Finite Group Databasebeta

← C2×#20273720 #27356311

Subgroups — C2×#20273720 #27356311

Order 13568.

1308 conjugacy classes of subgroups; only the 203 normal subgroups are shown.

normal subgroups only

This group has 1308 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×#16810937C212yesmaximal, complemented
3C2×#16810553C211yesmaximal, characteristic, complemented
4#20273720C218yesmaximal, 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
16C22×#11719919—21no—
17#16810937—24no—
18#16810937C2218yescomplemented
19C2×#13133151—22no—
20C2×#13133139—22no—
21#16810553—22no—
22#16810553C2214yescomplemented
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×#11719919—24no—
58C2×#11719919—44no—
59C2×#11719921—22no—
60C2×#11719921—22no—
61#13133151—28no—
62#13133139—28no—
63#13133124—21no—
64#13133124—24no—
65#13133124C2312yes—
66C23×C4×C53D411yescharacteristic
67C23×C4×C53D411yescharacteristic
68C23×C4×C53C2311yescharacteristic
69C22×C8×C53—22no—
70C2×C4⋊C4×C53—22no—
71C2×C4⋊C4×C53—22no—
72C2×C4⋊C4×C53D412yes—
73C2×C4⋊C4×C53C2312yes—
74C22⋊C8×C53—22no—
75C22⋊C8×C53C2314yes—
76C24×C53C2×D411yescharacteristic
77C23×D53—22no—
78C22×C4×C53—21no—
79C22×C4×C53—21no—
80C22×C4×C53—22no—
81C22×C4×C53—21no—
82C22×C4×C53—21no—
83C22×C4×C53—42no—
84C22×C4×C53—42no—
85C22×C4×C53D814yes—
86C22×C4×C53C2×D412yes—
87C22×C4×C53C2×D412yes—
88C22×C4×C53C2×D412yes—
89C22×C4×C53C2×D411yescharacteristic
90C22×C4×C53C2×D412yes—
91C22×C4×C53C2411yescharacteristic
92C2×C22⋊D53—24no—
93C2×C22⋊D53—24no—
94C2×C22⋊D53—44no—
95C2×C22⋊D53—44no—
96C22×Dic53—24no—
97C22×Dic53—22no—
98C22×Dic53—22no—
99C22×Dic53—22no—
100C22×Dic53—42no—
101C22×Dic53—42no—
102C22×Dic53—41no—
103C22×Dic53—41no—
104C22×Dic53—42no—
105C22×Dic53—42no—
106#11719945—48no—
107#11719945—416no—
108#11719945—48no—
109C2×D212—24no—
110C2×D212—24no—
111C2×D212—22no—
112C2×D212—44no—
113C2×D212—44no—
114C2×D212—82no—
115C2×D212C2×D414yes—
116C2×C4×D53—28no—
117C2×C4×D53—48no—
118C2×Dic106—24no—
119C2×Dic106—44no—
120C2×C8×C53—24no—
121C2×C8×C53—44no—
122C4⋊C4×C53—24no—
123C4⋊C4×C53—24no—
124#11719921—28no—
125#11719921—24no—
126#11719921—24no—
127C4⋊Dic53—24no—
128C4⋊Dic53—22no—
129C4⋊Dic53—24no—
130C4⋊Dic53—42no—
131C4⋊Dic53C2×D414yes—
132#11719919—48no—
133C23×C53—22no—
134C23×C53—22no—
135C23×C53#12011yescharacteristic
136C23×C53#12011yescharacteristic
137C23×C53C2×D812yes—
138C23×C53AGL(1,ℤ/8)12yes—
139C23×C53C22×D411yescharacteristic
140C22×D53—24no—
141C22×D53—44no—
142C22×D53—44no—
143C22×D53—44no—
144C2×C4×C53—22no—
145C2×C4×C53—22no—
146C2×C4×C53—24no—
147C2×C4×C53—24no—
148C2×C4×C53—24no—
149C2×C4×C53—22no—
150C2×C4×C53—41no—
151C2×C4×C53—42no—
152C2×C4×C53—41no—
153C2×C4×C53—42no—
154C2×C4×C53—42no—
155C2×C4×C53—81no—
156C2×C4×C53#12012yes—
157C2×C4×C53C2×D814yes—
158C2×C4×C53C22×D412yes—
159C22⋊D53—44no—
160C22⋊D53—44no—
161C22⋊D53—48no—
162C2×Dic53—24no—
