Subgroups — C₅×#11657 #11934377

Order 1280.

430 conjugacy classes of subgroups.

The subgroup lattice has 262 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×C₄×C₅×D₈	C₂	1	1	yes	maximal, characteristic
2	C₅×#2162	C₂	1	1	yes	maximal, characteristic, complemented
3	C₅×#1567	C₂	1	1	yes	maximal, characteristic, complemented
4	C₅×#1331	C₂	1	4	yes	maximal, complemented
5	C₂²×C₅×D₈	C₄	1	1	yes	characteristic
6	C₂×C₄×C₅×D₄	—	2	1	no	—
7	C₄×C₅×D₈	—	2	1	no	—
8	C₄×C₅×D₈	—	2	1	no	—
9	C₄×C₅×D₈	C₄	1	2	yes	—
10	C₄×C₅×D₈	C₂²	1	2	yes	—
11	C₂×C₅×#107	C₄	1	1	yes	characteristic
12	C₅×#423	—	2	1	no	—
13	C₂×C₅×#102	—	2	1	no	—
14	C₂×C₄×C₅×C₈	C₂²	1	1	yes	characteristic
15	C₅×C₄⋊C₁₆	—	2	1	no	—
16	C₅×C₂²⋊C₁₆	—	2	1	no	—
17	C₄×C₅×C₁₆	C₂²	1	2	yes	complemented
18	C₅×#335	—	2	1	no	—
19	C₅×#335	C₂²	1	2	yes	complemented
20	#11657	C₅	1	1	yes	maximal, characteristic, direct product factor, Sylow
21	C₂²×C₅×D₄	—	2	1	no	—
22	C₂³×C₄×C₅	—	4	1	no	—
23	C₂×C₅×D₈	—	2	4	no	—
24	C₂×C₅×D₈	—	2	1	no	—
25	C₂×C₅×D₈	C₂×C₄	1	2	yes	—
26	C₂×C₅×M₁₆	—	4	1	no	—
27	C₂²×C₅×C₈	C₂×C₄	1	1	yes	characteristic
28	C₂²×C₅×C₈	D₄	1	1	yes	characteristic
29	C₄×C₅×D₄	—	2	2	no	—
30	C₄×C₅×D₄	—	2	2	no	—
31	C₄×C₅×D₄	—	4	1	no	—
32	C₄×C₅×D₄	—	4	1	no	—
33	C₂×C₅×C₄⋊C₄	—	2	1	no	—
34	C₂×C₅×C₂²⋊C₄	—	4	1	no	—
35	C₂×C₄²×C₅	D₄	1	1	yes	characteristic
36	C₂×C₅×C₁₆	—	2	2	no	—
37	C₅×#107	—	2	1	no	—
38	C₅×#107	C₂×C₄	1	2	yes	—
39	C₅×C₄⋊C₈	—	2	2	no	—
40	C₅×C₄⋊C₈	—	4	1	no	—
41	C₅×#102	—	2	2	no	—
42	C₅×#102	—	2	2	no	—
43	C₄×C₅×C₈	C₂×C₄	1	1	yes	characteristic
44	C₄×C₅×C₈	D₄	1	1	yes	characteristic
45	C₄×C₅×C₈	D₄	1	1	yes	characteristic
46	C₄×C₅×C₈	C₂³	1	1	yes	characteristic
47	C₂×C₄×D₈	C₂×C₅	1	1	yes	characteristic
48	#2162	C₂×C₅	1	1	yes	characteristic, complemented
49	#1567	C₂×C₅	1	1	yes	characteristic, complemented
50	#1331	C₂×C₅	1	4	yes	complemented
51	C₂⁴×C₅	—	4	1	no	—
52	C₂×C₅×D₄	—	2	4	no	—
53	C₂×C₅×D₄	—	2	2	no	—
54	C₂×C₅×D₄	—	4	2	no	—
55	C₂×C₅×D₄	—	4	1	no	—
56	C₂²×C₄×C₅	—	4	1	no	—
57	C₂²×C₄×C₅	—	4	2	no	—
58	C₂²×C₄×C₅	—	4	2	no	—
59	C₂²×C₄×C₅	—	8	2	no	—
60	C₂²×C₄×C₅	—	8	2	no	—
61	C₂²×C₄×C₅	C₂²⋊C₄	1	1	yes	characteristic
62	C₂²×C₄×C₅	D₈	1	1	yes	characteristic
