Finite Group Databasebeta

← C2×#10624054 #13089037

Subgroups — C2×#10624054 #13089037

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×#67156C211yesmaximal, characteristic, complemented
2C2×#67151C211yesmaximal, characteristic, complemented
3C2×#67149C211yesmaximal, characteristic, complemented
4C2×C5×#624C211yesmaximal, characteristic, complemented
5C2×#67140C211yesmaximal, characteristic, complemented
6C2×C3×#1204C211yesmaximal, characteristic, complemented
7C2×C3×#1200C211yesmaximal, characteristic, complemented
8#10624054C218yesmaximal, direct product factor
9C2×#6771—31nomaximal
10C2×#6769—31nomaximal
11C22×C3⋊D15C2211yescharacteristic
12C22×C5×C3⋊D3C2211yescharacteristic
13C22×C3×D15C2211yescharacteristic
14C2×C3×Dic15C2211yescharacteristic
15C2×C3×C5×Dic3C2211yescharacteristic
16C2×C3×C5×Dic3C2211yescharacteristic
17C2×C32×Dic5C2211yescharacteristic
18#67156C2214yescomplemented
19#67151C2214yescomplemented
20#67149C2214yescomplemented
21C5×#624C2214yescomplemented
22#67140C2214yescomplemented
23C3×#1204C2214yescomplemented
24C3×#1200C2214yescomplemented
25C2×#3799—51nomaximal, Hall
26C2×D60—31no—
27C2×C4×D15—31no—
28C2×C4×C5×D3—31no—
29C2×C4×C5×D3—31no—
30C2×C3×C4×D5—31no—
31C2×C3×Dic10—31no—
32C2×#1204D311yescharacteristic, complemented
33C2×#1203—31no—
34C2×#1203—31no—
35C2×#1203—31no—
36C2×#1203—61no—
37C2×#1202—31no—
38C2×#1201—31no—
39C2×#1200—31no—
40C2×#1200D311yescharacteristic, complemented
41#6771—38no—
42#6769—38no—
43C22×C32×C5C2311yescharacteristic
44C2×C3⋊D15—22no—
45C2×C3⋊D15C2312yes—
46C2×C5×C3⋊D3—22no—
47C2×C5×C3⋊D3C2312yes—
48C2×C3×D15—22no—
49C2×C3×D15C2312yes—
50C3×Dic15C2312yes—
51C3×C5×Dic3C2312yes—
52C3×C5×Dic3C2312yes—
53C32×Dic5C2312yes—
54C2×#709—91noHall
55C2×C4⋊(C3⋊D3)—51no—
56C2×C3×C4×D3—51no—
57C2×C3×Dic6—51no—
58C2×#626—51no—
59C2×#626—51no—
60C2×#624—51no—
61C2×#624D511yescharacteristic, complemented
62#3799—58no—
63C22×D15—31no—
64C22×D15—31no—
65C22×D15—61no—
66C22×D15C2×D311yescharacteristic
67C22×C5×D3—31no—
68C22×C5×D3—31no—
69C22×C5×D3—61no—
70C22×C3×D5—31no—
71C2×C3×C4×C5—31no—
72C2×C3×C4×C5—31no—
73C2×Dic15C2×D311yescharacteristic
74D60—34no—
75C4×D15—34no—
76C2×C5×Dic3C2×D311yescharacteristic
77C2×C5×Dic3C2×D311yescharacteristic
78C4×C5×D3—34no—
79C4×C5×D3—34no—
80C2×C3×Dic5—21no—
81C2×C3×Dic5—31no—
82C2×C3×Dic5C2×D311yescharacteristic
83C2×C3×Dic5C2×D311yescharacteristic
84C3×C4×D5—34no—
85C3×Dic10—34no—
86#1204C2×D314yescomplemented
87#1203—34no—
88#1203—34no—
89#1203—34no—
90#1203—64no—
91#1202—34no—
92#1201—34no—
93#1200—34no—
94#1200C2×D314yescomplemented
95C2×#239—151no—
96C2×#235—151no—
97C2×C32×C5#4112yescomplemented
98C2×C32×C5C2411yescharacteristic
99C3⋊D15—22no—
100C5×C3⋊D3—22no—
101C3×D15—22no—
102C22×C4×C5—91no—
103C2×C22⋊D5—91no—
104C2×C22⋊D5—91no—
105#709—98no—
106C2×D20—91no—
107C2×C4×D5—31no—
108C2×C4×D5—91no—
109C2×Dic10—31no—
