Finite Group Databasebeta

← #20286264

Subgroups — #20286264

Order 6784.

478 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×#13133238C211yesmaximal, characteristic, complemented
2#16811284C211yesmaximal, characteristic, complemented
3#16811272C211yesmaximal, characteristic, complemented
4#16811016C211yesmaximal, characteristic, complemented
5#16810869C211yesmaximal, characteristic, complemented
6#16810861C211yesmaximal, characteristic, complemented
7#16810840C211yesmaximal, characteristic, complemented
8#16810831C211yesmaximal, characteristic, complemented
9#16810784C211yesmaximal, characteristic, complemented
10#16810782C211yesmaximal, characteristic, complemented
11#16810779C211yesmaximal, characteristic, complemented
12#16810768C211yesmaximal, characteristic, complemented
13#16810724C211yesmaximal, characteristic, complemented
14#16810692C211yesmaximal, characteristic, complemented
15C2×#137×C53C211yesmaximal, characteristic, complemented
16C2×#11719949C2211yescharacteristic
17C2×#11719947C2211yescharacteristic
18#13133238—24no—
19C2×#11719925C2211yescharacteristic
20C2×#11719925C2211yescharacteristic
21C2×#11719913C2211yescharacteristic
22C2×#11719924C2211yescharacteristic
23C2×#11719922C2211yescharacteristic
24C2×C8×D53C2211yescharacteristic
25#13133215C2211yescharacteristic
26#13133206—21no—
27#13133206C2211yescharacteristic
28#13133200—21no—
29#13133198—21no—
30#13133196C2211yescharacteristic
31#13133193—21no—
32#13133193C2211yescharacteristic
33#13133193C2211yescharacteristic
34#13133191—21no—
35#13133191C2211yescharacteristic
36#13133189—21no—
37Q8×Dic53C2211yescharacteristic
38D4×Dic53C2211yescharacteristic
39C4×C22⋊D53—21no—
40C4×C22⋊D53C2211yescharacteristic
41C2×#11719919—21no—
42#13133151C2211yescharacteristic
43#13133146C2211yescharacteristic
44#13133142C2211yescharacteristic
45#13133141C2211yescharacteristic
46#13133141C2211yescharacteristic
47#13133141C2211yescharacteristic
48#13133139C2211yescharacteristic
49#13133138C2211yescharacteristic
50C22⋊#88322C2211yescharacteristic
51#13133129C2211yescharacteristic
52#13133127C2211yescharacteristic
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
77C22×Dic53—21no—
78C22×Dic53—21no—
79#11719949—22no—
80#11719949—41no—
81#11719947—22no—
82#11719947—41no—
83C2×D212—21no—
84C2×D212C2311yescharacteristic
85C2×C4×D53—21no—
86C2×C4×D53C2311yescharacteristic
87C2×Dic106C2311yescharacteristic
88Q16×C53—22no—
89Q16×C53—22no—
90SD16×C53—22no—
91SD16×C53D414yescomplemented
92M16×C53—22no—
93C2×C8×C53C2311yescharacteristic
94C2×C8×C53C2311yescharacteristic
95C22⋊Dic53—21no—
96C22⋊Dic53—21no—
97C22⋊Dic53D411yescharacteristic
98#11719925—22no—
99#11719925—41no—
100#11719924—41no—
101#11719922—22no—
102#11719921—21no—
103#11719921—21no—
104#11719921—21no—
105#11719921—21no—
106#11719921—21no—
107#11719921D411yescharacteristic
108C4⋊Dic53—21no—
109C4⋊Dic53C2311yescharacteristic
110C4⋊Dic53C2311yescharacteristic
111#11719919—21no—
112#11719919—21no—
113#11719919—21no—
114#11719919—21no—
115#11719919—21no—
116#11719919D411yescharacteristic
117C4×Dic53—21no—
118C4×Dic53C2311yescharacteristic
119C2×#88322C2311yescharacteristic
120C2×#88322C2311yescharacteristic
121#11719913—22no—
122C8×D53—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—21no—
132Q8×C53—42no—
133Q8×C53C2×D412yes—
134D4×C53—21no—
135D4×C53—42no—
136D4×C53C2×D412yes—
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×Dic53—42no—
153C2×Dic53—42no—
154C2×Dic53C2×D411yescharacteristic
155C2×Dic53C2×D411yescharacteristic
156C2×Dic53C2×D411yescharacteristic
157D212—21no—
158D212—21no—
159D212—42no—
160C4×D53—21no—
161C4×D53—21no—
162C4×D53—42no—
163Dic106—21no—
164Dic106—21no—
165C8×C53—21no—
166C8×C53C2×D412yes—
167#88322—21no—
168#88322—21no—
169C22×C53—22no—
170C22×C53—21no—
171C22×C53—21no—
172C22×C53—21no—
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—21no—
184C4×C53—41no—
