Finite Group Databasebeta

← C2×#10604963 #13043253

Subgroups — C2×#10604963 #13043253

Order 1280.

1230 conjugacy classes of subgroups; only the 224 normal subgroups are shown.

normal subgroups only

This group has 1230 conjugacy classes of subgroups — too many to draw. Only its 224 normal subgroups are shown.

Dotted edges: containments with no intermediate normal subgroup — omitted non-normal subgroups may lie between.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#66150C211yesmaximal, characteristic
2C2×#65745C212yesmaximal
3#10604963C218yesmaximal, direct product factor
4C2×#64928C214yesmaximal, complemented
5C23×C5×Q8C411yescharacteristic, complemented
6C22×#709C2211yescharacteristic
7C2×#4130—22no—
8#66150—24no—
9#66150C414yescomplemented
10#66150C2214yes—
11C2×#4093C2212yes—
12C2×#4092—22no—
13C2×#4090C412yescomplemented
14C2×#4075—22no—
15C2×#4072C411yescharacteristic, complemented
16C2×#4011—22no—
17#65745—24no—
18#65745C2218yes—
19C2×#3999C2214yes—
20#64928C22116yescomplemented
21C2×#2129—51nomaximal, Sylow
22C22×C5×Q8—21no—
23C22×C5×Q8—22no—
24C22×C5×Q8—42no—
25C22×C5×Q8—42no—
26C22×C5×Q8C2×C412yes—
27C22×C5×Q8C2×C414yescomplemented
28C23×C4×C5—21no—
29C23×C4×C5C2×C411yescharacteristic
30C22×C22⋊D5D412yes—
31C2×#709—42no—
32C2×#709—42no—
33C2×#709C2×C412yes—
34C2×#709D414yes—
35C2×#709C2312yes—
36C22×D20C2311yescharacteristic
37C22×C4×D5D412yes—
38C22×Dic10C2×C411yescharacteristic
39C2×C22⋊AGL(1,𝔽5)—24no—
40#4130—28no—
41#4130—44no—
42C2×C4⋊AGL(1,𝔽5)—42no—
43C2×C4×AGL(1,𝔽5)—42no—
44C2×C22⋊Dic5—21no—
45#4093—24no—
46#4093C2×C414yes—
47#4093C2314yes—
48#4092—28no—
49#4092—28no—
50#4090—24no—
51#4090C2×C418yescomplemented
52#4075—28no—
53#4075—28no—
54C2×#685—24no—
55#4072—22no—
56#4072C2×C414yescomplemented
57C2×C4⋊Dic5—21no—
58C2×#683—24no—
59C2×C4×Dic5C2×C412yes—
60#4011—28no—
61#3999—24no—
62#3999C2318yes—
63C2×#536—51no—
64C2×#410—52no—
65#2129—58no—
66C2×#354—54no—
67C24×C5C22⋊C411yescharacteristic
68C23×D5C2×D412yes—
69C22×AGL(1,𝔽5)—44no—
70C2×C5×Q8—22no—
71C2×C5×Q8—24no—
72C2×C5×Q8—48no—
73C2×C5×Q8—48no—
74C2×C5×Q8—42no—
75C2×C5×Q8—42no—
76C2×C5×Q8—42no—
77C2×C5×Q8—42no—
78C2×C5×Q8C22×C414yes—
79C22×C4×C5—22no—
80C22×C4×C5—22no—
81C22×C4×C5—22no—
82C22×C4×C5—24no—
83C22×C4×C5—42no—
84C22×C4×C5—42no—
85C22×C4×C5C22⋊C412yes—
86C22×C4×C5C22⋊C411yescharacteristic
87C22×C4×C5C22×C411yescharacteristic
88C22×C4×C5C22×C412yes—
89C2×C22⋊D5—42no—
90C2×C22⋊D5—42no—
91C2×C22⋊D5C22⋊C414yes—
92C2×C22⋊D5C2×D414yes—
93C22×Dic5—22no—
94C22×Dic5C22⋊C412yes—
95#709—44no—
96#709—44no—
97#709—44no—
98#709—44no—
99C2×D20—41no—
100C2×D20—41no—
101C2×D20C22×C411yescharacteristic
102C2×D20C2×D412yes—
103C2×D20C2411yescharacteristic
104C2×C4×D5—22no—
105C2×C4×D5—22no—
106C2×C4×D5—24no—
107C2×C4×D5C22⋊C414yes—
108C2×C4×D5C2×D414yes—
109C2×Dic10—41no—
110C2×Dic10—41no—
111C2×Dic10C22⋊C412yes—
112C2×Dic10C22×C412yes—
113C22⋊AGL(1,𝔽5)—28no—
