Finite Group Databasebeta

← #20286708

Subgroups — #20286708

Order 6784.

420 conjugacy classes of subgroups.

The subgroup lattice has 376 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#16811282C211yesmaximal, characteristic, complemented
2#16811277C211yesmaximal, characteristic, complemented
3#16810983C211yesmaximal, characteristic, complemented
4#16810959C211yesmaximal, characteristic, complemented
5#16810937C211yesmaximal, characteristic, complemented
6#16810919C211yesmaximal, characteristic, complemented
7#16810869C211yesmaximal, characteristic, complemented
8#16810859C211yesmaximal, characteristic, complemented
9#16810839C211yesmaximal, characteristic, complemented
10#16810770C211yesmaximal, characteristic, complemented
11#16810764C211yesmaximal, characteristic, complemented
12#16810725C211yesmaximal, characteristic, complemented
13#16810625C211yesmaximal, characteristic, complemented
14C2×#13133142C211yesmaximal, characteristic, complemented
15C53×#435C211yesmaximal, characteristic, complemented
16C2×#11719945C2211yescharacteristic
17C2×#11719947—21no—
18C22×Dic106C2211yescharacteristic
19C2×#11719913C2211yescharacteristic
20C2×#11719913C2211yescharacteristic
21C2×D424C2211yescharacteristic
22#13133217C2211yescharacteristic
23#13133205C2211yescharacteristic
24#13133203—21no—
25#13133201—21no—
26#13133201—21no—
27#13133197—21no—
28#13133196C2211yescharacteristic
29#13133195—21no—
30#13133191C2211yescharacteristic
31#13133191C2211yescharacteristic
32#13133191C2211yescharacteristic
33#13133189—21no—
34C4×Dic106C2211yescharacteristic
35C4×D212C2211yescharacteristic
36#13133177—21no—
37C2×C4⋊Dic53C2211yescharacteristic
38C2×C4⋊Dic53C2211yescharacteristic
39#13133151C2211yescharacteristic
40#13133151C2211yescharacteristic
41#13133151C2211yescharacteristic
42#13133142C2211yescharacteristic
43#13133142C2211yescharacteristic
44#13133142C2211yescharacteristic
45#13133142C2211yescharacteristic
46#13133142C2211yescharacteristic
47#13133142C2211yescharacteristic
48#13133139C2211yescharacteristic
49#13133139C2211yescharacteristic
50#13133139C2211yescharacteristic
51#13133139C2211yescharacteristic
52C2×M16×C53C2211yescharacteristic
53C22×C8×C53C2211yescharacteristic
54C4×D4×C53C2211yescharacteristic
55C4⋊C8×C53C2211yescharacteristic
56C22⋊C8×C53C2211yescharacteristic
57C22⋊C8×C53C2211yescharacteristic
58#97×C53C2211yescharacteristic
59C2×D4×C53D411yescharacteristic
60C22×C4×C53C2311yescharacteristic
61C22×C4×C53C2311yescharacteristic
62C2×C22⋊D53—21no—
63C2×C22⋊D53—21no—
64C2×C22⋊D53—21no—
65C2×C22⋊D53—21no—
66C22×Dic53—21no—
67C22×Dic53—21no—
68C22×Dic53—21no—
69#11719947—44no—
70#11719945—22no—
71#11719945—41no—
72C2×D212C2311yescharacteristic
73C2×D212C2311yescharacteristic
74C2×C4×D53—21no—
75C2×C4×D53—21no—
76C2×Dic106—22no—
77C2×Dic106—21no—
78C2×Dic106—21no—
79C2×Dic106—41no—
80C2×Dic106C2311yescharacteristic
81C2×Dic106C2311yescharacteristic
82M16×C53—41no—
83C2×C8×C53—41no—
84C2×C8×C53C2311yescharacteristic
85C2×C8×C53C2311yescharacteristic
86C2×C8×C53C2311yescharacteristic
87C2×C8×C53C2311yescharacteristic
88C4⋊C4×C53D411yescharacteristic
89C22⋊C4×C53D411yescharacteristic
