Finite Group Databasebeta

← #20286901

Subgroups — #20286901

Order 6784.

392 conjugacy classes of subgroups.

The subgroup lattice has 370 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#16811338C211yesmaximal, characteristic, complemented
2#16810982C211yesmaximal, characteristic, complemented
3#16810961C211yesmaximal, characteristic, complemented
4#16810944C211yesmaximal, characteristic, complemented
5#16810908C211yesmaximal, characteristic, complemented
6#16810874C211yesmaximal, characteristic, complemented
7#16810854C211yesmaximal, characteristic, complemented
8#16810835C211yesmaximal, characteristic, complemented
9#16810768C211yesmaximal, characteristic, complemented
10#16810762C211yesmaximal, characteristic, complemented
11#16810721C211yesmaximal, characteristic, complemented
12#16810695C211yesmaximal, characteristic, complemented
13#16810632C211yesmaximal, characteristic, complemented
14#16810619C211yesmaximal, characteristic, complemented
15C53×#548C211yesmaximal, characteristic, complemented
16C2×#11719924C2211yescharacteristic
17C2×#11719924C2211yescharacteristic
18C2×#11719922C2211yescharacteristic
19C2×#11719917C2211yescharacteristic
20C22×#88322C2211yescharacteristic
21#13133217C2211yescharacteristic
22#13133203—21no—
23#13133203—21no—
24#13133191C2211yescharacteristic
25#13133191C2211yescharacteristic
26#13133188—21no—
27C4×D212C2211yescharacteristic
28#13133176C2211yescharacteristic
29#13133176C2211yescharacteristic
30#13133175—21no—
31#13133150C2211yescharacteristic
32#13133148C2211yescharacteristic
33#13133148C2211yescharacteristic
34C4⋊#88322C2211yescharacteristic
35#13133141C2211yescharacteristic
36#13133141C2211yescharacteristic
37#13133141C2211yescharacteristic
38#13133141C2211yescharacteristic
39#13133141C2211yescharacteristic
40#13133141C2211yescharacteristic
41#13133138C2211yescharacteristic
42#13133138C2211yescharacteristic
43#13133138C2211yescharacteristic
44#13133137C2211yescharacteristic
45C22⋊#88322C2211yescharacteristic
46C22⋊#88322C2211yescharacteristic
47#13133128C2211yescharacteristic
48C2×#41×C53C2211yescharacteristic
49C22×Q8×C53C2211yescharacteristic
50#124×C53—21no—
51#124×C53C2211yescharacteristic
52C22⋊Q8×C53—21no—
53C22⋊Q8×C53C2211yescharacteristic
54#121×C53—21no—
55#121×C53C2211yescharacteristic
56C4×Q8×C53C2211yescharacteristic
57C4×D4×C53—21no—
58C4×D4×C53C2211yescharacteristic
59#41×C53—22no—
60#41×C53—41no—
61C2×Q8×C53—21no—
62C2×Q8×C53—22no—
63C2×Q8×C53—41no—
64C2×Q8×C53C2311yescharacteristic
65C2×Q8×C53C2311yescharacteristic
66C2×D4×C53—21no—
67C2×D4×C53—21no—
68C2×D4×C53D411yescharacteristic
69C2×D4×C53C2311yescharacteristic
70C22×C4×C53—21no—
71C22×C4×C53—21no—
72C22×C4×C53C2311yescharacteristic
73C22×C4×C53C2311yescharacteristic
74C2×C22⋊D53—21no—
75C2×C22⋊D53—21no—
76C2×D212C2311yescharacteristic
77C2×C4×D53—21no—
78C2×Dic106—21no—
79C4⋊C4×C53—21no—
80C4⋊C4×C53—21no—
81C4⋊C4×C53D411yescharacteristic
82C4⋊C4×C53C2311yescharacteristic
83C22⋊C4×C53—21no—
84C22⋊C4×C53—21no—
85C22⋊C4×C53—21no—
86C22⋊C4×C53—21no—
87C22⋊C4×C53D411yescharacteristic
88C22⋊C4×C53D411yescharacteristic
