Subgroups — C₄×C₅×#217 #11750708

Order 960.

516 conjugacy classes of subgroups.

The subgroup lattice has 172 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂³×C₃×C₄×C₅	C₂	1	1	yes	maximal, characteristic
2	C₂×C₅×#217	C₂	1	2	yes	maximal
3	C₂×C₄×C₅×Dic₃	C₂	1	2	yes	maximal, complemented
4	C₅×C₃⋊#95	C₂	1	2	yes	maximal, complemented
5	C₄×C₅×C₂²⋊C₄	—	3	1	no	maximal, Hall
6	C₂⁴×C₃×C₅	C₂²	1	1	yes	characteristic
7	C₂²×C₃×C₄×C₅	—	2	4	no	—
8	C₂²×C₃×C₄×C₅	C₄	1	4	yes	complemented
9	C₂²×C₃×C₄×C₅	C₂²	1	1	yes	characteristic
10	C₂²×C₃×C₄×C₅	C₂²	1	1	yes	characteristic
11	C₂²×C₅×Dic₃	C₂²	1	4	yes	—
12	C₅×#217	C₄	1	8	yes	direct product factor
13	C₄×C₅×Dic₃	—	2	4	no	—
14	C₄×#217	C₅	1	1	yes	maximal, characteristic, direct product factor, Hall
15	C₂³×C₄×C₅	D₃	1	1	yes	characteristic
16	C₂×C₅×C₂²⋊C₄	—	3	2	no	—
17	C₂×C₄²×C₅	—	3	2	no	—
18	C₅×#95	—	3	2	no	—
19	C₂³×C₃×C₅	—	2	2	no	—
20	C₂³×C₃×C₅	—	2	2	no	—
21	C₂³×C₃×C₅	C₂×C₄	1	4	yes	—
22	C₂³×C₃×C₅	C₂×C₄	1	2	yes	—
23	C₂³×C₃×C₅	C₂³	1	1	yes	characteristic
24	C₂×C₃×C₄×C₅	—	2	2	no	—
25	C₂×C₃×C₄×C₅	—	2	8	no	—
26	C₂×C₃×C₄×C₅	C₂×C₄	1	2	yes	—
27	C₂×C₃×C₄×C₅	C₂×C₄	1	2	yes	—
28	C₂×C₃×C₄×C₅	D₄	1	4	yes	—
29	C₂×C₅×Dic₃	—	2	4	no	—
30	C₂×C₅×Dic₃	—	2	4	no	—
31	C₂×C₅×Dic₃	C₂×C₄	1	8	yes	complemented
32	C₂³×C₃×C₄	C₂×C₅	1	1	yes	characteristic
33	C₂×#217	C₂×C₅	1	2	yes	—
34	C₂×C₄×Dic₃	C₂×C₅	1	2	yes	complemented
35	C₃⋊#95	C₂×C₅	1	2	yes	complemented
36	C₂⁴×C₅	C₂×D₃	1	1	yes	characteristic
37	C₂²×C₄×C₅	—	2	4	no	—
38	C₂²×C₄×C₅	—	3	4	no	—
39	C₂²×C₄×C₅	Dic₃	1	4	yes	complemented
40	C₂²×C₄×C₅	C₂×D₃	1	1	yes	characteristic
41	C₂²×C₄×C₅	C₂×D₃	1	1	yes	characteristic
42	C₅×C₂²⋊C₄	—	3	8	no	—
43	C₄²×C₅	—	6	4	no	—
44	C₄×C₂²⋊C₄	—	3	1	no	Sylow
45	C₂²×C₃×C₅	—	2	2	no	—
46	C₂²×C₃×C₅	—	2	2	no	—
47	C₂²×C₃×C₅	—	2	8	no	—
48	C₂²×C₃×C₅	C₄²	1	4	yes	complemented
49	C₂²×C₃×C₅	C₂²×C₄	1	2	yes	—
50	C₂²×C₃×C₅	C₂²×C₄	1	1	yes	characteristic
51	C₂²×C₃×C₅	C₂×D₄	1	2	yes	—
52	C₂²×C₃×C₅	#41	1	2	yes	—
53	C₃×C₄×C₅	—	2	2	no	—
54	C₃×C₄×C₅	C₂²⋊C₄	1	4	yes	complemented
55	C₅×Dic₃	—	2	8	no	—
56	C₂⁴×C₃	C₂²×C₅	1	1	yes	characteristic
57	C₂²×C₃×C₄	—	2	4	no	—
58	C₂²×C₃×C₄	C₄×C₅	1	4	yes	complemented
59	C₂²×C₃×C₄	C₂²×C₅	1	1	yes	characteristic
60	C₂²×C₃×C₄	C₂²×C₅	1	1	yes	characteristic
61	C₂²×Dic₃	C₂²×C₅	1	4	yes	—
62	#217	C₄×C₅	1	8	yes	direct product factor
63	C₄×Dic₃	—	2	4	no	—
64	C₂³×C₅	—	2	2	no	—
65	C₂³×C₅	—	2	2	no	—
66	C₂³×C₅	C₄×D₃	1	4	yes	—
67	C₂³×C₅	C₂×Dic₃	1	2	yes	—
68	C₂³×C₅	C₂²×D₃	1	1	yes	characteristic
69	C₂×C₄×C₅	—	2	2	no	—
70	C₂×C₄×C₅	—	2	8	no	—
71	C₂×C₄×C₅	—	3	8	no	—
72	C₂×C₄×C₅	—	6	4	no	—
73	C₂×C₄×C₅	—	6	4	no	—
74	C₂×C₄×C₅	C₂×Dic₃	1	2	yes	—
75	C₂×C₄×C₅	C₂×Dic₃	1	2	yes	—
76	C₂×C₄×C₅	#67	1	4	yes	—
77	C₂³×C₄	C₅×D₃	1	1	yes	characteristic
78	C₂×C₂²⋊C₄	—	3	2	no	—
79	C₂×C₄²	—	3	2	no	—
80	#95	—	3	2	no	—
81	C₂×C₃×C₅	—	2	4	no	—
