Finite Group Databasebeta

← #20278656

Subgroups — #20278656

Order 6784.

590 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×#13133205C211yesmaximal, characteristic, complemented
2C2×#13133179C211yesmaximal, characteristic, complemented
3#16810541C211yesmaximal, characteristic, complemented
4#16810540C211yesmaximal, characteristic, complemented
5#16810536C211yesmaximal, characteristic, complemented
6#16810533C211yesmaximal, characteristic
7#16810527C211yesmaximal, characteristic, complemented
8#16810513C211yesmaximal, characteristic, complemented
9#16810513C211yesmaximal, characteristic, complemented
10#16810512C211yesmaximal, characteristic, complemented
11#16810501C211yesmaximal, characteristic, complemented
12#16810501C211yesmaximal, characteristic, complemented
13#16810485C211yesmaximal, characteristic, complemented
14C4×#11719921C211yesmaximal, characteristic, complemented
15C53×#387C211yesmaximal, characteristic, complemented
16C22×Dic106C2211yescharacteristic
17C22×D212C2211yescharacteristic
18C22×D212C2211yescharacteristic
19C22×D212C2211yescharacteristic
20C22×C4×D53C2211yescharacteristic
21C22×C4×D53C2211yescharacteristic
22#13133205—24no—
23#13133179—24no—
24C2×#11719919C2211yescharacteristic
25C2×#11719921C2211yescharacteristic
26C2×#11719921C2211yescharacteristic
27C2×#11719921C2211yescharacteristic
28C2×#11719921C2211yescharacteristic
29C2×#11719921C2211yescharacteristic
30C2×#11719921C2211yescharacteristic
31C2×#11719921C2211yescharacteristic
32C2×#11719921C2211yescharacteristic
33C2×#11719921C2211yescharacteristic
34C2×#11719921C2211yescharacteristic
35C2×#11719921C2211yescharacteristic
36C2×#11719921C2211yescharacteristic
37C2×#11719921C2211yescharacteristic
38C2×C4×Dic53C2211yescharacteristic
39C2×C4×Dic53C2211yescharacteristic
40#13133124C2211yescharacteristic
41#13133124C2211yescharacteristic
42#13133124C2211yescharacteristic
43#13133124C2211yescharacteristic
44#13133124C2211yescharacteristic
45#13133124C2211yescharacteristic
46C2×C4⋊C4×C53C2211yescharacteristic
47C2×C4⋊C4×C53C2211yescharacteristic
48C2×C42×C53C2211yescharacteristic
49#95×C53C2211yescharacteristic
50#95×C53C2211yescharacteristic
51#95×C53C2211yescharacteristic
52#95×C53C2211yescharacteristic
53C23×D53C2311yescharacteristic
54C23×D53C2311yescharacteristic
55C23×D53C2311yescharacteristic
56C22×C4×C53C2311yescharacteristic
57C22×C4×C53C2311yescharacteristic
58C22×C4×C53C2311yescharacteristic
59C22×C4×C53C2311yescharacteristic
60C22×C4×C53C2311yescharacteristic
61C22×C4×C53C2311yescharacteristic
62C22×C4×C53C2311yescharacteristic
63C22×Dic53C2311yescharacteristic
64C22×Dic53C2311yescharacteristic
65C22×Dic53C2311yescharacteristic
66C22×Dic53C2311yescharacteristic
67C22×Dic53C2311yescharacteristic
68C2×D212—22no—
69C2×D212—22no—
70C2×D212—22no—
71C2×D212—22no—
72C2×D212—22no—
73C2×D212—22no—
74C2×D212—41no—
75C2×D212—41no—
76C2×D212—41no—
77C2×C4×D53—22no—
78C2×C4×D53—22no—
