Finite Group Databasebeta

← C2×#20276781 #27538461

Subgroups — C2×#20276781 #27538461

Order 13568.

706 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×#16811064C211yesmaximal, characteristic, complemented
2C2×#16811059C211yesmaximal, characteristic
3C2×#16810741C211yesmaximal, characteristic, complemented
4#20276781C218yesmaximal, direct product factor
5C2×#16810444C211yesmaximal, characteristic, complemented
6C2×#16810439C211yesmaximal, characteristic, complemented
7C2×#16810436C211yesmaximal, characteristic, complemented
8C2×C53×#495C211yesmaximal, characteristic, complemented
9C2×#13133239C2211yescharacteristic
10C2×#13133223—21no—
11#16811064—22no—
12#16811064C2214yescomplemented
13#16811059—22no—
14#16811059C2214yes—
15C2×#13133163—21no—
16C2×#13133161—21no—
17C2×#13133160—21no—
18C2×#13133158—21no—
19C2×#13133157C2211yescharacteristic
20C2×#13133157C2211yescharacteristic
21#16810741—22no—
22#16810741C2214yescomplemented
23C2×#13133153C2211yescharacteristic
24C2×#13133145—21no—
25#16810444C2214yescomplemented
26#16810439C2214yescomplemented
27#16810436C2214yescomplemented
28C22×Q16×C53C2211yescharacteristic
29C22×SD16×C53—21no—
30C22×D8×C53C2211yescharacteristic
31C2×#124×C53—21no—
32C53×#495—22no—
33C53×#495C2214yescomplemented
34C2×C4×C8×C53C2211yescharacteristic
35C2×#11719945—21no—
36#13133239—21no—
37#13133239—21no—
38#13133239D412yes—
39#13133239C2312yes—
40C2×Dic212D411yescharacteristic
41C2×#11719913—21no—
42C2×D424D411yescharacteristic
43#13133223—24no—
44#13133223—42no—
45C2×#11719917—21no—
46C2×#11719912—41no—
47C2×C8×D53—41no—
48#13133163—24no—
49#13133161—24no—
50#13133160—24no—
51#13133158—24no—
52#13133157—21no—
53#13133157—21no—
54#13133157C2312yes—
55#13133157C2312yes—
56C2×#11719908—21no—
57C2×#11719908—21no—
58#13133153—21no—
59#13133153C2312yes—
60#13133145—24no—
61C22×Q8×C53—21no—
62C22×D4×C53—21no—
63C2×Q16×C53—21no—
64C2×Q16×C53—21no—
65C2×Q16×C53—21no—
66C2×Q16×C53—21no—
67C2×Q16×C53—21no—
68C2×Q16×C53C2312yes—
69C2×SD16×C53—24no—
70C2×SD16×C53—41no—
71C2×SD16×C53—41no—
72C2×SD16×C53—42no—
73C2×D8×C53—21no—
74C2×D8×C53—21no—
75C2×D8×C53—21no—
76C2×D8×C53—21no—
77C2×D8×C53—21no—
78C2×D8×C53C2312yes—
79C22×C8×C53D411yescharacteristic
80C22×C8×C53C2311yescharacteristic
81#124×C53—24no—
82#124×C53—42no—
83C2×C22⋊C4×C53—42no—
84C2×C42×C53D411yescharacteristic
85C4×C8×C53D412yes—
86C4×C8×C53C2312yes—
87C24×C53—41no—
88C2×Q8×C53—22no—
89C2×Q8×C53—41no—
90C2×Q8×C53—41no—
91C2×Q8×C53—41no—
92C2×Q8×C53—41no—
93C2×Q8×C53—41no—
94C2×D4×C53—22no—
95C2×D4×C53—41no—
96C2×D4×C53—41no—
97C2×D4×C53—41no—
