Finite Group Databasebeta

← #20286828

Subgroups — #20286828

Order 6784.

388 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#16811330C211yesmaximal, characteristic, complemented
2#16811282C211yesmaximal, characteristic
3#16810983C211yesmaximal, characteristic, complemented
4#16810954C211yesmaximal, characteristic, complemented
5#16810942C211yesmaximal, characteristic, complemented
6#16810909C211yesmaximal, characteristic, complemented
7#16810881C211yesmaximal, characteristic, complemented
8#16810852C211yesmaximal, characteristic, complemented
9#16810846C211yesmaximal, characteristic, complemented
10#16810787C211yesmaximal, characteristic, complemented
11#16810781C211yesmaximal, characteristic, complemented
12#16810731C211yesmaximal, characteristic, complemented
13#16810625C211yesmaximal, characteristic, complemented
14C2×#13133142C211yesmaximal, characteristic, complemented
15C53×#435C211yesmaximal, characteristic, complemented
16C2×#11719945C2211yescharacteristic
17C22×Dic106C2211yescharacteristic
18C2×Dic212C2211yescharacteristic
19C2×Dic212C2211yescharacteristic
20C2×#11719913C2211yescharacteristic
21#13133217C2211yescharacteristic
22#13133208—21no—
23#13133205C2211yescharacteristic
24#13133203—21no—
25#13133196C2211yescharacteristic
26#13133196C2211yescharacteristic
27#13133196C2211yescharacteristic
28#13133195—21no—
29#13133194—21no—
30#13133194—21no—
31#13133191C2211yescharacteristic
32#13133189—21no—
33C4×Dic106C2211yescharacteristic
34C4×Dic106C2211yescharacteristic
35D4×Dic53—21no—
36#13133177—21no—
37C2×C4⋊Dic53C2211yescharacteristic
38C2×C4⋊Dic53C2211yescharacteristic
39#13133149C2211yescharacteristic
40#13133149C2211yescharacteristic
41#13133149C2211yescharacteristic
42#13133142C2211yescharacteristic
43#13133142C2211yescharacteristic
44#13133142C2211yescharacteristic
45#13133142C2211yescharacteristic
46#13133142C2211yescharacteristic
47#13133142C2211yescharacteristic
48#13133142C2211yescharacteristic
49#13133139C2211yescharacteristic
50#13133139C2211yescharacteristic
51#13133139C2211yescharacteristic
52C2×M16×C53C2211yescharacteristic
53C22×C8×C53C2211yescharacteristic
54C4×D4×C53C2211yescharacteristic
55C4⋊C8×C53C2211yescharacteristic
56C22⋊C8×C53C2211yescharacteristic
57C22⋊C8×C53C2211yescharacteristic
58#97×C53C2211yescharacteristic
59C2×D4×C53D411yescharacteristic
60C22×C4×C53C2311yescharacteristic
61C22×C4×C53C2311yescharacteristic
62C2×C22⋊D53—21no—
63C2×C22⋊D53—21no—
64C22×Dic53—21no—
65C22×Dic53—21no—
66C22×Dic53—21no—
67#11719945—22no—
68#11719945—41no—
69C2×D212C2311yescharacteristic
70C2×C4×D53—21no—
71C2×Dic106—22no—
72C2×Dic106—21no—
73C2×Dic106—21no—
74C2×Dic106—41no—
75C2×Dic106C2311yescharacteristic
76C2×Dic106C2311yescharacteristic
77C2×Dic106C2311yescharacteristic
78M16×C53—41no—
79C2×C8×C53—41no—
80C2×C8×C53C2311yescharacteristic
81C2×C8×C53C2311yescharacteristic
82C2×C8×C53C2311yescharacteristic
83C2×C8×C53C2311yescharacteristic
84C4⋊C4×C53D411yescharacteristic
85C22⋊C4×C53D411yescharacteristic
86C22⋊C4×C53D411yescharacteristic
