Finite Group Databasebeta

← C2×#10624049 #13089032

Subgroups — C2×#10624049 #13089032

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×#67153C211yesmaximal, characteristic, complemented
3C2×#67150C211yesmaximal, characteristic, complemented
4C2×C5×#626C211yesmaximal, characteristic, complemented
5C2×C3⋊D3×Dic5C211yesmaximal, characteristic, complemented
6C2×C3×#1200C211yesmaximal, characteristic, complemented
7C2×C3×D3×Dic5C211yesmaximal, characteristic, complemented
8#10624049C218yesmaximal, direct product factor
9C2×#6782—31nomaximal
10C2×#6772—31nomaximal
11C22×C5×C3⋊D3C2211yescharacteristic
12C22×C3×D15C2211yescharacteristic
13C22×C3×C5×D3C2211yescharacteristic
14C2×C3⋊Dic15C2211yescharacteristic
15C2×C3×Dic15C2211yescharacteristic
16C2×C3×C5×Dic3C2211yescharacteristic
17C2×C32×Dic5C2211yescharacteristic
18#67156C2214yescomplemented
19#67153C2214yescomplemented
20#67150C2214yescomplemented
21C5×#626C2214yescomplemented
22C3⋊D3×Dic5C2214yescomplemented
23C3×#1200C2214yescomplemented
24C3×D3×Dic5C2214yescomplemented
25C2×#3798—51nomaximal, Hall
26C2×#1220—31no—
27C2×Dic30—31no—
28C2×C5×#67—31no—
29C2×C5×D12—31no—
30C22×C3×Dic5—31no—
31C2×C3×C4×D5—31no—
32C2×#1204—31no—
33C2×#1203—31no—
34C2×#1201—31no—
35C2×#1201—31no—
36C2×#1200D311yescharacteristic, complemented
37#6782—38no—
38C2×D3×Dic5—31no—
39C2×D3×Dic5—31no—
40C2×D3×Dic5—61no—
41C2×D3×Dic5D311yescharacteristic, complemented
42#6772—38no—
43C22×C32×C5C2311yescharacteristic
44C2×C5×C3⋊D3—22no—
45C2×C5×C3⋊D3C2312yes—
46C2×C3×D15—22no—
47C2×C3×D15C2312yes—
48C2×C3×C5×D3—22no—
49C2×C3×C5×D3C2312yes—
50C3⋊Dic15C2312yes—
51C3×Dic15C2312yes—
52C3×C5×Dic3C2312yes—
53C32×Dic5C2312yes—
54C2×#711—91noHall
55C2×C4×C3⋊D3—51no—
56C2×C3×C4×D3—51no—
57C2×C3×C4×D3—51no—
58C2×#627—51no—
59C2×#626—51no—
60C2×#626D511yescharacteristic, complemented
61C2×#625—51no—
62#3798—58no—
63C23×C3×C5—31no—
64C22×D15C2×D311yescharacteristic
65C22×C5×D3—31no—
66C22×C5×D3—31no—
67C22×C5×D3—61no—
68C22×C5×D3C2×D311yescharacteristic
69C22×C3×D5—31no—
70C2×C3×C4×C5—31no—
71#1220—34no—
72C2×Dic15—31no—
73C2×Dic15—31no—
74C2×Dic15—61no—
75C2×Dic15C2×D311yescharacteristic
76Dic30—34no—
77C5×#67—34no—
78C2×C5×Dic3C2×D311yescharacteristic
79C5×D12—34no—
80C2×C3×Dic5—21no—
81C2×C3×Dic5—31no—
82C2×C3×Dic5—34no—
83C2×C3×Dic5C2×D311yescharacteristic
84C2×C3×Dic5C2×D311yescharacteristic
85C3×C4×D5—34no—
86#1204—34no—
87#1203—34no—
88#1201—34no—
89#1201—34no—
90#1200C2×D314yescomplemented
91D3×Dic5—34no—
92D3×Dic5—34no—
93D3×Dic5—64no—
94D3×Dic5C2×D314yescomplemented
95C2×#235—151no—
96C2×#235—151no—
97C2×C32×C5#4112yescomplemented
98C2×C32×C5C2411yescharacteristic
99C5×C3⋊D3—22no—
100C3×D15—22no—
101C3×C5×D3—22no—
102C2×C5×D4—91no—
103C2×C22⋊D5—91no—
104C2×C22⋊D5—91no—
105C22×Dic5—31no—
106C22×Dic5—91no—
107#711—98no—
108C2×C4×D5—31no—
109C2×Dic10—91no—
110C22×C3⋊D3C2×D511yescharacteristic
