Finite Group Databasebeta

← #20279781

Subgroups — #20279781

Order 6784.

560 conjugacy classes of subgroups.

The subgroup lattice has 436 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#13133211C211yesmaximal, characteristic, complemented
2C2×#13133199C211yesmaximal, characteristic, complemented
3#16810556C211yesmaximal, characteristic, complemented
4#16810547C211yesmaximal, characteristic, complemented
5#16810538C211yesmaximal, characteristic, complemented
6#16810532C211yesmaximal, characteristic
7#16810524C211yesmaximal, characteristic, complemented
8#16810523C211yesmaximal, characteristic, complemented
9#16810523C211yesmaximal, characteristic, complemented
10#16810515C211yesmaximal, characteristic, complemented
11#16810511C211yesmaximal, characteristic, complemented
12#16810501C211yesmaximal, characteristic, complemented
13#16810474C211yesmaximal, characteristic, complemented
14C4×#11719921C211yesmaximal, characteristic, complemented
15C53×#388C211yesmaximal, characteristic, complemented
16C22×D212C2211yescharacteristic
17C22×C22⋊D53C2211yescharacteristic
18C22×C22⋊D53C2211yescharacteristic
19C22×C4×D53C2211yescharacteristic
20#13133211—24no—
21#13133199—24no—
22C2×C4⋊Dic53C2211yescharacteristic
23C2×#11719919C2211yescharacteristic
24C2×#11719919C2211yescharacteristic
25C2×#11719919C2211yescharacteristic
26C2×#11719919C2211yescharacteristic
27C2×#11719919C2211yescharacteristic
28C2×C22⋊Dic53C2211yescharacteristic
29C2×C22⋊Dic53C2211yescharacteristic
30C2×#11719921C2211yescharacteristic
31C2×#11719921C2211yescharacteristic
32C2×#11719921C2211yescharacteristic
33C2×#11719921C2211yescharacteristic
34C2×#11719921C2211yescharacteristic
35C2×#11719921C2211yescharacteristic
36C2×#11719921C2211yescharacteristic
37C2×#11719921C2211yescharacteristic
38C2×#11719921C2211yescharacteristic
39C2×C4×Dic53C2211yescharacteristic
40C2×C4×Dic53C2211yescharacteristic
41C2×C4×Dic53C2211yescharacteristic
42#13133124C2211yescharacteristic
43#13133124C2211yescharacteristic
44#13133124C2211yescharacteristic
45#13133124C2211yescharacteristic
46C2×C4⋊C4×C53C2211yescharacteristic
47C2×C22⋊C4×C53C2211yescharacteristic
48C2×C22⋊C4×C53C2211yescharacteristic
49C2×C22⋊C4×C53C2211yescharacteristic
50C2×C42×C53C2211yescharacteristic
51#95×C53C2211yescharacteristic
52#95×C53C2211yescharacteristic
53C24×C53C2311yescharacteristic
54C23×D53C2311yescharacteristic
55C23×D53C2311yescharacteristic
56C22×C4×C53C2311yescharacteristic
57C22×C4×C53C2311yescharacteristic
58C22×C4×C53C2311yescharacteristic
59C22×C4×C53C2311yescharacteristic
60C22×C4×C53C2311yescharacteristic
61C22×C4×C53C2311yescharacteristic
62C2×C22⋊D53—22no—
63C2×C22⋊D53—22no—
64C2×C22⋊D53—22no—
65C2×C22⋊D53—41no—
66C2×C22⋊D53—41no—
67C2×C22⋊D53—41no—
68C22×Dic53C2311yescharacteristic
69C22×Dic53C2311yescharacteristic
70C22×Dic53C2311yescharacteristic
71C22×Dic53C2311yescharacteristic
72C22×Dic53C2311yescharacteristic
73C22×Dic53C2311yescharacteristic
74C2×D212—22no—
75C2×D212—22no—
76C2×D212—41no—
77C2×C4×D53—22no—
78C2×C4×D53—22no—
79C2×C4×D53—41no—
80C4⋊C4×C53—41no—
81C22⋊C4×C53—22no—
82C22⋊C4×C53—22no—
83C22⋊C4×C53—41no—
84C42×C53—22no—
85C22⋊Dic53—22no—
86C22⋊Dic53—22no—
87#11719921—22no—
