Finite Group Databasebeta

← #20281032

Subgroups — #20281032

Order 6784.

544 conjugacy classes of subgroups.

The subgroup lattice has 388 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#13133216C211yesmaximal, characteristic, complemented
2C2×#13133216C211yesmaximal, characteristic, complemented
3C2×#13133197C211yesmaximal, characteristic
4C2×#13133195C211yesmaximal, characteristic
5#16810558C211yesmaximal, characteristic
6#16810557C211yesmaximal, characteristic, complemented
7#16810544C211yesmaximal, characteristic, complemented
8#16810543C211yesmaximal, characteristic, complemented
9#16810536C211yesmaximal, characteristic
10#16810515C211yesmaximal, characteristic
11#16810512C211yesmaximal, characteristic
12#16810506C211yesmaximal, characteristic, complemented
13#16810504C211yesmaximal, characteristic, complemented
14#16810494C211yesmaximal, characteristic
15C53×#398C211yesmaximal, characteristic, complemented
16C22×Dic106C2211yescharacteristic
17C22×Dic106C2211yescharacteristic
18C22×Dic106C2211yescharacteristic
19C22×C4×D53C2211yescharacteristic
20#13133216—24no—
21#13133216—24no—
22#13133197—24no—
23#13133195—24no—
24C2×C4⋊Dic53C2211yescharacteristic
25C2×C4⋊Dic53C2211yescharacteristic
26C2×C4⋊Dic53C2211yescharacteristic
27C2×#11719919C2211yescharacteristic
28C2×#11719919C2211yescharacteristic
29C2×#11719919C2211yescharacteristic
30C2×#11719919C2211yescharacteristic
31C2×#11719919C2211yescharacteristic
32C2×#11719919C2211yescharacteristic
33C2×#11719921C2211yescharacteristic
34C2×#11719921C2211yescharacteristic
35C2×#11719921C2211yescharacteristic
36C2×#11719921C2211yescharacteristic
37C2×#11719921C2211yescharacteristic
38C2×#11719921C2211yescharacteristic
39C2×C4×Dic53C2211yescharacteristic
40C2×C4×Dic53C2211yescharacteristic
41#13133124C2211yescharacteristic
42#13133124C2211yescharacteristic
43#13133124C2211yescharacteristic
44#13133124C2211yescharacteristic
45#13133124C2211yescharacteristic
46#13133124C2211yescharacteristic
47#13133124C2211yescharacteristic
48C2×C4⋊C4×C53C2211yescharacteristic
49C2×C4⋊C4×C53C2211yescharacteristic
50C2×C4⋊C4×C53C2211yescharacteristic
51C2×C4⋊C4×C53C2211yescharacteristic
52#95×C53C2211yescharacteristic
53#95×C53C2211yescharacteristic
54#95×C53C2211yescharacteristic
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
68C22×Dic53C2311yescharacteristic
69C22×Dic53C2311yescharacteristic
70C2×C4×D53—22no—
71C2×C4×D53—22no—
72C2×C4×D53—41no—
73C2×Dic106—22no—
74C2×Dic106—22no—
75C2×Dic106—22no—
76C2×Dic106—22no—
77C2×Dic106—22no—
78C2×Dic106—22no—
79C2×Dic106—41no—
80C2×Dic106—41no—
81C2×Dic106—41no—
82C4⋊C4×C53—22no—
83C4⋊C4×C53—22no—
84C4⋊C4×C53—22no—
85C4⋊C4×C53—22no—
86#11719921—22no—
87#11719921—22no—
88#11719921—22no—
89#11719921—22no—
90#11719921—41no—
91#11719921Q814yescomplemented
92C4⋊Dic53—22no—
93C4⋊Dic53—22no—
94C4⋊Dic53D414yes—
95#11719919—22no—
96#11719919—22no—
97#11719919—22no—
98#11719919—22no—
99#11719919—41no—
100#11719919D414yes—
101C4×Dic53—22no—
102C4×Dic53—22no—
103C23×C53C2411yescharacteristic
104C22×D53—21no—
105C22×D53—21no—
106C22×D53—21no—
107C22×D53—21no—
108C22×D53—21no—
109C22×D53—21no—
110C22×D53C2×Q812yes—
111C2×C4×C53—21no—
112C2×C4×C53—21no—
113C2×C4×C53—21no—
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×C53C2×D412yes—
130C2×C4×C53C2×D412yes—
131C2×C4×C53C2×Q812yes—
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×Dic53C2×D412yes—
149C2×Dic53C2×D412yes—
150C2×Dic53C2×D412yes—
151C2×Dic53C2×D412yes—
152C2×Dic53C2×Q812yes—
153C4×D53—44no—
154Dic106—44no—
155Dic106—44no—
156Dic106—44no—
157C22×C53C22×D411yescharacteristic
158C22×C53C22×D411yescharacteristic
159C22×C53C22×Q811yescharacteristic
160C22×C53H(5,𝔽2)11yescharacteristic
161C22×C532−1+411yescharacteristic
162C22×C532−1+411yescharacteristic
163C22×C532−1+411yescharacteristic
164C2×D53—22no—
165C2×D53—22no—
166C2×D53—22no—
167C2×D53—41no—
168C2×D53—41no—
169C2×D53—41no—
170C2×D53—41no—
171C4×C53—22no—
172C4×C53—22no—
173C4×C53—22no—
174C4×C53—41no—
175C4×C53—41no—
176C4×C53—41no—
177C4×C53—41no—
178Dic53—22no—
179Dic53—22no—
180Dic53—22no—
181Dic53—22no—
