Finite Group Databasebeta

← #20277396

Subgroups — #20277396

Order 6784.

556 conjugacy classes of subgroups.

The subgroup lattice has 374 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#13133195C211yesmaximal, characteristic
2C2×C4×Dic106C211yesmaximal, characteristic, complemented
3C2×#13133175C211yesmaximal, characteristic
4#16810527C211yesmaximal, characteristic, complemented
5#16810522C211yesmaximal, characteristic, complemented
6#16810520C211yesmaximal, characteristic, complemented
7#16810520C211yesmaximal, characteristic, complemented
8#16810515C211yesmaximal, characteristic
9#16810515C211yesmaximal, characteristic
10#16810514C211yesmaximal, characteristic
11#16810512C211yesmaximal, characteristic
12#16810504C211yesmaximal, characteristic, complemented
13#16810482C211yesmaximal, characteristic
14#16810474C211yesmaximal, characteristic, complemented
15C53×#387C211yesmaximal, characteristic, complemented
16C22×Dic106C2211yescharacteristic
17C22×C4×D53C2211yescharacteristic
18#13133195C418yescomplemented
19C4×Dic106—24no—
20#13133175—24no—
21C2×C4⋊Dic53C2211yescharacteristic
22C2×C4⋊Dic53C2211yescharacteristic
23C2×C4⋊Dic53C2211yescharacteristic
24C2×#11719919C2211yescharacteristic
25C2×#11719919C2211yescharacteristic
26C2×#11719919C2211yescharacteristic
27C2×#11719919C2211yescharacteristic
28C2×#11719919C2211yescharacteristic
29C2×#11719919C2211yescharacteristic
30C2×#11719921C2211yescharacteristic
31C2×#11719921C2211yescharacteristic
32C2×#11719921C2211yescharacteristic
33C2×#11719921C2211yescharacteristic
34C2×#11719921C2211yescharacteristic
35C2×#11719921C2211yescharacteristic
36C2×C4×Dic53C2211yescharacteristic
37C2×C4×Dic53C2211yescharacteristic
38C2×C4×Dic53C2211yescharacteristic
39C2×C4×Dic53C2211yescharacteristic
40C2×C4×Dic53C2211yescharacteristic
41#13133124C2211yescharacteristic
42#13133124C2211yescharacteristic
43#13133124C2211yescharacteristic
44#13133124C2211yescharacteristic
45#13133124C2211yescharacteristic
46#13133124C2211yescharacteristic
47C2×C4⋊C4×C53C2211yescharacteristic
48C2×C4⋊C4×C53C2211yescharacteristic
49C2×C42×C53C2211yescharacteristic
50#95×C53C2211yescharacteristic
51#95×C53C2211yescharacteristic
52#95×C53C2211yescharacteristic
53#95×C53C2211yescharacteristic
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
68C22×Dic53C2311yescharacteristic
69C2×C4×D53—22no—
70C2×C4×D53—22no—
71C2×C4×D53C2×C414yes—
72C2×Dic106—22no—
73C2×Dic106—22no—
74C2×Dic106C2×C414yes—
75C4⋊C4×C53—22no—
76C4⋊C4×C53C2×C414yes—
77C42×C53—22no—
78#11719921—22no—
79#11719921—22no—
80#11719921—41no—
81#11719921—41no—
82#11719921C2×C414yes—
83#11719921C2×C414yes—
84C4⋊Dic53—22no—
85C4⋊Dic53—22no—
86C4⋊Dic53—41no—
87#11719919—22no—
88#11719919—22no—
89#11719919—22no—
90#11719919—41no—
91#11719919C2×C414yes—
92#11719919C2×C414yes—
93C4×Dic53—22no—
94C4×Dic53—22no—
95C4×Dic53—41no—
