Subgroups — C₅×#13214 #11935934

Order 1280.

490 conjugacy classes of subgroups.

The subgroup lattice has 292 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×C₅×#495	C₂	1	1	yes	maximal, characteristic
2	C₂×C₅×#362	C₂	1	2	yes	maximal, complemented
3	C₂×C₅×#361	C₂	1	2	yes	maximal
4	C₄×C₅×M₃₂	C₂	1	1	yes	maximal, characteristic, complemented
5	C₅×#1848	C₂	1	1	yes	maximal, characteristic, complemented
6	C₂²×C₅×Q₁₆	C₂²	1	1	yes	characteristic
7	C₂²×C₅×SD₁₆	—	2	1	no	—
8	C₂²×C₅×D₈	C₂²	1	1	yes	characteristic
9	C₂×C₅×#124	—	2	1	no	—
10	C₂×C₅×M₃₂	C₂²	1	2	yes	—
11	C₅×#495	—	2	1	no	—
12	C₅×#495	—	2	1	no	—
13	C₅×#495	C₄	1	2	yes	—
14	C₅×#495	C₄	1	2	yes	—
15	C₂×C₅×#108	—	2	1	no	—
16	C₂×C₅×#108	C₂²	1	2	yes	—
17	C₅×#423	—	2	1	no	—
18	C₂×C₄×C₅×C₈	C₂²	1	1	yes	characteristic
19	C₅×#362	—	2	4	no	—
20	C₅×#361	—	2	4	no	—
21	C₅×#346	—	2	1	no	—
22	C₄×C₅×C₁₆	—	2	1	no	—
23	#13214	C₅	1	1	yes	maximal, characteristic, direct product factor, Sylow
24	C₂²×C₅×Q₈	—	2	1	no	—
25	C₂²×C₅×D₄	—	2	1	no	—
26	C₂×C₅×Q₁₆	—	2	2	no	—
27	C₂×C₅×Q₁₆	—	2	1	no	—
28	C₂×C₅×Q₁₆	—	2	2	no	—
29	C₂×C₅×Q₁₆	C₂×C₄	1	2	yes	—
30	C₂×C₅×SD₁₆	—	2	2	no	—
31	C₂×C₅×SD₁₆	—	2	2	no	—
32	C₂×C₅×SD₁₆	—	4	2	no	—
33	C₂×C₅×SD₁₆	—	4	2	no	—
34	C₂×C₅×D₈	—	2	2	no	—
35	C₂×C₅×D₈	—	2	1	no	—
36	C₂×C₅×D₈	—	2	2	no	—
37	C₂×C₅×D₈	C₂×C₄	1	2	yes	—
38	C₂×C₅×M₁₆	—	2	2	no	—
39	C₂²×C₅×C₈	D₄	1	1	yes	characteristic
40	C₂²×C₅×C₈	C₂³	1	1	yes	characteristic
41	C₅×#124	—	2	2	no	—
42	C₅×#124	—	2	2	no	—
43	C₅×#124	—	4	2	no	—
44	C₂×C₅×C₂²⋊C₄	—	4	2	no	—
45	C₂×C₄²×C₅	D₄	1	1	yes	characteristic
46	C₅×M₃₂	—	2	2	no	—
47	C₅×M₃₂	D₄	1	4	yes	—
48	C₂×C₅×C₁₆	—	2	2	no	—
49	C₅×#108	—	2	2	no	—
50	C₅×#108	—	2	2	no	—
51	C₅×#108	—	4	2	no	—
52	C₅×C₄⋊C₈	—	4	2	no	—
53	C₄×C₅×C₈	C₂×C₄	1	1	yes	characteristic
54	C₄×C₅×C₈	C₂×C₄	1	1	yes	characteristic
55	C₄×C₅×C₈	D₄	1	1	yes	characteristic
56	C₄×C₅×C₈	D₄	1	1	yes	characteristic
57	C₂×#495	C₂×C₅	1	1	yes	characteristic
58	C₂×#362	C₂×C₅	1	2	yes	complemented
59	C₂×#361	C₂×C₅	1	2	yes	—
60	C₄×M₃₂	C₂×C₅	1	1	yes	characteristic, complemented
61	#1848	C₂×C₅	1	1	yes	characteristic, complemented
62	C₂⁴×C₅	—	4	1	no	—
63	C₂×C₅×Q₈	—	2	2	no	—
64	C₂×C₅×Q₈	—	4	1	no	—
65	C₂×C₅×Q₈	—	4	1	no	—
66	C₂×C₅×Q₈	—	4	1	no	—
67	C₂×C₅×Q₈	—	4	1	no	—
68	C₂×C₅×Q₈	—	4	1	no	—
69	C₂×C₅×D₄	—	2	2	no	—
70	C₂×C₅×D₄	—	4	1	no	—
