Finite Group Databasebeta

← C2×#20276609 #27537528

Subgroups — C2×#20276609 #27537528

Order 13568.

646 conjugacy classes of subgroups.

The subgroup lattice has 360 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#16811064C211yesmaximal, characteristic
2C2×#16810744C211yesmaximal, characteristic
3C2×#16810730C211yesmaximal, characteristic, complemented
4#20276609C218yesmaximal, direct product factor
5C2×#16810447C211yesmaximal, characteristic, complemented
6C2×#16810434C211yesmaximal, characteristic
7C2×#16810400C211yesmaximal, characteristic, complemented
8C2×C53×#362C211yesmaximal, characteristic, complemented
9C2×#13133239C2211yescharacteristic
10C2×#13133225—21no—
11C2×Q8×Dic53—21no—
12#16811064C414yescomplemented
13#16811064C2214yes—
14C2×#13133163C412yescomplemented
15C2×#13133160C412yescomplemented
16C2×#13133157C2211yescharacteristic
17#16810744—22no—
18#16810744C2214yes—
19#16810730—22no—
20#16810730C2214yescomplemented
21C2×#13133149C2211yescharacteristic
22C2×#13133143—21no—
23C2×#13133141—21no—
24C2×#13133127C2211yescharacteristic
25#16810447C2214yescomplemented
26#16810434C2214yes—
27#16810400C2214yescomplemented
28C22×Q16×C53C2211yescharacteristic
29C2×M32×C53C2211yescharacteristic
30C2×#108×C53C2211yescharacteristic
31C53×#362C2214yescomplemented
32C2×#11719945—21no—
33#13133239—21no—
34#13133239—21no—
35#13133239C2×C412yes—
36#13133239C2312yes—
37C2×Dic212C2×C411yescharacteristic
38C2×#11719913—21no—
39C2×D424C2×C411yescharacteristic
40#13133225—24no—
41#13133225—42no—
42C2×#11719912—41no—
43C22×#88322D411yescharacteristic
44C2×C8×D53—41no—
45Q8×Dic53—24no—
46Q8×Dic53—42no—
47C2×C4⋊Dic53—21no—
48C2×C4⋊Dic53—41no—
49C2×C4×Dic53—41no—
50C2×C4×Dic53D411yescharacteristic
51#13133163C2×C418yescomplemented
52#13133160C2×C418yescomplemented
53#13133157C2×C412yes—
54#13133157C2312yes—
55C2×#11719908C2×C412yes—
56#13133149—21no—
57#13133149C2312yes—
58#13133143—24no—
59#13133141—24no—
60#13133127D412yes—
61#13133127C2312yes—
62C22×Q8×C53—21no—
63C2×Q16×C53—21no—
64C2×Q16×C53—21no—
65C2×Q16×C53—21no—
66C2×Q16×C53C2×C412yes—
67C2×Q16×C53C2×C412yes—
68C2×Q16×C53C2312yes—
69C2×M16×C53—21no—
70C22×C8×C53C2311yescharacteristic
71M32×C53—21no—
72M32×C53C2312yes—
73C2×C16×C53—21no—
74#108×C53—21no—
75#108×C53C2312yes—
76C2×Q8×C53—22no—
77C2×Q8×C53—22no—
78C2×Q8×C53—22no—
79C2×Q8×C53—41no—
80C2×Q8×C53—41no—
81C2×Q8×C53—41no—
82C22×C4×C53—41no—
83C22×C4×C53C2×D411yescharacteristic
84C2×C22⋊D53—41no—
85C22×Dic53—21no—
86C22×Dic53—41no—
87#11719945—22no—
88#11719945—22no—
89#11719945—41no—
90#11719945—41no—
91C2×D212—21no—
92C2×C4×D53—41no—
93C2×Dic106—21no—
94Q16×C53—22no—
95Q16×C53—22no—
96Q16×C53—22no—
97Q16×C53C22×C414yes—
98M16×C53—22no—
99M16×C53—41no—
100C2×C8×C53—41no—
101C2×C8×C53C22×C411yescharacteristic
102C2×C8×C53C22×C411yescharacteristic
103C2×C8×C53C2×D411yescharacteristic
104C2×C8×C53#4111yescharacteristic
105C2×C8×C53#4111yescharacteristic
106C2×C8×C53C2411yescharacteristic
107C4⋊Dic53—22no—
108C4⋊Dic53—44no—
109C4⋊Dic53—41no—
110C4×Dic53—21no—
111C4×Dic53—44no—
112C4×Dic53C2×D412yes—
