Subgroups — C₅×#12949 #11935669

Order 1280.

394 conjugacy classes of subgroups.

The subgroup lattice has 304 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₅×#2952	C₂	1	1	yes	maximal, characteristic
2	C₂×C₅×#365	C₂	1	1	yes	maximal, characteristic, complemented
3	C₅×#2148	C₂	1	1	yes	maximal, characteristic
4	C₅×#2143	C₂	1	1	yes	maximal, characteristic
5	C₅×#1953	C₂	1	1	yes	maximal, characteristic, complemented
6	C₅×#1383	C₂	1	1	yes	maximal, characteristic, complemented
7	C₅×#1378	C₂	1	1	yes	maximal, characteristic, complemented
8	C₂×C₅×#135	C₂²	1	1	yes	characteristic
9	C₅×#520	—	2	1	no	—
10	C₂×C₅×M₃₂	C₂²	1	1	yes	characteristic
11	C₂×C₅×M₃₂	C₂²	1	1	yes	characteristic
12	C₅×#442	—	2	1	no	—
13	C₅×#441	—	2	1	no	—
14	C₄×C₅×SD₁₆	—	2	2	no	—
15	C₅×#430	—	2	1	no	—
16	C₂×C₅×#108	C₂²	1	1	yes	characteristic
17	C₂×C₅×#108	C₂²	1	1	yes	characteristic
18	C₂×C₅×#106	C₂²	1	1	yes	characteristic
19	C₅×#417	—	2	1	no	—
20	C₅×#405	C₂²	1	1	yes	characteristic
21	C₅×#365	—	2	2	no	—
22	C₅×#362	—	2	1	no	—
23	C₅×#361	—	2	1	no	—
24	C₅×#359	C₄	1	4	yes	complemented
25	C₅×#344	—	2	1	no	—
26	#12949	C₅	1	1	yes	maximal, characteristic, direct product factor, Sylow
27	C₂×C₅×#41	—	2	1	no	—
28	C₅×#135	—	2	1	no	—
29	C₅×#135	—	2	1	no	—
30	C₅×#135	C₂×C₄	1	2	yes	—
31	C₅×#135	Q₈	1	2	yes	—
32	C₂×C₅×Q₁₆	D₄	1	1	yes	characteristic
33	C₂×C₅×SD₁₆	—	2	1	no	—
34	C₂×C₅×D₈	D₄	1	1	yes	characteristic
35	C₂×C₅×M₁₆	—	2	1	no	—
36	C₂×C₅×M₁₆	—	2	1	no	—
37	C₂×C₅×M₁₆	D₄	1	1	yes	characteristic
38	C₂²×C₅×C₈	C₂³	1	1	yes	characteristic
39	C₄×C₅×Q₈	—	2	2	no	—
40	C₄×C₅×D₄	—	2	2	no	—
41	C₄×C₅×D₄	—	4	2	no	—
42	C₅×#117	—	4	1	no	—
43	C₅×#117	D₄	1	1	yes	characteristic
44	C₂×C₅×C₄⋊C₄	—	2	1	no	—
45	C₂×C₅×C₄⋊C₄	—	4	1	no	—
46	C₅×M₃₂	—	2	1	no	—
47	C₅×M₃₂	—	2	1	no	—
48	C₅×M₃₂	—	2	1	no	—
49	C₅×M₃₂	—	2	1	no	—
50	C₂×C₅×C₁₆	—	2	1	no	—
51	C₂×C₅×C₁₆	C₂×C₄	1	2	yes	complemented
52	C₅×#108	—	2	1	no	—
53	C₅×#108	—	2	1	no	—
54	C₅×#108	—	2	1	no	—
55	C₅×#108	—	4	2	no	—
56	C₅×#108	C₂×C₄	1	2	yes	complemented
57	C₅×#106	—	2	1	no	—
58	C₅×#106	D₄	1	2	yes	—
59	C₅×#103	—	2	2	no	—
60	C₅×#102	—	2	2	no	—
61	C₅×#97	—	2	1	no	—
62	C₄×C₅×C₈	—	2	1	no	—
63	#2952	C₂×C₅	1	1	yes	characteristic
64	C₂×#365	C₂×C₅	1	1	yes	characteristic, complemented