89C42×C53—21no—
90C42×C53C2311yescharacteristic
91C22⋊Dic53—21no—
92C22⋊Dic53—21no—
93#11719924—22no—
94#11719924—41no—
95#11719922—41no—
96#11719921—21no—
97#11719921—21no—
98#11719921—21no—
99C4⋊Dic53—21no—
100C4⋊Dic53C2311yescharacteristic
101C4⋊Dic53C2311yescharacteristic
102C4⋊Dic53C2311yescharacteristic
103C4×Dic53—21no—
104C4×Dic53—21no—
105#11719917—41no—
106C2×#88322—41no—
107C2×#88322C2311yescharacteristic
108C2×#88322C2311yescharacteristic
109C2×#88322C2311yescharacteristic
110C2×#88322C2311yescharacteristic
111C23×C53—21no—
112C23×C53C2×D411yescharacteristic
113C23×C53C2×D411yescharacteristic
114C22×D53—21no—
115Q8×C53—21no—
116Q8×C53—21no—
117Q8×C53—21no—
118Q8×C53—42no—
119Q8×C53#4112yes—
120D4×C53—21no—
121D4×C53—21no—
122D4×C53—41no—
123D4×C53—42no—
124D4×C53—42no—
125C2×C4×C53—21no—
126C2×C4×C53—21no—
127C2×C4×C53—21no—
128C2×C4×C53—21no—
129C2×C4×C53—21no—
130C2×C4×C53—21no—
131C2×C4×C53—21no—
132C2×C4×C53—42no—
133C2×C4×C53—42no—
134C2×C4×C53C2×D411yescharacteristic
135C2×C4×C53C2×D411yescharacteristic
136C2×C4×C53C2×D411yescharacteristic
137C2×C4×C53C2×D411yescharacteristic
138C2×C4×C53C2411yescharacteristic
139C22⋊D53—81no—
140C22⋊D53—81no—
141C2×Dic53—21no—
142C2×Dic53—21no—
143C2×Dic53—21no—
144D212—21no—
145D212—21no—
146C4×D53—42no—
147Dic106—81no—
148#88322—21no—
149#88322—21no—
150#88322—21no—
151#88322—21no—
152C22×C53—21no—
153C22×C53—21no—
154C22×C53—21no—
155C22×C53—21no—
156C22×C53—21no—
157C22×C53—21no—
158C22×C53—41no—
159C22×C53—41no—
160C22×C53—41no—
161C22×C53C22×D411yescharacteristic
162C2×D53—41no—
163C2×D53—41no—
164C2×D53—41no—
165C4×C53—21no—
166C4×C53—21no—
167C4×C53—22no—
168C4×C53—21no—
169C4×C53—21no—
170C4×C53—41no—
171C4×C53—41no—
172C4×C53C2×#4111yescharacteristic
173C4×C53H(5,𝔽2)11yescharacteristic
174Dic53—41no—
175Dic53—41no—
176Dic53—41no—
177#3321—531nomaximal, Sylow
178C2×C53—41no—
179C2×C53—41no—
180C2×C53—81no—
181C2×C53#54611yescharacteristic
182C2×C53#57511yescharacteristic
183C2×C53#57711yescharacteristic
184D53—81no—
185#555—531no—
186#548D5311yescharacteristic, complemented
187#489—531no—
188#485—531no—
189#482—531no—
190#475—531no—
191#469—531no—
192#465—531no—
193#463—531no—
194#449—531no—
195#448—531no—
196#440—531no—
197#435—531no—
198#419—531no—
199#416—531no—
200C53#332111yescharacteristic, complemented, Sylow
201C2×#41C2×D5311yescharacteristic
202C22×Q8C2×D5311yescharacteristic
203C2×SD16—531no—
204C2×SD16—531no—
205C2×D8—531no—
206C2×M16—531no—
207C22×C8—531no—
208C4⋊Q8—531no—
209#124—21no—
210#124—1061no—
211#124—1061no—
212#124C2×D5311yescharacteristic
213C22⋊Q8—21no—
214C22⋊Q8C2×D5311yescharacteristic
215#121—21no—
216#121—531no—
217#121—531no—
218#121—1061no—
219#121C2×D5311yescharacteristic
220C4×Q8C2×D5311yescharacteristic
221C4×D4—21no—
222C4×D4—531no—
223C4×D4C2×D5311yescharacteristic
224#117—531no—
225#117—531no—
226#117—1061no—
227#107—531no—
228#106—531no—
229#106—531no—
230C4⋊C8—531no—
231#103—531no—
232#103—531no—
233#103—531no—
234#103—531no—
235#103—531no—
236#103—531no—
237#102—531no—
238#102—531no—
239#102—531no—
240#102—531no—
241C22⋊C8—531no—