113C2×#88322—21no—
114C2×#88322—21no—
115C2×#88322C2×D412yes—
116Dic212—21no—
117Dic212C22×C412yes—
118D424—21no—
119D424C22×C412yes—
120#11719913—22no—
121#11719913—22no—
122#11719912—44no—
123C8×D53—44no—
124C16×C53—22no—
125#11719908C22×C414yes—
126C23×C53AGL(1,ℤ/8)11yescharacteristic
127C22×D53—41no—
128Q8×C53—24no—
129Q8×C53—42no—
130Q8×C53—42no—
131Q8×C53—42no—
132C2×C4×C53—42no—
133C2×C4×C53—42no—
134C2×C4×C53—42no—
135C2×C4×C53C4×D411yescharacteristic
136C2×C4×C53C4×D411yescharacteristic
137C2×C4×C53AGL(1,ℤ/8)11yescharacteristic
138C2×C4×C53AGL(1,ℤ/8)11yescharacteristic
139C2×C4×C53AGL(1,ℤ/8)11yescharacteristic
140C2×C4×C53C22×D411yescharacteristic
141C22⋊D53—42no—
142C22⋊D53—42no—
143C2×Dic53—22no—
144C2×Dic53—42no—
145C2×Dic53—41no—
146C2×Dic53—42no—
147C2×Dic53—82no—
148D212—22no—
149D212—41no—
150C4×D53—42no—
151C4×D53—42no—
152Dic106—22no—
153Dic106—41no—
154C8×C53—42no—
155C8×C53C4×D411yescharacteristic
156C8×C53C4×D411yescharacteristic
157C8×C53C23×C411yescharacteristic
158C8×C53C2×#4111yescharacteristic
159#88322—22no—
160C2×#2180—531nomaximal, Sylow
161C22×C53—22no—
162C22×C53#44211yescharacteristic
163C22×C53#44211yescharacteristic
164C22×C53C2×AGL(1,ℤ/8)11yescharacteristic
165C2×D53—42no—
166C2×D53—82no—
167C4×C53—44no—
168C4×C53#44211yescharacteristic
169C4×C53#44211yescharacteristic
170C4×C53C2×C4×D411yescharacteristic
171C4×C53C2×AGL(1,ℤ/8)11yescharacteristic
172Dic53—42no—
173Dic53—42no—
174Dic53—82no—
175C2×#510—531no—
176C2×#444—531no—
177C2×#441—531no—
178#2180—538no—
179C2×#365—531no—
180C2×#362D5311yescharacteristic, complemented
181C2×#360—531no—
182C2×#347—531no—
183C2×C53—21no—
184C2×C53—21no—
185C2×C53#218012yescomplemented
186C2×C53C2×#44211yescharacteristic
187D53—82no—
188C2×#135—531no—
189C22×Q16C2×D5311yescharacteristic
190C2×#131—1061no—
191C2×C4×Q8—1061no—
192#510—534no—
193#510—534no—
194C2×Q32—532no—
195C2×SD32—532no—
196C2×M32—531no—
197C2×M32C2×D5311yescharacteristic
198#444—534no—
199#444—1062no—
200#441—534no—
201#441—1062no—
202C2×#108C2×D5311yescharacteristic
203C2×#106—531no—
204C2×#104—1061no—
205C2×#103—1061no—
206C2×#97—531no—
207#365—534no—
208#362C2×D5314yescomplemented
209#360—534no—
210#347—534no—
211C53C2×#218011yescharacteristic, complemented, Sylow
212C2×#41—1061no—
213C22×Q8—21no—
214#135—532no—
215#135—532no—
216#135—1061no—
217#135—1061no—
218C2×Q16—21no—
219C2×Q16—21no—
220C2×Q16—21no—
221C2×Q16—531no—
222C2×Q16C4×D5312yes—
223C2×Q16C4×D5312yes—
224C2×Q16C22×D5312yes—
225C2×SD16—1061no—
226C2×D8—531no—
227#131—1064no—
228#131—2122no—
229C2×M16—21no—
230C2×M16—2121no—
231C22×C8—531no—
232C22×C8—2121no—
233C22×C8C22×D5311yescharacteristic
234C4×Q8—1064no—
235C4×Q8—2122no—
236C2×C4⋊C4—1061no—
237C2×C4⋊C4—2121no—
238C2×C42—531no—
239C2×C42—2121no—
240Q32—538no—
241SD32—538no—
242M32—21no—
243M32—532no—
244M32—532no—
245M32C22×D5312yes—
246C2×C16—21no—
247C2×C16—532no—
248#108—21no—
249#108C22×D5312yes—
250#106—532no—
251#106—1061no—
252#104—1064no—
253#103—1064no—
254#97—532no—
255#97—532no—
256#41—1062no—
257#41—1062no—
258#41—2121no—
259#41—2121no—
