Finite Group Databasebeta

← C2×#10624053 #13089036

Subgroups — C2×#10624053 #13089036

Order 1440.

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

normal subgroups only

This group has 694 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×#67157C211yesmaximal, characteristic, complemented
2C2×#67150C211yesmaximal, characteristic, complemented
3C2×Dic3×D15C211yesmaximal, characteristic, complemented
4C2×C5×D3×Dic3C211yesmaximal, characteristic, complemented
5C2×#67141C211yesmaximal, characteristic, complemented
6C2×C3×#1203C211yesmaximal, characteristic, complemented
7C2×C3×#1200C211yesmaximal, characteristic, complemented
8#10624053C218yesmaximal, direct product factor
9C2×#6779—31nomaximal
10C2×#6771—31nomaximal
11C22×C3×D15C2211yescharacteristic
12C22×C3×D15C2211yescharacteristic
13C22×C3×C5×D3C2211yescharacteristic
14C2×C3⋊Dic15C2211yescharacteristic
15C2×C5×C3⋊Dic3C2211yescharacteristic
16C2×C3×C5×Dic3C2211yescharacteristic
17C2×C32×Dic5C2211yescharacteristic
18#67157C2214yescomplemented
19#67150C2214yescomplemented
20Dic3×D15C2214yescomplemented
21C5×D3×Dic3C2214yescomplemented
22#67141C2214yescomplemented
23C3×#1203C2214yescomplemented
24C3×#1200C2214yescomplemented
25C2×#3795—51nomaximal, Hall
26C2×#1220—31no—
27C2×C4×D15—31no—
28C22×C5×Dic3—31no—
29C2×C4×C5×D3—31no—
30C2×C3×C22⋊D5—31no—
31C2×C3×C4×D5—31no—
32C2×#1204—31no—
33C2×#1204—31no—
34C2×#1204—61no—
35C2×#1203D311yescharacteristic, complemented
36C2×#1202—31no—
37C2×#1202—31no—
38C2×#1201—31no—
39C2×#1200D311yescharacteristic, complemented
40#6779—38no—
41C2×D5×Dic3—31no—
42#6771—38no—
43C22×C32×C5C2311yescharacteristic
44C2×C3×D15—22no—
45C2×C3×D15—22no—
46C2×C3×D15C2312yes—
47C2×C3×D15C2312yes—
48C2×C3×C5×D3—22no—
49C2×C3×C5×D3C2312yes—
50C3⋊Dic15C2312yes—
51C5×C3⋊Dic3C2312yes—
52C3×C5×Dic3C2312yes—
53C32×Dic5C2312yes—
54C2×#709—91noHall
55C2×#634—51no—
56C2×C3×D12—51no—
57C2×C3×C4×D3—51no—
58C2×#625—51no—
59C2×#625—51no—
60C2×D3×Dic3—51no—
61C2×D3×Dic3D511yescharacteristic, complemented
62#3795—58no—
63C23×C3×C5—31no—
64C22×D15C2×D311yescharacteristic
65C22×D15C2×D311yescharacteristic
66C22×C5×D3C2×D311yescharacteristic
67C22×C3×D5—31no—
68C22×C3×D5—31no—
69C2×C3×C4×C5—31no—
70#1220—34no—
71C2×Dic15—31no—
72C2×Dic15—31no—
73C2×Dic15—61no—
74C4×D15—34no—
75C2×C5×Dic3—31no—
76C2×C5×Dic3—31no—
77C2×C5×Dic3—34no—
78C2×C5×Dic3—61no—
79C2×C5×Dic3C2×D311yescharacteristic
80C4×C5×D3—34no—
81C3×C22⋊D5—34no—
82C2×C3×Dic5—21no—
83C2×C3×Dic5C2×D311yescharacteristic
84C2×C3×Dic5C2×D311yescharacteristic
85C3×C4×D5—34no—
86#1204—34no—
87#1204—34no—
88#1204—64no—
89#1203C2×D314yescomplemented
90#1202—34no—
91#1202—34no—
92#1201—34no—
93#1200C2×D314yescomplemented
94D5×Dic3—34no—
95C2×#237—151no—
96C2×#235—151no—
97C2×C32×C5#4112yescomplemented
98C2×C32×C5C2411yescharacteristic
99C3×D15—22no—
100C3×D15—22no—
101C3×C5×D3—22no—
102C22×C4×C5—91no—
103C2×C22⋊D5—31no—
104C2×C22⋊D5—91no—
105#709—98no—
