Finite Group Databasebeta

← C2×#91498 #11754724

Subgroups — C2×#91498 #11754724

Order 960.

596 conjugacy classes of subgroups; only the 136 normal subgroups are shown.

normal subgroups only

This group has 596 conjugacy classes of subgroups — too many to draw. Only its 136 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×D4×D15C211yesmaximal, characteristic, complemented
2C2×#6766C211yesmaximal, characteristic, complemented
3C2×C5×#214C211yesmaximal, characteristic, complemented
4C2×C3×#686C211yesmaximal, characteristic, complemented
5#91498C218yesmaximal, direct product factor
6C2×#6660C211yesmaximal, characteristic, complemented
7C2×#6653C211yesmaximal, characteristic, complemented
8C2×#6651C211yesmaximal, characteristic, complemented
9C2×#4060—31nomaximal, Hall
10C23×D15—21no—
11C2×C3×C5×D4C2211yescharacteristic
12C2×#1220—21no—
13D4×D15—22no—
14D4×D15C2214yescomplemented
15C2×D60C2211yescharacteristic
16C2×C4×D15C2211yescharacteristic
17C2×C5×Dic6C2211yescharacteristic, complemented
18C2×C3×D20C2211yescharacteristic, complemented
19C2×#1202—21no—
20C2×D5×Dic3—21no—
21#6766—22no—
22#6766C2214yescomplemented
23C5×#214C2214yescomplemented
24C2×C5×C3⋊C8C2211yescharacteristic, complemented
25C3×#686C2214yescomplemented
26C2×C3×#168C2211yescharacteristic, complemented
27#6660C2214yescomplemented
28#6653C2214yescomplemented
29#6651C2214yescomplemented
30C2×#911—51nomaximal, Hall
31C2×#713—31no—
32C2×D4×D5—31no—
33C2×C5×SD16—31no—
34C2×#688—31no—
35C2×#686D311yescharacteristic, complemented
36#4060—38no—
37C2×D40—31no—
38C2×#676—31no—
39C23×C3×C5—21no—
40C22×D15—21no—
41C22×D15—24no—
42C22×D15—44no—
43C22×D15D411yescharacteristic
44C22×C3×D5—21no—
45C3×C5×D4—21no—
46C3×C5×D4C2312yes—
47C2×C3×C4×C5C2311yescharacteristic
48#1220—24no—
49C2×Dic15D411yescharacteristic
50D60—21no—
51D60C2312yes—
52C4×D15D412yes—
53C4×D15C2312yes—
54C2×C5×Dic3—21no—
55C5×Dic6—21no—
56C5×Dic6C2312yescomplemented
57C3×D20—21no—
58C3×D20C2312yescomplemented
59#1202—24no—
60D5×Dic3—24no—
61C3×#168C2312yescomplemented
62C5×C3⋊C8C2312yescomplemented
63C2×#237—51no—
64C2×D3×D4—51no—
65C2×C3×D8—51no—
66C2×#214D511yescharacteristic, complemented
67C2×#213—51no—
68#911—58no—
69C2×#204—51no—
70C2×ΓL(1,𝔽25)—51no—
71C23×D5—61no—
72C2×C5×Q8—31no—
73C2×C5×D4C2×D311yescharacteristic
74C2×C22⋊D5—61no—
75#713—34no—
76#713—62no—
77D4×D5—34no—
78D4×D5—62no—
79C2×D20—31no—
80C2×D20—61no—
81C2×D20C2×D311yescharacteristic, complemented
82C2×C4×D5—31no—
83C2×C4×D5—61no—
84C5×SD16—34no—
85C2×C5×C8—31no—
86#688—34no—
87#686C2×D314yescomplemented
88C2×#168C2×D311yescharacteristic, complemented
89D40—34no—
90#676—34no—
91C2×AGL(1,ℤ/8)—151noSylow
92C22×C3×C5—22no—
93C22×C3×C5—42no—
94C22×C3×C5C2×D411yescharacteristic
95C2×D15—22no—
96C2×D15—22no—
97C2×D15—48no—
98C2×D15—42no—
99C2×D15C2×D412yes—
100C2×C3×D5—22no—
101C2×C3×D5—42no—
102C3×C4×C5C2×D411yescharacteristic
103C3×C4×C5C2411yescharacteristic
104Dic15C2×D412yes—
105C5×Dic3—22no—
106C23×D3—101no—
107C2×C3×D4—51no—
108C2×C3×D4C2×D511yescharacteristic
109C2×#67—101no—
110C2×#67—101no—
111C22×Dic3—101no—
112#237—54no—
113#237—102no—
114D3×D4—54no—
115D3×D4—102no—
116C2×D12—51no—
117C2×C4×D3—51no—
118C2×Dic6C2×D511yescharacteristic, complemented
119C3×D8—54no—
120C2×C3×C8—51no—
121#214C2×D514yescomplemented
122#213—54no—
123C2×C3⋊C8C2×D511yescharacteristic, complemented
124#204—54no—
125ΓL(1,𝔽25)—54no—