96C4×Dic53—41no—
97C4×Dic53D414yes—
98C23×C53C2411yescharacteristic
99C22×D53—21no—
100C22×D53—21no—
101C22×D53—21no—
102C22×D53—21no—
103C22×D53—21no—
104C22×D53—21no—
105C22×D53C22×C412yes—
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×C53—21no—
120C2×C4×C53—21no—
121C2×C4×C53—21no—
122C2×C4×C53—21no—
123C2×C4×C53C22×C412yes—
124C2×C4×C53C22×C412yes—
125C2×C4×C53C22×C412yes—
126C2×C4×C53C2×D412yes—
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×Dic53—21no—
140C2×Dic53—21no—
141C2×Dic53—21no—
142C2×Dic53C22×C412yes—
143C2×Dic53C22×C412yes—
144C2×Dic53C22×C412yes—
145C2×Dic53C2×D412yes—
146C2×Dic53C2×D412yes—
147C2×Dic53#4112yes—
148C4×D53—28no—
149Dic106—28no—
150C22×C53C23×C411yescharacteristic
151C22×C53C22×D411yescharacteristic
152C22×C53C2×#4111yescharacteristic
153C22×C53H(5,𝔽2)11yescharacteristic
154C22×C532−1+411yescharacteristic
155C22×C532−1+411yescharacteristic
156C22×C532−1+411yescharacteristic
157C2×D53—22no—
158C2×D53—22no—
159C2×D53—22no—
160C2×D53—41no—
161C2×D53—41no—
162C2×D53—41no—
163C2×D53—41no—
164C4×C53—22no—
165C4×C53—22no—
166C4×C53—22no—
167C4×C53—22no—
168C4×C53—41no—
169C4×C53—41no—
170C4×C53—41no—
171Dic53—22no—
172Dic53—22no—
173Dic53—22no—
174Dic53—41no—
175Dic53—41no—
176Dic53—41no—
177Dic53C4×D414yes—
178#2323—531nomaximal, Sylow
179C2×C53C2×C4×D411yescharacteristic
180C2×C53#51911yescharacteristic
181C2×C53#52011yescharacteristic
182C2×C53#53611yescharacteristic
183C2×C53#53711yescharacteristic
184C2×C53#54111yescharacteristic
185C2×C53#54211yescharacteristic
186D53—42no—
187C2×C22⋊Q8—531no—
188C2×C4×Q8—531no—
189C2×#117—531no—
190#391—531no—
191#389—531no—
192#389—531no—
193#389—531no—
194#388—531no—
195#388—531no—
196#388—531no—
197#388—531no—
198#387—531no—
199#387D5311yescharacteristic, complemented
200#380—531no—
201C4×C4⋊C4—531no—
202C53#232311yescharacteristic, complemented, Sylow
203C22×Q8—531no—
204C23×C4—531no—
205C22⋊Q8—538no—
206C4×Q8—1064no—
207#117—1064no—
208C2×C4⋊C4—531no—
209C2×C4⋊C4—531no—
210C2×C4⋊C4—531no—
211C2×C4⋊C4—531no—
212C2×C4⋊C4—531no—
213C2×C4⋊C4—531no—
214C2×C4⋊C4—531no—
215C2×C4⋊C4—531no—
216C2×C4⋊C4—531no—
217C2×C4⋊C4C2×D5311yescharacteristic
218C2×C4⋊C4C2×D5311yescharacteristic
219C2×C22⋊C4—531no—
220C2×C22⋊C4—531no—
221C2×C22⋊C4—531no—
222C2×C22⋊C4—531no—
223C2×C22⋊C4—531no—
224C2×C22⋊C4—531no—
225C2×C42—531no—
226C2×C42—531no—
227C2×C42—531no—
228C2×C42—531no—
229C2×C42—531no—
230C2×C42C2×D5311yescharacteristic
231#95—531no—
232#95—531no—
233#95—531no—
234#95—531no—
235#95—531no—
236#95—531no—
237#95C2×D5311yescharacteristic
238#95C2×D5311yescharacteristic
239#95C2×D5311yescharacteristic
240#95C2×D5311yescharacteristic
241C24—531no—
242C2×Q8—534no—
243C2×Q8—1062no—
244C2×Q8—1062no—
245C22×C4—531no—
246C22×C4—531no—
247C22×C4—531no—
248C22×C4—531no—
249C22×C4—534no—
250C22×C4—531no—
251C22×C4—531no—
252C22×C4—531no—
