Finite Group Databasebeta

← #20286460

Subgroups — #20286460

Order 6784.

472 conjugacy classes of subgroups.

The subgroup lattice has 388 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C22×#11719925C211yesmaximal, characteristic, complemented
2C2×Q16×D53C211yesmaximal, characteristic, complemented
3Q8×C22⋊D53C211yesmaximal, characteristic
4#16811286C211yesmaximal, characteristic
5#16811001C211yesmaximal, characteristic, complemented
6#16810863C211yesmaximal, characteristic, complemented
7#16810862C211yesmaximal, characteristic, complemented
8#16810832C211yesmaximal, characteristic, complemented
9#16810831C211yesmaximal, characteristic, complemented
10#16810781C211yesmaximal, characteristic, complemented
11#16810780C211yesmaximal, characteristic, complemented
12#16810777C211yesmaximal, characteristic, complemented
13#16810715C211yesmaximal, characteristic, complemented
14#16810686C211yesmaximal, characteristic
15C53×#451C211yesmaximal, characteristic, complemented
16C22×Dic106C2211yescharacteristic
17C2×Q8×D53C2211yescharacteristic
18C2×Q8×D53C2211yescharacteristic
19C2×Dic212C2211yescharacteristic
20C2×Dic212C2211yescharacteristic
21C2×#11719925—24no—
22C2×#11719925C2211yescharacteristic
23C2×#11719925C2211yescharacteristic
24C2×#11719925C2211yescharacteristic
25C2×#11719925C2211yescharacteristic
26C2×#11719925C2211yescharacteristic
27Q16×D53—24no—
28C22×#88322C2211yescharacteristic
29C2×C8×D53C2211yescharacteristic
30#13133216—21no—
31#13133215—21no—
32#13133215C2211yescharacteristic
33#13133215C2211yescharacteristic
34#13133197C2211yescharacteristic
35#13133196—21no—
36#13133196C2211yescharacteristic
37#13133195—21no—
38#13133194—21no—
39#13133193—21no—
40#13133193C2211yescharacteristic
41Q8×Dic53C2211yescharacteristic
42#13133182C2211yescharacteristic
43C4×C22⋊D53—21no—
44C4×C22⋊D53C2211yescharacteristic
45#13133177—21no—
46#13133150C2211yescharacteristic
47#13133146C2211yescharacteristic
48#13133142C2211yescharacteristic
49#13133142C2211yescharacteristic
50#13133141C2211yescharacteristic
51#13133140C2211yescharacteristic
52#13133129C2211yescharacteristic
53C8×Dic53C2211yescharacteristic
54C22×Q8×C53C2211yescharacteristic
55C2×Q16×C53C2211yescharacteristic
56C2×Q16×C53C2211yescharacteristic
57C22⋊Q8×C53C2211yescharacteristic
58#103×C53C2211yescharacteristic
59#103×C53C2211yescharacteristic
60C22⋊C8×C53C2211yescharacteristic
61C2×Q8×C53—21no—
62C2×Q8×C53—22no—
63C2×Q8×C53—41no—
64C2×Q8×C53C2311yescharacteristic
65C2×Q8×C53C2311yescharacteristic
66C2×Q8×C53C2311yescharacteristic
67C22×C4×C53—21no—
68C22×C4×C53C2311yescharacteristic
69C2×C22⋊D53D411yescharacteristic
70C22×Dic53—21no—
71Q8×D53—22no—
72Q8×D53—22no—
73Q8×D53—41no—
74Q8×D53—41no—
75C2×C4×D53—21no—
76C2×C4×D53—21no—
77C2×C4×D53C2311yescharacteristic
78C2×Dic106—22no—
79C2×Dic106—21no—
80C2×Dic106—21no—
81C2×Dic106—21no—
82C2×Dic106—41no—
83C2×Dic106C2311yescharacteristic
84C2×Dic106C2311yescharacteristic
85C2×Dic106C2311yescharacteristic
86Q16×C53—22no—
87Q16×C53—41no—
88C2×C8×C53C2311yescharacteristic
89C2×C8×C53C2311yescharacteristic
90C4⋊C4×C53—21no—
91C4⋊C4×C53C2311yescharacteristic
92C22⋊C4×C53—21no—
93C22⋊Dic53—21no—
94C22⋊Dic53D411yescharacteristic
95#11719925—22no—
96#11719925—22no—
