Subgroups — C₄×C₅×#430 #11943178

Order 1280.

1034 conjugacy classes of subgroups.

The subgroup lattice has 240 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×C₅×#430	C₂	1	2	yes	maximal
2	C₂×C₄×C₅×M₁₆	C₂	1	2	yes	maximal
3	C₂×C₄×C₅×M₁₆	C₂	1	1	yes	maximal, characteristic
4	C₅×#1778	C₂	1	4	yes	maximal, complemented
5	C₄×C₅×#108	C₂	1	4	yes	maximal, complemented
6	C₅×#1738	C₂	1	2	yes	maximal
7	C₂²×C₅×M₁₆	C₂²	1	4	yes	—
8	C₂²×C₅×M₁₆	C₂²	1	1	yes	characteristic
9	C₂²×C₅×M₁₆	C₂²	1	1	yes	characteristic
10	C₂²×C₄²×C₅	C₂²	1	1	yes	characteristic
11	C₅×#430	C₄	1	16	yes	direct product factor
12	C₂×C₅×#108	C₂²	1	8	yes	—
13	C₄×C₅×M₁₆	—	2	2	no	—
14	C₄×C₅×M₁₆	—	2	2	no	—
15	C₄×C₅×M₁₆	C₄	1	8	yes	—
16	C₄×C₅×M₁₆	C₂²	1	4	yes	—
17	C₄×C₅×M₁₆	C₂²	1	4	yes	—
18	C₂×C₅×#97	—	2	2	no	—
19	C₂×C₅×#97	C₂²	1	2	yes	—
20	C₂×C₄×C₅×C₈	—	2	2	no	—
21	C₂×C₄×C₅×C₈	C₂²	1	2	yes	—
22	C₅×#341	C₂²	1	8	yes	—
23	C₄×#430	C₅	1	1	yes	maximal, characteristic, direct product factor, Sylow
24	C₂³×C₄×C₅	Q₈	1	2	yes	—
25	C₂³×C₄×C₅	C₂³	1	1	yes	characteristic
26	C₂×C₅×M₁₆	—	2	4	no	—
27	C₂×C₅×M₁₆	—	2	8	no	—
28	C₂×C₅×M₁₆	—	2	4	no	—
29	C₂×C₅×M₁₆	C₂×C₄	1	16	yes	—
30	C₂×C₅×M₁₆	C₂×C₄	1	4	yes	—
31	C₂×C₅×M₁₆	C₂×C₄	1	4	yes	—
32	C₂×C₅×M₁₆	C₂×C₄	1	8	yes	—
33	C₂×C₅×M₁₆	C₂³	1	8	yes	—
34	C₂²×C₅×C₈	—	2	4	no	—
35	C₂²×C₅×C₈	C₂³	1	2	yes	—
36	C₂²×C₅×C₈	C₂³	1	2	yes	—
37	C₂×C₄²×C₅	C₂×C₄	1	4	yes	—
38	C₂×C₄²×C₅	D₄	1	2	yes	—
39	C₂×C₄²×C₅	D₄	1	2	yes	—
40	C₂×C₄²×C₅	Q₈	1	2	yes	—
41	C₂×C₄²×C₅	C₂³	1	1	yes	characteristic
42	C₂×C₄²×C₅	C₂³	1	1	yes	characteristic
43	C₅×#108	C₂×C₄	1	32	yes	complemented
44	C₅×#97	—	2	4	no	—
45	C₅×#97	—	2	4	no	—
46	C₅×#97	C₂×C₄	1	8	yes	—
47	C₄×C₅×C₈	—	2	4	no	—
48	C₄×C₅×C₈	—	2	4	no	—
49	C₄×C₅×C₈	C₂×C₄	1	8	yes	—
50	C₂×#430	C₂×C₅	1	2	yes	—
51	C₂×C₄×M₁₆	C₂×C₅	1	2	yes	—
52	C₂×C₄×M₁₆	C₂×C₅	1	1	yes	characteristic
53	#1778	C₂×C₅	1	4	yes	complemented
54	C₄×#108	C₂×C₅	1	4	yes	complemented
55	#1738	C₂×C₅	1	2	yes	—
56	C₂⁴×C₅	C₂×Q₈	1	1	yes	characteristic
57	C₂²×C₄×C₅	—	4	4	no	—
58	C₂²×C₄×C₅	C₄⋊C₄	1	4	yes	—
59	C₂²×C₄×C₅	C₄⋊C₄	1	4	yes	—
60	C₂²×C₄×C₅	C₂²×C₄	1	2	yes	—
61	C₂²×C₄×C₅	C₂²×C₄	1	4	yes	—
62	C₂²×C₄×C₅	C₂×D₄	1	2	yes	—
63	C₂²×C₄×C₅	C₂×Q₈	1	2	yes	—
64	C₂²×C₄×C₅	C₂×Q₈	1	2	yes	—
65	C₂²×C₄×C₅	C₂×Q₈	1	1	yes	characteristic
66	C₂²×C₄×C₅	#41	1	4	yes	—
67	C₂²×C₄×C₅	C₂⁴	1	1	yes	characteristic
68	C₅×M₁₆	—	2	8	no	—
69	C₅×M₁₆	—	2	16	no	—
70	C₅×M₁₆	—	2	8	no	—
71	C₅×M₁₆	C₄²	1	16	yes	—
72	C₅×M₁₆	C₂²×C₄	1	16	yes	—
73	C₂×C₅×C₈	—	2	16	no	—
