Subgroups — C₃×C₅×#556 #11756645

Order 960.

424 conjugacy classes of subgroups.

The subgroup lattice has 212 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, complemented
3	C₃×C₅×#124	C₂	1	2	yes	maximal
4	C₃×C₅×C₂²⋊Q₈	C₂	1	2	yes	maximal, complemented
5	C₃×C₄×C₅×Q₈	C₂	1	2	yes	maximal, complemented
6	C₃×C₄×C₅×Q₈	C₂	1	1	yes	maximal, characteristic, complemented
7	C₃×C₄×C₅×D₄	C₂	1	1	yes	maximal, characteristic
8	C₃×C₅×#117	C₂	1	2	yes	maximal, complemented
9	C₅×#556	C₃	1	1	yes	maximal, characteristic, direct product factor, Hall
10	C₂×C₃×C₅×Q₈	C₂²	1	1	yes	characteristic
11	C₂×C₃×C₅×Q₈	C₂²	1	2	yes	—
12	C₂×C₃×C₅×D₄	C₂²	1	1	yes	characteristic
13	C₂²×C₃×C₄×C₅	C₂²	1	2	yes	—
14	C₃×C₅×C₄⋊C₄	C₂²	1	1	yes	characteristic
15	C₃×C₅×C₄⋊C₄	C₂²	1	2	yes	—
16	C₃×C₅×C₄⋊C₄	C₂²	1	1	yes	characteristic
17	C₃×C₅×C₄⋊C₄	C₂²	1	4	yes	—
18	C₃×C₅×C₄⋊C₄	C₂²	1	2	yes	—
19	C₃×C₅×C₄⋊C₄	C₂²	1	2	yes	—
20	C₃×C₅×C₂²⋊C₄	C₂²	1	4	yes	—
21	C₃×C₅×C₂²⋊C₄	C₂²	1	2	yes	—
22	C₃×C₅×C₂²⋊C₄	C₂²	1	4	yes	—
23	C₃×C₄²×C₅	C₂²	1	4	yes	—
24	C₃×C₄²×C₅	C₂²	1	1	yes	characteristic
25	C₃×C₄²×C₅	C₂²	1	2	yes	—
26	C₃×#556	C₅	1	1	yes	maximal, characteristic, direct product factor, Hall
27	C₅×C₄⋊Q₈	C₂×C₃	1	1	yes	characteristic, complemented
28	C₅×#126	C₂×C₃	1	4	yes	complemented
29	C₅×#124	C₂×C₃	1	2	yes	—
30	C₅×C₂²⋊Q₈	C₂×C₃	1	2	yes	complemented
31	C₄×C₅×Q₈	C₂×C₃	1	2	yes	complemented
32	C₄×C₅×Q₈	C₂×C₃	1	1	yes	characteristic, complemented
33	C₄×C₅×D₄	C₂×C₃	1	1	yes	characteristic
34	C₅×#117	C₂×C₃	1	2	yes	complemented
35	C₂³×C₃×C₅	C₂³	1	2	yes	—
36	C₃×C₅×Q₈	—	2	2	no	—
37	C₃×C₅×Q₈	—	2	4	no	—
38	C₃×C₅×D₄	—	2	2	no	—
39	C₂×C₃×C₄×C₅	—	2	4	no	—
40	C₂×C₃×C₄×C₅	C₂³	1	2	yes	—
41	C₂×C₃×C₄×C₅	C₂³	1	2	yes	—
42	C₂×C₃×C₄×C₅	C₂³	1	1	yes	characteristic
43	C₂×C₃×C₄×C₅	C₂³	1	4	yes	—
44	C₂×C₃×C₄×C₅	C₂³	1	2	yes	—
45	C₂×C₃×C₄×C₅	C₂³	1	1	yes	characteristic
46	C₂×C₃×C₄×C₅	C₂³	1	1	yes	characteristic
47	C₃×C₄⋊Q₈	C₂×C₅	1	1	yes	characteristic, complemented
48	C₃×#126	C₂×C₅	1	4	yes	complemented
49	C₃×#124	C₂×C₅	1	2	yes	—
50	C₃×C₂²⋊Q₈	C₂×C₅	1	2	yes	complemented
51	C₃×C₄×Q₈	C₂×C₅	1	2	yes	complemented
52	C₃×C₄×Q₈	C₂×C₅	1	1	yes	characteristic, complemented
