Finite Group Databasebeta

← C2×#20276784 #27538464

Subgroups — C2×#20276784 #27538464

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×#16811058C211yesmaximal, characteristic
3C2×#16811007C211yesmaximal, characteristic, complemented
4#20276784C218yesmaximal, direct product factor
5C2×#16810444C211yesmaximal, characteristic, complemented
6C2×#16810438C211yesmaximal, characteristic, complemented
7C2×#16810435C211yesmaximal, characteristic, complemented
8C2×C53×#443C211yesmaximal, characteristic, complemented
9C22×Dic212C2211yescharacteristic
10C22×#11719913—21no—
11C22×D424C2211yescharacteristic
12C2×#13133205—21no—
13#16811063—22no—
14#16811063C2214yescomplemented
15#16811058—22no—
16#16811058C2214yes—
17C2×#13133163—21no—
18C2×#13133161—21no—
19C2×#13133160—21no—
20C2×#13133158—21no—
21C2×#13133157C2211yescharacteristic
22C2×#13133157C2211yescharacteristic
23#16811007—22no—
24#16811007C2214yescomplemented
25#16810444C2214yescomplemented
26#16810438C2214yescomplemented
27#16810435C2214yescomplemented
28C2×#135×C53C2211yescharacteristic
29C2×#131×C53—21no—
30C53×#443—22no—
31C53×#443C2214yescomplemented
32C2×#108×C53C2211yescharacteristic
33C2×#104×C53—21no—
34C2×C4×C8×C53C2211yescharacteristic
35C22×Dic106—21no—
36C22×D212—21no—
37C2×Dic212—21no—
38C2×Dic212—21no—
39C2×Dic212—21no—
40C2×Dic212—21no—
41C2×Dic212—21no—
42C2×Dic212C2312yes—
43C2×#11719913—24no—
44C2×#11719913—41no—
45C2×#11719913—41no—
46C2×#11719913—42no—
47C2×D424—21no—
48C2×D424—21no—
49C2×D424—21no—
50C2×D424—21no—
51C2×D424—21no—
52C2×D424C2312yes—
53#13133205—24no—
54#13133205—42no—
55C2×#11719921—42no—
56#13133163—24no—
57#13133161—24no—
58#13133160—24no—
59#13133158—24no—
60#13133157—21no—
61#13133157—21no—
62#13133157C2312yes—
63#13133157C2312yes—
64C2×#11719908—21no—
65C2×#11719908—21no—
66C2×#41×C53—21no—
67#135×C53—21no—
68#135×C53—21no—
69#135×C53D412yes—
70#135×C53C2312yes—
71C2×Q16×C53D411yescharacteristic
72C2×SD16×C53—21no—
73C2×D8×C53D411yescharacteristic
74#131×C53—24no—
75#131×C53—42no—
76C2×M16×C53—21no—
77C2×M16×C53—41no—
78C22×C8×C53—41no—
79C22×C8×C53D411yescharacteristic
80C22×C8×C53C2311yescharacteristic
81C2×C42×C53D411yescharacteristic
82#108×C53—21no—
83#108×C53C2312yes—
84#104×C53—24no—
85C4×C8×C53D412yes—
86C4×C8×C53C2312yes—
87C23×D53—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—41no—
95C22×C4×C53—21no—
96C22×C4×C53—41no—
97C22×C4×C53C2×D411yescharacteristic
98C22×Dic53—41no—
99C2×D212—22no—
100C2×D212—41no—
101C2×D212—41no—
102C2×D212—41no—
103C2×D212—41no—
104C2×D212—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×C53C2×D412yes—
117M16×C53—22no—
118M16×C53—41no—
119M16×C53—44no—
120C2×C8×C53—41no—
121C2×C8×C53—44no—
122C2×C8×C53D811yescharacteristic
123C2×C8×C53D811yescharacteristic
124C2×C8×C53D812yes—
125C2×C8×C53C2×D411yescharacteristic
126C2×C8×C53C2×D411yescharacteristic
127C2×C8×C53C2×D411yescharacteristic
128C2×C8×C53C2×D412yes—
129C2×C8×C53#4111yescharacteristic
130C2×C8×C53#4111yescharacteristic
131C2×C8×C53C2411yescharacteristic
132C42×C53—21no—
133C42×C53C2×D412yes—
134#11719921—48no—
135Dic212—22no—
136Dic212—22no—
137Dic212—41no—
138Dic212—41no—
139D424—22no—
140D424—22no—
141D424—41no—
