Finite Group Databasebeta

← C2×#20276848 #27538664

Subgroups — C2×#20276848 #27538664

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×#13133160C211yesmaximal, characteristic, complemented
2C2×#16811041C211yesmaximal, characteristic, complemented
3C2×#16810895C211yesmaximal, characteristic, complemented
4C2×#16810894C211yesmaximal, characteristic
5#20276848C218yesmaximal, direct product factor
6C2×#16810452C211yesmaximal, characteristic, complemented
7C2×#16810437C211yesmaximal, characteristic, complemented
8C2×C53×#362C211yesmaximal, characteristic, complemented
9C2×#13133248—21no—
10C2×#13133245—21no—
11C22×D424C2211yescharacteristic
12C2×#13133223C2211yescharacteristic
13C2×#13133160—21no—
14C2×#13133160—21no—
15C2×#13133160—22no—
16C2×#13133160C2214yescomplemented
17#16811041—22no—
18#16811041C2214yescomplemented
19C2×#13133156—21no—
20C22×#11719908C2211yescharacteristic
21C2×C16×D53—21no—
22#16810895—22no—
23#16810895C2214yescomplemented
24#16810894—22no—
25#16810894C2214yes—
26C2×#13133153C2211yescharacteristic
27C2×#13133136—21no—
28C2×#13133133—21no—
29#16810452C2214yescomplemented
30#16810437C2214yescomplemented
31C22×Q16×C53C2211yescharacteristic
32C2×M32×C53C2211yescharacteristic
33C2×#108×C53C2211yescharacteristic
34C53×#362C2214yescomplemented
35C2×#11719945D411yescharacteristic
36C22×D212—21no—
37#13133248—24no—
38#13133248—42no—
39#13133245—24no—
40#13133245—42no—
41C2×#11719925—41no—
42C2×#11719913—41no—
43C2×#11719924—41no—
44C2×D424—21no—
45C2×D424—21no—
46C2×D424—21no—
47C2×D424—21no—
48C2×D424—21no—
49C2×D424—41no—
50C2×D424C2312yes—
51#13133223—21no—
52#13133223—21no—
53#13133223D412yes—
54#13133223C2312yes—
55C2×#11719917—21no—
56C2×#11719917D411yescharacteristic
57C2×#11719912—21no—
58C2×C8×D53—21no—
59#13133160—24no—
60#13133160—24no—
61#13133156—24no—
62C2×#11719908D411yescharacteristic
63C2×#11719908D411yescharacteristic
64C2×#11719908D412yes—
65C2×#11719908C2312yes—
66C16×D53—24no—
67#13133153—21no—
68#13133153C2312yes—
69#13133136—24no—
70#13133133—24no—
71C22×Q8×C53—21no—
72C2×Q16×C53—21no—
73C2×Q16×C53—21no—
74C2×Q16×C53—21no—
75C2×Q16×C53—21no—
76C2×Q16×C53—21no—
77C2×Q16×C53C2312yes—
78C2×M16×C53—21no—
79C22×C8×C53C2311yescharacteristic
80M32×C53—21no—
81M32×C53C2312yes—
82C2×C16×C53—21no—
83#108×C53—21no—
84#108×C53C2312yes—
85C23×D53—41no—
86C2×Q8×C53—22no—
87C2×Q8×C53—41no—
88C2×Q8×C53—41no—
89C2×Q8×C53—41no—
90C2×Q8×C53—41no—
91C2×Q8×C53—41no—
92C22×C4×C53—41no—
93C22×C4×C53C2×D411yescharacteristic
94C2×C22⋊D53—21no—
95#11719945—21no—
96#11719945—21no—
97#11719945D812yes—
98#11719945C2×D412yes—
99C2×D212—22no—
100C2×D212—41no—
101C2×D212—41no—
102C2×D212—41no—
103C2×D212—41no—
104C2×D212—41no—
105C2×D212D811yescharacteristic
106C2×C4×D53—21no—
107C2×Dic106D811yescharacteristic
108Q16×C53—22no—
109Q16×C53—22no—
110Q16×C53—41no—
111Q16×C53—41no—
112M16×C53—22no—
113M16×C53—41no—
114C2×C8×C53—41no—
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#11719917—21no—
124#11719917—22no—
125#11719917—41no—
126#11719917C2×D412yes—
127C2×#88322—21no—
128C2×#88322—41no—
129D424—22no—
130D424—22no—
131D424—44no—
132D424—41no—
133D424—41no—
134#11719913—44no—
135#11719912—22no—
136#11719912—22no—