110C22×C3⋊D3—51no—
111C22×C3⋊D3C2×D511yescharacteristic
112C22×C3×D3—51no—
113C2×C32×C4—51no—
114C4⋊(C3⋊D3)—54no—
115C2×C3×Dic3—51no—
116C2×C3×Dic3C2×D511yescharacteristic
117C2×C3×Dic3C2×D511yescharacteristic
118C3×C4×D3—54no—
119C3×Dic6—54no—
120#626—54no—
121#626—54no—
122#624—54no—
123#624C2×D514yescomplemented
124C22×C3×C5—21no—
125C22×C3×C5C22×D311yescharacteristic
126C22×C3×C5C22×D311yescharacteristic
127C2×D15—22no—
128C2×D15—32no—
129C2×D15—32no—
130C2×D15—62no—
131C2×D15—62no—
132C2×D15—62no—
133C2×D15—122no—
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—
143C3×C4×C5—32no—
144C3×C4×C5—32no—
145Dic15C22×D312yes—
146C3×Dic5—22no—
147C3×Dic5—32no—
148C3×Dic5C22×D312yes—
149C3×Dic5C22×D312yes—
150C5×Dic3C22×D312yes—
151C5×Dic3C22×D312yes—
152C2×C3×Q8—151no—
153C22×C3×C4—151no—
154C2×#67—151no—
155C2×#67—151no—
156#239—158no—
157#235—158no—
158C2×D12—151no—
159C2×D12—151no—
160C2×D12—151no—
161C2×D12—151no—
162C2×D12—301no—
163C2×C4×D3—31no—
164C2×C4×D3—31no—
165C2×C4×D3—51no—
166C2×C4×D3—151no—
167C2×C4×D3—151no—
168C2×Dic6—51no—
169C32×C5C2×#4111yescharacteristic, complemented, Hall
170C23×C5—91no—
171C22×D5—31no—
172C22×D5—91no—
173C2×C4×C5—31no—
174C2×C4×C5—31no—
175C2×C4×C5—94no—
176C22⋊D5—94no—
177C22⋊D5—94no—
178C2×Dic5—31no—
179C2×Dic5D3211yescharacteristic
180D20—94no—
181C4×D5—34no—
182C4×D5—94no—
183Dic10—34no—
184C22×C32C22×D511yescharacteristic
185C2×C3⋊D3—22no—
186C2×C3⋊D3—52no—
187C2×C3⋊D3—102no—
188C2×C3⋊D3C22×D512yes—
189C2×C3×D3—52no—
190C2×C3×D3—102no—
191C32×C4—52no—
192C3×Dic3—52no—
193C3×Dic3C22×D512yes—
194C3×Dic3C22×D512yes—
195C2×#41—451noSylow
196C2×C3×C5—21no—
197C2×C3×C5—22no—
198C2×C3×C5#23512yescomplemented
199C2×C3×C5#23912yescomplemented
200C2×C3×C5C23×D311yescharacteristic
201C2×C3×C5C23×D311yescharacteristic
202D15—22no—
203D15—62no—
204D15—62no—
205D15—122no—
206C3×D5—62no—
207C5×D3—62no—
208C5×D3—62no—
209C5×D3—122no—
210C23×C3—151no—
211C22×D3—31no—
212C22×D3—31no—
213C22×D3—51no—
214C22×D3—61no—
215C22×D3—151no—
216C22×D3—151no—
217C22×D3—301no—
218C3×Q8—154no—
219C2×C3×C4—31no—
220C2×C3×C4—31no—
221C2×C3×C4—51no—
222C2×C3×C4—51no—
223C2×C3×C4—101no—
224C2×C3×C4—154no—
225C2×C3×C4—151no—
226#67—154no—
227#67—154no—
228C2×Dic3—51no—
229C2×Dic3D3×D511yescharacteristic
230C2×Dic3D3×D511yescharacteristic
231D12—154no—
232D12—154no—
233D12—154no—
234D12—154no—
235D12—304no—
236C4×D3—34no—
237C4×D3—34no—
238C4×D3—54no—
239C4×D3—154no—
240C4×D3—154no—
241Dic6—54no—
242C22×C5—92no—
243C22×C5—182no—
244C22×C5C2×D3211yescharacteristic
245C2×D5—32no—
246C2×D5—62no—
247C2×D5—92no—
248C2×D5—182no—
249C4×C5—32no—
250C4×C5—32no—
251Dic5—32no—
252Dic5C2×D3212yes—
253C2×C32#70912yescomplemented
254C2×C32C23×D511yescharacteristic
255C3⋊D3—22no—
256C3⋊D3—102no—
257C3×D3—102no—
258#41—458no—
259C2×Q8—151no—
260C2×D4—451no—
261C2×D4—451no—
262C2×D4—451no—
263C22×C4—91no—
