Subgroups — C₃×C₅×#541 #11756630

Order 960.

384 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	C₃×C₅×C₄⋊Q₈	C₂	1	1	yes	maximal, characteristic, complemented
2	C₃×C₅×#126	C₂	1	4	yes	maximal
3	C₃×C₅×#125	C₂	1	1	yes	maximal, characteristic, complemented
4	C₃×C₅×#125	C₂	1	4	yes	maximal, complemented
5	C₃×C₅×C₂²⋊Q₈	C₂	1	2	yes	maximal
6	C₃×C₄×C₅×Q₈	C₂	1	2	yes	maximal, complemented
7	C₃×C₅×#117	C₂	1	1	yes	maximal, characteristic
8	C₅×#541	C₃	1	1	yes	maximal, characteristic, direct product factor, Hall
9	C₂×C₃×C₅×Q₈	C₂²	1	2	yes	—
10	C₂²×C₃×C₄×C₅	C₂²	1	1	yes	characteristic
11	C₃×C₅×C₄⋊C₄	C₂²	1	2	yes	—
12	C₃×C₅×C₄⋊C₄	C₂²	1	2	yes	—
13	C₃×C₅×C₄⋊C₄	C₂²	1	4	yes	—
14	C₃×C₅×C₄⋊C₄	C₂²	1	4	yes	—
15	C₃×C₅×C₄⋊C₄	C₂²	1	4	yes	—
16	C₃×C₅×C₄⋊C₄	C₂²	1	4	yes	—
17	C₃×C₅×C₂²⋊C₄	C₂²	1	2	yes	—
18	C₃×C₅×C₂²⋊C₄	C₂²	1	4	yes	—
19	C₃×C₄²×C₅	C₂²	1	1	yes	characteristic
20	C₃×C₄²×C₅	C₂²	1	1	yes	characteristic
21	C₃×C₄²×C₅	C₂²	1	4	yes	—
22	C₃×#541	C₅	1	1	yes	maximal, characteristic, direct product factor, Hall
23	C₅×C₄⋊Q₈	C₂×C₃	1	1	yes	characteristic, complemented
24	C₅×#126	C₂×C₃	1	4	yes	—
25	C₅×#125	C₂×C₃	1	1	yes	characteristic, complemented
26	C₅×#125	C₂×C₃	1	4	yes	complemented
27	C₅×C₂²⋊Q₈	C₂×C₃	1	2	yes	—
28	C₄×C₅×Q₈	C₂×C₃	1	2	yes	complemented
29	C₅×#117	C₂×C₃	1	1	yes	characteristic
30	C₂³×C₃×C₅	C₂³	1	1	yes	characteristic
31	C₃×C₅×Q₈	—	2	4	no	—
32	C₂×C₃×C₄×C₅	—	2	2	no	—
33	C₂×C₃×C₄×C₅	C₂³	1	2	yes	—
34	C₂×C₃×C₄×C₅	C₂³	1	2	yes	—
35	C₂×C₃×C₄×C₅	C₂³	1	4	yes	—
36	C₂×C₃×C₄×C₅	C₂³	1	4	yes	—
37	C₂×C₃×C₄×C₅	C₂³	1	1	yes	characteristic
38	C₂×C₃×C₄×C₅	C₂³	1	1	yes	characteristic
39	C₃×C₄⋊Q₈	C₂×C₅	1	1	yes	characteristic, complemented
40	C₃×#126	C₂×C₅	1	4	yes	—
41	C₃×#125	C₂×C₅	1	1	yes	characteristic, complemented
42	C₃×#125	C₂×C₅	1	4	yes	complemented
43	C₃×C₂²⋊Q₈	C₂×C₅	1	2	yes	—
44	C₃×C₄×Q₈	C₂×C₅	1	2	yes	complemented
45	C₃×#117	C₂×C₅	1	1	yes	characteristic
46	C₂×C₅×Q₈	C₂²×C₃	1	2	yes	—
47	C₂²×C₄×C₅	C₂²×C₃	1	1	yes	characteristic
48	C₅×C₄⋊C₄	C₂²×C₃	1	2	yes	—
49	C₅×C₄⋊C₄	C₂²×C₃	1	2	yes	—
50	C₅×C₄⋊C₄	C₂²×C₃	1	4	yes	—
51	C₅×C₄⋊C₄	C₂²×C₃	1	4	yes	—
52	C₅×C₄⋊C₄	C₂²×C₃	1	4	yes	—
53	C₅×C₄⋊C₄	C₂²×C₃	1	4	yes	—
54	C₅×C₂²⋊C₄	C₂²×C₃	1	2	yes	—
55	C₅×C₂²⋊C₄	C₂²×C₃	1	4	yes	—
56	C₄²×C₅	C₂²×C₃	1	1	yes	characteristic
57	C₄²×C₅	C₂²×C₃	1	1	yes	characteristic
58	C₄²×C₅	C₂²×C₃	1	4	yes	—
59	#541	C₃×C₅	1	1	yes	characteristic, direct product factor, Sylow
60	C₂²×C₃×C₅	—	2	1	no	—
61	C₂²×C₃×C₅	—	2	2	no	—
62	C₂²×C₃×C₅	C₂⁴	1	1	yes	characteristic
63	C₃×C₄×C₅	—	2	1	no	—
64	C₃×C₄×C₅	—	2	4	no	—
65	C₃×C₄×C₅	—	2	2	no	—
66	C₃×C₄×C₅	—	2	2	no	—
67	C₃×C₄×C₅	—	2	4	no	—
68	C₃×C₄×C₅	#41	1	2	yes	—
69	C₂×C₃×Q₈	C₂²×C₅	1	2	yes	—
70	C₂²×C₃×C₄	C₂²×C₅	1	1	yes	characteristic
71	C₃×C₄⋊C₄	C₂²×C₅	1	2	yes	—
72	C₃×C₄⋊C₄	C₂²×C₅	1	2	yes	—
73	C₃×C₄⋊C₄	C₂²×C₅	1	4	yes	—
74	C₃×C₄⋊C₄	C₂²×C₅	1	4	yes	—
75	C₃×C₄⋊C₄	C₂²×C₅	1	4	yes	—
76	C₃×C₄⋊C₄	C₂²×C₅	1	4	yes	—
77	C₃×C₂²⋊C₄	C₂²×C₅	1	2	yes	—
78	C₃×C₂²⋊C₄	C₂²×C₅	1	4	yes	—
