Finite Group Databasebeta

← #20277489

Subgroups — #20277489

Order 6784.

476 conjugacy classes of subgroups.

The subgroup lattice has 360 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#13133212C211yesmaximal, characteristic
2#16810519C211yesmaximal, characteristic, complemented
3#16810515C211yesmaximal, characteristic
4#16810514C211yesmaximal, characteristic
5#16810513C211yesmaximal, characteristic
6#16810509C211yesmaximal, characteristic, complemented
7#16810508C211yesmaximal, characteristic, complemented
8#16810504C211yesmaximal, characteristic, complemented
9#16810494C211yesmaximal, characteristic
10#16810494C211yesmaximal, characteristic
11#16810488C211yesmaximal, characteristic, complemented
12C4×#11719919C211yesmaximal, characteristic, complemented
13#16810474C211yesmaximal, characteristic, complemented
14#16810470C211yesmaximal, characteristic
15C53×#389C211yesmaximal, characteristic, complemented
16C22×C4×D53C2211yescharacteristic
17#13133212C418yescomplemented
18C2×C4⋊Dic53C2211yescharacteristic
19C2×C4⋊Dic53C2211yescharacteristic
20C2×#11719919C2211yescharacteristic
21C2×#11719919C2211yescharacteristic
22C2×#11719919C2211yescharacteristic
23C2×#11719919C2211yescharacteristic
24C2×#11719921C2211yescharacteristic
25C2×#11719921C2211yescharacteristic
26C2×#11719921C2211yescharacteristic
27C2×#11719921C2211yescharacteristic
28C2×#11719921C2211yescharacteristic
29C2×#11719921C2211yescharacteristic
30C2×C4×Dic53C2211yescharacteristic
31C2×C4×Dic53C2211yescharacteristic
32C2×C4×Dic53C2211yescharacteristic
33C2×C4×Dic53C2211yescharacteristic
34C2×C4×Dic53C2211yescharacteristic
35#13133124C2211yescharacteristic
36#13133124C2211yescharacteristic
37#13133124C2211yescharacteristic
38#13133124C2211yescharacteristic
39#13133124C2211yescharacteristic
40#13133124C2211yescharacteristic
41#13133124C2211yescharacteristic
42#13133124C2211yescharacteristic
43#13133124C2211yescharacteristic
44#13133124C2211yescharacteristic
45C2×C4⋊C4×C53C2211yescharacteristic
46C2×C4⋊C4×C53C2211yescharacteristic
47C2×C4⋊C4×C53C2211yescharacteristic
48C2×C4⋊C4×C53C2211yescharacteristic
49C2×C42×C53C2211yescharacteristic
50#95×C53C2211yescharacteristic
51#95×C53C2211yescharacteristic
52C23×D53C2311yescharacteristic
53C22×C4×C53C2311yescharacteristic
54C22×C4×C53C2311yescharacteristic
55C22×C4×C53C2311yescharacteristic
56C22×C4×C53C2311yescharacteristic
57C22×C4×C53C2311yescharacteristic
58C22×C4×C53C2311yescharacteristic
59C22×C4×C53C2311yescharacteristic
60C22×Dic53C2311yescharacteristic
61C22×Dic53C2311yescharacteristic
62C22×Dic53C2311yescharacteristic
63C22×Dic53C2311yescharacteristic
64C22×Dic53C2311yescharacteristic
65C22×Dic53C2311yescharacteristic
66C22×Dic53C2311yescharacteristic
67C2×C4×D53—22no—
68C2×C4×D53—41no—
69C2×C4×D53—41no—
70C4⋊C4×C53—41no—
71C4⋊C4×C53—41no—
72C4⋊C4×C53—41no—
73C4⋊C4×C53C2×C414yescomplemented
74C42×C53—41no—
75#11719921—22no—
76#11719921—41no—
77#11719921—41no—
78#11719921C2×C414yes—
79#11719921C2×C414yes—
80#11719921C2×C414yes—
81C4⋊Dic53—41no—
82C4⋊Dic53C2×C414yescomplemented
83#11719919—22no—
84#11719919—22no—
