Finite Group Databasebeta

← C2×#10624038 #13089021

Subgroups — C2×#10624038 #13089021

Order 1440.

662 conjugacy classes of subgroups; only the 172 normal subgroups are shown.

normal subgroups only

This group has 662 conjugacy classes of subgroups — too many to draw. Only its 172 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×#67154C211yesmaximal, characteristic
2C2×#67153C211yesmaximal, characteristic, complemented
3C2×Dic3×D15C211yesmaximal, characteristic
4C2×C5×#627C211yesmaximal, characteristic, complemented
5C2×#67141C211yesmaximal, characteristic, complemented
6C2×C3×#1204C211yesmaximal, characteristic, complemented
7C2×C3×#1200C211yesmaximal, characteristic
8#10624038C218yesmaximal, direct product factor
9C2×D15⋊Q8—31nomaximal
10C2×D5×Dic6—31nomaximal
11C22×C3×D15C2211yescharacteristic
12C2×C3⋊Dic15C2211yescharacteristic
13C2×C5×C3⋊Dic3C2211yescharacteristic
14C2×C3×Dic15C2211yescharacteristic
15C2×C3×C5×Dic3C2211yescharacteristic
16C2×C3×C5×Dic3C2211yescharacteristic
17C2×C32×Dic5C2211yescharacteristic
18#67154C2214yes—
19#67153C2214yescomplemented
20Dic3×D15C2214yes—
21C5×#627C2214yescomplemented
22#67141C2214yescomplemented
23C3×#1204C2214yescomplemented
24C3×#1200C2214yes—
25C2×D3×Dic6—51nomaximal, Hall
26C2×C4×D15—31no—
27C2×Dic30—31no—
28C2×C5×Dic6—31no—
29C2×C5×Dic6—31no—
30C2×C3×C4×D5—31no—
31C2×C3×Dic10—31no—
32C2×#1204—31no—
33C2×#1204—31no—
34C2×#1204—31no—
35C2×#1204—61no—
36C2×#1204D311yescharacteristic, complemented
37C2×#1200—31no—
38C2×#1200D311yescharacteristic
39C2×D5×Dic3—31no—
40C2×D5×Dic3—31no—
41D15⋊Q8—38no—
42D5×Dic6—38no—
43C22×C32×C5C2311yescharacteristic
44C2×C3×D15Q814yescomplemented
45C2×C3×D15C2312yes—
46C3⋊Dic15C2312yes—
47C5×C3⋊Dic3C2312yes—
48C3×Dic15C2312yes—
49C3×C5×Dic3C2312yes—
50C3×C5×Dic3C2312yes—
51C32×Dic5C2312yes—
52C2×Q8×D5—91noHall
53C2×#634—51no—
54C2×C3×C4×D3—51no—
55C2×C3×Dic6—51no—
56C2×#627—51no—
57C2×#627D511yescharacteristic, complemented
58C2×D3×Dic3—51no—
59C2×D3×Dic3—51no—
60D3×Dic6—58no—
61C22×D15C2×D311yescharacteristic
62C22×C3×D5—31no—
63C2×C3×C4×C5—31no—
64C2×C3×C4×C5—31no—
65C2×Dic15—31no—
66C2×Dic15—31no—
67C2×Dic15—61no—
68C2×Dic15C2×D311yescharacteristic
69C4×D15—34no—
70Dic30—34no—
71C2×C5×Dic3—31no—
72C2×C5×Dic3—31no—
73C2×C5×Dic3—61no—
74C2×C5×Dic3C2×D311yescharacteristic
75C2×C5×Dic3C2×D311yescharacteristic
76C5×Dic6—34no—
77C5×Dic6—34no—
78C2×C3×Dic5—21no—
79C2×C3×Dic5—31no—
80C2×C3×Dic5C2×D311yescharacteristic
81C2×C3×Dic5C2×D311yescharacteristic
82C3×C4×D5—34no—
83C3×Dic10—34no—
84#1204—34no—
85#1204—34no—
86#1204—34no—
87#1204—64no—
88#1204C2×D314yescomplemented
89#1200—34no—
90#1200C2×D314yes—
91D5×Dic3—34no—
92D5×Dic3—34no—
93C2×D3×Q8—151no—
94C22×Dic6—151no—
95C2×C32×C5C2×Q812yescomplemented
96C2×C32×C5C2411yescharacteristic
97C3×D15C2×Q814yescomplemented
98C2×C5×Q8—91no—
99Q8×D5—98no—
100C2×C4×D5—31no—
101C2×C4×D5—91no—
102C2×C4×D5—91no—
103C2×Dic10—31no—
104C2×Dic10—91no—
105C2×Dic10—91no—
106C22×C3×D3—51no—
107C2×C32×C4—51no—
108C2×C3⋊Dic3—51no—
109C2×C3⋊Dic3C2×D511yescharacteristic
110#634—54no—
111C2×C3×Dic3—51no—
112C2×C3×Dic3C2×D511yescharacteristic
113C2×C3×Dic3C2×D511yescharacteristic
114C3×C4×D3—54no—
115C3×Dic6—54no—
116#627—54no—
117#627C2×D514yescomplemented
118D3×Dic3—54no—
119D3×Dic3—54no—
120C22×C3×C5—21no—
121C22×C3×C5C22×D311yescharacteristic
122C22×C3×C5C22×D311yescharacteristic
123C2×D15Dic614yescomplemented
124C2×D15C22×D312yes—
125C2×C3×D5—34no—
126C2×C3×D5—32no—
127C3×C4×C5—32no—
128C3×C4×C5—32no—
129Dic15—32no—
130Dic15—32no—
131Dic15—62no—
132Dic15C22×D312yes—
133C3×Dic5—22no—
134C3×Dic5—32no—
135C3×Dic5C22×D312yes—
136C3×Dic5C22×D312yes—
137C5×Dic3—32no—
138C5×Dic3—32no—
139C5×Dic3—62no—
