Finite Group Databasebeta

← #20286796

Subgroups — #20286796

Order 6784.

410 conjugacy classes of subgroups.

The subgroup lattice has 376 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#16811330C211yesmaximal, characteristic, complemented
2#16810982C211yesmaximal, characteristic, complemented
3#16810956C211yesmaximal, characteristic, complemented
4#16810934C211yesmaximal, characteristic, complemented
5#16810903C211yesmaximal, characteristic, complemented
6#16810867C211yesmaximal, characteristic, complemented
7#16810862C211yesmaximal, characteristic, complemented
8#16810842C211yesmaximal, characteristic, complemented
9#16810785C211yesmaximal, characteristic, complemented
10#16810774C211yesmaximal, characteristic, complemented
11#16810723C211yesmaximal, characteristic, complemented
12#16810695C211yesmaximal, characteristic, complemented
13#16810624C211yesmaximal, characteristic, complemented
14C2×#13133138C211yesmaximal, characteristic, complemented
15C53×#546C211yesmaximal, characteristic
16C2×#11719925C2211yescharacteristic, complemented
17C2×#11719923C2211yescharacteristic
18C2×#11719923C2211yescharacteristic
19C2×#11719917C2211yescharacteristic, complemented
20C22×#88322C2211yescharacteristic
21#13133217C2211yescharacteristic, complemented
22#13133208—21no—
23#13133196C2211yescharacteristic
24#13133196C2211yescharacteristic, complemented
25#13133194—21no—
26#13133194—21no—
27C4×Dic106C2211yescharacteristic, complemented
28D4×Dic53—21no—
29C2×C4⋊Dic53C2211yescharacteristic
30C2×C4⋊Dic53C2211yescharacteristic, complemented
31#13133150C2211yescharacteristic, complemented
32#13133150C2211yescharacteristic, complemented
33#13133150C2211yescharacteristic, complemented
34C4⋊#88322C2211yescharacteristic, complemented
35#13133141C2211yescharacteristic, complemented
36#13133141C2211yescharacteristic, complemented
37#13133141C2211yescharacteristic, complemented
38#13133140C2211yescharacteristic, complemented
39#13133138C2211yescharacteristic
40#13133138C2211yescharacteristic
41#13133138C2211yescharacteristic
42#13133138C2211yescharacteristic
43#13133138C2211yescharacteristic
44#13133138C2211yescharacteristic
45C22⋊#88322C2211yescharacteristic
46C22⋊#88322C2211yescharacteristic, complemented
47#13133128C2211yescharacteristic, complemented
48C2×#41×C53C2211yescharacteristic
49C22×D4×C53C2211yescharacteristic
50#124×C53C2211yescharacteristic
51#123×C53—21no—
52C22⋊Q8×C53C2211yescharacteristic
53#121×C53—21no—
54#121×C53C2211yescharacteristic
55#120×C53—21no—
56C4×D4×C53C2211yescharacteristic
57C4×D4×C53C2211yescharacteristic
58C2×C22⋊C4×C53—21no—
59C24×C53—21no—
60#41×C53—22no—
61#41×C53—41no—
62C2×Q8×C53C2311yescharacteristic
63C2×D4×C53—22no—
64C2×D4×C53—21no—
65C2×D4×C53—21no—
66C2×D4×C53—21no—
67C2×D4×C53—41no—
68C2×D4×C53D411yescharacteristic
69C2×D4×C53C2311yescharacteristic
70C2×D4×C53C2311yescharacteristic
71C22×C4×C53—21no—
72C22×C4×C53—21no—
73C22×C4×C53C2311yescharacteristic
74C22×C4×C53C2311yescharacteristic
75C22×Dic53—21no—
76C22×Dic53—21no—
77C2×Dic106—21no—
78C2×Dic106C2311yescharacteristic
79C4⋊C4×C53—21no—
80C4⋊C4×C53D411yescharacteristic
81C4⋊C4×C53C2311yescharacteristic
82C22⋊C4×C53—21no—
83C22⋊C4×C53—21no—
84C22⋊C4×C53—21no—
85C22⋊C4×C53—21no—
86C22⋊C4×C53—21no—
87C22⋊C4×C53D411yescharacteristic
88C22⋊C4×C53D411yescharacteristic
89C42×C53C2311yescharacteristic
