Finite Group Databasebeta

← #20284166

Subgroups — #20284166

Order 6784.

496 conjugacy classes of subgroups.

The subgroup lattice has 396 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#13133237C211yesmaximal, characteristic, complemented
2#16811169C211yesmaximal, characteristic, complemented
3#41×Dic53C211yesmaximal, characteristic, complemented
4#16810847C211yesmaximal, characteristic, complemented
5#16810843C211yesmaximal, characteristic, complemented
6#16810840C211yesmaximal, characteristic, complemented
7#16810836C211yesmaximal, characteristic, complemented
8#16810831C211yesmaximal, characteristic, complemented
9#16810826C211yesmaximal, characteristic, complemented
10#16810821C211yesmaximal, characteristic, complemented
11#16810816C211yesmaximal, characteristic, complemented
12C2×#13133146C211yesmaximal, characteristic, complemented
13#16810620C211yesmaximal, characteristic, complemented
14#16810619C211yesmaximal, characteristic, complemented
15C2×#135×C53C211yesmaximal, characteristic, complemented
16C2×#11719945C2211yescharacteristic
17#13133237—24no—
18C2×#11719925C2211yescharacteristic
19C2×#11719923C2211yescharacteristic
20C2×#11719924C2211yescharacteristic
21C2×#11719922C2211yescharacteristic
22C22×#88322C2211yescharacteristic
23#13133215C2211yescharacteristic
24#13133213C2211yescharacteristic
25#13133206C2211yescharacteristic
26#13133204C2211yescharacteristic
27#13133189—21no—
28#13133189—21no—
29Q8×Dic53C2211yescharacteristic
30Q8×Dic53C2211yescharacteristic
31D4×Dic53—21no—
32D4×Dic53—21no—
33D4×Dic53C2211yescharacteristic
34D4×Dic53C2211yescharacteristic
35C4×C22⋊D53—21no—
36C4×C22⋊D53—21no—
37#13133176—21no—
38#13133176C2211yescharacteristic
39C2×C4×Dic53—21no—
40C2×C4×Dic53C2211yescharacteristic
41#13133146C2211yescharacteristic
42#13133146C2211yescharacteristic
43#13133146C2211yescharacteristic
44#13133146C2211yescharacteristic
45#13133142C2211yescharacteristic
46#13133142C2211yescharacteristic
47#13133141C2211yescharacteristic
48#13133141C2211yescharacteristic
49#13133139C2211yescharacteristic
50#13133139C2211yescharacteristic
51#13133138C2211yescharacteristic
52#13133138C2211yescharacteristic
53C2×#41×C53C2211yescharacteristic
54C2×#41×C53C2211yescharacteristic
55#135×C53—24no—
56C2×Q16×C53C2211yescharacteristic
57C2×SD16×C53C2211yescharacteristic
58C2×SD16×C53C2211yescharacteristic
59C2×D8×C53C2211yescharacteristic
60C22×C8×C53C2211yescharacteristic
61#41×C53—22no—
62#41×C53—22no—
63#41×C53—22no—
64#41×C53D414yescomplemented
65C2×Q8×C53C2311yescharacteristic
66C2×Q8×C53C2311yescharacteristic
67C2×D4×C53—21no—
68C2×D4×C53—21no—
69C2×D4×C53C2311yescharacteristic
70C2×D4×C53C2311yescharacteristic
71C22×C4×C53—21no—
72C22×C4×C53—21no—
73C22×C4×C53C2311yescharacteristic
74C2×C22⋊D53—21no—
75C2×C22⋊D53—21no—
76C2×C22⋊D53—21no—
77C22×Dic53—21no—
78C22×Dic53—21no—
79C22×Dic53D411yescharacteristic
80C22×Dic53D411yescharacteristic
81#11719945—22no—
82#11719945—22no—
83C2×D212C2311yescharacteristic
84C2×C4×D53—21no—
85C2×Dic106C2311yescharacteristic
86Q16×C53—22no—
87SD16×C53—22no—
88SD16×C53—22no—
89D8×C53—22no—
90C2×C8×C53—22no—
91C2×C8×C53C2311yescharacteristic
92C2×C8×C53C2311yescharacteristic
93C22⋊Dic53—21no—
94C22⋊Dic53—21no—
95C22⋊Dic53—21no—
96C22⋊Dic53—21no—
97C22⋊Dic53—21no—
