Subgroups — C₄×C₅×#364 #11935375

Order 1280.

398 conjugacy classes of subgroups.

The subgroup lattice has 200 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×C₅×#364	C₂	1	2	yes	maximal
2	C₄×C₅×M₃₂	C₂	1	2	yes	maximal
3	C₄²×C₅×C₈	C₂	1	1	yes	maximal, characteristic
4	C₅×#1376	C₂	1	2	yes	maximal
5	C₂×C₅×M₃₂	C₂²	1	4	yes	—
6	C₂×C₄×C₅×C₈	C₄	1	1	yes	characteristic
7	C₂×C₄×C₅×C₈	C₄	1	1	yes	characteristic
8	C₂×C₄×C₅×C₈	C₄	1	1	yes	characteristic
9	C₂×C₄×C₅×C₈	C₂²	1	1	yes	characteristic
10	C₂×C₄×C₅×C₈	C₂²	1	1	yes	characteristic
11	C₂×C₄×C₅×C₈	C₂²	1	1	yes	characteristic
12	C₄³×C₅	C₄	1	1	yes	characteristic
13	C₅×#364	C₄	1	8	yes	direct product factor
14	C₅×#346	—	2	2	no	—
15	C₄×C₅×C₁₆	—	2	2	no	—
16	C₄×#364	C₅	1	1	yes	maximal, characteristic, direct product factor, Sylow
17	C₂²×C₅×C₈	C₂×C₄	1	1	yes	characteristic
18	C₂²×C₅×C₈	C₂×C₄	1	1	yes	characteristic
19	C₂²×C₅×C₈	C₂×C₄	1	1	yes	characteristic
20	C₂²×C₅×C₈	C₂³	1	1	yes	characteristic
21	C₂×C₄²×C₅	—	2	2	no	—
22	C₂×C₄²×C₅	C₂×C₄	1	1	yes	characteristic
23	C₂×C₄²×C₅	C₂×C₄	1	1	yes	characteristic
24	C₂×C₄²×C₅	C₂×C₄	1	1	yes	characteristic
25	C₅×M₃₂	—	2	4	no	—
26	C₅×M₃₂	C₂×C₄	1	8	yes	—
27	C₂×C₅×C₁₆	—	2	4	no	—
28	C₄×C₅×C₈	—	2	2	no	—
29	C₄×C₅×C₈	—	2	2	no	—
30	C₄×C₅×C₈	C₈	1	4	yes	—
31	C₄×C₅×C₈	C₈	1	4	yes	—
32	C₄×C₅×C₈	C₂×C₄	1	4	yes	—
33	C₄×C₅×C₈	D₄	1	1	yes	characteristic
34	C₄×C₅×C₈	D₄	1	1	yes	characteristic
35	C₄×C₅×C₈	Q₈	1	1	yes	characteristic
36	C₄×C₅×C₈	Q₈	1	1	yes	characteristic
37	C₂×#364	C₂×C₅	1	2	yes	—
38	C₄×M₃₂	C₂×C₅	1	2	yes	—
39	C₄²×C₈	C₂×C₅	1	1	yes	characteristic
40	#1376	C₂×C₅	1	2	yes	—
41	C₂²×C₄×C₅	—	2	1	no	—
42	C₂²×C₄×C₅	—	2	1	no	—
43	C₂²×C₄×C₅	M₁₆	1	2	yes	—
44	C₂²×C₄×C₅	C₂²×C₄	1	1	yes	characteristic
45	C₂×C₅×C₈	—	2	1	no	—
46	C₂×C₅×C₈	—	2	1	no	—
47	C₂×C₅×C₈	C₄²	1	2	yes	—
48	C₂×C₅×C₈	C₄⋊C₄	1	1	yes	characteristic
49	C₂×C₅×C₈	C₄⋊C₄	1	1	yes	characteristic
50	C₂×C₅×C₈	C₂×C₈	1	2	yes	—
51	C₂×C₅×C₈	C₂×C₈	1	2	yes	—
52	C₂×C₅×C₈	C₂×C₈	1	2	yes	—
53	C₂×C₅×C₈	C₂×C₈	1	2	yes	—
54	C₂×C₅×C₈	M₁₆	1	2	yes	—
55	C₂×C₅×C₈	C₂²×C₄	1	2	yes	—
56	C₂×C₅×C₈	C₂×D₄	1	1	yes	characteristic
57	C₂×C₅×C₈	C₂×Q₈	1	1	yes	characteristic
58	C₂×C₅×C₈	#41	1	2	yes	—
59	C₄²×C₅	—	2	8	no	—
60	C₄²×C₅	—	2	1	no	—
61	C₄²×C₅	—	2	1	no	—
62	C₄²×C₅	C₄²	1	2	yes	—
63	C₄²×C₅	C₄⋊C₄	1	1	yes	characteristic
64	C₄²×C₅	C₄⋊C₄	1	1	yes	characteristic
65	C₄²×C₅	C₂×C₈	1	2	yes	—
66	C₄²×C₅	C₂×C₈	1	2	yes	—
67	C₅×C₁₆	—	2	8	no	—
68	C₂×M₃₂	C₂²×C₅	1	4	yes	—
69	C₂×C₄×C₈	C₄×C₅	1	1	yes	characteristic