53	C₃×C₄×D₄	C₂×C₅	1	1	yes	characteristic
54	C₃×#117	C₂×C₅	1	2	yes	complemented
55	C₂×C₅×Q₈	C₂²×C₃	1	1	yes	characteristic
56	C₂×C₅×Q₈	C₂²×C₃	1	2	yes	—
57	C₂×C₅×D₄	C₂²×C₃	1	1	yes	characteristic
58	C₂²×C₄×C₅	C₂²×C₃	1	2	yes	—
59	C₅×C₄⋊C₄	C₂²×C₃	1	1	yes	characteristic
60	C₅×C₄⋊C₄	C₂²×C₃	1	2	yes	—
61	C₅×C₄⋊C₄	C₂²×C₃	1	1	yes	characteristic
62	C₅×C₄⋊C₄	C₂²×C₃	1	4	yes	—
63	C₅×C₄⋊C₄	C₂²×C₃	1	2	yes	—
64	C₅×C₄⋊C₄	C₂²×C₃	1	2	yes	—
65	C₅×C₂²⋊C₄	C₂²×C₃	1	4	yes	—
66	C₅×C₂²⋊C₄	C₂²×C₃	1	2	yes	—
67	C₅×C₂²⋊C₄	C₂²×C₃	1	4	yes	—
68	C₄²×C₅	C₂²×C₃	1	4	yes	—
69	C₄²×C₅	C₂²×C₃	1	1	yes	characteristic
70	C₄²×C₅	C₂²×C₃	1	2	yes	—
71	#556	C₃×C₅	1	1	yes	characteristic, direct product factor, Sylow
72	C₂²×C₃×C₅	—	2	2	no	—
73	C₂²×C₃×C₅	—	2	2	no	—
74	C₂²×C₃×C₅	—	2	2	no	—
75	C₂²×C₃×C₅	C₂⁴	1	1	yes	characteristic
76	C₃×C₄×C₅	—	2	2	no	—
77	C₃×C₄×C₅	—	2	2	no	—
78	C₃×C₄×C₅	—	2	4	no	—
79	C₃×C₄×C₅	—	2	2	no	—
80	C₃×C₄×C₅	—	2	1	no	—
81	C₃×C₄×C₅	#41	1	2	yes	—
82	C₃×C₄×C₅	#41	1	2	yes	—
83	C₂×C₃×Q₈	C₂²×C₅	1	1	yes	characteristic
84	C₂×C₃×Q₈	C₂²×C₅	1	2	yes	—
85	C₂×C₃×D₄	C₂²×C₅	1	1	yes	characteristic
86	C₂²×C₃×C₄	C₂²×C₅	1	2	yes	—
87	C₃×C₄⋊C₄	C₂²×C₅	1	1	yes	characteristic
88	C₃×C₄⋊C₄	C₂²×C₅	1	2	yes	—
89	C₃×C₄⋊C₄	C₂²×C₅	1	1	yes	characteristic
90	C₃×C₄⋊C₄	C₂²×C₅	1	4	yes	—
91	C₃×C₄⋊C₄	C₂²×C₅	1	2	yes	—
92	C₃×C₄⋊C₄	C₂²×C₅	1	2	yes	—
93	C₃×C₂²⋊C₄	C₂²×C₅	1	4	yes	—
94	C₃×C₂²⋊C₄	C₂²×C₅	1	2	yes	—
95	C₃×C₂²⋊C₄	C₂²×C₅	1	4	yes	—
96	C₃×C₄²	C₂²×C₅	1	4	yes	—
97	C₃×C₄²	C₂²×C₅	1	1	yes	characteristic
98	C₃×C₄²	C₂²×C₅	1	2	yes	—
99	C₂³×C₅	C₂³×C₃	1	2	yes	—
100	C₅×Q₈	—	2	2	no	—
101	C₅×Q₈	—	2	4	no	—
102	C₅×D₄	—	2	2	no	—
103	C₂×C₄×C₅	—	2	4	no	—
104	C₂×C₄×C₅	C₂³×C₃	1	2	yes	—
105	C₂×C₄×C₅	C₂³×C₃	1	2	yes	—
106	C₂×C₄×C₅	C₂³×C₃	1	1	yes	characteristic
107	C₂×C₄×C₅	C₂³×C₃	1	4	yes	—
108	C₂×C₄×C₅	C₂³×C₃	1	2	yes	—
109	C₂×C₄×C₅	C₂³×C₃	1	1	yes	characteristic
110	C₂×C₄×C₅	C₂³×C₃	1	1	yes	characteristic
111	C₄⋊Q₈	C₂×C₃×C₅	1	1	yes	characteristic, complemented
112	#126	C₂×C₃×C₅	1	4	yes	complemented
113	#124	C₂×C₃×C₅	1	2	yes	—
114	C₂²⋊Q₈	C₂×C₃×C₅	1	2	yes	complemented
115	C₄×Q₈	C₂×C₃×C₅	1	2	yes	complemented
116	C₄×Q₈	C₂×C₃×C₅	1	1	yes	characteristic, complemented
