Finite Group Databasebeta

← #20287305

Subgroups — #20287305

Order 6784.

386 conjugacy classes of subgroups.

The subgroup lattice has 354 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#16811337C211yesmaximal, characteristic, complemented
2#16811273C211yesmaximal, characteristic, complemented
3#16811020C211yesmaximal, characteristic, complemented
4#16810945C211yesmaximal, characteristic, complemented
5#16810944C211yesmaximal, characteristic, complemented
6#16810940C211yesmaximal, characteristic, complemented
7#16810939C211yesmaximal, characteristic, complemented
8#16810911C211yesmaximal, characteristic, complemented
9#16810901C211yesmaximal, characteristic, complemented
10#16810873C211yesmaximal, characteristic, complemented
11D8×Dic53C211yesmaximal, characteristic, complemented
12#16810699C211yesmaximal, characteristic, complemented
13#16810665C211yesmaximal, characteristic, complemented
14#16810662C211yesmaximal, characteristic, complemented
15C53×#469C211yesmaximal, characteristic, complemented
16C2×#11719947—21no—
17C2×#11719912C2211yescharacteristic
18#13133216C2211yescharacteristic
19#13133216C2211yescharacteristic
20#13133204—21no—
21#13133203—21no—
22#13133199—21no—
23#13133198—21no—
24#13133194—21no—
25#13133189—21no—
26#13133188C2211yescharacteristic
27#13133188C2211yescharacteristic
28D4×Dic53C2211yescharacteristic
29D4×Dic53C2211yescharacteristic
30C4×C22⋊D53C2211yescharacteristic
31#13133176C2211yescharacteristic
32#13133175C2211yescharacteristic
33#13133173—21no—
34C2×C4⋊Dic53C2211yescharacteristic
35C4⋊C4×D53C2211yescharacteristic
36#13133151C2211yescharacteristic
37#13133151C2211yescharacteristic
38#13133150C2211yescharacteristic
39#13133149C2211yescharacteristic
40#13133149C2211yescharacteristic
41#13133148C2211yescharacteristic
42#13133148C2211yescharacteristic
43#13133148C2211yescharacteristic
44#13133146C2211yescharacteristic
45#13133138C2211yescharacteristic
46#13133138C2211yescharacteristic
47#13133138C2211yescharacteristic
48#13133138C2211yescharacteristic
49C22⋊#88322C2211yescharacteristic
50#13133129C2211yescharacteristic
51C8×Dic53C2211yescharacteristic
52C2×D8×C53C2211yescharacteristic
53C2×M16×C53C2211yescharacteristic
54#121×C53C2211yescharacteristic
55#121×C53C2211yescharacteristic
56#106×C53C2211yescharacteristic
57#102×C53C2211yescharacteristic
58#102×C53C2211yescharacteristic
59C2×D4×C53—21no—
60C2×D4×C53—21no—
61C2×D4×C53C2311yescharacteristic
62C2×D4×C53C2311yescharacteristic
63C22×C4×C53C2311yescharacteristic
64C2×C22⋊D53—21no—
65C2×C22⋊D53—21no—
66C2×C22⋊D53D411yescharacteristic
67C22×Dic53—21no—
68C22×Dic53—21no—
69C22×Dic53—21no—
70#11719947—44no—
71C2×C4×D53—21no—
72C2×C4×D53C2311yescharacteristic
73C2×Dic106—21no—
74C2×Dic106—21no—
75D8×C53—22no—
76M16×C53—22no—
77C2×C8×C53C2311yescharacteristic
78C2×C8×C53C2311yescharacteristic
79C4⋊C4×C53C2311yescharacteristic
80C4⋊C4×C53C2311yescharacteristic
81C22⋊C4×C53—21no—
82C22⋊C4×C53—21no—
83C22⋊Dic53—21no—
84C22⋊Dic53—21no—
