Finite Group Databasebeta

← #20290115

Subgroups — #20290115

Order 6784.

294 conjugacy classes of subgroups; only the 64 normal subgroups are shown.

normal subgroups only

This group has 294 conjugacy classes of subgroups — too many to draw. Only its 64 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#16811594C211yesmaximal, characteristic
2C4×C22⋊#6157C211yesmaximal, characteristic
3#16811477C214yesmaximal, complemented
4C2×#13133193C211yesmaximal, characteristic
5C2×#11719936—21no—
6C2×C22⋊#6157—21no—
7C2×C4×#6157—21no—
8#13133274C2212yes—
9C22⋊#88324—21no—
10(C2×#6157)⋊C4C2212yes—
11C22×C4×D53C2211yescharacteristic
12#13133193—21no—
13#13133193—21no—
14#13133193C412yescomplemented
15#13133193C2212yes—
16C2×C4⋊Dic53C411yescharacteristic, complemented
17C2×#11719919—21no—
18C2×#11719921—21no—
19C22×Q8×C53C411yescharacteristic, complemented
20C23×D53D411yescharacteristic
21C22×#6157—22no—
22C2×Q8×C53—21no—
23C2×Q8×C53—42no—
24C2×Q8×C53C2×C412yes—
25C22×C4×C53—21no—
26C22×C4×C53C2×C411yescharacteristic
27C22×Dic53—21no—
28C22×Dic53D411yescharacteristic
29C2×C4×D53—42no—
30C2×C4×D53C2×C411yescharacteristic
31C2×C4×D53D411yescharacteristic
32C2×C4×D53D411yescharacteristic
33C2×C4×D53C2311yescharacteristic
34C22⋊#6157—24no—
35C2×#88324—22no—
36C4×#6157—42no—
37#11719936—42no—
38#11719921—22no—
39#11719921—22no—
40C4⋊Dic53—21no—
41C4⋊Dic53C2×C412yes—
42#11719919—22no—
43#11719919—22no—
44C23×C53C22⋊C411yescharacteristic
45C22×D53—22no—
46C22×D53—22no—
47C22×D53—41no—
48C22×D53C22⋊C411yescharacteristic
49C22×D53C2×D411yescharacteristic
50C2×#6157—24no—
51C2×#6157—42no—
52C2×#6157—42no—
53Q8×C53—42no—
54Q8×C53—42no—
55C2×C4×C53—22no—
56C2×C4×C53—22no—
57C2×C4×C53—42no—
58C2×C4×C53C22⋊C411yescharacteristic
59C2×C4×C53C22×C411yescharacteristic
60C2×Dic53—22no—
61C2×Dic53—41no—
62C2×Dic53—42no—
63C2×Dic53C22⋊C411yescharacteristic
64C2×Dic53C2×D411yescharacteristic
65C4×D53—22no—
66C4×D53—22no—
67C4×D53—41no—
68C4×D53—41no—
69#88324—42no—
70C22×C53—21no—
71C22×C53—21no—
72C22×C53#10412yes—
73C22×C53C2×C22⋊C411yescharacteristic
74C2×D53—21no—
75C2×D53—21no—
76C2×D53—21no—
77C2×D53—21no—
78C2×D53—21no—
79C2×D53—44no—
80C2×D53#13712yes—
81#6157—44no—
82C4×C53—21no—
83C4×C53—42no—
84C4×C53#9912yes—
85Dic53—21no—
86Dic53—21no—
87Dic53—42no—
88#1508—531nomaximal, Sylow
89C2×C53—22no—
90C2×C53C2×#9911yescharacteristic
91C2×C53#41911yescharacteristic
92C2×C53C2×#10411yescharacteristic
93D53—22no—
94D53—41no—
95C2×C22⋊Q8—531no—
96#407—531no—
97C4×C22⋊C4—531no—
98#328—534no—
99C53#150811yescharacteristic, complemented, Sylow
100C22×Q8#615711yescharacteristic, complemented
101C23×C4—531no—
102C2×M16—1061no—
103C22⋊Q8—532no—
104C22⋊Q8—532no—
105C22⋊Q8—1061no—
106C22⋊Q8—1061no—
107C2×C4⋊C4—531no—
108C2×C4⋊C4—1061no—
109C2×C22⋊C4—1061no—
110C2×C22⋊C4—1061no—
111C2×C42—1061no—
112C22⋊C8—532no—
113C22⋊C8—1061no—
114#95—532no—
115C24—531no—
116C2×Q8—21no—
117C2×Q8—42no—
118C2×Q8C2×#615712yes—
119C22×C4—21no—
120C22×C4—531no—
121C22×C4—531no—
122C22×C4—531no—
123C22×C4—531no—
124C22×C4—531no—
125C22×C4—1062no—
126C22×C4—1061no—
127C22×C4—2122no—
128C22×C4C2×#615711yescharacteristic
129M16—2122no—
130C2×C8—1062no—
131C4⋊C4—532no—
132C4⋊C4—1062no—
133C4⋊C4—1062no—
134C4⋊C4—1061no—
135C22⋊C4—1062no—
136C22⋊C4—1062no—
137C22⋊C4—1064no—
138C42—2122no—
139C23—531no—
140C23—531no—
141C23—1062no—
142C23—1062no—
143C23—2121no—
144C23C22⋊#615711yescharacteristic
145Q8—42no—
146Q8—42no—
147C2×C4—22no—
148C2×C4—22no—
149C2×C4—42no—
150C2×C4—531no—
151C2×C4—531no—
152C2×C4—1062no—
153C2×C4—1062no—
154C2×C4—1064no—
155C2×C4—1062no—
156C2×C4—2121no—
157C2×C4—2121no—
158C2×C4—2121no—
159C2×C4—2122no—
160C2×C4—2122no—
161C2×C4—2122no—
162C2×C4C22⋊#615711yescharacteristic
163C2×C4C22×#615711yescharacteristic
164C8—2122no—
165C22—21no—
166C22—21no—
167C22—532no—
168C22—1061no—
169C22—1061no—
170C22—1061no—
171C22—1061no—
172C22—1061no—
173C22—2124no—
174C22#1313328212yes—
175C22C2×C22⋊#615711yescharacteristic
176C4—21no—
177C4—42no—
178C4—1061no—
179C4—1061no—
180C4—2124no—
181C4—2122no—
182C4#1313327512yes—
183C2—22no—
184C2—1062no—
185C2—2121no—
186C2C2×#1313327511yescharacteristic
187C2#1681161611yescharacteristic
188C2C2×#1313328211yescharacteristic
subgroupG/H
centerC22C2×C22⋊#6157
commutatorC2×C4×C53C22×C4
FrattiniC2×C4C22×#6157
FittingC22×Q8×C53C4
socleC22×C53C2×C22⋊C4
hypercenterC22×Q8#6157
hypocenterC53#1508
LCSG ▷ C2×C4×C53 ▷ C22×C53 ▷ C53
UCSC1 ◁ C22 ◁ C22×C4 ◁ C22×Q8

Complements

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

NH ≅ G/Ntype
3. #16811477C2semidirect
14. #13133193C4semidirect
16. C2×C4⋊Dic53C4semidirect
19. C22×Q8×C53C4semidirect
99. C53#1508semidirect
100. C22×Q8#6157semidirect