242C22⋊C8—531no—
243#97—531no—
244#41—22no—
245#41—41no—
246C2×Q8—21no—
247C2×Q8—22no—
248C2×Q8—41no—
249C2×Q8—1061no—
250C2×Q8C22×D5311yescharacteristic
251C2×Q8C22×D5311yescharacteristic
252C2×D4—21no—
253C2×D4—21no—
254C2×D4—531no—
255C2×D4—1061no—
256C2×D4—1061no—
257C2×D4C22⋊D5311yescharacteristic
258C2×D4C22×D5311yescharacteristic
259C22×C4—21no—
260C22×C4—21no—
261C22×C4—1061no—
262C22×C4C22×D5311yescharacteristic
263C22×C4C22×D5311yescharacteristic
264SD16—1062no—
265SD16—2121no—
266D8—2121no—
267M16—2121no—
268C2×C8—531no—
269C2×C8—531no—
270C2×C8—531no—
271C2×C8—531no—
272C2×C8—2121no—
273C4⋊C4—21no—
274C4⋊C4—21no—
275C4⋊C4—531no—
276C4⋊C4—531no—
277C4⋊C4—531no—
278C4⋊C4—1061no—
279C4⋊C4C22⋊D5311yescharacteristic
280C4⋊C4C22×D5311yescharacteristic
281C22⋊C4—21no—
282C22⋊C4—21no—
283C22⋊C4—21no—
284C22⋊C4—21no—
285C22⋊C4—1061no—
286C22⋊C4—1061no—
287C22⋊C4—1061no—
288C22⋊C4—1061no—
289C22⋊C4—1061no—
290C22⋊C4C22⋊D5311yescharacteristic
291C22⋊C4C22⋊D5311yescharacteristic
292C42—21no—
293C42—1061no—
294C42—1061no—
295C42C22×D5311yescharacteristic
296C23—21no—
297C23—1061no—
298C23C2×C22⋊D5311yescharacteristic
299C23C2×C22⋊D5311yescharacteristic
300Q8—21no—
301Q8—21no—
302Q8—21no—
303Q8—42no—
304Q8—4241no—
305Q8#1171994712yes—
306D4—21no—
307D4—21no—
308D4—41no—
309D4—42no—
310D4—42no—
311D4—1061no—
312D4—1061no—
313D4—4241no—
314D4—4241no—
315C2×C4—21no—
316C2×C4—21no—
317C2×C4—21no—
318C2×C4—21no—
319C2×C4—21no—
320C2×C4—21no—
321C2×C4—21no—
322C2×C4—42no—
323C2×C4—42no—
324C2×C4—1061no—
325C2×C4—1061no—
326C2×C4—1061no—
327C2×C4—2122no—
328C2×C4C2×C22⋊D5311yescharacteristic
329C2×C4C2×C22⋊D5311yescharacteristic
330C2×C4C2×C22⋊D5311yescharacteristic
331C2×C4C2×C22⋊D5311yescharacteristic
332C2×C4C23×D5311yescharacteristic
333C8—1061no—
334C8—1061no—
335C8—1061no—
336C8—1061no—
337C22—21no—
338C22—21no—
339C22—21no—
340C22—21no—
341C22—21no—
342C22—21no—
343C22—41no—
344C22—41no—
345C22—41no—
346C22—2121no—
347C22—2121no—
348C22—2121no—
349C22C22×C22⋊D5311yescharacteristic
350C4—21no—
351C4—21no—
352C4—22no—
353C4—21no—
354C4—21no—
355C4—41no—
356C4—41no—
357C4—2121no—
358C4—2121no—
359C4—2121no—
360C4C2×#1171994711yescharacteristic
361C4#1313326211yescharacteristic
362C2—41no—
363C2—41no—
364C2—81no—
365C2—4241no—
366C2#1681126711yescharacteristic
367C2#1681142111yescharacteristic
368C2#1681143111yescharacteristic
subgroupG/H
centerC22C22×C22⋊D53
commutatorC2×C4×C53C24
FrattiniC2×C4C23×D53
FittingC53×#548C2
socleC22×C53C22×D4
hypercenter#548D53
hypocenterC53#3321
LCSG ▷ C2×C4×C53 ▷ C2×C53 ▷ C53
UCSC1 ◁ C22 ◁ C4×D4 ◁ #548

Complements

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

NH ≅ G/Ntype
1. #16811338C2semidirect
2. #16810982C2semidirect
3. #16810961C2semidirect
4. #16810944C2semidirect
5. #16810908C2semidirect
6. #16810874C2semidirect
7. #16810854C2semidirect
8. #16810835C2semidirect
9. #16810768C2semidirect
10. #16810762C2semidirect
11. #16810721C2semidirect
12. #16810695C2semidirect
13. #16810632C2semidirect
14. #16810619C2semidirect
15. C53×#548C2semidirect
186. #548D53semidirect
200. C53#3321semidirect