Finite Group Databasebeta

← #20280960

Subgroups — #20280960

Order 6784.

618 conjugacy classes of subgroups.

The subgroup lattice has 452 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#13133204C211yesmaximal, characteristic, complemented
2C2×#13133198C211yesmaximal, characteristic, complemented
3C2×#13133194C211yesmaximal, characteristic, complemented
4#16810559C211yesmaximal, characteristic, complemented
5#16810546C211yesmaximal, characteristic, complemented
6#16810543C211yesmaximal, characteristic, complemented
7#16810537C211yesmaximal, characteristic, complemented
8#16810537C211yesmaximal, characteristic, complemented
9#16810537C211yesmaximal, characteristic, complemented
10#16810518C211yesmaximal, characteristic, complemented
11#16810516C211yesmaximal, characteristic, complemented
12#16810503C211yesmaximal, characteristic, complemented
13#16810496C211yesmaximal, characteristic, complemented
14#16810483C211yesmaximal, characteristic, complemented
15C2×#121×C53C211yesmaximal, characteristic
16C22×Dic106C2211yescharacteristic
17C23×Dic53C2211yescharacteristic
18#13133204—24no—
19#13133198—24no—
20#13133194—24no—
21C2×C4⋊Dic53C2211yescharacteristic
22C2×C4⋊Dic53C2211yescharacteristic
23C2×#11719919C2211yescharacteristic
24C2×#11719919C2211yescharacteristic
25C2×#11719919C2211yescharacteristic
26C2×#11719919C2211yescharacteristic
27C2×#11719919C2211yescharacteristic
28C2×#11719919C2211yescharacteristic
29C2×C22⋊Dic53C2211yescharacteristic
30C2×C22⋊Dic53C2211yescharacteristic
31C2×C22⋊Dic53C2211yescharacteristic
32C2×C22⋊Dic53C2211yescharacteristic
33C2×C22⋊Dic53C2211yescharacteristic
34C2×C22⋊Dic53C2211yescharacteristic
35C2×C22⋊Dic53C2211yescharacteristic
36C2×C22⋊Dic53C2211yescharacteristic
37C2×C22⋊Dic53C2211yescharacteristic
38C2×C22⋊Dic53C2211yescharacteristic
39C2×C22⋊Dic53C2211yescharacteristic
40C2×C4×Dic53C2211yescharacteristic
41#13133124C2211yescharacteristic
42#13133124C2211yescharacteristic
43#13133124C2211yescharacteristic
44#13133124C2211yescharacteristic
45#13133124C2211yescharacteristic
46#13133124C2211yescharacteristic
47C22×D4×C53C2211yescharacteristic
48C22×D4×C53C2211yescharacteristic
49C22×D4×C53C2211yescharacteristic
50C23×C4×C53C2211yescharacteristic
51#121×C53—24no—
52C2×C4⋊C4×C53C2211yescharacteristic
53C2×C22⋊C4×C53C2211yescharacteristic
54C2×C22⋊C4×C53C2211yescharacteristic
55C24×C53C2311yescharacteristic
56C24×C53C2311yescharacteristic
57C24×C53C2311yescharacteristic
58C2×D4×C53—22no—
59C2×D4×C53—22no—
60C2×D4×C53—22no—
61C2×D4×C53—22no—
62C2×D4×C53—22no—
63C2×D4×C53—22no—
64C2×D4×C53—41no—
65C2×D4×C53—41no—
66C2×D4×C53—41no—
67C22×C4×C53—22no—
68C22×C4×C53—22no—
69C22×C4×C53—41no—
70C22×C4×C53C2311yescharacteristic
71C22×C4×C53C2311yescharacteristic
72C22×C4×C53C2311yescharacteristic
73C22×C4×C53C2311yescharacteristic
74C22×Dic53—22no—
75C22×Dic53—22no—
76C22×Dic53—41no—
77C22×Dic53C2311yescharacteristic
78C22×Dic53C2311yescharacteristic
79C22×Dic53C2311yescharacteristic
80C22×Dic53C2311yescharacteristic
81C22×Dic53C2311yescharacteristic
82C22×Dic53C2311yescharacteristic
83C22×Dic53C2311yescharacteristic
84C22×Dic53C2311yescharacteristic
85C2×Dic106—22no—
86C2×Dic106—22no—
87C2×Dic106—41no—
88C4⋊C4×C53—22no—
89C22⋊C4×C53—22no—
90C22⋊C4×C53D414yes—
91C22⋊Dic53—22no—
92C22⋊Dic53—22no—
93C22⋊Dic53—22no—
94C22⋊Dic53—22no—