90C22⋊Dic53—21no—
91C22⋊Dic53—21no—
92#11719925—41no—
93#11719923—22no—
94#11719923—41no—
95C4⋊Dic53—21no—
96C4⋊Dic53—21no—
97C4⋊Dic53—21no—
98C4⋊Dic53C2311yescharacteristic
99C4⋊Dic53C2311yescharacteristic
100C4⋊Dic53C2311yescharacteristic
101#11719919—21no—
102#11719919—21no—
103#11719919—21no—
104C4×Dic53—21no—
105#11719917—41no—
106C2×#88322—41no—
107C2×#88322C2311yescharacteristic
108C2×#88322C2311yescharacteristic
109C2×#88322C2311yescharacteristic
110C2×#88322C2311yescharacteristic
111C23×C53—21no—
112C23×C53—21no—
113C23×C53—42no—
114C23×C53—42no—
115C23×C53—81no—
116C23×C53C2×D411yescharacteristic
117C23×C53C2×D411yescharacteristic
118Q8×C53—21no—
119Q8×C53—21no—
120D4×C53—22no—
121D4×C53—21no—
122D4×C53—21no—
123D4×C53—21no—
124D4×C53—42no—
125D4×C53—42no—
126D4×C53—81no—
127D4×C53#4112yes—
128C2×C4×C53—21no—
129C2×C4×C53—21no—
130C2×C4×C53—21no—
131C2×C4×C53—21no—
132C2×C4×C53—21no—
133C2×C4×C53—21no—
134C2×C4×C53—42no—
135C2×C4×C53—81no—
136C2×C4×C53C2×D411yescharacteristic
137C2×C4×C53C2×D411yescharacteristic
138C2×C4×C53C2×D411yescharacteristic
139C2×C4×C53C2×D411yescharacteristic
140C2×C4×C53C2411yescharacteristic
141C2×Dic53—21no—
142C2×Dic53—21no—
143C2×Dic53—21no—
144C2×Dic53—21no—
145C2×Dic53—81no—
146C2×Dic53—81no—
147Dic106—21no—
148Dic106—21no—
149Dic106—81no—
150#88322—21no—
151#88322—21no—
152#88322—21no—
153#88322—21no—
154C22×C53—21no—
155C22×C53—21no—
156C22×C53—21no—
157C22×C53—22no—
158C22×C53—21no—
159C22×C53—21no—
160C22×C53—41no—
161C22×C53—41no—
162C22×C53—41no—
163C22×C53—41no—
164C22×C53—41no—
165C22×C53—84no—
166C22×C53#13712yes—
167C22×C53C22×D411yescharacteristic
168C4×C53—21no—
169C4×C53—21no—
170C4×C53—21no—
171C4×C53—21no—
172C4×C53—41no—
173C4×C53—41no—
174C4×C53C2×#4111yescharacteristic
175C4×C53H(5,𝔽2)11yescharacteristic
176Dic53—41no—
177Dic53—41no—
178Dic53—41no—
179Dic53—41no—
180#3314—531nomaximal, Sylow
181C2×C53—22no—
182C2×C53—41no—
183C2×C53—42no—
184C2×C53—81no—
185C2×C53#54611yescharacteristic
186C2×C53C2×#13711yescharacteristic
187C2×C53#57611yescharacteristic
188#554—531no—
189#546D5311yescharacteristic
190#489—531no—
191#484—531no—
192#480—531no—
193#474—531no—
194#468—531no—
195#467—531no—
196#464—531no—
197#452—531no—
198#450—531no—
199#440—531no—
200#435—531no—
201#417—531no—
202C2×#102—531no—
203C53#331411yescharacteristic, complemented, Sylow
204C2×#41C2×D5311yescharacteristic
205C22×D4C2×D5311yescharacteristic
206C2×Q16—531no—
207C2×SD16—531no—
208C2×SD16—531no—
209C2×M16—531no—
210C22×C8—531no—
211C4⋊Q8—531no—
212#125—1061no—
213#124C2×D5311yescharacteristic
214#123—21no—
215C22⋊Q8—531no—
216C22⋊Q8—531no—
217C22⋊Q8—1061no—
218C22⋊Q8—1061no—
219C22⋊Q8C2×D5311yescharacteristic
220#121—21no—
221#121C2×D5311yescharacteristic
222#120—21no—
223C4×Q8—531no—
224C4×D4—1061no—
225C4×D4C2×D5311yescharacteristic
226C4×D4C2×D5311yescharacteristic
227C2×C4⋊C4—531no—
228C2×C4⋊C4—531no—
229C2×C22⋊C4—21no—
230#107—531no—
231#107—531no—
232#107—531no—
233C4⋊C8—531no—
234#103—531no—
235#103—531no—
236#103—531no—
237#103—531no—
238#102—531no—
239#102—531no—
240#102—531no—
241#102—531no—
242#102—531no—
243#102—531no—
244C22⋊C8—531no—
245C22⋊C8—531no—
246#97—531no—
247C24—21no—
248#41—22no—
249#41—41no—
250C2×Q8—531no—
251C2×Q8—1061no—
252C2×Q8C22×D5311yescharacteristic
253C2×D4—22no—
254C2×D4—21no—
255C2×D4—21no—
256C2×D4—21no—
