Finite Group Databasebeta

← C2×#20273706 #27356219

Subgroups — C2×#20273706 #27356219

Order 13568.

1136 conjugacy classes of subgroups.

The subgroup lattice has 424 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#16810951C212yesmaximal, complemented
2C2×#16810548C211yesmaximal, characteristic
3#20273706C218yesmaximal, direct product factor
4C2×#16810360C212yesmaximal, complemented
5C2×#16810347C211yesmaximal, characteristic
6C2×C53×#543C211yesmaximal, characteristic
7C22×C4⋊Dic53C2211yescharacteristic
8C22×C22⋊Dic53—21no—
9#16810951—24no—
10#16810951C2218yescomplemented
11C2×#13133148—22no—
12C2×#13133141—22no—
13C2×C22⋊#88322C2212yes—
14#16810548—22no—
15#16810548C2214yes—
16C2×#13133124—21no—
17C2×#13133124C2211yescharacteristic
18#16810360C2218yescomplemented
19#16810347C2214yes—
20C53×#543—24no—
21C53×#543—22no—
22C53×#543C2214yes—
23C2×C22⋊Q8×C53—22no—
24C2×C22⋊Q8×C53—21no—
25C2×C22⋊Q8×C53—22no—
26C2×C22⋊Q8×C53C2212yes—
27C22×C22⋊C4×C53—21no—
28C22×C22⋊C4×C53C2211yescharacteristic
29C23×Dic53—21no—
30C23×Dic53—21no—
31C22×#88322—22no—
32C2×C4⋊Dic53—21no—
33C2×C4⋊Dic53—22no—
34C2×C4⋊Dic53—22no—
35C2×C4⋊Dic53C2312yes—
36C2×C22⋊Dic53—24no—
37C2×C22⋊Dic53—44no—
38#13133148—28no—
39#13133141—28no—
40C22⋊#88322—22no—
41C22⋊#88322C2314yes—
42#13133124—21no—
43#13133124—24no—
44#13133124C2312yes—
45C25×C53D411yescharacteristic
46C22×Q8×C53—21no—
47C22×Q8×C53D412yes—
48C23×C4×C53—21no—
49C23×C4×C53—21no—
50C23×C4×C53D411yescharacteristic
51C23×C4×C53C2311yescharacteristic
52C22⋊Q8×C53—28no—
53C22⋊Q8×C53—28no—
54C22⋊Q8×C53—24no—
55C22⋊Q8×C53—28no—
56C22⋊Q8×C53—28no—
57C22⋊Q8×C53—24no—
58C22⋊Q8×C53—22no—
59C22⋊Q8×C53—22no—
60C22⋊Q8×C53D414yes—
61C22⋊Q8×C53C2314yes—
62C2×C4⋊C4×C53—21no—
63C2×C4⋊C4×C53—22no—
64C2×C4⋊C4×C53—22no—
65C2×C4⋊C4×C53D412yes—
66C2×C22⋊C4×C53—22no—
67C2×C22⋊C4×C53—24no—
68C2×C22⋊C4×C53—22no—
69C2×C22⋊C4×C53—24no—
70C2×C22⋊C4×C53—22no—
71C2×C22⋊C4×C53—22no—
72C2×C22⋊C4×C53D412yes—
73C2×C22⋊C4×C53C2312yes—
74C24×C53—21no—
75C24×C53—21no—
76C24×C53—22no—
77C24×C53—42no—
78C24×C53—42no—
79C24×C53SD1614yes—
80C24×C53C2×D411yescharacteristic
81C24×C53C2×D412yes—
82C2×Q8×C53—24no—
83C2×Q8×C53—22no—
84C2×Q8×C53—44no—
85C2×Q8×C53—82no—
86C2×Q8×C53C2×D414yes—
87C22×C4×C53—22no—
88C22×C4×C53—21no—
89C22×C4×C53—22no—
90C22×C4×C53—24no—
91C22×C4×C53—24no—
92C22×C4×C53—21no—
93C22×C4×C53—42no—
