Subgroups — C₂×C₅×#2164 #11956137

Order 1280.

610 conjugacy classes of subgroups.

The subgroup lattice has 232 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×C₅×#405	C₂	1	1	yes	maximal, characteristic
2	C₅×#2164	C₂	1	8	yes	maximal, direct product factor
3	C₂×C₅×C₄⋊C₁₆	C₂	1	2	yes	maximal, complemented
4	C₂×C₅×C₂²⋊C₁₆	C₂	1	2	yes	maximal
5	C₂×C₅×#346	C₂	1	1	yes	maximal, characteristic, complemented
6	C₂×C₄×C₅×C₁₆	C₂	1	1	yes	maximal, characteristic, complemented
7	C₂²×C₅×M₁₆	C₄	1	1	yes	characteristic
8	C₂³×C₅×C₈	C₂²	1	1	yes	characteristic
9	C₂×C₅×#117	C₄	1	1	yes	characteristic
10	C₂²×C₅×C₁₆	C₂²	1	2	yes	—
11	C₂²×C₅×C₁₆	C₂²	1	2	yes	—
12	C₅×#405	C₄	1	4	yes	—
13	C₅×#405	C₂²	1	4	yes	—
14	C₂×C₅×#97	C₄	1	1	yes	characteristic
15	C₂×C₅×#97	C₄	1	1	yes	characteristic
16	C₂×C₄×C₅×C₈	C₂²	1	1	yes	characteristic
17	C₂×C₄×C₅×C₈	C₂²	1	1	yes	characteristic
18	C₅×C₄⋊C₁₆	C₂²	1	8	yes	complemented
19	C₅×C₂²⋊C₁₆	C₂²	1	8	yes	—
20	C₅×#346	C₂²	1	4	yes	complemented
21	C₄×C₅×C₁₆	C₂²	1	4	yes	complemented
22	C₂×#2164	C₅	1	1	yes	maximal, characteristic, direct product factor, Sylow
23	C₂³×C₄×C₅	C₂×C₄	1	1	yes	characteristic
24	C₂×C₅×M₁₆	—	2	4	no	—
25	C₂×C₅×M₁₆	C₂×C₄	1	4	yes	—
26	C₂²×C₅×C₈	—	2	4	no	—
27	C₂²×C₅×C₈	C₂×C₄	1	1	yes	characteristic
28	C₂²×C₅×C₈	C₂×C₄	1	1	yes	characteristic
29	C₂²×C₅×C₈	C₂×C₄	1	2	yes	—
30	C₂²×C₅×C₈	C₂×C₄	1	1	yes	characteristic
31	C₂²×C₅×C₈	C₂³	1	2	yes	—
32	C₂²×C₅×C₈	C₂³	1	1	yes	characteristic
33	C₅×#117	C₈	1	4	yes	—
34	C₅×#117	C₂×C₄	1	4	yes	—
35	C₂×C₅×C₄⋊C₄	C₈	1	2	yes	—
36	C₂×C₅×C₂²⋊C₄	C₈	1	2	yes	—
37	C₂×C₄²×C₅	C₂×C₄	1	1	yes	characteristic
38	C₂×C₄²×C₅	C₂×C₄	1	1	yes	characteristic
39	C₂×C₅×C₁₆	—	2	4	no	—
40	C₂×C₅×C₁₆	—	2	4	no	—
41	C₂×C₅×C₁₆	C₂³	1	4	yes	—
42	C₂×C₅×C₁₆	C₂³	1	4	yes	—
43	C₅×#97	C₂×C₄	1	4	yes	—
44	C₅×#97	C₂×C₄	1	4	yes	—
45	C₄×C₅×C₈	C₂×C₄	1	2	yes	—
46	C₄×C₅×C₈	C₂×C₄	1	2	yes	—
47	C₄×C₅×C₈	C₂³	1	2	yes	—
48	C₄×C₅×C₈	C₂³	1	2	yes	—
49	C₂×#405	C₂×C₅	1	1	yes	characteristic
50	#2164	C₂×C₅	1	8	yes	direct product factor
51	C₂×C₄⋊C₁₆	C₂×C₅	1	2	yes	complemented
52	C₂×C₂²⋊C₁₆	C₂×C₅	1	2	yes	—
53	C₂×#346	C₂×C₅	1	1	yes	characteristic, complemented
54	C₂×C₄×C₁₆	C₂×C₅	1	1	yes	characteristic, complemented
55	C₂⁴×C₅	C₂×C₈	1	1	yes	characteristic
56	C₂²×C₄×C₅	—	2	2	no	—
57	C₂²×C₄×C₅	—	2	2	no	—
58	C₂²×C₄×C₅	C₂×C₈	1	2	yes	—
59	C₂²×C₄×C₅	C₂×C₈	1	2	yes	—
60	C₂²×C₄×C₅	C₂×C₈	1	1	yes	characteristic
61	C₂²×C₄×C₅	C₂×C₈	1	2	yes	—
62	C₂²×C₄×C₅	C₂²×C₄	1	2	yes	—
63	C₂²×C₄×C₅	C₂²×C₄	1	1	yes	characteristic
