Finite Group Databasebeta

← #20287057

Subgroups — #20287057

Order 6784.

420 conjugacy classes of subgroups.

The subgroup lattice has 360 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#16811274C211yesmaximal, characteristic, complemented
2#16811273C211yesmaximal, characteristic, complemented
3#16811027C211yesmaximal, characteristic, complemented
4#16810938C211yesmaximal, characteristic, complemented
5#16810937C211yesmaximal, characteristic, complemented
6#16810935C211yesmaximal, characteristic, complemented
7#16810902C211yesmaximal, characteristic, complemented
8#16810901C211yesmaximal, characteristic, complemented
9#16810858C211yesmaximal, characteristic, complemented
10#16810857C211yesmaximal, characteristic, complemented
11#16810703C211yesmaximal, characteristic, complemented
12#16810686C211yesmaximal, characteristic, complemented
13C2×#13133150C211yesmaximal, characteristic, complemented
14#107×D53C211yesmaximal, characteristic, complemented
15C53×#480C211yesmaximal, characteristic, complemented
16C2×#11719945—21no—
17C2×#11719947—21no—
18C2×#11719922C2211yescharacteristic
19C22×#88322C2211yescharacteristic
20C2×C8×D53C2211yescharacteristic
21#13133216C2211yescharacteristic
22#13133216C2211yescharacteristic
23#13133202—21no—
24#13133200—21no—
25#13133199—21no—
26#13133198—21no—
27#13133195—21no—
28#13133194—21no—
29#13133192C2211yescharacteristic
30#13133191C2211yescharacteristic
31#13133188C2211yescharacteristic
32D4×Dic53C2211yescharacteristic
33#13133179C2211yescharacteristic
34C4×C22⋊D53C2211yescharacteristic
35C2×C4⋊Dic53C2211yescharacteristic
36C4⋊C4×D53C2211yescharacteristic
37C4⋊C4×D53C2211yescharacteristic
38#13133151C2211yescharacteristic
39#13133151C2211yescharacteristic
40#13133150C2211yescharacteristic
41#13133150C2211yescharacteristic
42#13133150C2211yescharacteristic
43#13133150C2211yescharacteristic
44#13133150C2211yescharacteristic
45#13133146C2211yescharacteristic
46#13133139C2211yescharacteristic
47#13133139C2211yescharacteristic
48#13133138C2211yescharacteristic
49#13133137C2211yescharacteristic
50#13133129C2211yescharacteristic
51C8×Dic53C2211yescharacteristic
52#121×C53C2211yescharacteristic
53C2×C4⋊C4×C53C2211yescharacteristic
54#107×C53C2211yescharacteristic
55#107×C53C2211yescharacteristic
56#102×C53C2211yescharacteristic
57#102×C53C2211yescharacteristic
58C22⋊C8×C53C2211yescharacteristic
59C2×D4×C53—21no—
60C2×D4×C53C2311yescharacteristic
61C22×C4×C53—21no—
62C22×C4×C53C2311yescharacteristic
63C2×C22⋊D53—21no—
64C2×C22⋊D53—21no—
65C2×C22⋊D53D411yescharacteristic
66C22×Dic53—21no—
67C22×Dic53—21no—
68#11719947—44no—
69#11719945—44no—
70C2×D212—21no—
71C2×D212C2311yescharacteristic
72C2×C4×D53—21no—
73C2×C4×D53—21no—
74C2×C4×D53—21no—
75C2×C4×D53C2311yescharacteristic
76C2×Dic106—21no—
77C2×Dic106—21no—
78C2×C8×C53C2311yescharacteristic
79C2×C8×C53C2311yescharacteristic
80C4⋊C4×C53—21no—
81C4⋊C4×C53C2311yescharacteristic
82C4⋊C4×C53C2311yescharacteristic
83C4⋊C4×C53C2311yescharacteristic
84C22⋊C4×C53—21no—
85C22⋊Dic53—21no—
