Finite Group Databasebeta

← #20285612

Subgroups — #20285612

Order 6784.

402 conjugacy classes of subgroups.

The subgroup lattice has 394 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#16811201C211yesmaximal, characteristic, complemented
2#16810958C211yesmaximal, characteristic, complemented
3#16810956C211yesmaximal, characteristic, complemented
4#16810936C211yesmaximal, characteristic, complemented
5#16810934C211yesmaximal, characteristic, complemented
6#16810871C211yesmaximal, characteristic, complemented
7#16810867C211yesmaximal, characteristic, complemented
8#16810862C211yesmaximal, characteristic, complemented
9#16810857C211yesmaximal, characteristic, complemented
10#16810785C211yesmaximal, characteristic, complemented
11#16810783C211yesmaximal, characteristic, complemented
12#16810768C211yesmaximal, characteristic, complemented
13#16810766C211yesmaximal, characteristic, complemented
14#16810590C211yesmaximal, characteristic, complemented
15C53×#537C211yesmaximal, characteristic, complemented
16C2×#11719945C2211yescharacteristic
17C2×#11719925C2211yescharacteristic
18C2×#11719923C2211yescharacteristic
19C2×#11719924C2211yescharacteristic
20C2×#11719922C2211yescharacteristic
21C2×#11719917C2211yescharacteristic
22C22×#88322C2211yescharacteristic
23#13133196C2211yescharacteristic
24#13133196C2211yescharacteristic
25#13133193—21no—
26#13133191C2211yescharacteristic
27#13133191C2211yescharacteristic
28#13133189—21no—
29#13133189—21no—
30#13133188—21no—
31C2×C4⋊Dic53C2211yescharacteristic
32#13133150C2211yescharacteristic
33#13133150C2211yescharacteristic
34#13133150C2211yescharacteristic
35#13133150C2211yescharacteristic
36#13133141C2211yescharacteristic
37#13133141C2211yescharacteristic
38#13133140C2211yescharacteristic
39#13133140C2211yescharacteristic
40#13133138C2211yescharacteristic
41#13133138C2211yescharacteristic
42#13133137C2211yescharacteristic
43#13133137C2211yescharacteristic
44C22⋊#88322C2211yescharacteristic
45C22⋊#88322C2211yescharacteristic
46C22⋊#88322C2211yescharacteristic
47C22⋊#88322C2211yescharacteristic
48C2×#41×C53C2211yescharacteristic
49C2×#41×C53C2211yescharacteristic
50C22⋊Q8×C53—21no—
51C22⋊Q8×C53C2211yescharacteristic
52C22⋊Q8×C53C2211yescharacteristic
53#121×C53—21no—
54#121×C53—21no—
55#121×C53—21no—
56#121×C53C2211yescharacteristic
57#121×C53C2211yescharacteristic
58C2×C4⋊C4×C53C2211yescharacteristic
59#41×C53—41no—
60#41×C53—41no—
61#41×C53—41no—
62#41×C53—41no—
63C2×Q8×C53D411yescharacteristic
64C2×Q8×C53C2311yescharacteristic
65C2×D4×C53—21no—
66C2×D4×C53—21no—
67C2×D4×C53—21no—
68C2×D4×C53D411yescharacteristic
69C2×D4×C53D411yescharacteristic
70C2×D4×C53D411yescharacteristic
71C2×D4×C53C2311yescharacteristic
72C22×C4×C53—21no—
73C22×C4×C53—21no—
74C22×C4×C53C2311yescharacteristic
75C22×C4×C53C2311yescharacteristic
76C22×C4×C53C2311yescharacteristic
77C2×C22⋊D53—21no—
78C2×C22⋊D53—21no—
79C2×C22⋊D53—21no—
80C22×Dic53—21no—
81#11719945—41no—
82#11719945—41no—
83C2×D212C2311yescharacteristic
84C2×C4×D53—21no—
85C2×Dic106C2311yescharacteristic
86C4⋊C4×C53—21no—
87C4⋊C4×C53—21no—
88C4⋊C4×C53—21no—
89C4⋊C4×C53C2311yescharacteristic
90C4⋊C4×C53C2311yescharacteristic
91C22⋊C4×C53—21no—
92C22⋊C4×C53—21no—
93C22⋊C4×C53—21no—