64	C₅×M₁₆	—	2	8	no	—
65	C₂×C₅×C₈	—	2	2	no	—
66	C₂×C₅×C₈	—	2	2	no	—
67	C₂×C₅×C₈	—	2	4	no	—
68	C₂×C₅×C₈	—	2	4	no	—
69	C₂×C₅×C₈	M₁₆	1	4	yes	—
70	C₂×C₅×C₈	C₂²×C₄	1	2	yes	—
71	C₂×C₅×C₈	C₂²×C₄	1	2	yes	—
72	C₂×C₅×C₈	C₂²×C₄	1	1	yes	characteristic
73	C₂×C₅×C₈	C₂²×C₄	1	2	yes	—
74	C₂×C₅×C₈	#41	1	4	yes	—
75	C₂×C₅×C₈	C₂⁴	1	1	yes	characteristic
76	C₅×C₄⋊C₄	C₂×C₈	1	8	yes	—
77	C₅×C₂²⋊C₄	C₂×C₈	1	8	yes	—
78	C₄²×C₅	C₂×C₈	1	2	yes	—
79	C₄²×C₅	C₂×C₈	1	2	yes	—
80	C₄²×C₅	C₂²×C₄	1	2	yes	—
81	C₄²×C₅	C₂²×C₄	1	2	yes	—
82	C₅×C₁₆	—	2	4	no	—
83	C₅×C₁₆	—	2	4	no	—
84	C₂²×M₁₆	C₄×C₅	1	1	yes	characteristic
85	C₂³×C₈	C₂²×C₅	1	1	yes	characteristic
86	C₂×#117	C₄×C₅	1	1	yes	characteristic
87	C₂²×C₁₆	C₂²×C₅	1	2	yes	—
88	C₂²×C₁₆	C₂²×C₅	1	2	yes	—
89	#405	C₄×C₅	1	4	yes	—
90	#405	C₂²×C₅	1	4	yes	—
91	C₂×#97	C₄×C₅	1	1	yes	characteristic
92	C₂×#97	C₄×C₅	1	1	yes	characteristic
93	C₂×C₄×C₈	C₂²×C₅	1	1	yes	characteristic
94	C₂×C₄×C₈	C₂²×C₅	1	1	yes	characteristic
95	C₄⋊C₁₆	C₂²×C₅	1	8	yes	complemented
96	C₂²⋊C₁₆	C₂²×C₅	1	8	yes	—
97	#346	C₂²×C₅	1	4	yes	complemented
98	C₄×C₁₆	C₂²×C₅	1	4	yes	complemented
99	C₂³×C₅	—	2	2	no	—
100	C₂³×C₅	—	2	4	no	—
101	C₂³×C₅	C₂²×C₈	1	2	yes	—
102	C₂³×C₅	C₂²×C₈	1	1	yes	characteristic
103	C₂×C₄×C₅	—	2	2	no	—
104	C₂×C₄×C₅	—	2	4	no	—
105	C₂×C₄×C₅	—	2	4	no	—
106	C₂×C₄×C₅	—	2	2	no	—
107	C₂×C₄×C₅	—	2	2	no	—
108	C₂×C₄×C₅	—	2	4	no	—
109	C₂×C₄×C₅	#117	1	2	yes	—
110	C₂×C₄×C₅	C₂²×C₈	1	4	yes	—
111	C₂×C₄×C₅	C₂²×C₈	1	4	yes	—
112	C₂×C₄×C₅	C₂²×C₈	1	2	yes	—
113	C₂×C₄×C₅	C₂²×C₈	1	1	yes	characteristic
114	C₂×C₄×C₅	C₂×M₁₆	1	2	yes	—
115	C₂×C₄×C₅	C₂³×C₄	1	1	yes	characteristic
116	C₅×C₈	—	2	2	no	—
117	C₅×C₈	—	2	2	no	—
118	C₅×C₈	#117	1	2	yes	—
119	C₅×C₈	C₂×M₁₆	1	4	yes	—
120	C₅×C₈	C₂×#41	1	2	yes	—
121	C₂³×C₄	C₂×C₄×C₅	1	1	yes	characteristic
122	C₂×M₁₆	—	2	4	no	—
123	C₂×M₁₆	C₂×C₄×C₅	1	4	yes	—
124	C₂²×C₈	—	2	4	no	—
125	C₂²×C₈	C₂×C₄×C₅	1	1	yes	characteristic
126	C₂²×C₈	C₂×C₄×C₅	1	2	yes	—
127	C₂²×C₈	C₂×C₄×C₅	1	1	yes	characteristic
128	C₂²×C₈	C₂×C₄×C₅	1	1	yes	characteristic
129	C₂²×C₈	C₂³×C₅	1	2	yes	—
130	C₂²×C₈	C₂³×C₅	1	1	yes	characteristic
131	#117	C₅×C₈	1	4	yes	—
132	#117	C₂×C₄×C₅	1	4	yes	—
133	C₂×C₄⋊C₄	C₅×C₈	1	2	yes	—
134	C₂×C₂²⋊C₄	C₅×C₈	1	2	yes	—
135	C₂×C₄²	C₂×C₄×C₅	1	1	yes	characteristic
136	C₂×C₄²	C₂×C₄×C₅	1	1	yes	characteristic
137	C₂×C₁₆	—	2	4	no	—
138	C₂×C₁₆	—	2	4	no	—
139	C₂×C₁₆	C₂³×C₅	1	4	yes	—
140	C₂×C₁₆	C₂³×C₅	1	4	yes	—
141	#97	C₂×C₄×C₅	1	4	yes	—
142	#97	C₂×C₄×C₅	1	4	yes	—
143	C₄×C₈	C₂×C₄×C₅	1	2	yes	—
144	C₄×C₈	C₂×C₄×C₅	1	2	yes	—
