Finite Group Databasebeta

← #20286818

Subgroups — #20286818

Order 6784.

382 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#16811314C211yesmaximal, characteristic, complemented
2#16811276C211yesmaximal, characteristic
3#16810991C211yesmaximal, characteristic, complemented
4#16810962C211yesmaximal, characteristic, complemented
5#16810957C211yesmaximal, characteristic, complemented
6#16810924C211yesmaximal, characteristic, complemented
7#16810878C211yesmaximal, characteristic, complemented
8#16810861C211yesmaximal, characteristic, complemented
9#16810784C211yesmaximal, characteristic, complemented
10#16810784C211yesmaximal, characteristic, complemented
11#16810721C211yesmaximal, characteristic, complemented
12#16810693C211yesmaximal, characteristic, complemented
13#16810623C211yesmaximal, characteristic, complemented
14#16810618C211yesmaximal, characteristic, complemented
15C53×#462C211yesmaximal, characteristic, complemented
16C2×#11719949C2211yescharacteristic
17C2×#11719949C2211yescharacteristic
18C2×#11719924C2211yescharacteristic
19C2×#11719912C2211yescharacteristic
20C2×C8×D53C2211yescharacteristic
21#13133211—21no—
22#13133209C2211yescharacteristic
23#13133207—21no—
24#13133202—21no—
25#13133202—21no—
26#13133202—21no—
27#13133193C2211yescharacteristic
28#13133193C2211yescharacteristic
29#13133193C2211yescharacteristic
30#13133193C2211yescharacteristic
31#13133192—21no—
32C4×D212C2211yescharacteristic
33C4×D212C2211yescharacteristic
34#13133175C2211yescharacteristic
35#13133174—21no—
36C4⋊C4×D53—21no—
37C4⋊C4×D53C2211yescharacteristic
38#13133151C2211yescharacteristic
39#13133150C2211yescharacteristic
40#13133149C2211yescharacteristic
41#13133141C2211yescharacteristic
42#13133141C2211yescharacteristic
43#13133141C2211yescharacteristic
44#13133141C2211yescharacteristic
45#13133141C2211yescharacteristic
46#13133139C2211yescharacteristic
47#13133139C2211yescharacteristic
48#13133137C2211yescharacteristic
49#13133129C2211yescharacteristic
50#13133129C2211yescharacteristic
51#13133128C2211yescharacteristic
52C2×Q16×C53C2211yescharacteristic
53C2×Q16×C53C2211yescharacteristic
54C4⋊Q8×C53C2211yescharacteristic
55C4×Q8×C53C2211yescharacteristic
56C4⋊C8×C53C2211yescharacteristic
57#103×C53C2211yescharacteristic
58#103×C53C2211yescharacteristic
59C2×Q8×C53C2311yescharacteristic
60C2×Q8×C53C2311yescharacteristic
61C2×Q8×C53C2311yescharacteristic
62#11719949—22no—
63#11719949—22no—
64#11719949—41no—
65#11719949—41no—
66C2×D212—21no—
67C2×D212—21no—
68C2×D212D411yescharacteristic
69C2×D212C2311yescharacteristic
70C2×C4×D53—21no—
71C2×C4×D53—21no—
72C2×C4×D53—21no—
73C2×C4×D53—21no—
74C2×C4×D53C2311yescharacteristic
75C2×C4×D53C2311yescharacteristic
76Q16×C53—41no—
77Q16×C53—41no—
78C2×C8×C53C2311yescharacteristic
79C2×C8×C53C2311yescharacteristic
80C4⋊C4×C53—21no—
81C4⋊C4×C53—21no—
82C4⋊C4×C53—21no—
83C4⋊C4×C53C2311yescharacteristic
84C42×C53—21no—