88#11719921—22no—
89#11719921—22no—
90#11719921—22no—
91#11719921—22no—
92#11719921—22no—
93#11719921—22no—
94#11719921—41no—
95#11719921—41no—
96C4⋊Dic53—22no—
97#11719919—22no—
98#11719919—22no—
99#11719919—22no—
100#11719919—41no—
101#11719919D414yescomplemented
102C4×Dic53—22no—
103C4×Dic53—41no—
104C4×Dic53—41no—
105C23×C53—21no—
106C23×C53—21no—
107C23×C53—21no—
108C23×C53—21no—
109C23×C53—21no—
110C23×C53—21no—
111C23×C53—21no—
112C23×C53C2411yescharacteristic
113C22×D53—21no—
114C22×D53—21no—
115C22×D53—21no—
116C22×D53—21no—
117C22×D53—21no—
118C22×D53—21no—
119C22×D53—21no—
120C22×D53—21no—
121C22×D53—21no—
122C22×D53—21no—
123C22×D53—21no—
124C22×D53—21no—
125C22×D53—21no—
126C22×D53#4112yes—
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×C53—21no—
141C2×C4×C53—21no—
142C2×C4×C53—21no—
143C2×C4×C53C2×D412yes—
144C2×C4×C53#4112yes—
145C22⋊D53—44no—
146C22⋊D53—44no—
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×Dic53C2×D412yes—
163C2×Dic53#4112yes—
164C2×Dic53#4112yes—
165D212—44no—
166C4×D53—44no—
167C22×C53—41no—
168C22×C53—41no—
169C22×C53—41no—
170C22×C53—41no—
171C22×C53—41no—
172C22×C53—41no—
173C22×C53—41no—
174C22×C53C22×D411yescharacteristic
175C22×C53C2×#4111yescharacteristic
176C22×C53C2×#4111yescharacteristic
177C22×C53C2×#4111yescharacteristic
178C22×C53C2×#4111yescharacteristic
179C22×C53H(5,𝔽2)11yescharacteristic
180C22×C53H(5,𝔽2)11yescharacteristic
181C2×D53—22no—
182C2×D53—22no—
183C2×D53—22no—
184C2×D53—41no—
185C2×D53—41no—
186C2×D53—41no—
187C2×D53—41no—
188C2×D53—41no—
189C2×D53—41no—
190C2×D53—41no—
191C2×D53—41no—
192C2×D53—41no—
193C2×D53—41no—
194C2×D53—41no—
195C4×C53—22no—
196C4×C53—22no—
197C4×C53—41no—
198C4×C53—41no—
199C4×C53—41no—
200C4×C53—41no—
201Dic53—22no—
202Dic53—22no—
203Dic53—22no—
204Dic53—22no—
205Dic53—41no—
206Dic53—41no—
207#2554—531nomaximal, Sylow
208C2×C53—81no—
209C2×C53#52911yescharacteristic
210C2×C53#53211yescharacteristic
211C2×C53#53811yescharacteristic
212C2×C53#54611yescharacteristic
213C2×C53#55011yescharacteristic
214C2×C53#55111yescharacteristic
215C2×C53#55311yescharacteristic
216D53—42no—
217D53—81no—
218C2×#126—531no—
219C2×#123—531no—
220#398—531no—
221#396—531no—
222#394—531no—
223#392—531no—
224#390—531no—
225#390—531no—
226#390—531no—
227#388—531no—
228#388—531no—
229#388D5311yescharacteristic, complemented
230#386—531no—
231C4×C4⋊C4—531no—
232C4×C22⋊C4—531no—
233C53#255411yescharacteristic, complemented, Sylow
234C22×D4—531no—
235C22×D4—531no—
236C22×D4—531no—
237C23×C4—531no—
238#126—1064no—
239#123—1064no—
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⋊C4C2×D5311yescharacteristic
247C2×C22⋊C4—531no—
248C2×C22⋊C4—531no—
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⋊C4C2×D5311yescharacteristic
259C2×C22⋊C4C2×D5311yescharacteristic
260C2×C22⋊C4C2×D5311yescharacteristic
261C2×C42—531no—
262C2×C42—531no—
263C2×C42—531no—
264C2×C42C2×D5311yescharacteristic
265#95—531no—
266#95—531no—
267#95—531no—
268#95—531no—
269#95C2×D5311yescharacteristic
270#95C2×D5311yescharacteristic
271C24—531no—
272C24—531no—
273C24C22×D5311yescharacteristic
274C2×D4—1062no—
275C2×D4—1062no—
276C2×D4—1062no—
277C2×D4—1062no—
278C2×D4—1062no—
279C2×D4—2121no—
