Finite Group Databasebeta

← C2×#20274166 #27366393

Subgroups — C2×#20274166 #27366393

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×#16810974C211yesmaximal, characteristic, complemented
2C2×#16810913C211yesmaximal, characteristic
3C2×#16810903C211yesmaximal, characteristic
4#20274166C218yesmaximal, direct product factor
5C2×#16810376C211yesmaximal, characteristic, complemented
6C2×#16810371C211yesmaximal, characteristic, complemented
7C2×#16810370C211yesmaximal, characteristic, complemented
8C2×C8×D4×C53C211yesmaximal, characteristic
9C2×#13133217C2211yescharacteristic
10C2×#13133217C2211yescharacteristic
11#16810974—22no—
12#16810974C2214yescomplemented
13#16810913—22no—
14#16810913C2214yes—
15#16810903—22no—
16#16810903C2214yes—
17C2×#13133150—21no—
18C2×#13133148—21no—
19C2×C4⋊#88322C2211yescharacteristic
20C2×C4⋊#88322C2211yescharacteristic
21C2×#13133142—21no—
22C2×#13133142—21no—
23C2×#13133138—21no—
24C2×#13133138—21no—
25#16810376C2214yescomplemented
26#16810371C2214yescomplemented
27#16810370C2214yescomplemented
28C23×C8×C53—21no—
29C2×C4×D4×C53C2211yescharacteristic
30C8×D4×C53—22no—
31C8×D4×C53C2214yes—
32C2×C4⋊C8×C53C2211yescharacteristic
33C2×C22⋊C8×C53—21no—
34C2×C4×C8×C53C2211yescharacteristic
35C22×Dic106—21no—
36C22×Dic106—21no—
37C22×#88322—21no—
38C22×#88322—21no—
39#13133217—21no—
40#13133217—21no—
41#13133217—21no—
42#13133217—21no—
43#13133217—21no—
44#13133217—21no—
45#13133217C2312yes—
46#13133217C2312yes—
47C2×C4⋊Dic53—21no—
48C2×C4⋊Dic53—21no—
49C2×C4⋊Dic53—21no—
50C2×C4⋊Dic53—21no—
51#13133150—24no—
52#13133148—24no—
53C4⋊#88322—21no—
54C4⋊#88322—21no—
55C4⋊#88322C2312yes—
56C4⋊#88322C2312yes—
57#13133142—24no—
58#13133142—24no—
59#13133138—24no—
60#13133138—24no—
61C22×D4×C53D411yescharacteristic
62C23×C4×C53—21no—
63C22×C8×C53—24no—
64C22×C8×C53—21no—
65C22×C8×C53—44no—
66C22×C8×C53D411yescharacteristic
67C22×C8×C53D411yescharacteristic
68C4×D4×C53—21no—
69C4×D4×C53—21no—
70C4×D4×C53D412yes—
71C4×D4×C53C2312yes—
72C2×C4⋊C4×C53D411yescharacteristic
73C2×C22⋊C4×C53—21no—
74C2×C42×C53C2311yescharacteristic
75C4⋊C8×C53—21no—
76C4⋊C8×C53C2312yes—
77C22⋊C8×C53—24no—
78C4×C8×C53D412yes—
79C4×C8×C53C2312yes—
80C24×C53—21no—
81C2×D4×C53—21no—
82C2×D4×C53—22no—
83C2×D4×C53Q1614yes—
84C2×D4×C53C2×D412yes—
85C22×C4×C53—21no—
86C22×C4×C53—22no—
87C22×C4×C53—22no—
88C22×C4×C53—42no—
89C22×C4×C53—42no—
90C22×C4×C53C2×D411yescharacteristic
91C22×C4×C53C2×D411yescharacteristic
92C22×C4×C53#4111yescharacteristic
93C22×Dic53—41no—
94C22×Dic53—41no—
95C2×Dic106—22no—
96C2×Dic106—22no—
97C2×Dic106—42no—
98C2×Dic106—42no—
99C2×Dic106—41no—
100C2×Dic106—41no—
101C2×Dic106—82no—
102C2×C8×C53—22no—
103C2×C8×C53—22no—
104C2×C8×C53—22no—
105C2×C8×C53—48no—
106C2×C8×C53—42no—
107C2×C8×C53C2×D412yes—
108C2×C8×C53C2×D412yes—
109C4⋊C4×C53—21no—
110C4⋊C4×C53C2×D412yes—
111C22⋊C4×C53—22no—
112C22⋊C4×C53—22no—
113C42×C53C2×D411yescharacteristic
114C42×C53C2×D411yescharacteristic
115C42×C53#4111yescharacteristic
116C42×C53C2411yescharacteristic
117C4⋊Dic53—22no—
118C4⋊Dic53—22no—
119C4⋊Dic53—22no—
120C4⋊Dic53—22no—
121C4⋊Dic53—41no—
122C4⋊Dic53—41no—
123C4⋊Dic53—41no—
124C4⋊Dic53—41no—