94C22×C4×C53—42no—
95C22×C4×C53—82no—
96C22×C4×C53—82no—
97C22×C4×C53C2×D412yes—
98C22×C4×C53C2×D412yes—
99C22×C4×C53C2×D411yescharacteristic
100C22×C4×C53C2×D412yes—
101C22×C4×C53C2411yescharacteristic
102C22×Dic53—22no—
103C22×Dic53—22no—
104C22×Dic53—42no—
105C22×Dic53—42no—
106C22×Dic53—41no—
107C22×Dic53—41no—
108C22×Dic53—42no—
109C22×Dic53—42no—
110C22×Dic53—42no—
111C4⋊C4×C53—28no—
112C4⋊C4×C53—24no—
113C4⋊C4×C53—24no—
114C4⋊C4×C53—22no—
115C4⋊C4×C53—24no—
116C4⋊C4×C53C2×D414yes—
117C22⋊C4×C53—28no—
118C22⋊C4×C53—24no—
119C22⋊C4×C53—24no—
120C22⋊C4×C53—24no—
121C22⋊C4×C53—24no—
122C22⋊Dic53—48no—
123C4⋊Dic53—24no—
124C4⋊Dic53—42no—
125C2×#88322—24no—
126C2×#88322—44no—
127C23×C53—22no—
128C23×C53—22no—
129C23×C53—22no—
130C23×C53—24no—
131C23×C53—22no—
132C23×C53—24no—
133C23×C53—48no—
134C23×C53—42no—
135C23×C53—42no—
136C23×C53—42no—
137C23×C53—42no—
138C23×C53—84no—
139C23×C53—81no—
140C23×C53—81no—
141C23×C53#12011yescharacteristic
142C23×C53#12011yescharacteristic
143C23×C53C2×SD1612yes—
144C23×C53C2×SD1614yes—
145C23×C53#13712yes—
146C23×C53C22×D411yescharacteristic
147Q8×C53—44no—
148Q8×C53—82no—
149Q8×C53—84no—
150C2×C4×C53—24no—
151C2×C4×C53—24no—
152C2×C4×C53—24no—
153C2×C4×C53—22no—
154C2×C4×C53—42no—
155C2×C4×C53—41no—
156C2×C4×C53—41no—
157C2×C4×C53—42no—
158C2×C4×C53—44no—
159C2×C4×C53—44no—
160C2×C4×C53—42no—
161C2×C4×C53—81no—
162C2×C4×C53—84no—
163C2×C4×C53—84no—
164C2×C4×C53#12012yes—
165C2×C4×C53C22×D412yes—
166C2×Dic53—42no—
167C2×Dic53—42no—
168C2×Dic53—42no—
169C2×Dic53—42no—
170C2×Dic53—42no—
171C2×Dic53—42no—
172C2×Dic53—84no—
173C2×Dic53—82no—
174C2×Dic53—82no—
175#88322—44no—
176C2×#1608—531nomaximal, Sylow
177C22×C53—24no—
178C22×C53—24no—
179C22×C53—21no—
180C22×C53—21no—
181C22×C53—21no—
182C22×C53—21no—
183C22×C53—24no—
184C22×C53—44no—
185C22×C53—48no—
186C22×C53—42no—
187C22×C53—42no—
188C22×C53—82no—
189C22×C53—84no—
190C22×C53#44812yes—
191C22×C53#44812yes—
192C22×C53#45512yes—
193C22×C53#45812yes—
194C22×C53C2×#12011yescharacteristic
195C22×C53C22×SD1611yescharacteristic
196C22×C53C2×#13711yescharacteristic
197C4×C53—22no—
198C4×C53—22no—
199C4×C53—42no—
200C4×C53—44no—
201C4×C53—44no—
202Dic53—82no—
203Dic53—82no—
204C2×#543D5311yescharacteristic
205C2×#483—532no—
206C2×#397—531no—
207#1608—538no—
208C2×#328—532no—
209C2×#323—531no—
210C2×C53—21no—
211C2×C53—21no—
212C2×C53—22no—