111C22×C3×D3—51no—
112C22×C3×D3C2×D511yescharacteristic
113C2×C32×C4—51no—
114C2×C3⋊Dic3—51no—
115C4×C3⋊D3—54no—
116C2×C3×Dic3—51no—
117C2×C3×Dic3C2×D511yescharacteristic
118C3×C4×D3—54no—
119C3×C4×D3—54no—
120#627—54no—
121#626—54no—
122#626C2×D514yescomplemented
123#625—54no—
124C22×C3×C5—21no—
125C22×C3×C5—32no—
126C22×C3×C5—62no—
127C22×C3×C5C22×D311yescharacteristic
128C22×C3×C5C22×D311yescharacteristic
129C2×D15—22no—
130C2×D15C22×D312yes—
131C2×C5×D3—22no—
132C2×C5×D3—32no—
133C2×C5×D3—32no—
134C2×C5×D3—62no—
135C2×C5×D3—64no—
136C2×C5×D3—62no—
137C2×C5×D3—62no—
138C2×C5×D3C22×D312yes—
139C2×C3×D5—32no—
140C2×C3×D5—62no—
141C3×C4×C5—32no—
142Dic15—32no—
143Dic15—32no—
144Dic15—62no—
145Dic15C22×D312yes—
146C3×Dic5—22no—
147C3×Dic5—32no—
148C3×Dic5C22×D312yes—
149C3×Dic5C22×D312yes—
150C5×Dic3C22×D312yes—
151C22×C3×C4—151no—
152C22×C3×C4—151no—
153C2×#67—31no—
154C2×#67—151no—
155C2×#67—151no—
156C2×#67—151no—
157#235—158no—
158#235—158no—
159C2×D12—31no—
160C2×D12—151no—
161C2×C4×D3—51no—
162C2×C4×D3—51no—
163C2×C4×D3—151no—
164C2×C4×D3—151no—
165C2×C4×D3—301no—
166C2×Dic6—151no—
167C2×Dic6—151no—
168C32×C5C2×#4111yescharacteristic, complemented, Hall
169C23×C5—31no—
170C23×C5—91no—
171C22×D5—31no—
172C5×D4—94no—
173C2×C4×C5—31no—
174C22⋊D5—94no—
175C22⋊D5—94no—
176C2×Dic5—31no—
177C2×Dic5—34no—
178C2×Dic5—91no—
179C2×Dic5—94no—
180C2×Dic5D3211yescharacteristic
181C4×D5—34no—
182Dic10—94no—
183C22×C32C22×D511yescharacteristic
184C2×C3⋊D3—22no—
185C2×C3⋊D3C22×D512yes—
186C2×C3×D3—22no—
187C2×C3×D3—52no—
188C2×C3×D3—102no—
189C2×C3×D3C22×D512yes—
190C32×C4—52no—
191C3⋊Dic3—52no—
192C3×Dic3—52no—
193C3×Dic3C22×D512yes—
194C2×#41—451noSylow
195C2×C3×C5—21no—
196C2×C3×C5—22no—
197C2×C3×C5—62no—
198C2×C3×C5#23512yescomplemented
199C2×C3×C5#23512yescomplemented
200C2×C3×C5C23×D311yescharacteristic
201C2×C3×C5C23×D311yescharacteristic
202D15—22no—
203C3×D5—62no—
204C5×D3—22no—
205C5×D3—64no—
206C5×D3—62no—
207C5×D3—62no—
208C23×C3—31no—
209C23×C3—151no—
210C22×D3—31no—
211C22×D3—31no—
212C22×D3—51no—
213C22×D3—61no—
214C22×D3D3×D511yescharacteristic
215C2×C3×C4—31no—
216C2×C3×C4—51no—
217C2×C3×C4—51no—
218C2×C3×C4—101no—
219C2×C3×C4—154no—
220C2×C3×C4—154no—
221C2×C3×C4—151no—
222#67—34no—
223#67—154no—
224#67—154no—
225#67—154no—
226C2×Dic3—51no—
227C2×Dic3—151no—
228C2×Dic3—151no—
229C2×Dic3—301no—
230C2×Dic3D3×D511yescharacteristic
231D12—34no—
232D12—154no—
233C4×D3—54no—
234C4×D3—54no—
235C4×D3—154no—
236C4×D3—154no—
237C4×D3—304no—
238Dic6—154no—
239Dic6—154no—
240C22×C5—32no—
241C22×C5—62no—
242C22×C5—92no—
243C22×C5—182no—
244C22×C5C2×D3211yescharacteristic
245C2×D5—32no—
246C2×D5—62no—
247C4×C5—32no—
248Dic5—32no—
249Dic5—92no—
250Dic5C2×D3212yes—
251C2×C32#71112yescomplemented
252C2×C32C23×D511yescharacteristic
253C3⋊D3—22no—
254C3×D3—22no—
255C3×D3—102no—
256#41—458no—
257C2×Q8—451no—
