Finite Group Databasebeta

← C2×#20273740 #27356225

Subgroups — C2×#20273740 #27356225

Order 13568.

1180 conjugacy classes of subgroups.

The subgroup lattice has 428 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#16810956C212yesmaximal, complemented
2C2×#16810559C211yesmaximal, characteristic
3#20273740C218yesmaximal, direct product factor
4C2×#16810360C212yesmaximal, complemented
5C2×#16810347C211yesmaximal, characteristic
6C2×C53×#543C211yesmaximal, characteristic
7C22×C4⋊Dic53C2211yescharacteristic
8C22×#11719919—21no—
9C22×C22⋊Dic53—21no—
10#16810956—24no—
11#16810956C2218yescomplemented
12C2×#13133150—22no—
13C2×#13133141—22no—
14C2×C22⋊#88322C2212yes—
15#16810559—22no—
16#16810559C2214yes—
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×#11719919—24no—
37C2×#11719919—44no—
38C2×C22⋊Dic53—24no—
39C2×C22⋊Dic53—44no—
40#13133150—28no—
41#13133141—28no—
42C22⋊#88322—22no—
43C22⋊#88322C2314yes—
44#13133124—21no—
45#13133124C2312yes—
46C25×C53D411yescharacteristic
47C22×Q8×C53—21no—
48C22×Q8×C53D412yes—
49C23×C4×C53—21no—
50C23×C4×C53—21no—
51C23×C4×C53D411yescharacteristic
52C23×C4×C53C2311yescharacteristic
53C22⋊Q8×C53—28no—
54C22⋊Q8×C53—28no—
55C22⋊Q8×C53—24no—
56C22⋊Q8×C53—28no—
57C22⋊Q8×C53—28no—
58C22⋊Q8×C53—24no—
59C22⋊Q8×C53—22no—
60C22⋊Q8×C53—22no—
61C22⋊Q8×C53D414yes—
62C22⋊Q8×C53C2314yes—
63C2×C4⋊C4×C53—21no—
64C2×C4⋊C4×C53—22no—
65C2×C4⋊C4×C53—22no—
66C2×C4⋊C4×C53D412yes—
67C2×C22⋊C4×C53—22no—
68C2×C22⋊C4×C53—24no—
69C2×C22⋊C4×C53—22no—
70C2×C22⋊C4×C53—24no—
71C2×C22⋊C4×C53—22no—
72C2×C22⋊C4×C53—22no—
73C2×C22⋊C4×C53D412yes—
74C2×C22⋊C4×C53C2312yes—
75C24×C53—21no—
76C24×C53—21no—
77C24×C53—22no—
78C24×C53—42no—
79C24×C53—42no—
80C24×C53Q1614yes—
81C24×C53C2×D411yescharacteristic
82C24×C53C2×D412yes—
83C2×Q8×C53—24no—
84C2×Q8×C53—22no—
85C2×Q8×C53—44no—
86C2×Q8×C53—82no—
87C2×Q8×C53C2×D414yes—
88C22×C4×C53—22no—
89C22×C4×C53—21no—
90C22×C4×C53—22no—
91C22×C4×C53—24no—
92C22×C4×C53—24no—
93C22×C4×C53—21no—
94C22×C4×C53—42no—
95C22×C4×C53—42no—
96C22×C4×C53—82no—
97C22×C4×C53—82no—
98C22×C4×C53C2×D412yes—
99C22×C4×C53C2×D412yes—
100C22×C4×C53C2×D411yescharacteristic
101C22×C4×C53C2×D412yes—
102C22×C4×C53C2411yescharacteristic
103C22×Dic53—24no—
104C22×Dic53—22no—
105C22×Dic53—22no—
106C22×Dic53—42no—
107C22×Dic53—42no—
108C22×Dic53—42no—
109C22×Dic53—42no—
110C22×Dic53—41no—
111C22×Dic53—41no—
112C4⋊C4×C53—28no—
113C4⋊C4×C53—24no—
114C4⋊C4×C53—24no—
115C4⋊C4×C53—22no—
116C4⋊C4×C53—24no—
117C4⋊C4×C53C2×D414yes—
118C22⋊C4×C53—28no—
119C22⋊C4×C53—24no—
120C22⋊C4×C53—24no—
121C22⋊C4×C53—24no—
122C22⋊C4×C53—24no—
123C22⋊Dic53—48no—
124C4⋊Dic53—24no—
125C4⋊Dic53—42no—
126#11719919—48no—
127C2×#88322—24no—
128C2×#88322—44no—
129C23×C53—22no—
130C23×C53—22no—
131C23×C53—22no—
132C23×C53—24no—
133C23×C53—22no—
134C23×C53—24no—
135C23×C53—48no—
136C23×C53—42no—
137C23×C53—42no—
138C23×C53—42no—
139C23×C53—42no—
140C23×C53—84no—
141C23×C53—81no—
142C23×C53—81no—
143C23×C53#12011yescharacteristic
144C23×C53#12011yescharacteristic
