Finite Group Databasebeta

← #20277649

Subgroups — #20277649

Order 6784.

588 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×#13133177C211yesmaximal, characteristic
2#16810514C211yesmaximal, characteristic, complemented
3#16810512C211yesmaximal, characteristic, complemented
4#16810507C211yesmaximal, characteristic, complemented
5#16810505C211yesmaximal, characteristic, complemented
6#16810502C211yesmaximal, characteristic
7#16810500C211yesmaximal, characteristic
8#16810490C211yesmaximal, characteristic, complemented
9#16810485C211yesmaximal, characteristic, complemented
10#16810481C211yesmaximal, characteristic, complemented
11C22⋊C4×Dic53C211yesmaximal, characteristic, complemented
12C4×#11719921C211yesmaximal, characteristic, complemented
13#16810469C211yesmaximal, characteristic, complemented
14C2×#13133124C211yesmaximal, characteristic, complemented
15C4×C22⋊C4×C53C211yesmaximal, characteristic, complemented
16C22×C22⋊D53C2211yescharacteristic
17C23×Dic53C2211yescharacteristic
18C23×Dic53C2211yescharacteristic
19C22×C4×D53C2211yescharacteristic
20#13133177C418yescomplemented
21C2×#11719919C2211yescharacteristic
22C2×#11719919C2211yescharacteristic
23C2×#11719919C2211yescharacteristic
24C2×C22⋊Dic53C2211yescharacteristic
25C2×#11719921C2211yescharacteristic
26C2×#11719921C2211yescharacteristic
27C2×#11719921C2211yescharacteristic
28C2×#11719921C2211yescharacteristic
29C2×#11719921C2211yescharacteristic
30C2×C4×Dic53C2211yescharacteristic
31C2×C4×Dic53C2211yescharacteristic
32C2×C4×Dic53C2211yescharacteristic
33#13133124C2211yescharacteristic
34#13133124C2211yescharacteristic
35#13133124C2211yescharacteristic
36#13133124C2211yescharacteristic
37#13133124C2211yescharacteristic
38#13133124C2211yescharacteristic
39#13133124C2211yescharacteristic
40#13133124C2211yescharacteristic
41#13133124C2211yescharacteristic
42#13133124C2211yescharacteristic
43#13133124C2211yescharacteristic
44#13133124C2211yescharacteristic
45C23×C4×C53C2211yescharacteristic
46C2×C22⋊C4×C53C2211yescharacteristic
47C2×C22⋊C4×C53C2211yescharacteristic
48C2×C42×C53C2211yescharacteristic
49C2×C42×C53C2211yescharacteristic
50#95×C53C2211yescharacteristic
51#95×C53C2211yescharacteristic
52C24×C53C2311yescharacteristic
53C23×D53C2311yescharacteristic
54C22×C4×C53—22no—
55C22×C4×C53—22no—
56C22×C4×C53—41no—
57C22×C4×C53C2311yescharacteristic
58C22×C4×C53C2311yescharacteristic
59C22×C4×C53C2311yescharacteristic
60C22×C4×C53C2311yescharacteristic
61C22×C4×C53C2311yescharacteristic
62C22×C4×C53C2311yescharacteristic
63C2×C22⋊D53—22no—
64C2×C22⋊D53—22no—
65C2×C22⋊D53C2×C414yes—
66C22×Dic53—22no—
67C22×Dic53—22no—
68C22×Dic53—41no—
69C22×Dic53—41no—
70C22×Dic53—41no—
71C22×Dic53C2×C414yes—
72C22×Dic53C2311yescharacteristic
73C22×Dic53C2311yescharacteristic
74C22×Dic53C2311yescharacteristic
75C22×Dic53C2311yescharacteristic
76C22×Dic53C2311yescharacteristic
77C22×Dic53C2311yescharacteristic
78C22×Dic53C2311yescharacteristic
79C2×C4×D53—41no—
80C2×C4×D53—41no—
81C2×C4×D53C2×C414yescomplemented
82C22⋊C4×C53—22no—
83C22⋊C4×C53C2×C414yes—
