Finite Group Databasebeta

← #20290618

Subgroups — #20290618

Order 6784.

344 conjugacy classes of subgroups; only the 84 normal subgroups are shown.

normal subgroups only

This group has 344 conjugacy classes of subgroups — too many to draw. Only its 84 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
1D53⋊(C22⋊C8)C211yesmaximal, characteristic
2#16811565C211yesmaximal, characteristic
3#16811496C214yesmaximal, complemented
4C2×C4⋊C4×D53C211yesmaximal, characteristic
5C2×D53⋊C8C2212yes—
6C2×C4⋊#6157C2212yes—
7C2×C22⋊#6157—21no—
8#13133274—21no—
9C22⋊#88324—21no—
10(C2×#6157)⋊C4—21no—
11C22×C4×D53—21no—
12C22×C4×D53C2211yescharacteristic
13C2×C4⋊Dic53C411yescharacteristic, complemented
14C2×#11719919—21no—
15C4⋊C4×D53—21no—
16C4⋊C4×D53—21no—
17C4⋊C4×D53C412yescomplemented
18C4⋊C4×D53C2212yes—
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×D53—22no—
28C2×C4×D53—22no—
29C2×C4×D53—42no—
30C2×C4×D53—42no—
31C2×C4×D53C2×C411yescharacteristic
32C2×C4×D53D411yescharacteristic
33C2×C4×D53D411yescharacteristic
34C2×C4×D53C2311yescharacteristic
35C22⋊#6157—42no—
36C2×#88324—22no—
37C4⋊#6157—24no—
38D53⋊C8—24no—
39C4⋊C4×C53—21no—
40C4⋊C4×C53C2×C412yes—
41C4⋊Dic53—21no—
42C4⋊Dic53C2×C412yes—
43#11719919—22no—
44#11719919—22no—
45C23×C53C22⋊C411yescharacteristic
46C22×D53—22no—
47C22×D53—22no—
48C22×D53C22⋊C411yescharacteristic
49C22×D53D812yes—
50C22×D53SD1612yes—
51C22×D53C2×D411yescharacteristic
52C2×#6157—42no—
53C2×#6157—42no—
54C2×#6157—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—22no—
62C2×Dic53—42no—
63C2×Dic53C22⋊C411yescharacteristic
64C2×Dic53C2×D411yescharacteristic
65C4×D53—22no—
66C4×D53—22no—
67C4×D53—22no—
68C4×D53—44no—
69C4×D53—42no—
70C4×D53—42no—
71C4×D53#4114yes—
72#88324—42no—
73C22×C53—21no—
74C22×C53—21no—
75C22×C53#10212yes—
76C22×C53C2×C22⋊C411yescharacteristic
77C2×D53—21no—
78C2×D53—21no—
79C2×D53—24no—
80C2×D53—24no—
81C2×D53#10212yes—
82C2×D53#12312yes—
83C2×D53C2×D811yescharacteristic
84C2×D53C2×SD1611yescharacteristic
85C2×D53#13712yes—
86#6157—82no—
87C4×C53—21no—
88C4×C53—42no—
89C4×C53#11712yes—
90Dic53—21no—
91Dic53—42no—
92Dic53#12312yes—
93#1823—531nomaximal, Sylow
94C2×C53—22no—
95C2×C53#38111yescharacteristic
96C2×C53C2×#10211yescharacteristic
97C2×C53#41911yescharacteristic
98D53—22no—
99D53#48012yes—
100D53#48312yes—
101C22×C4⋊C4—531no—
102C2×C22⋊C8—531no—
103#380—531no—
104#340—534no—
105C53#182311yescharacteristic, complemented, Sylow
106C23×C4—531no—
107C23×C4—1061no—
108C22×C8—532no—
109C2×C4⋊C4—531no—
110C2×C4⋊C4—532no—
111C2×C4⋊C4—532no—
112C2×C4⋊C4—532no—
113C2×C4⋊C4—1061no—
114C2×C4⋊C4—1061no—
115C2×C4⋊C4—1061no—
116C2×C4⋊C4#615711yescharacteristic, complemented
117C2×C22⋊C4—1061no—
118C22⋊C8—1061no—
119C22⋊C8—1061no—
120#95—1061no—
121C24—531no—
122C22×C4—21no—
123C22×C4—531no—
124C22×C4—531no—
125C22×C4—531no—
126C22×C4—531no—
127C22×C4—531no—
128C22×C4—1064no—
129C22×C4—1062no—
130C22×C4—1062no—
131C22×C4—1061no—
132C22×C4—1062no—
133C22×C4—2122no—
134C22×C4—2122no—
135C22×C4C2×#615711yescharacteristic
136C2×C8—1062no—
137C2×C8—1064no—
138C4⋊C4—21no—
139C4⋊C4—532no—
140C4⋊C4—1062no—
141C4⋊C4—1062no—
142C4⋊C4—1064no—
143C4⋊C4—1061no—
144C4⋊C4C2×#615712yes—
145C22⋊C4—2122no—
146C23—531no—
147C23—532no—
148C23—532no—
149C23—531no—
150C23—1062no—
151C23—1062no—
152C23C22⋊#615711yescharacteristic
153C2×C4—22no—
154C2×C4—22no—
155C2×C4—42no—
156C2×C4—531no—
157C2×C4—534no—
158C2×C4—531no—
159C2×C4—1062no—
160C2×C4—1062no—
161C2×C4—1062no—
162C2×C4—1062no—
163C2×C4—1062no—
164C2×C4—2122no—
165C2×C4—2122no—
166C2×C4—2124no—
167C2×C4—2122no—
168C2×C4—2122no—
169C2×C4—2122no—
170C2×C4—2122no—
171C2×C4C22⋊#615711yescharacteristic
172C2×C4C22×#615711yescharacteristic
173C8—2122no—
174C22—21no—
175C22—21no—
176C22—532no—
177C22—531no—
178C22—531no—
179C22—532no—
180C22—532no—
181C22—1064no—
182C22—1064no—
183C22—1061no—
184C22—1061no—
185C22#1313327912yes—
186C22C2×C22⋊#615711yescharacteristic
187C4—21no—
188C4—42no—
189C4—532no—
190C4—1061no—
191C4—4242no—
192C4—2122no—
193C4#1313329612yes—
194C2—22no—
195C2—532no—
196C2—532no—
197C2—1062no—
198C2#1681156611yescharacteristic
199C2C2×#1313327911yescharacteristic
200C2#1681161611yescharacteristic
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#1823
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. #16811496C2semidirect
13. C2×C4⋊Dic53C4semidirect
17. C4⋊C4×D53C4semidirect
19. C2×C4⋊C4×C53C4semidirect
105. C53#1823semidirect
116. C2×C4⋊C4#6157semidirect