Finite Group Databasebeta

← C2×#10604688 #13028981

Subgroups — C2×#10604688 #13028981

Order 1280.

966 conjugacy classes of subgroups.

The subgroup lattice has 328 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#65754C211yesmaximal, characteristic
2C2×C4×#704C211yesmaximal, characteristic
3C22×C8×Dic5C211yesmaximal, characteristic
4#10604688C218yesmaximal, direct product factor
5C2×#64879C212yesmaximal, complemented
6C2×#170⋊C8C212yesmaximal, complemented
7C22×#704C412yes—
8C23×C5×C8C411yescharacteristic
9C22×C4×Dic5C2211yescharacteristic
10C23×#168C411yescharacteristic
11C2×C22⋊#170C412yes—
12#65754C414yes—
13#65754C2214yes—
14C2×#4004C412yes—
15C4×#704C414yes—
16C4×#704C2214yes—
17C2×#4002C412yes—
18C2×C4×#170C2212yes—
19C2×C4×#170C2212yes—
20C2×C8×Dic5C414yes—
21C2×C8×Dic5C411yescharacteristic
22C2×C8×Dic5C411yescharacteristic
23C2×C8×Dic5C2211yescharacteristic
24C2×C8×Dic5C2211yescharacteristic
25C2×C8×Dic5C2214yes—
26#170⋊C8C2218yescomplemented
27#64879C2218yescomplemented
28C2×#1451—51nomaximal, Sylow
29C23×C4×C5C2×C411yescharacteristic
30C23×Dic5C2×C411yescharacteristic
31C23×Dic5C2×C411yescharacteristic
32C2×#704—28no—
33C2×#704C2×C418yes—
34C22×#170C2×C414yes—
35C22×#170C2×C414yes—
36C22×C5×C8—24no—
37C22×C5×C8C2×C414yes—
38C22×C5×C8C2×C411yescharacteristic
39C22×C5×C8C2×C411yescharacteristic
40C2×C4×Dic5C2×C412yes—
41C2×C4×Dic5C2×C411yescharacteristic
42C2×C4×Dic5C2×C411yescharacteristic
43C2×C4×Dic5C2×C412yes—
44C2×C4×Dic5C2×C411yescharacteristic
45C2×C4×Dic5C2×C412yes—
46C2×C4×Dic5C2311yescharacteristic
47C2×C4×Dic5C2312yes—
48C22×#168—24no—
49C22×#168C2×C414yes—
50C22×#168C2×C411yescharacteristic
51C22×#168C2×C411yescharacteristic
52C22⋊#170C2×C418yes—
53#4004C2×C418yes—
54#4002C2×C418yes—
55C4×#170C2×C414yes—
56C4×#170C2×C414yes—
57C4×#170C2314yes—
58C4×#170C2314yes—
59C8×Dic5C2×C412yes—
60C8×Dic5C2×C414yes—
61C8×Dic5C2×C414yes—
62C8×Dic5C2×C412yes—
63C8×Dic5C2312yes—
64C8×Dic5C2312yes—
65C2×#431—51no—
66C2×C4×M16—51no—
67C22×C4×C8—51no—
68#1451—58no—
69C2×#322—52no—
70C2×#322—52no—
71C24×C5C4211yescharacteristic
72C22×C4×C5—22no—
73C22×C4×C5—22no—
74C22×C4×C5C4211yescharacteristic
75C22×C4×C5C4212yes—
76C22×C4×C5C22×C412yes—
77C22×C4×C5C22×C411yescharacteristic
78C22×Dic5—24no—
79C22×Dic5C4211yescharacteristic
80C22×Dic5C4212yes—
81C22×Dic5C4212yes—
82C22×Dic5C4211yescharacteristic
83C22×Dic5M1618yes—
84C22×Dic5C22×C412yes—
85C22×Dic5C22×C411yescharacteristic
86C22×Dic5C22×C412yes—
87C22×Dic5C22×C411yescharacteristic
88#704—216no—
89C2×#170—28no—
90C2×#170—28no—
91C2×#170C22×C418yes—
92C2×#170C22×C418yes—
93C2×C5×C8—22no—
94C2×C5×C8—28no—
95C2×C5×C8M1614yescomplemented
96C2×C5×C8C22×C412yes—
97C2×C5×C8C22×C412yes—
98C4×Dic5C4212yes—
99C4×Dic5C4212yes—
100C4×Dic5C4212yes—
