Subgroups — C₄×C₅×#365 #11935389

Order 1280.

330 conjugacy classes of subgroups.

The subgroup lattice has 198 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×C₅×#365	C₂	1	2	yes	maximal
2	C₄×C₅×M₃₂	C₂	1	1	yes	maximal, characteristic
3	C₄×C₅×#108	C₂	1	1	yes	maximal, characteristic
4	C₄×C₅×#106	C₂	1	1	yes	maximal, characteristic
5	C₅×#1383	C₂	1	2	yes	maximal
6	C₂×C₅×M₃₂	C₂²	1	1	yes	characteristic
7	C₂×C₅×M₃₂	C₂²	1	1	yes	characteristic
8	C₂×C₅×#108	C₂²	1	2	yes	—
9	C₂×C₅×#106	C₂²	1	2	yes	—
10	C₄×C₅×M₁₆	—	2	1	no	—
11	C₂×C₄×C₅×C₈	C₂²	1	1	yes	characteristic
12	C₄×C₅×C₄⋊C₄	—	2	1	no	—
13	C₅×#365	C₄	1	8	yes	direct product factor
14	C₅×#346	C₄	1	2	yes	complemented
15	C₄×C₅×C₁₆	C₄	1	2	yes	complemented
16	C₅×#341	—	2	1	no	—
17	C₅×#340	—	2	1	no	—
18	C₄×#365	C₅	1	1	yes	maximal, characteristic, direct product factor, Sylow
19	C₂×C₅×M₁₆	—	2	2	no	—
20	C₂²×C₅×C₈	D₄	1	1	yes	characteristic
21	C₂²×C₅×C₈	C₂³	1	1	yes	characteristic
22	C₂×C₅×C₄⋊C₄	—	2	2	no	—
23	C₂×C₄²×C₅	—	4	1	no	—
24	C₂×C₄²×C₅	D₄	1	1	yes	characteristic
25	C₅×M₃₂	—	2	2	no	—
26	C₅×M₃₂	C₂×C₄	1	4	yes	—
27	C₂×C₅×C₁₆	C₂×C₄	1	2	yes	—
28	C₂×C₅×C₁₆	C₂×C₄	1	2	yes	—
29	C₅×#108	—	2	2	no	—
30	C₅×#108	C₂×C₄	1	4	yes	—
31	C₅×#106	—	2	2	no	—
32	C₅×#106	C₂×C₄	1	4	yes	—
33	C₅×#97	—	4	1	no	—
34	C₄×C₅×C₈	—	4	1	no	—
35	C₄×C₅×C₈	C₂×C₄	1	1	yes	characteristic
36	C₄×C₅×C₈	C₂×C₄	1	1	yes	characteristic
37	C₄×C₅×C₈	Q₈	1	1	yes	characteristic
38	C₄×C₅×C₈	Q₈	1	1	yes	characteristic
39	C₅×#95	—	4	1	no	—
40	C₂×#365	C₂×C₅	1	2	yes	—
41	C₄×M₃₂	C₂×C₅	1	1	yes	characteristic
42	C₄×#108	C₂×C₅	1	1	yes	characteristic
43	C₄×#106	C₂×C₅	1	1	yes	characteristic
44	#1383	C₂×C₅	1	2	yes	—
45	C₂²×C₄×C₅	—	4	2	no	—
46	C₂²×C₄×C₅	SD₁₆	1	1	yes	characteristic
47	C₂²×C₄×C₅	SD₁₆	1	1	yes	characteristic
48	C₂²×C₄×C₅	C₂×D₄	1	1	yes	characteristic
49	C₅×M₁₆	—	2	4	no	—
50	C₅×M₁₆	—	4	2	no	—
51	C₂×C₅×C₈	—	2	2	no	—
52	C₂×C₅×C₈	—	4	2	no	—
53	C₂×C₅×C₈	C₄⋊C₄	1	1	yes	characteristic
54	C₂×C₅×C₈	C₄⋊C₄	1	1	yes	characteristic
55	C₂×C₅×C₈	C₂²×C₄	1	1	yes	characteristic
56	C₂×C₅×C₈	C₂²×C₄	1	2	yes	—
57	C₂×C₅×C₈	C₂×Q₈	1	1	yes	characteristic
58	C₂×C₅×C₈	#41	1	2	yes	—
59	C₅×C₄⋊C₄	—	2	4	no	—
60	C₅×C₄⋊C₄	—	4	2	no	—
61	C₄²×C₅	—	8	2	no	—
62	C₄²×C₅	C₄⋊C₄	1	1	yes	characteristic
63	C₄²×C₅	C₄⋊C₄	1	1	yes	characteristic
64	C₄²×C₅	SD₁₆	1	1	yes	characteristic
65	C₄²×C₅	SD₁₆	1	1	yes	characteristic
66	C₅×C₁₆	—	2	2	no	—
67	C₅×C₁₆	C₄²	1	4	yes	complemented
68	C₂×M₃₂	C₂²×C₅	1	1	yes	characteristic
