Finite Group Databasebeta

← #20278529

Subgroups — #20278529

Order 6784.

648 conjugacy classes of subgroups.

The subgroup lattice has 426 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#13133197C211yesmaximal, characteristic, complemented
2C2×#13133195C211yesmaximal, characteristic, complemented
3C2×C4×D212C211yesmaximal, characteristic
4C2×C4⋊C4×D53C211yesmaximal, characteristic, complemented
5#16810560C211yesmaximal, characteristic, complemented
6#16810555C211yesmaximal, characteristic, complemented
7#16810545C211yesmaximal, characteristic, complemented
8#16810544C211yesmaximal, characteristic, complemented
9#16810521C211yesmaximal, characteristic, complemented
10#16810501C211yesmaximal, characteristic
11#16810501C211yesmaximal, characteristic
12#16810497C211yesmaximal, characteristic, complemented
13#16810497C211yesmaximal, characteristic, complemented
14#16810497C211yesmaximal, characteristic, complemented
15C53×#387C211yesmaximal, characteristic, complemented
16C22×Dic106C2211yescharacteristic
17C22×D212C2211yescharacteristic
18C22×C4×D53C2211yescharacteristic
19C22×C4×D53C2211yescharacteristic
20C22×C4×D53C2211yescharacteristic
21C22×C4×D53C2211yescharacteristic
22#13133197—24no—
23#13133195—24no—
24C4×D212—24no—
25C2×C4⋊Dic53C2211yescharacteristic
26C2×C4⋊Dic53C2211yescharacteristic
27C2×C4⋊Dic53C2211yescharacteristic
28C2×C4⋊Dic53C2211yescharacteristic
29C2×C4⋊Dic53C2211yescharacteristic
30C2×C4⋊Dic53C2211yescharacteristic
31C2×#11719919C2211yescharacteristic
32C2×#11719919C2211yescharacteristic
33C2×#11719919C2211yescharacteristic
34C2×#11719919C2211yescharacteristic
35C4⋊C4×D53—24no—
36C2×#11719921C2211yescharacteristic
37C2×#11719921C2211yescharacteristic
38C2×#11719921C2211yescharacteristic
39C2×#11719921C2211yescharacteristic
40C2×#11719921C2211yescharacteristic
41C2×#11719921C2211yescharacteristic
42C2×#11719921C2211yescharacteristic
43C2×#11719921C2211yescharacteristic
44#13133124C2211yescharacteristic
45#13133124C2211yescharacteristic
46#13133124C2211yescharacteristic
47#13133124C2211yescharacteristic
48C2×C4⋊C4×C53C2211yescharacteristic
49C2×C4⋊C4×C53C2211yescharacteristic
50C2×C42×C53C2211yescharacteristic
51#95×C53C2211yescharacteristic
52#95×C53C2211yescharacteristic
53#95×C53C2211yescharacteristic
54#95×C53C2211yescharacteristic
55C23×D53C2311yescharacteristic
56C23×D53C2311yescharacteristic
57C22×C4×C53C2311yescharacteristic
58C22×C4×C53C2311yescharacteristic
59C22×C4×C53C2311yescharacteristic
60C22×C4×C53C2311yescharacteristic
61C22×C4×C53C2311yescharacteristic
62C22×C4×C53C2311yescharacteristic
63C22×C4×C53C2311yescharacteristic
64C22×Dic53C2311yescharacteristic
65C22×Dic53C2311yescharacteristic
66C22×Dic53C2311yescharacteristic
67C22×Dic53C2311yescharacteristic
68C22×Dic53C2311yescharacteristic
69C22×Dic53C2311yescharacteristic
70C2×D212—22no—
71C2×D212—22no—
72C2×D212Q814yescomplemented
73C2×C4×D53—22no—
74C2×C4×D53—22no—
75C2×C4×D53—22no—
76C2×C4×D53—22no—
77C2×C4×D53—22no—
78C2×C4×D53—22no—
79C2×C4×D53—22no—
80C2×C4×D53—22no—
81C2×C4×D53—22no—
82C2×C4×D53—41no—
83C2×C4×D53—41no—
84C2×C4×D53—41no—
85C2×Dic106—22no—
86C2×Dic106—22no—
87C2×Dic106—41no—
88C4⋊C4×C53—22no—
89C4⋊C4×C53D414yescomplemented
90C42×C53—22no—
91#11719921—22no—
92#11719921—22no—
93#11719921—22no—
94#11719921—22no—
95#11719921—22no—
96#11719921—22no—
97#11719921—41no—
98#11719921D414yes—
99C4⋊Dic53—22no—