98C2×D4×C53—41no—
99C2×D4×C53—41no—
100C22×C4×C53—21no—
101C22×C4×C53—41no—
102C22×C4×C53C2×D411yescharacteristic
103C2×C22⋊D53—41no—
104#11719945—22no—
105#11719945—22no—
106#11719945—41no—
107#11719945—41no—
108C2×D212—21no—
109C2×C4×D53—41no—
110C2×Dic106—21no—
111Q16×C53—22no—
112Q16×C53—22no—
113Q16×C53—41no—
114Q16×C53—41no—
115SD16×C53—44no—
116SD16×C53—44no—
117D8×C53—22no—
118D8×C53—22no—
119D8×C53—41no—
120D8×C53—41no—
121C2×C8×C53D811yescharacteristic
122C2×C8×C53D811yescharacteristic
123C2×C8×C53D812yes—
124C2×C8×C53C2×D411yescharacteristic
125C2×C8×C53C2×D411yescharacteristic
126C2×C8×C53C2×D411yescharacteristic
127C2×C8×C53C2×D412yes—
128C2×C8×C53#4111yescharacteristic
129C2×C8×C53#4111yescharacteristic
130C2×C8×C53C2411yescharacteristic
131C22⋊C4×C53—48no—
132C42×C53—21no—
133C42×C53C2×D412yes—
134#11719917—22no—
135#11719917—41no—
136C2×#88322—41no—
137Dic212—21no—
138Dic212C2×D412yes—
139D424—21no—
140D424C2×D412yes—
141#11719913—22no—
142#11719913—22no—
143#11719912—44no—
144C8×D53—44no—
145#11719908—22no—
146#11719908—22no—
147C23×C53—42no—
148C23×C53—84no—
149C23×C53—81no—
150C23×C53—81no—
151C23×C53AGL(1,ℤ/8)11yescharacteristic
152C22×D53—41no—
153Q8×C53—42no—
154Q8×C53—42no—
155Q8×C53—81no—
156Q8×C53—81no—
157D4×C53—42no—
158D4×C53—42no—
159D4×C53—81no—
160D4×C53—81no—
161C2×C4×C53—22no—
162C2×C4×C53—42no—
163C2×C4×C53—42no—
164C2×C4×C53—81no—
165C2×C4×C53—81no—
166C2×C4×C53#12111yescharacteristic
167C2×C4×C53#12111yescharacteristic
168C2×C4×C53C2×D811yescharacteristic
169C2×C4×C53C2×D811yescharacteristic
170C2×C4×C53AGL(1,ℤ/8)11yescharacteristic
171C2×C4×C53C22×D411yescharacteristic
172C22⋊D53—42no—
173C22⋊D53—42no—
174C2×Dic53—41no—
175D212—22no—
176D212—41no—
177C4×D53—42no—
178C4×D53—42no—
179Dic106—22no—
180Dic106—41no—
181C8×C53#12111yescharacteristic
182C8×C53#12111yescharacteristic
183C8×C53C2×D812yes—
184C8×C53C2×D812yes—
185C8×C53C22×D411yescharacteristic
186C8×C53C2×#4111yescharacteristic
187#88322—42no—
188C2×#2212—531nomaximal, Sylow
189C22×C53—22no—
190C22×C53—84no—
191C22×C53—82no—
192C22×C53—162no—
193C22×C53—162no—
194C22×C53#45911yescharacteristic
195C22×C53#45911yescharacteristic
196C22×C53C2×AGL(1,ℤ/8)11yescharacteristic
197C2×D53—42no—
198C2×D53—82no—
199C4×C53—42no—
200C4×C53—82no—
201C4×C53#45911yescharacteristic
202C4×C53#45911yescharacteristic
203C4×C53C2×#12111yescharacteristic
204C4×C53C22×D811yescharacteristic
205Dic53—42no—
206C2×#510—531no—
207C2×#509—531no—
208C2×#495D5311yescharacteristic, complemented
209C2×#443—531no—
210#2212—538no—
211C2×#364—531no—
212C2×#362—531no—
