Finite Group Databasebeta

← C2×#20276845 #27538661

Subgroups — C2×#20276845 #27538661

Order 13568.

658 conjugacy classes of subgroups.

The subgroup lattice has 408 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C22×#13133162C211yesmaximal, characteristic, complemented
2C2×#16810898C211yesmaximal, characteristic, complemented
3C2×#16810893C211yesmaximal, characteristic
4#20276845C218yesmaximal, direct product factor
5C2×#16810453C211yesmaximal, characteristic, complemented
6C2×#16810440C211yesmaximal, characteristic, complemented
7C2×#16810437C211yesmaximal, characteristic, complemented
8C2×C53×#504C211yesmaximal, characteristic, complemented
9C2×#13133247—21no—
10C2×#13133242—21no—
11C22×Dic212C2211yescharacteristic
12C22×D424C2211yescharacteristic
13C2×#13133162—21no—
14C2×#13133162—21no—
15C2×#13133162—22no—
16C2×#13133162C2214yescomplemented
17#16810898—22no—
18#16810898C2214yescomplemented
19#16810893—22no—
20#16810893C2214yes—
21C2×#13133153C2211yescharacteristic
22C2×#13133153C2211yescharacteristic
23C2×#13133135—21no—
24C2×#13133133—21no—
25#16810453C2214yescomplemented
26#16810440C2214yescomplemented
27#16810437C2214yescomplemented
28C2×#131×C53C2211yescharacteristic
29C53×#504—22no—
30C53×#504C2214yescomplemented
31C2×M32×C53—21no—
32C2×M32×C53C2211yescharacteristic
33C22×C16×C53—21no—
34C22×C16×C53C2211yescharacteristic
35C22×Dic106—21no—
36C22×D212—21no—
37#13133247—24no—
38#13133247—42no—
39#13133242—24no—
40#13133242—42no—
41C2×Dic212—21no—
42C2×Dic212—21no—
43C2×Dic212—21no—
44C2×Dic212—21no—
45C2×Dic212—21no—
46C2×Dic212C2312yes—
47C2×#11719925—41no—
48C2×#11719923—41no—
49C2×#11719924—41no—
50C2×D424—21no—
51C2×D424—21no—
52C2×D424—21no—
53C2×D424—21no—
54C2×D424—21no—
55C2×D424C2312yes—
56C2×#11719922—41no—
57C2×#11719917—21no—
58C2×#11719917—21no—
59#13133162—24no—
60#13133162—24no—
61#13133153—21no—
62#13133153—21no—
63#13133153C2312yes—
64#13133153C2312yes—
65#13133135—24no—
66#13133133—24no—
67C2×#41×C53D411yescharacteristic
68#131×C53—21no—
69#131×C53—21no—
70#131×C53D412yes—
71#131×C53C2312yes—
72C2×M16×C53—21no—
73C2×M16×C53D411yescharacteristic
74C22×C8×C53—21no—
75C22×C8×C53C2311yescharacteristic
76M32×C53—21no—
77M32×C53—24no—
78M32×C53C2312yes—
79C2×C16×C53—21no—
80C2×C16×C53—24no—
81C2×C16×C53D411yescharacteristic
82C2×C16×C53D411yescharacteristic
83C2×C16×C53D412yes—
84C2×C16×C53C2312yes—
85C23×D53—41no—
86#41×C53—21no—
87#41×C53—21no—
88#41×C53D812yes—
89#41×C53C2×D412yes—
90C2×Q8×C53D811yescharacteristic
91C2×D4×C53—21no—
92C2×D4×C53D811yescharacteristic
93C22×C4×C53—21no—
94C22×C4×C53C2×D411yescharacteristic
95C22×Dic53—41no—
96C2×D212—22no—
97C2×D212—41no—
98C2×D212—41no—
99C2×D212—41no—
100C2×D212—41no—
101C2×D212—41no—
102C2×Dic106—22no—
103C2×Dic106—41no—
104C2×Dic106—41no—
105C2×Dic106—41no—
106C2×Dic106—41no—
107C2×Dic106—41no—
108M16×C53—21no—
109M16×C53—22no—
110M16×C53—22no—
111M16×C53C2×D412yes—
112C2×C8×C53—21no—
113C2×C8×C53—22no—
114C2×C8×C53—22no—
115C2×C8×C53C2×D411yescharacteristic
116C2×C8×C53C2×D411yescharacteristic
117C2×C8×C53C2×D411yescharacteristic
118C2×C8×C53#4111yescharacteristic
119C2×C8×C53#4111yescharacteristic
120C2×C8×C53C2411yescharacteristic
121#11719925—44no—
122#11719924—44no—
123#11719923—44no—
124#11719922—44no—
125#11719917—22no—
126#11719917—22no—
127#11719917—41no—
128#11719917—41no—
129C2×#88322—41no—
130C2×#88322—41no—
