Subgroups — C₂×C₅×#3162 #11983377

Order 1280.

1902 conjugacy classes of subgroups.

The subgroup lattice has 210 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×C₅×#533	C₂	1	2	yes	maximal
2	C₅×#3162	C₂	1	16	yes	maximal, direct product factor
3	C₂×C₅×#501	C₂	1	8	yes	maximal, complemented
4	C₂×C₅×#427	C₂	1	4	yes	maximal
5	C₂²×C₅×#97	C₂	1	1	yes	maximal, characteristic
6	C₂³×C₅×C₈	C₂²	1	2	yes	—
7	C₅×#533	—	2	8	no	—
8	C₅×#533	C₂²	1	16	yes	—
9	C₂×C₅×C₄⋊Q₈	C₂²	1	4	yes	—
10	C₂×C₅×#125	C₂²	1	4	yes	—
11	C₂×C₅×C₂²⋊Q₈	—	2	4	no	—
12	C₂×C₄×C₅×Q₈	—	2	4	no	—
13	C₂×C₅×#117	C₂²	1	4	yes	—
14	C₂²×C₄²×C₅	C₂²	1	1	yes	characteristic
15	C₅×#501	C₂²	1	64	yes	complemented
16	C₅×#427	C₂²	1	32	yes	—
17	C₂×C₅×#107	C₂²	1	8	yes	—
18	C₂×C₅×#106	C₂²	1	8	yes	—
19	C₂×C₅×#97	C₂²	1	4	yes	—
20	C₂×C₅×#97	C₂²	1	8	yes	—
21	C₂×#3162	C₅	1	1	yes	maximal, characteristic, direct product factor, Sylow
22	C₂²×C₅×Q₈	—	2	4	no	—
23	C₂³×C₄×C₅	D₄	1	2	yes	—
24	C₂³×C₄×C₅	C₂³	1	1	yes	characteristic
25	C₂²×C₅×C₈	Q₈	1	16	yes	complemented
26	C₂²×C₅×C₈	C₂³	1	4	yes	—
27	C₂²×C₅×C₈	C₂³	1	8	yes	—
28	C₅×C₄⋊Q₈	—	2	8	no	—
29	C₅×C₄⋊Q₈	C₂³	1	16	yes	—
30	C₅×#125	—	2	8	no	—
31	C₅×#125	C₂³	1	16	yes	—
32	C₅×C₂²⋊Q₈	—	2	32	no	—
33	C₄×C₅×Q₈	—	2	32	no	—
34	C₅×#117	—	2	8	no	—
35	C₅×#117	C₂³	1	16	yes	—
36	C₂×C₅×C₄⋊C₄	—	2	8	no	—
37	C₂×C₅×C₄⋊C₄	—	2	4	no	—
38	C₂×C₅×C₄⋊C₄	C₂³	1	8	yes	—
39	C₂×C₅×C₂²⋊C₄	—	2	4	no	—
40	C₂×C₄²×C₅	—	2	4	no	—
41	C₂×C₄²×C₅	D₄	1	4	yes	—
42	C₂×C₄²×C₅	D₄	1	2	yes	—
43	C₂×C₄²×C₅	C₂³	1	2	yes	—
44	C₂×C₄²×C₅	C₂³	1	4	yes	—
45	C₅×#107	C₂³	1	32	yes	—
46	C₅×#106	C₂³	1	32	yes	—
47	C₅×#97	C₂³	1	16	yes	—
48	C₂×#533	C₂×C₅	1	2	yes	—
49	#3162	C₂×C₅	1	16	yes	direct product factor
50	C₂×#501	C₂×C₅	1	8	yes	complemented
51	C₂×#427	C₂×C₅	1	4	yes	—
52	C₂²×#97	C₂×C₅	1	1	yes	characteristic
53	C₂⁴×C₅	C₂×D₄	1	1	yes	characteristic
54	C₂×C₅×Q₈	—	2	16	no	—
55	C₂×C₅×Q₈	—	4	16	no	—
56	C₂²×C₄×C₅	—	2	8	no	—
57	C₂²×C₄×C₅	—	4	4	no	—
58	C₂²×C₄×C₅	C₂×D₄	1	4	yes	—
59	C₂²×C₄×C₅	C₂×D₄	1	8	yes	—
60	C₂²×C₄×C₅	C₂×D₄	1	2	yes	—
61	C₂²×C₄×C₅	C₂×D₄	1	1	yes	characteristic
62	C₂²×C₄×C₅	C₂×Q₈	1	8	yes	—
