Finite Group Databasebeta

← C2×#10617094 #13045171

Subgroups — C2×#10617094 #13045171

Order 1280.

1978 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
1C22×#4033C211yesmaximal, characteristic
2C2×#65759C212yesmaximal
3C2×#65757C212yesmaximal
4C2×#65754C212yesmaximal
5#10617094C2116yesmaximal, direct product factor
6C2×#65707C218yesmaximal, complemented
7C23×Dic10C411yescharacteristic
8C22×C5×C4⋊C4C411yescharacteristic
9C22×#683C411yescharacteristic
10C22×#683C411yescharacteristic
11C22×C4×Dic5C2211yescharacteristic
12C22×C4×Dic5C2211yescharacteristic
13C22×C4×Dic5C2211yescharacteristic
14C2×#4033C414yes—
15C2×#4033C418yes—
16C2×#4033C2218yes—
17C2×#4033C2214yes—
18C2×C22⋊#170C2214yes—
19#65759C22116yes—
20#65757C22116yes—
21#65754C22116yes—
22C2×#4004C2214yes—
23C2×#4004C2214yes—
24C2×#4002C2214yes—
25C2×#4002C2214yes—
26C2×C4⋊#170C2214yes—
27C2×C4×#170C2214yes—
28#65707C22164yescomplemented
29C2×#2983—51nomaximal, Sylow
30C23×C4×C5C2×C411yescharacteristic
31C23×C4×C5C2×C411yescharacteristic
32C23×C4×C5C2×C411yescharacteristic
33C23×Dic5C2×C411yescharacteristic
34C23×Dic5C2×C411yescharacteristic
35C23×Dic5C2×C411yescharacteristic
36C23×Dic5C2311yescharacteristic
37C22×Dic10—28no—
38C22×Dic10C2×C418yes—
39C22×Dic10C2×C414yes—
40C22×#170C2318yes—
41C2×C5×C4⋊C4C2×C414yes—
42C2×C5×C4⋊C4C2×C418yes—
43C2×#683C2×C414yes—
44C2×#683C2×C418yes—
45C2×#683C2×C414yes—
46C2×#683C2×C418yes—
47C2×C4×Dic5C2×C412yes—
48C2×C4×Dic5C2×C414yes—
49C2×C4×Dic5C2×C412yes—
50C2×C4×Dic5C2×C414yes—
51C2×C4×Dic5C2×C412yes—
52C2×C4×Dic5C2×C414yes—
53C2×C4×Dic5C2312yes—
54C2×C4×Dic5C2312yes—
55C2×C4×Dic5C2312yes—
56C2×C4×Dic5C2314yes—
57C2×C4×Dic5C2314yes—
58C2×C4×Dic5C2314yes—
59#4033C2×C4132yes—
60#4033C2×C4116yes—
61#4033C23116yes—
62C22⋊#170C23116yes—
63#4004C23116yes—
64#4004C23116yes—
65#4002C23116yes—
66#4002C23116yes—
67C4⋊#170C23116yes—
68C4×#170C23116yes—
69C22×C4×Q8—51no—
70#2983—516no—
71C2×#446—58no—
72C2×#432—52no—
73C2×#432—52no—
74C2×#431—52no—
75C24×C5C22×C411yescharacteristic
76C22×C4×C5—24no—
77C22×C4×C5—42no—
78C22×C4×C5—42no—
79C22×C4×C5C22×C414yes—
80C22×C4×C5C22×C412yes—
81C22×C4×C5C22×C412yes—
82C22×C4×C5C22×C412yes—
83C22×C4×C5C22×C414yes—
84C22×C4×C5C22×C414yes—
85C22×Dic5—24no—
86C22×Dic5—24no—
87C22×Dic5—42no—
88C22×Dic5M1618yes—
89C22×Dic5C22×C414yes—
90C22×Dic5C22×C412yes—
91C22×Dic5C22×C412yes—
92C22×Dic5C22×C412yes—
93C22×Dic5C22×C411yescharacteristic
94C22×Dic5C22×C414yes—
95C22×Dic5C22×C414yes—
96C22×Dic5C22×C412yes—
97C22×Dic5C2412yes—
98C22×Dic5C2411yescharacteristic
99C2×Dic10—232no—
