Finite Group Databasebeta

← #20286983

Subgroups — #20286983

Order 6784.

380 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#16811313C211yesmaximal, characteristic, complemented
2#16810977C211yesmaximal, characteristic, complemented
3#16810956C211yesmaximal, characteristic, complemented
4#16810944C211yesmaximal, characteristic, complemented
5#16810930C211yesmaximal, characteristic, complemented
6#16810879C211yesmaximal, characteristic, complemented
7#16810867C211yesmaximal, characteristic, complemented
8#16810842C211yesmaximal, characteristic, complemented
9#16810785C211yesmaximal, characteristic, complemented
10#16810779C211yesmaximal, characteristic, complemented
11C4×#11719925C211yesmaximal, characteristic, complemented
12#16810701C211yesmaximal, characteristic, complemented
13#16810632C211yesmaximal, characteristic, complemented
14#16810624C211yesmaximal, characteristic, complemented
15C53×#548C211yesmaximal, characteristic
16C2×#11719925C2211yescharacteristic
17C2×#11719925C2211yescharacteristic
18C2×#11719923C2211yescharacteristic
19C2×#11719917C2211yescharacteristic
20C2×#11719917C2211yescharacteristic
21#13133210—21no—
22#13133209C2211yescharacteristic
23#13133207—21no—
24#13133198—21no—
25#13133196C2211yescharacteristic
26#13133196C2211yescharacteristic
27C4×Dic106C2211yescharacteristic
28#13133176C2211yescharacteristic
29#13133173—21no—
30C2×C4⋊Dic53C2211yescharacteristic
31#13133150C2211yescharacteristic
32#13133150C2211yescharacteristic
33#13133148C2211yescharacteristic
34C4⋊#88322C2211yescharacteristic
35#13133141C2211yescharacteristic
36#13133141C2211yescharacteristic
37#13133141C2211yescharacteristic
38#13133141C2211yescharacteristic
39#13133141C2211yescharacteristic
40#13133141C2211yescharacteristic
41#13133140C2211yescharacteristic
42#13133138C2211yescharacteristic
43#13133138C2211yescharacteristic
44#13133138C2211yescharacteristic
45C22⋊#88322C2211yescharacteristic
46C22⋊#88322C2211yescharacteristic
47C4×#88322C2211yescharacteristic
48C2×#41×C53C2211yescharacteristic
49C22×Q8×C53C2211yescharacteristic
50#124×C53—21no—
51#124×C53C2211yescharacteristic
52C22⋊Q8×C53—21no—
53C22⋊Q8×C53C2211yescharacteristic
54#121×C53—21no—
55#121×C53C2211yescharacteristic
56C4×Q8×C53C2211yescharacteristic
57C4×D4×C53—21no—
58C4×D4×C53C2211yescharacteristic
59#41×C53—22no—
60#41×C53—41no—
61C2×Q8×C53—21no—
62C2×Q8×C53—22no—
63C2×Q8×C53—41no—
64C2×Q8×C53C2311yescharacteristic
65C2×Q8×C53C2311yescharacteristic
66C2×D4×C53—21no—
67C2×D4×C53—21no—
68C2×D4×C53D411yescharacteristic
69C2×D4×C53C2311yescharacteristic
70C22×C4×C53—21no—
71C22×C4×C53—21no—
72C22×C4×C53C2311yescharacteristic
73C22×C4×C53C2311yescharacteristic
74C22×Dic53—21no—
75C2×Dic106C2311yescharacteristic
76C4⋊C4×C53—21no—
77C4⋊C4×C53—21no—
78C4⋊C4×C53D411yescharacteristic
79C4⋊C4×C53C2311yescharacteristic
80C22⋊C4×C53—21no—
81C22⋊C4×C53—21no—
82C22⋊C4×C53—21no—
