Finite Group Databasebeta

← #20290742

Subgroups — #20290742

Order 6784.

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

normal subgroups only

This group has 338 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
1C22×#11719940C211yesmaximal, characteristic
2C2×#13133278C212yesmaximal, complemented
3#16811593C211yesmaximal, characteristic
4#16811503C212yesmaximal, complemented
5C2×#13133194C211yesmaximal, characteristic
6C2×#11719940—21no—
7C2×#11719940—24no—
8C2×#11719940C2212yes—
9C2×#11719940C2212yes—
10C22×#88324—21no—
11#13133278—24no—
12C22⋊#88324—21no—
13C22⋊#88324—21no—
14C22×Dic106C2211yescharacteristic
15C23×Dic53C2211yescharacteristic
16#13133194—21no—
17#13133194—21no—
18#13133194—21no—
19#13133194—21no—
20C2×C4⋊Dic53C411yescharacteristic
21C2×#11719919C411yescharacteristic
22C2×#11719919C2211yescharacteristic
23C2×C22⋊Dic53C411yescharacteristic
24C2×C22⋊C4×C53C411yescharacteristic
25C24×C53C2×C411yescharacteristic
26C22×C4×C53C2×C411yescharacteristic
27C22×C4×C53C2×C411yescharacteristic
28C22×Dic53—22no—
29C22×Dic53—22no—
30C22×Dic53C2×C411yescharacteristic
31C22×Dic53C2×C411yescharacteristic
32C22×Dic53C2×C411yescharacteristic
33C22×Dic53D411yescharacteristic
34C22×Dic53D411yescharacteristic
35C22×Dic53D411yescharacteristic
36C22×Dic53D411yescharacteristic
37C22×Dic53C2311yescharacteristic
38C2×Dic106—21no—
39C2×Dic106—21no—
40C2×Dic106—22no—
41C2×Dic106—22no—
42#11719940—24no—
43#11719940—22no—
44#11719940—22no—
45#11719940—22no—
46#11719940D414yes—
47C2×#88324—24no—
48C2×#88324—22no—
49C2×#88324—22no—
50C22⋊C4×C53—21no—
51C22⋊C4×C53—21no—
52C22⋊Dic53—21no—
53C22⋊Dic53—21no—
54C4⋊Dic53—21no—
55C4⋊Dic53—21no—
56#11719919—21no—
57#11719919—21no—
58#11719919—21no—
59#11719919—21no—
60C23×C53—22no—
61C23×C53—22no—
62C23×C53—41no—
63C23×C53C22⋊C411yescharacteristic
64C23×C53C22⋊C411yescharacteristic
65C23×C53C22×C411yescharacteristic
66C2×C4×C53—21no—
67C2×C4×C53—21no—
68C2×C4×C53—22no—
69C2×C4×C53—41no—
70C2×Dic53—22no—
71C2×Dic53—21no—
72C2×Dic53—21no—
73C2×Dic53—22no—
74C2×Dic53—22no—
75C2×Dic53—21no—
76C2×Dic53—21no—
77C2×Dic53—21no—
78C2×Dic53—21no—
79C2×Dic53—22no—
80C2×Dic53—41no—
81C2×Dic53C22⋊C411yescharacteristic
82C2×Dic53C22⋊C411yescharacteristic
83C2×Dic53C2×D412yes—
84C2×Dic53C2×D411yescharacteristic
85C2×Dic53C2×D411yescharacteristic
86C2×Dic53#4112yes—
87Dic106—41no—
88Dic106—42no—
89Dic106—41no—
90#88324—24no—
91#88324—42no—
92C22×C53—21no—
93C22×C53—21no—
94C22×C53—21no—
95C22×C53—21no—
96C22×C53—21no—
97C22×C53—21no—
98C22×C53—21no—
99C22×C53—44no—
100C22×C53#10112yescomplemented
101C22×C53C2×C22⋊C411yescharacteristic
102C22×C53C4×D412yes—
103C4×C53—41no—
104C4×C53—41no—
105Dic53—21no—
106Dic53—21no—
107Dic53—41no—
108Dic53—41no—
109Dic53#12011yescharacteristic
110Dic53#12112yes—
111Dic53#12311yescharacteristic
