Finite Group Databasebeta

← C2×#20285365 #28253926

Subgroups — C2×#20285365 #28253926

Order 13568.

2098 conjugacy classes of subgroups.

The subgroup lattice has 370 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#16811228C212yesmaximal, complemented
2#20285365C2116yesmaximal, direct product factor
3C2×#16811025C211yesmaximal, characteristic, complemented
4C2×#16811000C211yesmaximal, characteristic, complemented
5C2×#16810988C212yesmaximal, complemented
6C2×#16810876C212yesmaximal, complemented
7C2×#16810859C212yesmaximal, complemented
8C2×#16810736C212yesmaximal, complemented
9C2×C4×D424C212yesmaximal, complemented
10C2×C53×#405C211yesmaximal, characteristic, complemented
11C22×D424C2212yes—
12C22×D424C2212yes—
13#16811228—28no—
14#16811228C22116yescomplemented
15C2×#13133214C2212yes—
16C2×#13133209C2212yes—
17C2×#13133202—24no—
18C2×#13133191C2214yes—
19C2×#13133190—24no—
20C2×C4×D212C2214yes—
21#16811025C2218yescomplemented
22#16811000C2218yescomplemented
23#16810988C22116yescomplemented
24#16810876C22116yescomplemented
25#16810859C22116yescomplemented
26#16810736C22116yescomplemented
27C4×D424C22116yescomplemented
28C2×#13133151C2212yes—
29C2×#13133149C2212yes—
30C2×#13133139C2214yes—
31C2×#13133139C2214yes—
32C22×M16×C53C2211yescharacteristic
33C23×C8×C53C2211yescharacteristic
34C2×#117×C53C2211yescharacteristic
35C53×#405C2218yescomplemented
36C2×#97×C53C2211yescharacteristic
37C2×#97×C53C2211yescharacteristic
38C2×C4×C8×C53C2211yescharacteristic
39C2×C4×C8×C53C2211yescharacteristic
40C22×D212—24no—
41C22×D212C2314yes—
42C22×C22⋊D53—24no—
43C22×C4×D53—24no—
44C2×D424—28no—
45C2×D424—28no—
46C2×D424C2318yes—
47C2×D424C2318yes—
48#13133214—24no—
49#13133214C2318yes—
50#13133209—24no—
51#13133209C2318yes—
52#13133202—232no—
53#13133191—28no—
54#13133191C23116yes—
55#13133190—232no—
56C4×D212—28no—
57C4×D212C23116yes—
58C2×C4⋊Dic53C2314yes—
59C2×#11719919—24no—
60C2×#11719921—24no—
61C2×#11719921—24no—
62#13133151C2318yes—
63#13133149C2318yes—
64#13133139C23116yes—
65#13133139C23116yes—
66C23×C4×C53C2311yescharacteristic
67C2×M16×C53—24no—
68C2×M16×C53C2314yes—
69C22×C8×C53—24no—
70C22×C8×C53C2314yes—
71C22×C8×C53C2311yescharacteristic
72C22×C8×C53C2311yescharacteristic
73C22×C8×C53C2311yescharacteristic
74C22×C8×C53C2311yescharacteristic
75#117×C53D414yes—
76#117×C53C2314yes—
77C2×C4⋊C4×C53D412yes—
78C2×C22⋊C4×C53D412yes—
79C2×C42×C53C2311yescharacteristic
80C2×C42×C53C2311yescharacteristic
81#97×C53C2314yes—
82#97×C53C2314yes—
83C4×C8×C53C2314yes—
84C4×C8×C53C2314yes—
85C24×C53C2×D411yescharacteristic
86C23×D53—24no—
87C22×C4×C53—22no—
88C22×C4×C53—22no—
89C22×C4×C53C2×D412yes—
90C22×C4×C53C2×D412yes—
91C22×C4×C53C2×D411yescharacteristic
92C22×C4×C53C2×D412yes—
93C22×C4×C53C2412yes—
94C22×C4×C53C2411yescharacteristic
95C2×C22⋊D53—216no—
96C2×C22⋊D53—88no—
97C22×Dic53—24no—
98C2×D212—28no—
99C2×D212—28no—
100C2×D212—24no—
101C2×D212—216no—
102C2×D212—88no—
103C2×D212C2418yes—
104C2×C4×D53—216no—
105C2×C4×D53—88no—
106M16×C53—28no—
107C2×C8×C53—22no—
108C2×C8×C53—22no—
109C2×C8×C53—28no—
110C2×C8×C53#4114yes—
111C2×C8×C53#4114yes—
112C2×C8×C53C2412yes—
113C2×C8×C53C2412yes—
114C2×C8×C53C2412yes—
115C2×C8×C53C2412yes—
116C4⋊C4×C53C2×D418yes—
