Subgroups — C₅×#12988 #11935708

Order 1280.

466 conjugacy classes of subgroups.

The subgroup lattice has 286 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₅×#3407	C₂	1	1	yes	maximal, characteristic, complemented
2	C₅×#2932	C₂	1	1	yes	maximal, characteristic
3	C₅×#2117	C₂	1	2	yes	maximal, complemented
4	C₅×#2115	C₂	1	1	yes	maximal, characteristic, complemented
5	C₅×#1375	C₂	1	2	yes	maximal, complemented
6	C₂×C₅×#131	C₂²	1	1	yes	characteristic
7	C₂×C₅×#131	C₂²	1	1	yes	characteristic
8	C₅×#532	C₄	1	1	yes	characteristic
9	C₅×#504	—	2	4	no	—
10	C₂×C₅×M₃₂	—	2	2	no	—
11	C₂×C₅×M₃₂	C₂²	1	2	yes	—
12	C₂×C₅×M₃₂	C₂²	1	2	yes	—
13	C₅×#435	—	2	2	no	—
14	C₅×#435	—	2	2	no	—
15	C₅×#423	C₄	1	1	yes	characteristic
16	C₅×#408	C₄	1	1	yes	characteristic
17	C₅×#408	C₄	1	1	yes	characteristic
18	C₂×C₅×#97	C₂²	1	1	yes	characteristic
19	C₅×#350	—	2	4	no	—
20	C₅×#349	—	2	2	no	—
21	#12988	C₅	1	1	yes	maximal, characteristic, direct product factor, Sylow
22	C₂×C₅×#41	C₂×C₄	1	1	yes	characteristic
23	C₂×C₅×#41	C₂×C₄	1	1	yes	characteristic
24	C₅×#131	—	2	2	no	—
25	C₅×#131	—	2	2	no	—
26	C₅×#131	—	2	2	no	—
27	C₅×#131	—	2	2	no	—
28	C₂×C₅×M₁₆	—	2	1	no	—
29	C₂×C₅×M₁₆	C₂×C₄	1	1	yes	characteristic
30	C₂×C₅×M₁₆	C₂×C₄	1	1	yes	characteristic
31	C₂×C₅×M₁₆	D₄	1	2	yes	—
32	C₂²×C₅×C₈	—	2	1	no	—
33	C₂²×C₅×C₈	C₂×C₄	1	1	yes	characteristic
34	C₂²×C₅×C₈	D₄	1	2	yes	—
35	C₂²×C₅×C₈	C₂³	1	1	yes	characteristic
36	C₅×C₄⋊Q₈	C₈	1	1	yes	characteristic
37	C₅×#127	C₈	1	1	yes	characteristic
38	C₅×#124	C₈	1	1	yes	characteristic
39	C₅×#124	C₈	1	1	yes	characteristic
40	C₅×#121	—	2	2	no	—
41	C₄×C₅×D₄	—	2	2	no	—
42	C₂×C₄²×C₅	C₂×C₄	1	1	yes	characteristic
43	C₅×M₃₂	—	2	4	no	—
44	C₅×M₃₂	—	2	2	no	—
45	C₅×M₃₂	—	2	2	no	—
46	C₅×M₃₂	—	2	2	no	—
47	C₅×M₃₂	—	2	2	no	—
48	C₂×C₅×C₁₆	—	2	4	no	—
49	C₂×C₅×C₁₆	—	2	2	no	—
50	C₂×C₅×C₁₆	D₄	1	4	yes	complemented
51	C₅×C₄⋊C₈	—	2	2	no	—
52	C₅×C₂²⋊C₈	—	2	2	no	—
53	C₅×C₂²⋊C₈	—	2	2	no	—
54	C₅×#97	—	2	1	no	—
55	C₅×#97	—	2	1	no	—
56	#3407	C₂×C₅	1	1	yes	characteristic, complemented
57	#2932	C₂×C₅	1	1	yes	characteristic
58	#2117	C₂×C₅	1	2	yes	complemented
59	#2115	C₂×C₅	1	1	yes	characteristic, complemented
60	#1375	C₂×C₅	1	2	yes	complemented
61	C₅×#41	—	2	2	no	—
62	C₅×#41	—	2	2	no	—
63	C₅×#41	—	2	2	no	—
64	C₅×#41	—	2	2	no	—
