Finite Group Databasebeta

← #20290151

Subgroups — #20290151

Order 6784.

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

normal subgroups only

This group has 340 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
1D53⋊(C22⋊C8)C211yesmaximal, characteristic, complemented
2#16811565C211yesmaximal, characteristic, complemented
3#16811474C214yesmaximal, complemented
4C2×#13133188C211yesmaximal, characteristic
5C2×D53⋊C8—21no—
6C2×C4⋊#6157—21no—
7C2×C22⋊#6157—21no—
8#13133274C2212yescomplemented
9C22⋊#88324—21no—
10(C2×#6157)⋊C4C2212yescomplemented
11C22×C22⋊D53—21no—
12C22×C4×D53C2211yescharacteristic
13#13133188—21no—
14#13133188—21no—
15#13133188C412yescomplemented
16#13133188C2212yes—
17C2×C4⋊Dic53C411yescharacteristic, complemented
18C2×C22⋊Dic53—21no—
19C22×D4×C53C411yescharacteristic, complemented
20C24×C53—21no—
21C23×D53D411yescharacteristic
22C22×#6157—22no—
23C2×D4×C53—21no—
24C2×D4×C53—42no—
25C2×D4×C53C2×C412yes—
26C22×C4×C53C2×C411yescharacteristic
27C2×C22⋊D53—22no—
28C2×C22⋊D53—22no—
29C2×C22⋊D53—42no—
30C2×C22⋊D53—42no—
31C22×Dic53—21no—
32C22×Dic53D411yescharacteristic
33C2×C4×D53—42no—
34C2×C4×D53C2×C411yescharacteristic
35C2×C4×D53D411yescharacteristic
36C2×C4×D53D411yescharacteristic
37C2×C4×D53C2311yescharacteristic
38C22⋊#6157—42no—
39C2×#88324—22no—
40C4⋊#6157—42no—
41D53⋊C8—42no—
42C22⋊Dic53—22no—
43C22⋊Dic53—22no—
44C4⋊Dic53—21no—
45C4⋊Dic53C2×C412yes—
46C23×C53—22no—
47C23×C53—42no—
48C23×C53—42no—
49C23×C53—42no—
50C23×C53C22⋊C411yescharacteristic
51C22×D53—41no—
52C22×D53—41no—
53C22×D53C22⋊C411yescharacteristic
54C22×D53D812yes—
55C22×D53SD1612yes—
56C22×D53C2×D411yescharacteristic
57C2×#6157—42no—
58C2×#6157—42no—
59C2×#6157—42no—
60D4×C53—42no—
61D4×C53—42no—
62C2×C4×C53—22no—
63C2×C4×C53C22⋊C411yescharacteristic
64C2×C4×C53C22×C411yescharacteristic
65C22⋊D53—44no—
66C22⋊D53—42no—
67C22⋊D53—42no—
68C2×Dic53—22no—
69C2×Dic53—41no—
70C2×Dic53—42no—
71C2×Dic53C22⋊C411yescharacteristic
72C2×Dic53C2×D411yescharacteristic
73C4×D53—22no—
74C4×D53—22no—
75C4×D53—41no—
76C4×D53—41no—
77#88324—42no—
78C22×C53—21no—
79C22×C53—21no—
80C22×C53—42no—
81C22×C53—42no—
82C22×C53—42no—
83C22×C53—84no—
84C22×C53#10212yes—
85C22×C53C2×C22⋊C411yescharacteristic
86C2×D53—21no—
87C2×D53—21no—
88C2×D53—21no—
89C2×D53—21no—
90C2×D53—42no—
91C2×D53—42no—
92C2×D53#10212yes—
93C2×D53C2×D811yescharacteristic
94C2×D53C2×SD1611yescharacteristic
95#6157—82no—
96C4×C53—21no—
97C4×C53#9912yes—
98Dic53—21no—
99Dic53—21no—
100Dic53—42no—
101#1519—531nomaximal, Sylow
102C2×C53—22no—
103C2×C53—82no—
104C2×C53C2×#9911yescharacteristic
105C2×C53C2×#10211yescharacteristic