86C22⋊Dic53—21no—
87C22⋊Dic53D411yescharacteristic
88#11719922—22no—
89#11719921—21no—
90#11719921—21no—
91#11719921—21no—
92#11719921D411yescharacteristic
93C4⋊Dic53—21no—
94C4⋊Dic53C2311yescharacteristic
95C4⋊Dic53C2311yescharacteristic
96C4⋊Dic53C2311yescharacteristic
97#11719919—21no—
98#11719919—21no—
99#11719919—21no—
100#11719919—21no—
101#11719919D411yescharacteristic
102C4×Dic53C2311yescharacteristic
103C2×#88322—22no—
104C2×#88322C2311yescharacteristic
105C2×#88322C2311yescharacteristic
106C8×D53—22no—
107C23×C53—21no—
108C23×C53C2×D411yescharacteristic
109C22×D53—21no—
110C22×D53C2×D411yescharacteristic
111D4×C53—21no—
112D4×C53—21no—
113D4×C53—42no—
114C2×C4×C53—21no—
115C2×C4×C53—21no—
116C2×C4×C53—21no—
117C2×C4×C53—21no—
118C2×C4×C53—21no—
119C2×C4×C53—42no—
120C2×C4×C53C2×D411yescharacteristic
121C2×C4×C53C2411yescharacteristic
122C22⋊D53—42no—
123C22⋊D53—42no—
124C22⋊D53D814yescomplemented
125C2×Dic53—21no—
126C2×Dic53—21no—
127C2×Dic53—21no—
128C2×Dic53—42no—
129C2×Dic53—42no—
130C2×Dic53C2×D411yescharacteristic
131C2×Dic53C2×D411yescharacteristic
132C2×Dic53C2×D411yescharacteristic
133D212—21no—
134D212—21no—
135D212—42no—
136C4×D53—21no—
137C4×D53—21no—
138C4×D53—42no—
139C4×D53—42no—
140C4×D53—42no—
141Dic106—42no—
142Dic106—42no—
143C8×C53—21no—
144C8×C53—21no—
145#88322—21no—
146#88322#4112yescomplemented
147C22×C53—21no—
148C22×C53—21no—
149C22×C53—41no—
150C22×C53—41no—
151C22×C53—41no—
152C22×C53C2×D812yes—
153C22×C53C22×D411yescharacteristic
154C2×D53—21no—
155C2×D53—21no—
156C2×D53—41no—
157C2×D53—41no—
158C2×D53—41no—
159C2×D53C2×D812yes—
160C4×C53—21no—
161C4×C53—41no—
162C4×C53—41no—
163C4×C53—41no—
164C4×C53C2×#4111yescharacteristic
165C4×C532−1+411yescharacteristic
166Dic53—21no—
167Dic53—21no—
168Dic53—41no—
169Dic53—41no—
170Dic53—41no—
171Dic53C2×D812yes—
172#3334—531nomaximal, Sylow
173C2×C53—22no—
174C2×C53—81no—
175C2×C53#54711yescharacteristic
176C2×C53C22×D811yescharacteristic
177C2×C53#57811yescharacteristic
178D53—22no—
179D53—81no—
180#547—531no—
181#547—531no—
182#500—531no—
183#480—531no—
184#480—531no—
185#480—531no—
186#480D5311yescharacteristic, complemented
187#474—531no—
188#474—531no—
189#466—531no—
190#466—531no—
191C4×D8—531no—
192C8×D4—531no—
193C2×#107—531no—
194C2×#107—531no—
195C53#333411yescharacteristic, complemented, Sylow
196C2×#41—1061no—
197C2×#41—1061no—
198C2×D8—531no—
199C22×C8—531no—
200C22×C8—531no—
201C4⋊Q8—531no—
202C4⋊Q8—531no—
203#123—1061no—
204#123—1061no—
205#123—1061no—
206#123—1061no—
207C22⋊Q8—1061no—
208C22⋊Q8—1061no—
209#121—531no—
210#121—531no—
211#121—531no—
212#121C2×D5311yescharacteristic
213C4×D4—531no—
214C4×D4—531no—
215C4×D4—531no—
216C2×C4⋊C4—531no—
217C2×C4⋊C4—531no—
218C2×C4⋊C4—531no—
219C2×C4⋊C4C2×D5311yescharacteristic
220#107—531no—
221#107—531no—
222#107—531no—
223#107—531no—
224#107—531no—
225#107—531no—
226#107—531no—
227#107C2×D5311yescharacteristic
228#107C2×D5311yescharacteristic
229C4⋊C8—531no—
230#102—531no—
231#102—531no—
232#102—531no—
233#102—531no—
234#102C2×D5311yescharacteristic
