Finite Group Databasebeta

← #20281141

Subgroups — #20281141

Order 6784.

512 conjugacy classes of subgroups.

The subgroup lattice has 402 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#13133202C211yesmaximal, characteristic
2C2×#13133195C211yesmaximal, characteristic, complemented
3#16810557C211yesmaximal, characteristic, complemented
4#16810556C211yesmaximal, characteristic, complemented
5#16810550C211yesmaximal, characteristic, complemented
6#16810541C211yesmaximal, characteristic
7#16810536C211yesmaximal, characteristic, complemented
8#16810527C211yesmaximal, characteristic, complemented
9#16810515C211yesmaximal, characteristic, complemented
10#16810512C211yesmaximal, characteristic, complemented
11#16810501C211yesmaximal, characteristic
12#16810497C211yesmaximal, characteristic, complemented
13#16810485C211yesmaximal, characteristic, complemented
14#16810480C211yesmaximal, characteristic, complemented
15C53×#400C211yesmaximal, characteristic, complemented
16C22×Dic106C2211yescharacteristic
17C22×D212C2211yescharacteristic
18C22×C4×D53C2211yescharacteristic
19C22×C4×D53C2211yescharacteristic
20#13133202—24no—
21#13133195—24no—
22C2×C4⋊Dic53C2211yescharacteristic
23C2×C4⋊Dic53C2211yescharacteristic
24C2×#11719919C2211yescharacteristic
25C2×#11719919C2211yescharacteristic
26C2×#11719919C2211yescharacteristic
27C2×#11719919C2211yescharacteristic
28C2×#11719921C2211yescharacteristic
29C2×#11719921C2211yescharacteristic
30C2×#11719921C2211yescharacteristic
31C2×#11719921C2211yescharacteristic
32C2×#11719921C2211yescharacteristic
33C2×#11719921C2211yescharacteristic
34C2×#11719921C2211yescharacteristic
35C2×#11719921C2211yescharacteristic
36C2×#11719921C2211yescharacteristic
37C2×#11719921C2211yescharacteristic
38C2×C4×Dic53C2211yescharacteristic
39#13133124C2211yescharacteristic
40#13133124C2211yescharacteristic
41#13133124C2211yescharacteristic
42#13133124C2211yescharacteristic
43#13133124C2211yescharacteristic
44#13133124C2211yescharacteristic
45#13133124C2211yescharacteristic
46C2×C4⋊C4×C53C2211yescharacteristic
47C2×C4⋊C4×C53C2211yescharacteristic
48#95×C53C2211yescharacteristic
49#95×C53C2211yescharacteristic
50#95×C53C2211yescharacteristic
51#95×C53C2211yescharacteristic
52#95×C53C2211yescharacteristic
53C23×D53C2311yescharacteristic
54C23×D53C2311yescharacteristic
55C22×C4×C53C2311yescharacteristic
56C22×C4×C53C2311yescharacteristic
57C22×C4×C53C2311yescharacteristic
58C22×C4×C53C2311yescharacteristic
59C22×C4×C53C2311yescharacteristic
60C22×C4×C53C2311yescharacteristic
61C22×C4×C53C2311yescharacteristic
62C22×Dic53C2311yescharacteristic
63C22×Dic53C2311yescharacteristic
64C22×Dic53C2311yescharacteristic
65C22×Dic53C2311yescharacteristic
66C22×Dic53C2311yescharacteristic
67C22×Dic53C2311yescharacteristic
68C2×D212—22no—
69C2×D212—22no—
70C2×D212—41no—
71C2×C4×D53—22no—
72C2×C4×D53—22no—
73C2×C4×D53—22no—
74C2×C4×D53—22no—
75C2×C4×D53—41no—
76C2×C4×D53—41no—
77C2×Dic106—22no—
78C2×Dic106—22no—
79C2×Dic106—41no—
80C4⋊C4×C53—22no—
81C4⋊C4×C53—22no—
82#11719921—22no—
83#11719921—22no—
84#11719921—22no—
85#11719921—22no—
86#11719921—22no—
87#11719921—41no—
88#11719921—41no—
89#11719921—41no—
90#11719921—41no—
91#11719921D414yes—
