Finite Group Databasebeta

← C2×#20273315 #27351997

Subgroups — C2×#20273315 #27351997

Order 13568.

1050 conjugacy classes of subgroups.

The subgroup lattice has 480 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#125×D53C211yesmaximal, characteristic
2C2×#16810678C211yesmaximal, characteristic
3C2×#16810575C211yesmaximal, characteristic
4#20273315C218yesmaximal, direct product factor
5C2×#16810365C211yesmaximal, characteristic, complemented
6C2×#16810364C211yesmaximal, characteristic, complemented
7C2×#16810364C211yesmaximal, characteristic, complemented
8C2×C53×#330C211yesmaximal, characteristic, complemented
9C22×#11719912—21no—
10C2×#13133209C2211yescharacteristic
11C2×#13133208C411yescharacteristic
12C2×#13133208C411yescharacteristic
13C2×#13133207—21no—
14C2×#13133207—21no—
15#125×D53—22no—
16#125×D53—22no—
17#125×D53C414yes—
18#125×D53C2214yes—
19C2×C4⋊C4×D53—21no—
20C2×C4⋊C4×D53—21no—
21C2×C4⋊C4×D53C411yescharacteristic
22C2×C4⋊C4×D53C411yescharacteristic
23C2×C42×D53C2211yescharacteristic
24#16810678—22no—
25#16810678C2214yes—
26C2×#13133146—21no—
27C2×#13133129—21no—
28#16810575—22no—
29#16810575C2214yes—
30C2×#13133128C2211yescharacteristic
31C2×#13133128C2211yescharacteristic
32C2×C8×Dic53—21no—
33#16810365C2214yescomplemented
34#16810364C2214yescomplemented
35#16810364C2214yescomplemented
36C2×#125×C53C2211yescharacteristic
37C2×#97×C53C2211yescharacteristic
38C2×#97×C53C2211yescharacteristic
39C53×#330C2214yescomplemented
40C22×C4×D53—21no—
41C22×C4×D53—21no—
42C22×C4×D53C2×C411yescharacteristic
43C22×C4×D53D411yescharacteristic
44C22×C4×D53D411yescharacteristic
45C2×#11719912—28no—
46C2×#11719912—24no—
47C22×#88322—21no—
48C22×#88322—21no—
49#13133209—21no—
50#13133209—21no—
51#13133209C2×C412yes—
52#13133209C2312yes—
53#13133208—22no—
54#13133208—22no—
55#13133208C2×C414yes—
56#13133208C2×C414yes—
57#13133207—24no—
58#13133207—24no—
59#13133207—24no—
60#13133207—24no—
61C2×C4⋊Dic53—21no—
62C2×C4⋊Dic53—21no—
63C2×C4⋊Dic53C2×C411yescharacteristic
64C2×C4⋊Dic53C2×C411yescharacteristic
65C2×#11719919—21no—
66C2×#11719919—21no—
67C2×#11719919—21no—
68C2×#11719919—21no—
69C2×#11719919—21no—
70C2×#11719919—21no—
71C4⋊C4×D53—24no—
72C4⋊C4×D53—24no—
73C4⋊C4×D53—22no—
74C4⋊C4×D53—22no—
75C4⋊C4×D53—24no—
76C4⋊C4×D53—24no—
77C4⋊C4×D53C2×C414yes—
78C4⋊C4×D53C2×C414yes—
79C2×C4×Dic53C2×C411yescharacteristic
80C2×C4×Dic53D411yescharacteristic
81C2×C4×Dic53D411yescharacteristic
82C42×D53C2×C412yes—
83C42×D53D412yes—
84C42×D53D412yes—
85C42×D53C2312yes—
86#13133146—24no—
87#13133129—24no—
88#13133128—21no—
89#13133128—21no—
90#13133128C2312yes—
91#13133128C2312yes—
92C8×Dic53—24no—
93C22×C8×C53—21no—
94C22×C8×C53—21no—
95#125×C53—21no—
96#125×C53—21no—
97#125×C53C2×C412yes—
98#125×C53C2312yes—
99C2×C4⋊C4×C53—21no—
100C2×C4⋊C4×C53—21no—
101C2×C4⋊C4×C53C2×C411yescharacteristic
102C2×C4⋊C4×C53C2×C411yescharacteristic
