Finite Group Databasebeta

← #20276692

Subgroups — #20276692

Order 6784.

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

normal subgroups only

This group has 206 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
2#16811064C211yesmaximal, characteristic, complemented
3#16810895C211yesmaximal, characteristic, complemented
4#16810439C211yesmaximal, characteristic, complemented
5#16810431C211yesmaximal, characteristic, complemented
6#16810406C211yesmaximal, characteristic, complemented
7C2×Q32×C53C211yesmaximal, characteristic, complemented
8#13133265—21no—
9#13133249—22no—
10#13133248—21no—
11#13133248—21no—
12#13133240—21no—
13#13133239C2211yescharacteristic
14Q16×D53—21no—
15#13133223C2211yescharacteristic
16#13133163—21no—
17#13133160—21no—
18#13133157C2211yescharacteristic
19#13133153C2211yescharacteristic
20#13133136—21no—
21C2×Q16×C53C2211yescharacteristic
22C2×Q16×C53C2211yescharacteristic
23Q32×C53—22no—
24C2×C16×C53C2211yescharacteristic
25C2×Q8×C53—21no—
26C2×Q8×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×C53—21no—
36Q16×C53—21no—
37Q16×C53D412yes—
38C2×C8×C53C2311yescharacteristic
39#11719925—22no—
40#11719925—41no—
41#11719924—22no—
42#11719924—41no—
43#11719917—21no—
44#11719917D411yescharacteristic
45Dic212D411yescharacteristic
46D424D411yescharacteristic
47#11719913—21no—
48#11719912—21no—
49C8×D53—21no—
50C16×C53—21no—
51#11719908—21no—
52Q8×C53—22no—
53Q8×C53—41no—
54Q8×C53—41no—
55Q8×C53—41no—
56C2×C4×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×C53AGL(1,ℤ/8)11yescharacteristic
76C2×D53—21no—
77C2×D53—41no—
78C4×C53—42no—
79C4×C53—81no—
80C4×C53#12011yescharacteristic
81C4×C53C2×D811yescharacteristic
82Dic53—21no—
83Dic53—41no—
84#2195—531nomaximal, Sylow
85C2×C53—21no—
86C2×C53#44711yescharacteristic
87D53—41no—
88D53—81no—
89#578—531no—
90#510—531no—
91C2×Q32D5311yescharacteristic, complemented
92#473—531no—
93#362—531no—
94#359—531no—
95#350—531no—
96C53#219511yescharacteristic, complemented, Sylow
972−1+4—1061no—
98#137—1061no—
99#137—1062no—
100#137—1061no—
101#135—531no—
102#135—1061no—
103C2×Q16—1061no—
104C2×Q16C2×D5311yescharacteristic
105C2×Q16C2×D5311yescharacteristic
106#131—531no—
107Q32—22no—
108Q32—1061no—
109SD32—1061no—
110M32—531no—
111C2×C16C2×D5311yescharacteristic
112#108—531no—
113#101—1061no—
114#41—531no—
115#41—1062no—
116#41—1061no—
117#41—2122no—
118#41—2121no—
119C2×Q8—21no—
120C2×Q8—21no—
121C2×Q8—1062no—
122C2×Q8—2121no—
123Q16—21no—
124Q16—21no—
125Q16—21no—
126Q16—531no—
127Q16—1062no—
128Q16—2121no—
129Q16C22⋊D5312yes—
130SD16—1062no—
131SD16—1061no—
132SD16—2121no—
133D8—531no—
134M16—531no—
135M16—1061no—
136M16—1061no—
137C2×C8—1061no—
138C2×C8C22×D5311yescharacteristic
139C16—21no—
140C16—1061no—
141Q8—22no—
142Q8—41no—
143Q8—41no—
144Q8—41no—
145Q8—531no—
146Q8—1061no—
147Q8—2122no—
148D4—531no—
149D4—1061no—
150D4—1061no—
151D4—2122no—
152D4—2121no—
153C2×C4—41no—
154C2×C4—41no—
155C2×C4—1061no—
156C2×C4—2122no—
157C2×C4—2122no—
158C2×C4—2121no—
159C2×C4D4×D5311yescharacteristic
160C8—1061no—
161C8—2121no—
162C8D4×D5311yescharacteristic
163C8C2×C22⋊D5311yescharacteristic
164C22—1061no—
165C22—2121no—
166C22#1313324111yescharacteristic
167C4—42no—
168C4—81no—
169C4—1061no—
170C4—2121no—
171C4#1313318511yescharacteristic
172C4D8×D5311yescharacteristic
173C2—21no—
174C2—4241no—
175C2—2121no—
176C2#1681075511yescharacteristic
subgroupG/H
centerC2#16810755
commutatorC2×C8×C53C23
FrattiniC2×C8C22×D53
FittingC2×Q32×C53C2
socleC2×C53#447
hypercenterC2×Q32D53
hypocenterC53#2195
LCSG ▷ C2×C8×C53 ▷ C4×C53 ▷ C2×C53 ▷ C53
UCSC1 ◁ C2 ◁ C2×C4 ◁ C2×C8 ◁ C2×Q32

Complements

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

NH ≅ G/Ntype
2. #16811064C2semidirect
3. #16810895C2semidirect
4. #16810439C2semidirect
5. #16810431C2semidirect
6. #16810406C2semidirect
7. C2×Q32×C53C2semidirect
91. C2×Q32D53semidirect
96. C53#2195semidirect