Finite Group Databasebeta

← #20284835

Subgroups — #20284835

Order 6784.

424 conjugacy classes of subgroups.

The subgroup lattice has 360 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#16811170C211yesmaximal, characteristic
2C4×Q8×D53C211yesmaximal, characteristic, complemented
3#16810938C211yesmaximal, characteristic, complemented
4#16810935C211yesmaximal, characteristic, complemented
5#16810917C211yesmaximal, characteristic, complemented
6#16810916C211yesmaximal, characteristic, complemented
7#16810822C211yesmaximal, characteristic, complemented
8#16810821C211yesmaximal, characteristic, complemented
9#16810763C211yesmaximal, characteristic, complemented
10#16810760C211yesmaximal, characteristic, complemented
11#16810725C211yesmaximal, characteristic, complemented
12#16810723C211yesmaximal, characteristic, complemented
13#16810721C211yesmaximal, characteristic, complemented
14#16810678C211yesmaximal, characteristic, complemented
15C53×#440C211yesmaximal, characteristic, complemented
16C2×#11719945—21no—
17C2×Q8×D53C2211yescharacteristic
18C2×#11719913C2211yescharacteristic
19C2×#11719923C2211yescharacteristic
20C2×#11719924C2211yescharacteristic
21#13133216C2211yescharacteristic
22#13133213C2211yescharacteristic
23#13133203—21no—
24#13133192C2211yescharacteristic
25#13133190—21no—
26#13133188C2211yescharacteristic
27Q8×Dic53C2211yescharacteristic
28C4×Dic106—21no—
29C4×Dic106C2211yescharacteristic
30#13133182—21no—
31#13133182C2211yescharacteristic
32C4×D212C2211yescharacteristic
33C4×C22⋊D53—21no—
34C4⋊C4×D53—21no—
35C4⋊C4×D53C2211yescharacteristic
36C42×D53—21no—
37C42×D53C2211yescharacteristic
38#13133151C2211yescharacteristic
39#13133150C2211yescharacteristic
40#13133150C2211yescharacteristic
41#13133146C2211yescharacteristic
42#13133146C2211yescharacteristic
43#13133142C2211yescharacteristic
44#13133141C2211yescharacteristic
45#13133140C2211yescharacteristic
46#13133139C2211yescharacteristic
47#13133138C2211yescharacteristic
48#13133137C2211yescharacteristic
49#13133129C2211yescharacteristic
50#13133129C2211yescharacteristic
51#13133128C2211yescharacteristic
52C2×SD16×C53C2211yescharacteristic
53C4×Q8×C53C2211yescharacteristic
54C4×D4×C53C2211yescharacteristic
55#107×C53C2211yescharacteristic
56#103×C53C2211yescharacteristic
57#102×C53C2211yescharacteristic
58#97×C53C2211yescharacteristic
59C2×Q8×C53C2311yescharacteristic
60C2×D4×C53C2311yescharacteristic
61C22×C4×C53—21no—
62C2×C22⋊D53—21no—
63C2×C22⋊D53—21no—
64C2×C22⋊D53—21no—
65Q8×D53—22no—
66Q8×D53—22no—
67#11719945—44no—
68C2×D212—21no—
69C2×D212C2311yescharacteristic
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×Dic106—21no—
78C2×Dic106C2311yescharacteristic
79SD16×C53—22no—
80C2×C8×C53C2311yescharacteristic
81C2×C8×C53C2311yescharacteristic
82C4⋊C4×C53—21no—
83C4⋊C4×C53C2311yescharacteristic
84C4⋊C4×C53C2311yescharacteristic
85C22⋊C4×C53—21no—
86C42×C53—21no—
87C42×C53C2311yescharacteristic
