Finite Group Databasebeta

← C2×#20276833 #27538647

Subgroups — C2×#20276833 #27538647

Order 13568.

678 conjugacy classes of subgroups.

The subgroup lattice has 418 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C22×#13133158C211yesmaximal, characteristic, complemented
2C2×#16811040C211yesmaximal, characteristic, complemented
3C2×#16810893C211yesmaximal, characteristic, complemented
4#20276833C218yesmaximal, direct product factor
5C2×#16810452C211yesmaximal, characteristic, complemented
6C2×#16810436C211yesmaximal, characteristic, complemented
7C2×#16810435C211yesmaximal, characteristic, complemented
8C2×C53×#472C211yesmaximal, characteristic, complemented
9C2×#13133242—21no—
10C22×D424C2211yescharacteristic
11C2×#13133158—21no—
12C2×#13133158—21no—
13C2×#13133158—22no—
14C2×#13133158C2214yescomplemented
15#16811040—22no—
16#16811040C2214yescomplemented
17C2×#13133157—21no—
18C2×#13133157C2211yescharacteristic
19C22×#11719908—21no—
20C22×#11719908C2211yescharacteristic
21#16810893—22no—
22#16810893C2214yescomplemented
23C2×#13133153C2211yescharacteristic
24C2×#13133133—21no—
25#16810452C2214yescomplemented
26#16810436C2214yescomplemented
27#16810435C2214yescomplemented
28C2×AGL(1,ℤ/8)×C53—21no—
29C22×D8×C53C2211yescharacteristic
30C2×#131×C53C2211yescharacteristic
31C53×#472—22no—
32C53×#472C2214yescomplemented
33C2×#108×C53C2211yescharacteristic
34C2×#100×C53—21no—
35C22×D212—21no—
36#13133242—24no—
37#13133242—42no—
38C2×#11719924—41no—
39C2×D424—21no—
40C2×D424—21no—
41C2×D424—21no—
42C2×D424—21no—
43C2×D424—21no—
44C2×D424C2312yes—
45C2×#11719922—41no—
46C2×#11719917—21no—
47#13133158—24no—
48#13133158—24no—
49#13133157—21no—
50#13133157—24no—
51#13133157C2312yes—
52C2×#11719908—21no—
53C2×#11719908—24no—
54C2×#11719908D411yescharacteristic
55C2×#11719908D411yescharacteristic
56C2×#11719908D412yes—
57C2×#11719908C2312yes—
58#13133153—21no—
59#13133153C2312yes—
60#13133133—24no—
61C2×#41×C53D411yescharacteristic
62C22×D4×C53—21no—
63AGL(1,ℤ/8)×C53—24no—
64AGL(1,ℤ/8)×C53—42no—
65C2×SD16×C53—41no—
66C2×D8×C53—21no—
67C2×D8×C53—21no—
68C2×D8×C53—21no—
69C2×D8×C53—21no—
70C2×D8×C53—21no—
71C2×D8×C53—41no—
72C2×D8×C53C2312yes—
73#131×C53—21no—
74#131×C53—21no—
75#131×C53D412yes—
76#131×C53C2312yes—
77C2×M16×C53—21no—
78C2×M16×C53—21no—
79C2×M16×C53D411yescharacteristic
80C22×C8×C53—21no—
81C22×C8×C53C2311yescharacteristic
82#108×C53—21no—
83#108×C53C2312yes—
84#100×C53—24no—
85C24×C53—41no—
86C23×D53—41no—
87#41×C53—21no—
88#41×C53—21no—
89#41×C53D812yes—
90#41×C53C2×D412yes—
91C2×Q8×C53D811yescharacteristic
92C2×D4×C53—21no—
93C2×D4×C53—22no—
94C2×D4×C53—41no—
95C2×D4×C53—41no—
96C2×D4×C53—41no—
97C2×D4×C53—41no—
98C2×D4×C53—41no—
99C2×D4×C53D811yescharacteristic
100C22×C4×C53—21no—
101C22×C4×C53C2×D411yescharacteristic
102C2×D212—22no—
103C2×D212—41no—
104C2×D212—41no—
105C2×D212—41no—
106C2×D212—41no—
107C2×D212—41no—
108SD16×C53—44no—
109D8×C53—22no—
110D8×C53—22no—
111D8×C53—44no—
112D8×C53—41no—
113D8×C53—41no—
114M16×C53—21no—
115M16×C53—22no—
116M16×C53—22no—
117M16×C53—22no—
118M16×C53—41no—
119M16×C53C2×D412yes—
120C2×C8×C53—21no—
121C2×C8×C53—22no—
122C2×C8×C53—22no—
123C2×C8×C53—41no—
124C2×C8×C53C2×D411yescharacteristic
125C2×C8×C53C2×D411yescharacteristic
126C2×C8×C53C2×D411yescharacteristic
127C2×C8×C53#4111yescharacteristic
128C2×C8×C53#4111yescharacteristic
129C2×C8×C53C2411yescharacteristic
130#11719924—44no—
131#11719922—44no—
