Finite Group Databasebeta

← #20280862

Subgroups — #20280862

Order 6784.

498 conjugacy classes of subgroups.

The subgroup lattice has 400 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#13133198C211yesmaximal, characteristic
2C2×#13133194C211yesmaximal, characteristic, complemented
3#16810564C211yesmaximal, characteristic, complemented
4#16810559C211yesmaximal, characteristic, complemented
5#16810551C211yesmaximal, characteristic, complemented
6#16810548C211yesmaximal, characteristic, complemented
7#16810543C211yesmaximal, characteristic, complemented
8#16810537C211yesmaximal, characteristic
9#16810535C211yesmaximal, characteristic, complemented
10#16810520C211yesmaximal, characteristic, complemented
11#16810519C211yesmaximal, characteristic, complemented
12#16810516C211yesmaximal, characteristic, complemented
13#16810511C211yesmaximal, characteristic, complemented
14C22⋊C4×Dic53C211yesmaximal, characteristic, complemented
15C53×#394C211yesmaximal, characteristic
16C22×Dic106C2211yescharacteristic
17C23×Dic53C2211yescharacteristic
18#13133198—24no—
19#13133194—24no—
20C2×C4⋊Dic53C2211yescharacteristic
21C2×C4⋊Dic53C2211yescharacteristic
22C2×#11719919C2211yescharacteristic
23C2×#11719919C2211yescharacteristic
24C2×#11719919C2211yescharacteristic
25C2×#11719919C2211yescharacteristic
26C2×#11719919C2211yescharacteristic
27C2×#11719919C2211yescharacteristic
28C2×C22⋊Dic53C2211yescharacteristic
29C2×C22⋊Dic53C2211yescharacteristic
30C2×C22⋊Dic53C2211yescharacteristic
31C2×C22⋊Dic53C2211yescharacteristic
32C2×C22⋊Dic53C2211yescharacteristic
33C2×C22⋊Dic53C2211yescharacteristic
34C2×C22⋊Dic53C2211yescharacteristic
35C2×C4×Dic53C2211yescharacteristic
36C2×C4×Dic53C2211yescharacteristic
37#13133124C2211yescharacteristic
38#13133124C2211yescharacteristic
39#13133124C2211yescharacteristic
40#13133124C2211yescharacteristic
41#13133124C2211yescharacteristic
42#13133124C2211yescharacteristic
43#13133124C2211yescharacteristic
44#13133124C2211yescharacteristic
45#13133124C2211yescharacteristic
46C22×D4×C53C2211yescharacteristic
47C2×C4⋊C4×C53C2211yescharacteristic
48C2×C22⋊C4×C53C2211yescharacteristic
49C2×C22⋊C4×C53C2211yescharacteristic
50C2×C22⋊C4×C53C2211yescharacteristic
51C2×C22⋊C4×C53C2211yescharacteristic
52#95×C53C2211yescharacteristic
53C24×C53C2311yescharacteristic
54C24×C53C2311yescharacteristic
55C2×D4×C53—22no—
56C2×D4×C53—41no—
57C2×D4×C53—41no—
58C22×C4×C53C2311yescharacteristic
59C22×C4×C53C2311yescharacteristic
60C22×C4×C53C2311yescharacteristic
61C22×C4×C53C2311yescharacteristic
62C22×C4×C53C2311yescharacteristic
63C22×Dic53—41no—
64C22×Dic53—41no—
65C22×Dic53D414yes—
66C22×Dic53C2311yescharacteristic
67C22×Dic53C2311yescharacteristic
68C22×Dic53C2311yescharacteristic
69C22×Dic53C2311yescharacteristic
70C22×Dic53C2311yescharacteristic
71C22×Dic53C2311yescharacteristic
72C22×Dic53C2311yescharacteristic
73C22×Dic53C2311yescharacteristic