145C23×C53C2×Q1612yes—
146C23×C53C2×Q1614yes—
147C23×C53#13712yes—
148C23×C53C22×D411yescharacteristic
149Q8×C53—44no—
150Q8×C53—82no—
151Q8×C53—84no—
152C2×C4×C53—24no—
153C2×C4×C53—24no—
154C2×C4×C53—24no—
155C2×C4×C53—22no—
156C2×C4×C53—42no—
157C2×C4×C53—41no—
158C2×C4×C53—41no—
159C2×C4×C53—42no—
160C2×C4×C53—44no—
161C2×C4×C53—44no—
162C2×C4×C53—42no—
163C2×C4×C53—81no—
164C2×C4×C53—84no—
165C2×C4×C53—84no—
166C2×C4×C53#12012yes—
167C2×C4×C53C22×D412yes—
168C2×Dic53—24no—
169C2×Dic53—48no—
170C2×Dic53—42no—
171C2×Dic53—42no—
172C2×Dic53—42no—
173C2×Dic53—42no—
174C2×Dic53—42no—
175C2×Dic53—82no—
176C2×Dic53—82no—
177#88322—44no—
178C2×#1614—531nomaximal, Sylow
179C22×C53—24no—
180C22×C53—24no—
181C22×C53—21no—
182C22×C53—21no—
183C22×C53—21no—
184C22×C53—21no—
185C22×C53—24no—
186C22×C53—44no—
187C22×C53—48no—
188C22×C53—42no—
189C22×C53—42no—
190C22×C53—82no—
191C22×C53—84no—
192C22×C53#45112yes—
193C22×C53#45112yes—
194C22×C53#45312yes—
195C22×C53#45812yes—
196C22×C53C2×#12011yescharacteristic
197C22×C53C22×Q1611yescharacteristic
198C22×C53C2×#13711yescharacteristic
199C4×C53—22no—
200C4×C53—22no—
201C4×C53—42no—
202C4×C53—44no—
203C4×C53—44no—
204Dic53—44no—
205Dic53—82no—
206C2×#543D5311yescharacteristic
207C2×#484—532no—
208C2×#399—531no—
209#1614—538no—
210C2×#328—532no—
211C2×#323—531no—
212C2×C53—21no—
213C2×C53—21no—
214C2×C53—22no—
215C2×C53—44no—
216C2×C53#161414yescomplemented
217C2×C53C2×#45111yescharacteristic
218C2×C53C2×#45311yescharacteristic
219C2×C53C2×#45811yescharacteristic
220#543—24no—
221#543—22no—
222#543C2×D5314yes—
223C2×C22⋊Q8—22no—
224C2×C22⋊Q8—21no—
225C2×C22⋊Q8—22no—
226C2×C22⋊Q8C2×D5312yes—
227C22×C4⋊C4—531no—
228C22×C4⋊C4—1061no—
229C22×C22⋊C4—21no—
230C22×C22⋊C4—1061no—
231C22×C22⋊C4C2×D5311yescharacteristic
232#484—538no—
233#484—1064no—
234C2×#107—1062no—
235C2×#103—1062no—
236C2×C22⋊C8—532no—
237#399—534no—
238#399—1062no—
239C2×#95—531no—
240#328—538no—
241#323—534no—
242C53C2×#161411yescharacteristic, complemented, Sylow
243C25C22⋊D5311yescharacteristic
244C22×Q8—21no—
245C22×Q8C22⋊D5312yes—
246C23×C4—21no—
247C23×C4—21no—
248C23×C4—1061no—
249C23×C4—1061no—
250C23×C4C22⋊D5311yescharacteristic
251C23×C4C22×D5311yescharacteristic
252C22×C8—1062no—
253C22⋊Q8—28no—
254C22⋊Q8—28no—
255C22⋊Q8—24no—
256C22⋊Q8—28no—
257C22⋊Q8—24no—
258C22⋊Q8—22no—
259C22⋊Q8—22no—
260C22⋊Q8—28no—
261C22⋊Q8C22⋊D5314yes—
262C22⋊Q8C22×D5314yes—
263C2×C4⋊C4—21no—
264C2×C4⋊C4—22no—
265C2×C4⋊C4—22no—
266C2×C4⋊C4—532no—
267C2×C4⋊C4—1062no—
268C2×C4⋊C4—1062no—
269C2×C4⋊C4—1061no—
270C2×C4⋊C4—1064no—
271C2×C4⋊C4—2124no—
272C2×C4⋊C4C22⋊D5312yes—
273C2×C22⋊C4—22no—
274C2×C22⋊C4—24no—
275C2×C22⋊C4—22no—
276C2×C22⋊C4—24no—
277C2×C22⋊C4—22no—
278C2×C22⋊C4—22no—
279C2×C22⋊C4—1064no—
280C2×C22⋊C4—2124no—
281C2×C22⋊C4C22⋊D5312yes—
282C2×C22⋊C4C22×D5312yes—
283#107—1068no—
284#103—1068no—
285C22⋊C8—534no—
286C22⋊C8—1062no—
287#95—532no—
288#95—1061no—
289C24—21no—
290C24—21no—
291C24—22no—
292C24—42no—
293C24—42no—
294C24#1171992514yes—
295C24C2×C22⋊D5311yescharacteristic