63	C₂²×C₄×C₅	SD₁₆	1	1	yes	characteristic
64	C₅×D₈	—	2	4	no	—
65	C₅×D₈	—	2	2	no	—
66	C₅×D₈	M₁₆	1	4	yes	complemented
67	C₅×M₁₆	—	8	2	no	—
68	C₂×C₅×C₈	—	2	2	no	—
69	C₂×C₅×C₈	—	4	1	no	—
70	C₂×C₅×C₈	—	4	2	no	—
71	C₂×C₅×C₈	C₂²⋊C₄	1	1	yes	characteristic
72	C₂×C₅×C₈	C₂²⋊C₄	1	1	yes	characteristic
73	C₂×C₅×C₈	C₂²×C₄	1	1	yes	characteristic
74	C₂×C₅×C₈	C₂×D₄	1	1	yes	characteristic
75	C₅×C₄⋊C₄	—	2	2	no	—
76	C₅×C₄⋊C₄	—	4	1	no	—
77	C₅×C₂²⋊C₄	—	4	2	no	—
78	C₅×C₂²⋊C₄	—	4	2	no	—
79	C₄²×C₅	C₂²⋊C₄	1	1	yes	characteristic
80	C₄²×C₅	D₈	1	1	yes	characteristic
81	C₄²×C₅	SD₁₆	1	1	yes	characteristic
82	C₄²×C₅	C₂×D₄	1	1	yes	characteristic
83	C₅×C₁₆	—	4	2	no	—
84	C₂²×D₈	C₄×C₅	1	1	yes	characteristic
85	C₂×C₄×D₄	—	2	1	no	—
86	C₄×D₈	—	2	1	no	—
87	C₄×D₈	—	2	1	no	—
88	C₄×D₈	C₄×C₅	1	2	yes	—
89	C₄×D₈	C₂²×C₅	1	2	yes	—
90	C₂×#107	C₄×C₅	1	1	yes	characteristic
91	#423	—	2	1	no	—
92	C₂×#102	—	2	1	no	—
93	C₂×C₄×C₈	C₂²×C₅	1	1	yes	characteristic
94	C₄⋊C₁₆	—	2	1	no	—
95	C₂²⋊C₁₆	—	2	1	no	—
96	C₄×C₁₆	C₂²×C₅	1	2	yes	complemented
97	#335	—	2	1	no	—
98	#335	C₂²×C₅	1	2	yes	complemented
99	C₂³×C₅	—	4	4	no	—
100	C₂³×C₅	—	4	2	no	—
101	C₂³×C₅	—	8	2	no	—
102	C₂³×C₅	—	8	2	no	—
103	C₂³×C₅	#102	1	1	yes	characteristic
104	C₅×D₄	—	2	4	no	—
105	C₅×D₄	—	4	4	no	—
106	C₅×D₄	—	4	2	no	—
107	C₂×C₄×C₅	—	2	1	no	—
108	C₂×C₄×C₅	—	2	1	no	—
109	C₂×C₄×C₅	—	4	1	no	—
110	C₂×C₄×C₅	—	4	2	no	—
111	C₂×C₄×C₅	—	8	2	no	—
112	C₂×C₄×C₅	—	8	4	no	—
113	C₂×C₄×C₅	—	8	2	no	—
114	C₂×C₄×C₅	—	8	2	no	—
115	C₂×C₄×C₅	#102	1	1	yes	characteristic
116	C₂×C₄×C₅	#102	1	1	yes	characteristic
117	C₂×C₄×C₅	#102	1	1	yes	characteristic
118	C₂×C₄×C₅	D₁₆	1	2	yes	—
119	C₂×C₄×C₅	SD₃₂	1	2	yes	—
120	C₂×C₄×C₅	C₂×C₂²⋊C₄	1	1	yes	characteristic
121	C₂×C₄×C₅	C₂×D₈	1	1	yes	characteristic
122	C₂×C₄×C₅	C₂×SD₁₆	1	1	yes	characteristic
123	C₅×C₈	—	2	1	no	—
124	C₅×C₈	—	2	1	no	—
125	C₅×C₈	—	2	1	no	—
126	C₅×C₈	—	8	2	no	—
127	C₅×C₈	C₂×M₁₆	1	2	yes	—
128	C₂²×D₄	—	2	1	no	—
129	C₂³×C₄	—	4	1	no	—
130	C₂×D₈	—	2	4	no	—
131	C₂×D₈	—	2	1	no	—
132	C₂×D₈	C₂×C₄×C₅	1	2	yes	—
133	C₂×M₁₆	—	4	1	no	—
134	C₂²×C₈	C₂×C₄×C₅	1	1	yes	characteristic
135	C₂²×C₈	C₅×D₄	1	1	yes	characteristic
136	C₄×D₄	—	2	2	no	—
137	C₄×D₄	—	2	2	no	—
138	C₄×D₄	—	4	1	no	—
