Finite Group Databasebeta

← #20283783

Subgroups — #20283783

Order 6784.

394 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#16811170C211yesmaximal, characteristic
2#16810697C211yesmaximal, characteristic, complemented
3#16810696C211yesmaximal, characteristic, complemented
4#16810695C211yesmaximal, characteristic, complemented
5#16810694C211yesmaximal, characteristic, complemented
6#16810693C211yesmaximal, characteristic, complemented
7#16810692C211yesmaximal, characteristic, complemented
8#16810691C211yesmaximal, characteristic, complemented
9#16810679C211yesmaximal, characteristic, complemented
10#16810678C211yesmaximal, characteristic, complemented
11#16810591C211yesmaximal, characteristic, complemented
12#16810589C211yesmaximal, characteristic, complemented
13#16810588C211yesmaximal, characteristic, complemented
14#16810588C211yesmaximal, characteristic, complemented
15C53×#435C211yesmaximal, characteristic, complemented
16C2×#11719945C2211yescharacteristic
17C2×#11719945C2211yescharacteristic
18C2×#11719917C2211yescharacteristic
19C2×#11719912C2211yescharacteristic
20C2×#11719912C2211yescharacteristic
21C22×#88322C2211yescharacteristic
22C2×C8×D53C2211yescharacteristic
23C2×C8×D53C2211yescharacteristic
24#13133216C411yescharacteristic
25#13133213C411yescharacteristic
26#13133203C411yescharacteristic
27#13133203C411yescharacteristic
28#13133192C411yescharacteristic
29#13133190C411yescharacteristic
30#13133190C411yescharacteristic
31#13133188C411yescharacteristic
32C4×D212C2211yescharacteristic
33C4×C22⋊D53C2211yescharacteristic
34C4×C22⋊D53C2211yescharacteristic
35C42×D53C2211yescharacteristic
36C4⋊#88322C2211yescharacteristic
37#13133146C2211yescharacteristic
38#13133146C2211yescharacteristic
39C22⋊#88322C2211yescharacteristic
40C22⋊#88322C2211yescharacteristic
41#13133129C2211yescharacteristic
42#13133129C2211yescharacteristic
43#13133129C2211yescharacteristic
44#13133129C2211yescharacteristic
45#13133129C2211yescharacteristic
46#13133129C2211yescharacteristic
47#13133129C2211yescharacteristic
48#13133129C2211yescharacteristic
49#13133128C2211yescharacteristic
50#13133127C2211yescharacteristic
51#13133127C2211yescharacteristic
52C2×M16×C53C2211yescharacteristic
53C22×C8×C53C2211yescharacteristic
54C4×D4×C53C2211yescharacteristic
55C4⋊C8×C53C2211yescharacteristic
56C22⋊C8×C53C2211yescharacteristic
57C22⋊C8×C53C2211yescharacteristic
58#97×C53C2211yescharacteristic
59C2×D4×C53C2×C411yescharacteristic
60C22×C4×C53C2311yescharacteristic
61C22×C4×C53C2311yescharacteristic
62C2×C22⋊D53C2×C411yescharacteristic
63C2×C22⋊D53C2×C411yescharacteristic
64C2×C22⋊D53C2×C411yescharacteristic
65C2×C22⋊D53C2×C411yescharacteristic
66C2×C22⋊D53C2×C411yescharacteristic
67C2×C22⋊D53C2×C411yescharacteristic
68#11719945—41no—
69#11719945—41no—
70#11719945—41no—
71#11719945—41no—
72C2×D212C2×C411yescharacteristic
73C2×D212C2×C411yescharacteristic
74C2×D212C2×C411yescharacteristic
75C2×C4×D53C2×C411yescharacteristic
76C2×C4×D53C2×C411yescharacteristic
77C2×C4×D53C2311yescharacteristic
78C2×C4×D53C2311yescharacteristic
79C2×C4×D53C2311yescharacteristic
80C2×Dic106C2×C411yescharacteristic
81C2×Dic106C2×C411yescharacteristic
82M16×C53—41no—
83C2×C8×C53—41no—
84C2×C8×C53C2311yescharacteristic
85C2×C8×C53C2311yescharacteristic
86C2×C8×C53C2311yescharacteristic
87C2×C8×C53C2311yescharacteristic
88C4⋊C4×C53C2×C411yescharacteristic
