Subgroups — C₃×C₅×#550 #11756639

Order 960.

624 conjugacy classes of subgroups.

The subgroup lattice has 156 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×C₃×C₅×#41	C₂	1	2	yes	maximal, complemented
2	C₃×C₅×#127	C₂	1	3	yes	maximal, complemented
3	C₃×C₅×#121	C₂	1	6	yes	maximal, complemented
4	C₃×C₄×C₅×Q₈	C₂	1	1	yes	maximal, characteristic, complemented
5	C₃×C₄×C₅×D₄	C₂	1	3	yes	maximal, complemented
6	C₅×#550	C₃	1	1	yes	maximal, characteristic, direct product factor, Hall
7	C₃×C₅×#41	—	2	8	no	—
8	C₂×C₃×C₅×Q₈	C₂²	1	1	yes	characteristic
9	C₂×C₃×C₅×D₄	C₂²	1	3	yes	—
10	C₂×C₃×C₅×D₄	C₂²	1	6	yes	—
11	C₂×C₃×C₅×D₄	C₂²	1	6	yes	—
12	C₂²×C₃×C₄×C₅	C₂²	1	6	yes	—
13	C₃×C₅×C₄⋊C₄	C₂²	1	1	yes	characteristic
14	C₃×C₅×C₄⋊C₄	C₂²	1	3	yes	—
15	C₃×C₅×C₂²⋊C₄	C₂²	1	6	yes	—
16	C₃×C₄²×C₅	C₂²	1	3	yes	—
17	C₃×#550	C₅	1	1	yes	maximal, characteristic, direct product factor, Hall
18	C₂×C₅×#41	C₂×C₃	1	2	yes	complemented
19	C₅×#127	C₂×C₃	1	3	yes	complemented
20	C₅×#121	C₂×C₃	1	6	yes	complemented
21	C₄×C₅×Q₈	C₂×C₃	1	1	yes	characteristic, complemented
22	C₄×C₅×D₄	C₂×C₃	1	3	yes	complemented
23	C₂³×C₃×C₅	C₂³	1	6	yes	—
24	C₃×C₅×Q₈	D₄	1	4	yes	complemented
25	C₃×C₅×D₄	—	2	12	no	—
26	C₃×C₅×D₄	—	2	6	no	—
27	C₃×C₅×D₄	—	4	6	no	—
28	C₂×C₃×C₄×C₅	—	2	12	no	—
29	C₂×C₃×C₄×C₅	C₂³	1	3	yes	—
30	C₂×C₃×C₄×C₅	C₂³	1	2	yes	—
31	C₂×C₃×C₄×C₅	C₂³	1	3	yes	—
32	C₂×C₃×C₄×C₅	C₂³	1	1	yes	characteristic
33	C₂×C₃×#41	C₂×C₅	1	2	yes	complemented
34	C₃×#127	C₂×C₅	1	3	yes	complemented
35	C₃×#121	C₂×C₅	1	6	yes	complemented
36	C₃×C₄×Q₈	C₂×C₅	1	1	yes	characteristic, complemented
37	C₃×C₄×D₄	C₂×C₅	1	3	yes	complemented
38	C₅×#41	—	2	8	no	—
39	C₂×C₅×Q₈	C₂²×C₃	1	1	yes	characteristic
40	C₂×C₅×D₄	C₂²×C₃	1	3	yes	—
41	C₂×C₅×D₄	C₂²×C₃	1	6	yes	—
42	C₂×C₅×D₄	C₂²×C₃	1	6	yes	—
43	C₂²×C₄×C₅	C₂²×C₃	1	6	yes	—
44	C₅×C₄⋊C₄	C₂²×C₃	1	1	yes	characteristic
45	C₅×C₄⋊C₄	C₂²×C₃	1	3	yes	—
46	C₅×C₂²⋊C₄	C₂²×C₃	1	6	yes	—
47	C₄²×C₅	C₂²×C₃	1	3	yes	—
48	#550	C₃×C₅	1	1	yes	characteristic, direct product factor, Sylow
49	C₂²×C₃×C₅	—	2	6	no	—
50	C₂²×C₃×C₅	—	2	6	no	—
51	C₂²×C₃×C₅	—	2	6	no	—
52	C₂²×C₃×C₅	C₂⁴	1	1	yes	characteristic
53	C₃×C₄×C₅	—	2	2	no	—
54	C₃×C₄×C₅	—	2	3	no	—
55	C₃×C₄×C₅	C₂×D₄	1	6	yes	—
56	C₃×C₄×C₅	#41	1	2	yes	complemented
57	C₃×#41	—	2	8	no	—
58	C₂×C₃×Q₈	C₂²×C₅	1	1	yes	characteristic
59	C₂×C₃×D₄	C₂²×C₅	1	3	yes	—
60	C₂×C₃×D₄	C₂²×C₅	1	6	yes	—
61	C₂×C₃×D₄	C₂²×C₅	1	6	yes	—
62	C₂²×C₃×C₄	C₂²×C₅	1	6	yes	—
63	C₃×C₄⋊C₄	C₂²×C₅	1	1	yes	characteristic
64	C₃×C₄⋊C₄	C₂²×C₅	1	3	yes	—
65	C₃×C₂²⋊C₄	C₂²×C₅	1	6	yes	—
66	C₃×C₄²	C₂²×C₅	1	3	yes	—
67	C₂³×C₅	C₂³×C₃	1	6	yes	—
68	C₅×Q₈	C₃×D₄	1	4	yes	complemented
69	C₅×D₄	—	2	12	no	—
70	C₅×D₄	—	2	6	no	—
71	C₅×D₄	—	4	6	no	—
72	C₂×C₄×C₅	—	2	12	no	—
