Finite Group Databasebeta

← C4×#16810414 #27299232

Subgroups — C4×#16810414 #27299232

Order 13568.

576 conjugacy classes of subgroups; only the 127 normal subgroups are shown.

normal subgroups only

This group has 576 conjugacy classes of subgroups — too many to draw. Only its 127 normal subgroups are shown.

Dotted edges: containments with no intermediate normal subgroup — omitted non-normal subgroups may lie between.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C4×#13133198C211yesmaximal, characteristic
2C2×#16810414C212yesmaximal
3C4×#13133132C211yesmaximal, characteristic
4#20272834C212yesmaximal
5C4×#100×C53C211yesmaximal, characteristic
6C2×#13133198C2211yescharacteristic
7C22×C4×Dic53C411yescharacteristic, complemented
8C2×#13133132C2212yes—
9#16810511—21no—
10#16810504—21no—
11#16810503C2211yescharacteristic
12#16810495C411yescharacteristic, complemented
13#16810484C411yescharacteristic, complemented
14C4×#11719919—21no—
15C4×C22⋊Dic53—21no—
16C4×C22⋊Dic53—21no—
17C4×C22⋊Dic53C411yescharacteristic, complemented
18#16810414C418yesdirect product factor
19#16810390—21no—
20C2×C4×D4×C53C2211yescharacteristic
21C2×#100×C53C2212yes—
22C4×M16×C53—21no—
23C53×#343—21no—
24C23×Dic53C2×C411yescharacteristic
25C23×Dic53C2×C411yescharacteristic
26#13133198—22no—
27#13133198C2×C414yes—
28C2×#11719919—21no—
29C2×#11719919—21no—
30C2×C22⋊Dic53—21no—
31C2×C22⋊Dic53—21no—
32C2×C22⋊Dic53—22no—
33C2×C22⋊Dic53C2×C411yescharacteristic
34C2×C22⋊Dic53C2×C411yescharacteristic
35C2×C4×Dic53—21no—
36C2×C4×Dic53—21no—
37C2×C4×Dic53—21no—
38C2×C4×Dic53—41no—
39C2×C4×Dic53—41no—
40C2×C4×Dic53—41no—
41#13133132—22no—
42#13133132C2×C414yes—
43#13133124—21no—
44#13133124—21no—
45#13133124—21no—
46#13133124—21no—
47#13133124—21no—
48#13133124—41no—
49C22×D4×C53C2311yescharacteristic
50C23×C4×C53C2×C411yescharacteristic
51C23×C4×C53D411yescharacteristic
52C2×M16×C53—22no—
53C4×D4×C53—41no—
54C4×D4×C53—41no—
55C2×C4⋊C4×C53D411yescharacteristic
56C2×C22⋊C4×C53C2×C411yescharacteristic
57C2×C22⋊C4×C53D411yescharacteristic
58C2×C42×C53D411yescharacteristic
59#100×C53—22no—
60#100×C53C2×C414yes—
61#97×C53—41no—
62C4×C8×C53—41no—
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—42no—
74C22×C4×C53—42no—
75C22×C4×C53C22⋊C411yescharacteristic
76C22×C4×C53C22⋊C411yescharacteristic
77C22×C4×C53C22⋊C411yescharacteristic
78C22×C4×C53C22⋊C411yescharacteristic
79C22×C4×C53C2×D411yescharacteristic
80C22×Dic53—21no—
81C22×Dic53—21no—
82C22×Dic53—21no—
83C22×Dic53—21no—
84C22×Dic53—21no—
85C22×Dic53—42no—
86C22×Dic53—41no—
87C22×Dic53—41no—
88C22×Dic53—41no—
89C22×Dic53—42no—
90C22×Dic53C4212yescomplemented
91M16×C53—24no—
92M16×C53—42no—
93C2×C8×C53—42no—
94C4⋊C4×C53—41no—
95C22⋊C4×C53—41no—
96C22⋊C4×C53—41no—
97C42×C53—21no—
98C42×C53—21no—
99C22⋊Dic53—21no—
100C22⋊Dic53—24no—
