Finite Group Databasebeta

← #20278468

Subgroups — #20278468

Order 6784.

610 conjugacy classes of subgroups.

The subgroup lattice has 442 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#13133190C211yesmaximal, characteristic
2C2×#13133179C211yesmaximal, characteristic, complemented
3#16810561C211yesmaximal, characteristic, complemented
4#16810552C211yesmaximal, characteristic, complemented
5#16810539C211yesmaximal, characteristic, complemented
6#16810511C211yesmaximal, characteristic, complemented
7#16810508C211yesmaximal, characteristic, complemented
8#16810501C211yesmaximal, characteristic, complemented
9#16810499C211yesmaximal, characteristic, complemented
10#16810499C211yesmaximal, characteristic, complemented
11#16810499C211yesmaximal, characteristic, complemented
12#16810497C211yesmaximal, characteristic, complemented
13#16810485C211yesmaximal, characteristic, complemented
14#95×D53C211yesmaximal, characteristic, complemented
15C53×#397C211yesmaximal, characteristic, complemented
16C22×D212C2211yescharacteristic
17C22×C22⋊D53C2211yescharacteristic
18C22×C22⋊D53C2211yescharacteristic
19C22×C4×D53C2211yescharacteristic
20C22×C4×D53C2211yescharacteristic
21C22×C4×D53C2211yescharacteristic
22C22×C4×D53C2211yescharacteristic
23#13133190—24no—
24#13133179—24no—
25C2×C4⋊Dic53C2211yescharacteristic
26C2×#11719919C2211yescharacteristic
27C2×#11719919C2211yescharacteristic
28C2×C22⋊Dic53C2211yescharacteristic
29C2×C22⋊Dic53C2211yescharacteristic
30C2×#11719921C2211yescharacteristic
31C2×#11719921C2211yescharacteristic
32C2×#11719921C2211yescharacteristic
33C2×#11719921C2211yescharacteristic
34C2×#11719921C2211yescharacteristic
35C2×#11719921C2211yescharacteristic
36C2×C4×Dic53C2211yescharacteristic
37#13133124C2211yescharacteristic
38#13133124C2211yescharacteristic
39#13133124C2211yescharacteristic
40#13133124C2211yescharacteristic
41#13133124C2211yescharacteristic
42#13133124C2211yescharacteristic
43#13133124C2211yescharacteristic
44#13133124C2211yescharacteristic
45#13133124C2211yescharacteristic
46C2×C4⋊C4×C53C2211yescharacteristic
47C2×C22⋊C4×C53C2211yescharacteristic
48C2×C22⋊C4×C53C2211yescharacteristic
49C2×C22⋊C4×C53C2211yescharacteristic
50#95×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—22no—
66C2×C22⋊D53—22no—
67C2×C22⋊D53—41no—
68C22×Dic53C2311yescharacteristic
69C22×Dic53C2311yescharacteristic
70C22×Dic53C2311yescharacteristic
71C22×Dic53C2311yescharacteristic
72C22×Dic53C2311yescharacteristic
73C22×Dic53C2311yescharacteristic
74C2×D212—22no—
75C2×D212—22no—
76C2×D212D414yes—
77C2×C4×D53—22no—
78C2×C4×D53—22no—
79C2×C4×D53—22no—
80C2×C4×D53—22no—
81C2×C4×D53—22no—
82C2×C4×D53—22no—
83C2×C4×D53—22no—
84C2×C4×D53—41no—
85C2×C4×D53—41no—
86C2×C4×D53—41no—
87C2×C4×D53—41no—
88C2×C4×D53—41no—
89C4⋊C4×C53—22no—
90C22⋊C4×C53—22no—
91C22⋊C4×C53—41no—
92C22⋊C4×C53—41no—
93C22⋊Dic53—22no—
94C22⋊Dic53—41no—