114C22⋊AGL(1,𝔽5)—44no—
115C4⋊AGL(1,𝔽5)—48no—
116C4×AGL(1,𝔽5)—48no—
117C22⋊Dic5—24no—
118#685—28no—
119#685—28no—
120C4⋊Dic5—24no—
121#683—216no—
122C4×Dic5—22no—
123C4×Dic5C22×C414yes—
124C22×#41—51no—
125C23×Q8AGL(1,𝔽5)11yescharacteristic, complemented
126#536—54no—
127#536—54no—
128#536—104no—
129C2×C4⋊Q8—52no—
130C2×#124—52no—
131C2×#123—102no—
132C2×C22⋊Q8—102no—
133C2×#117—51no—
134C2×#117—102no—
135#410—58no—
136#410—104no—
137C2×#99—54no—
138C2×#99—102no—
139#354—516no—
140C23×C5—21no—
141C23×C5—21no—
142C23×C5—42no—
143C23×C5C2×C22⋊C411yescharacteristic
144C23×C5C2×C22⋊C412yes—
145C22×D5—22no—
146C22×D5—22no—
147C22×D5—44no—
148C22×D5C2×C22⋊C412yes—
149C22×D5C22×D412yes—
150C2×AGL(1,𝔽5)—48no—
151C2×AGL(1,𝔽5)—88no—
152C5×Q8—44no—
153C5×Q8—44no—
154C5×Q8—48no—
155C2×C4×C5—21no—
156C2×C4×C5—21no—
157C2×C4×C5—24no—
158C2×C4×C5—24no—
159C2×C4×C5—24no—
160C2×C4×C5—48no—
161C2×C4×C5—42no—
162C2×C4×C5—42no—
163C2×C4×C5#9912yes—
164C2×C4×C5#9912yes—
165C2×C4×C5C2×C22⋊C411yescharacteristic
166C2×C4×C5C2×C22⋊C412yes—
167C2×C4×C5C23×C411yescharacteristic
168C22⋊D5—44no—
169C22⋊D5—44no—
170C2×Dic5—22no—
171C2×Dic5—22no—
172C2×Dic5—24no—
173C2×Dic5—44no—
174C2×Dic5C2×C22⋊C414yes—
175D20—41no—
176D20—42no—
177D20—41no—
178C4×D5—24no—
179C4×D5—24no—
180C4×D5—24no—
181C4×D5—24no—
182Dic10—42no—
183Dic10—42no—
184C2×#41—54no—
185C2×#41—52no—
186C2×#41—52no—
187C2×#41—202no—
188C2×#41—202no—
189C22×Q8—22no—
190C22×Q8—21no—
191C22×Q8—42no—
192C22×Q8—42no—
193C22×Q8—51no—
194C22×Q8C2×AGL(1,𝔽5)12yes—
195C22×Q8C2×AGL(1,𝔽5)14yescomplemented
196C22×D4—52no—
197C22×D4—51no—
198C23×C4—21no—
199C23×C4—52no—
200C23×C4C2×AGL(1,𝔽5)11yescharacteristic
201C4⋊Q8—58no—
202C4⋊Q8—104no—
203#124—54no—
204#124—54no—
205#124—104no—
206#123—108no—
207#123—108no—
208C22⋊Q8—108no—
209C22⋊Q8—108no—
210#117—54no—
211#117—102no—
212#117—108no—
213#117—204no—
214C2×C4⋊C4—104no—
215C2×C4⋊C4—101no—
216C2×C4⋊C4—202no—
217C2×C22⋊C4—104no—
218C2×C22⋊C4—101no—
219C2×C22⋊C4—104no—
220C2×C42—52no—
221C2×C42—202no—
222#99—58no—
223#99—104no—
224#99—108no—
225C22×C5—22no—
226C22×C5—22no—
227C22×C5—42no—
228C22×C5—42no—
229C22×C5—42no—
230C22×C5C2×#9911yescharacteristic
231C22×C5C2×#9911yescharacteristic
232C22×C5C22×C22⋊C411yescharacteristic
233C2×D5—22no—
234C2×D5—22no—
235C2×D5—44no—
236C2×D5—42no—
237C2×D5—42no—
238C2×D5—82no—
239AGL(1,𝔽5)—88no—
240C4×C5—21no—
241C4×C5—21no—
242C4×C5—44no—
243C4×C5C2×#9914yes—
244Dic5—24no—
245Dic5—44no—
246C24—52no—
247C24C22⋊AGL(1,𝔽5)11yescharacteristic
248#41—204no—
249#41—204no—
250#41—204no—
251#41—204no—
252C2×Q8—24no—
253C2×Q8—22no—
254C2×Q8—48no—
255C2×Q8—48no—
256C2×Q8—42no—
257C2×Q8—42no—
258C2×Q8—42no—
259C2×Q8—42no—
260C2×Q8—52no—
261C2×Q8—52no—
262C2×Q8—201no—
263C2×Q8—201no—
264C2×Q8C22×AGL(1,𝔽5)14yes—
265C2×D4—54no—
266C2×D4—54no—
267C2×D4—52no—
268C2×D4—51no—
269C2×D4—51no—
270C2×D4—201no—
271C2×D4—201no—
272C2×D4—202no—
273C2×D4—202no—