101C22⋊Dic53—24no—
102C22⋊Dic53—42no—
103C22⋊Dic53—42no—
104C22⋊Dic53—41no—
105C22⋊Dic53C4212yescomplemented
106#11719919—24no—
107#11719919—42no—
108C4×Dic53—42no—
109C4×Dic53—42no—
110C4×Dic53—41no—
111C4×Dic53—41no—
112C4×Dic53—82no—
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—21no—
131C2×C4×C53—21no—
132C2×C4×C53—21no—
133C2×C4×C53—42no—
134C2×C4×C53—42no—
135C2×C4×C53—42no—
136C2×C4×C53—41no—
137C2×C4×C53—41no—
138C2×C4×C53—41no—
139C2×C4×C53—81no—
140C2×C4×C53#9911yescharacteristic
141C2×C4×C53#9911yescharacteristic
142C2×C4×C53C4×D412yes—
143C2×Dic53—22no—
144C2×Dic53—22no—
145C2×Dic53—22no—
146C2×Dic53—41no—
147C2×Dic53—42no—
148C2×Dic53—41no—
149C2×Dic53—41no—
150C2×Dic53—42no—
151C2×Dic53—41no—
152C2×Dic53—41no—
153C2×Dic53—41no—
154C2×Dic53—42no—
155C2×Dic53—42no—
156C2×Dic53—41no—
157C2×Dic53—41no—
158C2×Dic53—44no—
159C2×Dic53—41no—
160C2×Dic53—82no—
161C2×Dic53—82no—
162C8×C53—44no—
163C4×#352—531nomaximal, Sylow
164C22×C53—21no—
165C22×C53—21no—
166C22×C53—21no—
167C22×C53—21no—
168C22×C53—22no—
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—41no—
184C4×C53—41no—
185C4×C53—41no—
186C4×C53#35212yescomplemented
187Dic53—42no—
188Dic53—42no—
189Dic53—42no—
190Dic53—81no—
191Dic53—84no—
192C4×#123—531no—
193C2×#352—532no—
194C4×#100D5311yescharacteristic
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×#123—531no—
206C2×C4×D4C2×D5311yescharacteristic
207C22×C42—531no—
208C2×#100C2×D5312yes—
209C2×#99—532no—
210C4×M16—21no—
211#388—1061no—
212#387—1061no—
213#386—531no—
214#385—531no—
215#381—531no—
216C4×C4⋊C4—1061no—
217C4×C22⋊C4—531no—
218C4×C22⋊C4—1061no—
219C4×C22⋊C4—1061no—
220#352—538no—
221#343—21no—
222#342—1061no—
223C53C4×#35211yescharacteristic, complemented, Sylow
224C22×D4C22×D5311yescharacteristic
225C23×C4—531no—
226C23×C4—531no—
227C23×C4C4×D5311yescharacteristic
228C23×C4D21211yescharacteristic
229C2×M16—22no—
230#123—534no—
231#123—1062no—
232C4×D4—41no—
233C4×D4—41no—
234C2×C4⋊C4—1061no—
235C2×C4⋊C4—1061no—
236C2×C4⋊C4C22⋊D5311yescharacteristic
237C2×C22⋊C4—531no—
238C2×C22⋊C4—531no—
239C2×C22⋊C4—1061no—
240C2×C22⋊C4—1062no—
241C2×C22⋊C4—1061no—
242C2×C22⋊C4C4×D5311yescharacteristic
243C2×C22⋊C4D21211yescharacteristic
244C2×C42—1061no—
245C2×C42—1061no—
246C2×C42—1061no—
247C2×C42—2121no—
248C2×C42—2121no—
249C2×C42—2121no—
250C2×C42C22⋊D5311yescharacteristic
251#100—22no—
252#100C4×D5314yes—
253#99—534no—
254#99—1062no—
255#97—41no—
256C4×C8—41no—
257#95—1061no—
258#95—1061no—
259#95—1061no—
260#95—1061no—
261#95—1061no—
262#95—2121no—
263C24C2×C4×D5311yescharacteristic
264C24C2×D21211yescharacteristic
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—42no—
274C22×C4—42no—
275C22×C4—532no—
276C22×C4—1061no—
277C22×C4—1061no—
278C22×C4—1061no—
279C22×C4—1061no—
280C22×C4—1061no—
281C22×C4—2122no—
