Finite Group Databasebeta

← C2×#20276789 #27538470

Subgroups — C2×#20276789 #27538470

Order 13568.

790 conjugacy classes of subgroups.

The subgroup lattice has 438 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#16811058C211yesmaximal, characteristic, complemented
2C2×#16810998C211yesmaximal, characteristic, complemented
3C2×#16810740C211yesmaximal, characteristic, complemented
4#20276789C218yesmaximal, direct product factor
5C2×#16810443C211yesmaximal, characteristic, complemented
6C2×#16810437C211yesmaximal, characteristic, complemented
7C2×#16810436C211yesmaximal, characteristic, complemented
8C2×C53×#509C211yesmaximal, characteristic, complemented
9C22×D424C2211yescharacteristic
10C22×#11719922—21no—
11C2×#13133224—21no—
12C2×#13133213—21no—
13#16811058—22no—
14#16811058C2214yescomplemented
15C2×#13133160—21no—
16C2×#13133158—21no—
17C2×#13133157C2211yescharacteristic
18#16810998—22no—
19#16810998C2214yescomplemented
20#16810740—22no—
21#16810740C2214yescomplemented
22C2×#13133153C2211yescharacteristic
23C2×#13133144—21no—
24C2×C8×Dic53C2211yescharacteristic
25#16810443C2214yescomplemented
26#16810437C2214yescomplemented
27#16810436C2214yescomplemented
28C2×#135×C53C2211yescharacteristic
29C22×D8×C53C2211yescharacteristic
30C53×#509—22no—
31C53×#509C2214yescomplemented
32C2×SD32×C53—21no—
33C2×D16×C53—21no—
34C2×M32×C53C2211yescharacteristic
35C22×D212—21no—
36C22×C22⋊D53—42no—
37C2×D424—21no—
38C2×D424—21no—
39C2×D424—21no—
40C2×D424—21no—
41C2×D424—21no—
42C2×D424C2312yes—
43C2×#11719922—24no—
44C2×#11719922—41no—
45C2×#11719922—41no—
46C2×#11719922—42no—
47#13133224—24no—
48#13133224—42no—
49C2×#11719917—21no—
50C2×#11719917—41no—
51C22×#88322—41no—
52C22×#88322D411yescharacteristic
53#13133213—24no—
54#13133213—42no—
55C2×C4×Dic53D411yescharacteristic
56#13133160—24no—
57#13133158—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×C53D412yescomplemented
71#135×C53C2312yes—
72C2×Q16×C53D411yescharacteristic, complemented
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, complemented
80C2×D8×C53C2312yes—
81C22×C8×C53C2311yescharacteristic
82SD32×C53—24no—
83D16×C53—24no—
84M32×C53—21no—
85M32×C53C2312yes—
86C2×C16×C53—21no—
87C24×C53—41no—
88C23×D53—41no—
89#41×C53—22no—
90#41×C53—22no—
91#41×C53—41no—
92#41×C53—41no—
93C2×Q8×C53—21no—
94C2×D4×C53—21no—
95C2×D4×C53—22no—
96C2×D4×C53—41no—
97C2×D4×C53—41no—
98C2×D4×C53—41no—
99C2×D4×C53—41no—
100C2×D4×C53—41no—
101C2×D4×C53—41no—
102C22×C4×C53—41no—
103C22×C4×C53C2×D411yescharacteristic
104C2×C22⋊D53—48no—
105C2×C22⋊D53—88no—
106C22×Dic53—21no—
107C2×D212—22no—
108C2×D212—41no—
109C2×D212—41no—
110C2×D212—41no—
111C2×D212—41no—
112C2×D212—41no—
113Q16×C53—21no—
114Q16×C53C2×D412yescomplemented
115SD16×C53—22no—
116SD16×C53—22no—
117D8×C53—21no—
118D8×C53—22no—
119D8×C53—22no—
120D8×C53—41no—
121D8×C53—41no—
122D8×C53C2×D412yescomplemented
123C2×C8×C53C2×D411yescharacteristic
124C2×C8×C53C2×D411yescharacteristic
125C2×C8×C53C2×D411yescharacteristic
126C2×C8×C53#4111yescharacteristic
127C2×C8×C53#4111yescharacteristic
128C2×C8×C53C2411yescharacteristic
129#11719922—44no—
130#11719922—44no—
131C4×Dic53—21no—
132C4×Dic53C2×D412yes—
133#11719917—22no—
134#11719917—41no—
135#11719917—44no—
136C2×#88322—41no—
137C2×#88322—44no—
138C2×#88322D811yescharacteristic
139C2×#88322D811yescharacteristic
140C2×#88322D812yes—
141C2×#88322C2×D412yes—
142D424—22no—
143D424—22no—
144D424—41no—
145D424—41no—
146C16×C53—22no—
147#11719908—22no—
148C23×C53—41no—
149C23×C53—42no—
