Finite Group Databasebeta

← C2×#91489 #11754714

Subgroups — C2×#91489 #11754714

Order 960.

596 conjugacy classes of subgroups; only the 144 normal subgroups are shown.

normal subgroups only

This group has 596 conjugacy classes of subgroups — too many to draw. Only its 144 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×C3×D4×D5C211yesmaximal, characteristic, complemented
2C2×#6772C211yesmaximal, characteristic, complemented
3C2×#6716C211yesmaximal, characteristic, complemented
4C2×C5×#213C211yesmaximal, characteristic, complemented
5#91489C218yesmaximal, direct product factor
6C2×#6657C211yesmaximal, characteristic, complemented
7C2×#6652C211yesmaximal, characteristic, complemented
8C2×#6649C211yesmaximal, characteristic, complemented
9C2×#4057—31nomaximal, Hall
10C23×C3×D5—21no—
11C2×C3×C5×D4C2211yescharacteristic
12C2×Dic30C2211yescharacteristic, complemented
13C2×C5×D12C2211yescharacteristic, complemented
14C2×C3×C22⋊D5—21no—
15C3×D4×D5—22no—
16C3×D4×D5C2214yescomplemented
17C2×C3×D20C2211yescharacteristic
18C2×C3×C4×D5C2211yescharacteristic
19C2×#1201—21no—
20C2×D3×Dic5—21no—
21#6772—22no—
22#6772C2214yescomplemented
23#6716C2214yescomplemented
24C2×C3⋊#168C2211yescharacteristic, complemented
25C5×#213C2214yescomplemented
26C2×C5×C3⋊C8C2211yescharacteristic, complemented
27#6657C2214yescomplemented
28#6652C2214yescomplemented
29#6649C2214yescomplemented
30C2×#932—51nomaximal, Hall
31C2×#711—31no—
32C2×D4×D5D311yescharacteristic, complemented
33C2×C5×D8—31no—
34C2×#687—31no—
35C2×#686—31no—
36#4057—38no—
37C2×#677—31no—
38C2×#676—31no—
39C23×C3×C5—21no—
40C22×C5×D3—21no—
41C22×C3×D5—21no—
42C22×C3×D5—24no—
43C22×C3×D5—44no—
44C22×C3×D5D411yescharacteristic
45C3×C5×D4—21no—
46C3×C5×D4C2312yes—
47C2×C3×C4×C5C2311yescharacteristic
48C2×Dic15—21no—
49Dic30—21no—
50Dic30C2312yescomplemented
51C5×D12—21no—
52C5×D12C2312yescomplemented
53C3×C22⋊D5—24no—
54C2×C3×Dic5D411yescharacteristic
55C3×D20—21no—
56C3×D20C2312yes—
57C3×C4×D5D412yes—
58C3×C4×D5C2312yes—
59#1201—24no—
60D3×Dic5—24no—
61C3⋊#168C2312yescomplemented
62C5×C3⋊C8C2312yescomplemented
63C22×C3×D4—51no—
64C2×#235—51no—
65C2×#214—51no—
66C2×#214—51no—
67#932—58no—
68C2×#213—51no—
69C2×#213D511yescharacteristic, complemented
70C2×#208—51no—
71C23×D5—21no—
72C2×C5×D4—31no—
73C2×C5×D4C2×D311yescharacteristic
74C2×C22⋊D5—21no—
75C2×C22⋊D5—61no—
76C22×Dic5—61no—
77#711—34no—
78#711—62no—
79D4×D5—22no—
80D4×D5C2×D314yescomplemented
81C2×D20C2×D311yescharacteristic
82C2×C4×D5C2×D311yescharacteristic
83C2×Dic10—31no—
84C5×D8—34no—
85C2×C5×C8—31no—
86#687—34no—
87#686—34no—
88C2×#168—31no—
