Subgroups — C₅×#20364 #11943084

Order 1280.

1286 conjugacy classes of subgroups.

The subgroup lattice has 188 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	2	yes	maximal
2	C₂×C₄³×C₅	C₂	1	1	yes	maximal, characteristic
3	C₅×#1745	C₂	1	4	yes	maximal, complemented
4	C₅×#1744	C₂	1	4	yes	maximal, complemented
5	C₅×#1728	C₂	1	4	yes	maximal
6	C₂²×C₅×M₁₆	C₂²	1	4	yes	—
7	C₂²×C₄²×C₅	C₄	1	4	yes	—
8	C₂²×C₄²×C₅	C₂²	1	2	yes	—
9	C₂²×C₄²×C₅	C₂²	1	1	yes	characteristic
10	C₄×C₅×M₁₆	C₄	1	16	yes	complemented
11	C₂×C₅×#97	C₂²	1	4	yes	—
12	C₂×C₄×C₅×C₈	C₂²	1	4	yes	—
13	C₄³×C₅	C₄	1	4	yes	—
14	C₄³×C₅	C₂²	1	2	yes	—
15	C₄³×C₅	C₂²	1	2	yes	—
16	C₅×#336	C₂²	1	16	yes	—
17	#20364	C₅	1	1	yes	maximal, characteristic, direct product factor, Sylow
18	C₂³×C₄×C₅	C₂×C₄	1	4	yes	—
19	C₂³×C₄×C₅	C₂×C₄	1	2	yes	—
20	C₂³×C₄×C₅	C₂³	1	1	yes	characteristic
21	C₂×C₅×M₁₆	—	2	16	no	—
22	C₂×C₅×M₁₆	C₂×C₄	1	16	yes	—
23	C₂²×C₅×C₈	C₂³	1	8	yes	—
24	C₂×C₄²×C₅	—	2	8	no	—
25	C₂×C₄²×C₅	—	2	16	no	—
26	C₂×C₄²×C₅	C₂×C₄	1	4	yes	—
27	C₂×C₄²×C₅	C₂×C₄	1	8	yes	—
28	C₂×C₄²×C₅	C₂×C₄	1	4	yes	—
29	C₂×C₄²×C₅	C₂×C₄	1	2	yes	—
30	C₂×C₄²×C₅	C₂×C₄	1	8	yes	—
31	C₂×C₄²×C₅	C₂×C₄	1	4	yes	—
32	C₂×C₄²×C₅	C₂³	1	4	yes	—
33	C₂×C₄²×C₅	C₂³	1	1	yes	characteristic
34	C₂×C₄²×C₅	C₂³	1	1	yes	characteristic
35	C₅×#97	C₂×C₄	1	16	yes	complemented
36	C₄×C₅×C₈	C₂×C₄	1	16	yes	complemented
37	C₂×C₄×M₁₆	C₂×C₅	1	2	yes	—
38	C₂×C₄³	C₂×C₅	1	1	yes	characteristic
39	#1745	C₂×C₅	1	4	yes	complemented
40	#1744	C₂×C₅	1	4	yes	complemented
41	#1728	C₂×C₅	1	4	yes	—
42	C₂⁴×C₅	C₂²×C₄	1	1	yes	characteristic
43	C₂²×C₄×C₅	—	2	8	no	—
44	C₂²×C₄×C₅	—	2	16	no	—
45	C₂²×C₄×C₅	—	2	8	no	—
46	C₂²×C₄×C₅	C₄²	1	8	yes	—
47	C₂²×C₄×C₅	C₂²×C₄	1	2	yes	—
48	C₂²×C₄×C₅	C₂²×C₄	1	8	yes	—
49	C₂²×C₄×C₅	C₂²×C₄	1	1	yes	characteristic
50	C₂²×C₄×C₅	C₂²×C₄	1	2	yes	—
51	C₂²×C₄×C₅	C₂²×C₄	1	4	yes	—
52	C₂²×C₄×C₅	#41	1	8	yes	—
53	C₂²×C₄×C₅	C₂⁴	1	1	yes	characteristic
54	C₅×M₁₆	—	2	32	no	—
55	C₂×C₅×C₈	—	2	16	no	—
56	C₂×C₅×C₈	C₂²×C₄	1	16	yes	—
57	C₄²×C₅	—	2	8	no	—
58	C₄²×C₅	—	2	16	no	—
59	C₄²×C₅	—	2	8	no	—
60	C₄²×C₅	—	2	8	no	—
61	C₄²×C₅	C₄²	1	8	yes	—
62	C₄²×C₅	M₁₆	1	16	yes	complemented
