Finite Group Databasebeta

← C2×#20276607 #27537526

Subgroups — C2×#20276607 #27537526

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×#16810745C211yesmaximal, characteristic
2C2×#16810730C211yesmaximal, characteristic, complemented
3#20276607C218yesmaximal, direct product factor
4C2×#16810446C211yesmaximal, characteristic, complemented
5C2×#16810439C211yesmaximal, characteristic, complemented
6C2×#16810433C211yesmaximal, characteristic
7C2×#16810400C211yesmaximal, characteristic, complemented
8C2×C53×#510C211yesmaximal, characteristic
9C2×#13133224—21no—
10C2×Q8×Dic53—21no—
11C2×#13133157C2211yescharacteristic
12#16810745—22no—
13#16810745C2214yes—
14#16810730—22no—
15#16810730C2214yescomplemented
16C2×#13133153C2211yescharacteristic
17C2×#13133149C2211yescharacteristic
18C2×#13133144—21no—
19C2×#13133141—21no—
20C2×#13133127C2211yescharacteristic
21#16810446C2214yescomplemented
22#16810439C2214yescomplemented
23#16810433C2214yes—
24#16810400C2214yescomplemented
25C2×#135×C53C2211yescharacteristic
26C22×Q16×C53C2211yescharacteristic
27C53×#510C414yescomplemented
28C53×#510C2214yes—
29C2×Q32×C53C412yescomplemented
30C2×SD32×C53C412yescomplemented
31C2×M32×C53C2211yescharacteristic
32#13133224—24no—
33#13133224—42no—
34C2×#11719917—21no—
35C2×#11719917—41no—
36C22×#88322—41no—
37C22×#88322D411yescharacteristic
38Q8×Dic53—24no—
39Q8×Dic53—42no—
40C2×C4⋊Dic53—21no—
41C2×C4⋊Dic53—41no—
42C2×C4×Dic53—41no—
43C2×C4×Dic53D411yescharacteristic
44#13133157—21no—
45#13133157C2312yes—
46C2×#11719908—21no—
47#13133153—21no—
48#13133153C2312yes—
49#13133149—21no—
50#13133149C2312yes—
51#13133144—24no—
52#13133141—24no—
53#13133127D412yes—
54#13133127C2312yes—
55C2×#41×C53—21no—
56C22×Q8×C53—21no—
57#135×C53—21no—
58#135×C53—21no—
59#135×C53C2×C412yes—
60#135×C53C2312yes—
61C2×Q16×C53—21no—
62C2×Q16×C53—21no—
63C2×Q16×C53—21no—
64C2×Q16×C53C2×C412yes—
65C2×Q16×C53C2×C411yescharacteristic
66C2×Q16×C53C2×C412yes—
67C2×Q16×C53C2312yes—
68C2×SD16×C53—21no—
69C2×D8×C53C2×C411yescharacteristic
70C22×C8×C53C2311yescharacteristic
71Q32×C53C2×C418yescomplemented
72SD32×C53C2×C418yescomplemented
73M32×C53C2×C412yes—
74M32×C53C2312yes—
75C2×C16×C53C2×C412yes—
76#41×C53—22no—
77#41×C53—22no—
78#41×C53—41no—
79#41×C53—41no—
80C2×Q8×C53—21no—
81C2×Q8×C53—22no—
82C2×Q8×C53—22no—
83C2×Q8×C53—22no—
84C2×Q8×C53—41no—
85C2×Q8×C53—41no—
86C2×Q8×C53—41no—
87C2×D4×C53—21no—
88C2×D4×C53—41no—
89C22×C4×C53—41no—
90C22×C4×C53—41no—
91C22×C4×C53C2×D411yescharacteristic
92C22×Dic53—21no—
93C22×Dic53—41no—
94Q16×C53—22no—
95Q16×C53—22no—
96Q16×C53—21no—
97Q16×C53—22no—
98Q16×C53C22×C414yes—
99Q16×C53C22×C412yes—
100SD16×C53—22no—
101SD16×C53—22no—
102D8×C53—21no—
103D8×C53C22×C412yes—
104C2×C8×C53C22×C411yescharacteristic
105C2×C8×C53C22×C411yescharacteristic
106C2×C8×C53C2×D411yescharacteristic
107C2×C8×C53#4111yescharacteristic
108C2×C8×C53#4111yescharacteristic
109C2×C8×C53C2411yescharacteristic
110C4⋊Dic53—22no—
111C4⋊Dic53—44no—
