Finite Group Databasebeta

← #20281339

Subgroups — #20281339

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×#13133212C211yesmaximal, characteristic, complemented
2#16810562C211yesmaximal, characteristic, complemented
3#16810556C211yesmaximal, characteristic, complemented
4#16810545C211yesmaximal, characteristic, complemented
5#16810539C211yesmaximal, characteristic
6#16810539C211yesmaximal, characteristic
7#16810534C211yesmaximal, characteristic, complemented
8#16810530C211yesmaximal, characteristic, complemented
9#16810522C211yesmaximal, characteristic, complemented
10#16810517C211yesmaximal, characteristic, complemented
11#16810513C211yesmaximal, characteristic, complemented
12#16810511C211yesmaximal, characteristic, complemented
13#16810502C211yesmaximal, characteristic
14#16810483C211yesmaximal, characteristic, complemented
15C2×C22⋊Q8×C53C211yesmaximal, characteristic, complemented
16C22×C22⋊D53C2211yescharacteristic
17#13133212—24no—
18C2×C4⋊Dic53C2211yescharacteristic
19C2×C4⋊Dic53C2211yescharacteristic
20C2×C4⋊Dic53C2211yescharacteristic
21C2×C4⋊Dic53C2211yescharacteristic
22C2×#11719919C2211yescharacteristic
23C2×#11719919C2211yescharacteristic
24C2×C22⋊Dic53C2211yescharacteristic
25C2×C22⋊Dic53C2211yescharacteristic
26C2×C22⋊Dic53C2211yescharacteristic
27C2×#11719921C2211yescharacteristic
28C2×#11719921C2211yescharacteristic
29C2×#11719921C2211yescharacteristic
30C2×#11719921C2211yescharacteristic
31C2×#11719921C2211yescharacteristic
32C2×#11719921C2211yescharacteristic
33C2×C4×Dic53C2211yescharacteristic
34C2×C4×Dic53C2211yescharacteristic
35#13133124C2211yescharacteristic
36#13133124C2211yescharacteristic
37#13133124C2211yescharacteristic
38#13133124C2211yescharacteristic
39#13133124C2211yescharacteristic
40#13133124C2211yescharacteristic
41#13133124C2211yescharacteristic
42#13133124C2211yescharacteristic
43#13133124C2211yescharacteristic
44#13133124C2211yescharacteristic
45C22×Q8×C53C2211yescharacteristic
46C23×C4×C53C2211yescharacteristic
47C22⋊Q8×C53—24no—
48C2×C4⋊C4×C53C2211yescharacteristic
49C2×C4⋊C4×C53C2211yescharacteristic
50C2×C4⋊C4×C53C2211yescharacteristic
51C2×C22⋊C4×C53C2211yescharacteristic
52C2×C22⋊C4×C53C2211yescharacteristic
53C24×C53C2311yescharacteristic
54C23×D53C2311yescharacteristic
55C2×Q8×C53—22no—
56C2×Q8×C53—22no—
57C2×Q8×C53—41no—
58C22×C4×C53—22no—
59C22×C4×C53—22no—
60C22×C4×C53—41no—
61C22×C4×C53C2311yescharacteristic
62C22×C4×C53C2311yescharacteristic
63C22×C4×C53C2311yescharacteristic
64C22×C4×C53C2311yescharacteristic
65C22×C4×C53C2311yescharacteristic
66C22×C4×C53C2311yescharacteristic
67C2×C22⋊D53—22no—
68C2×C22⋊D53—41no—
69C2×C22⋊D53—41no—
70C22×Dic53C2311yescharacteristic
71C22×Dic53C2311yescharacteristic
72C22×Dic53C2311yescharacteristic
73C22×Dic53C2311yescharacteristic
74C22×Dic53C2311yescharacteristic
75C22×Dic53C2311yescharacteristic