125C2×#88322—22no—
126C2×#88322—22no—
127C2×#88322—42no—
128C2×#88322—42no—
129C23×C53—22no—
130C23×C53—24no—
131C23×C53—42no—
132C23×C53—42no—
133C23×C53#12111yescharacteristic
134D4×C53—22no—
135D4×C53—24no—
136D4×C53C2×Q1614yes—
137C2×C4×C53—22no—
138C2×C4×C53—22no—
139C2×C4×C53—42no—
140C2×C4×C53—42no—
141C2×C4×C53—42no—
142C2×C4×C53—44no—
143C2×C4×C53#12111yescharacteristic
144C2×C4×C53#12111yescharacteristic
145C2×C4×C53#12111yescharacteristic
146C2×C4×C53C2×Q1612yes—
147C2×C4×C53#13512yes—
148C2×C4×C53#13512yes—
149C2×C4×C53AGL(1,ℤ/8)12yes—
150C2×C4×C53C22×D411yescharacteristic
151C2×C4×C53C22×D411yescharacteristic
152C2×C4×C53C2×#4111yescharacteristic
153C2×Dic53—42no—
154C2×Dic53—42no—
155C2×Dic53—82no—
156C2×Dic53—82no—
157Dic106—42no—
158Dic106—42no—
159Dic106—82no—
160Dic106—82no—
161Dic106—82no—
162C8×C53—22no—
163C8×C53—22no—
164C8×C53—42no—
165#88322—42no—
166#88322—42no—
167C2×#1683—531nomaximal, Sylow
168C22×C53—24no—
169C22×C53—48no—
170C22×C53—42no—
171C22×C53—42no—
172C22×C53#46312yes—
173C22×C53#46712yes—
174C22×C53#47112yes—
175C22×C53C2×#12111yescharacteristic
176C4×C53—21no—
177C4×C53—21no—
178C4×C53—42no—
179C4×C53#46312yes—
180C4×C53#46712yes—
181C4×C53C22×Q1611yescharacteristic
182C4×C53C2×#13511yescharacteristic
183C4×C53C2×#13511yescharacteristic
184C4×C53C2×AGL(1,ℤ/8)11yescharacteristic
185Dic53—82no—
186Dic53—82no—
187C2×#487—531no—
188C2×#476—531no—
189C2×#474—531no—
190C2×C8×D4D5311yescharacteristic
191#1683—538no—
192C2×#335—531no—
193C2×#333—531no—
194C2×#332—531no—
195C2×C53—44no—
196C2×C53#168314yescomplemented
197C2×C53C2×#46311yescharacteristic
198C2×C53C2×#46711yescharacteristic
199C2×C53C2×#47111yescharacteristic
200C23×C8—21no—
201C2×C4⋊Q8—531no—
202C2×C4⋊Q8—531no—
203C2×C4×D4C2×D5311yescharacteristic
204#487—534no—
205#487—1062no—
206#476—534no—
207#476—1062no—
208#474—534no—
209#474—1062no—
210C8×D4—22no—
211C8×D4C2×D5314yes—
212C2×#107—1061no—
213C2×#106—1061no—
214C2×C4⋊C8—531no—
215C2×C4⋊C8—531no—
216C2×C4⋊C8C2×D5311yescharacteristic
217C2×#103—1061no—
218C2×#103—1061no—
219C2×#102—1061no—
220C2×#102—1061no—
221C2×C22⋊C8—21no—
222C2×C4×C8C2×D5311yescharacteristic
223#335—534no—
224#333—534no—
225#332—534no—
226C53C2×#168311yescharacteristic, complemented, Sylow
227C22×Q8—1061no—
228C22×Q8—1061no—
229C22×D4D21211yescharacteristic
230C23×C4—21no—
231C22×C8—21no—
232C22×C8—24no—
233C22×C8—44no—
234C22×C8—1061no—
235C22×C8—1061no—
236C22×C8C22⋊D5311yescharacteristic
237C22×C8C22⋊D5311yescharacteristic
238C4⋊Q8—532no—
239C4⋊Q8—532no—
240C4⋊Q8—1061no—
241C4⋊Q8—1061no—
242C4⋊Q8—1061no—
243C4⋊Q8—1061no—
244C4⋊Q8—1061no—
245C4⋊Q8—1061no—
246C4×D4—21no—
247C4×D4—21no—
248C4×D4D21212yes—
249C4×D4C22×D5312yes—
250C2×C4⋊C4—1061no—
251C2×C4⋊C4—1061no—
252C2×C4⋊C4—1061no—
253C2×C4⋊C4—1061no—
254C2×C4⋊C4D21211yescharacteristic
255C2×C22⋊C4—21no—
256C2×C42C22×D5311yescharacteristic
257#107—1064no—
258#106—1064no—
259C4⋊C8—21no—
260C4⋊C8—532no—
261C4⋊C8—532no—
262C4⋊C8—1061no—
263C4⋊C8—1061no—
264C4⋊C8C22×D5312yes—
265#103—1064no—
266#103—1064no—
267#102—1064no—
268#102—1064no—
269C22⋊C8—24no—
270C4×C8C22⋊D5312yes—
271C4×C8C22×D5312yes—
272C24—21no—
