Subgroups — C₄×C₅×#366 #11935392

Order 1280.

366 conjugacy classes of subgroups.

The subgroup lattice has 156 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×C₅×#366	C₂	1	2	yes	maximal
2	C₂×C₄×C₅×C₁₆	C₂	1	1	yes	maximal, characteristic
3	C₄×C₅×#107	C₂	1	2	yes	maximal
4	C₅×#1380	C₂	1	2	yes	maximal
5	C₂²×C₅×C₁₆	C₂²	1	1	yes	characteristic
6	C₂²×C₅×C₁₆	C₂²	1	1	yes	characteristic
7	C₂×C₅×#107	C₂²	1	4	yes	—
8	C₂×C₄×C₅×C₈	C₂²	1	1	yes	characteristic
9	C₄×C₅×C₄⋊C₄	—	2	2	no	—
10	C₅×#366	C₄	1	8	yes	direct product factor
11	C₄×C₅×C₁₆	C₄	1	4	yes	complemented
12	C₅×#340	—	2	2	no	—
13	C₄×#366	C₅	1	1	yes	maximal, characteristic, direct product factor, Sylow
14	C₂²×C₅×C₈	D₄	1	1	yes	characteristic
15	C₂²×C₅×C₈	C₂³	1	1	yes	characteristic
16	C₂×C₅×C₄⋊C₄	—	2	4	no	—
17	C₂×C₄²×C₅	—	4	2	no	—
18	C₂×C₄²×C₅	D₄	1	1	yes	characteristic
19	C₂×C₅×C₁₆	—	2	2	no	—
20	C₂×C₅×C₁₆	C₂×C₄	1	2	yes	—
21	C₂×C₅×C₁₆	C₂×C₄	1	2	yes	—
22	C₂×C₅×C₁₆	C₂×C₄	1	4	yes	—
23	C₅×#107	—	2	4	no	—
24	C₅×#107	C₂×C₄	1	8	yes	—
25	C₄×C₅×C₈	C₂×C₄	1	2	yes	—
26	C₄×C₅×C₈	Q₈	1	2	yes	—
27	C₅×#95	—	4	2	no	—
28	C₂×#366	C₂×C₅	1	2	yes	—
29	C₂×C₄×C₁₆	C₂×C₅	1	1	yes	characteristic
30	C₄×#107	C₂×C₅	1	2	yes	—
31	#1380	C₂×C₅	1	2	yes	—
32	C₂²×C₄×C₅	—	4	4	no	—
33	C₂²×C₄×C₅	D₈	1	1	yes	characteristic
34	C₂²×C₄×C₅	D₈	1	1	yes	characteristic
35	C₂²×C₄×C₅	C₂×D₄	1	1	yes	characteristic
36	C₂×C₅×C₈	—	2	2	no	—
37	C₂×C₅×C₈	C₄⋊C₄	1	2	yes	—
38	C₂×C₅×C₈	C₂²×C₄	1	1	yes	characteristic
39	C₂×C₅×C₈	C₂²×C₄	1	2	yes	—
40	C₂×C₅×C₈	C₂×Q₈	1	1	yes	characteristic
41	C₂×C₅×C₈	#41	1	2	yes	—
42	C₅×C₄⋊C₄	—	2	8	no	—
43	C₅×C₄⋊C₄	—	4	4	no	—
44	C₄²×C₅	—	8	4	no	—
45	C₄²×C₅	C₄⋊C₄	1	2	yes	—
46	C₄²×C₅	Q₁₆	1	2	yes	—
47	C₅×C₁₆	—	2	2	no	—
48	C₅×C₁₆	C₄²	1	4	yes	complemented
49	C₂²×C₁₆	C₂²×C₅	1	1	yes	characteristic
50	C₂²×C₁₆	C₂²×C₅	1	1	yes	characteristic
51	C₂×#107	C₂²×C₅	1	4	yes	—
52	C₂×C₄×C₈	C₂²×C₅	1	1	yes	characteristic
53	C₄×C₄⋊C₄	—	2	2	no	—
54	#366	C₄×C₅	1	8	yes	direct product factor