137C8×D53—22no—
138C8×D53—22no—
139C16×C53—22no—
140#11719908C2×D412yes—
141#11719908C2×D412yes—
142C23×C53#13511yescharacteristic
143C22×D53—21no—
144C22×D53—42no—
145C22×D53—81no—
146C22×D53—81no—
147C22×D53—162no—
148Q8×C53—42no—
149Q8×C53—42no—
150Q8×C53—81no—
151Q8×C53—81no—
152C2×C4×C53—42no—
153C2×C4×C53—81no—
154C2×C4×C53—81no—
155C2×C4×C53#12111yescharacteristic
156C2×C4×C53#12111yescharacteristic
157C2×C4×C53C2×D811yescharacteristic
158C2×C4×C53C2×D811yescharacteristic
159C2×C4×C53#13511yescharacteristic
160C2×C4×C53C22×D411yescharacteristic
161C22⋊D53—22no—
162C22⋊D53—22no—
163C2×Dic53—21no—
164D212—21no—
165D212—42no—
166D212—42no—
167D212—81no—
168D212—81no—
169D212C2×D812yes—
170C4×D53—22no—
171C4×D53—22no—
172Dic106—21no—
173Dic106C2×D812yes—
174C8×C53—42no—
175C8×C53#12111yescharacteristic
176C8×C53#12111yescharacteristic
177C8×C53C22×D411yescharacteristic
178C8×C53C2×#4111yescharacteristic
179#88322—22no—
180#88322—42no—
181C2×#2223—531nomaximal, Sylow
182C22×C53—22no—
183C22×C53#46611yescharacteristic
184C22×C53#46611yescharacteristic
185C22×C53C2×#13511yescharacteristic
186C2×D53—22no—
187C2×D53—42no—
188C2×D53—82no—
189C2×D53—162no—
190C2×D53—162no—
191C2×D53—162no—
192C4×C53—82no—
193C4×C53#46611yescharacteristic
194C4×C53#46611yescharacteristic
195C4×C53C2×#12111yescharacteristic
196C4×C53C22×D811yescharacteristic
197Dic53—22no—
198C22×SD32—531no—
199C2×#504—531no—
200C2×#473—531no—
201C2×#472—531no—
202#2223—538no—
203C2×#368—531no—
204C2×#362D5311yescharacteristic, complemented
205C2×#361—531no—
206C2×C53—21no—
207C2×C53—21no—
208C2×C53#222312yescomplemented
209C2×C53C2×#46611yescharacteristic
210D53—42no—
211D53—162no—
212C2×#137—1061no—
213C2×AGL(1,ℤ/8)—1061no—
214C22×Q16C2×D5311yescharacteristic
215C22×D8—531no—
216C2×#131—531no—
217C2×SD32—534no—
218C2×SD32—1061no—
219C2×SD32—1061no—
220C2×SD32—1062no—
221#504—534no—
222#504—1062no—
223C2×M32—1061no—
224C2×M32C2×D5311yescharacteristic
225C22×C16—531no—
226C22×C16—1061no—
227#473—534no—
228#473—1062no—
229#472—534no—
230#472—1062no—
231C2×#108—531no—
232C2×#108C2×D5311yescharacteristic
233C2×#101—1061no—
234C2×#100—1061no—
235#368—534no—
236#362C2×D5314yescomplemented
237#361—534no—
238C53C2×#222311yescharacteristic, complemented, Sylow
239C2×#41—531no—
240C22×Q8—21no—
241C22×D4—1061no—
242#137—1064no—
243#137—2122no—
244AGL(1,ℤ/8)—1064no—
245AGL(1,ℤ/8)—2122no—
246C2×Q16—21no—
247C2×Q16—21no—
248C2×Q16—21no—
249C2×Q16—21no—
250C2×Q16—21no—
251C2×Q16—2121no—
252C2×Q16C22×D5312yes—
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—532no—
263#131—532no—
264#131—1061no—
265#131—1061no—
266C2×M16—21no—
267C2×M16—531no—
268C2×M16—1061no—
269C2×M16—1061no—
270C22×C8—1061no—
271C22×C8C22×D5311yescharacteristic
272SD32—1064no—
273SD32—1064no—
274M32—21no—
275M32—1064no—
276M32C22×D5312yes—
277C2×C16—21no—
278C2×C16—531no—
279C2×C16—531no—
280C2×C16—532no—
281C2×C16—532no—
282C2×C16—1064no—
283#108—21no—
284#108—532no—
285#108—1061no—
286#108C22×D5312yes—
287#101—1064no—
288#100—1064no—
289C24—2121no—
290#41—532no—
291#41—532no—
292#41—1061no—
293#41—1061no—
294C2×Q8—22no—
295C2×Q8—41no—
296C2×Q8—41no—
297C2×Q8—41no—
298C2×Q8—41no—
299C2×Q8—41no—
300C2×Q8—531no—
