Finite Group Databasebeta

← #20286886

Subgroups — #20286886

Order 6784.

388 conjugacy classes of subgroups.

The subgroup lattice has 362 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#16811319C211yesmaximal, characteristic
2#16810991C211yesmaximal, characteristic, complemented
3#16810961C211yesmaximal, characteristic, complemented
4#16810956C211yesmaximal, characteristic, complemented
5#16810930C211yesmaximal, characteristic, complemented
6#16810871C211yesmaximal, characteristic, complemented
7#16810854C211yesmaximal, characteristic, complemented
8#16810825C211yesmaximal, characteristic, complemented
9#16810762C211yesmaximal, characteristic, complemented
10#16810762C211yesmaximal, characteristic, complemented
11#16810721C211yesmaximal, characteristic, complemented
12#16810695C211yesmaximal, characteristic, complemented
13#16810632C211yesmaximal, characteristic, complemented
14C2×#13133141C211yesmaximal, characteristic, complemented
15D4×Q8×C53C211yesmaximal, characteristic, complemented
16C2×#11719924C2211yescharacteristic
17C2×#11719924C2211yescharacteristic
18C2×#11719924C2211yescharacteristic
19C2×#11719917C2211yescharacteristic
20C22×#88322C2211yescharacteristic
21#13133212—21no—
22#13133209C2211yescharacteristic
23#13133199—21no—
24#13133191C2211yescharacteristic
25#13133191C2211yescharacteristic
26#13133189—21no—
27C4×D212C2211yescharacteristic
28#13133177—21no—
29#13133176C2211yescharacteristic
30C2×C4⋊Dic53C2211yescharacteristic
31#13133150C2211yescharacteristic
32#13133150C2211yescharacteristic
33#13133148C2211yescharacteristic
34C4⋊#88322C2211yescharacteristic
35#13133141C2211yescharacteristic
36#13133141C2211yescharacteristic
37#13133141C2211yescharacteristic
38#13133141C2211yescharacteristic
39#13133141C2211yescharacteristic
40#13133141C2211yescharacteristic
41#13133141C2211yescharacteristic
42#13133141C2211yescharacteristic
43#13133141C2211yescharacteristic
44#13133137C2211yescharacteristic
45C22⋊#88322C2211yescharacteristic
46C22⋊#88322C2211yescharacteristic
47#13133128C2211yescharacteristic
48C22×Q8×C53C2211yescharacteristic
49C22×Q8×C53C2211yescharacteristic
50C4⋊Q8×C53—21no—
51C4⋊Q8×C53C2211yescharacteristic
52C22⋊Q8×C53—21no—
53C22⋊Q8×C53—21no—
54C22⋊Q8×C53C2211yescharacteristic
55C22⋊Q8×C53C2211yescharacteristic
56C4×Q8×C53C2211yescharacteristic
57C4×D4×C53—21no—
58C4×D4×C53C2211yescharacteristic
59C2×Q8×C53—21no—
60C2×Q8×C53—21no—
61C2×Q8×C53—22no—
62C2×Q8×C53—22no—
63C2×Q8×C53—41no—
64C2×Q8×C53—41no—
65C2×Q8×C53C2311yescharacteristic
66C2×Q8×C53C2311yescharacteristic
67C2×Q8×C53C2311yescharacteristic
68C2×D4×C53D411yescharacteristic
69C22×C4×C53—21no—
70C22×C4×C53—21no—
71C22×C4×C53C2311yescharacteristic
72C22×C4×C53C2311yescharacteristic
73C2×C22⋊D53—21no—
74C2×C22⋊D53—21no—
75C22×Dic53—21no—
76C2×D212C2311yescharacteristic
77C2×C4×D53—21no—
78C4⋊C4×C53—21no—
79C4⋊C4×C53—21no—
80C4⋊C4×C53—21no—
81C4⋊C4×C53—21no—
82C4⋊C4×C53—21no—
83C4⋊C4×C53D411yescharacteristic
84C4⋊C4×C53C2311yescharacteristic
85C22⋊C4×C53—21no—
86C22⋊C4×C53—21no—
87C22⋊C4×C53D411yescharacteristic
