Finite Group Databasebeta

← #20284243

Subgroups — #20284243

Order 6784.

458 conjugacy classes of subgroups.

The subgroup lattice has 372 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#13133246C211yesmaximal, characteristic
2C2×Q16×D53C211yesmaximal, characteristic, complemented
3#16811193C211yesmaximal, characteristic
4#16811099C211yesmaximal, characteristic, complemented
5#16810844C211yesmaximal, characteristic, complemented
6#16810843C211yesmaximal, characteristic, complemented
7#16810842C211yesmaximal, characteristic, complemented
8#16810834C211yesmaximal, characteristic, complemented
9#16810833C211yesmaximal, characteristic, complemented
10#16810832C211yesmaximal, characteristic, complemented
11#16810831C211yesmaximal, characteristic, complemented
12#16810648C211yesmaximal, characteristic, complemented
13#16810630C211yesmaximal, characteristic, complemented
14#16810616C211yesmaximal, characteristic
15C53×#464C211yesmaximal, characteristic, complemented
16C2×#11719947C2211yescharacteristic
17C2×Q8×D53C2211yescharacteristic
18C2×Q8×D53C2211yescharacteristic
19#13133246—24no—
20C2×Dic212C2211yescharacteristic
21C2×Dic212C2211yescharacteristic
22C2×#11719925C2211yescharacteristic
23C2×#11719925C2211yescharacteristic
24C2×#11719925C2211yescharacteristic
25Q16×D53—24no—
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—
36Q8×Dic53C2211yescharacteristic
37C4×Dic106—21no—
38C4×Dic106C2211yescharacteristic
39#13133182—21no—
40#13133182C2211yescharacteristic
41#13133175—21no—
42#13133175C2211yescharacteristic
43#13133174—21no—
44#13133174C2211yescharacteristic
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—
66Q8×D53—22no—
67Q8×D53—22no—
68Q8×D53—22no—
69Q8×D53D414yes—
70#11719947—22no—
71#11719947—22no—
72C2×C4×D53—21no—
73C2×C4×D53—21no—
74C2×C4×D53C2311yescharacteristic
75C2×Dic106—21no—
76C2×Dic106—21no—
77C2×Dic106C2311yescharacteristic
78C2×Dic106C2311yescharacteristic
79C2×Dic106C2311yescharacteristic
80Q16×C53—22no—
81SD16×C53—22no—
82C2×C8×C53C2311yescharacteristic
83C2×C8×C53C2311yescharacteristic
84C4⋊C4×C53—21no—
85C4⋊C4×C53C2311yescharacteristic
86C22⋊C4×C53—21no—
87C22⋊C4×C53—21no—
88C42×C53—21no—
89C42×C53C2311yescharacteristic
90C22⋊Dic53—21no—
91C22⋊Dic53—21no—
92#11719925—22no—
93#11719925—22no—
94#11719925—22no—
95#11719923—22no—
96#11719921—21no—
97#11719921D411yescharacteristic
98#11719921D411yescharacteristic
99C4⋊Dic53—21no—
100C4⋊Dic53—21no—
101C4⋊Dic53—21no—
102C4⋊Dic53C2311yescharacteristic
103#11719919—21no—
104#11719919—21no—
105#11719919—21no—
106#11719919—21no—
107#11719919D411yescharacteristic
108#11719919D411yescharacteristic
109C4×Dic53—21no—
110C4×Dic53—21no—