100C4⋊Dic53—22no—
101C4⋊Dic53—22no—
102C4⋊Dic53—22no—
103C4⋊Dic53—22no—
104C4⋊Dic53—41no—
105#11719919—22no—
106#11719919—22no—
107#11719919—22no—
108#11719919—22no—
109C23×C53C2411yescharacteristic
110C22×D53—21no—
111C22×D53—21no—
112C22×D53—21no—
113C22×D53—21no—
114C22×D53—21no—
115C22×D53—21no—
116C22×D53—21no—
117C22×D53—21no—
118C22×D53—21no—
119C22×D53—21no—
120C22×D53C2×D412yes—
121C22×D53C2×Q812yes—
122C22×D53C2×Q812yes—
123C22×D53#4112yes—
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×C53—21no—
137C2×C4×C53—21no—
138C2×C4×C53—21no—
139C2×C4×C53—21no—
140C2×C4×C53C2×D412yes—
141C2×C4×C53C2×D412yes—
142C2×C4×C53C2×D412yes—
143C2×C4×C53C2×D412yes—
144C2×C4×C53C2×Q812yes—
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×Dic53—21no—
156C2×Dic53—21no—
157C2×Dic53—21no—
158C2×Dic53—21no—
159C2×Dic53—21no—
160C2×Dic53—21no—
161C2×Dic53C2×D412yes—
162C2×Dic53#4112yes—
163D212—28no—
164C4×D53—28no—
165C4×D53—44no—
166C4×D53—44no—
167C4×D53—44no—
168Dic106—44no—
169C22×C53C22×D411yescharacteristic
170C22×C53C22×D411yescharacteristic
171C22×C53C22×Q811yescharacteristic
172C22×C53C2×#4111yescharacteristic
173C22×C53C2×#4111yescharacteristic
174C22×C53H(5,𝔽2)11yescharacteristic
175C22×C532−1+411yescharacteristic
176C2×D53—22no—
177C2×D53—22no—
178C2×D53—22no—
179C2×D53—22no—
180C2×D53—22no—
181C2×D53—22no—
182C2×D53—22no—
183C2×D53—22no—
184C2×D53—22no—
185C2×D53—41no—
186C2×D53—41no—
187C2×D53—41no—
188C2×D53—41no—
189C2×D53C22⋊Q814yescomplemented
190C4×C53—22no—
191C4×C53—22no—
192C4×C53—22no—
193C4×C53—22no—
194C4×C53—22no—
195C4×C53—41no—
196C4×C53—41no—
197Dic53—22no—
198Dic53—22no—
199Dic53—41no—
200Dic53—41no—
201Dic53—41no—
202Dic53—41no—
203#2449—531nomaximal, Sylow
204C2×C53C2×C22⋊Q811yescharacteristic
205C2×C53#52511yescharacteristic
206C2×C53#53911yescharacteristic
207C2×C53D4211yescharacteristic
208C2×C53#54711yescharacteristic
209C2×C53D4×Q811yescharacteristic
210C2×C53#55411yescharacteristic
211D53—24no—
212D53—42no—
213C2×C22⋊Q8—531no—
214C2×C22⋊Q8—531no—
215C2×C4×D4—531no—
216C22×C4⋊C4—531no—
217#399—531no—
218#398—531no—
219#396—531no—
220#395—531no—
221#389—531no—
222#387D5311yescharacteristic, complemented
223#386—531no—
224#386—531no—
225#385—531no—
226#385—531no—
227#385—531no—
228C53#244911yescharacteristic, complemented, Sylow
229C22×Q8—531no—
230C22×D4—531no—
231C23×C4—531no—
232C23×C4—531no—
233C23×C4—531no—
234C23×C4—531no—
235C22⋊Q8—1064no—
236C22⋊Q8—1064no—
237C4×D4—1064no—
238C2×C4⋊C4—531no—
239C2×C4⋊C4—531no—
240C2×C4⋊C4—531no—
241C2×C4⋊C4—531no—
242C2×C4⋊C4—531no—
243C2×C4⋊C4—531no—
244C2×C4⋊C4—531no—
245C2×C4⋊C4—531no—
246C2×C4⋊C4—531no—
247C2×C4⋊C4—531no—
248C2×C4⋊C4—1064no—
249C2×C4⋊C4C2×D5311yescharacteristic
250C2×C4⋊C4C2×D5311yescharacteristic
251C2×C22⋊C4—531no—
252C2×C22⋊C4—531no—
253C2×C22⋊C4—531no—
254C2×C22⋊C4—531no—
255C2×C22⋊C4—531no—
256C2×C22⋊C4—531no—
257C2×C22⋊C4—531no—
258C2×C22⋊C4—531no—
259C2×C42C2×D5311yescharacteristic
260#95—531no—
261#95—531no—
262#95—531no—
263#95—531no—
264#95C2×D5311yescharacteristic
265#95C2×D5311yescharacteristic
266#95C2×D5311yescharacteristic
267#95C2×D5311yescharacteristic
268C24—531no—
269C24—531no—
270C2×Q8—1062no—
271C2×Q8—1062no—
272C2×Q8—2121no—
273C2×D4—534no—
274C2×D4—1062no—
275C2×D4—1062no—
276C22×C4—531no—
277C22×C4—531no—
278C22×C4—531no—
279C22×C4—531no—
280C22×C4—531no—
281C22×C4—531no—
282C22×C4—1062no—
283C22×C4—1062no—
