Finite Group Databasebeta

← #20280352

Subgroups — #20280352

Order 6784.

470 conjugacy classes of subgroups.

The subgroup lattice has 362 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#13133212C211yesmaximal, characteristic
2C2×C4×Dic106C211yesmaximal, characteristic, complemented
3#16810565C211yesmaximal, characteristic, complemented
4#16810558C211yesmaximal, characteristic
5#16810544C211yesmaximal, characteristic, complemented
6#16810536C211yesmaximal, characteristic
7#16810520C211yesmaximal, characteristic, complemented
8#16810519C211yesmaximal, characteristic, complemented
9#16810515C211yesmaximal, characteristic
10#16810515C211yesmaximal, characteristic
11#16810514C211yesmaximal, characteristic
12#16810512C211yesmaximal, characteristic
13#16810504C211yesmaximal, characteristic, complemented
14#16810468C211yesmaximal, characteristic, complemented
15C53×#387C211yesmaximal, characteristic, complemented
16C22×Dic106C2211yescharacteristic
17#13133212—24no—
18C4×Dic106—24no—
19C2×C4⋊Dic53C2211yescharacteristic
20C2×C4⋊Dic53C2211yescharacteristic
21C2×C4⋊Dic53C2211yescharacteristic
22C2×#11719919C2211yescharacteristic
23C2×#11719919C2211yescharacteristic
24C2×#11719919C2211yescharacteristic
25C2×#11719919C2211yescharacteristic
26C2×#11719919C2211yescharacteristic
27C2×#11719921C2211yescharacteristic
28C2×#11719921C2211yescharacteristic
29C2×#11719921C2211yescharacteristic
30C2×#11719921C2211yescharacteristic
31C2×#11719921C2211yescharacteristic
32C2×#11719921C2211yescharacteristic
33C2×#11719921C2211yescharacteristic
34C2×C4×Dic53C2211yescharacteristic
35C2×C4×Dic53C2211yescharacteristic
36C2×C4×Dic53C2211yescharacteristic
37C2×C4×Dic53C2211yescharacteristic
38#13133124C2211yescharacteristic
39#13133124C2211yescharacteristic
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
54C22×C4×C53C2311yescharacteristic
55C22×C4×C53C2311yescharacteristic
56C22×C4×C53C2311yescharacteristic
57C22×C4×C53C2311yescharacteristic
58C22×C4×C53C2311yescharacteristic
59C22×C4×C53C2311yescharacteristic
60C22×C4×C53C2311yescharacteristic
61C22×Dic53C2311yescharacteristic
62C22×Dic53C2311yescharacteristic
63C22×Dic53C2311yescharacteristic
64C22×Dic53C2311yescharacteristic
65C22×Dic53C2311yescharacteristic
66C22×Dic53C2311yescharacteristic
67C22×Dic53C2311yescharacteristic
68C2×Dic106—22no—
69C2×Dic106—22no—
70C2×Dic106—22no—
71C4⋊C4×C53—22no—
72C4⋊C4×C53—22no—
73C42×C53—22no—
74#11719921—22no—
75#11719921—22no—
76#11719921—22no—
77#11719921—22no—
78#11719921—22no—
79#11719921—22no—
80#11719921—22no—
81C4⋊Dic53—22no—
82C4⋊Dic53—22no—
83C4⋊Dic53—22no—
84#11719919—22no—
85#11719919—22no—
86#11719919—22no—
87#11719919—22no—
88#11719919—22no—
89C4×Dic53—22no—
90C4×Dic53—22no—
91C4×Dic53—22no—
92C4×Dic53D414yes—
93C23×C53C2411yescharacteristic