235#102C2×D5311yescharacteristic
236C22⋊C8—531no—
237C22⋊C8C2×D5311yescharacteristic
238C4×C8—531no—
239#41—2124no—
240#41—2124no—
241C2×Q8—1061no—
242C2×Q8—1061no—
243C2×D4—21no—
244C2×D4—531no—
245C2×D4—531no—
246C2×D4—1061no—
247C2×D4—1061no—
248C2×D4—1061no—
249C2×D4C22×D5311yescharacteristic
250C22×C4—21no—
251C22×C4—531no—
252C22×C4—1061no—
253C22×C4—1061no—
254C22×C4—1061no—
255C22×C4—1061no—
256C22×C4—1061no—
257C22×C4C22×D5311yescharacteristic
258D8—1062no—
259C2×C8—531no—
260C2×C8—531no—
261C2×C8—1062no—
262C2×C8—1062no—
263C2×C8C22×D5311yescharacteristic
264C2×C8C22×D5311yescharacteristic
265C4⋊C4—21no—
266C4⋊C4—531no—
267C4⋊C4—531no—
268C4⋊C4—531no—
269C4⋊C4—531no—
270C4⋊C4—1061no—
271C4⋊C4—1061no—
272C4⋊C4—1061no—
273C4⋊C4—1061no—
274C4⋊C4—1061no—
275C4⋊C4C22×D5311yescharacteristic
276C4⋊C4C22×D5311yescharacteristic
277C4⋊C4C22×D5311yescharacteristic
278C22⋊C4—21no—
279C22⋊C4—531no—
280C22⋊C4—531no—
281C22⋊C4—1061no—
282C22⋊C4—1061no—
283C22⋊C4—1061no—
284C22⋊C4—1061no—
285C22⋊C4—1061no—
286C42—531no—
287C23—21no—
288C23—531no—
289C23—1061no—
290C23D4×D5311yescharacteristic
291Q8—2122no—
292Q8—2122no—
293D4—21no—
294D4—21no—
295D4—42no—
296D4—534no—
297D4—1061no—
298D4—1061no—
299D4—2122no—
300D4—2122no—
301D4—2122no—
302C2×C4—21no—
303C2×C4—21no—
304C2×C4—21no—
305C2×C4—21no—
306C2×C4—21no—
307C2×C4—42no—
308C2×C4—531no—
309C2×C4—531no—
310C2×C4—531no—
311C2×C4—1061no—
312C2×C4—1061no—
313C2×C4—1061no—
314C2×C4—1061no—
315C2×C4—1061no—
316C2×C4—2122no—
317C2×C4—2122no—
318C2×C4—2122no—
319C2×C4—2122no—
320C2×C4—2122no—
321C2×C4D4×D5311yescharacteristic
322C2×C4C23×D5311yescharacteristic
323C8—21no—
324C8—21no—
325C8—532no—
326C8—1061no—
327C22—21no—
328C22—21no—
329C22—41no—
330C22—41no—
331C22—41no—
332C22—532no—
333C22—1061no—
334C22—1061no—
335C22—2121no—
336C22—2121no—
337C22—2121no—
338C22D8×D5312yes—
339C22C2×D4×D5311yescharacteristic
340C4—21no—
341C4—41no—
342C4—41no—
343C4—41no—
344C4—532no—
345C4—1061no—
346C4—1061no—
347C4—2121no—
348C4—2121no—
349C4—2121no—
350C4#41×D5311yescharacteristic
351C4#1313326411yescharacteristic
352C2—22no—
353C2—81no—
354C2—1062no—
355C2—4241no—
356C2#1681127111yescharacteristic
357C2C2×D8×D5311yescharacteristic
358C2#1681143911yescharacteristic
subgroupG/H
centerC22C2×D4×D53
commutatorC2×C4×C53C24
FrattiniC2×C4C23×D53
FittingC53×#480C2
socleC22×C53C22×D4
hypercenter#480D53
hypocenterC53#3334
LCSG ▷ C2×C4×C53 ▷ C2×C53 ▷ C53
UCSC1 ◁ C22 ◁ C22×C4 ◁ #480

Complements

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

NH ≅ G/Ntype
1. #16811274C2semidirect
2. #16811273C2semidirect
3. #16811027C2semidirect
4. #16810938C2semidirect
5. #16810937C2semidirect
6. #16810935C2semidirect
7. #16810902C2semidirect
8. #16810901C2semidirect
9. #16810858C2semidirect
10. #16810857C2semidirect
11. #16810703C2semidirect
12. #16810686C2semidirect
13. C2×#13133150C2semidirect
14. #107×D53C2semidirect
15. C53×#480C2semidirect
124. C22⋊D53D8semidirect
146. #88322#41semidirect
186. #480D53semidirect
195. C53#3334semidirect