Finite Group Databasebeta

← #20284012

Subgroups — #20284012

Order 6784.

700 conjugacy classes of subgroups; only the 149 normal subgroups are shown.

normal subgroups only

This group has 700 conjugacy classes of subgroups — too many to draw. Only its 149 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×#13133264C211yesmaximal, characteristic
2C2×#13133250C211yesmaximal, characteristic, complemented
3C22×Dic212C211yesmaximal, characteristic, complemented
4#16811194C211yesmaximal, characteristic, complemented
5#16810787C214yesmaximal, complemented
6#16810781C214yesmaximal, complemented
7#16810633C211yesmaximal, characteristic, complemented
8#16810620C211yesmaximal, characteristic, complemented
9C53×#408C211yesmaximal, characteristic, complemented
10#13133264—24no—
11#13133264—24no—
12C2×#11719945C2212yes—
13C2×#11719945C2211yescharacteristic
14C2×#11719947—22no—
15C2×#11719947—21no—
16C22×Dic106C2212yes—
17C22×Dic106C2211yescharacteristic
18C22×Dic106C2211yescharacteristic
19C2×Q8×D53—21no—
20#13133250—24no—
21C2×Dic212—24no—
22C2×Dic212C2212yes—
23C2×Dic212C2214yes—
24C2×#11719913C2212yes—
25#13133215—21no—
26#13133204—21no—
27#13133198—22no—
28#13133196C2214yes—
29#13133176C2211yescharacteristic
30#13133142C2212yes—
31#13133142C2214yes—
32#13133139C2212yes—
33C2×#41×C53C2211yescharacteristic
34C2×M16×C53C2211yescharacteristic
35C22×C8×C53C2211yescharacteristic
36C22⋊C8×C53C2214yes—
37#41×C53—22no—
38#41×C53—22no—
39C2×Q8×C53D411yescharacteristic
40C2×D4×C53D412yes—
41C2×D4×C53D411yescharacteristic
42C22×C4×C53C2312yes—
43C22×C4×C53C2311yescharacteristic
44C2×C22⋊D53—22no—
45C2×C22⋊D53—21no—
46C22×Dic53—22no—
47C22×Dic53—21no—
48C22×Dic53—21no—
49Q8×D53—28no—
50#11719947—216no—
51#11719947—28no—
52#11719945—22no—
53#11719945—24no—
54#11719945—24no—
55#11719945D414yes—
56C2×D212C2311yescharacteristic
57C2×C4×D53—22no—
58C2×C4×D53—21no—
59C2×Dic106—22no—
60C2×Dic106—21no—
61C2×Dic106—22no—
62C2×Dic106—24no—
63C2×Dic106—24no—
64C2×Dic106—22no—
65C2×Dic106—22no—
66C2×Dic106—21no—
67C2×Dic106D414yes—
68C2×Dic106C2312yes—
69C2×Dic106C2311yescharacteristic
70C2×Dic106C2312yes—
71M16×C53—22no—
72C2×C8×C53—22no—
73C2×C8×C53C2312yes—
74C2×C8×C53C2312yes—
75C22⋊Dic53—21no—
76C22⋊Dic53—24no—
77C4⋊Dic53C2312yes—
78#11719919—24no—
79C4×Dic53—21no—
80Dic212—28no—
81Dic212—24no—
82#11719913—24no—
83C23×C53C2×D412yes—
84C23×C53C2×D411yescharacteristic
85C22×D53—21no—
86Q8×C53—22no—
87D4×C53—24no—
88D4×C53—22no—
89C2×C4×C53—22no—
90C2×C4×C53—22no—
91C2×C4×C53C2×D412yes—
92C2×C4×C53C2×D411yescharacteristic
93C2×C4×C53C2×D412yes—
94C2×C4×C53C2×D412yes—
95C2×C4×C53C2411yescharacteristic
96C22⋊D53—28no—
97C22⋊D53—24no—
98C2×Dic53—28no—
99C2×Dic53—24no—
100C2×Dic53—21no—
101C2×Dic53—22no—
102C2×Dic53—22no—
103C2×Dic53—22no—
104C2×Dic53—42no—
105D212—21no—
106D212C2×D412yes—
107C4×D53—28no—
108C4×D53—24no—
109Dic106—28no—
110Dic106—24no—
111Dic106—22no—
112Dic106—21no—
113Dic106—22no—
114Dic106—22no—
115Dic106—42no—
116Dic106C2×D412yes—
117Dic106C2×D414yes—
118C8×C53—22no—
119C8×C53—22no—
120C22×C53—22no—
121C22×C53—22no—
122C22×C53—24no—
123C22×C53#12012yes—
124C22×C53C22×D411yescharacteristic
125C2×D53—22no—
126C2×D53—42no—
127C4×C53—22no—
128C4×C53#12012yes—
129C4×C53C22×D411yescharacteristic
130C4×C53C22×D411yescharacteristic