85#11719919—22no—
86#11719919C2×C414yescomplemented
87C4×Dic53—22no—
88C4×Dic53—41no—
89C4×Dic53—41no—
90C4×Dic53—41no—
91C4×Dic53C2×C414yescomplemented
92C23×C53C2411yescharacteristic
93C22×D53—21no—
94C22×D53—21no—
95C22×D53—21no—
96C22×D53—21no—
97C22×D53—21no—
98C22×D53—21no—
99C22×D53C22×C412yes—
100C2×C4×C53—21no—
101C2×C4×C53—21no—
102C2×C4×C53—21no—
103C2×C4×C53—21no—
104C2×C4×C53—21no—
105C2×C4×C53—21no—
106C2×C4×C53—21no—
107C2×C4×C53—21no—
108C2×C4×C53—21no—
109C2×C4×C53—21no—
110C2×C4×C53—21no—
111C2×C4×C53—21no—
112C2×C4×C53—21no—
113C2×C4×C53—21no—
114C2×C4×C53—21no—
115C2×C4×C53—21no—
116C2×C4×C53—21no—
117C2×C4×C53C22×C412yes—
118C2×C4×C53C22×C412yes—
119C2×C4×C53C22×C412yes—
120C2×C4×C53#4112yes—
121C2×Dic53—21no—
122C2×Dic53—21no—
123C2×Dic53—21no—
124C2×Dic53—21no—
125C2×Dic53—21no—
126C2×Dic53—21no—
127C2×Dic53—21no—
128C2×Dic53—21no—
129C2×Dic53—21no—
130C2×Dic53—21no—
131C2×Dic53—21no—
132C2×Dic53—21no—
133C2×Dic53—21no—
134C2×Dic53—21no—
135C2×Dic53—21no—
136C2×Dic53—21no—
137C2×Dic53—21no—
138C2×Dic53C22×C412yes—
139C2×Dic53C22×C412yes—
140C2×Dic53C22×C412yes—
141C2×Dic53#4112yes—
142C4×D53—44no—
143C22×C53C23×C411yescharacteristic
144C22×C53C2×#4111yescharacteristic
145C22×C53C2×#4111yescharacteristic
146C22×C53H(5,𝔽2)11yescharacteristic
147C22×C53H(5,𝔽2)11yescharacteristic
148C22×C532−1+411yescharacteristic
149C22×C532−1+411yescharacteristic
150C2×D53—22no—
151C2×D53—22no—
152C2×D53—22no—
153C2×D53—41no—
154C2×D53—41no—
155C2×D53—41no—
156C2×D53—41no—
157C4×C53—22no—
158C4×C53—22no—
159C4×C53—22no—
160C4×C53—22no—
161C4×C53—41no—
162C4×C53—41no—
163C4×C53—41no—
164Dic53—22no—
165Dic53—22no—
166Dic53—22no—
167Dic53—22no—
168Dic53—41no—
169Dic53—41no—
170Dic53—41no—
171#2334—531nomaximal, Sylow
172C2×C53C4×#4111yescharacteristic
173C2×C53#52011yescharacteristic
174C2×C53#52011yescharacteristic
175C2×C53#53811yescharacteristic
176C2×C53#53911yescharacteristic
177C2×C53#54111yescharacteristic
178C2×C53#54211yescharacteristic
179D53—42no—
180C2×#126—531no—
181#389—531no—
182#389D5311yescharacteristic, complemented
183#388—531no—
184#388—531no—
185#388—531no—
186#387—531no—
187#387—531no—
188#387—531no—
189#385—531no—
190#385—531no—
191#382—531no—
192C4×C4⋊C4—531no—
193C4×C4⋊C4—531no—
194C4×C22⋊C4—531no—
195C53#233411yescharacteristic, complemented, Sylow
196C23×C4—531no—
197#126—538no—
198C2×C4⋊C4—531no—
199C2×C4⋊C4—531no—
200C2×C4⋊C4—531no—
201C2×C4⋊C4—531no—
202C2×C4⋊C4—531no—
203C2×C4⋊C4—531no—
204C2×C4⋊C4C2×D5311yescharacteristic
205C2×C4⋊C4C2×D5311yescharacteristic
206C2×C4⋊C4C2×D5311yescharacteristic
207C2×C4⋊C4C2×D5311yescharacteristic
208C2×C22⋊C4—531no—
209C2×C22⋊C4—531no—
210C2×C22⋊C4—531no—
211C2×C22⋊C4—531no—
212C2×C22⋊C4—531no—
213C2×C22⋊C4—531no—
214C2×C42—531no—
215C2×C42—531no—
216C2×C42—531no—
217C2×C42—531no—
218C2×C42—531no—
219C2×C42C2×D5311yescharacteristic
220#95—531no—
221#95—531no—
222#95—531no—
223#95—531no—
224#95—531no—
225#95—531no—
226#95—531no—
227#95—531no—
228#95—531no—
229#95—531no—
230#95C2×D5311yescharacteristic
231#95C2×D5311yescharacteristic
232C24—531no—
233C22×C4—531no—
234C22×C4—531no—