85C22⋊Dic53—21no—
86C22⋊Dic53—21no—
87C22⋊Dic53—21no—
88C22⋊Dic53D411yescharacteristic
89#11719921—21no—
90#11719921D411yescharacteristic
91C4⋊Dic53—21no—
92C4⋊Dic53C2311yescharacteristic
93C4⋊Dic53C2311yescharacteristic
94C4⋊Dic53C2311yescharacteristic
95C4⋊Dic53C2311yescharacteristic
96#11719919—21no—
97#11719919—21no—
98#11719919—21no—
99#11719919D411yescharacteristic
100C4×Dic53—21no—
101C4×Dic53—21no—
102C4×Dic53C2311yescharacteristic
103C2×#88322C2311yescharacteristic
104C2×#88322C2311yescharacteristic
105#11719912—22no—
106C23×C53—21no—
107C23×C53—21no—
108C23×C53C2×D411yescharacteristic
109C22×D53C2×D411yescharacteristic
110D4×C53—21no—
111D4×C53—21no—
112D4×C53—21no—
113D4×C53—21no—
114D4×C53—42no—
115D4×C53—81no—
116C2×C4×C53—21no—
117C2×C4×C53—21no—
118C2×C4×C53—21no—
119C2×C4×C53—21no—
120C2×C4×C53C2×D411yescharacteristic
121C2×C4×C53C2411yescharacteristic
122C22⋊D53—22no—
123C22⋊D53—42no—
124C22⋊D53—81no—
125C2×Dic53—21no—
126C2×Dic53—21no—
127C2×Dic53—21no—
128C2×Dic53—21no—
129C2×Dic53—42no—
130C2×Dic53—42no—
131C2×Dic53—42no—
132C2×Dic53C2×D411yescharacteristic
133C2×Dic53C2×D411yescharacteristic
134C2×Dic53C2×D411yescharacteristic
135C4×D53—21no—
136C4×D53—21no—
137C4×D53—42no—
138Dic106—42no—
139Dic106—81no—
140C8×C53—21no—
141C8×C53#4112yescomplemented
142#88322—21no—
143#88322—21no—
144C22×C53—21no—
145C22×C53—21no—
146C22×C53—21no—
147C22×C53—41no—
148C22×C53—41no—
149C22×C53—41no—
150C22×C53—41no—
151C22×C53—41no—
152C22×C53—41no—
153C22×C53C22×D411yescharacteristic
154C2×D53—21no—
155C2×D53—21no—
156C2×D53—21no—
157C4×C53—21no—
158C4×C53—41no—
159C4×C53—41no—
160C4×C53C2×#4111yescharacteristic
161C4×C532−1+411yescharacteristic
162Dic53—21no—
163Dic53—21no—
164Dic53—41no—
165Dic53—41no—
166Dic53—41no—
167Dic53—41no—
168Dic53AGL(1,ℤ/8)12yes—
169#3354—531nomaximal, Sylow
170C2×C53—41no—
171C2×C53—81no—
172C2×C53—81no—
173C2×C53#54711yescharacteristic
174C2×C53C2×AGL(1,ℤ/8)11yescharacteristic
175C2×C53#57711yescharacteristic
176D53—41no—
177#555—531no—
178#547—531no—
179#498—531no—
180#482—531no—
181#482—531no—
182#481—531no—
183#481—531no—
184#476—531no—
185#474—531no—
186#469—531no—
187#469D5311yescharacteristic, complemented
188C4×D8—531no—
189#436—531no—
190#428—531no—
191#428—531no—
192C53#335411yescharacteristic, complemented, Sylow
193C2×#41—1061no—
194C2×D8C2×D5311yescharacteristic
195C2×M16—531no—
196C2×M16C2×D5311yescharacteristic
197C4⋊Q8—531no—
198C4⋊Q8—531no—
199#124—1061no—
200#124—1061no—
201#123—1061no—
202#123—1061no—
203C22⋊Q8—1061no—
204#121—531no—
205#121—531no—
206#121—1061no—
207#121C2×D5311yescharacteristic
208#121C2×D5311yescharacteristic
209C4×D4—531no—
210C4×D4—531no—
211C4×D4—531no—
212#117—531no—
213#117—531no—
214#117—1061no—
215C2×C4⋊C4—531no—
216C2×C4⋊C4—531no—
217#107—531no—
218#107—531no—
219#107—531no—
220#106—531no—
221#106—531no—
222#106—531no—
223#106—531no—
224#106—531no—
225#106C2×D5311yescharacteristic
226C4⋊C8—531no—
227#102—531no—
228#102—531no—
229#102—531no—
230#102—531no—