132#11719917—22no—
133#11719917—41no—
134C2×#88322—41no—
135D424—22no—
136D424—22no—
137D424—41no—
138D424—41no—
139#11719908—22no—
140#11719908C2×D412yes—
141#11719908C2×D412yes—
142C23×C53—21no—
143C23×C53—42no—
144C23×C53—81no—
145C23×C53—81no—
146C23×C53—162no—
147C23×C53#13511yescharacteristic
148C22×D53—42no—
149C22×D53—81no—
150C22×D53—81no—
151C22×D53—162no—
152Q8×C53—21no—
153Q8×C53C2×D812yes—
154D4×C53—22no—
155D4×C53—22no—
156D4×C53—21no—
157D4×C53—42no—
158D4×C53—42no—
159D4×C53—81no—
160D4×C53—81no—
161D4×C53C2×D812yes—
162C2×C4×C53—21no—
163C2×C4×C53—22no—
164C2×C4×C53—22no—
165C2×C4×C53#12111yescharacteristic
166C2×C4×C53#12111yescharacteristic
167C2×C4×C53C2×D811yescharacteristic
168C2×C4×C53C2×D811yescharacteristic
169C2×C4×C53#13511yescharacteristic
170C2×C4×C53C22×D411yescharacteristic
171D212—42no—
172D212—42no—
173D212—81no—
174D212—81no—
175C8×C53—22no—
176C8×C53—42no—
177C8×C53#12111yescharacteristic
178C8×C53#12111yescharacteristic
179C8×C53C22×D411yescharacteristic
180C8×C53C2×#4111yescharacteristic
181#88322—42no—
182C2×#2221—531nomaximal, Sylow
183C22×C53—22no—
184C22×C53—22no—
185C22×C53—42no—
186C22×C53—82no—
187C22×C53—162no—
188C22×C53—162no—
189C22×C53—162no—
190C22×C53#46611yescharacteristic
191C22×C53#46611yescharacteristic
192C22×C53C2×#13511yescharacteristic
193C2×D53—82no—
194C2×D53—162no—
195C2×D53—162no—
196C2×D53—162no—
197C4×C53—22no—
198C4×C53#46611yescharacteristic
199C4×C53#46611yescharacteristic
200C4×C53C2×#12111yescharacteristic
201C4×C53C22×D811yescharacteristic
202C22×D16—531no—
203C2×#504—531no—
204C2×#472—531no—
205C2×#472D5311yescharacteristic, complemented
206#2221—538no—
207C2×#368—531no—
208C2×#361—531no—
209C2×#361—531no—
210C2×C53—21no—
211C2×C53—21no—
212C2×C53—42no—
213C2×C53—162no—
214C2×C53#222112yescomplemented
215C2×C53C2×#46611yescharacteristic
216D53—162no—
217C2×AGL(1,ℤ/8)—21no—
218C2×AGL(1,ℤ/8)—1061no—
219C22×D8—531no—
220C22×D8C2×D5311yescharacteristic
221C2×#131C2×D5311yescharacteristic
222C2×D16—534no—
223C2×D16—1061no—
224C2×D16—1061no—
225C2×D16—1062no—
226#504—534no—
227#504—1062no—
228C2×M32—531no—
229C2×M32—1061no—
230C22×C16—531no—
231C22×C16—1061no—
232#472—22no—
233#472—534no—
234#472—1062no—
235#472C2×D5314yescomplemented
236C2×#108—531no—
237C2×#108C2×D5311yescharacteristic
238C2×#100—21no—
239C2×#100—1061no—
240#368—534no—
241#361—534no—
242#361—534no—
243C53C2×#222111yescharacteristic, complemented, Sylow
244C2×#41C22⋊D5311yescharacteristic
245C22×D4—21no—
246C22×D4—1061no—
247AGL(1,ℤ/8)—24no—
248AGL(1,ℤ/8)—42no—
249AGL(1,ℤ/8)—1064no—
250AGL(1,ℤ/8)—2122no—
251C2×SD16—41no—
252C2×SD16—2121no—
253C2×D8—21no—
254C2×D8—21no—
255C2×D8—21no—
256C2×D8—21no—
257C2×D8—21no—
258C2×D8—41no—
259C2×D8—532no—
260C2×D8—1061no—
261C2×D8—1061no—
262C2×D8—1061no—
263C2×D8—1061no—
264C2×D8—1061no—
265C2×D8—2121no—
266C2×D8C22×D5312yes—
267#131—21no—
268#131—21no—
269#131C22⋊D5312yes—
270#131C22×D5312yes—
271C2×M16—21no—
272C2×M16—21no—
273C2×M16—1061no—
274C2×M16C22⋊D5311yescharacteristic
275C22×C8—21no—
276C22×C8C22×D5311yescharacteristic
277D16—1064no—
278D16—1064no—
279M32—532no—
280M32—1064no—
281M32—1061no—
282C2×C16—531no—
283C2×C16—531no—
284C2×C16—532no—
285C2×C16—532no—
286C2×C16—1064no—
287C2×C16—1061no—
288#108—21no—
289#108—532no—
290#108—1061no—
291#108C22×D5312yes—
292#100—24no—
293#100—1064no—
294C24—41no—
295C24—2121no—
296#41—21no—
297#41—21no—
298#41#1171992212yes—
299#41C2×C22⋊D5312yes—