213C2×#361—531no—
214C2×C53—21no—
215C2×C53—21no—
216C2×C53—162no—
217C2×C53#221212yescomplemented
218C2×C53C2×#45911yescharacteristic
219D53—82no—
220C2×#135—531no—
221C22×Q16C2×D5311yescharacteristic
222C22×SD16—21no—
223C22×D8C2×D5311yescharacteristic
224C2×#131—1061no—
225C2×#124—21no—
226#510—534no—
227#510—1062no—
228#509—534no—
229#509—1062no—
230C2×Q32—1061no—
231C2×SD32—1061no—
232C2×SD32—1061no—
233C2×D16—1061no—
234C2×M32—531no—
235C2×M32—531no—
236#495—22no—
237#495C2×D5314yescomplemented
238#443—534no—
239#443—1062no—
240C2×#108—531no—
241C2×#104—1061no—
242C2×C4×C8C2×D5311yescharacteristic
243#364—534no—
244#362—534no—
245#361—534no—
246C53C2×#221211yescharacteristic, complemented, Sylow
247C2×#41—1061no—
248C22×Q8—21no—
249C22×D4—21no—
250#135—532no—
251#135—532no—
252#135—1061no—
253#135—1061no—
254C2×Q16—21no—
255C2×Q16—21no—
256C2×Q16—21no—
257C2×Q16—21no—
258C2×Q16—21no—
259C2×Q16—531no—
260C2×Q16C22×D5312yes—
261C2×SD16—24no—
262C2×SD16—41no—
263C2×SD16—41no—
264C2×SD16—42no—
265C2×SD16—1061no—
266C2×D8—21no—
267C2×D8—21no—
268C2×D8—21no—
269C2×D8—21no—
270C2×D8—21no—
271C2×D8—531no—
272C2×D8C22×D5312yes—
273#131—1064no—
274#131—2122no—
275C2×M16—1061no—
276C2×M16—2121no—
277C22×C8—2121no—
278C22×C8C22⋊D5311yescharacteristic
279C22×C8C22×D5311yescharacteristic
280#124—24no—
281#124—42no—
282C2×C22⋊C4—42no—
283C2×C42C22⋊D5311yescharacteristic
284Q32—1064no—
285SD32—1064no—
286SD32—1064no—
287D16—1064no—
288M32—532no—
289M32—532no—
290M32—1061no—
291M32—1061no—
292C2×C16—1061no—
293C2×C16—1061no—
294#108—532no—
295#108—1061no—
296#104—1064no—
297C4×C8C22⋊D5312yes—
298C4×C8C22×D5312yes—
299C24—41no—
300#41—1062no—
301#41—1062no—
302#41—2121no—
303#41—2121no—
304C2×Q8—22no—
305C2×Q8—41no—
306C2×Q8—41no—
307C2×Q8—41no—
308C2×Q8—41no—
309C2×Q8—41no—
310C2×Q8—1061no—
311C2×D4—22no—
312C2×D4—41no—
313C2×D4—41no—
314C2×D4—41no—
315C2×D4—41no—
316C2×D4—41no—
317C2×D4—1061no—
318C2×D4—2121no—
319C22×C4—21no—
320C22×C4—41no—
321C22×C4—2121no—
322C22×C4C2×C22⋊D5311yescharacteristic
323Q16—22no—
324Q16—22no—
325Q16—41no—
326Q16—41no—
327Q16—532no—
328Q16—1061no—
329SD16—44no—
330SD16—44no—
331SD16—1062no—
332SD16—1062no—
333D8—22no—
334D8—22no—
335D8—41no—
336D8—41no—
337D8—532no—
338D8—1061no—
339M16—1062no—
340M16—2121no—
341M16—2124no—
342C2×C8—2121no—
343C2×C8—2124no—
344C2×C8#1171992211yescharacteristic
345C2×C8#1171992211yescharacteristic
346C2×C8#1171992212yes—
347C2×C8D4×D5311yescharacteristic
348C2×C8D4×D5311yescharacteristic
349C2×C8#1171994711yescharacteristic
350C2×C8#1171994711yescharacteristic
351C2×C8C2×C22⋊D5311yescharacteristic