274C22×C4—22no—
275C22×C4—22no—
276C22×C4—22no—
277C22×C4—24no—
278C22×C4—42no—
279C22×C4—42no—
280C22×C4—52no—
281C22×C4—54no—
282C22×C4—54no—
283C22×C4—102no—
284C22×C4—102no—
285C22×C4—104no—
286C22×C4—102no—
287C22×C4—204no—
288C22×C4C22⋊AGL(1,𝔽5)12yes—
289C22×C4C22⋊AGL(1,𝔽5)11yescharacteristic
290C22×C4C22×AGL(1,𝔽5)11yescharacteristic
291C22×C4C22×AGL(1,𝔽5)12yes—
292C4⋊C4—1016no—
293C4⋊C4—104no—
294C4⋊C4—208no—
295C22⋊C4—108no—
296C22⋊C4—108no—
297C22⋊C4—104no—
298C22⋊C4—108no—
299C22⋊C4—204no—
300C42—54no—
301C42—102no—
302C42—208no—
303C2×C5—21no—
304C2×C5—21no—
305C2×C5—42no—
306C2×C5#212912yescomplemented
307C2×C5C22×#9911yescharacteristic
308D5—42no—
309D5—42no—
310C23—21no—
311C23—21no—
312C23—42no—
313C23—52no—
314C23—52no—
315C23—102no—
316C23—102no—
317C23—204no—
318C23C2×C22⋊AGL(1,𝔽5)11yescharacteristic
319C23C2×C22⋊AGL(1,𝔽5)12yes—
320Q8—44no—
321Q8—44no—
322Q8—48no—
323Q8—202no—
324Q8—202no—
325D4—202no—
326D4—201no—
327D4—201no—
328D4—204no—
329D4—204no—
330C2×C4—21no—
331C2×C4—21no—
332C2×C4—24no—
333C2×C4—24no—
334C2×C4—24no—
335C2×C4—48no—
336C2×C4—42no—
337C2×C4—42no—
338C2×C4—54no—
339C2×C4—102no—
340C2×C4—102no—
341C2×C4—104no—
342C2×C4—104no—
343C2×C4—104no—
344C2×C4—104no—
345C2×C4—104no—
346C2×C4—204no—
347C2×C4—208no—
348C2×C4—408no—
349C2×C4#401112yes—
350C2×C4#401112yes—
351C2×C4C2×C22⋊AGL(1,𝔽5)11yescharacteristic
352C2×C4C2×C22⋊AGL(1,𝔽5)12yes—
353C2×C4C23×AGL(1,𝔽5)11yescharacteristic
354C5C2×#212911yescharacteristic, complemented, Sylow
355C22—22no—
356C22—22no—
357C22—42no—
358C22—42no—
359C22—42no—
360C22—102no—
361C22—102no—
362C22—202no—
363C22—202no—
364C22—204no—
365C22—402no—
366C22C2×#401111yescharacteristic
367C22C2×#401111yescharacteristic
368C22C22×C22⋊AGL(1,𝔽5)11yescharacteristic
369C4—21no—
370C4—21no—
371C4—44no—
372C4—104no—
373C4—204no—
374C4—408no—
375C4C2×#401114yes—
376C2—21no—
377C2—21no—
378C2—42no—
379C2—202no—
380C2—202no—
381C2#1060496312yesdirect product factor
382C2C22×#401111yescharacteristic
subgroupG/H
centerC22C2×#4011
commutatorC2×C4×C5C23×C4
FrattiniC2×C4C23×AGL(1,𝔽5)
FittingC23×C5×Q8C4
socleC22×C5C2×#99
prime radicalC22×#709C22
cosocleC2×D20C24
hypercenterC23×Q8AGL(1,𝔽5)
hypocenterC5C2×#2129
LCSG ▷ C2×C4×C5 ▷ C22×C5 ▷ C2×C5 ▷ C5
UCSC1 ◁ C22 ◁ C22×C4 ◁ C23×C4 ◁ C23×Q8

Complements

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

NH ≅ G/Ntype
3. #10604963C2direct (N × H)
4. C2×#64928C2semidirect
5. C23×C5×Q8C4semidirect
9. #66150C4semidirect
13. C2×#4090C4semidirect
15. C2×#4072C4semidirect
20. #64928C22semidirect
27. C22×C5×Q8C2×C4semidirect
51. #4090C2×C4semidirect
56. #4072C2×C4semidirect
125. C23×Q8AGL(1,𝔽5)semidirect
195. C22×Q8C2×AGL(1,𝔽5)semidirect
306. C2×C5#2129semidirect
354. C5C2×#2129semidirect
381. C2#10604963direct (N × H)

Maximal subgroups

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

subgroupindex
C2×#661502
C2×#657452
#106049632
C2×#649282
C2×#21295