280C2×D4—2121no—
281C2×D4—2121no—
282C2×D4—2121no—
283C22×C4—531no—
284C22×C4—531no—
285C22×C4—531no—
286C22×C4—531no—
287C22×C4—531no—
288C22×C4—531no—
289C22×C4—1062no—
290C22×C4—1062no—
291C22×C4—2121no—
292C22×C4C22×D5311yescharacteristic
293C22×C4C22×D5311yescharacteristic
294C22×C4C22×D5311yescharacteristic
295C22×C4C22×D5311yescharacteristic
296C22×C4C22×D5311yescharacteristic
297C22×C4C22×D5311yescharacteristic
298C4⋊C4—41no—
299C4⋊C4—534no—
300C4⋊C4—1062no—
301C4⋊C4—1062no—
302C4⋊C4—1062no—
303C4⋊C4—1062no—
304C4⋊C4—2121no—
305C22⋊C4—22no—
306C22⋊C4—22no—
307C22⋊C4—41no—
308C22⋊C4—1062no—
309C22⋊C4—1062no—
310C22⋊C4—1062no—
311C22⋊C4—1062no—
312C22⋊C4—1062no—
313C22⋊C4—1062no—
314C22⋊C4—1062no—
315C22⋊C4—1062no—
316C22⋊C4—1062no—
317C22⋊C4—2121no—
318C22⋊C4—2121no—
319C42—22no—
320C42—1062no—
321C42—2121no—
322C42—2121no—
323C23—21no—
324C23—21no—
325C23—21no—
326C23—21no—
327C23—21no—
328C23—21no—
329C23—21no—
330C23—532no—
331C23—1061no—
332C23—1061no—
333C23—1061no—
334C23—1061no—
335C23—1061no—
336C23—1061no—
337C23—1061no—
338C23—1061no—
339C23—1061no—
340C23—1061no—
341C23—1061no—
342C23—1061no—
343C23—1061no—
344C23C23×D5311yescharacteristic
345D4—2124no—
346D4—2124no—
347D4—2124no—
348C2×C4—21no—
349C2×C4—21no—
350C2×C4—21no—
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—21no—
362C2×C4—21no—
363C2×C4—21no—
364C2×C4—532no—
365C2×C4—532no—
366C2×C4—532no—
367C2×C4—532no—
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—2124no—
383C2×C4#1171994512yes—
384C2×C4D4×D5312yes—
385C22—41no—
386C22—41no—
387C22—41no—
388C22—41no—
389C22—41no—
390C22—41no—
391C22—41no—
392C22—1062no—
393C22—1062no—
394C22—1062no—
395C22—2121no—
396C22—2121no—
397C22—2121no—
398C22—2121no—
399C22—2121no—
400C22—2121no—
401C22—2121no—
402C22—2121no—
403C22—2121no—
404C22—2121no—
405C22—2121no—
406C22C2×D4×D5311yescharacteristic
407C22#41×D5311yescharacteristic
408C22#41×D5311yescharacteristic
409C22#41×D5311yescharacteristic
410C22C2×#1171994511yescharacteristic
411C22#1313326211yescharacteristic
412C22#1313326311yescharacteristic
413C4—22no—
414C4—22no—
415C4—41no—
416C4—41no—
417C4—41no—
418C4—41no—
419C4—1062no—
420C4—1062no—
421C4—1062no—
422C4—1062no—
423C4—2121no—
424C4—2121no—
425C2—81no—
426C2—4241no—
427C2—2122no—
428C2#1681115611yescharacteristic
429C2#1681117011yescharacteristic
430C2#1681120711yescharacteristic
431C2#1681126511yescharacteristic
432C2#1681129111yescharacteristic
433C2#1681130211yescharacteristic
434C2#1681131711yescharacteristic
subgroupG/H
center, FrattiniC23C23×D53
commutator, socleC23×C53C24
FittingC53×#388C2
hypercenter#388D53
hypocenterC53#2554
LCSG ▷ C23×C53 ▷ C53
UCSC1 ◁ C23 ◁ #388

Complements

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

NH ≅ G/Ntype
1. C2×#13133211C2semidirect
2. C2×#13133199C2semidirect
3. #16810556C2semidirect
4. #16810547C2semidirect
5. #16810538C2semidirect
7. #16810524C2semidirect
8. #16810523C2semidirect
9. #16810523C2semidirect
10. #16810515C2semidirect
11. #16810511C2semidirect
12. #16810501C2semidirect
13. #16810474C2semidirect
14. C4×#11719921C2semidirect
15. C53×#388C2semidirect
101. #11719919D4semidirect
229. #388D53semidirect
233. C53#2554semidirect