Finite Group Databasebeta

← #20284266

Subgroups — #20284266

Order 6784.

472 conjugacy classes of subgroups.

The subgroup lattice has 384 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#13133246C211yesmaximal, characteristic, complemented
2C2×#13133236C211yesmaximal, characteristic
3#16811193C211yesmaximal, characteristic, complemented
4#16811102C211yesmaximal, characteristic, complemented
5#16810844C211yesmaximal, characteristic, complemented
6#16810842C211yesmaximal, characteristic, complemented
7#16810836C211yesmaximal, characteristic, complemented
8#16810834C211yesmaximal, characteristic, complemented
9#16810832C211yesmaximal, characteristic, complemented
10#16810828C211yesmaximal, characteristic, complemented
11#16810826C211yesmaximal, characteristic, complemented
12#16810648C211yesmaximal, characteristic, complemented
13#16810621C211yesmaximal, characteristic, complemented
14#103×D53C211yesmaximal, characteristic, complemented
15C53×#463C211yesmaximal, characteristic, complemented
16C2×#11719947C2211yescharacteristic
17C2×#11719947C2211yescharacteristic
18C2×Q8×D53C2211yescharacteristic
19#13133246—24no—
20#13133236—24no—
21C2×Dic212C2211yescharacteristic
22C2×Dic212C2211yescharacteristic
23C2×#11719925C2211yescharacteristic
24C2×#11719923C2211yescharacteristic
25C2×#11719923C2211yescharacteristic
26C2×#11719923C2211yescharacteristic
27C2×#11719912C2211yescharacteristic
28C2×C8×D53C2211yescharacteristic
29#13133217C2211yescharacteristic
30#13133215C2211yescharacteristic
31#13133204C2211yescharacteristic
32#13133199—21no—
33#13133199—21no—
34#13133194—21no—
35#13133194—21no—
36C4×Dic106C2211yescharacteristic
37#13133182C2211yescharacteristic
38D4×Dic53C2211yescharacteristic
39C4×C22⋊D53—21no—
40#13133177—21no—
41#13133174C2211yescharacteristic
42#13133173—21no—
43C2×#11719919—21no—
44C4⋊C4×D53C2211yescharacteristic
45C4⋊#88322C2211yescharacteristic
46#13133146C2211yescharacteristic
47#13133146C2211yescharacteristic
48#13133142C2211yescharacteristic
49#13133142C2211yescharacteristic
50#13133141C2211yescharacteristic
51#13133140C2211yescharacteristic
52#13133140C2211yescharacteristic
53#13133138C2211yescharacteristic
54C2×SD16×C53C2211yescharacteristic
55C2×D8×C53C2211yescharacteristic
56#124×C53C2211yescharacteristic
57C4×D4×C53C2211yescharacteristic
58C4⋊C8×C53C2211yescharacteristic
59#103×C53C2211yescharacteristic
60#102×C53C2211yescharacteristic
61C2×Q8×C53C2311yescharacteristic
62C2×D4×C53C2311yescharacteristic
63C2×D4×C53C2311yescharacteristic
64C22×C4×C53—21no—
65C2×C22⋊D53—21no—
66C2×C22⋊D53—21no—
67C22×Dic53—21no—
68C22×Dic53—21no—
69C22×Dic53—21no—
70Q8×D53—22no—
71Q8×D53—22no—
72#11719947—22no—
73#11719947—22no—
74#11719947—22no—
75#11719947D414yes—
76C2×C4×D53—21no—
77C2×C4×D53—21no—
78C2×C4×D53C2311yescharacteristic
79C2×Dic106—21no—
80C2×Dic106C2311yescharacteristic
81C2×Dic106C2311yescharacteristic
82C2×Dic106C2311yescharacteristic
83SD16×C53—22no—
84D8×C53—22no—
85C2×C8×C53C2311yescharacteristic
86C2×C8×C53C2311yescharacteristic
87C4⋊C4×C53C2311yescharacteristic
88C22⋊C4×C53—21no—
89C22⋊C4×C53—21no—
90C22⋊C4×C53—21no—
91C42×C53C2311yescharacteristic