76C22×Dic53C2311yescharacteristic
77C4⋊C4×C53—22no—
78C4⋊C4×C53—22no—
79C4⋊C4×C53D414yescomplemented
80C22⋊C4×C53—22no—
81C22⋊C4×C53—22no—
82C22⋊Dic53—22no—
83C22⋊Dic53—22no—
84C22⋊Dic53—41no—
85#11719921—22no—
86#11719921—22no—
87#11719921—22no—
88#11719921—22no—
89#11719921—41no—
90#11719921—41no—
91C4⋊Dic53—22no—
92C4⋊Dic53—41no—
93C4⋊Dic53—41no—
94C4⋊Dic53—41no—
95#11719919—22no—
96#11719919—41no—
97C4×Dic53—22no—
98C4×Dic53—41no—
99C23×C53—21no—
100C23×C53—21no—
101C23×C53—21no—
102C23×C53—21no—
103C23×C53—21no—
104C23×C53—21no—
105C23×C53#4112yes—
106C23×C53C2411yescharacteristic
107C22×D53—21no—
108C22×D53—21no—
109C22×D53—21no—
110C22×D53—21no—
111C22×D53—21no—
112C22×D53—21no—
113C22×D53—21no—
114Q8×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—44no—
130C2×C4×C53C2×D412yes—
131C2×C4×C53C2×D412yes—
132C2×C4×C53C2×D412yes—
133C2×C4×C53#4112yes—
134C22⋊D53—44no—
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—21no—
152C2×Dic53—21no—
153C2×Dic53—21no—
154C2×Dic53—21no—
155C2×Dic53—21no—
156C22×C53—22no—
157C22×C53—22no—
158C22×C53—22no—
159C22×C53—41no—
160C22×C53—41no—
161C22×C53—41no—
162C22×C53—41no—
163C22×C53C22×D411yescharacteristic
164C22×C53C2×#4111yescharacteristic
165C22×C53C2×#4111yescharacteristic
166C22×C53H(5,𝔽2)11yescharacteristic
167C22×C53H(5,𝔽2)11yescharacteristic
168C22×C532−1+411yescharacteristic
169C22×C532−1+411yescharacteristic
170C2×D53—41no—
171C2×D53—41no—
172C2×D53—41no—
173C2×D53—41no—
174C2×D53—41no—
175C2×D53—41no—
176C2×D53—41no—
177C4×C53—22no—
178C4×C53—22no—
179C4×C53—22no—
180C4×C53—22no—
181C4×C53—41no—
182C4×C53—41no—
183Dic53—41no—
184Dic53—41no—
185Dic53—41no—
186Dic53—41no—
187Dic53—41no—
188Dic53—41no—
189Dic53—41no—
190#2702—531nomaximal, Sylow
191C2×C53—42no—
192C2×C53#53711yescharacteristic
193C2×C53#53811yescharacteristic
194C2×C53#54211yescharacteristic
195C2×C53#54611yescharacteristic
196C2×C53#54811yescharacteristic
197C2×C53#55211yescharacteristic
198C2×C53#56311yescharacteristic
199D53—81no—
200C2×#126—531no—
201C2×C22⋊Q8D5311yescharacteristic, complemented
202#400—531no—
203#398—531no—
204#396—531no—
205#394—531no—
206#394—531no—
207#393—531no—
208#391—531no—
209#389—531no—
210#388—531no—
211#388—531no—
212#388—531no—
213#386—531no—
214#380—531no—
215C53#270211yescharacteristic, complemented, Sylow
216C22×Q8C2×D5311yescharacteristic
217C22×D4—531no—
218C23×C4C2×D5311yescharacteristic
219#126—1064no—
220C22⋊Q8—24no—
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×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⋊C4—531no—
238C2×C22⋊C4—531no—
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#95—531no—
253C24—531no—
254C24C22×D5311yescharacteristic
255C2×Q8—22no—
256C2×Q8—22no—
257C2×Q8—41no—
258C2×D4—1062no—
259C2×D4—2121no—
260C2×D4—2121no—
261C22×C4—22no—
262C22×C4—22no—
263C22×C4—41no—
