Finite Group Databasebeta

← #20290855

Subgroups — #20290855

Order 6784.

408 conjugacy classes of subgroups; only the 74 normal subgroups are shown.

normal subgroups only

This group has 408 conjugacy classes of subgroups — too many to draw. Only its 74 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
1C2×#13133276C214yesmaximal, complemented
2#16811565C211yesmaximal, characteristic, complemented
3C4×C22⋊#6157C211yesmaximal, characteristic, complemented
4C2×#13133192C211yesmaximal, characteristic
5C2×C4⋊#6157—21no—
6C2×C22⋊#6157C2212yes—
7C2×C22⋊#6157C2212yes—
8C2×C4×#6157—21no—
9#13133276—28no—
10(C2×#6157)⋊C4—21no—
11(C2×#6157)⋊C4—21no—
12C22×D212C411yescharacteristic, complemented
13C22×D212C2212yes—
14C22×C4×D53C2211yescharacteristic
15#13133192—21no—
16#13133192—21no—
17#13133192—21no—
18#13133192—21no—
19C2×#11719921C412yescomplemented
20C2×C4⋊C4×C53C411yescharacteristic, complemented
21C23×D53C2×C412yes—
22C23×D53C2311yescharacteristic
23C22×#6157—22no—
24C22×#6157—22no—
25C22×C4×C53C2×C412yes—
26C22×C4×C53C2×C411yescharacteristic
27C22×Dic53C2×C411yescharacteristic
28C2×D212—22no—
29C2×D212—22no—
30C2×D212—24no—
31C2×D212—24no—
32C2×D212—21no—
33C2×D212—21no—
34C2×D212—42no—
35C2×C4×D53—42no—
36C2×C4×D53D411yescharacteristic
37C2×C4×D53D411yescharacteristic
38C2×C4×D53D411yescharacteristic
39C2×C4×D53D411yescharacteristic
40C22⋊#6157—22no—
41C22⋊#6157—24no—
42C22⋊#6157D414yescomplemented
43C4⋊#6157—42no—
44C4×#6157—42no—
45C4⋊C4×C53—21no—
46C4⋊C4×C53—21no—
47#11719921—22no—
48#11719921—22no—
49C23×C53C22×C411yescharacteristic
50C22×D53—22no—
51C22×D53—24no—
52C22×D53—41no—
53C22×D53—41no—
54C22×D53—42no—
55C22×D53—42no—
56C22×D53C2×D411yescharacteristic
57C22×D53C2×D411yescharacteristic
58C22×D53C2×D412yes—
59C22×D53#4112yes—
60C2×#6157—24no—
61C2×#6157—42no—
62C2×#6157—42no—
63C2×#6157—42no—
64C2×#6157—42no—
65C2×#6157—42no—
66C2×C4×C53—22no—
67C2×C4×C53—22no—
68C2×C4×C53—24no—
69C2×C4×C53C22⋊C411yescharacteristic
70C2×C4×C53C22⋊C411yescharacteristic
71C2×Dic53—41no—
72C2×Dic53C22⋊C411yescharacteristic
73C2×Dic53C22⋊C411yescharacteristic
74D212—42no—
75D212—44no—
76D212—42no—
77D212—42no—
78D212—42no—
79C4×D53—22no—
80C4×D53—22no—
81C4×D53—41no—
82C4×D53—41no—
83C22×C53—21no—
84C22×C53—21no—
85C22×C53C2×C22⋊C411yescharacteristic
86C22×C53C4×D412yes—
87C2×D53—21no—
88C2×D53—21no—
89C2×D53—21no—
90C2×D53—21no—
91C2×D53—42no—
92C2×D53—42no—
93C2×D53—42no—
94C2×D53—42no—
95C2×D53—42no—
96C2×D53—84no—
97C2×D53#12112yes—
98C2×D53#12411yescharacteristic
99C2×D53#12711yescharacteristic
100#6157—44no—
101#6157—82no—
102C4×C53—21no—
103C4×C53—42no—
104C4×C53#9912yescomplemented
105Dic53—21no—
106Dic53—21no—
107#1964—531nomaximal, Sylow
108C2×C53—22no—
109C2×C53#39011yescharacteristic