97#11719925—22no—
98#11719925—22no—
99#11719925D414yescomplemented
100#11719921—21no—
101#11719921—21no—
102#11719921D411yescharacteristic
103C4⋊Dic53—21no—
104C4⋊Dic53—21no—
105C4⋊Dic53C2311yescharacteristic
106#11719919—21no—
107#11719919—21no—
108#11719919—21no—
109#11719919—21no—
110#11719919—21no—
111#11719919—21no—
112#11719919—21no—
113#11719919D411yescharacteristic
114C4×Dic53—21no—
115C4×Dic53—21no—
116C4×Dic53C2311yescharacteristic
117C2×#88322—22no—
118C2×#88322C2311yescharacteristic
119C2×#88322C2311yescharacteristic
120Dic212—22no—
121Dic212—41no—
122C8×D53—22no—
123C23×C53C2×D411yescharacteristic
124C22×D53C2×D411yescharacteristic
125Q8×C53—21no—
126Q8×C53—21no—
127Q8×C53—21no—
128Q8×C53—21no—
129Q8×C53—21no—
130Q8×C53—42no—
131Q8×C53C2×D412yes—
132C2×C4×C53—21no—
133C2×C4×C53—21no—
134C2×C4×C53—21no—
135C2×C4×C53—21no—
136C2×C4×C53—21no—
137C2×C4×C53—21no—
138C2×C4×C53—42no—
139C2×C4×C53C2×D411yescharacteristic
140C2×C4×C53C2411yescharacteristic
141C22⋊D53Q1614yescomplemented
142C2×Dic53—21no—
143C2×Dic53—21no—
144C2×Dic53—21no—
145C2×Dic53—21no—
146C2×Dic53—42no—
147C2×Dic53C2×D411yescharacteristic
148C2×Dic53C2×D411yescharacteristic
149C2×Dic53C2×D411yescharacteristic
150C4×D53—21no—
151C4×D53—21no—
152C4×D53—42no—
153C4×D53—42no—
154Dic106—21no—
155Dic106—21no—
156Dic106—21no—
157Dic106—21no—
158Dic106—21no—
159Dic106—42no—
160Dic106—42no—
161Dic106—42no—
162Dic106C2×D412yes—
163C8×C53—21no—
164C8×C53—21no—
165#88322—21no—
166#88322C2×D412yes—
167C22×C53—21no—
168C22×C53—21no—
169C22×C53C2×Q1612yescomplemented
170C22×C53C22×D411yescharacteristic
171C2×D53—21no—
172C2×D53—21no—
173C2×D53C2×Q1612yescomplemented
174C4×C53—21no—
175C4×C53—22no—
176C4×C53—41no—
177C4×C53—41no—
178C4×C53—41no—
179C4×C53C22×D411yescharacteristic
180C4×C53H(5,𝔽2)11yescharacteristic
181Dic53—22no—
182Dic53—21no—
183Dic53—21no—
184Dic53—41no—
185Dic53—41no—
186Dic53—41no—
187Dic53C2×Q1612yescomplemented
188D4×Q16—531nomaximal, Sylow
189C2×C53—22no—
190C2×C53D4211yescharacteristic
191C2×C53C22×Q1611yescharacteristic
192C2×C53#57811yescharacteristic
193D53—22no—
194C22×Q16—531no—
195C22×Q16—531no—
196D4×Q8—531no—
197D4×Q8—531no—
198#494—531no—
199#467—531no—
200#467—531no—
201#462—531no—
202#462—531no—
203#451—531no—
204#451—531no—
205#451—531no—
206#451D5311yescharacteristic, complemented
207C4×Q16—531no—
208C8×D4—531no—
209C53D4×Q1611yescharacteristic, complemented, Sylow
210C22×Q8—531no—
211C22×Q8—531no—
212C22×Q8—531no—
213C22×Q8C2×D5311yescharacteristic
214C2×Q16—531no—
215C2×Q16—531no—
216C2×Q16—531no—
217C2×Q16—531no—
218C2×Q16—531no—
219C2×Q16—531no—
220C2×Q16—531no—
221C2×Q16—1064no—
222C2×Q16—1064no—
223C2×Q16C2×D5311yescharacteristic
224C2×Q16C2×D5311yescharacteristic
225C22×C8—531no—
226C22×C8—531no—
227C4⋊Q8—531no—
228C4⋊Q8—531no—
229C4⋊Q8—1061no—
230C4⋊Q8—1061no—
231C22⋊Q8—531no—
232C22⋊Q8—531no—
233C22⋊Q8—531no—
234C22⋊Q8—1061no—
235C22⋊Q8—1061no—
236C22⋊Q8—1061no—
237C22⋊Q8—1061no—
238C22⋊Q8C2×D5311yescharacteristic
239C4×Q8—531no—
240C4×Q8—531no—
241C4×D4—531no—
242C4×D4—1061no—
243C4×D4—1061no—
244#107—531no—
245C4⋊C8—531no—
246#103—531no—
247#103—531no—
248#103—531no—
249#103—531no—
250#103C2×D5311yescharacteristic
251#103C2×D5311yescharacteristic