95#11719921—22no—
96#11719921—22no—
97#11719921—22no—
98#11719921—22no—
99#11719921—41no—
100#11719921—41no—
101C4⋊Dic53—22no—
102#11719919—22no—
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×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×D53C2×D412yes—
124C22×D53C2×D412yes—
125C22×D53#4112yes—
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×C53—21no—
144C2×C4×C53C2×D412yes—
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×Dic53—21no—
162C2×Dic53—21no—
163C2×Dic53#4112yes—
164C2×Dic53#4112yes—
165D212—28no—
166C4×D53—44no—
167C4×D53—44no—
168C4×D53—44no—
169C4×D53—44no—
170C22×C53—41no—
171C22×C53—41no—
172C22×C53—41no—
173C22×C53—41no—
174C22×C53—41no—
175C22×C53—41no—
176C22×C53—41no—
177C22×C53C22×D411yescharacteristic
178C22×C53C2×#4111yescharacteristic
179C22×C53C2×#4111yescharacteristic
180C22×C53C2×#4111yescharacteristic
181C22×C53C2×#4111yescharacteristic
182C22×C53H(5,𝔽2)11yescharacteristic
183C22×C53H(5,𝔽2)11yescharacteristic
184C2×D53—22no—
185C2×D53—22no—
186C2×D53—22no—
187C2×D53—22no—
188C2×D53—22no—
189C2×D53—22no—
190C2×D53—22no—
191C2×D53—22no—
192C2×D53—22no—
193C2×D53—41no—
194C2×D53—41no—
195C2×D53—41no—
196C2×D53—41no—
197C2×D53#12314yes—
198C4×C53—22no—
199C4×C53—41no—
200C4×C53—41no—
201C4×C53—41no—
202C4×C53—41no—
203C4×C53—41no—
204Dic53—22no—
205Dic53—22no—
206Dic53—41no—
207Dic53—41no—
208Dic53—41no—
209Dic53—41no—
210#2444—531nomaximal, Sylow
211C2×C53—81no—
212C2×C53C2×#12311yescharacteristic
213C2×C53#52911yescharacteristic
214C2×C53#53811yescharacteristic
215C2×C53#54611yescharacteristic
216C2×C53#54611yescharacteristic
217C2×C53#55311yescharacteristic
218C2×C53#55311yescharacteristic
219D53—24no—
220D53—42no—
221C2×#121—531no—
222C2×C4×D4—531no—
223#400—531no—
224#397—531no—
225#397D5311yescharacteristic, complemented
226#394—531no—
227#388—531no—
228#387—531no—
229#386—531no—
230#386—531no—
231#386—531no—
232#386—531no—
233#385—531no—
234#381—531no—
235C2×#95—531no—
236C53#244411yescharacteristic, complemented, Sylow
237C22×D4—531no—
238C22×D4—531no—
239C22×D4—531no—
240C23×C4—531no—
241C23×C4—531no—
242C23×C4—531no—
243C23×C4—531no—
244#121—1064no—
245C4×D4—1064no—
246C2×C4⋊C4—531no—
247C2×C4⋊C4—531no—
248C2×C4⋊C4—531no—
249C2×C4⋊C4C2×D5311yescharacteristic
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—
262#95—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#95C2×D5311yescharacteristic
272#95C2×D5311yescharacteristic
273#95C2×D5311yescharacteristic
274C24—531no—
275C24—531no—
276C24C22×D5311yescharacteristic
277C2×D4—534no—
278C2×D4—1062no—
279C2×D4—1062no—
280C2×D4—1062no—
281C2×D4—1062no—
282C2×D4—1062no—
283C2×D4—1062no—
284C2×D4—1062no—
285C2×D4—2121no—
286C22×C4—531no—
287C22×C4—531no—
288C22×C4—531no—
289C22×C4—531no—
290C22×C4—531no—
291C22×C4—531no—
292C22×C4—1062no—
293C22×C4—1062no—
294C22×C4—1062no—
295C22×C4—1062no—