103C2×C42×C53C2311yescharacteristic
104#97×C53—21no—
105#97×C53—21no—
106#97×C53C2312yes—
107#97×C53C2312yes—
108C23×D53C22⋊C411yescharacteristic
109C22×C4×C53—21no—
110C22×C4×C53—21no—
111C22×C4×C53C22×C411yescharacteristic
112C22×C4×C53C2×D411yescharacteristic
113C22×C4×C53C2×D411yescharacteristic
114C22×Dic53—21no—
115C22×Dic53—21no—
116C22×Dic53C22⋊C411yescharacteristic
117C22×Dic53C22⋊C411yescharacteristic
118C22×Dic53C22⋊C411yescharacteristic
119C2×C4×D53—24no—
120C2×C4×D53—24no—
121C2×C4×D53—24no—
122C2×C4×D53—24no—
123C2×C4×D53—42no—
124C2×C4×D53—44no—
125C2×C4×D53—44no—
126C2×C4×D53C22⋊C412yes—
127C2×C4×D53C22⋊C412yes—
128C2×C4×D53C22⋊C412yes—
129C2×C4×D53C22×C412yes—
130C2×C4×D53C2×D412yes—
131C2×C4×D53C2×D412yes—
132C2×C8×C53—24no—
133C2×C8×C53—22no—
134C2×C8×C53—22no—
135C2×C8×C53—42no—
136C4⋊C4×C53—22no—
137C4⋊C4×C53—22no—
138C4⋊C4×C53—21no—
139C4⋊C4×C53—21no—
140C4⋊C4×C53—22no—
141C4⋊C4×C53—22no—
142C4⋊C4×C53C22×C412yes—
143C4⋊C4×C53C22×C412yes—
144C42×C53C22×C411yescharacteristic
145C42×C53C2×D411yescharacteristic
146C42×C53C2×D411yescharacteristic
147C42×C53C2411yescharacteristic
148C4⋊Dic53—21no—
149C4⋊Dic53—21no—
150C4⋊Dic53—24no—
151C4⋊Dic53—24no—
152C4⋊Dic53C22×C412yes—
153C4⋊Dic53C22×C412yes—
154#11719919—22no—
155#11719919—22no—
156#11719919—22no—
157#11719919—22no—
158#11719919—24no—
159#11719919—24no—
160#11719919—24no—
161#11719919—24no—
162C4×Dic53C22⋊C412yes—
163C4×Dic53C22⋊C412yes—
164C4×Dic53C22⋊C412yes—
165C4×Dic53C22×C412yes—
166C4×Dic53C2×D412yes—
167C4×Dic53C2×D412yes—
168C2×#88322—24no—
169C2×#88322—22no—
170C2×#88322—22no—
171C2×#88322—42no—
172#11719912—216no—
173C23×C53C2×C22⋊C411yescharacteristic
174C22×D53—22no—
175C22×D53—22no—
176C22×D53#10414yes—
177C22×D53C2×C22⋊C412yes—
178C2×C4×C53—22no—
179C2×C4×C53—22no—
180C2×C4×C53—22no—
181C2×C4×C53—22no—
182C2×C4×C53—22no—
183C2×C4×C53—42no—
184C2×C4×C53—42no—
185C2×C4×C53C2×C22⋊C411yescharacteristic
186C2×C4×C53C2×C22⋊C411yescharacteristic
187C2×C4×C53C2×C22⋊C411yescharacteristic
188C2×C4×C53C23×C411yescharacteristic
189C2×C4×C53C22×D411yescharacteristic
190C2×C4×C53C22×D411yescharacteristic
191C2×Dic53—22no—
192C2×Dic53—22no—
193C2×Dic53—22no—
194C2×Dic53—22no—
195C2×Dic53—42no—
196C2×Dic53—42no—
197C2×Dic53#10114yes—
198C2×Dic53C2×C22⋊C412yes—
199C2×Dic53C2×C22⋊C412yes—
200C2×Dic53C2×C22⋊C412yes—
201C4×D53—24no—
202C4×D53—24no—
203C4×D53—24no—
204C4×D53—24no—
205C4×D53—42no—
206C4×D53—42no—
207C4×D53—48no—
208C4×D53—48no—
209C8×C53—24no—
210C8×C53—42no—
211#88322—24no—
212#88322—42no—
213C2×#1528—531nomaximal, Sylow
214C22×C53C2×#10112yes—
215C22×C53C2×#10412yes—
216C22×C53#42112yes—
217C22×C53C22×C22⋊C411yescharacteristic
218C2×D53—22no—
219C2×D53—22no—
220C2×D53—28no—
221C2×D53C2×#10414yes—
222C4×C53—21no—
223C4×C53—21no—
224C4×C53—21no—
225C4×C53—21no—
226C4×C53—21no—
227C4×C53—21no—
228C4×C53—42no—
229C4×C53—42no—