213C2×C53—44no—
214C2×C53#160814yescomplemented
215C2×C53C2×#44811yescharacteristic
216C2×C53C2×#45511yescharacteristic
217C2×C53C2×#45811yescharacteristic
218#543—24no—
219#543—22no—
220#543C2×D5314yes—
221C2×C22⋊Q8—22no—
222C2×C22⋊Q8—21no—
223C2×C22⋊Q8—22no—
224C2×C22⋊Q8C2×D5312yes—
225C22×C4⋊C4—531no—
226C22×C22⋊C4—21no—
227C22×C22⋊C4—1061no—
228C22×C22⋊C4C2×D5311yescharacteristic
229#483—538no—
230#483—1064no—
231C2×#106—1062no—
232C2×#103—1062no—
233C2×C22⋊C8—532no—
234#397—534no—
235#397—1062no—
236C2×#95—531no—
237C2×#95—1061no—
238#328—538no—
239#323—534no—
240C53C2×#160811yescharacteristic, complemented, Sylow
241C25C22⋊D5311yescharacteristic
242C22×Q8—21no—
243C22×Q8C22⋊D5312yes—
244C23×C4—21no—
245C23×C4—21no—
246C23×C4—1061no—
247C23×C4—1061no—
248C23×C4C22⋊D5311yescharacteristic
249C23×C4C22×D5311yescharacteristic
250C22×C8—1062no—
251C22⋊Q8—28no—
252C22⋊Q8—28no—
253C22⋊Q8—24no—
254C22⋊Q8—28no—
255C22⋊Q8—24no—
256C22⋊Q8—22no—
257C22⋊Q8—22no—
258C22⋊Q8—28no—
259C22⋊Q8C22⋊D5314yes—
260C22⋊Q8C22×D5314yes—
261C2×C4⋊C4—21no—
262C2×C4⋊C4—22no—
263C2×C4⋊C4—22no—
264C2×C4⋊C4—532no—
265C2×C4⋊C4—1062no—
266C2×C4⋊C4—1062no—
267C2×C4⋊C4—1061no—
268C2×C4⋊C4C22⋊D5312yes—
269C2×C22⋊C4—22no—
270C2×C22⋊C4—24no—
271C2×C22⋊C4—22no—
272C2×C22⋊C4—24no—
273C2×C22⋊C4—22no—
274C2×C22⋊C4—22no—
275C2×C22⋊C4—1064no—
276C2×C22⋊C4—2124no—
277C2×C22⋊C4C22⋊D5312yes—
278C2×C22⋊C4C22×D5312yes—
279#106—1068no—
280#103—1068no—
281C22⋊C8—534no—
282C22⋊C8—1062no—
283#95—532no—
284#95—1061no—
285#95—1064no—
286C24—21no—
287C24—21no—
288C24—22no—
289C24—42no—
290C24—42no—
291C24#1171992414yes—
292C24C2×C22⋊D5311yescharacteristic
293C24C2×C22⋊D5312yes—
294C2×Q8—24no—
295C2×Q8—22no—
296C2×Q8—44no—
297C2×Q8—82no—
298C2×Q8C2×C22⋊D5314yes—
299C22×C4—22no—
300C22×C4—21no—
301C22×C4—22no—
302C22×C4—24no—
303C22×C4—24no—
304C22×C4—21no—
305C22×C4—42no—
306C22×C4—42no—
307C22×C4—82no—
308C22×C4—82no—
309C22×C4—1062no—
310C22×C4—1062no—
311C22×C4—2122no—
312C22×C4—2122no—
313C22×C4—2121no—
314C22×C4—2121no—
315C22×C4—2122no—
316C22×C4—2122no—
317C22×C4—2122no—
318C22×C4C2×C22⋊D5312yes—
319C22×C4C2×C22⋊D5312yes—
320C22×C4C2×C22⋊D5311yescharacteristic
321C22×C4C2×C22⋊D5312yes—
322C22×C4C23×D5311yescharacteristic
323C2×C8—1064no—
324C2×C8—2124no—
325C4⋊C4—24no—
326C4⋊C4—28no—
327C4⋊C4—24no—
328C4⋊C4—22no—
329C4⋊C4—24no—
330C4⋊C4—1064no—
331C4⋊C4—2122no—
