Subgroups — C₅×#20366 #11943086

Order 1280.

1078 conjugacy classes of subgroups.

The subgroup lattice has 216 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×C₄×C₅×M₁₆	C₂	1	1	yes	maximal, characteristic
2	C₂²×C₄×C₅×C₈	C₂	1	1	yes	maximal, characteristic
3	C₄×C₅×#117	C₂	1	1	yes	maximal, characteristic
4	C₅×#1746	C₂	1	4	yes	maximal, complemented
5	C₅×#1744	C₂	1	4	yes	maximal, complemented
6	C₅×#1729	C₂	1	4	yes	maximal
7	C₂²×C₅×M₁₆	C₂²	1	2	yes	—
8	C₂³×C₅×C₈	C₂²	1	2	yes	—
9	C₂×C₅×#117	C₂²	1	2	yes	—
10	C₂²×C₄²×C₅	C₂²	1	1	yes	characteristic
11	C₄×C₅×M₁₆	C₄	1	8	yes	complemented
12	C₂×C₅×#97	C₂²	1	2	yes	—
13	C₂×C₄×C₅×C₈	C₄	1	8	yes	complemented
14	C₂×C₄×C₅×C₈	C₂²	1	2	yes	—
15	C₂×C₄×C₅×C₈	C₂²	1	2	yes	—
16	C₂×C₄×C₅×C₈	C₂²	1	2	yes	—
17	C₄×C₅×C₄⋊C₄	C₄	1	4	yes	—
18	C₄×C₅×C₂²⋊C₄	C₄	1	4	yes	—
19	C₅×#376	C₂²	1	2	yes	—
20	C₄³×C₅	C₂²	1	2	yes	—
21	C₅×#336	C₂²	1	16	yes	—
22	#20366	C₅	1	1	yes	maximal, characteristic, direct product factor, Sylow
23	C₂³×C₄×C₅	C₂×C₄	1	2	yes	—
24	C₂³×C₄×C₅	C₂³	1	1	yes	characteristic
25	C₂×C₅×M₁₆	—	2	8	no	—
26	C₂×C₅×M₁₆	C₂×C₄	1	8	yes	—
27	C₂²×C₅×C₈	—	2	8	no	—
28	C₂²×C₅×C₈	C₂×C₄	1	8	yes	—
29	C₂²×C₅×C₈	C₂³	1	4	yes	—
30	C₂²×C₅×C₈	C₂³	1	4	yes	—
31	C₅×#117	—	2	8	no	—
32	C₂×C₅×C₄⋊C₄	C₂×C₄	1	4	yes	—
33	C₂×C₅×C₂²⋊C₄	C₂×C₄	1	4	yes	—
34	C₂×C₄²×C₅	C₂×C₄	1	8	yes	—
35	C₂×C₄²×C₅	C₂×C₄	1	2	yes	—
36	C₂×C₄²×C₅	C₂×C₄	1	4	yes	—
37	C₂×C₄²×C₅	C₂×C₄	1	4	yes	—
38	C₂×C₄²×C₅	C₂³	1	4	yes	—
39	C₂×C₄²×C₅	C₂³	1	1	yes	characteristic
40	C₂×C₄²×C₅	C₂³	1	1	yes	characteristic
41	C₅×#97	C₂×C₄	1	8	yes	—
42	C₄×C₅×C₈	C₂×C₄	1	8	yes	—
43	C₄×C₅×C₈	C₂×C₄	1	8	yes	—
44	C₄×C₅×C₈	C₂×C₄	1	8	yes	—
45	C₅×#95	C₂×C₄	1	8	yes	—
46	C₂×C₄×M₁₆	C₂×C₅	1	1	yes	characteristic
47	C₂²×C₄×C₈	C₂×C₅	1	1	yes	characteristic
48	C₄×#117	C₂×C₅	1	1	yes	characteristic
49	#1746	C₂×C₅	1	4	yes	complemented
50	#1744	C₂×C₅	1	4	yes	complemented