88C22⋊C4×C53D411yescharacteristic
89C42×C53—21no—
90C42×C53C2311yescharacteristic
91C22⋊Dic53—21no—
92#11719924—22no—
93#11719924—41no—
94#11719924—41no—
95#11719921—21no—
96#11719921—21no—
97#11719921—21no—
98C4⋊Dic53—21no—
99C4⋊Dic53C2311yescharacteristic
100C4⋊Dic53C2311yescharacteristic
101C4⋊Dic53C2311yescharacteristic
102#11719919—21no—
103#11719919—21no—
104C4×Dic53—21no—
105#11719917—41no—
106C2×#88322—41no—
107C2×#88322C2311yescharacteristic
108C2×#88322C2311yescharacteristic
109C2×#88322C2311yescharacteristic
110C2×#88322C2311yescharacteristic
111C23×C53C2×D411yescharacteristic
112C23×C53C2×D411yescharacteristic
113C22×D53—21no—
114Q8×C53—21no—
115Q8×C53—21no—
116Q8×C53—21no—
117Q8×C53—21no—
118Q8×C53—21no—
119Q8×C53—42no—
120Q8×C53—42no—
121Q8×C53#4112yes—
122D4×C53—22no—
123C2×C4×C53—21no—
124C2×C4×C53—21no—
125C2×C4×C53—21no—
126C2×C4×C53—21no—
127C2×C4×C53—21no—
128C2×C4×C53—21no—
129C2×C4×C53—21no—
130C2×C4×C53—21no—
131C2×C4×C53—42no—
132C2×C4×C53—42no—
133C2×C4×C53C2×D411yescharacteristic
134C2×C4×C53C2×D411yescharacteristic
135C2×C4×C53C2×D411yescharacteristic
136C2×C4×C53C2×D411yescharacteristic
137C2×C4×C53C2411yescharacteristic
138C22⋊D53—42no—
139C22⋊D53—81no—
140C2×Dic53—21no—
141C2×Dic53—21no—
142C2×Dic53—21no—
143C2×Dic53—81no—
144D212—21no—
145D212—21no—
146C4×D53—81no—
147#88322—21no—
148#88322—21no—
149#88322—21no—
150#88322—21no—
151C22×C53—21no—
152C22×C53—21no—
153C22×C53—21no—
154C22×C53—21no—
155C22×C53—21no—
156C22×C53#13712yes—
157C22×C53C22×D411yescharacteristic
158C2×D53—41no—
159C2×D53—41no—
160C2×D53—41no—
161C4×C53—21no—
162C4×C53—21no—
163C4×C53—22no—
164C4×C53—21no—
165C4×C53—21no—
166C4×C53—41no—
167C4×C53—41no—
168C4×C53—41no—
169C4×C53C2×#4111yescharacteristic
170C4×C53H(5,𝔽2)11yescharacteristic
171Dic53—41no—
172Dic53—41no—
173Dic53—41no—
174#3320—531nomaximal, Sylow
175C2×C53—22no—
176C2×C53—41no—
177C2×C53#54611yescharacteristic
178C2×C53C2×#13711yescharacteristic
179C2×C53#57711yescharacteristic
180D53—81no—
181#553—531no—
182D4×Q8D5311yescharacteristic, complemented
183#491—531no—
184#485—531no—
185#484—531no—
186#479—531no—
187#468—531no—
188#465—531no—
189#461—531no—
190#448—531no—
191#448—531no—
192#440—531no—
193#435—531no—
194#419—531no—
195C2×#103—531no—
196C53#332011yescharacteristic, complemented, Sylow
197C22×Q8C2×D5311yescharacteristic
198C22×Q8C2×D5311yescharacteristic
199C2×SD16—531no—
200C2×SD16—531no—
201C2×SD16—531no—
202C2×M16—531no—
203C22×C8—531no—
204C4⋊Q8—21no—
205C4⋊Q8C2×D5311yescharacteristic
206#126—1061no—
207#125—531no—
208#123—1061no—
209C22⋊Q8—21no—
210C22⋊Q8—21no—
211C22⋊Q8C2×D5311yescharacteristic
212C22⋊Q8C2×D5311yescharacteristic
213#121—531no—
214#121—531no—
215#121—1061no—
216C4×Q8C2×D5311yescharacteristic
217C4×D4—21no—
218C4×D4—531no—
219C4×D4—1061no—
220C4×D4C2×D5311yescharacteristic
221#117—531no—
222C2×C4⋊C4—531no—
223#107—531no—
224#107—531no—
225#106—531no—
226C4⋊C8—531no—
227#103—531no—
228#103—531no—
229#103—531no—
230#103—531no—
231#103—531no—
232#103—531no—
233#103—531no—
234#103—531no—
235#103—531no—
236#102—531no—