252C22⋊C8—531no—
253C22⋊C8C2×D5311yescharacteristic
254C4×C8—531no—
255C2×Q8—21no—
256C2×Q8—22no—
257C2×Q8—41no—
258C2×Q8—531no—
259C2×Q8—531no—
260C2×Q8—531no—
261C2×Q8—1061no—
262C2×Q8—1061no—
263C2×Q8—1062no—
264C2×Q8—1061no—
265C2×Q8—1062no—
266C2×Q8—1062no—
267C2×Q8—2121no—
268C2×Q8—2121no—
269C2×Q8—2121no—
270C2×Q8C22×D5311yescharacteristic
271C2×Q8C22×D5311yescharacteristic
272C2×Q8C22×D5311yescharacteristic
273C2×D4—531no—
274C22×C4—21no—
275C22×C4—531no—
276C22×C4—1061no—
277C22×C4—1061no—
278C22×C4—1061no—
279C22×C4C22×D5311yescharacteristic
280Q16—22no—
281Q16—41no—
282Q16—534no—
283Q16—1062no—
284Q16—1062no—
285Q16—1062no—
286Q16—1062no—
287Q16—1062no—
288Q16—2121no—
289C2×C8—531no—
290C2×C8—531no—
291C2×C8—1062no—
292C2×C8—1062no—
293C2×C8C22×D5311yescharacteristic
294C2×C8C22×D5311yescharacteristic
295C4⋊C4—21no—
296C4⋊C4—531no—
297C4⋊C4—531no—
298C4⋊C4—1061no—
299C4⋊C4—1061no—
300C4⋊C4—1061no—
301C4⋊C4—1061no—
302C4⋊C4—1061no—
303C4⋊C4—1061no—
304C4⋊C4—1061no—
305C4⋊C4—1061no—
306C4⋊C4—1061no—
307C4⋊C4C22×D5311yescharacteristic
308C22⋊C4—21no—
309C22⋊C4—531no—
310C22⋊C4—531no—
311C22⋊C4—1061no—
312C22⋊C4—1061no—
313C22⋊C4—1061no—
314C42—531no—
315C42—1061no—
316C42—1061no—
317C23—531no—
318C23D4×D5311yescharacteristic
319Q8—21no—
320Q8—21no—
321Q8—21no—
322Q8—21no—
323Q8—21no—
324Q8—42no—
325Q8—532no—
326Q8—1061no—
327Q8—1061no—
328Q8—1061no—
329Q8—1061no—
330Q8—1061no—
331Q8—2122no—
332Q8—2122no—
333Q8—2122no—
334Q8D4×D5312yes—
335D4—534no—
336C2×C4—21no—
337C2×C4—21no—
338C2×C4—21no—
339C2×C4—21no—
340C2×C4—21no—
341C2×C4—21no—
342C2×C4—42no—
343C2×C4—531no—
344C2×C4—531no—
345C2×C4—531no—
346C2×C4—1061no—
347C2×C4—1061no—
348C2×C4—1061no—
349C2×C4—1061no—
350C2×C4—1061no—
351C2×C4—1061no—
352C2×C4—2122no—
353C2×C4—2122no—
354C2×C4—2122no—
355C2×C4D4×D5311yescharacteristic
356C2×C4C23×D5311yescharacteristic
357C8—21no—
358C8—21no—
359C8—532no—
360C8—1061no—
361C22—21no—
362C22—21no—
363C22—532no—
364C22—1061no—
365C22—1061no—
366C22Q16×D5312yescomplemented
367C22C2×D4×D5311yescharacteristic
368C4—21no—
369C4—22no—
370C4—41no—
371C4—41no—
372C4—41no—
373C4—532no—
374C4—1061no—
375C4—1061no—
376C4—1062no—
377C4—2121no—
378C4—2121no—
379C4—2121no—
380C4C2×D4×D5311yescharacteristic
381C4#1313326211yescharacteristic
382C2—22no—
383C2—1062no—
384C2#1681125511yescharacteristic
385C2C2×Q16×D5311yescharacteristic
386C2#1681143911yescharacteristic
subgroupG/H
centerC22C2×D4×D53
commutatorC2×C4×C53C24
FrattiniC2×C4C23×D53
FittingC53×#451C2
socleC22×C53C22×D4
hypercenter#451D53
hypocenterC53D4×Q16
LCSG ▷ C2×C4×C53 ▷ C2×C53 ▷ C53
UCSC1 ◁ C22 ◁ C22×C4 ◁ #451

Complements

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

NH ≅ G/Ntype
1. C22×#11719925C2semidirect
2. C2×Q16×D53C2semidirect
5. #16811001C2semidirect
6. #16810863C2semidirect
7. #16810862C2semidirect
8. #16810832C2semidirect
9. #16810831C2semidirect
10. #16810781C2semidirect
11. #16810780C2semidirect
12. #16810777C2semidirect
13. #16810715C2semidirect
15. C53×#451C2semidirect
99. #11719925D4semidirect
141. C22⋊D53Q16semidirect
169. C22×C53C2×Q16semidirect
173. C2×D53C2×Q16semidirect
187. Dic53C2×Q16semidirect
206. #451D53semidirect
209. C53D4×Q16semidirect
366. C22Q16×D53semidirect