145	C₄×C₈	C₂³×C₅	1	2	yes	—
146	C₄×C₈	C₂³×C₅	1	2	yes	—
147	C₂²×C₅	—	2	4	no	—
148	C₂²×C₅	—	2	2	no	—
149	C₂²×C₅	—	4	4	no	—
150	C₂²×C₅	#431	1	2	yes	—
151	C₂²×C₅	#504	1	4	yes	—
152	C₂²×C₅	C₂³×C₈	1	1	yes	characteristic
153	C₄×C₅	—	2	4	no	—
154	C₄×C₅	—	2	4	no	—
155	C₄×C₅	—	2	2	no	—
156	C₄×C₅	#431	1	2	yes	—
157	C₄×C₅	C₂×#117	1	1	yes	characteristic
158	C₄×C₅	C₂²×M₁₆	1	1	yes	characteristic
159	C₂⁴	C₂×C₅×C₈	1	1	yes	characteristic
160	C₂²×C₄	—	2	2	no	—
161	C₂²×C₄	—	2	2	no	—
162	C₂²×C₄	C₂×C₅×C₈	1	2	yes	—
163	C₂²×C₄	C₂×C₅×C₈	1	2	yes	—
164	C₂²×C₄	C₂×C₅×C₈	1	1	yes	characteristic
165	C₂²×C₄	C₂×C₅×C₈	1	2	yes	—
166	C₂²×C₄	C₂²×C₄×C₅	1	2	yes	—
167	C₂²×C₄	C₂²×C₄×C₅	1	1	yes	characteristic
168	M₁₆	—	2	8	no	—
169	C₂×C₈	—	2	2	no	—
170	C₂×C₈	—	2	2	no	—
171	C₂×C₈	—	2	4	no	—
172	C₂×C₈	—	2	4	no	—
173	C₂×C₈	C₅×M₁₆	1	4	yes	—
174	C₂×C₈	C₂²×C₄×C₅	1	2	yes	—
175	C₂×C₈	C₂²×C₄×C₅	1	2	yes	—
176	C₂×C₈	C₂²×C₄×C₅	1	1	yes	characteristic
177	C₂×C₈	C₂²×C₄×C₅	1	2	yes	—
178	C₂×C₈	C₅×#41	1	4	yes	—
179	C₂×C₈	C₂⁴×C₅	1	1	yes	characteristic
180	C₄⋊C₄	C₂×C₅×C₈	1	8	yes	—
181	C₂²⋊C₄	C₂×C₅×C₈	1	8	yes	—
182	C₄²	C₂×C₅×C₈	1	2	yes	—
183	C₄²	C₂×C₅×C₈	1	2	yes	—
184	C₄²	C₂²×C₄×C₅	1	2	yes	—
185	C₄²	C₂²×C₄×C₅	1	2	yes	—
186	C₁₆	—	2	4	no	—
187	C₁₆	—	2	4	no	—
188	C₂×C₅	—	4	2	no	—
189	C₂×C₅	#2164	1	4	yes	direct product factor
190	C₂×C₅	C₂×#431	1	1	yes	characteristic
191	C₂×C₅	C₂×#504	1	2	yes	—
192	C₂³	—	2	2	no	—
193	C₂³	—	2	4	no	—
194	C₂³	C₂²×C₅×C₈	1	2	yes	—
195	C₂³	C₂²×C₅×C₈	1	1	yes	characteristic
196	C₂×C₄	—	2	2	no	—
197	C₂×C₄	—	2	4	no	—
198	C₂×C₄	—	2	4	no	—
199	C₂×C₄	—	2	4	no	—
200	C₂×C₄	—	2	2	no	—
201	C₂×C₄	—	2	2	no	—
202	C₂×C₄	C₅×#117	1	2	yes	—
203	C₂×C₄	C₂²×C₅×C₈	1	4	yes	—
204	C₂×C₄	C₂²×C₅×C₈	1	4	yes	—
205	C₂×C₄	C₂²×C₅×C₈	1	2	yes	—
206	C₂×C₄	C₂²×C₅×C₈	1	1	yes	characteristic
207	C₂×C₄	C₂×C₅×M₁₆	1	2	yes	—
208	C₂×C₄	C₂³×C₄×C₅	1	1	yes	characteristic
209	C₈	—	2	2	no	—
210	C₈	—	2	2	no	—
211	C₈	C₅×#117	1	2	yes	—
212	C₈	C₂×C₅×M₁₆	1	4	yes	—
213	C₈	C₂×C₅×#41	1	2	yes	—
214	C₅	C₂×#2164	1	1	yes	characteristic, direct product factor, Sylow
215	C₂²	—	2	4	no	—
216	C₂²	—	2	2	no	—
217	C₂²	—	4	4	no	—
218	C₂²	C₅×#431	1	2	yes	—
219	C₂²	C₅×#504	1	4	yes	—
220	C₂²	C₂³×C₅×C₈	1	1	yes	characteristic
221	C₄	—	2	4	no	—
222	C₄	—	2	4	no	—
223	C₄	—	2	2	no	—
224	C₄	C₅×#431	1	2	yes	—
225	C₄	C₂×C₅×#117	1	1	yes	characteristic
226	C₄	C₂²×C₅×M₁₆	1	1	yes	characteristic
227	C₂	—	4	2	no	—
228	C₂	C₅×#2164	1	4	yes	direct product factor
229	C₂	C₂×C₅×#431	1	1	yes	characteristic
230	C₂	C₂×C₅×#504	1	2	yes	—