65	#2148	C₂×C₅	1	1	yes	characteristic
66	#2143	C₂×C₅	1	1	yes	characteristic
67	#1953	C₂×C₅	1	1	yes	characteristic, complemented
68	#1383	C₂×C₅	1	1	yes	characteristic, complemented
69	#1378	C₂×C₅	1	1	yes	characteristic, complemented
70	C₅×#41	—	2	2	no	—
71	C₅×#41	—	2	2	no	—
72	C₅×#41	—	4	1	no	—
73	C₅×#41	—	4	1	no	—
74	C₂×C₅×Q₈	—	2	1	no	—
75	C₂×C₅×D₄	—	2	1	no	—
76	C₂×C₅×D₄	—	4	1	no	—
77	C₂²×C₄×C₅	—	4	2	no	—
78	C₂²×C₄×C₅	—	4	1	no	—
79	C₂²×C₄×C₅	—	4	1	no	—
80	C₂²×C₄×C₅	C₂×D₄	1	1	yes	characteristic
81	C₅×Q₁₆	—	2	1	no	—
82	C₅×Q₁₆	C₄⋊C₄	1	2	yes	—
83	C₅×SD₁₆	—	2	2	no	—
84	C₅×SD₁₆	—	2	2	no	—
85	C₅×D₈	—	2	1	no	—
86	C₅×D₈	C₄⋊C₄	1	2	yes	—
87	C₅×M₁₆	—	2	2	no	—
88	C₅×M₁₆	—	2	1	no	—
89	C₅×M₁₆	—	2	1	no	—
90	C₅×M₁₆	—	4	1	no	—
91	C₅×M₁₆	—	4	1	no	—
92	C₅×M₁₆	—	4	1	no	—
93	C₂×C₅×C₈	—	2	1	no	—
94	C₂×C₅×C₈	—	4	1	no	—
95	C₂×C₅×C₈	—	4	1	no	—
96	C₂×C₅×C₈	C₂²×C₄	1	1	yes	characteristic
97	C₂×C₅×C₈	C₂×D₄	1	1	yes	characteristic
98	C₂×C₅×C₈	C₂×D₄	1	1	yes	characteristic
99	C₂×C₅×C₈	C₂×Q₈	1	1	yes	characteristic
100	C₂×C₅×C₈	#41	1	1	yes	characteristic
101	C₂×C₅×C₈	#41	1	1	yes	characteristic
102	C₅×C₄⋊C₄	—	2	1	no	—
103	C₅×C₄⋊C₄	—	2	2	no	—
104	C₅×C₄⋊C₄	—	4	2	no	—
105	C₅×C₄⋊C₄	—	4	1	no	—
106	C₅×C₂²⋊C₄	—	4	2	no	—
107	C₅×C₂²⋊C₄	SD₁₆	1	2	yes	—
108	C₄²×C₅	—	2	1	no	—
109	C₄²×C₅	—	4	2	no	—
110	C₅×C₁₆	—	2	2	no	—
111	C₅×C₁₆	—	4	1	no	—
112	C₂×#135	C₂²×C₅	1	1	yes	characteristic
113	#520	—	2	1	no	—
114	C₂×M₃₂	C₂²×C₅	1	1	yes	characteristic
115	C₂×M₃₂	C₂²×C₅	1	1	yes	characteristic
116	#442	—	2	1	no	—
117	#441	—	2	1	no	—
118	C₄×SD₁₆	—	2	2	no	—
119	#430	—	2	1	no	—
120	C₂×#108	C₂²×C₅	1	1	yes	characteristic
121	C₂×#108	C₂²×C₅	1	1	yes	characteristic
122	C₂×#106	C₂²×C₅	1	1	yes	characteristic
123	#417	—	2	1	no	—
124	#405	C₂²×C₅	1	1	yes	characteristic
125	#365	—	2	2	no	—
126	#362	—	2	1	no	—
127	#361	—	2	1	no	—
128	#359	C₄×C₅	1	4	yes	complemented
129	#344	—	2	1	no	—
130	C₂³×C₅	—	4	1	no	—
131	C₂³×C₅	C₂×SD₁₆	1	1	yes	characteristic
132	C₅×Q₈	—	2	2	no	—
133	C₅×Q₈	—	4	1	no	—
134	C₅×D₄	—	2	2	no	—
135	C₅×D₄	—	4	2	no	—
136	C₅×D₄	—	4	2	no	—
137	C₅×D₄	—	4	1	no	—
138	C₂×C₄×C₅	—	2	2	no	—
139	C₂×C₄×C₅	—	4	1	no	—
140	C₂×C₄×C₅	—	4	2	no	—
141	C₂×C₄×C₅	—	4	2	no	—
142	C₂×C₄×C₅	—	4	2	no	—
143	C₂×C₄×C₅	—	16	2	no	—