51	#1729	C₂×C₅	1	4	yes	—
52	C₂⁴×C₅	C₂²×C₄	1	1	yes	characteristic
53	C₂²×C₄×C₅	—	2	8	no	—
54	C₂²×C₄×C₅	C₄²	1	8	yes	—
55	C₂²×C₄×C₅	C₂²×C₄	1	8	yes	—
56	C₂²×C₄×C₅	C₂²×C₄	1	2	yes	—
57	C₂²×C₄×C₅	C₂²×C₄	1	1	yes	characteristic
58	C₂²×C₄×C₅	C₂²×C₄	1	2	yes	—
59	C₂²×C₄×C₅	C₂²×C₄	1	2	yes	—
60	C₂²×C₄×C₅	C₂²×C₄	1	2	yes	—
61	C₂²×C₄×C₅	#41	1	8	yes	—
62	C₂²×C₄×C₅	C₂⁴	1	1	yes	characteristic
63	C₅×M₁₆	—	2	16	no	—
64	C₂×C₅×C₈	—	2	16	no	—
65	C₂×C₅×C₈	—	2	8	no	—
66	C₂×C₅×C₈	—	2	8	no	—
67	C₂×C₅×C₈	C₂²×C₄	1	8	yes	—
68	C₂×C₅×C₈	C₂²×C₄	1	8	yes	—
69	C₅×C₄⋊C₄	—	2	8	no	—
70	C₅×C₂²⋊C₄	—	2	8	no	—
71	C₄²×C₅	—	2	16	no	—
72	C₄²×C₅	—	2	8	no	—
73	C₄²×C₅	C₄²	1	8	yes	—
74	C₄²×C₅	C₂²×C₄	1	2	yes	—
75	C₄²×C₅	C₂²×C₄	1	2	yes	—
76	C₄²×C₅	C₂²×C₄	1	2	yes	—
77	C₄²×C₅	C₂²×C₄	1	2	yes	—
78	C₂²×M₁₆	C₂²×C₅	1	2	yes	—
79	C₂³×C₈	C₂²×C₅	1	2	yes	—
80	C₂×#117	C₂²×C₅	1	2	yes	—
81	C₂²×C₄²	C₂²×C₅	1	1	yes	characteristic
82	C₄×M₁₆	C₄×C₅	1	8	yes	complemented
83	C₂×#97	C₂²×C₅	1	2	yes	—
84	C₂×C₄×C₈	C₄×C₅	1	8	yes	complemented
85	C₂×C₄×C₈	C₂²×C₅	1	2	yes	—
86	C₂×C₄×C₈	C₂²×C₅	1	2	yes	—
87	C₂×C₄×C₈	C₂²×C₅	1	2	yes	—
88	C₄×C₄⋊C₄	C₄×C₅	1	4	yes	—
89	C₄×C₂²⋊C₄	C₄×C₅	1	4	yes	—
90	#376	C₂²×C₅	1	2	yes	—
91	C₄³	C₂²×C₅	1	2	yes	—
92	#336	C₂²×C₅	1	16	yes	—
93	C₂³×C₅	—	2	4	no	—
94	C₂³×C₅	C₂×C₄²	1	2	yes	—
95	C₂³×C₅	#117	1	4	yes	—
96	C₂³×C₅	C₂³×C₄	1	1	yes	characteristic
97	C₂×C₄×C₅	—	2	4	no	—
98	C₂×C₄×C₅	—	2	16	no	—
99	C₂×C₄×C₅	—	2	8	no	—
100	C₂×C₄×C₅	—	2	16	no	—
101	C₂×C₄×C₅	C₂×C₄²	1	4	yes	—
102	C₂×C₄×C₅	C₂×C₄²	1	2	yes	—
103	C₂×C₄×C₅	C₂×C₄²	1	4	yes	—
104	C₂×C₄×C₅	#117	1	8	yes	—
105	C₂×C₄×C₅	#117	1	8	yes	—
106	C₂×C₄×C₅	#117	1	4	yes	—
107	C₂×C₄×C₅	#131	1	8	yes	—
108	C₂×C₄×C₅	C₂³×C₄	1	1	yes	characteristic
109	C₂×C₄×C₅	C₂³×C₄	1	1	yes	characteristic
110	C₂×C₄×C₅	C₂×#41	1	4	yes	—
111	C₅×C₈	—	2	8	no	—
112	C₅×C₈	—	2	8	no	—
113	C₂³×C₄	C₂×C₄×C₅	1	2	yes	—
114	C₂³×C₄	C₂³×C₅	1	1	yes	characteristic