85C42×C53C2311yescharacteristic
86#11719924—22no—
87#11719921—21no—
88#11719921—21no—
89#11719921—21no—
90#11719921—21no—
91#11719921—21no—
92#11719921—21no—
93#11719921D411yescharacteristic
94#11719921D411yescharacteristic
95C4⋊Dic53D411yescharacteristic
96C4⋊Dic53C2311yescharacteristic
97C4⋊Dic53C2311yescharacteristic
98C4⋊Dic53C2311yescharacteristic
99#11719919—21no—
100#11719919—21no—
101#11719919—21no—
102#11719919—21no—
103#11719919—21no—
104#11719919—21no—
105#11719919—21no—
106C4×Dic53—21no—
107C2×#88322C2311yescharacteristic
108C2×#88322C2311yescharacteristic
109#11719912—41no—
110C8×D53—41no—
111C22×D53—21no—
112C22×D53C2×D411yescharacteristic
113C22×D53C2×D411yescharacteristic
114Q8×C53—21no—
115Q8×C53—21no—
116Q8×C53—21no—
117Q8×C53—21no—
118Q8×C53—21no—
119Q8×C53—21no—
120C2×C4×C53—21no—
121C2×C4×C53—21no—
122C2×C4×C53—21no—
123C2×C4×C53—21no—
124C2×C4×C53C2×D411yescharacteristic
125C2×C4×C53C2×D411yescharacteristic
126C2×C4×C53C2411yescharacteristic
127C2×Dic53—21no—
128C2×Dic53—21no—
129C2×Dic53—21no—
130C2×Dic53C2×D411yescharacteristic
131C2×Dic53C2×D411yescharacteristic
132D212—21no—
133D212—41no—
134D212—42no—
135D212—42no—
136D212#4112yes—
137C4×D53—21no—
138C4×D53—21no—
139C4×D53—21no—
140C4×D53—21no—
141C4×D53—42no—
142C4×D53—42no—
143C4×D53—42no—
144C4×D53—42no—
145C8×C53—21no—
146C8×C53—21no—
147#88322—21no—
148#88322—21no—
149C22×C53C22×D411yescharacteristic
150C2×D53—21no—
151C2×D53—21no—
152C2×D53—22no—
153C2×D53—21no—
154C2×D53—21no—
155C2×D53—21no—
156C2×D53—21no—
157C2×D53—41no—
158C2×D53—41no—
159C4×C53—21no—
160C4×C53—21no—
161C4×C53—41no—
162C4×C53—41no—
163C4×C53—41no—
164C4×C53—41no—
165C4×C53C2×#4111yescharacteristic
166C4×C53H(5,𝔽2)11yescharacteristic
167Dic53—21no—
168Dic53—21no—
169Dic53—41no—
170Dic53—41no—
171Dic53—41no—
172#3315—531nomaximal, Sylow
173C2×C53#54611yescharacteristic
174C2×C53#57511yescharacteristic
175C2×C53#57811yescharacteristic
176D53—41no—
177D53—41no—
178D53—42no—
179#552—531no—
180#547—531no—
181#491—531no—
182#485—531no—
183#484—531no—
184#478—531no—
185#470—531no—
186#467—531no—
187#462D5311yescharacteristic, complemented
188#452—531no—
189#452—531no—
190#440—531no—
191#435—531no—
192#417—531no—
193#416—531no—
194C53#331511yescharacteristic, complemented, Sylow
195C2×#41—531no—
196C2×#41—531no—
197C2×Q16C2×D5311yescharacteristic
198C2×Q16C2×D5311yescharacteristic
199C2×SD16—531no—
200C2×M16—531no—
201C22×C8—531no—
202C4⋊Q8C2×D5311yescharacteristic
203#126—1061no—
204#125—531no—
205#125—1061no—
206#123—1061no—
207#123—1061no—
208#123—1061no—
209C22⋊Q8—531no—
210C22⋊Q8—531no—
211C22⋊Q8—531no—
212C22⋊Q8—531no—
213#121—1061no—
214C4×Q8C2×D5311yescharacteristic
215C4×D4—531no—
216C4×D4—531no—
217#117—531no—
218#117—1061no—
219C2×C4⋊C4—531no—
220C2×C4⋊C4—1061no—
221#107—531no—
222#107—531no—
223#106—531no—
224C4⋊C8C2×D5311yescharacteristic
225#103—531no—
226#103—531no—
227#103—531no—