65	C₂×C₅×Q₈	C₂×C₈	1	1	yes	characteristic
66	C₂×C₅×Q₈	C₂×C₈	1	1	yes	characteristic
67	C₂×C₅×D₄	—	2	1	no	—
68	C₂×C₅×D₄	—	2	2	no	—
69	C₂×C₅×D₄	C₂×C₈	1	1	yes	characteristic
70	C₂×C₅×D₄	C₂×C₈	1	1	yes	characteristic
71	C₂×C₅×D₄	M₁₆	1	2	yes	—
72	C₂²×C₄×C₅	—	2	1	no	—
73	C₂²×C₄×C₅	—	2	1	no	—
74	C₂²×C₄×C₅	C₂²⋊C₄	1	2	yes	—
75	C₂²×C₄×C₅	C₂²×C₄	1	1	yes	characteristic
76	C₅×M₁₆	—	2	2	no	—
77	C₅×M₁₆	—	2	4	no	—
78	C₅×M₁₆	—	2	2	no	—
79	C₅×M₁₆	—	4	1	no	—
80	C₅×M₁₆	—	4	1	no	—
81	C₂×C₅×C₈	—	2	2	no	—
82	C₂×C₅×C₈	—	2	2	no	—
83	C₂×C₅×C₈	—	2	1	no	—
84	C₂×C₅×C₈	—	2	1	no	—
85	C₂×C₅×C₈	—	2	2	no	—
86	C₂×C₅×C₈	—	4	2	no	—
87	C₂×C₅×C₈	C₂²⋊C₄	1	2	yes	—
88	C₂×C₅×C₈	C₂×D₄	1	2	yes	—
89	C₂×C₅×C₈	C₂×D₄	1	1	yes	characteristic
90	C₂×C₅×C₈	C₂×D₄	1	1	yes	characteristic
91	C₂×C₅×C₈	#41	1	2	yes	—
92	C₅×C₄⋊C₄	—	2	2	no	—
93	C₅×C₂²⋊C₄	—	2	2	no	—
94	C₅×C₂²⋊C₄	—	2	2	no	—
95	C₄²×C₅	—	2	1	no	—
96	C₄²×C₅	C₂×C₈	1	1	yes	characteristic
97	C₄²×C₅	C₂×C₈	1	1	yes	characteristic
98	C₅×C₁₆	—	2	4	no	—
99	C₅×C₁₆	—	4	2	no	—
100	C₂×#131	C₂²×C₅	1	1	yes	characteristic
101	C₂×#131	C₂²×C₅	1	1	yes	characteristic
102	#532	C₄×C₅	1	1	yes	characteristic
103	#504	—	2	4	no	—
104	C₂×M₃₂	—	2	2	no	—
105	C₂×M₃₂	C₂²×C₅	1	2	yes	—
106	C₂×M₃₂	C₂²×C₅	1	2	yes	—
107	#435	—	2	2	no	—
108	#435	—	2	2	no	—
109	#423	C₄×C₅	1	1	yes	characteristic
110	#408	C₄×C₅	1	1	yes	characteristic
111	#408	C₄×C₅	1	1	yes	characteristic
112	C₂×#97	C₂²×C₅	1	1	yes	characteristic
113	#350	—	2	4	no	—
114	#349	—	2	2	no	—
115	C₂³×C₅	—	2	1	no	—
116	C₂³×C₅	C₂²⋊C₈	1	2	yes	—
117	C₂³×C₅	C₂×M₁₆	1	1	yes	characteristic
118	C₅×Q₈	—	2	2	no	—
119	C₅×Q₈	—	2	2	no	—
120	C₅×D₄	—	2	4	no	—
121	C₅×D₄	—	2	2	no	—
122	C₅×D₄	—	2	2	no	—
123	C₅×D₄	—	4	2	no	—
124	C₅×D₄	—	8	2	no	—
125	C₂×C₄×C₅	—	2	2	no	—
126	C₂×C₄×C₅	—	2	2	no	—
127	C₂×C₄×C₅	—	2	1	no	—
128	C₂×C₄×C₅	—	2	1	no	—
129	C₂×C₄×C₅	—	2	1	no	—
130	C₂×C₄×C₅	—	4	1	no	—
131	C₂×C₄×C₅	—	4	1	no	—
132	C₂×C₄×C₅	—	8	1	no	—
133	C₂×C₄×C₅	C₂²⋊C₈	1	2	yes	—
134	C₂×C₄×C₅	C₂×C₂²⋊C₄	1	1	yes	characteristic
135	C₂×C₄×C₅	C₄×D₄	1	2	yes	—
136	C₂×C₄×C₅	C₂²×C₈	1	1	yes	characteristic
137	C₂×C₄×C₅	#131	1	2	yes	—
138	C₅×C₈	—	2	2	no	—
139	C₅×C₈	—	4	1	no	—
140	C₅×C₈	—	4	1	no	—
141	C₅×C₈	#120	1	1	yes	characteristic
142	C₅×C₈	#121	1	2	yes	—
143	C₅×C₈	#123	1	1	yes	characteristic
144	C₂×#41	C₂×C₄×C₅	1	1	yes	characteristic