253C22×C4—1062no—
254C22×C4—1062no—
255C22×C4C22×D5311yescharacteristic
256C22×C4C22×D5311yescharacteristic
257C22×C4C22×D5311yescharacteristic
258C22×C4C22×D5311yescharacteristic
259C22×C4C22×D5311yescharacteristic
260C22×C4C22×D5311yescharacteristic
261C22×C4C22×D5311yescharacteristic
262C4⋊C4—22no—
263C4⋊C4—534no—
264C4⋊C4—534no—
265C4⋊C4—1062no—
266C4⋊C4—1062no—
267C4⋊C4—1062no—
268C4⋊C4—1062no—
269C4⋊C4—1062no—
270C4⋊C4—2121no—
271C4⋊C4—2121no—
272C4⋊C4C4×D5314yes—
273C22⋊C4—534no—
274C22⋊C4—534no—
275C22⋊C4—1062no—
276C22⋊C4—1062no—
277C22⋊C4—2121no—
278C22⋊C4—2121no—
279C42—22no—
280C42—534no—
281C42—1062no—
282C42—1062no—
283C42—2121no—
284C42—2121no—
285C23—532no—
286C23—1061no—
287C23—1061no—
288C23—1061no—
289C23—1061no—
290C23—1061no—
291C23—1061no—
292C23C23×D5311yescharacteristic
293Q8—1068no—
294C2×C4—21no—
295C2×C4—21no—
296C2×C4—21no—
297C2×C4—21no—
298C2×C4—21no—
299C2×C4—21no—
300C2×C4—21no—
301C2×C4—21no—
302C2×C4—21no—
303C2×C4—21no—
304C2×C4—21no—
305C2×C4—21no—
306C2×C4—21no—
307C2×C4—21no—
308C2×C4—21no—
309C2×C4—21no—
310C2×C4—21no—
311C2×C4—532no—
312C2×C4—532no—
313C2×C4—532no—
314C2×C4—532no—
315C2×C4—532no—
316C2×C4—532no—
317C2×C4—1068no—
318C2×C4—1061no—
319C2×C4—1061no—
320C2×C4—1061no—
321C2×C4—1061no—
322C2×C4—1061no—
323C2×C4—1061no—
324C2×C4—1061no—
325C2×C4—1061no—
326C2×C4—1061no—
327C2×C4—1061no—
328C2×C4—1061no—
329C2×C4—1061no—
330C2×C4—1061no—
331C2×C4—1061no—
332C2×C4—1061no—
333C2×C4C2×C4×D5312yes—
334C2×C4C2×C4×D5312yes—
335C2×C4C2×C4×D5312yes—
336C2×C4D4×D5312yes—
337C22—1062no—
338C22—1062no—
339C22—1062no—
340C22—2121no—
341C22—2121no—
342C22—2121no—
343C22—2121no—
344C22C22×C4×D5311yescharacteristic
345C22C2×D4×D5311yescharacteristic
346C22#41×D5311yescharacteristic
347C22#1313326311yescharacteristic
348C22#1313326411yescharacteristic
349C22#1313326411yescharacteristic
350C22#1313326511yescharacteristic
351C4—22no—
352C4—22no—
353C4—22no—
354C4—22no—
355C4—41no—
356C4—41no—
357C4—41no—
358C4—534no—
359C4—1062no—
360C4—1062no—
361C4—1062no—
362C4—2121no—
363C4—2121no—
364C4—2121no—
365C2—2122no—
366C2C4×D4×D5311yescharacteristic
367C2#1681109911yescharacteristic
368C2#1681110311yescharacteristic
369C2#1681119511yescharacteristic
370C2#1681120311yescharacteristic
371C2#1681123211yescharacteristic
372C2#1681123911yescharacteristic
subgroupG/H
center, FrattiniC23C23×D53
commutatorC22×C53C23×C4
FittingC53×#387C2
socleC23×C53C24
hypercenter#387D53
hypocenterC53#2323
LCSG ▷ C22×C53 ▷ C53
UCSC1 ◁ C23 ◁ #387

Complements

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

NH ≅ G/Ntype
2. C2×C4×Dic106C2semidirect
4. #16810527C2semidirect
5. #16810522C2semidirect
6. #16810520C2semidirect
7. #16810520C2semidirect
12. #16810504C2semidirect
14. #16810474C2semidirect
15. C53×#387C2semidirect
18. #13133195C4semidirect
199. #387D53semidirect
202. C53#2323semidirect