Finite Group Databasebeta

← #20286122

Subgroups — #20286122

Order 6784.

358 conjugacy classes of subgroups.

The subgroup lattice has 358 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#16811238C211yesmaximal, characteristic, complemented
2#16811227C211yesmaximal, characteristic, complemented
3#16810931C211yesmaximal, characteristic, complemented
4#16810923C211yesmaximal, characteristic, complemented
5#16810911C211yesmaximal, characteristic, complemented
6#16810901C211yesmaximal, characteristic, complemented
7#16810873C211yesmaximal, characteristic, complemented
8#16810868C211yesmaximal, characteristic, complemented
9#16810856C211yesmaximal, characteristic, complemented
10#16810851C211yesmaximal, characteristic, complemented
11#16810836C211yesmaximal, characteristic, complemented
12#16810836C211yesmaximal, characteristic, complemented
13#16810820C211yesmaximal, characteristic, complemented
14#16810648C211yesmaximal, characteristic, complemented
15C53×#459C211yesmaximal, characteristic, complemented
16C2×#11719923C2211yescharacteristic
17C2×#11719923C2211yescharacteristic
18C2×#11719912C2211yescharacteristic
19C2×C8×D53C2211yescharacteristic
20#13133216C2211yescharacteristic
21#13133210—21no—
22#13133209C2211yescharacteristic
23#13133204C2211yescharacteristic
24#13133203—21no—
25#13133200—21no—
26#13133198—21no—
27#13133198—21no—
28#13133197C2211yescharacteristic
29#13133196—21no—
30#13133189—21no—
31#13133189—21no—
32#13133188C2211yescharacteristic
33#13133188C2211yescharacteristic
34#13133188C2211yescharacteristic
35C4×Dic106C2211yescharacteristic
36D4×Dic53C2211yescharacteristic
37D4×Dic53C2211yescharacteristic
38#13133174C2211yescharacteristic
39#13133151C2211yescharacteristic
40#13133150C2211yescharacteristic
41#13133149C2211yescharacteristic
42#13133148C2211yescharacteristic
43C4⋊#88322C2211yescharacteristic
44#13133146C2211yescharacteristic
45#13133146C2211yescharacteristic
46#13133142C2211yescharacteristic
47#13133142C2211yescharacteristic
48#13133138C2211yescharacteristic
49#13133138C2211yescharacteristic
50#13133138C2211yescharacteristic
51#13133138C2211yescharacteristic
52C2×D8×C53C2211yescharacteristic
53C2×D8×C53C2211yescharacteristic
54#127×C53C2211yescharacteristic
55C4×D4×C53C2211yescharacteristic
56C4⋊C8×C53C2211yescharacteristic
57#102×C53C2211yescharacteristic
58#102×C53C2211yescharacteristic
59C2×D4×C53—21no—
60C2×D4×C53—21no—
61C2×D4×C53C2311yescharacteristic
62C2×D4×C53C2311yescharacteristic
63C2×D4×C53C2311yescharacteristic
64C22×C4×C53—21no—
65C2×C22⋊D53—21no—
66C2×C22⋊D53—21no—
67C2×C22⋊D53—21no—
68C22×Dic53—21no—
69C22×Dic53—21no—
70C2×C4×D53C2311yescharacteristic
71C2×Dic106—21no—
72C2×Dic106C2311yescharacteristic
73D8×C53—41no—
74D8×C53—41no—
75C2×C8×C53C2311yescharacteristic
76C2×C8×C53C2311yescharacteristic
77C4⋊C4×C53C2311yescharacteristic
78C22⋊C4×C53—21no—
79C42×C53C2311yescharacteristic
80C22⋊Dic53—21no—
81C22⋊Dic53—21no—
82C22⋊Dic53—21no—
83C22⋊Dic53—21no—