264C22×C4—151no—
265C22×C4—451no—
266C3×C5—21no—
267C3×C5C2×#23511yescharacteristic, complemented
268C3×C5C2×#23911yescharacteristic, complemented
269C22×C3—21no—
270C22×C3—152no—
271C22×C3—302no—
272C22×C3C2×D3×D511yescharacteristic
273C22×C3C2×D3×D511yescharacteristic
274C2×D3—32no—
275C2×D3—32no—
276C2×D3—52no—
277C2×D3—62no—
278C2×D3—62no—
279C2×D3—62no—
280C2×D3—102no—
281C2×D3—122no—
282C2×D3—152no—
283C2×D3—152no—
284C2×D3—302no—
285C2×D3—302no—
286C2×D3—302no—
287C2×D3—602no—
288C3×C4—32no—
289C3×C4—32no—
290C3×C4—52no—
291C3×C4—52no—
292C3×C4—102no—
293C3×C4—152no—
294Dic3—52no—
295Dic3C2×D3×D512yes—
296Dic3C2×D3×D512yes—
297C2×C5—182no—
298C2×C5#379912yescomplemented
299C2×C5C22×D3211yescharacteristic
300D5—62no—
301D5—182no—
302C32C2×#70911yescharacteristic, complemented, Sylow
303C23—91no—
304C23—151no—
305C23—451no—
306Q8—154no—
307D4—454no—
308D4—454no—
309D4—454no—
310C2×C4—31no—
311C2×C4—31no—
312C2×C4—51no—
313C2×C4—94no—
314C2×C4—154no—
315C2×C4—151no—
316C2×C4—454no—
317C2×C3—22no—
318C2×C3—21no—
319C2×C3—302no—
320C2×C3#676912yescomplemented
321C2×C3#677112yescomplemented
322C2×C3C22×D3×D511yescharacteristic
323C2×C3C22×D3×D511yescharacteristic
324D3—62no—
325D3—62no—
326D3—102no—
327D3—122no—
328D3—302no—
329D3—302no—
330D3—602no—
331C5C2×#379911yescharacteristic, complemented, Sylow
332C22—92no—
333C22—152no—
334C22—182no—
335C22—302no—
336C22—452no—
337C22—902no—
338C22D32×D511yescharacteristic
339C4—32no—
340C4—32no—
341C4—52no—
342C4—152no—
343C3—21no—
344C3C2×#676911yescharacteristic, complemented
345C3C2×#677111yescharacteristic, complemented
346C2—182no—
347C2—302no—
348C2—902no—
349C2#1062405412yesdirect 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×#67156C2semidirect
2. C2×#67151C2semidirect
3. C2×#67149C2semidirect
4. C2×C5×#624C2semidirect
5. C2×#67140C2semidirect
6. C2×C3×#1204C2semidirect
7. C2×C3×#1200C2semidirect
8. #10624054C2direct (N × H)
18. #67156C22semidirect
19. #67151C22semidirect
20. #67149C22semidirect
21. C5×#624C22semidirect
22. #67140C22semidirect
23. C3×#1204C22semidirect
24. C3×#1200C22semidirect
32. C2×#1204D3semidirect
40. C2×#1200D3semidirect
61. C2×#624D5semidirect
86. #1204C2×D3semidirect
94. #1200C2×D3semidirect
97. C2×C32×C5#41semidirect
123. #624C2×D5semidirect
169. C32×C5C2×#41semidirect
198. C2×C3×C5#235semidirect
199. C2×C3×C5#239semidirect
253. C2×C32#709semidirect
267. C3×C5C2×#235semidirect
268. C3×C5C2×#239semidirect
298. C2×C5#3799semidirect
302. C32C2×#709semidirect
320. C2×C3#6769semidirect
321. C2×C3#6771semidirect
331. C5C2×#3799semidirect
344. C3C2×#6769semidirect
345. C3C2×#6771semidirect
349. C2#10624054direct (N × H)

Maximal subgroups

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

subgroupindex
C2×#671562
C2×#671512
C2×#671492
C2×C5×#6242
C2×#671402
C2×C3×#12042
C2×C3×#12002
#106240542
C2×#67713
C2×#67693
C2×#37995