74	C₂×C₅×C₈	—	2	8	no	—
75	C₂×C₅×C₈	—	2	4	no	—
76	C₂×C₅×C₈	C₂²×C₄	1	4	yes	—
77	C₂×C₅×C₈	C₂²×C₄	1	4	yes	—
78	C₂×C₅×C₈	C₂²×C₄	1	8	yes	—
79	C₄²×C₅	C₄⋊C₄	1	4	yes	—
80	C₄²×C₅	C₄⋊C₄	1	4	yes	—
81	C₄²×C₅	C₂²×C₄	1	2	yes	—
82	C₄²×C₅	C₂²×C₄	1	2	yes	—
83	C₄²×C₅	C₂×D₄	1	2	yes	—
84	C₄²×C₅	C₂×D₄	1	1	yes	characteristic
85	C₄²×C₅	C₂×D₄	1	1	yes	characteristic
86	C₂²×M₁₆	C₂²×C₅	1	4	yes	—
87	C₂²×M₁₆	C₂²×C₅	1	1	yes	characteristic
88	C₂²×M₁₆	C₂²×C₅	1	1	yes	characteristic
89	C₂²×C₄²	C₂²×C₅	1	1	yes	characteristic
90	#430	C₄×C₅	1	16	yes	direct product factor
91	C₂×#108	C₂²×C₅	1	8	yes	—
92	C₄×M₁₆	—	2	2	no	—
93	C₄×M₁₆	—	2	2	no	—
94	C₄×M₁₆	C₄×C₅	1	8	yes	—
95	C₄×M₁₆	C₂²×C₅	1	4	yes	—
96	C₄×M₁₆	C₂²×C₅	1	4	yes	—
97	C₂×#97	—	2	2	no	—
98	C₂×#97	C₂²×C₅	1	2	yes	—
99	C₂×C₄×C₈	—	2	2	no	—
100	C₂×C₄×C₈	C₂²×C₅	1	2	yes	—
101	#341	C₂²×C₅	1	8	yes	—
102	C₂³×C₅	—	4	2	no	—
103	C₂³×C₅	C₂×C₄⋊C₄	1	2	yes	—
104	C₂³×C₅	C₄×Q₈	1	4	yes	—
105	C₂³×C₅	C₂²×Q₈	1	1	yes	characteristic
106	C₂×C₄×C₅	—	2	8	no	—
107	C₂×C₄×C₅	—	2	4	no	—
108	C₂×C₄×C₅	—	4	4	no	—
109	C₂×C₄×C₅	—	4	2	no	—
110	C₂×C₄×C₅	C₂×C₄²	1	4	yes	—
111	C₂×C₄×C₅	C₂×C₄⋊C₄	1	2	yes	—
112	C₂×C₄×C₅	C₂×C₄⋊C₄	1	4	yes	—
113	C₂×C₄×C₅	C₂×C₄⋊C₄	1	2	yes	—
114	C₂×C₄×C₅	C₂×C₄⋊C₄	1	2	yes	—
115	C₂×C₄×C₅	#117	1	4	yes	—
116	C₂×C₄×C₅	C₄×D₄	1	8	yes	—
117	C₂×C₄×C₅	C₄×Q₈	1	4	yes	—
118	C₂×C₄×C₅	C₂³×C₄	1	1	yes	characteristic
119	C₂×C₄×C₅	C₂³×C₄	1	2	yes	—
120	C₂×C₄×C₅	C₂²×D₄	1	1	yes	characteristic
121	C₂×C₄×C₅	C₂×#41	1	2	yes	—
122	C₅×C₈	—	2	4	no	—
123	C₅×C₈	—	2	16	no	—
124	C₅×C₈	C₂×C₄²	1	8	yes	—
125	C₂³×C₄	C₅×Q₈	1	2	yes	—
126	C₂³×C₄	C₂³×C₅	1	1	yes	characteristic
127	C₂×M₁₆	—	2	4	no	—
128	C₂×M₁₆	—	2	8	no	—
129	C₂×M₁₆	—	2	4	no	—
130	C₂×M₁₆	C₂×C₄×C₅	1	16	yes	—
131	C₂×M₁₆	C₂×C₄×C₅	1	4	yes	—
132	C₂×M₁₆	C₂×C₄×C₅	1	4	yes	—
133	C₂×M₁₆	C₂×C₄×C₅	1	8	yes	—
134	C₂×M₁₆	C₂³×C₅	1	8	yes	—
135	C₂²×C₈	—	2	4	no	—
136	C₂²×C₈	C₂³×C₅	1	2	yes	—
137	C₂²×C₈	C₂³×C₅	1	2	yes	—
138	C₂×C₄²	C₂×C₄×C₅	1	4	yes	—
139	C₂×C₄²	C₅×D₄	1	2	yes	—
140	C₂×C₄²	C₅×D₄	1	2	yes	—
141	C₂×C₄²	C₅×Q₈	1	2	yes	—
142	C₂×C₄²	C₂³×C₅	1	1	yes	characteristic
143	C₂×C₄²	C₂³×C₅	1	1	yes	characteristic
144	#108	C₂×C₄×C₅	1	32	yes	complemented
145	#97	—	2	4	no	—
146	#97	—	2	4	no	—
147	#97	C₂×C₄×C₅	1	8	yes	—
148	C₄×C₈	—	2	4	no	—
149	C₄×C₈	—	2	4	no	—
150	C₄×C₈	C₂×C₄×C₅	1	8	yes	—
151	C₂²×C₅	—	2	4	no	—
152	C₂²×C₅	—	2	2	no	—
153	C₂²×C₅	—	4	4	no	—
154	C₂²×C₅	C₄×C₄⋊C₄	1	4	yes	—
155	C₂²×C₅	C₂²×C₄⋊C₄	1	1	yes	characteristic
156	C₂²×C₅	C₂×C₄×Q₈	1	2	yes	—
157	C₄×C₅	—	2	4	no	—