264C22×C4—531no—
265C22×C4—531no—
266C22×C4—531no—
267C22×C4—531no—
268C22×C4—531no—
269C22×C4—531no—
270C22×C4—531no—
271C22×C4C22×D5311yescharacteristic
272C22×C4C22×D5311yescharacteristic
273C22×C4C22×D5311yescharacteristic
274C22×C4C22×D5311yescharacteristic
275C22×C4C22×D5311yescharacteristic
276C22×C4C22×D5311yescharacteristic
277C4⋊C4—22no—
278C4⋊C4—22no—
279C4⋊C4—1062no—
280C4⋊C4—1062no—
281C4⋊C4—2121no—
282C4⋊C4—2121no—
283C4⋊C4—2121no—
284C4⋊C4—2121no—
285C4⋊C4C22⋊D5314yescomplemented
286C22⋊C4—22no—
287C22⋊C4—22no—
288C22⋊C4—1062no—
289C22⋊C4—1062no—
290C22⋊C4—1062no—
291C22⋊C4—1062no—
292C22⋊C4—1062no—
293C22⋊C4—1062no—
294C22⋊C4—2121no—
295C22⋊C4—2121no—
296C22⋊C4—2121no—
297C42—1062no—
298C42—2121no—
299C23—21no—
300C23—21no—
301C23—21no—
302C23—21no—
303C23—21no—
304C23—21no—
305C23—1061no—
306C23—1061no—
307C23—1061no—
308C23—1061no—
309C23—1061no—
310C23—1061no—
311C23—1061no—
312C23#1171994912yes—
313C23C23×D5311yescharacteristic
314Q8—44no—
315D4—2124no—
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—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—1061no—
348C2×C4—1061no—
349C2×C4—1061no—
350C2×C4—1061no—
351C2×C4—1061no—
352C2×C4#1171994712yes—
353C2×C4C2×C22⋊D5312yes—
354C2×C4C2×C22⋊D5312yes—
355C2×C4C2×C22⋊D5312yes—
356C22—22no—
357C22—22no—
358C22—22no—
359C22—41no—
360C22—41no—
361C22—41no—
362C22—41no—
363C22—2121no—
364C22—2121no—
365C22—2121no—
366C22—2121no—
367C22—2121no—
368C22—2121no—
369C22—2121no—
370C22C22×C22⋊D5311yescharacteristic
371C22C2×#1171994711yescharacteristic
372C22C2×#1171994911yescharacteristic
373C22#1313326211yescharacteristic
374C22#1313326311yescharacteristic
375C22#1313326411yescharacteristic
376C22#1313326511yescharacteristic
377C4—22no—
378C4—22no—
379C4—22no—
380C4—22no—
381C4—41no—
382C4—41no—
383C4—2121no—
384C4—2121no—
385C4—2121no—
386C4—2121no—
387C4—2121no—
388C4—2121no—
389C4—2121no—
390C2—42no—
391C2—4241no—
392C2#1681120111yescharacteristic
393C2#1681121311yescharacteristic
394C2#1681124611yescharacteristic
395C2#1681126711yescharacteristic
396C2#1681128411yescharacteristic
397C2#1681131511yescharacteristic
398C2#1681137411yescharacteristic
subgroupG/H
center, FrattiniC23C23×D53
commutator, socleC23×C53C24
FittingC2×C22⋊Q8×C53C2
hypercenterC2×C22⋊Q8D53
hypocenterC53#2702
LCSG ▷ C23×C53 ▷ C53
UCSC1 ◁ C23 ◁ C2×C22⋊Q8

Complements

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

NH ≅ G/Ntype
1. C2×#13133212C2semidirect
2. #16810562C2semidirect
3. #16810556C2semidirect
4. #16810545C2semidirect
7. #16810534C2semidirect
8. #16810530C2semidirect
9. #16810522C2semidirect
10. #16810517C2semidirect
11. #16810513C2semidirect
12. #16810511C2semidirect
14. #16810483C2semidirect
15. C2×C22⋊Q8×C53C2semidirect
79. C4⋊C4×C53D4semidirect
201. C2×C22⋊Q8D53semidirect
215. C53#2702semidirect
285. C4⋊C4C22⋊D53semidirect