82	C₂×C₃×C₅	C₂×C₄²	1	1	yes	characteristic
83	C₂×C₃×C₅	C₂×C₂²⋊C₄	1	1	yes	characteristic
84	C₂×C₃×C₅	#117	1	1	yes	characteristic
85	C₂×C₃×C₅	C₄×D₄	1	4	yes	—
86	C₂³×C₃	—	2	2	no	—
87	C₂³×C₃	—	2	2	no	—
88	C₂³×C₃	C₂×C₄×C₅	1	4	yes	—
89	C₂³×C₃	C₂×C₄×C₅	1	2	yes	—
90	C₂³×C₃	C₂³×C₅	1	1	yes	characteristic
91	C₂×C₃×C₄	—	2	2	no	—
92	C₂×C₃×C₄	—	2	8	no	—
93	C₂×C₃×C₄	C₂×C₄×C₅	1	2	yes	—
94	C₂×C₃×C₄	C₂×C₄×C₅	1	2	yes	—
95	C₂×C₃×C₄	C₅×D₄	1	4	yes	—
96	C₂×Dic₃	—	2	4	no	—
97	C₂×Dic₃	—	2	4	no	—
98	C₂×Dic₃	C₂×C₄×C₅	1	8	yes	complemented
99	C₂²×C₅	—	2	2	no	—
100	C₂²×C₅	—	2	8	no	—
101	C₂²×C₅	—	2	2	no	—
102	C₂²×C₅	C₄×Dic₃	1	4	yes	complemented
103	C₂²×C₅	C₂×C₄×D₃	1	2	yes	—
104	C₂²×C₅	#235	1	2	yes	—
105	C₂²×C₅	C₂²×Dic₃	1	1	yes	characteristic
106	C₂²×C₅	C₂×#67	1	2	yes	—
107	C₄×C₅	—	2	2	no	—
108	C₄×C₅	—	6	8	no	—
109	C₄×C₅	#217	1	4	yes	direct product factor
110	C₂⁴	C₂×C₅×D₃	1	1	yes	characteristic
111	C₂²×C₄	—	2	4	no	—
112	C₂²×C₄	—	3	4	no	—
113	C₂²×C₄	C₅×Dic₃	1	4	yes	complemented
114	C₂²×C₄	C₂×C₅×D₃	1	1	yes	characteristic
115	C₂²×C₄	C₂×C₅×D₃	1	1	yes	characteristic
116	C₂²⋊C₄	—	3	8	no	—
117	C₄²	—	6	4	no	—
118	C₃×C₅	C₄×C₂²⋊C₄	1	1	yes	characteristic, complemented, Hall
119	C₂²×C₃	—	2	2	no	—
120	C₂²×C₃	—	2	2	no	—
121	C₂²×C₃	—	2	8	no	—
122	C₂²×C₃	C₄²×C₅	1	4	yes	complemented
123	C₂²×C₃	C₂²×C₄×C₅	1	2	yes	—
124	C₂²×C₃	C₂²×C₄×C₅	1	1	yes	characteristic
125	C₂²×C₃	C₂×C₅×D₄	1	2	yes	—
126	C₂²×C₃	C₅×#41	1	2	yes	—
127	C₃×C₄	—	2	2	no	—
128	C₃×C₄	C₅×C₂²⋊C₄	1	4	yes	complemented
129	Dic₃	—	2	8	no	—
130	C₂×C₅	—	2	4	no	—
131	C₂×C₅	C₂×C₄×Dic₃	1	1	yes	characteristic
132	C₂×C₅	#926	1	1	yes	characteristic
133	C₂×C₅	C₄×#67	1	4	yes	—
134	C₂×C₅	C₂×#217	1	1	yes	characteristic
135	C₂³	—	2	2	no	—
136	C₂³	—	2	2	no	—
137	C₂³	C₄×C₅×D₃	1	4	yes	—
138	C₂³	C₂×C₅×Dic₃	1	2	yes	—
139	C₂³	C₂²×C₅×D₃	1	1	yes	characteristic
140	C₂×C₄	—	2	2	no	—
141	C₂×C₄	—	2	8	no	—
142	C₂×C₄	—	3	8	no	—
143	C₂×C₄	—	6	4	no	—
144	C₂×C₄	—	6	4	no	—
145	C₂×C₄	C₂×C₅×Dic₃	1	2	yes	—
146	C₂×C₄	C₂×C₅×Dic₃	1	2	yes	—
147	C₂×C₄	C₅×#67	1	4	yes	—
148	C₂×C₃	—	2	4	no	—
149	C₂×C₃	C₂×C₄²×C₅	1	1	yes	characteristic
150	C₂×C₃	C₂×C₅×C₂²⋊C₄	1	1	yes	characteristic
151	C₂×C₃	C₅×#117	1	1	yes	characteristic
152	C₂×C₃	C₄×C₅×D₄	1	4	yes	—
153	C₅	C₄×#217	1	1	yes	characteristic, direct product factor, Sylow
154	C₂²	—	2	2	no	—
155	C₂²	—	2	8	no	—
156	C₂²	—	2	2	no	—
157	C₂²	C₄×C₅×Dic₃	1	4	yes	complemented
158	C₂²	C₂×C₄×C₅×D₃	1	2	yes	—
159	C₂²	C₅×#235	1	2	yes	—
160	C₂²	C₂²×C₅×Dic₃	1	1	yes	characteristic
161	C₂²	C₂×C₅×#67	1	2	yes	—
162	C₄	—	2	2	no	—
163	C₄	—	6	8	no	—
164	C₄	C₅×#217	1	4	yes	direct product factor
165	C₃	C₄×C₅×C₂²⋊C₄	1	1	yes	characteristic, complemented, Sylow
166	C₂	—	2	4	no	—
167	C₂	C₂×C₄×C₅×Dic₃	1	1	yes	characteristic
168	C₂	C₅×#926	1	1	yes	characteristic
169	C₂	C₄×C₅×#67	1	4	yes	—
170	C₂	C₂×C₅×#217	1	1	yes	characteristic

