Finite Group Databasebeta

← #20285796

Subgroups — #20285796

Order 6784.

364 conjugacy classes of subgroups.

The subgroup lattice has 356 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#16811233C211yesmaximal, characteristic, complemented
2#16811194C211yesmaximal, characteristic
3#16811014C211yesmaximal, characteristic, complemented
4#16811001C211yesmaximal, characteristic, complemented
5#16810990C211yesmaximal, characteristic, complemented
6#16810976C211yesmaximal, characteristic, complemented
7#16810844C211yesmaximal, characteristic, complemented
8#16810843C211yesmaximal, characteristic, complemented
9#16810832C211yesmaximal, characteristic, complemented
10#16810831C211yesmaximal, characteristic, complemented
11#16810785C211yesmaximal, characteristic
12#16810781C211yesmaximal, characteristic, complemented
13#16810778C211yesmaximal, characteristic, complemented
14#16810681C211yesmaximal, characteristic, complemented
15C53×#452C211yesmaximal, characteristic
16C22×Dic106C2211yescharacteristic
17C2×Dic212C2211yescharacteristic
18C2×Dic212C2211yescharacteristic
19C2×#11719925C2211yescharacteristic
20C2×#11719925C2211yescharacteristic
21C2×#11719925C2211yescharacteristic
22C2×#11719923C2211yescharacteristic
23#13133215C2211yescharacteristic
24#13133215C2211yescharacteristic
25#13133215C2211yescharacteristic
26#13133210—21no—
27#13133210—21no—
28#13133208—21no—
29#13133208C2211yescharacteristic
30#13133207—21no—
31#13133204C2211yescharacteristic
32#13133204C2211yescharacteristic
33#13133198—21no—
34#13133198—21no—
35#13133196—21no—
36#13133196—21no—
37#13133196C2211yescharacteristic
38Q8×Dic53C2211yescharacteristic
39#13133182C2211yescharacteristic
40#13133176C2211yescharacteristic
41#13133146C2211yescharacteristic
42#13133146C2211yescharacteristic
43#13133142C2211yescharacteristic
44#13133142C2211yescharacteristic
45#13133141C2211yescharacteristic
46#13133140C2211yescharacteristic
47#13133140C2211yescharacteristic
48#13133138C2211yescharacteristic
49C22⋊#88322C2211yescharacteristic
50#13133127C2211yescharacteristic
51C8×Dic53C2211yescharacteristic
52C2×#41×C53C2211yescharacteristic
53C2×Q16×C53C2211yescharacteristic
54C2×SD16×C53C2211yescharacteristic
55C22⋊Q8×C53C2211yescharacteristic
56#103×C53C2211yescharacteristic
57#102×C53C2211yescharacteristic
58C22⋊C8×C53C2211yescharacteristic
59#41×C53—41no—
60#41×C53—41no—
61C2×Q8×C53C2311yescharacteristic
62C2×Q8×C53C2311yescharacteristic
63C2×D4×C53—21no—
64C2×D4×C53C2311yescharacteristic
65C22×C4×C53—21no—
66C22×C4×C53C2311yescharacteristic
67C22×Dic53—21no—
68C2×Dic106—21no—
69C2×Dic106—41no—
70C2×Dic106—41no—
71C2×Dic106C2311yescharacteristic
72C2×Dic106C2311yescharacteristic
73C2×Dic106C2311yescharacteristic
74Q16×C53—41no—
75SD16×C53—41no—
76C2×C8×C53C2311yescharacteristic
77C2×C8×C53C2311yescharacteristic
78C4⋊C4×C53—21no—
79C4⋊C4×C53C2311yescharacteristic
