Finite Group Databasebeta

← #20284268

Subgroups — #20284268

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×#13133249C211yesmaximal, characteristic, complemented
2C2×#13133238C211yesmaximal, characteristic
3#16811193C211yesmaximal, characteristic, complemented
4#16811105C211yesmaximal, characteristic, complemented
5#16810843C211yesmaximal, characteristic, complemented
6#16810837C211yesmaximal, characteristic, complemented
7#16810835C211yesmaximal, characteristic, complemented
8#16810833C211yesmaximal, characteristic, complemented
9#16810831C211yesmaximal, characteristic, complemented
10#16810829C211yesmaximal, characteristic, complemented
11#16810827C211yesmaximal, characteristic, complemented
12#16810648C211yesmaximal, characteristic, complemented
13#16810621C211yesmaximal, characteristic, complemented
14#103×D53C211yesmaximal, characteristic, complemented
15C53×#464C211yesmaximal, characteristic, complemented
16C2×#11719949C2211yescharacteristic
17C2×#11719947C2211yescharacteristic
18C2×Q8×D53C2211yescharacteristic
19#13133249—24no—
20#13133238—24no—
21C2×#11719925C2211yescharacteristic
22C2×#11719925C2211yescharacteristic
23C2×#11719913C2211yescharacteristic
24C2×#11719913C2211yescharacteristic
25C2×#11719924C2211yescharacteristic
26C2×#11719922C2211yescharacteristic
27C2×#11719912C2211yescharacteristic
28C2×C8×D53C2211yescharacteristic
29#13133217C2211yescharacteristic
30#13133215C2211yescharacteristic
31#13133204C2211yescharacteristic
32#13133199—21no—
33#13133199—21no—
34#13133194—21no—
35#13133194—21no—
36Q8×Dic53C2211yescharacteristic
37C4×D212—21no—
38C4×D212C2211yescharacteristic
39#13133179—21no—
40#13133179C2211yescharacteristic
41#13133174—21no—
42#13133174C2211yescharacteristic
43C4⋊C4×D53—21no—
44C4⋊C4×D53C2211yescharacteristic
45C4⋊#88322C2211yescharacteristic
46#13133146C2211yescharacteristic
47#13133146C2211yescharacteristic
48#13133142C2211yescharacteristic
49#13133142C2211yescharacteristic
50#13133141C2211yescharacteristic
51#13133140C2211yescharacteristic
52#13133140C2211yescharacteristic
53#13133138C2211yescharacteristic
54C2×Q16×C53C2211yescharacteristic
55C2×SD16×C53C2211yescharacteristic
56#124×C53C2211yescharacteristic
57C4×Q8×C53C2211yescharacteristic
58C4⋊C8×C53C2211yescharacteristic
59#103×C53C2211yescharacteristic
60#102×C53C2211yescharacteristic
61C2×Q8×C53C2311yescharacteristic
62C2×Q8×C53C2311yescharacteristic
63C2×D4×C53C2311yescharacteristic
64C2×C22⋊D53—21no—
65C22×Dic53—21no—
66#11719949—22no—
67#11719949D414yes—
68Q8×D53—22no—
69Q8×D53—22no—
70#11719947—22no—
71#11719947—22no—
72C2×D212—21no—
73C2×D212C2311yescharacteristic
74C2×C4×D53—21no—
75C2×C4×D53—21no—
76C2×C4×D53—21no—
77C2×C4×D53—21no—
78C2×C4×D53—21no—
79C2×C4×D53C2311yescharacteristic
80C2×Dic106—21no—
81C2×Dic106C2311yescharacteristic
82C2×Dic106C2311yescharacteristic
83Q16×C53—22no—
84SD16×C53—22no—
85C2×C8×C53C2311yescharacteristic
86C2×C8×C53C2311yescharacteristic
87C4⋊C4×C53—21no—
88C4⋊C4×C53C2311yescharacteristic
89C22⋊C4×C53—21no—
90C22⋊C4×C53—21no—
91C42×C53—21no—
92C42×C53C2311yescharacteristic
93C22⋊Dic53—21no—
94C22⋊Dic53—21no—
95#11719925—22no—
96#11719925—22no—
97#11719924—22no—
98#11719922—22no—
99#11719921—21no—
100#11719921—21no—
101#11719921D411yescharacteristic
102#11719921D411yescharacteristic
103C4⋊Dic53—21no—
104C4⋊Dic53—21no—
105C4⋊Dic53—21no—
106C4⋊Dic53C2311yescharacteristic
107#11719919—21no—
108#11719919—21no—
109#11719919—21no—
110#11719919D411yescharacteristic
111#11719919D411yescharacteristic
112C4×Dic53—21no—
113C4×Dic53C2311yescharacteristic
114C2×#88322C2311yescharacteristic
115C2×#88322C2311yescharacteristic
116#11719913—22no—
117#11719913—22no—
118#11719912—22no—
119C8×D53—22no—
120C23×C53—21no—
121C22×D53—21no—
122C22×D53C2×D411yescharacteristic
123Q8×C53—21no—
124Q8×C53—21no—
125Q8×C53—21no—
126Q8×C53C2×D412yes—
127D4×C53—21no—
128D4×C53—21no—
129C2×C4×C53—21no—
130C2×C4×C53—21no—
131C2×C4×C53—21no—
132C2×C4×C53C2×D411yescharacteristic
133C2×C4×C53C2×D411yescharacteristic
134C2×C4×C53C2411yescharacteristic
135C22⋊D53—42no—
136C2×Dic53—21no—
137C2×Dic53—21no—
