Finite Group Databasebeta

← C2×#20273831 #27359702

Subgroups — C2×#20273831 #27359702

Order 13568.

1722 conjugacy classes of subgroups; only the 251 normal subgroups are shown.

normal subgroups only

This group has 1722 conjugacy classes of subgroups — too many to draw. Only its 251 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
1C2×#16810995C211yesmaximal, characteristic, complemented
2C2×#16810913C212yesmaximal, complemented
3#20273831C218yesmaximal, direct product factor
4C2×#16810367C211yesmaximal, characteristic, complemented
5C2×D4⋊#88322C212yesmaximal, complemented
6C2×D42×C53C211yesmaximal, characteristic
7C22×#11719923—22no—
8C2×#13133217C2211yescharacteristic, complemented
9#16810995—22no—
10#16810995C2214yescomplemented
11#16810913—24no—
12#16810913C2218yescomplemented
13C2×#13133148—22no—
14C2×C4⋊#88322C2212yescomplemented
15C2×#13133138—22no—
16C2×C4×#88322C2211yescharacteristic, complemented
17#16810367C2214yescomplemented
18D4⋊#88322C2218yescomplemented
19C23×D4×C53—22no—
20D42×C53—22no—
21D42×C53—24no—
22D42×C53C2214yes—
23C2×#127×C53C2211yescharacteristic
24C2×#121×C53—22no—
25C2×#120×C53—22no—
26C2×C4×D4×C53C2212yes—
27C22×Dic106—21no—
28C2×#11719923—28no—
29C2×#11719923—48no—
30C22×#88322—21no—
31C22×#88322—22no—
32#13133217—21no—
33#13133217—22no—
34#13133217C2312yescomplemented
35C2×C4⋊Dic53—22no—
36#13133148—28no—
37C4⋊#88322—22no—
38C4⋊#88322C2314yescomplemented
39#13133138—28no—
40C4×#88322—21no—
41C4×#88322C2312yescomplemented
42C25×C53—22no—
43C22×D4×C53—24no—
44C22×D4×C53—28no—
45C22×D4×C53—28no—
46C22×D4×C53—22no—
47C22×D4×C53—48no—
48C22×D4×C53—48no—
49C22×D4×C53D412yes—
50C22×D4×C53D412yes—
51C23×C4×C53—22no—
52#127×C53—22no—
53#127×C53D412yes—
54#127×C53C2312yes—
55#121×C53—28no—
56#121×C53—28no—
57#120×C53—28no—
58#120×C53—28no—
59C4×D4×C53—22no—
60C4×D4×C53—22no—
61C4×D4×C53D414yes—
62C4×D4×C53C2314yes—
63C2×C4⋊C4×C53D412yes—
64C2×C22⋊C4×C53—22no—
65C2×C22⋊C4×C53—22no—
66C2×C42×C53C2311yescharacteristic
67C24×C53—22no—
68C24×C53—28no—
69C24×C53—22no—
70C24×C53—416no—
71C24×C53—84no—
72C2×D4×C53—216no—
73C2×D4×C53—22no—
74C2×D4×C53—24no—
75C2×D4×C53—22no—
76C2×D4×C53—24no—
77C2×D4×C53—24no—
78C2×D4×C53—44no—
79C2×D4×C53—416no—
80C2×D4×C53—432no—
81C2×D4×C53—416no—
82C2×D4×C53—416no—
83C2×D4×C53—44no—
84C2×D4×C53—44no—
85C2×D4×C53SD1618yescomplemented
86C2×D4×C53C2×D414yes—
87C2×D4×C53C2×D414yes—
88C22×C4×C53—22no—
89C22×C4×C53—24no—
90C22×C4×C53—24no—
91C22×C4×C53—44no—
92C22×C4×C53—44no—
93C22×C4×C53C2×D411yescharacteristic
94C22×C4×C53C2×D412yes—
95C22×Dic53—41no—
96C2×Dic106—22no—
97C2×Dic106—44no—
98C2×Dic106—41no—
99C4⋊C4×C53—22no—
100C4⋊C4×C53C2×D414yes—
101C22⋊C4×C53—28no—
102C22⋊C4×C53—24no—
103C22⋊C4×C53—24no—
104C42×C53C2×D411yescharacteristic
105C42×C53C2×D412yes—
106C42×C53C2411yescharacteristic
107#11719923—416no—
108C4⋊Dic53—24no—