63	C₂²×C₄×C₅	C₂⁴	1	1	yes	characteristic
64	C₂²×C₄×C₅	C₂⁴	1	2	yes	—
65	C₂×C₅×C₈	C₂×Q₈	1	16	yes	—
66	C₂×C₅×C₈	C₂×Q₈	1	32	yes	complemented
67	C₂×C₅×C₈	C₂⁴	1	8	yes	—
68	C₅×C₄⋊C₄	—	2	32	no	—
69	C₅×C₄⋊C₄	—	2	16	no	—
70	C₅×C₄⋊C₄	—	2	8	no	—
71	C₅×C₄⋊C₄	C₂⁴	1	16	yes	—
72	C₅×C₂²⋊C₄	—	2	16	no	—
73	C₄²×C₅	—	2	16	no	—
74	C₄²×C₅	C₂×D₄	1	8	yes	—
75	C₄²×C₅	C₂×D₄	1	4	yes	—
76	C₄²×C₅	C₂⁴	1	4	yes	—
77	C₂³×C₈	C₂²×C₅	1	2	yes	—
78	#533	—	2	8	no	—
79	#533	C₂²×C₅	1	16	yes	—
80	C₂×C₄⋊Q₈	C₂²×C₅	1	4	yes	—
81	C₂×#125	C₂²×C₅	1	4	yes	—
82	C₂×C₂²⋊Q₈	—	2	4	no	—
83	C₂×C₄×Q₈	—	2	4	no	—
84	C₂×#117	C₂²×C₅	1	4	yes	—
85	C₂²×C₄²	C₂²×C₅	1	1	yes	characteristic
86	#501	C₂²×C₅	1	64	yes	complemented
87	#427	C₂²×C₅	1	32	yes	—
88	C₂×#107	C₂²×C₅	1	8	yes	—
89	C₂×#106	C₂²×C₅	1	8	yes	—
90	C₂×#97	C₂²×C₅	1	4	yes	—
91	C₂×#97	C₂²×C₅	1	8	yes	—
92	C₂³×C₅	—	2	4	no	—
93	C₂³×C₅	C₄⋊Q₈	1	4	yes	—
94	C₂³×C₅	C₂²×D₄	1	2	yes	—
95	C₂³×C₅	C₂²×D₄	1	1	yes	characteristic
96	C₅×Q₈	—	4	32	no	—
97	C₂×C₄×C₅	—	2	8	no	—
98	C₂×C₄×C₅	—	2	16	no	—
99	C₂×C₄×C₅	—	4	8	no	—
100	C₂×C₄×C₅	—	4	16	no	—
101	C₂×C₄×C₅	C₄⋊Q₈	1	8	yes	—
102	C₂×C₄×C₅	C₄⋊Q₈	1	4	yes	—
103	C₂×C₄×C₅	C₂²×D₄	1	8	yes	—
104	C₂×C₄×C₅	C₂²×D₄	1	2	yes	—
105	C₂×C₄×C₅	C₂²×D₄	1	1	yes	characteristic
106	C₂×C₄×C₅	C₂²×Q₈	1	4	yes	—
107	C₂×C₄×C₅	C₂²×Q₈	1	8	yes	—
108	C₂×C₄×C₅	C₂⁵	1	1	yes	characteristic
109	C₅×C₈	C₂²×Q₈	1	16	yes	—
110	C₂²×Q₈	—	2	4	no	—
111	C₂³×C₄	C₅×D₄	1	2	yes	—
112	C₂³×C₄	C₂³×C₅	1	1	yes	characteristic
113	C₂²×C₈	C₅×Q₈	1	16	yes	complemented
114	C₂²×C₈	C₂³×C₅	1	4	yes	—
115	C₂²×C₈	C₂³×C₅	1	8	yes	—
116	C₄⋊Q₈	—	2	8	no	—
117	C₄⋊Q₈	C₂³×C₅	1	16	yes	—
118	#125	—	2	8	no	—
119	#125	C₂³×C₅	1	16	yes	—
120	C₂²⋊Q₈	—	2	32	no	—
121	C₄×Q₈	—	2	32	no	—
122	#117	—	2	8	no	—
123	#117	C₂³×C₅	1	16	yes	—
124	C₂×C₄⋊C₄	—	2	8	no	—
125	C₂×C₄⋊C₄	—	2	4	no	—
126	C₂×C₄⋊C₄	C₂³×C₅	1	8	yes	—
127	C₂×C₂²⋊C₄	—	2	4	no	—
128	C₂×C₄²	—	2	4	no	—
129	C₂×C₄²	C₅×D₄	1	2	yes	—
130	C₂×C₄²	C₅×D₄	1	4	yes	—
131	C₂×C₄²	C₂³×C₅	1	2	yes	—
132	C₂×C₄²	C₂³×C₅	1	4	yes	—
133	#107	C₂³×C₅	1	32	yes	—
134	#106	C₂³×C₅	1	32	yes	—
135	#97	C₂³×C₅	1	16	yes	—
136	C₂²×C₅	—	2	8	no	—
137	C₂²×C₅	—	2	4	no	—
