Finite Group Databasebeta

← C2×#10624230 #13089426

Subgroups — C2×#10624230 #13089426

Order 1440.

710 conjugacy classes of subgroups; only the 159 normal subgroups are shown.

normal subgroups only

This group has 710 conjugacy classes of subgroups — too many to draw. Only its 159 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×C5×#638C211yesmaximal, characteristic, complemented
2C2×C3×#1220C211yesmaximal, characteristic, complemented
3C22×C3×Dic15C211yesmaximal, characteristic, complemented
4C2×#67158C211yesmaximal, characteristic, complemented
5C2×#67156C211yesmaximal, characteristic, complemented
6C2×#67155C211yesmaximal, characteristic, complemented
7#10624230C218yesmaximal, direct product factor
8C2×#67154C211yesmaximal, characteristic, complemented
9C2×#6782—31nomaximal
10C2×#6780—31nomaximal
11C23×C32×C5C2211yescharacteristic
12C22×C5×C3⋊D3C2211yescharacteristic
13C22×C3×D15C2211yescharacteristic
14C5×#638C2214yescomplemented
15C2×C5×C3⋊Dic3C2211yescharacteristic
16C3×#1220C2214yescomplemented
17C2×C3×Dic15C2211yescharacteristic
18C2×C3×Dic15C2211yescharacteristic
19C2×C3×Dic15C2211yescharacteristic
20C2×C3×Dic15C2214yescomplemented
21#67158C2214yescomplemented
22#67156C2214yescomplemented
23#67155C2214yescomplemented
24#67154C2214yescomplemented
25C2×#3804—51nomaximal, Hall
26C2×#1220D311yescharacteristic, complemented
27C22×Dic15D311yescharacteristic, complemented
28C2×C5×#67—31no—
29C2×C5×#67—31no—
30C2×C5×#67—61no—
31C2×C3×C22⋊D5—31no—
32C22×C3×Dic5—31no—
33C2×#1204—31no—
34C2×#1204—31no—
35C2×#1203—31no—
36C2×#1201—31no—
37C2×#1200—31no—
38#6782—38no—
39C2×D3×Dic5—31no—
40C2×D3×Dic5—31no—
41#6780—38no—
42C2×D5×Dic3—31no—
43C22×C32×C5—22no—
44C22×C32×C5C2312yes—
45C22×C32×C5C2311yescharacteristic
46C2×C5×C3⋊D3—22no—
47C2×C5×C3⋊D3C2312yes—
48C2×C3×D15—22no—
49C2×C3×D15C2312yes—
50C5×C3⋊Dic3C2312yes—
51C3×Dic15C2312yes—
52C3×Dic15C2312yes—
53C3×Dic15C2312yes—
54C2×#711—91noHall
55C2×#638D511yescharacteristic, complemented
56C2×C3×#67—51no—
57C22×C3×Dic3—51no—
58C2×#627—51no—
59C2×#626—51no—
60C2×#624—51no—
61#3804—58no—
62C2×D3×Dic3—51no—
63C23×C3×C5—21no—
64C23×C3×C5C2×D311yescharacteristic
65C23×C3×C5C2×D311yescharacteristic
66C22×D15C2×D311yescharacteristic
67C22×C5×D3—31no—
68C22×C5×D3—31no—
69C22×C5×D3—61no—
70C22×C3×D5—31no—
71#1220C2×D314yescomplemented
72C2×Dic15C2×D311yescharacteristic
73C2×Dic15C2×D311yescharacteristic
74C2×Dic15C2×D311yescharacteristic
75C2×Dic15C2×D314yescomplemented
76C5×#67—34no—
77C5×#67—34no—
78C5×#67—64no—
79C2×C5×Dic3—31no—
80C2×C5×Dic3—31no—
81C2×C5×Dic3—61no—
82C3×C22⋊D5—34no—
83C2×C3×Dic5—31no—
84C2×C3×Dic5—31no—
85C2×C3×Dic5—31no—
86C2×C3×Dic5—34no—
87#1204—34no—
88#1204—34no—
89#1203—34no—
90#1201—34no—
91#1200—34no—
92D3×Dic5—34no—
93D3×Dic5—34no—
94D5×Dic3—34no—
95C2×#237—151no—
96C2×#235—151no—
97C2×C32×C5—22no—
98C2×C32×C5#4112yescomplemented
99C2×C32×C5C2411yescharacteristic
100C5×C3⋊D3—22no—
101C3×D15—22no—
102C2×C5×D4—91no—
103C2×C22⋊D5—31no—
104C2×C22⋊D5—91no—
105C22×Dic5—31no—
106C22×Dic5—91no—
107#711—98no—
108C2×C4×D5—91no—
109C2×Dic10—91no—
110C23×C32C2×D511yescharacteristic
111C22×C3⋊D3C2×D511yescharacteristic
112C22×C3×D3—51no—
113#638C2×D514yescomplemented
114C2×C3⋊Dic3C2×D511yescharacteristic
115C3×#67—54no—
116C2×C3×Dic3—51no—
117C2×C3×Dic3—51no—
118C2×C3×Dic3—51no—
119C2×C3×Dic3—54no—
120#627—54no—
121#626—54no—
122#624—54no—
123D3×Dic3—54no—
124C22×C3×C5—22no—
125C22×C3×C5—22no—
126C22×C3×C5—21no—
127C22×C3×C5—22no—
128C22×C3×C5—42no—
129C22×C3×C5C22×D312yes—
130C22×C3×C5C22×D312yes—
131C22×C3×C5C22×D311yescharacteristic
132C22×C3×C5C22×D311yescharacteristic
133C2×D15—22no—
134C2×D15C22×D312yes—
135C2×C5×D3—32no—
136C2×C5×D3—32no—
137C2×C5×D3—62no—
138C2×C5×D3—62no—
139C2×C5×D3—62no—
140C2×C5×D3—122no—
141C2×C3×D5—32no—
142C2×C3×D5—62no—
143Dic15C22×D312yes—
144Dic15C22×D312yes—
145Dic15C22×D312yes—
146C3×Dic5—32no—
147C3×Dic5—32no—
148C3×Dic5—32no—
149C5×Dic3—32no—
150C5×Dic3—32no—
151C5×Dic3—62no—
152C2×C3×D4—151no—
153C22×C3×C4—151no—
154C2×#67—31no—
155C2×#67—31no—