89#677—34no—
90#676—34no—
91C2×AGL(1,ℤ/8)—151noSylow
92C22×C3×C5—22no—
93C22×C3×C5—42no—
94C22×C3×C5C2×D411yescharacteristic
95C2×C5×D3—22no—
96C2×C5×D3—42no—
97C2×C3×D5—22no—
98C2×C3×D5—22no—
99C2×C3×D5—48no—
100C2×C3×D5—42no—
101C2×C3×D5C2×D412yes—
102C3×C4×C5C2×D411yescharacteristic
103C3×C4×C5C2411yescharacteristic
104Dic15—22no—
105C3×Dic5C2×D412yes—
106C24×C3—101no—
107C2×C3×D4—54no—
108C2×C3×D4—51no—
109C2×C3×D4—102no—
110C2×C3×D4—101no—
111C2×C3×D4C2×D511yescharacteristic
112C22×C3×C4—51no—
113C2×#67—101no—
114#235—54no—
115#235—102no—
116C2×D12C2×D511yescharacteristic, complemented
117C2×C4×D3—101no—
118C2×Dic6—51no—
119#214—54no—
120#214—54no—
121#213—54no—
122#213C2×D514yescomplemented
123#208—54no—
124C2×C3⋊C8—51no—
125C2×C3⋊C8C2×D511yescharacteristic, complemented
126C23×C5—21no—
127C23×C5—61no—
128C22×D5—21no—
129C22×D5—24no—
130C22×D5—44no—
131C22×D5#6711yescharacteristic
132C5×D4—21no—
133C5×D4—32no—
134C5×D4—61no—
135C5×D4C22×D312yes—
136C2×C4×C5C22×D311yescharacteristic
137C22⋊D5—24no—
138C22⋊D5—64no—
139C2×Dic5—61no—
140C2×Dic5—64no—
141C2×Dic5#6711yescharacteristic
142D20—21no—
143D20C22×D312yes—
144C4×D5#6712yes—
145C4×D5C22×D312yes—
146Dic10—32no—
147Dic10—61no—
148C5×C8—32no—
149#168—32no—
150C2×#41—151no—
151C22×D4—51no—
152AGL(1,ℤ/8)—158no—
153C2×SD16—151no—
154C2×SD16—151no—
155C2×D8—31no—
156C2×D8—151no—
157C2×M16—151no—
158C2×C3×C5—42no—
159C2×C3×C5AGL(1,ℤ/8)12yescomplemented
160C2×C3×C5C22×D411yescharacteristic
161C3×D5—22no—
162C3×D5—42no—
163C5×D3—42no—
164C23×C3—21no—
165C23×C3—51no—
166C23×C3—104no—
167C23×C3—101no—
168C23×C3—204no—
169C22×D3—21no—
170C3×D4—21no—
171C3×D4—52no—
172C3×D4—104no—
173C3×D4—101no—
174C3×D4C22×D512yes—
175C2×C3×C4—51no—
176C2×C3×C4—52no—
177C2×C3×C4—52no—
178C2×C3×C4C22×D511yescharacteristic
179#67—104no—
180C2×Dic3—101no—
181D12—21no—
182D12C22×D512yescomplemented
183C4×D3—104no—
184Dic6—52no—
185Dic6—101no—
186C3⋊C8—52no—
187C3⋊C8C22×D512yescomplemented
188C22×C5—22no—
189C22×C5—42no—
190C22×C5—62no—
191C22×C5—122no—
192C22×C5C2×#6711yescharacteristic
193C2×D5—22no—
194C2×D5—22no—
195C2×D5—48no—
196C2×D5—42no—
197C2×D5C2×#6712yes—
198C4×C5C2×#6711yescharacteristic
199C4×C5C23×D311yescharacteristic
200Dic5—62no—
201Dic5C2×#6712yes—
202C24—101no—
203#41—154no—
204#41—302no—
205C2×Q8—151no—
206C2×D4—31no—
207C2×D4—54no—
208C2×D4—51no—
209C2×D4—102no—
210C2×D4—101no—
211C2×D4—301no—
212C2×D4D3×D511yescharacteristic
213C22×C4—51no—
214C22×C4—301no—
215SD16—154no—
216SD16—154no—
217D8—34no—
218D8—154no—
219M16—154no—
220C2×C8—31no—
221C2×C8—151no—
222C3×C5C2×AGL(1,ℤ/8)11yescharacteristic, complemented, Hall