142D424—41no—
143#11719913—44no—
144#11719913—44no—
145#11719908—22no—
146#11719908—22no—
147C23×C53—41no—
148C23×C53AGL(1,ℤ/8)11yescharacteristic
149C22×D53—42no—
150C22×D53—84no—
151C22×D53—81no—
152C22×D53—81no—
153Q8×C53—22no—
154Q8×C53—41no—
155D4×C53—22no—
156D4×C53—42no—
157D4×C53—42no—
158D4×C53—41no—
159C2×C4×C53—22no—
160C2×C4×C53—42no—
161C2×C4×C53—41no—
162C2×C4×C53—42no—
163C2×C4×C53—42no—
164C2×C4×C53#12111yescharacteristic
165C2×C4×C53#12111yescharacteristic
166C2×C4×C53C2×D811yescharacteristic
167C2×C4×C53C2×D811yescharacteristic
168C2×C4×C53AGL(1,ℤ/8)11yescharacteristic
169C2×C4×C53C22×D411yescharacteristic
170C2×Dic53—42no—
171C2×Dic53—81no—
172C2×Dic53—81no—
173D212—42no—
174D212—42no—
175D212—81no—
176D212—81no—
177Dic106—42no—
178Dic106—42no—
179Dic106—81no—
180Dic106—81no—
181C8×C53—42no—
182C8×C53#12111yescharacteristic
183C8×C53#12111yescharacteristic
184C8×C53C2×D812yes—
185C8×C53C2×D812yes—
186C8×C53C22×D411yescharacteristic
187C8×C53C2×#4111yescharacteristic
188C2×#2212—531nomaximal, Sylow
189C22×C53—22no—
190C22×C53—42no—
191C22×C53—82no—
192C22×C53#45911yescharacteristic
193C22×C53#45911yescharacteristic
194C22×C53C2×AGL(1,ℤ/8)11yescharacteristic
195C2×D53—84no—
196C2×D53—82no—
197C2×D53—162no—
198C2×D53—162no—
199C4×C53—42no—
200C4×C53—42no—
201C4×C53#45911yescharacteristic
202C4×C53#45911yescharacteristic
203C4×C53C2×#12111yescharacteristic
204C4×C53C22×D811yescharacteristic
205Dic53—82no—
206C2×#510—531no—
207C2×#509—531no—
208C2×#495—531no—
209C2×#443D5311yescharacteristic, complemented
210#2212—538no—
211C2×#364—531no—
212C2×#362—531no—
213C2×#361—531no—
214C2×C53—21no—
215C2×C53—21no—
216C2×C53—82no—
217C2×C53#221212yescomplemented
218C2×C53C2×#45911yescharacteristic
219D53—162no—
220C2×#135C2×D5311yescharacteristic
221C22×Q16—531no—
222C22×SD16—1061no—
223C22×D8—531no—
224C2×#131—21no—
225C2×#124—1061no—
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—534no—
237#495—1062no—
238#443—22no—
239#443C2×D5314yescomplemented
240C2×#108C2×D5311yescharacteristic
241C2×#104—21no—
242C2×C4×C8C2×D5311yescharacteristic
243#364—534no—
244#362—534no—
245#361—534no—
246C53C2×#221211yescharacteristic, complemented, Sylow
247C2×#41—21no—
248C22×Q8—1061no—
249C22×D4—1061no—
250#135—21no—
251#135—21no—
252#135D21212yes—
253#135C22×D5312yes—
254C2×Q16—532no—
255C2×Q16—1061no—
256C2×Q16—1061no—
257C2×Q16—1061no—
258C2×Q16—1061no—
259C2×Q16—1061no—
260C2×Q16D21211yescharacteristic
261C2×SD16—21no—
262C2×SD16—1064no—
263C2×SD16—2121no—
264C2×SD16—2121no—
265C2×SD16—2122no—
266C2×D8—532no—
267C2×D8—1061no—
268C2×D8—1061no—
269C2×D8—1061no—
270C2×D8—1061no—
271C2×D8—1061no—
272C2×D8D21211yescharacteristic
273#131—24no—
274#131—42no—
275C2×M16—21no—
276C2×M16—41no—
277C22×C8—41no—
278C22×C8C22⋊D5311yescharacteristic
279C22×C8C22×D5311yescharacteristic
280#124—1064no—
281#124—2122no—
282C2×C22⋊C4—2122no—
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—21no—
295#108C22×D5312yes—
296#104—24no—
297C4×C8C22⋊D5312yes—
298C4×C8C22×D5312yes—
299C24—2121no—
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—41no—
313C2×D4—1062no—
314C2×D4—2121no—
315C2×D4—2121no—
316C2×D4—2121no—
317C2×D4—2121no—
318C2×D4—2121no—