101C4×Dic5C22×C412yes—
102C4×Dic5C22×C411yescharacteristic
103C4×Dic5C22×C411yescharacteristic
104C4×Dic5C22×C411yescharacteristic
105C4×Dic5C22×C412yes—
106C4×Dic5C22×C412yes—
107C4×Dic5C2411yescharacteristic
108C2×#168—22no—
109C2×#168—28no—
110C2×#168M1614yescomplemented
111C2×#168C22×C412yes—
112C2×#168C22×C412yes—
113C22×M16—52no—
114C23×C8—51no—
115C23×C8AGL(1,𝔽5)11yescharacteristic
116C22×C42—51no—
117#431—54no—
118#431—54no—
119C2×C4⋊C8—52no—
120C2×C22⋊C8—52no—
121C4×M16—54no—
122C4×M16—54no—
123C2×#97—52no—
124C2×C4×C8—52no—
125C2×C4×C8—52no—
126C2×C4×C8—54no—
127C2×C4×C8—51no—
128C2×C4×C8—51no—
129C2×C4×C8—51no—
130C2×C4×C8—51no—
131C2×C4×C8—54no—
132#322—58no—
133#322—58no—
134C23×C5—22no—
135C23×C5—22no—
136C23×C5#9714yes—
137C23×C5C2×C4212yes—
138C23×C5C2×C4211yescharacteristic
139C2×C4×C5—22no—
140C2×C4×C5—22no—
141C2×C4×C5—22no—
142C2×C4×C5—24no—
143C2×C4×C5C2×C4212yes—
144C2×C4×C5C2×C4211yescharacteristic
145C2×C4×C5C2×M1612yes—
146C2×C4×C5C2×M1612yes—
147C2×C4×C5C23×C411yescharacteristic
148C2×Dic5—22no—
149C2×Dic5—24no—
150C2×Dic5—24no—
151C2×Dic5#9718yes—
152C2×Dic5#9714yes—
153C2×Dic5C2×C4212yes—
154C2×Dic5C2×C4211yescharacteristic
155C2×Dic5C2×C4212yes—
156C2×Dic5C2×C4211yescharacteristic
157C2×Dic5C2×M1614yes—
158C2×Dic5C2×M1618yes—
159C2×Dic5#13114yes—
160C2×Dic5C23×C411yescharacteristic
161C2×Dic5C23×C411yescharacteristic
162#170—28no—
163#170—28no—
164C5×C8—22no—
165C5×C8C2×M1614yescomplemented
166#168—22no—
167#168C2×M1614yescomplemented
168C23×C4—51no—
169C23×C4—51no—
170C23×C4C2×AGL(1,𝔽5)11yescharacteristic
171C2×M16—58no—
172C2×M16—108no—
173C22×C8—24no—
174C22×C8—51no—
175C22×C8—54no—
176C22×C8—54no—
177C22×C8—54no—
178C22×C8—51no—
179C22×C8—104no—
180C22×C8C2×AGL(1,𝔽5)14yes—
181C22×C8C2×AGL(1,𝔽5)11yescharacteristic
182C22×C8C2×AGL(1,𝔽5)11yescharacteristic
183C2×C42—51no—
184C2×C42—52no—
185C2×C42—51no—
186C2×C42—52no—
187C2×C42—52no—
188C2×C42—52no—
189C2×C42—51no—
190C2×C42—51no—
191C4⋊C8—58no—
192C22⋊C8—58no—
193#97—58no—
194C4×C8—54no—
195C4×C8—54no—
196C4×C8—54no—
197C4×C8—54no—
198C4×C8—52no—
199C4×C8—52no—
200C4×C8—54no—
201C4×C8—52no—
202C4×C8—52no—
203C4×C8—54no—
204C22×C5—22no—
205C22×C5—22no—
206C22×C5—28no—
207C22×C5C2×#9714yes—
208C22×C5C2×#9712yes—
209C22×C5C4×M1612yes—
210C22×C5#40512yes—
211C22×C5C22×C4211yescharacteristic
212C4×C5—22no—
213C4×C5C4×M1612yes—
214C4×C5C22×M1611yescharacteristic
215C4×C5C22×M1611yescharacteristic
216Dic5—22no—
217Dic5C2×#9712yes—
218Dic5C2×#9714yes—
219Dic5#40512yes—
220Dic5C22×M1612yes—
221Dic5C2×#13112yes—
222C24C4×AGL(1,𝔽5)11yescharacteristic
223C22×C4—22no—
224C22×C4—22no—
225C22×C4—51no—
226C22×C4—51no—
227C22×C4—52no—
228C22×C4—52no—
229C22×C4—51no—
230C22×C4—58no—
231C22×C4—52no—
232C22×C4—52no—
233C22×C4—51no—
234C22×C4—104no—