117	C₄×D₄	C₂×C₃×C₅	1	1	yes	characteristic
118	#117	C₂×C₃×C₅	1	2	yes	complemented
119	C₂×C₃×C₅	—	4	2	no	—
120	C₂×C₃×C₅	C₂×#41	1	1	yes	characteristic
121	C₂×C₃×C₅	C₂×#41	1	1	yes	characteristic
122	C₂×C₃×C₅	2₋¹⁺⁴	1	1	yes	characteristic
123	C₂³×C₃	C₂³×C₅	1	2	yes	—
124	C₃×Q₈	—	2	2	no	—
125	C₃×Q₈	—	2	4	no	—
126	C₃×D₄	—	2	2	no	—
127	C₂×C₃×C₄	—	2	4	no	—
128	C₂×C₃×C₄	C₂³×C₅	1	2	yes	—
129	C₂×C₃×C₄	C₂³×C₅	1	2	yes	—
130	C₂×C₃×C₄	C₂³×C₅	1	1	yes	characteristic
131	C₂×C₃×C₄	C₂³×C₅	1	4	yes	—
132	C₂×C₃×C₄	C₂³×C₅	1	2	yes	—
133	C₂×C₃×C₄	C₂³×C₅	1	1	yes	characteristic
134	C₂×C₃×C₄	C₂³×C₅	1	1	yes	characteristic
135	C₂²×C₅	—	2	2	no	—
136	C₂²×C₅	—	2	2	no	—
137	C₂²×C₅	—	2	2	no	—
138	C₂²×C₅	C₂⁴×C₃	1	1	yes	characteristic
139	C₄×C₅	—	2	2	no	—
140	C₄×C₅	—	2	2	no	—
141	C₄×C₅	—	2	4	no	—
142	C₄×C₅	—	2	2	no	—
143	C₄×C₅	—	2	1	no	—
144	C₄×C₅	C₃×#41	1	2	yes	—
145	C₄×C₅	C₃×#41	1	2	yes	—
146	C₂×Q₈	C₂²×C₃×C₅	1	1	yes	characteristic
147	C₂×Q₈	C₂²×C₃×C₅	1	2	yes	—
148	C₂×D₄	C₂²×C₃×C₅	1	1	yes	characteristic
149	C₂²×C₄	C₂²×C₃×C₅	1	2	yes	—
150	C₄⋊C₄	C₂²×C₃×C₅	1	1	yes	characteristic
151	C₄⋊C₄	C₂²×C₃×C₅	1	2	yes	—
152	C₄⋊C₄	C₂²×C₃×C₅	1	1	yes	characteristic
153	C₄⋊C₄	C₂²×C₃×C₅	1	4	yes	—
154	C₄⋊C₄	C₂²×C₃×C₅	1	2	yes	—
155	C₄⋊C₄	C₂²×C₃×C₅	1	2	yes	—
156	C₂²⋊C₄	C₂²×C₃×C₅	1	4	yes	—
157	C₂²⋊C₄	C₂²×C₃×C₅	1	2	yes	—
158	C₂²⋊C₄	C₂²×C₃×C₅	1	4	yes	—
159	C₄²	C₂²×C₃×C₅	1	4	yes	—
160	C₄²	C₂²×C₃×C₅	1	1	yes	characteristic
161	C₄²	C₂²×C₃×C₅	1	2	yes	—
162	C₃×C₅	#556	1	1	yes	characteristic, direct product factor, Hall
163	C₂²×C₃	—	2	2	no	—
164	C₂²×C₃	—	2	2	no	—
165	C₂²×C₃	—	2	2	no	—
166	C₂²×C₃	C₂⁴×C₅	1	1	yes	characteristic
167	C₃×C₄	—	2	2	no	—
168	C₃×C₄	—	2	2	no	—
169	C₃×C₄	—	2	4	no	—
170	C₃×C₄	—	2	2	no	—
171	C₃×C₄	—	2	1	no	—
172	C₃×C₄	C₅×#41	1	2	yes	—
173	C₃×C₄	C₅×#41	1	2	yes	—
174	C₂×C₅	—	4	2	no	—
175	C₂×C₅	C₂×C₃×#41	1	1	yes	characteristic
176	C₂×C₅	C₂×C₃×#41	1	1	yes	characteristic
177	C₂×C₅	C₃×2₋¹⁺⁴	1	1	yes	characteristic
178	C₂³	C₂³×C₃×C₅	1	2	yes	—
179	Q₈	—	2	2	no	—
180	Q₈	—	2	4	no	—
181	D₄	—	2	2	no	—
182	C₂×C₄	—	2	4	no	—
183	C₂×C₄	C₂³×C₃×C₅	1	2	yes	—
184	C₂×C₄	C₂³×C₃×C₅	1	2	yes	—
185	C₂×C₄	C₂³×C₃×C₅	1	1	yes	characteristic