80C22⋊C4×C53—21no—
81C22⋊Dic53—21no—
82C22⋊Dic53—21no—
83C22⋊Dic53—21no—
84C22⋊Dic53—21no—
85C22⋊Dic53—21no—
86C22⋊Dic53D411yescharacteristic
87C22⋊Dic53D411yescharacteristic
88#11719925—41no—
89#11719925—41no—
90#11719925—41no—
91#11719923—41no—
92C4⋊Dic53—21no—
93C4⋊Dic53—21no—
94C4⋊Dic53D411yescharacteristic
95C4⋊Dic53D411yescharacteristic
96C4⋊Dic53C2311yescharacteristic
97#11719919—21no—
98#11719919—21no—
99#11719919—21no—
100#11719919—21no—
101#11719919—21no—
102#11719919—21no—
103#11719919—21no—
104#11719919—21no—
105#11719919—21no—
106C4×Dic53—21no—
107C4×Dic53—21no—
108C4×Dic53C2311yescharacteristic
109C4×Dic53C2311yescharacteristic
110C2×#88322C2311yescharacteristic
111C2×#88322C2311yescharacteristic
112Dic212—41no—
113Dic212—41no—
114C23×C53—21no—
115C23×C53C2×D411yescharacteristic
116Q8×C53—21no—
117Q8×C53—21no—
118Q8×C53—21no—
119Q8×C53—21no—
120D4×C53—21no—
121D4×C53—21no—
122D4×C53—42no—
123C2×C4×C53—21no—
124C2×C4×C53—21no—
125C2×C4×C53—21no—
126C2×C4×C53—21no—
127C2×C4×C53—21no—
128C2×C4×C53—42no—
129C2×C4×C53C2×D411yescharacteristic
130C2×C4×C53C2411yescharacteristic
131C2×Dic53—21no—
132C2×Dic53—21no—
133C2×Dic53—21no—
134C2×Dic53—21no—
135C2×Dic53—42no—
136C2×Dic53C2×D411yescharacteristic
137C2×Dic53C2×D411yescharacteristic
138C2×Dic53C2×D411yescharacteristic
139C2×Dic53C2×D411yescharacteristic
140Dic106—21no—
141Dic106—21no—
142Dic106—21no—
143Dic106—21no—
144Dic106—21no—
145Dic106—21no—
146Dic106—42no—
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
158C4×C53—21no—
159C4×C53—41no—
160C4×C53—41no—
161C4×C53—41no—
162C4×C53H(5,𝔽2)11yescharacteristic
163C4×C53H(5,𝔽2)11yescharacteristic
164Dic53—21no—
165Dic53—21no—
166Dic53—21no—
167Dic53—21no—
168Dic53—41no—
169Dic53—41no—
170Dic53—41no—
171Dic53—41no—
172#3223—531nomaximal, Sylow
173C2×C53—41no—
174C2×C53—81no—
175C2×C53#53511yescharacteristic
176C2×C53#57811yescharacteristic
177C2×C53#57811yescharacteristic
178#541—531no—
179#536—531no—
180#497—531no—
181#494—531no—
182#491—531no—
183#488—531no—
184#464—531no—
185#464—531no—
186#462—531no—
187#462—531no—
188#452—531no—
189#452D5311yescharacteristic
190#451—531no—
191#451—531no—
192#433—531no—
193C53#322311yescharacteristic, complemented, Sylow
194C2×#41C2×D5311yescharacteristic
195C22×Q8—531no—
196C2×Q16—531no—
197C2×Q16—531no—
198C2×Q16—531no—
199C2×Q16—531no—
200C2×Q16—531no—
201C2×Q16C2×D5311yescharacteristic
202C2×SD16—531no—
203C2×SD16C2×D5311yescharacteristic
204C4⋊Q8—531no—
205C4⋊Q8—531no—
206C4⋊Q8—531no—
207#126—1061no—
208#126—1061no—
209#125—531no—
210#125—1061no—
211#125—1061no—
212#124—531no—
213#124—531no—
214#123—1061no—
215#123—1061no—
216C22⋊Q8—531no—
217C22⋊Q8—1061no—
218C22⋊Q8—1061no—
219C22⋊Q8C2×D5311yescharacteristic
220C4×Q8—531no—
221C4×Q8—531no—
222#117—531no—
223C4⋊C8—531no—
224C4⋊C8—531no—
225#103—531no—
226#103—531no—
227#103—531no—
228#103—531no—