258C2×D4—91no—
259C2×D4—451no—
260C2×D4—451no—
261C22×C4—151no—
262C22×C4—151no—
263C22×C4—451no—
264C3×C5—21no—
265C3×C5C2×#23511yescharacteristic, complemented
266C3×C5C2×#23511yescharacteristic, complemented
267C22×C3—21no—
268C22×C3—32no—
269C22×C3—62no—
270C22×C3—152no—
271C22×C3—302no—
272C22×C3C2×D3×D511yescharacteristic
273C22×C3C2×D3×D511yescharacteristic
274C2×D3—22no—
275C2×D3—32no—
276C2×D3—32no—
277C2×D3—52no—
278C2×D3—62no—
279C2×D3—64no—
280C2×D3—62no—
281C2×D3—62no—
282C2×D3—102no—
283C2×D3C2×D3×D512yes—
284C3×C4—32no—
285C3×C4—52no—
286C3×C4—52no—
287C3×C4—102no—
288C3×C4—152no—
289Dic3—52no—
290Dic3—152no—
291Dic3—152no—
292Dic3—302no—
293Dic3C2×D3×D512yes—
294C2×C5—62no—
295C2×C5—182no—
296C2×C5#379812yescomplemented
297C2×C5C22×D3211yescharacteristic
298D5—62no—
299C32C2×#71111yescharacteristic, complemented, Sylow
300C23—31no—
301C23—91no—
302C23—151no—
303Q8—454no—
304D4—94no—
305D4—454no—
306D4—454no—
307C2×C4—31no—
308C2×C4—51no—
309C2×C4—154no—
310C2×C4—154no—
311C2×C4—151no—
312C2×C4—454no—
313C2×C4—451no—
314C2×C3—22no—
315C2×C3—21no—
316C2×C3—62no—
317C2×C3—302no—
318C2×C3#677212yescomplemented
319C2×C3#678212yescomplemented
320C2×C3C22×D3×D511yescharacteristic
321C2×C3C22×D3×D511yescharacteristic
322D3—22no—
323D3—64no—
324D3—62no—
325D3—62no—
326D3—102no—
327C5C2×#379811yescharacteristic, complemented, Sylow
328C22—32no—
329C22—62no—
330C22—92no—
331C22—152no—
332C22—182no—
333C22—302no—
334C22D32×D511yescharacteristic
335C4—32no—
336C4—52no—
337C4—152no—
338C4—452no—
339C3—21no—
340C3C2×#677211yescharacteristic, complemented
341C3C2×#678211yescharacteristic, complemented
342C2—62no—
343C2—182no—
344C2—302no—
345C2#1062404912yesdirect product factor
346C2C2×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×#67153C2semidirect
3. C2×#67150C2semidirect
4. C2×C5×#626C2semidirect
5. C2×C3⋊D3×Dic5C2semidirect
6. C2×C3×#1200C2semidirect
7. C2×C3×D3×Dic5C2semidirect
8. #10624049C2direct (N × H)
18. #67156C22semidirect
19. #67153C22semidirect
20. #67150C22semidirect
21. C5×#626C22semidirect
22. C3⋊D3×Dic5C22semidirect
23. C3×#1200C22semidirect
24. C3×D3×Dic5C22semidirect
36. C2×#1200D3semidirect
41. C2×D3×Dic5D3semidirect
60. C2×#626D5semidirect
90. #1200C2×D3semidirect
94. D3×Dic5C2×D3semidirect
97. C2×C32×C5#41semidirect
122. #626C2×D5semidirect
168. C32×C5C2×#41semidirect
198. C2×C3×C5#235semidirect
199. C2×C3×C5#235semidirect
251. C2×C32#711semidirect
265. C3×C5C2×#235semidirect
266. C3×C5C2×#235semidirect
296. C2×C5#3798semidirect
299. C32C2×#711semidirect
318. C2×C3#6772semidirect
319. C2×C3#6782semidirect
327. C5C2×#3798semidirect
340. C3C2×#6772semidirect
341. C3C2×#6782semidirect
345. C2#10624049direct (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×#671532
C2×#671502
C2×C5×#6262
C2×C3⋊D3×Dic52
C2×C3×#12002
C2×C3×D3×Dic52
#106240492
C2×#67823
C2×#67723
C2×#37985