106C2×D20—91no—
107C2×C4×D5—31no—
108C2×C4×D5—91no—
109C2×Dic10—91no—
110C22×C3×D3—51no—
111C22×C3×D3—51no—
112C22×C3×D3C2×D511yescharacteristic
113C2×C32×C4—51no—
114C2×C3⋊Dic3—51no—
115C2×C3⋊Dic3C2×D511yescharacteristic
116#634—54no—
117C2×C3×Dic3C2×D511yescharacteristic
118C3×D12—54no—
119C3×C4×D3—54no—
120#625—54no—
121#625—54no—
122D3×Dic3—54no—
123D3×Dic3C2×D514yescomplemented
124C22×C3×C5—21no—
125C22×C3×C5—32no—
126C22×C3×C5—62no—
127C22×C3×C5C22×D311yescharacteristic
128C22×C3×C5C22×D311yescharacteristic
129C2×D15—22no—
130C2×D15—22no—
131C2×D15C22×D312yes—
132C2×D15C22×D312yes—
133C2×C5×D3—22no—
134C2×C5×D3C22×D312yes—
135C2×C3×D5—32no—
136C2×C3×D5—32no—
137C2×C3×D5—62no—
138C2×C3×D5—62no—
139C3×C4×C5—32no—
140Dic15—32no—
141Dic15—32no—
142Dic15—62no—
143C3×Dic5—22no—
144C3×Dic5C22×D312yes—
145C3×Dic5C22×D312yes—
146C5×Dic3—32no—
147C5×Dic3—32no—
148C5×Dic3—62no—
149C5×Dic3C22×D312yes—
150C2×C3×D4—151no—
151C22×C3×C4—151no—
152C2×#67—151no—
153C2×#67—151no—
154C2×#67—151no—
155C2×#67—151no—
156C22×Dic3—31no—
157C22×Dic3—151no—
158#237—158no—
159#235—158no—
160C2×D12—51no—
161C2×C4×D3—31no—
162C2×C4×D3—51no—
163C2×C4×D3—151no—
164C2×Dic6—151no—
165C2×Dic6—151no—
166C2×Dic6—301no—
167C32×C5C2×#4111yescharacteristic, complemented, Hall
168C23×C5—31no—
169C22×D5—31no—
170C22×D5—31no—
171C2×C4×C5—31no—
172C2×C4×C5—94no—
173C2×C4×C5—91no—
174C22⋊D5—34no—
175C22⋊D5—94no—
176C2×Dic5—91no—
177C2×Dic5D3211yescharacteristic
178D20—94no—
179C4×D5—34no—
180C4×D5—94no—
181Dic10—94no—
182C22×C32C22×D511yescharacteristic
183C2×C3×D3—22no—
184C2×C3×D3—52no—
185C2×C3×D3—52no—
186C2×C3×D3—102no—
187C2×C3×D3—102no—
188C2×C3×D3C22×D512yes—
189C32×C4—52no—
190C3⋊Dic3—52no—
191C3⋊Dic3C22×D512yes—
192C3×Dic3C22×D512yes—
193C2×#41—451noSylow
194C2×C3×C5—21no—
195C2×C3×C5—22no—
196C2×C3×C5—62no—
197C2×C3×C5#23512yescomplemented
198C2×C3×C5#23712yescomplemented
199C2×C3×C5C23×D311yescharacteristic
200C2×C3×C5C23×D311yescharacteristic
201D15—22no—
202D15—22no—
203C3×D5—62no—
204C3×D5—62no—
205C5×D3—22no—
206C23×C3—31no—
207C23×C3—151no—
208C23×C3—151no—
209C22×D3—51no—
210C22×D3—51no—
211C22×D3D3×D511yescharacteristic
212C3×D4—154no—
213C2×C3×C4—31no—
214C2×C3×C4—51no—
215C2×C3×C4—51no—
216C2×C3×C4—101no—
217C2×C3×C4—154no—
218#67—154no—
219#67—154no—
220#67—154no—
221#67—154no—
222C2×Dic3—31no—
223C2×Dic3—31no—
224C2×Dic3—34no—
225C2×Dic3—61no—
226C2×Dic3—151no—
227C2×Dic3—151no—
228C2×Dic3—154no—
229C2×Dic3—301no—
230C2×Dic3D3×D511yescharacteristic
231D12—54no—
232C4×D3—34no—
233C4×D3—54no—
234C4×D3—154no—
235Dic6—154no—
236Dic6—154no—
237Dic6—304no—
238C22×C5—32no—
239C22×C5—62no—
240C22×C5C2×D3211yescharacteristic
241C2×D5—32no—
242C2×D5—32no—
243C2×D5—62no—
244C2×D5—62no—
245C4×C5—32no—
246C4×C5—92no—
247Dic5—92no—
248Dic5C2×D3212yes—
249C2×C32#70912yescomplemented
250C2×C32C23×D511yescharacteristic
251C3×D3—22no—
252C3×D3—102no—
253C3×D3—102no—
254#41—458no—
255C2×Q8—451no—
