Finite Group Databasebeta

← C2×#20276782 #27538462

Subgroups — C2×#20276782 #27538462

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×#16811063C211yesmaximal, characteristic, complemented
2C2×#16811006C211yesmaximal, characteristic, complemented
3C2×#16810740C211yesmaximal, characteristic, complemented
4#20276782C218yesmaximal, direct product factor
5C2×#16810443C211yesmaximal, characteristic, complemented
6C2×#16810440C211yesmaximal, characteristic, complemented
7C2×#16810436C211yesmaximal, characteristic, complemented
8C2×C53×#509C211yesmaximal, characteristic
9C22×Dic212C2211yescharacteristic
10C22×#11719923—21no—
11C2×#13133224—21no—
12C2×#13133204—21no—
13#16811063—22no—
14#16811063C2214yescomplemented
15C2×#13133163—21no—
16C2×#13133161—21no—
17C2×#13133157C2211yescharacteristic
18#16811006—22no—
19#16811006C2214yescomplemented
20#16810740—22no—
21#16810740C2214yescomplemented
22C2×#13133153C2211yescharacteristic
23C2×#13133144—21no—
24C2×C8×Dic53C2211yescharacteristic
25#16810443C2214yescomplemented
26#16810440C2214yescomplemented
27#16810436C2214yescomplemented
28C2×#135×C53C2211yescharacteristic
29C22×D8×C53C2211yescharacteristic
30C53×#509—22no—
31C53×#509C2214yes—
32C2×SD32×C53—21no—
33C2×D16×C53—21no—
34C2×M32×C53C2211yescharacteristic
35C22×Dic106—21no—
36C2×Dic212—21no—
37C2×Dic212—21no—
38C2×Dic212—21no—
39C2×Dic212—21no—
40C2×Dic212—21no—
41C2×Dic212C2312yes—
42C2×#11719923—24no—
43C2×#11719923—41no—
44C2×#11719923—41no—
45C2×#11719923—42no—
46#13133224—24no—
47#13133224—42no—
48C2×#11719917—21no—
49C2×#11719917—41no—
50C22×#88322—41no—
51C22×#88322D411yescharacteristic
52#13133204—24no—
53#13133204—42no—
54C2×C22⋊Dic53—42no—
55C2×C4×Dic53D411yescharacteristic
56#13133163—24no—
57#13133161—24no—
58#13133157—21no—
59#13133157C2312yes—
60C2×#11719908—21no—
61#13133153—21no—
62#13133153C2312yes—
63#13133144—24no—
64C8×Dic53D412yes—
65C8×Dic53C2312yes—
66C2×#41×C53—21no—
67C22×D4×C53—21no—
68#135×C53—21no—
69#135×C53—21no—
70#135×C53D412yes—
71#135×C53C2312yes—
72C2×Q16×C53D411yescharacteristic
73C2×SD16×C53—21no—
74C2×D8×C53—21no—
75C2×D8×C53—21no—
76C2×D8×C53—21no—
77C2×D8×C53—21no—
78C2×D8×C53—21no—
79C2×D8×C53D411yescharacteristic
80C2×D8×C53C2312yes—
81C22×C8×C53C2311yescharacteristic
82SD32×C53—24no—
83D16×C53—24no—
84M32×C53—21no—
85M32×C53C2312yes—
86C2×C16×C53—21no—
87C24×C53—41no—
88#41×C53—22no—
89#41×C53—22no—
90#41×C53—41no—
91#41×C53—41no—
92C2×Q8×C53—21no—
93C2×D4×C53—21no—
94C2×D4×C53—22no—
95C2×D4×C53—41no—
96C2×D4×C53—41no—
97C2×D4×C53—41no—
98C2×D4×C53—41no—
99C2×D4×C53—41no—
100C2×D4×C53—41no—
101C22×C4×C53—41no—
102C22×C4×C53C2×D411yescharacteristic
103C22×Dic53—21no—
104C22×Dic53—41no—
105C2×Dic106—22no—
106C2×Dic106—41no—
107C2×Dic106—41no—
108C2×Dic106—41no—
109C2×Dic106—41no—
110C2×Dic106—41no—
111Q16×C53—21no—
112Q16×C53C2×D412yes—
113SD16×C53—22no—
114SD16×C53—22no—
115D8×C53—21no—
116D8×C53—22no—
117D8×C53—22no—
118D8×C53—41no—
119D8×C53—41no—
120D8×C53C2×D412yes—
121C2×C8×C53C2×D411yescharacteristic
122C2×C8×C53C2×D411yescharacteristic
123C2×C8×C53C2×D411yescharacteristic
124C2×C8×C53#4111yescharacteristic
125C2×C8×C53#4111yescharacteristic
126C2×C8×C53C2411yescharacteristic
127C22⋊Dic53—48no—
128#11719923—44no—
129#11719923—44no—
130C4×Dic53—21no—
131C4×Dic53C2×D412yes—
132#11719917—22no—