79C2×C4×D53—22no—
80C2×C4×D53—22no—
81C2×C4×D53—41no—
82C2×C4×D53—41no—
83C2×Dic106—22no—
84C2×Dic106—22no—
85C2×Dic106—41no—
86C4⋊C4×C53—22no—
87C4⋊C4×C53—22no—
88C42×C53—22no—
89#11719921—22no—
90#11719921—22no—
91#11719921—22no—
92#11719921—22no—
93#11719921—22no—
94#11719921—22no—
95#11719921—22no—
96#11719921—22no—
97#11719921—41no—
98#11719921—41no—
99#11719921—41no—
100#11719921—41no—
101#11719921D414yescomplemented
102#11719919—41no—
103C4×Dic53—22no—
104C4×Dic53—41no—
105C23×C53C2411yescharacteristic
106C22×D53—21no—
107C22×D53—21no—
108C22×D53—21no—
109C22×D53—21no—
110C22×D53—21no—
111C22×D53—21no—
112C22×D53—21no—
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×D53C2×D412yes—
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×C53—21no—
145C2×C4×C53C2×D412yes—
146C2×C4×C53#4112yes—
147C2×C4×C53#4112yes—
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×Dic53#4112yes—
163D212—28no—
164D212—44no—
165D212—44no—
166C4×D53—44no—
167C4×D53—44no—
168Dic106—44no—
169C22×C53C22×D411yescharacteristic
170C22×C53C2×#4111yescharacteristic
171C22×C53C2×#4111yescharacteristic
172C22×C53C2×#4111yescharacteristic
173C22×C53C2×#4111yescharacteristic
174C22×C53H(5,𝔽2)11yescharacteristic
175C22×C53H(5,𝔽2)11yescharacteristic
176C2×D53—22no—
177C2×D53—22no—
178C2×D53—22no—
179C2×D53—22no—
180C2×D53—22no—
181C2×D53—22no—
182C2×D53—41no—
183C2×D53—41no—
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—
195C2×D53—41no—
196C2×D53—41no—
197C4×C53—22no—
198C4×C53—22no—
199C4×C53—22no—
200C4×C53—41no—
201C4×C53—41no—
202C4×C53—41no—
203C4×C53—41no—
204Dic53—22no—
205Dic53—22no—
206Dic53—41no—
207Dic53—41no—
208Dic53—41no—
209#2459—531nomaximal, Sylow
210C2×C53#52511yescharacteristic
211C2×C53#52911yescharacteristic
212C2×C53#53811yescharacteristic
213C2×C53#54611yescharacteristic
214C2×C53#54611yescharacteristic
215C2×C53#55511yescharacteristic
216C2×C53#55911yescharacteristic
217D53—42no—
218D53—42no—
219D53—81no—
220C2×#124—531no—
221C2×C4×D4—531no—
222#394—531no—
223#394—531no—
224#393—531no—
225#392—531no—
226#391—531no—
227#388—531no—
228#388—531no—
229#388—531no—
230#387D5311yescharacteristic, complemented
231#386—531no—
232#386—531no—
233#381—531no—
234C4×C22⋊C4—531no—
235C53#245911yescharacteristic, complemented, Sylow
236C22×Q8—531no—
237C22×D4—531no—
238C22×D4—531no—
239C22×D4—531no—
240C23×C4—531no—
241C23×C4—531no—
242#124—1064no—
243C4×D4—1064no—
244C2×C4⋊C4—531no—
245C2×C4⋊C4C2×D5311yescharacteristic
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⋊C4—531no—
259C2×C22⋊C4—531no—
260C2×C42—531no—
261C2×C42—531no—
262C2×C42C2×D5311yescharacteristic
263#95—531no—
264#95—531no—
265#95—531no—
266#95—531no—
267#95—531no—
268#95—531no—
269#95C2×D5311yescharacteristic
270#95C2×D5311yescharacteristic
271#95C2×D5311yescharacteristic
272#95C2×D5311yescharacteristic
273C24—531no—
274C24—531no—
275C24—531no—
276C2×Q8—1062no—
277C2×Q8—1062no—
278C2×Q8—2121no—