185C4×C53—41no—
186C4×C53C22×D411yescharacteristic
187C4×C53H(5,𝔽2)11yescharacteristic
188Dic53—21no—
189Dic53—21no—
190Dic53—21no—
191Dic53—41no—
192Dic53—41no—
193Dic53—41no—
194#3269—531nomaximal, Sylow
195C2×C53—41no—
196C2×C53—42no—
197C2×C53D4211yescharacteristic
198C2×C53#57511yescharacteristic
199C2×C53#57811yescharacteristic
200D53—41no—
201D53—81no—
202C2×#137D5311yescharacteristic, complemented
203C2×#135—531no—
204#548—531no—
205#547—531no—
206#497—531no—
207#468—531no—
208#467—531no—
209#463—531no—
210#462—531no—
211#452—531no—
212#452—531no—
213#451—531no—
214#449—531no—
215#440—531no—
216#435—531no—
217C53#326911yescharacteristic, complemented, Sylow
218C2×#41—531no—
219C2×#41—531no—
220C2×#41C2×D5311yescharacteristic
221C22×Q8C2×D5311yescharacteristic
222#137—24no—
223#135—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×M16C2×D5311yescharacteristic
234C22×C8—531no—
235C4⋊Q8—531no—
236#124—531no—
237#124—1061no—
238#123—1061no—
239#123—1061no—
240C22⋊Q8—531no—
241C22⋊Q8—531no—
242C22⋊Q8—531no—
243C22⋊Q8—1061no—
244#121—531no—
245#121—1061no—
246#121—1061no—
247C4×Q8—531no—
248C4×D4—531no—
249C4×D4—531no—
250C4×D4—1061no—
251C2×C4⋊C4—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—
262#97—531no—
263#41—22no—
264#41—41no—
265#41—1062no—
266#41—1062no—
267#41—2121no—
268#41—2121no—
269C2×Q8—21no—
270C2×Q8—22no—
271C2×Q8—41no—
272C2×Q8—531no—
273C2×Q8C22×D5311yescharacteristic
274C2×Q8C22×D5311yescharacteristic
275C2×Q8C22×D5311yescharacteristic
276C2×D4—21no—
277C2×D4—531no—
278C2×D4—531no—
279C2×D4—1061no—
280C2×D4—1061no—
281C2×D4—1061no—
282C2×D4C22×D5311yescharacteristic
283C22×C4—21no—
284C22×C4—21no—
285C22×C4—531no—
286C22×C4—1061no—
287C22×C4—1061no—
288C22×C4—1061no—
289C22×C4C22×D5311yescharacteristic
290Q16—22no—
291Q16—22no—
292Q16—1062no—
293Q16—2121no—
294SD16—22no—
295SD16—1062no—
296SD16—2121no—
297SD16C22⋊D5314yescomplemented
298D8—1062no—
299M16—22no—
300C2×C8—531no—
301C2×C8—531no—
302C2×C8—1062no—
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—
312C4⋊C4—1061no—
313C4⋊C4—1061no—
314C22⋊C4—531no—
315C22⋊C4—531no—
316C22⋊C4—1061no—
317C22⋊C4—1061no—
318C22⋊C4—1061no—
319C22⋊C4—1061no—
320C22⋊C4—1061no—
321C22⋊C4—1061no—
322C22⋊C4—1061no—
323C42—531no—
324C42—1061no—
325C23—21no—
326C23—531no—
327C23—1061no—
328C23D4×D5311yescharacteristic
329Q8—21no—
330Q8—21no—
331Q8—21no—
332Q8—21no—
333Q8—21no—
334Q8—42no—
335Q8—1061no—
336Q8—1061no—
337Q8C2×C22⋊D5312yes—
338D4—21no—
339D4—42no—
340D4—1061no—
341D4—1061no—
342D4—2122no—
343D4—2121no—
344D4—2122no—
345D4—2122no—
346D4C2×C22⋊D5312yes—
347C2×C4—21no—
348C2×C4—21no—
349C2×C4—21no—
350C2×C4—21no—
351C2×C4—21no—
352C2×C4—42no—
353C2×C4—42no—
354C2×C4—531no—
355C2×C4—531no—
356C2×C4—531no—
357C2×C4—1061no—
358C2×C4—1061no—
359C2×C4—1061no—
360C2×C4—1061no—
361C2×C4—1061no—
362C2×C4—2122no—
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—22no—
375C22—41no—
376C22—41no—
377C22—1061no—
378C22—1061no—
379C22—1061no—
380C22—2121no—
381C22—2121no—
382C22—2121no—
383C22C2×D4×D5311yescharacteristic
384C4—21no—
385C4—22no—
386C4—41no—
387C4—41no—
388C4—1061no—
389C4—1061no—
390C4—1061no—
391C4—2121no—
392C4—2121no—
393C4—2121no—
394C4C22×C22⋊D5311yescharacteristic
395C4#1313326211yescharacteristic
396C2—41no—
397C2—42no—
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#3269
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×#13133238C2semidirect
2. #16811284C2semidirect
3. #16811272C2semidirect
4. #16811016C2semidirect
5. #16810869C2semidirect
6. #16810861C2semidirect
7. #16810840C2semidirect
8. #16810831C2semidirect
9. #16810784C2semidirect
10. #16810782C2semidirect
11. #16810779C2semidirect
12. #16810768C2semidirect
13. #16810724C2semidirect
14. #16810692C2semidirect
15. C2×#137×C53C2semidirect
91. SD16×C53D4semidirect
202. C2×#137D53semidirect
217. C53#3269semidirect
297. SD16C22⋊D53semidirect