186	C₂×C₄	C₂³×C₃×C₅	1	4	yes	—
187	C₂×C₄	C₂³×C₃×C₅	1	2	yes	—
188	C₂×C₄	C₂³×C₃×C₅	1	1	yes	characteristic
189	C₂×C₄	C₂³×C₃×C₅	1	1	yes	characteristic
190	C₂×C₃	—	4	2	no	—
191	C₂×C₃	C₂×C₅×#41	1	1	yes	characteristic
192	C₂×C₃	C₂×C₅×#41	1	1	yes	characteristic
193	C₂×C₃	C₅×2₋¹⁺⁴	1	1	yes	characteristic
194	C₅	C₃×#556	1	1	yes	characteristic, direct product factor, Sylow
195	C₂²	—	2	2	no	—
196	C₂²	—	2	2	no	—
197	C₂²	—	2	2	no	—
198	C₂²	C₂⁴×C₃×C₅	1	1	yes	characteristic
199	C₄	—	2	2	no	—
200	C₄	—	2	2	no	—
201	C₄	—	2	4	no	—
202	C₄	—	2	2	no	—
203	C₄	—	2	1	no	—
204	C₄	C₃×C₅×#41	1	2	yes	—
205	C₄	C₃×C₅×#41	1	2	yes	—
206	C₃	C₅×#556	1	1	yes	characteristic, direct product factor, Sylow
207	C₂	—	4	2	no	—
208	C₂	C₂×C₃×C₅×#41	1	1	yes	characteristic
209	C₂	C₂×C₃×C₅×#41	1	1	yes	characteristic
210	C₂	C₃×C₅×2₋¹⁺⁴	1	1	yes	characteristic

	subgroup	G/H
center, socle	C₂²×C₃×C₅	C₂⁴
commutator, Frattini, cosocle	C₂²	C₂⁴×C₃×C₅
prime radical	C₂×C₃×C₅×D₄	C₂²
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
2. C₃×C₅×#126	C₂	semidirect
4. C₃×C₅×C₂²⋊Q₈	C₂	semidirect
5. C₃×C₄×C₅×Q₈	C₂	semidirect
6. C₃×C₄×C₅×Q₈	C₂	semidirect
8. C₃×C₅×#117	C₂	semidirect
9. C₅×#556	C₃	direct (N × H)
26. C₃×#556	C₅	direct (N × H)
27. C₅×C₄⋊Q₈	C₂×C₃	semidirect
28. C₅×#126	C₂×C₃	semidirect
30. C₅×C₂²⋊Q₈	C₂×C₃	semidirect
31. C₄×C₅×Q₈	C₂×C₃	semidirect
32. C₄×C₅×Q₈	C₂×C₃	semidirect
34. C₅×#117	C₂×C₃	semidirect
47. C₃×C₄⋊Q₈	C₂×C₅	semidirect
48. C₃×#126	C₂×C₅	semidirect
50. C₃×C₂²⋊Q₈	C₂×C₅	semidirect
51. C₃×C₄×Q₈	C₂×C₅	semidirect
52. C₃×C₄×Q₈	C₂×C₅	semidirect
54. C₃×#117	C₂×C₅	semidirect
71. #556	C₃×C₅	direct (N × H)
111. C₄⋊Q₈	C₂×C₃×C₅	semidirect
112. #126	C₂×C₃×C₅	semidirect
114. C₂²⋊Q₈	C₂×C₃×C₅	semidirect
115. C₄×Q₈	C₂×C₃×C₅	semidirect
116. C₄×Q₈	C₂×C₃×C₅	semidirect
118. #117	C₂×C₃×C₅	semidirect
162. C₃×C₅	#556	direct (N × H)
194. C₅	C₃×#556	direct (N × H)
206. C₃	C₅×#556	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×#124	2
C₃×C₅×C₂²⋊Q₈	2
C₃×C₄×C₅×Q₈	2
C₃×C₄×C₅×D₄	2
C₃×C₅×#117	2
C₅×#556	3
C₃×#556	5

Subgroups — C3×C5×#556 #11756645

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₃×C₅×#556 #11756645