Finite Group Databasebeta

← #20277388

Subgroups — #20277388

Order 6784.

632 conjugacy classes of subgroups.

The subgroup lattice has 414 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#13133202C211yesmaximal, characteristic
2C2×C4×D212C211yesmaximal, characteristic
3C2×C4⋊C4×D53C211yesmaximal, characteristic, complemented
4#16810522C211yesmaximal, characteristic, complemented
5#16810520C211yesmaximal, characteristic, complemented
6#16810515C211yesmaximal, characteristic, complemented
7#16810512C211yesmaximal, characteristic, complemented
8#16810507C211yesmaximal, characteristic, complemented
9#16810501C211yesmaximal, characteristic
10#16810497C211yesmaximal, characteristic, complemented
11#16810497C211yesmaximal, characteristic, complemented
12#16810494C211yesmaximal, characteristic, complemented
13#16810485C211yesmaximal, characteristic, complemented
14#16810470C211yesmaximal, characteristic, complemented
15C53×#387C211yesmaximal, characteristic, complemented
16C22×D212C2211yescharacteristic
17C22×C4×D53C2211yescharacteristic
18C22×C4×D53C2211yescharacteristic
19C22×C4×D53C2211yescharacteristic
20C22×C4×D53C2211yescharacteristic
21#13133202C418yescomplemented
22C4×D212—24no—
23C2×C4⋊Dic53C2211yescharacteristic
24C2×C4⋊Dic53C2211yescharacteristic
25C2×C4⋊Dic53C2211yescharacteristic
26C2×#11719919C2211yescharacteristic
27C2×#11719919C2211yescharacteristic
28C2×#11719919C2211yescharacteristic
29C2×#11719919C2211yescharacteristic
30C4⋊C4×D53—24no—
31C2×#11719921C2211yescharacteristic
32C2×#11719921C2211yescharacteristic
33C2×#11719921C2211yescharacteristic
34C2×#11719921C2211yescharacteristic
35C2×#11719921C2211yescharacteristic
36C2×#11719921C2211yescharacteristic
37C2×#11719921C2211yescharacteristic
38C2×#11719921C2211yescharacteristic
39C2×C4×Dic53C2211yescharacteristic
40C2×C4×Dic53C2211yescharacteristic
41#13133124C2211yescharacteristic
42#13133124C2211yescharacteristic
43#13133124C2211yescharacteristic
44#13133124C2211yescharacteristic
45#13133124C2211yescharacteristic
46#13133124C2211yescharacteristic
47C2×C4⋊C4×C53C2211yescharacteristic
48C2×C4⋊C4×C53C2211yescharacteristic
49C2×C42×C53C2211yescharacteristic
50#95×C53C2211yescharacteristic
51#95×C53C2211yescharacteristic
52#95×C53C2211yescharacteristic
53#95×C53C2211yescharacteristic
54C23×D53C2311yescharacteristic
55C23×D53C2311yescharacteristic
56C22×C4×C53C2311yescharacteristic
57C22×C4×C53C2311yescharacteristic
58C22×C4×C53C2311yescharacteristic
59C22×C4×C53C2311yescharacteristic
60C22×C4×C53C2311yescharacteristic
61C22×C4×C53C2311yescharacteristic
62C22×C4×C53C2311yescharacteristic
