Finite Group Databasebeta

← #20286970

Subgroups — #20286970

Order 6784.

392 conjugacy classes of subgroups.

The subgroup lattice has 370 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#16811319C211yesmaximal, characteristic, complemented
2#16811282C211yesmaximal, characteristic, complemented
3#16810978C211yesmaximal, characteristic, complemented
4#16810954C211yesmaximal, characteristic, complemented
5#16810948C211yesmaximal, characteristic, complemented
6#16810931C211yesmaximal, characteristic, complemented
7#16810876C211yesmaximal, characteristic, complemented
8#16810869C211yesmaximal, characteristic, complemented
9#16810839C211yesmaximal, characteristic, complemented
10#16810770C211yesmaximal, characteristic, complemented
11#16810764C211yesmaximal, characteristic, complemented
12C4×#11719913C211yesmaximal, characteristic, complemented
13#16810633C211yesmaximal, characteristic, complemented
14#16810625C211yesmaximal, characteristic, complemented
15C53×#436C211yesmaximal, characteristic, complemented
16C2×#11719945C2211yescharacteristic
17C22×Dic106C2211yescharacteristic
18C2×#11719913C2211yescharacteristic
19C2×#11719913C2211yescharacteristic
20C2×D424C2211yescharacteristic
21#13133212—21no—
22#13133209C2211yescharacteristic
23#13133205C2211yescharacteristic
24#13133203—21no—
25#13133199—21no—
26#13133196C2211yescharacteristic
27#13133195—21no—
28#13133191C2211yescharacteristic
29#13133191C2211yescharacteristic
30#13133191C2211yescharacteristic
31#13133189—21no—
32#13133189—21no—
33C4×Dic106C2211yescharacteristic
34C4×D212C2211yescharacteristic
35#13133177—21no—
36#13133177—21no—
37#13133176C2211yescharacteristic
38C2×C4⋊Dic53C2211yescharacteristic
39#13133151C2211yescharacteristic
40#13133149C2211yescharacteristic
41#13133149C2211yescharacteristic
42#13133142C2211yescharacteristic
43#13133142C2211yescharacteristic
44#13133142C2211yescharacteristic
45#13133142C2211yescharacteristic
46#13133142C2211yescharacteristic
47#13133142C2211yescharacteristic
48#13133139C2211yescharacteristic
49#13133139C2211yescharacteristic
50#13133139C2211yescharacteristic
51#13133139C2211yescharacteristic
52C2×M16×C53C2211yescharacteristic
53C2×M16×C53C2211yescharacteristic
54C4×D4×C53C2211yescharacteristic
55C4⋊C8×C53C2211yescharacteristic
56C22⋊C8×C53C2211yescharacteristic
57C22⋊C8×C53C2211yescharacteristic
58C4×C8×C53C2211yescharacteristic
59C2×D4×C53D411yescharacteristic
60C22×C4×C53C2311yescharacteristic
61C22×C4×C53C2311yescharacteristic
62C2×C22⋊D53—21no—
63C2×C22⋊D53—21no—
64C2×C22⋊D53—21no—
65C2×C22⋊D53—21no—
66C22×Dic53—21no—
67C22×Dic53—21no—
68#11719945—22no—
69#11719945—41no—
70C2×D212C2311yescharacteristic
71C2×D212C2311yescharacteristic
72C2×C4×D53—21no—
73C2×C4×D53—21no—
74C2×Dic106—22no—
75C2×Dic106—21no—
76C2×Dic106—41no—
77C2×Dic106C2311yescharacteristic
78C2×Dic106C2311yescharacteristic
79M16×C53—41no—
80M16×C53—41no—
81C2×C8×C53C2311yescharacteristic
82C2×C8×C53C2311yescharacteristic
83C2×C8×C53C2311yescharacteristic
84C2×C8×C53C2311yescharacteristic
85C4⋊C4×C53D411yescharacteristic
86C22⋊C4×C53D411yescharacteristic