150C23×C53—84no—
151C23×C53—81no—
152C23×C53—81no—
153C23×C53AGL(1,ℤ/8)11yescharacteristic
154C22×D53—42no—
155C22×D53—84no—
156C22×D53—81no—
157C22×D53—81no—
158Q8×C53—22no—
159Q8×C53—41no—
160D4×C53—22no—
161D4×C53—42no—
162D4×C53—42no—
163D4×C53—42no—
164D4×C53—42no—
165D4×C53—41no—
166D4×C53—81no—
167D4×C53—81no—
168C2×C4×C53—41no—
169C2×C4×C53—42no—
170C2×C4×C53—42no—
171C2×C4×C53#12111yescharacteristic
172C2×C4×C53#12111yescharacteristic
173C2×C4×C53C2×D811yescharacteristic
174C2×C4×C53C2×D811yescharacteristic
175C2×C4×C53AGL(1,ℤ/8)11yescharacteristic
176C2×C4×C53C22×D411yescharacteristic
177C22⋊D53—816no—
178C2×Dic53—24no—
179C2×Dic53—22no—
180D212—42no—
181D212—42no—
182D212—81no—
183D212—81no—
184C8×C53#12111yescharacteristic
185C8×C53#12111yescharacteristic
186C8×C53C22×D411yescharacteristic
187C8×C53C2×#4111yescharacteristic
188#88322—42no—
189#88322C2×D812yes—
190#88322C2×D812yes—
191C2×#2213—531nomaximal, Sylow
192C22×C53—22no—
193C22×C53—42no—
194C22×C53—84no—
195C22×C53—82no—
196C22×C53—82no—
197C22×C53—162no—
198C22×C53—162no—
199C22×C53#45911yescharacteristic
200C22×C53#45911yescharacteristic
201C22×C53C2×AGL(1,ℤ/8)11yescharacteristic
202C2×D53—84no—
203C2×D53—82no—
204C2×D53—162no—
205C2×D53—162no—
206C4×C53—42no—
207C4×C53#45911yescharacteristic
208C4×C53#45911yescharacteristic
209C4×C53C2×#12111yescharacteristic
210C4×C53C22×D811yescharacteristic
211Dic53—24no—
212C2×#509—531no—
213C2×#509D5311yescharacteristic, complemented
214C2×#493—531no—
215C2×#443—531no—
216#2213—538no—
217C2×#364—531no—
218C2×#361—531no—
219C2×#361—531no—
220C2×C53—21no—
221C2×C53—21no—
222C2×C53—82no—
223C2×C53—162no—
224C2×C53#221312yescomplemented
225C2×C53C2×#45911yescharacteristic
226D53—162no—
227C2×#135C2×D5311yescharacteristic
228C22×D8—531no—
229C22×D8—1061no—
230C22×D8C2×D5311yescharacteristic
231C2×#131—1061no—
232C2×#127—1061no—
233#509—22no—
234#509—534no—
235#509—1062no—
236#509C2×D5314yescomplemented
237C2×SD32—21no—
238C2×SD32—1061no—
239C2×D16—21no—
240C2×D16—1061no—
241C2×M32—531no—
242C2×M32C2×D5311yescharacteristic
243#493—534no—
244#493—1062no—
245#443—534no—
246#443—1062no—
247C2×#108—531no—
248C2×#104—1061no—
249C2×C4×C8—531no—
250#364—534no—
251#361—534no—
252#361—534no—
253C53C2×#221311yescharacteristic, complemented, Sylow
254C2×#41—21no—
255C22×D4—21no—
256C22×D4—1061no—
257C22×D4—2122no—
258#135—21no—
259#135—21no—
260#135C22⋊D5312yescomplemented
261#135C22×D5312yes—
262C2×Q16C22⋊D5311yescharacteristic, complemented
263C2×SD16—21no—
264C2×D8—21no—
265C2×D8—21no—
266C2×D8—21no—
267C2×D8—21no—
268C2×D8—21no—
269C2×D8—532no—
270C2×D8—1064no—
271C2×D8—1061no—
272C2×D8—1061no—
273C2×D8—1061no—
274C2×D8—1061no—
275C2×D8—1061no—
276C2×D8—2121no—
277C2×D8—2121no—
278C2×D8—2122no—
279C2×D8C22⋊D5311yescharacteristic, complemented
280C2×D8C22×D5312yes—
281#131—1064no—
282#131—2122no—
283C2×M16—1061no—
284C2×M16—2121no—
285C22×C8—531no—
286C22×C8—2121no—
287C22×C8C22×D5311yescharacteristic
288#127—1064no—
289#127—2122no—
290C2×C42—531no—
291SD32—24no—
292SD32—1064no—
293D16—24no—
294D16—1064no—
295M32—21no—
296M32—532no—
297M32—1061no—
298M32C22×D5312yes—
299C2×C16—21no—
300C2×C16—1061no—
301#108—532no—
302#108—1061no—
303#104—1064no—
304C4×C8—532no—
305C4×C8—532no—
306C24—41no—
307C24—2121no—
308#41—22no—
309#41—22no—
310#41—41no—
311#41—41no—
312C2×Q8—21no—
313C2×D4—21no—
314C2×D4—22no—
315C2×D4—41no—
316C2×D4—41no—
317C2×D4—41no—
318C2×D4—41no—
319C2×D4—41no—
320C2×D4—41no—
321C2×D4—1062no—
322C2×D4—4248no—