279C2×D4—1062no—
280C2×D4—1062no—
281C2×D4—1062no—
282C2×D4—1062no—
283C2×D4—1062no—
284C2×D4—1062no—
285C2×D4—2121no—
286C2×D4—2121no—
287C2×D4—2121no—
288C22×C4—531no—
289C22×C4—531no—
290C22×C4—531no—
291C22×C4—531no—
292C22×C4—531no—
293C22×C4—1062no—
294C22×C4—1062no—
295C22×C4—1062no—
296C22×C4—1062no—
297C22×C4—2121no—
298C22×C4—2121no—
299C22×C4C22×D5311yescharacteristic
300C22×C4C22×D5311yescharacteristic
301C22×C4C22×D5311yescharacteristic
302C22×C4C22×D5311yescharacteristic
303C22×C4C22×D5311yescharacteristic
304C22×C4C22×D5311yescharacteristic
305C22×C4C22×D5311yescharacteristic
306C4⋊C4—22no—
307C4⋊C4—22no—
308C4⋊C4—2121no—
309C22⋊C4—534no—
310C22⋊C4—1062no—
311C22⋊C4—1062no—
312C22⋊C4—1062no—
313C22⋊C4—1062no—
314C22⋊C4—1062no—
315C22⋊C4—1062no—
316C22⋊C4—1062no—
317C22⋊C4—1062no—
318C22⋊C4—2121no—
319C22⋊C4—2121no—
320C22⋊C4—2121no—
321C22⋊C4—2121no—
322C42—22no—
323C42—1062no—
324C42—2121no—
325C23—532no—
326C23—532no—
327C23—1061no—
328C23—1061no—
329C23—1061no—
330C23—1061no—
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—
344C23—1061no—
345C23—1061no—
346C23C23×D5311yescharacteristic
347Q8—2124no—
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—21no—
369C2×C4—532no—
370C2×C4—532no—
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—2124no—
385C2×C4—2124no—
386C2×C4#1171994512yes—
387C2×C4D4×D5312yes—
388C2×C4#1171994912yes—
389C22—1062no—
390C22—1062no—
391C22—1062no—
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—
406C22—2121no—
407C22—2121no—
408C22—2121no—
409C22—2121no—
410C22C2×D4×D5311yescharacteristic
411C22#41×D5311yescharacteristic
412C22#41×D5311yescharacteristic
413C22C2×#1171994511yescharacteristic
414C22C2×#1171994911yescharacteristic
415C22#1313326311yescharacteristic
416C22#1313326311yescharacteristic
417C4—22no—
418C4—22no—
419C4—22no—
420C4—41no—
421C4—41no—
422C4—41no—
423C4—41no—
424C4—1062no—
425C4—1062no—
426C4—2121no—
427C4—2121no—
428C4—2121no—
429C2—4241no—
430C2—2122no—
431C2—2122no—
432C2#1681113211yescharacteristic
433C2#1681115811yescharacteristic
434C2#1681121211yescharacteristic
435C2#1681126411yescharacteristic
436C2#1681126511yescharacteristic
437C2#1681133611yescharacteristic
438C2#1681135811yescharacteristic
subgroupG/H
center, FrattiniC23C23×D53
commutator, socleC23×C53C24
FittingC53×#387C2
hypercenter#387D53
hypocenterC53#2459
LCSG ▷ C23×C53 ▷ C53
UCSC1 ◁ C23 ◁ #387

Complements

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

NH ≅ G/Ntype
1. C2×#13133205C2semidirect
2. C2×#13133179C2semidirect
3. #16810541C2semidirect
4. #16810540C2semidirect
5. #16810536C2semidirect
7. #16810527C2semidirect
8. #16810513C2semidirect
9. #16810513C2semidirect
10. #16810512C2semidirect
11. #16810501C2semidirect
12. #16810501C2semidirect
13. #16810485C2semidirect
14. C4×#11719921C2semidirect
15. C53×#387C2semidirect
101. #11719921D4semidirect
230. #387D53semidirect
235. C53#2459semidirect