87C42×C53C2311yescharacteristic
88C22⋊Dic53—21no—
89C22⋊Dic53—21no—
90C22⋊Dic53—21no—
91#11719921—21no—
92#11719921—21no—
93#11719921—21no—
94C4⋊Dic53—21no—
95C4⋊Dic53—21no—
96C4⋊Dic53—21no—
97C4⋊Dic53C2311yescharacteristic
98C4⋊Dic53C2311yescharacteristic
99C4⋊Dic53C2311yescharacteristic
100C4⋊Dic53C2311yescharacteristic
101#11719919—21no—
102#11719919—21no—
103#11719919—21no—
104#11719919—21no—
105#11719919—21no—
106C4×Dic53—21no—
107C4×Dic53—21no—
108Dic212—22no—
109Dic212—41no—
110#11719913—41no—
111C23×C53C2×D411yescharacteristic
112C23×C53C2×D411yescharacteristic
113C22×D53—21no—
114D4×C53—22no—
115C2×C4×C53—21no—
116C2×C4×C53—21no—
117C2×C4×C53—21no—
118C2×C4×C53—21no—
119C2×C4×C53C2×D411yescharacteristic
120C2×C4×C53C2×D411yescharacteristic
121C2×C4×C53C2×D411yescharacteristic
122C2×C4×C53C2×D411yescharacteristic
123C2×C4×C53C2411yescharacteristic
124C22⋊D53—42no—
125C22⋊D53—42no—
126C2×Dic53—21no—
127C2×Dic53—21no—
128C2×Dic53—21no—
129C2×Dic53—21no—
130C2×Dic53—21no—
131C2×Dic53—21no—
132C2×Dic53—21no—
133C2×Dic53—42no—
134C2×Dic53—81no—
135C2×Dic53—81no—
136D212—21no—
137D212—21no—
138C4×D53—42no—
139Dic106—21no—
140Dic106—21no—
141Dic106—21no—
142Dic106—21no—
143Dic106—21no—
144Dic106—42no—
145Dic106—81no—
146Dic106#4112yes—
147C8×C53—21no—
148C8×C53—21no—
149C8×C53—21no—
150C8×C53—21no—
151C22×C53—21no—
152C22×C53—21no—
153C22×C53—21no—
154C22×C53—21no—
155C22×C53—21no—
156C22×C53#13712yes—
157C22×C53C22×D411yescharacteristic
158C2×D53—41no—
159C2×D53—41no—
160C2×D53—41no—
161C4×C53—21no—
162C4×C53—21no—
163C4×C53—21no—
164C4×C53—21no—
165C4×C53C2×#4111yescharacteristic
166C4×C53H(5,𝔽2)11yescharacteristic
167Dic53—22no—
168Dic53—41no—
169Dic53—41no—
170Dic53—41no—
171Dic53—41no—
172Dic53—41no—
173Dic53—41no—
174#3316—531nomaximal, Sylow
175C2×C53—22no—
176C2×C53—41no—
177C2×C53#54611yescharacteristic
178C2×C53C2×#13711yescharacteristic
179C2×C53#57711yescharacteristic
180D53—81no—
181#554—531no—
182#548—531no—
183#489—531no—
184#483—531no—
185#481—531no—
186#475—531no—
187#470—531no—
188#465—531no—
189#464—531no—
190#452—531no—
191#451—531no—
192#441—531no—
193#435D5311yescharacteristic, complemented
194#417—531no—
195C2×#103—531no—
196C53#331611yescharacteristic, complemented, Sylow
197C2×#41—531no—
198C22×Q8—531no—
199C2×Q16—531no—
200C2×Q16—531no—
201C2×SD16—531no—
202C2×M16C2×D5311yescharacteristic
203C22×C8C2×D5311yescharacteristic
204C4⋊Q8—531no—
205#125—1061no—
206#124—531no—
207#124—1061no—
208C22⋊Q8—531no—
209C22⋊Q8—531no—
210C22⋊Q8—531no—
211C22⋊Q8—1061no—
212C22⋊Q8—1061no—
213C22⋊Q8—1061no—
214#121—531no—
215#121—1061no—
216C4×Q8—531no—
217C4×Q8—531no—
218C4×D4—1061no—
219C4×D4—1061no—
220C4×D4C2×D5311yescharacteristic
221C2×C4⋊C4—531no—
222C2×C4⋊C4—531no—
223#106—531no—
224#106—531no—
225#106—531no—
226C4⋊C8C2×D5311yescharacteristic
227#103—531no—
228#103—531no—
229#103—531no—
230#103—531no—
231#103—531no—
232#103—531no—
233#103—531no—
234#102—531no—
235#102—531no—
236#102—531no—
237C22⋊C8C2×D5311yescharacteristic
238C22⋊C8C2×D5311yescharacteristic