140C5×Dic3C22×D312yes—
141C5×Dic3C22×D312yes—
142C2×C3×Q8—151no—
143C22×C3×C4—151no—
144C22×Dic3—151no—
145C22×Dic3—151no—
146D3×Q8—158no—
147C2×C4×D3—51no—
148C2×C4×D3—151no—
149C2×C4×D3—151no—
150C2×Dic6—31no—
151C2×Dic6—31no—
152C2×Dic6—51no—
153C2×Dic6—158no—
154C2×Dic6—151no—
155C2×Dic6—151no—
156C2×Dic6—151no—
157C2×Dic6—151no—
158C2×Dic6—301no—
159C32×C5C22×Q811yescharacteristic, complemented, Hall
160C22×D5—31no—
161C5×Q8—94no—
162C2×C4×C5—31no—
163C2×C4×C5—31no—
164C2×C4×C5—91no—
165C2×Dic5—31no—
166C2×Dic5—91no—
167C2×Dic5D3211yescharacteristic
168C4×D5—34no—
169C4×D5—94no—
170C4×D5—94no—
171Dic10—34no—
172Dic10—94no—
173Dic10—94no—
174C22×C32C22×D511yescharacteristic
175C2×C3×D3—54no—
176C2×C3×D3—52no—
177C32×C4—52no—
178C3⋊Dic3—52no—
179C3⋊Dic3C22×D512yes—
180C3×Dic3—52no—
181C3×Dic3C22×D512yes—
182C3×Dic3C22×D512yes—
183C22×Q8—451noSylow
184C2×C3×C5—21no—
185C2×C3×C5—22no—
186C2×C3×C5C2×Dic612yescomplemented
187C2×C3×C5D3×Q812yescomplemented
188C2×C3×C5C23×D311yescharacteristic
189C2×C3×C5C23×D311yescharacteristic
190D15C2×Dic614yescomplemented
191C3×D5—34no—
192C23×C3—151no—
193C22×D3—51no—
194C3×Q8—154no—
195C2×C3×C4—31no—
196C2×C3×C4—31no—
197C2×C3×C4—51no—
198C2×C3×C4—51no—
199C2×C3×C4—101no—
200C2×C3×C4—154no—
201C2×C3×C4—151no—
202C2×Dic3—31no—
203C2×Dic3—31no—
204C2×Dic3—51no—
205C2×Dic3—61no—
206C2×Dic3—154no—
207C2×Dic3—151no—
208C2×Dic3—151no—
209C2×Dic3—154no—
210C2×Dic3—301no—
211C2×Dic3D3×D511yescharacteristic
212C2×Dic3D3×D511yescharacteristic
213C4×D3—54no—
214C4×D3—154no—
215C4×D3—154no—
216Dic6—34no—
217Dic6—34no—
218Dic6—54no—
219Dic6—154no—
220Dic6—154no—
221Dic6—154no—
222Dic6—154no—
223Dic6—304no—
224C22×C5C2×D3211yescharacteristic
225C2×D5—32no—
226C2×D5—34no—
227C4×C5—32no—
228C4×C5—32no—
229C4×C5—92no—
230Dic5—32no—
231Dic5—92no—
232Dic5C2×D3212yes—
233C2×C32Q8×D512yescomplemented
234C2×C32C23×D511yescharacteristic
235C3×D3—54no—
236C2×Q8—91no—
237C2×Q8—151no—
238C2×Q8—451no—
239C2×Q8—451no—
240C2×Q8—458no—
241C22×C4—151no—
242C22×C4—451no—
243C22×C4—451no—
244C3×C5—21no—
245C3×C5C22×Dic611yescharacteristic, complemented
246C3×C5C2×D3×Q811yescharacteristic, complemented
247C22×C3—21no—
248C22×C3—154no—
249C22×C3—152no—
250C22×C3C2×D3×D511yescharacteristic
251C22×C3C2×D3×D511yescharacteristic
252C2×D3—54no—
253C2×D3—52no—
254C3×C4—32no—
255C3×C4—32no—
256C3×C4—52no—
257C3×C4—52no—
258C3×C4—102no—
259C3×C4—152no—
260Dic3—32no—
261Dic3—32no—
262Dic3—52no—
263Dic3—62no—
264Dic3—152no—
265Dic3—152no—
266Dic3—302no—
267Dic3C2×D3×D512yes—
268Dic3C2×D3×D512yes—
269C2×C5D3×Dic612yescomplemented
270C2×C5C22×D3211yescharacteristic
271D5—34no—
272C32C2×Q8×D511yescharacteristic, complemented, Sylow
273C23—151no—
274Q8—94no—
275Q8—154no—
276Q8—454no—
277Q8—454no—
278C2×C4—31no—
279C2×C4—31no—
280C2×C4—51no—
281C2×C4—91no—
282C2×C4—154no—
283C2×C4—151no—
284C2×C4—454no—
285C2×C4—454no—
286C2×C4—451no—
287C2×C3—22no—
288C2×C3—21no—
289C2×C3—154no—
290C2×C3D5×Dic612yescomplemented
291C2×C3D15⋊Q812yescomplemented
292C2×C3C22×D3×D511yescharacteristic
293C2×C3C22×D3×D511yescharacteristic
294D3—54no—
295C5C2×D3×Dic611yescharacteristic, complemented, Sylow
296C22—154no—
297C22—152no—
298C22D32×D511yescharacteristic
299C4—32no—
300C4—32no—
301C4—52no—
302C4—92no—
303C4—152no—
304C4—452no—
305C3—21no—
306C3C2×D5×Dic611yescharacteristic, complemented
307C3C2×D15⋊Q811yescharacteristic, complemented
308C2—154no—
309C2#1062403812yesdirect product factor
310C2C2×D32×D511yescharacteristic
subgroupG/H
center, hypercenterC22D32×D5
commutator, cosocleC2×C32×C5C24
FrattiniC2C2×D32×D5
Fitting, socleC22×C32×C5C23
prime radicalC22×C3×D15C22
hypocenterC32×C5C22×Q8
LCSG ▷ C2×C32×C5 ▷ C32×C5
UCSC1 ◁ C22

