Subgroups — C₂²×C₅×#334 #11940503

Order 1280.

942 conjugacy classes of subgroups.

The subgroup lattice has 178 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂²×C₅×C₄⋊C₈	C₂	1	2	yes	maximal
2	C₂²×C₄×C₅×C₈	C₂	1	1	yes	maximal, characteristic
3	C₂×C₅×#334	C₂	1	12	yes	maximal, direct product factor
4	C₂³×C₅×C₈	—	2	2	no	—
5	C₂³×C₅×C₈	C₄	1	1	yes	characteristic
6	C₂³×C₅×C₈	C₄	1	1	yes	characteristic
7	C₂²×C₄²×C₅	C₂²	1	1	yes	characteristic
8	C₂×C₅×C₄⋊C₈	—	2	6	no	—
9	C₂×C₅×C₄⋊C₈	C₂²	1	12	yes	—
10	C₂×C₄×C₅×C₈	C₄	1	6	yes	—
11	C₂×C₄×C₅×C₈	C₂²	1	6	yes	—
12	C₅×#334	C₂²	1	16	yes	direct product factor
13	C₂²×#334	C₅	1	1	yes	maximal, characteristic, direct product factor, Sylow
14	C₂³×C₄×C₅	C₂×C₄	1	1	yes	characteristic
15	C₂³×C₄×C₅	D₄	1	1	yes	characteristic
16	C₂³×C₄×C₅	Q₈	1	1	yes	characteristic
17	C₂²×C₅×C₈	—	2	4	no	—
18	C₂²×C₅×C₈	—	2	12	no	—
19	C₂²×C₅×C₈	—	4	8	no	—
20	C₂²×C₅×C₈	C₈	1	8	yes	complemented
21	C₂²×C₅×C₈	C₂×C₄	1	6	yes	—
22	C₂²×C₅×C₈	C₂×C₄	1	6	yes	—
23	C₂×C₄²×C₅	C₂×C₄	1	3	yes	—
24	C₂×C₄²×C₅	D₄	1	3	yes	—
25	C₂×C₄²×C₅	Q₈	1	3	yes	—
26	C₂×C₄²×C₅	C₂³	1	3	yes	—
27	C₅×C₄⋊C₈	—	2	12	no	—
28	C₅×C₄⋊C₈	C₂³	1	8	yes	—
29	C₄×C₅×C₈	C₂×C₄	1	12	yes	—
30	C₄×C₅×C₈	C₂³	1	4	yes	—
31	C₂²×C₄⋊C₈	C₂×C₅	1	2	yes	—
32	C₂²×C₄×C₈	C₂×C₅	1	1	yes	characteristic
33	C₂×#334	C₂×C₅	1	12	yes	direct product factor
34	C₂⁴×C₅	C₄⋊C₄	1	1	yes	characteristic
35	C₂²×C₄×C₅	—	2	4	no	—
36	C₂²×C₄×C₅	C₄⋊C₄	1	3	yes	—
37	C₂²×C₄×C₅	C₄⋊C₄	1	3	yes	—
38	C₂²×C₄×C₅	C₄⋊C₄	1	3	yes	—
39	C₂²×C₄×C₅	C₂×C₈	1	4	yes	—
40	C₂²×C₄×C₅	M₁₆	1	4	yes	—
41	C₂²×C₄×C₅	SD₁₆	1	8	yes	—
42	C₂²×C₄×C₅	C₂²×C₄	1	3	yes	—
43	C₂²×C₄×C₅	C₂×D₄	1	3	yes	—
44	C₂²×C₄×C₅	C₂×Q₈	1	3	yes	—
45	C₂×C₅×C₈	—	2	12	no	—
46	C₂×C₅×C₈	—	2	8	no	—
47	C₂×C₅×C₈	—	4	24	no	—
48	C₂×C₅×C₈	C₂×C₈	1	24	yes	complemented
49	C₂×C₅×C₈	C₂²×C₄	1	4	yes	—
50	C₂×C₅×C₈	C₂²×C₄	1	4	yes	—
51	C₄²×C₅	C₄⋊C₄	1	6	yes	—
52	C₄²×C₅	C₂²×C₄	1	3	yes	—
53	C₄²×C₅	C₂×D₄	1	3	yes	—
54	C₄²×C₅	C₂×Q₈	1	3	yes	—
55	C₄²×C₅	C₂⁴	1	1	yes	characteristic
56	C₂³×C₈	—	2	2	no	—
57	C₂³×C₈	C₄×C₅	1	1	yes	characteristic
58	C₂³×C₈	C₄×C₅	1	1	yes	characteristic
59	C₂²×C₄²	C₂²×C₅	1	1	yes	characteristic
60	C₂×C₄⋊C₈	—	2	6	no	—
61	C₂×C₄⋊C₈	C₂²×C₅	1	12	yes	—
62	C₂×C₄×C₈	C₄×C₅	1	6	yes	—
63	C₂×C₄×C₈	C₂²×C₅	1	6	yes	—
64	#334	C₂²×C₅	1	16	yes	direct product factor
65	C₂³×C₅	C₄⋊C₈	1	4	yes	—
66	C₂³×C₅	#106	1	4	yes	—
67	C₂³×C₅	#108	1	4	yes	—
68	C₂³×C₅	C₂×C₄⋊C₄	1	3	yes	—
69	C₂×C₄×C₅	—	2	6	no	—
70	C₂×C₄×C₅	—	2	6	no	—
71	C₂×C₄×C₅	C₄⋊C₈	1	12	yes	—
72	C₂×C₄×C₅	#106	1	12	yes	—
73	C₂×C₄×C₅	C₂×C₄⋊C₄	1	3	yes	—
74	C₂×C₄×C₅	C₂×C₄⋊C₄	1	3	yes	—
75	C₂×C₄×C₅	C₂×C₄⋊C₄	1	3	yes	—
76	C₂×C₄×C₅	C₂²×C₈	1	6	yes	—
77	C₂×C₄×C₅	C₂×M₁₆	1	6	yes	—
78	C₂×C₄×C₅	C₂×SD₁₆	1	12	yes	—
79	C₂×C₄×C₅	C₂³×C₄	1	1	yes	characteristic
80	C₂×C₄×C₅	C₂²×D₄	1	1	yes	characteristic
81	C₂×C₄×C₅	C₂²×Q₈	1	1	yes	characteristic
82	C₅×C₈	—	2	4	no	—
83	C₅×C₈	—	4	8	no	—
84	C₅×C₈	C₂²×C₈	1	8	yes	complemented
85	C₂³×C₄	C₂×C₄×C₅	1	1	yes	characteristic
