Finite Group Databasebeta

← #20278271

Subgroups — #20278271

Order 6784.

666 conjugacy classes of subgroups.

The subgroup lattice has 454 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#13133202C211yesmaximal, characteristic, complemented
2C2×#13133199C211yesmaximal, characteristic, complemented
3C2×#13133190C211yesmaximal, characteristic
4C2×C4×D212C211yesmaximal, characteristic, complemented
5#16810556C211yesmaximal, characteristic, complemented
6#16810550C211yesmaximal, characteristic, complemented
7#16810546C211yesmaximal, characteristic, complemented
8#16810539C211yesmaximal, characteristic, complemented
9#16810521C211yesmaximal, characteristic, complemented
10#16810499C211yesmaximal, characteristic, complemented
11#16810499C211yesmaximal, characteristic, complemented
12#16810497C211yesmaximal, characteristic, complemented
13#16810482C211yesmaximal, characteristic, complemented
14#95×D53C211yesmaximal, characteristic, complemented
15C53×#388C211yesmaximal, characteristic, complemented
16C22×D212C2211yescharacteristic
17C22×C22⋊D53C2211yescharacteristic
18C22×C22⋊D53C2211yescharacteristic
19C22×C4×D53C2211yescharacteristic
20C22×C4×D53C2211yescharacteristic
21C22×C4×D53C2211yescharacteristic
22C22×C4×D53C2211yescharacteristic
23#13133202—24no—
24#13133199—24no—
25#13133190—24no—
26C4×D212—24no—
27C2×C4⋊Dic53C2211yescharacteristic
28C2×C4⋊Dic53C2211yescharacteristic
29C2×C4⋊Dic53C2211yescharacteristic
30C2×#11719919C2211yescharacteristic
31C2×#11719919C2211yescharacteristic
32C2×#11719919C2211yescharacteristic
33C2×#11719919C2211yescharacteristic
34C2×C22⋊Dic53C2211yescharacteristic
35C2×C22⋊Dic53C2211yescharacteristic
36C2×#11719921C2211yescharacteristic
37C2×#11719921C2211yescharacteristic
38C2×#11719921C2211yescharacteristic
39C2×#11719921C2211yescharacteristic
40C2×#11719921C2211yescharacteristic
41C2×#11719921C2211yescharacteristic
42#13133124C2211yescharacteristic
43#13133124C2211yescharacteristic
44#13133124C2211yescharacteristic
45#13133124C2211yescharacteristic
46#13133124C2211yescharacteristic
47#13133124C2211yescharacteristic
48C2×C4⋊C4×C53C2211yescharacteristic
49C2×C22⋊C4×C53C2211yescharacteristic
50C2×C22⋊C4×C53C2211yescharacteristic
51C2×C22⋊C4×C53C2211yescharacteristic
52C2×C42×C53C2211yescharacteristic
53#95×C53C2211yescharacteristic
54#95×C53C2211yescharacteristic
55C24×C53C2311yescharacteristic
56C23×D53C2311yescharacteristic
57C23×D53C2311yescharacteristic
58C22×C4×C53C2311yescharacteristic
59C22×C4×C53C2311yescharacteristic
60C22×C4×C53C2311yescharacteristic
61C22×C4×C53C2311yescharacteristic
62C22×C4×C53C2311yescharacteristic
63C22×C4×C53C2311yescharacteristic
64C2×C22⋊D53—22no—