112C4⋊Dic53—41no—
113C4×Dic53—21no—
114C4×Dic53—44no—
115C4×Dic53C2×D412yes—
116#11719917—22no—
117#11719917—44no—
118#11719917—41no—
119C2×#88322—21no—
120C2×#88322—21no—
121C2×#88322—44no—
122C2×#88322—41no—
123C2×#88322C2×D412yes—
124C16×C53C22×C414yes—
125#11719908—22no—
126C23×C53—41no—
127C23×C53AGL(1,ℤ/8)11yescharacteristic
128Q8×C53—24no—
129Q8×C53—22no—
130Q8×C53—42no—
131Q8×C53—42no—
132Q8×C53—42no—
133Q8×C53—41no—
134D4×C53—22no—
135D4×C53—42no—
136D4×C53—42no—
137D4×C53—41no—
138C2×C4×C53—42no—
139C2×C4×C53—41no—
140C2×C4×C53—42no—
141C2×C4×C53—42no—
142C2×C4×C53—42no—
143C2×C4×C53—42no—
144C2×C4×C53C4×D411yescharacteristic
145C2×C4×C53C4×D411yescharacteristic
146C2×C4×C53AGL(1,ℤ/8)11yescharacteristic
147C2×C4×C53AGL(1,ℤ/8)11yescharacteristic
148C2×C4×C53AGL(1,ℤ/8)11yescharacteristic
149C2×C4×C53C22×D411yescharacteristic
150C2×Dic53—22no—
151C2×Dic53—42no—
152C2×Dic53—42no—
153C2×Dic53—82no—
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—44no—
169C4×C53#44211yescharacteristic
170C4×C53#44211yescharacteristic
171C4×C53C2×C4×D411yescharacteristic
172C4×C53C2×AGL(1,ℤ/8)11yescharacteristic
173Dic53—42no—
174Dic53—82no—
175C2×#510D5311yescharacteristic
176C2×#444—531no—
177C2×#441—531no—
178#2180—538no—
179C2×#365—531no—
180C2×#362—531no—
181C2×#360—531no—
182C2×#347—531no—
183C2×C53—21no—
184C2×C53—21no—
185C2×C53—82no—
186C2×C53#218012yescomplemented
187C2×C53C2×#44211yescharacteristic
188C2×#135C2×D5311yescharacteristic
189C22×Q16C2×D5311yescharacteristic
190C2×#131—1061no—
191C2×C4×Q8—1061no—
192#510Dic5314yescomplemented
193#510C2×D5314yes—
194C2×Q32Dic5312yescomplemented
195C2×SD32Dic5312yescomplemented
196C2×M32—531no—
197C2×M32C2×D5311yescharacteristic
198#444—534no—
199#444—1062no—
200#441—534no—
201#441—1062no—
202C2×#108—531no—
203C2×#106—531no—
204C2×#104—1061no—
205C2×#103—1061no—
206C2×#97—531no—
207#365—534no—
208#362—534no—
209#360—534no—
210#347—534no—
211C53C2×#218011yescharacteristic, complemented, Sylow
212C2×#41—21no—
213C22×Q8—21no—
214#135—21no—
215#135—21no—
216#135C2×Dic5312yes—
217#135C22×D5312yes—
218C2×Q16—21no—
219C2×Q16—21no—
220C2×Q16—21no—
221C2×Q16C2×Dic5312yes—
222C2×Q16C2×Dic5312yes—
223C2×Q16C2×Dic5311yescharacteristic
224C2×Q16C22×D5312yes—
225C2×SD16—21no—
226C2×D8C2×Dic5311yescharacteristic
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—1061no—
237C2×C4⋊C4—2121no—
238C2×C42—531no—
239C2×C42—2121no—
240Q32C2×Dic5318yescomplemented
241SD32C2×Dic5318yescomplemented
242M32—532no—
243M32—1061no—
244M32C2×Dic5312yes—
245M32C22×D5312yes—
246C2×C16—1061no—
247C2×C16C2×Dic5312yes—
248#108—532no—
249#108—1061no—
250#106—532no—
251#106—1061no—
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—22no—
262C2×Q8—22no—
263C2×Q8—22no—
264C2×Q8—41no—
265C2×Q8—41no—
266C2×Q8—41no—
267C2×D4—21no—
268C2×D4—41no—
269C22×C4—41no—
270C22×C4—41no—
271C22×C4—1061no—
272C22×C4—2121no—
273C22×C4D4×D5311yescharacteristic
274Q16—22no—
275Q16—22no—