86	C₂³×C₄	C₅×D₄	1	1	yes	characteristic
87	C₂³×C₄	C₅×Q₈	1	1	yes	characteristic
88	C₂²×C₈	—	2	4	no	—
89	C₂²×C₈	—	2	12	no	—
90	C₂²×C₈	—	4	8	no	—
91	C₂²×C₈	C₅×C₈	1	8	yes	complemented
92	C₂²×C₈	C₂×C₄×C₅	1	6	yes	—
93	C₂²×C₈	C₂×C₄×C₅	1	6	yes	—
94	C₂×C₄²	C₂×C₄×C₅	1	3	yes	—
95	C₂×C₄²	C₅×D₄	1	3	yes	—
96	C₂×C₄²	C₅×Q₈	1	3	yes	—
97	C₂×C₄²	C₂³×C₅	1	3	yes	—
98	C₄⋊C₈	—	2	12	no	—
99	C₄⋊C₈	C₂³×C₅	1	8	yes	—
100	C₄×C₈	C₂×C₄×C₅	1	12	yes	—
101	C₄×C₈	C₂³×C₅	1	4	yes	—
102	C₂²×C₅	#334	1	16	yes	direct product factor
103	C₂²×C₅	C₂×C₄⋊C₈	1	6	yes	—
104	C₂²×C₅	C₂×#106	1	6	yes	—
105	C₂²×C₅	C₂×#108	1	6	yes	—
106	C₂²×C₅	C₂²×C₄⋊C₄	1	1	yes	characteristic
107	C₄×C₅	—	2	3	no	—
108	C₄×C₅	—	2	1	no	—
109	C₄×C₅	C₂×C₄⋊C₈	1	6	yes	—
110	C₄×C₅	C₂×#106	1	6	yes	—
111	C₄×C₅	C₂³×C₈	1	1	yes	characteristic
112	C₄×C₅	C₂²×M₁₆	1	1	yes	characteristic
113	C₄×C₅	C₂²×SD₁₆	1	2	yes	—
114	C₂⁴	C₅×C₄⋊C₄	1	1	yes	characteristic
115	C₂²×C₄	—	2	4	no	—
116	C₂²×C₄	C₅×C₄⋊C₄	1	3	yes	—
117	C₂²×C₄	C₅×C₄⋊C₄	1	3	yes	—
118	C₂²×C₄	C₅×C₄⋊C₄	1	3	yes	—
119	C₂²×C₄	C₂×C₅×C₈	1	4	yes	—
120	C₂²×C₄	C₅×M₁₆	1	4	yes	—
121	C₂²×C₄	C₅×SD₁₆	1	8	yes	—
122	C₂²×C₄	C₂²×C₄×C₅	1	3	yes	—
123	C₂²×C₄	C₂×C₅×D₄	1	3	yes	—
124	C₂²×C₄	C₂×C₅×Q₈	1	3	yes	—
125	C₂×C₈	—	2	12	no	—
126	C₂×C₈	—	2	8	no	—
127	C₂×C₈	—	4	24	no	—
128	C₂×C₈	C₂×C₅×C₈	1	24	yes	complemented
129	C₂×C₈	C₂²×C₄×C₅	1	4	yes	—
130	C₂×C₈	C₂²×C₄×C₅	1	4	yes	—
131	C₄²	C₅×C₄⋊C₄	1	6	yes	—
132	C₄²	C₂²×C₄×C₅	1	3	yes	—
133	C₄²	C₂×C₅×D₄	1	3	yes	—
134	C₄²	C₂×C₅×Q₈	1	3	yes	—
135	C₄²	C₂⁴×C₅	1	1	yes	characteristic
136	C₂×C₅	C₂×#334	1	12	yes	direct product factor
137	C₂×C₅	C₂²×C₄⋊C₈	1	1	yes	characteristic
138	C₂×C₅	C₂²×#106	1	1	yes	characteristic
139	C₂×C₅	C₂²×#108	1	1	yes	characteristic
140	C₂³	C₅×C₄⋊C₈	1	4	yes	—
141	C₂³	C₅×#106	1	4	yes	—
142	C₂³	C₅×#108	1	4	yes	—
143	C₂³	C₂×C₅×C₄⋊C₄	1	3	yes	—
144	C₂×C₄	—	2	6	no	—
145	C₂×C₄	—	2	6	no	—
146	C₂×C₄	C₅×C₄⋊C₈	1	12	yes	—
147	C₂×C₄	C₅×#106	1	12	yes	—
148	C₂×C₄	C₂×C₅×C₄⋊C₄	1	3	yes	—
149	C₂×C₄	C₂×C₅×C₄⋊C₄	1	3	yes	—
150	C₂×C₄	C₂×C₅×C₄⋊C₄	1	3	yes	—
151	C₂×C₄	C₂²×C₅×C₈	1	6	yes	—
152	C₂×C₄	C₂×C₅×M₁₆	1	6	yes	—
153	C₂×C₄	C₂×C₅×SD₁₆	1	12	yes	—
154	C₂×C₄	C₂³×C₄×C₅	1	1	yes	characteristic
155	C₂×C₄	C₂²×C₅×D₄	1	1	yes	characteristic
156	C₂×C₄	C₂²×C₅×Q₈	1	1	yes	characteristic
157	C₈	—	2	4	no	—
158	C₈	—	4	8	no	—
159	C₈	C₂²×C₅×C₈	1	8	yes	complemented
160	C₅	C₂²×#334	1	1	yes	characteristic, direct product factor, Sylow
161	C₂²	C₅×#334	1	16	yes	direct product factor
162	C₂²	C₂×C₅×C₄⋊C₈	1	6	yes	—
163	C₂²	C₂×C₅×#106	1	6	yes	—
164	C₂²	C₂×C₅×#108	1	6	yes	—
165	C₂²	C₂²×C₅×C₄⋊C₄	1	1	yes	characteristic
166	C₄	—	2	3	no	—
167	C₄	—	2	1	no	—
168	C₄	C₂×C₅×C₄⋊C₈	1	6	yes	—
169	C₄	C₂×C₅×#106	1	6	yes	—
170	C₄	C₂³×C₅×C₈	1	1	yes	characteristic
171	C₄	C₂²×C₅×M₁₆	1	1	yes	characteristic
172	C₄	C₂²×C₅×SD₁₆	1	2	yes	—
173	C₂	C₂×C₅×#334	1	12	yes	direct product factor
174	C₂	C₂²×C₅×C₄⋊C₈	1	1	yes	characteristic
175	C₂	C₂²×C₅×#106	1	1	yes	characteristic
176	C₂	C₂²×C₅×#108	1	1	yes	characteristic