117C22⋊C4×C53C2×D418yes—
118C42×C53C2×D412yes—
119C42×C53C2×D412yes—
120C42×C53C2412yes—
121C42×C53C2412yes—
122#11719921—216no—
123#11719921—216no—
124C4⋊Dic53—24no—
125C4⋊Dic53C2418yes—
126#11719919—216no—
127D424—216no—
128D424—216no—
129C23×C53—22no—
130C23×C53—24no—
131C23×C53C22×D412yes—
132C23×C53C22×D411yescharacteristic
133C22×D53—28no—
134C22×D53—48no—
135C22×D53—48no—
136C22×D53—48no—
137C2×C4×C53—22no—
138C2×C4×C53—24no—
139C2×C4×C53—24no—
140C2×C4×C53—24no—
141C2×C4×C53—22no—
142C2×C4×C53—22no—
143C2×C4×C53C22×D414yes—
144C2×C4×C53C22×D414yes—
145C2×C4×C53C22×D412yes—
146C2×C4×C53C22×D411yescharacteristic
147C2×C4×C53C2×#4112yes—
148C2×C4×C53C2×#4112yes—
149C2×C4×C53C2511yescharacteristic
150C22⋊D53—816no—
151C2×Dic53—28no—
152C2×Dic53—48no—
153D212—28no—
154D212—28no—
155D212—48no—
156D212—816no—
157C4×D53—816no—
158C8×C53—22no—
159C8×C53—22no—
160C8×C53C2×#4114yes—
161C8×C53C2×#4114yes—
162C2×#3176—531nomaximal, Sylow
163C22×C53—22no—
164C22×C53—24no—
165C22×C53—44no—
166C22×C53#53212yes—
167C22×C53#57614yes—
168C22×C53C23×D411yescharacteristic
169C2×D53—48no—
170C2×D53—48no—
171C2×D53—48no—
172C2×D53—816no—
173C4×C53—22no—
174C4×C53—24no—
175C4×C53—24no—
176C4×C53#53212yes—
177C4×C53C22×#4111yescharacteristic
178C4×C53C22×#4111yescharacteristic
179Dic53—48no—
180C2×#540—532no—
181#3176—5316no—
182C2×#499—531no—
183C2×#493—531no—
184C2×#490—532no—
185C2×#469—532no—
186C2×#466—532no—
187C2×#442—532no—
188C2×C4×D8—532no—
189C2×#405D5311yescharacteristic, complemented
190C2×C53—42no—
191C2×C53#317614yescomplemented
192C2×C53C2×#53211yescharacteristic
193C2×C53C2×#57612yes—
194D53—88no—
195C22×D8—532no—
196C22×D8—532no—
197C22×M16C2×D5311yescharacteristic
198C23×C8C2×D5311yescharacteristic
199#540—5316no—
200#540—1068no—
201C2×#127—532no—
202C2×#125—532no—
203C2×#123—1064no—
204C2×#121—534no—
205C2×#121—1064no—
206C2×C4×D4—534no—
207C2×#117C2×D5311yescharacteristic
208#499—538no—
209#493—538no—
210#490—5316no—
211#469—5316no—
212#466—5316no—
213#442—5316no—
214C4×D8—5316no—
215C2×#107—532no—
216C2×#106—532no—
217C2×#102—534no—
218C2×#102—534no—
219#405C2×D5318yescomplemented
220C2×#97C2×D5311yescharacteristic
221C2×#97C2×D5311yescharacteristic
222C2×C4×C8C2×D5311yescharacteristic
223C2×C4×C8C2×D5311yescharacteristic
224C53C2×#317611yescharacteristic, complemented, Sylow
225C22×D4—534no—
226C22×D4—1064no—
227C22×D4—1064no—
228C23×C4—1064no—
229C23×C4C22×D5311yescharacteristic
230C2×D8—538no—
231C2×D8—538no—
232C2×D8—1068no—
233C2×D8—1068no—
234C2×M16—24no—
235C2×M16C22×D5314yes—
236C22×C8—24no—
237C22×C8C22×D5311yescharacteristic
238C22×C8C22×D5314yes—
239C22×C8C22×D5311yescharacteristic
240C22×C8C22×D5311yescharacteristic
241C22×C8C22×D5311yescharacteristic
242#127—538no—
243#127—1064no—
244#125—538no—
245#125—1064no—
246#123—10632no—
247#121—5316no—
248#121—1068no—
249#121—10632no—
250C4×D4—5316no—
251C4×D4—1068no—
252#117D21214yes—
253#117C22×D5314yes—
254C2×C4⋊C4—534no—
255C2×C4⋊C4—1064no—
256C2×C4⋊C4D21212yes—
257C2×C22⋊C4—1064no—
258C2×C22⋊C4—1064no—
259C2×C22⋊C4D21212yes—
260C2×C42C22×D5311yescharacteristic
261C2×C42C22×D5311yescharacteristic
262#107—538no—
263#106—538no—
264#102—5316no—
265#102—5316no—
266#97C22×D5314yes—