109C4⋊Dic53—42no—
110C2×#88322—24no—
111C2×#88322—22no—
112C2×#88322—24no—
113C2×#88322—44no—
114C23×C53—24no—
115C23×C53—28no—
116C23×C53—24no—
117C23×C53—432no—
118C23×C53—44no—
119C23×C53—44no—
120C23×C53—48no—
121C23×C53—44no—
122C23×C53—88no—
123C23×C53—832no—
124C23×C53#12011yescharacteristic
125D4×C53—24no—
126D4×C53—28no—
127D4×C53—44no—
128D4×C53—432no—
129D4×C53—44no—
130D4×C53—44no—
131D4×C53—48no—
132D4×C53—82no—
133D4×C53C2×SD1618yescomplemented
134C2×C4×C53—22no—
135C2×C4×C53—24no—
136C2×C4×C53—44no—
137C2×C4×C53—44no—
138C2×C4×C53—44no—
139C2×C4×C53—48no—
140C2×C4×C53#12011yescharacteristic
141C2×C4×C53#12012yes—
142C2×C4×C53C2×SD1614yes—
143C2×C4×C53AGL(1,ℤ/8)14yes—
144C2×C4×C53C22×D411yescharacteristic
145C2×C4×C53C22×D412yes—
146C2×Dic53—42no—
147C2×Dic53—82no—
148Dic106—44no—
149Dic106—82no—
150#88322—24no—
151#88322—44no—
152C2×#1629—531nomaximal, Sylow
153C22×C53—28no—
154C22×C53—416no—
155C22×C53—44no—
156C22×C53—44no—
157C22×C53—48no—
158C22×C53—44no—
159C22×C53—832no—
160C22×C53—88no—
161C22×C53#45014yes—
162C22×C53#45312yes—
163C22×C53C2×#12011yescharacteristic
164C4×C53—21no—
165C4×C53—21no—
166C4×C53—44no—
167C4×C53#45014yes—
168C4×C53C22×SD1612yes—
169C4×C53C2×AGL(1,ℤ/8)12yes—
170Dic53—82no—
171C2×D42D5311yescharacteristic
172C2×#492—531no—
173C2×#476—532no—
174#1629—538no—
175C2×#331—531no—
176C2×D4⋊C8—532no—
177C2×C53—48no—
178C2×C53—84no—
179C2×C53#162914yescomplemented
180C2×C53C2×#45012yes—
181C2×C53C2×#45311yescharacteristic
182C23×D4—22no—
183C22×SD16—1062no—
184D42—22no—
185D42—24no—
186D42C2×D5314yes—
187C2×C4⋊Q8—531no—
188C2×#127C2×D5311yescharacteristic
189C2×#121—22no—
190C2×#120—22no—
191C2×C4×D4C2×D5312yes—
192#492—534no—
193#492—1062no—
194#476—538no—
195#476—1064no—
196C2×#106—1062no—
197C2×C4⋊C8—532no—
198C2×#102—1062no—
199C2×C4×C8—531no—
200#331—534no—
201D4⋊C8—538no—
202C53C2×#162911yescharacteristic, complemented, Sylow
203C25—22no—
204C22×Q8—1061no—
205C22×D4—24no—
206C22×D4—28no—
207C22×D4—28no—
208C22×D4—22no—
209C22×D4—48no—
210C22×D4—48no—
211C22×D4C22⋊D5312yes—
212C22×D4C22⋊D5312yes—
213C23×C4—22no—
214C2×SD16—1068no—
215C2×SD16—2128no—
216C22×C8—1061no—
217C22×C8—1062no—
218C4⋊Q8—532no—
219C4⋊Q8—1062no—
220C4⋊Q8—1061no—
221#127—22no—
222#127C22⋊D5312yes—
223#127C22×D5312yes—
224#121—28no—
225#121—28no—
226#120—28no—
227#120—28no—
228C4×D4—22no—
229C4×D4—22no—
230C4×D4C22⋊D5314yes—
231C4×D4C22×D5314yes—
232C2×C4⋊C4—1062no—
233C2×C4⋊C4C22⋊D5312yes—
234C2×C22⋊C4—22no—
235C2×C22⋊C4—22no—
236C2×C42C22×D5311yescharacteristic
237#106—1068no—
238C4⋊C8—534no—
239C4⋊C8—1062no—
240#102—1068no—
241C4×C8—532no—
242C4×C8—1061no—
243C24—22no—
244C24—28no—
245C24—22no—
246C24—416no—
247C24—84no—
248C2×Q8—1062no—
249C2×Q8—2124no—
250C2×Q8—2121no—
251C2×D4—216no—
252C2×D4—22no—
253C2×D4—24no—
254C2×D4—22no—
255C2×D4—24no—
256C2×D4—24no—
257C2×D4—44no—