231#102C2×D5311yescharacteristic
232#102C2×D5311yescharacteristic
233C22⋊C8—531no—
234C22⋊C8—531no—
235C4×C8—531no—
236#41—2124no—
237C2×Q8—1061no—
238C2×Q8—1061no—
239C2×D4—21no—
240C2×D4—21no—
241C2×D4—531no—
242C2×D4—1061no—
243C2×D4—1061no—
244C2×D4C22×D5311yescharacteristic
245C2×D4C22×D5311yescharacteristic
246C22×C4—531no—
247C22×C4—1061no—
248C22×C4—1061no—
249C22×C4—1061no—
250C22×C4—1061no—
251C22×C4C22×D5311yescharacteristic
252D8—22no—
253M16—22no—
254M16—1062no—
255C2×C8—531no—
256C2×C8—531no—
257C2×C8C22×D5311yescharacteristic
258C2×C8C22×D5311yescharacteristic
259C4⋊C4—531no—
260C4⋊C4—531no—
261C4⋊C4—531no—
262C4⋊C4—531no—
263C4⋊C4—531no—
264C4⋊C4—1061no—
265C4⋊C4—1061no—
266C4⋊C4—1061no—
267C4⋊C4—1061no—
268C4⋊C4C22×D5311yescharacteristic
269C4⋊C4C22×D5311yescharacteristic
270C22⋊C4—21no—
271C22⋊C4—21no—
272C22⋊C4—531no—
273C22⋊C4—531no—
274C22⋊C4—1061no—
275C22⋊C4—1061no—
276C22⋊C4—1061no—
277C22⋊C4—1061no—
278C22⋊C4—1061no—
279C22⋊C4—1061no—
280C42—531no—
281C42—1061no—
282C42—1061no—
283C23—21no—
284C23—21no—
285C23—531no—
286C23D4×D5311yescharacteristic
287Q8—4241no—
288Q8—2122no—
289D4—21no—
290D4—21no—
291D4—21no—
292D4—21no—
293D4—42no—
294D4—81no—
295D4—1062no—
296D4—4241no—
297D4—2122no—
298C2×C4—21no—
299C2×C4—21no—
300C2×C4—21no—
301C2×C4—21no—
302C2×C4—531no—
303C2×C4—531no—
304C2×C4—531no—
305C2×C4—1061no—
306C2×C4—1061no—
307C2×C4—1061no—
308C2×C4—1061no—
309C2×C4—1061no—
310C2×C4—1061no—
311C2×C4—2122no—
312C2×C4—2122no—
313C2×C4—2122no—
314C2×C4—2122no—
315C2×C4D4×D5311yescharacteristic
316C2×C4C23×D5311yescharacteristic
317C8—21no—
318C8—1061no—
319C8—1061no—
320C8#1171994712yescomplemented
321C22—21no—
322C22—21no—
323C22—21no—
324C22—41no—
325C22—41no—
326C22—41no—
327C22—41no—
328C22—41no—
329C22—41no—
330C22—1061no—
331C22—1061no—
332C22—1061no—
333C22C2×D4×D5311yescharacteristic
334C4—21no—
335C4—41no—
336C4—41no—
337C4—532no—
338C4—1061no—
339C4—1061no—
340C4—2121no—
341C4—2121no—
342C4—2121no—
343C4—2121no—
344C4C2×#1171994711yescharacteristic
345C4#1313326411yescharacteristic
346C2—41no—
347C2—81no—
348C2—81no—
349C2—2121no—
350C2#1681127311yescharacteristic
351C2AGL(1,ℤ/8)×D5311yescharacteristic
352C2#1681143311yescharacteristic
subgroupG/H
centerC22C2×D4×D53
commutatorC2×C4×C53C24
FrattiniC2×C4C23×D53
FittingC53×#469C2
socleC22×C53C22×D4
hypercenter#469D53
hypocenterC53#3354
LCSG ▷ C2×C4×C53 ▷ C2×C53 ▷ C53
UCSC1 ◁ C22 ◁ C22×C4 ◁ #469

Complements

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

NH ≅ G/Ntype
1. #16811337C2semidirect
2. #16811273C2semidirect
3. #16811020C2semidirect
4. #16810945C2semidirect
5. #16810944C2semidirect
6. #16810940C2semidirect
7. #16810939C2semidirect
8. #16810911C2semidirect
9. #16810901C2semidirect
10. #16810873C2semidirect
11. D8×Dic53C2semidirect
12. #16810699C2semidirect
13. #16810665C2semidirect
14. #16810662C2semidirect
15. C53×#469C2semidirect
141. C8×C53#41semidirect
187. #469D53semidirect
192. C53#3354semidirect
320. C8#11719947semidirect