138	C₂²×C₅	C₂×C₄⋊Q₈	1	4	yes	—
139	C₂²×C₅	C₂×C₄⋊Q₈	1	2	yes	—
140	C₂²×C₅	#575	1	4	yes	—
141	C₂²×C₅	C₂³×D₄	1	1	yes	characteristic
142	C₄×C₅	—	2	8	no	—
143	C₄×C₅	—	4	16	no	—
144	C₄×C₅	C₂×C₄⋊Q₈	1	2	yes	—
145	C₄×C₅	C₂×C₄⋊Q₈	1	4	yes	—
146	C₄×C₅	C₂³×Q₈	1	2	yes	—
147	C₂⁴	C₂×C₅×D₄	1	1	yes	characteristic
148	C₂×Q₈	—	2	16	no	—
149	C₂×Q₈	—	4	16	no	—
150	C₂²×C₄	—	2	8	no	—
151	C₂²×C₄	—	4	4	no	—
152	C₂²×C₄	C₂×C₅×D₄	1	4	yes	—
153	C₂²×C₄	C₂×C₅×D₄	1	8	yes	—
154	C₂²×C₄	C₂×C₅×D₄	1	1	yes	characteristic
155	C₂²×C₄	C₂×C₅×D₄	1	2	yes	—
156	C₂²×C₄	C₂×C₅×Q₈	1	8	yes	—
157	C₂²×C₄	C₂⁴×C₅	1	1	yes	characteristic
158	C₂²×C₄	C₂⁴×C₅	1	2	yes	—
159	C₂×C₈	C₂×C₅×Q₈	1	16	yes	—
160	C₂×C₈	C₂×C₅×Q₈	1	32	yes	complemented
161	C₂×C₈	C₂⁴×C₅	1	8	yes	—
162	C₄⋊C₄	—	2	32	no	—
163	C₄⋊C₄	—	2	16	no	—
164	C₄⋊C₄	—	2	8	no	—
165	C₄⋊C₄	C₂⁴×C₅	1	16	yes	—
166	C₂²⋊C₄	—	2	16	no	—
167	C₄²	—	2	16	no	—
168	C₄²	C₂×C₅×D₄	1	8	yes	—
169	C₄²	C₂×C₅×D₄	1	4	yes	—
170	C₄²	C₂⁴×C₅	1	4	yes	—
171	C₂×C₅	—	2	4	no	—
172	C₂×C₅	#3162	1	4	yes	direct product factor
173	C₂×C₅	C₂²×C₄⋊Q₈	1	1	yes	characteristic
174	C₂×C₅	C₂×#575	1	2	yes	—
175	C₂³	—	2	4	no	—
176	C₂³	C₅×C₄⋊Q₈	1	4	yes	—
177	C₂³	C₂²×C₅×D₄	1	2	yes	—
178	C₂³	C₂²×C₅×D₄	1	1	yes	characteristic
179	Q₈	—	4	32	no	—
180	C₂×C₄	—	2	8	no	—
181	C₂×C₄	—	2	16	no	—
182	C₂×C₄	—	4	8	no	—
183	C₂×C₄	—	4	16	no	—
184	C₂×C₄	C₅×C₄⋊Q₈	1	8	yes	—
185	C₂×C₄	C₅×C₄⋊Q₈	1	4	yes	—
186	C₂×C₄	C₂²×C₅×D₄	1	8	yes	—
187	C₂×C₄	C₂²×C₅×D₄	1	2	yes	—
188	C₂×C₄	C₂²×C₅×D₄	1	1	yes	characteristic
189	C₂×C₄	C₂²×C₅×Q₈	1	4	yes	—
190	C₂×C₄	C₂²×C₅×Q₈	1	8	yes	—
191	C₂×C₄	C₂⁵×C₅	1	1	yes	characteristic
192	C₈	C₂²×C₅×Q₈	1	16	yes	—
193	C₅	C₂×#3162	1	1	yes	characteristic, direct product factor, Sylow
194	C₂²	—	2	8	no	—
195	C₂²	—	2	4	no	—
196	C₂²	C₂×C₅×C₄⋊Q₈	1	4	yes	—
197	C₂²	C₂×C₅×C₄⋊Q₈	1	2	yes	—
198	C₂²	C₅×#575	1	4	yes	—
199	C₂²	C₂³×C₅×D₄	1	1	yes	characteristic
200	C₄	—	2	8	no	—
201	C₄	—	4	16	no	—
202	C₄	C₂×C₅×C₄⋊Q₈	1	4	yes	—
203	C₄	C₂×C₅×C₄⋊Q₈	1	2	yes	—
204	C₄	C₂³×C₅×Q₈	1	2	yes	—
205	C₂	—	2	4	no	—
206	C₂	C₅×#3162	1	4	yes	direct product factor
207	C₂	C₂²×C₅×C₄⋊Q₈	1	1	yes	characteristic
208	C₂	C₂×C₅×#575	1	2	yes	—