110C2×C53C2×#9911yescharacteristic
111C2×C53#41011yescharacteristic
112D53—22no—
113D53—21no—
114D53—21no—
115D53—82no—
116C2×#121—531no—
117C2×#99—534no—
118#380—531no—
119C4×C22⋊C4—531no—
120C53#196411yescharacteristic, complemented, Sylow
121C22×D4—532no—
122C22×D4—531no—
123C23×C4—531no—
124#121—1061no—
125#121—1061no—
126#121—1061no—
127#121—1061no—
128C2×C4⋊C4—1061no—
129C2×C4⋊C4#615711yescharacteristic, complemented
130C2×C22⋊C4—532no—
131C2×C22⋊C4—532no—
132C2×C22⋊C4—532no—
133C2×C42—1061no—
134#99—1068no—
135#95—1061no—
136#95—1061no—
137C24—531no—
138C24—532no—
139C2×D4—1064no—
140C2×D4—1062no—
141C2×D4—1062no—
142C2×D4—1064no—
143C2×D4—1061no—
144C2×D4—1061no—
145C2×D4—2122no—
146C22×C4—531no—
147C22×C4—531no—
148C22×C4—531no—
149C22×C4—531no—
150C22×C4—531no—
151C22×C4—1062no—
152C22×C4—1062no—
153C22×C4—2122no—
154C22×C4C2×#615712yes—
155C22×C4C2×#615711yescharacteristic
156C4⋊C4—21no—
157C4⋊C4—21no—
158C4⋊C4—2122no—
159C22⋊C4—534no—
160C22⋊C4—1062no—
161C22⋊C4—1062no—
162C22⋊C4—1062no—
163C22⋊C4—1064no—
164C42—2122no—
165C23—531no—
166C23—531no—
167C23—532no—
168C23—532no—
169C23—1064no—
170C23—1062no—
171C23—2121no—
172C23—2121no—
173C23—2122no—
174C23—2122no—
175C23C22×#615711yescharacteristic
176D4—2122no—
177D4—2124no—
178D4—2122no—
179D4—2122no—
180D4—2122no—
181C2×C4—22no—
182C2×C4—22no—
183C2×C4—24no—
184C2×C4—531no—
185C2×C4—531no—
186C2×C4—1062no—
187C2×C4—1062no—
188C2×C4—1064no—
189C2×C4—2121no—
190C2×C4—2121no—
191C2×C4—2121no—
192C2×C4—2122no—
193C2×C4—2122no—
194C2×C4—2122no—
195C2×C4—2122no—
196C2×C4—2122no—
197C2×C4C22⋊#615711yescharacteristic
198C2×C4C22⋊#615711yescharacteristic
199C22—21no—
200C22—21no—
201C22—532no—
202C22—531no—
203C22—531no—
204C22—1061no—
205C22—1061no—
206C22—1061no—
207C22—1061no—
208C22—4244no—
209C22—2122no—
210C22—2122no—
211C22—2122no—
212C22—2122no—
213C22—2122no—
214C22C2×C22⋊#615711yescharacteristic
215C22D4×#615712yes—
216C4—21no—
217C4—42no—
218C4—1061no—
219C4—1061no—
220C4—4242no—
221C4—2124no—
222C4#1313327612yescomplemented
223C2—22no—
224C2—1061no—
225C2—1062no—
226C2—1061no—
227C2—4242no—
228C2#1681157511yescharacteristic
229C2C2×#1313327611yescharacteristic
230C2#1681160411yescharacteristic
subgroupG/H
centerC22C2×C22⋊#6157
commutatorC23×C53C22×C4
FrattiniC23C22×#6157
FittingC2×C4⋊C4×C53C4
socleC22×C53C2×C22⋊C4
hypercenterC2×C4⋊C4#6157
hypocenterC53#1964
LCSG ▷ C23×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
1. C2×#13133276C2semidirect
2. #16811565C2semidirect
3. C4×C22⋊#6157C2semidirect
12. C22×D212C4semidirect
19. C2×#11719921C4semidirect
20. C2×C4⋊C4×C53C4semidirect
42. C22⋊#6157D4semidirect
104. C4×C53#99semidirect
120. C53#1964semidirect
129. C2×C4⋊C4#6157semidirect
222. C4#13133276semidirect