273C2×Q8—1062no—
274C2×Q8—1062no—
275C2×Q8—4242no—
276C2×Q8—2122no—
277C2×Q8—2122no—
278C2×Q8—2121no—
279C2×Q8—2121no—
280C2×D4—21no—
281C2×D4—22no—
282C2×D4Dic21214yes—
283C2×D4C2×D21212yes—
284C22×C4—21no—
285C22×C4—22no—
286C22×C4—22no—
287C22×C4—42no—
288C22×C4—42no—
289C22×C4—2121no—
290C22×C4—2121no—
291C22×C4C2×D21211yescharacteristic
292C22×C4#1171994511yescharacteristic
293C22×C4C2×C22⋊D5311yescharacteristic
294C2×C8—22no—
295C2×C8—22no—
296C2×C8—22no—
297C2×C8—48no—
298C2×C8—42no—
299C2×C8—1062no—
300C2×C8—1062no—
301C2×C8—2122no—
302C2×C8—2122no—
303C2×C8C2×C22⋊D5312yes—
304C2×C8C2×C22⋊D5312yes—
305C4⋊C4—21no—
306C4⋊C4—1062no—
307C4⋊C4—1062no—
308C4⋊C4—1062no—
309C4⋊C4—1062no—
310C4⋊C4—2121no—
311C4⋊C4—2121no—
312C4⋊C4—2121no—
313C4⋊C4—2121no—
314C4⋊C4C2×D21212yes—
315C22⋊C4—22no—
316C22⋊C4—22no—
317C42C2×D21211yescharacteristic
318C42#1171994511yescharacteristic
319C42C2×C22⋊D5311yescharacteristic
320C42C23×D5311yescharacteristic
321C23—22no—
322C23—24no—
323C23—42no—
324C23—42no—
325C23#1313319111yescharacteristic
326Q8—4242no—
327Q8—4242no—
328Q8—4242no—
329Q8—2122no—
330Q8—2122no—
331D4—24no—
332D4—22no—
333D4C2×Dic21214yes—
334C2×C4—22no—
335C2×C4—22no—
336C2×C4—42no—
337C2×C4—44no—
338C2×C4—42no—
339C2×C4—42no—
340C2×C4—4242no—
341C2×C4—4242no—
342C2×C4—2122no—
343C2×C4—2122no—
344C2×C4#1313319111yescharacteristic
345C2×C4#1313319111yescharacteristic
346C2×C4#1313319111yescharacteristic
347C2×C4C2×Dic21212yes—
348C2×C4#1313323712yes—
349C2×C4#1313323912yes—
350C2×C4#1313324312yes—
351C2×C4C22×C22⋊D5311yescharacteristic
352C2×C4C22×D21211yescharacteristic
353C2×C4C2×#1171994511yescharacteristic
354C8—22no—
355C8—22no—
356C8—42no—
357C8—2122no—
358C8—2122no—
359C22—24no—
360C22—42no—
361C22—48no—
362C22—42no—
363C22#1681083812yes—
364C22#1681086412yes—
365C22#1681088612yes—
366C22C2×#1313319111yescharacteristic
367C4—21no—
368C4—21no—
369C4—42no—
370C4—4242no—
371C4—4242no—
372C4#1681083812yes—
373C4#1681086412yes—
374C4C22×Dic21211yescharacteristic
375C4C2×#1313323711yescharacteristic
376C4C2×#1313323911yescharacteristic
377C4C2×#1313324311yescharacteristic
378C2—44no—
379C2#2027416614yesdirect product factor
380C2C2×#1681083811yescharacteristic
381C2C2×#1681086411yescharacteristic
382C2C2×#1681088611yescharacteristic
subgroupG/H
centerC23#13133191
commutator, cosocleC42×C53C24
FrattiniC42C23×D53
FittingC2×C8×D4×C53C2
socleC23×C53#121
prime radicalC22×D4×C53D4
hypercenterC2×C8×D4D53
hypocenterC53C2×#1683
LCSG ▷ C42×C53 ▷ C22×C53 ▷ C53
UCSC1 ◁ C23 ◁ C2×C42 ◁ C2×C8×D4

Complements

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

NH ≅ G/Ntype
1. C2×#16810974C2semidirect
4. #20274166C2direct (N × H)
5. C2×#16810376C2semidirect
6. C2×#16810371C2semidirect
7. C2×#16810370C2semidirect
12. #16810974C22semidirect
25. #16810376C22semidirect
26. #16810371C22semidirect
27. #16810370C22semidirect
196. C2×C53#1683semidirect
226. C53C2×#1683semidirect
379. C2#20274166direct (N × H)

Maximal subgroups

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

subgroupindex
C2×#168109742
C2×#168109132
C2×#168109032
#202741662
C2×#168103762
C2×#168103712
C2×#168103702
C2×C8×D4×C532
C2×#168353