163C2×Dic53—24no—
164C2×Dic53—48no—
165C2×Dic53—42no—
166C2×Dic53—42no—
167C2×Dic53—44no—
168C2×Dic53—42no—
169C2×Dic53—42no—
170C2×Dic53—42no—
171C2×Dic53—82no—
172C2×Dic53—82no—
173D212—44no—
174D212—48no—
175D212—82no—
176D212—84no—
177C4×D53—416no—
178Dic106—48no—
179C8×C53—44no—
180C2×#1610—531nomaximal, Sylow
181C22×C53—24no—
182C22×C53—24no—
183C22×C53—21no—
184C22×C53—21no—
185C22×C53—21no—
186C22×C53—21no—
187C22×C53#44712yes—
188C22×C53#44712yes—
189C22×C53#45612yes—
190C22×C53#45812yes—
191C22×C53C2×#12011yescharacteristic
192C22×C53C22×D811yescharacteristic
193C22×C53C2×AGL(1,ℤ/8)11yescharacteristic
194C2×D53—44no—
195C2×D53—44no—
196C2×D53—44no—
197C2×D53—88no—
198C4×C53—24no—
199C4×C53—22no—
200C4×C53—22no—
201C4×C53—42no—
202Dic53—44no—
203Dic53—44no—
204Dic53—82no—
205C2×#537—531no—
206C2×#480—532no—
207C2×#398—531no—
208#1610—538no—
209C2×#327—532no—
210C2×#324D5311yescharacteristic, complemented
211C2×C53—21no—
212C2×C53—21no—
213C2×C53—22no—
214C2×C53#161014yescomplemented
215C2×C53C2×#44711yescharacteristic
216C2×C53C2×#45611yescharacteristic
217C2×C53C2×#45811yescharacteristic
218D53—84no—
219C22×#41—1061no—
220#537—534no—
221#537—1064no—
222#537—1062no—
223C2×C22⋊Q8—1062no—
224C2×#121—532no—
225C2×#121—1062no—
226C2×#121—1061no—
227C22×C4⋊C4—531no—
228C22×C4⋊C4—1061no—
229C22×C4⋊C4C2×D5311yescharacteristic
230#480—538no—
231#480—1064no—
232C2×#107—1062no—
233C2×#102—1062no—
234C2×C22⋊C8C2×D5312yes—
235#398—534no—
236#398—1062no—
237C2×#95—531no—
238C2×#95—1061no—
239#327—538no—
240#324C2×D5314yescomplemented
241C53C2×#161011yescharacteristic, complemented, Sylow
242C2×#41—1064no—
243C2×#41—1064no—
244C2×#41—2124no—
245C2×#41—2124no—
246C22×Q8—1061no—
247C22×D4—532no—
248C22×D4—1061no—
249C22×D4—1061no—
250C22×D4—1062no—
251C23×C4—1062no—
252C23×C4—1061no—
253C23×C4—1061no—
254C23×C4D21211yescharacteristic
255C23×C4D21211yescharacteristic
256C23×C4C22×D5311yescharacteristic
257C22×C8—22no—
258C22⋊Q8—1068no—
259C22⋊Q8—1068no—
260#121—534no—
261#121—534no—
262#121—1068no—
263#121—1068no—
264#121—1064no—
265#121—1062no—
266#121—1064no—
267#121—1062no—
268C2×C4⋊C4—22no—
269C2×C4⋊C4—22no—
270C2×C4⋊C4—532no—
271C2×C4⋊C4—532no—
272C2×C4⋊C4—1061no—
273C2×C4⋊C4—1062no—
274C2×C4⋊C4—1062no—
275C2×C4⋊C4—1061no—
276C2×C4⋊C4—1064no—
277C2×C4⋊C4—2124no—
278C2×C4⋊C4D21212yes—
279C2×C4⋊C4C22×D5312yes—
280C2×C22⋊C4—1062no—
281C2×C22⋊C4—1062no—
282#107—1068no—
283#102—1068no—
284C22⋊C8—22no—
285C22⋊C8C22×D5314yes—
286#95—532no—
287#95—1061no—
288#95—1064no—
289C24—1062no—
290C24C2×D21211yescharacteristic
291#41—21216no—
292#41—2128no—
293#41—2128no—
294C2×Q8—1064no—
295C2×Q8—2124no—
296C2×D4—534no—
297C2×D4—1064no—
298C2×D4—1064no—
299C2×D4—1064no—
300C2×D4—1064no—
301C2×D4—1062no—
302C2×D4—4242no—
303C2×D4—2124no—
304C2×D4—2124no—
305C2×D4—2124no—
306C2×D4—2124no—
307C22×C4—21no—
308C22×C4—21no—
309C22×C4—22no—