94C22×D53—21no—
95C22×D53—21no—
96C22×D53—21no—
97C22×D53—21no—
98C22×D53—21no—
99C22×D53—21no—
100C22×D53—21no—
101C2×C4×C53—21no—
102C2×C4×C53—21no—
103C2×C4×C53—21no—
104C2×C4×C53—21no—
105C2×C4×C53—21no—
106C2×C4×C53—21no—
107C2×C4×C53—21no—
108C2×C4×C53—21no—
109C2×C4×C53—21no—
110C2×C4×C53—21no—
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×C53C2×D412yes—
120C2×C4×C53#4112yes—
121C2×C4×C53#4112yes—
122C2×Dic53—21no—
123C2×Dic53—21no—
124C2×Dic53—21no—
125C2×Dic53—21no—
126C2×Dic53—21no—
127C2×Dic53—21no—
128C2×Dic53—21no—
129C2×Dic53—21no—
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×Dic53C2×D412yes—
140C2×Dic53C2×D412yes—
141C2×Dic53#4112yes—
142C2×Dic53#4112yes—
143Dic106—28no—
144C22×C53C22×D411yescharacteristic
145C22×C53C2×#4111yescharacteristic
146C22×C53C2×#4111yescharacteristic
147C22×C53C2×#4111yescharacteristic
148C22×C53C2×#4111yescharacteristic
149C22×C532−1+411yescharacteristic
150C22×C532−1+411yescharacteristic
151C2×D53—41no—
152C2×D53—41no—
153C2×D53—41no—
154C2×D53—41no—
155C2×D53—41no—
156C2×D53—41no—
157C2×D53—41no—
158C4×C53—22no—
159C4×C53—22no—
160C4×C53—22no—
161C4×C53—41no—
162C4×C53—41no—
163C4×C53—41no—
164C4×C53—41no—
165Dic53—22no—
166Dic53—22no—
167Dic53—22no—
168Dic53—22no—
169Dic53—41no—
170Dic53—41no—
171Dic53—41no—
172#2604—531nomaximal, Sylow
173C2×C53#52511yescharacteristic
174C2×C53#53211yescharacteristic
175C2×C53#53611yescharacteristic
176C2×C53#55211yescharacteristic
177C2×C53#55211yescharacteristic
178C2×C53#55611yescharacteristic
179C2×C53#55611yescharacteristic
180D53—81no—
181C2×#126—531no—
182C2×C4×Q8—531no—
183#400—531no—
184#399—531no—
185#395—531no—
186#393—531no—
187#389—531no—
188#389—531no—
189#388—531no—
190#388—531no—
191#388—531no—
192#388—531no—
193#387—531no—
194#387D5311yescharacteristic, complemented
195#376—531no—
196C53#260411yescharacteristic, complemented, Sylow
197C22×Q8—531no—
198#126—1064no—
199C4×Q8—1064no—
200C2×C4⋊C4—531no—
201C2×C4⋊C4—531no—
202C2×C4⋊C4—531no—
203C2×C4⋊C4—531no—
204C2×C4⋊C4—531no—
205C2×C4⋊C4—531no—
206C2×C4⋊C4—531no—
207C2×C4⋊C4—531no—
208C2×C4⋊C4C2×D5311yescharacteristic
209C2×C4⋊C4C2×D5311yescharacteristic
210C2×C22⋊C4—531no—
211C2×C22⋊C4—531no—
212C2×C22⋊C4—531no—
213C2×C22⋊C4—531no—
214C2×C22⋊C4—531no—
215C2×C22⋊C4—531no—
216C2×C22⋊C4—531no—
217C2×C42—531no—
218C2×C42—531no—
219C2×C42—531no—
220C2×C42—531no—
221C2×C42C2×D5311yescharacteristic
222#95—531no—
223#95—531no—
224#95—531no—
225#95—531no—
226#95—531no—
227#95—531no—
228#95—531no—
229#95—531no—
230#95C2×D5311yescharacteristic
231#95C2×D5311yescharacteristic
232#95C2×D5311yescharacteristic
233#95C2×D5311yescharacteristic
234C24—531no—
235C2×Q8—1062no—
236C2×Q8—1062no—
237C2×Q8—1062no—
238C22×C4—531no—
239C22×C4—531no—
240C22×C4—531no—
241C22×C4—531no—
242C22×C4—531no—
