Finite Group Databasebeta

← C2×#20276604 #27537523

Subgroups — C2×#20276604 #27537523

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×#16810746C211yesmaximal, characteristic
3#20276604C218yesmaximal, direct product factor
4C2×#16810445C211yesmaximal, characteristic, complemented
5C2×#16810439C211yesmaximal, characteristic, complemented
6C2×#16810432C211yesmaximal, characteristic
7C2×#16810401C211yesmaximal, characteristic, complemented
8C2×C53×#441C211yesmaximal, characteristic, complemented
9C2×#13133239C2211yescharacteristic
10C2×#13133223—21no—
11#16811064C414yescomplemented
12#16811064C2214yes—
13C2×#13133163C412yescomplemented
14C2×#13133160C412yescomplemented
15C2×#13133157C2211yescharacteristic
16C2×#13133157C2211yescharacteristic
17#16810746—22no—
18#16810746C2214yes—
19C2×#13133153C2211yescharacteristic
20C2×#13133145—21no—
21#16810445C2214yescomplemented
22#16810439C2214yescomplemented
23#16810432C2214yes—
24#16810401C2214yescomplemented
25C22×Q16×C53C2211yescharacteristic
26C2×C4×Q8×C53—21no—
27C53×#441—22no—
28C53×#441C2214yescomplemented
29C2×#106×C53C2211yescharacteristic
30C2×#103×C53—21no—
31C2×#97×C53C2211yescharacteristic
32C2×#11719945—21no—
33#13133239—21no—
34#13133239—21no—
35#13133239C2×C412yes—
36#13133239C2312yes—
37C2×Dic212C2×C411yescharacteristic
38C2×#11719913—21no—
39C2×D424C2×C411yescharacteristic
40#13133223—24no—
41#13133223—42no—
42C2×#11719917—21no—
43C2×#11719912—41no—
44C2×C8×D53—41no—
45#13133163C2×C418yescomplemented
46#13133160C2×C418yescomplemented
47#13133157—21no—
48#13133157C2×C412yes—
49#13133157C2312yes—
50#13133157C2312yes—
51C2×#11719908—21no—
52C2×#11719908C2×C412yes—
53#13133153—21no—
54#13133153C2312yes—
55#13133145—24no—
56C22×Q8×C53—21no—
57C2×Q16×C53—21no—
58C2×Q16×C53—21no—
59C2×Q16×C53—21no—
60C2×Q16×C53C2×C412yes—
61C2×Q16×C53C2×C412yes—
62C2×Q16×C53C2312yes—
63C22×C8×C53D411yescharacteristic
64C22×C8×C53C2311yescharacteristic
65C4×Q8×C53—24no—
66C4×Q8×C53—42no—
67C2×C4⋊C4×C53—21no—
68C2×C4⋊C4×C53—41no—
69C2×C42×C53—41no—
70C2×C42×C53D411yescharacteristic
71#106×C53—21no—
72#106×C53C2312yes—
73#103×C53—24no—
74#97×C53D412yes—
75#97×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—21no—
83C22×C4×C53—41no—
84C22×C4×C53—41no—
85C22×C4×C53C2×D411yescharacteristic
86C2×C22⋊D53—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—
98C2×C8×C53—21no—
99C2×C8×C53—21no—
100C2×C8×C53C22×C411yescharacteristic
101C2×C8×C53C22×C411yescharacteristic
102C2×C8×C53C2×D412yes—
103C2×C8×C53C2×D411yescharacteristic
104C2×C8×C53#4111yescharacteristic
105C2×C8×C53#4111yescharacteristic
106C2×C8×C53C2411yescharacteristic
107C4⋊C4×C53—22no—
108C4⋊C4×C53—44no—
109C4⋊C4×C53—41no—
110C42×C53—21no—
111C42×C53—44no—
112C42×C53C2×D412yes—
113#11719917—22no—
114#11719917—41no—
115C2×#88322—41no—
116Dic212—21no—
117Dic212C22×C412yes—
118D424—21no—
119D424C22×C412yes—
120#11719913—22no—
121#11719913—22no—
122#11719912—44no—
123C8×D53—44no—
124#11719908—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—22no—
133C2×C4×C53—42no—
134C2×C4×C53—42no—
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
145C22⋊D53—42no—
146C22⋊D53—42no—
147C2×Dic53—41no—
148D212—22no—
149D212—41no—
150C4×D53—42no—
151C4×D53—42no—
152Dic106—22no—
153Dic106—41no—
154C8×C53—22no—
155C8×C53C4×D411yescharacteristic
156C8×C53C4×D411yescharacteristic
157C8×C53C23×C411yescharacteristic
158C8×C53C2×#4111yescharacteristic
159#88322—42no—
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—42no—
168C4×C53—44no—
169C4×C53—82no—
170C4×C53#44211yescharacteristic
171C4×C53#44211yescharacteristic
172C4×C53C2×C4×D411yescharacteristic
173C4×C53C2×AGL(1,ℤ/8)11yescharacteristic
174Dic53—42no—
175C2×#510—531no—
176C2×#444—531no—
177C2×#441D5311yescharacteristic, complemented
178#2180—538no—
179C2×#365—531no—
180C2×#362—531no—
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—21no—
192#510—534no—
