Finite Group Databasebeta

← #20281117

Subgroups — #20281117

Order 6784.

562 conjugacy classes of subgroups.

The subgroup lattice has 440 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#13133204C211yesmaximal, characteristic, complemented
2#16810551C211yesmaximal, characteristic, complemented
3#16810543C211yesmaximal, characteristic, complemented
4#16810537C211yesmaximal, characteristic, complemented
5#16810537C211yesmaximal, characteristic, complemented
6#16810537C211yesmaximal, characteristic, complemented
7#16810535C211yesmaximal, characteristic, complemented
8#16810511C211yesmaximal, characteristic, complemented
9#16810504C211yesmaximal, characteristic, complemented
10#16810503C211yesmaximal, characteristic, complemented
11#16810503C211yesmaximal, characteristic, complemented
12#16810496C211yesmaximal, characteristic, complemented
13#16810486C211yesmaximal, characteristic, complemented
14#16810481C211yesmaximal, characteristic, complemented
15C2×#121×C53C211yesmaximal, characteristic
16C22×Dic106C2211yescharacteristic
17C23×Dic53C2211yescharacteristic
18#13133204—24no—
19C2×#11719919C2211yescharacteristic
20C2×#11719919C2211yescharacteristic
21C2×#11719919C2211yescharacteristic
22C2×#11719919C2211yescharacteristic
23C2×C22⋊Dic53C2211yescharacteristic
24C2×C22⋊Dic53C2211yescharacteristic
25C2×C22⋊Dic53C2211yescharacteristic
26C2×C22⋊Dic53C2211yescharacteristic
27C2×C22⋊Dic53C2211yescharacteristic
28C2×C22⋊Dic53C2211yescharacteristic
29C2×C22⋊Dic53C2211yescharacteristic
30C2×C22⋊Dic53C2211yescharacteristic
31C2×C22⋊Dic53C2211yescharacteristic
32C2×C22⋊Dic53C2211yescharacteristic
33C2×C22⋊Dic53C2211yescharacteristic
34C2×C4×Dic53C2211yescharacteristic
35#13133124C2211yescharacteristic
36#13133124C2211yescharacteristic
37#13133124C2211yescharacteristic
38#13133124C2211yescharacteristic
39#13133124C2211yescharacteristic
40#13133124C2211yescharacteristic
41#13133124C2211yescharacteristic
42#13133124C2211yescharacteristic
43#13133124C2211yescharacteristic
44#13133124C2211yescharacteristic
45C22×D4×C53C2211yescharacteristic
46C22×D4×C53C2211yescharacteristic
47C22×D4×C53C2211yescharacteristic
48C23×C4×C53C2211yescharacteristic
49#121×C53—24no—
50C2×C4⋊C4×C53C2211yescharacteristic
51C2×C22⋊C4×C53C2211yescharacteristic
52C2×C22⋊C4×C53C2211yescharacteristic
53C24×C53C2311yescharacteristic
54C24×C53C2311yescharacteristic
55C24×C53C2311yescharacteristic
56C2×D4×C53—22no—
57C2×D4×C53—22no—
58C2×D4×C53—22no—
59C2×D4×C53—41no—
60C2×D4×C53—41no—
61C2×D4×C53—41no—
62C2×D4×C53—41no—
63C2×D4×C53—41no—
64C2×D4×C53D414yes—
65C22×C4×C53—22no—
66C22×C4×C53—22no—
67C22×C4×C53—41no—
68C22×C4×C53C2311yescharacteristic
69C22×C4×C53C2311yescharacteristic
70C22×C4×C53C2311yescharacteristic
71C22×C4×C53C2311yescharacteristic
72C22×Dic53—22no—
73C22×Dic53—41no—
74C22×Dic53—41no—
75C22×Dic53C2311yescharacteristic
76C22×Dic53C2311yescharacteristic
77C22×Dic53C2311yescharacteristic
78C22×Dic53C2311yescharacteristic
79C22×Dic53C2311yescharacteristic
80C22×Dic53C2311yescharacteristic
81C22×Dic53C2311yescharacteristic
