Finite Group Databasebeta

← #20275254

Subgroups — #20275254

Order 6784.

380 conjugacy classes of subgroups; only the 79 normal subgroups are shown.

normal subgroups only

This group has 380 conjugacy classes of subgroups — too many to draw. Only its 79 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
1#16811182C211yesmaximal, characteristic
2#16811096C211yesmaximal, characteristic, complemented
3C2×#13133132C211yesmaximal, characteristic, complemented
4C2×#13133131C211yesmaximal, characteristic, complemented
5#16810390C211yesmaximal, characteristic, complemented
6#16810389C211yesmaximal, characteristic, complemented
7C2×#99×C53C211yesmaximal, characteristic, complemented
8C22×C22⋊D53C2211yescharacteristic
9#13133200C412yescomplemented
10#13133198—21no—
11#13133189—22no—
12#13133186C412yescomplemented
13#13133185—21no—
14C4×C22⋊D53—22no—
15#13133177—22no—
16#13133173—21no—
17C2×C22⋊Dic53C2211yescharacteristic
18C2×C22⋊Dic53C2211yescharacteristic
19C2×C22⋊Dic53C2211yescharacteristic
20C22⋊C4×D53—21no—
21#13133132—22no—
22#13133131—22no—
23C22×D4×C53C2211yescharacteristic
24C2×C22⋊C4×C53C2211yescharacteristic
25C2×C22⋊C4×C53C2211yescharacteristic
26#99×C53—22no—
27C24×C53C2×C411yescharacteristic, complemented
28C24×C53C2311yescharacteristic
29C23×D53D411yescharacteristic
30C2×D4×C53—21no—
31C2×D4×C53—21no—
32C2×D4×C53—21no—
33C2×D4×C53—21no—
34C2×D4×C53—21no—
35C2×D4×C53—21no—
36C22×C4×C53—21no—
37C22×C4×C53—21no—
38C22×C4×C53C2×C411yescharacteristic, complemented
39C2×C22⋊D53—21no—
40C2×C22⋊D53—21no—
41C2×C22⋊D53—21no—
42C2×C22⋊D53—21no—
43C2×C22⋊D53—22no—
44C2×C22⋊D53C2×C412yes—
45C2×C22⋊D53D412yes—
46C22×Dic53—21no—
47C22×Dic53—21no—
48C22×Dic53—21no—
49C22×Dic53D411yescharacteristic
50C2×C4×D53—22no—
51C22⋊C4×C53—21no—
52C22⋊C4×C53—21no—
53C22⋊C4×C53—21no—
54C22⋊C4×C53D412yescomplemented
55C22⋊Dic53—21no—
56C22⋊Dic53—21no—
57C22⋊Dic53—22no—
58C22⋊Dic53—21no—
59C22⋊Dic53—21no—
60C22⋊Dic53C2×C412yescomplemented
61C22⋊Dic53D412yescomplemented
62#11719921—22no—
63#11719921—22no—
64#11719919—22no—
65#11719919—22no—
66C4×Dic53—22no—
67C23×C53—21no—
68C23×C53—21no—
69C23×C53—21no—
70C23×C53—81no—
71C23×C53—81no—
72C23×C53C22×C411yescharacteristic
73C23×C53C2×D411yescharacteristic
74C23×C53C2×D411yescharacteristic
75C23×C53C2×D411yescharacteristic
76C23×C53C2×D411yescharacteristic
77C23×C53#4111yescharacteristic
78C23×C53#4111yescharacteristic
79C22×D53—21no—
80C22×D53—21no—
81C22×D53—81no—
82C22×D53C22⋊C412yes—
83D4×C53—41no—
84D4×C53—41no—
85D4×C53—41no—
86D4×C53—41no—
87C2×C4×C53—21no—
88C2×C4×C53—21no—
89C2×C4×C53—21no—
90C2×C4×C53—22no—
91C2×C4×C53—41no—
92C2×C4×C53—81no—
93C2×C4×C53—81no—
94C22⋊D53—22no—
95C22⋊D53—22no—
96C22⋊D53—22no—
97C22⋊D53—22no—
98C22⋊D53—82no—
99C2×Dic53—21no—
100C2×Dic53—21no—
101C2×Dic53—22no—
102C2×Dic53—22no—
103C2×Dic53—41no—
104C2×Dic53—81no—
105C2×Dic53—81no—
106C2×Dic53—81no—
107C2×Dic53C22⋊C412yes—
108C4×D53—82no—
109C22×C53—41no—
110C22×C53—41no—
111C22×C53—41no—
112C22×C53—41no—
113C22×C53—41no—
114C22×C53—41no—
115C22×C53—41no—
116C22×C53—42no—
117C22×C53—41no—
118C22×C53—81no—
119C22×C53—81no—
120C22×C53C2×C22⋊C411yescharacteristic
121C22×C53C4×D411yescharacteristic
122C22×C53C4×D411yescharacteristic
123C22×C53#12011yescharacteristic
124C22×C53#12111yescharacteristic
125C22×C53#12111yescharacteristic
126C22×C53#12311yescharacteristic
127C2×D53—22no—
128C2×D53—42no—
129C2×D53—42no—
130C2×D53—81no—
131C4×C53—42no—
132C4×C53—41no—
133C4×C53—81no—
134Dic53—22no—
135Dic53—42no—
136Dic53—42no—
137Dic53—81no—
138#1913—531nomaximal, Sylow
139C2×C53—21no—
140C2×C53—21no—
141C2×C53—21no—
142C2×C53—21no—
143C2×C53—21no—