106C2×C53#42111yescharacteristic
107D53—21no—
108D53—22no—
109D53—21no—
110C2×#121—531no—
111C2×C22⋊C8—531no—
112#380—531no—
113#327—534no—
114C53#151911yescharacteristic, complemented, Sylow
115C22×D4—1061no—
116C22×D4#615711yescharacteristic, complemented
117C23×C4—531no—
118C22×C8—1061no—
119#121—532no—
120#121—532no—
121#121—1061no—
122#121—1061no—
123C2×C4⋊C4—531no—
124C2×C4⋊C4—1061no—
125C2×C22⋊C4—1061no—
126C2×C22⋊C4—1061no—
127C22⋊C8—532no—
128C22⋊C8—1061no—
129#95—532no—
130C24—21no—
131C24—531no—
132C2×D4—21no—
133C2×D4—42no—
134C2×D4—1062no—
135C2×D4—1062no—
136C2×D4—2122no—
137C2×D4—2122no—
138C2×D4C2×#615712yes—
139C22×C4—531no—
140C22×C4—531no—
141C22×C4—531no—
142C22×C4—531no—
143C22×C4—531no—
144C22×C4—1061no—
145C22×C4—1062no—
146C22×C4—2122no—
147C22×C4C2×#615711yescharacteristic
148C2×C8—1062no—
149C2×C8—2122no—
150C4⋊C4—532no—
151C4⋊C4—1061no—
152C4⋊C4—2122no—
153C22⋊C4—1062no—
154C22⋊C4—1062no—
155C22⋊C4—2122no—
156C23—22no—
157C23—42no—
158C23—42no—
159C23—42no—
160C23—531no—
161C23—531no—
162C23—532no—
163C23—532no—
164C23—2121no—
165C23—2121no—
166C23C22⋊#615711yescharacteristic
167D4—42no—
168D4—42no—
169D4—2122no—
170D4—2124no—
171D4—2122no—
172C2×C4—22no—
173C2×C4—531no—
174C2×C4—531no—
175C2×C4—1062no—
176C2×C4—1062no—
177C2×C4—1062no—
178C2×C4—2121no—
179C2×C4—2121no—
180C2×C4—2121no—
181C2×C4—2122no—
182C2×C4—2122no—
183C2×C4—2122no—
184C2×C4—2122no—
185C2×C4C22⋊#615711yescharacteristic
186C2×C4C22×#615711yescharacteristic
187C8—2122no—
188C22—21no—
189C22—21no—
190C22—42no—
191C22—42no—
192C22—42no—
193C22—84no—
194C22—531no—
195C22—532no—
196C22—531no—
197C22—1061no—
198C22—1061no—
199C22—1061no—
200C22—1061no—
201C22—2122no—
202C22—2122no—
203C22#1313327912yes—
204C22C2×C22⋊#615711yescharacteristic
205C4—21no—
206C4—1061no—
207C4—1061no—
208C4—4242no—
209C4—2122no—
210C4#1313327512yes—
211C2—22no—
212C2—82no—
213C2—1061no—
214C2—1062no—
215C2—1061no—
216C2C2×#1313327511yescharacteristic
217C2C2×#1313327911yescharacteristic
218C2#1681162111yescharacteristic
subgroupG/H
centerC22C2×C22⋊#6157
commutatorC2×C4×C53C22×C4
FrattiniC2×C4C22×#6157
FittingC22×D4×C53C4
socleC22×C53C2×C22⋊C4
hypercenterC22×D4#6157
hypocenterC53#1519
LCSG ▷ C2×C4×C53 ▷ C22×C53 ▷ C53
UCSC1 ◁ C22 ◁ C22×C4 ◁ C22×D4

Complements

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

NH ≅ G/Ntype
1. D53⋊(C22⋊C8)C2semidirect
2. #16811565C2semidirect
3. #16811474C2semidirect
8. #13133274C22semidirect
10. (C2×#6157)⋊C4C22semidirect
15. #13133188C4semidirect
17. C2×C4⋊Dic53C4semidirect
19. C22×D4×C53C4semidirect
114. C53#1519semidirect
116. C22×D4#6157semidirect