228#103—531no—
229#103—531no—
230#103C2×D5311yescharacteristic
231#103C2×D5311yescharacteristic
232#102—531no—
233#102—531no—
234#102—531no—
235C22⋊C8—531no—
236C22⋊C8—531no—
237#97—531no—
238#41—1062no—
239#41—1062no—
240#41—2121no—
241#41—2121no—
242C2×Q8C22×D5311yescharacteristic
243C2×Q8C22×D5311yescharacteristic
244C2×Q8C22×D5311yescharacteristic
245C2×D4—531no—
246C2×D4—531no—
247C2×D4—1061no—
248C2×D4—1061no—
249C22×C4—531no—
250C22×C4—531no—
251C22×C4—1061no—
252C22×C4—1061no—
253C22×C4—1061no—
254C22×C4—1061no—
255Q16—41no—
256Q16—41no—
257SD16—1062no—
258M16—2121no—
259C2×C8—531no—
260C2×C8—531no—
261C2×C8—2121no—
262C2×C8C22×D5311yescharacteristic
263C2×C8C22×D5311yescharacteristic
264C4⋊C4—21no—
265C4⋊C4—21no—
266C4⋊C4—21no—
267C4⋊C4—531no—
268C4⋊C4—531no—
269C4⋊C4—531no—
270C4⋊C4—531no—
271C4⋊C4—1061no—
272C4⋊C4—1061no—
273C4⋊C4—1061no—
274C4⋊C4—1061no—
275C4⋊C4—1061no—
276C4⋊C4—1061no—
277C4⋊C4—1061no—
278C4⋊C4C22×D5311yescharacteristic
279C22⋊C4—531no—
280C22⋊C4—531no—
281C22⋊C4—1061no—
282C22⋊C4—1061no—
283C22⋊C4—1061no—
284C22⋊C4—1061no—
285C22⋊C4—1061no—
286C22⋊C4—1061no—
287C42—21no—
288C42—1061no—
289C42C22×D5311yescharacteristic
290C23—531no—
291C23—531no—
292C23—1061no—
293Q8—21no—
294Q8—21no—
295Q8—21no—
296Q8—21no—
297Q8—21no—
298Q8—21no—
299D4—532no—
300D4—1061no—
301D4—2121no—
302D4—2122no—
303D4—2122no—
304C2×C4—21no—
305C2×C4—21no—
306C2×C4—21no—
307C2×C4—21no—
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—1061no—
317C2×C4—2122no—
318C2×C4—2122no—
319C2×C4—2122no—
320C2×C4—2122no—
321C2×C4D4×D5311yescharacteristic
322C2×C4D4×D5311yescharacteristic
323C2×C4C23×D5311yescharacteristic
324C8—21no—
325C8—21no—
326C8—1061no—
327C8—1061no—
328C22—1061no—
329C22—1061no—
330C22—1062no—
331C22—1061no—
332C22—1061no—
333C22—1061no—
334C22—1061no—
335C22—2121no—
336C22—2121no—
337C22C2×D4×D5311yescharacteristic
338C4—21no—
339C4—21no—
340C4—41no—
341C4—41no—
342C4—41no—
343C4—41no—
344C4—1061no—
345C4—1061no—
346C4—2121no—
347C4—2121no—
348C4—2121no—
349C4#41×D5311yescharacteristic
350C4#1313326211yescharacteristic
351C2—2121no—
352C2—2121no—
353C2—2122no—
354C2#1681127011yescharacteristic
355C2#1681142211yescharacteristic
356C2#1681143611yescharacteristic
subgroupG/H
centerC22C2×D4×D53
commutatorC2×C4×C53C24
FrattiniC2×C4C23×D53
FittingC53×#462C2
socleC22×C53C22×D4
hypercenter#462D53
hypocenterC53#3315
LCSG ▷ C2×C4×C53 ▷ C2×C53 ▷ C53
UCSC1 ◁ C22 ◁ C42 ◁ #462

Complements

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

NH ≅ G/Ntype
1. #16811314C2semidirect
3. #16810991C2semidirect
4. #16810962C2semidirect
5. #16810957C2semidirect
6. #16810924C2semidirect
7. #16810878C2semidirect
8. #16810861C2semidirect
9. #16810784C2semidirect
10. #16810784C2semidirect
11. #16810721C2semidirect
12. #16810693C2semidirect
13. #16810623C2semidirect
14. #16810618C2semidirect
15. C53×#462C2semidirect
187. #462D53semidirect
194. C53#3315semidirect