89C22⋊C4×C53C2×C411yescharacteristic
90C22⋊C4×C53C2×C411yescharacteristic
91C42×C53C2311yescharacteristic
92C22⋊Dic53C2×C411yescharacteristic
93C22⋊Dic53C2×C411yescharacteristic
94#11719921C2×C411yescharacteristic
95#11719921C2×C411yescharacteristic
96#11719921C2×C411yescharacteristic
97#11719921C2×C411yescharacteristic
98C4⋊Dic53C2×C411yescharacteristic
99#11719919C2×C411yescharacteristic
100#11719919C2×C411yescharacteristic
101C4×Dic53C2×C411yescharacteristic
102C4×Dic53C2×C411yescharacteristic
103C4×Dic53C2311yescharacteristic
104#11719917—41no—
105C2×#88322—41no—
106C2×#88322C2311yescharacteristic
107C2×#88322C2311yescharacteristic
108C2×#88322C2311yescharacteristic
109C2×#88322C2311yescharacteristic
110#11719912—41no—
111#11719912—41no—
112C8×D53—41no—
113C8×D53—41no—
114C23×C53C22×C411yescharacteristic
115C23×C53C22×C411yescharacteristic
116C22×D53C22×C411yescharacteristic
117C22×D53C22×C411yescharacteristic
118C22×D53C22×C411yescharacteristic
119D4×C53—41no—
120C2×C4×C53—21no—
121C2×C4×C53—21no—
122C2×C4×C53—21no—
123C2×C4×C53—21no—
124C2×C4×C53C22×C411yescharacteristic
125C2×C4×C53C22×C411yescharacteristic
126C2×C4×C53C22×C411yescharacteristic
127C2×C4×C53C22×C411yescharacteristic
128C2×C4×C53C2411yescharacteristic
129C22⋊D53—41no—
130C22⋊D53—41no—
131C22⋊D53—41no—
132C22⋊D53—41no—
133C22⋊D53—41no—
134C22⋊D53—41no—
135C2×Dic53C22×C411yescharacteristic
136C2×Dic53C22×C411yescharacteristic
137C2×Dic53C22×C411yescharacteristic
138C2×Dic53C22×C411yescharacteristic
139C2×Dic53C22×C411yescharacteristic
140D212—41no—
141D212—41no—
142D212—41no—
143C4×D53—21no—
144C4×D53—21no—
145C4×D53—21no—
146C4×D53—21no—
147C4×D53—21no—
148C4×D53—21no—
149C4×D53—41no—
150C4×D53—41no—
151Dic106—41no—
152Dic106—41no—
153C8×C53—21no—
154C8×C53—21no—
155C8×C53—21no—
156C8×C53—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×C53C23×C411yescharacteristic
168C2×D53—21no—
169C2×D53—21no—
170C2×D53—21no—
171C2×D53—21no—
172C2×D53—21no—
173C2×D53—21no—
174C2×D53—21no—
175C2×D53—21no—
176C2×D53—21no—
177C4×C53—21no—
178C4×C53—21no—
179C4×C53—21no—
180C4×C53—21no—
181C4×C53H(5,𝔽2)11yescharacteristic
182C4×C53H(5,𝔽2)11yescharacteristic
183Dic53—21no—
184Dic53—21no—
185Dic53—21no—
186Dic53—21no—
187Dic53—21no—
188#2980—531nomaximal, Sylow
189C2×C53—41no—
190C2×C53—41no—
191C2×C53#51811yescharacteristic
192C2×C53#56811yescharacteristic
193C2×C53#56811yescharacteristic
194D53—41no—
195D53—41no—
196D53—41no—
197#532—531no—
198#435—531no—
199#435—531no—
200#435—531no—
201#435—531no—
202#435—531no—
203#435—531no—
204#435—531no—
205#435D5311yescharacteristic, complemented
206#432—531no—
207#432—531no—
208#408—531no—
209#408—531no—
210#408—531no—
211#408—531no—
212C53#298011yescharacteristic, complemented, Sylow
213C2×#41—531no—
214C2×#41—531no—
215C2×M16—531no—
216C2×M16—531no—
217C2×M16—531no—
218C2×M16C2×D5311yescharacteristic
219C22×C8—531no—
220C22×C8—531no—
221C22×C8—531no—
222C22×C8C2×D5311yescharacteristic
223C4⋊Q8—531no—
224#127—531no—
225#124—531no—
226#124—531no—
227#121—531no—
228#121—531no—
229#121—531no—
230#121—531no—
231C4×D4—531no—
232C4×D4—531no—
233C4×D4—531no—
234C4×D4C2×D5311yescharacteristic
235C2×C42—531no—
236C4⋊C8—531no—
237C4⋊C8—531no—
238C4⋊C8—531no—
239C4⋊C8C2×D5311yescharacteristic
240C22⋊C8—531no—
241C22⋊C8—531no—
242C22⋊C8—531no—
243C22⋊C8—531no—
244C22⋊C8—531no—
245C22⋊C8—531no—
246C22⋊C8—531no—
