Finite Group Databasebeta

← C2×#20276603 #27537522

Subgroups — C2×#20276603 #27537522

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×#16811063C211yesmaximal, characteristic
2C2×#16810745C211yesmaximal, characteristic
3C2×#16810729C211yesmaximal, characteristic, complemented
4#20276603C218yesmaximal, direct product factor
5C2×#16810445C211yesmaximal, characteristic, complemented
6C2×#16810440C211yesmaximal, characteristic, complemented
7C2×#16810400C211yesmaximal, characteristic, complemented
8C2×C53×#360C211yesmaximal, characteristic
9C22×Dic212C2211yescharacteristic
10C2×#13133224—21no—
11C2×#13133182—21no—
12#16811063C414yescomplemented
13#16811063C2214yes—
14C2×#13133163C412yescomplemented
15C2×#13133161C412yescomplemented
16C2×#13133157C2211yescharacteristic
17#16810745—22no—
18#16810745C2214yes—
19#16810729—22no—
20#16810729C2214yescomplemented
21C2×#13133153C2211yescharacteristic
22C2×#13133144—21no—
23C2×#13133140—21no—
24C2×#13133127C2211yescharacteristic
25#16810445C2214yescomplemented
26#16810440C2214yescomplemented
27#16810400C2214yescomplemented
28C2×#135×C53C2211yescharacteristic
29C2×M32×C53C2211yescharacteristic
30C2×#106×C53C2211yescharacteristic
31C53×#360C2214yes—
32C22×Dic106—21no—
33C2×Dic212—21no—
34C2×Dic212—21no—
35C2×Dic212—21no—
36C2×Dic212C2×C412yes—
37C2×Dic212C2×C412yes—
38C2×Dic212C2312yes—
39#13133224—24no—
40#13133224—42no—
41C2×#11719917—21no—
42C2×#11719917—41no—
43C22×#88322—41no—
44C22×#88322D411yescharacteristic
45#13133182—24no—
46#13133182—42no—
47C2×#11719919—41no—
48C2×C4×Dic53—41no—
49C2×C4×Dic53D411yescharacteristic
50#13133163C2×C418yescomplemented
51#13133161C2×C418yescomplemented
52#13133157C2×C412yes—
53#13133157C2312yes—
54C2×#11719908C2×C412yes—
55#13133153—21no—
56#13133153C2312yes—
57#13133144—24no—
58#13133140—24no—
59#13133127D412yes—
60#13133127C2312yes—
61C2×#41×C53—21no—
62#135×C53—21no—
63#135×C53—21no—
64#135×C53C2×C412yes—
65#135×C53C2312yes—
66C2×Q16×C53C2×C411yescharacteristic
67C2×SD16×C53—21no—
68C2×D8×C53C2×C411yescharacteristic
69C22×C8×C53C2311yescharacteristic
70C2×C4⋊C4×C53—21no—
71M32×C53—21no—
72M32×C53C2312yes—
73C2×C16×C53—21no—
74#106×C53—21no—
75#106×C53C2312yes—
76#41×C53—22no—
77#41×C53—22no—
78#41×C53—41no—
79#41×C53—41no—
80C2×Q8×C53—21no—
81C2×D4×C53—21no—
82C2×D4×C53—41no—
83C22×C4×C53—41no—
84C22×C4×C53—41no—
85C22×C4×C53C2×D411yescharacteristic
86C22×Dic53—21no—
87C22×Dic53—41no—
88C2×Dic106—22no—
89C2×Dic106—22no—
90C2×Dic106—22no—
91C2×Dic106—41no—
92C2×Dic106—41no—
93C2×Dic106—41no—
94Q16×C53—21no—
95Q16×C53C22×C412yes—
96SD16×C53—22no—
97SD16×C53—22no—
98D8×C53—21no—
99D8×C53C22×C412yes—
100C2×C8×C53C22×C411yescharacteristic
101C2×C8×C53C22×C411yescharacteristic
102C2×C8×C53C2×D411yescharacteristic
103C2×C8×C53#4111yescharacteristic
104C2×C8×C53#4111yescharacteristic
105C2×C8×C53C2411yescharacteristic
106C4⋊C4×C53—22no—
107C4⋊C4×C53—41no—
108#11719919—44no—
109C4×Dic53—21no—
110C4×Dic53—44no—
111C4×Dic53C2×D412yes—