131Dic53—22no—
132Dic53—24no—
133Dic53—42no—
134Dic53—42no—
135#3011—531nomaximal, Sylow
136C2×C53—22no—
137C2×C53—42no—
138C2×C53C2×#12011yescharacteristic
139C2×C53#57812yes—
140D53—42no—
141C2×2−1+4—531no—
142C2×#137—531no—
143C22×Q16—531no—
144#536—531no—
145#452—534no—
146#451—534no—
147#419—531no—
148#416—531no—
149#408D5311yescharacteristic, complemented
150C53#301111yescharacteristic, complemented, Sylow
1512−1+4—1064no—
1522−1+4—1064no—
153C2×#41—531no—
154C2×#41—532no—
155C2×#41—1062no—
156C2×#41—1061no—
157C2×#41C2×D5311yescharacteristic
158C22×Q8—531no—
159C22×Q8—531no—
160C22×Q8—532no—
161C22×Q8—1061no—
162#137—1064no—
163C2×Q16—534no—
164C2×Q16—532no—
165C2×Q16—1064no—
166C2×SD16—532no—
167C2×M16C2×D5311yescharacteristic
168C22×C8C2×D5311yescharacteristic
169C4⋊Q8—1061no—
170#124—1061no—
171#123—1062no—
172C22⋊Q8—534no—
173#117—531no—
174#103—532no—
175#103—534no—
176#102—532no—
177C22⋊C8C2×D5314yes—
178#41—22no—
179#41—22no—
180#41—534no—
181#41—1064no—
182#41—1064no—
183#41—10616no—
184#41—1068no—
185#41—1062no—
186C2×Q8—534no—
187C2×Q8—532no—
188C2×Q8—531no—
189C2×Q8—532no—
190C2×Q8—1064no—
191C2×Q8—1064no—
192C2×Q8—1068no—
193C2×Q8—1062no—
194C2×Q8—1061no—
195C2×Q8—1061no—
196C2×Q8—1062no—
197C2×Q8—1062no—
198C2×Q8—1062no—
199C2×Q8D21211yescharacteristic
200C2×D4—531no—
201C2×D4—1062no—
202C2×D4—1061no—
203C2×D4D21212yes—
204C2×D4D21211yescharacteristic
205C22×C4—1062no—
206C22×C4—1061no—
207C22×C4—1062no—
208C22×C4—1061no—
209C22×C4—1061no—
210C22×C4C22×D5312yes—
211C22×C4C22×D5311yescharacteristic
212Q16—1068no—
213Q16—1064no—
214SD16—1064no—
215M16—22no—
216C2×C8—22no—
217C2×C8C22×D5312yes—
218C2×C8C22×D5312yes—
219C4⋊C4—532no—
220C4⋊C4—1064no—
221C22⋊C4—1061no—
222C22⋊C4—1064no—
223C42—1061no—
224C23—1061no—
225C23C2×D21212yes—
226C23C2×D21211yescharacteristic
227Q8—22no—
228Q8—534no—
229Q8—532no—
230Q8—1068no—
231Q8—1064no—
232Q8—1062no—
233Q8—1061no—
234Q8—1062no—
235Q8—1062no—
236Q8—2122no—
237D4—24no—
238D4—22no—
239D4—532no—
240D4—1068no—
241D4—1064no—
242D4—1061no—
243C2×C4—22no—
244C2×C4—22no—
245C2×C4—1068no—
246C2×C4—1068no—
247C2×C4—1064no—
248C2×C4—1062no—
249C2×C4—1061no—
250C2×C4—1064no—
251C2×C4—1062no—
252C2×C4—1062no—
253C2×C4—2122no—
254C2×C4C2×D21212yes—
255C2×C4C2×D21211yescharacteristic
256C2×C4D4×D5312yes—
257C2×C4D4×D5312yes—
258C2×C4C23×D5311yescharacteristic
259C8—22no—
260C8—22no—
261C22—22no—
262C22—22no—
263C22—24no—
264C22—1062no—
265C22—2122no—
266C22#1313318712yes—
267C22C22×D21211yescharacteristic
268C4—22no—
269C4—1064no—
270C4—1062no—
271C4—2122no—
272C4—2122no—
273C4#1313318712yes—
274C4C2×D4×D5311yescharacteristic
275C4C2×D4×D5311yescharacteristic
276C2—22no—
277C2—42no—
278C2—2122no—
279C2C2×#1313318711yescharacteristic
280C2#1681144012yes—
subgroupG/H
centerC22C22×D212
commutatorC2×C4×C53C24
FrattiniC2×C4C23×D53
FittingC53×#408C2
socleC22×C53C22×D4
hypercenter#408D53
hypocenterC53#3011
LCSG ▷ C2×C4×C53 ▷ C2×C53 ▷ C53
UCSC1 ◁ C22 ◁ C2×#41 ◁ #408

Complements

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

NH ≅ G/Ntype
2. C2×#13133250C2semidirect
3. C22×Dic212C2semidirect
4. #16811194C2semidirect
5. #16810787C2semidirect
6. #16810781C2semidirect
7. #16810633C2semidirect
8. #16810620C2semidirect
9. C53×#408C2semidirect
149. #408D53semidirect
150. C53#3011semidirect