139	C₄×D₄	—	4	1	no	—
140	C₂×C₄⋊C₄	—	2	1	no	—
141	C₂×C₂²⋊C₄	—	4	1	no	—
142	C₂×C₄²	C₅×D₄	1	1	yes	characteristic
143	C₂×C₁₆	—	2	2	no	—
144	#107	—	2	1	no	—
145	#107	C₂×C₄×C₅	1	2	yes	—
146	C₄⋊C₈	—	2	2	no	—
147	C₄⋊C₈	—	4	1	no	—
148	#102	—	2	2	no	—
149	#102	—	2	2	no	—
150	C₄×C₈	C₂×C₄×C₅	1	1	yes	characteristic
151	C₄×C₈	C₅×D₄	1	1	yes	characteristic
152	C₄×C₈	C₅×D₄	1	1	yes	characteristic
153	C₄×C₈	C₂³×C₅	1	1	yes	characteristic
154	C₂²×C₅	—	2	1	no	—
155	C₂²×C₅	—	2	1	no	—
156	C₂²×C₅	—	4	4	no	—
157	C₂²×C₅	—	8	8	no	—
158	C₂²×C₅	—	8	2	no	—
159	C₂²×C₅	—	8	2	no	—
160	C₂²×C₅	#357	1	2	yes	—
161	C₂²×C₅	C₂×#102	1	1	yes	characteristic
162	C₄×C₅	—	2	1	no	—
163	C₄×C₅	—	2	1	no	—
164	C₄×C₅	—	2	1	no	—
165	C₄×C₅	—	8	2	no	—
166	C₄×C₅	#357	1	2	yes	—
167	C₄×C₅	#407	1	1	yes	characteristic
168	C₄×C₅	#421	1	1	yes	characteristic
169	C₄×C₅	C₂×D₁₆	1	1	yes	characteristic
170	C₄×C₅	C₂×SD₃₂	1	1	yes	characteristic
171	C₂⁴	—	4	1	no	—
172	C₂×D₄	—	2	4	no	—
173	C₂×D₄	—	2	2	no	—
174	C₂×D₄	—	4	2	no	—
175	C₂×D₄	—	4	1	no	—
176	C₂²×C₄	—	4	1	no	—
177	C₂²×C₄	—	4	2	no	—
178	C₂²×C₄	—	4	2	no	—
179	C₂²×C₄	—	8	2	no	—
180	C₂²×C₄	—	8	2	no	—
181	C₂²×C₄	C₅×C₂²⋊C₄	1	1	yes	characteristic
182	C₂²×C₄	C₅×D₈	1	1	yes	characteristic
183	C₂²×C₄	C₅×SD₁₆	1	1	yes	characteristic
184	D₈	—	2	4	no	—
185	D₈	—	2	2	no	—
186	D₈	C₅×M₁₆	1	4	yes	complemented
187	M₁₆	—	8	2	no	—
188	C₂×C₈	—	2	2	no	—
189	C₂×C₈	—	4	1	no	—
190	C₂×C₈	—	4	2	no	—
191	C₂×C₈	C₅×C₂²⋊C₄	1	1	yes	characteristic
192	C₂×C₈	C₅×C₂²⋊C₄	1	1	yes	characteristic
193	C₂×C₈	C₂²×C₄×C₅	1	1	yes	characteristic
194	C₂×C₈	C₂×C₅×D₄	1	1	yes	characteristic
195	C₄⋊C₄	—	2	2	no	—
196	C₄⋊C₄	—	4	1	no	—
197	C₂²⋊C₄	—	4	2	no	—
198	C₂²⋊C₄	—	4	2	no	—
199	C₄²	C₅×C₂²⋊C₄	1	1	yes	characteristic
200	C₄²	C₅×D₈	1	1	yes	characteristic
201	C₄²	C₅×SD₁₆	1	1	yes	characteristic
202	C₄²	C₂×C₅×D₄	1	1	yes	characteristic
203	C₁₆	—	4	2	no	—
204	C₂×C₅	—	2	2	no	—
205	C₂×C₅	—	8	4	no	—
206	C₂×C₅	#1486	1	1	yes	characteristic
207	C₂×C₅	C₂×#357	1	1	yes	characteristic
208	C₂×C₅	#2143	1	1	yes	characteristic
209	C₂³	—	4	4	no	—
210	C₂³	—	4	2	no	—
211	C₂³	—	8	2	no	—
212	C₂³	—	8	2	no	—
213	C₂³	C₅×#102	1	1	yes	characteristic
214	D₄	—	2	4	no	—