73	C₂×C₄×C₅	C₂³×C₃	1	3	yes	—
74	C₂×C₄×C₅	C₂³×C₃	1	2	yes	—
75	C₂×C₄×C₅	C₂³×C₃	1	3	yes	—
76	C₂×C₄×C₅	C₂³×C₃	1	1	yes	characteristic
77	C₂×#41	C₂×C₃×C₅	1	2	yes	complemented
78	#127	C₂×C₃×C₅	1	3	yes	complemented
79	#121	C₂×C₃×C₅	1	6	yes	complemented
80	C₄×Q₈	C₂×C₃×C₅	1	1	yes	characteristic, complemented
81	C₄×D₄	C₂×C₃×C₅	1	3	yes	complemented
82	C₂×C₃×C₅	—	4	6	no	—
83	C₂×C₃×C₅	C₂²×D₄	1	1	yes	characteristic
84	C₂×C₃×C₅	C₂×#41	1	1	yes	characteristic
85	C₂×C₃×C₅	H(5,𝔽₂)	1	1	yes	characteristic
86	C₂³×C₃	C₂³×C₅	1	6	yes	—
87	C₃×Q₈	C₅×D₄	1	4	yes	complemented
88	C₃×D₄	—	2	12	no	—
89	C₃×D₄	—	2	6	no	—
90	C₃×D₄	—	4	6	no	—
91	C₂×C₃×C₄	—	2	12	no	—
92	C₂×C₃×C₄	C₂³×C₅	1	3	yes	—
93	C₂×C₃×C₄	C₂³×C₅	1	2	yes	—
94	C₂×C₃×C₄	C₂³×C₅	1	3	yes	—
95	C₂×C₃×C₄	C₂³×C₅	1	1	yes	characteristic
96	C₂²×C₅	—	2	6	no	—
97	C₂²×C₅	—	2	6	no	—
98	C₂²×C₅	—	2	6	no	—
99	C₂²×C₅	C₂⁴×C₃	1	1	yes	characteristic
100	C₄×C₅	—	2	2	no	—
101	C₄×C₅	—	2	3	no	—
102	C₄×C₅	C₂×C₃×D₄	1	6	yes	—
103	C₄×C₅	C₃×#41	1	2	yes	complemented
104	#41	—	2	8	no	—
105	C₂×Q₈	C₂²×C₃×C₅	1	1	yes	characteristic
106	C₂×D₄	C₂²×C₃×C₅	1	3	yes	—
107	C₂×D₄	C₂²×C₃×C₅	1	6	yes	—
108	C₂×D₄	C₂²×C₃×C₅	1	6	yes	—
109	C₂²×C₄	C₂²×C₃×C₅	1	6	yes	—
110	C₄⋊C₄	C₂²×C₃×C₅	1	1	yes	characteristic
111	C₄⋊C₄	C₂²×C₃×C₅	1	3	yes	—
112	C₂²⋊C₄	C₂²×C₃×C₅	1	6	yes	—
113	C₄²	C₂²×C₃×C₅	1	3	yes	—
114	C₃×C₅	#550	1	1	yes	characteristic, direct product factor, Hall
115	C₂²×C₃	—	2	6	no	—
116	C₂²×C₃	—	2	6	no	—
117	C₂²×C₃	—	2	6	no	—
118	C₂²×C₃	C₂⁴×C₅	1	1	yes	characteristic
119	C₃×C₄	—	2	2	no	—
120	C₃×C₄	—	2	3	no	—
121	C₃×C₄	C₂×C₅×D₄	1	6	yes	—
122	C₃×C₄	C₅×#41	1	2	yes	complemented
123	C₂×C₅	—	4	6	no	—
124	C₂×C₅	C₂²×C₃×D₄	1	1	yes	characteristic
125	C₂×C₅	C₂×C₃×#41	1	1	yes	characteristic
126	C₂×C₅	C₃×H(5,𝔽₂)	1	1	yes	characteristic
127	C₂³	C₂³×C₃×C₅	1	6	yes	—
128	Q₈	C₃×C₅×D₄	1	4	yes	complemented
129	D₄	—	2	12	no	—
130	D₄	—	2	6	no	—
131	D₄	—	4	6	no	—
132	C₂×C₄	—	2	12	no	—
133	C₂×C₄	C₂³×C₃×C₅	1	3	yes	—
134	C₂×C₄	C₂³×C₃×C₅	1	2	yes	—
135	C₂×C₄	C₂³×C₃×C₅	1	3	yes	—
136	C₂×C₄	C₂³×C₃×C₅	1	1	yes	characteristic
137	C₂×C₃	—	4	6	no	—
138	C₂×C₃	C₂²×C₅×D₄	1	1	yes	characteristic
139	C₂×C₃	C₂×C₅×#41	1	1	yes	characteristic
140	C₂×C₃	C₅×H(5,𝔽₂)	1	1	yes	characteristic
141	C₅	C₃×#550	1	1	yes	characteristic, direct product factor, Sylow
142	C₂²	—	2	6	no	—
143	C₂²	—	2	6	no	—
144	C₂²	—	2	6	no	—
145	C₂²	C₂⁴×C₃×C₅	1	1	yes	characteristic
146	C₄	—	2	2	no	—
147	C₄	—	2	3	no	—
148	C₄	C₂×C₃×C₅×D₄	1	6	yes	—
149	C₄	C₃×C₅×#41	1	2	yes	complemented
150	C₃	C₅×#550	1	1	yes	characteristic, direct product factor, Sylow
151	C₂	—	4	6	no	—
152	C₂	C₂²×C₃×C₅×D₄	1	1	yes	characteristic
153	C₂	C₂×C₃×C₅×#41	1	1	yes	characteristic
154	C₂	C₃×C₅×H(5,𝔽₂)	1	1	yes	characteristic