84C42×C53—22no—
85C42×C53—41no—
86C22⋊Dic53—22no—
87#11719921—22no—
88#11719921—22no—
89#11719921—41no—
90#11719921—41no—
91#11719921C2×C414yescomplemented
92#11719919—41no—
93#11719919—41no—
94#11719919C2×C414yescomplemented
95C4×Dic53—22no—
96C4×Dic53—41no—
97C4×Dic53C2×C414yescomplemented
98C23×C53—21no—
99C23×C53—21no—
100C23×C53—21no—
101C23×C53—21no—
102C23×C53C22×C412yes—
103C23×C53#4112yes—
104C23×C53#4112yes—
105C23×C53C2411yescharacteristic
106C22×D53—21no—
107C22×D53—21no—
108C22×D53—21no—
109C22×D53—21no—
110C22×D53—21no—
111C22×D53—21no—
112C22×D53C22×C412yes—
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—44no—
129C2×C4×C53C22×C412yes—
130C2×C4×C53C22×C412yes—
131C2×C4×C53#4112yes—
132C22⋊D53—28no—
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—44no—
151C2×Dic53—44no—
152C2×Dic53C22×C412yes—
153C2×Dic53C22×C412yes—
154C2×Dic53C22×C412yes—
155C2×Dic53#4112yes—
156C4×D53—44no—
157C22×C53—22no—
158C22×C53—22no—
159C22×C53—22no—
160C22×C53—22no—
161C22×C53—22no—
162C22×C53—22no—
163C22×C53#11714yes—
164C22×C53C23×C411yescharacteristic
165C22×C53C2×#4111yescharacteristic
166C22×C53C2×#4111yescharacteristic
167C22×C53C2×#4111yescharacteristic
168C22×C53C2×#4111yescharacteristic
169C22×C53H(5,𝔽2)11yescharacteristic
170C22×C53H(5,𝔽2)11yescharacteristic
171C2×D53—22no—
172C2×D53—22no—
173C2×D53—22no—
174C2×D53—41no—
175C2×D53—41no—
176C2×D53—41no—
177C2×D53—41no—
178C4×C53—22no—
179C4×C53—22no—
180C4×C53—22no—
181C4×C53—41no—
182C4×C53—41no—
183C4×C53—41no—
184Dic53—22no—
185Dic53—22no—
186Dic53—22no—
187Dic53—22no—
188Dic53—41no—
189Dic53—41no—
190Dic53—41no—
191#2353—531nomaximal, Sylow
192C2×C53—24no—
193C2×C53C2×#11711yescharacteristic
194C2×C53C4×#4111yescharacteristic
195C2×C53#51811yescharacteristic
196C2×C53#55111yescharacteristic
197C2×C53#55111yescharacteristic
198C2×C53#55311yescharacteristic
199C2×C53#55311yescharacteristic
200D53—42no—
201C2×C4×D4—531no—
202#388—531no—
203#388—531no—
204#387—531no—
205#387—531no—
206#386—531no—
207#386—531no—
208#383—531no—
209#381—531no—
210#380—531no—
211C4×C22⋊C4—531no—
212C4×C22⋊C4—531no—
213C4×C22⋊C4D5311yescharacteristic, complemented
214#376—531no—
215C2×#95—531no—
216C53#235311yescharacteristic, complemented, Sylow
217C22×D4—531no—
218C23×C4—531no—
219C23×C4—531no—
220C23×C4—531no—
221C23×C4C2×D5311yescharacteristic
222C4×D4—538no—
223C2×C4⋊C4—531no—
224C2×C4⋊C4—531no—
225C2×C4⋊C4—531no—
226C2×C22⋊C4—531no—
227C2×C22⋊C4—531no—
228C2×C22⋊C4—531no—
229C2×C22⋊C4—531no—
230C2×C22⋊C4—531no—
231C2×C22⋊C4—531no—
232C2×C22⋊C4C2×D5311yescharacteristic
233C2×C22⋊C4C2×D5311yescharacteristic
234C2×C42—531no—
235C2×C42—531no—
236C2×C42—531no—
237C2×C42C2×D5311yescharacteristic
238C2×C42C2×D5311yescharacteristic
239#95—531no—
240#95—531no—
241#95—531no—
242#95—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#95C2×D5311yescharacteristic
252#95C2×D5311yescharacteristic
253C24—531no—
254C24C22×D5311yescharacteristic
255C2×D4—534no—
256C2×D4—1062no—
257C2×D4—1062no—
258C22×C4—22no—
259C22×C4—22no—
260C22×C4—41no—
261C22×C4—531no—