79	C₃×C₄²	C₂²×C₅	1	1	yes	characteristic
80	C₃×C₄²	C₂²×C₅	1	1	yes	characteristic
81	C₃×C₄²	C₂²×C₅	1	4	yes	—
82	C₂³×C₅	C₂³×C₃	1	1	yes	characteristic
83	C₅×Q₈	—	2	4	no	—
84	C₂×C₄×C₅	—	2	2	no	—
85	C₂×C₄×C₅	C₂³×C₃	1	2	yes	—
86	C₂×C₄×C₅	C₂³×C₃	1	2	yes	—
87	C₂×C₄×C₅	C₂³×C₃	1	4	yes	—
88	C₂×C₄×C₅	C₂³×C₃	1	4	yes	—
89	C₂×C₄×C₅	C₂³×C₃	1	1	yes	characteristic
90	C₂×C₄×C₅	C₂³×C₃	1	1	yes	characteristic
91	C₄⋊Q₈	C₂×C₃×C₅	1	1	yes	characteristic, complemented
92	#126	C₂×C₃×C₅	1	4	yes	—
93	#125	C₂×C₃×C₅	1	1	yes	characteristic, complemented
94	#125	C₂×C₃×C₅	1	4	yes	complemented
95	C₂²⋊Q₈	C₂×C₃×C₅	1	2	yes	—
96	C₄×Q₈	C₂×C₃×C₅	1	2	yes	complemented
97	#117	C₂×C₃×C₅	1	1	yes	characteristic
98	C₂×C₃×C₅	—	4	1	no	—
99	C₂×C₃×C₅	C₂×#41	1	1	yes	characteristic
100	C₂×C₃×C₅	2₋¹⁺⁴	1	2	yes	—
101	C₂³×C₃	C₂³×C₅	1	1	yes	characteristic
102	C₃×Q₈	—	2	4	no	—
103	C₂×C₃×C₄	—	2	2	no	—
104	C₂×C₃×C₄	C₂³×C₅	1	2	yes	—
105	C₂×C₃×C₄	C₂³×C₅	1	2	yes	—
106	C₂×C₃×C₄	C₂³×C₅	1	4	yes	—
107	C₂×C₃×C₄	C₂³×C₅	1	4	yes	—
108	C₂×C₃×C₄	C₂³×C₅	1	1	yes	characteristic
109	C₂×C₃×C₄	C₂³×C₅	1	1	yes	characteristic
110	C₂²×C₅	—	2	1	no	—
111	C₂²×C₅	—	2	2	no	—
112	C₂²×C₅	C₂⁴×C₃	1	1	yes	characteristic
113	C₄×C₅	—	2	1	no	—
114	C₄×C₅	—	2	4	no	—
115	C₄×C₅	—	2	2	no	—
116	C₄×C₅	—	2	2	no	—
117	C₄×C₅	—	2	4	no	—
118	C₄×C₅	C₃×#41	1	2	yes	—
119	C₂×Q₈	C₂²×C₃×C₅	1	2	yes	—
120	C₂²×C₄	C₂²×C₃×C₅	1	1	yes	characteristic
121	C₄⋊C₄	C₂²×C₃×C₅	1	2	yes	—
122	C₄⋊C₄	C₂²×C₃×C₅	1	2	yes	—
123	C₄⋊C₄	C₂²×C₃×C₅	1	4	yes	—
124	C₄⋊C₄	C₂²×C₃×C₅	1	4	yes	—
125	C₄⋊C₄	C₂²×C₃×C₅	1	4	yes	—
126	C₄⋊C₄	C₂²×C₃×C₅	1	4	yes	—
127	C₂²⋊C₄	C₂²×C₃×C₅	1	2	yes	—
128	C₂²⋊C₄	C₂²×C₃×C₅	1	4	yes	—
129	C₄²	C₂²×C₃×C₅	1	1	yes	characteristic
130	C₄²	C₂²×C₃×C₅	1	1	yes	characteristic
131	C₄²	C₂²×C₃×C₅	1	4	yes	—
132	C₃×C₅	#541	1	1	yes	characteristic, direct product factor, Hall
133	C₂²×C₃	—	2	1	no	—
134	C₂²×C₃	—	2	2	no	—
135	C₂²×C₃	C₂⁴×C₅	1	1	yes	characteristic
136	C₃×C₄	—	2	1	no	—
137	C₃×C₄	—	2	4	no	—
138	C₃×C₄	—	2	2	no	—
139	C₃×C₄	—	2	2	no	—
140	C₃×C₄	—	2	4	no	—
141	C₃×C₄	C₅×#41	1	2	yes	—
142	C₂×C₅	—	4	1	no	—
143	C₂×C₅	C₂×C₃×#41	1	1	yes	characteristic
144	C₂×C₅	C₃×2₋¹⁺⁴	1	2	yes	—
145	C₂³	C₂³×C₃×C₅	1	1	yes	characteristic
146	Q₈	—	2	4	no	—
147	C₂×C₄	—	2	2	no	—
148	C₂×C₄	C₂³×C₃×C₅	1	2	yes	—
149	C₂×C₄	C₂³×C₃×C₅	1	2	yes	—
150	C₂×C₄	C₂³×C₃×C₅	1	4	yes	—
151	C₂×C₄	C₂³×C₃×C₅	1	4	yes	—
152	C₂×C₄	C₂³×C₃×C₅	1	1	yes	characteristic
153	C₂×C₄	C₂³×C₃×C₅	1	1	yes	characteristic
154	C₂×C₃	—	4	1	no	—
155	C₂×C₃	C₂×C₅×#41	1	1	yes	characteristic
156	C₂×C₃	C₅×2₋¹⁺⁴	1	2	yes	—
157	C₅	C₃×#541	1	1	yes	characteristic, direct product factor, Sylow
158	C₂²	—	2	1	no	—
159	C₂²	—	2	2	no	—
160	C₂²	C₂⁴×C₃×C₅	1	1	yes	characteristic
161	C₄	—	2	1	no	—
162	C₄	—	2	4	no	—
163	C₄	—	2	2	no	—
164	C₄	—	2	2	no	—
165	C₄	—	2	4	no	—
166	C₄	C₃×C₅×#41	1	2	yes	—
167	C₃	C₅×#541	1	1	yes	characteristic, direct product factor, Sylow
168	C₂	—	4	1	no	—
169	C₂	C₂×C₃×C₅×#41	1	1	yes	characteristic
170	C₂	C₃×C₅×2₋¹⁺⁴	1	2	yes	—