144	C₂×C₄×C₅	—	16	1	no	—
145	C₂×C₄×C₅	C₄×D₄	1	1	yes	characteristic
146	C₂×C₄×C₅	#120	1	1	yes	characteristic
147	C₂×C₄×C₅	C₂²⋊Q₈	1	1	yes	characteristic
148	C₂×C₄×C₅	#135	1	1	yes	characteristic
149	C₂×C₄×C₅	AGL(1,ℤ/8)	1	1	yes	characteristic
150	C₂×C₄×C₅	AGL(1,ℤ/8)	1	1	yes	characteristic
151	C₅×C₈	—	4	2	no	—
152	C₅×C₈	—	4	1	no	—
153	C₅×C₈	—	8	1	no	—
154	C₅×C₈	C₂×C₄⋊C₄	1	1	yes	characteristic
155	C₅×C₈	C₄×D₄	1	1	yes	characteristic
156	C₅×C₈	C₂²⋊Q₈	1	1	yes	characteristic
157	C₅×C₈	#123	1	1	yes	characteristic
158	C₂×#41	—	2	1	no	—
159	#135	—	2	1	no	—
160	#135	—	2	1	no	—
161	#135	C₂×C₄×C₅	1	2	yes	—
162	#135	C₅×Q₈	1	2	yes	—
163	C₂×Q₁₆	C₅×D₄	1	1	yes	characteristic
164	C₂×SD₁₆	—	2	1	no	—
165	C₂×D₈	C₅×D₄	1	1	yes	characteristic
166	C₂×M₁₆	—	2	1	no	—
167	C₂×M₁₆	—	2	1	no	—
168	C₂×M₁₆	C₅×D₄	1	1	yes	characteristic
169	C₂²×C₈	C₂³×C₅	1	1	yes	characteristic
170	C₄×Q₈	—	2	2	no	—
171	C₄×D₄	—	2	2	no	—
172	C₄×D₄	—	4	2	no	—
173	#117	—	4	1	no	—
174	#117	C₅×D₄	1	1	yes	characteristic
175	C₂×C₄⋊C₄	—	2	1	no	—
176	C₂×C₄⋊C₄	—	4	1	no	—
177	M₃₂	—	2	1	no	—
178	M₃₂	—	2	1	no	—
179	M₃₂	—	2	1	no	—
180	M₃₂	—	2	1	no	—
181	C₂×C₁₆	—	2	1	no	—
182	C₂×C₁₆	C₂×C₄×C₅	1	2	yes	complemented
183	#108	—	2	1	no	—
184	#108	—	2	1	no	—
185	#108	—	2	1	no	—
186	#108	—	4	2	no	—
187	#108	C₂×C₄×C₅	1	2	yes	complemented
188	#106	—	2	1	no	—
189	#106	C₅×D₄	1	2	yes	—
190	#103	—	2	2	no	—
191	#102	—	2	2	no	—
192	#97	—	2	1	no	—
193	C₄×C₈	—	2	1	no	—
194	C₂²×C₅	—	4	1	no	—
195	C₂²×C₅	—	4	2	no	—
196	C₂²×C₅	—	8	1	no	—
197	C₂²×C₅	—	8	1	no	—
198	C₂²×C₅	C₄×SD₁₆	1	1	yes	characteristic
199	C₂²×C₅	#450	1	1	yes	characteristic
200	C₂²×C₅	#476	1	1	yes	characteristic
201	C₄×C₅	—	4	2	no	—
202	C₄×C₅	—	4	2	no	—
203	C₄×C₅	—	8	2	no	—
204	C₄×C₅	#385	1	1	yes	characteristic
205	C₄×C₅	#442	1	1	yes	characteristic
206	C₄×C₅	#449	1	1	yes	characteristic
207	C₄×C₅	#478	1	1	yes	characteristic
208	#41	—	2	2	no	—
209	#41	—	2	2	no	—
210	#41	—	4	1	no	—
211	#41	—	4	1	no	—
212	C₂×Q₈	—	2	1	no	—
213	C₂×D₄	—	2	1	no	—
214	C₂×D₄	—	4	1	no	—
215	C₂²×C₄	—	4	2	no	—
216	C₂²×C₄	—	4	1	no	—
217	C₂²×C₄	—	4	1	no	—
218	C₂²×C₄	C₂×C₅×D₄	1	1	yes	characteristic
219	Q₁₆	—	2	1	no	—
220	Q₁₆	C₅×C₄⋊C₄	1	2	yes	—
221	SD₁₆	—	2	2	no	—
222	SD₁₆	—	2	2	no	—