112#11719917—22no—
113#11719917—44no—
114#11719917—41no—
115C2×#88322—21no—
116C2×#88322—21no—
117C2×#88322—44no—
118C2×#88322—41no—
119C2×#88322C2×D412yes—
120Dic212—22no—
121Dic212—22no—
122Dic212—22no—
123Dic212C22×C414yes—
124C16×C53—22no—
125#11719908C22×C414yes—
126C23×C53—41no—
127C23×C53AGL(1,ℤ/8)11yescharacteristic
128Q8×C53—22no—
129Q8×C53—41no—
130D4×C53—22no—
131D4×C53—42no—
132D4×C53—42no—
133D4×C53—41no—
134C2×C4×C53—41no—
135C2×C4×C53—42no—
136C2×C4×C53—42no—
137C2×C4×C53—42no—
138C2×C4×C53—82no—
139C2×C4×C53C4×D411yescharacteristic
140C2×C4×C53C4×D411yescharacteristic
141C2×C4×C53AGL(1,ℤ/8)11yescharacteristic
142C2×C4×C53AGL(1,ℤ/8)11yescharacteristic
143C2×C4×C53AGL(1,ℤ/8)11yescharacteristic
144C2×C4×C53C22×D411yescharacteristic
145C2×Dic53—22no—
146C2×Dic53—42no—
147C2×Dic53—42no—
148C2×Dic53—42no—
149C2×Dic53—42no—
150Dic106—24no—
151Dic106—42no—
152Dic106—42no—
153Dic106—42no—
154C8×C53C4×D411yescharacteristic
155C8×C53C4×D411yescharacteristic
156C8×C53C23×C411yescharacteristic
157C8×C53C2×#4111yescharacteristic
158#88322—22no—
159#88322—42no—
160C2×#2180—531nomaximal, Sylow
161C22×C53—22no—
162C22×C53—42no—
163C22×C53—82no—
164C22×C53#44211yescharacteristic
165C22×C53#44211yescharacteristic
166C22×C53C2×AGL(1,ℤ/8)11yescharacteristic
167C4×C53—42no—
168C4×C53—82no—
169C4×C53#44211yescharacteristic
170C4×C53#44211yescharacteristic
171C4×C53C2×C4×D411yescharacteristic
172C4×C53C2×AGL(1,ℤ/8)11yescharacteristic
173Dic53—42no—
174Dic53—44no—
175C2×#510—531no—
176C2×#444—531no—
177C2×#441—531no—
178#2180—538no—
179C2×#365—531no—
180C2×#362—531no—
181C2×#360D5311yescharacteristic
182C2×#347—531no—
183C2×C53—21no—
184C2×C53—21no—
185C2×C53—82no—
186C2×C53#218012yescomplemented
187C2×C53C2×#44211yescharacteristic
188C2×#135C2×D5311yescharacteristic
189C22×Q16—531no—
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×#108—531no—
203C2×#106C2×D5311yescharacteristic
204C2×#104—1061no—
205C2×#103—1061no—
206C2×#97—531no—
207#365—534no—
208#362—534no—
209#360C2×D5314yes—
210#347—534no—
211C53C2×#218011yescharacteristic, complemented, Sylow
212C2×#41—21no—
213C22×Q8—1061no—
214#135—21no—
215#135—21no—
216#135C4×D5312yes—
217#135C22×D5312yes—
218C2×Q16—532no—
219C2×Q16—532no—
220C2×Q16—532no—
221C2×Q16—1061no—
222C2×Q16—1061no—
223C2×Q16—1061no—
224C2×Q16C4×D5311yescharacteristic
225C2×SD16—21no—
226C2×D8C4×D5311yescharacteristic
227#131—1064no—
228#131—2122no—
229C2×M16—1061no—
230C2×M16—2121no—
231C22×C8—531no—
232C22×C8—2121no—
233C22×C8C22×D5311yescharacteristic
234C4×Q8—1064no—
235C4×Q8—2122no—
236C2×C4⋊C4—21no—
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—532no—
249#108—1061no—
250#106—21no—
251#106C22×D5312yes—
252#104—1064no—
253#103—1064no—
254#97—532no—
255#97—532no—
256#41—22no—
257#41—22no—
258#41—41no—
259#41—41no—
260C2×Q8—21no—
261C2×Q8—1062no—
262C2×Q8—1062no—
263C2×Q8—1062no—
264C2×Q8—2121no—
265C2×Q8—2121no—
266C2×Q8—2121no—
267C2×D4—21no—
268C2×D4—41no—
