Finite Group Databasebeta

← #20287002

Subgroups — #20287002

Order 6784.

386 conjugacy classes of subgroups.

The subgroup lattice has 362 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#16811337C211yesmaximal, characteristic
2Q8×C22⋊D53C211yesmaximal, characteristic, complemented
3#16810972C211yesmaximal, characteristic, complemented
4#16810965C211yesmaximal, characteristic, complemented
5#16810962C211yesmaximal, characteristic, complemented
6#16810916C211yesmaximal, characteristic, complemented
7#16810867C211yesmaximal, characteristic, complemented
8#16810866C211yesmaximal, characteristic, complemented
9#16810826C211yesmaximal, characteristic, complemented
10#16810760C211yesmaximal, characteristic, complemented
11SD16×Dic53C211yesmaximal, characteristic, complemented
12#16810698C211yesmaximal, characteristic, complemented
13#16810632C211yesmaximal, characteristic, complemented
14#16810630C211yesmaximal, characteristic, complemented
15C53×#448C211yesmaximal, characteristic, complemented
16C2×Q8×D53C2211yescharacteristic
17C2×#11719923C2211yescharacteristic
18C2×#11719917C2211yescharacteristic
19C2×#11719912C2211yescharacteristic
20#13133216C2211yescharacteristic
21#13133215—21no—
22#13133215C2211yescharacteristic
23#13133204—21no—
24#13133203—21no—
25#13133196—21no—
26#13133196C2211yescharacteristic
27#13133193—21no—
28#13133193C2211yescharacteristic
29#13133189—21no—
30#13133188C2211yescharacteristic
31Q8×Dic53C2211yescharacteristic
32D4×Dic53C2211yescharacteristic
33C4×C22⋊D53—21no—
34C4×C22⋊D53C2211yescharacteristic
35#13133176C2211yescharacteristic
36#13133175C2211yescharacteristic
37#13133173—21no—
38#13133151C2211yescharacteristic
39#13133150C2211yescharacteristic
40#13133149C2211yescharacteristic
41#13133146C2211yescharacteristic
42#13133142C2211yescharacteristic
43#13133142C2211yescharacteristic
44#13133141C2211yescharacteristic
45#13133141C2211yescharacteristic
46#13133141C2211yescharacteristic
47#13133141C2211yescharacteristic
48#13133140C2211yescharacteristic
49#13133138C2211yescharacteristic
50#13133129C2211yescharacteristic
51C8×Dic53C2211yescharacteristic
52C22×Q8×C53C2211yescharacteristic
53C2×SD16×C53C2211yescharacteristic
54C2×SD16×C53C2211yescharacteristic
55#121×C53C2211yescharacteristic
56#103×C53C2211yescharacteristic
57#103×C53C2211yescharacteristic
58C22⋊C8×C53C2211yescharacteristic
59C2×Q8×C53—21no—
60C2×Q8×C53—22no—
61C2×Q8×C53—41no—
62C2×Q8×C53C2311yescharacteristic
63C2×Q8×C53C2311yescharacteristic
64C2×D4×C53—21no—
65C2×D4×C53C2311yescharacteristic
66C22×C4×C53—21no—
67C22×C4×C53C2311yescharacteristic
68C2×C22⋊D53—21no—
69C2×C22⋊D53D411yescharacteristic
70C22×Dic53—21no—
71Q8×D53—22no—
72Q8×D53—41no—
73C2×C4×D53—21no—
74C2×C4×D53C2311yescharacteristic
75C2×Dic106—21no—
76C2×Dic106—21no—
77C2×Dic106C2311yescharacteristic
78SD16×C53—22no—
79SD16×C53—41no—
80C2×C8×C53C2311yescharacteristic
81C2×C8×C53C2311yescharacteristic
82C4⋊C4×C53C2311yescharacteristic
83C22⋊C4×C53—21no—
84C22⋊Dic53—21no—