	subgroup	G/H
center	C₂³×C₄×C₅	D₄
commutator	C₄	C₂³×C₅×C₈
Frattini, cosocle	C₄²	C₂⁴×C₅
socle, prime radical	C₂⁴×C₅	C₄⋊C₄
LCS	G ▷ C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³×C₄×C₅ ◁ C₂²×C₄²×C₅ ◁ G

Complements

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

N	H ≅ G/N	type
3. C₂×C₅×#334	C₂	direct (N × H)
12. C₅×#334	C₂²	direct (N × H)
13. C₂²×#334	C₅	direct (N × H)
20. C₂²×C₅×C₈	C₈	semidirect
33. C₂×#334	C₂×C₅	direct (N × H)
48. C₂×C₅×C₈	C₂×C₈	semidirect
64. #334	C₂²×C₅	direct (N × H)
84. C₅×C₈	C₂²×C₈	semidirect
91. C₂²×C₈	C₅×C₈	semidirect
102. C₂²×C₅	#334	direct (N × H)
128. C₂×C₈	C₂×C₅×C₈	semidirect
136. C₂×C₅	C₂×#334	direct (N × H)
159. C₈	C₂²×C₅×C₈	semidirect
160. C₅	C₂²×#334	direct (N × H)
161. C₂²	C₅×#334	direct (N × H)
173. C₂	C₂×C₅×#334	direct (N × H)

Maximal subgroups

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

subgroup	index
C₂²×C₅×C₄⋊C₈	2
C₂²×C₄×C₅×C₈	2
C₂×C₅×#334	2
C₂²×#334	5

Subgroups — C22×C5×#334 #11940503

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₂²×C₅×#334 #11940503