243C22×C4—531no—
244C22×C4—531no—
245C22×C4C22×D5311yescharacteristic
246C22×C4C22×D5311yescharacteristic
247C22×C4C22×D5311yescharacteristic
248C22×C4C22×D5311yescharacteristic
249C22×C4C22×D5311yescharacteristic
250C22×C4C22×D5311yescharacteristic
251C22×C4C22×D5311yescharacteristic
252C4⋊C4—22no—
253C4⋊C4—22no—
254C4⋊C4—1062no—
255C4⋊C4—1062no—
256C4⋊C4—1062no—
257C4⋊C4—1062no—
258C4⋊C4—1062no—
259C4⋊C4—1062no—
260C4⋊C4—1062no—
261C4⋊C4—1062no—
262C22⋊C4—1062no—
263C22⋊C4—1062no—
264C22⋊C4—1062no—
265C22⋊C4—1062no—
266C22⋊C4—1062no—
267C22⋊C4—1062no—
268C22⋊C4—1062no—
269C42—22no—
270C42—534no—
271C42—1062no—
272C42—1062no—
273C42—1062no—
274C23—1061no—
275C23—1061no—
276C23—1061no—
277C23—1061no—
278C23—1061no—
279C23—1061no—
280C23—1061no—
281C23C23×D5311yescharacteristic
282Q8—1068no—
283C2×C4—21no—
284C2×C4—21no—
285C2×C4—21no—
286C2×C4—21no—
287C2×C4—21no—
288C2×C4—21no—
289C2×C4—21no—
290C2×C4—21no—
291C2×C4—21no—
292C2×C4—21no—
293C2×C4—21no—
294C2×C4—21no—
295C2×C4—21no—
296C2×C4—21no—
297C2×C4—21no—
298C2×C4—21no—
299C2×C4—21no—
300C2×C4—21no—
301C2×C4—532no—
302C2×C4—532no—
303C2×C4—532no—
304C2×C4—532no—
305C2×C4—1061no—
306C2×C4—1061no—
307C2×C4—1061no—
308C2×C4—1061no—
309C2×C4—1061no—
310C2×C4—1061no—
311C2×C4—1061no—
312C2×C4—1061no—
313C2×C4—1061no—
314C2×C4—1061no—
315C2×C4—1061no—
316C2×C4—1061no—
317C2×C4—1061no—
318C2×C4—1061no—
319C2×C4—1061no—
320C2×C4—1061no—
321C2×C4—1061no—
322C2×C4#1171994512yes—
323C2×C4D4×D5312yes—
324C2×C4#1171994912yes—
325C22—2121no—
326C22—2121no—
327C22—2121no—
328C22—2121no—
329C22—2121no—
330C22—2121no—
331C22—2121no—
332C22C2×D4×D5311yescharacteristic
333C22#41×D5311yescharacteristic
334C22#41×D5311yescharacteristic
335C22C2×#1171994511yescharacteristic
336C22C2×#1171994911yescharacteristic
337C22#1313326411yescharacteristic
338C22#1313326411yescharacteristic
339C4—22no—
340C4—22no—
341C4—22no—
342C4—41no—
343C4—41no—
344C4—41no—
345C4—41no—
346C4—1062no—
347C4—1062no—
348C4—1062no—
349C4—1062no—
350C4—2121no—
351C4—2121no—
352C4—2121no—
353C2—4241no—
354C2#1681113611yescharacteristic
355C2#1681117011yescharacteristic
356C2#1681119511yescharacteristic
357C2#1681130611yescharacteristic
358C2#1681131111yescharacteristic
359C2#1681134011yescharacteristic
360C2#1681134611yescharacteristic
subgroupG/H
center, FrattiniC23C23×D53
commutator, socleC23×C53C24
FittingC53×#387C2
hypercenter#387D53
hypocenterC53#2604
LCSG ▷ C23×C53 ▷ C53
UCSC1 ◁ C23 ◁ #387

Complements

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

NH ≅ G/Ntype
2. C2×C4×Dic106C2semidirect
3. #16810565C2semidirect
5. #16810544C2semidirect
7. #16810520C2semidirect
8. #16810519C2semidirect
13. #16810504C2semidirect
14. #16810468C2semidirect
15. C53×#387C2semidirect
194. #387D53semidirect
196. C53#2604semidirect