215	D₄	—	4	4	no	—
216	D₄	—	4	2	no	—
217	C₂×C₄	—	2	1	no	—
218	C₂×C₄	—	2	1	no	—
219	C₂×C₄	—	4	1	no	—
220	C₂×C₄	—	4	2	no	—
221	C₂×C₄	—	8	2	no	—
222	C₂×C₄	—	8	4	no	—
223	C₂×C₄	—	8	2	no	—
224	C₂×C₄	—	8	2	no	—
225	C₂×C₄	C₅×#102	1	1	yes	characteristic
226	C₂×C₄	C₅×#102	1	1	yes	characteristic
227	C₂×C₄	C₅×#102	1	1	yes	characteristic
228	C₂×C₄	C₅×D₁₆	1	2	yes	—
229	C₂×C₄	C₅×SD₃₂	1	2	yes	—
230	C₂×C₄	C₂×C₅×C₂²⋊C₄	1	1	yes	characteristic
231	C₂×C₄	C₂×C₅×D₈	1	1	yes	characteristic
232	C₂×C₄	C₂×C₅×SD₁₆	1	1	yes	characteristic
233	C₈	—	2	1	no	—
234	C₈	—	2	1	no	—
235	C₈	—	2	1	no	—
236	C₈	—	8	2	no	—
237	C₈	C₂×C₅×M₁₆	1	2	yes	—
238	C₅	#11657	1	1	yes	characteristic, direct product factor, Sylow
239	C₂²	—	2	1	no	—
240	C₂²	—	2	1	no	—
241	C₂²	—	4	4	no	—
242	C₂²	—	8	8	no	—
243	C₂²	—	8	2	no	—
244	C₂²	—	8	2	no	—
245	C₂²	C₅×#357	1	2	yes	—
246	C₂²	C₂×C₅×#102	1	1	yes	characteristic
247	C₄	—	2	1	no	—
248	C₄	—	2	1	no	—
249	C₄	—	2	1	no	—
250	C₄	—	8	2	no	—
251	C₄	C₅×#357	1	2	yes	—
252	C₄	C₅×#407	1	1	yes	characteristic
253	C₄	C₅×#421	1	1	yes	characteristic
254	C₄	C₂×C₅×D₁₆	1	1	yes	characteristic
255	C₄	C₂×C₅×SD₃₂	1	1	yes	characteristic
256	C₂	—	2	2	no	—
257	C₂	—	8	4	no	—
258	C₂	C₅×#1486	1	1	yes	characteristic
259	C₂	C₂×C₅×#357	1	1	yes	characteristic
260	C₂	C₅×#2143	1	1	yes	characteristic

	subgroup	G/H
center	C₂×C₄×C₅	C₂×D₈
commutator	C₂×C₈	C₂²×C₄×C₅
Frattini, cosocle	C₄×C₈	C₂³×C₅
socle	C₂²×C₅	C₂×#102
prime radical	C₂²×C₅×D₈	C₄
LCS	G ▷ C₂×C₈ ▷ C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂×C₄×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. C₅×#2162	C₂	semidirect
3. C₅×#1567	C₂	semidirect
4. C₅×#1331	C₂	semidirect
17. C₄×C₅×C₁₆	C₂²	semidirect
19. C₅×#335	C₂²	semidirect
20. #11657	C₅	direct (N × H)
48. #2162	C₂×C₅	semidirect
49. #1567	C₂×C₅	semidirect
50. #1331	C₂×C₅	semidirect
66. C₅×D₈	M₁₆	semidirect
96. C₄×C₁₆	C₂²×C₅	semidirect
98. #335	C₂²×C₅	semidirect
186. D₈	C₅×M₁₆	semidirect
238. C₅	#11657	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×C₄×C₅×D₈	2
C₅×#2162	2
C₅×#1567	2
C₅×#1331	2
#11657	5

Subgroups — C5×#11657 #11934377

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₅×#11657 #11934377