257C2×D4—41no—
258C2×D4C22⋊D5311yescharacteristic
259C2×D4C22×D5311yescharacteristic
260C2×D4C22×D5311yescharacteristic
261C22×C4—21no—
262C22×C4—21no—
263C22×C4—1061no—
264C22×C4—1061no—
265C22×C4C22×D5311yescharacteristic
266C22×C4C22×D5311yescharacteristic
267Q16—2121no—
268SD16—1062no—
269SD16—2121no—
270M16—2121no—
271C2×C8—531no—
272C2×C8—531no—
273C2×C8—531no—
274C2×C8—531no—
275C2×C8—2121no—
276C4⋊C4—21no—
277C4⋊C4—531no—
278C4⋊C4—531no—
279C4⋊C4—531no—
280C4⋊C4—1061no—
281C4⋊C4—1061no—
282C4⋊C4—1061no—
283C4⋊C4—1061no—
284C4⋊C4—1061no—
285C4⋊C4—1061no—
286C4⋊C4C22⋊D5311yescharacteristic
287C4⋊C4C22×D5311yescharacteristic
288C22⋊C4—21no—
289C22⋊C4—21no—
290C22⋊C4—21no—
291C22⋊C4—21no—
292C22⋊C4—21no—
293C22⋊C4—1061no—
294C22⋊C4—1061no—
295C22⋊C4C22⋊D5311yescharacteristic
296C22⋊C4C22⋊D5311yescharacteristic
297C42—1061no—
298C42C22×D5311yescharacteristic
299C23—21no—
300C23—21no—
301C23—42no—
302C23—42no—
303C23—81no—
304C23C2×C22⋊D5311yescharacteristic
305C23C2×C22⋊D5311yescharacteristic
306Q8—21no—
307Q8—21no—
308Q8—1061no—
309Q8—1061no—
310Q8—4241no—
311D4—22no—
312D4—21no—
313D4—21no—
314D4—21no—
315D4—42no—
316D4—42no—
317D4—81no—
318D4#1171994712yes—
319C2×C4—21no—
320C2×C4—21no—
321C2×C4—21no—
322C2×C4—21no—
323C2×C4—21no—
324C2×C4—21no—
325C2×C4—42no—
326C2×C4—81no—
327C2×C4—1061no—
328C2×C4—1061no—
329C2×C4—1061no—
330C2×C4—1061no—
331C2×C4—4241no—
332C2×C4—4241no—
333C2×C4C2×C22⋊D5311yescharacteristic
334C2×C4C2×C22⋊D5311yescharacteristic
335C2×C4C2×C22⋊D5311yescharacteristic
336C2×C4C2×C22⋊D5311yescharacteristic
337C2×C4C23×D5311yescharacteristic
338C8—1061no—
339C8—1061no—
340C8—1061no—
341C8—1061no—
342C22—21no—
343C22—21no—
344C22—21no—
345C22—22no—
346C22—21no—
347C22—21no—
348C22—41no—
349C22—41no—
350C22—41no—
351C22—41no—
352C22—41no—
353C22—84no—
354C22#1313324712yes—
355C22C22×C22⋊D5311yescharacteristic
356C4—21no—
357C4—21no—
358C4—21no—
359C4—21no—
360C4—41no—
361C4—41no—
362C4—2121no—
363C4—2121no—
364C4—2121no—
365C4—2121no—
366C4C2×#1171994711yescharacteristic
367C4#1313326211yescharacteristic
368C2—22no—
369C2—41no—
370C2—42no—
371C2—81no—
372C2#1681126711yescharacteristic
373C2C2×#1313324711yescharacteristic
374C2#1681142611yescharacteristic
subgroupG/H
centerC22C22×C22⋊D53
commutatorC2×C4×C53C24
FrattiniC2×C4C23×D53
FittingC53×#546C2
socleC22×C53C22×D4
hypercenter#546D53
hypocenterC53#3314
LCSG ▷ C2×C4×C53 ▷ C2×C53 ▷ C53
UCSC1 ◁ C22 ◁ C4×D4 ◁ #546

Complements

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

NH ≅ G/Ntype
1. #16811330C2semidirect
2. #16810982C2semidirect
3. #16810956C2semidirect
4. #16810934C2semidirect
5. #16810903C2semidirect
6. #16810867C2semidirect
7. #16810862C2semidirect
8. #16810842C2semidirect
9. #16810785C2semidirect
10. #16810774C2semidirect
11. #16810723C2semidirect
12. #16810695C2semidirect
13. #16810624C2semidirect
14. C2×#13133138C2semidirect
16. C2×#11719925C22semidirect
19. C2×#11719917C22semidirect
21. #13133217C22semidirect
24. #13133196C22semidirect
27. C4×Dic106C22semidirect
30. C2×C4⋊Dic53C22semidirect
31. #13133150C22semidirect
32. #13133150C22semidirect
33. #13133150C22semidirect
34. C4⋊#88322C22semidirect
35. #13133141C22semidirect
36. #13133141C22semidirect
37. #13133141C22semidirect
38. #13133140C22semidirect
46. C22⋊#88322C22semidirect
47. #13133128C22semidirect
203. C53#3314semidirect