83C22⋊C4×C53—21no—
84C22⋊C4×C53D411yescharacteristic
85C22⋊C4×C53D411yescharacteristic
86C42×C53—21no—
87C42×C53C2311yescharacteristic
88C22⋊Dic53—21no—
89C22⋊Dic53—21no—
90C22⋊Dic53—21no—
91#11719925—22no—
92#11719925—41no—
93#11719923—41no—
94C4⋊Dic53—21no—
95C4⋊Dic53C2311yescharacteristic
96C4⋊Dic53C2311yescharacteristic
97C4⋊Dic53C2311yescharacteristic
98#11719919—21no—
99#11719919—21no—
100#11719919—21no—
101#11719919—21no—
102#11719919—21no—
103C4×Dic53—21no—
104C4×Dic53—21no—
105#11719917—41no—
106#11719917—41no—
107C2×#88322C2311yescharacteristic
108C2×#88322C2311yescharacteristic
109C2×#88322C2311yescharacteristic
110C2×#88322C2311yescharacteristic
111C23×C53—21no—
112C23×C53C2×D411yescharacteristic
113C23×C53C2×D411yescharacteristic
114Q8×C53—21no—
115Q8×C53—21no—
116Q8×C53—21no—
117Q8×C53—42no—
118Q8×C53#4112yes—
119D4×C53—21no—
120D4×C53—21no—
121D4×C53—41no—
122D4×C53—42no—
123D4×C53—42no—
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
138C2×Dic53—21no—
139C2×Dic53—21no—
140C2×Dic53—21no—
141C2×Dic53—21no—
142C2×Dic53—42no—
143Dic106—21no—
144Dic106—21no—
145#88322—21no—
146#88322—21no—
147#88322—21no—
148#88322—21no—
149C22×C53—21no—
150C22×C53—21no—
151C22×C53—21no—
152C22×C53—21no—
153C22×C53—21no—
154C22×C53—21no—
155C22×C53—41no—
156C22×C53—41no—
157C22×C53—41no—
158C22×C53C22×D411yescharacteristic
159C4×C53—21no—
160C4×C53—21no—
161C4×C53—22no—
162C4×C53—21no—
163C4×C53—21no—
164C4×C53—41no—
165C4×C53—41no—
166C4×C53C2×#4111yescharacteristic
167C4×C53H(5,𝔽2)11yescharacteristic
168Dic53—41no—
169Dic53—41no—
170Dic53—41no—
171Dic53—41no—
172#3327—531nomaximal, Sylow
173C2×C53—41no—
174C2×C53—41no—
175C2×C53—81no—
176C2×C53#54611yescharacteristic
177C2×C53#57511yescharacteristic
178C2×C53#57811yescharacteristic
179#552—531no—
180#548D5311yescharacteristic
181#488—531no—
182#484—531no—
183#482—531no—
184#479—531no—
185#470—531no—
186#468—531no—
187#464—531no—
188#452—531no—
189#451—531no—
190C4×Q16—531no—
191#436—531no—
192#419—531no—
193#417—531no—
194C53#332711yescharacteristic, complemented, Sylow
195C2×#41C2×D5311yescharacteristic
196C22×Q8C2×D5311yescharacteristic
197C2×Q16—531no—
198C2×Q16—531no—
199C2×SD16—531no—
200C2×M16—531no—
201C2×M16—531no—
202#126—1061no—
203#125—531no—
204#125—1061no—
205#124—21no—
206#124C2×D5311yescharacteristic
207#123—1061no—
208C22⋊Q8—21no—
209C22⋊Q8—531no—
210C22⋊Q8—531no—
211C22⋊Q8C2×D5311yescharacteristic
212#121—21no—
213#121C2×D5311yescharacteristic
214C4×Q8—531no—
215C4×Q8C2×D5311yescharacteristic
216C4×D4—21no—
217C4×D4C2×D5311yescharacteristic
218#117—531no—
219#117—1061no—
220C2×C4⋊C4—531no—
221#107—531no—
222#107—531no—
223#106—531no—
224C4⋊C8—531no—
225#103—531no—
226#103—531no—
227#103—531no—
228#103—531no—
229#103—531no—
230#103—531no—
231#103—531no—
232#102—531no—
233#102—531no—
234#102—531no—
235C22⋊C8—531no—
236C22⋊C8—531no—
237C4×C8—531no—
238#41—22no—
239#41—41no—
