Finite Group Databasebeta

← #20283531

Subgroups — #20283531

Order 6784.

484 conjugacy classes of subgroups.

The subgroup lattice has 356 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#13133249C211yesmaximal, characteristic
2C4×#11719949C211yesmaximal, characteristic
3C4×Q8×D53C211yesmaximal, characteristic, complemented
4#16810728C211yesmaximal, characteristic, complemented
5#16810725C211yesmaximal, characteristic, complemented
6#16810721C211yesmaximal, characteristic, complemented
7Q16×Dic53C211yesmaximal, characteristic, complemented
8#16810715C211yesmaximal, characteristic, complemented
9#16810710C211yesmaximal, characteristic, complemented
10#16810708C211yesmaximal, characteristic, complemented
11#16810662C211yesmaximal, characteristic, complemented
12#16810630C211yesmaximal, characteristic, complemented
13#16810623C211yesmaximal, characteristic
14#16810575C211yesmaximal, characteristic, complemented
15C53×#441C211yesmaximal, characteristic, complemented
16C2×#11719949C2211yescharacteristic
17C2×Q8×D53C2211yescharacteristic
18#13133249C418yescomplemented
19C2×#11719925C2211yescharacteristic
20C2×#11719913C2211yescharacteristic
21C2×#11719924C2211yescharacteristic
22C2×#11719912C2211yescharacteristic
23C2×#11719912C2211yescharacteristic
24Q8×Dic53C2211yescharacteristic
25Q8×Dic53C2211yescharacteristic
26C4×Dic106—21no—
27C4×Dic106C2211yescharacteristic
28#13133182—21no—
29#13133182C2211yescharacteristic
30C4×D212—21no—
31C4×D212C2211yescharacteristic
32#13133179—21no—
33#13133179C2211yescharacteristic
34#13133175—21no—
35#13133175C2211yescharacteristic
36C4⋊C4×D53—21no—
37C4⋊C4×D53C2211yescharacteristic
38C42×D53—21no—
39C42×D53—21no—
40C42×D53C2211yescharacteristic
41#13133151C2211yescharacteristic
42#13133150C2211yescharacteristic
43#13133148C2211yescharacteristic
44#13133142C2211yescharacteristic
45#13133141C2211yescharacteristic
46#13133141C2211yescharacteristic
47#13133140C2211yescharacteristic
48#13133139C2211yescharacteristic
49#13133137C2211yescharacteristic
50#13133128C2211yescharacteristic
51C8×Dic53C2211yescharacteristic
52C8×Dic53C2211yescharacteristic
53C2×Q16×C53C2211yescharacteristic
54C4×Q8×C53C2211yescharacteristic
55C4×Q8×C53C2211yescharacteristic
56#106×C53C2211yescharacteristic
57#103×C53C2211yescharacteristic
58#103×C53C2211yescharacteristic
59#97×C53C2211yescharacteristic
60C2×Q8×C53C2311yescharacteristic
61C2×Q8×C53C2311yescharacteristic
62#11719949—22no—
63#11719949C2×C414yes—
64Q8×D53—22no—
65Q8×D53C2×C414yescomplemented
66C2×D212—21no—
67C2×D212C2311yescharacteristic
68C2×C4×D53—21no—
69C2×C4×D53—21no—
70C2×C4×D53—21no—
71C2×C4×D53—21no—
72C2×C4×D53—21no—
73C2×C4×D53D411yescharacteristic
74C2×C4×D53D411yescharacteristic
75C2×C4×D53C2311yescharacteristic
76C2×Dic106—21no—
77C2×Dic106C2311yescharacteristic
78Q16×C53C2×C414yescomplemented
79C2×C8×C53C2311yescharacteristic
80C2×C8×C53C2311yescharacteristic
81C4⋊C4×C53—21no—
82C4⋊C4×C53—21no—
83C4⋊C4×C53C2311yescharacteristic
84C4⋊C4×C53C2311yescharacteristic
85C42×C53—21no—
86C42×C53—21no—
87C42×C53C2311yescharacteristic
