Finite Group Databasebeta

← C2×#10624043 #13089026

Subgroups — C2×#10624043 #13089026

Order 1440.

710 conjugacy classes of subgroups; only the 156 normal subgroups are shown.

normal subgroups only

This group has 710 conjugacy classes of subgroups — too many to draw. Only its 156 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×#67155C211yesmaximal, characteristic, complemented
2C2×#67153C211yesmaximal, characteristic, complemented
3C2×D3×Dic15C211yesmaximal, characteristic, complemented
4C2×C5×#626C211yesmaximal, characteristic, complemented
5C2×#67138C211yesmaximal, characteristic, complemented
6C2×C3×#1201C211yesmaximal, characteristic, complemented
7C2×C3×D5×Dic3C211yesmaximal, characteristic, complemented
8#10624043C218yesmaximal, direct product factor
9C2×#6780—31nomaximal
10C2×#6772—31nomaximal
11C22×C5×C3⋊D3C2211yescharacteristic
12C22×C3×C5×D3C2211yescharacteristic
13C22×C32×D5C2211yescharacteristic
14C2×C3⋊Dic15C2211yescharacteristic
15C2×C3×Dic15C2211yescharacteristic
16C2×C3×Dic15C2211yescharacteristic
17C2×C3×C5×Dic3C2211yescharacteristic
18#67155C2214yescomplemented
19#67153C2214yescomplemented
20D3×Dic15C2214yescomplemented
21C5×#626C2214yescomplemented
22#67138C2214yescomplemented
23C3×#1201C2214yescomplemented
24C3×D5×Dic3C2214yescomplemented
25C2×#3804—51nomaximal, Hall
26C22×Dic15—31no—
27C2×Dic30—31no—
28C2×C5×#67—31no—
29C2×C5×D12—31no—
30C2×C3×C22⋊D5—31no—
31C2×C3×C4×D5—31no—
32C2×#1204—31no—
33C2×#1201—31no—
34C2×#1201—31no—
35C2×#1201—61no—
36C2×#1201D311yescharacteristic, complemented
37C2×D3×Dic5—31no—
38C2×D3×Dic5—31no—
39C2×D3×Dic5—31no—
40#6780—38no—
41C2×D5×Dic3D311yescharacteristic, complemented
42#6772—38no—
43C22×C32×C5C2311yescharacteristic
44C2×C5×C3⋊D3—22no—
45C2×C5×C3⋊D3C2312yes—
46C2×C3×C5×D3—22no—
47C2×C3×C5×D3C2312yes—
48C2×C32×D5—22no—
49C2×C32×D5C2312yes—
50C3⋊Dic15C2312yes—
51C3×Dic15C2312yes—
52C3×Dic15C2312yes—
53C3×C5×Dic3C2312yes—
54C2×#711—91noHall
55C2×#638—51no—
56C2×C3×#67—51no—
57C22×C3×Dic3—51no—
58C2×#627—51no—
59C2×#626D511yescharacteristic, complemented
60C2×#624—51no—
61#3804—58no—
62C2×D3×Dic3—51no—
63C23×C3×C5—31no—
64C22×C5×D3—31no—
65C22×C5×D3—31no—
66C22×C5×D3—61no—
67C22×C5×D3C2×D311yescharacteristic
68C22×C3×D5—21no—
69C22×C3×D5C2×D311yescharacteristic
70C22×C3×D5C2×D311yescharacteristic
71C2×C3×C4×C5—31no—
72C2×Dic15—31no—
73C2×Dic15—31no—
74C2×Dic15—34no—
75C2×Dic15—61no—
76C2×Dic15C2×D311yescharacteristic
77C2×Dic15C2×D311yescharacteristic
78Dic30—34no—
79C5×#67—34no—
80C2×C5×Dic3C2×D311yescharacteristic
81C5×D12—34no—
82C3×C22⋊D5—34no—
83C2×C3×Dic5—31no—
84C2×C3×Dic5—31no—
85C3×C4×D5—34no—
86#1204—34no—
87#1201—34no—
88#1201—34no—
89#1201—64no—
90#1201C2×D314yescomplemented
91D3×Dic5—34no—
92D3×Dic5—34no—
93D3×Dic5—34no—
94D5×Dic3C2×D314yescomplemented
95C2×#237—151no—
96C2×#235—151no—
97C2×C32×C5#4112yescomplemented
98C2×C32×C5C2411yescharacteristic
99C5×C3⋊D3—22no—
100C3×C5×D3—22no—
101C32×D5—22no—
102C2×C5×D4—91no—
103C2×C22⋊D5—31no—
104C2×C22⋊D5—91no—
105C22×Dic5—91no—
106C22×Dic5—91no—
107#711—98no—
108C2×C4×D5—31no—
109C2×Dic10—91no—
110C23×C32—51no—
111C22×C3⋊D3C2×D511yescharacteristic
112C22×C3×D3C2×D511yescharacteristic
113#638—54no—
114C2×C3⋊Dic3—51no—
115C3×#67—54no—
116C2×C3×Dic3—51no—
117C2×C3×Dic3—51no—
118C2×C3×Dic3—54no—
119C2×C3×Dic3C2×D511yescharacteristic
120#627—54no—
121#626C2×D514yescomplemented
122#624—54no—
123D3×Dic3—54no—
124C22×C3×C5—21no—
125C22×C3×C5—32no—
126C22×C3×C5—62no—
127C22×C3×C5C22×D311yescharacteristic
128C22×C3×C5C22×D311yescharacteristic
129C2×C5×D3—22no—
130C2×C5×D3—32no—
131C2×C5×D3—32no—
132C2×C5×D3—62no—
133C2×C5×D3—62no—
134C2×C5×D3—62no—
135C2×C5×D3—122no—
136C2×C5×D3C22×D312yes—
137C2×C3×D5—22no—
138C2×C3×D5—22no—
139C2×C3×D5—22no—
140C2×C3×D5—42no—
141C2×C3×D5C22×D312yes—
142C2×C3×D5C22×D312yes—
143C3×C4×C5—32no—
144Dic15—32no—