65C2×C22⋊D53—22no—
66C2×C22⋊D53—22no—
67C2×C22⋊D53—22no—
68C2×C22⋊D53—22no—
69C2×C22⋊D53—41no—
70C22×Dic53C2311yescharacteristic
71C22×Dic53C2311yescharacteristic
72C22×Dic53C2311yescharacteristic
73C22×Dic53C2311yescharacteristic
74C22×Dic53C2311yescharacteristic
75C22×Dic53C2311yescharacteristic
76C2×D212—22no—
77C2×D212—22no—
78C2×D212D414yes—
79C2×C4×D53—22no—
80C2×C4×D53—22no—
81C2×C4×D53—22no—
82C2×C4×D53—22no—
83C2×C4×D53—22no—
84C2×C4×D53—22no—
85C2×C4×D53—22no—
86C2×C4×D53—22no—
87C2×C4×D53—22no—
88C2×C4×D53—41no—
89C2×C4×D53—41no—
90C2×C4×D53—41no—
91C4⋊C4×C53—22no—
92C22⋊C4×C53—22no—
93C22⋊C4×C53—22no—
94C22⋊C4×C53—22no—
95C42×C53—22no—
96C22⋊Dic53—22no—
97C22⋊Dic53—41no—
98#11719921—22no—
99#11719921—22no—
100#11719921—22no—
101#11719921—22no—
102#11719921—22no—
103#11719921D414yescomplemented
104C4⋊Dic53—22no—
105C4⋊Dic53—22no—
106C4⋊Dic53—22no—
107#11719919—22no—
108#11719919—22no—
109#11719919—22no—
110#11719919—41no—
111C23×C53—21no—
112C23×C53—21no—
113C23×C53—21no—
114C23×C53—21no—
115C23×C53—21no—
116C23×C53—21no—
117C23×C53—21no—
118C23×C53C2411yescharacteristic
119C22×D53—21no—
120C22×D53—21no—
121C22×D53—21no—
122C22×D53—21no—
123C22×D53—21no—
124C22×D53—21no—
125C22×D53—21no—
126C22×D53—21no—
127C22×D53—21no—
128C22×D53—21no—
129C22×D53C2×D412yes—
130C22×D53C2×D412yes—
131C22×D53C2×D412yes—
132C22×D53#4112yes—
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×C53—21no—
141C2×C4×C53—21no—
142C2×C4×C53—21no—
143C2×C4×C53—21no—
144C2×C4×C53—21no—
145C2×C4×C53—21no—
146C2×C4×C53—21no—
147C2×C4×C53—21no—
148C2×C4×C53C2×D412yes—
149C2×C4×C53C2×D412yes—
150C2×C4×C53#4112yes—
151C22⋊D53—44no—
152C22⋊D53—44no—
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×Dic53—21no—
162C2×Dic53—21no—
163C2×Dic53—21no—
164C2×Dic53—21no—
165C2×Dic53—21no—
166C2×Dic53—21no—
167C2×Dic53—21no—
168C2×Dic53—21no—
169C2×Dic53C2×D412yes—
170C2×Dic53#4112yes—
171D212—28no—
172C4×D53—44no—
173C4×D53—44no—
174C4×D53—44no—
175C4×D53—44no—
176C22×C53—41no—
177C22×C53—41no—
178C22×C53—41no—
179C22×C53—41no—
180C22×C53—41no—
181C22×C53—41no—
182C22×C53—41no—
183C22×C53C22×D411yescharacteristic
184C22×C53C22×D411yescharacteristic
185C22×C53C2×#4111yescharacteristic
186C22×C53C2×#4111yescharacteristic
187C22×C53C2×#4111yescharacteristic
188C22×C53H(5,𝔽2)11yescharacteristic
189C22×C532−1+411yescharacteristic
190C2×D53—22no—
191C2×D53—22no—
192C2×D53—22no—
193C2×D53—22no—
194C2×D53—22no—
195C2×D53—22no—
196C2×D53—22no—
197C2×D53—22no—
198C2×D53—22no—
199C2×D53—41no—
200C2×D53—41no—
201C2×D53—41no—
202C2×D53—41no—
203C2×D53#12114yes—
204C4×C53—22no—
205C4×C53—22no—
206C4×C53—22no—
207C4×C53—41no—
208C4×C53—41no—