	subgroup	G/H
center	C₂²×C₄×C₅	C₂×D₄
commutator, Frattini, cosocle	C₂×C₄	C₂⁵×C₅
socle	C₂³×C₅	C₂²×D₄
prime radical	C₂⁴×C₅	C₂×D₄
LCS	G ▷ C₂×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂²×C₄×C₅ ◁ C₂²×C₄²×C₅ ◁ G

Complements

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

N	H ≅ G/N	type
2. C₅×#3162	C₂	direct (N × H)
3. C₂×C₅×#501	C₂	semidirect
15. C₅×#501	C₂²	semidirect
21. C₂×#3162	C₅	direct (N × H)
25. C₂²×C₅×C₈	Q₈	semidirect
49. #3162	C₂×C₅	direct (N × H)
50. C₂×#501	C₂×C₅	semidirect
66. C₂×C₅×C₈	C₂×Q₈	semidirect
86. #501	C₂²×C₅	semidirect
113. C₂²×C₈	C₅×Q₈	semidirect
160. C₂×C₈	C₂×C₅×Q₈	semidirect
172. C₂×C₅	#3162	direct (N × H)
193. C₅	C₂×#3162	direct (N × H)
206. C₂	C₅×#3162	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×#533	2
C₅×#3162	2
C₂×C₅×#501	2
C₂×C₅×#427	2
C₂²×C₅×#97	2
C₂×#3162	5

Subgroups — C2×C5×#3162 #11983377

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₂×C₅×#3162 #11983377