156C2×#67—51no—
157C2×#67—61no—
158C2×#67—151no—
159C22×Dic3—51no—
160C22×Dic3—151no—
161#237—158no—
162#235—158no—
163C2×D12—151no—
164C2×C4×D3—151no—
165C2×C4×D3—151no—
166C2×C4×D3—151no—
167C2×Dic6—151no—
168C2×Dic6—151no—
169C32×C5C2×#4111yescharacteristic, complemented, Hall
170C23×C5—91no—
171C23×C5D3211yescharacteristic
172C22×D5—31no—
173C5×D4—94no—
174C2×C4×C5—91no—
175C22⋊D5—34no—
176C22⋊D5—94no—
177C2×Dic5—31no—
178C2×Dic5—31no—
179C2×Dic5—31no—
180C2×Dic5—34no—
181C2×Dic5—94no—
182C4×D5—94no—
183Dic10—94no—
184C22×C32—22no—
185C22×C32C22×D512yes—
186C22×C32C22×D511yescharacteristic
187C2×C3⋊D3—22no—
188C2×C3⋊D3C22×D512yes—
189C2×C3×D3—52no—
190C2×C3×D3—102no—
191C3⋊Dic3C22×D512yes—
192C3×Dic3—52no—
193C3×Dic3—52no—
194C3×Dic3—52no—
195C2×#41—451noSylow
196C2×C3×C5—22no—
197C2×C3×C5—22no—
198C2×C3×C5—21no—
199C2×C3×C5—22no—
200C2×C3×C5—42no—
201C2×C3×C5#23512yescomplemented
202C2×C3×C5#23712yescomplemented
203C2×C3×C5C23×D311yescharacteristic
204C2×C3×C5C23×D311yescharacteristic
205D15—22no—
206C3×D5—62no—
207C5×D3—62no—
208C5×D3—62no—
209C5×D3—122no—
210C23×C3—21no—
211C23×C3—151no—
212C23×C3D3×D511yescharacteristic
213C23×C3D3×D511yescharacteristic
214C22×D3—31no—
215C22×D3—31no—
216C22×D3—51no—
217C22×D3—61no—
218C3×D4—154no—
219C2×C3×C4—151no—
220C2×C3×C4—151no—
221C2×C3×C4—151no—
222C2×C3×C4—154no—
223#67—34no—
224#67—34no—
225#67—54no—
226#67—64no—
227#67—154no—
228C2×Dic3—31no—
229C2×Dic3—31no—
230C2×Dic3—51no—
231C2×Dic3—51no—
232C2×Dic3—51no—
233C2×Dic3—54no—
234C2×Dic3—61no—
235C2×Dic3—154no—
236D12—154no—
237C4×D3—154no—
238C4×D3—154no—
239C4×D3—154no—
240Dic6—154no—
241Dic6—154no—
242C22×C5—22no—
243C22×C5—92no—
244C22×C5—182no—
245C22×C5C2×D3212yes—
246C22×C5C2×D3211yescharacteristic
247C2×D5—32no—
248C2×D5—62no—
249C4×C5—92no—
250Dic5—32no—
251Dic5—32no—
252Dic5—32no—
253C2×C32—22no—
254C2×C32#71112yescomplemented
255C2×C32C23×D511yescharacteristic
256C3⋊D3—22no—
257C3×D3—102no—
258#41—458no—
259C2×Q8—451no—
260C2×D4—91no—
261C2×D4—151no—
262C2×D4—451no—
263C22×C4—151no—
264C22×C4—451no—
265C22×C4—451no—
266C3×C5—21no—
267C3×C5C2×#23511yescharacteristic, complemented
268C3×C5C2×#23711yescharacteristic, complemented
269C22×C3—22no—
270C22×C3—22no—
271C22×C3—22no—
272C22×C3—21no—
273C22×C3—42no—
274C22×C3—152no—
275C22×C3—302no—
276C22×C3C2×D3×D512yes—
277C22×C3C2×D3×D512yes—
278C22×C3C2×D3×D511yescharacteristic
279C22×C3C2×D3×D511yescharacteristic
280C2×D3—32no—
281C2×D3—32no—
282C2×D3—52no—
283C2×D3—62no—
284C2×D3—62no—
285C2×D3—62no—
286C2×D3—102no—
287C2×D3—122no—
288C3×C4—152no—
289C3×C4—152no—
290C3×C4—152no—
291Dic3—32no—
292Dic3—32no—
293Dic3—52no—
294Dic3—52no—
295Dic3—52no—
296Dic3—62no—
297C2×C5—22no—
298C2×C5—182no—
299C2×C5#380412yescomplemented
300C2×C5C22×D3211yescharacteristic
301D5—62no—
302C32C2×#71111yescharacteristic, complemented, Sylow
303C23—91no—
304C23—151no—
305C23(C3⋊D3)⋊D511yescharacteristic
306Q8—454no—
307D4—94no—
308D4—154no—
309D4—454no—
310C2×C4—91no—
311C2×C4—154no—
312C2×C4—151no—
313C2×C4—151no—
314C2×C4—151no—
315C2×C4—454no—
316C2×C4—454no—
317C2×C3—22no—
318C2×C3—22no—
319C2×C3—21no—
320C2×C3—22no—
321C2×C3—42no—
322C2×C3—302no—
323C2×C3#678012yescomplemented
324C2×C3#678212yescomplemented
325C2×C3C22×D3×D511yescharacteristic
326C2×C3C22×D3×D511yescharacteristic
327D3—62no—
328D3—62no—
329D3—102no—
330D3—122no—
331C5C2×#380411yescharacteristic, complemented, Sylow
332C22—22no—
333C22—92no—
334C22—152no—
335C22—182no—
336C22—302no—
337C22C2×(C3⋊D3)⋊D512yes—
338C22C2×(C3⋊D3)⋊D511yescharacteristic
339C4—92no—
340C4—152no—
341C4—152no—
342C4—152no—
343C3—21no—
344C3C2×#678011yescharacteristic, complemented
345C3C2×#678211yescharacteristic, complemented
346C2—22no—
347C2—182no—
348C2—302no—
349C2#1062423012yesdirect product factor
350C2C22×(C3⋊D3)⋊D511yescharacteristic
subgroupG/H
centerC22C2×(C3⋊D3)⋊D5
commutator, cosocleC2×C32×C5C24
FrattiniC2C22×(C3⋊D3)⋊D5
FittingC23×C32×C5C22
socleC22×C32×C5C23
hypercenterC23(C3⋊D3)⋊D5
hypocenterC32×C5C2×#41
LCSG ▷ C2×C32×C5 ▷ C32×C5
UCSC1 ◁ C22 ◁ C23

