Finite Group Databasebeta

← C2×#20273754 #27356317

Subgroups — C2×#20273754 #27356317

Order 13568.

1264 conjugacy classes of subgroups; only the 203 normal subgroups are shown.

normal subgroups only

This group has 1264 conjugacy classes of subgroups — too many to draw. Only its 203 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×#16811204C211yesmaximal, characteristic
2C2×#16810942C212yesmaximal, complemented
3C2×#16810562C211yesmaximal, characteristic, complemented
4#20273754C218yesmaximal, direct product factor
5C2×#16810358C212yesmaximal, complemented
6C2×C53×#324C211yesmaximal, characteristic, complemented
7C22×#11719945—21no—
8#16811204—24no—
9#16811204—22no—
10#16811204C2214yes—
11C2×#13133197—22no—
12C2×#13133192—22no—
13C2×#13133191—21no—
14C2×#13133191C2212yes—
15C22×C4⋊Dic53C2211yescharacteristic
16#16810942—24no—
17#16810942C2218yescomplemented
18C2×#13133149—22no—
19C2×#13133139—22no—
20#16810562—22no—
21#16810562C2214yescomplemented
22C2×#13133124—21no—
23C2×#13133124—21no—
24C2×#13133124C2211yescharacteristic
25#16810358C2218yescomplemented
26C22×C4⋊C4×C53C2211yescharacteristic
27C2×C22⋊C8×C53C2212yes—
28C53×#324C2214yescomplemented
29C2×#11719945—24no—
30C2×#11719945—24no—
31C2×#11719945—44no—
32C2×#11719945—44no—
33C22×Dic106—21no—
34C22×D212—21no—
35C22×D212—21no—
36C22×D212D412yes—
37C22×C22⋊D53—22no—
38C23×Dic53—21no—
39C23×Dic53—21no—
40C22×C4×D53—22no—
41#13133197—28no—
42#13133197—28no—
43#13133192—28no—
44#13133192—28no—
45#13133191—24no—
46#13133191—24no—
47#13133191—22no—
48#13133191—22no—
49#13133191D414yes—
50#13133191C2314yes—
51C2×C4⋊Dic53—21no—
52C2×C4⋊Dic53—21no—
53C2×C4⋊Dic53—22no—
54C2×C4⋊Dic53—22no—
55C2×C4⋊Dic53D412yes—
56C2×C4⋊Dic53C2312yes—
57C2×#11719921—22no—
58C2×#11719921—22no—
59#13133149—28no—
60#13133139—28no—
61#13133124—21no—
62#13133124—24no—
63#13133124—24no—
64#13133124C2312yes—
65C23×C4×C53D411yescharacteristic
66C23×C4×C53D411yescharacteristic
67C23×C4×C53C2311yescharacteristic
68C22×C8×C53—22no—
69C2×C4⋊C4×C53—22no—
70C2×C4⋊C4×C53—22no—
71C2×C4⋊C4×C53D412yes—
72C2×C4⋊C4×C53C2312yes—
73C22⋊C8×C53—22no—
74C22⋊C8×C53C2314yes—
75C24×C53C2×D411yescharacteristic
76C23×D53—22no—
77C22×C4×C53—21no—
78C22×C4×C53—21no—
79C22×C4×C53—22no—
80C22×C4×C53—21no—
81C22×C4×C53—21no—
82C22×C4×C53—42no—
83C22×C4×C53—42no—
84C22×C4×C53SD1614yes—
85C22×C4×C53C2×D412yes—
86C22×C4×C53C2×D412yes—
87C22×C4×C53C2×D412yes—
88C22×C4×C53C2×D411yescharacteristic
89C22×C4×C53C2×D412yes—
90C22×C4×C53C2411yescharacteristic
91C2×C22⋊D53—24no—
92C2×C22⋊D53—24no—
93C2×C22⋊D53—44no—
94C2×C22⋊D53—44no—
95C22×Dic53—22no—
96C22×Dic53—22no—
97C22×Dic53—22no—
98C22×Dic53—42no—
99C22×Dic53—42no—
100C22×Dic53—44no—
101C22×Dic53—41no—
102C22×Dic53—41no—
103C22×Dic53—42no—
104#11719945—48no—
105#11719945—416no—
106#11719945—48no—
107C2×D212—24no—
108C2×D212—24no—
109C2×D212—22no—
110C2×D212—44no—
111C2×D212—44no—
112C2×D212—82no—
113C2×D212C2×D414yes—
114C2×C4×D53—28no—
115C2×C4×D53—48no—
116C2×Dic106—24no—
117C2×Dic106—44no—
118C2×C8×C53—24no—
119C2×C8×C53—44no—
120C4⋊C4×C53—24no—
121C4⋊C4×C53—24no—
122#11719921—28no—
123#11719921—24no—
124#11719921—24no—
125C4⋊Dic53—24no—
126C4⋊Dic53—22no—
127C4⋊Dic53—24no—
128C4⋊Dic53—42no—
129C4⋊Dic53C2×D414yes—