71	C₂×C₅×D₄	—	4	1	no	—
72	C₂×C₅×D₄	—	4	1	no	—
73	C₂×C₅×D₄	—	4	1	no	—
74	C₂×C₅×D₄	—	4	1	no	—
75	C₂²×C₄×C₅	—	4	1	no	—
76	C₂²×C₄×C₅	SD₁₆	1	2	yes	—
77	C₂²×C₄×C₅	C₂×D₄	1	1	yes	characteristic
78	C₅×Q₁₆	—	2	4	no	—
79	C₅×Q₁₆	—	4	2	no	—
80	C₅×SD₁₆	—	4	4	no	—
81	C₅×SD₁₆	—	4	4	no	—
82	C₅×D₈	—	2	4	no	—
83	C₅×D₈	—	4	2	no	—
84	C₅×M₁₆	—	4	2	no	—
85	C₅×M₁₆	—	4	2	no	—
86	C₂×C₅×C₈	—	4	2	no	—
87	C₂×C₅×C₈	C₂²⋊C₄	1	1	yes	characteristic
88	C₂×C₅×C₈	C₂²⋊C₄	1	1	yes	characteristic
89	C₂×C₅×C₈	D₈	1	2	yes	—
90	C₂×C₅×C₈	SD₁₆	1	2	yes	—
91	C₂×C₅×C₈	C₂²×C₄	1	1	yes	characteristic
92	C₂×C₅×C₈	C₂×D₄	1	2	yes	—
93	C₂×C₅×C₈	C₂×D₄	1	1	yes	characteristic
94	C₂×C₅×C₈	#41	1	2	yes	—
95	C₅×C₂²⋊C₄	—	4	4	no	—
96	C₅×C₂²⋊C₄	—	4	4	no	—
97	C₄²×C₅	C₂²⋊C₄	1	1	yes	characteristic
98	C₄²×C₅	C₂²⋊C₄	1	1	yes	characteristic
99	C₄²×C₅	D₈	1	2	yes	—
100	C₅×C₁₆	—	2	4	no	—
101	C₂²×Q₁₆	C₂²×C₅	1	1	yes	characteristic
102	C₂²×SD₁₆	—	2	1	no	—
103	C₂²×D₈	C₂²×C₅	1	1	yes	characteristic
104	C₂×#124	—	2	1	no	—
105	C₂×M₃₂	C₂²×C₅	1	2	yes	—
106	#495	—	2	1	no	—
107	#495	—	2	1	no	—
108	#495	C₄×C₅	1	2	yes	—
109	#495	C₄×C₅	1	2	yes	—
110	C₂×#108	—	2	1	no	—
111	C₂×#108	C₂²×C₅	1	2	yes	—
112	#423	—	2	1	no	—
113	C₂×C₄×C₈	C₂²×C₅	1	1	yes	characteristic
114	#362	—	2	4	no	—
115	#361	—	2	4	no	—
116	#346	—	2	1	no	—
117	C₄×C₁₆	—	2	1	no	—
118	C₂³×C₅	—	4	2	no	—
119	C₂³×C₅	—	8	4	no	—
120	C₂³×C₅	—	8	1	no	—
121	C₂³×C₅	—	8	1	no	—
122	C₂³×C₅	C₂×SD₁₆	1	1	yes	characteristic
123	C₅×Q₈	—	4	2	no	—
124	C₅×Q₈	—	4	2	no	—
125	C₅×Q₈	—	8	1	no	—
126	C₅×Q₈	—	8	1	no	—
127	C₅×D₄	—	4	2	no	—
128	C₅×D₄	—	4	2	no	—
129	C₅×D₄	—	8	1	no	—
130	C₅×D₄	—	8	1	no	—
131	C₂×C₄×C₅	—	4	2	no	—
132	C₂×C₄×C₅	—	4	2	no	—
133	C₂×C₄×C₅	—	8	1	no	—
134	C₂×C₄×C₅	—	8	1	no	—
135	C₂×C₄×C₅	#102	1	2	yes	—
136	C₂×C₄×C₅	#102	1	2	yes	—
137	C₂×C₄×C₅	C₂×C₂²⋊C₄	1	1	yes	characteristic
138	C₂×C₄×C₅	#124	1	1	yes	characteristic
139	C₂×C₄×C₅	#127	1	1	yes	characteristic
140	C₂×C₄×C₅	C₂×D₈	1	1	yes	characteristic
141	C₂×C₄×C₅	C₂×D₈	1	1	yes	characteristic
142	C₂×C₄×C₅	C₂×SD₁₆	1	1	yes	characteristic
143	C₅×C₈	—	8	2	no	—
144	C₅×C₈	#102	1	2	yes	—
145	C₅×C₈	#102	1	2	yes	—
146	C₅×C₈	C₄×D₄	1	2	yes	—
147	C₅×C₈	#121	1	2	yes	—
148	C₂²×Q₈	—	2	1	no	—
149	C₂²×D₄	—	2	1	no	—
150	C₂×Q₁₆	—	2	2	no	—
151	C₂×Q₁₆	—	2	1	no	—
152	C₂×Q₁₆	—	2	2	no	—
153	C₂×Q₁₆	C₂×C₄×C₅	1	2	yes	—