55	C₄×C₁₆	C₄×C₅	1	4	yes	complemented
56	#340	—	2	2	no	—
57	C₂³×C₅	C₂×D₈	1	1	yes	characteristic
58	C₂×C₄×C₅	—	2	2	no	—
59	C₂×C₄×C₅	—	4	8	no	—
60	C₂×C₄×C₅	—	8	4	no	—
61	C₂×C₄×C₅	—	8	4	no	—
62	C₂×C₄×C₅	#107	1	2	yes	—
63	C₂×C₄×C₅	#107	1	2	yes	—
64	C₂×C₄×C₅	D₁₆	1	2	yes	—
65	C₂×C₄×C₅	Q₃₂	1	2	yes	—
66	C₂×C₄×C₅	C₂×C₄⋊C₄	1	1	yes	characteristic
67	C₂×C₄×C₅	C₄×D₄	1	2	yes	—
68	C₂×C₄×C₅	C₂×Q₁₆	1	1	yes	characteristic
69	C₂×C₄×C₅	#135	1	2	yes	—
70	C₅×C₈	—	2	2	no	—
71	C₅×C₈	C₂×C₄²	1	1	yes	characteristic
72	C₅×C₈	#117	1	1	yes	characteristic
73	C₅×C₈	C₄×Q₈	1	2	yes	—
74	C₂²×C₈	C₅×D₄	1	1	yes	characteristic
75	C₂²×C₈	C₂³×C₅	1	1	yes	characteristic
76	C₂×C₄⋊C₄	—	2	4	no	—
77	C₂×C₄²	—	4	2	no	—
78	C₂×C₄²	C₅×D₄	1	1	yes	characteristic
79	C₂×C₁₆	—	2	2	no	—
80	C₂×C₁₆	C₂×C₄×C₅	1	2	yes	—
81	C₂×C₁₆	C₂×C₄×C₅	1	2	yes	—
82	C₂×C₁₆	C₂×C₄×C₅	1	4	yes	—
83	#107	—	2	4	no	—
84	#107	C₂×C₄×C₅	1	8	yes	—
85	C₄×C₈	C₂×C₄×C₅	1	2	yes	—
86	C₄×C₈	C₅×Q₈	1	2	yes	—
87	#95	—	4	2	no	—
88	C₂²×C₅	C₂×#107	1	1	yes	characteristic
89	C₂²×C₅	C₄×D₈	1	2	yes	—
90	C₂²×C₅	C₂×D₁₆	1	1	yes	characteristic
91	C₂²×C₅	C₂×Q₃₂	1	1	yes	characteristic
92	C₂²×C₅	#508	1	2	yes	—
93	C₄×C₅	—	2	2	no	—
94	C₄×C₅	—	8	8	no	—
95	C₄×C₅	#366	1	4	yes	direct product factor
96	C₄×C₅	C₄×C₄⋊C₄	1	1	yes	characteristic
97	C₄×C₅	#427	1	1	yes	characteristic
98	C₄×C₅	C₄×Q₁₆	1	2	yes	—
99	C₂²×C₄	—	4	4	no	—
100	C₂²×C₄	C₅×D₈	1	1	yes	characteristic
101	C₂²×C₄	C₅×D₈	1	1	yes	characteristic
102	C₂²×C₄	C₂×C₅×D₄	1	1	yes	characteristic
103	C₂×C₈	—	2	2	no	—
104	C₂×C₈	C₅×C₄⋊C₄	1	2	yes	—
105	C₂×C₈	C₂²×C₄×C₅	1	1	yes	characteristic
106	C₂×C₈	C₂²×C₄×C₅	1	2	yes	—
107	C₂×C₈	C₂×C₅×Q₈	1	1	yes	characteristic
108	C₂×C₈	C₅×#41	1	2	yes	—
109	C₄⋊C₄	—	2	8	no	—
110	C₄⋊C₄	—	4	4	no	—
111	C₄²	—	8	4	no	—
112	C₄²	C₅×C₄⋊C₄	1	2	yes	—
113	C₄²	C₅×Q₁₆	1	2	yes	—
114	C₁₆	—	2	2	no	—
115	C₁₆	C₄²×C₅	1	4	yes	complemented
116	C₂×C₅	C₄×#107	1	1	yes	characteristic
117	C₂×C₅	C₂×#366	1	1	yes	characteristic
118	C₂×C₅	#2158	1	1	yes	characteristic
