Finite Group Databasebeta

← #20285428

Subgroups — #20285428

Order 6784.

402 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
1#16811175C211yesmaximal, characteristic, complemented
2#16811168C211yesmaximal, characteristic
3#16811027C211yesmaximal, characteristic, complemented
4#16810994C211yesmaximal, characteristic, complemented
5#16810972C211yesmaximal, characteristic, complemented
6#16810968C211yesmaximal, characteristic, complemented
7#16810871C211yesmaximal, characteristic, complemented
8#16810869C211yesmaximal, characteristic, complemented
9#16810867C211yesmaximal, characteristic, complemented
10#16810726C211yesmaximal, characteristic, complemented
11#16810724C211yesmaximal, characteristic, complemented
12#16810722C211yesmaximal, characteristic, complemented
13#16810720C211yesmaximal, characteristic, complemented
14M16×Dic53C211yesmaximal, characteristic, complemented
15C53×#468C211yesmaximal, characteristic, complemented
16C2×#11719947—21no—
17C2×#11719913C2211yescharacteristic
18C2×#11719923C2211yescharacteristic
19C2×#11719924C2211yescharacteristic
20C2×#11719917C2211yescharacteristic
21#13133216C2211yescharacteristic
22#13133216C2211yescharacteristic
23#13133215C2211yescharacteristic
24#13133213C2211yescharacteristic
25#13133207—21no—
26#13133203—21no—
27#13133196C2211yescharacteristic
28#13133194—21no—
29#13133191C2211yescharacteristic
30#13133189—21no—
31Q8×Dic53C2211yescharacteristic
32#13133182—21no—
33#13133182C2211yescharacteristic
34D4×Dic53C2211yescharacteristic
35#13133179C2211yescharacteristic
36#13133177—21no—
37#13133173—21no—
38C2×C4×Dic53C2211yescharacteristic
39#13133151C2211yescharacteristic
40#13133150C2211yescharacteristic
41#13133150C2211yescharacteristic
42#13133142C2211yescharacteristic
43#13133141C2211yescharacteristic
44#13133140C2211yescharacteristic
45#13133139C2211yescharacteristic
46#13133138C2211yescharacteristic
47#13133137C2211yescharacteristic
48#13133127C2211yescharacteristic
49#13133127C2211yescharacteristic
50C8×Dic53C2211yescharacteristic
51C8×Dic53C2211yescharacteristic
52C2×SD16×C53C2211yescharacteristic
53C2×M16×C53C2211yescharacteristic
54C22⋊Q8×C53C2211yescharacteristic
55#121×C53C2211yescharacteristic
56#107×C53C2211yescharacteristic
57#103×C53C2211yescharacteristic
58#102×C53C2211yescharacteristic
59C2×Q8×C53C2311yescharacteristic
60C2×D4×C53—21no—
61C2×D4×C53C2311yescharacteristic
62C22×C4×C53C2311yescharacteristic
63C2×C22⋊D53—21no—
64C2×C22⋊D53—21no—
65C2×C22⋊D53—21no—
66C22×Dic53—21no—
67C22×Dic53D411yescharacteristic
68C22×Dic53D411yescharacteristic
69#11719947—44no—
70C2×D212C2311yescharacteristic
71C2×C4×D53—21no—
72C2×Dic106—21no—
73C2×Dic106—21no—
74C2×Dic106C2311yescharacteristic
75SD16×C53—22no—
76M16×C53—22no—
77C2×C8×C53C2311yescharacteristic
78C2×C8×C53C2311yescharacteristic
79C4⋊C4×C53—21no—
80C4⋊C4×C53C2311yescharacteristic
81C4⋊C4×C53C2311yescharacteristic
82C22⋊C4×C53—21no—
83C22⋊C4×C53—21no—
84C22⋊Dic53—21no—
85C22⋊Dic53—21no—
86#11719924—22no—
