Finite Group Databasebeta

← #20285398

Subgroups — #20285398

Order 6784.

374 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#16811244C211yesmaximal, characteristic, complemented
2#16811228C211yesmaximal, characteristic, complemented
3#16811032C211yesmaximal, characteristic, complemented
4#16811012C211yesmaximal, characteristic, complemented
5#16810978C211yesmaximal, characteristic, complemented
6#16810970C211yesmaximal, characteristic, complemented
7#16810876C211yesmaximal, characteristic, complemented
8#16810869C211yesmaximal, characteristic, complemented
9#16810859C211yesmaximal, characteristic, complemented
10#16810852C211yesmaximal, characteristic, complemented
11#16810736C211yesmaximal, characteristic, complemented
12#16810725C211yesmaximal, characteristic, complemented
13C4×#11719913C211yesmaximal, characteristic, complemented
14C4×D424C211yesmaximal, characteristic, complemented
15C53×#405C211yesmaximal, characteristic, complemented
16C2×#11719913C2211yescharacteristic
17C2×#11719913C2211yescharacteristic
18C2×D424C2211yescharacteristic
19C2×D424C2211yescharacteristic
20#13133217C2211yescharacteristic
21#13133214C2211yescharacteristic
22#13133212—21no—
23#13133209C2211yescharacteristic
24#13133205C2211yescharacteristic
25#13133203—21no—
26#13133202—21no—
27#13133202—21no—
28#13133199—21no—
29#13133196C2211yescharacteristic
30#13133195—21no—
31#13133191C2211yescharacteristic
32#13133191C2211yescharacteristic
33#13133191C2211yescharacteristic
34#13133190—21no—
35#13133190—21no—
36C4×Dic106C2211yescharacteristic
37C4×D212C2211yescharacteristic
38C4×D212C2211yescharacteristic
39C4×D212C2211yescharacteristic
40#13133151C2211yescharacteristic
41#13133151C2211yescharacteristic
42#13133149C2211yescharacteristic
43#13133149C2211yescharacteristic
44#13133142C2211yescharacteristic
45#13133142C2211yescharacteristic
46#13133139C2211yescharacteristic
47#13133139C2211yescharacteristic
48#13133139C2211yescharacteristic
49#13133139C2211yescharacteristic
50#13133139C2211yescharacteristic
51#13133139C2211yescharacteristic
52C2×M16×C53C2211yescharacteristic
53C22×C8×C53C2211yescharacteristic
54#117×C53C2211yescharacteristic
55#97×C53C2211yescharacteristic
56#97×C53C2211yescharacteristic
57C4×C8×C53C2211yescharacteristic
58C4×C8×C53C2211yescharacteristic
59C22×C4×C53C2311yescharacteristic
60C2×C22⋊D53—21no—
61C2×C22⋊D53—21no—
62C2×C22⋊D53—21no—
63C2×D212—21no—
64C2×D212—21no—
65C2×D212C2311yescharacteristic
66C2×D212C2311yescharacteristic
67C2×D212C2311yescharacteristic
68C2×C4×D53—21no—
69C2×C4×D53—21no—
70C2×C4×D53—21no—
71C2×Dic106—21no—
72C2×Dic106C2311yescharacteristic
73M16×C53—22no—
74C2×C8×C53—22no—
75C2×C8×C53C2311yescharacteristic
76C2×C8×C53C2311yescharacteristic
77C2×C8×C53C2311yescharacteristic
78C2×C8×C53C2311yescharacteristic
79C4⋊C4×C53D411yescharacteristic
80C4⋊C4×C53D411yescharacteristic
81C22⋊C4×C53D411yescharacteristic
82C22⋊C4×C53D411yescharacteristic
83C42×C53C2311yescharacteristic
84C42×C53C2311yescharacteristic