63	C₄²×C₅	C₂²×C₄	1	4	yes	—
64	C₄²×C₅	C₂²×C₄	1	4	yes	—
65	C₂²×M₁₆	C₂²×C₅	1	4	yes	—
66	C₂²×C₄²	C₄×C₅	1	4	yes	—
67	C₂²×C₄²	C₂²×C₅	1	2	yes	—
68	C₂²×C₄²	C₂²×C₅	1	1	yes	characteristic
69	C₄×M₁₆	C₄×C₅	1	16	yes	complemented
70	C₂×#97	C₂²×C₅	1	4	yes	—
71	C₂×C₄×C₈	C₂²×C₅	1	4	yes	—
72	C₄³	C₄×C₅	1	4	yes	—
73	C₄³	C₂²×C₅	1	2	yes	—
74	C₄³	C₂²×C₅	1	2	yes	—
75	#336	C₂²×C₅	1	16	yes	—
76	C₂³×C₅	—	2	4	no	—
77	C₂³×C₅	C₂×C₄²	1	2	yes	—
78	C₂³×C₅	#117	1	4	yes	—
79	C₂³×C₅	C₂³×C₄	1	1	yes	characteristic
80	C₂×C₄×C₅	—	2	32	no	—
81	C₂×C₄×C₅	—	2	8	no	—
82	C₂×C₄×C₅	—	2	4	no	—
83	C₂×C₄×C₅	—	2	8	no	—
84	C₂×C₄×C₅	—	2	16	no	—
85	C₂×C₄×C₅	C₂×C₄²	1	4	yes	—
86	C₂×C₄×C₅	C₂×C₄²	1	2	yes	—
87	C₂×C₄×C₅	C₂×C₄²	1	4	yes	—
88	C₂×C₄×C₅	#117	1	16	yes	—
89	C₂×C₄×C₅	#117	1	4	yes	—
90	C₂×C₄×C₅	C₂×M₁₆	1	16	yes	—
91	C₂×C₄×C₅	C₂×M₁₆	1	8	yes	—
92	C₂×C₄×C₅	C₂³×C₄	1	2	yes	—
93	C₂×C₄×C₅	C₂×#41	1	4	yes	—
94	C₅×C₈	—	2	16	no	—
95	C₂³×C₄	C₂×C₄×C₅	1	4	yes	—
96	C₂³×C₄	C₂×C₄×C₅	1	2	yes	—
97	C₂³×C₄	C₂³×C₅	1	1	yes	characteristic
98	C₂×M₁₆	—	2	16	no	—
99	C₂×M₁₆	C₂×C₄×C₅	1	16	yes	—
100	C₂²×C₈	C₂³×C₅	1	8	yes	—
101	C₂×C₄²	—	2	8	no	—
102	C₂×C₄²	—	2	16	no	—
103	C₂×C₄²	C₂×C₄×C₅	1	4	yes	—
104	C₂×C₄²	C₂×C₄×C₅	1	8	yes	—
105	C₂×C₄²	C₂×C₄×C₅	1	4	yes	—
106	C₂×C₄²	C₂×C₄×C₅	1	2	yes	—
107	C₂×C₄²	C₂×C₄×C₅	1	8	yes	—
108	C₂×C₄²	C₂×C₄×C₅	1	4	yes	—
109	C₂×C₄²	C₂³×C₅	1	4	yes	—
110	C₂×C₄²	C₂³×C₅	1	1	yes	characteristic
111	C₂×C₄²	C₂³×C₅	1	1	yes	characteristic
112	#97	C₂×C₄×C₅	1	16	yes	complemented
113	C₄×C₈	C₂×C₄×C₅	1	16	yes	complemented
114	C₂²×C₅	—	2	4	no	—
115	C₂²×C₅	—	2	8	no	—
116	C₂²×C₅	#376	1	4	yes	—
117	C₂²×C₅	C₂²×C₄²	1	1	yes	characteristic
118	C₂²×C₅	C₂×#117	1	2	yes	—
119	C₂²×C₅	C₂²×M₁₆	1	4	yes	—
120	C₄×C₅	—	2	4	no	—
121	C₄×C₅	—	2	16	no	—
122	C₄×C₅	#376	1	4	yes	—
123	C₄×C₅	C₄×M₁₆	1	8	yes	—
124	C₄×C₅	C₂×#117	1	4	yes	—
125	C₂⁴	C₂²×C₄×C₅	1	1	yes	characteristic
126	C₂²×C₄	—	2	8	no	—
127	C₂²×C₄	—	2	16	no	—
128	C₂²×C₄	—	2	8	no	—
129	C₂²×C₄	C₄²×C₅	1	8	yes	—
130	C₂²×C₄	C₂²×C₄×C₅	1	2	yes	—
131	C₂²×C₄	C₂²×C₄×C₅	1	8	yes	—
132	C₂²×C₄	C₂²×C₄×C₅	1	1	yes	characteristic
133	C₂²×C₄	C₂²×C₄×C₅	1	2	yes	—
134	C₂²×C₄	C₂²×C₄×C₅	1	4	yes	—