237C22⋊C8—531no—
238C22⋊C8—531no—
239#97—531no—
240C2×Q8—21no—
241C2×Q8—21no—
242C2×Q8—22no—
243C2×Q8—22no—
244C2×Q8—41no—
245C2×Q8—41no—
246C2×Q8C22×D5311yescharacteristic
247C2×Q8C22×D5311yescharacteristic
248C2×Q8C22×D5311yescharacteristic
249C2×D4—531no—
250C2×D4—1061no—
251C2×D4—1061no—
252C2×D4C22⋊D5311yescharacteristic
253C22×C4—21no—
254C22×C4—21no—
255C22×C4—1061no—
256C22×C4—1061no—
257C22×C4C22×D5311yescharacteristic
258C22×C4C22×D5311yescharacteristic
259SD16—1062no—
260SD16—2121no—
261SD16—2121no—
262M16—2121no—
263C2×C8—531no—
264C2×C8—531no—
265C2×C8—531no—
266C2×C8—531no—
267C2×C8—2121no—
268C4⋊C4—21no—
269C4⋊C4—21no—
270C4⋊C4—21no—
271C4⋊C4—21no—
272C4⋊C4—21no—
273C4⋊C4—531no—
274C4⋊C4—531no—
275C4⋊C4—531no—
276C4⋊C4—1061no—
277C4⋊C4—1061no—
278C4⋊C4—1061no—
279C4⋊C4C22⋊D5311yescharacteristic
280C4⋊C4C22×D5311yescharacteristic
281C22⋊C4—21no—
282C22⋊C4—21no—
283C22⋊C4—1061no—
284C22⋊C4—1061no—
285C22⋊C4—1061no—
286C22⋊C4—1061no—
287C22⋊C4C22⋊D5311yescharacteristic
288C22⋊C4C22⋊D5311yescharacteristic
289C42—21no—
290C42—1061no—
291C42C22×D5311yescharacteristic
292C23—1061no—
293C23C2×C22⋊D5311yescharacteristic
294C23C2×C22⋊D5311yescharacteristic
295Q8—21no—
296Q8—21no—
297Q8—21no—
298Q8—21no—
299Q8—21no—
300Q8—42no—
301Q8—42no—
302Q8#1171994712yes—
303D4—22no—
304D4—1061no—
305D4—1061no—
306D4—4241no—
307D4—2122no—
308C2×C4—21no—
309C2×C4—21no—
310C2×C4—21no—
311C2×C4—21no—
312C2×C4—21no—
313C2×C4—21no—
314C2×C4—21no—
315C2×C4—21no—
316C2×C4—42no—
317C2×C4—42no—
318C2×C4—1061no—
319C2×C4—1061no—
320C2×C4—1061no—
321C2×C4—4241no—
322C2×C4—4241no—
323C2×C4C2×C22⋊D5311yescharacteristic
324C2×C4C2×C22⋊D5311yescharacteristic
325C2×C4C2×C22⋊D5311yescharacteristic
326C2×C4C2×C22⋊D5311yescharacteristic
327C2×C4C23×D5311yescharacteristic
328C8—1061no—
329C8—1061no—
330C8—1061no—
331C8—1061no—
332C22—21no—
333C22—21no—
334C22—21no—
335C22—21no—
336C22—21no—
337C22—2121no—
338C22—2121no—
339C22—2121no—
340C22#1313324812yes—
341C22C22×C22⋊D5311yescharacteristic
342C4—21no—
343C4—21no—
344C4—22no—
345C4—21no—
346C4—21no—
347C4—41no—
348C4—41no—
349C4—41no—
350C4—2121no—
351C4—2121no—
352C4—2121no—
353C4C2×#1171994711yescharacteristic
354C4#1313326211yescharacteristic
355C2—22no—
356C2—41no—
357C2—4241no—
358C2#1681126711yescharacteristic
359C2C2×#1313324811yescharacteristic
360C2#1681143111yescharacteristic
subgroupG/H
centerC22C22×C22⋊D53
commutatorC2×C4×C53C24
FrattiniC2×C4C23×D53
FittingD4×Q8×C53C2
socleC22×C53C22×D4
hypercenterD4×Q8D53
hypocenterC53#3320
LCSG ▷ C2×C4×C53 ▷ C2×C53 ▷ C53
UCSC1 ◁ C22 ◁ C4×D4 ◁ D4×Q8

Complements

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

NH ≅ G/Ntype
2. #16810991C2semidirect
3. #16810961C2semidirect
4. #16810956C2semidirect
5. #16810930C2semidirect
6. #16810871C2semidirect
7. #16810854C2semidirect
8. #16810825C2semidirect
9. #16810762C2semidirect
10. #16810762C2semidirect
11. #16810721C2semidirect
12. #16810695C2semidirect
13. #16810632C2semidirect
14. C2×#13133141C2semidirect
15. D4×Q8×C53C2semidirect
182. D4×Q8D53semidirect
196. C53#3320semidirect