Finite Group Databasebeta

← #20276678

Subgroups — #20276678

Order 6784.

210 conjugacy classes of subgroups; only the 47 normal subgroups are shown.

normal subgroups only

This group has 210 conjugacy classes of subgroups — too many to draw. Only its 47 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#16811436C211yesmaximal, characteristic, complemented
2#16811059C211yesmaximal, characteristic, complemented
3#16810890C211yesmaximal, characteristic, complemented
4#16810439C211yesmaximal, characteristic, complemented
5#16810431C211yesmaximal, characteristic, complemented
6#16810406C211yesmaximal, characteristic, complemented
7C2×SD32×C53C211yesmaximal, characteristic, complemented
8#13133265—21no—
9#13133249—22no—
10#13133248—21no—
11#13133243—21no—
12#13133240—21no—
13#13133239C2211yescharacteristic
14Q16×D53—21no—
15#13133223C2211yescharacteristic
16#13133161—21no—
17#13133158—21no—
18#13133157C2211yescharacteristic
19#13133153C2211yescharacteristic
20#13133134—21no—
21C2×Q16×C53C2211yescharacteristic
22C2×D8×C53C2211yescharacteristic
23SD32×C53—22no—
24C2×C16×C53C2211yescharacteristic
25C2×Q8×C53—21no—
26C2×D4×C53—21no—
27#11719949—22no—
28#11719949—41no—
29Q8×D53—22no—
30Q8×D53—41no—
31#11719945—21no—
32#11719945—42no—
33#11719945D411yescharacteristic
34Q16×C53—21no—
35Q16×C53D412yes—
36D8×C53—21no—
37D8×C53—21no—
38C2×C8×C53C2311yescharacteristic
39#11719925—22no—
40#11719924—22no—
41#11719923—41no—
42#11719922—41no—
43#11719917—21no—
44#11719917D411yescharacteristic
45Dic212D411yescharacteristic
46D424D411yescharacteristic
47#11719913—21no—
48#11719912—21no—
49C8×D53—21no—
50C16×C53—21no—
51#11719908—21no—
52C23×C53—41no—
53Q8×C53—22no—
54Q8×C53—41no—
55D4×C53—41no—
56D4×C53—41no—
57C2×C4×C53—41no—
58C2×C4×C53C2×D411yescharacteristic
59C22⋊D53—21no—
60C22⋊D53—41no—
61D212—21no—
62D212—42no—
63D212D811yescharacteristic
64C4×D53—21no—
65C4×D53—42no—
66C4×D53—42no—
67C4×D53—41no—
68Dic106—21no—
69Dic106—42no—
70Dic106D811yescharacteristic
71C8×C53C2×D411yescharacteristic
72C8×C53C2×D411yescharacteristic
73#88322—21no—
74#88322—41no—
75C22×C53—81no—
76C22×C53—161no—
77C22×C53AGL(1,ℤ/8)11yescharacteristic
78C2×D53—21no—
79C2×D53—41no—
80C4×C53—42no—
81C4×C53#12011yescharacteristic
82C4×C53C2×D811yescharacteristic
83Dic53—21no—
84Dic53—41no—
85#2193—531nomaximal, Sylow
86C2×C53—21no—
87C2×C53—161no—
88C2×C53#44711yescharacteristic
89D53—41no—
90D53—81no—
91#578—531no—
92#509—531no—
93C2×SD32D5311yescharacteristic, complemented
94#472—531no—
95#362—531no—
96#359—531no—
97#350—531no—
98C53#219311yescharacteristic, complemented, Sylow
992−1+4—1061no—
100#137—1061no—
101#137—1062no—
102AGL(1,ℤ/8)—1061no—
103#135—531no—
104#135—1061no—
105C2×Q16—1061no—
106C2×Q16C2×D5311yescharacteristic
107C2×D8C2×D5311yescharacteristic
108#131—531no—
109SD32—22no—
110SD32—1061no—
111D16—1061no—
112M32—531no—
113C2×C16C2×D5311yescharacteristic
114#108—531no—
115#100—1061no—
116#41—531no—
117#41—1062no—
118#41—1061no—
119#41—2122no—
120#41—2121no—
121C2×Q8—21no—
122C2×Q8—1062no—
123C2×Q8—2121no—
124C2×D4—21no—
125Q16—21no—
126Q16—531no—
127Q16—1062no—
128Q16C22⋊D5312yes—
129SD16—1062no—
130SD16—1061no—
131SD16—2121no—
132D8—21no—
133D8—21no—
134D8—531no—
135D8—2121no—
136M16—531no—
137M16—1061no—
138M16—1061no—
139C2×C8—1061no—
140C2×C8C22×D5311yescharacteristic
141C16—21no—
142C16—1061no—
143C23—41no—
144Q8—22no—
145Q8—41no—
146Q8—531no—
147Q8—1061no—
148Q8—2122no—
149D4—41no—
150D4—41no—
151D4—531no—
152D4—1061no—
153D4—1061no—
154D4—2122no—
155D4—2121no—
156C2×C4—41no—
157C2×C4—1061no—
158C2×C4—2122no—
159C2×C4—2122no—
160C2×C4—2121no—
161C2×C4D4×D5311yescharacteristic
162C8—1061no—
163C8—2121no—
164C8D4×D5311yescharacteristic
165C8C2×C22⋊D5311yescharacteristic
166C22—81no—
167C22—161no—
168C22—1061no—
169C22—2121no—
170C22#1313324111yescharacteristic
171C4—42no—
172C4—1061no—
173C4—2121no—
174C4#1313318511yescharacteristic
175C4D8×D5311yescharacteristic
176C2—21no—
177C2—161no—
178C2—4241no—
179C2—2121no—
180C2#1681075511yescharacteristic
subgroupG/H
centerC2#16810755
commutatorC2×C8×C53C23
FrattiniC2×C8C22×D53
FittingC2×SD32×C53C2
socleC2×C53#447
hypercenterC2×SD32D53
hypocenterC53#2193
LCSG ▷ C2×C8×C53 ▷ C4×C53 ▷ C2×C53 ▷ C53
UCSC1 ◁ C2 ◁ C2×C4 ◁ C2×C8 ◁ C2×SD32

Complements

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

NH ≅ G/Ntype
1. #16811436C2semidirect
2. #16811059C2semidirect
3. #16810890C2semidirect
4. #16810439C2semidirect
5. #16810431C2semidirect
6. #16810406C2semidirect
7. C2×SD32×C53C2semidirect
93. C2×SD32D53semidirect
98. C53#2193semidirect