63C22×Dic53C2311yescharacteristic
64C22×Dic53C2311yescharacteristic
65C22×Dic53C2311yescharacteristic
66C22×Dic53C2311yescharacteristic
67C22×Dic53C2311yescharacteristic
68C22×Dic53C2311yescharacteristic
69C2×D212—22no—
70C2×D212—22no—
71C2×D212C2×C414yes—
72C2×C4×D53—22no—
73C2×C4×D53—22no—
74C2×C4×D53—22no—
75C2×C4×D53—22no—
76C2×C4×D53—41no—
77C2×C4×D53—41no—
78C2×C4×D53—41no—
79C2×C4×D53—41no—
80C2×C4×D53—41no—
81C2×C4×D53—41no—
82C2×C4×D53C2×C414yescomplemented
83C2×C4×D53D414yes—
84C4⋊C4×C53—22no—
85C4⋊C4×C53C2×C414yescomplemented
86C42×C53—22no—
87#11719921—22no—
88#11719921—22no—
89#11719921—22no—
90#11719921—41no—
91#11719921—41no—
92#11719921C2×C414yescomplemented
93#11719921C2×C414yescomplemented
94#11719921C2×C414yescomplemented
95C4⋊Dic53—22no—
96C4⋊Dic53—22no—
97C4⋊Dic53—41no—
98#11719919—22no—
99#11719919—22no—
100#11719919—41no—
101#11719919C2×C414yescomplemented
102C4×Dic53—41no—
103C4×Dic53—41no—
104C23×C53C2411yescharacteristic
105C22×D53—21no—
106C22×D53—21no—
107C22×D53—21no—
108C22×D53—21no—
109C22×D53—21no—
110C22×D53—21no—
111C22×D53—21no—
112C22×D53—21no—
113C22×D53—21no—
114C22×D53—21no—
115C22×D53C22×C412yes—
116C22×D53C22×C412yes—
117C22×D53C2×D412yes—
118C22×D53#4112yes—
119C2×C4×C53—21no—
120C2×C4×C53—21no—
121C2×C4×C53—21no—
122C2×C4×C53—21no—
123C2×C4×C53—21no—
124C2×C4×C53—21no—
125C2×C4×C53—21no—
126C2×C4×C53—21no—
127C2×C4×C53—21no—
128C2×C4×C53—21no—
129C2×C4×C53—21no—
130C2×C4×C53—21no—
131C2×C4×C53—21no—
132C2×C4×C53—21no—
133C2×C4×C53—21no—
134C2×C4×C53—21no—
135C2×C4×C53—21no—
136C2×C4×C53C22×C412yes—
137C2×C4×C53C22×C412yes—
138C2×C4×C53C22×C412yes—
139C2×C4×C53C2×D412yes—
140C2×Dic53—21no—
141C2×Dic53—21no—
142C2×Dic53—21no—
143C2×Dic53—21no—
144C2×Dic53—21no—
145C2×Dic53—21no—
146C2×Dic53—21no—
147C2×Dic53—21no—
148C2×Dic53—21no—
149C2×Dic53—21no—
150C2×Dic53—21no—
151C2×Dic53—21no—
152C2×Dic53—21no—
153C2×Dic53—21no—
154C2×Dic53—21no—
155C2×Dic53C22×C412yes—
156C2×Dic53C22×C412yes—
157C2×Dic53C2×D412yes—
158D212—28no—
159C4×D53—28no—
160C4×D53—44no—
161C4×D53—44no—
162C4×D53—44no—
163C22×C53C23×C411yescharacteristic
164C22×C53C22×D411yescharacteristic
165C22×C53C2×#4111yescharacteristic
166C22×C53H(5,𝔽2)11yescharacteristic
167C22×C53H(5,𝔽2)11yescharacteristic
168C22×C532−1+411yescharacteristic
169C22×C532−1+411yescharacteristic
170C2×D53—22no—
171C2×D53—22no—
172C2×D53—22no—
173C2×D53—22no—
174C2×D53—22no—
175C2×D53—22no—
176C2×D53—22no—
177C2×D53—22no—
178C2×D53—22no—
179C2×D53—41no—
180C2×D53—41no—
181C2×D53—41no—
182C2×D53—41no—
183C2×D53C4×D414yes—
184C4×C53—22no—
185C4×C53—22no—
186C4×C53—22no—
187C4×C53—22no—
188C4×C53—41no—