230Dic53—22no—
231Dic53—22no—
232Dic53—42no—
233Dic53—42no—
234Dic53C2×#10114yes—
235C22×#125—531no—
236C2×#432—531no—
237C2×C4×M16—531no—
238#1528—538no—
239C2×#330—531no—
240C2×#330—531no—
241C2×#330—531no—
242C2×#330D5311yescharacteristic, complemented
243C2×C53#152814yescomplemented
244C2×C53C22×#10111yescharacteristic
245C2×C53C22×#10411yescharacteristic
246C2×C53C2×#42111yescharacteristic
247D53—24no—
248C22×M16—1061no—
249C2×#125—534no—
250C2×#125—534no—
251C2×#125—531no—
252C2×#125—531no—
253C2×#125—531no—
254C2×#125—1061no—
255C2×#125—1061no—
256C2×#125—1062no—
257C2×#125—1062no—
258C2×#125C2×D5311yescharacteristic
259C22×C4⋊C4—531no—
260C22×C4⋊C4—531no—
261C22×C4⋊C4—1061no—
262C22×C4⋊C4—1061no—
263C22×C42—531no—
264#432—534no—
265#432—1062no—
266C2×C4⋊C8—1061no—
267C2×C22⋊C8—1061no—
268C4×M16—534no—
269C4×M16—1062no—
270C2×#97—531no—
271C2×#97—531no—
272C2×#97C2×D5311yescharacteristic
273C2×#97C2×D5311yescharacteristic
274C2×C4×C8—1061no—
275#330—534no—
276#330—534no—
277#330—534no—
278#330C2×D5314yescomplemented
279C53C2×#152811yescharacteristic, complemented, Sylow
280C23×C4—531no—
281C23×C4—531no—
282C23×C4—531no—
283C23×C4—1061no—
284C23×C4—1061no—
285C2×M16—1064no—
286C2×M16—1068no—
287C22×C8—21no—
288C22×C8—21no—
289C22×C8—1061no—
290C22×C8—1061no—
291#125—21no—
292#125—21no—
293#125—534no—
294#125—534no—
295#125—532no—
296#125—532no—
297#125—1064no—
298#125—1064no—
299#125—1064no—
300#125—1064no—
301#125—1062no—
302#125—1062no—
303#125—1061no—
304#125—1061no—
305#125C4×D5312yes—
306#125C22×D5312yes—
307C2×C4⋊C4—21no—
308C2×C4⋊C4—21no—
309C2×C4⋊C4—531no—
310C2×C4⋊C4—531no—
311C2×C4⋊C4—534no—
312C2×C4⋊C4—534no—
313C2×C4⋊C4—1062no—
314C2×C4⋊C4—1062no—
315C2×C4⋊C4—1064no—
316C2×C4⋊C4—1064no—
317C2×C4⋊C4—1064no—
318C2×C4⋊C4—1064no—
319C2×C4⋊C4—1061no—
320C2×C4⋊C4—1061no—
321C2×C4⋊C4—1061no—
322C2×C4⋊C4—1061no—
323C2×C4⋊C4—1061no—
324C2×C4⋊C4—1061no—
325C2×C4⋊C4—1061no—
326C2×C4⋊C4—1061no—
327C2×C4⋊C4C4×D5311yescharacteristic
328C2×C4⋊C4C4×D5311yescharacteristic
329C2×C42—531no—
330C2×C42—531no—
331C2×C42—531no—
332C2×C42—532no—
333C2×C42—532no—
334C2×C42—532no—
335C2×C42—532no—
336C2×C42C22×D5311yescharacteristic
337C4⋊C8—1064no—
338C22⋊C8—1064no—
339#97—21no—
340#97—21no—
341#97—532no—
342#97—532no—
343#97—1061no—
344#97—1061no—
345#97C22×D5312yes—
346#97C22×D5312yes—
347C4×C8—1064no—
348C24—531no—
349C22×C4—21no—
350C22×C4—21no—
351C22×C4—531no—
352C22×C4—531no—
353C22×C4—531no—
354C22×C4—532no—
355C22×C4—532no—
356C22×C4—532no—
357C22×C4—532no—
358C22×C4—532no—
359C22×C4—532no—
360C22×C4—1064no—
361C22×C4—1064no—
362C22×C4—1064no—
363C22×C4—1064no—
364C22×C4—1061no—
365C22×C4—1061no—
366C22×C4—2122no—
367C22×C4—2124no—
368C22×C4—2124no—
369C22×C4C2×C4×D5311yescharacteristic
370C22×C4D4×D5311yescharacteristic
371C22×C4D4×D5311yescharacteristic
372M16—10616no—
373C2×C8—24no—
374C2×C8—22no—
375C2×C8—22no—