352C2×C8C2×C22⋊D5312yes—
353C2×C8C23×D5311yescharacteristic
354C22⋊C4—48no—
355C42—21no—
356C42C2×C22⋊D5312yes—
357C16—1062no—
358C16—1062no—
359C23—42no—
360C23—84no—
361C23—81no—
362C23—81no—
363C23—2121no—
364C23#1313324311yescharacteristic
365Q8—42no—
366Q8—42no—
367Q8—81no—
368Q8—81no—
369Q8—1062no—
370Q8—2121no—
371D4—42no—
372D4—42no—
373D4—81no—
374D4—81no—
375D4—1062no—
376D4—2122no—
377D4—2122no—
378D4—2121no—
379C2×C4—22no—
380C2×C4—42no—
381C2×C4—42no—
382C2×C4—81no—
383C2×C4—81no—
384C2×C4—2121no—
385C2×C4—2122no—
386C2×C4—2122no—
387C2×C4#1313318811yescharacteristic
388C2×C4#1313318811yescharacteristic
389C2×C4C2×#1171992211yescharacteristic
390C2×C4C2×#1171992211yescharacteristic
391C2×C4#1313324311yescharacteristic
392C2×C4C22×C22⋊D5311yescharacteristic
393C8—2122no—
394C8#1313318811yescharacteristic
395C8#1313318811yescharacteristic
396C8C2×#1171992212yes—
397C8C2×#1171992212yes—
398C8C2×D4×D5311yescharacteristic
399C8C2×#1171994711yescharacteristic
400C22—22no—
401C22—84no—
402C22—82no—
403C22—162no—
404C22—162no—
405C22—4242no—
406C22—2122no—
407C22#1681081511yescharacteristic
408C22#1681081511yescharacteristic
409C22C2×#1313324311yescharacteristic
410C4—42no—
411C4—82no—
412C4—2122no—
413C4#1681081511yescharacteristic
414C4#1681081511yescharacteristic
415C4C2×#1313318811yescharacteristic
416C4C22×#1171992211yescharacteristic
417C2—21no—
418C2—21no—
419C2—162no—
420C2—4242no—
421C2#2027678112yesdirect product factor
422C2C2×#1681081511yescharacteristic
subgroupG/H
centerC22#16810815
commutator, cosocleC2×C8×C53C24
FrattiniC2×C8C23×D53
FittingC2×C53×#495C2
socleC22×C53#459
prime radicalC2×#16811059C2
hypercenterC2×#495D53
hypocenterC53C2×#2212
LCSG ▷ C2×C8×C53 ▷ C4×C53 ▷ C2×C53 ▷ C53
UCSC1 ◁ C22 ◁ C22×C4 ◁ C2×C4×C8 ◁ C2×#495

Complements

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

NH ≅ G/Ntype
1. C2×#16811064C2semidirect
3. C2×#16810741C2semidirect
4. #20276781C2direct (N × H)
5. C2×#16810444C2semidirect
6. C2×#16810439C2semidirect
7. C2×#16810436C2semidirect
8. C2×C53×#495C2semidirect
12. #16811064C22semidirect
22. #16810741C22semidirect
25. #16810444C22semidirect
26. #16810439C22semidirect
27. #16810436C22semidirect
33. C53×#495C22semidirect
208. C2×#495D53semidirect
217. C2×C53#2212semidirect
237. #495C2×D53semidirect
246. C53C2×#2212semidirect
421. C2#20276781direct (N × H)

Maximal subgroups

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

subgroupindex
C2×#168110642
C2×#168110592
C2×#168107412
#202767812
C2×#168104442
C2×#168104392
C2×#168104362
C2×C53×#4952
C2×#221253