	subgroup	G/H
center, socle	C₂²×C₃×C₅	C₂⁴
commutator, Frattini, cosocle	C₂²	C₂⁴×C₃×C₅
prime radical	C₂³×C₃×C₅	C₂³
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. C₃×C₅×C₄⋊Q₈	C₂	semidirect
3. C₃×C₅×#125	C₂	semidirect
4. C₃×C₅×#125	C₂	semidirect
6. C₃×C₄×C₅×Q₈	C₂	semidirect
8. C₅×#541	C₃	direct (N × H)
22. C₃×#541	C₅	direct (N × H)
23. C₅×C₄⋊Q₈	C₂×C₃	semidirect
25. C₅×#125	C₂×C₃	semidirect
26. C₅×#125	C₂×C₃	semidirect
28. C₄×C₅×Q₈	C₂×C₃	semidirect
39. C₃×C₄⋊Q₈	C₂×C₅	semidirect
41. C₃×#125	C₂×C₅	semidirect
42. C₃×#125	C₂×C₅	semidirect
44. C₃×C₄×Q₈	C₂×C₅	semidirect
59. #541	C₃×C₅	direct (N × H)
91. C₄⋊Q₈	C₂×C₃×C₅	semidirect
93. #125	C₂×C₃×C₅	semidirect
94. #125	C₂×C₃×C₅	semidirect
96. C₄×Q₈	C₂×C₃×C₅	semidirect
132. C₃×C₅	#541	direct (N × H)
157. C₅	C₃×#541	direct (N × H)
167. C₃	C₅×#541	direct (N × H)

Maximal subgroups

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

subgroup	index
C₃×C₅×C₄⋊Q₈	2
C₃×C₅×#126	2
C₃×C₅×#125	2
C₃×C₅×C₂²⋊Q₈	2
C₃×C₄×C₅×Q₈	2
C₃×C₅×#117	2
C₅×#541	3
C₃×#541	5

Subgroups — C3×C5×#541 #11756630

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₃×C₅×#541 #11756630