Subgroups — C₅×#33497 #11956217

Order 1280.

486 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	C₂×C₅×C₈×Q₈	C₂	1	1	yes	maximal, characteristic
2	C₅×#2183	C₂	1	8	yes	maximal, complemented
3	C₅×#2163	C₂	1	4	yes	maximal
4	C₅×#2162	C₂	1	2	yes	maximal
5	C₂×C₄×C₅×Q₈	C₄	1	1	yes	characteristic
6	C₅×C₈×Q₈	C₄	1	4	yes	—
7	C₅×C₈×Q₈	C₂²	1	4	yes	—
8	C₂×C₅×C₄⋊C₈	C₄	1	2	yes	—
9	C₂×C₅×C₄⋊C₈	C₄	1	1	yes	characteristic
10	C₂×C₄×C₅×C₈	C₂²	1	1	yes	characteristic
11	C₂×C₄×C₅×C₈	C₂²	1	2	yes	—
12	C₅×C₄⋊C₁₆	C₂²	1	4	yes	—
13	C₅×C₄⋊C₁₆	C₂²	1	8	yes	—
14	C₅×C₂²⋊C₁₆	C₂²	1	4	yes	—
15	C₅×#346	C₂²	1	8	yes	—
16	C₄×C₅×C₁₆	C₂²	1	4	yes	—
17	#33497	C₅	1	1	yes	maximal, characteristic, direct product factor, Sylow
18	C₂²×C₅×Q₈	C₈	1	1	yes	characteristic
19	C₂²×C₅×C₈	C₂×C₄	1	1	yes	characteristic
20	C₂²×C₅×C₈	C₂×C₄	1	2	yes	—
21	C₂²×C₅×C₈	C₂³	1	1	yes	characteristic
22	C₄×C₅×Q₈	C₈	1	4	yes	—
23	C₄×C₅×Q₈	C₂×C₄	1	4	yes	—
24	C₂×C₅×C₄⋊C₄	C₈	1	2	yes	—
25	C₂×C₅×C₄⋊C₄	C₈	1	1	yes	characteristic
26	C₂×C₄²×C₅	C₂×C₄	1	2	yes	—
27	C₂×C₄²×C₅	C₂×C₄	1	1	yes	characteristic
28	C₂×C₅×C₁₆	C₂³	1	8	yes	—
29	C₅×C₄⋊C₈	C₂×C₄	1	8	yes	—
30	C₅×C₄⋊C₈	C₂×C₄	1	4	yes	—
31	C₄×C₅×C₈	C₂×C₄	1	2	yes	—
32	C₄×C₅×C₈	C₂×C₄	1	4	yes	—
33	C₄×C₅×C₈	C₂³	1	2	yes	—
34	C₄×C₅×C₈	C₂³	1	4	yes	—
35	C₂×C₈×Q₈	C₂×C₅	1	1	yes	characteristic
36	#2183	C₂×C₅	1	8	yes	complemented
37	#2163	C₂×C₅	1	4	yes	—
38	#2162	C₂×C₅	1	2	yes	—
39	C₂×C₅×Q₈	—	2	4	no	—
40	C₂×C₅×Q₈	C₂×C₈	1	4	yes	—
41	C₂²×C₄×C₅	C₂×C₈	1	2	yes	—
42	C₂²×C₄×C₅	C₂×C₈	1	2	yes	—
43	C₂²×C₄×C₅	C₂×C₈	1	1	yes	characteristic
44	C₂²×C₄×C₅	C₂×C₈	1	1	yes	characteristic
45	C₂²×C₄×C₅	C₂²×C₄	1	1	yes	characteristic
46	C₂×C₅×C₈	—	2	2	no	—
47	C₂×C₅×C₈	—	4	1	no	—
48	C₂×C₅×C₈	—	4	2	no	—
49	C₂×C₅×C₈	C₂²×C₄	1	2	yes	—
50	C₂×C₅×C₈	C₂²×C₄	1	4	yes	—
51	C₂×C₅×C₈	C₂²×C₄	1	1	yes	characteristic
52	C₂×C₅×C₈	C₂⁴	1	1	yes	characteristic
53	C₅×C₄⋊C₄	C₂×C₈	1	8	yes	—
54	C₅×C₄⋊C₄	C₂×C₈	1	4	yes	—
55	C₄²×C₅	C₂×C₈	1	4	yes	—
56	C₄²×C₅	C₂×C₈	1	2	yes	—
57	C₄²×C₅	C₂²×C₄	1	4	yes	—
58	C₄²×C₅	C₂²×C₄	1	2	yes	—