323C2×D4—2128no—
324C2×D4—2121no—
325C2×D4—2121no—
326C2×D4—2121no—
327C2×D4—2121no—
328C2×D4—2121no—
329C22×C4—41no—
330C22×C4—1061no—
331C22×C4D4×D5311yescharacteristic
332Q16—21no—
333Q16C2×C22⋊D5312yescomplemented
334SD16—22no—
335SD16—22no—
336D8—21no—
337D8—22no—
338D8—22no—
339D8—41no—
340D8—41no—
341D8—1062no—
342D8—1062no—
343D8—2124no—
344D8—2124no—
345D8—2121no—
346D8—2121no—
347D8C2×C22⋊D5312yescomplemented
348M16—1062no—
349M16—2124no—
350M16—2121no—
351C2×C8—531no—
352C2×C8—531no—
353C2×C8—532no—
354C2×C8—532no—
355C2×C8—2124no—
356C2×C8—2121no—
357C2×C8D4×D5311yescharacteristic
358C2×C8#1171994711yescharacteristic
359C2×C8#1171994711yescharacteristic
360C2×C8C2×C22⋊D5311yescharacteristic
361C2×C8C2×C22⋊D5311yescharacteristic
362C2×C8C23×D5311yescharacteristic
363C42—532no—
364C42—1061no—
365C16—22no—
366C16—1062no—
367C23—41no—
368C23—42no—
369C23—84no—
370C23—81no—
371C23—81no—
372C23—4244no—
373C23—4241no—
374C23—4241no—
375C23—2122no—
376C23#1313324111yescharacteristic
377Q8—22no—
378Q8—41no—
379D4—22no—
380D4—42no—
381D4—42no—
382D4—41no—
383D4—42no—
384D4—42no—
385D4—81no—
386D4—81no—
387D4—42416no—
388D4—4241no—
389D4—4241no—
390D4—2122no—
391D4—2122no—
392C2×C4—41no—
393C2×C4—42no—
394C2×C4—42no—
395C2×C4—1064no—
396C2×C4—1062no—
397C2×C4#1313318911yescharacteristic
398C2×C4#1313318911yescharacteristic
399C2×C4D8×D5311yescharacteristic
400C2×C4D8×D5311yescharacteristic
401C2×C4#1313324111yescharacteristic
402C2×C4C2×D4×D5311yescharacteristic
403C8—532no—
404C8—532no—
405C8—2122no—
406C8#1313318911yescharacteristic
407C8#1313318911yescharacteristic
408C8C22×C22⋊D5311yescharacteristic
409C8C2×#1171994711yescharacteristic
410C22—22no—
411C22—42no—
412C22—84no—
413C22—82no—
414C22—82no—
415C22—162no—
416C22—162no—
417C22—8482no—
418C22—8482no—
419C22—4244no—
420C22—4242no—
421C22#1681081611yescharacteristic
422C22#1681081611yescharacteristic
423C22C2×#1313324111yescharacteristic
424C4—42no—
425C4—1064no—
426C4#1681081611yescharacteristic
427C4#1681081611yescharacteristic
428C4C2×#1313318911yescharacteristic
429C4C2×D8×D5311yescharacteristic
430C2—21no—
431C2—21no—
432C2—82no—
433C2—162no—
434C2—8482no—
435C2#2027678912yesdirect product factor
436C2C2×#1681081611yescharacteristic
subgroupG/H
centerC22#16810816
commutator, cosocleC2×C8×C53C24
FrattiniC2×C8C23×D53
FittingC2×C53×#509C2
socleC22×C53#459
hypercenterC2×#509D53
hypocenterC53C2×#2213
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×#16811058C2semidirect
2. C2×#16810998C2semidirect
3. C2×#16810740C2semidirect
4. #20276789C2direct (N × H)
5. C2×#16810443C2semidirect
6. C2×#16810437C2semidirect
7. C2×#16810436C2semidirect
8. C2×C53×#509C2semidirect
14. #16811058C22semidirect
19. #16810998C22semidirect
21. #16810740C22semidirect
25. #16810443C22semidirect
26. #16810437C22semidirect
27. #16810436C22semidirect
31. C53×#509C22semidirect
70. #135×C53D4semidirect
72. C2×Q16×C53D4semidirect
79. C2×D8×C53D4semidirect
114. Q16×C53C2×D4semidirect
122. D8×C53C2×D4semidirect
213. C2×#509D53semidirect
224. C2×C53#2213semidirect
236. #509C2×D53semidirect
253. C53C2×#2213semidirect
260. #135C22⋊D53semidirect
262. C2×Q16C22⋊D53semidirect
279. C2×D8C22⋊D53semidirect
333. Q16C2×C22⋊D53semidirect
347. D8C2×C22⋊D53semidirect
435. C2#20276789direct (N × H)

Maximal subgroups

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

subgroupindex
C2×#168110582
C2×#168109982
C2×#168107402
#202767892
C2×#168104432
C2×#168104372
C2×#168104362
C2×C53×#5092
C2×#221353