82C22×Dic53C2311yescharacteristic
83C2×Dic106—22no—
84C2×Dic106—41no—
85C2×Dic106—41no—
86C4⋊C4×C53—22no—
87C22⋊C4×C53—22no—
88C22⋊C4×C53—22no—
89C22⋊Dic53—22no—
90C22⋊Dic53—22no—
91C22⋊Dic53—22no—
92C22⋊Dic53—22no—
93C22⋊Dic53—22no—
94C22⋊Dic53—22no—
95C22⋊Dic53—41no—
96C22⋊Dic53—41no—
97C22⋊Dic53—41no—
98C22⋊Dic53—41no—
99C22⋊Dic53—41no—
100#11719919—22no—
101#11719919—41no—
102#11719919—41no—
103#11719919—41no—
104C4×Dic53—22no—
105C23×C53—21no—
106C23×C53—21no—
107C23×C53—21no—
108C23×C53—21no—
109C23×C53—21no—
110C23×C53—21no—
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×C53C2×D412yes—
125C23×C53C2×D412yes—
126C23×C53C2411yescharacteristic
127D4×C53—44no—
128D4×C53—44no—
129D4×C53—44no—
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×C53—44no—
141C2×C4×C53C2×D412yes—
142C2×C4×C53#4112yes—
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×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—44no—
167C2×Dic53#4112yes—
168Dic106—44no—
169C22×C53—22no—
170C22×C53—22no—
171C22×C53—22no—
172C22×C53—22no—
173C22×C53—22no—
174C22×C53—22no—
175C22×C53—41no—
176C22×C53—41no—
177C22×C53—41no—
178C22×C53—41no—
179C22×C53—41no—
180C22×C53—41no—
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×C53C22×D411yescharacteristic
191C22×C53C2×#4111yescharacteristic
192C22×C53C2×#4111yescharacteristic
193C22×C53H(5,𝔽2)11yescharacteristic
194C22×C53H(5,𝔽2)11yescharacteristic
195C22×C53H(5,𝔽2)11yescharacteristic
196C22×C532−1+411yescharacteristic
197C4×C53—22no—
198C4×C53—22no—
199C4×C53—41no—
200C4×C53—41no—
201Dic53—22no—
202Dic53—41no—
203Dic53—41no—
204Dic53—41no—
205Dic53—41no—
206Dic53—41no—
207Dic53—41no—
208Dic53—41no—
209#2686—531nomaximal, Sylow
210C2×C53—42no—
211C2×C53—42no—
212C2×C53—81no—
213C2×C53#53411yescharacteristic
214C2×C53#53811yescharacteristic
215C2×C53#54211yescharacteristic
216C2×C53#54611yescharacteristic
217C2×C53#54711yescharacteristic
218C2×C53#55511yescharacteristic
219C2×C53#56211yescharacteristic
220C2×#124—531no—
221C2×#121D5311yescharacteristic
222#397—531no—
223#395—531no—
224#394—531no—
225#394—531no—
226#394—531no—
227#393—531no—
228#388—531no—
229#387—531no—
230#386—531no—
231#386—531no—
232#385—531no—
233#381—531no—
234#380—531no—
235C53#268611yescharacteristic, complemented, Sylow
236C22×Q8—531no—
237C22×D4C2×D5311yescharacteristic
238C22×D4C2×D5311yescharacteristic
239C22×D4C2×D5311yescharacteristic
240C23×C4—531no—
241C23×C4C2×D5311yescharacteristic
242#124—1064no—
243#121—24no—
244C2×C4⋊C4—531no—
245C2×C4⋊C4—531no—
246C2×C4⋊C4—531no—
247C2×C4⋊C4—531no—
248C2×C4⋊C4C2×D5311yescharacteristic
249C2×C22⋊C4—531no—
250C2×C22⋊C4—531no—
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×C22⋊C4—531no—
260C2×C22⋊C4C2×D5311yescharacteristic
261C2×C22⋊C4C2×D5311yescharacteristic
262C2×C42—531no—
263#95—531no—
264#95—531no—
265#95—531no—
266#95—531no—
267#95—531no—
268#95—531no—
269#95—531no—
270#95—531no—
271#95—531no—
272#95—531no—
273C24C22×D5311yescharacteristic
274C24C22×D5311yescharacteristic
275C24C22×D5311yescharacteristic
276C2×Q8—1062no—
277C2×Q8—2121no—
278C2×Q8—2121no—