223	D₈	—	2	1	no	—
224	D₈	C₅×C₄⋊C₄	1	2	yes	—
225	M₁₆	—	2	2	no	—
226	M₁₆	—	2	1	no	—
227	M₁₆	—	2	1	no	—
228	M₁₆	—	4	1	no	—
229	M₁₆	—	4	1	no	—
230	M₁₆	—	4	1	no	—
231	C₂×C₈	—	2	1	no	—
232	C₂×C₈	—	4	1	no	—
233	C₂×C₈	—	4	1	no	—
234	C₂×C₈	C₂²×C₄×C₅	1	1	yes	characteristic
235	C₂×C₈	C₂×C₅×D₄	1	1	yes	characteristic
236	C₂×C₈	C₂×C₅×D₄	1	1	yes	characteristic
237	C₂×C₈	C₂×C₅×Q₈	1	1	yes	characteristic
238	C₂×C₈	C₅×#41	1	1	yes	characteristic
239	C₂×C₈	C₅×#41	1	1	yes	characteristic
240	C₄⋊C₄	—	2	1	no	—
241	C₄⋊C₄	—	2	2	no	—
242	C₄⋊C₄	—	4	2	no	—
243	C₄⋊C₄	—	4	1	no	—
244	C₂²⋊C₄	—	4	2	no	—
245	C₂²⋊C₄	C₅×SD₁₆	1	2	yes	—
246	C₄²	—	2	1	no	—
247	C₄²	—	4	2	no	—
248	C₁₆	—	2	2	no	—
249	C₁₆	—	4	1	no	—
250	C₂×C₅	—	2	1	no	—
251	C₂×C₅	—	2	1	no	—
252	C₂×C₅	—	2	1	no	—
253	C₂×C₅	—	8	2	no	—
254	C₂×C₅	#1864	1	1	yes	characteristic
255	C₂³	—	4	1	no	—
256	C₂³	C₂×C₅×SD₁₆	1	1	yes	characteristic
257	Q₈	—	2	2	no	—
258	Q₈	—	4	1	no	—
259	D₄	—	2	2	no	—
260	D₄	—	4	2	no	—
261	D₄	—	4	2	no	—
262	D₄	—	4	1	no	—
263	C₂×C₄	—	2	2	no	—
264	C₂×C₄	—	4	1	no	—
265	C₂×C₄	—	4	2	no	—
266	C₂×C₄	—	4	2	no	—
267	C₂×C₄	—	4	2	no	—
268	C₂×C₄	—	16	2	no	—
269	C₂×C₄	—	16	1	no	—
270	C₂×C₄	C₄×C₅×D₄	1	1	yes	characteristic
271	C₂×C₄	C₅×#120	1	1	yes	characteristic
272	C₂×C₄	C₅×C₂²⋊Q₈	1	1	yes	characteristic
273	C₂×C₄	C₅×#135	1	1	yes	characteristic
274	C₂×C₄	C₅×AGL(1,ℤ/8)	1	1	yes	characteristic
275	C₂×C₄	C₅×AGL(1,ℤ/8)	1	1	yes	characteristic
276	C₈	—	4	2	no	—
277	C₈	—	4	1	no	—
278	C₈	—	8	1	no	—
279	C₈	C₂×C₅×C₄⋊C₄	1	1	yes	characteristic
280	C₈	C₄×C₅×D₄	1	1	yes	characteristic
281	C₈	C₅×C₂²⋊Q₈	1	1	yes	characteristic
282	C₈	C₅×#123	1	1	yes	characteristic
283	C₅	#12949	1	1	yes	characteristic, direct product factor, Sylow
284	C₂²	—	4	1	no	—
285	C₂²	—	4	2	no	—
286	C₂²	—	8	1	no	—
287	C₂²	—	8	1	no	—
288	C₂²	C₄×C₅×SD₁₆	1	1	yes	characteristic
289	C₂²	C₅×#450	1	1	yes	characteristic
290	C₂²	C₅×#476	1	1	yes	characteristic
291	C₄	—	4	2	no	—
292	C₄	—	4	2	no	—
293	C₄	—	8	2	no	—
294	C₄	C₅×#385	1	1	yes	characteristic
295	C₄	C₅×#442	1	1	yes	characteristic
296	C₄	C₅×#449	1	1	yes	characteristic
297	C₄	C₅×#478	1	1	yes	characteristic
298	C₂	—	2	1	no	—
299	C₂	—	2	1	no	—
300	C₂	—	2	1	no	—
301	C₂	—	8	2	no	—
302	C₂	C₅×#1864	1	1	yes	characteristic