284C22×C4—1062no—
285C22×C4—1062no—
286C22×C4—1062no—
287C22×C4—1062no—
288C22×C4—1062no—
289C22×C4—1062no—
290C22×C4—1062no—
291C22×C4—2121no—
292C22×C4—2121no—
293C22×C4—2121no—
294C22×C4C22×D5311yescharacteristic
295C22×C4C22×D5311yescharacteristic
296C22×C4C22×D5311yescharacteristic
297C22×C4C22×D5311yescharacteristic
298C22×C4C22×D5311yescharacteristic
299C22×C4C22×D5311yescharacteristic
300C22×C4C22×D5311yescharacteristic
301C4⋊C4—22no—
302C4⋊C4—1062no—
303C4⋊C4—1062no—
304C4⋊C4—1062no—
305C4⋊C4—1062no—
306C4⋊C4—1062no—
307C4⋊C4—1062no—
308C4⋊C4—1062no—
309C4⋊C4—1062no—
310C4⋊C4—1062no—
311C4⋊C4—2121no—
312C4⋊C4D21214yescomplemented
313C22⋊C4—534no—
314C22⋊C4—1062no—
315C22⋊C4—1062no—
316C22⋊C4—1062no—
317C22⋊C4—1062no—
318C22⋊C4—1062no—
319C22⋊C4—1062no—
320C22⋊C4—2121no—
321C42—22no—
322C23—532no—
323C23—532no—
324C23—532no—
325C23—532no—
326C23—1061no—
327C23—1061no—
328C23—1061no—
329C23—1061no—
330C23—1061no—
331C23—1061no—
332C23—1061no—
333C23—1061no—
334C23—1061no—
335C23—1061no—
336C23C23×D5311yescharacteristic
337Q8—2124no—
338D4—1068no—
339C2×C4—21no—
340C2×C4—21no—
341C2×C4—21no—
342C2×C4—21no—
343C2×C4—21no—
344C2×C4—21no—
345C2×C4—21no—
346C2×C4—21no—
347C2×C4—21no—
348C2×C4—21no—
349C2×C4—21no—
350C2×C4—21no—
351C2×C4—21no—
352C2×C4—21no—
353C2×C4—21no—
354C2×C4—21no—
355C2×C4—532no—
356C2×C4—532no—
357C2×C4—1068no—
358C2×C4—1061no—
359C2×C4—1061no—
360C2×C4—1061no—
361C2×C4—1061no—
362C2×C4—1061no—
363C2×C4—1061no—
364C2×C4—1061no—
365C2×C4—1061no—
366C2×C4—1061no—
367C2×C4—1061no—
368C2×C4—1061no—
369C2×C4—1061no—
370C2×C4—1061no—
371C2×C4—1061no—
372C2×C4—1061no—
373C2×C4—1061no—
374C2×C4—2124no—
375C2×C4—2124no—
376C2×C4—2124no—
377C2×C4C2×D21212yes—
378C2×C4C2×D21212yes—
379C2×C4C2×D21212yes—
380C2×C4D4×D5312yes—
381C2×C4Q8×D5312yes—
382C22—534no—
383C22—1062no—
384C22—1062no—
385C22—1062no—
386C22—1062no—
387C22—1062no—
388C22—1062no—
389C22—1062no—
390C22—1062no—
391C22—1062no—
392C22—2121no—
393C22—2121no—
394C22—2121no—
395C22—2121no—
396C22C2×D4×D5311yescharacteristic
397C22C22×D21211yescharacteristic
398C22C2×Q8×D5311yescharacteristic
399C22#41×D5311yescharacteristic
400C22#41×D5311yescharacteristic
401C22#1313326311yescharacteristic
402C22#1313326411yescharacteristic
403C4—22no—
404C4—22no—
405C4—22no—
406C4—22no—
407C4—22no—
408C4—41no—
409C4—41no—
410C4—1062no—
411C4—1062no—
412C4—2121no—
413C4—2121no—
414C4—2121no—
415C4—2121no—
416C2—1064no—
417C2—2122no—
418C2C22⋊Q8×D5311yescharacteristic
419C2#1681113511yescharacteristic
420C2#1681121411yescharacteristic
421C2D4×D21211yescharacteristic
422C2#1681127111yescharacteristic
423C2Q8×D21211yescharacteristic
424C2#1681132911yescharacteristic
subgroupG/H
center, FrattiniC23C23×D53
commutator, socleC23×C53C24
FittingC53×#387C2
hypercenter#387D53
hypocenterC53#2449
LCSG ▷ C23×C53 ▷ C53
UCSC1 ◁ C23 ◁ #387

Complements

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

NH ≅ G/Ntype
1. C2×#13133197C2semidirect
2. C2×#13133195C2semidirect
4. C2×C4⋊C4×D53C2semidirect
5. #16810560C2semidirect
6. #16810555C2semidirect
7. #16810545C2semidirect
8. #16810544C2semidirect
9. #16810521C2semidirect
12. #16810497C2semidirect
13. #16810497C2semidirect
14. #16810497C2semidirect
15. C53×#387C2semidirect
72. C2×D212Q8semidirect
89. C4⋊C4×C53D4semidirect
189. C2×D53C22⋊Q8semidirect
222. #387D53semidirect
228. C53#2449semidirect
312. C4⋊C4D212semidirect