95C22⋊Dic53—22no—
96C22⋊Dic53—22no—
97C22⋊Dic53—41no—
98C22⋊Dic53—41no—
99C22⋊Dic53—41no—
100C22⋊Dic53—41no—
101C22⋊Dic53D414yescomplemented
102C4⋊Dic53—22no—
103C4⋊Dic53—41no—
104#11719919—22no—
105#11719919—22no—
106#11719919—22no—
107#11719919—22no—
108#11719919—22no—
109#11719919—41no—
110C4×Dic53—22no—
111C23×C53—21no—
112C23×C53—21no—
113C23×C53—21no—
114C23×C53—21no—
115C23×C53—21no—
116C23×C53—21no—
117C23×C53—21no—
118C23×C53—21no—
119C23×C53—21no—
120C23×C53—21no—
121C23×C53—21no—
122C23×C53—21no—
123C23×C53—21no—
124C23×C53—21no—
125C23×C53—21no—
126C23×C53—21no—
127C23×C53—21no—
128C23×C53—21no—
129C23×C53—21no—
130C23×C53C2×D412yes—
131C23×C53C2×D412yes—
132C23×C53C2411yescharacteristic
133D4×C53—44no—
134D4×C53—44no—
135D4×C53—44no—
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—44no—
146C2×C4×C53C2×D412yes—
147C2×C4×C53C2×D412yes—
148C2×C4×C53#4112yes—
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×Dic53—21no—
162C2×Dic53—21no—
163C2×Dic53—21no—
164C2×Dic53—21no—
165C2×Dic53—21no—
166C2×Dic53—21no—
167C2×Dic53—21no—
168C2×Dic53—21no—
169C2×Dic53—21no—
170C2×Dic53—21no—
171C2×Dic53—44no—
172C2×Dic53C2×D412yes—
173C2×Dic53C2×D412yes—
174Dic106—44no—
175C22×C53—22no—
176C22×C53—22no—
177C22×C53—22no—
178C22×C53—22no—
179C22×C53—22no—
180C22×C53—22no—
181C22×C53—41no—
182C22×C53—41no—
183C22×C53—41no—
184C22×C53—41no—
185C22×C53—41no—
186C22×C53—41no—
187C22×C53—41no—
188C22×C53—41no—
189C22×C53—41no—
190C22×C53—41no—
191C22×C53—41no—
192C22×C53—41no—
193C22×C53—41no—
194C22×C53—41no—
195C22×C53—41no—
196C22×C53C22×D411yescharacteristic
197C22×C53C22×D411yescharacteristic
198C22×C53C2×#4111yescharacteristic
199C22×C53H(5,𝔽2)11yescharacteristic
200C22×C53H(5,𝔽2)11yescharacteristic
201C22×C532−1+411yescharacteristic
202C22×C532−1+411yescharacteristic
203C4×C53—22no—
204C4×C53—22no—
205C4×C53—22no—
206C4×C53—41no—
207Dic53—22no—
208Dic53—22no—
209Dic53—41no—
210Dic53—41no—
211Dic53—41no—
212Dic53—41no—
213Dic53—41no—
214Dic53—41no—
215#2672—531nomaximal, Sylow
216C2×C53—42no—
217C2×C53—42no—
218C2×C53—81no—
219C2×C53#53611yescharacteristic
220C2×C53#53711yescharacteristic
221C2×C53#54211yescharacteristic
222C2×C53D4211yescharacteristic
223C2×C53#54611yescharacteristic
224C2×C53#54711yescharacteristic
225C2×C53#56211yescharacteristic
226C2×#124—531no—
227C2×#123—531no—
228C2×C22⋊Q8—531no—
229C2×#121D5311yescharacteristic
230#399—531no—
231#396—531no—
232#395—531no—
233#394—531no—
234#394—531no—
235#394—531no—
236#389—531no—
237#388—531no—
238#386—531no—
239#385—531no—
240#380—531no—
241C53#267211yescharacteristic, complemented, Sylow
242C22×Q8—531no—
243C22×D4C2×D5311yescharacteristic
244C22×D4C2×D5311yescharacteristic
245C22×D4C2×D5311yescharacteristic
246C23×C4—531no—
247C23×C4C2×D5311yescharacteristic
248#124—1064no—
249#123—1064no—
250C22⋊Q8—1064no—
251#121—24no—
252C2×C4⋊C4—531no—
253C2×C4⋊C4—531no—
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⋊C4C2×D5311yescharacteristic
261C2×C22⋊C4—531no—
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⋊C4—531no—
271C2×C22⋊C4—531no—
272C2×C22⋊C4C2×D5311yescharacteristic
273C2×C22⋊C4C2×D5311yescharacteristic
274C2×C42—531no—
275#95—531no—
276#95—531no—
277#95—531no—
278#95—531no—
279#95—531no—
280#95—531no—
281C24C22×D5311yescharacteristic
282C24C22×D5311yescharacteristic