158	C₄×C₅	—	2	2	no	—
159	C₄×C₅	C₄×C₄⋊C₄	1	4	yes	—
160	C₄×C₅	#430	1	4	yes	direct product factor
161	C₄×C₅	C₂²×C₄²	1	1	yes	characteristic
162	C₄×C₅	C₂×#117	1	1	yes	characteristic
163	C₄×C₅	C₂×C₄×D₄	1	2	yes	—
164	C₂⁴	C₂×C₅×Q₈	1	1	yes	characteristic
165	C₂²×C₄	—	4	4	no	—
166	C₂²×C₄	C₅×C₄⋊C₄	1	4	yes	—
167	C₂²×C₄	C₅×C₄⋊C₄	1	4	yes	—
168	C₂²×C₄	C₂²×C₄×C₅	1	2	yes	—
169	C₂²×C₄	C₂²×C₄×C₅	1	4	yes	—
170	C₂²×C₄	C₂×C₅×D₄	1	2	yes	—
171	C₂²×C₄	C₂×C₅×Q₈	1	2	yes	—
172	C₂²×C₄	C₂×C₅×Q₈	1	2	yes	—
173	C₂²×C₄	C₂×C₅×Q₈	1	1	yes	characteristic
174	C₂²×C₄	C₅×#41	1	4	yes	—
175	C₂²×C₄	C₂⁴×C₅	1	1	yes	characteristic
176	M₁₆	—	2	8	no	—
177	M₁₆	—	2	16	no	—
178	M₁₆	—	2	8	no	—
179	M₁₆	C₄²×C₅	1	16	yes	—
180	M₁₆	C₂²×C₄×C₅	1	16	yes	—
181	C₂×C₈	—	2	16	no	—
182	C₂×C₈	—	2	8	no	—
183	C₂×C₈	—	2	4	no	—
184	C₂×C₈	C₂²×C₄×C₅	1	4	yes	—
185	C₂×C₈	C₂²×C₄×C₅	1	4	yes	—
186	C₂×C₈	C₂²×C₄×C₅	1	8	yes	—
187	C₄²	C₅×C₄⋊C₄	1	4	yes	—
188	C₄²	C₅×C₄⋊C₄	1	4	yes	—
189	C₄²	C₂²×C₄×C₅	1	2	yes	—
190	C₄²	C₂²×C₄×C₅	1	2	yes	—
191	C₄²	C₂×C₅×D₄	1	1	yes	characteristic
192	C₄²	C₂×C₅×D₄	1	2	yes	—
193	C₄²	C₂×C₅×D₄	1	1	yes	characteristic
194	C₂×C₅	—	2	2	no	—
195	C₂×C₅	—	2	4	no	—
196	C₂×C₅	C₂×C₄×C₄⋊C₄	1	1	yes	characteristic
197	C₂×C₅	C₂×#430	1	2	yes	—
198	C₂³	—	4	2	no	—
199	C₂³	C₂×C₅×C₄⋊C₄	1	2	yes	—
200	C₂³	C₄×C₅×Q₈	1	4	yes	—
201	C₂³	C₂²×C₅×Q₈	1	1	yes	characteristic
202	C₂×C₄	—	2	8	no	—
203	C₂×C₄	—	2	4	no	—
204	C₂×C₄	—	4	4	no	—
205	C₂×C₄	—	4	2	no	—
206	C₂×C₄	C₂×C₄²×C₅	1	4	yes	—
207	C₂×C₄	C₂×C₅×C₄⋊C₄	1	2	yes	—
208	C₂×C₄	C₂×C₅×C₄⋊C₄	1	4	yes	—
209	C₂×C₄	C₂×C₅×C₄⋊C₄	1	2	yes	—
210	C₂×C₄	C₂×C₅×C₄⋊C₄	1	2	yes	—
211	C₂×C₄	C₅×#117	1	4	yes	—
212	C₂×C₄	C₄×C₅×D₄	1	8	yes	—
213	C₂×C₄	C₄×C₅×Q₈	1	4	yes	—
214	C₂×C₄	C₂³×C₄×C₅	1	1	yes	characteristic
215	C₂×C₄	C₂³×C₄×C₅	1	2	yes	—
216	C₂×C₄	C₂²×C₅×D₄	1	1	yes	characteristic
217	C₂×C₄	C₂×C₅×#41	1	2	yes	—
218	C₈	—	2	4	no	—
219	C₈	—	2	16	no	—
220	C₈	C₂×C₄²×C₅	1	8	yes	—
221	C₅	C₄×#430	1	1	yes	characteristic, direct product factor, Sylow
222	C₂²	—	2	4	no	—
223	C₂²	—	2	2	no	—
224	C₂²	—	4	4	no	—
225	C₂²	C₄×C₅×C₄⋊C₄	1	4	yes	—
226	C₂²	C₂²×C₅×C₄⋊C₄	1	1	yes	characteristic
227	C₂²	C₂×C₄×C₅×Q₈	1	2	yes	—
228	C₄	—	2	4	no	—
229	C₄	—	2	2	no	—
230	C₄	C₄×C₅×C₄⋊C₄	1	4	yes	—
231	C₄	C₅×#430	1	4	yes	direct product factor
232	C₄	C₂²×C₄²×C₅	1	1	yes	characteristic
233	C₄	C₂×C₅×#117	1	1	yes	characteristic
234	C₄	C₂×C₄×C₅×D₄	1	2	yes	—
235	C₂	—	2	4	no	—
236	C₂	—	2	2	no	—
237	C₂	C₂×C₄×C₅×C₄⋊C₄	1	1	yes	characteristic
238	C₂	C₂×C₅×#430	1	2	yes	—