70	C₂×C₄×C₈	C₄×C₅	1	1	yes	characteristic
71	C₂×C₄×C₈	C₄×C₅	1	1	yes	characteristic
72	C₂×C₄×C₈	C₂²×C₅	1	1	yes	characteristic
73	C₂×C₄×C₈	C₂²×C₅	1	1	yes	characteristic
74	C₂×C₄×C₈	C₂²×C₅	1	1	yes	characteristic
75	C₄³	C₄×C₅	1	1	yes	characteristic
76	#364	C₄×C₅	1	8	yes	direct product factor
77	#346	—	2	2	no	—
78	C₄×C₁₆	—	2	2	no	—
79	C₂³×C₅	C₂×M₁₆	1	1	yes	characteristic
80	C₂×C₄×C₅	—	2	4	no	—
81	C₂×C₄×C₅	—	2	4	no	—
82	C₂×C₄×C₅	—	2	2	no	—
83	C₂×C₄×C₅	—	2	2	no	—
84	C₂×C₄×C₅	—	2	4	no	—
85	C₂×C₄×C₅	C₄⋊C₈	1	2	yes	—
86	C₂×C₄×C₅	C₄⋊C₈	1	2	yes	—
87	C₂×C₄×C₅	C₂×C₄²	1	1	yes	characteristic
88	C₂×C₄×C₅	C₂×C₄⋊C₄	1	1	yes	characteristic
89	C₂×C₄×C₅	#117	1	1	yes	characteristic
90	C₂×C₄×C₅	C₂²×C₈	1	1	yes	characteristic
91	C₂×C₄×C₅	C₂²×C₈	1	1	yes	characteristic
92	C₂×C₄×C₅	C₂×M₁₆	1	1	yes	characteristic
93	C₅×C₈	—	2	2	no	—
94	C₅×C₈	C₄×C₈	1	4	yes	—
95	C₅×C₈	C₄⋊C₈	1	2	yes	—
96	C₅×C₈	C₄⋊C₈	1	2	yes	—
97	C₅×C₈	C₄×D₄	1	2	yes	—
98	C₅×C₈	C₄×Q₈	1	2	yes	—
99	C₂²×C₈	C₂×C₄×C₅	1	1	yes	characteristic
100	C₂²×C₈	C₂×C₄×C₅	1	1	yes	characteristic
101	C₂²×C₈	C₂×C₄×C₅	1	1	yes	characteristic
102	C₂²×C₈	C₂³×C₅	1	1	yes	characteristic
103	C₂×C₄²	—	2	2	no	—
104	C₂×C₄²	C₂×C₄×C₅	1	1	yes	characteristic
105	C₂×C₄²	C₂×C₄×C₅	1	1	yes	characteristic
106	C₂×C₄²	C₂×C₄×C₅	1	1	yes	characteristic
107	M₃₂	—	2	4	no	—
108	M₃₂	C₂×C₄×C₅	1	8	yes	—
109	C₂×C₁₆	—	2	4	no	—
110	C₄×C₈	—	2	2	no	—
111	C₄×C₈	—	2	2	no	—
112	C₄×C₈	C₅×C₈	1	4	yes	—
113	C₄×C₈	C₅×C₈	1	4	yes	—
114	C₄×C₈	C₂×C₄×C₅	1	4	yes	—
115	C₄×C₈	C₅×D₄	1	1	yes	characteristic
116	C₄×C₈	C₅×D₄	1	1	yes	characteristic
117	C₄×C₈	C₅×Q₈	1	1	yes	characteristic
118	C₄×C₈	C₅×Q₈	1	1	yes	characteristic
119	C₂²×C₅	—	2	2	no	—
120	C₂²×C₅	C₄×M₁₆	1	1	yes	characteristic
121	C₂²×C₅	C₂×C₄⋊C₈	1	1	yes	characteristic
122	C₂²×C₅	#431	1	1	yes	characteristic
123	C₄×C₅	—	2	4	no	—
124	C₄×C₅	—	2	4	no	—
125	C₄×C₅	—	2	2	no	—
126	C₄×C₅	#364	1	4	yes	direct product factor
127	C₄×C₅	C₄×C₄⋊C₄	1	1	yes	characteristic
128	C₄×C₅	C₂×C₄×C₈	1	1	yes	characteristic
129	C₄×C₅	C₂×C₄⋊C₈	1	1	yes	characteristic
130	C₄×C₅	#431	1	1	yes	characteristic
131	C₂²×C₄	—	2	1	no	—
132	C₂²×C₄	—	2	1	no	—
133	C₂²×C₄	C₅×M₁₆	1	2	yes	—
134	C₂²×C₄	C₂²×C₄×C₅	1	1	yes	characteristic
135	C₂×C₈	—	2	1	no	—
136	C₂×C₈	—	2	1	no	—
137	C₂×C₈	C₄²×C₅	1	2	yes	—
138	C₂×C₈	C₅×C₄⋊C₄	1	1	yes	characteristic
139	C₂×C₈	C₅×C₄⋊C₄	1	1	yes	characteristic
140	C₂×C₈	C₂×C₅×C₈	1	2	yes	—
141	C₂×C₈	C₂×C₅×C₈	1	2	yes	—
142	C₂×C₈	C₂×C₅×C₈	1	2	yes	—
143	C₂×C₈	C₂×C₅×C₈	1	2	yes	—