260C2×Q8—22no—
261C2×Q8—22no—
262C2×Q8—22no—
263C2×Q8—41no—
264C2×Q8—41no—
265C2×Q8—41no—
266C2×Q8—1061no—
267C2×D4—1061no—
268C2×D4—2121no—
269C22×C4—41no—
270C22×C4—1061no—
271C22×C4—2121no—
272C22×C4—2121no—
273C22×C4D4×D5311yescharacteristic
274Q16—22no—
275Q16—22no—
276Q16—22no—
277Q16—532no—
278Q16—1061no—
279Q16C2×C4×D5314yes—
280SD16—1062no—
281SD16—1062no—
282D8—532no—
283D8—1061no—
284M16—22no—
285M16—41no—
286M16—2124no—
287C2×C8—41no—
288C2×C8—532no—
289C2×C8—1061no—
290C2×C8—1061no—
291C2×C8—2124no—
292C2×C8C2×C4×D5311yescharacteristic
293C2×C8C2×C4×D5311yescharacteristic
294C2×C8D4×D5311yescharacteristic
295C2×C8#1171994711yescharacteristic
296C2×C8#1171994711yescharacteristic
297C2×C8C23×D5311yescharacteristic
298C4⋊C4—1062no—
299C4⋊C4—2124no—
300C4⋊C4—2121no—
301C42—532no—
302C42—1061no—
303C42—2124no—
304C16—22no—
305C16—534no—
306C23—2121no—
307C23#1313324411yescharacteristic
308Q8—24no—
309Q8—42no—
310Q8—42no—
311Q8—42no—
312Q8—1062no—
313Q8—2121no—
314D4—1062no—
315D4—2122no—
316D4—2122no—
317D4—2121no—
318C2×C4—42no—
319C2×C4—42no—
320C2×C4—42no—
321C2×C4—1062no—
322C2×C4—4242no—
323C2×C4—2122no—
324C2×C4—2122no—
325C2×C4—2121no—
326C2×C4—2122no—
327C2×C4—2122no—
328C2×C4#1313317711yescharacteristic
329C2×C4#1313317711yescharacteristic
330C2×C4#1313324111yescharacteristic
331C2×C4#1313324111yescharacteristic
332C2×C4#1313324411yescharacteristic
333C2×C4C2×D4×D5311yescharacteristic
334C8—42no—
335C8—1062no—
336C8#1313317711yescharacteristic
337C8#1313317711yescharacteristic
338C8C22×C4×D5311yescharacteristic
339C8C2×#1171994711yescharacteristic
340C22—22no—
341C22—4242no—
342C22—2122no—
343C22#1681073211yescharacteristic
344C22#1681073211yescharacteristic
345C22C2×#1313324411yescharacteristic
346C4—44no—
347C4—4242no—
348C4—2122no—
349C4—2122no—
350C4#1681073211yescharacteristic
351C4#1681073211yescharacteristic
352C4C2×#1313317711yescharacteristic
353C4C2×#1313324111yescharacteristic
354C2—21no—
355C2—21no—
356C2—4242no—
357C2#2027660912yesdirect product factor
358C2C2×#1681073211yescharacteristic
subgroupG/H
centerC22#16810732
commutatorC8×C53C23×C4
FrattiniC2×C8C23×D53
FittingC2×C53×#362C2
socleC22×C53#442
prime radicalC2×#13133239C22
cosocleC2×C8×C53C24
hypercenterC2×#362D53
hypocenterC53C2×#2180
LCSG ▷ 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
3. C2×#16810730C2semidirect
4. #20276609C2direct (N × H)
5. C2×#16810447C2semidirect
7. C2×#16810400C2semidirect
8. C2×C53×#362C2semidirect
12. #16811064C4semidirect
14. C2×#13133163C4semidirect
15. C2×#13133160C4semidirect
20. #16810730C22semidirect
25. #16810447C22semidirect
27. #16810400C22semidirect
31. C53×#362C22semidirect
51. #13133163C2×C4semidirect
52. #13133160C2×C4semidirect
180. C2×#362D53semidirect
185. C2×C53#2180semidirect
208. #362C2×D53semidirect
211. C53C2×#2180semidirect
357. C2#20276609direct (N × H)

Maximal subgroups

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

subgroupindex
C2×#168110642
C2×#168107442
C2×#168107302
#202766092
C2×#168104472
C2×#168104342
C2×#168104002
C2×C53×#3622
C2×#218053