Finite Group Databasebeta

← C2×#20274163 #27366390

Subgroups — C2×#20274163 #27366390

Order 13568.

726 conjugacy classes of subgroups.

The subgroup lattice has 384 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#16810910C211yesmaximal, characteristic
2C2×#16810900C211yesmaximal, characteristic
3C2×C8×D212C211yesmaximal, characteristic
4#20274163C218yesmaximal, direct product factor
5C2×#16810376C211yesmaximal, characteristic, complemented
6C2×#16810372C211yesmaximal, characteristic, complemented
7C2×#16810369C211yesmaximal, characteristic, complemented
8C2×C53×#487C211yesmaximal, characteristic, complemented
9C22×C8×D53—21no—
10C2×C4×D212C2211yescharacteristic
11#16810910—22no—
12#16810910C2214yes—
13#16810900—22no—
14#16810900C2214yes—
15C8×D212—22no—
16C8×D212C2214yes—
17C2×#13133150—21no—
18C2×#13133148—21no—
19C2×C4⋊#88322C2211yescharacteristic
20C2×C4⋊#88322C2211yescharacteristic
21C2×C4⋊#88322C2211yescharacteristic
22C2×#13133137—21no—
23C2×#13133137—21no—
24C2×#13133129—21no—
25#16810376C2214yescomplemented
26#16810372C2214yescomplemented
27#16810369C2214yescomplemented
28C2×C4⋊Q8×C53C2211yescharacteristic
29C2×C4⋊Q8×C53C2211yescharacteristic
30C53×#487—22no—
31C53×#487C2214yescomplemented
32C2×#103×C53—21no—
33C2×#103×C53—21no—
34C2×C4×C8×C53C2211yescharacteristic
35C22×D212D411yescharacteristic
36C22×C4×D53—21no—
37C22×#88322—21no—
38C22×#88322—21no—
39C22×#88322—21no—
40C2×C8×D53—24no—
41C2×C8×D53—44no—
42C4×D212—21no—
43C4×D212—21no—
44C4×D212D412yes—
45C4×D212C2312yes—
46C2×C4⋊Dic53D411yescharacteristic
47C2×#11719921—21no—
48#13133150—24no—
49#13133148—24no—
50C4⋊#88322—21no—
51C4⋊#88322—21no—
52C4⋊#88322—21no—
53C4⋊#88322C2312yes—
54C4⋊#88322C2312yes—
55C4⋊#88322C2312yes—
56#13133137—24no—
57#13133137—24no—
58#13133129—24no—
59C22×Q8×C53—21no—
60C22×Q8×C53—21no—
61C22×C8×C53D411yescharacteristic
62C22×C8×C53D411yescharacteristic
63C4⋊Q8×C53—21no—
64C4⋊Q8×C53—21no—
65C4⋊Q8×C53—21no—
66C4⋊Q8×C53—21no—
67C4⋊Q8×C53—21no—
68C4⋊Q8×C53—21no—
69C4⋊Q8×C53C2312yes—
70C4⋊Q8×C53C2312yes—
71C2×C4⋊C4×C53—21no—
72C2×C4⋊C4×C53—21no—
73C2×C4⋊C4×C53—21no—
74C2×C4⋊C4×C53—21no—
75C2×C42×C53C2311yescharacteristic
76#103×C53—24no—
77#103×C53—24no—
78C4×C8×C53D412yes—
79C4×C8×C53C2312yes—
80C23×D53—21no—
81C2×Q8×C53—22no—
82C2×Q8×C53—22no—
83C2×Q8×C53—41no—
84C2×Q8×C53—41no—
85C2×Q8×C53—42no—
86C2×Q8×C53—42no—
87C2×Q8×C53—82no—
88C22×C4×C53—41no—
89C22×C4×C53—41no—
90C22×C4×C53C2×D411yescharacteristic
91C22×C4×C53C2×D411yescharacteristic
92C22×C4×C53#4111yescharacteristic
93C22×Dic53—21no—
94C2×D212—21no—
95C2×D212—22no—
96C2×D212Q1614yes—
97C2×D212C2×D412yes—
98C2×C4×D53—22no—
99C2×C4×D53—22no—
100C2×C4×D53—42no—
101C2×C4×D53—42no—
102C2×C8×C53—22no—
103C2×C8×C53—22no—
104C2×C8×C53C2×D412yes—
105C2×C8×C53C2×D412yes—
106C4⋊C4×C53—22no—
107C4⋊C4×C53—22no—
108C4⋊C4×C53—22no—
109C4⋊C4×C53—22no—
110C4⋊C4×C53—41no—
111C4⋊C4×C53—41no—
112C4⋊C4×C53—41no—
113C4⋊C4×C53—41no—
114C42×C53C2×D411yescharacteristic
115C42×C53C2×D411yescharacteristic
116C42×C53#4111yescharacteristic
117C42×C53C2411yescharacteristic
118#11719921—22no—
119#11719921—22no—
120C4⋊Dic53—21no—
121C4⋊Dic53C2×D412yes—
122C2×#88322—22no—