88#11719925C2×C414yescomplemented
89#11719924C2×C414yescomplemented
90#11719921—21no—
91#11719921—21no—
92#11719921—21no—
93C4⋊Dic53—21no—
94C4⋊Dic53—21no—
95C4⋊Dic53C2311yescharacteristic
96C4⋊Dic53C2311yescharacteristic
97#11719919—21no—
98#11719919—21no—
99#11719919—21no—
100C4×Dic53—21no—
101C4×Dic53—21no—
102C4×Dic53—21no—
103C4×Dic53—21no—
104C4×Dic53—21no—
105C4×Dic53D411yescharacteristic
106C4×Dic53D411yescharacteristic
107C4×Dic53C2311yescharacteristic
108C2×#88322C2311yescharacteristic
109C2×#88322C2311yescharacteristic
110#11719913C2×C414yescomplemented
111#11719912—22no—
112#11719912C2×C414yescomplemented
113C22×D53—21no—
114C22×D53C2×D411yescharacteristic
115Q8×C53—21no—
116Q8×C53—21no—
117Q8×C53C22×C412yes—
118Q8×C53C22×C412yes—
119C2×C4×C53—21no—
120C2×C4×C53—21no—
121C2×C4×C53—21no—
122C2×C4×C53—21no—
123C2×C4×C53C2×D411yescharacteristic
124C2×C4×C53C2×D411yescharacteristic
125C2×C4×C53C2411yescharacteristic
126C2×Dic53—21no—
127C2×Dic53—21no—
128C2×Dic53—21no—
129C2×Dic53C2×D411yescharacteristic
130C2×Dic53C2×D411yescharacteristic
131C2×Dic53C2×D411yescharacteristic
132D212—24no—
133D212—21no—
134D212C22×C412yes—
135C4×D53—24no—
136C4×D53—24no—
137C4×D53—22no—
138C4×D53—41no—
139C4×D53—42no—
140C4×D53—42no—
141C4×D53—42no—
142C4×D53C22×C412yes—
143C4×D53#4112yes—
144Dic106—21no—
145Dic106—24no—
146Dic106C22×C412yes—
147C8×C53—21no—
148C8×C53C22×C412yes—
149#88322—21no—
150#88322C22×C412yes—
151C22×C53C22×D411yescharacteristic
152C2×D53—21no—
153C2×D53—21no—
154C2×D53—22no—
155C2×D53—41no—
156C2×D53—41no—
157C2×D53C4×D412yes—
158C4×C53—22no—
159C4×C53—22no—
160C4×C53—21no—
161C4×C53—21no—
162C4×C53—41no—
163C4×C53—41no—
164C4×C53C23×C411yescharacteristic
165C4×C53C2×#4111yescharacteristic
166Dic53—22no—
167Dic53—21no—
168Dic53—41no—
169Dic53—41no—
170Dic53C4×D412yes—
171Dic53#13712yescomplemented
172C4×#137—531nomaximal, Sylow
173C2×C53C2×C4×D411yescharacteristic
174C2×C53C2×#13711yescharacteristic
175C2×C53#57511yescharacteristic
176D53—22no—
177D53—42no—
178C2×#137—531no—
179C4×#41—531no—
180C2×C4×Q8—531no—
181#441—531no—
182#441D5311yescharacteristic, complemented
183#440—531no—
184#440—531no—
185C4×Q16—531no—
186C4×Q16—531no—
187C4×SD16—531no—
188C4×SD16—531no—
189#428—531no—
190#419—531no—
191#417—531no—
192C4×M16—531no—
193C53C4×#13711yescharacteristic, complemented, Sylow
194C2×#41—531no—
195C22×Q8—531no—
196#137—538no—
197C2×Q16—531no—
198C2×Q16C2×D5311yescharacteristic
199C2×SD16—531no—
200C2×SD16—531no—
201C2×M16—531no—
202C2×M16—531no—
203C4×Q8—531no—
204C4×Q8—531no—
205C4×Q8—531no—
206C4×Q8—531no—
207C4×Q8—1061no—
208C4×Q8—1061no—
209C4×Q8C2×D5311yescharacteristic
210C4×Q8C2×D5311yescharacteristic
211C4×D4—531no—
212C4×D4—531no—
213C4×D4—1061no—
214C4×D4—1061no—
215#117—531no—
216#117—1061no—
217C2×C4⋊C4—531no—
218C2×C4⋊C4—1061no—
219C2×C42—531no—
220C2×C42—1061no—
221C2×C42—1061no—
222#107—531no—
223#107—531no—
224#106—531no—
225#106C2×D5311yescharacteristic
226#103—531no—
227#103—531no—
228#103—531no—
229#103—531no—
230#103C2×D5311yescharacteristic
231#103C2×D5311yescharacteristic
232#102—531no—
233#102—531no—
