Finite Group Databasebeta

← C4×#16810413 #27299229

Subgroups — C4×#16810413 #27299229

Order 13568.

576 conjugacy classes of subgroups.

The subgroup lattice has 400 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#16810413C212yesmaximal
2C4×#13133134C211yesmaximal, characteristic
3C4×#13133132C211yesmaximal, characteristic
4#20272833C212yesmaximal
5C4×#123×C53C211yesmaximal, characteristic
6C2×#13133134C2212yes—
7C2×#13133132C2212yes—
8C4×#11719917—21no—
9C4×C22⋊Dic53—21no—
10#16810413C418yesdirect product factor
11#16810392—21no—
12#16810390—21no—
13C2×#123×C53C2211yescharacteristic
14C2×C4×D4×C53C2211yescharacteristic
15C22×C42×C53C411yescharacteristic, complemented
16C53×#388—21no—
17C53×#387—21no—
18C53×#386C2211yescharacteristic
19C53×#385C411yescharacteristic, complemented
20C53×#381C411yescharacteristic, complemented
21C4×C4⋊C4×C53—21no—
22C4×C22⋊C4×C53—21no—
23C4×C22⋊C4×C53C411yescharacteristic, complemented
24C2×#11719917—22no—
25C2×C22⋊Dic53—22no—
26C2×C4×Dic53—41no—
27#13133134—22no—
28#13133134C2×C414yes—
29#13133132—22no—
30#13133132C2×C414yes—
31#13133128—41no—
32C4×#88322—41no—
33#13133124—41no—
34C22×D4×C53C2311yescharacteristic
35C23×C4×C53C2×C411yescharacteristic
36C23×C4×C53C2×C411yescharacteristic
37C23×C4×C53C2×C411yescharacteristic
38C23×C4×C53D411yescharacteristic
39#123×C53—22no—
40#123×C53C2×C414yes—
41C4×D4×C53—41no—
42C4×D4×C53—41no—
43C2×C4⋊C4×C53—21no—
44C2×C4⋊C4×C53—21no—
45C2×C4⋊C4×C53D411yescharacteristic
46C2×C22⋊C4×C53—21no—
47C2×C22⋊C4×C53—21no—
48C2×C22⋊C4×C53C2×C411yescharacteristic
49C2×C22⋊C4×C53C2×C411yescharacteristic
50C2×C22⋊C4×C53C2×C411yescharacteristic
51C2×C22⋊C4×C53D411yescharacteristic
52C2×C42×C53—21no—
53C2×C42×C53—21no—
54C2×C42×C53—21no—
55C2×C42×C53—41no—
56C2×C42×C53—41no—
57C2×C42×C53D411yescharacteristic
58#95×C53—21no—
59#95×C53—21no—
60#95×C53—21no—
61#95×C53—21no—
62#95×C53—21no—
63C24×C53C22×C411yescharacteristic
64C24×C53C2×D411yescharacteristic
65C2×D4×C53—41no—
66C2×D4×C53—41no—
67C2×D4×C53C22×C412yes—
68C2×D4×C53#4112yes—
69C22×C4×C53—21no—
70C22×C4×C53—21no—
71C22×C4×C53—21no—
72C22×C4×C53—21no—
73C22×C4×C53—21no—
74C22×C4×C53—21no—
75C22×C4×C53—21no—
76C22×C4×C53—21no—
77C22×C4×C53—21no—
78C22×C4×C53—42no—
79C22×C4×C53—42no—
80C22×C4×C53—42no—
81C22×C4×C53—41no—
82C22×C4×C53—41no—
83C22×C4×C53—41no—
84C22×C4×C53C4212yescomplemented
85C22×C4×C53C22⋊C411yescharacteristic
86C22×C4×C53C22⋊C411yescharacteristic
