Finite Group Databasebeta

← #20284916

Subgroups — #20284916

Order 6784.

396 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
1#16811228C211yesmaximal, characteristic, complemented
2#16811103C211yesmaximal, characteristic, complemented
3#16810942C211yesmaximal, characteristic, complemented
4#16810937C211yesmaximal, characteristic, complemented
5#16810931C211yesmaximal, characteristic, complemented
6#16810924C211yesmaximal, characteristic, complemented
7#16810824C211yesmaximal, characteristic, complemented
8#16810819C211yesmaximal, characteristic, complemented
9#16810770C211yesmaximal, characteristic, complemented
10#16810770C211yesmaximal, characteristic, complemented
11#16810736C211yesmaximal, characteristic, complemented
12#16810725C211yesmaximal, characteristic, complemented
13C4×#11719913C211yesmaximal, characteristic, complemented
14C4×D424C211yesmaximal, characteristic, complemented
15C53×#433C211yesmaximal, characteristic, complemented
16C2×#11719945C2211yescharacteristic
17C2×#11719913C2211yescharacteristic
18C2×#11719913C2211yescharacteristic
19C2×D424C2211yescharacteristic
20C2×D424C2211yescharacteristic
21#13133214C2211yescharacteristic
22#13133209C2211yescharacteristic
23#13133202—21no—
24#13133202—21no—
25#13133191C2211yescharacteristic
26#13133191C2211yescharacteristic
27#13133190—21no—
28#13133190—21no—
29C4×Dic106C2211yescharacteristic
30C4×Dic106C2211yescharacteristic
31C4×D212C2211yescharacteristic
32C4×D212C2211yescharacteristic
33C4×D212C2211yescharacteristic
34C4×D212C2211yescharacteristic
35#13133177—21no—
36#13133177—21no—
37#13133175—21no—
38C2×C4⋊Dic53C2211yescharacteristic
39C4⋊C4×D53—21no—
40#13133151C2211yescharacteristic
41#13133151C2211yescharacteristic
42#13133149C2211yescharacteristic
43#13133149C2211yescharacteristic
44#13133142C2211yescharacteristic
45#13133142C2211yescharacteristic
46#13133139C2211yescharacteristic
47#13133139C2211yescharacteristic
48#13133139C2211yescharacteristic
49#13133139C2211yescharacteristic
50#13133139C2211yescharacteristic
51#13133139C2211yescharacteristic
52#117×C53C2211yescharacteristic
53C4⋊C8×C53C2211yescharacteristic
54C4⋊C8×C53C2211yescharacteristic
55C22⋊C8×C53C2211yescharacteristic
56C22⋊C8×C53C2211yescharacteristic
57#97×C53C2211yescharacteristic
58C4×C8×C53C2211yescharacteristic
59C22×C4×C53C2311yescharacteristic
60C2×C22⋊D53—21no—
61C2×C22⋊D53—21no—
62C2×C22⋊D53—21no—
63C22×Dic53—21no—
64#11719945—22no—
65#11719945—22no—
66C2×D212—21no—
67C2×D212—21no—
68C2×D212C2311yescharacteristic
69C2×D212C2311yescharacteristic
70C2×D212C2311yescharacteristic
71C2×C4×D53—21no—
72C2×C4×D53—21no—
73C2×C4×D53—21no—
74C2×C4×D53—21no—
75C2×C4×D53—21no—
76C2×Dic106C2311yescharacteristic
77C2×C8×C53C2311yescharacteristic
78C2×C8×C53C2311yescharacteristic
79C2×C8×C53C2311yescharacteristic
80C2×C8×C53C2311yescharacteristic
81C4⋊C4×C53D411yescharacteristic
82C4⋊C4×C53D411yescharacteristic
83C22⋊C4×C53D411yescharacteristic
84C22⋊C4×C53D411yescharacteristic
85C42×C53C2311yescharacteristic