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—
121C22×D53C2×D411yescharacteristic
122Q8×C53—21no—
123Q8×C53—21no—
124Q8×C53—21no—
125Q8×C53C2×D412yes—
126D4×C53—21no—
127D4×C53—21no—
128C2×C4×C53—21no—
129C2×C4×C53—21no—
130C2×C4×C53—21no—
131C2×C4×C53C2×D411yescharacteristic
132C2×C4×C53C2×D411yescharacteristic
133C2×C4×C53C2411yescharacteristic
134C22⋊D53—42no—
135C2×Dic53—21no—
136C2×Dic53—21no—
137C2×Dic53—21no—
138C2×Dic53—21no—
139C2×Dic53—42no—
140C2×Dic53C2×D411yescharacteristic
141C2×Dic53C2×D411yescharacteristic
142C2×Dic53C2×D411yescharacteristic
143C4×D53—24no—
144C4×D53—42no—
145C4×D53C2×D412yes—
146C4×D53#4112yes—
147Dic106—21no—
148Dic106—24no—
149Dic106—21no—
150Dic106—21no—
151Dic106—21no—
152Dic106—21no—
153Dic106—42no—
154Dic106C2×D412yes—
155C8×C53—21no—
156C8×C53—21no—
157#88322—21no—
158#88322—21no—
159C22×C53—41no—
160C22×C53—41no—
161C22×C53—41no—
162C22×C53C22×D411yescharacteristic
163C2×D53—21no—
164C2×D53—21no—
165C2×D53#12112yes—
166C4×C53—22no—
167C4×C53—21no—
168C4×C53—21no—
169C4×C53—41no—
170C4×C53—41no—
171C4×C53C22×D411yescharacteristic
172C4×C53C2×#4111yescharacteristic
173Dic53—22no—
174Dic53—21no—
175Dic53—21no—
176Dic53—41no—
177Dic53—41no—
178Dic53—41no—
179Dic53#12112yes—
180#3044—531nomaximal, Sylow
181C2×C53—81no—
182C2×C53C2×#12111yescharacteristic
183C2×C53#57811yescharacteristic
184C2×C53#57811yescharacteristic
185D53—22no—
186C2×#137—531no—
187C22×Q16—531no—
188#536—531no—
189#519—531no—
190#464—531no—
191#464—531no—
192#464—531no—
193#464D5311yescharacteristic, complemented
194#462—531no—
195#462—531no—
196#462—531no—
197#462—531no—
198#424—531no—
199#419—531no—
200#416—531no—
201C53#304411yescharacteristic, complemented, Sylow
202C2×#41—531no—
203C22×Q8—531no—
204C22×Q8—531no—
205#137—1064no—
206C2×Q16—531no—
207C2×Q16—531no—
208C2×Q16—531no—
209C2×Q16—531no—
210C2×Q16—531no—
211C2×Q16—1064no—
212C2×Q16C2×D5311yescharacteristic
213C2×SD16—531no—
214C2×SD16C2×D5311yescharacteristic
215C2×M16—531no—
216C22×C8—531no—
217C4⋊Q8—531no—
218C4⋊Q8—531no—
219#124—531no—
220#124C2×D5311yescharacteristic
221#123—1061no—
222#123—1061no—
223C22⋊Q8—1061no—
224C22⋊Q8—1061no—
225C4×Q8—531no—
226C4×Q8—531no—
227C4×Q8—531no—
228C4×Q8—1061no—
229C4×Q8—1061no—
230C4×Q8C2×D5311yescharacteristic
231#117—531no—
232#117—531no—
233#117—1061no—
234#117—1061no—
235C4⋊C8—531no—
236C4⋊C8—531no—
237C4⋊C8—531no—
238C4⋊C8C2×D5311yescharacteristic
239#103—531no—
240#103—531no—
241#103—531no—
242#103—531no—
243#103—531no—
244#103C2×D5311yescharacteristic
245#102—531no—
246#102C2×D5311yescharacteristic
247#41—1062no—
248#41—1062no—
249C2×Q8—534no—
250C2×Q8—531no—
251C2×Q8—531no—