84C22⋊Dic53—21no—
85C22⋊Dic53—21no—
86C22⋊Dic53—21no—
87#11719923—41no—
88#11719923—41no—
89#11719921—21no—
90#11719921D411yescharacteristic
91#11719921D411yescharacteristic
92C4⋊Dic53—21no—
93C4⋊Dic53C2311yescharacteristic
94C4⋊Dic53C2311yescharacteristic
95C4⋊Dic53C2311yescharacteristic
96#11719919—21no—
97#11719919—21no—
98#11719919—21no—
99#11719919—21no—
100#11719919D411yescharacteristic
101#11719919D411yescharacteristic
102C4×Dic53—21no—
103C4×Dic53C2311yescharacteristic
104C2×#88322C2311yescharacteristic
105C2×#88322C2311yescharacteristic
106#11719912—41no—
107C8×D53—41no—
108C23×C53—21no—
109C23×C53—21no—
110C23×C53—21no—
111C22×D53C2×D411yescharacteristic
112D4×C53—21no—
113D4×C53—21no—
114D4×C53—21no—
115D4×C53—21no—
116D4×C53—21no—
117D4×C53—21no—
118D4×C53—81no—
119D4×C53—81no—
120C2×C4×C53—21no—
121C2×C4×C53—81no—
122C2×C4×C53C2×D411yescharacteristic
123C2×C4×C53C2×D411yescharacteristic
124C2×C4×C53C2411yescharacteristic
125C22⋊D53—81no—
126C22⋊D53—81no—
127C22⋊D53—81no—
128C2×Dic53—21no—
129C2×Dic53—21no—
130C2×Dic53—21no—
131C2×Dic53—21no—
132C2×Dic53—81no—
133C2×Dic53—81no—
134C2×Dic53C2×D411yescharacteristic
135C2×Dic53C2×D411yescharacteristic
136C2×Dic53C2×D411yescharacteristic
137C4×D53—21no—
138C4×D53—21no—
139Dic106—21no—
140Dic106—21no—
141Dic106—81no—
142C8×C53—21no—
143C8×C53—21no—
144#88322—21no—
145#88322—21no—
146C22×C53—41no—
147C22×C53—41no—
148C22×C53—41no—
149C22×C53—41no—
150C22×C53—41no—
151C22×C53—41no—
152C22×C53—41no—
153C22×C53—41no—
154C22×C53—41no—
155C22×C53C22×D411yescharacteristic
156C2×D53—21no—
157C2×D53—21no—
158C2×D53—21no—
159C4×C53—21no—
160C4×C53—21no—
161C4×C53—41no—
162C4×C53H(5,𝔽2)11yescharacteristic
163C4×C532−1+411yescharacteristic
164Dic53—21no—
165Dic53—21no—
166Dic53—21no—
167Dic53—41no—
168Dic53—41no—
169Dic53—41no—
170Dic53—41no—
171#3257—531nomaximal, Sylow
172C2×C53—81no—
173C2×C53—81no—
174C2×C53—81no—
175C2×C53#53711yescharacteristic
176C2×C53#57611yescharacteristic
177C2×C53#57711yescharacteristic
178D53—41no—
179#542—531no—
180#540—531no—
181#479—531no—
182#478—531no—
183#476—531no—
184#474—531no—
185#469—531no—
186#468—531no—
187#466—531no—
188#465—531no—
189#463—531no—
190#463—531no—
191#460—531no—
192#459D5311yescharacteristic, complemented
193#424—531no—
194C53#325711yescharacteristic, complemented, Sylow
195C2×SD16—531no—
196C2×SD16—531no—
197C2×D8C2×D5311yescharacteristic
198C2×D8C2×D5311yescharacteristic
199C2×M16—531no—
200C22×C8—531no—
201C4⋊Q8—531no—
202#127C2×D5311yescharacteristic
203#126—1061no—
204#125—531no—
205#124—531no—
206#124—1061no—
207#123—1061no—
208#123—1061no—
209#123—1061no—
210C22⋊Q8—531no—
211C22⋊Q8—1061no—
212#121—531no—
213#121—531no—
214#121—531no—
215#121—1061no—
216#121—1061no—
217C4×Q8—531no—
218C4×D4—531no—
219C4×D4—531no—
220C4×D4C2×D5311yescharacteristic
221#117—531no—
222#107—531no—
223#107—531no—
224#106—531no—
225#106—531no—
226C4⋊C8—531no—
227C4⋊C8—531no—
228C4⋊C8—531no—
229C4⋊C8C2×D5311yescharacteristic