	subgroup	G/H
center, socle	C₂²×C₃×C₅	C₂⁴
commutator, Frattini, cosocle	C₂²	C₂⁴×C₃×C₅
LCS	G ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂²×C₃×C₅ ◁ G

Complements

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

N	H ≅ G/N	type
1. C₂×C₃×C₅×#41	C₂	semidirect
2. C₃×C₅×#127	C₂	semidirect
3. C₃×C₅×#121	C₂	semidirect
4. C₃×C₄×C₅×Q₈	C₂	semidirect
5. C₃×C₄×C₅×D₄	C₂	semidirect
6. C₅×#550	C₃	direct (N × H)
17. C₃×#550	C₅	direct (N × H)
18. C₂×C₅×#41	C₂×C₃	semidirect
19. C₅×#127	C₂×C₃	semidirect
20. C₅×#121	C₂×C₃	semidirect
21. C₄×C₅×Q₈	C₂×C₃	semidirect
22. C₄×C₅×D₄	C₂×C₃	semidirect
24. C₃×C₅×Q₈	D₄	semidirect
33. C₂×C₃×#41	C₂×C₅	semidirect
34. C₃×#127	C₂×C₅	semidirect
35. C₃×#121	C₂×C₅	semidirect
36. C₃×C₄×Q₈	C₂×C₅	semidirect
37. C₃×C₄×D₄	C₂×C₅	semidirect
48. #550	C₃×C₅	direct (N × H)
56. C₃×C₄×C₅	#41	semidirect
68. C₅×Q₈	C₃×D₄	semidirect
77. C₂×#41	C₂×C₃×C₅	semidirect
78. #127	C₂×C₃×C₅	semidirect
79. #121	C₂×C₃×C₅	semidirect
80. C₄×Q₈	C₂×C₃×C₅	semidirect
81. C₄×D₄	C₂×C₃×C₅	semidirect
87. C₃×Q₈	C₅×D₄	semidirect
103. C₄×C₅	C₃×#41	semidirect
114. C₃×C₅	#550	direct (N × H)
122. C₃×C₄	C₅×#41	semidirect
128. Q₈	C₃×C₅×D₄	semidirect
141. C₅	C₃×#550	direct (N × H)
149. C₄	C₃×C₅×#41	semidirect
150. C₃	C₅×#550	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×C₃×C₅×#41	2
C₃×C₅×#127	2
C₃×C₅×#121	2
C₃×C₄×C₅×Q₈	2
C₃×C₄×C₅×D₄	2
C₅×#550	3
C₃×#550	5

Subgroups — C3×C5×#550 #11756639

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₃×C₅×#550 #11756639