69	C₂×M₃₂	C₂²×C₅	1	1	yes	characteristic
70	C₂×#108	C₂²×C₅	1	2	yes	—
71	C₂×#106	C₂²×C₅	1	2	yes	—
72	C₄×M₁₆	—	2	1	no	—
73	C₂×C₄×C₈	C₂²×C₅	1	1	yes	characteristic
74	C₄×C₄⋊C₄	—	2	1	no	—
75	#365	C₄×C₅	1	8	yes	direct product factor
76	#346	C₄×C₅	1	2	yes	complemented
77	C₄×C₁₆	C₄×C₅	1	2	yes	complemented
78	#341	—	2	1	no	—
79	#340	—	2	1	no	—
80	C₂³×C₅	C₂×SD₁₆	1	1	yes	characteristic
81	C₂×C₄×C₅	—	2	2	no	—
82	C₂×C₄×C₅	—	4	1	no	—
83	C₂×C₄×C₅	—	4	4	no	—
84	C₂×C₄×C₅	—	8	2	no	—
85	C₂×C₄×C₅	—	8	2	no	—
86	C₂×C₄×C₅	#106	1	1	yes	characteristic
87	C₂×C₄×C₅	#106	1	1	yes	characteristic
88	C₂×C₄×C₅	#106	1	1	yes	characteristic
89	C₂×C₄×C₅	#106	1	1	yes	characteristic
90	C₂×C₄×C₅	C₂×C₄⋊C₄	1	1	yes	characteristic
91	C₂×C₄×C₅	C₄×D₄	1	2	yes	—
92	C₂×C₄×C₅	C₂×SD₁₆	1	1	yes	characteristic
93	C₂×C₄×C₅	#135	1	2	yes	—
94	C₅×C₈	—	2	1	no	—
95	C₅×C₈	—	2	1	no	—
96	C₅×C₈	—	4	4	no	—
97	C₅×C₈	C₂×C₄²	1	1	yes	characteristic
98	C₅×C₈	#117	1	1	yes	characteristic
99	C₅×C₈	C₄×Q₈	1	2	yes	—
100	C₂×M₁₆	—	2	2	no	—
101	C₂²×C₈	C₅×D₄	1	1	yes	characteristic
102	C₂²×C₈	C₂³×C₅	1	1	yes	characteristic
103	C₂×C₄⋊C₄	—	2	2	no	—
104	C₂×C₄²	—	4	1	no	—
105	C₂×C₄²	C₅×D₄	1	1	yes	characteristic
106	M₃₂	—	2	2	no	—
107	M₃₂	C₂×C₄×C₅	1	4	yes	—
108	C₂×C₁₆	C₂×C₄×C₅	1	2	yes	—
109	C₂×C₁₆	C₂×C₄×C₅	1	2	yes	—
110	#108	—	2	2	no	—
111	#108	C₂×C₄×C₅	1	4	yes	—
112	#106	—	2	2	no	—
113	#106	C₂×C₄×C₅	1	4	yes	—
114	#97	—	4	1	no	—
115	C₄×C₈	—	4	1	no	—
116	C₄×C₈	C₂×C₄×C₅	1	1	yes	characteristic
117	C₄×C₈	C₂×C₄×C₅	1	1	yes	characteristic
118	C₄×C₈	C₅×Q₈	1	1	yes	characteristic
119	C₄×C₈	C₅×Q₈	1	1	yes	characteristic
120	#95	—	4	1	no	—
121	C₂²×C₅	—	2	1	no	—
122	C₂²×C₅	—	2	1	no	—
123	C₂²×C₅	C₂×#106	1	1	yes	characteristic
124	C₂²×C₅	C₄×SD₁₆	1	2	yes	—
125	C₄×C₅	—	2	1	no	—
126	C₄×C₅	—	2	1	no	—
127	C₄×C₅	—	2	1	no	—
128	C₄×C₅	—	8	4	no	—
129	C₄×C₅	#365	1	2	yes	direct product factor
130	C₄×C₅	C₄×C₄⋊C₄	1	1	yes	characteristic
131	C₄×C₅	#427	1	1	yes	characteristic
132	C₄×C₅	C₄×SD₁₆	1	2	yes	—
133	C₂²×C₄	—	4	2	no	—
134	C₂²×C₄	C₅×SD₁₆	1	1	yes	characteristic
135	C₂²×C₄	C₅×SD₁₆	1	1	yes	characteristic
136	C₂²×C₄	C₂×C₅×D₄	1	1	yes	characteristic
137	M₁₆	—	2	4	no	—
138	M₁₆	—	4	2	no	—
139	C₂×C₈	—	2	2	no	—
140	C₂×C₈	—	4	2	no	—
141	C₂×C₈	C₅×C₄⋊C₄	1	1	yes	characteristic
142	C₂×C₈	C₅×C₄⋊C₄	1	1	yes	characteristic
143	C₂×C₈	C₂²×C₄×C₅	1	1	yes	characteristic
144	C₂×C₈	C₂²×C₄×C₅	1	2	yes	—
145	C₂×C₈	C₂×C₅×Q₈	1	1	yes	characteristic
146	C₂×C₈	C₅×#41	1	2	yes	—