230#103—531no—
231#103—531no—
232#102—531no—
233#102—531no—
234#102—531no—
235#102—531no—
236#102C2×D5311yescharacteristic
237#102C2×D5311yescharacteristic
238C2×Q8—531no—
239C2×Q8—1061no—
240C2×D4—21no—
241C2×D4—21no—
242C2×D4—1061no—
243C2×D4—1061no—
244C2×D4—1061no—
245C2×D4C22×D5311yescharacteristic
246C2×D4C22×D5311yescharacteristic
247C2×D4C22×D5311yescharacteristic
248C22×C4—21no—
249C22×C4—531no—
250C22×C4—1061no—
251C22×C4—1061no—
252SD16—2121no—
253SD16—2121no—
254D8—41no—
255D8—41no—
256M16—2121no—
257C2×C8—531no—
258C2×C8—531no—
259C2×C8—2121no—
260C2×C8C22×D5311yescharacteristic
261C2×C8C22×D5311yescharacteristic
262C4⋊C4—531no—
263C4⋊C4—531no—
264C4⋊C4—531no—
265C4⋊C4—531no—
266C4⋊C4—531no—
267C4⋊C4—1061no—
268C4⋊C4—1061no—
269C4⋊C4—1061no—
270C4⋊C4—1061no—
271C4⋊C4—1061no—
272C4⋊C4C22×D5311yescharacteristic
273C22⋊C4—21no—
274C22⋊C4—531no—
275C22⋊C4—531no—
276C22⋊C4—1061no—
277C22⋊C4—1061no—
278C22⋊C4—1061no—
279C22⋊C4—1061no—
280C22⋊C4—1061no—
281C22⋊C4—1061no—
282C22⋊C4—1061no—
283C22⋊C4—1061no—
284C42—531no—
285C42—1061no—
286C42C22×D5311yescharacteristic
287C23—21no—
288C23—21no—
289C23—21no—
290C23—531no—
291Q8—1061no—
292Q8—1061no—
293Q8—4241no—
294D4—21no—
295D4—21no—
296D4—21no—
297D4—21no—
298D4—21no—
299D4—21no—
300D4—81no—
301D4—81no—
302D4—4241no—
303D4—4241no—
304D4—4241no—
305C2×C4—21no—
306C2×C4—81no—
307C2×C4—531no—
308C2×C4—531no—
309C2×C4—531no—
310C2×C4—1061no—
311C2×C4—1061no—
312C2×C4—1061no—
313C2×C4—1061no—
314C2×C4—1061no—
315C2×C4—1061no—
316C2×C4—4241no—
317C2×C4—4241no—
318C2×C4D4×D5311yescharacteristic
319C2×C4D4×D5311yescharacteristic
320C2×C4C23×D5311yescharacteristic
321C8—21no—
322C8—21no—
323C8—1061no—
324C8—1061no—
325C22—41no—
326C22—41no—
327C22—41no—
328C22—41no—
329C22—41no—
330C22—41no—
331C22—41no—
332C22—41no—
333C22—41no—
334C22—1061no—
335C22—1061no—
336C22—1061no—
337C22C2×D4×D5311yescharacteristic
338C4—21no—
339C4—21no—
340C4—41no—
341C4—1061no—
342C4—1061no—
343C4—1061no—
344C4—2121no—
345C4—2121no—
346C4—2121no—
347C4—2121no—
348C4#1313326211yescharacteristic
349C4#1313326411yescharacteristic
350C2—81no—
351C2—81no—
352C2—81no—
353C2—2121no—
354C2#1681120211yescharacteristic
355C2#1681142411yescharacteristic
356C2#1681143311yescharacteristic
subgroupG/H
centerC22C2×D4×D53
commutatorC2×C4×C53C24
FrattiniC2×C4C23×D53
FittingC53×#459C2
socleC22×C53C22×D4
hypercenter#459D53
hypocenterC53#3257
LCSG ▷ C2×C4×C53 ▷ C2×C53 ▷ C53
UCSC1 ◁ C22 ◁ C42 ◁ #459

Complements

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

NH ≅ G/Ntype
1. #16811238C2semidirect
2. #16811227C2semidirect
3. #16810931C2semidirect
4. #16810923C2semidirect
5. #16810911C2semidirect
6. #16810901C2semidirect
7. #16810873C2semidirect
8. #16810868C2semidirect
9. #16810856C2semidirect
10. #16810851C2semidirect
11. #16810836C2semidirect
12. #16810836C2semidirect
13. #16810820C2semidirect
14. #16810648C2semidirect
15. C53×#459C2semidirect
192. #459D53semidirect
194. C53#3257semidirect