115	C₂×M₁₆	—	2	8	no	—
116	C₂×M₁₆	C₂×C₄×C₅	1	8	yes	—
117	C₂²×C₈	—	2	8	no	—
118	C₂²×C₈	C₂×C₄×C₅	1	8	yes	—
119	C₂²×C₈	C₂³×C₅	1	4	yes	—
120	C₂²×C₈	C₂³×C₅	1	4	yes	—
121	#117	—	2	8	no	—
122	C₂×C₄⋊C₄	C₂×C₄×C₅	1	4	yes	—
123	C₂×C₂²⋊C₄	C₂×C₄×C₅	1	4	yes	—
124	C₂×C₄²	C₂×C₄×C₅	1	8	yes	—
125	C₂×C₄²	C₂×C₄×C₅	1	2	yes	—
126	C₂×C₄²	C₂×C₄×C₅	1	4	yes	—
127	C₂×C₄²	C₂×C₄×C₅	1	4	yes	—
128	C₂×C₄²	C₂³×C₅	1	4	yes	—
129	C₂×C₄²	C₂³×C₅	1	1	yes	characteristic
130	C₂×C₄²	C₂³×C₅	1	1	yes	characteristic
131	#97	C₂×C₄×C₅	1	8	yes	—
132	C₄×C₈	C₂×C₄×C₅	1	8	yes	—
133	C₄×C₈	C₂×C₄×C₅	1	8	yes	—
134	C₄×C₈	C₂×C₄×C₅	1	8	yes	—
135	#95	C₂×C₄×C₅	1	8	yes	—
136	C₂²×C₅	—	2	4	no	—
137	C₂²×C₅	—	2	8	no	—
138	C₂²×C₅	#376	1	4	yes	—
139	C₂²×C₅	C₂²×C₄²	1	1	yes	characteristic
140	C₂²×C₅	C₂×#117	1	2	yes	—
141	C₂²×C₅	C₂×#131	1	4	yes	—
142	C₄×C₅	—	2	4	no	—
143	C₄×C₅	—	4	8	no	—
144	C₄×C₅	#376	1	4	yes	—
145	C₄×C₅	#405	1	8	yes	—
146	C₄×C₅	C₂×#117	1	2	yes	—
147	C₄×C₅	C₂×#117	1	2	yes	—
148	C₂⁴	C₂²×C₄×C₅	1	1	yes	characteristic
149	C₂²×C₄	—	2	8	no	—
150	C₂²×C₄	C₄²×C₅	1	8	yes	—
151	C₂²×C₄	C₂²×C₄×C₅	1	8	yes	—
152	C₂²×C₄	C₂²×C₄×C₅	1	2	yes	—
153	C₂²×C₄	C₂²×C₄×C₅	1	1	yes	characteristic
154	C₂²×C₄	C₂²×C₄×C₅	1	2	yes	—
155	C₂²×C₄	C₂²×C₄×C₅	1	2	yes	—
156	C₂²×C₄	C₂²×C₄×C₅	1	2	yes	—
157	C₂²×C₄	C₅×#41	1	8	yes	—
158	C₂²×C₄	C₂⁴×C₅	1	1	yes	characteristic
159	M₁₆	—	2	16	no	—
160	C₂×C₈	—	2	16	no	—
161	C₂×C₈	—	2	8	no	—
162	C₂×C₈	—	2	8	no	—
163	C₂×C₈	C₂²×C₄×C₅	1	8	yes	—
164	C₂×C₈	C₂²×C₄×C₅	1	8	yes	—
165	C₄⋊C₄	—	2	8	no	—
166	C₂²⋊C₄	—	2	8	no	—
167	C₄²	—	2	16	no	—
168	C₄²	—	2	8	no	—
169	C₄²	C₄²×C₅	1	8	yes	—
170	C₄²	C₂²×C₄×C₅	1	2	yes	—
171	C₄²	C₂²×C₄×C₅	1	2	yes	—
172	C₄²	C₂²×C₄×C₅	1	2	yes	—
173	C₄²	C₂²×C₄×C₅	1	2	yes	—
174	C₂×C₅	—	2	4	no	—
175	C₂×C₅	C₂×#376	1	1	yes	characteristic
176	C₂×C₅	C₂×#405	1	2	yes	—
177	C₂×C₅	#2922	1	4	yes	—
178	C₂³	—	2	4	no	—