130C23×C53—22no—
131C23×C53—22no—
132C23×C53#12011yescharacteristic
133C23×C53#12011yescharacteristic
134C23×C53C2×SD1612yes—
135C23×C53AGL(1,ℤ/8)12yes—
136C23×C53C22×D411yescharacteristic
137C22×D53—24no—
138C22×D53—44no—
139C22×D53—44no—
140C22×D53—44no—
141C2×C4×C53—22no—
142C2×C4×C53—22no—
143C2×C4×C53—24no—
144C2×C4×C53—24no—
145C2×C4×C53—24no—
146C2×C4×C53—22no—
147C2×C4×C53—41no—
148C2×C4×C53—42no—
149C2×C4×C53—41no—
150C2×C4×C53—42no—
151C2×C4×C53—42no—
152C2×C4×C53—81no—
153C2×C4×C53#12012yes—
154C2×C4×C53C2×SD1614yes—
155C2×C4×C53C22×D412yes—
156C22⋊D53—44no—
157C22⋊D53—44no—
158C22⋊D53—48no—
159C2×Dic53—24no—
160C2×Dic53—44no—
161C2×Dic53—42no—
162C2×Dic53—42no—
163C2×Dic53—44no—
164C2×Dic53—42no—
165C2×Dic53—42no—
166C2×Dic53—84no—
167C2×Dic53—82no—
168C2×Dic53—82no—
169D212—44no—
170D212—48no—
171D212—82no—
172D212—84no—
173C4×D53—416no—
174Dic106—48no—
175C8×C53—44no—
176C2×#1616—531nomaximal, Sylow
177C22×C53—24no—
178C22×C53—24no—
179C22×C53—21no—
180C22×C53—21no—
181C22×C53—21no—
182C22×C53—21no—
183C22×C53#45012yes—
184C22×C53#45012yes—
185C22×C53#45412yes—
186C22×C53#45812yes—
187C22×C53C2×#12011yescharacteristic
188C22×C53C22×SD1611yescharacteristic
189C22×C53C2×AGL(1,ℤ/8)11yescharacteristic
190C2×D53—44no—
191C2×D53—44no—
192C2×D53—44no—
193C2×D53—88no—
194C4×C53—24no—
195C4×C53—22no—
196C4×C53—22no—
197C4×C53—42no—
198Dic53—44no—
199Dic53—82no—
200Dic53—82no—
201C2×#537—531no—
202C2×#481—532no—
203C2×#400—531no—
204#1616—538no—
205C2×#327—532no—
206C2×#324D5311yescharacteristic, complemented
207C2×C53—21no—
208C2×C53—21no—
209C2×C53—22no—
210C2×C53#161614yescomplemented
211C2×C53C2×#45011yescharacteristic
212C2×C53C2×#45411yescharacteristic
213C2×C53C2×#45811yescharacteristic
214D53—84no—
215C22×#41—1061no—
216#537—534no—
217#537—1064no—
218#537—1062no—
219C2×C22⋊Q8—1062no—
220C2×#121—532no—
221C2×#121—1062no—
222C2×#121—1061no—
223C22×C4⋊C4—531no—
224C22×C4⋊C4C2×D5311yescharacteristic
225#481—538no—
226#481—1064no—
227C2×#106—1062no—
228C2×#102—1062no—
229C2×C22⋊C8C2×D5312yes—
230#400—534no—
231#400—1062no—
232C2×#95—531no—
233C2×#95—1061no—
234C2×#95—1061no—
235#327—538no—
236#324C2×D5314yescomplemented
237C53C2×#161611yescharacteristic, complemented, Sylow
238C2×#41—1064no—
239C2×#41—1064no—
240C2×#41—2124no—
241C2×#41—2124no—
242C22×Q8—1061no—
243C22×D4—532no—
244C22×D4—1061no—
245C22×D4—1061no—
246C22×D4—1062no—
247C23×C4—1062no—
248C23×C4—1061no—
249C23×C4—1061no—
250C23×C4D21211yescharacteristic
251C23×C4D21211yescharacteristic
252C23×C4C22×D5311yescharacteristic
253C22×C8—22no—
254C22⋊Q8—1068no—
255C22⋊Q8—1068no—
256#121—534no—
257#121—534no—
258#121—1068no—
259#121—1068no—
260#121—1064no—
261#121—1062no—
262#121—1064no—
263#121—1062no—
264C2×C4⋊C4—22no—
265C2×C4⋊C4—22no—
266C2×C4⋊C4—532no—
267C2×C4⋊C4—532no—
268C2×C4⋊C4—1061no—
269C2×C4⋊C4—1062no—
270C2×C4⋊C4—1062no—
271C2×C4⋊C4—1061no—
272C2×C4⋊C4D21212yes—
273C2×C4⋊C4C22×D5312yes—
274C2×C22⋊C4—1062no—
275C2×C22⋊C4—1062no—
276#106—1068no—
277#102—1068no—
278C22⋊C8—22no—
279C22⋊C8C22×D5314yes—
280#95—532no—
281#95—1061no—
282#95—1064no—
283#95—1064no—
284C24—1062no—
285C24C2×D21211yescharacteristic
286#41—21216no—
287#41—2128no—
288#41—2128no—
289C2×Q8—1064no—
290C2×Q8—2124no—