92C22⋊Dic53—21no—
93C22⋊Dic53—21no—
94C22⋊Dic53—21no—
95#11719925—22no—
96#11719923—22no—
97#11719923—22no—
98#11719923—22no—
99#11719921D411yescharacteristic
100#11719921D411yescharacteristic
101C4⋊Dic53—21no—
102C4⋊Dic53—21no—
103C4⋊Dic53C2311yescharacteristic
104#11719919—21no—
105#11719919—21no—
106#11719919—21no—
107#11719919—21no—
108#11719919—21no—
109#11719919D411yescharacteristic
110#11719919D411yescharacteristic
111C4×Dic53—21no—
112C4×Dic53—21no—
113C4×Dic53C2311yescharacteristic
114C2×#88322C2311yescharacteristic
115C2×#88322C2311yescharacteristic
116Dic212—22no—
117Dic212—22no—
118#11719912—22no—
119C8×D53—22no—
120C23×C53—21no—
121C23×C53—21no—
122C22×D53C2×D411yescharacteristic
123Q8×C53—21no—
124Q8×C53—21no—
125D4×C53—21no—
126D4×C53—21no—
127D4×C53—21no—
128D4×C53C2×D412yes—
129C2×C4×C53—21no—
130C2×C4×C53—21no—
131C2×C4×C53—81no—
132C2×C4×C53C2×D411yescharacteristic
133C2×C4×C53C2×D411yescharacteristic
134C2×C4×C53C2411yescharacteristic
135C22⋊D53—24no—
136C22⋊D53—42no—
137C2×Dic53—21no—
138C2×Dic53—24no—
139C2×Dic53—21no—
140C2×Dic53—21no—
141C2×Dic53—21no—
142C2×Dic53—42no—
143C2×Dic53—81no—
144C2×Dic53C2×D411yescharacteristic
145C2×Dic53C2×D411yescharacteristic
146C2×Dic53C2×D411yescharacteristic
147C4×D53—42no—
148C4×D53—81no—
149C4×D53C2×D412yes—
150C4×D53#4112yes—
151Dic106—21no—
152Dic106—21no—
153Dic106—21no—
154Dic106—21no—
155Dic106—21no—
156Dic106—42no—
157Dic106C2×D412yes—
158C8×C53—21no—
159C8×C53—21no—
160#88322—21no—
161#88322—21no—
162C22×C53—22no—
163C22×C53—41no—
164C22×C53—41no—
165C22×C53—41no—
166C22×C53—41no—
167C22×C53—41no—
168C22×C53C22×D411yescharacteristic
169C2×D53—21no—
170C2×D53—21no—
171C2×D53#12112yes—
172C4×C53—21no—
173C4×C53—21no—
174C4×C53—41no—
175C4×C53—41no—
176C4×C53C22×D411yescharacteristic
177C4×C53C2×#4111yescharacteristic
178Dic53—22no—
179Dic53—21no—
180Dic53—21no—
181Dic53—41no—
182Dic53—41no—
183Dic53—41no—
184Dic53#12112yes—
185#3046—531nomaximal, Sylow
186C2×C53—42no—
187C2×C53—81no—
188C2×C53C2×#12111yescharacteristic
189C2×C53#57711yescharacteristic
190C2×C53#57811yescharacteristic
191D53—22no—
192C2×#137—531no—
193C2×#135—531no—
194#536—531no—
195#520—531no—
196#464—531no—
197#464—531no—
198#463—531no—
199#463D5311yescharacteristic, complemented
200#462—531no—
201#462—531no—
202#461—531no—
203#461—531no—
204#424—531no—
205#417—531no—
206C2×#103—531no—
207C53#304611yescharacteristic, complemented, Sylow
208C2×#41—531no—
209C2×#41—531no—
210C22×Q8—531no—
211#137—1064no—
212#135—1064no—
213C2×Q16—531no—
214C2×Q16—531no—
215C2×Q16—531no—
216C2×SD16—531no—
217C2×SD16—531no—
218C2×SD16—531no—
219C2×SD16C2×D5311yescharacteristic
220C2×D8C2×D5311yescharacteristic
221C2×M16—531no—
222C22×C8—531no—
223C4⋊Q8—531no—
224C4⋊Q8—531no—
225#124—531no—
226#124C2×D5311yescharacteristic
227#123—1061no—
228#123—1061no—
229C22⋊Q8—1061no—
230C22⋊Q8—1061no—
231C4×Q8—531no—
232C4×Q8—531no—
233C4×D4—531no—
234C4×D4—1061no—
235C4×D4—1061no—
236C4×D4C2×D5311yescharacteristic
237#117—531no—
238#117—1061no—
239C2×C4⋊C4—531no—
240C2×C4⋊C4—1061no—
241C4⋊C8—531no—
242C4⋊C8—531no—
243C4⋊C8—531no—
244C4⋊C8C2×D5311yescharacteristic