376C2×C8—42no—
377C2×C8—1064no—
378C2×C8—1062no—
379C2×C8—1062no—
380C2×C8—2122no—
381C4⋊C4—22no—
382C4⋊C4—22no—
383C4⋊C4—21no—
384C4⋊C4—21no—
385C4⋊C4—22no—
386C4⋊C4—22no—
387C4⋊C4—532no—
388C4⋊C4—532no—
389C4⋊C4—1064no—
390C4⋊C4—1064no—
391C4⋊C4—1064no—
392C4⋊C4—1064no—
393C4⋊C4—1064no—
394C4⋊C4—1064no—
395C4⋊C4—1061no—
396C4⋊C4—1061no—
397C4⋊C4—1062no—
398C4⋊C4—1062no—
399C4⋊C4—1062no—
400C4⋊C4—1062no—
401C4⋊C4C2×C4×D5312yes—
402C4⋊C4C2×C4×D5312yes—
403C42—532no—
404C42—532no—
405C42—532no—
406C42—532no—
407C42—532no—
408C42—532no—
409C42C2×C4×D5311yescharacteristic
410C42D4×D5311yescharacteristic
411C42D4×D5311yescharacteristic
412C42C23×D5311yescharacteristic
413C23—532no—
414C23—534no—
415C23—1062no—
416C23—1062no—
417C23C22⋊C4×D5311yescharacteristic
418C2×C4—22no—
419C2×C4—22no—
420C2×C4—22no—
421C2×C4—22no—
422C2×C4—22no—
423C2×C4—42no—
424C2×C4—42no—
425C2×C4—532no—
426C2×C4—534no—
427C2×C4—532no—
428C2×C4—532no—
429C2×C4—1064no—
430C2×C4—1064no—
431C2×C4—1064no—
432C2×C4—1064no—
433C2×C4—1062no—
434C2×C4—1062no—
435C2×C4—1062no—
436C2×C4—1062no—
437C2×C4—2122no—
438C2×C4—2122no—
439C2×C4—2128no—
440C2×C4—2128no—
441C2×C4—2122no—
442C2×C4—2122no—
443C2×C4C22⋊C4×D5311yescharacteristic
444C2×C4C22⋊C4×D5311yescharacteristic
445C2×C4C22⋊C4×D5311yescharacteristic
446C2×C4C22×C4×D5311yescharacteristic
447C2×C4C2×D4×D5311yescharacteristic
448C2×C4C2×D4×D5311yescharacteristic
449C8—24no—
450C8—42no—
451C8—1064no—
452C8—2122no—
453C22—534no—
454C22—1062no—
455C22—1062no—
456C22—1068no—
457C22#101×D5312yes—
458C22#104×D5312yes—
459C22#1681063812yes—
460C22C2×C22⋊C4×D5311yescharacteristic
461C4—21no—
462C4—21no—
463C4—21no—
464C4—21no—
465C4—21no—
466C4—21no—
467C4—42no—
468C4—42no—
469C4—534no—
470C4—1062no—
471C4—1062no—
472C4—2122no—
473C4—2122no—
474C2—1064no—
475C2#2027331514yesdirect product factor
476C2C2×#101×D5311yescharacteristic
477C2C2×#104×D5311yescharacteristic
478C2C2×#1681063811yescharacteristic
subgroupG/H
centerC23C22⋊C4×D53
commutatorC2×C4×C53C23×C4
FrattiniC42C23×D53
FittingC2×C53×#330C2
socleC23×C53C2×C22⋊C4
prime radicalC23×D53C22⋊C4
cosocleC42×C53C24
hypercenterC2×#330D53
hypocenterC53C2×#1528
LCSG ▷ C2×C4×C53 ▷ C22×C53 ▷ C53
UCSC1 ◁ C23 ◁ C2×C42 ◁ C2×#330

Complements

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

NH ≅ G/Ntype
4. #20273315C2direct (N × H)
5. C2×#16810365C2semidirect
6. C2×#16810364C2semidirect
7. C2×#16810364C2semidirect
8. C2×C53×#330C2semidirect
33. #16810365C22semidirect
34. #16810364C22semidirect
35. #16810364C22semidirect
39. C53×#330C22semidirect
242. C2×#330D53semidirect
243. C2×C53#1528semidirect
278. #330C2×D53semidirect
279. C53C2×#1528semidirect
475. C2#20273315direct (N × H)

Maximal subgroups

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

subgroupindex
C2×#125×D532
C2×#168106782
C2×#168105752
#202733152
C2×#168103652
C2×#168103642
C2×C53×#3302
C2×#152853