258C2×D4—432no—
259C2×D4—416no—
260C2×D4—416no—
261C2×D4—416no—
262C2×D4—44no—
263C2×D4—44no—
264C2×D4#1171992318yescomplemented
265C2×D4C2×C22⋊D5314yes—
266C2×D4C2×C22⋊D5314yes—
267C22×C4—22no—
268C22×C4—24no—
269C22×C4—24no—
270C22×C4—44no—
271C22×C4—44no—
272C22×C4—2121no—
273C22×C4C2×C22⋊D5311yescharacteristic
274C22×C4C2×C22⋊D5312yes—
275SD16—21216no—
276C2×C8—1064no—
277C2×C8—1062no—
278C2×C8—1064no—
279C2×C8—2124no—
280C4⋊C4—22no—
281C4⋊C4—1064no—
282C4⋊C4—2122no—
283C4⋊C4C2×C22⋊D5314yes—
284C22⋊C4—28no—
285C22⋊C4—24no—
286C22⋊C4—24no—
287C42C2×C22⋊D5311yescharacteristic
288C42C2×C22⋊D5312yes—
289C42C23×D5311yescharacteristic
290C23—28no—
291C23—24no—
292C23—24no—
293C23—432no—
294C23—44no—
295C23—44no—
296C23—48no—
297C23—44no—
298C23—88no—
299C23—832no—
300C23#1313318611yescharacteristic
301Q8—4242no—
302Q8—2124no—
303D4—28no—
304D4—24no—
305D4—44no—
306D4—432no—
307D4—48no—
308D4—44no—
309D4—44no—
310D4—82no—
311D4C2×#1171992318yescomplemented
312C2×C4—22no—
313C2×C4—24no—
314C2×C4—44no—
315C2×C4—48no—
316C2×C4—44no—
317C2×C4—44no—
318C2×C4—4242no—
319C2×C4—2122no—
320C2×C4#1313318611yescharacteristic
321C2×C4#1313318612yes—
322C2×C4C2×#1171992314yes—
323C2×C4#1313324314yes—
324C2×C4C22×C22⋊D5311yescharacteristic
325C2×C4C22×C22⋊D5312yes—
326C8—1064no—
327C8—2124no—
328C22—28no—
329C22—44no—
330C22—416no—
331C22—44no—
332C22—44no—
333C22—48no—
334C22—832no—
335C22—88no—
336C22#1681077414yes—
337C22#1681079112yes—
338C22C2×#1313318611yescharacteristic
339C4—21no—
340C4—21no—
341C4—44no—
342C4—4242no—
343C4#1681077414yes—
344C4C22×#1171992312yes—
345C4C2×#1313324312yes—
346C2—48no—
347C2—84no—
348C2#2027383114yesdirect product factor
349C2C2×#1681077412yes—
350C2C2×#1681079111yescharacteristic
subgroupG/H
centerC23#13133186
commutator, cosocleC42×C53C24
FrattiniC42C23×D53
Fitting, prime radicalC2×D42×C53C2
socleC23×C53#120
hypercenterC2×D42D53
hypocenterC53C2×#1629
LCSG ▷ C42×C53 ▷ C22×C53 ▷ C53
UCSC1 ◁ C23 ◁ C2×C42 ◁ C2×D42

Complements

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

NH ≅ G/Ntype
1. C2×#16810995C2semidirect
2. C2×#16810913C2semidirect
3. #20273831C2direct (N × H)
4. C2×#16810367C2semidirect
5. C2×D4⋊#88322C2semidirect
8. C2×#13133217C22semidirect
10. #16810995C22semidirect
12. #16810913C22semidirect
14. C2×C4⋊#88322C22semidirect
16. C2×C4×#88322C22semidirect
17. #16810367C22semidirect
18. D4⋊#88322C22semidirect
34. #13133217C23semidirect
38. C4⋊#88322C23semidirect
41. C4×#88322C23semidirect
85. C2×D4×C53SD16semidirect
133. D4×C53C2×SD16semidirect
179. C2×C53#1629semidirect
202. C53C2×#1629semidirect
264. C2×D4#11719923semidirect
311. D4C2×#11719923semidirect
348. C2#20273831direct (N × H)

Maximal subgroups

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

subgroupindex
C2×#168109952
C2×#168109132
#202738312
C2×#168103672
C2×D4⋊#883222
C2×D42×C532
C2×#162953