126C23×C5—21no—
127C22×D5—21no—
128C22×D5—31no—
129C22×D5—64no—
130C22×D5—61no—
131C22×D5—124no—
132C5×Q8—32no—
133C5×Q8—61no—
134C5×D4—21no—
135C5×D4C22×D312yes—
136C2×C4×C5—61no—
137C2×C4×C5C22×D311yescharacteristic
138C22⋊D5—64no—
139C2×Dic5—31no—
140D20—21no—
141D20—32no—
142D20—64no—
143D20—61no—
144D20C22×D312yescomplemented
145C4×D5—32no—
146C4×D5—32no—
147C4×D5—64no—
148C5×C8—32no—
149#168C22×D312yescomplemented
150C2×#41—151no—
151C22×D4—151no—
152AGL(1,ℤ/8)—158no—
153C2×SD16—31no—
154C2×SD16—151no—
155C2×D8—51no—
156C2×D8—151no—
157C2×M16—151no—
158C2×C3×C5—42no—
159C2×C3×C5AGL(1,ℤ/8)12yescomplemented
160C2×C3×C5C22×D411yescharacteristic
161D15—22no—
162D15—42no—
163C3×D5—42no—
164C23×C3—21no—
165C23×C3—101no—
166C22×D3—51no—
167C22×D3—104no—
168C22×D3—101no—
169C22×D3—204no—
170C3×D4—21no—
171C3×D4—52no—
172C3×D4—101no—
173C3×D4C22×D512yes—
174C2×C3×C4C22×D511yescharacteristic
175#67—104no—
176#67—104no—
177C2×Dic3—21no—
178C2×Dic3—51no—
179C2×Dic3—104no—
180D12—52no—
181D12—101no—
182C4×D3—52no—
183C4×D3—52no—
184Dic6—21no—
185Dic6C22×D512yescomplemented
186C3×C8—52no—
187C3⋊C8C22×D512yescomplemented
188C22×C5—22no—
189C22×C5—42no—
190C22×C5D3×D411yescharacteristic
191C2×D5—22no—
192C2×D5—32no—
193C2×D5—42no—
194C2×D5—62no—
195C2×D5—62no—
196C2×D5—128no—
197C2×D5—122no—
198C4×C5—62no—
199C4×C5D3×D411yescharacteristic
200C4×C5C23×D311yescharacteristic
201Dic5—32no—
202C24—301no—
203#41—154no—
204#41—302no—
205C2×Q8—31no—
206C2×D4—51no—
207C2×D4—154no—
208C2×D4—151no—
209C2×D4—301no—
210C2×D4—302no—
211C2×D4—301no—
212C2×D4D3×D511yescharacteristic
213C22×C4—151no—
214C22×C4—301no—
215SD16—34no—
216SD16—154no—
217D8—54no—
218D8—154no—
219M16—154no—
220C2×C8—31no—
221C2×C8—51no—
222C3×C5C2×AGL(1,ℤ/8)11yescharacteristic, complemented, Hall
223C22×C3—22no—
224C22×C3—42no—
225C22×C3—102no—
226C22×C3—202no—
227C22×C3D4×D511yescharacteristic
228C2×D3—52no—
229C2×D3—102no—
230C2×D3—102no—
231C2×D3—208no—
232C2×D3—202no—
233C3×C4D4×D511yescharacteristic
234C3×C4C23×D511yescharacteristic
235Dic3—22no—
236Dic3—52no—
237C2×C5—42no—
238C2×C5#91112yescomplemented
239C2×C5C2×D3×D411yescharacteristic
240D5—42no—
241D5—62no—
242D5—122no—
243C23—21no—
244C23—101no—
245C23—151no—
246C23—304no—
247C23—301no—
248C23—604no—
249Q8—32no—
250Q8—61no—
251D4—21no—
252D4—52no—
253D4—101no—
254D4—152no—
255D4—304no—
256D4—304no—
257D4—301no—
258D4C2×D3×D512yes—
259C2×C4—61no—
260C2×C4—151no—
261C2×C4—152no—
262C2×C4—152no—
263C2×C4—304no—
264C2×C4C2×D3×D511yescharacteristic
265C8—32no—
266C8—52no—
267C2×C3—42no—
268C2×C3—202no—
269C2×C3#406012yescomplemented
270C2×C3C2×D4×D511yescharacteristic
271D3—102no—
272D3—202no—
273C5C2×#91111yescharacteristic, complemented, Sylow
274C22—22no—
275C22—42no—
276C22—102no—
277C22—152no—
278C22—202no—
279C22—302no—
280C22—302no—
281C22—608no—
282C22—602no—
283C22#679011yescharacteristic
284C4—62no—
285C4—152no—
286C4#679011yescharacteristic
287C4C22×D3×D511yescharacteristic
288C3C2×#406011yescharacteristic, complemented, Sylow
289C2—42no—
290C2—202no—
291C2—302no—
292C2—602no—
293C2#9149812yesdirect product factor
294C2C2×#679011yescharacteristic
subgroupG/H
centerC22#6790
commutator, cosocleC3×C4×C5C24
FrattiniC4C22×D3×D5
FittingC2×C3×C5×D4C22
socleC22×C3×C5C2×D4
hypercenterC2×D4D3×D5
hypocenterC3×C5C2×AGL(1,ℤ/8)
LCSG ▷ C3×C4×C5 ▷ C2×C3×C5 ▷ C3×C5
UCSC1 ◁ C22 ◁ C2×C4 ◁ C2×D4

