Finite Group Databasebeta

← #20276671

Subgroups — #20276671

Order 6784.

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

normal subgroups only

This group has 220 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#16811424C211yesmaximal, characteristic, complemented
2#16811059C211yesmaximal, characteristic, complemented
3#16810890C211yesmaximal, characteristic, complemented
4#16810436C211yesmaximal, characteristic, complemented
5#16810431C211yesmaximal, characteristic, complemented
6#16810406C211yesmaximal, characteristic, complemented
7C2×D16×C53C211yesmaximal, characteristic, complemented
8#13133262—21no—
9#13133243—21no—
10#13133243—21no—
11#13133241—22no—
12#13133239C2211yescharacteristic
13#13133236—21no—
14D8×D53—21no—
15#13133223C2211yescharacteristic
16#13133161—21no—
17#13133158—21no—
18#13133157C2211yescharacteristic, complemented
19#13133153C2211yescharacteristic, complemented
20#13133134—21no—
21C2×D8×C53C2211yescharacteristic
22C2×D8×C53C2211yescharacteristic, complemented
23D16×C53—22no—
24C2×C16×C53C2211yescharacteristic, complemented
25C2×D4×C53—21no—
26C2×D4×C53—21no—
27C2×C22⋊D53—42no—
28#11719947—22no—
29#11719947—41no—
30D4×D53—22no—
31D4×D53—41no—
32#11719945—21no—
33#11719945D411yescharacteristic
34D8×C53—21no—
35D8×C53—21no—
36D8×C53—21no—
37D8×C53D412yes—
38C2×C8×C53C2311yescharacteristic
39#11719923—22no—
40#11719923—41no—
41#11719922—22no—
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—
53C23×C53—41no—
54C22×D53—42no—
55D4×C53—22no—
56D4×C53—41no—
57D4×C53—41no—
58D4×C53—41no—
59C2×C4×C53C2×D411yescharacteristic
60C22⋊D53—21no—
61C22⋊D53—42no—
62C22⋊D53—42no—
63C22⋊D53—41no—
64C2×Dic53—42no—
65D212—21no—
66D212D811yescharacteristic
67C4×D53—21no—
68C4×D53—41no—
69Dic106—21no—
70Dic106D811yescharacteristic
71C8×C53C2×D411yescharacteristic
72C8×C53C2×D411yescharacteristic
73#88322—21no—
74#88322—41no—
75C22×C53—42no—
76C22×C53—81no—
77C22×C53—161no—
78C22×C53—161no—
79C22×C53AGL(1,ℤ/8)11yescharacteristic
80C2×D53—21no—
81C2×D53—41no—
82C2×D53—162no—
83C4×C53#12011yescharacteristic
84C4×C53C2×D811yescharacteristic
85Dic53—21no—
86Dic53—41no—
87#2192—531nomaximal, Sylow
88C2×C53—21no—
89C2×C53—82no—
90C2×C53—161no—
91C2×C53#44711yescharacteristic
92D53—41no—
93D53—81no—
94#576—531no—
95#509—531no—
96C2×D16D5311yescharacteristic, complemented
97#472—531no—
98#361—531no—
99#359—531no—
100#350—531no—
101C53#219211yescharacteristic, complemented, Sylow
102H(5,𝔽2)—1061no—
103AGL(1,ℤ/8)—1061no—
104AGL(1,ℤ/8)—1062no—
105AGL(1,ℤ/8)—1061no—
106#135—531no—
107#135—1061no—
108C2×D8—1061no—
109C2×D8C2×D5311yescharacteristic
110C2×D8C2×D5311yescharacteristic, complemented
111#131—531no—
112SD32—1061no—
113D16—22no—
114D16—1061no—
115M32—531no—
116C2×C16C2×D5311yescharacteristic, complemented
117#108—531no—
118#100—1061no—
119#41—531no—
120#41—1062no—
121#41—1061no—
122#41—2121no—
123C2×D4—21no—
124C2×D4—21no—
125C2×D4—1062no—
126C2×D4—2121no—
127C2×D4—2122no—
128Q16—531no—
129SD16—1062no—
130SD16—1061no—
131SD16—2121no—
132D8—21no—
133D8—21no—
134D8—21no—
135D8—531no—
136D8—1062no—
137D8—2121no—
138D8C22⋊D5312yes—
139M16—531no—
140M16—1061no—
141M16—1061no—
142C2×C8—1061no—
143C2×C8C22×D5311yescharacteristic
144C16—21no—
145C16—1061no—
146C23—41no—
147C23—41no—
148C23—2122no—
149Q8—531no—
150Q8—1061no—
151D4—22no—
152D4—41no—
153D4—41no—
154D4—41no—
155D4—531no—
156D4—1061no—
157D4—1061no—
158D4—2122no—
159D4—2122no—
160D4—2121no—
161C2×C4—1061no—
162C2×C4—2122no—
163C2×C4—2121no—
164C2×C4D4×D5311yescharacteristic
165C8—1061no—
166C8—2121no—
167C8D4×D5311yescharacteristic
168C8C2×C22⋊D5311yescharacteristic
169C22—42no—
170C22—81no—
171C22—161no—
172C22—161no—
173C22—8482no—
174C22—1061no—
175C22—2121no—
176C22#1313324111yescharacteristic
177C4—1061no—
178C4—2121no—
179C4#1313318511yescharacteristic
180C4D8×D5311yescharacteristic
181C2—21no—
182C2—82no—
183C2—161no—
184C2—4241no—
185C2—2121no—
186C2#1681075511yescharacteristic
subgroupG/H
centerC2#16810755
commutatorC2×C8×C53C23
FrattiniC2×C8C22×D53
FittingC2×D16×C53C2
socleC2×C53#447
hypercenterC2×D16D53
hypocenterC53#2192
LCSG ▷ C2×C8×C53 ▷ C4×C53 ▷ C2×C53 ▷ C53
UCSC1 ◁ C2 ◁ C2×C4 ◁ C2×C8 ◁ C2×D16

Complements

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

NH ≅ G/Ntype
1. #16811424C2semidirect
2. #16811059C2semidirect
3. #16810890C2semidirect
4. #16810436C2semidirect
5. #16810431C2semidirect
6. #16810406C2semidirect
7. C2×D16×C53C2semidirect
18. #13133157C22semidirect
19. #13133153C22semidirect
22. C2×D8×C53C22semidirect
24. C2×C16×C53C22semidirect
96. C2×D16D53semidirect
101. C53#2192semidirect
110. C2×D8C2×D53semidirect
116. C2×C16C2×D53semidirect