Finite Group Databasebeta

← #20273901

Subgroups — #20273901

Order 6784.

212 conjugacy classes of subgroups; only the 57 normal subgroups are shown.

normal subgroups only

This group has 212 conjugacy classes of subgroups — too many to draw. Only its 57 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#16810969C211yesmaximal, characteristic
2#16810908C211yesmaximal, characteristic, complemented
3#16810833C211yesmaximal, characteristic, complemented
4#16810367C211yesmaximal, characteristic
5Q8⋊#88322C211yesmaximal, characteristic, complemented
6Q8⋊#88322C211yesmaximal, characteristic, complemented
7C53×#559C211yesmaximal, characteristic
8C2×#11719925—21no—
9#13133217C2211yescharacteristic
10#13133148—21no—
11C4⋊#88322C2211yescharacteristic
12C4⋊#88322C2211yescharacteristic
13#13133141—21no—
14#13133140—21no—
15#13133138—22no—
16C4×#88322C2211yescharacteristic
17#127×C53C2211yescharacteristic
18#124×C53—21no—
19#124×C53—21no—
20#123×C53—22no—
21C4×Q8×C53C2211yescharacteristic
22C4×Q8×C53C2211yescharacteristic
23C4×D4×C53—21no—
24C4×D4×C53—21no—
25C2×Q8×C53D411yescharacteristic
26C2×Q8×C53D411yescharacteristic
27C2×D4×C53—21no—
28C2×D4×C53—21no—
29C2×D4×C53D412yes—
30C22×C4×C53—22no—
31C2×Dic106—21no—
32C4⋊C4×C53—21no—
33C4⋊C4×C53—21no—
34C4⋊C4×C53D411yescharacteristic
35C4⋊C4×C53D411yescharacteristic
36C22⋊C4×C53—22no—
37C22⋊C4×C53—22no—
38C22⋊C4×C53—22no—
39C42×C53—21no—
40C42×C53—21no—
41C42×C53C2311yescharacteristic
42#11719925—42no—
43C4⋊Dic53—21no—
44C4⋊Dic53—21no—
45C2×#88322—21no—
46C2×#88322—21no—
47C2×#88322—21no—
48C23×C53—22no—
49Q8×C53—21no—
50Q8×C53—21no—
51Q8×C53—21no—
52Q8×C53—21no—
53D4×C53—42no—
54D4×C53—41no—
55D4×C53—41no—
56D4×C53—41no—
57D4×C53—41no—
58C2×C4×C53—21no—
59C2×C4×C53—21no—
60C2×C4×C53—21no—
61C2×C4×C53—21no—
62C2×C4×C53—82no—
63C2×C4×C53C2×D411yescharacteristic
64C2×C4×C53C2×D411yescharacteristic
65C2×C4×C53C2×D411yescharacteristic
66C2×Dic53—41no—
67Dic106—41no—
68Dic106—41no—
69#88322—41no—
70#88322—41no—
71#88322—41no—
72C22×C53—42no—
73C22×C53—42no—
74C22×C53—42no—
75C22×C53#12011yescharacteristic
76C4×C53—21no—
77C4×C53—41no—
78C4×C53—41no—
79C4×C53—41no—
80C4×C53—41no—
81C4×C53#13511yescharacteristic
82C4×C53#13511yescharacteristic
83C4×C53AGL(1,ℤ/8)11yescharacteristic
84C4×C53#13711yescharacteristic
85Dic53—81no—
86#1640—531nomaximal, Sylow
87C2×C53—82no—
88C2×C53#44911yescharacteristic
89C2×C53#45211yescharacteristic
90C2×C53#45511yescharacteristic
91#559D5311yescharacteristic
92#486—531no—
93#475—531no—
94#462—531no—
95#331—531no—
96Q8⋊C8—531no—
97Q8⋊C8—531no—
98C53#164011yescharacteristic, complemented, Sylow
99C2×Q16—1061no—
100C4⋊Q8—531no—
101#127C2×D5311yescharacteristic
102#124—21no—
103#124—21no—
104#123—22no—
105C4×Q8C2×D5311yescharacteristic
106C4×Q8C2×D5311yescharacteristic
107C4×D4—21no—
108C4×D4—21no—
109#106—1061no—
110C4⋊C8—531no—
111C4⋊C8—531no—
112#103—1061no—
113#103—1061no—
114#102—1062no—
115C4×C8—531no—
116C2×Q8—1061no—
117C2×Q8C22⋊D5311yescharacteristic
118C2×Q8C22⋊D5311yescharacteristic
119C2×D4—21no—
120C2×D4—21no—
121C2×D4C22⋊D5312yes—
122C22×C4—22no—
123Q16—2122no—
124C2×C8—1061no—
125C2×C8—1061no—
126C2×C8—1061no—
127C4⋊C4—21no—
128C4⋊C4—21no—
129C4⋊C4—1061no—
130C4⋊C4—1061no—
131C4⋊C4C22⋊D5311yescharacteristic
132C4⋊C4C22⋊D5311yescharacteristic
133C22⋊C4—22no—
134C22⋊C4—22no—
135C22⋊C4—22no—
136C42—21no—
137C42—21no—
138C42C22×D5311yescharacteristic
139C23—22no—
140Q8—21no—
141Q8—21no—
142Q8—21no—
143Q8—21no—
144Q8—2121no—
145Q8—2121no—
146D4—42no—
147D4—41no—
148D4—41no—
149D4—41no—
150D4—41no—
151C2×C4—21no—
152C2×C4—21no—
153C2×C4—21no—
154C2×C4—21no—
155C2×C4—82no—
156C2×C4—2121no—
157C2×C4C2×C22⋊D5311yescharacteristic
158C2×C4C2×C22⋊D5311yescharacteristic
159C2×C4C2×C22⋊D5311yescharacteristic
160C8—2121no—
161C8—2121no—
162C8—2121no—
163C22—42no—
164C22—42no—
165C22—42no—
166C22#1313318611yescharacteristic
167C4—21no—
168C4—41no—
169C4—41no—
170C4—41no—
171C4—41no—
172C4—4241no—
173C4#1313323711yescharacteristic
174C4#1313323711yescharacteristic
175C4#1313324211yescharacteristic
176C4#1313324711yescharacteristic
177C2—82no—
178C2#1681076811yescharacteristic
179C2#1681078511yescharacteristic
180C2#1681080111yescharacteristic
subgroupG/H
centerC22#13133186
commutatorC42×C53C23
FrattiniC42C22×D53
FittingC53×#559C2
socleC22×C53#120
hypercenter#559D53
hypocenterC53#1640
LCSG ▷ C42×C53 ▷ C22×C53 ▷ C53
UCSC1 ◁ C22 ◁ C42 ◁ #559

Complements

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

NH ≅ G/Ntype
2. #16810908C2semidirect
3. #16810833C2semidirect
5. Q8⋊#88322C2semidirect
6. Q8⋊#88322C2semidirect
98. C53#1640semidirect