235C22×C4—531no—
236C22×C4—531no—
237C22×C4—531no—
238C22×C4—531no—
239C22×C4—531no—
240C22×C4—1062no—
241C22×C4—2121no—
242C22×C4—2121no—
243C22×C4C22×D5311yescharacteristic
244C22×C4C22×D5311yescharacteristic
245C22×C4C22×D5311yescharacteristic
246C22×C4C22×D5311yescharacteristic
247C22×C4C22×D5311yescharacteristic
248C22×C4C22×D5311yescharacteristic
249C22×C4C22×D5311yescharacteristic
250C4⋊C4—41no—
251C4⋊C4—41no—
252C4⋊C4—41no—
253C4⋊C4—534no—
254C4⋊C4—534no—
255C4⋊C4—1062no—
256C4⋊C4—1062no—
257C4⋊C4—1062no—
258C4⋊C4—2121no—
259C4⋊C4C4×D5314yescomplemented
260C22⋊C4—534no—
261C22⋊C4—534no—
262C22⋊C4—534no—
263C22⋊C4—1062no—
264C22⋊C4—2121no—
265C22⋊C4—2121no—
266C42—41no—
267C42—534no—
268C42—1062no—
269C42—2121no—
270C42—2121no—
271C42—2121no—
272C23—532no—
273C23—1061no—
274C23—1061no—
275C23—1061no—
276C23—1061no—
277C23—1061no—
278C23—1061no—
279C23C23×D5311yescharacteristic
280C2×C4—21no—
281C2×C4—21no—
282C2×C4—21no—
283C2×C4—21no—
284C2×C4—21no—
285C2×C4—21no—
286C2×C4—21no—
287C2×C4—21no—
288C2×C4—21no—
289C2×C4—21no—
290C2×C4—21no—
291C2×C4—21no—
292C2×C4—21no—
293C2×C4—21no—
294C2×C4—21no—
295C2×C4—21no—
296C2×C4—21no—
297C2×C4—532no—
298C2×C4—532no—
299C2×C4—532no—
300C2×C4—532no—
301C2×C4—1061no—
302C2×C4—1061no—
303C2×C4—1061no—
304C2×C4—1061no—
305C2×C4—1061no—
306C2×C4—1061no—
307C2×C4—1061no—
308C2×C4—1061no—
309C2×C4—1061no—
310C2×C4—1061no—
311C2×C4—1061no—
312C2×C4—1061no—
313C2×C4—1061no—
314C2×C4—1061no—
315C2×C4—1061no—
316C2×C4—1061no—
317C2×C4—1061no—
318C2×C4—2124no—
319C2×C4C2×C4×D5312yes—
320C2×C4C2×C4×D5312yes—
321C2×C4C2×C4×D5312yes—
322C2×C4#1171994912yes—
323C22—1062no—
324C22—1062no—
325C22—1062no—
326C22—2121no—
327C22—2121no—
328C22—2121no—
329C22—2121no—
330C22C22×C4×D5311yescharacteristic
331C22#41×D5311yescharacteristic
332C22C2×#1171994911yescharacteristic
333C22#1313326211yescharacteristic
334C22#1313326211yescharacteristic
335C22#1313326411yescharacteristic
336C22#1313326511yescharacteristic
337C4—22no—
338C4—22no—
339C4—22no—
340C4—22no—
341C4—41no—
342C4—41no—
343C4—41no—
344C4—1062no—
345C4—1062no—
346C4—1062no—
347C4—1062no—
348C4—2121no—
349C4—2121no—
350C4—2121no—
351C2—2122no—
352C2C4×#1171994911yescharacteristic
353C2#1681110211yescharacteristic
354C2#1681110411yescharacteristic
355C2#1681120511yescharacteristic
356C2#1681122211yescharacteristic
357C2#1681123211yescharacteristic
358C2#1681124711yescharacteristic
subgroupG/H
center, FrattiniC23C23×D53
commutatorC22×C53C23×C4
FittingC53×#389C2
socleC23×C53C24
hypercenter#389D53
hypocenterC53#2334
LCSG ▷ C22×C53 ▷ C53
UCSC1 ◁ C23 ◁ #389

Complements

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

NH ≅ G/Ntype
2. #16810519C2semidirect
6. #16810509C2semidirect
7. #16810508C2semidirect
8. #16810504C2semidirect
11. #16810488C2semidirect
12. C4×#11719919C2semidirect
13. #16810474C2semidirect
15. C53×#389C2semidirect
17. #13133212C4semidirect
73. C4⋊C4×C53C2×C4semidirect
82. C4⋊Dic53C2×C4semidirect
86. #11719919C2×C4semidirect
91. C4×Dic53C2×C4semidirect
182. #389D53semidirect
195. C53#2334semidirect
259. C4⋊C4C4×D53semidirect