123C2×#88322—22no—
124C2×#88322—22no—
125C2×#88322—42no—
126C2×#88322—42no—
127C2×#88322—42no—
128C8×D53—48no—
129C23×C53#12111yescharacteristic
130C22×D53—22no—
131C22×D53—24no—
132C22×D53—42no—
133C22×D53—42no—
134Q8×C53—42no—
135Q8×C53—42no—
136Q8×C53—82no—
137Q8×C53—82no—
138Q8×C53—82no—
139C2×C4×C53—22no—
140C2×C4×C53—42no—
141C2×C4×C53—42no—
142C2×C4×C53—82no—
143C2×C4×C53—82no—
144C2×C4×C53#12111yescharacteristic
145C2×C4×C53#12111yescharacteristic
146C2×C4×C53#12111yescharacteristic
147C2×C4×C53C2×Q1612yes—
148C2×C4×C53#13512yes—
149C2×C4×C53#13512yes—
150C2×C4×C53AGL(1,ℤ/8)12yes—
151C2×C4×C53C22×D411yescharacteristic
152C2×C4×C53C22×D411yescharacteristic
153C2×C4×C53C2×#4111yescharacteristic
154C2×Dic53—22no—
155C2×Dic53—42no—
156D212—22no—
157D212—24no—
158D212C2×Q1614yes—
159C4×D53—42no—
160C4×D53—42no—
161C4×D53—44no—
162C8×C53—22no—
163C8×C53—22no—
164#88322—42no—
165#88322—42no—
166#88322—42no—
167C2×#1683—531nomaximal, Sylow
168C22×C53#46312yes—
169C22×C53#46712yes—
170C22×C53#47112yes—
171C22×C53C2×#12111yescharacteristic
172C2×D53—24no—
173C2×D53—48no—
174C2×D53—42no—
175C2×D53—42no—
176C4×C53—21no—
177C4×C53—21no—
178C4×C53—82no—
179C4×C53—82no—
180C4×C53#46312yes—
181C4×C53#46712yes—
182C4×C53C22×Q1611yescharacteristic
183C4×C53C2×#13511yescharacteristic
184C4×C53C2×#13511yescharacteristic
185C4×C53C2×AGL(1,ℤ/8)11yescharacteristic
186Dic53—42no—
187C2×#487D5311yescharacteristic, complemented
188C2×#476—531no—
189C2×#474—531no—
190C2×C8×D4—531no—
191#1683—538no—
192C2×#335—531no—
193C2×#333—531no—
194C2×#332—531no—
195C2×C53#168314yescomplemented
196C2×C53C2×#46311yescharacteristic
197C2×C53C2×#46711yescharacteristic
198C2×C53C2×#47111yescharacteristic
199D53—44no—
200C23×C8—1061no—
201C2×C4⋊Q8C2×D5311yescharacteristic
202C2×C4⋊Q8C2×D5311yescharacteristic
203C2×C4×D4—531no—
204#487—22no—
205#487C2×D5314yescomplemented
206#476—534no—
207#476—1062no—
208#474—534no—
209#474—1062no—
210C8×D4—534no—
211C8×D4—1062no—
212C2×#107—1061no—
213C2×#106—1061no—
214C2×C4⋊C8—531no—
215C2×C4⋊C8—531no—
216C2×C4⋊C8—531no—
217C2×#103—21no—
218C2×#103—21no—
219C2×#102—1061no—
220C2×#102—1061no—
221C2×C22⋊C8—1061no—
222C2×C4×C8C2×D5311yescharacteristic
223#335—534no—
224#333—534no—
225#332—534no—
226C53C2×#168311yescharacteristic, complemented, Sylow
227C22×Q8—21no—
228C22×Q8—21no—
229C22×D4—531no—
230C23×C4—1061no—
231C22×C8—1061no—
232C22×C8—1061no—
233C22×C8—1061no—
234C22×C8—1064no—
235C22×C8—2124no—
236C22×C8C22⋊D5311yescharacteristic
237C22×C8C22⋊D5311yescharacteristic
238C4⋊Q8—21no—
239C4⋊Q8—21no—
240C4⋊Q8—21no—
241C4⋊Q8—21no—
242C4⋊Q8—21no—
243C4⋊Q8—21no—
244C4⋊Q8C22×D5312yes—
245C4⋊Q8C22×D5312yes—
246C4×D4—532no—
247C4×D4—532no—
248C4×D4—1061no—
249C4×D4—1061no—
250C2×C4⋊C4—21no—
251C2×C4⋊C4—21no—
252C2×C4⋊C4—21no—
253C2×C4⋊C4—21no—
254C2×C4⋊C4—531no—
255C2×C22⋊C4—1061no—
256C2×C42C22×D5311yescharacteristic
257#107—1064no—
258#106—1064no—
259C4⋊C8—532no—
260C4⋊C8—532no—
261C4⋊C8—532no—
262C4⋊C8—1061no—
263C4⋊C8—1061no—
264C4⋊C8—1061no—
265#103—24no—
266#103—24no—
267#102—1064no—
268#102—1064no—
269C22⋊C8—1064no—
270C4×C8C22⋊D5312yes—
271C4×C8C22×D5312yes—
272C24—1061no—
273C2×Q8—22no—