59	C₅×C₁₆	—	2	8	no	—
60	C₂×C₄×Q₈	C₄×C₅	1	1	yes	characteristic
61	C₈×Q₈	C₄×C₅	1	4	yes	—
62	C₈×Q₈	C₂²×C₅	1	4	yes	—
63	C₂×C₄⋊C₈	C₄×C₅	1	2	yes	—
64	C₂×C₄⋊C₈	C₄×C₅	1	1	yes	characteristic
65	C₂×C₄×C₈	C₂²×C₅	1	1	yes	characteristic
66	C₂×C₄×C₈	C₂²×C₅	1	2	yes	—
67	C₄⋊C₁₆	C₂²×C₅	1	4	yes	—
68	C₄⋊C₁₆	C₂²×C₅	1	8	yes	—
69	C₂²⋊C₁₆	C₂²×C₅	1	4	yes	—
70	#346	C₂²×C₅	1	8	yes	—
71	C₄×C₁₆	C₂²×C₅	1	4	yes	—
72	C₂³×C₅	C₂²×C₈	1	1	yes	characteristic
73	C₅×Q₈	—	2	8	no	—
74	C₂×C₄×C₅	—	2	4	no	—
75	C₂×C₄×C₅	—	2	1	no	—
76	C₂×C₄×C₅	—	2	1	no	—
77	C₂×C₄×C₅	—	4	2	no	—
78	C₂×C₄×C₅	—	4	1	no	—
79	C₂×C₄×C₅	M₃₂	1	4	yes	—
80	C₂×C₄×C₅	C₂²×C₈	1	1	yes	characteristic
81	C₂×C₄×C₅	C₂²×C₈	1	4	yes	—
82	C₂×C₄×C₅	C₂²×C₈	1	2	yes	—
83	C₂×C₄×C₅	C₂²×C₈	1	4	yes	—
84	C₂×C₄×C₅	C₂²×C₈	1	2	yes	—
85	C₂×C₄×C₅	C₂³×C₄	1	1	yes	characteristic
86	C₅×C₈	—	2	1	no	—
87	C₅×C₈	—	2	2	no	—
88	C₅×C₈	—	2	4	no	—
89	C₅×C₈	2₋¹⁺⁴	1	2	yes	—
90	C₂²×Q₈	C₅×C₈	1	1	yes	characteristic
91	C₂²×C₈	C₂×C₄×C₅	1	1	yes	characteristic
92	C₂²×C₈	C₂×C₄×C₅	1	2	yes	—
93	C₂²×C₈	C₂³×C₅	1	1	yes	characteristic
94	C₄×Q₈	C₅×C₈	1	4	yes	—
95	C₄×Q₈	C₂×C₄×C₅	1	4	yes	—
96	C₂×C₄⋊C₄	C₅×C₈	1	2	yes	—
97	C₂×C₄⋊C₄	C₅×C₈	1	1	yes	characteristic
98	C₂×C₄²	C₂×C₄×C₅	1	2	yes	—
99	C₂×C₄²	C₂×C₄×C₅	1	1	yes	characteristic
100	C₂×C₁₆	C₂³×C₅	1	8	yes	—
101	C₄⋊C₈	C₂×C₄×C₅	1	8	yes	—
102	C₄⋊C₈	C₂×C₄×C₅	1	4	yes	—
103	C₄×C₈	C₂×C₄×C₅	1	2	yes	—
104	C₄×C₈	C₂×C₄×C₅	1	4	yes	—
105	C₄×C₈	C₂³×C₅	1	2	yes	—
106	C₄×C₈	C₂³×C₅	1	4	yes	—
107	C₂²×C₅	—	2	1	no	—
108	C₂²×C₅	—	2	1	no	—
109	C₂²×C₅	C₂×M₃₂	1	2	yes	—
110	C₂²×C₅	C₂³×C₈	1	1	yes	characteristic
111	C₄×C₅	—	2	4	no	—
112	C₄×C₅	—	2	4	no	—
113	C₄×C₅	—	2	2	no	—
114	C₄×C₅	—	2	1	no	—
115	C₄×C₅	C₂×M₃₂	1	4	yes	—
116	C₄×C₅	#519	1	1	yes	characteristic
117	C₄×C₅	#568	1	1	yes	characteristic
118	C₂×Q₈	—	2	4	no	—
119	C₂×Q₈	C₂×C₅×C₈	1	4	yes	—
120	C₂²×C₄	C₂×C₅×C₈	1	2	yes	—
121	C₂²×C₄	C₂×C₅×C₈	1	2	yes	—
122	C₂²×C₄	C₂×C₅×C₈	1	1	yes	characteristic
123	C₂²×C₄	C₂×C₅×C₈	1	1	yes	characteristic
124	C₂²×C₄	C₂²×C₄×C₅	1	1	yes	characteristic