262C22×C4—531no—
263C22×C4—531no—
264C22×C4—531no—
265C22×C4—534no—
266C22×C4—534no—
267C22×C4—531no—
268C22×C4—531no—
269C22×C4—531no—
270C22×C4—1062no—
271C22×C4—1062no—
272C22×C4—2121no—
273C22×C4—2121no—
274C22×C4—2121no—
275C22×C4—2121no—
276C22×C4—2121no—
277C22×C4C22×D5311yescharacteristic
278C22×C4C22×D5311yescharacteristic
279C22×C4C22×D5311yescharacteristic
280C22×C4C22×D5311yescharacteristic
281C22×C4C22×D5311yescharacteristic
282C22×C4C22×D5311yescharacteristic
283C4⋊C4—534no—
284C4⋊C4—2121no—
285C4⋊C4—2121no—
286C22⋊C4—22no—
287C22⋊C4—534no—
288C22⋊C4—1062no—
289C22⋊C4—1062no—
290C22⋊C4—1062no—
291C22⋊C4—2121no—
292C22⋊C4—2121no—
293C22⋊C4C4×D5314yes—
294C42—22no—
295C42—41no—
296C42—534no—
297C42—1062no—
298C42—2121no—
299C23—21no—
300C23—21no—
301C23—21no—
302C23—21no—
303C23—532no—
304C23—1061no—
305C23—1061no—
306C23—1061no—
307C23—1061no—
308C23—1061no—
309C23—1061no—
310C23C2×C4×D5312yes—
311C23#1171994512yes—
312C23#1171994512yes—
313C23C23×D5311yescharacteristic
314D4—1068no—
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—21no—
327C2×C4—21no—
328C2×C4—21no—
329C2×C4—21no—
330C2×C4—44no—
331C2×C4—532no—
332C2×C4—532no—
333C2×C4—532no—
334C2×C4—532no—
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—2124no—
353C2×C4—2124no—
354C2×C4—2124no—
355C2×C4C2×C4×D5312yes—
356C2×C4C2×C4×D5312yes—
357C2×C4#1171994712yes—
358C22—22no—
359C22—22no—
360C22—22no—
361C22—22no—
362C22—22no—
363C22—22no—
364C22—1062no—
365C22—1062no—
366C22—1062no—
367C22—2121no—
368C22—2121no—
369C22—2121no—
370C22—2121no—
371C22#1313317414yes—
372C22C22×C4×D5311yescharacteristic
373C22#41×D5311yescharacteristic
374C22C2×#1171994711yescharacteristic
375C22C2×#1171994511yescharacteristic
376C22C2×#1171994511yescharacteristic
377C22#1313326211yescharacteristic
378C22#1313326311yescharacteristic
379C4—22no—
380C4—22no—
381C4—22no—
382C4—41no—
383C4—41no—
384C4—41no—
385C4—1062no—
386C4—1062no—
387C4—1062no—
388C4—1062no—
389C4—2121no—
390C4—2121no—
391C4—2121no—
392C2—24no—
393C2—2122no—
394C2C2×#1313317411yescharacteristic
395C2C4×#1171994711yescharacteristic
396C2#1681109511yescharacteristic
397C2#1681129711yescharacteristic
398C2#1681129911yescharacteristic
399C2#1681131911yescharacteristic
400C2#1681132211yescharacteristic
subgroupG/H
center, FrattiniC23C23×D53
commutatorC22×C53C23×C4
FittingC4×C22⋊C4×C53C2
socleC23×C53C24
hypercenterC4×C22⋊C4D53
hypocenterC53#2353
LCSG ▷ C22×C53 ▷ C53
UCSC1 ◁ C23 ◁ C4×C22⋊C4

Complements

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

NH ≅ G/Ntype
2. #16810514C2semidirect
3. #16810512C2semidirect
4. #16810507C2semidirect
5. #16810505C2semidirect
8. #16810490C2semidirect
9. #16810485C2semidirect
10. #16810481C2semidirect
11. C22⋊C4×Dic53C2semidirect
12. C4×#11719921C2semidirect
13. #16810469C2semidirect
14. C2×#13133124C2semidirect
15. C4×C22⋊C4×C53C2semidirect
20. #13133177C4semidirect
81. C2×C4×D53C2×C4semidirect
91. #11719921C2×C4semidirect
94. #11719919C2×C4semidirect
97. C4×Dic53C2×C4semidirect
213. C4×C22⋊C4D53semidirect
216. C53#2353semidirect