85C22⋊Dic53—21no—
86C22⋊Dic53—21no—
87C22⋊Dic53—21no—
88C22⋊Dic53D411yescharacteristic
89#11719923—41no—
90#11719921—21no—
91#11719921—21no—
92#11719921D411yescharacteristic
93C4⋊Dic53—21no—
94C4⋊Dic53C2311yescharacteristic
95C4⋊Dic53C2311yescharacteristic
96C4⋊Dic53C2311yescharacteristic
97#11719919—21no—
98#11719919—21no—
99#11719919—21no—
100#11719919—21no—
101#11719919—21no—
102#11719919D411yescharacteristic
103C4×Dic53—21no—
104C4×Dic53—21no—
105C4×Dic53—21no—
106C4×Dic53C2311yescharacteristic
107#11719917—41no—
108C2×#88322C2311yescharacteristic
109C2×#88322C2311yescharacteristic
110#11719912—41no—
111C23×C53—21no—
112C23×C53C2×D411yescharacteristic
113C22×D53C2×D411yescharacteristic
114Q8×C53—21no—
115Q8×C53—21no—
116Q8×C53—21no—
117Q8×C53—42no—
118Q8×C53#4112yes—
119D4×C53—21no—
120D4×C53—21no—
121D4×C53—81no—
122C2×C4×C53—21no—
123C2×C4×C53—21no—
124C2×C4×C53—21no—
125C2×C4×C53—21no—
126C2×C4×C53—21no—
127C2×C4×C53—42no—
128C2×C4×C53C2×D411yescharacteristic
129C2×C4×C53C2411yescharacteristic
130C22⋊D53—22no—
131C22⋊D53—81no—
132C2×Dic53—21no—
133C2×Dic53—21no—
134C2×Dic53—21no—
135C2×Dic53—21no—
136C2×Dic53—42no—
137C2×Dic53C2×D411yescharacteristic
138C2×Dic53C2×D411yescharacteristic
139C2×Dic53C2×D411yescharacteristic
140C4×D53—21no—
141C4×D53—21no—
142C4×D53—42no—
143Dic106—21no—
144Dic106—21no—
145Dic106—42no—
146Dic106—81no—
147C8×C53—21no—
148C8×C53—21no—
149#88322—21no—
150#88322—21no—
151C22×C53—21no—
152C22×C53—21no—
153C22×C53—21no—
154C22×C53—41no—
155C22×C53—41no—
156C22×C53—41no—
157C22×C53C22×D411yescharacteristic
158C2×D53—21no—
159C2×D53—21no—
160C2×D53—21no—
161C4×C53—22no—
162C4×C53—21no—
163C4×C53—41no—
164C4×C53—41no—
165C4×C53C2×#4111yescharacteristic
166C4×C53H(5,𝔽2)11yescharacteristic
167Dic53—21no—
168Dic53—21no—
169Dic53—41no—
170Dic53—41no—
171Dic53—41no—
172Dic53—41no—
173Dic53#13712yes—
174#3329—531nomaximal, Sylow
175C2×C53—41no—
176C2×C53—81no—
177C2×C53#54611yescharacteristic
178C2×C53C2×#13711yescharacteristic
179C2×C53#57711yescharacteristic
180D53—41no—
181#555—531no—
182D4×Q8—531no—
183#487—531no—
184#485—531no—
185#485—531no—
186#477—531no—
187#468—531no—
188#468—531no—
189#461—531no—
190#448—531no—
191#448D5311yescharacteristic, complemented
192C4×SD16—531no—
193#436—531no—
194#419—531no—
195#419—531no—
196C53#332911yescharacteristic, complemented, Sylow
197C22×Q8—531no—
198C22×Q8C2×D5311yescharacteristic
199C2×SD16—531no—
200C2×SD16C2×D5311yescharacteristic
201C2×SD16C2×D5311yescharacteristic
202C2×M16—531no—
203C2×M16—531no—
204C4⋊Q8—531no—
205C4⋊Q8—531no—
206C4⋊Q8—1061no—
207#124—1061no—
208#124—1061no—
209C22⋊Q8—531no—
210C22⋊Q8—531no—
211C22⋊Q8—1061no—
212C22⋊Q8—1061no—
213#121—531no—
214#121—1061no—
215#121C2×D5311yescharacteristic
216C4×Q8—531no—
217C4×D4—531no—
218C4×D4—531no—
219C4×D4—1061no—
220#117—531no—
221#117—531no—
222#117—1061no—
223#107—531no—
224#107—531no—
225#106—531no—
226C4⋊C8—531no—
227#103—531no—
228#103—531no—
229#103—531no—
230#103—531no—
231#103—531no—
232#103—531no—