182Dic53—22no—
183Dic53—41no—
184Dic53—41no—
185#2677—531nomaximal, Sylow
186C2×C53#53611yescharacteristic
187C2×C53#53611yescharacteristic
188C2×C53#54411yescharacteristic
189C2×C53D4211yescharacteristic
190C2×C53D4×Q811yescharacteristic
191C2×C53D4×Q811yescharacteristic
192C2×C53#56311yescharacteristic
193D53—42no—
194C2×C4⋊Q8—531no—
195C2×C4⋊Q8—531no—
196C2×C22⋊Q8—531no—
197C2×C22⋊Q8—531no—
198#399—531no—
199#398—531no—
200#398D5311yescharacteristic, complemented
201#395—531no—
202#395—531no—
203#393—531no—
204#388—531no—
205#388—531no—
206#387—531no—
207#387—531no—
208#385—531no—
209C53#267711yescharacteristic, complemented, Sylow
210C22×Q8—531no—
211C22×Q8—531no—
212C22×Q8—531no—
213C23×C4—531no—
214C4⋊Q8—1064no—
215C4⋊Q8—1064no—
216C22⋊Q8—1064no—
217C22⋊Q8—1064no—
218C2×C4⋊C4—531no—
219C2×C4⋊C4—531no—
220C2×C4⋊C4—531no—
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⋊C4C2×D5311yescharacteristic
228C2×C4⋊C4C2×D5311yescharacteristic
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×C42—531no—
238C2×C42—531no—
239#95—531no—
240#95—531no—
241#95—531no—
242#95—531no—
243#95—531no—
244#95—531no—
245#95—531no—
246#95C2×D5311yescharacteristic
247#95C2×D5311yescharacteristic
248#95C2×D5311yescharacteristic
249C24—531no—
250C2×Q8—1062no—
251C2×Q8—1062no—
252C2×Q8—1062no—
253C2×Q8—1062no—
254C2×Q8—1062no—
255C2×Q8—1062no—
256C2×Q8—2121no—
257C2×Q8—2121no—
258C2×Q8—2121no—
259C22×C4—531no—
260C22×C4—531no—
261C22×C4—531no—
262C22×C4—531no—
263C22×C4—531no—
264C22×C4—531no—
265C22×C4—531no—
266C22×C4—1062no—
267C22×C4—1062no—
268C22×C4—2121no—
269C22×C4C22×D5311yescharacteristic
270C22×C4C22×D5311yescharacteristic
271C22×C4C22×D5311yescharacteristic
272C22×C4C22×D5311yescharacteristic
273C22×C4C22×D5311yescharacteristic
274C22×C4C22×D5311yescharacteristic
275C22×C4C22×D5311yescharacteristic
276C4⋊C4—22no—
277C4⋊C4—22no—
278C4⋊C4—22no—
279C4⋊C4—22no—
280C4⋊C4—534no—
281C4⋊C4—534no—
282C4⋊C4—1062no—
283C4⋊C4—1062no—
284C4⋊C4—1062no—
285C4⋊C4—1062no—
286C4⋊C4—1062no—
287C4⋊C4—1062no—
288C4⋊C4—2121no—
289C22⋊C4—534no—
290C22⋊C4—1062no—
291C22⋊C4—1062no—
292C22⋊C4—1062no—
293C22⋊C4—1062no—
294C22⋊C4—2121no—
295C42—1062no—
296C42—1062no—
297C23—532no—
298C23—1061no—
299C23—1061no—
300C23—1061no—
301C23—1061no—
302C23—1061no—
303C23—1061no—
304C23C23×D5311yescharacteristic
305Q8—2124no—
306Q8—2124no—
307Q8—2124no—
308C2×C4—21no—
309C2×C4—21no—
310C2×C4—21no—
311C2×C4—21no—
312C2×C4—21no—
313C2×C4—21no—
314C2×C4—21no—
315C2×C4—21no—
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—532no—
327C2×C4—532no—
328C2×C4—532no—
329C2×C4—532no—
330C2×C4—532no—
331C2×C4—1061no—
332C2×C4—1061no—
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—2124no—
348C2×C4D4×D5312yes—
349C2×C4D4×D5312yes—
350C2×C4Q8×D5312yes—
351C22—1062no—
352C22—1062no—
353C22—1062no—
354C22—2121no—
355C22—2121no—
356C22—2121no—
357C22—2121no—
358C22C2×D4×D5311yescharacteristic
359C22C2×D4×D5311yescharacteristic
360C22C2×Q8×D5311yescharacteristic
361C22#1313326311yescharacteristic
362C22#1313326411yescharacteristic
363C22#1313326411yescharacteristic
364C22#1313326511yescharacteristic
365C4—22no—
366C4—22no—
367C4—22no—
368C4—41no—
369C4—41no—
370C4—41no—
371C4—41no—
372C4—1062no—
373C4—1062no—
374C4—1062no—
375C4—1062no—
376C4—1062no—
377C4—2121no—
378C4—2121no—
379C2—2122no—
380C2#1681119511yescharacteristic
381C2#1681119611yescharacteristic
382C2#1681125311yescharacteristic
383C2#1681125711yescharacteristic
384C2#1681128611yescharacteristic
385C2#1681128811yescharacteristic
386C2#1681137711yescharacteristic
subgroupG/H
center, FrattiniC23C23×D53
commutator, socleC23×C53C24
FittingC53×#398C2
hypercenter#398D53
hypocenterC53#2677
LCSG ▷ C23×C53 ▷ C53
UCSC1 ◁ C23 ◁ #398

Complements

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

NH ≅ G/Ntype
1. C2×#13133216C2semidirect
2. C2×#13133216C2semidirect
6. #16810557C2semidirect
7. #16810544C2semidirect
8. #16810543C2semidirect
12. #16810506C2semidirect
13. #16810504C2semidirect
15. C53×#398C2semidirect
91. #11719921Q8semidirect
200. #398D53semidirect
209. C53#2677semidirect