133#11719917—41no—
134#11719917—44no—
135C2×#88322—41no—
136C2×#88322—44no—
137C2×#88322D811yescharacteristic
138C2×#88322D811yescharacteristic
139C2×#88322D812yes—
140C2×#88322C2×D412yes—
141Dic212—22no—
142Dic212—22no—
143Dic212—41no—
144Dic212—41no—
145C16×C53—22no—
146#11719908—22no—
147C23×C53—41no—
148C23×C53—42no—
149C23×C53—84no—
150C23×C53—81no—
151C23×C53—81no—
152C23×C53AGL(1,ℤ/8)11yescharacteristic
153Q8×C53—22no—
154Q8×C53—41no—
155D4×C53—22no—
156D4×C53—42no—
157D4×C53—42no—
158D4×C53—42no—
159D4×C53—42no—
160D4×C53—41no—
161D4×C53—81no—
162D4×C53—81no—
163C2×C4×C53—41no—
164C2×C4×C53—42no—
165C2×C4×C53—42no—
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
172C2×Dic53—22no—
173C2×Dic53—42no—
174C2×Dic53—42no—
175C2×Dic53—81no—
176C2×Dic53—81no—
177Dic106—42no—
178Dic106—42no—
179Dic106—81no—
180Dic106—81no—
181C8×C53#12111yescharacteristic
182C8×C53#12111yescharacteristic
183C8×C53C22×D411yescharacteristic
184C8×C53C2×#4111yescharacteristic
185#88322—42no—
186#88322C2×D812yes—
187#88322C2×D812yes—
188C2×#2212—531nomaximal, Sylow
189C22×C53—22no—
190C22×C53—42no—
191C22×C53—84no—
192C22×C53—82no—
193C22×C53—82no—
194C22×C53—162no—
195C22×C53—162no—
196C22×C53#45911yescharacteristic
197C22×C53#45911yescharacteristic
198C22×C53C2×AGL(1,ℤ/8)11yescharacteristic
199C4×C53—42no—
200C4×C53#45911yescharacteristic
201C4×C53#45911yescharacteristic
202C4×C53C2×#12111yescharacteristic
203C4×C53C22×D811yescharacteristic
204Dic53—42no—
205Dic53—82no—
206C2×#510—531no—
207C2×#509D5311yescharacteristic
208C2×#495—531no—
209C2×#443—531no—
210#2212—538no—
211C2×#364—531no—
212C2×#362—531no—
213C2×#361—531no—
214C2×C53—21no—
215C2×C53—21no—
216C2×C53—82no—
217C2×C53—162no—
218C2×C53#221212yescomplemented
219C2×C53C2×#45911yescharacteristic
220C2×#135C2×D5311yescharacteristic
221C22×Q16—531no—
222C22×SD16—1061no—
223C22×D8C2×D5311yescharacteristic
224C2×#131—1061no—
225C2×#124—1061no—
226#510—534no—
227#510—1062no—
228#509—22no—
229#509C2×D5314yes—
230C2×Q32—1061no—
231C2×SD32—21no—
232C2×SD32—1061no—
233C2×D16—21no—
234C2×M32—531no—
235C2×M32C2×D5311yescharacteristic
236#495—534no—
237#495—1062no—
238#443—534no—
239#443—1062no—
240C2×#108—531no—
241C2×#104—1061no—
242C2×C4×C8—531no—
243#364—534no—
244#362—534no—
245#361—534no—
246C53C2×#221211yescharacteristic, complemented, Sylow
247C2×#41—21no—
248C22×Q8—1061no—
249C22×D4—21no—
250#135—21no—
251#135—21no—
252#135C22⋊D5312yes—
253#135C22×D5312yes—
254C2×Q16—532no—
255C2×Q16—1061no—
256C2×Q16—1061no—
257C2×Q16—1061no—
258C2×Q16—1061no—
259C2×Q16—1061no—
260C2×Q16C22⋊D5311yescharacteristic
261C2×SD16—21no—
262C2×SD16—1064no—
263C2×SD16—2121no—
264C2×SD16—2121no—
265C2×SD16—2122no—
266C2×D8—21no—
267C2×D8—21no—
268C2×D8—21no—
269C2×D8—21no—
270C2×D8—21no—
271C2×D8C22⋊D5311yescharacteristic
272C2×D8C22×D5312yes—
273#131—1064no—
274#131—2122no—
275C2×M16—1061no—
276C2×M16—2121no—
277C22×C8—531no—
278C22×C8—2121no—
279C22×C8C22×D5311yescharacteristic
280#124—1064no—
281#124—2122no—
282C2×C22⋊C4—2122no—
283C2×C42—531no—
284Q32—1064no—
285SD32—24no—
286SD32—1064no—
287D16—24no—
288M32—21no—
289M32—532no—
290M32—1061no—
291M32C22×D5312yes—
292C2×C16—21no—
293C2×C16—1061no—
294#108—532no—
295#108—1061no—
296#104—1064no—
297C4×C8—532no—
298C4×C8—532no—
299C24—41no—
300#41—22no—
301#41—22no—
302#41—41no—
303#41—41no—