Complements

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

NH ≅ G/Ntype
1. C2×C5×#638C2semidirect
2. C2×C3×#1220C2semidirect
3. C22×C3×Dic15C2semidirect
4. C2×#67158C2semidirect
5. C2×#67156C2semidirect
6. C2×#67155C2semidirect
7. #10624230C2direct (N × H)
8. C2×#67154C2semidirect
14. C5×#638C22semidirect
16. C3×#1220C22semidirect
20. C2×C3×Dic15C22semidirect
21. #67158C22semidirect
22. #67156C22semidirect
23. #67155C22semidirect
24. #67154C22semidirect
26. C2×#1220D3semidirect
27. C22×Dic15D3semidirect
55. C2×#638D5semidirect
71. #1220C2×D3semidirect
75. C2×Dic15C2×D3semidirect
98. C2×C32×C5#41semidirect
113. #638C2×D5semidirect
169. C32×C5C2×#41semidirect
201. C2×C3×C5#235semidirect
202. C2×C3×C5#237semidirect
254. C2×C32#711semidirect
267. C3×C5C2×#235semidirect
268. C3×C5C2×#237semidirect
299. C2×C5#3804semidirect
302. C32C2×#711semidirect
323. C2×C3#6780semidirect
324. C2×C3#6782semidirect
331. C5C2×#3804semidirect
344. C3C2×#6780semidirect
345. C3C2×#6782semidirect
349. C2#10624230direct (N × H)

Maximal subgroups

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

subgroupindex
C2×C5×#6382
C2×C3×#12202
C22×C3×Dic152
C2×#671582
C2×#671562
C2×#671552
#106242302
C2×#671542
C2×#67823
C2×#67803
C2×#38045