	subgroup	G/H
center, socle	C₂×C₅	#1864
commutator	C₂×C₈	C₂²×C₄×C₅
Frattini, cosocle	C₂²×C₈	C₂³×C₅
prime radical	C₂×C₅×#135	C₂²
LCS	G ▷ C₂×C₈ ▷ C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂×C₅ ◁ C₂²×C₄×C₅ ◁ C₅×#405 ◁ G

Complements

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

N	H ≅ G/N	type
2. C₂×C₅×#365	C₂	semidirect
5. C₅×#1953	C₂	semidirect
6. C₅×#1383	C₂	semidirect
7. C₅×#1378	C₂	semidirect
24. C₅×#359	C₄	semidirect
26. #12949	C₅	direct (N × H)
51. C₂×C₅×C₁₆	C₂×C₄	semidirect
56. C₅×#108	C₂×C₄	semidirect
64. C₂×#365	C₂×C₅	semidirect
67. #1953	C₂×C₅	semidirect
68. #1383	C₂×C₅	semidirect
69. #1378	C₂×C₅	semidirect
128. #359	C₄×C₅	semidirect
182. C₂×C₁₆	C₂×C₄×C₅	semidirect
187. #108	C₂×C₄×C₅	semidirect
283. C₅	#12949	direct (N × H)

Maximal subgroups

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

subgroup	index
C₅×#2952	2
C₂×C₅×#365	2
C₅×#2148	2
C₅×#2143	2
C₅×#1953	2
C₅×#1383	2
C₅×#1378	2
#12949	5

Subgroups — C5×#12949 #11935669

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₅×#12949 #11935669