233#103—531no—
234#103C2×D5311yescharacteristic
235#103C2×D5311yescharacteristic
236#102—531no—
237C22⋊C8—531no—
238C22⋊C8C2×D5311yescharacteristic
239C4×C8—531no—
240C2×Q8—21no—
241C2×Q8—22no—
242C2×Q8—41no—
243C2×Q8—531no—
244C2×Q8—1061no—
245C2×Q8—1061no—
246C2×Q8—1062no—
247C2×Q8—2121no—
248C2×Q8C22×D5311yescharacteristic
249C2×Q8C22×D5311yescharacteristic
250C2×D4—21no—
251C2×D4—531no—
252C2×D4—1061no—
253C2×D4C22×D5311yescharacteristic
254C22×C4—21no—
255C22×C4—531no—
256C22×C4—1061no—
257C22×C4—1061no—
258C22×C4C22×D5311yescharacteristic
259SD16—22no—
260SD16—41no—
261SD16—2121no—
262M16—2121no—
263M16—2121no—
264C2×C8—531no—
265C2×C8—531no—
266C2×C8C22×D5311yescharacteristic
267C2×C8C22×D5311yescharacteristic
268C4⋊C4—531no—
269C4⋊C4—531no—
270C4⋊C4—531no—
271C4⋊C4—531no—
272C4⋊C4—1061no—
273C4⋊C4—1061no—
274C4⋊C4—1061no—
275C4⋊C4—1061no—
276C4⋊C4—1061no—
277C4⋊C4—1061no—
278C4⋊C4C22×D5311yescharacteristic
279C22⋊C4—21no—
280C22⋊C4—531no—
281C22⋊C4—531no—
282C22⋊C4—1061no—
283C22⋊C4—1061no—
284C22⋊C4—1061no—
285C22⋊C4—1061no—
286C22⋊C4—1061no—
287C22⋊C4—1061no—
288C42—531no—
289C42—1061no—
290C42—1061no—
291C42—1061no—
292C23—21no—
293C23—531no—
294C23D4×D5311yescharacteristic
295Q8—21no—
296Q8—21no—
297Q8—21no—
298Q8—42no—
299Q8—1061no—
300Q8—1061no—
301Q8—4241no—
302Q8—2122no—
303Q8#1171994712yes—
304D4—21no—
305D4—21no—
306D4—81no—
307D4—1062no—
308D4—4241no—
309C2×C4—21no—
310C2×C4—21no—
311C2×C4—21no—
312C2×C4—21no—
313C2×C4—21no—
314C2×C4—42no—
315C2×C4—531no—
316C2×C4—531no—
317C2×C4—531no—
318C2×C4—1061no—
319C2×C4—1061no—
320C2×C4—1061no—
321C2×C4—1061no—
322C2×C4—1061no—
323C2×C4—1061no—
324C2×C4—2122no—
325C2×C4—2122no—
326C2×C4D4×D5311yescharacteristic
327C2×C4C23×D5311yescharacteristic
328C8—21no—
329C8—21no—
330C8—1061no—
331C8—1061no—
332C22—21no—
333C22—21no—
334C22—21no—
335C22—41no—
336C22—41no—
337C22—41no—
338C22—1061no—
339C22—1061no—
340C22—1061no—
341C22C2×D4×D5311yescharacteristic
342C4—21no—
343C4—22no—
344C4—41no—
345C4—41no—
346C4—532no—
347C4—1061no—
348C4—1061no—
349C4—2121no—
350C4—2121no—
351C4—2121no—
352C4—2121no—
353C4C2×#1171994711yescharacteristic
354C4#1313326211yescharacteristic
355C2—41no—
356C2—81no—
357C2—2121no—
358C2#1681126011yescharacteristic
359C2#137×D5311yescharacteristic
360C2#1681142911yescharacteristic
subgroupG/H
centerC22C2×D4×D53
commutatorC2×C4×C53C24
FrattiniC2×C4C23×D53
FittingC53×#448C2
socleC22×C53C22×D4
hypercenter#448D53
hypocenterC53#3329
LCSG ▷ C2×C4×C53 ▷ C2×C53 ▷ C53
UCSC1 ◁ C22 ◁ C22×C4 ◁ #448

Complements

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

NH ≅ G/Ntype
2. Q8×C22⋊D53C2semidirect
3. #16810972C2semidirect
4. #16810965C2semidirect
5. #16810962C2semidirect
6. #16810916C2semidirect
7. #16810867C2semidirect
8. #16810866C2semidirect
9. #16810826C2semidirect
10. #16810760C2semidirect
11. SD16×Dic53C2semidirect
12. #16810698C2semidirect
13. #16810632C2semidirect
14. #16810630C2semidirect
15. C53×#448C2semidirect
191. #448D53semidirect
196. C53#3329semidirect