234#97—531no—
235#97C2×D5311yescharacteristic
236C4×C8—531no—
237C4×C8—531no—
238#41—534no—
239#41—1062no—
240C2×Q8—534no—
241C2×Q8—531no—
242C2×Q8—1061no—
243C2×Q8—1062no—
244C2×Q8C22×D5311yescharacteristic
245C2×Q8C22×D5311yescharacteristic
246C2×D4—531no—
247C2×D4—1061no—
248C22×C4—531no—
249C22×C4—531no—
250C22×C4—531no—
251C22×C4—1061no—
252C22×C4—1061no—
253C22×C4—1061no—
254C22×C4—1061no—
255C22×C4—1061no—
256Q16—534no—
257Q16C4×D5314yescomplemented
258SD16—534no—
259SD16—534no—
260M16—534no—
261M16—1062no—
262C2×C8—531no—
263C2×C8—531no—
264C2×C8C22×D5311yescharacteristic
265C2×C8C22×D5311yescharacteristic
266C4⋊C4—21no—
267C4⋊C4—21no—
268C4⋊C4—531no—
269C4⋊C4—531no—
270C4⋊C4—1061no—
271C4⋊C4—1061no—
272C4⋊C4—1061no—
273C4⋊C4—1061no—
274C4⋊C4—1061no—
275C4⋊C4C22×D5311yescharacteristic
276C4⋊C4C22×D5311yescharacteristic
277C22⋊C4—1061no—
278C22⋊C4—1061no—
279C22⋊C4—1061no—
280C42—21no—
281C42—21no—
282C42—531no—
283C42—531no—
284C42—531no—
285C42—1061no—
286C42—1061no—
287C42—1061no—
288C42—1061no—
289C42—1061no—
290C42C22×D5311yescharacteristic
291C23—531no—
292C23—1061no—
293Q8—21no—
294Q8—21no—
295Q8—532no—
296Q8—1064no—
297Q8—1061no—
298Q8C2×C4×D5312yes—
299Q8C2×C4×D5312yes—
300D4—532no—
301D4—1064no—
302D4—1061no—
303C2×C4—21no—
304C2×C4—21no—
305C2×C4—21no—
306C2×C4—21no—
307C2×C4—531no—
308C2×C4—531no—
309C2×C4—532no—
310C2×C4—532no—
311C2×C4—531no—
312C2×C4—1062no—
313C2×C4—1064no—
314C2×C4—1064no—
315C2×C4—1061no—
316C2×C4—1061no—
317C2×C4—1061no—
318C2×C4—2121no—
319C2×C4—2122no—
320C2×C4—2122no—
321C2×C4—2122no—
322C2×C4D4×D5311yescharacteristic
323C2×C4D4×D5311yescharacteristic
324C2×C4C23×D5311yescharacteristic
325C8—21no—
326C8—532no—
327C8—1061no—
328C8C2×C4×D5312yes—
329C22—532no—
330C22—1061no—
331C22—1061no—
332C22—1062no—
333C22—2121no—
334C22—2121no—
335C22C2×D4×D5311yescharacteristic
336C4—22no—
337C4—22no—
338C4—21no—
339C4—21no—
340C4—41no—
341C4—41no—
342C4—532no—
343C4—532no—
344C4—1062no—
345C4—1061no—
346C4—2121no—
347C4—2121no—
348C4C22×C4×D5311yescharacteristic
349C4#41×D5311yescharacteristic
350C2—1062no—
351C2—2122no—
352C2C4×D4×D5311yescharacteristic
353C2#137×D5311yescharacteristic
354C2#1681142211yescharacteristic
subgroupG/H
centerC22C2×D4×D53
commutatorC4×C53C23×C4
FrattiniC2×C4C23×D53
FittingC53×#441C2
socleC22×C53C22×D4
hypercenter#441D53
hypocenterC53C4×#137
LCSG ▷ C4×C53 ▷ C2×C53 ▷ C53
UCSC1 ◁ C22 ◁ C42 ◁ #441

Complements

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

NH ≅ G/Ntype
3. C4×Q8×D53C2semidirect
4. #16810728C2semidirect
5. #16810725C2semidirect
6. #16810721C2semidirect
7. Q16×Dic53C2semidirect
8. #16810715C2semidirect
9. #16810710C2semidirect
10. #16810708C2semidirect
11. #16810662C2semidirect
12. #16810630C2semidirect
14. #16810575C2semidirect
15. C53×#441C2semidirect
18. #13133249C4semidirect
65. Q8×D53C2×C4semidirect
78. Q16×C53C2×C4semidirect
88. #11719925C2×C4semidirect
89. #11719924C2×C4semidirect
110. #11719913C2×C4semidirect
112. #11719912C2×C4semidirect
171. Dic53#137semidirect
182. #441D53semidirect
193. C53C4×#137semidirect
257. Q16C4×D53semidirect