86C42×C53C2311yescharacteristic
87#11719921—21no—
88#11719921—21no—
89#11719921—21no—
90#11719921—21no—
91#11719921—21no—
92#11719921—21no—
93C4⋊Dic53—21no—
94C4⋊Dic53C2311yescharacteristic
95C4⋊Dic53C2311yescharacteristic
96C4⋊Dic53C2311yescharacteristic
97C4⋊Dic53C2311yescharacteristic
98#11719919—21no—
99#11719919—21no—
100#11719919—21no—
101#11719919—21no—
102C4×Dic53—21no—
103C4×Dic53—21no—
104D424—22no—
105D424—22no—
106#11719913—22no—
107#11719913—22no—
108C23×C53C2×D411yescharacteristic
109C22×D53—21no—
110C22×D53—21no—
111C22×D53—21no—
112C2×C4×C53—21no—
113C2×C4×C53—21no—
114C2×C4×C53C2×D411yescharacteristic
115C2×C4×C53C2×D411yescharacteristic
116C2×C4×C53C2×D411yescharacteristic
117C2×C4×C53C2×D411yescharacteristic
118C2×C4×C53C2×D411yescharacteristic
119C2×C4×C53C2411yescharacteristic
120C22⋊D53—24no—
121C22⋊D53—81no—
122C22⋊D53—81no—
123C2×Dic53—21no—
124C2×Dic53—21no—
125C2×Dic53—21no—
126C2×Dic53—21no—
127C2×Dic53—21no—
128C2×Dic53—81no—
129D212—21no—
130D212—21no—
131D212—21no—
132D212—21no—
133D212—21no—
134D212—81no—
135D212—81no—
136D212#4112yes—
137C4×D53—24no—
138C4×D53—81no—
139C4×D53—81no—
140C4×D53—81no—
141C4×D53—81no—
142Dic106—21no—
143Dic106#4112yes—
144C8×C53—21no—
145C8×C53—21no—
146C8×C53—21no—
147C8×C53—21no—
148C22×C53—21no—
149C22×C53—21no—
150C22×C53—21no—
151C22×C53C22×D411yescharacteristic
152C2×D53—22no—
153C2×D53—41no—
154C2×D53—41no—
155C2×D53—41no—
156C2×D53—41no—
157C2×D53—41no—
158C2×D53—41no—
159C2×D53—41no—
160C2×D53—41no—
161C4×C53—21no—
162C4×C53—21no—
163C4×C53—21no—
164C4×C53—21no—
165C4×C53—21no—
166C4×C53C2×#4111yescharacteristic
167C4×C53C2×#4111yescharacteristic
168Dic53—22no—
169Dic53—41no—
170Dic53—41no—
171Dic53—41no—
172Dic53—41no—
173#3119—531nomaximal, Sylow
174C2×C53—41no—
175C2×C53#52511yescharacteristic
176C2×C53#57611yescharacteristic
177C2×C53#57711yescharacteristic
178D53—42no—
179D53—81no—
180D53—81no—
181#540—531no—
182#520—531no—
183#481—531no—
184#480—531no—
185#479—531no—
186#478—531no—
187#460—531no—
188#459—531no—
189#449—531no—
190#449—531no—
191#442—531no—
192#440—531no—
193C4×SD16—531no—
194C4×D8—531no—
195#433D5311yescharacteristic, complemented
196C53#311911yescharacteristic, complemented, Sylow
197C2×#41—531no—
198C2×SD16—531no—
199C2×SD16—531no—
200C2×D8—531no—
201C2×D8—531no—
202#127—531no—
203#125—531no—
204#123—1061no—
205#123—1061no—
206#121—531no—
207#121—531no—
208#121—1061no—
209#121—1061no—
210C4×Q8—531no—
211C4×Q8—531no—
212C4×D4—531no—
213C4×D4—531no—
214C4×D4—531no—
215C4×D4—531no—
216C4×D4—1061no—
217C4×D4—1061no—
218#117—1061no—
219#117C2×D5311yescharacteristic
220C2×C4⋊C4—531no—
221C2×C4⋊C4—1061no—
222#107—531no—
223#107—531no—
224#106—531no—
225#106—531no—
226C4⋊C8C2×D5311yescharacteristic
227C4⋊C8C2×D5311yescharacteristic
228#103—531no—
229#103—531no—
230#102—531no—
231#102—531no—
232#102—531no—
233#102—531no—
234#102—531no—
235#102—531no—
236C22⋊C8C2×D5311yescharacteristic
237C22⋊C8C2×D5311yescharacteristic
238#97C2×D5311yescharacteristic
239C4×C8C2×D5311yescharacteristic