145	C₂×#41	C₂×C₄×C₅	1	1	yes	characteristic
146	#131	—	2	2	no	—
147	#131	—	2	2	no	—
148	#131	—	2	2	no	—
149	#131	—	2	2	no	—
150	C₂×M₁₆	—	2	1	no	—
151	C₂×M₁₆	C₂×C₄×C₅	1	1	yes	characteristic
152	C₂×M₁₆	C₂×C₄×C₅	1	1	yes	characteristic
153	C₂×M₁₆	C₅×D₄	1	2	yes	—
154	C₂²×C₈	—	2	1	no	—
155	C₂²×C₈	C₂×C₄×C₅	1	1	yes	characteristic
156	C₂²×C₈	C₅×D₄	1	2	yes	—
157	C₂²×C₈	C₂³×C₅	1	1	yes	characteristic
158	C₄⋊Q₈	C₅×C₈	1	1	yes	characteristic
159	#127	C₅×C₈	1	1	yes	characteristic
160	#124	C₅×C₈	1	1	yes	characteristic
161	#124	C₅×C₈	1	1	yes	characteristic
162	#121	—	2	2	no	—
163	C₄×D₄	—	2	2	no	—
164	C₂×C₄²	C₂×C₄×C₅	1	1	yes	characteristic
165	M₃₂	—	2	4	no	—
166	M₃₂	—	2	2	no	—
167	M₃₂	—	2	2	no	—
168	M₃₂	—	2	2	no	—
169	M₃₂	—	2	2	no	—
170	C₂×C₁₆	—	2	4	no	—
171	C₂×C₁₆	—	2	2	no	—
172	C₂×C₁₆	C₅×D₄	1	4	yes	complemented
173	C₄⋊C₈	—	2	2	no	—
174	C₂²⋊C₈	—	2	2	no	—
175	C₂²⋊C₈	—	2	2	no	—
176	#97	—	2	1	no	—
177	#97	—	2	1	no	—
178	C₂²×C₅	—	2	2	no	—
179	C₂²×C₅	—	4	1	no	—
180	C₂²×C₅	—	4	2	no	—
181	C₂²×C₅	—	4	2	no	—
182	C₂²×C₅	—	4	1	no	—
183	C₂²×C₅	C₂×C₂²⋊C₈	1	1	yes	characteristic
184	C₂²×C₅	#435	1	2	yes	—
185	C₄×C₅	—	2	2	no	—
186	C₄×C₅	—	4	1	no	—
187	C₄×C₅	—	4	1	no	—
188	C₄×C₅	—	4	1	no	—
189	C₄×C₅	#386	1	1	yes	characteristic
190	C₄×C₅	#408	1	1	yes	characteristic
191	C₄×C₅	C₈×D₄	1	2	yes	—
192	#41	—	2	2	no	—
193	#41	—	2	2	no	—
194	#41	—	2	2	no	—
195	#41	—	2	2	no	—
196	C₂×Q₈	C₂×C₅×C₈	1	1	yes	characteristic
197	C₂×Q₈	C₂×C₅×C₈	1	1	yes	characteristic
198	C₂×D₄	—	2	1	no	—
199	C₂×D₄	—	2	2	no	—
200	C₂×D₄	C₂×C₅×C₈	1	1	yes	characteristic
201	C₂×D₄	C₂×C₅×C₈	1	1	yes	characteristic
202	C₂×D₄	C₅×M₁₆	1	2	yes	—
203	C₂²×C₄	—	2	1	no	—
204	C₂²×C₄	—	2	1	no	—
205	C₂²×C₄	C₅×C₂²⋊C₄	1	2	yes	—
206	C₂²×C₄	C₂²×C₄×C₅	1	1	yes	characteristic
207	M₁₆	—	2	2	no	—
208	M₁₆	—	2	4	no	—
209	M₁₆	—	2	2	no	—
210	M₁₆	—	4	1	no	—
211	M₁₆	—	4	1	no	—
212	C₂×C₈	—	2	2	no	—
213	C₂×C₈	—	2	2	no	—
214	C₂×C₈	—	2	1	no	—
215	C₂×C₈	—	2	1	no	—
216	C₂×C₈	—	2	2	no	—
217	C₂×C₈	—	4	2	no	—
218	C₂×C₈	C₅×C₂²⋊C₄	1	2	yes	—
219	C₂×C₈	C₂×C₅×D₄	1	2	yes	—
220	C₂×C₈	C₂×C₅×D₄	1	1	yes	characteristic
221	C₂×C₈	C₂×C₅×D₄	1	1	yes	characteristic
222	C₂×C₈	C₅×#41	1	2	yes	—
223	C₄⋊C₄	—	2	2	no	—
224	C₂²⋊C₄	—	2	2	no	—
225	C₂²⋊C₄	—	2	2	no	—
226	C₄²	—	2	1	no	—
227	C₄²	C₂×C₅×C₈	1	1	yes	characteristic
228	C₄²	C₂×C₅×C₈	1	1	yes	characteristic