87C22⋊C4×C53D411yescharacteristic
88C42×C53C2311yescharacteristic
89C22⋊Dic53—21no—
90C22⋊Dic53—21no—
91#11719921—21no—
92#11719921—21no—
93#11719921—21no—
94#11719921—21no—
95#11719921—21no—
96#11719921—21no—
97C4⋊Dic53—21no—
98C4⋊Dic53C2311yescharacteristic
99C4⋊Dic53C2311yescharacteristic
100C4⋊Dic53C2311yescharacteristic
101C4⋊Dic53C2311yescharacteristic
102#11719919—21no—
103#11719919—21no—
104#11719919—21no—
105#11719919—21no—
106C4×Dic53—21no—
107C4×Dic53—21no—
108D424—41no—
109#11719913—22no—
110#11719913—41no—
111C23×C53C2×D411yescharacteristic
112C23×C53C2×D411yescharacteristic
113C22×D53—21no—
114C22×D53—21no—
115D4×C53—41no—
116C2×C4×C53—21no—
117C2×C4×C53—21no—
118C2×C4×C53—21no—
119C2×C4×C53—21no—
120C2×C4×C53C2×D411yescharacteristic
121C2×C4×C53C2×D411yescharacteristic
122C2×C4×C53C2×D411yescharacteristic
123C2×C4×C53C2×D411yescharacteristic
124C2×C4×C53C2411yescharacteristic
125C22⋊D53—42no—
126C22⋊D53—42no—
127C22⋊D53—42no—
128C22⋊D53—81no—
129C2×Dic53—21no—
130C2×Dic53—21no—
131C2×Dic53—21no—
132C2×Dic53—21no—
133C2×Dic53—21no—
134C2×Dic53—21no—
135C2×Dic53—42no—
136C2×Dic53—81no—
137D212—21no—
138D212—21no—
139D212—21no—
140D212—21no—
141C4×D53—42no—
142C4×D53—81no—
143Dic106—21no—
144Dic106—21no—
145Dic106—21no—
146Dic106—42no—
147Dic106#4112yes—
148C8×C53—21no—
149C8×C53—21no—
150C8×C53—21no—
151C8×C53—21no—
152C22×C53—21no—
153C22×C53—21no—
154C22×C53—21no—
155C22×C53—21no—
156C22×C53—21no—
157C22×C53—21no—
158C22×C53C22×D411yescharacteristic
159C2×D53—41no—
160C2×D53—41no—
161C2×D53—41no—
162C2×D53—41no—
163C2×D53—41no—
164C2×D53—41no—
165C4×C53—21no—
166C4×C53—21no—
167C4×C53—21no—
168C4×C53—21no—
169C4×C53C2×#4111yescharacteristic
170C4×C53H(5,𝔽2)11yescharacteristic
171Dic53—22no—
172Dic53—41no—
173Dic53—41no—
174Dic53—41no—
175Dic53—41no—
176Dic53—41no—
177#3326—531nomaximal, Sylow
178C2×C53—41no—
179C2×C53—41no—
180C2×C53#54611yescharacteristic
181C2×C53#57511yescharacteristic
182C2×C53#57711yescharacteristic
183D53—81no—
184D53—81no—
185#553—531no—
186#548—531no—
187#488—531no—
188#483—531no—
189#482—531no—
190#479—531no—
191#469—531no—
192#468—531no—
193#463—531no—
194#449—531no—
195#448—531no—
196C4×SD16—531no—
197#436D5311yescharacteristic, complemented
198#419—531no—
199#417—531no—
200C53#332611yescharacteristic, complemented, Sylow
201C2×#41—531no—
202C22×Q8—531no—
203C2×SD16—531no—
204C2×SD16—531no—
205C2×D8—531no—
206C2×M16C2×D5311yescharacteristic
207C2×M16C2×D5311yescharacteristic
208#126—1061no—
209#125—531no—
210#124—531no—
211#124—1061no—
212#123—1061no—
213C22⋊Q8—531no—
214C22⋊Q8—1061no—
215#121—531no—
216#121—531no—
217#121—531no—
218#121—1061no—
219#121—1061no—
220C4×Q8—531no—
221C4×D4—531no—
222C4×D4—1061no—
223C4×D4—1061no—
224C4×D4C2×D5311yescharacteristic
225#117—531no—
226C2×C4⋊C4—531no—
227#107—531no—
228#106—531no—
229#106—531no—
230C4⋊C8C2×D5311yescharacteristic
231#103—531no—
232#103—531no—
233#103—531no—
234#103—531no—
235#103—531no—
236#103—531no—
237#102—531no—
238#102—531no—
239#102—531no—
240#102—531no—
241C22⋊C8C2×D5311yescharacteristic
242C22⋊C8C2×D5311yescharacteristic