189C4×C53—41no—
190C4×C53—41no—
191Dic53—22no—
192Dic53—22no—
193Dic53—22no—
194Dic53—41no—
195Dic53—41no—
196Dic53—41no—
197#2322—531nomaximal, Sylow
198C2×C53C2×C4×D411yescharacteristic
199C2×C53#52011yescharacteristic
200C2×C53#52011yescharacteristic
201C2×C53#53411yescharacteristic
202C2×C53#53611yescharacteristic
203C2×C53#53911yescharacteristic
204C2×C53#53911yescharacteristic
205D53—24no—
206D53—42no—
207C2×#123—531no—
208C2×C4×D4—531no—
209C22×C4⋊C4—531no—
210#389—531no—
211#389—531no—
212#388—531no—
213#388—531no—
214#387—531no—
215#387D5311yescharacteristic, complemented
216#386—531no—
217#385—531no—
218#385—531no—
219#385—531no—
220#381—531no—
221C4×C22⋊C4—531no—
222C53#232211yescharacteristic, complemented, Sylow
223C22×D4—531no—
224C23×C4—531no—
225C23×C4—531no—
226C23×C4—531no—
227C23×C4—531no—
228#123—538no—
229C4×D4—1064no—
230C2×C4⋊C4—531no—
231C2×C4⋊C4—531no—
232C2×C4⋊C4—531no—
233C2×C4⋊C4—531no—
234C2×C4⋊C4—531no—
235C2×C4⋊C4—531no—
236C2×C4⋊C4—531no—
237C2×C4⋊C4—1064no—
238C2×C4⋊C4C2×D5311yescharacteristic
239C2×C4⋊C4C2×D5311yescharacteristic
240C2×C22⋊C4—531no—
241C2×C22⋊C4—531no—
242C2×C22⋊C4—531no—
243C2×C22⋊C4—531no—
244C2×C22⋊C4—531no—
245C2×C22⋊C4—531no—
246C2×C22⋊C4—531no—
247C2×C22⋊C4—531no—
248C2×C42—531no—
249C2×C42—531no—
250C2×C42C2×D5311yescharacteristic
251#95—531no—
252#95—531no—
253#95—531no—
254#95—531no—
255#95—531no—
256#95—531no—
257#95C2×D5311yescharacteristic
258#95C2×D5311yescharacteristic
259#95C2×D5311yescharacteristic
260#95C2×D5311yescharacteristic
261C24—531no—
262C24—531no—
263C2×D4—534no—
264C2×D4—1062no—
265C2×D4—1062no—
266C22×C4—531no—
267C22×C4—531no—
268C22×C4—531no—
269C22×C4—534no—
270C22×C4—534no—
271C22×C4—531no—
272C22×C4—531no—
273C22×C4—531no—
274C22×C4—1062no—
275C22×C4—1062no—
276C22×C4—1062no—
277C22×C4—1062no—
278C22×C4—2121no—
279C22×C4—2121no—
280C22×C4—2121no—
281C22×C4—2121no—
282C22×C4—2121no—
283C22×C4—2121no—
284C22×C4C22×D5311yescharacteristic
285C22×C4C22×D5311yescharacteristic
286C22×C4C22×D5311yescharacteristic
287C22×C4C22×D5311yescharacteristic
288C22×C4C22×D5311yescharacteristic
289C22×C4C22×D5311yescharacteristic
290C22×C4C22×D5311yescharacteristic
291C4⋊C4—22no—
292C4⋊C4—534no—
293C4⋊C4—1062no—
294C4⋊C4—1062no—
295C4⋊C4—1062no—
296C4⋊C4—1062no—
297C4⋊C4—2121no—
298C4⋊C4—2121no—
299C4⋊C4C4×D5314yescomplemented
300C22⋊C4—534no—
301C22⋊C4—534no—
302C22⋊C4—534no—
303C22⋊C4—1062no—
304C22⋊C4—1062no—
305C22⋊C4—1062no—
306C22⋊C4—2121no—
307C22⋊C4—2121no—
308C42—22no—
309C42—2121no—
310C42—2121no—
311C23—532no—
312C23—532no—
313C23—532no—
314C23—532no—
315C23—1061no—
316C23—1061no—