296C22×C4—1062no—
297C22×C4—1062no—
298C22×C4—1062no—
299C22×C4—2121no—
300C22×C4—2121no—
301C22×C4—2121no—
302C22×C4—2121no—
303C22×C4—2121no—
304C22×C4C22×D5311yescharacteristic
305C22×C4C22×D5311yescharacteristic
306C22×C4C22×D5311yescharacteristic
307C22×C4C22×D5311yescharacteristic
308C22×C4C22×D5311yescharacteristic
309C22×C4C22×D5311yescharacteristic
310C4⋊C4—22no—
311C4⋊C4—1062no—
312C4⋊C4—1062no—
313C4⋊C4—2121no—
314C22⋊C4—22no—
315C22⋊C4—41no—
316C22⋊C4—41no—
317C22⋊C4—1062no—
318C22⋊C4—1062no—
319C22⋊C4—1062no—
320C22⋊C4—1062no—
321C22⋊C4—1062no—
322C22⋊C4—2121no—
323C22⋊C4—2121no—
324C22⋊C4—2121no—
325C42—1062no—
326C23—21no—
327C23—21no—
328C23—21no—
329C23—21no—
330C23—21no—
331C23—21no—
332C23—21no—
333C23—532no—
334C23—532no—
335C23—532no—
336C23—532no—
337C23—1061no—
338C23—1061no—
339C23—1061no—
340C23—1061no—
341C23—1061no—
342C23—1061no—
343C23—1061no—
344C23—1061no—
345C23—1061no—
346C23—1061no—
347C23C23×D5311yescharacteristic
348D4—1068no—
349D4—2124no—
350D4—2124no—
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—21no—
365C2×C4—21no—
366C2×C4—21no—
367C2×C4—21no—
368C2×C4—532no—
369C2×C4—532no—
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—2124no—
388C2×C4—2124no—
389C2×C4—2124no—
390C2×C4D4×D5312yes—
391C22—41no—
392C22—41no—
393C22—41no—
394C22—41no—
395C22—41no—
396C22—41no—
397C22—41no—
398C22—534no—
399C22—1062no—
400C22—1062no—
401C22—1062no—
402C22—1062no—
403C22—1062no—
404C22—1062no—
405C22—1062no—
406C22—1062no—
407C22—1062no—
408C22—2121no—
409C22—2121no—
410C22—2121no—
411C22—2121no—
412C22C2×D4×D5311yescharacteristic
413C22#41×D5311yescharacteristic
414C22#41×D5311yescharacteristic
415C22#41×D5311yescharacteristic
416C22#41×D5311yescharacteristic
417C22#1313326211yescharacteristic
418C22#1313326311yescharacteristic
419C4—22no—
420C4—41no—
421C4—41no—
422C4—41no—
423C4—41no—
424C4—41no—
425C4—1062no—
426C4—1062no—
427C4—2121no—
428C4—2121no—
429C4—2121no—
430C4—2121no—
431C2—81no—
432C2—1064no—
433C2—2122no—
434C2#123×D5311yescharacteristic
435C2#1681115611yescharacteristic
436C2#1681120911yescharacteristic
437C2#1681126611yescharacteristic
438C2#1681127011yescharacteristic
439C2#1681131811yescharacteristic
440C2#1681132411yescharacteristic
subgroupG/H
center, FrattiniC23C23×D53
commutator, socleC23×C53C24
FittingC53×#397C2
hypercenter#397D53
hypocenterC53#2444
LCSG ▷ C23×C53 ▷ C53
UCSC1 ◁ C23 ◁ #397

Complements

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

NH ≅ G/Ntype
2. C2×#13133179C2semidirect
3. #16810561C2semidirect
4. #16810552C2semidirect
5. #16810539C2semidirect
6. #16810511C2semidirect
7. #16810508C2semidirect
8. #16810501C2semidirect
9. #16810499C2semidirect
10. #16810499C2semidirect
11. #16810499C2semidirect
12. #16810497C2semidirect
13. #16810485C2semidirect
14. #95×D53C2semidirect
15. C53×#397C2semidirect
225. #397D53semidirect
236. C53#2444semidirect