Finite Group Databasebeta

← #20286366

Subgroups — #20286366

Order 6784.

476 conjugacy classes of subgroups.

The subgroup lattice has 404 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#13133249C211yesmaximal, characteristic, complemented
2#16811284C211yesmaximal, characteristic, complemented
3#16811280C211yesmaximal, characteristic, complemented
4#16811004C211yesmaximal, characteristic, complemented
5#16810869C211yesmaximal, characteristic, complemented
6#16810866C211yesmaximal, characteristic, complemented
7#16810847C211yesmaximal, characteristic, complemented
8#16810843C211yesmaximal, characteristic, complemented
9#16810779C211yesmaximal, characteristic, complemented
10#16810777C211yesmaximal, characteristic, complemented
11#16810768C211yesmaximal, characteristic, complemented
12#16810767C211yesmaximal, characteristic, complemented
13Q16×Dic53C211yesmaximal, characteristic, complemented
14#16810699C211yesmaximal, characteristic, complemented
15C2×#137×C53C211yesmaximal, characteristic, complemented
16C2×#11719949C2211yescharacteristic
17C2×Q8×D53C2211yescharacteristic
18#13133249—24no—
19C2×#11719925C2211yescharacteristic
20C2×#11719925C2211yescharacteristic
21C2×#11719913C2211yescharacteristic
22C2×#11719924C2211yescharacteristic
23C2×#11719922C2211yescharacteristic
24C2×#11719912C2211yescharacteristic
25#13133206—21no—
26#13133206C2211yescharacteristic
27#13133204—21no—
28#13133204C2211yescharacteristic
29#13133196—21no—
30#13133196C2211yescharacteristic
31#13133193—21no—
32#13133193C2211yescharacteristic
33#13133191—21no—
34#13133191C2211yescharacteristic
35#13133188—21no—
36#13133188C2211yescharacteristic
37Q8×Dic53C2211yescharacteristic
38Q8×Dic53C2211yescharacteristic
39C4×C22⋊D53—21no—
40C4×C22⋊D53—21no—
41C4×C22⋊D53C2211yescharacteristic
42#13133151C2211yescharacteristic
43#13133146C2211yescharacteristic
44#13133142C2211yescharacteristic
45#13133141C2211yescharacteristic
46#13133141C2211yescharacteristic
47#13133141C2211yescharacteristic
48#13133139C2211yescharacteristic
49#13133138C2211yescharacteristic
50C22⋊#88322C2211yescharacteristic
51#13133129C2211yescharacteristic
52C8×Dic53C2211yescharacteristic
53C2×#41×C53C2211yescharacteristic
54C22×Q8×C53C2211yescharacteristic
55#137×C53—24no—
56C2×Q16×C53C2211yescharacteristic
57C2×Q16×C53C2211yescharacteristic
58C2×SD16×C53C2211yescharacteristic
59C2×SD16×C53C2211yescharacteristic
60C2×M16×C53C2211yescharacteristic
61#41×C53—22no—
62#41×C53—41no—
63C2×Q8×C53—21no—
64C2×Q8×C53—22no—
65C2×Q8×C53—41no—
66C2×Q8×C53C2311yescharacteristic
67C2×Q8×C53C2311yescharacteristic
68C2×Q8×C53C2311yescharacteristic
69C2×D4×C53—21no—
70C2×D4×C53C2311yescharacteristic
71C22×C4×C53—21no—
72C22×C4×C53—21no—
73C22×C4×C53C2311yescharacteristic
74C2×C22⋊D53—21no—
75C2×C22⋊D53—21no—
76C2×C22⋊D53D411yescharacteristic
77#11719949—22no—
78#11719949—41no—
79Q8×D53—22no—
80Q8×D53—41no—
81C2×D212—21no—
82C2×D212C2311yescharacteristic
83C2×C4×D53—21no—
84C2×C4×D53—21no—
85C2×C4×D53C2311yescharacteristic
86C2×Dic106—21no—
87C2×Dic106C2311yescharacteristic
88Q16×C53—22no—
89Q16×C53D414yescomplemented
90SD16×C53—22no—
91SD16×C53—22no—
92M16×C53—22no—
93C2×C8×C53C2311yescharacteristic
94C2×C8×C53C2311yescharacteristic
95C22⋊Dic53—21no—
96C22⋊Dic53—21no—
97C22⋊Dic53—21no—