74C2×Dic106—22no—
75C2×Dic106—41no—
76C2×Dic106—41no—
77C4⋊C4×C53—41no—
78C22⋊C4×C53—22no—
79C22⋊C4×C53—22no—
80C22⋊C4×C53—22no—
81C22⋊C4×C53—41no—
82C22⋊Dic53—22no—
83C22⋊Dic53—22no—
84C22⋊Dic53—22no—
85C22⋊Dic53—22no—
86C22⋊Dic53—22no—
87C22⋊Dic53—22no—
88C22⋊Dic53—41no—
89C4⋊Dic53—22no—
90C4⋊Dic53—22no—
91#11719919—22no—
92#11719919—22no—
93#11719919—22no—
94#11719919—22no—
95#11719919—22no—
96#11719919—41no—
97C4×Dic53—41no—
98C4×Dic53—41no—
99C23×C53—21no—
100C23×C53—21no—
101C23×C53—21no—
102C23×C53—21no—
103C23×C53—21no—
104C23×C53—21no—
105C23×C53—21no—
106C23×C53—21no—
107C23×C53—21no—
108C23×C53—21no—
109C23×C53—21no—
110C23×C53—21no—
111C23×C53—21no—
112C23×C53C2×D412yes—
113C23×C53C2411yescharacteristic
114D4×C53—44no—
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#4112yes—
130C2×Dic53—21no—
131C2×Dic53—21no—
132C2×Dic53—21no—
133C2×Dic53—21no—
134C2×Dic53—21no—
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×Dic53—44no—
152C2×Dic53C2×D412yes—
153C2×Dic53C2×D412yes—
154C2×Dic53#4112yes—
155Dic106—44no—
156C22×C53—22no—
157C22×C53—22no—
158C22×C53—22no—
159C22×C53—41no—
160C22×C53—41no—
161C22×C53—41no—
162C22×C53—41no—
163C22×C53—41no—
164C22×C53—41no—
165C22×C53—41no—
166C22×C53—41no—
167C22×C53—41no—
168C22×C53—41no—
169C22×C53—41no—
170C22×C53C22×D411yescharacteristic
171C22×C53C2×#4111yescharacteristic
172C22×C53C2×#4111yescharacteristic
173C22×C53H(5,𝔽2)11yescharacteristic
174C22×C53H(5,𝔽2)11yescharacteristic
175C22×C532−1+411yescharacteristic
176C22×C532−1+411yescharacteristic
177C4×C53—22no—
178C4×C53—41no—
179C4×C53—41no—
180C4×C53—41no—
181C4×C53—41no—
182Dic53—22no—
183Dic53—22no—
184Dic53—22no—
185Dic53—41no—
186Dic53—41no—
187Dic53—41no—
188Dic53—41no—
189Dic53—41no—
190#2663—531nomaximal, Sylow
191C2×C53—42no—
192C2×C53—81no—
193C2×C53#53211yescharacteristic
194C2×C53#53411yescharacteristic
195C2×C53#53611yescharacteristic
196C2×C53#53911yescharacteristic
197C2×C53#53911yescharacteristic
198C2×C53#54211yescharacteristic
199C2×C53#54211yescharacteristic
200C2×#123—531no—
201C2×C22⋊Q8—531no—
202#400—531no—
203#399—531no—
204#397—531no—
205#397—531no—
206#395—531no—
207#394—531no—
208#394D5311yescharacteristic
209#393—531no—
210#389—531no—
211#389—531no—
212#388—531no—
213#388—531no—
214C4×C22⋊C4—531no—
215C53#266311yescharacteristic, complemented, Sylow
216C22×Q8—531no—
217C22×D4C2×D5311yescharacteristic
218C23×C4—531no—
219#123—1064no—
220C22⋊Q8—1064no—
221C2×C4⋊C4—531no—
222C2×C4⋊C4—531no—
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×C22⋊C4—531no—
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⋊C4C2×D5311yescharacteristic
238C2×C22⋊C4C2×D5311yescharacteristic
239C2×C22⋊C4C2×D5311yescharacteristic
240C2×C22⋊C4C2×D5311yescharacteristic
241C2×C42—531no—
242C2×C42—531no—
243#95—531no—
244#95—531no—
245#95—531no—
246#95—531no—
247#95—531no—
248#95—531no—
249#95—531no—
250#95—531no—
251#95—531no—