85C22⋊Dic53—21no—
86#11719921—21no—
87#11719921—21no—
88#11719921—21no—
89#11719921—21no—
90#11719921—21no—
91#11719921—21no—
92#11719921—21no—
93#11719921—21no—
94C4⋊Dic53—21no—
95C4⋊Dic53C2311yescharacteristic
96C4⋊Dic53C2311yescharacteristic
97C4⋊Dic53C2311yescharacteristic
98C4⋊Dic53C2311yescharacteristic
99#11719919—21no—
100#11719919—21no—
101#11719919—21no—
102#11719919—21no—
103C4×Dic53—21no—
104D424—22no—
105D424—22no—
106#11719913—22no—
107#11719913—22no—
108C23×C53C2×D411yescharacteristic
109C22×D53—21no—
110C22×D53—21no—
111C22×D53—21no—
112C2×C4×C53—21no—
113C2×C4×C53—21no—
114C2×C4×C53C2×D411yescharacteristic
115C2×C4×C53C2×D411yescharacteristic
116C2×C4×C53C2×D411yescharacteristic
117C2×C4×C53C2×D411yescharacteristic
118C2×C4×C53C2×D411yescharacteristic
119C2×C4×C53C2411yescharacteristic
120C22⋊D53—81no—
121C22⋊D53—81no—
122C22⋊D53—81no—
123C2×Dic53—21no—
124C2×Dic53—21no—
125C2×Dic53—21no—
126C2×Dic53—21no—
127C2×Dic53—21no—
128D212—21no—
129D212—21no—
130D212—21no—
131D212—21no—
132D212—21no—
133D212—21no—
134D212—81no—
135D212—81no—
136C4×D53—81no—
137C4×D53—81no—
138C4×D53—81no—
139Dic106—21no—
140Dic106—21no—
141Dic106—81no—
142C8×C53—21no—
143C8×C53—21no—
144C8×C53#4112yes—
145C8×C53#4112yes—
146C22×C53—21no—
147C22×C53—21no—
148C22×C53—21no—
149C22×C53C22×D411yescharacteristic
150C2×D53—41no—
151C2×D53—41no—
152C2×D53—41no—
153C2×D53—41no—
154C2×D53—41no—
155C2×D53—41no—
156C2×D53—41no—
157C2×D53—41no—
158C2×D53—41no—
159C4×C53—21no—
160C4×C53—21no—
161C4×C53—21no—
162C4×C53—21no—
163C4×C53—21no—
164C4×C53C2×#4111yescharacteristic
165C4×C53C2×#4111yescharacteristic
166Dic53—41no—
167Dic53—41no—
168Dic53—41no—
169Dic53—41no—
170Dic53—41no—
171#3179—531nomaximal, Sylow
172C2×C53—41no—
173C2×C53#53211yescharacteristic
174C2×C53#57611yescharacteristic
175C2×C53#57711yescharacteristic
176D53—81no—
177D53—81no—
178D53—81no—
179#542—531no—
180#540—531no—
181#501—531no—
182#496—531no—
183#488—531no—
184#486—531no—
185#469—531no—
186#468—531no—
187#466—531no—
188#465—531no—
189#442—531no—
190#440—531no—
191C4×SD16—531no—
192C4×D8—531no—
193#405D5311yescharacteristic, complemented
194C53#317911yescharacteristic, complemented, Sylow
195C2×SD16—531no—
196C2×SD16—531no—
197C2×D8—531no—
198C2×D8—531no—
199C2×M16C2×D5311yescharacteristic
200C22×C8C2×D5311yescharacteristic
201C4⋊Q8—531no—
202#127—531no—
203#126—1061no—
204#125—531no—
205#124—531no—
206#124—1061no—
207#123—1061no—
208#123—1061no—
209#123—1061no—
210C22⋊Q8—531no—
211C22⋊Q8—1061no—
212#121—531no—
213#121—531no—
214#121—531no—
215#121—1061no—
216#121—1061no—
217C4×Q8—531no—
218C4×D4—531no—
219C4×D4—531no—
220C4×D4—531no—
221#117C2×D5311yescharacteristic
222#107—531no—
223#107—531no—
224#106—531no—
225#106—531no—
226#103—531no—
227#103—531no—
228#102—531no—
229#102—531no—
230#102—531no—
231#102—531no—
232#102—531no—
233#102—531no—
234#97C2×D5311yescharacteristic
235#97C2×D5311yescharacteristic
236C4×C8C2×D5311yescharacteristic
237C4×C8C2×D5311yescharacteristic
238C2×Q8—531no—