131Dic212—22no—
132Dic212—22no—
133Dic212—41no—
134Dic212—41no—
135D424—22no—
136D424—22no—
137D424—41no—
138D424—41no—
139C16×C53—22no—
140C16×C53C2×D412yes—
141C16×C53C2×D412yes—
142C23×C53—21no—
143C23×C53#13511yescharacteristic
144C22×D53—42no—
145C22×D53—81no—
146C22×D53—81no—
147C22×D53—162no—
148Q8×C53—21no—
149Q8×C53C2×D812yes—
150D4×C53—22no—
151D4×C53—22no—
152D4×C53—21no—
153D4×C53C2×D812yes—
154C2×C4×C53—21no—
155C2×C4×C53—22no—
156C2×C4×C53—22no—
157C2×C4×C53#12111yescharacteristic
158C2×C4×C53#12111yescharacteristic
159C2×C4×C53C2×D811yescharacteristic
160C2×C4×C53C2×D811yescharacteristic
161C2×C4×C53#13511yescharacteristic
162C2×C4×C53C22×D411yescharacteristic
163C2×Dic53—42no—
164C2×Dic53—81no—
165C2×Dic53—81no—
166D212—42no—
167D212—42no—
168D212—81no—
169D212—81no—
170Dic106—42no—
171Dic106—42no—
172Dic106—81no—
173Dic106—81no—
174C8×C53—22no—
175C8×C53#12111yescharacteristic
176C8×C53#12111yescharacteristic
177C8×C53C22×D411yescharacteristic
178C8×C53C2×#4111yescharacteristic
179#88322—42no—
180#88322—42no—
181C2×#2223—531nomaximal, Sylow
182C22×C53—22no—
183C22×C53—22no—
184C22×C53—42no—
185C22×C53#46611yescharacteristic
186C22×C53#46611yescharacteristic
187C22×C53C2×#13511yescharacteristic
188C2×D53—82no—
189C2×D53—162no—
190C2×D53—162no—
191C2×D53—162no—
192C4×C53—22no—
193C4×C53#46611yescharacteristic
194C4×C53#46611yescharacteristic
195C4×C53C2×#12111yescharacteristic
196C4×C53C22×D811yescharacteristic
197Dic53—82no—
198C22×SD32—531no—
199C2×#504D5311yescharacteristic, complemented
200C2×#473—531no—
201C2×#472—531no—
202#2223—538no—
203C2×#368—531no—
204C2×#362—531no—
205C2×#361—531no—
206C2×C53—21no—
207C2×C53—21no—
208C2×C53—42no—
209C2×C53#222312yescomplemented
210C2×C53C2×#46611yescharacteristic
211D53—162no—
212C2×#137—1061no—
213C2×AGL(1,ℤ/8)—1061no—
214C22×Q16—531no—
215C22×D8—531no—
216C2×#131C2×D5311yescharacteristic
217C2×SD32—534no—
218C2×SD32—1061no—
219C2×SD32—1061no—
220C2×SD32—1062no—
221#504—22no—
222#504C2×D5314yescomplemented
223C2×M32—21no—
224C2×M32C2×D5311yescharacteristic
225C22×C16—21no—
226C22×C16C2×D5311yescharacteristic
227#473—534no—
228#473—1062no—
229#472—534no—
230#472—1062no—
231C2×#108—531no—
232C2×#108—531no—
233C2×#101—1061no—
234C2×#100—1061no—
235#368—534no—
236#362—534no—
237#361—534no—
238C53C2×#222311yescharacteristic, complemented, Sylow
239C2×#41D21211yescharacteristic
240C22×Q8—1061no—
241C22×D4—1061no—
242#137—1064no—
243#137—2122no—
244AGL(1,ℤ/8)—1064no—
245AGL(1,ℤ/8)—2122no—
246C2×Q16—532no—
247C2×Q16—1061no—
248C2×Q16—1061no—
249C2×Q16—1061no—
250C2×Q16—1061no—
251C2×Q16—1061no—
252C2×Q16—2121no—
253C2×SD16—2121no—
254C2×SD16—2121no—
255C2×D8—532no—
256C2×D8—1061no—
257C2×D8—1061no—
258C2×D8—1061no—
259C2×D8—1061no—
260C2×D8—1061no—
261C2×D8—2121no—
262#131—21no—
263#131—21no—
264#131D21212yes—
265#131C22×D5312yes—
266C2×M16—21no—
267C2×M16—1061no—
268C2×M16—1061no—
269C2×M16D21211yescharacteristic
270C22×C8—21no—
271C22×C8C22×D5311yescharacteristic
272SD32—1064no—
273SD32—1064no—
274M32—21no—
275M32—24no—
276M32C22×D5312yes—
277C2×C16—21no—
278C2×C16—24no—
279C2×C16C22⋊D5311yescharacteristic
280C2×C16C22⋊D5311yescharacteristic
281C2×C16C22⋊D5312yes—
282C2×C16C22×D5312yes—
283#108—532no—
284#108—532no—
285#108—1061no—
286#108—1061no—
287#101—1064no—
288#100—1064no—
289C24—2121no—
290#41—21no—
291#41—21no—
292#41D42412yes—
293#41C2×D21212yes—