252#95C2×D5311yescharacteristic
253C24C22×D5311yescharacteristic
254C24C22×D5311yescharacteristic
255C2×Q8—1062no—
256C2×Q8—2121no—
257C2×Q8—2121no—
258C2×D4—22no—
259C2×D4—41no—
260C2×D4—41no—
261C22×C4—531no—
262C22×C4—531no—
263C22×C4—531no—
264C22×C4—534no—
265C22×C4—531no—
266C22×C4—531no—
267C22×C4—531no—
268C22×C4—531no—
269C22×C4—531no—
270C22×C4—2121no—
271C22×C4—2121no—
272C22×C4C22×D5311yescharacteristic
273C22×C4C22×D5311yescharacteristic
274C22×C4C22×D5311yescharacteristic
275C22×C4C22×D5311yescharacteristic
276C22×C4C22×D5311yescharacteristic
277C4⋊C4—41no—
278C4⋊C4—1062no—
279C4⋊C4—1062no—
280C4⋊C4—1062no—
281C4⋊C4—1062no—
282C4⋊C4—1062no—
283C4⋊C4—1062no—
284C4⋊C4—1062no—
285C4⋊C4—2121no—
286C22⋊C4—22no—
287C22⋊C4—22no—
288C22⋊C4—22no—
289C22⋊C4—41no—
290C22⋊C4—1062no—
291C22⋊C4—1062no—
292C22⋊C4—1062no—
293C22⋊C4—1062no—
294C22⋊C4—1062no—
295C22⋊C4—1062no—
296C22⋊C4—2121no—
297C42—2121no—
298C42—2121no—
299C23—21no—
300C23—21no—
301C23—21no—
302C23—21no—
303C23—21no—
304C23—21no—
305C23—21no—
306C23—21no—
307C23—21no—
308C23—21no—
309C23—21no—
310C23—21no—
311C23—21no—
312C23D4×D5312yes—
313C23C23×D5311yescharacteristic
314Q8—2124no—
315D4—44no—
316C2×C4—21no—
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—532no—
331C2×C4—532no—
332C2×C4—532no—
333C2×C4—1061no—
334C2×C4—1061no—
335C2×C4—1061no—
336C2×C4—1061no—
337C2×C4—1061no—
338C2×C4—1061no—
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—2124no—
355C2×C4#1171994712yes—
356C22—22no—
357C22—22no—
358C22—22no—
359C22—41no—
360C22—41no—
361C22—41no—
362C22—41no—
363C22—41no—
364C22—41no—
365C22—41no—
366C22—41no—
367C22—41no—
368C22—41no—
369C22—41no—
370C22C2×D4×D5311yescharacteristic
371C22#41×D5311yescharacteristic
372C22C2×#1171994711yescharacteristic
373C22#1313326211yescharacteristic
374C22#1313326211yescharacteristic
375C22#1313326411yescharacteristic
376C22#1313326411yescharacteristic
377C4—22no—
378C4—41no—
379C4—41no—
380C4—41no—
381C4—41no—
382C4—1062no—
383C4—1062no—
384C4—1062no—
385C4—2121no—
386C4—2121no—
387C4—2121no—
388C4—2121no—
389C4—2121no—
390C2—42no—
391C2—81no—
392C2#1681116811yescharacteristic
393C2#1681118311yescharacteristic
394C2#1681119511yescharacteristic
395C2#1681121611yescharacteristic
396C2#1681121811yescharacteristic
397C2#1681123711yescharacteristic
398C2#1681124011yescharacteristic
subgroupG/H
center, FrattiniC23C23×D53
commutator, socleC23×C53C24
FittingC53×#394C2
hypercenter#394D53
hypocenterC53#2663
LCSG ▷ C23×C53 ▷ C53
UCSC1 ◁ C23 ◁ #394

Complements

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

NH ≅ G/Ntype
2. C2×#13133194C2semidirect
3. #16810564C2semidirect
4. #16810559C2semidirect
5. #16810551C2semidirect
6. #16810548C2semidirect
7. #16810543C2semidirect
9. #16810535C2semidirect
10. #16810520C2semidirect
11. #16810519C2semidirect
12. #16810516C2semidirect
13. #16810511C2semidirect
14. C22⋊C4×Dic53C2semidirect
215. C53#2663semidirect