240C2×Q8—21no—
241C2×Q8—22no—
242C2×Q8—41no—
243C2×Q8—531no—
244C2×Q8C22×D5311yescharacteristic
245C2×Q8C22×D5311yescharacteristic
246C2×D4—21no—
247C2×D4—21no—
248C2×D4C22⋊D5311yescharacteristic
249C2×D4C22×D5311yescharacteristic
250C22×C4—21no—
251C22×C4—21no—
252C22×C4—1061no—
253C22×C4C22×D5311yescharacteristic
254C22×C4C22×D5311yescharacteristic
255Q16—1062no—
256Q16—2121no—
257SD16—2121no—
258M16—2121no—
259M16—2121no—
260C2×C8—531no—
261C2×C8—531no—
262C2×C8—531no—
263C2×C8—531no—
264C4⋊C4—21no—
265C4⋊C4—21no—
266C4⋊C4—531no—
267C4⋊C4—531no—
268C4⋊C4—531no—
269C4⋊C4—1061no—
270C4⋊C4—1061no—
271C4⋊C4—1061no—
272C4⋊C4—1061no—
273C4⋊C4—1061no—
274C4⋊C4—1061no—
275C4⋊C4C22⋊D5311yescharacteristic
276C4⋊C4C22×D5311yescharacteristic
277C22⋊C4—21no—
278C22⋊C4—21no—
279C22⋊C4—21no—
280C22⋊C4—21no—
281C22⋊C4—1061no—
282C22⋊C4—1061no—
283C22⋊C4—1061no—
284C22⋊C4C22⋊D5311yescharacteristic
285C22⋊C4C22⋊D5311yescharacteristic
286C42—21no—
287C42—1061no—
288C42—1061no—
289C42C22×D5311yescharacteristic
290C23—21no—
291C23C2×C22⋊D5311yescharacteristic
292C23C2×C22⋊D5311yescharacteristic
293Q8—21no—
294Q8—21no—
295Q8—21no—
296Q8—42no—
297Q8—1061no—
298Q8—1061no—
299Q8#1171994712yes—
300D4—21no—
301D4—21no—
302D4—41no—
303D4—42no—
304D4—42no—
305C2×C4—21no—
306C2×C4—21no—
307C2×C4—21no—
308C2×C4—21no—
309C2×C4—21no—
310C2×C4—21no—
311C2×C4—21no—
312C2×C4—42no—
313C2×C4—42no—
314C2×C4—1061no—
315C2×C4—1061no—
316C2×C4—1061no—
317C2×C4—1061no—
318C2×C4—2122no—
319C2×C4C2×C22⋊D5311yescharacteristic
320C2×C4C2×C22⋊D5311yescharacteristic
321C2×C4C2×C22⋊D5311yescharacteristic
322C2×C4C2×C22⋊D5311yescharacteristic
323C2×C4C23×D5311yescharacteristic
324C8—1061no—
325C8—1061no—
326C8—1061no—
327C8—1061no—
328C22—21no—
329C22—21no—
330C22—21no—
331C22—21no—
332C22—21no—
333C22—21no—
334C22—41no—
335C22—41no—
336C22—41no—
337C22C22×C22⋊D5311yescharacteristic
338C4—21no—
339C4—21no—
340C4—22no—
341C4—21no—
342C4—21no—
343C4—41no—
344C4—41no—
345C4—2121no—
346C4—2121no—
347C4—2121no—
348C4—2121no—
349C4C2×#1171994711yescharacteristic
350C4#1313326211yescharacteristic
351C2—41no—
352C2—41no—
353C2—81no—
354C2#1681126711yescharacteristic
355C2#1681142111yescharacteristic
356C2#1681143811yescharacteristic
subgroupG/H
centerC22C22×C22⋊D53
commutatorC2×C4×C53C24
FrattiniC2×C4C23×D53
FittingC53×#548C2
socleC22×C53C22×D4
hypercenter#548D53
hypocenterC53#3327
LCSG ▷ C2×C4×C53 ▷ C2×C53 ▷ C53
UCSC1 ◁ C22 ◁ C4×D4 ◁ #548

Complements

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

NH ≅ G/Ntype
1. #16811313C2semidirect
2. #16810977C2semidirect
3. #16810956C2semidirect
4. #16810944C2semidirect
5. #16810930C2semidirect
6. #16810879C2semidirect
7. #16810867C2semidirect
8. #16810842C2semidirect
9. #16810785C2semidirect
10. #16810779C2semidirect
11. C4×#11719925C2semidirect
12. #16810701C2semidirect
13. #16810632C2semidirect
14. #16810624C2semidirect
194. C53#3327semidirect