229#103—531no—
230#103C2×D5311yescharacteristic
231#102—531no—
232#102C2×D5311yescharacteristic
233C22⋊C8—531no—
234C22⋊C8C2×D5311yescharacteristic
235#97—531no—
236C4×C8—531no—
237#41—41no—
238#41—41no—
239C2×Q8—531no—
240C2×Q8—531no—
241C2×Q8—531no—
242C2×Q8—1061no—
243C2×Q8—2121no—
244C2×Q8—2121no—
245C2×Q8C22×D5311yescharacteristic
246C2×Q8C22×D5311yescharacteristic
247C2×D4—21no—
248C2×D4C22×D5311yescharacteristic
249C22×C4—21no—
250C22×C4—1061no—
251C22×C4C22×D5311yescharacteristic
252Q16—41no—
253Q16—2121no—
254Q16—2121no—
255Q16—2121no—
256Q16—2121no—
257Q16—2121no—
258SD16—41no—
259SD16—2121no—
260C2×C8—531no—
261C2×C8—531no—
262C2×C8C22×D5311yescharacteristic
263C2×C8C22×D5311yescharacteristic
264C4⋊C4—21no—
265C4⋊C4—531no—
266C4⋊C4—531no—
267C4⋊C4—531no—
268C4⋊C4—1061no—
269C4⋊C4—1061no—
270C4⋊C4—1061no—
271C4⋊C4—1061no—
272C4⋊C4—1061no—
273C4⋊C4—1061no—
274C4⋊C4—1061no—
275C4⋊C4—1061no—
276C4⋊C4—1061no—
277C4⋊C4—1061no—
278C4⋊C4—1061no—
279C4⋊C4C22×D5311yescharacteristic
280C22⋊C4—21no—
281C22⋊C4—531no—
282C22⋊C4—531no—
283C22⋊C4—1061no—
284C22⋊C4—1061no—
285C22⋊C4—1061no—
286C22⋊C4—1061no—
287C22⋊C4—1061no—
288C42—531no—
289C42—531no—
290C42—1061no—
291C42—1061no—
292C23—21no—
293C23D4×D5311yescharacteristic
294Q8—21no—
295Q8—21no—
296Q8—21no—
297Q8—21no—
298Q8—1061no—
299Q8—1061no—
300Q8—1061no—
301Q8—1061no—
302Q8—1061no—
303Q8—1061no—
304Q8—2122no—
305D4—21no—
306D4—21no—
307D4—42no—
308C2×C4—21no—
309C2×C4—21no—
310C2×C4—21no—
311C2×C4—21no—
312C2×C4—21no—
313C2×C4—42no—
314C2×C4—531no—
315C2×C4—531no—
316C2×C4—531no—
317C2×C4—531no—
318C2×C4—1061no—
319C2×C4—1061no—
320C2×C4—1061no—
321C2×C4—1061no—
322C2×C4—2122no—
323C2×C4D4×D5311yescharacteristic
324C2×C4C23×D5311yescharacteristic
325C8—21no—
326C8—21no—
327C8—1061no—
328C8—1061no—
329C22—21no—
330C22—21no—
331C22—21no—
332C22—41no—
333C22—41no—
334C22—41no—
335C22C2×D4×D5311yescharacteristic
336C4—21no—
337C4—41no—
338C4—41no—
339C4—41no—
340C4—1061no—
341C4—1061no—
342C4—1061no—
343C4—1061no—
344C4—2121no—
345C4—2121no—
346C4—2121no—
347C4—2121no—
348C4#1313326211yescharacteristic
349C4#1313326211yescharacteristic
350C2—41no—
351C2—81no—
352C2#1681118811yescharacteristic
353C2#1681143711yescharacteristic
354C2#1681143911yescharacteristic
subgroupG/H
centerC22C2×D4×D53
commutatorC2×C4×C53C24
FrattiniC2×C4C23×D53
FittingC53×#452C2
socleC22×C53C22×D4
hypercenter#452D53
hypocenterC53#3223
LCSG ▷ C2×C4×C53 ▷ C2×C53 ▷ C53
UCSC1 ◁ C22 ◁ C22×C4 ◁ #452

Complements

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

NH ≅ G/Ntype
1. #16811233C2semidirect
3. #16811014C2semidirect
4. #16811001C2semidirect
5. #16810990C2semidirect
6. #16810976C2semidirect
7. #16810844C2semidirect
8. #16810843C2semidirect
9. #16810832C2semidirect
10. #16810831C2semidirect
12. #16810781C2semidirect
13. #16810778C2semidirect
14. #16810681C2semidirect
193. C53#3223semidirect