193#510—534no—
194C2×Q32—532no—
195C2×SD32—532no—
196C2×M32—531no—
197C2×M32—531no—
198#444—534no—
199#444—1062no—
200#441—22no—
201#441C2×D5314yescomplemented
202C2×#108—531no—
203C2×#106C2×D5311yescharacteristic
204C2×#104—1061no—
205C2×#103—21no—
206C2×#97C2×D5311yescharacteristic
207#365—534no—
208#362—534no—
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—1061no—
230C2×M16—2121no—
231C22×C8—2121no—
232C22×C8C22⋊D5311yescharacteristic
233C22×C8C22×D5311yescharacteristic
234C4×Q8—24no—
235C4×Q8—42no—
236C2×C4⋊C4—21no—
237C2×C4⋊C4—41no—
238C2×C42—41no—
239C2×C42C22⋊D5311yescharacteristic
240Q32—538no—
241SD32—538no—
242M32—532no—
243M32—532no—
244M32—532no—
245M32—1061no—
246C2×C16—532no—
247C2×C16—1061no—
248#108—532no—
249#108—1061no—
250#106—21no—
251#106C22×D5312yes—
252#104—1064no—
253#103—24no—
254#97C22⋊D5312yes—
255#97C22×D5312yes—
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—21no—
270C22×C4—41no—
271C22×C4—41no—
272C22×C4—2121no—
273C22×C4C2×C22⋊D5311yescharacteristic
274Q16—22no—
275Q16—22no—
276Q16—22no—
277Q16—532no—
278Q16—1061no—
279Q16C2×C4×D5314yes—
280SD16—1062no—
281SD16—1062no—
282D8—532no—
283D8—1061no—
284M16—1062no—
285M16—2124no—
286M16—2121no—
287C2×C8—21no—
288C2×C8—21no—
289C2×C8—2124no—
290C2×C8—2121no—
291C2×C8C2×C4×D5311yescharacteristic
292C2×C8C2×C4×D5311yescharacteristic
293C2×C8#1171994511yescharacteristic
294C2×C8#1171994511yescharacteristic
295C2×C8C2×C22⋊D5312yes—
296C2×C8C2×C22⋊D5311yescharacteristic
297C2×C8C23×D5311yescharacteristic
298C4⋊C4—22no—
299C4⋊C4—44no—
300C4⋊C4—41no—
301C42—21no—
302C42—44no—
303C42C2×C22⋊D5312yes—
304C16—534no—
305C16—1062no—
306C23—2121no—
307C23#1313324311yescharacteristic
308Q8—24no—
309Q8—42no—
310Q8—42no—
311Q8—42no—
312Q8—1062no—
313Q8—2121no—
314D4—1062no—
315D4—2122no—
316D4—2122no—
317D4—2121no—
318C2×C4—22no—
319C2×C4—42no—
320C2×C4—42no—
321C2×C4—42no—
322C2×C4—42no—
323C2×C4—42no—
324C2×C4—82no—
325C2×C4—2121no—
326C2×C4—2122no—
327C2×C4—2122no—
328C2×C4C4×C22⋊D5311yescharacteristic
329C2×C4C4×C22⋊D5311yescharacteristic
330C2×C4#1313324211yescharacteristic
331C2×C4#1313324211yescharacteristic
332C2×C4#1313324311yescharacteristic
333C2×C4C22×C22⋊D5311yescharacteristic
334C8—22no—
335C8—2122no—
336C8C4×C22⋊D5311yescharacteristic
337C8C4×C22⋊D5311yescharacteristic
338C8C22×C4×D5311yescharacteristic
339C8C2×#1171994511yescharacteristic
340C22—22no—
341C22—4242no—
342C22—2122no—
343C22#1681073311yescharacteristic
344C22#1681073311yescharacteristic
345C22C2×#1313324311yescharacteristic
346C4—42no—
347C4—44no—
348C4—82no—
349C4—2122no—
350C4#1681073311yescharacteristic
351C4#1681073311yescharacteristic
352C4C2×C4×C22⋊D5311yescharacteristic
353C4C2×#1313324211yescharacteristic
354C2—21no—
355C2—21no—
356C2—4242no—
357C2#2027660412yesdirect product factor
358C2C2×#1681073311yescharacteristic
subgroupG/H
centerC22#16810733
commutatorC8×C53C23×C4
FrattiniC2×C8C23×D53
FittingC2×C53×#441C2
socleC22×C53#442
prime radicalC2×#13133239C22
cosocleC2×C8×C53C24
hypercenterC2×#441D53
hypocenterC53C2×#2180
LCSG ▷ C8×C53 ▷ C4×C53 ▷ C2×C53 ▷ C53
UCSC1 ◁ C22 ◁ C22×C4 ◁ C2×#97 ◁ C2×#441

Complements

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

NH ≅ G/Ntype
3. #20276604C2direct (N × H)
4. C2×#16810445C2semidirect
5. C2×#16810439C2semidirect
7. C2×#16810401C2semidirect
8. C2×C53×#441C2semidirect
11. #16811064C4semidirect
13. C2×#13133163C4semidirect
14. C2×#13133160C4semidirect
21. #16810445C22semidirect
22. #16810439C22semidirect
24. #16810401C22semidirect
28. C53×#441C22semidirect
45. #13133163C2×C4semidirect
46. #13133160C2×C4semidirect
177. C2×#441D53semidirect
185. C2×C53#2180semidirect
201. #441C2×D53semidirect
211. C53C2×#2180semidirect
357. C2#20276604direct (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×#168107462
#202766042
C2×#168104452
C2×#168104392
C2×#168104322
C2×#168104012
C2×C53×#4412
C2×#218053