283C24C22×D5311yescharacteristic
284C2×Q8—1062no—
285C2×Q8—1062no—
286C2×Q8—2121no—
287C2×D4—22no—
288C2×D4—22no—
289C2×D4—22no—
290C2×D4—22no—
291C2×D4—22no—
292C2×D4—22no—
293C2×D4—41no—
294C2×D4—41no—
295C2×D4—41no—
296C22×C4—22no—
297C22×C4—22no—
298C22×C4—41no—
299C22×C4—531no—
300C22×C4—531no—
301C22×C4—531no—
302C22×C4—531no—
303C22×C4—531no—
304C22×C4—531no—
305C22×C4—531no—
306C22×C4—531no—
307C22×C4—1062no—
308C22×C4—1062no—
309C22×C4—2121no—
310C22×C4C22×D5311yescharacteristic
311C22×C4C22×D5311yescharacteristic
312C22×C4C22×D5311yescharacteristic
313C22×C4C22×D5311yescharacteristic
314C4⋊C4—22no—
315C4⋊C4—1062no—
316C4⋊C4—1062no—
317C4⋊C4—1062no—
318C4⋊C4—1062no—
319C4⋊C4—1062no—
320C4⋊C4—1062no—
321C4⋊C4—2121no—
322C4⋊C4—2121no—
323C22⋊C4—22no—
324C22⋊C4—534no—
325C22⋊C4—1062no—
326C22⋊C4—1062no—
327C22⋊C4—1062no—
328C22⋊C4—1062no—
329C22⋊C4—1062no—
330C22⋊C4—1062no—
331C22⋊C4—2121no—
332C22⋊C4—2121no—
333C22⋊C4—2121no—
334C22⋊C4—2121no—
335C22⋊C4C22⋊D5314yes—
336C42—1062no—
337C23—21no—
338C23—21no—
339C23—21no—
340C23—21no—
341C23—21no—
342C23—21no—
343C23—21no—
344C23—21no—
345C23—21no—
346C23—21no—
347C23—21no—
348C23—21no—
349C23—21no—
350C23—21no—
351C23—21no—
352C23—21no—
353C23—21no—
354C23—21no—
355C23—21no—
356C23D4×D5312yes—
357C23C2×C22⋊D5312yes—
358C23C23×D5311yescharacteristic
359Q8—2124no—
360D4—44no—
361D4—44no—
362D4—44no—
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—44no—
373C2×C4—532no—
374C2×C4—532no—
375C2×C4—1061no—
376C2×C4—1061no—
377C2×C4—1061no—
378C2×C4—1061no—
379C2×C4—1061no—
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—1061no—
397C2×C4—2124no—
398C2×C4#1171994712yes—
399C2×C4C2×C22⋊D5312yes—
400C2×C4C2×C22⋊D5312yes—
401C22—22no—
402C22—22no—
403C22—22no—
404C22—22no—
405C22—22no—
406C22—22no—
407C22—41no—
408C22—41no—
409C22—41no—
410C22—41no—
411C22—41no—
412C22—41no—
413C22—41no—
414C22—41no—
415C22—41no—
416C22—41no—
417C22—41no—
418C22—41no—
419C22—41no—
420C22—41no—
421C22—41no—
422C22C2×D4×D5311yescharacteristic
423C22C22×C22⋊D5311yescharacteristic
424C22C2×#1171994711yescharacteristic
425C22#1313326211yescharacteristic
426C22#1313326211yescharacteristic
427C22#1313326411yescharacteristic
428C22#1313326411yescharacteristic
429C4—22no—
430C4—22no—
431C4—22no—
432C4—41no—
433C4—1062no—
434C4—1062no—
435C4—2121no—
436C4—2121no—
437C4—2121no—
438C4—2121no—
439C4—2121no—
440C4—2121no—
441C2—42no—
442C2—42no—
443C2—81no—
444C2#1681119411yescharacteristic
445C2#1681120011yescharacteristic
446C2#1681124011yescharacteristic
447C2D4×C22⋊D5311yescharacteristic
448C2#1681126711yescharacteristic
449C2#1681127311yescharacteristic
450C2#1681136811yescharacteristic
subgroupG/H
center, FrattiniC23C23×D53
commutator, socleC23×C53C24
FittingC2×#121×C53C2
hypercenterC2×#121D53
hypocenterC53#2672
LCSG ▷ C23×C53 ▷ C53
UCSC1 ◁ C23 ◁ C2×#121

Complements

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

NH ≅ G/Ntype
1. C2×#13133204C2semidirect
2. C2×#13133198C2semidirect
3. C2×#13133194C2semidirect
4. #16810559C2semidirect
5. #16810546C2semidirect
6. #16810543C2semidirect
7. #16810537C2semidirect
8. #16810537C2semidirect
9. #16810537C2semidirect
10. #16810518C2semidirect
11. #16810516C2semidirect
12. #16810503C2semidirect
13. #16810496C2semidirect
14. #16810483C2semidirect
101. C22⋊Dic53D4semidirect
241. C53#2672semidirect