135	C₂²×C₄	C₅×#41	1	8	yes	—
136	C₂²×C₄	C₂⁴×C₅	1	1	yes	characteristic
137	M₁₆	—	2	32	no	—
138	C₂×C₈	—	2	16	no	—
139	C₂×C₈	C₂²×C₄×C₅	1	16	yes	—
140	C₄²	—	2	8	no	—
141	C₄²	—	2	16	no	—
142	C₄²	—	2	8	no	—
143	C₄²	—	2	8	no	—
144	C₄²	C₄²×C₅	1	8	yes	—
145	C₄²	C₅×M₁₆	1	16	yes	complemented
146	C₄²	C₂²×C₄×C₅	1	4	yes	—
147	C₄²	C₂²×C₄×C₅	1	4	yes	—
148	C₂×C₅	—	2	4	no	—
149	C₂×C₅	C₂×#376	1	1	yes	characteristic
150	C₂×C₅	C₂×C₄×M₁₆	1	2	yes	—
151	C₂×C₅	#2920	1	4	yes	—
152	C₂³	—	2	4	no	—
153	C₂³	C₂×C₄²×C₅	1	2	yes	—
154	C₂³	C₅×#117	1	4	yes	—
155	C₂³	C₂³×C₄×C₅	1	1	yes	characteristic
156	C₂×C₄	—	2	32	no	—
157	C₂×C₄	—	2	8	no	—
158	C₂×C₄	—	2	4	no	—
159	C₂×C₄	—	2	8	no	—
160	C₂×C₄	—	2	16	no	—
161	C₂×C₄	C₂×C₄²×C₅	1	4	yes	—
162	C₂×C₄	C₂×C₄²×C₅	1	2	yes	—
163	C₂×C₄	C₂×C₄²×C₅	1	4	yes	—
164	C₂×C₄	C₅×#117	1	4	yes	—
165	C₂×C₄	C₅×#117	1	16	yes	—
166	C₂×C₄	C₂×C₅×M₁₆	1	16	yes	—
167	C₂×C₄	C₂×C₅×M₁₆	1	8	yes	—
168	C₂×C₄	C₂³×C₄×C₅	1	2	yes	—
169	C₂×C₄	C₂×C₅×#41	1	4	yes	—
170	C₈	—	2	16	no	—
171	C₅	#20364	1	1	yes	characteristic, direct product factor, Sylow
172	C₂²	—	2	4	no	—
173	C₂²	—	2	8	no	—
174	C₂²	C₅×#376	1	4	yes	—
175	C₂²	C₂²×C₄²×C₅	1	1	yes	characteristic
176	C₂²	C₂×C₅×#117	1	2	yes	—
177	C₂²	C₂²×C₅×M₁₆	1	4	yes	—
178	C₄	—	2	4	no	—
179	C₄	—	2	16	no	—
180	C₄	C₅×#376	1	4	yes	—
181	C₄	C₄×C₅×M₁₆	1	8	yes	—
182	C₄	C₂×C₅×#117	1	4	yes	—
183	C₂	—	2	4	no	—
184	C₂	C₂×C₅×#376	1	1	yes	characteristic
185	C₂	C₂×C₄×C₅×M₁₆	1	2	yes	—
186	C₂	C₅×#2920	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
3. C₅×#1745	C₂	semidirect
4. C₅×#1744	C₂	semidirect
10. C₄×C₅×M₁₆	C₄	semidirect
17. #20364	C₅	direct (N × H)
35. C₅×#97	C₂×C₄	semidirect
36. C₄×C₅×C₈	C₂×C₄	semidirect
39. #1745	C₂×C₅	semidirect
40. #1744	C₂×C₅	semidirect
62. C₄²×C₅	M₁₆	semidirect
69. C₄×M₁₆	C₄×C₅	semidirect
112. #97	C₂×C₄×C₅	semidirect
113. C₄×C₈	C₂×C₄×C₅	semidirect
145. C₄²	C₅×M₁₆	semidirect
171. C₅	#20364	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
C₂×C₄³×C₅	2
C₅×#1745	2
C₅×#1744	2
C₅×#1728	2
#20364	5

Subgroups — C5×#20364 #11943084

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₅×#20364 #11943084