	subgroup	G/H
center	C₄²×C₅	C₂×D₄
commutator	C₄	C₂²×C₄²×C₅
Frattini, cosocle	C₂²×C₄	C₂⁴×C₅
socle	C₂²×C₅	C₂²×C₄⋊C₄
prime radical	C₂⁴×C₅	C₂×Q₈
LCS	G ▷ C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₄²×C₅ ◁ C₂²×C₄²×C₅ ◁ G

Complements

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

N	H ≅ G/N	type
4. C₅×#1778	C₂	semidirect
5. C₄×C₅×#108	C₂	semidirect
11. C₅×#430	C₄	direct (N × H)
23. C₄×#430	C₅	direct (N × H)
43. C₅×#108	C₂×C₄	semidirect
53. #1778	C₂×C₅	semidirect
54. C₄×#108	C₂×C₅	semidirect
90. #430	C₄×C₅	direct (N × H)
144. #108	C₂×C₄×C₅	semidirect
160. C₄×C₅	#430	direct (N × H)
221. C₅	C₄×#430	direct (N × H)
231. C₄	C₅×#430	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×#430	2
C₂×C₄×C₅×M₁₆	2
C₅×#1778	2
C₄×C₅×#108	2
C₅×#1738	2
C₄×#430	5

Subgroups — C4×C5×#430 #11943178

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₄×C₅×#430 #11943178