87#11719923—22no—
88#11719921—21no—
89#11719921—21no—
90C4⋊Dic53—21no—
91C4⋊Dic53C2311yescharacteristic
92C4⋊Dic53C2311yescharacteristic
93#11719919—21no—
94#11719919—21no—
95#11719919—21no—
96#11719919—21no—
97#11719919—21no—
98#11719919—21no—
99C4×Dic53—21no—
100C4×Dic53—21no—
101C4×Dic53D411yescharacteristic
102C4×Dic53D411yescharacteristic
103C4×Dic53C2311yescharacteristic
104C4×Dic53C2311yescharacteristic
105#11719917—22no—
106C2×#88322C2311yescharacteristic
107C2×#88322C2311yescharacteristic
108#11719913—22no—
109C23×C53—21no—
110C23×C53C2×D411yescharacteristic
111C22×D53—21no—
112Q8×C53—21no—
113Q8×C53—21no—
114D4×C53—21no—
115D4×C53—21no—
116D4×C53—42no—
117C2×C4×C53—21no—
118C2×C4×C53—21no—
119C2×C4×C53—21no—
120C2×C4×C53—21no—
121C2×C4×C53—21no—
122C2×C4×C53C2×D411yescharacteristic
123C2×C4×C53C2411yescharacteristic
124C22⋊D53—42no—
125C22⋊D53—42no—
126C22⋊D53—42no—
127C2×Dic53—22no—
128C2×Dic53—22no—
129C2×Dic53—21no—
130C2×Dic53—21no—
131C2×Dic53—21no—
132C2×Dic53—42no—
133C2×Dic53C2×D411yescharacteristic
134C2×Dic53C2×D411yescharacteristic
135C2×Dic53C2×D411yescharacteristic
136C2×Dic53C2×D411yescharacteristic
137D212—21no—
138D212—21no—
139C4×D53—42no—
140Dic106—21no—
141Dic106—21no—
142Dic106—42no—
143Dic106—42no—
144C8×C53—21no—
145C8×C53#4112yescomplemented
146#88322—21no—
147#88322#4112yescomplemented
148C22×C53—21no—
149C22×C53—21no—
150C22×C53—21no—
151C22×C53—41no—
152C22×C53—41no—
153C22×C53—41no—
154C22×C53C22×D411yescharacteristic
155C2×D53—41no—
156C2×D53—41no—
157C2×D53—41no—
158C4×C53—21no—
159C4×C53—41no—
160C4×C53—41no—
161C4×C53—41no—
162C4×C53C2×#4111yescharacteristic
163C4×C53C2×#4111yescharacteristic
164Dic53—21no—
165Dic53—21no—
166Dic53—41no—
167Dic53—41no—
168Dic53—41no—
169Dic53AGL(1,ℤ/8)12yes—
170Dic53#13712yes—
171#3182—531nomaximal, Sylow
172C2×C53—41no—
173C2×C53—81no—
174C2×C53#53211yescharacteristic
175C2×C53C2×AGL(1,ℤ/8)11yescharacteristic
176C2×C53C2×#13711yescharacteristic
177D53—81no—
178#533—531no—
179#532—531no—
180#500—531no—
181#492—531no—
182#487—531no—
183#486—531no—
184#468—531no—
185#468—531no—
186#468—531no—
187#468D5311yescharacteristic, complemented
188#440—531no—
189#440—531no—
190#440—531no—
191#440—531no—
192C4×M16—531no—
193C53#318211yescharacteristic, complemented, Sylow
194C2×#41—1061no—
195C2×SD16—531no—
196C2×SD16—531no—
197C2×SD16—531no—
198C2×SD16C2×D5311yescharacteristic
199C2×M16—531no—
200C2×M16C2×D5311yescharacteristic
201C4⋊Q8—531no—
202C4⋊Q8—531no—
203C4⋊Q8—531no—
204#127—531no—
205#125—1061no—
206#124—1061no—
207C22⋊Q8—531no—
208C22⋊Q8—1061no—
209C22⋊Q8C2×D5311yescharacteristic
210#121—531no—
211#121—1061no—
212#121C2×D5311yescharacteristic
213C4×Q8—531no—
214C4×Q8—531no—
215C4×Q8—1061no—
216C4×D4—531no—
217C4×D4—531no—
218C4×D4—1061no—
219#117—1061no—
220C2×C42—531no—
221#107—531no—
222#107—531no—
223#107—531no—
224#107C2×D5311yescharacteristic
225#103—531no—
226#103—531no—
227#103—531no—
228#103C2×D5311yescharacteristic
229#102—531no—
230#102—531no—