145Dic15—32no—
146Dic15—62no—
147Dic15C22×D312yes—
148Dic15C22×D312yes—
149C3×Dic5—32no—
150C3×Dic5—32no—
151C5×Dic3C22×D312yes—
152C2×C3×D4—151no—
153C22×C3×C4—151no—
154C2×#67—31no—
155C2×#67—51no—
156C2×#67—151no—
157C2×#67—151no—
158C2×#67—301no—
159C22×Dic3—51no—
160C22×Dic3—151no—
161#237—158no—
162#235—158no—
163C2×D12—31no—
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—31no—
171C23×C5—91no—
172C22×D5D3211yescharacteristic
173C5×D4—94no—
174C2×C4×C5—31no—
175C22⋊D5—34no—
176C22⋊D5—94no—
177C2×Dic5—31no—
178C2×Dic5—31no—
179C2×Dic5—94no—
180C2×Dic5—94no—
181C2×Dic5—91no—
182C4×D5—34no—
183Dic10—94no—
184C22×C32—52no—
185C22×C32—102no—
186C22×C32C22×D511yescharacteristic
187C2×C3⋊D3—22no—
188C2×C3⋊D3C22×D512yes—
189C2×C3×D3—22no—
190C2×C3×D3C22×D512yes—
191C3⋊Dic3—52no—
192C3×Dic3—52no—
193C3×Dic3—52no—
194C3×Dic3C22×D512yes—
195C2×#41—451noSylow
196C2×C3×C5—21no—
197C2×C3×C5—22no—
198C2×C3×C5—62no—
199C2×C3×C5#23512yescomplemented
200C2×C3×C5#23712yescomplemented
201C2×C3×C5C23×D311yescharacteristic
202C2×C3×C5C23×D311yescharacteristic
203C3×D5—22no—
204C3×D5—22no—
205C3×D5—42no—
206C5×D3—22no—
207C5×D3—62no—
208C5×D3—62no—
209C5×D3—122no—
210C23×C3—31no—
211C23×C3—51no—
212C23×C3—51no—
213C23×C3—101no—
214C22×D3—31no—
215C22×D3—31no—
216C22×D3—61no—
217C22×D3D3×D511yescharacteristic
218C3×D4—154no—
219C2×C3×C4—31no—
220C2×C3×C4—154no—
221C2×C3×C4—151no—
222C2×C3×C4—151no—
223#67—34no—
224#67—54no—
225#67—154no—
226#67—154no—
227#67—304no—
228C2×Dic3—51no—
229C2×Dic3—51no—
230C2×Dic3—54no—
231C2×Dic3—154no—
232C2×Dic3—151no—
233C2×Dic3—151no—
234C2×Dic3—301no—
235C2×Dic3D3×D511yescharacteristic
236D12—34no—
237C4×D3—154no—
238C4×D3—154no—
239C4×D3—154no—
240Dic6—154no—
241Dic6—154no—
242C22×C5—32no—
243C22×C5—62no—
244C22×C5—92no—
245C22×C5—182no—
246C22×C5C2×D3211yescharacteristic
247C2×D5—22no—
248C2×D5C2×D3212yes—
249C4×C5—32no—
250Dic5—32no—
251Dic5—32no—
252Dic5—92no—
253C2×C32—102no—
254C2×C32#71112yescomplemented
255C2×C32C23×D511yescharacteristic
256C3⋊D3—22no—
257C3×D3—22no—
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—21no—
270C22×C3—32no—
271C22×C3—52no—
272C22×C3—52no—
273C22×C3—62no—
274C22×C3—102no—
275C22×C3—102no—
276C22×C3—102no—
277C22×C3—202no—
278C22×C3C2×D3×D511yescharacteristic
279C22×C3C2×D3×D511yescharacteristic
280C2×D3—22no—
281C2×D3—32no—
282C2×D3—32no—
283C2×D3—62no—
284C2×D3—62no—
285C2×D3—62no—
286C2×D3—122no—
287C2×D3C2×D3×D512yes—
288C3×C4—32no—
289C3×C4—152no—
290C3×C4—152no—
291Dic3—52no—
292Dic3—52no—
293Dic3—152no—
294Dic3—152no—
295Dic3—302no—
296Dic3C2×D3×D512yes—
297C2×C5—62no—
298C2×C5—182no—
299C2×C5#380412yescomplemented
300C2×C5C22×D3211yescharacteristic
301D5—22no—
302C32C2×#71111yescharacteristic, complemented, Sylow
303C23—31no—
304C23—51no—
305C23—91no—
306Q8—454no—
307D4—94no—
308D4—154no—
309D4—454no—
310C2×C4—31no—
311C2×C4—154no—
312C2×C4—151no—
313C2×C4—151no—
314C2×C4—454no—
315C2×C4—454no—
316C2×C4—451no—
317C2×C3—22no—
318C2×C3—21no—
319C2×C3—62no—
320C2×C3—102no—
321C2×C3—102no—
322C2×C3—202no—
323C2×C3#677212yescomplemented
324C2×C3#678012yescomplemented
325C2×C3C22×D3×D511yescharacteristic
326C2×C3C22×D3×D511yescharacteristic
327D3—22no—
328D3—62no—
329D3—62no—
330D3—122no—
331C5C2×#380411yescharacteristic, complemented, Sylow
332C22—32no—
333C22—52no—
334C22—62no—
335C22—92no—
336C22—102no—
337C22—182no—
338C22D32×D511yescharacteristic
339C4—32no—
340C4—152no—
341C4—152no—
342C4—452no—
343C3—21no—
344C3C2×#677211yescharacteristic, complemented
345C3C2×#678011yescharacteristic, complemented
346C2—62no—
347C2—102no—
348C2—182no—
349C2#1062404312yesdirect product factor
350C2C2×D32×D511yescharacteristic
subgroupG/H
center, hypercenterC22D32×D5
commutator, cosocleC2×C32×C5C24
FrattiniC2C2×D32×D5
Fitting, socleC22×C32×C5C23
hypocenterC32×C5C2×#41
LCSG ▷ C2×C32×C5 ▷ C32×C5
UCSC1 ◁ C22