247C22⋊C8—531no—
248C22⋊C8—531no—
249C22⋊C8—531no—
250C22⋊C8C2×D5311yescharacteristic
251C22⋊C8C2×D5311yescharacteristic
252#97—531no—
253#97—531no—
254#97—531no—
255#97C2×D5311yescharacteristic
256#41—2121no—
257#41—2121no—
258#41—2121no—
259#41—2121no—
260C2×Q8—531no—
261C2×Q8—531no—
262C2×D4—531no—
263C2×D4—531no—
264C2×D4—531no—
265C2×D4—531no—
266C2×D4—531no—
267C2×D4—531no—
268C2×D4—531no—
269C2×D4—531no—
270C2×D4—531no—
271C2×D4C4×D5311yescharacteristic
272C22×C4—531no—
273C22×C4—531no—
274C22×C4—531no—
275C22×C4—531no—
276C22×C4—531no—
277C22×C4C22×D5311yescharacteristic
278C22×C4C22×D5311yescharacteristic
279M16—41no—
280M16—2121no—
281M16—2121no—
282M16—2121no—
283C2×C8—41no—
284C2×C8—531no—
285C2×C8—531no—
286C2×C8—531no—
287C2×C8—531no—
288C2×C8—2121no—
289C2×C8—2121no—
290C2×C8—2121no—
291C2×C8C22×D5311yescharacteristic
292C2×C8C22×D5311yescharacteristic
293C2×C8C22×D5311yescharacteristic
294C2×C8C22×D5311yescharacteristic
295C4⋊C4—531no—
296C4⋊C4—531no—
297C4⋊C4—531no—
298C4⋊C4C4×D5311yescharacteristic
299C22⋊C4—531no—
300C22⋊C4—531no—
301C22⋊C4—531no—
302C22⋊C4—531no—
303C22⋊C4—531no—
304C22⋊C4—531no—
305C22⋊C4C4×D5311yescharacteristic
306C22⋊C4C4×D5311yescharacteristic
307C42—531no—
308C42—531no—
309C42—531no—
310C42C22×D5311yescharacteristic
311C23—531no—
312C23—531no—
313C23—531no—
314C23C2×C4×D5311yescharacteristic
315C23C2×C4×D5311yescharacteristic
316Q8—2121no—
317Q8—2121no—
318D4—41no—
319D4—2121no—
320D4—2121no—
321D4—2121no—
322D4—2121no—
323D4—2121no—
324D4—2121no—
325D4—2121no—
326D4—2121no—
327D4—2121no—
328C2×C4—21no—
329C2×C4—21no—
330C2×C4—21no—
331C2×C4—21no—
332C2×C4—531no—
333C2×C4—531no—
334C2×C4—531no—
335C2×C4—531no—
336C2×C4—531no—
337C2×C4—1061no—
338C2×C4—1061no—
339C2×C4—1061no—
340C2×C4—1061no—
341C2×C4—1061no—
342C2×C4—1061no—
343C2×C4—2121no—
344C2×C4—2121no—
345C2×C4C2×C4×D5311yescharacteristic
346C2×C4C2×C4×D5311yescharacteristic
347C2×C4C2×C4×D5311yescharacteristic
348C2×C4C2×C4×D5311yescharacteristic
349C2×C4C23×D5311yescharacteristic
350C8—21no—
351C8—21no—
352C8—21no—
353C8—21no—
354C8—1061no—
355C8—1061no—
356C8—1061no—
357C8—1061no—
358C22—21no—
359C22—21no—
360C22—21no—
361C22—21no—
362C22—21no—
363C22—21no—
364C22—1061no—
365C22—1061no—
366C22—1061no—
367C22—1061no—
368C22—1061no—
369C22—1061no—
370C22—1061no—
371C22—1061no—
372C22—1061no—
373C22C22×C4×D5311yescharacteristic
374C4—21no—
375C4—21no—
376C4—21no—
377C4—21no—
378C4—1061no—
379C4—1061no—
380C4—1061no—
381C4—1061no—
382C4—1061no—
383C4#1313326211yescharacteristic
384C4#1313326311yescharacteristic
385C2—41no—
386C2—41no—
387C2—2121no—
388C2—2121no—
389C2—2121no—
390C2#1681109511yescharacteristic
391C2#1681138811yescharacteristic
392C2#1681138911yescharacteristic
subgroupG/H
center, FrattiniC2×C4C23×D53
commutator, socleC22×C53C23×C4
FittingC53×#435C2
hypercenter#435D53
hypocenterC53#2980
LCSG ▷ C22×C53 ▷ C53
UCSC1 ◁ C2×C4 ◁ #435

Complements

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

NH ≅ G/Ntype
2. #16810697C2semidirect
3. #16810696C2semidirect
4. #16810695C2semidirect
5. #16810694C2semidirect
6. #16810693C2semidirect
7. #16810692C2semidirect
8. #16810691C2semidirect
9. #16810679C2semidirect
10. #16810678C2semidirect
11. #16810591C2semidirect
12. #16810589C2semidirect
13. #16810588C2semidirect
14. #16810588C2semidirect
15. C53×#435C2semidirect
205. #435D53semidirect
212. C53#2980semidirect