301C2×D4—531no—
302C2×D4—1061no—
303C2×D4—1062no—
304C2×D4—2121no—
305C2×D4—2121no—
306C2×D4—2121no—
307C2×D4—2121no—
308C2×D4—2121no—
309C22×C4—41no—
310C22×C4—1061no—
311C22×C4D4×D5311yescharacteristic
312Q16—22no—
313Q16—22no—
314Q16—41no—
315Q16—41no—
316Q16—2124no—
317SD16—2124no—
318SD16—2124no—
319D8—1062no—
320D8—1062no—
321D8—2124no—
322D8—2121no—
323D8—2121no—
324M16—22no—
325M16—41no—
326M16—532no—
327M16—1061no—
328M16—1062no—
329M16—1062no—
330M16—1062no—
331M16—2121no—
332C2×C8—41no—
333C2×C8—1061no—
334C2×C8—1062no—
335C2×C8—1062no—
336C2×C8—2121no—
337C2×C8#1171994511yescharacteristic
338C2×C8#1171994511yescharacteristic
339C2×C8D4×D5311yescharacteristic
340C2×C8D4×D5311yescharacteristic
341C2×C8D4×D5311yescharacteristic
342C2×C8C23×D5311yescharacteristic
343C16—22no—
344C16—532no—
345C16—532no—
346C23—8482no—
347C23—1061no—
348C23—4241no—
349C23—4241no—
350C23—2122no—
351C23#1313323811yescharacteristic
352Q8—42no—
353Q8—42no—
354Q8—81no—
355Q8—81no—
356Q8—532no—
357Q8—1061no—
358D4—532no—
359D4—1062no—
360D4—1062no—
361D4—1061no—
362D4—4241no—
363D4—4241no—
364D4—2122no—
365D4—2122no—
366C2×C4—42no—
367C2×C4—81no—
368C2×C4—81no—
369C2×C4—1061no—
370C2×C4—1062no—
371C2×C4—1062no—
372C2×C4#1313319011yescharacteristic
373C2×C4#1313319011yescharacteristic
374C2×C4D8×D5311yescharacteristic
375C2×C4D8×D5311yescharacteristic
376C2×C4#1313323811yescharacteristic
377C2×C4C2×D4×D5311yescharacteristic
378C8—42no—
379C8—1062no—
380C8—2122no—
381C8#1313319011yescharacteristic
382C8#1313319011yescharacteristic
383C8C2×D4×D5311yescharacteristic
384C8C2×#1171994511yescharacteristic
385C22—22no—
386C22—8482no—
387C22—8482no—
388C22—8482no—
389C22—1062no—
390C22—4242no—
391C22—2122no—
392C22#1681085811yescharacteristic
393C22#1681085811yescharacteristic
394C22C2×#1313323811yescharacteristic
395C4—82no—
396C4—1062no—
397C4#1681085811yescharacteristic
398C4#1681085811yescharacteristic
399C4C2×#1313319011yescharacteristic
400C4C2×D8×D5311yescharacteristic
401C2—21no—
402C2—21no—
403C2—8482no—
404C2—2122no—
405C2#2027684812yesdirect product factor
406C2C2×#1681085811yescharacteristic
subgroupG/H
centerC22#16810858
commutator, cosocleC2×C8×C53C24
FrattiniC2×C8C23×D53
FittingC2×C53×#362C2
socleC22×C53#466
prime radicalC2×#16810894C2
hypercenterC2×#362D53
hypocenterC53C2×#2223
LCSG ▷ C2×C8×C53 ▷ C4×C53 ▷ C2×C53 ▷ C53
UCSC1 ◁ C22 ◁ C22×C4 ◁ C22×C8 ◁ C2×#362

Complements

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

NH ≅ G/Ntype
1. C22×#13133160C2semidirect
2. C2×#16811041C2semidirect
3. C2×#16810895C2semidirect
5. #20276848C2direct (N × H)
6. C2×#16810452C2semidirect
7. C2×#16810437C2semidirect
8. C2×C53×#362C2semidirect
16. C2×#13133160C22semidirect
18. #16811041C22semidirect
23. #16810895C22semidirect
29. #16810452C22semidirect
30. #16810437C22semidirect
34. C53×#362C22semidirect
204. C2×#362D53semidirect
208. C2×C53#2223semidirect
236. #362C2×D53semidirect
238. C53C2×#2223semidirect
405. C2#20276848direct (N × H)

Maximal subgroups

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

subgroupindex
C22×#131331602
C2×#168110412
C2×#168108952
C2×#168108942
#202768482
C2×#168104522
C2×#168104372
C2×C53×#3622
C2×#222353