	subgroup	G/H
center	C₂²×C₄×C₅	C₂×D₃
commutator	C₂×C₃	C₂×C₄²×C₅
Frattini	C₂³	C₂²×C₅×D₃
Fitting	C₂³×C₃×C₄×C₅	C₂
socle	C₂³×C₃×C₅	C₂³
prime radical	C₂⁴×C₃×C₅	C₂²
cosocle	C₂³×C₃	C₂³×C₅
hypercenter	C₂³×C₄×C₅	D₃
hypocenter	C₃	C₄×C₅×C₂²⋊C₄
LCS	G ▷ C₂×C₃ ▷ C₃
UCS	C₁ ◁ C₂²×C₄×C₅ ◁ C₂³×C₄×C₅

Complements

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

N	H ≅ G/N	type
3. C₂×C₄×C₅×Dic₃	C₂	semidirect
4. C₅×C₃⋊#95	C₂	semidirect
8. C₂²×C₃×C₄×C₅	C₄	semidirect
12. C₅×#217	C₄	direct (N × H)
14. C₄×#217	C₅	direct (N × H)
31. C₂×C₅×Dic₃	C₂×C₄	semidirect
34. C₂×C₄×Dic₃	C₂×C₅	semidirect
35. C₃⋊#95	C₂×C₅	semidirect
39. C₂²×C₄×C₅	Dic₃	semidirect
48. C₂²×C₃×C₅	C₄²	semidirect
54. C₃×C₄×C₅	C₂²⋊C₄	semidirect
58. C₂²×C₃×C₄	C₄×C₅	semidirect
62. #217	C₄×C₅	direct (N × H)
98. C₂×Dic₃	C₂×C₄×C₅	semidirect
102. C₂²×C₅	C₄×Dic₃	semidirect
109. C₄×C₅	#217	direct (N × H)
113. C₂²×C₄	C₅×Dic₃	semidirect
118. C₃×C₅	C₄×C₂²⋊C₄	semidirect
122. C₂²×C₃	C₄²×C₅	semidirect
128. C₃×C₄	C₅×C₂²⋊C₄	semidirect
153. C₅	C₄×#217	direct (N × H)
157. C₂²	C₄×C₅×Dic₃	semidirect
164. C₄	C₅×#217	direct (N × H)
165. C₃	C₄×C₅×C₂²⋊C₄	semidirect

Maximal subgroups

By isomorphism type — maximal subgroups of the same type appear once, whether or not they are normal.

subgroup	index
C₂³×C₃×C₄×C₅	2
C₂×C₅×#217	2
C₂×C₄×C₅×Dic₃	2
C₅×C₃⋊#95	2
C₄×C₅×C₂²⋊C₄	3
C₄×#217	5

Subgroups — C4×C5×#217 #11750708

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₄×C₅×#217 #11750708