100C2×Dic10—216no—
101C2×Dic10C22×C4116yes—
102C2×#170—216no—
103C2×#170C24116yes—
104C5×C4⋊C4C22×C4116yes—
105#683C22×C4116yes—
106#683C22×C4116yes—
107C4×Dic5C22×C418yes—
108C4×Dic5C22×C418yes—
109C4×Dic5C22×C418yes—
110C4×Dic5C22×C414yes—
111C4×Dic5C22×C414yes—
112C4×Dic5C22×C414yes—
113C4×Dic5C2414yes—
114C4×Dic5C2414yes—
115C4×Dic5C2414yes—
116C23×Q8—51no—
117C2×C4×Q8—58no—
118C2×C4×Q8—54no—
119C2×C4×Q8—54no—
120C2×C4×Q8—58no—
121C22×C4⋊C4—51no—
122C22×C4⋊C4—51no—
123C22×C4⋊C4AGL(1,𝔽5)11yescharacteristic
124C22×C42—51no—
125C22×C42—51no—
126C22×C42—51no—
127#446—564no—
128#432—516no—
129#432—516no—
130#431—516no—
131C2×C4⋊C8—54no—
132C2×C4⋊C8—54no—
133C2×C4⋊C8—54no—
134C2×C22⋊C8—54no—
135C2×#97—54no—
136C2×#97—54no—
137C2×C4×C8—54no—
138C23×C5—22no—
139C23×C5—22no—
140C23×C5C2×M1614yes—
141C23×C5C23×C412yes—
142C23×C5C23×C411yescharacteristic
143C2×C4×C5—28no—
144C2×C4×C5—24no—
145C2×C4×C5—24no—
146C2×C4×C5—24no—
147C2×C4×C5—44no—
148C2×C4×C5—44no—
149C2×C4×C5C23×C414yes—
150C2×C4×C5C23×C414yes—
151C2×C4×C5C23×C414yes—
152C2×Dic5—22no—
153C2×Dic5—28no—
154C2×Dic5—24no—
155C2×Dic5—24no—
156C2×Dic5—24no—
157C2×Dic5—24no—
158C2×Dic5—44no—
159C2×Dic5C2×M16116yes—
160C2×Dic5C2×M1618yes—
161C2×Dic5C23×C412yes—
162C2×Dic5C23×C414yes—
163C2×Dic5C23×C414yes—
164C2×Dic5C23×C411yescharacteristic
165C2×Dic5C23×C414yes—
166C2×Dic52−1+414yes—
167C2×Dic5C2511yescharacteristic
168Dic10—232no—
169#170—216no—
170C22×Q8—58no—
171C22×Q8—54no—
172C22×Q8—108no—
173C23×C4—51no—
174C23×C4—51no—
175C23×C4—51no—
176C23×C4—51no—
177C23×C4C2×AGL(1,𝔽5)11yescharacteristic
178C23×C4C2×AGL(1,𝔽5)11yescharacteristic
179C23×C4C2×AGL(1,𝔽5)11yescharacteristic
180C22×C8—58no—
181C4×Q8—516no—
182C4×Q8—516no—
183C4×Q8—532no—
184C2×C4⋊C4—58no—
185C2×C4⋊C4—54no—
186C2×C4⋊C4—58no—
187C2×C4⋊C4—54no—
188C2×C4⋊C4C2×AGL(1,𝔽5)18yes—
189C2×C4⋊C4C2×AGL(1,𝔽5)14yes—
190C2×C42—52no—
191C2×C42—52no—
192C2×C42—52no—
193C2×C42—54no—
194C2×C42—54no—
195C2×C42—54no—
196C2×C42—54no—
197C2×C42—52no—
198C2×C42—54no—
199C2×C42—54no—
200C2×C42—52no—
201C2×C42—52no—
202C4⋊C8—516no—
203C4⋊C8—516no—
204C4⋊C8—516no—
205C22⋊C8—516no—
206#97—516no—
207#97—516no—
208C4×C8—516no—
209C22×C5—22no—
210C22×C5—28no—
211C22×C5—22no—
212C22×C5#51912yes—
213C22×C5C22×M1614yes—
214C22×C5C22×M1612yes—
215C22×C5#56812yes—
216C22×C5C24×C411yescharacteristic
217C4×C5—24no—
218C4×C5—24no—
219C4×C5—24no—
220Dic5—22no—
221Dic5—24no—
222Dic5—24no—
223Dic5#51912yes—
224Dic5C22×M1618yes—
225Dic5C2×2−1+412yes—
226C24C22×AGL(1,𝔽5)11yescharacteristic
227C2×Q8—516no—
228C2×Q8—1032no—