144C2×C53—21no—
145C2×C53—21no—
146C2×C53—81no—
147C2×C53#38611yescharacteristic
148D53—42no—
149#534—531no—
150#518—531no—
151C2×#99—531no—
152C2×#99—531no—
153C2×#99D5311yescharacteristic, complemented
154#342—531no—
155#342—531no—
156C53#191311yescharacteristic, complemented, Sylow
157C22×D4—531no—
158C22×D4C2×D5311yescharacteristic
159#123—532no—
160#123—1061no—
161#121—1062no—
162#120—532no—
163#120—1061no—
164C4×D4—1062no—
165C4×D4—1062no—
166#117—1061no—
167C2×C22⋊C4—531no—
168C2×C22⋊C4—531no—
169C2×C22⋊C4—531no—
170C2×C22⋊C4—1061no—
171C2×C22⋊C4C2×D5311yescharacteristic
172C2×C22⋊C4C2×D5311yescharacteristic
173#99—22no—
174#99—1062no—
175#99—1062no—
176C24—531no—
177C24C4×D5311yescharacteristic, complemented
178C24C22×D5311yescharacteristic
179C2×D4—21no—
180C2×D4—21no—
181C2×D4—21no—
182C2×D4—21no—
183C2×D4—21no—
184C2×D4—21no—
185C2×D4—532no—
186C2×D4—532no—
187C2×D4—1061no—
188C2×D4—1061no—
189C2×D4—1061no—
190C2×D4—1061no—
191C2×D4—1062no—
192C22×C4—21no—
193C22×C4—21no—
194C22×C4—531no—
195C22×C4—1062no—
196C22×C4—1061no—
197C22×C4—1061no—
198C22×C4—1061no—
199C22×C4C4×D5311yescharacteristic, complemented
200C4⋊C4—1062no—
201C4⋊C4—1062no—
202C22⋊C4—21no—
203C22⋊C4—21no—
204C22⋊C4—21no—
205C22⋊C4—532no—
206C22⋊C4—532no—
207C22⋊C4—1062no—
208C22⋊C4—1062no—
209C22⋊C4—1062no—
210C22⋊C4—1061no—
211C22⋊C4—1061no—
212C22⋊C4—1061no—
213C22⋊C4—1061no—
214C22⋊C4C22⋊D5312yescomplemented
215C42—1062no—
216C23—21no—
217C23—21no—
218C23—21no—
219C23—81no—
220C23—81no—
221C23—532no—
222C23—1061no—
223C23—1061no—
224C23—4241no—
225C23C2×C4×D5311yescharacteristic
226C23#1171994511yescharacteristic
227C23D4×D5311yescharacteristic
228C23D4×D5311yescharacteristic
229C23D4×D5311yescharacteristic
230C23#1171994711yescharacteristic
231C23C2×C22⋊D5311yescharacteristic
232D4—41no—
233D4—41no—
234D4—41no—
235D4—41no—
236D4—1062no—
237D4—1062no—
238D4—1062no—
239D4—1062no—
240D4—4242no—
241C2×C4—21no—
242C2×C4—21no—
243C2×C4—21no—
244C2×C4—22no—
245C2×C4—41no—
246C2×C4—81no—
247C2×C4—81no—
248C2×C4—532no—
249C2×C4—1061no—
250C2×C4—1061no—
251C2×C4—1062no—
252C2×C4—1062no—
253C2×C4—4242no—
254C2×C4—4241no—
255C2×C4—4241no—
256C2×C4—4241no—
257C2×C4—2121no—
258C22—41no—
259C22—41no—
260C22—41no—
261C22—41no—
262C22—41no—
263C22—41no—
264C22—41no—
265C22—42no—
266C22—41no—
267C22—81no—
268C22—81no—
269C22—1062no—
270C22—4241no—
271C22—2122no—
272C22—2122no—
273C22C22⋊C4×D5311yescharacteristic
274C22#1313317711yescharacteristic
275C22C4×C22⋊D5311yescharacteristic
276C22#1313318511yescharacteristic
277C22#1313318811yescharacteristic
278C22#1313319011yescharacteristic
279C22#1313319911yescharacteristic
280C4—42no—
281C4—41no—
282C4—81no—
283C4—1062no—
284C4—4241no—
285C4—2122no—
286C4—2122no—
287C2—21no—
288C2—21no—
289C2—21no—
290C2—21no—
291C2—21no—
292C2—21no—
293C2—21no—
294C2—81no—
295C2—2122no—
296C2#1681049911yescharacteristic
subgroupG/H
centerC2#16810499
commutatorC23×C53C22×C4
FrattiniC24C22×D53
FittingC2×#99×C53C2
socleC2×C53#386
hypercenterC2×#99D53
hypocenterC53#1913
LCSG ▷ C23×C53 ▷ C2×C53 ▷ C53
UCSC1 ◁ C2 ◁ C24 ◁ C2×#99

Complements

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

NH ≅ G/Ntype
2. #16811096C2semidirect
3. C2×#13133132C2semidirect
4. C2×#13133131C2semidirect
5. #16810390C2semidirect
6. #16810389C2semidirect
7. C2×#99×C53C2semidirect
9. #13133200C4semidirect
12. #13133186C4semidirect
27. C24×C53C2×C4semidirect
38. C22×C4×C53C2×C4semidirect
54. C22⋊C4×C53D4semidirect
60. C22⋊Dic53C2×C4semidirect
61. C22⋊Dic53D4semidirect
153. C2×#99D53semidirect
156. C53#1913semidirect
177. C24C4×D53semidirect
199. C22×C4C4×D53semidirect
214. C22⋊C4C22⋊D53semidirect