310C22×C4—21no—
311C22×C4—21no—
312C22×C4—42no—
313C22×C4—42no—
314C22×C4—1068no—
315C22×C4—1064no—
316C22×C4—1062no—
317C22×C4—1062no—
318C22×C4—1062no—
319C22×C4—2128no—
320C22×C4—2121no—
321C22×C4—2121no—
322C22×C4—2122no—
323C22×C4—2122no—
324C22×C4—2122no—
325C22×C4—2122no—
326C22×C4D42414yes—
327C22×C4C2×D21212yes—
328C22×C4C2×D21212yes—
329C22×C4C2×D21211yescharacteristic
330C22×C4D4×D5312yes—
331C22×C4D4×D5312yes—
332C22×C4C23×D5311yescharacteristic
333C2×C8—24no—
334C2×C8—44no—
335C4⋊C4—24no—
336C4⋊C4—24no—
337C4⋊C4—534no—
338C4⋊C4—1064no—
339C4⋊C4—1062no—
340C4⋊C4—1064no—
341C4⋊C4—2122no—
342C4⋊C4—2128no—
343C22⋊C4—1068no—
344C22⋊C4—1064no—
345C22⋊C4—1064no—
346C23—22no—
347C23—22no—
348C23—1064no—
349C23—2124no—
350C23—2124no—
351C23—2124no—
352C23#1313318711yescharacteristic
353C23#1313318711yescharacteristic
354C23C2×D42412yes—
355C23#1313324512yes—
356C23C22×D21211yescharacteristic
357Q8—2128no—
358D4—4242no—
359D4—4244no—
360D4—2124no—
361D4—2128no—
362D4—2128no—
363D4—2124no—
364D4—2124no—
365C2×C4—24no—
366C2×C4—24no—
367C2×C4—24no—
368C2×C4—22no—
369C2×C4—22no—
370C2×C4—22no—
371C2×C4—41no—
372C2×C4—42no—
373C2×C4—41no—
374C2×C4—42no—
375C2×C4—42no—
376C2×C4—81no—
377C2×C4—1064no—
378C2×C4—1064no—
379C2×C4—4242no—
380C2×C4—4242no—
381C2×C4—21216no—
382C2×C4—2122no—
383C2×C4—2122no—
384C2×C4—2122no—
385C2×C4—2128no—
386C2×C4—2122no—
387C2×C4—2124no—
388C2×C4—2122no—
389C2×C4#1313318712yes—
390C2×C4C2×D42414yes—
391C2×C4C2×D4×D5312yes—
392C8—44no—
393C22—24no—
394C22—24no—
395C22—21no—
396C22—21no—
397C22—21no—
398C22—21no—
399C22—4248no—
400C22—2124no—
401C22—2124no—
402C22—2124no—
403C22#1681075912yes—
404C22#1681075912yes—
405C22#1681080812yes—
406C22#1681081412yes—
407C22C2×#1313318711yescharacteristic
408C22C22×D42411yescharacteristic
409C22C2×#1313324511yescharacteristic
410C4—22no—
411C4—22no—
412C4—24no—
413C4—42no—
414C4—4242no—
415C4—2124no—
416C4—2124no—
417C2—21no—
418C2—22no—
419C2—21no—
420C2—4244no—
421C2#2027372014yesdirect product factor
422C2C2×#1681075911yescharacteristic
423C2C2×#1681080811yescharacteristic
424C2C2×#1681081411yescharacteristic
subgroupG/H
centerC23#13133187
commutator, cosocleC22×C4×C53C24
FrattiniC22×C4C23×D53
FittingC2×C53×#324C2
socleC23×C53#120
prime radicalC2×#16811204C2
hypercenterC2×#324D53
hypocenterC53C2×#1610
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×#16810937C2semidirect
3. C2×#16810553C2semidirect
4. #20273720C2direct (N × H)
5. C2×#16810358C2semidirect
6. C2×C53×#324C2semidirect
18. #16810937C22semidirect
22. #16810553C22semidirect
25. #16810358C22semidirect
28. C53×#324C22semidirect
210. C2×#324D53semidirect
214. C2×C53#1610semidirect
240. #324C2×D53semidirect
241. C53C2×#1610semidirect
421. C2#20273720direct (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×#168109372
C2×#168105532
#202737202
C2×#168103582
C2×C53×#3242
C2×#161053