147	C₄⋊C₄	—	2	4	no	—
148	C₄⋊C₄	—	4	2	no	—
149	C₄²	—	8	2	no	—
150	C₄²	C₅×C₄⋊C₄	1	1	yes	characteristic
151	C₄²	C₅×C₄⋊C₄	1	1	yes	characteristic
152	C₄²	C₅×SD₁₆	1	1	yes	characteristic
153	C₄²	C₅×SD₁₆	1	1	yes	characteristic
154	C₁₆	—	2	2	no	—
155	C₁₆	C₄²×C₅	1	4	yes	complemented
156	C₂×C₅	—	2	2	no	—
157	C₂×C₅	C₄×#106	1	1	yes	characteristic
158	C₂×C₅	C₂×#365	1	1	yes	characteristic
159	C₂×C₅	#2155	1	1	yes	characteristic
160	C₂³	C₂×C₅×SD₁₆	1	1	yes	characteristic
161	C₂×C₄	—	2	2	no	—
162	C₂×C₄	—	4	1	no	—
163	C₂×C₄	—	4	4	no	—
164	C₂×C₄	—	8	2	no	—
165	C₂×C₄	—	8	2	no	—
166	C₂×C₄	C₅×#106	1	1	yes	characteristic
167	C₂×C₄	C₅×#106	1	1	yes	characteristic
168	C₂×C₄	C₅×#106	1	1	yes	characteristic
169	C₂×C₄	C₅×#106	1	1	yes	characteristic
170	C₂×C₄	C₂×C₅×C₄⋊C₄	1	1	yes	characteristic
171	C₂×C₄	C₄×C₅×D₄	1	2	yes	—
172	C₂×C₄	C₂×C₅×SD₁₆	1	1	yes	characteristic
173	C₂×C₄	C₅×#135	1	2	yes	—
174	C₈	—	2	1	no	—
175	C₈	—	2	1	no	—
176	C₈	—	4	4	no	—
177	C₈	C₂×C₄²×C₅	1	1	yes	characteristic
178	C₈	C₅×#117	1	1	yes	characteristic
179	C₈	C₄×C₅×Q₈	1	2	yes	—
180	C₅	C₄×#365	1	1	yes	characteristic, direct product factor, Sylow
181	C₂²	—	2	1	no	—
182	C₂²	—	2	1	no	—
183	C₂²	C₂×C₅×#106	1	1	yes	characteristic
184	C₂²	C₄×C₅×SD₁₆	1	2	yes	—
185	C₄	—	2	1	no	—
186	C₄	—	2	1	no	—
187	C₄	—	2	1	no	—
188	C₄	—	8	4	no	—
189	C₄	C₅×#365	1	2	yes	direct product factor
190	C₄	C₄×C₅×C₄⋊C₄	1	1	yes	characteristic
191	C₄	C₅×#427	1	1	yes	characteristic
192	C₄	C₄×C₅×SD₁₆	1	2	yes	—
193	C₂	—	2	2	no	—
194	C₂	C₄×C₅×#106	1	1	yes	characteristic
195	C₂	C₂×C₅×#365	1	1	yes	characteristic
196	C₂	C₅×#2155	1	1	yes	characteristic

	subgroup	G/H
center	C₂×C₄×C₅	#106
commutator	C₈	C₂×C₄²×C₅
Frattini, cosocle	C₂²×C₈	C₂³×C₅
socle	C₂²×C₅	C₂×#106
prime radical	C₂³×C₅	C₂×SD₁₆
LCS	G ▷ C₈ ▷ C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂×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
13. C₅×#365	C₄	direct (N × H)
14. C₅×#346	C₄	semidirect
15. C₄×C₅×C₁₆	C₄	semidirect
18. C₄×#365	C₅	direct (N × H)
67. C₅×C₁₆	C₄²	semidirect
75. #365	C₄×C₅	direct (N × H)
76. #346	C₄×C₅	semidirect
77. C₄×C₁₆	C₄×C₅	semidirect
129. C₄×C₅	#365	direct (N × H)
155. C₁₆	C₄²×C₅	semidirect
180. C₅	C₄×#365	direct (N × H)
189. C₄	C₅×#365	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×#365	2
C₄×C₅×M₃₂	2
C₄×C₅×#108	2
C₄×C₅×#106	2
C₅×#1383	2
C₄×#365	5

Subgroups — C4×C5×#365 #11935389

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₄×C₅×#365 #11935389