296C24C2×C22⋊D5312yes—
297C2×Q8—24no—
298C2×Q8—22no—
299C2×Q8—44no—
300C2×Q8—82no—
301C2×Q8C2×C22⋊D5314yes—
302C22×C4—22no—
303C22×C4—21no—
304C22×C4—22no—
305C22×C4—24no—
306C22×C4—24no—
307C22×C4—21no—
308C22×C4—42no—
309C22×C4—42no—
310C22×C4—82no—
311C22×C4—82no—
312C22×C4—1064no—
313C22×C4—1062no—
314C22×C4—1062no—
315C22×C4—2122no—
316C22×C4—2122no—
317C22×C4—2121no—
318C22×C4—2121no—
319C22×C4—2122no—
320C22×C4—2122no—
321C22×C4C2×C22⋊D5312yes—
322C22×C4C2×C22⋊D5312yes—
323C22×C4C2×C22⋊D5311yescharacteristic
324C22×C4C2×C22⋊D5312yes—
325C22×C4C23×D5311yescharacteristic
326C2×C8—1064no—
327C2×C8—2124no—
328C4⋊C4—24no—
329C4⋊C4—28no—
330C4⋊C4—24no—
331C4⋊C4—22no—
332C4⋊C4—24no—
333C4⋊C4—1064no—
334C4⋊C4—2128no—
335C4⋊C4—2122no—
336C4⋊C4C2×C22⋊D5314yes—
337C22⋊C4—28no—
338C22⋊C4—24no—
339C22⋊C4—24no—
340C22⋊C4—24no—
341C22⋊C4—24no—
342C22⋊C4—2128no—
343C23—22no—
344C23—22no—
345C23—24no—
346C23—22no—
347C23—24no—
348C23—22no—
349C23—42no—
350C23—42no—
351C23—48no—
352C23—42no—
353C23—42no—
354C23—84no—
355C23—81no—
356C23—81no—
357C23#1313318611yescharacteristic
358C23#1313318611yescharacteristic
359C23C2×#1171992512yes—
360C23C2×#1171992514yes—
361C23#1313324812yes—
362C23C22×C22⋊D5311yescharacteristic
363Q8—44no—
364Q8—82no—
365Q8—84no—
366C2×C4—24no—
367C2×C4—24no—
368C2×C4—22no—
369C2×C4—24no—
370C2×C4—42no—
371C2×C4—41no—
372C2×C4—41no—
373C2×C4—42no—
374C2×C4—44no—
375C2×C4—44no—
376C2×C4—42no—
377C2×C4—81no—
378C2×C4—84no—
379C2×C4—84no—
380C2×C4—1064no—
381C2×C4—4242no—
382C2×C4—4242no—
383C2×C4—2122no—
384C2×C4—2122no—
385C2×C4—2122no—
386C2×C4—2128no—
387C2×C4—2122no—
388C2×C4—2122no—
389C2×C4#1313318612yes—
390C2×C4C22×C22⋊D5312yes—
391C8—2124no—
392C22—24no—
393C22—24no—
394C22—21no—
395C22—21no—
396C22—21no—
397C22—21no—
398C22—24no—
399C22—44no—
400C22—48no—
401C22—42no—
402C22—42no—
403C22—82no—
404C22—84no—
405C22#1681077912yes—
406C22#1681077912yes—
407C22#1681079112yes—
408C22#1681081312yes—
409C22C2×#1313318611yescharacteristic
410C22C22×#1171992511yescharacteristic
411C22C2×#1313324811yescharacteristic
412C4—22no—
413C4—22no—
414C4—42no—
415C4—44no—
416C4—44no—
417C4—4242no—
418C4—2124no—
419C2—21no—
420C2—21no—
421C2—22no—
422C2—44no—
423C2#2027374014yesdirect product factor
424C2C2×#1681077911yescharacteristic
425C2C2×#1681079111yescharacteristic
426C2C2×#1681081311yescharacteristic
subgroupG/H
centerC23#13133186
commutator, cosocleC22×C4×C53C24
FrattiniC22×C4C23×D53
FittingC2×C53×#543C2
socleC23×C53#120
prime radicalC25×C53D4
hypercenterC2×#543D53
hypocenterC53C2×#1614
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×#16810956C2semidirect
3. #20273740C2direct (N × H)
4. C2×#16810360C2semidirect
11. #16810956C22semidirect
18. #16810360C22semidirect
216. C2×C53#1614semidirect
242. C53C2×#1614semidirect
423. C2#20273740direct (N × H)

Maximal subgroups

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

subgroupindex
C2×#168109562
C2×#168105592
#202737402
C2×#168103602
C2×#168103472
C2×C53×#5432
C2×#161453