2×C412yes—
29#41×C101—22no—
30#41×C101—22no—
31C2×Q8×C101D411yescharacteristic
32C2×D4×C101—24no—
33C2×D4×C101—21no—
34C2×D4×C101—21no—
35C2×D4×C101—22no—
36C2×D4×C101—22no—
37C2×D4×C101C2×C412yes—
38C2×D4×C101D411yescharacteristic
39C22×C4×C101D412yes—
40C22×C4×C101C2311yescharacteristic
41C4⋊C4×C101—21no—
42C22⋊C4×C101—24no—
43C22⋊C4×C101C2×C412yes—
44C42×C101—21no—
45C23×C101—22no—
46C23×C101—24no—
47C23×C101—82no—
48C23×C101C22⋊C412yes—
49C23×C101C22×C411yescharacteristic
50Q8×C101—22no—
51D4×C101—24no—
52D4×C101—22no—
53D4×C101—41no—
54D4×C101—42no—
55D4×C101—41no—
56D4×C101—84no—
57C2×C4×C101—22no—
58C2×C4×C101—22no—
59C2×C4×C101—22no—
60C2×C4×C101C22⋊C412yes—
61C2×C4×C101C2×D412yes—
62C2×C4×C101C2×D411yescharacteristic
63C2×C4×C101C2×D411yescharacteristic
64C2×C4×C101#4112yes—
65#11717579—24no—
66#11717579—42no—
67C22×C101—22no—
68C22×C101—41no—
69C22×C101—42no—
70C22×C101—44no—
71C22×C101—42no—
72C22×C101—84no—
73C22×C101C2×C22⋊C411yescharacteristic
74C22×C101C4×D412yes—
75C4×C101—22no—
76C4×C101—42no—
77C4×C101#12011yescharacteristic
78C4×C101#12112yes—
79C4×C101#12311yescharacteristic
80C2×C101—21no—
81C2×C101—22no—
82C2×C101—42no—
83C2×C101—82no—
84C2×C101#38611yescharacteristic
85#1914—1011nomaximal, Sylow
86C101#191411yescharacteristic, complemented, Sylow
87#568—1011no—
88#535D10111yescharacteristic
89#413—1011no—
90C2×#100—1012no—
91#344—1012no—
92C2×#41C2×D10111yescharacteristic
93C22×D4C2×D10111yescharacteristic
94#131—2024no—
95C2×M16—1012no—
96C2×M16—1012no—
97C2×M16—2022no—
98#127—21no—
99#124—21no—
100#121—22no—
101#120Dic10114yes—
102#117C2×D10111yescharacteristic
103#101—2021no—
104#100—2024no—
105#100—2021no—
106C24C2×Dic10112yes—
107#41—22no—
108#41—22no—
109C2×Q8C22⋊D10111yescharacteristic
110C2×D4—24no—
111C2×D4—22no—
112C2×D4—21no—
113C2×D4—21no—
114C2×D4—22no—
115C2×D4C2×Dic10112yes—
116C2×D4C22⋊D10111yescharacteristic
117C22×C4C22⋊D10112yes—
118C22×C4C22×D10111yescharacteristic
119M16—1014no—
120M16—2022no—
121M16—2022no—
122M16—2024no—
123M16—2022no—
124C2×C8—2022no—
125C2×C8—2024no—
126C2×C8—2022no—
127C4⋊C4—21no—
128C22⋊C4—24no—
129C22⋊C4C2×Dic10112yes—
130C42—21no—
131C23—22no—
132C23—24no—
133C23—82no—
134C23C22⋊Dic10112yes—
135C23C22×Dic10111yescharacteristic
136Q8—22no—
137D4—24no—
138D4—22no—
139D4—41no—
140D4—42no—
141D4—41no—
142D4—84no—
143C2×C4—22no—
144C2×C4—22no—
145C2×C4—22no—
146C2×C4C22⋊Dic10112yes—
147C2×C4D4×D10112yes—
148C2×C4C2×C22⋊D10111yescharacteristic
149C2×C4C2×C22⋊D10111yescharacteristic
150C2×C4#1310807312yes—
151C8—4042no—
152C8—2024no—
153C22—22no—
154C22—41no—
155C22—42no—
156C22—44no—
157C22—42no—
158C22—84no—
159C22C2×C22⋊Dic10111yescharacteristic
160C22D4×Dic10112yes—
161C4—22no—
162C4—42no—
163C4#1569222311yescharacteristic
164C4#1569222712yes—
165C4#1569223611yescharacteristic
166C2—21no—
167C2—22no—
168C2—42no—
169C2—82no—
170C2#2026505711yescharacteristic
subgroupG/H
centerC2#20265057
commutatorC23×C101C22×C4
FrattiniC22×C4C22×D101
Fitting, prime radical#535×C101C2
socleC2×C101#386
cosocleC22×C4×C101C23
hypercenter#535D101
hypocenterC101#1914
LCSG ▷ C23×C101 ▷ C2×C101 ▷ C101
UCSC1 ◁ C2 ◁ C22×C4 ◁ #535

Complements

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

NH ≅ G/Ntype
1. #20265944C2semidirect
2. #20265161C2semidirect
3. C2×#15692172C2semidirect
4. #20264947C2semidirect
24. #13108058D4semidirect
86. C101#1914semidirect

Maximal subgroups

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

subgroupindex
#202659442
#202651612
C2×#156921722
#202649472
#535×C1012
#1914101