98#11719925—22no—
99#11719924—22no—
100#11719923—22no—
101#11719922—22no—
102#11719921—21no—
103#11719921—21no—
104C4⋊Dic53—21no—
105C4⋊Dic53—21no—
106C4⋊Dic53C2311yescharacteristic
107C4⋊Dic53C2311yescharacteristic
108#11719919—21no—
109#11719919—21no—
110C4×Dic53—21no—
111C4×Dic53—21no—
112C4×Dic53—21no—
113C4×Dic53D411yescharacteristic
114C4×Dic53D411yescharacteristic
115C4×Dic53C2311yescharacteristic
116C4×Dic53C2311yescharacteristic
117C2×#88322—22no—
118C2×#88322C2311yescharacteristic
119C2×#88322C2311yescharacteristic
120C23×C53—21no—
121C23×C53—21no—
122C23×C53C2×D411yescharacteristic
123C22×D53—21no—
124Q8×C53—21no—
125Q8×C53—21no—
126Q8×C53—21no—
127Q8×C53C2×D412yes—
128D4×C53—24no—
129D4×C53—21no—
130D4×C53—21no—
131D4×C53—21no—
132D4×C53—42no—
133D4×C53C2×D412yes—
134C2×C4×C53—21no—
135C2×C4×C53—24no—
136C2×C4×C53—21no—
137C2×C4×C53—42no—
138C2×C4×C53C2×D411yescharacteristic
139C2×C4×C53C2×D412yes—
140C2×C4×C53#4112yes—
141C2×C4×C53C2411yescharacteristic
142C22⋊D53—42no—
143C22⋊D53—42no—
144C22⋊D53—42no—
145C2×Dic53—22no—
146C2×Dic53—21no—
147C2×Dic53—21no—
148C2×Dic53—21no—
149C2×Dic53—41no—
150C2×Dic53—42no—
151C2×Dic53—42no—
152C2×Dic53C2×D411yescharacteristic
153C2×Dic53C2×D411yescharacteristic
154C2×Dic53C2×D411yescharacteristic
155C2×Dic53C2×D411yescharacteristic
156D212—21no—
157D212—21no—
158C4×D53—42no—
159Dic106—21no—
160Dic106—21no—
161C8×C53—21no—
162C8×C53—21no—
163#88322—21no—
164#88322—21no—
165C22×C53—21no—
166C22×C53—21no—
167C22×C53—22no—
168C22×C53—41no—
169C22×C53—41no—
170C22×C53—41no—
171C22×C53—41no—
172C22×C53—41no—
173C22×C53#12112yes—
174C22×C53C22×D411yescharacteristic
175C2×D53—41no—
176C2×D53—41no—
177C2×D53—41no—
178C4×C53—22no—
179C4×C53—41no—
180C4×C53#12112yes—
181C4×C53C22×D411yescharacteristic
182C4×C53C2×#4111yescharacteristic
183Dic53—21no—
184Dic53—21no—
185Dic53—21no—
186Dic53—41no—
187Dic53—41no—
188Dic53—41no—
189Dic53#13512yescomplemented
190#3035—531nomaximal, Sylow
191C2×C53—22no—
192C2×C53—42no—
193C2×C53—81no—
194C2×C53C2×#12111yescharacteristic
195C2×C53C2×#13511yescharacteristic
196C2×C53#57511yescharacteristic
197D53—81no—
198C2×#135—531no—
199C2×#135D5311yescharacteristic, complemented
200#532—531no—
201C4×#41—531no—
202#464—531no—
203#464—531no—
204#463—531no—
205#463—531no—
206#462—531no—
207#461—531no—
208#460—531no—
209#459—531no—
210C2×C4⋊C8—531no—
211#416—531no—
212#416—531no—
213C53#303511yescharacteristic, complemented, Sylow
214C2×#41—531no—
215C2×#41C2×D5311yescharacteristic
216C2×#41C2×D5311yescharacteristic
217#135—24no—
218#135—1064no—
219C2×Q16—531no—
220C2×Q16C2×D5311yescharacteristic
221C2×SD16—531no—
222C2×SD16—531no—
223C2×SD16C2×D5311yescharacteristic
224C2×SD16C2×D5311yescharacteristic
225C2×D8—531no—
226C2×D8C2×D5311yescharacteristic
227C22×C8—531no—
228C22×C8C2×D5311yescharacteristic
229C4⋊Q8—531no—
230#127—531no—
231#124—531no—
232#124—531no—
233#121—1061no—
234#121—1061no—
235C4×Q8—531no—
236C4×Q8—531no—
237C4×D4—531no—
238C4×D4—531no—
239C4×D4—1061no—
240C4×D4—1061no—
241C4×D4—1061no—
242C4×D4—1061no—
243#117—531no—
244#117—1061no—
245C2×C42—531no—
246C2×C42—1061no—
247C4⋊C8—531no—
248C4⋊C8—531no—
249C4⋊C8—531no—
250C4⋊C8—531no—
251#103—531no—
252#103—531no—
253#103—531no—
254#103—531no—
255#102—531no—