138C2×Dic53—21no—
139C2×Dic53—42no—
140C2×Dic53C2×D411yescharacteristic
141C2×Dic53C2×D411yescharacteristic
142C2×Dic53C2×D411yescharacteristic
143D212—24no—
144D212—21no—
145D212C2×D412yes—
146C4×D53—24no—
147C4×D53—42no—
148C4×D53—81no—
149C4×D53—81no—
150C4×D53—81no—
151C4×D53C2×D412yes—
152C4×D53#4112yes—
153Dic106—21no—
154Dic106—21no—
155Dic106—21no—
156Dic106—21no—
157Dic106—42no—
158C8×C53—21no—
159C8×C53—21no—
160#88322—21no—
161#88322—21no—
162C22×C53—41no—
163C22×C53—41no—
164C22×C53—41no—
165C22×C53C22×D411yescharacteristic
166C2×D53—21no—
167C2×D53—21no—
168C2×D53—22no—
169C2×D53—41no—
170C2×D53—41no—
171C2×D53#12112yes—
172C4×C53—22no—
173C4×C53—21no—
174C4×C53—21no—
175C4×C53—41no—
176C4×C53—41no—
177C4×C53C22×D411yescharacteristic
178C4×C53C2×#4111yescharacteristic
179Dic53—21no—
180Dic53—21no—
181Dic53—41no—
182Dic53—41no—
183Dic53—41no—
184Dic53#12112yes—
185#3046—531nomaximal, Sylow
186C2×C53—81no—
187C2×C53C2×#12111yescharacteristic
188C2×C53#57711yescharacteristic
189C2×C53#57811yescharacteristic
190D53—22no—
191D53—42no—
192C2×#137—531no—
193C2×#135—531no—
194#536—531no—
195#520—531no—
196#464—531no—
197#464D5311yescharacteristic, complemented
198#463—531no—
199#463—531no—
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×Q16C2×D5311yescharacteristic
216C2×SD16—531no—
217C2×SD16—531no—
218C2×SD16—531no—
219C2×SD16C2×D5311yescharacteristic
220C2×D8—531no—
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×Q8C2×D5311yescharacteristic
233C4×D4—531no—
234C4×D4—531no—
235C4×D4—1061no—
236C4×D4—1061no—
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—1061no—
260C2×Q8—1062no—
261C2×Q8—1062no—
262C2×Q8C22×D5311yescharacteristic
263C2×Q8C22×D5311yescharacteristic
264C2×D4—531no—
265C2×D4—1061no—
266C2×D4—1061no—
267C2×D4C22×D5311yescharacteristic
268C22×C4—531no—
269C22×C4—1061no—
270C22×C4—1061no—
271C22×C4—1061no—
272C22×C4—1061no—
273C22×C4—1061no—
274C22×C4—1061no—
275Q16—22no—
276Q16—1062no—
277Q16—1062no—
278SD16—22no—
279SD16—1062no—
280SD16—1062no—
281SD16—1062no—
282D8—1062no—
283M16—1062no—
284C2×C8—531no—
285C2×C8—531no—
286C2×C8—1062no—
287C2×C8C22×D5311yescharacteristic
288C2×C8C22×D5311yescharacteristic
289C4⋊C4—21no—
290C4⋊C4—531no—
291C4⋊C4—531no—
292C4⋊C4—531no—
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—531no—
303C22⋊C4—531no—
304C22⋊C4—1061no—
305C22⋊C4—1061no—
306C22⋊C4—1061no—
307C22⋊C4—1061no—
308C42—21no—
309C42—531no—
310C42—1061no—
311C42C22×D5311yescharacteristic
312C23—21no—
313C23—531no—
314C23—1061no—
315Q8—21no—
316Q8—21no—
317Q8—21no—
318Q8—1061no—
319Q8—1061no—
320Q8—1061no—
321Q8—1061no—
322Q8—2122no—
323Q8D4×D5312yes—
324D4—21no—
325D4—21no—
326D4—532no—
327D4—1064no—
328D4—1061no—
329D4—2122no—
330C2×C4—21no—
331C2×C4—21no—
332C2×C4—21no—
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—4241no—
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—41no—
355C22—41no—
356C22—41no—
357C22—532no—
358C22—1061no—
359C22—1061no—
360C22—1062no—
361C22—2121no—
362C22—2121no—
363C22C2×D4×D5311yescharacteristic
364C4—22no—
365C4—21no—
366C4—21no—
367C4—41no—
368C4—41no—
369C4—532no—
370C4—1061no—
371C4—1061no—
372C4—2121no—
373C4—2121no—
374C4—2121no—
375C4C2×D4×D5311yescharacteristic
376C4#41×D5311yescharacteristic
377C2—81no—
378C2—1062no—
379C2—2122no—
380C2#121×D5311yescharacteristic
381C2#1681143011yescharacteristic
382C2#1681143911yescharacteristic
subgroupG/H
centerC22C2×D4×D53
commutatorC2×C4×C53C24
FrattiniC2×C4C23×D53
FittingC53×#464C2
socleC22×C53C22×D4
hypercenter#464D53
hypocenterC53#3046
LCSG ▷ C2×C4×C53 ▷ C2×C53 ▷ C53
UCSC1 ◁ C22 ◁ C42 ◁ #464

Complements

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

NH ≅ G/Ntype
1. C2×#13133249C2semidirect
3. #16811193C2semidirect
4. #16811105C2semidirect
5. #16810843C2semidirect
6. #16810837C2semidirect
7. #16810835C2semidirect
8. #16810833C2semidirect
9. #16810831C2semidirect
10. #16810829C2semidirect
11. #16810827C2semidirect
12. #16810648C2semidirect
13. #16810621C2semidirect
14. #103×D53C2semidirect
15. C53×#464C2semidirect
197. #464D53semidirect
207. C53#3046semidirect