256C2×D4—151no—
257C2×D4—451no—
258C2×D4—451no—
259C22×C4—91no—
260C22×C4—151no—
261C22×C4—451no—
262C3×C5—21no—
263C3×C5C2×#23511yescharacteristic, complemented
264C3×C5C2×#23711yescharacteristic, complemented
265C22×C3—21no—
266C22×C3—32no—
267C22×C3—62no—
268C22×C3—152no—
269C22×C3—152no—
270C22×C3—302no—
271C22×C3—302no—
272C22×C3C2×D3×D511yescharacteristic
273C22×C3C2×D3×D511yescharacteristic
274C2×D3—22no—
275C2×D3—52no—
276C2×D3—52no—
277C2×D3—102no—
278C2×D3—102no—
279C2×D3C2×D3×D512yes—
280C3×C4—32no—
281C3×C4—52no—
282C3×C4—52no—
283C3×C4—102no—
284Dic3—32no—
285Dic3—32no—
286Dic3—62no—
287Dic3—152no—
288Dic3—152no—
289Dic3—302no—
290Dic3C2×D3×D512yes—
291C2×C5—62no—
292C2×C5#379512yescomplemented
293C2×C5C22×D3211yescharacteristic
294D5—62no—
295D5—62no—
296C32C2×#70911yescharacteristic, complemented, Sylow
297C23—31no—
298C23—151no—
299C23—151no—
300Q8—454no—
301D4—154no—
302D4—454no—
303D4—454no—
304C2×C4—31no—
305C2×C4—51no—
306C2×C4—94no—
307C2×C4—91no—
308C2×C4—154no—
309C2×C4—454no—
310C2×C4—451no—
311C2×C3—22no—
312C2×C3—21no—
313C2×C3—62no—
314C2×C3—302no—
315C2×C3—302no—
316C2×C3#677112yescomplemented
317C2×C3#677912yescomplemented
318C2×C3C22×D3×D511yescharacteristic
319C2×C3C22×D3×D511yescharacteristic
320D3—22no—
321D3—102no—
322D3—102no—
323C5C2×#379511yescharacteristic, complemented, Sylow
324C22—32no—
325C22—62no—
326C22—152no—
327C22—152no—
328C22—302no—
329C22—302no—
330C22D32×D511yescharacteristic
331C4—32no—
332C4—52no—
333C4—92no—
334C4—452no—
335C3—21no—
336C3C2×#677111yescharacteristic, complemented
337C3C2×#677911yescharacteristic, complemented
338C2—62no—
339C2—302no—
340C2—302no—
341C2#1062405312yesdirect product factor
342C2C2×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×#67157C2semidirect
2. C2×#67150C2semidirect
3. C2×Dic3×D15C2semidirect
4. C2×C5×D3×Dic3C2semidirect
5. C2×#67141C2semidirect
6. C2×C3×#1203C2semidirect
7. C2×C3×#1200C2semidirect
8. #10624053C2direct (N × H)
18. #67157C22semidirect
19. #67150C22semidirect
20. Dic3×D15C22semidirect
21. C5×D3×Dic3C22semidirect
22. #67141C22semidirect
23. C3×#1203C22semidirect
24. C3×#1200C22semidirect
35. C2×#1203D3semidirect
39. C2×#1200D3semidirect
61. C2×D3×Dic3D5semidirect
89. #1203C2×D3semidirect
93. #1200C2×D3semidirect
97. C2×C32×C5#41semidirect
123. D3×Dic3C2×D5semidirect
167. C32×C5C2×#41semidirect
197. C2×C3×C5#235semidirect
198. C2×C3×C5#237semidirect
249. C2×C32#709semidirect
263. C3×C5C2×#235semidirect
264. C3×C5C2×#237semidirect
292. C2×C5#3795semidirect
296. C32C2×#709semidirect
316. C2×C3#6771semidirect
317. C2×C3#6779semidirect
323. C5C2×#3795semidirect
336. C3C2×#6771semidirect
337. C3C2×#6779semidirect
341. C2#10624053direct (N × H)

Maximal subgroups

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

subgroupindex
C2×#671572
C2×#671502
C2×Dic3×D152
C2×C5×D3×Dic32
C2×#671412
C2×C3×#12032
C2×C3×#12002
#106240532
C2×#67793
C2×#67713
C2×#37955