179	C₂³	C₂×C₄²×C₅	1	2	yes	—
180	C₂³	C₅×#117	1	4	yes	—
181	C₂³	C₂³×C₄×C₅	1	1	yes	characteristic
182	C₂×C₄	—	2	4	no	—
183	C₂×C₄	—	2	16	no	—
184	C₂×C₄	—	2	8	no	—
185	C₂×C₄	—	2	16	no	—
186	C₂×C₄	C₂×C₄²×C₅	1	4	yes	—
187	C₂×C₄	C₂×C₄²×C₅	1	2	yes	—
188	C₂×C₄	C₂×C₄²×C₅	1	4	yes	—
189	C₂×C₄	C₅×#117	1	4	yes	—
190	C₂×C₄	C₅×#117	1	8	yes	—
191	C₂×C₄	C₅×#117	1	8	yes	—
192	C₂×C₄	C₅×#131	1	8	yes	—
193	C₂×C₄	C₂³×C₄×C₅	1	1	yes	characteristic
194	C₂×C₄	C₂³×C₄×C₅	1	1	yes	characteristic
195	C₂×C₄	C₂×C₅×#41	1	4	yes	—
196	C₈	—	2	8	no	—
197	C₈	—	2	8	no	—
198	C₅	#20366	1	1	yes	characteristic, direct product factor, Sylow
199	C₂²	—	2	4	no	—
200	C₂²	—	2	8	no	—
201	C₂²	C₅×#376	1	4	yes	—
202	C₂²	C₂²×C₄²×C₅	1	1	yes	characteristic
203	C₂²	C₂×C₅×#117	1	2	yes	—
204	C₂²	C₂×C₅×#131	1	4	yes	—
205	C₄	—	2	4	no	—
206	C₄	—	4	8	no	—
207	C₄	C₅×#376	1	4	yes	—
208	C₄	C₅×#405	1	8	yes	—
209	C₄	C₂×C₅×#117	1	2	yes	—
210	C₄	C₂×C₅×#117	1	2	yes	—
211	C₂	—	2	4	no	—
212	C₂	C₂×C₅×#376	1	1	yes	characteristic
213	C₂	C₂×C₅×#405	1	2	yes	—
214	C₂	C₅×#2922	1	4	yes	—

	subgroup	G/H
center	C₂×C₄²×C₅	C₂³
commutator	C₂²	C₂²×C₄²×C₅
Frattini, cosocle	C₂²×C₄	C₂⁴×C₅
socle	C₂³×C₅	C₂³×C₄
prime radical	C₂⁴×C₅	C₂²×C₄
LCS	G ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂×C₄²×C₅ ◁ G

Complements

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

N	H ≅ G/N	type
4. C₅×#1746	C₂	semidirect
5. C₅×#1744	C₂	semidirect
11. C₄×C₅×M₁₆	C₄	semidirect
13. C₂×C₄×C₅×C₈	C₄	semidirect
22. #20366	C₅	direct (N × H)
49. #1746	C₂×C₅	semidirect
50. #1744	C₂×C₅	semidirect
82. C₄×M₁₆	C₄×C₅	semidirect
84. C₂×C₄×C₈	C₄×C₅	semidirect
198. C₅	#20366	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×C₄×C₅×M₁₆	2
C₂²×C₄×C₅×C₈	2
C₄×C₅×#117	2
C₅×#1746	2
C₅×#1744	2
C₅×#1729	2
#20366	5

Subgroups — C5×#20366 #11943086

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₅×#20366 #11943086