154	C₂×SD₁₆	—	2	2	no	—
155	C₂×SD₁₆	—	2	2	no	—
156	C₂×SD₁₆	—	4	2	no	—
157	C₂×SD₁₆	—	4	2	no	—
158	C₂×D₈	—	2	2	no	—
159	C₂×D₈	—	2	1	no	—
160	C₂×D₈	—	2	2	no	—
161	C₂×D₈	C₂×C₄×C₅	1	2	yes	—
162	C₂×M₁₆	—	2	2	no	—
163	C₂²×C₈	C₅×D₄	1	1	yes	characteristic
164	C₂²×C₈	C₂³×C₅	1	1	yes	characteristic
165	#124	—	2	2	no	—
166	#124	—	2	2	no	—
167	#124	—	4	2	no	—
168	C₂×C₂²⋊C₄	—	4	2	no	—
169	C₂×C₄²	C₅×D₄	1	1	yes	characteristic
170	M₃₂	—	2	2	no	—
171	M₃₂	C₅×D₄	1	4	yes	—
172	C₂×C₁₆	—	2	2	no	—
173	#108	—	2	2	no	—
174	#108	—	2	2	no	—
175	#108	—	4	2	no	—
176	C₄⋊C₈	—	4	2	no	—
177	C₄×C₈	C₂×C₄×C₅	1	1	yes	characteristic
178	C₄×C₈	C₂×C₄×C₅	1	1	yes	characteristic
179	C₄×C₈	C₅×D₄	1	1	yes	characteristic
180	C₄×C₈	C₅×D₄	1	1	yes	characteristic
181	C₂²×C₅	—	2	2	no	—
182	C₂²×C₅	—	8	4	no	—
183	C₂²×C₅	—	8	2	no	—
184	C₂²×C₅	—	16	2	no	—
185	C₂²×C₅	—	16	2	no	—
186	C₂²×C₅	C₂×#102	1	1	yes	characteristic
187	C₂²×C₅	#486	1	1	yes	characteristic
188	C₂²×C₅	#492	1	1	yes	characteristic
189	C₄×C₅	—	2	2	no	—
190	C₄×C₅	—	2	2	no	—
191	C₄×C₅	—	8	2	no	—
192	C₄×C₅	#390	1	1	yes	characteristic
193	C₄×C₅	C₂×#102	1	1	yes	characteristic
194	C₄×C₅	#486	1	1	yes	characteristic
195	C₄×C₅	#493	1	1	yes	characteristic
196	C₂⁴	—	4	1	no	—
197	C₂×Q₈	—	2	2	no	—
198	C₂×Q₈	—	4	1	no	—
199	C₂×Q₈	—	4	1	no	—
200	C₂×Q₈	—	4	1	no	—
201	C₂×Q₈	—	4	1	no	—
202	C₂×Q₈	—	4	1	no	—
203	C₂×D₄	—	2	2	no	—
204	C₂×D₄	—	4	1	no	—
205	C₂×D₄	—	4	1	no	—
206	C₂×D₄	—	4	1	no	—
207	C₂×D₄	—	4	1	no	—
208	C₂×D₄	—	4	1	no	—
209	C₂²×C₄	—	4	1	no	—
210	C₂²×C₄	C₅×SD₁₆	1	2	yes	—
211	C₂²×C₄	C₂×C₅×D₄	1	1	yes	characteristic
212	Q₁₆	—	2	4	no	—
213	Q₁₆	—	4	2	no	—
214	SD₁₆	—	4	4	no	—
215	SD₁₆	—	4	4	no	—
216	D₈	—	2	4	no	—
217	D₈	—	4	2	no	—
218	M₁₆	—	4	2	no	—
219	M₁₆	—	4	2	no	—
220	C₂×C₈	—	4	2	no	—
221	C₂×C₈	C₅×C₂²⋊C₄	1	1	yes	characteristic
222	C₂×C₈	C₅×C₂²⋊C₄	1	1	yes	characteristic
223	C₂×C₈	C₅×D₈	1	2	yes	—
224	C₂×C₈	C₅×SD₁₆	1	2	yes	—
225	C₂×C₈	C₂²×C₄×C₅	1	1	yes	characteristic
226	C₂×C₈	C₂×C₅×D₄	1	2	yes	—
227	C₂×C₈	C₂×C₅×D₄	1	1	yes	characteristic
228	C₂×C₈	C₅×#41	1	2	yes	—
229	C₂²⋊C₄	—	4	4	no	—
230	C₂²⋊C₄	—	4	4	no	—
231	C₄²	C₅×C₂²⋊C₄	1	1	yes	characteristic
232	C₄²	C₅×C₂²⋊C₄	1	1	yes	characteristic
233	C₄²	C₅×D₈	1	2	yes	—
234	C₁₆	—	2	4	no	—
235	C₂×C₅	—	2	1	no	—
236	C₂×C₅	—	2	1	no	—
237	C₂×C₅	—	16	2	no	—
238	C₂×C₅	#1957	1	1	yes	characteristic