112#1901—531nomaximal, Sylow
113C2×C53—22no—
114C2×C53—22no—
115C2×C53—41no—
116C2×C53#38611yescharacteristic
117C2×C53C2×#10111yescharacteristic
118C2×C53#41311yescharacteristic
119C22×M16—531no—
120C2×C22⋊Q8—531no—
121C2×#101—532no—
122#407—531no—
123#343—532no—
124C53#190111yescharacteristic, complemented, Sylow
125C22×Q8—531no—
126C23×C4—531no—
127C2×M16—532no—
128C2×M16—532no—
129C2×M16—1064no—
130C2×M16—1061no—
131C22×C8—1061no—
132C22⋊Q8—1061no—
133C22⋊Q8—1061no—
134C22⋊Q8—1061no—
135C22⋊Q8—1061no—
136C2×C4⋊C4—531no—
137C2×C4⋊C4—531no—
138C2×C4⋊C4—531no—
139C2×C22⋊C4—531no—
140C2×C22⋊C4#615711yescharacteristic
141#101—1064no—
142C22⋊C8—1061no—
143C22⋊C8—1061no—
144C24C2×#615711yescharacteristic
145C2×Q8—1062no—
146C2×Q8—1061no—
147C2×Q8—1061no—
148C2×Q8—1062no—
149C22×C4—531no—
150C22×C4—531no—
151C22×C4—531no—
152C22×C4—531no—
153C22×C4—531no—
154C22×C4—531no—
155C22×C4—531no—
156C22×C4—531no—
157C22×C4—1062no—
158C22×C4—1062no—
159C22×C4C2×#615711yescharacteristic
160C22×C4C2×#615711yescharacteristic
161M16—534no—
162M16—1062no—
163M16—1064no—
164M16—1062no—
165M16—1062no—
166C2×C8—1062no—
167C2×C8—1064no—
168C2×C8—1062no—
169C4⋊C4—1061no—
170C4⋊C4—1061no—
171C4⋊C4—1061no—
172C4⋊C4—1061no—
173C4⋊C4—1061no—
174C4⋊C4—1061no—
175C22⋊C4—21no—
176C22⋊C4—21no—
177C22⋊C4—1061no—
178C22⋊C4—1061no—
179C23—22no—
180C23—22no—
181C23—41no—
182C23C22⋊#615711yescharacteristic
183C23C22⋊#615711yescharacteristic
184C23C22×#615711yescharacteristic
185Q8—2121no—
186Q8—2122no—
187Q8—2121no—
188C2×C4—21no—
189C2×C4—21no—
190C2×C4—22no—
191C2×C4—41no—
192C2×C4—531no—
193C2×C4—531no—
194C2×C4—532no—
195C2×C4—532no—
196C2×C4—531no—
197C2×C4—531no—
198C2×C4—1061no—
199C2×C4—1061no—
200C2×C4—1062no—
201C2×C4—1062no—
202C2×C4—1061no—
203C2×C4—1061no—
204C2×C4—1062no—
205C2×C4—1061no—
206C2×C4—1061no—
207C2×C4—1062no—
208C2×C4—2121no—
209C8—1064no—
210C8—2122no—
211C22—21no—
212C22—21no—
213C22—21no—
214C22—21no—
215C22—21no—
216C22—21no—
217C22—21no—
218C22—44no—
219C22#1313327812yescomplemented
220C22C2×C22⋊#615711yescharacteristic
221C22D4×#615712yes—
222C4—41no—
223C4—41no—
224C4—531no—
225C4—532no—
226C4—531no—
227C4—1061no—
228C4—1061no—
229C4—2121no—
230C4—2121no—
231C2—22no—
232C2—22no—
233C2—41no—
234C2#1681157111yescharacteristic
235C2C2×#1313327811yescharacteristic
236C2#1681161011yescharacteristic
subgroupG/H
centerC22C2×C22⋊#6157
commutatorC23×C53C22×C4
FrattiniC23C22×#6157
FittingC2×C22⋊C4×C53C4
socleC22×C53C2×C22⋊C4
hypercenterC2×C22⋊C4#6157
hypocenterC53#1901
LCSG ▷ C23×C53 ▷ C2×C53 ▷ C53
UCSC1 ◁ C22 ◁ C24 ◁ C2×C22⋊C4

Complements

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

NH ≅ G/Ntype
2. C2×#13133278C2semidirect
4. #16811503C2semidirect
100. C22×C53#101semidirect
124. C53#1901semidirect
219. C22#13133278semidirect