239#97C2×D5311yescharacteristic
240#41—1062no—
241#41—2121no—
242C2×Q8—531no—
243C2×Q8—531no—
244C2×Q8—531no—
245C2×Q8—1062no—
246C2×Q8—1061no—
247C2×Q8—1061no—
248C2×Q8—2121no—
249C2×D4—531no—
250C2×D4—1061no—
251C2×D4—1061no—
252C2×D4D21211yescharacteristic
253C22×C4—1061no—
254C22×C4—1061no—
255C22×C4—1061no—
256C22×C4—1061no—
257C22×C4C22×D5311yescharacteristic
258C22×C4C22×D5311yescharacteristic
259Q16—1062no—
260Q16—2121no—
261SD16—2121no—
262M16—41no—
263C2×C8—41no—
264C2×C8C22×D5311yescharacteristic
265C2×C8C22×D5311yescharacteristic
266C2×C8C22×D5311yescharacteristic
267C2×C8C22×D5311yescharacteristic
268C4⋊C4—531no—
269C4⋊C4—531no—
270C4⋊C4—531no—
271C4⋊C4—531no—
272C4⋊C4—1061no—
273C4⋊C4—1061no—
274C4⋊C4—1061no—
275C4⋊C4—1061no—
276C4⋊C4—1061no—
277C4⋊C4—1061no—
278C4⋊C4—1061no—
279C4⋊C4—1061no—
280C4⋊C4D21211yescharacteristic
281C22⋊C4—1061no—
282C22⋊C4—1061no—
283C22⋊C4—1061no—
284C22⋊C4—1061no—
285C22⋊C4—1061no—
286C22⋊C4—1061no—
287C22⋊C4D21211yescharacteristic
288C22⋊C4D21211yescharacteristic
289C42—1061no—
290C42—1061no—
291C42C22×D5311yescharacteristic
292C23—1061no—
293C23C2×D21211yescharacteristic
294C23C2×D21211yescharacteristic
295Q8—532no—
296Q8—1061no—
297Q8—1061no—
298Q8—1061no—
299Q8—1061no—
300Q8—1061no—
301Q8—4241no—
302Q8—2122no—
303D4—22no—
304D4—1061no—
305D4—1061no—
306D4—2122no—
307D4—2122no—
308C2×C4—21no—
309C2×C4—21no—
310C2×C4—21no—
311C2×C4—21no—
312C2×C4—1061no—
313C2×C4—1061no—
314C2×C4—1061no—
315C2×C4—1061no—
316C2×C4—1061no—
317C2×C4—1061no—
318C2×C4—1061no—
319C2×C4—4241no—
320C2×C4—4241no—
321C2×C4—2122no—
322C2×C4—2122no—
323C2×C4C2×D21211yescharacteristic
324C2×C4C2×D21211yescharacteristic
325C2×C4C2×D21211yescharacteristic
326C2×C4C2×D21211yescharacteristic
327C2×C4C23×D5311yescharacteristic
328C8—21no—
329C8—21no—
330C8—21no—
331C8—21no—
332C22—21no—
333C22—21no—
334C22—21no—
335C22—21no—
336C22—21no—
337C22—2121no—
338C22—2121no—
339C22—2121no—
340C22#1313325012yes—
341C22C22×D21211yescharacteristic
342C4—21no—
343C4—21no—
344C4—21no—
345C4—21no—
346C4—1062no—
347C4—2121no—
348C4—2121no—
349C4—2121no—
350C4—2121no—
351C4—2121no—
352C4—2121no—
353C4#41×D5311yescharacteristic
354C4#1313326211yescharacteristic
355C2—22no—
356C2—41no—
357C2—4241no—
358C2#1681126811yescharacteristic
359C2C2×#1313325011yescharacteristic
360C2#1681143511yescharacteristic
subgroupG/H
centerC22C22×D212
commutatorC2×C4×C53C24
FrattiniC2×C4C23×D53
FittingC53×#435C2
socleC22×C53C22×D4
hypercenter#435D53
hypocenterC53#3316
LCSG ▷ C2×C4×C53 ▷ C2×C53 ▷ C53
UCSC1 ◁ C22 ◁ C4×D4 ◁ #435

Complements

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

NH ≅ G/Ntype
1. #16811330C2semidirect
3. #16810983C2semidirect
4. #16810954C2semidirect
5. #16810942C2semidirect
6. #16810909C2semidirect
7. #16810881C2semidirect
8. #16810852C2semidirect
9. #16810846C2semidirect
10. #16810787C2semidirect
11. #16810781C2semidirect
12. #16810731C2semidirect
13. #16810625C2semidirect
14. C2×#13133142C2semidirect
15. C53×#435C2semidirect
193. #435D53semidirect
196. C53#3316semidirect