282C22×C4—2122no—
283C22×C4—2121no—
284C22×C4—2121no—
285C22×C4—2121no—
286C22×C4#1171992111yescharacteristic
287C22×C4#1171992111yescharacteristic
288C22×C4#1171992111yescharacteristic
289C22×C4#1171992111yescharacteristic
290C22×C4C2×C22⋊D5311yescharacteristic
291M16—24no—
292M16—42no—
293C2×C8—42no—
294C4⋊C4—41no—
295C4⋊C4—1064no—
296C4⋊C4—2122no—
297C22⋊C4—41no—
298C22⋊C4—41no—
299C22⋊C4—532no—
300C22⋊C4—1061no—
301C22⋊C4—1064no—
302C22⋊C4—1064no—
303C22⋊C4—2122no—
304C22⋊C4—2122no—
305C22⋊C4—2121no—
306C42—21no—
307C42—21no—
308C42—4242no—
309C42—2122no—
310C42—2122no—
311C42—2121no—
312C42—2121no—
313C23—21no—
314C23—21no—
315C23—21no—
316C23—21no—
317C23—41no—
318C23—41no—
319C23—41no—
320C23—41no—
321C23C42×D5311yescharacteristic
322C23C2×#1171992111yescharacteristic
323C23#1313317411yescharacteristic
324C23C4×D21212yes—
325D4—42no—
326D4—42no—
327C2×C4—21no—
328C2×C4—21no—
329C2×C4—22no—
330C2×C4—21no—
331C2×C4—21no—
332C2×C4—21no—
333C2×C4—42no—
334C2×C4—42no—
335C2×C4—42no—
336C2×C4—41no—
337C2×C4—41no—
338C2×C4—41no—
339C2×C4—81no—
340C2×C4—1062no—
341C2×C4—1062no—
342C2×C4—1062no—
343C2×C4—4242no—
344C2×C4—4242no—
345C2×C4—2121no—
346C2×C4—2124no—
347C2×C4—2121no—
348C2×C4—2121no—
349C2×C4—2121no—
350C2×C4—2121no—
351C2×C4—2122no—
352C2×C4—2121no—
353C2×C4—2122no—
354C2×C4—2121no—
355C2×C4—2121no—
356C2×C4—2121no—
357C2×C4—2122no—
358C2×C4—2122no—
359C2×C4#1313313111yescharacteristic
360C2×C4#1313313111yescharacteristic
361C2×C4C4×C22⋊D5312yes—
362C8—44no—
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×#1171992111yescharacteristic
378C22C2×#1313313111yescharacteristic
379C22#1681059611yescharacteristic
380C4—22no—
381C4—21no—
382C4—41no—
383C4—41no—
384C4—41no—
385C4—4241no—
386C4—4244no—
387C4—2122no—
388C4—2122no—
389C4—2122no—
390C4#1681041412yesdirect product factor
391C2—21no—
392C2—21no—
393C2—42no—
394C2—41no—
395C2—41no—
396C2C4×#1313313111yescharacteristic
397C2C2×#1681041411yescharacteristic
398C2#2027635711yescharacteristic
subgroupG/H
centerC2×C4#13133131
commutatorC23×C53C2×C42
FrattiniC22×D4C22×D53
FittingC4×#100×C53C2
socleC22×C53C2×#99
prime radical, cosocleC22×D4×C53C23
hypercenterC4×#100D53
hypocenterC53C4×#352
LCSG ▷ C23×C53 ▷ C22×C53 ▷ C2×C53 ▷ C53
UCSC1 ◁ C2×C4 ◁ C22×C4 ◁ C2×C4×D4 ◁ C4×#100

Complements

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

NH ≅ G/Ntype
7. C22×C4×Dic53C4semidirect
12. #16810495C4semidirect
13. #16810484C4semidirect
17. C4×C22⋊Dic53C4semidirect
18. #16810414C4direct (N × H)
90. C22×Dic53C42semidirect
105. C22⋊Dic53C42semidirect
186. C4×C53#352semidirect
223. C53C4×#352semidirect
390. C4#16810414direct (N × H)

Maximal subgroups

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

subgroupindex
C4×#131331982
C2×#168104142
C4×#131331322
#202728342
C4×#100×C532
C4×#35253