279C2×D4—22no—
280C2×D4—22no—
281C2×D4—22no—
282C2×D4—41no—
283C2×D4—41no—
284C2×D4—41no—
285C2×D4—41no—
286C2×D4—41no—
287C2×D4C22⋊D5314yes—
288C22×C4—22no—
289C22×C4—22no—
290C22×C4—41no—
291C22×C4—531no—
292C22×C4—531no—
293C22×C4—531no—
294C22×C4—531no—
295C22×C4—531no—
296C22×C4—531no—
297C22×C4—531no—
298C22×C4—531no—
299C22×C4—1062no—
300C22×C4—2121no—
301C22×C4—2121no—
302C22×C4C22×D5311yescharacteristic
303C22×C4C22×D5311yescharacteristic
304C22×C4C22×D5311yescharacteristic
305C22×C4C22×D5311yescharacteristic
306C4⋊C4—22no—
307C4⋊C4—1062no—
308C4⋊C4—2121no—
309C4⋊C4—2121no—
310C4⋊C4—2121no—
311C22⋊C4—22no—
312C22⋊C4—22no—
313C22⋊C4—1062no—
314C22⋊C4—1062no—
315C22⋊C4—1062no—
316C22⋊C4—1062no—
317C22⋊C4—1062no—
318C22⋊C4—1062no—
319C22⋊C4—2121no—
320C22⋊C4—2121no—
321C22⋊C4—2121no—
322C22⋊C4—2121no—
323C22⋊C4—2121no—
324C42—1062no—
325C23—21no—
326C23—21no—
327C23—21no—
328C23—21no—
329C23—21no—
330C23—21no—
331C23—21no—
332C23—21no—
333C23—21no—
334C23—21no—
335C23—21no—
336C23—21no—
337C23—21no—
338C23—21no—
339C23—21no—
340C23—21no—
341C23—21no—
342C23—21no—
343C23—21no—
344C23C2×C22⋊D5312yes—
345C23C2×C22⋊D5312yes—
346C23C23×D5311yescharacteristic
347Q8—2124no—
348D4—44no—
349D4—44no—
350D4—44no—
351C2×C4—21no—
352C2×C4—21no—
353C2×C4—21no—
354C2×C4—21no—
355C2×C4—21no—
356C2×C4—21no—
357C2×C4—21no—
358C2×C4—21no—
359C2×C4—21no—
360C2×C4—21no—
361C2×C4—44no—
362C2×C4—532no—
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—1061no—
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—2124no—
387C2×C4#1171994712yes—
388C2×C4C2×C22⋊D5312yes—
389C22—22no—
390C22—22no—
391C22—22no—
392C22—22no—
393C22—22no—
394C22—22no—
395C22—41no—
396C22—41no—
397C22—41no—
398C22—41no—
399C22—41no—
400C22—41no—
401C22—41no—
402C22—41no—
403C22—41no—
404C22—41no—
405C22—41no—
406C22—41no—
407C22—41no—
408C22—41no—
409C22—41no—
410C22C22×C22⋊D5311yescharacteristic
411C22#41×D5311yescharacteristic
412C22C2×#1171994711yescharacteristic
413C22#1313326211yescharacteristic
414C22#1313326211yescharacteristic
415C22#1313326211yescharacteristic
416C22#1313326411yescharacteristic
417C4—22no—
418C4—22no—
419C4—41no—
420C4—41no—
421C4—1062no—
422C4—2121no—
423C4—2121no—
424C4—2121no—
425C4—2121no—
426C4—2121no—
427C4—2121no—
428C4—2121no—
429C2—42no—
430C2—42no—
431C2—81no—
432C2#1681118211yescharacteristic
433C2#1681120511yescharacteristic
434C2#1681124011yescharacteristic
435C2#1681126711yescharacteristic
436C2#1681127211yescharacteristic
437C2#1681133711yescharacteristic
438C2#1681136811yescharacteristic
subgroupG/H
center, FrattiniC23C23×D53
commutator, socleC23×C53C24
FittingC2×#121×C53C2
hypercenterC2×#121D53
hypocenterC53#2686
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. #16810551C2semidirect
3. #16810543C2semidirect
4. #16810537C2semidirect
5. #16810537C2semidirect
6. #16810537C2semidirect
7. #16810535C2semidirect
8. #16810511C2semidirect
9. #16810504C2semidirect
10. #16810503C2semidirect
11. #16810503C2semidirect
12. #16810496C2semidirect
13. #16810486C2semidirect
14. #16810481C2semidirect
235. C53#2686semidirect