87C22×C4×C53C22⋊C411yescharacteristic
88C22×C4×C53C22⋊C411yescharacteristic
89C22×C4×C53C2×D411yescharacteristic
90C22×Dic53—42no—
91C4⋊C4×C53—24no—
92C4⋊C4×C53—41no—
93C4⋊C4×C53—42no—
94C22⋊C4×C53—21no—
95C22⋊C4×C53—24no—
96C22⋊C4×C53—41no—
97C22⋊C4×C53—41no—
98C22⋊C4×C53—42no—
99C22⋊C4×C53—41no—
100C22⋊C4×C53C4212yescomplemented
101C42×C53—21no—
102C42×C53—21no—
103C42×C53—42no—
104C42×C53—42no—
105C42×C53—41no—
106C42×C53—41no—
107C22⋊Dic53—24no—
108C22⋊Dic53—42no—
109C4×Dic53—82no—
110#11719917—24no—
111#11719917—42no—
112C2×#88322—42no—
113C23×C53—21no—
114C23×C53—21no—
115C23×C53—21no—
116C23×C53—21no—
117C23×C53—41no—
118C23×C53—41no—
119C23×C53—41no—
120C23×C53—41no—
121C23×C53C2×C4211yescharacteristic
122C23×C53C2×C22⋊C411yescharacteristic
123C23×C53#11711yescharacteristic
124C23×C53C4×D412yes—
125D4×C53—42no—
126D4×C53—42no—
127C2×C4×C53—21no—
128C2×C4×C53—21no—
129C2×C4×C53—22no—
130C2×C4×C53—22no—
131C2×C4×C53—22no—
132C2×C4×C53—22no—
133C2×C4×C53—21no—
134C2×C4×C53—21no—
135C2×C4×C53—21no—
136C2×C4×C53—42no—
137C2×C4×C53—41no—
138C2×C4×C53—42no—
139C2×C4×C53—41no—
140C2×C4×C53—41no—
141C2×C4×C53—42no—
142C2×C4×C53—41no—
143C2×C4×C53—41no—
144C2×C4×C53—41no—
145C2×C4×C53—42no—
146C2×C4×C53—42no—
147C2×C4×C53—42no—
148C2×C4×C53—41no—
149C2×C4×C53—41no—
150C2×C4×C53—41no—
151C2×C4×C53—42no—
152C2×C4×C53—41no—
153C2×C4×C53—41no—
154C2×C4×C53—41no—
155C2×C4×C53—81no—
156C2×C4×C53#9911yescharacteristic
157C2×C4×C53#9911yescharacteristic
158C2×C4×C53C4×D412yes—
159C2×Dic53—44no—
160C2×Dic53—82no—
161C2×Dic53—82no—
162#88322—44no—
163C4×#352—531nomaximal, Sylow
164C22×C53—21no—
165C22×C53—21no—
166C22×C53—22no—
167C22×C53—21no—
168C22×C53—21no—
169C22×C53—41no—
170C22×C53—41no—
171C22×C53—42no—
172C22×C53—42no—
173C22×C53—41no—
174C22×C53—41no—
175C22×C53—41no—
176C22×C53—41no—
177C22×C53—81no—
178C22×C53C4×C22⋊C411yescharacteristic
179C22×C53C2×#9911yescharacteristic
180C22×C53#41011yescharacteristic
181C4×C53—21no—
182C4×C53—22no—
183C4×C53—42no—
184C4×C53—41no—
185C4×C53—41no—
186C4×C53—41no—
187C4×C53—42no—
188C4×C53—42no—
189C4×C53—81no—
190C4×C53#35212yescomplemented
191Dic53—84no—
192C4×#123D5311yescharacteristic
193C2×#352—532no—
194C4×#100—531no—
195C4×#99—531no—
196#1390—532no—
197C2×C53—21no—
198C2×C53—21no—
199C2×C53—42no—
200C2×C53—41no—
201C2×C53—41no—
202C2×C53C4×#9911yescharacteristic
203C2×C53C2×#35211yescharacteristic
204C2×C53#212011yescharacteristic
205C2×#123C2×D5311yescharacteristic