304C2×Q8—21no—
305C2×Q8—1062no—
306C2×Q8—2121no—
307C2×Q8—2121no—
308C2×Q8—2121no—
309C2×Q8—2121no—
310C2×Q8—2121no—
311C2×D4—21no—
312C2×D4—22no—
313C2×D4—41no—
314C2×D4—41no—
315C2×D4—41no—
316C2×D4—41no—
317C2×D4—41no—
318C2×D4—41no—
319C22×C4—41no—
320C22×C4—1061no—
321C22×C4—2121no—
322C22×C4D4×D5311yescharacteristic
323Q16—21no—
324Q16—1062no—
325Q16—1062no—
326Q16—2121no—
327Q16—2121no—
328Q16C2×C22⋊D5312yes—
329SD16—22no—
330SD16—22no—
331SD16—2124no—
332SD16—2124no—
333D8—21no—
334D8—22no—
335D8—22no—
336D8—41no—
337D8—41no—
338D8C2×C22⋊D5312yes—
339M16—1062no—
340M16—2124no—
341M16—2121no—
342C2×C8—531no—
343C2×C8—531no—
344C2×C8—532no—
345C2×C8—532no—
346C2×C8—2124no—
347C2×C8—2121no—
348C2×C8D4×D5311yescharacteristic
349C2×C8#1171994711yescharacteristic
350C2×C8#1171994711yescharacteristic
351C2×C8C2×C22⋊D5311yescharacteristic
352C2×C8C2×C22⋊D5311yescharacteristic
353C2×C8C23×D5311yescharacteristic
354C22⋊C4—2128no—
355C42—532no—
356C42—1061no—
357C16—22no—
358C16—1062no—
359C23—41no—
360C23—42no—
361C23—84no—
362C23—81no—
363C23—81no—
364C23#1313324111yescharacteristic
365Q8—22no—
366Q8—41no—
367Q8—4241no—
368Q8—4241no—
369Q8—2122no—
370Q8—2122no—
371D4—22no—
372D4—42no—
373D4—42no—
374D4—41no—
375D4—42no—
376D4—42no—
377D4—81no—
378D4—81no—
379C2×C4—41no—
380C2×C4—42no—
381C2×C4—42no—
382C2×C4—1062no—
383C2×C4—4241no—
384C2×C4—4241no—
385C2×C4—2122no—
386C2×C4—2122no—
387C2×C4#1313318911yescharacteristic
388C2×C4#1313318911yescharacteristic
389C2×C4D8×D5311yescharacteristic
390C2×C4D8×D5311yescharacteristic
391C2×C4#1313324111yescharacteristic
392C2×C4C2×D4×D5311yescharacteristic
393C8—532no—
394C8—532no—
395C8—2122no—
396C8#1313318911yescharacteristic
397C8#1313318911yescharacteristic
398C8C22×C22⋊D5311yescharacteristic
399C8C2×#1171994711yescharacteristic
400C22—22no—
401C22—42no—
402C22—84no—
403C22—82no—
404C22—82no—
405C22—162no—
406C22—162no—
407C22#1681081611yescharacteristic
408C22#1681081611yescharacteristic
409C22C2×#1313324111yescharacteristic
410C4—42no—
411C4—4242no—
412C4—2122no—
413C4#1681081611yescharacteristic
414C4#1681081611yescharacteristic
415C4C2×#1313318911yescharacteristic
416C4C2×D8×D5311yescharacteristic
417C2—21no—
418C2—21no—
419C2—82no—
420C2—162no—
421C2#2027678212yesdirect product factor
422C2C2×#1681081611yescharacteristic
subgroupG/H
centerC22#16810816
commutator, cosocleC2×C8×C53C24
FrattiniC2×C8C23×D53
Fitting, prime radicalC2×C53×#509C2
socleC22×C53#459
hypercenterC2×#509D53
hypocenterC53C2×#2212
LCSG ▷ C2×C8×C53 ▷ C4×C53 ▷ C2×C53 ▷ C53
UCSC1 ◁ C22 ◁ C22×C4 ◁ C22×C8 ◁ C2×#509

Complements

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

NH ≅ G/Ntype
1. C2×#16811063C2semidirect
2. C2×#16811006C2semidirect
3. C2×#16810740C2semidirect
4. #20276782C2direct (N × H)
5. C2×#16810443C2semidirect
6. C2×#16810440C2semidirect
7. C2×#16810436C2semidirect
14. #16811063C22semidirect
19. #16811006C22semidirect
21. #16810740C22semidirect
25. #16810443C22semidirect
26. #16810440C22semidirect
27. #16810436C22semidirect
218. C2×C53#2212semidirect
246. C53C2×#2212semidirect
421. C2#20276782direct (N × H)

Maximal subgroups

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

subgroupindex
C2×#168110632
C2×#168110062
C2×#168107402
#202767822
C2×#168104432
C2×#168104402
C2×#168104362
C2×C53×#5092
C2×#221253