Complements

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

NH ≅ G/Ntype
1. C2×#67155C2semidirect
2. C2×#67153C2semidirect
3. C2×D3×Dic15C2semidirect
4. C2×C5×#626C2semidirect
5. C2×#67138C2semidirect
6. C2×C3×#1201C2semidirect
7. C2×C3×D5×Dic3C2semidirect
8. #10624043C2direct (N × H)
18. #67155C22semidirect
19. #67153C22semidirect
20. D3×Dic15C22semidirect
21. C5×#626C22semidirect
22. #67138C22semidirect
23. C3×#1201C22semidirect
24. C3×D5×Dic3C22semidirect
36. C2×#1201D3semidirect
41. C2×D5×Dic3D3semidirect
59. C2×#626D5semidirect
90. #1201C2×D3semidirect
94. D5×Dic3C2×D3semidirect
97. C2×C32×C5#41semidirect
121. #626C2×D5semidirect
169. C32×C5C2×#41semidirect
199. C2×C3×C5#235semidirect
200. 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#6772semidirect
324. C2×C3#6780semidirect
331. C5C2×#3804semidirect
344. C3C2×#6772semidirect
345. C3C2×#6780semidirect
349. C2#10624043direct (N × H)

Maximal subgroups

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

subgroupindex
C2×#671552
C2×#671532
C2×D3×Dic152
C2×C5×#6262
C2×#671382
C2×C3×#12012
C2×C3×D5×Dic32
#106240432
C2×#67803
C2×#67723
C2×#38045