119	C₂×C₅	C₄×D₁₆	1	2	yes	—
120	C₂×C₅	C₄×Q₃₂	1	2	yes	—
121	C₂³	C₂×C₅×D₈	1	1	yes	characteristic
122	C₂×C₄	—	2	2	no	—
123	C₂×C₄	—	4	8	no	—
124	C₂×C₄	—	8	4	no	—
125	C₂×C₄	—	8	4	no	—
126	C₂×C₄	C₅×#107	1	2	yes	—
127	C₂×C₄	C₅×#107	1	2	yes	—
128	C₂×C₄	C₅×D₁₆	1	2	yes	—
129	C₂×C₄	C₅×Q₃₂	1	2	yes	—
130	C₂×C₄	C₂×C₅×C₄⋊C₄	1	1	yes	characteristic
131	C₂×C₄	C₄×C₅×D₄	1	2	yes	—
132	C₂×C₄	C₂×C₅×Q₁₆	1	1	yes	characteristic
133	C₂×C₄	C₅×#135	1	2	yes	—
134	C₈	—	2	2	no	—
135	C₈	C₂×C₄²×C₅	1	1	yes	characteristic
136	C₈	C₅×#117	1	1	yes	characteristic
137	C₈	C₄×C₅×Q₈	1	2	yes	—
138	C₅	C₄×#366	1	1	yes	characteristic, direct product factor, Sylow
139	C₂²	C₂×C₅×#107	1	1	yes	characteristic
140	C₂²	C₄×C₅×D₈	1	2	yes	—
141	C₂²	C₂×C₅×D₁₆	1	1	yes	characteristic
142	C₂²	C₂×C₅×Q₃₂	1	1	yes	characteristic
143	C₂²	C₅×#508	1	2	yes	—
144	C₄	—	2	2	no	—
145	C₄	—	8	8	no	—
146	C₄	C₅×#366	1	4	yes	direct product factor
147	C₄	C₄×C₅×C₄⋊C₄	1	1	yes	characteristic
148	C₄	C₅×#427	1	1	yes	characteristic
149	C₄	C₄×C₅×Q₁₆	1	2	yes	—
150	C₂	C₄×C₅×#107	1	1	yes	characteristic
151	C₂	C₂×C₅×#366	1	1	yes	characteristic
152	C₂	C₅×#2158	1	1	yes	characteristic
153	C₂	C₄×C₅×D₁₆	1	2	yes	—
154	C₂	C₄×C₅×Q₃₂	1	2	yes	—

	subgroup	G/H
center	C₂²×C₄×C₅	D₈
commutator	C₈	C₂×C₄²×C₅
Frattini, cosocle	C₂²×C₈	C₂³×C₅
socle, prime radical	C₂³×C₅	C₂×D₈
LCS	G ▷ C₈ ▷ C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂²×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
10. C₅×#366	C₄	direct (N × H)
11. C₄×C₅×C₁₆	C₄	semidirect
13. C₄×#366	C₅	direct (N × H)
48. C₅×C₁₆	C₄²	semidirect
54. #366	C₄×C₅	direct (N × H)
55. C₄×C₁₆	C₄×C₅	semidirect
95. C₄×C₅	#366	direct (N × H)
115. C₁₆	C₄²×C₅	semidirect
138. C₅	C₄×#366	direct (N × H)
146. C₄	C₅×#366	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×#366	2
C₂×C₄×C₅×C₁₆	2
C₄×C₅×#107	2
C₅×#1380	2
C₄×#366	5

Subgroups — C4×C5×#366 #11935392

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₄×C₅×#366 #11935392