144	C₂×C₈	C₅×M₁₆	1	2	yes	—
145	C₂×C₈	C₂²×C₄×C₅	1	2	yes	—
146	C₂×C₈	C₂×C₅×D₄	1	1	yes	characteristic
147	C₂×C₈	C₂×C₅×Q₈	1	1	yes	characteristic
148	C₂×C₈	C₅×#41	1	2	yes	—
149	C₄²	—	2	8	no	—
150	C₄²	—	2	1	no	—
151	C₄²	—	2	1	no	—
152	C₄²	C₄²×C₅	1	2	yes	—
153	C₄²	C₅×C₄⋊C₄	1	1	yes	characteristic
154	C₄²	C₅×C₄⋊C₄	1	1	yes	characteristic
155	C₄²	C₂×C₅×C₈	1	2	yes	—
156	C₄²	C₂×C₅×C₈	1	2	yes	—
157	C₁₆	—	2	8	no	—
158	C₂×C₅	—	2	1	no	—
159	C₂×C₅	—	2	1	no	—
160	C₂×C₅	C₄×C₄⋊C₈	1	1	yes	characteristic
161	C₂×C₅	C₂×#364	1	2	yes	—
162	C₂³	C₂×C₅×M₁₆	1	1	yes	characteristic
163	C₂×C₄	—	2	4	no	—
164	C₂×C₄	—	2	4	no	—
165	C₂×C₄	—	2	2	no	—
166	C₂×C₄	—	2	2	no	—
167	C₂×C₄	—	2	4	no	—
168	C₂×C₄	C₅×C₄⋊C₈	1	2	yes	—
169	C₂×C₄	C₅×C₄⋊C₈	1	2	yes	—
170	C₂×C₄	C₂×C₄²×C₅	1	1	yes	characteristic
171	C₂×C₄	C₂×C₅×C₄⋊C₄	1	1	yes	characteristic
172	C₂×C₄	C₅×#117	1	1	yes	characteristic
173	C₂×C₄	C₂²×C₅×C₈	1	1	yes	characteristic
174	C₂×C₄	C₂²×C₅×C₈	1	1	yes	characteristic
175	C₂×C₄	C₂×C₅×M₁₆	1	1	yes	characteristic
176	C₈	—	2	2	no	—
177	C₈	C₄×C₅×C₈	1	4	yes	—
178	C₈	C₅×C₄⋊C₈	1	2	yes	—
179	C₈	C₅×C₄⋊C₈	1	2	yes	—
180	C₈	C₄×C₅×D₄	1	2	yes	—
181	C₈	C₄×C₅×Q₈	1	2	yes	—
182	C₅	C₄×#364	1	1	yes	characteristic, direct product factor, Sylow
183	C₂²	—	2	2	no	—
184	C₂²	C₄×C₅×M₁₆	1	1	yes	characteristic
185	C₂²	C₂×C₅×C₄⋊C₈	1	1	yes	characteristic
186	C₂²	C₅×#431	1	1	yes	characteristic
187	C₄	—	2	4	no	—
188	C₄	—	2	4	no	—
189	C₄	—	2	2	no	—
190	C₄	C₅×#364	1	4	yes	direct product factor
191	C₄	C₄×C₅×C₄⋊C₄	1	1	yes	characteristic
192	C₄	C₂×C₄×C₅×C₈	1	1	yes	characteristic
193	C₄	C₂×C₅×C₄⋊C₈	1	1	yes	characteristic
194	C₄	C₅×#431	1	1	yes	characteristic
195	C₂	—	2	1	no	—
196	C₂	—	2	1	no	—
197	C₂	C₄×C₅×C₄⋊C₈	1	1	yes	characteristic
198	C₂	C₂×C₅×#364	1	2	yes	—

	subgroup	G/H
center	C₄×C₅×C₈	D₄
commutator	C₄	C₂×C₄×C₅×C₈
Frattini, cosocle	C₂²×C₈	C₂³×C₅
socle	C₂²×C₅	C₂×C₄⋊C₈
prime radical	C₂³×C₅	C₂×M₁₆
LCS	G ▷ C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₄×C₅×C₈ ◁ C₂×C₄×C₅×C₈ ◁ G

Complements

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

N	H ≅ G/N	type
13. C₅×#364	C₄	direct (N × H)
16. C₄×#364	C₅	direct (N × H)
76. #364	C₄×C₅	direct (N × H)
126. C₄×C₅	#364	direct (N × H)
182. C₅	C₄×#364	direct (N × H)
190. C₄	C₅×#364	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×#364	2
C₄×C₅×M₃₂	2
C₄²×C₅×C₈	2
C₅×#1376	2
C₄×#364	5

Subgroups — C4×C5×#364 #11935375

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₄×C₅×#364 #11935375