Finite Group Databasebeta

← #20276685

Subgroups — #20276685

Order 6784.

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

normal subgroups only

This group has 216 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
2#16811064C211yesmaximal, characteristic, complemented
3#16810895C211yesmaximal, characteristic, complemented
4#16810436C211yesmaximal, characteristic, complemented
5#16810431C211yesmaximal, characteristic, complemented
6#16810406C211yesmaximal, characteristic, complemented
7C2×SD32×C53C211yesmaximal, characteristic, complemented
8#13133262—21no—
9#13133248—21no—
10#13133243—21no—
11#13133241—22no—
12#13133239C2211yescharacteristic
13#13133236—21no—
14D8×D53—21no—
15#13133223C2211yescharacteristic
16#13133163—21no—
17#13133160—21no—
18#13133157C2211yescharacteristic, complemented
19#13133153C2211yescharacteristic, complemented
20#13133136—21no—
21C2×Q16×C53C2211yescharacteristic, complemented
22C2×D8×C53C2211yescharacteristic
23SD32×C53—22no—
24C2×C16×C53C2211yescharacteristic, complemented
25C2×Q8×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
34Q16×C53—21no—
35Q16×C53—21no—
36D8×C53—21no—
37D8×C53D412yes—
38C2×C8×C53C2311yescharacteristic
39#11719925—41no—
40#11719924—41no—
41#11719923—22no—
42#11719922—22no—
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—
53C22×D53—42no—
54Q8×C53—41no—
55Q8×C53—41no—
56D4×C53—22no—
57D4×C53—41no—
58C2×C4×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—161no—
77C22×C53AGL(1,ℤ/8)11yescharacteristic
78C2×D53—21no—
79C2×D53—41no—
80C2×D53—162no—
81C4×C53—81no—
82C4×C53#12011yescharacteristic
83C4×C53C2×D811yescharacteristic
84Dic53—21no—
85Dic53—41no—
86#2194—531nomaximal, Sylow
87C2×C53—21no—
88C2×C53—82no—
89C2×C53#44711yescharacteristic
90D53—41no—
91D53—81no—
92#576—531no—
93#510—531no—
94C2×SD32D5311yescharacteristic, complemented
95#473—531no—
96#361—531no—
97#359—531no—
98#350—531no—
99C53#219411yescharacteristic, complemented, Sylow
100H(5,𝔽2)—1061no—
101#137—1061no—
102AGL(1,ℤ/8)—1061no—
103AGL(1,ℤ/8)—1062no—
104#135—531no—
105#135—1061no—
106C2×Q16C2×D5311yescharacteristic, complemented
107C2×D8—1061no—
108C2×D8C2×D5311yescharacteristic
109#131—531no—
110Q32—1061no—
111SD32—22no—
112SD32—1061no—
113M32—531no—
114C2×C16C2×D5311yescharacteristic, complemented
115#108—531no—
116#101—1061no—
117#41—531no—
118#41—1062no—
119#41—1061no—
120#41—2121no—
121C2×Q8—21no—
122C2×D4—21no—
123C2×D4—1062no—
124C2×D4—2121no—
125C2×D4—2122no—
126Q16—21no—
127Q16—21no—
128Q16—531no—
129Q16—2121no—
130SD16—1062no—
131SD16—1061no—
132SD16—2121no—
133D8—21no—
134D8—531no—
135D8—1062no—
136D8C22⋊D5312yes—
137M16—531no—
138M16—1061no—
139M16—1061no—
140C2×C8—1061no—
141C2×C8C22×D5311yescharacteristic
142C16—21no—
143C16—1061no—
144C23—41no—
145C23—2122no—
146Q8—41no—
147Q8—41no—
148Q8—531no—
149Q8—1061no—
150D4—22no—
151D4—41no—
152D4—531no—
153D4—1061no—
154D4—1061no—
155D4—2122no—
156D4—2122no—
157D4—2121no—
158C2×C4—41no—
159C2×C4—1061no—
160C2×C4—2122no—
161C2×C4—2121no—
162C2×C4D4×D5311yescharacteristic
163C8—1061no—
164C8—2121no—
165C8D4×D5311yescharacteristic
166C8C2×C22⋊D5311yescharacteristic
167C22—42no—
168C22—161no—
169C22—8482no—
170C22—1061no—
171C22—2121no—
172C22#1313324111yescharacteristic
173C4—81no—
174C4—1061no—
175C4—2121no—
176C4#1313318511yescharacteristic
177C4D8×D5311yescharacteristic
178C2—21no—
179C2—82no—
180C2—4241no—
181C2—2121no—
182C2#1681075511yescharacteristic
subgroupG/H
centerC2#16810755
commutatorC2×C8×C53C23
FrattiniC2×C8C22×D53
FittingC2×SD32×C53C2
socleC2×C53#447
hypercenterC2×SD32D53
hypocenterC53#2194
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
2. #16811064C2semidirect
3. #16810895C2semidirect
4. #16810436C2semidirect
5. #16810431C2semidirect
6. #16810406C2semidirect
7. C2×SD32×C53C2semidirect
18. #13133157C22semidirect
19. #13133153C22semidirect
21. C2×Q16×C53C22semidirect
24. C2×C16×C53C22semidirect
94. C2×SD32D53semidirect
99. C53#2194semidirect
106. C2×Q16C2×D53semidirect
114. C2×C16C2×D53semidirect