269C22×C4—41no—
270C22×C4—41no—
271C22×C4—1061no—
272C22×C4—2121no—
273C22×C4D4×D5311yescharacteristic
274Q16—21no—
275Q16—534no—
276Q16—1062no—
277Q16—1062no—
278Q16—1062no—
279Q16C2×C4×D5312yes—
280SD16—22no—
281SD16—22no—
282D8—21no—
283D8C2×C4×D5312yes—
284M16—1062no—
285M16—2124no—
286M16—2121no—
287C2×C8—532no—
288C2×C8—1061no—
289C2×C8—1061no—
290C2×C8—2124no—
291C2×C8—2121no—
292C2×C8C2×C4×D5311yescharacteristic
293C2×C8C2×C4×D5311yescharacteristic
294C2×C8D4×D5311yescharacteristic
295C2×C8#1171994711yescharacteristic
296C2×C8#1171994711yescharacteristic
297C2×C8C23×D5311yescharacteristic
298C4⋊C4—22no—
299C4⋊C4—41no—
300C4⋊C4—2124no—
301C42—532no—
302C42—1061no—
303C42—2124no—
304C16—22no—
305C16—534no—
306C23—41no—
307C23#1313324111yescharacteristic
308Q8—22no—
309Q8—41no—
310Q8—1064no—
311Q8—2122no—
312Q8—2122no—
313Q8—2122no—
314D4—22no—
315D4—42no—
316D4—42no—
317D4—41no—
318C2×C4—41no—
319C2×C4—42no—
320C2×C4—42no—
321C2×C4—42no—
322C2×C4—82no—
323C2×C4—1062no—
324C2×C4—2122no—
325C2×C4—2122no—
326C2×C4—2122no—
327C2×C4—2122no—
328C2×C4#1313317711yescharacteristic
329C2×C4#1313317711yescharacteristic
330C2×C4#1313324111yescharacteristic
331C2×C4#1313324411yescharacteristic
332C2×C4#1313324411yescharacteristic
333C2×C4C2×D4×D5311yescharacteristic
334C8—1062no—
335C8—2122no—
336C8#1313317711yescharacteristic
337C8#1313317711yescharacteristic
338C8C22×C4×D5311yescharacteristic
339C8C2×#1171994711yescharacteristic
340C22—22no—
341C22—42no—
342C22—82no—
343C22#1681073211yescharacteristic
344C22#1681073211yescharacteristic
345C22C2×#1313324111yescharacteristic
346C4—42no—
347C4—82no—
348C4—2122no—
349C4—2124no—
350C4#1681073211yescharacteristic
351C4#1681073211yescharacteristic
352C4C2×#1313317711yescharacteristic
353C4C2×#1313324411yescharacteristic
354C2—21no—
355C2—21no—
356C2—82no—
357C2#2027660312yesdirect product factor
358C2C2×#1681073211yescharacteristic
subgroupG/H
centerC22#16810732
commutatorC8×C53C23×C4
FrattiniC2×C8C23×D53
FittingC2×C53×#360C2
socleC22×C53#442
prime radicalC2×#135×C53C22
cosocleC2×C8×C53C24
hypercenterC2×#360D53
hypocenterC53C2×#2180
LCSG ▷ C8×C53 ▷ C4×C53 ▷ C2×C53 ▷ C53
UCSC1 ◁ C22 ◁ C22×C4 ◁ C22×C8 ◁ C2×#360

Complements

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

NH ≅ G/Ntype
3. C2×#16810729C2semidirect
4. #20276603C2direct (N × H)
5. C2×#16810445C2semidirect
6. C2×#16810440C2semidirect
7. C2×#16810400C2semidirect
12. #16811063C4semidirect
14. C2×#13133163C4semidirect
15. C2×#13133161C4semidirect
20. #16810729C22semidirect
25. #16810445C22semidirect
26. #16810440C22semidirect
27. #16810400C22semidirect
50. #13133163C2×C4semidirect
51. #13133161C2×C4semidirect
186. C2×C53#2180semidirect
211. C53C2×#2180semidirect
357. C2#20276603direct (N × H)

Maximal subgroups

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

subgroupindex
C2×#168110632
C2×#168107452
C2×#168107292
#202766032
C2×#168104452
C2×#168104402
C2×#168104002
C2×C53×#3602
C2×#218053