223C22×C3—22no—
224C22×C3—42no—
225C22×C3—52no—
226C22×C3—102no—
227C22×C3—102no—
228C22×C3—208no—
229C22×C3—202no—
230C22×C3D4×D511yescharacteristic
231C2×D3—22no—
232C2×D3—42no—
233C3×C4—52no—
234C3×C4D4×D511yescharacteristic
235C3×C4C23×D511yescharacteristic
236Dic3—102no—
237C2×C5—42no—
238C2×C5—122no—
239C2×C5#93212yescomplemented
240C2×C5C22×#6711yescharacteristic
241D5—22no—
242D5—42no—
243C23—21no—
244C23—51no—
245C23—61no—
246C23—104no—
247C23—101no—
248C23—204no—
249Q8—152no—
250Q8—301no—
251D4—21no—
252D4—32no—
253D4—52no—
254D4—61no—
255D4—104no—
256D4—101no—
257D4—304no—
258D4C2×D3×D512yes—
259C2×C4—51no—
260C2×C4—52no—
261C2×C4—52no—
262C2×C4—304no—
263C2×C4—301no—
264C2×C4C2×D3×D511yescharacteristic
265C8—32no—
266C8—152no—
267C2×C3—42no—
268C2×C3—102no—
269C2×C3—202no—
270C2×C3#405712yescomplemented
271C2×C3C2×D4×D511yescharacteristic
272D3—42no—
273C5C2×#93211yescharacteristic, complemented, Sylow
274C22—22no—
275C22—42no—
276C22—52no—
277C22—62no—
278C22—102no—
279C22—102no—
280C22—122no—
281C22—208no—
282C22—202no—
283C22D5×#6711yescharacteristic
284C4—52no—
285C4—302no—
286C4D5×#6711yescharacteristic
287C4C22×D3×D511yescharacteristic
288C3C2×#405711yescharacteristic, complemented, Sylow
289C2—42no—
290C2—102no—
291C2—122no—
292C2—202no—
293C2#9148912yesdirect product factor
294C2C2×D5×#6711yescharacteristic
subgroupG/H
centerC22D5×#67
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×C3×D4×D5C2semidirect
2. C2×#6772C2semidirect
3. C2×#6716C2semidirect
4. C2×C5×#213C2semidirect
5. #91489C2direct (N × H)
6. C2×#6657C2semidirect
7. C2×#6652C2semidirect
8. C2×#6649C2semidirect
12. C2×Dic30C22semidirect
13. C2×C5×D12C22semidirect
16. C3×D4×D5C22semidirect
22. #6772C22semidirect
23. #6716C22semidirect
24. C2×C3⋊#168C22semidirect
25. C5×#213C22semidirect
26. C2×C5×C3⋊C8C22semidirect
27. #6657C22semidirect
28. #6652C22semidirect
29. #6649C22semidirect
32. C2×D4×D5D3semidirect
50. Dic30C23semidirect
52. C5×D12C23semidirect
61. C3⋊#168C23semidirect
62. C5×C3⋊C8C23semidirect
69. C2×#213D5semidirect
80. D4×D5C2×D3semidirect
116. C2×D12C2×D5semidirect
122. #213C2×D5semidirect
125. C2×C3⋊C8C2×D5semidirect
159. C2×C3×C5AGL(1,ℤ/8)semidirect
182. D12C22×D5semidirect
187. C3⋊C8C22×D5semidirect
222. C3×C5C2×AGL(1,ℤ/8)semidirect
239. C2×C5#932semidirect
270. C2×C3#4057semidirect
273. C5C2×#932semidirect
288. C3C2×#4057semidirect
293. C2#91489direct (N × H)

Maximal subgroups

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

subgroupindex
C2×C3×D4×D52
C2×#67722
C2×#67162
C2×C5×#2132
#914892
C2×#66572
C2×#66522
C2×#66492
C2×#40573
C2×#9325