209C4×C53—41no—
210Dic53—22no—
211Dic53—22no—
212Dic53—41no—
213Dic53—41no—
214Dic53—41no—
215Dic53—41no—
216#2425—531nomaximal, Sylow
217C2×C53—81no—
218C2×C53C2×#12111yescharacteristic
219C2×C53#52911yescharacteristic
220C2×C53#53911yescharacteristic
221C2×C53#54611yescharacteristic
222C2×C53#54611yescharacteristic
223C2×C53#54711yescharacteristic
224C2×C53#54711yescharacteristic
225D53—24no—
226D53—42no—
227C2×#123—531no—
228C2×#123—531no—
229C2×#121—531no—
230C2×C4×D4—531no—
231#398—531no—
232#397—531no—
233#396—531no—
234#394—531no—
235#389—531no—
236#388D5311yescharacteristic, complemented
237#386—531no—
238#386—531no—
239#385—531no—
240#380—531no—
241C2×#95—531no—
242C53#242511yescharacteristic, complemented, Sylow
243C22×D4—531no—
244C22×D4—531no—
245C22×D4—531no—
246C23×C4—531no—
247C23×C4—531no—
248C23×C4—531no—
249C23×C4—531no—
250#123—1064no—
251#123—1064no—
252#121—1064no—
253C4×D4—1064no—
254C2×C4⋊C4—531no—
255C2×C4⋊C4—531no—
256C2×C4⋊C4—531no—
257C2×C4⋊C4—531no—
258C2×C4⋊C4—531no—
259C2×C4⋊C4—531no—
260C2×C4⋊C4—531no—
261C2×C4⋊C4C2×D5311yescharacteristic
262C2×C22⋊C4—531no—
263C2×C22⋊C4—531no—
264C2×C22⋊C4—531no—
265C2×C22⋊C4—531no—
266C2×C22⋊C4—531no—
267C2×C22⋊C4—531no—
268C2×C22⋊C4—531no—
269C2×C22⋊C4—531no—
270C2×C22⋊C4C2×D5311yescharacteristic
271C2×C22⋊C4C2×D5311yescharacteristic
272C2×C22⋊C4C2×D5311yescharacteristic
273C2×C42C2×D5311yescharacteristic
274#95—531no—
275#95—531no—
276#95—531no—
277#95—531no—
278#95—531no—
279#95—531no—
280#95C2×D5311yescharacteristic
281#95C2×D5311yescharacteristic
282C24—531no—
283C24—531no—
284C24C22×D5311yescharacteristic
285C2×D4—534no—
286C2×D4—1062no—
287C2×D4—1062no—
288C2×D4—1062no—
289C2×D4—1062no—
290C2×D4—1062no—
291C2×D4—1062no—
292C2×D4—1062no—
293C2×D4—2121no—
294C22×C4—531no—
295C22×C4—531no—
296C22×C4—531no—
297C22×C4—531no—
298C22×C4—531no—
299C22×C4—531no—
300C22×C4—1062no—
301C22×C4—1062no—
302C22×C4—1062no—
303C22×C4—1062no—
304C22×C4—1062no—
305C22×C4—1062no—
306C22×C4—1062no—
307C22×C4—1062no—
308C22×C4—1062no—
309C22×C4—2121no—
310C22×C4—2121no—
311C22×C4—2121no—
312C22×C4C22×D5311yescharacteristic
313C22×C4C22×D5311yescharacteristic
314C22×C4C22×D5311yescharacteristic
315C22×C4C22×D5311yescharacteristic
316C22×C4C22×D5311yescharacteristic
317C22×C4C22×D5311yescharacteristic
318C4⋊C4—22no—
319C4⋊C4—1062no—
320C4⋊C4—1062no—
321C4⋊C4—1062no—
322C4⋊C4—1062no—
323C4⋊C4—1062no—
324C4⋊C4—1062no—
325C4⋊C4—2121no—
326C22⋊C4—22no—
327C22⋊C4—22no—
328C22⋊C4—22no—
329C22⋊C4—534no—
330C22⋊C4—1062no—
331C22⋊C4—1062no—
332C22⋊C4—1062no—
333C22⋊C4—1062no—
334C22⋊C4—1062no—
335C22⋊C4—1062no—
336C22⋊C4—2121no—
337C42—22no—
338C23—21no—
339C23—21no—
340C23—21no—