88C22⋊Dic53—21no—
89#11719924—22no—
90#11719923—22no—
91#11719921—21no—
92#11719921—21no—
93C4⋊Dic53—21no—
94C4⋊Dic53C2311yescharacteristic
95C4⋊Dic53C2311yescharacteristic
96#11719919—21no—
97#11719919—21no—
98#11719919—21no—
99#11719919—21no—
100C4×Dic53—21no—
101C4×Dic53—21no—
102C4×Dic53—21no—
103C4×Dic53D411yescharacteristic
104C4×Dic53D411yescharacteristic
105C4×Dic53C2311yescharacteristic
106C2×#88322C2311yescharacteristic
107C2×#88322C2311yescharacteristic
108#11719913—22no—
109C23×C53—21no—
110C22×D53—21no—
111C22×D53C2×D411yescharacteristic
112Q8×C53—21no—
113Q8×C53#4112yescomplemented
114D4×C53—21no—
115D4×C53—21no—
116C2×C4×C53—21no—
117C2×C4×C53—21no—
118C2×C4×C53—21no—
119C2×C4×C53—42no—
120C2×C4×C53C2×D411yescharacteristic
121C2×C4×C53C2×D411yescharacteristic
122C2×C4×C53C2411yescharacteristic
123C22⋊D53—42no—
124C22⋊D53—42no—
125C22⋊D53—42no—
126C2×Dic53—21no—
127C2×Dic53—21no—
128C2×Dic53—21no—
129C2×Dic53C2×D411yescharacteristic
130C2×Dic53C2×D411yescharacteristic
131C2×Dic53C2×D411yescharacteristic
132D212—21no—
133D212—21no—
134D212—42no—
135C4×D53—24no—
136C4×D53—22no—
137C4×D53—21no—
138C4×D53—21no—
139C4×D53—22no—
140C4×D53—42no—
141C4×D53—42no—
142Dic106—24no—
143Dic106—21no—
144Dic106—42no—
145Dic106#4112yescomplemented
146C8×C53—21no—
147C8×C53—21no—
148#88322—21no—
149#88322—21no—
150C22×C53—41no—
151C22×C53—41no—
152C22×C53—41no—
153C22×C53C22×D411yescharacteristic
154C2×D53—21no—
155C2×D53—21no—
156C2×D53—41no—
157C2×D53—41no—
158C2×D53—41no—
159C2×D53AGL(1,ℤ/8)12yes—
160C4×C53—22no—
161C4×C53—21no—
162C4×C53—21no—
163C4×C53—41no—
164C4×C53—41no—
165C4×C53C2×#4111yescharacteristic
166C4×C53C2×#4111yescharacteristic
167Dic53—21no—
168Dic53—22no—
169Dic53—21no—
170Dic53—41no—
171Dic53—41no—
172Dic53#13712yes—
173#3112—531nomaximal, Sylow
174C2×C53—81no—
175C2×C53#52511yescharacteristic
176C2×C53C2×AGL(1,ℤ/8)11yescharacteristic
177C2×C53C2×#13711yescharacteristic
178D53—22no—
179D53—81no—
180#532—531no—
181C2×C4×Q8—531no—
182#480—531no—
183#480—531no—
184#477—531no—
185#477—531no—
186#460—531no—
187#460—531no—
188#448—531no—
189#448—531no—
190#440—531no—
191#440—531no—
192#440—531no—
193#440D5311yescharacteristic, complemented
194#432—531no—
195C53#311211yescharacteristic, complemented, Sylow
196C2×#41—1061no—
197C22×Q8—531no—
198C2×SD16—531no—
199C2×SD16—531no—
200C2×SD16—531no—
201C2×SD16C2×D5311yescharacteristic
202C4⋊Q8—531no—
203#127—531no—
204#124—1061no—
205#121—531no—
206#121—531no—
207#121—1061no—
208C4×Q8—531no—
209C4×Q8—531no—
210C4×Q8—531no—
211C4×Q8—1061no—
212C4×Q8—1061no—
213C4×Q8C2×D5311yescharacteristic
214C4×D4—531no—
215C4×D4—1061no—
216C4×D4C2×D5311yescharacteristic
217C2×C4⋊C4—531no—
218C2×C4⋊C4—1061no—
219C2×C42—531no—
220C2×C42—1061no—
221#107—531no—
222#107—531no—
223#107—531no—
224#107C2×D5311yescharacteristic
225C4⋊C8—531no—
226C4⋊C8—531no—
227#103—531no—
228#103—531no—
229#103—531no—
230#103C2×D5311yescharacteristic