98C22⋊Dic53—21no—
99C22⋊Dic53D411yescharacteristic
100#11719925—22no—
101#11719925—41no—
102#11719924—22no—
103#11719922—41no—
104#11719921—21no—
105#11719921—21no—
106#11719921—21no—
107#11719921—21no—
108#11719921D411yescharacteristic
109C4⋊Dic53—21no—
110C4⋊Dic53—21no—
111C4⋊Dic53C2311yescharacteristic
112C4⋊Dic53C2311yescharacteristic
113#11719919—21no—
114#11719919—21no—
115#11719919D411yescharacteristic
116C4×Dic53—21no—
117C4×Dic53—21no—
118C4×Dic53C2311yescharacteristic
119C2×#88322C2311yescharacteristic
120C2×#88322C2311yescharacteristic
121#11719913—22no—
122#11719912—22no—
123C23×C53—21no—
124C23×C53C2×D411yescharacteristic
125C22×D53—21no—
126C22×D53C2×D411yescharacteristic
127Q8×C53—21no—
128Q8×C53—21no—
129Q8×C53—21no—
130Q8×C53—21no—
131Q8×C53—42no—
132Q8×C53C2×D412yes—
133Q8×C53C2×D412yes—
134D4×C53—21no—
135D4×C53—21no—
136D4×C53—42no—
137C2×C4×C53—21no—
138C2×C4×C53—21no—
139C2×C4×C53—21no—
140C2×C4×C53—21no—
141C2×C4×C53—21no—
142C2×C4×C53—42no—
143C2×C4×C53—42no—
144C2×C4×C53C2×D411yescharacteristic
145C2×C4×C53C2411yescharacteristic
146C22⋊D53—41no—
147C22⋊D53—42no—
148C22⋊D53—42no—
149C2×Dic53—21no—
150C2×Dic53—21no—
151C2×Dic53—21no—
152C2×Dic53C2×D411yescharacteristic
153C2×Dic53C2×D411yescharacteristic
154C2×Dic53C2×D411yescharacteristic
155D212—21no—
156D212—21no—
157D212—42no—
158C4×D53—21no—
159C4×D53—21no—
160C4×D53—42no—
161C4×D53—42no—
162Dic106—21no—
163Dic106—21no—
164Dic106—42no—
165C8×C53—21no—
166C8×C53C2×D412yes—
167#88322—21no—
168#88322—21no—
169C22×C53—21no—
170C22×C53—21no—
171C22×C53—21no—
172C22×C53—41no—
173C22×C53—41no—
174C22×C53—41no—
175C22×C53C22×D411yescharacteristic
176C2×D53—21no—
177C2×D53—21no—
178C2×D53—21no—
179C2×D53—41no—
180C2×D53—41no—
181C2×D53—41no—
182C4×C53—22no—
183C4×C53—22no—
184C4×C53—21no—
185C4×C53—41no—
186C4×C53C22×D411yescharacteristic
187C4×C53H(5,𝔽2)11yescharacteristic
188Dic53—21no—
189Dic53—21no—
190Dic53—21no—
191Dic53—41no—
192Dic53—41no—
193Dic53—41no—
194#3277—531nomaximal, Sylow
195C2×C53—41no—
196C2×C53—81no—
197C2×C53D4211yescharacteristic
198C2×C53#57511yescharacteristic
199C2×C53#57811yescharacteristic
200D53—41no—
201D53—81no—
202C2×#137—531no—
203C2×#137D5311yescharacteristic, complemented
204#548—531no—
205#548—531no—
206#495—531no—
207#468—531no—
208#468—531no—
209#464—531no—
210#464—531no—
211#451—531no—
212#451—531no—
213#449—531no—
214#449—531no—
215C4×Q16—531no—
216#436—531no—
217C53#327711yescharacteristic, complemented, Sylow
218C2×#41—531no—
219C2×#41C2×D5311yescharacteristic
220C22×Q8—531no—
221C22×Q8C2×D5311yescharacteristic
222#137—24no—
223#137—1064no—
224C2×Q16—531no—
225C2×Q16—531no—
226C2×Q16C2×D5311yescharacteristic
227C2×Q16C2×D5311yescharacteristic
228C2×SD16—531no—
229C2×SD16—531no—
230C2×SD16C2×D5311yescharacteristic
231C2×SD16C2×D5311yescharacteristic
232C2×D8—531no—
233C2×M16—531no—
234C2×M16C2×D5311yescharacteristic
235#124—531no—
236#124—531no—
237#124—1061no—
238#124—1061no—
239C22⋊Q8—531no—
240C22⋊Q8—531no—
241C22⋊Q8—1061no—
242C22⋊Q8—1061no—
243#121—531no—
244#121—531no—
245#121—1061no—
246#121—1061no—
247C4×Q8—531no—
248C4×Q8—531no—
249C4×D4—531no—
250C4×D4—1061no—
251C4×D4—1061no—
252#107—531no—
253C4⋊C8—531no—
254#103—531no—
255#103—531no—
256#103—531no—
257#103—531no—
258#102—531no—
259#102—531no—
