Finite Group Databasebeta

← #20290998

Subgroups — #20290998

Order 6784.

238 conjugacy classes of subgroups; only the 52 normal subgroups are shown.

normal subgroups only

This group has 238 conjugacy classes of subgroups — too many to draw. Only its 52 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#16811604C212yesmaximal
2#16811530C212yesmaximal, complemented
3#16811529C212yesmaximal, complemented
4#16811198C211yesmaximal, characteristic
5#13133296—22no—
6#13133276—22no—
7#13133276C2214yes—
8C2×#11719949C2211yescharacteristic
9C2×Q8×D53C411yescharacteristic, complemented
10#13133215C411yescharacteristic, complemented
11#13133206C2211yescharacteristic
12#13133205C2211yescharacteristic
13#13133202—22no—
14#13133195—22no—
15#13133174C411yescharacteristic, complemented
16C4⋊Q8×C53C411yescharacteristic, complemented
17C2×Q8×C53C2×C411yescharacteristic
18C2×Q8×C53C2×C411yescharacteristic
19#11719949—42no—
20Q8×D53—42no—
21C2×D212D412yes—
22C2×D212C2311yescharacteristic
23C2×C4×D53—21no—
24C2×C4×D53C2×C411yescharacteristic
25C2×C4×D53D412yes—
26C2×Dic106—21no—
27C2×Dic106C2×C411yescharacteristic
28C22⋊#6157—24no—
29C4⋊#6157—42no—
30C4×#6157—42no—
31C4⋊C4×C53—22no—
32C42×C53C2×C411yescharacteristic
33#11719921—21no—
34#11719921—22no—
35#11719921—22no—
36#11719919—22no—
37#11719919—21no—
38C4×Dic53C2×C411yescharacteristic
39C22×D53C22⋊C411yescharacteristic
40C22×D53C2×D412yes—
41C2×#6157—44no—
42Q8×C53—41no—
43Q8×C53—41no—
44C2×C4×C53—21no—
45C2×C4×C53—21no—
46C2×C4×C53C22⋊C412yes—
47C2×C4×C53C22×C411yescharacteristic
48C2×Dic53—21no—
49C2×Dic53—21no—
50C2×Dic53C22⋊C411yescharacteristic
51D212—42no—
52D212—41no—
53C4×D53—22no—
54C4×D53—22no—
55C4×D53—22no—
56C4×D53—42no—
57Dic106—42no—
58Dic106—41no—
59C22×C53C2×C22⋊C411yescharacteristic
60C2×D53—21no—
61C2×D53—22no—
62C2×D53—41no—
63C2×D53—42no—
64#6157—84no—
65C4×C53—21no—
66C4×C53—22no—
67C4×C53—41no—
68C4×C53—41no—
69Dic53—41no—
70Dic53—41no—
71Dic53#9912yes—
72#2129—531nomaximal, Sylow
73C2×C53—21no—
74C2×C53C2×#9911yescharacteristic
75D53—42no—
76D53—41no—
77#536—531no—
78#410—532no—
79#354—532no—
80#354—532no—
81C53#212911yescharacteristic, complemented, Sylow
82C2×#41—531no—
83C22×Q8—531no—
84C4⋊Q8—531no—
85C4⋊Q8#615711yescharacteristic, complemented
86#124—531no—
87#124—531no—
88#123—1062no—
89C22⋊Q8—1062no—
90#117—531no—
91#117—1062no—
92#99—534no—
93#99—1062no—
94#41—2122no—
95C2×Q8—531no—
96C2×Q8—1061no—
97C2×Q8—2122no—
98C2×Q8C2×#615711yescharacteristic
99C2×Q8C2×#615711yescharacteristic
100C2×D4—532no—
101C2×D4—531no—
102C22×C4—532no—
103C22×C4—531no—
104C22×C4—1061no—
105C4⋊C4—22no—
106C4⋊C4—1061no—
107C4⋊C4—1062no—
108C4⋊C4—2122no—
109C22⋊C4—1061no—
110C22⋊C4—1062no—
111C22⋊C4—1062no—
112C22⋊C4—1064no—
113C42—531no—
114C42—2122no—
115C42C2×#615711yescharacteristic
116C23—531no—
117C23—532no—
118Q8—41no—
119Q8—41no—
120Q8—2121no—
121Q8—2122no—
122D4—2122no—
123D4—2121no—
124C2×C4—21no—
125C2×C4—21no—
126C2×C4—531no—
127C2×C4—1062no—
128C2×C4—1062no—
129C2×C4—1062no—
130C2×C4—1061no—
131C2×C4—1061no—
132C2×C4—2122no—
133C2×C4—2124no—
134C2×C4C22⋊#615712yes—
135C2×C4C22×#615711yescharacteristic
136C22—1062no—
137C22—1061no—
138C22—2122no—
139C22—2121no—
140C22C2×C22⋊#615711yescharacteristic
141C4—22no—
142C4—21no—
143C4—41no—
144C4—41no—
145C4—532no—
146C4—4244no—
147C4—2121no—
148C4—2121no—
149C2—21no—
150C2—2122no—
151C2—2121no—
152C2D53⋊#9911yescharacteristic
subgroupG/H
centerC2D53⋊#99
commutatorC2×C4×C53C22×C4
FrattiniC2×C4C22×#6157
FittingC4⋊Q8×C53C4
socleC2×C53C2×#99
hypercenterC4⋊Q8#6157
hypocenterC53#2129
LCSG ▷ C2×C4×C53 ▷ C22×C53 ▷ C2×C53 ▷ C53
UCSC1 ◁ C2 ◁ C22 ◁ C2×Q8 ◁ C4⋊Q8

Complements

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

NH ≅ G/Ntype
2. #16811530C2semidirect
3. #16811529C2semidirect
9. C2×Q8×D53C4semidirect
10. #13133215C4semidirect
15. #13133174C4semidirect
16. C4⋊Q8×C53C4semidirect
81. C53#2129semidirect
85. C4⋊Q8#6157semidirect