252C2×Q8—531no—
253C2×Q8—1061no—
254C2×Q8—1061no—
255C2×Q8—1062no—
256C2×Q8—1062no—
257C2×Q8—1062no—
258C2×Q8C22×D5311yescharacteristic
259C2×Q8C22×D5311yescharacteristic
260C2×D4—1061no—
261C2×D4C22×D5311yescharacteristic
262C22×C4—531no—
263C22×C4—1061no—
264C22×C4—1061no—
265C22×C4—1061no—
266Q16—22no—
267Q16—1062no—
268Q16—1062no—
269Q16—1062no—
270Q16—1062no—
271Q16—1062no—
272SD16—22no—
273SD16—1062no—
274M16—1062no—
275C2×C8—531no—
276C2×C8—531no—
277C2×C8—1062no—
278C2×C8C22×D5311yescharacteristic
279C2×C8C22×D5311yescharacteristic
280C4⋊C4—21no—
281C4⋊C4—531no—
282C4⋊C4—531no—
283C4⋊C4—531no—
284C4⋊C4—1061no—
285C4⋊C4—1061no—
286C4⋊C4—1061no—
287C4⋊C4—1061no—
288C4⋊C4—1061no—
289C4⋊C4—1061no—
290C4⋊C4—1061no—
291C4⋊C4C22×D5311yescharacteristic
292C22⋊C4—21no—
293C22⋊C4—21no—
294C22⋊C4—531no—
295C22⋊C4—531no—
296C22⋊C4—1061no—
297C22⋊C4—1061no—
298C22⋊C4—1061no—
299C42—21no—
300C42—531no—
301C42—1061no—
302C42—1061no—
303C42—1061no—
304C42—1061no—
305C42C22×D5311yescharacteristic
306C23—21no—
307C23—531no—
308Q8—21no—
309Q8—21no—
310Q8—21no—
311Q8—532no—
312Q8—1064no—
313Q8—1061no—
314Q8—1061no—
315Q8—1061no—
316Q8—1061no—
317Q8—1061no—
318Q8—2122no—
319Q8D4×D5312yes—
320D4—21no—
321D4—21no—
322D4—2122no—
323C2×C4—21no—
324C2×C4—21no—
325C2×C4—21no—
326C2×C4—531no—
327C2×C4—532no—
328C2×C4—532no—
329C2×C4—531no—
330C2×C4—531no—
331C2×C4—1064no—
332C2×C4—1061no—
333C2×C4—1061no—
334C2×C4—1061no—
335C2×C4—1061no—
336C2×C4—2122no—
337C2×C4—2122no—
338C2×C4D4×D5311yescharacteristic
339C2×C4D4×D5311yescharacteristic
340C2×C4C23×D5311yescharacteristic
341C8—21no—
342C8—21no—
343C8—1061no—
344C8—1061no—
345C22—41no—
346C22—41no—
347C22—41no—
348C22—532no—
349C22—1061no—
350C22—1061no—
351C22C2×D4×D5311yescharacteristic
352C4—22no—
353C4—21no—
354C4—21no—
355C4—41no—
356C4—41no—
357C4—532no—
358C4—1062no—
359C4—1061no—
360C4—1061no—
361C4—2121no—
362C4—2121no—
363C4—2121no—
364C4C2×D4×D5311yescharacteristic
365C4#41×D5311yescharacteristic
366C2—81no—
367C2—1062no—
368C2#121×D5311yescharacteristic
369C2#1681143711yescharacteristic
370C2#1681143911yescharacteristic
subgroupG/H
centerC22C2×D4×D53
commutatorC2×C4×C53C24
FrattiniC2×C4C23×D53
FittingC53×#464C2
socleC22×C53C22×D4
hypercenter#464D53
hypocenterC53#3044
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
2. C2×Q16×D53C2semidirect
4. #16811099C2semidirect
5. #16810844C2semidirect
6. #16810843C2semidirect
7. #16810842C2semidirect
8. #16810834C2semidirect
9. #16810833C2semidirect
10. #16810832C2semidirect
11. #16810831C2semidirect
12. #16810648C2semidirect
13. #16810630C2semidirect
15. C53×#464C2semidirect
193. #464D53semidirect
201. C53#3044semidirect