125	C₂×C₈	—	2	2	no	—
126	C₂×C₈	—	4	1	no	—
127	C₂×C₈	—	4	2	no	—
128	C₂×C₈	C₂²×C₄×C₅	1	1	yes	characteristic
129	C₂×C₈	C₂²×C₄×C₅	1	2	yes	—
130	C₂×C₈	C₂²×C₄×C₅	1	4	yes	—
131	C₂×C₈	C₂⁴×C₅	1	1	yes	characteristic
132	C₄⋊C₄	C₂×C₅×C₈	1	8	yes	—
133	C₄⋊C₄	C₂×C₅×C₈	1	4	yes	—
134	C₄²	C₂×C₅×C₈	1	4	yes	—
135	C₄²	C₂×C₅×C₈	1	2	yes	—
136	C₄²	C₂²×C₄×C₅	1	4	yes	—
137	C₄²	C₂²×C₄×C₅	1	2	yes	—
138	C₁₆	—	2	8	no	—
139	C₂×C₅	—	2	2	no	—
140	C₂×C₅	#2982	1	1	yes	characteristic
141	C₂×C₅	C₂²×M₃₂	1	1	yes	characteristic
142	C₂×C₅	#3407	1	1	yes	characteristic
143	C₂³	C₂²×C₅×C₈	1	1	yes	characteristic
144	Q₈	—	2	8	no	—
145	C₂×C₄	—	2	4	no	—
146	C₂×C₄	—	2	1	no	—
147	C₂×C₄	—	2	1	no	—
148	C₂×C₄	—	4	2	no	—
149	C₂×C₄	—	4	1	no	—
150	C₂×C₄	C₅×M₃₂	1	4	yes	—
151	C₂×C₄	C₂²×C₅×C₈	1	1	yes	characteristic
152	C₂×C₄	C₂²×C₅×C₈	1	4	yes	—
153	C₂×C₄	C₂²×C₅×C₈	1	2	yes	—
154	C₂×C₄	C₂²×C₅×C₈	1	4	yes	—
155	C₂×C₄	C₂²×C₅×C₈	1	2	yes	—
156	C₂×C₄	C₂³×C₄×C₅	1	1	yes	characteristic
157	C₈	—	2	1	no	—
158	C₈	—	2	2	no	—
159	C₈	—	2	4	no	—
160	C₈	C₅×2₋¹⁺⁴	1	2	yes	—
161	C₅	#33497	1	1	yes	characteristic, direct product factor, Sylow
162	C₂²	—	2	1	no	—
163	C₂²	—	2	1	no	—
164	C₂²	C₂×C₅×M₃₂	1	2	yes	—
165	C₂²	C₂³×C₅×C₈	1	1	yes	characteristic
166	C₄	—	2	1	no	—
167	C₄	—	2	4	no	—
168	C₄	—	2	4	no	—
169	C₄	—	2	2	no	—
170	C₄	C₂×C₅×M₃₂	1	4	yes	—
171	C₄	C₅×#519	1	1	yes	characteristic
172	C₄	C₅×#568	1	1	yes	characteristic
173	C₂	—	2	2	no	—
174	C₂	C₅×#2982	1	1	yes	characteristic
175	C₂	C₂²×C₅×M₃₂	1	1	yes	characteristic
176	C₂	C₅×#3407	1	1	yes	characteristic

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

Complements

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

N	H ≅ G/N	type
2. C₅×#2183	C₂	semidirect
17. #33497	C₅	direct (N × H)
36. #2183	C₂×C₅	semidirect
161. C₅	#33497	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×C₈×Q₈	2
C₅×#2183	2
C₅×#2163	2
C₅×#2162	2
#33497	5

Subgroups — C5×#33497 #11956217

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₅×#33497 #11956217