294C2×Q8—1062no—
295C2×Q8—2121no—
296C2×Q8—2121no—
297C2×Q8—2121no—
298C2×Q8—2121no—
299C2×Q8—2121no—
300C2×Q8D42411yescharacteristic
301C2×D4—21no—
302C2×D4—1062no—
303C2×D4—2121no—
304C2×D4—2121no—
305C2×D4—2121no—
306C2×D4—2121no—
307C2×D4—2121no—
308C2×D4D42411yescharacteristic
309C22×C4—21no—
310C22×C4—2121no—
311C22×C4C2×D21211yescharacteristic
312Q16—1062no—
313Q16—1062no—
314Q16—2124no—
315Q16—2121no—
316Q16—2121no—
317SD16—2124no—
318SD16—2124no—
319D8—1062no—
320D8—1062no—
321D8—2124no—
322D8—2121no—
323D8—2121no—
324M16—21no—
325M16—22no—
326M16—22no—
327M16—1062no—
328M16—1062no—
329M16—2121no—
330M16—2121no—
331M16C2×D21212yes—
332C2×C8—21no—
333C2×C8—22no—
334C2×C8—22no—
335C2×C8—2121no—
336C2×C8—2121no—
337C2×C8C2×D21211yescharacteristic
338C2×C8#1171994511yescharacteristic
339C2×C8#1171994511yescharacteristic
340C2×C8C2×C22⋊D5311yescharacteristic
341C2×C8C2×C22⋊D5311yescharacteristic
342C2×C8C23×D5311yescharacteristic
343C16—22no—
344C16C2×C22⋊D5312yes—
345C16C2×C22⋊D5312yes—
346C23—21no—
347C23—8482no—
348C23—4241no—
349C23—4241no—
350C23—2122no—
351C23#1313323911yescharacteristic
352Q8—21no—
353Q8—4241no—
354Q8—4241no—
355Q8—2122no—
356Q8—2122no—
357Q8C2×D42412yes—
358D4—22no—
359D4—22no—
360D4—21no—
361D4—4241no—
362D4—4241no—
363D4—2122no—
364D4—2122no—
365D4C2×D42412yes—
366C2×C4—21no—
367C2×C4—22no—
368C2×C4—22no—
369C2×C4—4241no—
370C2×C4—4241no—
371C2×C4—2122no—
372C2×C4#1313319111yescharacteristic
373C2×C4#1313319111yescharacteristic
374C2×C4C2×D42411yescharacteristic
375C2×C4C2×D42411yescharacteristic
376C2×C4#1313323911yescharacteristic
377C2×C4C22×D21211yescharacteristic
378C8—22no—
379C8—2122no—
380C8—2122no—
381C8#1313319111yescharacteristic
382C8#1313319111yescharacteristic
383C8C22×C22⋊D5311yescharacteristic
384C8C2×#1171994511yescharacteristic
385C22—22no—
386C22—22no—
387C22—42no—
388C22—8482no—
389C22—8482no—
390C22—8482no—
391C22—4242no—
392C22#1681085911yescharacteristic
393C22#1681085911yescharacteristic
394C22C2×#1313323911yescharacteristic
395C4—22no—
396C4—4242no—
397C4#1681085911yescharacteristic
398C4#1681085911yescharacteristic
399C4C2×#1313319111yescharacteristic
400C4C22×D42411yescharacteristic
401C2—21no—
402C2—21no—
403C2—42no—
404C2—8482no—
405C2#2027684512yesdirect product factor
406C2C2×#1681085911yescharacteristic
subgroupG/H
centerC22#16810859
commutator, cosocleC2×C8×C53C24
FrattiniC2×C8C23×D53
FittingC2×C53×#504C2
socleC22×C53#466
prime radicalC2×#16810893C2
hypercenterC2×#504D53
hypocenterC53C2×#2223
LCSG ▷ C2×C8×C53 ▷ C4×C53 ▷ C2×C53 ▷ C53
UCSC1 ◁ C22 ◁ C22×C4 ◁ C2×#131 ◁ C2×#504

Complements

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

NH ≅ G/Ntype
1. C22×#13133162C2semidirect
2. C2×#16810898C2semidirect
4. #20276845C2direct (N × H)
5. C2×#16810453C2semidirect
6. C2×#16810440C2semidirect
7. C2×#16810437C2semidirect
8. C2×C53×#504C2semidirect
16. C2×#13133162C22semidirect
18. #16810898C22semidirect
25. #16810453C22semidirect
26. #16810440C22semidirect
27. #16810437C22semidirect
30. C53×#504C22semidirect
199. C2×#504D53semidirect
209. C2×C53#2223semidirect
222. #504C2×D53semidirect
238. C53C2×#2223semidirect
405. C2#20276845direct (N × H)

Maximal subgroups

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

subgroupindex
C22×#131331622
C2×#168108982
C2×#168108932
#202768452
C2×#168104532
C2×#168104402
C2×#168104372
C2×C53×#5042
C2×#222353