Finite Group Databasebeta

← #20290621

Subgroups — #20290621

Order 6784.

306 conjugacy classes of subgroups; only the 70 normal subgroups are shown.

normal subgroups only

This group has 306 conjugacy classes of subgroups — too many to draw. Only its 70 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#16811493C214yesmaximal, complemented
4C2×#13133175C211yesmaximal, characteristic
5C2×#11719936C2212yes—
6C2×C22⋊#6157—21no—
7C2×C4×#6157C2212yes—
8#13133274—21no—
9C22⋊#88324—21no—
10(C2×#6157)⋊C4—21no—
11C22×C4×D53C2211yescharacteristic
12#13133175—21no—
13#13133175—21no—
14#13133175C412yescomplemented
15#13133175C2212yes—
16C2×C4⋊Dic53C411yescharacteristic, complemented
17C2×#11719921—21no—
18C2×C4×Dic53—21no—
19C2×C4⋊C4×C53C411yescharacteristic, complemented
20C23×D53D411yescharacteristic
21C22×#6157—22no—
22C22×C4×C53—21no—
23C22×C4×C53C2×C411yescharacteristic
24C22×Dic53—21no—
25C22×Dic53D411yescharacteristic
26C2×C4×D53—24no—
27C2×C4×D53C2×C411yescharacteristic
28C2×C4×D53D411yescharacteristic
29C2×C4×D53D411yescharacteristic
30C2×C4×D53C2311yescharacteristic
31C22⋊#6157—24no—
32C2×#88324—22no—
33C4×#6157—24no—
34#11719936—24no—
35C4⋊C4×C53—21no—
36C4⋊C4×C53C2×C412yes—
37#11719921—22no—
38#11719921—22no—
39C4⋊Dic53—21no—
40C4⋊Dic53C2×C412yes—
41C4×Dic53—22no—
42C4×Dic53—22no—
43C23×C53C22⋊C411yescharacteristic
44C22×D53—22no—
45C22×D53—22no—
46C22×D53—41no—
47C22×D53C22⋊C411yescharacteristic
48C22×D53C2×D411yescharacteristic
49C2×#6157—24no—
50C2×#6157—42no—
51C2×#6157—42no—
52C2×C4×C53—22no—
53C2×C4×C53—22no—
54C2×C4×C53—42no—
55C2×C4×C53C22⋊C411yescharacteristic
56C2×C4×C53C22×C411yescharacteristic
57C2×Dic53—22no—
58C2×Dic53—22no—
59C2×Dic53—42no—
60C2×Dic53C22⋊C411yescharacteristic
61C2×Dic53C2×D411yescharacteristic
62C4×D53—22no—
63C4×D53—22no—
64C4×D53—22no—
65C4×D53#4114yes—
66#88324—42no—
67C22×C53—21no—
68C22×C53—21no—
69C22×C53#10412yes—
70C22×C53C2×C22⋊C411yescharacteristic
71C2×D53—21no—
72C2×D53—21no—
73C2×D53—21no—
74C2×D53—21no—
75C2×D53—21no—
76C2×D53—44no—
77C2×D53#12312yes—
78#6157—44no—
79C4×C53—21no—
80C4×C53—42no—
81C4×C53#11712yes—
82Dic53—21no—
83Dic53—42no—
84Dic53#12312yes—
85#1826—531nomaximal, Sylow
86C2×C53—22no—
87C2×C53#38111yescharacteristic
88C2×C53C2×#10411yescharacteristic
89C2×C53#42111yescharacteristic
90D53—22no—
91D53—41no—
92C2×#117—531no—
93#407—531no—
94C4×C22⋊C4—531no—
95#339—534no—
96C53#182611yescharacteristic, complemented, Sylow
97C23×C4—531no—
98C2×M16—532no—
99#117—532no—
100#117—532no—
101#117—1061no—
102#117—1061no—
103C2×C4⋊C4—531no—
104C2×C4⋊C4#615711yescharacteristic, complemented
105C2×C22⋊C4—1061no—
106C2×C22⋊C4—1061no—
107C2×C42—532no—
108C2×C42—1061no—
109C22⋊C8—1061no—
110C22⋊C8—1061no—
111#95—1061no—
112C24—531no—
113C22×C4—21no—
114C22×C4—531no—
115C22×C4—531no—
116C22×C4—531no—
117C22×C4—531no—
118C22×C4—531no—
119C22×C4—1064no—
120C22×C4—1062no—
121C22×C4—1061no—
122C22×C4C2×#615711yescharacteristic
123M16—1064no—
124C2×C8—1062no—
125C4⋊C4—21no—
126C4⋊C4—532no—
127C4⋊C4—1061no—
128C4⋊C4C2×#615712yes—
129C22⋊C4—1062no—
130C22⋊C4—1062no—
131C22⋊C4—1064no—
132C42—1062no—
133C42—1062no—
134C42—1064no—
135C23—531no—
136C23—531no—
137C23—1062no—
138C23—1062no—
139C23—2121no—
140C23C22⋊#615711yescharacteristic
141C2×C4—22no—
142C2×C4—22no—
143C2×C4—42no—
144C2×C4—531no—
145C2×C4—534no—
146C2×C4—531no—
147C2×C4—1062no—
148C2×C4—1062no—
149C2×C4—1062no—
150C2×C4—1062no—
151C2×C4—1064no—
152C2×C4—1062no—
153C2×C4—2122no—
154C2×C4—2122no—
155C2×C4—2122no—
156C2×C4C22⋊#615711yescharacteristic
157C2×C4C22×#615711yescharacteristic
158C8—2122no—
159C22—21no—
160C22—21no—
161C22—532no—
162C22—1061no—
163C22—1061no—
164C22—1061no—
165C22—1061no—
166C22—1061no—
167C22—2124no—
168C22#1313328112yes—
169C22C2×C22⋊#615711yescharacteristic
170C4—21no—
171C4—42no—
172C4—532no—
173C4—1061no—
174C4—2124no—
175C4—2122no—
176C4#1313329612yes—
177C2—22no—
178C2—1062no—
179C2—2121no—
180C2#1681156611yescharacteristic
181C2C2×#1313328111yescharacteristic
182C2#1681162211yescharacteristic
subgroupG/H
centerC22C2×C22⋊#6157
commutatorC2×C4×C53C22×C4
FrattiniC2×C4C22×#6157
FittingC2×C4⋊C4×C53C4
socleC22×C53C2×C22⋊C4
hypercenterC2×C4⋊C4#6157
hypocenterC53#1826
LCSG ▷ C2×C4×C53 ▷ C2×C53 ▷ C53
UCSC1 ◁ C22 ◁ C22×C4 ◁ C2×C4⋊C4

Complements

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

NH ≅ G/Ntype
3. #16811493C2semidirect
14. #13133175C4semidirect
16. C2×C4⋊Dic53C4semidirect
19. C2×C4⋊C4×C53C4semidirect
96. C53#1826semidirect
104. C2×C4⋊C4#6157semidirect