317C23—1061no—
318C23—1061no—
319C23—1061no—
320C23—1061no—
321C23—1061no—
322C23—1061no—
323C23—1061no—
324C23—1061no—
325C23C23×D5311yescharacteristic
326D4—1068no—
327C2×C4—21no—
328C2×C4—21no—
329C2×C4—21no—
330C2×C4—21no—
331C2×C4—21no—
332C2×C4—21no—
333C2×C4—21no—
334C2×C4—21no—
335C2×C4—21no—
336C2×C4—21no—
337C2×C4—21no—
338C2×C4—21no—
339C2×C4—21no—
340C2×C4—21no—
341C2×C4—21no—
342C2×C4—21no—
343C2×C4—21no—
344C2×C4—532no—
345C2×C4—532no—
346C2×C4—532no—
347C2×C4—1068no—
348C2×C4—1061no—
349C2×C4—1061no—
350C2×C4—1061no—
351C2×C4—1061no—
352C2×C4—1061no—
353C2×C4—1061no—
354C2×C4—1061no—
355C2×C4—1061no—
356C2×C4—1061no—
357C2×C4—1061no—
358C2×C4—1061no—
359C2×C4—1061no—
360C2×C4—1061no—
361C2×C4—1061no—
362C2×C4—1061no—
363C2×C4—2124no—
364C2×C4—2124no—
365C2×C4—2124no—
366C2×C4C2×C4×D5312yes—
367C2×C4C2×C4×D5312yes—
368C2×C4C2×C4×D5312yes—
369C2×C4D4×D5312yes—
370C22—534no—
371C22—1062no—
372C22—1062no—
373C22—1062no—
374C22—1062no—
375C22—1062no—
376C22—1062no—
377C22—1062no—
378C22—1062no—
379C22—1062no—
380C22—2121no—
381C22—2121no—
382C22—2121no—
383C22—2121no—
384C22C22×C4×D5311yescharacteristic
385C22C2×D4×D5311yescharacteristic
386C22#41×D5311yescharacteristic
387C22#1313326311yescharacteristic
388C22#1313326311yescharacteristic
389C22#1313326411yescharacteristic
390C22#1313326511yescharacteristic
391C4—22no—
392C4—22no—
393C4—22no—
394C4—22no—
395C4—41no—
396C4—41no—
397C4—41no—
398C4—1062no—
399C4—1062no—
400C4—1062no—
401C4—2121no—
402C4—2121no—
403C4—2121no—
404C2—1064no—
405C2—2122no—
406C2C4×D4×D5311yescharacteristic
407C2#1681110311yescharacteristic
408C2#1681110511yescharacteristic
409C2#1681118411yescharacteristic
410C2#1681119611yescharacteristic
411C2#1681121411yescharacteristic
412C2#1681122111yescharacteristic
subgroupG/H
center, FrattiniC23C23×D53
commutatorC22×C53C23×C4
FittingC53×#387C2
socleC23×C53C24
hypercenter#387D53
hypocenterC53#2322
LCSG ▷ C22×C53 ▷ C53
UCSC1 ◁ C23 ◁ #387

Complements

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

NH ≅ G/Ntype
3. C2×C4⋊C4×D53C2semidirect
4. #16810522C2semidirect
5. #16810520C2semidirect
6. #16810515C2semidirect
7. #16810512C2semidirect
8. #16810507C2semidirect
10. #16810497C2semidirect
11. #16810497C2semidirect
12. #16810494C2semidirect
13. #16810485C2semidirect
14. #16810470C2semidirect
15. C53×#387C2semidirect
21. #13133202C4semidirect
82. C2×C4×D53C2×C4semidirect
85. C4⋊C4×C53C2×C4semidirect
92. #11719921C2×C4semidirect
93. #11719921C2×C4semidirect
94. #11719921C2×C4semidirect
101. #11719919C2×C4semidirect
215. #387D53semidirect
222. C53#2322semidirect
299. C4⋊C4C4×D53semidirect