231#102—531no—
232#102C2×D5311yescharacteristic
233#97—531no—
234#97—531no—
235C4×C8—531no—
236C4×C8—531no—
237#41—2124no—
238C2×Q8—531no—
239C2×Q8—1061no—
240C2×Q8—1061no—
241C2×Q8C22×D5311yescharacteristic
242C2×D4—21no—
243C2×D4—531no—
244C2×D4—1061no—
245C2×D4—1061no—
246C2×D4—1061no—
247C2×D4C22×D5311yescharacteristic
248C22×C4—531no—
249C22×C4—531no—
250C22×C4—1061no—
251C22×C4—1061no—
252C22×C4C22×D5311yescharacteristic
253SD16—22no—
254SD16—1062no—
255SD16—1062no—
256SD16—1062no—
257M16—22no—
258M16—1062no—
259C2×C8—531no—
260C2×C8—531no—
261C2×C8C22×D5311yescharacteristic
262C2×C8C22×D5311yescharacteristic
263C4⋊C4—21no—
264C4⋊C4—531no—
265C4⋊C4—531no—
266C4⋊C4—1061no—
267C4⋊C4—1061no—
268C4⋊C4—1061no—
269C4⋊C4—1061no—
270C4⋊C4—1061no—
271C4⋊C4—1061no—
272C4⋊C4—1061no—
273C4⋊C4C22×D5311yescharacteristic
274C4⋊C4C22×D5311yescharacteristic
275C22⋊C4—21no—
276C22⋊C4—21no—
277C22⋊C4—1061no—
278C22⋊C4—1061no—
279C22⋊C4—1061no—
280C22⋊C4—1061no—
281C42—531no—
282C42—531no—
283C42—531no—
284C42—531no—
285C42—1061no—
286C42—1061no—
287C23—21no—
288C23—1061no—
289C23D4×D5311yescharacteristic
290Q8—21no—
291Q8—21no—
292Q8—1061no—
293Q8—1061no—
294Q8—2122no—
295Q8—2122no—
296D4—21no—
297D4—21no—
298D4—42no—
299D4—1061no—
300D4—1061no—
301D4—2122no—
302D4—2122no—
303D4—2122no—
304C2×C4—21no—
305C2×C4—21no—
306C2×C4—21no—
307C2×C4—21no—
308C2×C4—21no—
309C2×C4—531no—
310C2×C4—531no—
311C2×C4—531no—
312C2×C4—531no—
313C2×C4—1062no—
314C2×C4—1062no—
315C2×C4—1061no—
316C2×C4—1061no—
317C2×C4—1061no—
318C2×C4—2122no—
319C2×C4—2122no—
320C2×C4D4×D5311yescharacteristic
321C2×C4C23×D5311yescharacteristic
322C8—21no—
323C8—532no—
324C8—1061no—
325C8#1171994712yescomplemented
326C22—21no—
327C22—21no—
328C22—21no—
329C22—41no—
330C22—41no—
331C22—41no—
332C22—2121no—
333C22—2121no—
334C22—2121no—
335C22C2×D4×D5311yescharacteristic
336C4—21no—
337C4—41no—
338C4—41no—
339C4—41no—
340C4—532no—
341C4—532no—
342C4—1061no—
343C4—1061no—
344C4—2121no—
345C4—2121no—
346C4—2121no—
347C4#41×D5311yescharacteristic
348C4C2×#1171994711yescharacteristic
349C2—41no—
350C2—81no—
351C2—4241no—
352C2#1681116811yescharacteristic
353C2AGL(1,ℤ/8)×D5311yescharacteristic
354C2#137×D5311yescharacteristic
subgroupG/H
centerC22C2×D4×D53
commutatorC2×C4×C53C24
FrattiniC2×C4C23×D53
FittingC53×#468C2
socleC22×C53C22×D4
hypercenter#468D53
hypocenterC53#3182
LCSG ▷ C2×C4×C53 ▷ C2×C53 ▷ C53
UCSC1 ◁ C22 ◁ C22×C4 ◁ #468

Complements

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

NH ≅ G/Ntype
1. #16811175C2semidirect
3. #16811027C2semidirect
4. #16810994C2semidirect
5. #16810972C2semidirect
6. #16810968C2semidirect
7. #16810871C2semidirect
8. #16810869C2semidirect
9. #16810867C2semidirect
10. #16810726C2semidirect
11. #16810724C2semidirect
12. #16810722C2semidirect
13. #16810720C2semidirect
14. M16×Dic53C2semidirect
15. C53×#468C2semidirect
145. C8×C53#41semidirect
147. #88322#41semidirect
187. #468D53semidirect
193. C53#3182semidirect
325. C8#11719947semidirect