239	C₂×C₅	#2148	1	2	yes	—
240	C₂³	—	4	2	no	—
241	C₂³	—	8	4	no	—
242	C₂³	—	8	1	no	—
243	C₂³	—	8	1	no	—
244	C₂³	C₂×C₅×SD₁₆	1	1	yes	characteristic
245	Q₈	—	4	2	no	—
246	Q₈	—	4	2	no	—
247	Q₈	—	8	1	no	—
248	Q₈	—	8	1	no	—
249	D₄	—	4	2	no	—
250	D₄	—	4	2	no	—
251	D₄	—	8	1	no	—
252	D₄	—	8	1	no	—
253	C₂×C₄	—	4	2	no	—
254	C₂×C₄	—	4	2	no	—
255	C₂×C₄	—	8	1	no	—
256	C₂×C₄	—	8	1	no	—
257	C₂×C₄	C₅×#102	1	2	yes	—
258	C₂×C₄	C₅×#102	1	2	yes	—
259	C₂×C₄	C₂×C₅×C₂²⋊C₄	1	1	yes	characteristic
260	C₂×C₄	C₅×#124	1	1	yes	characteristic
261	C₂×C₄	C₅×#127	1	1	yes	characteristic
262	C₂×C₄	C₂×C₅×D₈	1	1	yes	characteristic
263	C₂×C₄	C₂×C₅×D₈	1	1	yes	characteristic
264	C₂×C₄	C₂×C₅×SD₁₆	1	1	yes	characteristic
265	C₈	—	8	2	no	—
266	C₈	C₅×#102	1	2	yes	—
267	C₈	C₅×#102	1	2	yes	—
268	C₈	C₄×C₅×D₄	1	2	yes	—
269	C₈	C₅×#121	1	2	yes	—
270	C₅	#13214	1	1	yes	characteristic, direct product factor, Sylow
271	C₂²	—	2	2	no	—
272	C₂²	—	8	4	no	—
273	C₂²	—	8	2	no	—
274	C₂²	—	16	2	no	—
275	C₂²	—	16	2	no	—
276	C₂²	C₂×C₅×#102	1	1	yes	characteristic
277	C₂²	C₅×#486	1	1	yes	characteristic
278	C₂²	C₅×#492	1	1	yes	characteristic
279	C₄	—	2	2	no	—
280	C₄	—	2	2	no	—
281	C₄	—	8	2	no	—
282	C₄	C₅×#390	1	1	yes	characteristic
283	C₄	C₂×C₅×#102	1	1	yes	characteristic
284	C₄	C₅×#486	1	1	yes	characteristic
285	C₄	C₅×#493	1	1	yes	characteristic
286	C₂	—	2	1	no	—
287	C₂	—	2	1	no	—
288	C₂	—	16	2	no	—
289	C₂	C₅×#1957	1	1	yes	characteristic
290	C₂	C₅×#2148	1	2	yes	—

	subgroup	G/H
center, socle	C₂²×C₅	C₂×#102
commutator	C₂×C₈	C₂²×C₄×C₅
Frattini, cosocle	C₂²×C₈	C₂³×C₅
prime radical	C₂²×C₅×D₈	C₂²
LCS	G ▷ C₂×C₈ ▷ C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂²×C₅ ◁ C₂×C₄²×C₅ ◁ C₂×C₄×C₅×C₈ ◁ G

Complements

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

N	H ≅ G/N	type
2. C₂×C₅×#362	C₂	semidirect
4. C₄×C₅×M₃₂	C₂	semidirect
5. C₅×#1848	C₂	semidirect
23. #13214	C₅	direct (N × H)
58. C₂×#362	C₂×C₅	semidirect
60. C₄×M₃₂	C₂×C₅	semidirect
61. #1848	C₂×C₅	semidirect
270. C₅	#13214	direct (N × H)

Maximal subgroups

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

subgroup	index
C₂×C₅×#495	2
C₂×C₅×#362	2
C₂×C₅×#361	2
C₄×C₅×M₃₂	2
C₅×#1848	2
#13214	5

Subgroups — C5×#13214 #11935934

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₅×#13214 #11935934