94C22⋊C4×C53—21no—
95C22⋊Dic53—21no—
96C22⋊Dic53—21no—
97#11719925—41no—
98#11719924—41no—
99#11719923—41no—
100#11719922—41no—
101#11719921—21no—
102#11719921—21no—
103C4⋊Dic53—21no—
104C4⋊Dic53C2311yescharacteristic
105C4⋊Dic53C2311yescharacteristic
106#11719919—21no—
107#11719919—21no—
108#11719917—41no—
109C2×#88322—41no—
110C2×#88322C2311yescharacteristic
111C2×#88322C2311yescharacteristic
112C2×#88322C2311yescharacteristic
113C2×#88322C2311yescharacteristic
114C23×C53—21no—
115C23×C53C2×D411yescharacteristic
116C23×C53C2×D411yescharacteristic
117C23×C53C2×D411yescharacteristic
118C22×D53—21no—
119Q8×C53—21no—
120Q8×C53—21no—
121Q8×C53—41no—
122D4×C53—21no—
123D4×C53—21no—
124D4×C53—41no—
125D4×C53—41no—
126D4×C53—41no—
127D4×C53—42no—
128D4×C53—81no—
129D4×C53—81no—
130C2×C4×C53—21no—
131C2×C4×C53—21no—
132C2×C4×C53—21no—
133C2×C4×C53—21no—
134C2×C4×C53—21no—
135C2×C4×C53—21no—
136C2×C4×C53—21no—
137C2×C4×C53—21no—
138C2×C4×C53—21no—
139C2×C4×C53—42no—
140C2×C4×C53—81no—
141C2×C4×C53C2×D411yescharacteristic
142C2×C4×C53C2×D411yescharacteristic
143C2×C4×C53C2×D411yescharacteristic
144C2×C4×C53C2411yescharacteristic
145C22⋊D53—42no—
146C22⋊D53—81no—
147C22⋊D53—81no—
148C2×Dic53—21no—
149C2×Dic53—21no—
150C2×Dic53—21no—
151C2×Dic53—81no—
152D212—21no—
153D212—21no—
154C4×D53—42no—
155Dic106—21no—
156Dic106—21no—
157#88322—21no—
158#88322—21no—
159#88322—21no—
160#88322—21no—
161C22×C53—21no—
162C22×C53—21no—
163C22×C53—21no—
164C22×C53—21no—
165C22×C53—21no—
166C22×C53—21no—
167C22×C53—21no—
168C22×C53—21no—
169C22×C53—21no—
170C22×C53—41no—
171C22×C53—41no—
172C22×C53—41no—
173C22×C53C22×D411yescharacteristic
174C2×D53—41no—
175C2×D53—41no—
176C2×D53—41no—
177C4×C53—21no—
178C4×C53—21no—
179C4×C53—21no—
180C4×C53—41no—
181C4×C53—41no—
182C4×C53—41no—
183C4×C53H(5,𝔽2)11yescharacteristic
184C4×C53H(5,𝔽2)11yescharacteristic
185Dic53—41no—
186Dic53—41no—
187Dic53—41no—
188#3203—531nomaximal, Sylow
189C2×C53—41no—
190C2×C53—41no—
191C2×C53—41no—
192C2×C53—81no—
193C2×C53#53411yescharacteristic
194C2×C53#57611yescharacteristic
195C2×C53#57811yescharacteristic
196D53—81no—
197#537—531no—
198#537D5311yescharacteristic, complemented
199#484—531no—
200#484—531no—
201#480—531no—
202#480—531no—
203#468—531no—
204#468—531no—
205#467—531no—
206#466—531no—
207#452—531no—
208#452—531no—
209#449—531no—
210#449—531no—
211#408—531no—
212C53#320311yescharacteristic, complemented, Sylow
213C2×#41—531no—
214C2×#41C2×D5311yescharacteristic
215C2×#41C2×D5311yescharacteristic
216C2×Q16—531no—
217C2×SD16—531no—
218C2×SD16—531no—
219C2×D8—531no—
220C2×M16—531no—
221C22×C8—531no—
222C22⋊Q8—21no—
223C22⋊Q8—531no—
224C22⋊Q8—531no—
225C22⋊Q8—1061no—
226C22⋊Q8C2×D5311yescharacteristic
227C22⋊Q8C2×D5311yescharacteristic
228#121—21no—
229#121—21no—
230#121—21no—
231#121—531no—
232#121—531no—
233#121—1061no—
234#121—1061no—
235#121—1061no—
236#121C2×D5311yescharacteristic
237#121C2×D5311yescharacteristic
238C2×C4⋊C4—531no—
239C2×C4⋊C4C2×D5311yescharacteristic
240#107—531no—
241#107—531no—
242#107—531no—
243#107—531no—
244#103—531no—
245#103—531no—
246#103—531no—
247#103—531no—
248#102—531no—
249#102—531no—
250#102—531no—
251#102—531no—
252C22⋊C8—531no—
253C22⋊C8—531no—
254C22⋊C8—531no—