260C22⋊C8—531no—
261C22⋊C8—531no—
262C4×C8—531no—
263#41—22no—
264#41—41no—
265#41—1062no—
266#41—2121no—
267C2×Q8—21no—
268C2×Q8—22no—
269C2×Q8—41no—
270C2×Q8—531no—
271C2×Q8—1061no—
272C2×Q8—1062no—
273C2×Q8—2121no—
274C2×Q8C22×D5311yescharacteristic
275C2×Q8C22×D5311yescharacteristic
276C2×Q8C22×D5311yescharacteristic
277C2×D4—21no—
278C2×D4—531no—
279C2×D4—531no—
280C2×D4—1061no—
281C2×D4—1061no—
282C2×D4—1061no—
283C2×D4C22×D5311yescharacteristic
284C22×C4—21no—
285C22×C4—21no—
286C22×C4—531no—
287C22×C4—1061no—
288C22×C4—1061no—
289C22×C4C22×D5311yescharacteristic
290Q16—22no—
291Q16—1062no—
292Q16—2121no—
293Q16C22⋊D5314yescomplemented
294SD16—22no—
295SD16—22no—
296SD16—1062no—
297SD16—1062no—
298D8—2121no—
299M16—22no—
300M16—1062no—
301C2×C8—531no—
302C2×C8—531no—
303C2×C8C22×D5311yescharacteristic
304C2×C8C22×D5311yescharacteristic
305C4⋊C4—531no—
306C4⋊C4—531no—
307C4⋊C4—531no—
308C4⋊C4—1061no—
309C4⋊C4—1061no—
310C4⋊C4—1061no—
311C4⋊C4—1061no—
312C22⋊C4—531no—
313C22⋊C4—531no—
314C22⋊C4—1061no—
315C22⋊C4—1061no—
316C22⋊C4—1061no—
317C22⋊C4—1061no—
318C22⋊C4—1061no—
319C22⋊C4—1061no—
320C22⋊C4—1061no—
321C22⋊C4—1061no—
322C42—531no—
323C42—1061no—
324C42—1061no—
325C23—21no—
326C23—531no—
327C23—1061no—
328C23D4×D5311yescharacteristic
329Q8—21no—
330Q8—21no—
331Q8—21no—
332Q8—21no—
333Q8—42no—
334Q8—1061no—
335Q8—1061no—
336Q8—2122no—
337Q8C2×C22⋊D5312yes—
338Q8C2×C22⋊D5312yes—
339D4—21no—
340D4—21no—
341D4—42no—
342D4—1061no—
343D4—1061no—
344D4—2122no—
345D4—2121no—
346D4—2122no—
347D4—2122no—
348C2×C4—21no—
349C2×C4—21no—
350C2×C4—21no—
351C2×C4—21no—
352C2×C4—21no—
353C2×C4—42no—
354C2×C4—42no—
355C2×C4—531no—
356C2×C4—531no—
357C2×C4—531no—
358C2×C4—1061no—
359C2×C4—1061no—
360C2×C4—1061no—
361C2×C4—1061no—
362C2×C4—1061no—
363C2×C4—2122no—
364C2×C4—2122no—
365C2×C4D4×D5311yescharacteristic
366C2×C4C23×D5311yescharacteristic
367C8—21no—
368C8—1061no—
369C8—1061no—
370C8C2×C22⋊D5312yes—
371C22—21no—
372C22—21no—
373C22—21no—
374C22—41no—
375C22—41no—
376C22—41no—
377C22—1061no—
378C22—1061no—
379C22—1061no—
380C22—2121no—
381C22—2121no—
382C22—2121no—
383C22C2×D4×D5311yescharacteristic
384C4—21no—
385C4—22no—
386C4—22no—
387C4—41no—
388C4—1061no—
389C4—1061no—
390C4—1061no—
391C4—2121no—
392C4—2121no—
393C4—2121no—
394C4C22×C22⋊D5311yescharacteristic
395C4#1313326211yescharacteristic
396C2—41no—
397C2—81no—
398C2—4241no—
399C2—2121no—
400C2D4×C22⋊D5311yescharacteristic
401C2#1681142211yescharacteristic
402C2#1681143911yescharacteristic
subgroupG/H
centerC22C2×D4×D53
commutatorC2×C4×C53C24
FrattiniC2×C4C23×D53
FittingC2×#137×C53C2
socleC22×C53C22×D4
hypercenterC2×#137D53
hypocenterC53#3277
LCSG ▷ C2×C4×C53 ▷ C2×C53 ▷ C53
UCSC1 ◁ C22 ◁ C22×C4 ◁ C2×#137

Complements

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

NH ≅ G/Ntype
1. C2×#13133249C2semidirect
2. #16811284C2semidirect
3. #16811280C2semidirect
4. #16811004C2semidirect
5. #16810869C2semidirect
6. #16810866C2semidirect
7. #16810847C2semidirect
8. #16810843C2semidirect
9. #16810779C2semidirect
10. #16810777C2semidirect
11. #16810768C2semidirect
12. #16810767C2semidirect
13. Q16×Dic53C2semidirect
14. #16810699C2semidirect
15. C2×#137×C53C2semidirect
89. Q16×C53D4semidirect
203. C2×#137D53semidirect
217. C53#3277semidirect
293. Q16C22⋊D53semidirect