300C2×Q8#1171992211yescharacteristic
301C2×D4—21no—
302C2×D4—22no—
303C2×D4—41no—
304C2×D4—41no—
305C2×D4—41no—
306C2×D4—41no—
307C2×D4—41no—
308C2×D4—1062no—
309C2×D4—2121no—
310C2×D4—2121no—
311C2×D4—2121no—
312C2×D4—2121no—
313C2×D4—2121no—
314C2×D4#1171992211yescharacteristic
315C22×C4—21no—
316C22×C4C2×C22⋊D5311yescharacteristic
317SD16—44no—
318SD16—2124no—
319D8—22no—
320D8—22no—
321D8—44no—
322D8—41no—
323D8—41no—
324D8—1062no—
325D8—1062no—
326D8—2124no—
327D8—2121no—
328D8—2121no—
329M16—21no—
330M16—22no—
331M16—22no—
332M16—22no—
333M16—41no—
334M16—1062no—
335M16—2121no—
336M16C2×C22⋊D5312yes—
337C2×C8—21no—
338C2×C8—22no—
339C2×C8—22no—
340C2×C8—41no—
341C2×C8—2121no—
342C2×C8D4×D5311yescharacteristic
343C2×C8D4×D5311yescharacteristic
344C2×C8#1171994711yescharacteristic
345C2×C8#1171994711yescharacteristic
346C2×C8C2×C22⋊D5311yescharacteristic
347C2×C8C23×D5311yescharacteristic
348C16—532no—
349C16—532no—
350C16—1062no—
351C23—21no—
352C23—42no—
353C23—81no—
354C23—81no—
355C23—162no—
356C23—8482no—
357C23—4241no—
358C23—4241no—
359C23—2122no—
360C23#1313323711yescharacteristic
361Q8—21no—
362Q8C2×#1171992212yes—
363D4—22no—
364D4—22no—
365D4—21no—
366D4—42no—
367D4—42no—
368D4—81no—
369D4—81no—
370D4—4241no—
371D4—4241no—
372D4—2122no—
373D4—2122no—
374D4C2×#1171992212yes—
375C2×C4—21no—
376C2×C4—22no—
377C2×C4—22no—
378C2×C4#1313318911yescharacteristic
379C2×C4#1313318911yescharacteristic
380C2×C4C2×#1171992211yescharacteristic
381C2×C4C2×#1171992211yescharacteristic
382C2×C4#1313323711yescharacteristic
383C2×C4C22×C22⋊D5311yescharacteristic
384C8—22no—
385C8—42no—
386C8—2122no—
387C8#1313318911yescharacteristic
388C8#1313318911yescharacteristic
389C8C2×D4×D5311yescharacteristic
390C8C2×#1171994711yescharacteristic
391C22—22no—
392C22—22no—
393C22—42no—
394C22—82no—
395C22—162no—
396C22—162no—
397C22—162no—
398C22—8482no—
399C22—8482no—
400C22—8482no—
401C22—4242no—
402C22#1681085711yescharacteristic
403C22#1681085711yescharacteristic
404C22C2×#1313323711yescharacteristic
405C4—22no—
406C4#1681085711yescharacteristic
407C4#1681085711yescharacteristic
408C4C2×#1313318911yescharacteristic
409C4C22×#1171992211yescharacteristic
410C2—21no—
411C2—21no—
412C2—42no—
413C2—162no—
414C2—8482no—
415C2#2027683312yesdirect product factor
416C2C2×#1681085711yescharacteristic
subgroupG/H
centerC22#16810857
commutator, cosocleC2×C8×C53C24
FrattiniC2×C8C23×D53
FittingC2×C53×#472C2
socleC22×C53#466
hypercenterC2×#472D53
hypocenterC53C2×#2221
LCSG ▷ C2×C8×C53 ▷ C4×C53 ▷ C2×C53 ▷ C53
UCSC1 ◁ C22 ◁ C22×C4 ◁ C2×#131 ◁ C2×#472

Complements

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

NH ≅ G/Ntype
1. C22×#13133158C2semidirect
2. C2×#16811040C2semidirect
3. C2×#16810893C2semidirect
4. #20276833C2direct (N × H)
5. C2×#16810452C2semidirect
6. C2×#16810436C2semidirect
7. C2×#16810435C2semidirect
8. C2×C53×#472C2semidirect
14. C2×#13133158C22semidirect
16. #16811040C22semidirect
22. #16810893C22semidirect
25. #16810452C22semidirect
26. #16810436C22semidirect
27. #16810435C22semidirect
32. C53×#472C22semidirect
205. C2×#472D53semidirect
214. C2×C53#2221semidirect
235. #472C2×D53semidirect
243. C53C2×#2221semidirect
415. C2#20276833direct (N × H)

Maximal subgroups

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

subgroupindex
C22×#131331582
C2×#168110402
C2×#168108932
#202768332
C2×#168104522
C2×#168104362
C2×#168104352
C2×C53×#4722
C2×#222153