245#103—531no—
246#103—531no—
247#103—531no—
248#103—531no—
249#103—531no—
250#103C2×D5311yescharacteristic
251#102—531no—
252#102C2×D5311yescharacteristic
253#41—534no—
254#41—1062no—
255#41—1062no—
256#41—1062no—
257C2×Q8—531no—
258C2×Q8—531no—
259C2×Q8—531no—
260C2×Q8—1061no—
261C2×Q8—1062no—
262C2×Q8—1062no—
263C2×Q8C22×D5311yescharacteristic
264C2×D4—1061no—
265C2×D4—1061no—
266C2×D4C22×D5311yescharacteristic
267C2×D4C22×D5311yescharacteristic
268C22×C4—21no—
269C22×C4—531no—
270C22×C4—1061no—
271C22×C4—1061no—
272C22×C4—1061no—
273C22×C4—1061no—
274C22×C4—1061no—
275Q16—1062no—
276Q16—1062no—
277Q16—1062no—
278SD16—22no—
279SD16—1062no—
280SD16—1062no—
281SD16—1062no—
282D8—22no—
283M16—1062no—
284C2×C8—531no—
285C2×C8—531no—
286C2×C8—1062no—
287C2×C8C22×D5311yescharacteristic
288C2×C8C22×D5311yescharacteristic
289C4⋊C4—531no—
290C4⋊C4—531no—
291C4⋊C4—531no—
292C4⋊C4—1061no—
293C4⋊C4—1061no—
294C4⋊C4—1061no—
295C4⋊C4—1061no—
296C4⋊C4—1061no—
297C4⋊C4—1061no—
298C4⋊C4—1061no—
299C4⋊C4C22×D5311yescharacteristic
300C22⋊C4—21no—
301C22⋊C4—21no—
302C22⋊C4—21no—
303C22⋊C4—531no—
304C22⋊C4—531no—
305C22⋊C4—1061no—
306C22⋊C4—1061no—
307C22⋊C4—1061no—
308C42—531no—
309C42—1061no—
310C42—1061no—
311C42C22×D5311yescharacteristic
312C23—21no—
313C23—21no—
314C23—531no—
315Q8—21no—
316Q8—21no—
317Q8—532no—
318Q8—1061no—
319Q8—1061no—
320Q8—1061no—
321Q8—1061no—
322Q8—1061no—
323Q8—2122no—
324D4—21no—
325D4—21no—
326D4—21no—
327D4—1064no—
328D4—2122no—
329D4D4×D5312yes—
330C2×C4—21no—
331C2×C4—21no—
332C2×C4—81no—
333C2×C4—531no—
334C2×C4—532no—
335C2×C4—532no—
336C2×C4—531no—
337C2×C4—531no—
338C2×C4—1064no—
339C2×C4—1061no—
340C2×C4—1061no—
341C2×C4—1061no—
342C2×C4—1061no—
343C2×C4—4241no—
344C2×C4—4241no—
345C2×C4—2122no—
346C2×C4—2122no—
347C2×C4D4×D5311yescharacteristic
348C2×C4D4×D5311yescharacteristic
349C2×C4C23×D5311yescharacteristic
350C8—21no—
351C8—21no—
352C8—1061no—
353C8—1061no—
354C22—22no—
355C22—41no—
356C22—41no—
357C22—41no—
358C22—41no—
359C22—41no—
360C22—532no—
361C22—1061no—
362C22—1061no—
363C22C2×D4×D5311yescharacteristic
364C4—21no—
365C4—21no—
366C4—41no—
367C4—41no—
368C4—532no—
369C4—1062no—
370C4—1061no—
371C4—1061no—
372C4—2121no—
373C4—2121no—
374C4—2121no—
375C4C2×D4×D5311yescharacteristic
376C4#41×D5311yescharacteristic
377C2—42no—
378C2—81no—
379C2—1062no—
380C2#121×D5311yescharacteristic
381C2#1681143311yescharacteristic
382C2#1681143711yescharacteristic
subgroupG/H
centerC22C2×D4×D53
commutatorC2×C4×C53C24
FrattiniC2×C4C23×D53
FittingC53×#463C2
socleC22×C53C22×D4
hypercenter#463D53
hypocenterC53#3046
LCSG ▷ C2×C4×C53 ▷ C2×C53 ▷ C53
UCSC1 ◁ C22 ◁ C42 ◁ #463

Complements

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

NH ≅ G/Ntype
1. C2×#13133246C2semidirect
3. #16811193C2semidirect
4. #16811102C2semidirect
5. #16810844C2semidirect
6. #16810842C2semidirect
7. #16810836C2semidirect
8. #16810834C2semidirect
9. #16810832C2semidirect
10. #16810828C2semidirect
11. #16810826C2semidirect
12. #16810648C2semidirect
13. #16810621C2semidirect
14. #103×D53C2semidirect
15. C53×#463C2semidirect
199. #463D53semidirect
207. C53#3046semidirect