90C22⋊C4×C53D411yescharacteristic
91C42×C53C2311yescharacteristic
92C22⋊Dic53—21no—
93#11719921—21no—
94#11719921—21no—
95#11719921—21no—
96#11719921—21no—
97#11719921—21no—
98#11719921—21no—
99C4⋊Dic53—21no—
100C4⋊Dic53—21no—
101C4⋊Dic53—21no—
102C4⋊Dic53C2311yescharacteristic
103C4⋊Dic53C2311yescharacteristic
104C4⋊Dic53C2311yescharacteristic
105C4⋊Dic53C2311yescharacteristic
106#11719919—21no—
107#11719919—21no—
108C4×Dic53—21no—
109D424—41no—
110#11719913—22no—
111#11719913—41no—
112C23×C53C2×D411yescharacteristic
113C23×C53C2×D411yescharacteristic
114C22×D53—21no—
115C22×D53—21no—
116D4×C53—22no—
117C2×C4×C53—21no—
118C2×C4×C53—21no—
119C2×C4×C53—21no—
120C2×C4×C53—21no—
121C2×C4×C53C2×D411yescharacteristic
122C2×C4×C53C2×D411yescharacteristic
123C2×C4×C53C2×D411yescharacteristic
124C2×C4×C53C2×D411yescharacteristic
125C2×C4×C53C2411yescharacteristic
126C22⋊D53—42no—
127C22⋊D53—42no—
128C22⋊D53—42no—
129C22⋊D53—42no—
130C2×Dic53—21no—
131C2×Dic53—21no—
132C2×Dic53—21no—
133C2×Dic53—21no—
134C2×Dic53—21no—
135C2×Dic53—21no—
136C2×Dic53—42no—
137C2×Dic53—42no—
138C2×Dic53—42no—
139D212—21no—
140D212—21no—
141D212—21no—
142D212—21no—
143C4×D53—42no—
144C4×D53—42no—
145Dic106—21no—
146Dic106—21no—
147Dic106—21no—
148Dic106—42no—
149Dic106—42no—
150Dic106#4112yescomplemented
151C8×C53—21no—
152C8×C53—21no—
153C8×C53—21no—
154C8×C53—21no—
155C22×C53—21no—
156C22×C53—21no—
157C22×C53—21no—
158C22×C53—21no—
159C22×C53—21no—
160C22×C53AGL(1,ℤ/8)12yes—
161C22×C53C22×D411yescharacteristic
162C2×D53—41no—
163C2×D53—41no—
164C2×D53—41no—
165C2×D53—41no—
166C2×D53—41no—
167C2×D53—41no—
168C4×C53—21no—
169C4×C53—21no—
170C4×C53—21no—
171C4×C53—21no—
172C4×C53C2×#4111yescharacteristic
173C4×C53H(5,𝔽2)11yescharacteristic
174Dic53—22no—
175Dic53—41no—
176Dic53—41no—
177Dic53—41no—
178Dic53—41no—
179Dic53—41no—
180#3308—531nomaximal, Sylow
181C2×C53—22no—
182C2×C53—41no—
183C2×C53#54611yescharacteristic
184C2×C53C2×AGL(1,ℤ/8)11yescharacteristic
185C2×C53#57811yescharacteristic
186D53—81no—
187D53—81no—
188#548—531no—
189#547—531no—
190#489—531no—
191#484—531no—
192#480—531no—
193#477—531no—
194#468—531no—
195#466—531no—
196#463—531no—
197#449—531no—
198#448—531no—
199#440—531no—
200#435D5311yescharacteristic, complemented
201#417—531no—
202C2×#103—531no—
203C53#330811yescharacteristic, complemented, Sylow
204C2×#41—531no—
205C2×#41—1061no—
206C22×Q8—531no—
207C2×SD16—531no—
208C2×SD16—531no—
209C2×D8—531no—
210C2×M16C2×D5311yescharacteristic
211C22×C8C2×D5311yescharacteristic
212C4⋊Q8—531no—
213#124—531no—
214#124—1061no—
215#123—1061no—
216#123—1061no—
217C22⋊Q8—531no—
218C22⋊Q8—1061no—
219C22⋊Q8—1061no—
220#121—531no—
221#121—531no—
222#121—531no—
223#121—1061no—
224C4×Q8—531no—
225C4×D4—531no—
226C4×D4—1061no—
227C4×D4C2×D5311yescharacteristic
228C2×C4⋊C4—531no—
229C2×C4⋊C4—531no—
230#107—531no—
231#107—531no—
232#107—531no—
233C4⋊C8C2×D5311yescharacteristic
234#103—531no—
235#103—531no—
236#103—531no—
237#103—531no—
238#103—531no—
239#103—531no—
240#102—531no—
241#102—531no—
242#102—531no—
243#102—531no—
244C22⋊C8C2×D5311yescharacteristic
245C22⋊C8C2×D5311yescharacteristic
246#97C2×D5311yescharacteristic