341C23—21no—
342C23—21no—
343C23—21no—
344C23—21no—
345C23—532no—
346C23—532no—
347C23—532no—
348C23—532no—
349C23—1061no—
350C23—1061no—
351C23—1061no—
352C23—1061no—
353C23—1061no—
354C23—1061no—
355C23—1061no—
356C23—1061no—
357C23—1061no—
358C23—1061no—
359C23C23×D5311yescharacteristic
360D4—1068no—
361D4—2124no—
362D4—2124no—
363C2×C4—21no—
364C2×C4—21no—
365C2×C4—21no—
366C2×C4—21no—
367C2×C4—21no—
368C2×C4—21no—
369C2×C4—21no—
370C2×C4—21no—
371C2×C4—21no—
372C2×C4—21no—
373C2×C4—21no—
374C2×C4—21no—
375C2×C4—21no—
376C2×C4—21no—
377C2×C4—21no—
378C2×C4—532no—
379C2×C4—532no—
380C2×C4—1061no—
381C2×C4—1061no—
382C2×C4—1061no—
383C2×C4—1061no—
384C2×C4—1061no—
385C2×C4—1061no—
386C2×C4—1061no—
387C2×C4—1061no—
388C2×C4—1061no—
389C2×C4—1061no—
390C2×C4—1061no—
391C2×C4—1061no—
392C2×C4—1061no—
393C2×C4—1061no—
394C2×C4—1061no—
395C2×C4—1061no—
396C2×C4—2124no—
397C2×C4—2124no—
398C2×C4—2124no—
399C2×C4—2124no—
400C2×C4#1171994512yes—
401C2×C4D4×D5312yes—
402C2×C4D4×D5312yes—
403C22—41no—
404C22—41no—
405C22—41no—
406C22—41no—
407C22—41no—
408C22—41no—
409C22—41no—
410C22—534no—
411C22—1062no—
412C22—1062no—
413C22—1062no—
414C22—1062no—
415C22—1062no—
416C22—1062no—
417C22—1062no—
418C22—1062no—
419C22—1062no—
420C22—2121no—
421C22—2121no—
422C22—2121no—
423C22—2121no—
424C22C2×D4×D5311yescharacteristic
425C22C2×D4×D5311yescharacteristic
426C22#41×D5311yescharacteristic
427C22#41×D5311yescharacteristic
428C22C2×#1171994511yescharacteristic
429C22#1313326311yescharacteristic
430C22#1313326411yescharacteristic
431C4—22no—
432C4—22no—
433C4—22no—
434C4—41no—
435C4—41no—
436C4—41no—
437C4—1062no—
438C4—1062no—
439C4—2121no—
440C4—2121no—
441C4—2121no—
442C4—2121no—
443C2—81no—
444C2—1064no—
445C2—2122no—
446C2#121×D5311yescharacteristic
447C2#1681115111yescharacteristic
448C2#1681121411yescharacteristic
449C2#1681126611yescharacteristic
450C2#1681126911yescharacteristic
451C2#1681127111yescharacteristic
452C2#1681127811yescharacteristic
subgroupG/H
center, FrattiniC23C23×D53
commutator, socleC23×C53C24
FittingC53×#388C2
hypercenter#388D53
hypocenterC53#2425
LCSG ▷ C23×C53 ▷ C53
UCSC1 ◁ C23 ◁ #388

Complements

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

NH ≅ G/Ntype
1. C2×#13133202C2semidirect
2. C2×#13133199C2semidirect
4. C2×C4×D212C2semidirect
5. #16810556C2semidirect
6. #16810550C2semidirect
7. #16810546C2semidirect
8. #16810539C2semidirect
9. #16810521C2semidirect
10. #16810499C2semidirect
11. #16810499C2semidirect
12. #16810497C2semidirect
13. #16810482C2semidirect
14. #95×D53C2semidirect
15. C53×#388C2semidirect
103. #11719921D4semidirect
236. #388D53semidirect
242. C53#2425semidirect