255C22⋊C8—531no—
256#41—41no—
257#41—41no—
258#41—41no—
259#41—41no—
260#41—2121no—
261#41—2121no—
262C2×Q8—531no—
263C2×Q8C22⋊D5311yescharacteristic
264C2×Q8C22×D5311yescharacteristic
265C2×D4—21no—
266C2×D4—21no—
267C2×D4—21no—
268C2×D4—531no—
269C2×D4—1061no—
270C2×D4—1061no—
271C2×D4—1061no—
272C2×D4C22⋊D5311yescharacteristic
273C2×D4C22⋊D5311yescharacteristic
274C2×D4C22⋊D5311yescharacteristic
275C2×D4C22×D5311yescharacteristic
276C22×C4—21no—
277C22×C4—21no—
278C22×C4—1061no—
279C22×C4—1061no—
280C22×C4C22×D5311yescharacteristic
281C22×C4C22×D5311yescharacteristic
282C22×C4C22×D5311yescharacteristic
283Q16—2121no—
284SD16—2121no—
285SD16—2121no—
286D8—2121no—
287M16—2121no—
288C2×C8—531no—
289C2×C8—531no—
290C2×C8—531no—
291C2×C8—531no—
292C2×C8—2121no—
293C4⋊C4—21no—
294C4⋊C4—21no—
295C4⋊C4—21no—
296C4⋊C4—531no—
297C4⋊C4—531no—
298C4⋊C4—1061no—
299C4⋊C4—1061no—
300C4⋊C4—1061no—
301C4⋊C4C22×D5311yescharacteristic
302C4⋊C4C22×D5311yescharacteristic
303C22⋊C4—21no—
304C22⋊C4—21no—
305C22⋊C4—21no—
306C22⋊C4—21no—
307C22⋊C4—1061no—
308C22⋊C4—1061no—
309C22⋊C4—1061no—
310C22⋊C4—1061no—
311C23—21no—
312C23—1061no—
313C23C2×C22⋊D5311yescharacteristic
314C23C2×C22⋊D5311yescharacteristic
315C23C2×C22⋊D5311yescharacteristic
316Q8—21no—
317Q8—21no—
318Q8—41no—
319Q8—1061no—
320Q8—1061no—
321D4—21no—
322D4—21no—
323D4—41no—
324D4—41no—
325D4—41no—
326D4—42no—
327D4—81no—
328D4—81no—
329D4—1061no—
330D4—1061no—
331D4—4241no—
332D4—4241no—
333D4—2122no—
334C2×C4—21no—
335C2×C4—21no—
336C2×C4—21no—
337C2×C4—21no—
338C2×C4—21no—
339C2×C4—21no—
340C2×C4—21no—
341C2×C4—21no—
342C2×C4—21no—
343C2×C4—42no—
344C2×C4—81no—
345C2×C4—1061no—
346C2×C4—1061no—
347C2×C4—1061no—
348C2×C4—4241no—
349C2×C4—2122no—
350C2×C4C2×C22⋊D5311yescharacteristic
351C2×C4C2×C22⋊D5311yescharacteristic
352C2×C4C2×C22⋊D5311yescharacteristic
353C2×C4C23×D5311yescharacteristic
354C8—1061no—
355C8—1061no—
356C8—1061no—
357C8—1061no—
358C22—21no—
359C22—21no—
360C22—21no—
361C22—21no—
362C22—21no—
363C22—21no—
364C22—21no—
365C22—21no—
366C22—21no—
367C22—41no—
368C22—41no—
369C22—41no—
370C22—2121no—
371C22—2121no—
372C22—2121no—
373C22C22×C22⋊D5311yescharacteristic
374C4—21no—
375C4—21no—
376C4—21no—
377C4—41no—
378C4—41no—
379C4—41no—
380C4—2121no—
381C4—2121no—
382C4—2121no—
383C4#1313326211yescharacteristic
384C4#1313326211yescharacteristic
385C2—41no—
386C2—41no—
387C2—41no—
388C2—81no—
389C2—4241no—
390C2#1681118211yescharacteristic
391C2#1681142611yescharacteristic
392C2#1681143811yescharacteristic
subgroupG/H
centerC22C22×C22⋊D53
commutatorC2×C4×C53C24
FrattiniC2×C4C23×D53
FittingC53×#537C2
socleC22×C53C22×D4
hypercenter#537D53
hypocenterC53#3203
LCSG ▷ C2×C4×C53 ▷ C2×C53 ▷ C53
UCSC1 ◁ C22 ◁ C2×#41 ◁ #537

Complements

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

NH ≅ G/Ntype
1. #16811201C2semidirect
2. #16810958C2semidirect
3. #16810956C2semidirect
4. #16810936C2semidirect
5. #16810934C2semidirect
6. #16810871C2semidirect
7. #16810867C2semidirect
8. #16810862C2semidirect
9. #16810857C2semidirect
10. #16810785C2semidirect
11. #16810783C2semidirect
12. #16810768C2semidirect
13. #16810766C2semidirect
14. #16810590C2semidirect
15. C53×#537C2semidirect
198. #537D53semidirect
212. C53#3203semidirect