247#41—1062no—
248#41—2121no—
249#41—2124no—
250C2×Q8—531no—
251C2×Q8—531no—
252C2×Q8—1062no—
253C2×Q8—1061no—
254C2×Q8—1061no—
255C2×Q8—2121no—
256C2×D4—531no—
257C2×D4—531no—
258C2×D4—1061no—
259C2×D4—1061no—
260C2×D4—1061no—
261C2×D4—1061no—
262C2×D4D21211yescharacteristic
263C22×C4—1061no—
264C22×C4—1061no—
265C22×C4—1061no—
266C22×C4—1061no—
267C22×C4—1061no—
268C22×C4C22×D5311yescharacteristic
269C22×C4C22×D5311yescharacteristic
270SD16—1062no—
271SD16—2121no—
272D8—2121no—
273M16—41no—
274C2×C8—41no—
275C2×C8C22×D5311yescharacteristic
276C2×C8C22×D5311yescharacteristic
277C2×C8C22×D5311yescharacteristic
278C2×C8C22×D5311yescharacteristic
279C4⋊C4—531no—
280C4⋊C4—531no—
281C4⋊C4—531no—
282C4⋊C4—531no—
283C4⋊C4—1061no—
284C4⋊C4—1061no—
285C4⋊C4—1061no—
286C4⋊C4—1061no—
287C4⋊C4—1061no—
288C4⋊C4D21211yescharacteristic
289C22⋊C4—1061no—
290C22⋊C4—1061no—
291C22⋊C4—1061no—
292C22⋊C4—1061no—
293C22⋊C4—1061no—
294C22⋊C4—1061no—
295C22⋊C4—1061no—
296C22⋊C4D21211yescharacteristic
297C22⋊C4D21211yescharacteristic
298C42—1061no—
299C42C22×D5311yescharacteristic
300C23—1061no—
301C23—1061no—
302C23C2×D21211yescharacteristic
303C23C2×D21211yescharacteristic
304Q8—532no—
305Q8—1061no—
306Q8—1061no—
307Q8—1061no—
308Q8—2122no—
309Q8—2122no—
310D4—22no—
311D4—1061no—
312D4—1061no—
313D4—1061no—
314D4—1061no—
315D4—2122no—
316D4—2122no—
317D4—2122no—
318D4—2122no—
319C2×C4—21no—
320C2×C4—21no—
321C2×C4—21no—
322C2×C4—21no—
323C2×C4—1061no—
324C2×C4—1061no—
325C2×C4—1061no—
326C2×C4—1061no—
327C2×C4—1061no—
328C2×C4—1061no—
329C2×C4—2122no—
330C2×C4—2122no—
331C2×C4—2122no—
332C2×C4—2122no—
333C2×C4—2122no—
334C2×C4C2×D21211yescharacteristic
335C2×C4C2×D21211yescharacteristic
336C2×C4C2×D21211yescharacteristic
337C2×C4C2×D21211yescharacteristic
338C2×C4C23×D5311yescharacteristic
339C8—21no—
340C8—21no—
341C8—21no—
342C8—21no—
343C22—21no—
344C22—21no—
345C22—21no—
346C22—21no—
347C22—21no—
348C22—2121no—
349C22—2121no—
350C22—2121no—
351C22—2121no—
352C22—2121no—
353C22—2121no—
354C22#1313324512yes—
355C22C22×D21211yescharacteristic
356C4—21no—
357C4—21no—
358C4—21no—
359C4—21no—
360C4—1062no—
361C4—2121no—
362C4—2121no—
363C4—2121no—
364C4—2121no—
365C4—2121no—
366C4#41×D5311yescharacteristic
367C4#1313326211yescharacteristic
368C2—22no—
369C2—41no—
370C2—4241no—
371C2—4241no—
372C2#1681126811yescharacteristic
373C2C2×#1313324511yescharacteristic
374C2#1681144011yescharacteristic
subgroupG/H
centerC22C22×D212
commutatorC2×C4×C53C24
FrattiniC2×C4C23×D53
FittingC53×#435C2
socleC22×C53C22×D4
hypercenter#435D53
hypocenterC53#3308
LCSG ▷ C2×C4×C53 ▷ C2×C53 ▷ C53
UCSC1 ◁ C22 ◁ C4×D4 ◁ #435

Complements

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

NH ≅ G/Ntype
1. #16811282C2semidirect
2. #16811277C2semidirect
3. #16810983C2semidirect
4. #16810959C2semidirect
5. #16810937C2semidirect
6. #16810919C2semidirect
7. #16810869C2semidirect
8. #16810859C2semidirect
9. #16810839C2semidirect
10. #16810770C2semidirect
11. #16810764C2semidirect
12. #16810725C2semidirect
13. #16810625C2semidirect
14. C2×#13133142C2semidirect
15. C53×#435C2semidirect
150. Dic106#41semidirect
200. #435D53semidirect
203. C53#3308semidirect