239C2×Q8—1061no—
240C2×D4—531no—
241C2×D4—531no—
242C2×D4—531no—
243C2×D4—1061no—
244C2×D4—1061no—
245C2×D4—1061no—
246C2×D4—1061no—
247C2×D4—1061no—
248C22×C4—1061no—
249C22×C4—1061no—
250C22×C4—1061no—
251C22×C4C22×D5311yescharacteristic
252SD16—1062no—
253SD16—1062no—
254D8—1062no—
255D8—1062no—
256M16—22no—
257C2×C8—22no—
258C2×C8C22×D5311yescharacteristic
259C2×C8C22×D5311yescharacteristic
260C2×C8C22×D5311yescharacteristic
261C2×C8C22×D5311yescharacteristic
262C4⋊C4—531no—
263C4⋊C4—531no—
264C4⋊C4—531no—
265C4⋊C4—531no—
266C4⋊C4—1061no—
267C4⋊C4—1061no—
268C4⋊C4—1061no—
269C4⋊C4—1061no—
270C4⋊C4—1061no—
271C4⋊C4D21211yescharacteristic
272C4⋊C4D21211yescharacteristic
273C22⋊C4—1061no—
274C22⋊C4—1061no—
275C22⋊C4—1061no—
276C22⋊C4—1061no—
277C22⋊C4—1061no—
278C22⋊C4—1061no—
279C22⋊C4—1061no—
280C22⋊C4—1061no—
281C22⋊C4—1061no—
282C22⋊C4D21211yescharacteristic
283C22⋊C4D21211yescharacteristic
284C42—1061no—
285C42C22×D5311yescharacteristic
286C42C22×D5311yescharacteristic
287C23—1061no—
288C23—1061no—
289C23—1061no—
290C23C2×D21211yescharacteristic
291Q8—1061no—
292Q8—1061no—
293Q8—4241no—
294D4—1061no—
295D4—1061no—
296D4—1061no—
297D4—1061no—
298D4—1061no—
299D4—1061no—
300D4—4241no—
301D4—4241no—
302D4—4241no—
303D4—4241no—
304D4—4241no—
305C2×C4—21no—
306C2×C4—21no—
307C2×C4—1061no—
308C2×C4—1061no—
309C2×C4—1061no—
310C2×C4—1061no—
311C2×C4—1061no—
312C2×C4—4241no—
313C2×C4—4241no—
314C2×C4—4241no—
315C2×C4C2×D21211yescharacteristic
316C2×C4C2×D21211yescharacteristic
317C2×C4C2×D21211yescharacteristic
318C2×C4C2×D21211yescharacteristic
319C2×C4C2×D21211yescharacteristic
320C2×C4C23×D5311yescharacteristic
321C8—21no—
322C8—21no—
323C8#1171994512yes—
324C8#1171994512yes—
325C22—21no—
326C22—21no—
327C22—21no—
328C22—2121no—
329C22—2121no—
330C22—2121no—
331C22—2121no—
332C22—2121no—
333C22—2121no—
334C22—2121no—
335C22—2121no—
336C22—2121no—
337C22C22×D21211yescharacteristic
338C4—21no—
339C4—21no—
340C4—21no—
341C4—21no—
342C4—21no—
343C4—2121no—
344C4—2121no—
345C4—2121no—
346C4—2121no—
347C4—2121no—
348C4C2×#1171994511yescharacteristic
349C4C2×#1171994511yescharacteristic
350C2—41no—
351C2—4241no—
352C2—4241no—
353C2—4241no—
354C2#1681117311yescharacteristic
355C2#1681142811yescharacteristic
356C2#1681143511yescharacteristic
subgroupG/H
centerC22C22×D212
commutatorC2×C4×C53C24
FrattiniC2×C4C23×D53
FittingC53×#405C2
socleC22×C53C22×D4
hypercenter#405D53
hypocenterC53#3179
LCSG ▷ C2×C4×C53 ▷ C2×C53 ▷ C53
UCSC1 ◁ C22 ◁ #117 ◁ #405

Complements

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

NH ≅ G/Ntype
1. #16811244C2semidirect
2. #16811228C2semidirect
3. #16811032C2semidirect
4. #16811012C2semidirect
5. #16810978C2semidirect
6. #16810970C2semidirect
7. #16810876C2semidirect
8. #16810869C2semidirect
9. #16810859C2semidirect
10. #16810852C2semidirect
11. #16810736C2semidirect
12. #16810725C2semidirect
13. C4×#11719913C2semidirect
14. C4×D424C2semidirect
15. C53×#405C2semidirect
193. #405D53semidirect
194. C53#3179semidirect