Finite Group Databasebeta

← #20276689

Subgroups — #20276689

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#16811066C211yesmaximal, characteristic, complemented
3C2×#13133161C211yesmaximal, characteristic, complemented
4#16810899C211yesmaximal, characteristic, complemented
5#16810429C211yesmaximal, characteristic, complemented
6#16810408C211yesmaximal, characteristic, complemented
7C53×#361C211yesmaximal, characteristic, complemented
8#13133262—21no—
9#13133250—21no—
10#13133243—21no—
11#13133241—22no—
12#13133239C2211yescharacteristic
13#13133236—21no—
14C2×Dic212C2211yescharacteristic, complemented
15D8×D53—21no—
16#13133223C2211yescharacteristic
17Dic424—21no—
18#13133162—21no—
19#13133161—22no—
20C2×#11719908C2211yescharacteristic, complemented
21#13133135—21no—
22C2×D8×C53C2211yescharacteristic
23M32×C53C2211yescharacteristic, complemented
24#108×C53C2211yescharacteristic, complemented
25C2×D4×C53—21no—
26C2×C22⋊D53—42no—
27#11719947—22no—
28#11719947—41no—
29D4×D53—22no—
30D4×D53—41no—
31#11719945—21no—
32#11719945D411yescharacteristic
33C2×Dic106—21no—
34D8×C53—21no—
35D8×C53D412yes—
36M16×C53—21no—
37C2×C8×C53C2311yescharacteristic
38#11719923—22no—
39#11719922—22no—
40#11719917D411yescharacteristic
41Dic212—21no—
42Dic212—21no—
43Dic212—41no—
44Dic212D411yescharacteristic
45D424D411yescharacteristic
46#11719913—21no—
47#11719913—41no—
48#11719912—21no—
49C8×D53—21no—
50C16×C53—21no—
51#11719908—21no—
52C23×C53—41no—
53C22×D53—42no—
54D4×C53—22no—
55D4×C53—41no—
56C2×C4×C53C2×D411yescharacteristic
57C22⋊D53—21no—
58C22⋊D53—42no—
59C22⋊D53—42no—
60C22⋊D53—41no—
61C2×Dic53—42no—
62C2×Dic53—41no—
63D212—21no—
64D212D811yescharacteristic
65C4×D53—21no—
66C4×D53—41no—
67Dic106—21no—
68Dic106—41no—
69Dic106—41no—
70Dic106D811yescharacteristic
71C8×C53—41no—
72C8×C53C2×D411yescharacteristic
73C8×C53C2×D411yescharacteristic
74#88322—21no—
75C22×C53—42no—
76C22×C53—161no—
77C22×C53AGL(1,ℤ/8)11yescharacteristic
78C2×D53—21no—
79C2×D53—41no—
80C2×D53—162no—
81C4×C53#12011yescharacteristic
82C4×C53C2×D811yescharacteristic
83Dic53—21no—
84Dic53—41no—
85Dic53—81no—
86#2194—531nomaximal, Sylow
87C2×C53—21no—
88C2×C53—82no—
89C2×C53#44711yescharacteristic
90D53—41no—
91D53—81no—
92#576—531no—
93#510—531no—
94C2×SD32—531no—
95#473—531no—
96#361D5311yescharacteristic, complemented
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×Q16—531no—
107C2×D8—1061no—
108C2×D8C2×D5311yescharacteristic
109#131—531no—
110Q32—1061no—
111SD32—1061no—
112SD32—1062no—
113M32C2×D5311yescharacteristic, complemented
114C2×C16—531no—
115#108C2×D5311yescharacteristic, complemented
116#101—1061no—
117#41—531no—
118#41—1062no—
119#41—1061no—
120#41—2121no—
121C2×Q8—1061no—
122C2×D4—21no—
123C2×D4—1062no—
124C2×D4—2121no—
125C2×D4—2122no—
126Q16—531no—
127Q16—1061no—
128Q16—1061no—
129Q16—2121no—
130SD16—1062no—
131SD16—1061no—
132SD16—2121no—
133D8—21no—
134D8—531no—
135D8—1062no—
136D8D21212yes—
137M16—21no—
138M16—531no—
139M16—1061no—
140C2×C8—1061no—
141C2×C8C22×D5311yescharacteristic
142C16—21no—
143C16—1061no—
144C23—41no—
145C23—2122no—
146Q8—531no—
147Q8—1061no—
148Q8—2121no—
149Q8—2121no—
150D4—22no—
151D4—41no—
152D4—531no—
153D4—1061no—
154D4—1061no—
155D4—2122no—
156D4—2122no—
157D4—2121no—
158C2×C4—1061no—
159C2×C4—2122no—
160C2×C4—2121no—
161C2×C4—2121no—
162C2×C4D4×D5311yescharacteristic
163C8—41no—
164C8—1061no—
165C8C2×D21211yescharacteristic
166C8D4×D5311yescharacteristic
167C22—42no—
168C22—161no—
169C22—8482no—
170C22—1061no—
171C22—2121no—
172C22#1313324411yescharacteristic
173C4—1061no—
174C4—4241no—
175C4—2121no—
176C4#1313318711yescharacteristic
177C4D8×D5311yescharacteristic
178C2—21no—
179C2—82no—
180C2—4241no—
181C2—2121no—
182C2#1681075811yescharacteristic
subgroupG/H
centerC2#16810758
commutatorC2×C8×C53C23
FrattiniC2×C8C22×D53
FittingC53×#361C2
socleC2×C53#447
hypercenter#361D53
hypocenterC53#2194
LCSG ▷ C2×C8×C53 ▷ C4×C53 ▷ C2×C53 ▷ C53
UCSC1 ◁ C2 ◁ C2×C4 ◁ C2×C8 ◁ #361

Complements

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

NH ≅ G/Ntype
2. #16811066C2semidirect
3. C2×#13133161C2semidirect
4. #16810899C2semidirect
5. #16810429C2semidirect
6. #16810408C2semidirect
7. C53×#361C2semidirect
14. C2×Dic212C22semidirect
20. C2×#11719908C22semidirect
23. M32×C53C22semidirect
24. #108×C53C22semidirect
96. #361D53semidirect
99. C53#2194semidirect
113. M32C2×D53semidirect
115. #108C2×D53semidirect