276Q16—21no—
277Q16—22no—
278Q16C22×Dic5314yes—
279Q16C22×Dic5312yes—
280SD16—22no—
281SD16—22no—
282D8—21no—
283D8C22×Dic5312yes—
284M16—1062no—
285M16—2124no—
286M16—2121no—
287C2×C8—532no—
288C2×C8—1061no—
289C2×C8—1061no—
290C2×C8—2124no—
291C2×C8—2121no—
292C2×C8D4×D5311yescharacteristic
293C2×C8#1171994711yescharacteristic
294C2×C8#1171994711yescharacteristic
295C2×C8C22×Dic5311yescharacteristic
296C2×C8C22×Dic5311yescharacteristic
297C2×C8C23×D5311yescharacteristic
298C4⋊C4—1062no—
299C4⋊C4—2124no—
300C4⋊C4—2121no—
301C42—532no—
302C42—1061no—
303C42—2124no—
304C16—1062no—
305C16C22×Dic5314yes—
306C23—41no—
307C23#1313324111yescharacteristic
308Q8—22no—
309Q8—24no—
310Q8—42no—
311Q8—42no—
312Q8—42no—
313Q8—41no—
314D4—22no—
315D4—42no—
316D4—42no—
317D4—41no—
318C2×C4—42no—
319C2×C4—42no—
320C2×C4—41no—
321C2×C4—42no—
322C2×C4—42no—
323C2×C4—42no—
324C2×C4—1062no—
325C2×C4—4242no—
326C2×C4—2122no—
327C2×C4—2122no—
328C2×C4D4×Dic5311yescharacteristic
329C2×C4D4×Dic5311yescharacteristic
330C2×C4#1313324111yescharacteristic
331C2×C4#1313324111yescharacteristic
332C2×C4#1313324111yescharacteristic
333C2×C4C2×D4×D5311yescharacteristic
334C8—1062no—
335C8—2122no—
336C8D4×Dic5311yescharacteristic
337C8D4×Dic5311yescharacteristic
338C8C23×Dic5311yescharacteristic
339C8C2×#1171994711yescharacteristic
340C22—22no—
341C22—42no—
342C22—82no—
343C22#1681073511yescharacteristic
344C22#1681073511yescharacteristic
345C22C2×#1313324111yescharacteristic
346C4—42no—
347C4—44no—
348C4—4242no—
349C4—2122no—
350C4#1681073511yescharacteristic
351C4#1681073511yescharacteristic
352C4C2×D4×Dic5311yescharacteristic
353C4C2×#1313324111yescharacteristic
354C2—21no—
355C2—21no—
356C2—82no—
357C2#2027660712yesdirect product factor
358C2C2×#1681073511yescharacteristic
subgroupG/H
centerC22#16810735
commutatorC8×C53C23×C4
FrattiniC2×C8C23×D53
FittingC2×C53×#510C2
socleC22×C53#442
prime radicalC2×#135×C53C22
cosocleC2×C8×C53C24
hypercenterC2×#510D53
hypocenterC53C2×#2180
LCSG ▷ C8×C53 ▷ C4×C53 ▷ C2×C53 ▷ C53
UCSC1 ◁ C22 ◁ C22×C4 ◁ C22×C8 ◁ C2×#510

Complements

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

NH ≅ G/Ntype
2. C2×#16810730C2semidirect
3. #20276607C2direct (N × H)
4. C2×#16810446C2semidirect
5. C2×#16810439C2semidirect
7. C2×#16810400C2semidirect
15. #16810730C22semidirect
21. #16810446C22semidirect
22. #16810439C22semidirect
24. #16810400C22semidirect
27. C53×#510C4semidirect
29. C2×Q32×C53C4semidirect
30. C2×SD32×C53C4semidirect
71. Q32×C53C2×C4semidirect
72. SD32×C53C2×C4semidirect
186. C2×C53#2180semidirect
192. #510Dic53semidirect
194. C2×Q32Dic53semidirect
195. C2×SD32Dic53semidirect
211. C53C2×#2180semidirect
240. Q32C2×Dic53semidirect
241. SD32C2×Dic53semidirect
357. C2#20276607direct (N × H)

Maximal subgroups

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

subgroupindex
C2×#168107452
C2×#168107302
#202766072
C2×#168104462
C2×#168104392
C2×#168104332
C2×#168104002
C2×C53×#5102
C2×#218053