240#41—1062no—
241#41—1062no—
242C2×Q8—531no—
243C2×D4—531no—
244C2×D4—531no—
245C2×D4—531no—
246C2×D4—1061no—
247C2×D4—1061no—
248C2×D4—1061no—
249C2×D4—1061no—
250C2×D4—1061no—
251C22×C4—1061no—
252C22×C4—1061no—
253C22×C4—1061no—
254C22×C4—1061no—
255C22×C4—1061no—
256C22×C4—1061no—
257C22×C4C22×D5311yescharacteristic
258SD16—1062no—
259SD16—1062no—
260D8—1062no—
261D8—1062no—
262C2×C8C22×D5311yescharacteristic
263C2×C8C22×D5311yescharacteristic
264C2×C8C22×D5311yescharacteristic
265C2×C8C22×D5311yescharacteristic
266C4⋊C4—531no—
267C4⋊C4—531no—
268C4⋊C4—531no—
269C4⋊C4—531no—
270C4⋊C4—1061no—
271C4⋊C4—1061no—
272C4⋊C4—1061no—
273C4⋊C4—1061no—
274C4⋊C4—1061no—
275C4⋊C4D21211yescharacteristic
276C4⋊C4D21211yescharacteristic
277C22⋊C4—1061no—
278C22⋊C4—1061no—
279C22⋊C4—1061no—
280C22⋊C4—1061no—
281C22⋊C4—1061no—
282C22⋊C4—1061no—
283C22⋊C4D21211yescharacteristic
284C22⋊C4D21211yescharacteristic
285C42—1061no—
286C42—1061no—
287C42C22×D5311yescharacteristic
288C42C22×D5311yescharacteristic
289C23—1061no—
290C23—1061no—
291C23—1061no—
292C23C2×D21211yescharacteristic
293Q8—532no—
294Q8—1061no—
295D4—532no—
296D4—1064no—
297D4—1061no—
298D4—1061no—
299D4—1061no—
300D4—1061no—
301D4—1061no—
302D4—4241no—
303D4—4241no—
304D4—4241no—
305D4—4241no—
306C2×C4—21no—
307C2×C4—21no—
308C2×C4—1064no—
309C2×C4—1061no—
310C2×C4—1061no—
311C2×C4—1061no—
312C2×C4—1061no—
313C2×C4—1061no—
314C2×C4—4241no—
315C2×C4—4241no—
316C2×C4—4241no—
317C2×C4—4241no—
318C2×C4—4241no—
319C2×C4C2×D21211yescharacteristic
320C2×C4C2×D21211yescharacteristic
321C2×C4C2×D21211yescharacteristic
322C2×C4C2×D21211yescharacteristic
323C2×C4C2×D21211yescharacteristic
324C2×C4C23×D5311yescharacteristic
325C8—21no—
326C8—21no—
327C8—21no—
328C8—21no—
329C22—21no—
330C22—21no—
331C22—21no—
332C22—1062no—
333C22—2121no—
334C22—2121no—
335C22—2121no—
336C22—2121no—
337C22—2121no—
338C22—2121no—
339C22—2121no—
340C22—2121no—
341C22C22×D21211yescharacteristic
342C4—21no—
343C4—21no—
344C4—21no—
345C4—21no—
346C4—21no—
347C4—1062no—
348C4—2121no—
349C4—2121no—
350C4—2121no—
351C4—2121no—
352C4#41×D5311yescharacteristic
353C4#41×D5311yescharacteristic
354C2—41no—
355C2—4241no—
356C2—4241no—
357C2—2122no—
358C2#1681113511yescharacteristic
359C2#1681142811yescharacteristic
360C2#1681143511yescharacteristic
subgroupG/H
centerC22C22×D212
commutatorC2×C4×C53C24
FrattiniC2×C4C23×D53
FittingC53×#433C2
socleC22×C53C22×D4
hypercenter#433D53
hypocenterC53#3119
LCSG ▷ C2×C4×C53 ▷ C2×C53 ▷ C53
UCSC1 ◁ C22 ◁ #117 ◁ #433

Complements

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

NH ≅ G/Ntype
1. #16811228C2semidirect
2. #16811103C2semidirect
3. #16810942C2semidirect
4. #16810937C2semidirect
5. #16810931C2semidirect
6. #16810924C2semidirect
7. #16810824C2semidirect
8. #16810819C2semidirect
9. #16810770C2semidirect
10. #16810770C2semidirect
11. #16810736C2semidirect
12. #16810725C2semidirect
13. C4×#11719913C2semidirect
14. C4×D424C2semidirect
15. C53×#433C2semidirect
195. #433D53semidirect
196. C53#3119semidirect