206C2×C4×D4C2×D5311yescharacteristic
207C22×C42Dic5311yescharacteristic, complemented
208C2×#100—532no—
209C2×#99—532no—
210C4×M16—1061no—
211#388—21no—
212#387—21no—
213#386C2×D5311yescharacteristic
214#385Dic5311yescharacteristic, complemented
215#381Dic5311yescharacteristic, complemented
216C4×C4⋊C4—21no—
217C4×C22⋊C4—21no—
218C4×C22⋊C4—1061no—
219C4×C22⋊C4Dic5311yescharacteristic, complemented
220#352—538no—
221#343—1061no—
222#342—1061no—
223C53C4×#35211yescharacteristic, complemented, Sylow
224C22×D4C22×D5311yescharacteristic
225C23×C4C2×Dic5311yescharacteristic
226C23×C4C2×Dic5311yescharacteristic
227C23×C4C2×Dic5311yescharacteristic
228C23×C4C22⋊D5311yescharacteristic
229C2×M16—1062no—
230#123—22no—
231#123C4×D5314yes—
232C4×D4—41no—
233C4×D4—41no—
234C2×C4⋊C4—21no—
235C2×C4⋊C4—21no—
236C2×C4⋊C4C22⋊D5311yescharacteristic
237C2×C22⋊C4—21no—
238C2×C22⋊C4—21no—
239C2×C22⋊C4—1062no—
240C2×C22⋊C4C2×Dic5311yescharacteristic
241C2×C22⋊C4C2×Dic5311yescharacteristic
242C2×C22⋊C4C2×Dic5311yescharacteristic
243C2×C22⋊C4C22⋊D5311yescharacteristic
244C2×C42—21no—
245C2×C42—21no—
246C2×C42—21no—
247C2×C42—41no—
248C2×C42—41no—
249C2×C42—2121no—
250C2×C42C22⋊D5311yescharacteristic
251#100—534no—
252#100—1062no—
253#99—534no—
254#99—1062no—
255#97—2121no—
256C4×C8—2121no—
257#95—21no—
258#95—21no—
259#95—21no—
260#95—21no—
261#95—21no—
262#95—2121no—
263C24C22×Dic5311yescharacteristic
264C24C2×C22⋊D5311yescharacteristic
265C2×D4—41no—
266C2×D4—41no—
267C2×D4C2×C4×D5312yes—
268C2×D4#1171994512yes—
269C22×C4—21no—
270C22×C4—21no—
271C22×C4—21no—
272C22×C4—21no—
273C22×C4—21no—
274C22×C4—21no—
275C22×C4—21no—
276C22×C4—21no—
277C22×C4—21no—
278C22×C4—42no—
279C22×C4—42no—
280C22×C4—42no—
281C22×C4—41no—
282C22×C4—41no—
283C22×C4—41no—
284C22×C4—2122no—
285C22×C4C4×Dic5312yescomplemented
286C22×C4C22⋊Dic5311yescharacteristic
287C22×C4C22⋊Dic5311yescharacteristic
288C22×C4C22⋊Dic5311yescharacteristic
289C22×C4C22⋊Dic5311yescharacteristic
290C22×C4C2×C22⋊D5311yescharacteristic
291M16—1064no—
292M16—2122no—
293C2×C8—2122no—
294C4⋊C4—24no—
295C4⋊C4—41no—
296C4⋊C4—42no—
297C22⋊C4—21no—
298C22⋊C4—24no—
299C22⋊C4—41no—
300C22⋊C4—41no—
301C22⋊C4—42no—
302C22⋊C4—41no—
303C22⋊C4—1064no—
304C22⋊C4—2122no—
305C22⋊C4C4×Dic5312yescomplemented
306C42—21no—
307C42—21no—
308C42—42no—
309C42—42no—
310C42—41no—
311C42—41no—
312C42—4242no—
313C23—21no—
314C23—21no—
315C23—21no—
316C23—21no—
317C23—41no—
318C23—41no—