92C4⋊Dic53—41no—
93C4⋊Dic53—41no—
94#11719919—22no—
95#11719919—22no—
96#11719919—22no—
97#11719919—41no—
98C4×Dic53—41no—
99C23×C53C2411yescharacteristic
100C22×D53—21no—
101C22×D53—21no—
102C22×D53—21no—
103C22×D53—21no—
104C22×D53—21no—
105C22×D53—21no—
106C22×D53—21no—
107C22×D53—21no—
108C22×D53—21no—
109C22×D53—21no—
110C22×D53—21no—
111C22×D53—21no—
112C22×D53C2×D412yes—
113C22×D53#4112yes—
114C2×C4×C53—21no—
115C2×C4×C53—21no—
116C2×C4×C53—21no—
117C2×C4×C53—21no—
118C2×C4×C53—21no—
119C2×C4×C53—21no—
120C2×C4×C53—21no—
121C2×C4×C53—21no—
122C2×C4×C53—21no—
123C2×C4×C53—21no—
124C2×C4×C53—21no—
125C2×C4×C53—21no—
126C2×C4×C53—21no—
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×C53C2×D412yes—
135C2×Dic53—21no—
136C2×Dic53—21no—
137C2×Dic53—21no—
138C2×Dic53—21no—
139C2×Dic53—21no—
140C2×Dic53—21no—
141C2×Dic53—21no—
142C2×Dic53—21no—
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×Dic53C2×D412yes—
152C2×Dic53#4112yes—
153D212—44no—
154C4×D53—44no—
155C4×D53—44no—
156Dic106—44no—
157C22×C53C22×D411yescharacteristic
158C22×C53C2×#4111yescharacteristic
159C22×C53C2×#4111yescharacteristic
160C22×C53H(5,𝔽2)11yescharacteristic
161C22×C53H(5,𝔽2)11yescharacteristic
162C22×C532−1+411yescharacteristic
163C22×C532−1+411yescharacteristic
164C2×D53—22no—
165C2×D53—22no—
166C2×D53—22no—
167C2×D53—22no—
168C2×D53—22no—
169C2×D53—22no—
170C2×D53—41no—
171C2×D53—41no—
172C2×D53—41no—
173C2×D53—41no—
174C2×D53—41no—
175C2×D53—41no—
176C2×D53—41no—
177C2×D53—41no—
178C4×C53—22no—
179C4×C53—41no—
180C4×C53—41no—
181C4×C53—41no—
182C4×C53—41no—
183C4×C53—41no—
184C4×C53—41no—
185Dic53—22no—
186Dic53—22no—
187Dic53—41no—
188Dic53—41no—
189Dic53—41no—
190Dic53—41no—
191#2687—531nomaximal, Sylow
192C2×C53#53611yescharacteristic
193C2×C53#53911yescharacteristic
194C2×C53#54211yescharacteristic
195C2×C53#54611yescharacteristic
196C2×C53#54611yescharacteristic
197C2×C53#55211yescharacteristic
198C2×C53#56211yescharacteristic
199D53—42no—
200D53—42no—
201C2×#123—531no—
202C2×C22⋊Q8—531no—
203#400D5311yescharacteristic, complemented
204#398—531no—
205#398—531no—
206#397—531no—
207#394—531no—
208#393—531no—
209#391—531no—
210#388—531no—
211#388—531no—
212#386—531no—
213#385—531no—
214#381—531no—
215#380—531no—
216C53#268711yescharacteristic, complemented, Sylow
217C22×Q8—531no—
218C22×D4—531no—
219C23×C4—531no—
220C23×C4—531no—
221#123—1064no—
222C22⋊Q8—1064no—
223C2×C4⋊C4—531no—
224C2×C4⋊C4—531no—
225C2×C4⋊C4—531no—
226C2×C4⋊C4—531no—
227C2×C4⋊C4—531no—
228C2×C4⋊C4—531no—
229C2×C4⋊C4C2×D5311yescharacteristic
230C2×C4⋊C4C2×D5311yescharacteristic
231C2×C22⋊C4—531no—
232C2×C22⋊C4—531no—
233C2×C22⋊C4—531no—
234C2×C22⋊C4—531no—
235C2×C22⋊C4—531no—
236C2×C22⋊C4—531no—
237C2×C22⋊C4—531no—
238C2×C22⋊C4—531no—
239C2×C22⋊C4—531no—
240C2×C22⋊C4—531no—
241C2×C42—531no—
242#95—531no—
243#95—531no—
244#95—531no—
245#95—531no—
246#95—531no—
247#95—531no—
248#95—531no—
249#95C2×D5311yescharacteristic
250#95C2×D5311yescharacteristic
251#95C2×D5311yescharacteristic
252#95C2×D5311yescharacteristic
253#95C2×D5311yescharacteristic
254C24—531no—
255C24—531no—
256C2×Q8—1062no—
257C2×Q8—1062no—
258C2×Q8—2121no—
259C2×D4—1062no—
260C2×D4—1062no—
261C2×D4—2121no—