256#102—531no—
257#102—531no—
258#102—531no—
259#41—22no—
260#41—22no—
261#41—22no—
262#41—1062no—
263#41—1062no—
264#41C22⋊D5314yescomplemented
265C2×Q8—531no—
266C2×Q8C22×D5311yescharacteristic
267C2×Q8C22×D5311yescharacteristic
268C2×D4—21no—
269C2×D4—21no—
270C2×D4—531no—
271C2×D4—1061no—
272C2×D4—1061no—
273C2×D4—1061no—
274C2×D4C22×D5311yescharacteristic
275C2×D4C22×D5311yescharacteristic
276C22×C4—21no—
277C22×C4—21no—
278C22×C4—531no—
279C22×C4—531no—
280C22×C4—1061no—
281C22×C4—1061no—
282C22×C4—1061no—
283C22×C4C22×D5311yescharacteristic
284Q16—22no—
285Q16—1062no—
286SD16—22no—
287SD16—22no—
288SD16—1062no—
289SD16—1062no—
290D8—22no—
291D8—1062no—
292C2×C8—22no—
293C2×C8—531no—
294C2×C8—531no—
295C2×C8—1062no—
296C2×C8C22×D5311yescharacteristic
297C2×C8C22×D5311yescharacteristic
298C4⋊C4—531no—
299C4⋊C4—531no—
300C4⋊C4—1061no—
301C4⋊C4—1061no—
302C4⋊C4—1061no—
303C4⋊C4—1061no—
304C22⋊C4—1061no—
305C22⋊C4—1061no—
306C22⋊C4—1061no—
307C22⋊C4—1061no—
308C22⋊C4—1061no—
309C22⋊C4—1061no—
310C22⋊C4—1061no—
311C42—531no—
312C42—531no—
313C42—531no—
314C42—531no—
315C42—1061no—
316C42—1061no—
317C42—1061no—
318C23—21no—
319C23—21no—
320C23—1061no—
321C23D4×D5311yescharacteristic
322Q8—21no—
323Q8—21no—
324Q8—21no—
325Q8—1061no—
326Q8—1061no—
327Q8C2×C22⋊D5312yes—
328D4—24no—
329D4—21no—
330D4—21no—
331D4—21no—
332D4—42no—
333D4—1061no—
334D4—1061no—
335D4—2122no—
336D4—2122no—
337D4—2122no—
338D4C2×C22⋊D5312yes—
339C2×C4—21no—
340C2×C4—24no—
341C2×C4—21no—
342C2×C4—42no—
343C2×C4—531no—
344C2×C4—531no—
345C2×C4—531no—
346C2×C4—531no—
347C2×C4—1062no—
348C2×C4—1061no—
349C2×C4—1061no—
350C2×C4—1061no—
351C2×C4—2122no—
352C2×C4—2121no—
353C2×C4—2122no—
354C2×C4—2122no—
355C2×C4D4×D5311yescharacteristic
356C2×C4#1171994712yes—
357C2×C4C2×C22⋊D5312yes—
358C2×C4C23×D5311yescharacteristic
359C8—21no—
360C8—21no—
361C8—1061no—
362C8—1061no—
363C22—21no—
364C22—21no—
365C22—22no—
366C22—41no—
367C22—41no—
368C22—41no—
369C22—41no—
370C22—41no—
371C22—2121no—
372C22—2121no—
373C22—2121no—
374C22#1313318912yes—
375C22C2×D4×D5311yescharacteristic
376C4—22no—
377C4—41no—
378C4—532no—
379C4—1061no—
380C4—1061no—
381C4—1061no—
382C4—2121no—
383C4—2121no—
384C4—2121no—
385C4#1313318912yes—
386C4C22×C22⋊D5311yescharacteristic
387C4C2×#1171994711yescharacteristic
388C2—22no—
389C2—42no—
390C2—81no—
391C2—4241no—
392C2C2×#1313318911yescharacteristic
393C2#135×D5311yescharacteristic
394C2#1681142011yescharacteristic
subgroupG/H
centerC2×C4D4×D53
commutatorC2×C4×C53C24
FrattiniC2×C4C23×D53
FittingC2×#135×C53C2
socleC22×C53C22×D4
hypercenterC2×#135D53
hypocenterC53#3035
LCSG ▷ C2×C4×C53 ▷ C2×C53 ▷ C53
UCSC1 ◁ C2×C4 ◁ C22×C4 ◁ C2×#135

Complements

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

NH ≅ G/Ntype
1. C2×#13133237C2semidirect
2. #16811169C2semidirect
3. #41×Dic53C2semidirect
4. #16810847C2semidirect
5. #16810843C2semidirect
6. #16810840C2semidirect
7. #16810836C2semidirect
8. #16810831C2semidirect
9. #16810826C2semidirect
10. #16810821C2semidirect
11. #16810816C2semidirect
12. C2×#13133146C2semidirect
13. #16810620C2semidirect
14. #16810619C2semidirect
15. C2×#135×C53C2semidirect
64. #41×C53D4semidirect
189. Dic53#135semidirect
199. C2×#135D53semidirect
213. C53#3035semidirect
264. #41C22⋊D53semidirect