319C22×C4—21no—
320C22×C4—41no—
321C22×C4—2121no—
322C22×C4C2×C22⋊D5311yescharacteristic
323Q16—21no—
324Q16—1062no—
325Q16—1062no—
326Q16—2121no—
327Q16—2121no—
328Q16C2×D21212yes—
329SD16—22no—
330SD16—22no—
331SD16—2124no—
332SD16—2124no—
333D8—21no—
334D8—1062no—
335D8—1062no—
336D8—2121no—
337D8—2121no—
338D8C2×D21212yes—
339M16—22no—
340M16—41no—
341M16—44no—
342C2×C8—41no—
343C2×C8—44no—
344C2×C8#1171992211yescharacteristic
345C2×C8#1171992211yescharacteristic
346C2×C8#1171992212yes—
347C2×C8C2×D21211yescharacteristic
348C2×C8C2×D21211yescharacteristic
349C2×C8#1171994511yescharacteristic
350C2×C8#1171994511yescharacteristic
351C2×C8C2×C22⋊D5311yescharacteristic
352C2×C8C2×C22⋊D5312yes—
353C2×C8C23×D5311yescharacteristic
354C22⋊C4—2128no—
355C42—21no—
356C42C2×C22⋊D5312yes—
357C16—1062no—
358C16—1062no—
359C23—41no—
360C23—4244no—
361C23—4241no—
362C23—4241no—
363C23—2122no—
364C23#1313324211yescharacteristic
365Q8—22no—
366Q8—41no—
367Q8—4241no—
368Q8—4241no—
369Q8—2122no—
370Q8—2122no—
371D4—22no—
372D4—42no—
373D4—42no—
374D4—41no—
375D4—4241no—
376D4—4241no—
377D4—2122no—
378D4—2122no—
379C2×C4—22no—
380C2×C4—42no—
381C2×C4—41no—
382C2×C4—42no—
383C2×C4—42no—
384C2×C4—4241no—
385C2×C4—4241no—
386C2×C4—2122no—
387C2×C4#1313319111yescharacteristic
388C2×C4#1313319111yescharacteristic
389C2×C4C2×#1171992211yescharacteristic
390C2×C4C2×#1171992211yescharacteristic
391C2×C4#1313324211yescharacteristic
392C2×C4C22×C22⋊D5311yescharacteristic
393C8—42no—
394C8#1313319111yescharacteristic
395C8#1313319111yescharacteristic
396C8C2×#1171992212yes—
397C8C2×#1171992212yes—
398C8C22×D21211yescharacteristic
399C8C2×#1171994511yescharacteristic
400C22—22no—
401C22—42no—
402C22—82no—
403C22—8482no—
404C22—8482no—
405C22—4244no—
406C22—4242no—
407C22#1681081811yescharacteristic
408C22#1681081811yescharacteristic
409C22C2×#1313324211yescharacteristic
410C4—42no—
411C4—42no—
412C4—4242no—
413C4#1681081811yescharacteristic
414C4#1681081811yescharacteristic
415C4C2×#1313319111yescharacteristic
416C4C22×#1171992211yescharacteristic
417C2—21no—
418C2—21no—
419C2—82no—
420C2—8482no—
421C2#2027678412yesdirect product factor
422C2C2×#1681081811yescharacteristic
subgroupG/H
centerC22#16810818
commutator, cosocleC2×C8×C53C24
FrattiniC2×C8C23×D53
FittingC2×C53×#443C2
socleC22×C53#459
prime radicalC2×#16811058C2
hypercenterC2×#443D53
hypocenterC53C2×#2212
LCSG ▷ C2×C8×C53 ▷ C4×C53 ▷ C2×C53 ▷ C53
UCSC1 ◁ C22 ◁ C22×C4 ◁ C2×C4×C8 ◁ C2×#443

Complements

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

NH ≅ G/Ntype
1. C2×#16811063C2semidirect
3. C2×#16811007C2semidirect
4. #20276784C2direct (N × H)
5. C2×#16810444C2semidirect
6. C2×#16810438C2semidirect
7. C2×#16810435C2semidirect
8. C2×C53×#443C2semidirect
14. #16811063C22semidirect
24. #16811007C22semidirect
25. #16810444C22semidirect
26. #16810438C22semidirect
27. #16810435C22semidirect
31. C53×#443C22semidirect
209. C2×#443D53semidirect
217. C2×C53#2212semidirect
239. #443C2×D53semidirect
246. C53C2×#2212semidirect
421. C2#20276784direct (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×#168110582
C2×#168110072
#202767842
C2×#168104442
C2×#168104382
C2×#168104352
C2×C53×#4432
C2×#221253