229	C₁₆	—	2	4	no	—
230	C₁₆	—	4	2	no	—
231	C₂×C₅	—	2	1	no	—
232	C₂×C₅	—	2	2	no	—
233	C₂×C₅	—	4	2	no	—
234	C₂×C₅	—	8	1	no	—
235	C₂×C₅	#1869	1	1	yes	characteristic
236	C₂³	—	2	1	no	—
237	C₂³	C₅×C₂²⋊C₈	1	2	yes	—
238	C₂³	C₂×C₅×M₁₆	1	1	yes	characteristic
239	Q₈	—	2	2	no	—
240	Q₈	—	2	2	no	—
241	D₄	—	2	4	no	—
242	D₄	—	2	2	no	—
243	D₄	—	2	2	no	—
244	D₄	—	4	2	no	—
245	D₄	—	8	2	no	—
246	C₂×C₄	—	2	2	no	—
247	C₂×C₄	—	2	2	no	—
248	C₂×C₄	—	2	1	no	—
249	C₂×C₄	—	2	1	no	—
250	C₂×C₄	—	2	1	no	—
251	C₂×C₄	—	4	1	no	—
252	C₂×C₄	—	4	1	no	—
253	C₂×C₄	—	8	1	no	—
254	C₂×C₄	C₅×C₂²⋊C₈	1	2	yes	—
255	C₂×C₄	C₂×C₅×C₂²⋊C₄	1	1	yes	characteristic
256	C₂×C₄	C₄×C₅×D₄	1	2	yes	—
257	C₂×C₄	C₂²×C₅×C₈	1	1	yes	characteristic
258	C₂×C₄	C₅×#131	1	2	yes	—
259	C₈	—	2	2	no	—
260	C₈	—	4	1	no	—
261	C₈	—	4	1	no	—
262	C₈	C₅×#120	1	1	yes	characteristic
263	C₈	C₅×#121	1	2	yes	—
264	C₈	C₅×#123	1	1	yes	characteristic
265	C₅	#12988	1	1	yes	characteristic, direct product factor, Sylow
266	C₂²	—	2	2	no	—
267	C₂²	—	4	1	no	—
268	C₂²	—	4	2	no	—
269	C₂²	—	4	2	no	—
270	C₂²	—	4	1	no	—
271	C₂²	C₂×C₅×C₂²⋊C₈	1	1	yes	characteristic
272	C₂²	C₅×#435	1	2	yes	—
273	C₄	—	2	2	no	—
274	C₄	—	4	1	no	—
275	C₄	—	4	1	no	—
276	C₄	—	4	1	no	—
277	C₄	C₅×#386	1	1	yes	characteristic
278	C₄	C₅×#408	1	1	yes	characteristic
279	C₄	C₅×C₈×D₄	1	2	yes	—
280	C₂	—	2	1	no	—
281	C₂	—	2	2	no	—
282	C₂	—	4	2	no	—
283	C₂	—	8	1	no	—
284	C₂	C₅×#1869	1	1	yes	characteristic

	subgroup	G/H
center	C₄×C₅	#386
commutator	C₂×C₄	C₂²×C₅×C₈
Frattini, cosocle	C₂²×C₈	C₂³×C₅
socle	C₂×C₅	#1869
prime radical	C₅×#532	C₄
LCS	G ▷ C₂×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₄×C₅ ◁ C₂²×C₅×C₈ ◁ G

Complements

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

N	H ≅ G/N	type
1. C₅×#3407	C₂	semidirect
3. C₅×#2117	C₂	semidirect
4. C₅×#2115	C₂	semidirect
5. C₅×#1375	C₂	semidirect
21. #12988	C₅	direct (N × H)
50. C₂×C₅×C₁₆	D₄	semidirect
56. #3407	C₂×C₅	semidirect
58. #2117	C₂×C₅	semidirect
59. #2115	C₂×C₅	semidirect
60. #1375	C₂×C₅	semidirect
172. C₂×C₁₆	C₅×D₄	semidirect
265. C₅	#12988	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×#3407	2
C₅×#2932	2
C₅×#2117	2
C₅×#2115	2
C₅×#1375	2
#12988	5

Subgroups — C5×#12988 #11935708

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₅×#12988 #11935708