267#97C22×D5314yes—
268C4×C8C22×D5314yes—
269C4×C8C22×D5314yes—
270C24—1064no—
271C24C2×D21211yescharacteristic
272C2×D4—538no—
273C2×D4—10616no—
274C2×D4—10616no—
275C2×D4—1064no—
276C2×D4—1068no—
277C2×D4—1068no—
278C2×D4—4248no—
279C2×D4—4248no—
280C22×C4—22no—
281C22×C4—22no—
282C22×C4—1064no—
283C22×C4—10616no—
284C22×C4—4248no—
285C22×C4C2×D21212yes—
286C22×C4C2×D21212yes—
287C22×C4C2×D21212yes—
288C22×C4C2×D21211yescharacteristic
289C22×C4C23×D5312yes—
290C22×C4C23×D5311yescharacteristic
291D8—10616no—
292D8—10616no—
293M16—28no—
294C2×C8—22no—
295C2×C8—22no—
296C2×C8—28no—
297C2×C8#1171994514yes—
298C2×C8#1171994514yes—
299C2×C8C23×D5312yes—
300C2×C8C23×D5312yes—
301C2×C8C23×D5312yes—
302C2×C8C23×D5312yes—
303C4⋊C4—538no—
304C4⋊C4—1064no—
305C4⋊C4—10616no—
306C4⋊C4C2×D21218yes—
307C22⋊C4—10616no—
308C22⋊C4—10616no—
309C22⋊C4C2×D21218yes—
310C42C2×D21212yes—
311C42C2×D21212yes—
312C42C23×D5312yes—
313C42C23×D5312yes—
314C23—24no—
315C23—22no—
316C23—1068no—
317C23—2128no—
318C23—2128no—
319C23—2128no—
320C23C22×D21212yes—
321C23C22×D21211yescharacteristic
322D4—1068no—
323D4—1068no—
324D4—42416no—
325D4—42416no—
326D4—2128no—
327C2×C4—24no—
328C2×C4—24no—
329C2×C4—22no—
330C2×C4—24no—
331C2×C4—22no—
332C2×C4—22no—
333C2×C4—1068no—
334C2×C4—42416no—
335C2×C4—2128no—
336C2×C4C22×D21214yes—
337C2×C4C22×D21214yes—
338C2×C4C22×D21212yes—
339C2×C4C22×D21211yescharacteristic
340C2×C4C2×#1171994512yes—
341C2×C4C2×#1171994512yes—
342C2×C4C24×D5311yescharacteristic
343C8—22no—
344C8—22no—
345C8C2×#1171994514yes—
346C8C2×#1171994514yes—
347C22—24no—
348C22—22no—
349C22—44no—
350C22—42416no—
351C22—2128no—
352C22—2128no—
353C22—2128no—
354C22#1681117312yes—
355C22#1681142814yes—
356C22C23×D21211yescharacteristic
357C4—24no—
358C4—24no—
359C4—22no—
360C4—2128no—
361C4#1681117312yes—
362C4C22×#1171994511yescharacteristic
363C4C22×#1171994511yescharacteristic
364C2—42no—
365C2—4248no—
366C2#2028536514yesdirect product factor
367C2C2×#1681117311yescharacteristic
368C2C2×#1681142812yes—
subgroupG/H
centerC23C22×D212
commutator, cosocleC2×C4×C53C25
FrattiniC2×C4C24×D53
FittingC2×C53×#405C2
socleC23×C53C22×D4
hypercenterC2×#405D53
hypocenterC53C2×#3176
LCSG ▷ C2×C4×C53 ▷ C2×C53 ▷ C53
UCSC1 ◁ C23 ◁ C2×#117 ◁ C2×#405

Complements

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

NH ≅ G/Ntype
1. C2×#16811228C2semidirect
2. #20285365C2direct (N × H)
3. C2×#16811025C2semidirect
4. C2×#16811000C2semidirect
5. C2×#16810988C2semidirect
6. C2×#16810876C2semidirect
7. C2×#16810859C2semidirect
8. C2×#16810736C2semidirect
9. C2×C4×D424C2semidirect
10. C2×C53×#405C2semidirect
14. #16811228C22semidirect
21. #16811025C22semidirect
22. #16811000C22semidirect
23. #16810988C22semidirect
24. #16810876C22semidirect
25. #16810859C22semidirect
26. #16810736C22semidirect
27. C4×D424C22semidirect
35. C53×#405C22semidirect
189. C2×#405D53semidirect
191. C2×C53#3176semidirect
219. #405C2×D53semidirect
224. C53C2×#3176semidirect
366. C2#20285365direct (N × H)

Maximal subgroups

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

subgroupindex
C2×#168112282
#202853652
C2×#168110252
C2×#168110002
C2×#168109882
C2×#168108762
C2×#168108592
C2×#168107362
C2×C4×D4242
C2×C53×#4052
C2×#317653