332C4⋊C4C2×C22⋊D5314yes—
333C22⋊C4—28no—
334C22⋊C4—24no—
335C22⋊C4—24no—
336C22⋊C4—24no—
337C22⋊C4—24no—
338C22⋊C4—2128no—
339C23—22no—
340C23—22no—
341C23—24no—
342C23—22no—
343C23—24no—
344C23—22no—
345C23—42no—
346C23—42no—
347C23—48no—
348C23—42no—
349C23—42no—
350C23—84no—
351C23—81no—
352C23—81no—
353C23#1313318611yescharacteristic
354C23#1313318611yescharacteristic
355C23C2×#1171992412yes—
356C23C2×#1171992414yes—
357C23#1313324812yes—
358C23C22×C22⋊D5311yescharacteristic
359Q8—44no—
360Q8—82no—
361Q8—84no—
362C2×C4—24no—
363C2×C4—24no—
364C2×C4—22no—
365C2×C4—24no—
366C2×C4—42no—
367C2×C4—41no—
368C2×C4—41no—
369C2×C4—42no—
370C2×C4—44no—
371C2×C4—44no—
372C2×C4—42no—
373C2×C4—81no—
374C2×C4—84no—
375C2×C4—84no—
376C2×C4—4244no—
377C2×C4—4242no—
378C2×C4—4242no—
379C2×C4—2122no—
380C2×C4—2122no—
381C2×C4—2122no—
382C2×C4—2122no—
383C2×C4—2122no—
384C2×C4—2122no—
385C2×C4#1313318612yes—
386C2×C4C22×C22⋊D5312yes—
387C8—2124no—
388C22—24no—
389C22—24no—
390C22—21no—
391C22—21no—
392C22—21no—
393C22—21no—
394C22—24no—
395C22—44no—
396C22—48no—
397C22—42no—
398C22—42no—
399C22—84no—
400C22—82no—
401C22#1681076212yes—
402C22#1681076212yes—
403C22#1681080112yes—
404C22#1681081312yes—
405C22C2×#1313318611yescharacteristic
406C22C22×#1171992411yescharacteristic
407C22C2×#1313324811yescharacteristic
408C4—22no—
409C4—22no—
410C4—42no—
411C4—44no—
412C4—44no—
413C4—4242no—
414C4—4242no—
415C2—21no—
416C2—22no—
417C2—21no—
418C2—44no—
419C2#2027370614yesdirect product factor
420C2C2×#1681076211yescharacteristic
421C2C2×#1681080111yescharacteristic
422C2C2×#1681081311yescharacteristic
subgroupG/H
centerC23#13133186
commutator, cosocleC22×C4×C53C24
FrattiniC22×C4C23×D53
FittingC2×C53×#543C2
socleC23×C53#120
prime radicalC25×C53D4
hypercenterC2×#543D53
hypocenterC53C2×#1608
LCSG ▷ C22×C4×C53 ▷ C22×C53 ▷ C53
UCSC1 ◁ C23 ◁ C23×C4 ◁ C2×#543

Complements

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

NH ≅ G/Ntype
1. C2×#16810951C2semidirect
3. #20273706C2direct (N × H)
4. C2×#16810360C2semidirect
10. #16810951C22semidirect
18. #16810360C22semidirect
214. C2×C53#1608semidirect
240. C53C2×#1608semidirect
419. C2#20273706direct (N × H)

Maximal subgroups

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

subgroupindex
C2×#168109512
C2×#168105482
#202737062
C2×#168103602
C2×#168103472
C2×C53×#5432
C2×#160853