231#102—531no—
232#102—531no—
233#102—531no—
234#102C2×D5311yescharacteristic
235C22⋊C8—531no—
236C22⋊C8—531no—
237#97—531no—
238#97C2×D5311yescharacteristic
239#41—2124no—
240C2×Q8—531no—
241C2×Q8—1061no—
242C2×Q8—1061no—
243C2×Q8—1062no—
244C2×Q8—1062no—
245C2×Q8C22×D5311yescharacteristic
246C2×D4—531no—
247C2×D4—1061no—
248C2×D4—1061no—
249C2×D4—1061no—
250C2×D4—1061no—
251C2×D4C22×D5311yescharacteristic
252C22×C4—21no—
253C22×C4—531no—
254C22×C4—531no—
255C22×C4—531no—
256C22×C4—1061no—
257C22×C4—1061no—
258C22×C4—1061no—
259SD16—22no—
260SD16—1062no—
261SD16—1062no—
262SD16—1062no—
263C2×C8—531no—
264C2×C8—531no—
265C2×C8C22×D5311yescharacteristic
266C2×C8C22×D5311yescharacteristic
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—21no—
278C22⋊C4—1061no—
279C22⋊C4—1061no—
280C22⋊C4—1061no—
281C42—21no—
282C42—531no—
283C42—531no—
284C42—531no—
285C42—1061no—
286C42—1061no—
287C42—1061no—
288C42C22×D5311yescharacteristic
289C23—21no—
290C23—531no—
291C23—1061no—
292Q8—21no—
293Q8—532no—
294Q8—1064no—
295Q8—1061no—
296Q8—2122no—
297Q8#1171994512yescomplemented
298D4—21no—
299D4—21no—
300D4—1061no—
301D4—1061no—
302D4—2122no—
303D4—2122no—
304D4—2122no—
305D4—2122no—
306C2×C4—21no—
307C2×C4—21no—
308C2×C4—21no—
309C2×C4—42no—
310C2×C4—531no—
311C2×C4—531no—
312C2×C4—531no—
313C2×C4—1062no—
314C2×C4—1062no—
315C2×C4—1064no—
316C2×C4—1061no—
317C2×C4—1061no—
318C2×C4—1061no—
319C2×C4—1061no—
320C2×C4—1061no—
321C2×C4—2122no—
322C2×C4—2122no—
323C2×C4D4×D5311yescharacteristic
324C2×C4D4×D5311yescharacteristic
325C2×C4C23×D5311yescharacteristic
326C8—21no—
327C8—21no—
328C8—1061no—
329C8—1061no—
330C22—41no—
331C22—41no—
332C22—41no—
333C22—532no—
334C22—1061no—
335C22—1061no—
336C22—2121no—
337C22—2121no—
338C22—2121no—
339C22C2×D4×D5311yescharacteristic
340C4—22no—
341C4—21no—
342C4—21no—
343C4—41no—
344C4—41no—
345C4—532no—
346C4—1061no—
347C4—1062no—
348C4—1061no—
349C4—2121no—
350C4—2121no—
351C4#41×D5311yescharacteristic
352C4C2×#1171994511yescharacteristic
353C2—81no—
354C2—1062no—
355C2—4241no—
356C2#1681113211yescharacteristic
357C2AGL(1,ℤ/8)×D5311yescharacteristic
358C2#137×D5311yescharacteristic
subgroupG/H
centerC22C2×D4×D53
commutatorC2×C4×C53C24
FrattiniC2×C4C23×D53
FittingC53×#440C2
socleC22×C53C22×D4
hypercenter#440D53
hypocenterC53#3112
LCSG ▷ C2×C4×C53 ▷ C2×C53 ▷ C53
UCSC1 ◁ C22 ◁ C42 ◁ #440

Complements

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

NH ≅ G/Ntype
2. C4×Q8×D53C2semidirect
3. #16810938C2semidirect
4. #16810935C2semidirect
5. #16810917C2semidirect
6. #16810916C2semidirect
7. #16810822C2semidirect
8. #16810821C2semidirect
9. #16810763C2semidirect
10. #16810760C2semidirect
11. #16810725C2semidirect
12. #16810723C2semidirect
13. #16810721C2semidirect
14. #16810678C2semidirect
15. C53×#440C2semidirect
113. Q8×C53#41semidirect
145. Dic106#41semidirect
193. #440D53semidirect
195. C53#3112semidirect
297. Q8#11719945semidirect