319C23—41no—
320C23—41no—
321C23C2×C4×Dic5311yescharacteristic
322C23C2×C22⋊Dic5311yescharacteristic
323C23#1313317611yescharacteristic
324C23C4×C22⋊D5312yes—
325D4—42no—
326D4—42no—
327C2×C4—21no—
328C2×C4—21no—
329C2×C4—22no—
330C2×C4—22no—
331C2×C4—21no—
332C2×C4—21no—
333C2×C4—21no—
334C2×C4—22no—
335C2×C4—22no—
336C2×C4—42no—
337C2×C4—41no—
338C2×C4—41no—
339C2×C4—42no—
340C2×C4—41no—
341C2×C4—42no—
342C2×C4—41no—
343C2×C4—41no—
344C2×C4—41no—
345C2×C4—42no—
346C2×C4—42no—
347C2×C4—41no—
348C2×C4—42no—
349C2×C4—41no—
350C2×C4—41no—
351C2×C4—42no—
352C2×C4—41no—
353C2×C4—41no—
354C2×C4—41no—
355C2×C4—81no—
356C2×C4—4242no—
357C2×C4—4242no—
358C2×C4—2124no—
359C2×C4#1313313211yescharacteristic
360C2×C4#1313313211yescharacteristic
361C2×C4C4×C22⋊D5312yes—
362C8—2124no—
363C22—21no—
364C22—21no—
365C22—21no—
366C22—21no—
367C22—22no—
368C22—41no—
369C22—41no—
370C22—42no—
371C22—42no—
372C22—41no—
373C22—41no—
374C22—41no—
375C22—41no—
376C22—81no—
377C22C4×C22⋊Dic5311yescharacteristic
378C22C2×#1313313211yescharacteristic
379C22#1681059711yescharacteristic
380C4—22no—
381C4—21no—
382C4—42no—
383C4—42no—
384C4—41no—
385C4—41no—
386C4—42no—
387C4—41no—
388C4—81no—
389C4—4244no—
390C4#1681041312yesdirect product factor
391C2—21no—
392C2—21no—
393C2—42no—
394C2—41no—
395C2—41no—
396C2C4×#1313313211yescharacteristic
397C2C2×#1681041311yescharacteristic
398C2#2027635611yescharacteristic
subgroupG/H
centerC2×C4#13133132
commutatorC23×C53C2×C42
FrattiniC22×D4C22×D53
FittingC4×#123×C53C2
socleC22×C53C2×#99
prime radical, cosocleC22×D4×C53C23
hypercenterC4×#123D53
hypocenterC53C4×#352
LCSG ▷ C23×C53 ▷ C22×C53 ▷ C2×C53 ▷ C53
UCSC1 ◁ C2×C4 ◁ C22×C4 ◁ C2×C4×D4 ◁ C4×#123

Complements

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

NH ≅ G/Ntype
10. #16810413C4direct (N × H)
15. C22×C42×C53C4semidirect
19. C53×#385C4semidirect
20. C53×#381C4semidirect
23. C4×C22⋊C4×C53C4semidirect
84. C22×C4×C53C42semidirect
100. C22⋊C4×C53C42semidirect
190. C4×C53#352semidirect
207. C22×C42Dic53semidirect
214. #385Dic53semidirect
215. #381Dic53semidirect
219. C4×C22⋊C4Dic53semidirect
223. C53C4×#352semidirect
285. C22×C4C4×Dic53semidirect
305. C22⋊C4C4×Dic53semidirect
390. C4#16810413direct (N × H)

Maximal subgroups

By isomorphism type — maximal subgroups of the same type appear once, whether or not they are normal.

subgroupindex
C2×#168104132
C4×#131331342
C4×#131331322
#202728332
C4×#123×C532
C4×#35253