262C22×C4—531no—
263C22×C4—531no—
264C22×C4—531no—
265C22×C4—531no—
266C22×C4—531no—
267C22×C4—531no—
268C22×C4—1062no—
269C22×C4—1062no—
270C22×C4—1062no—
271C22×C4—1062no—
272C22×C4—2121no—
273C22×C4—2121no—
274C22×C4C22×D5311yescharacteristic
275C22×C4C22×D5311yescharacteristic
276C22×C4C22×D5311yescharacteristic
277C22×C4C22×D5311yescharacteristic
278C22×C4C22×D5311yescharacteristic
279C22×C4C22×D5311yescharacteristic
280C22×C4C22×D5311yescharacteristic
281C4⋊C4—22no—
282C4⋊C4—22no—
283C4⋊C4—1062no—
284C4⋊C4—1062no—
285C4⋊C4—1062no—
286C4⋊C4—2121no—
287C4⋊C4—2121no—
288C4⋊C4—2121no—
289C22⋊C4—534no—
290C22⋊C4—1062no—
291C22⋊C4—1062no—
292C22⋊C4—1062no—
293C22⋊C4—1062no—
294C22⋊C4—1062no—
295C22⋊C4—2121no—
296C22⋊C4—2121no—
297C22⋊C4—2121no—
298C22⋊C4—2121no—
299C42—2121no—
300C23—532no—
301C23—532no—
302C23—1061no—
303C23—1061no—
304C23—1061no—
305C23—1061no—
306C23—1061no—
307C23—1061no—
308C23—1061no—
309C23—1061no—
310C23—1061no—
311C23—1061no—
312C23—1061no—
313C23—1061no—
314C23C23×D5311yescharacteristic
315Q8—2124no—
316D4—2124no—
317C2×C4—21no—
318C2×C4—21no—
319C2×C4—21no—
320C2×C4—21no—
321C2×C4—21no—
322C2×C4—21no—
323C2×C4—21no—
324C2×C4—21no—
325C2×C4—21no—
326C2×C4—21no—
327C2×C4—21no—
328C2×C4—21no—
329C2×C4—21no—
330C2×C4—21no—
331C2×C4—21no—
332C2×C4—21no—
333C2×C4—21no—
334C2×C4—21no—
335C2×C4—21no—
336C2×C4—21no—
337C2×C4—532no—
338C2×C4—532no—
339C2×C4—1061no—
340C2×C4—1061no—
341C2×C4—1061no—
342C2×C4—1061no—
343C2×C4—1061no—
344C2×C4—1061no—
345C2×C4—1061no—
346C2×C4—1061no—
347C2×C4—1061no—
348C2×C4—1061no—
349C2×C4—1061no—
350C2×C4—1061no—
351C2×C4—1061no—
352C2×C4—1061no—
353C2×C4—1061no—
354C2×C4—1061no—
355C2×C4—2124no—
356C2×C4—2124no—
357C2×C4D4×D5312yes—
358C22—1062no—
359C22—1062no—
360C22—1062no—
361C22—1062no—
362C22—1062no—
363C22—1062no—
364C22—2121no—
365C22—2121no—
366C22—2121no—
367C22—2121no—
368C22—2121no—
369C22—2121no—
370C22—2121no—
371C22—2121no—
372C22C2×D4×D5311yescharacteristic
373C22#41×D5311yescharacteristic
374C22#41×D5311yescharacteristic
375C22#1313326311yescharacteristic
376C22#1313326311yescharacteristic
377C22#1313326411yescharacteristic
378C22#1313326511yescharacteristic
379C4—22no—
380C4—41no—
381C4—41no—
382C4—41no—
383C4—41no—
384C4—41no—
385C4—41no—
386C4—1062no—
387C4—1062no—
388C4—2121no—
389C4—2121no—
390C4—2121no—
391C4—2121no—
392C2—2122no—
393C2—2122no—
394C2#1681119611yescharacteristic
395C2#1681121411yescharacteristic
396C2#1681123911yescharacteristic
397C2#1681126511yescharacteristic
398C2#1681126611yescharacteristic
399C2#1681130511yescharacteristic
400C2#1681137111yescharacteristic
subgroupG/H
center, FrattiniC23C23×D53
commutator, socleC23×C53C24
FittingC53×#400C2
hypercenter#400D53
hypocenterC53#2687
LCSG ▷ C23×C53 ▷ C53
UCSC1 ◁ C23 ◁ #400

Complements

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

NH ≅ G/Ntype
2. C2×#13133195C2semidirect
3. #16810557C2semidirect
4. #16810556C2semidirect
5. #16810550C2semidirect
7. #16810536C2semidirect
8. #16810527C2semidirect
9. #16810515C2semidirect
10. #16810512C2semidirect
12. #16810497C2semidirect
13. #16810485C2semidirect
14. #16810480C2semidirect
15. C53×#400C2semidirect
203. #400D53semidirect
216. C53#2687semidirect