	subgroup	G/H
center	C₂²×C₅×C₈	C₂³
commutator	C₂²	C₂³×C₅×C₈
Frattini, cosocle	C₂×C₈	C₂⁴×C₅
socle	C₂³×C₅	C₂²×C₈
prime radical	C₂⁴×C₅	C₂×C₈
LCS	G ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂²×C₅×C₈ ◁ G

Complements

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

N	H ≅ G/N	type
2. C₅×#2164	C₂	direct (N × H)
3. C₂×C₅×C₄⋊C₁₆	C₂	semidirect
5. C₂×C₅×#346	C₂	semidirect
6. C₂×C₄×C₅×C₁₆	C₂	semidirect
18. C₅×C₄⋊C₁₆	C₂²	semidirect
20. C₅×#346	C₂²	semidirect
21. C₄×C₅×C₁₆	C₂²	semidirect
22. C₂×#2164	C₅	direct (N × H)
50. #2164	C₂×C₅	direct (N × H)
51. C₂×C₄⋊C₁₆	C₂×C₅	semidirect
53. C₂×#346	C₂×C₅	semidirect
54. C₂×C₄×C₁₆	C₂×C₅	semidirect
95. C₄⋊C₁₆	C₂²×C₅	semidirect
97. #346	C₂²×C₅	semidirect
98. C₄×C₁₆	C₂²×C₅	semidirect
189. C₂×C₅	#2164	direct (N × H)
214. C₅	C₂×#2164	direct (N × H)
228. C₂	C₅×#2164	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×#405	2
C₅×#2164	2
C₂×C₅×C₄⋊C₁₆	2
C₂×C₅×C₂²⋊C₁₆	2
C₂×C₅×#346	2
C₂×C₄×C₅×C₁₆	2
C₂×#2164	5

Subgroups — C2×C5×#2164 #11956137

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₂×C₅×#2164 #11956137