Complements

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

NH ≅ G/Ntype
1. C2×D4×D15C2semidirect
2. C2×#6766C2semidirect
3. C2×C5×#214C2semidirect
4. C2×C3×#686C2semidirect
5. #91498C2direct (N × H)
6. C2×#6660C2semidirect
7. C2×#6653C2semidirect
8. C2×#6651C2semidirect
14. D4×D15C22semidirect
17. C2×C5×Dic6C22semidirect
18. C2×C3×D20C22semidirect
22. #6766C22semidirect
23. C5×#214C22semidirect
24. C2×C5×C3⋊C8C22semidirect
25. C3×#686C22semidirect
26. C2×C3×#168C22semidirect
27. #6660C22semidirect
28. #6653C22semidirect
29. #6651C22semidirect
35. C2×#686D3semidirect
56. C5×Dic6C23semidirect
58. C3×D20C23semidirect
61. C3×#168C23semidirect
62. C5×C3⋊C8C23semidirect
66. C2×#214D5semidirect
81. C2×D20C2×D3semidirect
87. #686C2×D3semidirect
88. C2×#168C2×D3semidirect
118. C2×Dic6C2×D5semidirect
121. #214C2×D5semidirect
123. C2×C3⋊C8C2×D5semidirect
144. D20C22×D3semidirect
149. #168C22×D3semidirect
159. C2×C3×C5AGL(1,ℤ/8)semidirect
185. Dic6C22×D5semidirect
187. C3⋊C8C22×D5semidirect
222. C3×C5C2×AGL(1,ℤ/8)semidirect
238. C2×C5#911semidirect
269. C2×C3#4060semidirect
273. C5C2×#911semidirect
288. C3C2×#4060semidirect
293. C2#91498direct (N × H)

Maximal subgroups

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

subgroupindex
C2×D4×D152
C2×#67662
C2×C5×#2142
C2×C3×#6862
#914982
C2×#66602
C2×#66532
C2×#66512
C2×#40603
C2×#9115