274C2×Q8—22no—
275C2×Q8—41no—
276C2×Q8—41no—
277C2×Q8—42no—
278C2×Q8—42no—
279C2×Q8—82no—
280C2×D4—534no—
281C2×D4—532no—
282C2×D4—1061no—
283C2×D4—1062no—
284C22×C4—41no—
285C22×C4—41no—
286C22×C4—1061no—
287C22×C4—1062no—
288C22×C4—1062no—
289C22×C4—2122no—
290C22×C4—2122no—
291C22×C4D4×D5311yescharacteristic
292C22×C4#1171994711yescharacteristic
293C22×C4C2×C22⋊D5311yescharacteristic
294C2×C8—22no—
295C2×C8—22no—
296C2×C8—1062no—
297C2×C8—1062no—
298C2×C8—1062no—
299C2×C8—2128no—
300C2×C8—2122no—
301C2×C8—2122no—
302C2×C8—2122no—
303C2×C8C2×C22⋊D5312yes—
304C2×C8C2×C22⋊D5312yes—
305C4⋊C4—22no—
306C4⋊C4—22no—
307C4⋊C4—22no—
308C4⋊C4—22no—
309C4⋊C4—41no—
310C4⋊C4—41no—
311C4⋊C4—41no—
312C4⋊C4—41no—
313C4⋊C4—532no—
314C4⋊C4—1061no—
315C22⋊C4—1062no—
316C22⋊C4—1062no—
317C42D4×D5311yescharacteristic
318C42#1171994711yescharacteristic
319C42C2×C22⋊D5311yescharacteristic
320C42C23×D5311yescharacteristic
321C23—1062no—
322C23—1064no—
323C23—2122no—
324C23—2122no—
325C23#1313318811yescharacteristic
326Q8—42no—
327Q8—42no—
328Q8—82no—
329Q8—82no—
330Q8—82no—
331D4—534no—
332D4—1064no—
333D4—1062no—
334C2×C4—22no—
335C2×C4—42no—
336C2×C4—42no—
337C2×C4—82no—
338C2×C4—82no—
339C2×C4—1062no—
340C2×C4—2122no—
341C2×C4—2124no—
342C2×C4—2122no—
343C2×C4—2122no—
344C2×C4#1313318811yescharacteristic
345C2×C4#1313318811yescharacteristic
346C2×C4#1313318811yescharacteristic
347C2×C4Q16×D5312yes—
348C2×C4#1313323712yes—
349C2×C4#1313324012yes—
350C2×C4#1313324212yes—
351C2×C4C2×D4×D5311yescharacteristic
352C2×C4C22×C22⋊D5311yescharacteristic
353C2×C4C2×#1171994711yescharacteristic
354C8—22no—
355C8—22no—
356C8—2122no—
357C8—2122no—
358C8—2122no—
359C22—1064no—
360C22—2122no—
361C22—2128no—
362C22—2122no—
363C22#1681083512yes—
364C22#1681086112yes—
365C22#1681088312yes—
366C22C2×#1313318811yescharacteristic
367C4—21no—
368C4—21no—
369C4—82no—
370C4—82no—
371C4—2122no—
372C4#1681083512yes—
373C4#1681086112yes—
374C4C2×Q16×D5311yescharacteristic
375C4C2×#1313323711yescharacteristic
376C4C2×#1313324011yescharacteristic
377C4C2×#1313324211yescharacteristic
378C2—2124no—
379C2#2027416314yesdirect product factor
380C2C2×#1681083511yescharacteristic
381C2C2×#1681086111yescharacteristic
382C2C2×#1681088311yescharacteristic
subgroupG/H
centerC23#13133188
commutator, cosocleC42×C53C24
FrattiniC42C23×D53
FittingC2×C53×#487C2
socleC23×C53#121
prime radicalC22×D212D4
hypercenterC2×#487D53
hypocenterC53C2×#1683
LCSG ▷ C42×C53 ▷ C22×C53 ▷ C53
UCSC1 ◁ C23 ◁ C2×C42 ◁ C2×#487

Complements

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

NH ≅ G/Ntype
4. #20274163C2direct (N × H)
5. C2×#16810376C2semidirect
6. C2×#16810372C2semidirect
7. C2×#16810369C2semidirect
8. C2×C53×#487C2semidirect
25. #16810376C22semidirect
26. #16810372C22semidirect
27. #16810369C22semidirect
31. C53×#487C22semidirect
187. C2×#487D53semidirect
195. C2×C53#1683semidirect
205. #487C2×D53semidirect
226. C53C2×#1683semidirect
379. C2#20274163direct (N × H)

Maximal subgroups

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

subgroupindex
C2×#168109102
C2×#168109002
C2×C8×D2122
#202741632
C2×#168103762
C2×#168103722
C2×#168103692
C2×C53×#4872
C2×#168353