291C2×D4—534no—
292C2×D4—1064no—
293C2×D4—1064no—
294C2×D4—1064no—
295C2×D4—1064no—
296C2×D4—1062no—
297C2×D4—4242no—
298C2×D4—2124no—
299C2×D4—2124no—
300C2×D4—2124no—
301C2×D4—2124no—
302C22×C4—21no—
303C22×C4—21no—
304C22×C4—22no—
305C22×C4—21no—
306C22×C4—21no—
307C22×C4—42no—
308C22×C4—42no—
309C22×C4—1068no—
310C22×C4—1062no—
311C22×C4—1062no—
312C22×C4—1062no—
313C22×C4—2128no—
314C22×C4—2122no—
315C22×C4—2122no—
316C22×C4—2124no—
317C22×C4—2121no—
318C22×C4—2121no—
319C22×C4—2122no—
320C22×C4#1171991314yes—
321C22×C4C2×D21212yes—
322C22×C4C2×D21212yes—
323C22×C4C2×D21211yescharacteristic
324C22×C4D4×D5312yes—
325C22×C4D4×D5312yes—
326C22×C4C23×D5311yescharacteristic
327C2×C8—24no—
328C2×C8—44no—
329C4⋊C4—24no—
330C4⋊C4—24no—
331C4⋊C4—534no—
332C4⋊C4—1064no—
333C4⋊C4—1062no—
334C4⋊C4—1064no—
335C4⋊C4—2122no—
336C22⋊C4—1068no—
337C22⋊C4—1064no—
338C22⋊C4—1064no—
339C23—22no—
340C23—22no—
341C23—1064no—
342C23—2124no—
343C23—2124no—
344C23—2124no—
345C23#1313318711yescharacteristic
346C23#1313318711yescharacteristic
347C23C2×#1171991312yes—
348C23#1313324512yes—
349C23C22×D21211yescharacteristic
350Q8—2128no—
351D4—4242no—
352D4—4244no—
353D4—2124no—
354D4—2128no—
355D4—2128no—
356D4—2124no—
357D4—2124no—
358C2×C4—24no—
359C2×C4—24no—
360C2×C4—24no—
361C2×C4—22no—
362C2×C4—22no—
363C2×C4—22no—
364C2×C4—41no—
365C2×C4—42no—
366C2×C4—41no—
367C2×C4—42no—
368C2×C4—42no—
369C2×C4—81no—
370C2×C4—1064no—
371C2×C4—4244no—
372C2×C4—4242no—
373C2×C4—4242no—
374C2×C4—21216no—
375C2×C4—2122no—
376C2×C4—2122no—
377C2×C4—2124no—
378C2×C4—2122no—
379C2×C4—2122no—
380C2×C4—2124no—
381C2×C4#1313318712yes—
382C2×C4C2×#1171991314yes—
383C2×C4C2×D4×D5312yes—
384C8—44no—
385C22—24no—
386C22—24no—
387C22—21no—
388C22—21no—
389C22—21no—
390C22—21no—
391C22—4248no—
392C22—2124no—
393C22—2124no—
394C22—2124no—
395C22#1681077612yes—
396C22#1681077612yes—
397C22#1681079812yes—
398C22#1681081412yes—
399C22C2×#1313318711yescharacteristic
400C22C22×#1171991311yescharacteristic
401C22C2×#1313324511yescharacteristic
402C4—22no—
403C4—22no—
404C4—24no—
405C4—42no—
406C4—4242no—
407C4—4242no—
408C4—2124no—
409C2—21no—
410C2—22no—
411C2—21no—
412C2—4244no—
413C2#2027375414yesdirect product factor
414C2C2×#1681077611yescharacteristic
415C2C2×#1681079811yescharacteristic
416C2C2×#1681081411yescharacteristic
subgroupG/H
centerC23#13133187
commutator, cosocleC22×C4×C53C24
FrattiniC22×C4C23×D53
FittingC2×C53×#324C2
socleC23×C53#120
prime radicalC2×#16811204C2
hypercenterC2×#324D53
hypocenterC53C2×#1616
LCSG ▷ C22×C4×C53 ▷ C22×C53 ▷ C53
UCSC1 ◁ C23 ◁ C23×C4 ◁ C2×#324

Complements

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

NH ≅ G/Ntype
2. C2×#16810942C2semidirect
3. C2×#16810562C2semidirect
4. #20273754C2direct (N × H)
5. C2×#16810358C2semidirect
6. C2×C53×#324C2semidirect
17. #16810942C22semidirect
21. #16810562C22semidirect
25. #16810358C22semidirect
28. C53×#324C22semidirect
206. C2×#324D53semidirect
210. C2×C53#1616semidirect
236. #324C2×D53semidirect
237. C53C2×#1616semidirect
413. C2#20273754direct (N × H)

Maximal subgroups

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

subgroupindex
C2×#168112042
C2×#168109422
C2×#168105622
#202737542
C2×#168103582
C2×C53×#3242
C2×#161653