243C4×C8C2×D5311yescharacteristic
244#41—1062no—
245#41—2121no—
246C2×Q8—531no—
247C2×Q8—531no—
248C2×Q8—1062no—
249C2×Q8—1061no—
250C2×Q8—2121no—
251C2×D4—531no—
252C2×D4—531no—
253C2×D4—1061no—
254C2×D4—1061no—
255C2×D4—1061no—
256C2×D4—1061no—
257C2×D4D21211yescharacteristic
258C22×C4—1061no—
259C22×C4—1061no—
260C22×C4—1061no—
261C22×C4—1061no—
262C22×C4C22×D5311yescharacteristic
263C22×C4C22×D5311yescharacteristic
264SD16—1062no—
265SD16—2121no—
266D8—2121no—
267M16—41no—
268M16—41no—
269C2×C8C22×D5311yescharacteristic
270C2×C8C22×D5311yescharacteristic
271C2×C8C22×D5311yescharacteristic
272C2×C8C22×D5311yescharacteristic
273C4⋊C4—531no—
274C4⋊C4—531no—
275C4⋊C4—531no—
276C4⋊C4—531no—
277C4⋊C4—1061no—
278C4⋊C4—1061no—
279C4⋊C4—1061no—
280C4⋊C4—1061no—
281C4⋊C4—1061no—
282C4⋊C4D21211yescharacteristic
283C22⋊C4—1061no—
284C22⋊C4—1061no—
285C22⋊C4—1061no—
286C22⋊C4—1061no—
287C22⋊C4—1061no—
288C22⋊C4—1061no—
289C22⋊C4—1061no—
290C22⋊C4—1061no—
291C22⋊C4D21211yescharacteristic
292C22⋊C4D21211yescharacteristic
293C42—1061no—
294C42—1061no—
295C42C22×D5311yescharacteristic
296C23—1061no—
297C23—1061no—
298C23C2×D21211yescharacteristic
299C23C2×D21211yescharacteristic
300Q8—532no—
301Q8—1061no—
302Q8—1061no—
303Q8—1061no—
304Q8—2122no—
305D4—41no—
306D4—1061no—
307D4—1061no—
308D4—1061no—
309D4—1061no—
310D4—4241no—
311D4—2122no—
312D4—2122no—
313D4—2122no—
314C2×C4—21no—
315C2×C4—21no—
316C2×C4—21no—
317C2×C4—21no—
318C2×C4—1061no—
319C2×C4—1061no—
320C2×C4—1061no—
321C2×C4—1061no—
322C2×C4—1061no—
323C2×C4—1061no—
324C2×C4—4241no—
325C2×C4—4241no—
326C2×C4—2122no—
327C2×C4—2122no—
328C2×C4C2×D21211yescharacteristic
329C2×C4C2×D21211yescharacteristic
330C2×C4C2×D21211yescharacteristic
331C2×C4C2×D21211yescharacteristic
332C2×C4C23×D5311yescharacteristic
333C8—21no—
334C8—21no—
335C8—21no—
336C8—21no—
337C22—21no—
338C22—21no—
339C22—21no—
340C22—21no—
341C22—21no—
342C22—21no—
343C22—2121no—
344C22—2121no—
345C22—2121no—
346C22—2121no—
347C22—2121no—
348C22—2121no—
349C22C22×D21211yescharacteristic
350C4—21no—
351C4—21no—
352C4—21no—
353C4—21no—
354C4—1062no—
355C4—2121no—
356C4—2121no—
357C4—2121no—
358C4—2121no—
359C4—2121no—
360C4#41×D5311yescharacteristic
361C4#1313326211yescharacteristic
362C2—41no—
363C2—41no—
364C2—4241no—
365C2—4241no—
366C2#1681126811yescharacteristic
367C2#1681142311yescharacteristic
368C2#1681143511yescharacteristic
subgroupG/H
centerC22C22×D212
commutatorC2×C4×C53C24
FrattiniC2×C4C23×D53
FittingC53×#436C2
socleC22×C53C22×D4
hypercenter#436D53
hypocenterC53#3326
LCSG ▷ C2×C4×C53 ▷ C2×C53 ▷ C53
UCSC1 ◁ C22 ◁ C4×D4 ◁ #436

Complements

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

NH ≅ G/Ntype
1. #16811319C2semidirect
2. #16811282C2semidirect
3. #16810978C2semidirect
4. #16810954C2semidirect
5. #16810948C2semidirect
6. #16810931C2semidirect
7. #16810876C2semidirect
8. #16810869C2semidirect
9. #16810839C2semidirect
10. #16810770C2semidirect
11. #16810764C2semidirect
12. C4×#11719913C2semidirect
13. #16810633C2semidirect
14. #16810625C2semidirect
15. C53×#436C2semidirect
197. #436D53semidirect
200. C53#3326semidirect