Complements

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

NH ≅ G/Ntype
2. C2×#67153C2semidirect
4. C2×C5×#627C2semidirect
5. C2×#67141C2semidirect
6. C2×C3×#1204C2semidirect
8. #10624038C2direct (N × H)
19. #67153C22semidirect
21. C5×#627C22semidirect
22. #67141C22semidirect
23. C3×#1204C22semidirect
36. C2×#1204D3semidirect
44. C2×C3×D15Q8semidirect
57. C2×#627D5semidirect
88. #1204C2×D3semidirect
95. C2×C32×C5C2×Q8semidirect
97. C3×D15C2×Q8semidirect
117. #627C2×D5semidirect
123. C2×D15Dic6semidirect
159. C32×C5C22×Q8semidirect
186. C2×C3×C5C2×Dic6semidirect
187. C2×C3×C5D3×Q8semidirect
190. D15C2×Dic6semidirect
233. C2×C32Q8×D5semidirect
245. C3×C5C22×Dic6semidirect
246. C3×C5C2×D3×Q8semidirect
269. C2×C5D3×Dic6semidirect
272. C32C2×Q8×D5semidirect
290. C2×C3D5×Dic6semidirect
291. C2×C3D15⋊Q8semidirect
295. C5C2×D3×Dic6semidirect
306. C3C2×D5×Dic6semidirect
307. C3C2×D15⋊Q8semidirect
309. C2#10624038direct (N × H)

Maximal subgroups

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

subgroupindex
C2×#671542
C2×#671532
C2×Dic3×D152
C2×C5×#6272
C2×#671412
C2×C3×#12042
C2×C3×#12002
#106240382
C2×D15⋊Q83
C2×D5×Dic63
C2×D3×Dic65