235C22×C4C4×AGL(1,𝔽5)11yescharacteristic
236C22×C4C4×AGL(1,𝔽5)12yes—
237C22×C4C22×AGL(1,𝔽5)12yes—
238C22×C4C22×AGL(1,𝔽5)11yescharacteristic
239M16—1016no—
240C2×C8—22no—
241C2×C8—28no—
242C2×C8—54no—
243C2×C8—52no—
244C2×C8—58no—
245C2×C8—58no—
246C2×C8—52no—
247C2×C8—108no—
248C2×C8—108no—
249C2×C8—108no—
250C2×C8—102no—
251C2×C8#70414yescomplemented
252C2×C8C22×AGL(1,𝔽5)12yes—
253C2×C8C22×AGL(1,𝔽5)12yes—
254C42—51no—
255C42—51no—
256C42—52no—
257C42—52no—
258C42—52no—
259C42—52no—
260C42—52no—
261C42—51no—
262C42—51no—
263C42—52no—
264C2×C5—24no—
265C2×C5#145114yescomplemented
266C2×C5C22×#9711yescharacteristic
267C2×C5C2×C4×M1611yescharacteristic
268C2×C5C2×#40511yescharacteristic
269C23—22no—
270C23—22no—
271C23#399214yes—
272C23C2×C4×AGL(1,𝔽5)12yes—
273C23C2×C4×AGL(1,𝔽5)11yescharacteristic
274C2×C4—22no—
275C2×C4—22no—
276C2×C4—24no—
277C2×C4—22no—
278C2×C4—54no—
279C2×C4—58no—
280C2×C4—58no—
281C2×C4—54no—
282C2×C4—54no—
283C2×C4—51no—
284C2×C4—51no—
285C2×C4—51no—
286C2×C4—52no—
287C2×C4—51no—
288C2×C4—52no—
289C2×C4—104no—
290C2×C4—104no—
291C2×C4—102no—
292C2×C4D5⋊M1612yes—
293C2×C4C2×C4×AGL(1,𝔽5)12yes—
294C2×C4C2×C4×AGL(1,𝔽5)11yescharacteristic
295C2×C4C2×#70412yes—
296C2×C4C23×AGL(1,𝔽5)11yescharacteristic
297C8—22no—
298C8—54no—
299C8—108no—
300C8—108no—
301C8—102no—
302C8C2×#70414yescomplemented
303C5C2×#145111yescharacteristic, complemented, Sylow
304C22—22no—
305C22—22no—
306C22—28no—
307C22C2×#399214yes—
308C22C2×#399212yes—
309C22#6571812yes—
310C22C4×#70412yes—
311C22C22×C4×AGL(1,𝔽5)11yescharacteristic
312C4—22no—
313C4—52no—
314C4—52no—
315C4—54no—
316C4—52no—
317C4—52no—
318C4—102no—
319C4C4×#70412yes—
320C4C2×D5⋊M1611yescharacteristic
321C4C22×#70411yescharacteristic
322C2—24no—
323C2#1060468814yesdirect product factor
324C2C22×#399211yescharacteristic
325C2C2×#6571811yescharacteristic
326C2C2×C4×#70411yescharacteristic
subgroupG/H
centerC22×C4C22×AGL(1,𝔽5)
commutatorC22×C5C22×C42
FrattiniC2×C4C23×AGL(1,𝔽5)
FittingC23×C5×C8C4
socleC23×C5C2×C42
prime radicalC24×C5C42
cosocleC4×Dic5C24
hypercenterC23×C8AGL(1,𝔽5)
hypocenterC5C2×#1451
LCSG ▷ C22×C5 ▷ C5
UCSC1 ◁ C22×C4 ◁ C23×C8

Complements

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

NH ≅ G/Ntype
4. #10604688C2direct (N × H)
5. C2×#64879C2semidirect
6. C2×#170⋊C8C2semidirect
26. #170⋊C8C22semidirect
27. #64879C22semidirect
95. C2×C5×C8M16semidirect
110. C2×#168M16semidirect
165. C5×C8C2×M16semidirect
167. #168C2×M16semidirect
251. C2×C8#704semidirect
265. C2×C5#1451semidirect
302. C8C2×#704semidirect
303. C5C2×#1451semidirect
323. C2#10604688direct (N × H)

Maximal subgroups

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

subgroupindex
C2×#657542
C2×C4×#7042
C22×C8×Dic52
#106046882
C2×#648792
C2×#170⋊C82
C2×#14515