229C2×Q8—1016no—
230C22×C4—24no—
231C22×C4—42no—
232C22×C4—42no—
233C22×C4—58no—
234C22×C4—54no—
235C22×C4—51no—
236C22×C4—52no—
237C22×C4—52no—
238C22×C4—52no—
239C22×C4—54no—
240C22×C4—51no—
241C22×C4—52no—
242C22×C4—52no—
243C22×C4—54no—
244C22×C4—104no—
245C22×C4—104no—
246C22×C4—202no—
247C22×C4C22×AGL(1,𝔽5)14yes—
248C22×C4C22×AGL(1,𝔽5)12yes—
249C22×C4C22×AGL(1,𝔽5)12yes—
250C22×C4C22×AGL(1,𝔽5)12yes—
251C22×C4C22×AGL(1,𝔽5)14yes—
252C22×C4C22×AGL(1,𝔽5)14yes—
253C2×C8—516no—
254C2×C8—1016no—
255C4⋊C4—516no—
256C4⋊C4—516no—
257C4⋊C4C22×AGL(1,𝔽5)116yes—
258C42—54no—
259C42—54no—
260C42—54no—
261C42—54no—
262C42—54no—
263C42—54no—
264C42—58no—
265C42—58no—
266C42—58no—
267C2×C5—24no—
268C2×C5#298314yescomplemented
269C2×C5C2×#51911yescharacteristic
270C2×C5C23×M1611yescharacteristic
271C2×C5C2×#56811yescharacteristic
272C23—22no—
273C23—22no—
274C23D5⋊M1614yes—
275C23C23×AGL(1,𝔽5)12yes—
276C23C23×AGL(1,𝔽5)11yescharacteristic
277Q8—1032no—
278C2×C4—28no—
279C2×C4—24no—
280C2×C4—24no—
281C2×C4—24no—
282C2×C4—44no—
283C2×C4—44no—
284C2×C4—516no—
285C2×C4—58no—
286C2×C4—54no—
287C2×C4—51no—
288C2×C4—51no—
289C2×C4—54no—
290C2×C4—52no—
291C2×C4—54no—
292C2×C4—54no—
293C2×C4—104no—
294C2×C4—104no—
295C2×C4—108no—
296C2×C4—104no—
297C2×C4—104no—
298C2×C4—102no—
299C2×C4—204no—
300C2×C4C23×AGL(1,𝔽5)14yes—
301C2×C4C23×AGL(1,𝔽5)14yes—
302C2×C4C23×AGL(1,𝔽5)14yes—
303C8—1016no—
304C5C2×#298311yescharacteristic, complemented, Sylow
305C22—28no—
306C22—22no—
307C22—22no—
308C22C2×D5⋊M1614yes—
309C22C2×D5⋊M1612yes—
310C22#6625612yes—
311C22#6626212yes—
312C22C24×AGL(1,𝔽5)11yescharacteristic
313C4—24no—
314C4—24no—
315C4—24no—
316C4—58no—
317C4—52no—
318C4—52no—
319C4—104no—
320C4—104no—
321C4—102no—
322C2—24no—
323C2#1061709414yesdirect product factor
324C2C22×D5⋊M1611yescharacteristic
325C2C2×#6625611yescharacteristic
326C2C2×#6626211yescharacteristic
subgroupG/H
centerC23C23×AGL(1,𝔽5)
commutatorC22×C5C24×C4
FrattiniC22C24×AGL(1,𝔽5)
FittingC22×C5×C4⋊C4C4
socleC23×C5C23×C4
prime radicalC24×C5C22×C4
cosocleC2×Dic5C25
hypercenterC22×C4⋊C4AGL(1,𝔽5)
hypocenterC5C2×#2983
LCSG ▷ C22×C5 ▷ C5
UCSC1 ◁ C23 ◁ C22×C4⋊C4

Complements

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

NH ≅ G/Ntype
5. #10617094C2direct (N × H)
6. C2×#65707C2semidirect
28. #65707C22semidirect
268. C2×C5#2983semidirect
304. C5C2×#2983semidirect
323. C2#10617094direct (N × H)

Maximal subgroups

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

subgroupindex
C22×#40332
C2×#657592
C2×#657572
C2×#657542
#106170942
C2×#657072
C2×#29835