Subgroups — C₃×#66055 #11755732

Order 960.

400 conjugacy classes of subgroups.

The subgroup lattice has 196 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
2	C₃×Q₈×Dic₅	C₂	1	2	yes	maximal
3	C₃×#4090	C₂	1	2	yes	maximal
4	C₃×#4036	C₂	1	2	yes	maximal
5	C₃×#4035	C₂	1	4	yes	maximal
6	C₃×#4033	C₂	1	2	yes	maximal
7	C₃×C₄×Dic₁₀	C₂	1	1	yes	maximal, characteristic
8	C₃×C₄×Dic₁₀	C₂	1	1	yes	maximal, characteristic
9	#66055	C₃	1	1	yes	maximal, characteristic, direct product factor, Hall
10	C₂×C₃×C₅×Q₈	C₂²	1	2	yes	—
11	C₂×C₃×Dic₁₀	C₂²	1	1	yes	characteristic
12	C₂×C₃×Dic₁₀	C₂²	1	1	yes	characteristic
13	C₃×C₅×C₄⋊C₄	C₂²	1	2	yes	—
14	C₃×C₅×C₄⋊C₄	C₂²	1	2	yes	—
15	C₃×C₄²×C₅	C₂²	1	1	yes	characteristic
16	C₃×C₄⋊Dic₅	C₂²	1	1	yes	characteristic
17	C₃×C₄⋊Dic₅	C₂²	1	4	yes	—
18	C₃×C₄⋊Dic₅	C₂²	1	1	yes	characteristic
19	C₃×#683	C₂²	1	4	yes	—
20	C₃×#683	C₂²	1	4	yes	—
21	C₃×#683	C₂²	1	2	yes	—
22	C₃×#683	C₂²	1	2	yes	—
23	C₃×C₄×Dic₅	C₂²	1	2	yes	—
24	C₃×C₄×Dic₅	C₂²	1	4	yes	—
25	C₃×C₄×Dic₅	C₂²	1	2	yes	—
26	C₃×#557	—	5	1	no	maximal, Hall
27	C₅×C₄⋊Q₈	C₂×C₃	1	1	yes	characteristic
28	Q₈×Dic₅	C₂×C₃	1	2	yes	—
29	#4090	C₂×C₃	1	2	yes	—
30	#4036	C₂×C₃	1	2	yes	—
31	#4035	C₂×C₃	1	4	yes	—
32	#4033	C₂×C₃	1	2	yes	—
33	C₄×Dic₁₀	C₂×C₃	1	1	yes	characteristic
34	C₄×Dic₁₀	C₂×C₃	1	1	yes	characteristic
35	C₃×C₅×Q₈	—	2	4	no	—
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	C₂×C₃×C₄×C₅	C₂³	1	1	yes	characteristic
40	C₂×C₃×Dic₅	C₂³	1	2	yes	—
41	C₂×C₃×Dic₅	C₂³	1	2	yes	—
42	C₂×C₃×Dic₅	C₂³	1	2	yes	—
43	C₂×C₃×Dic₅	C₂³	1	2	yes	—
44	C₃×Dic₁₀	—	2	2	no	—
45	C₃×Dic₁₀	Q₈	1	4	yes	complemented
46	C₃×C₄⋊Q₈	—	5	2	no	—
47	C₃×C₄⋊Q₈	D₅	1	1	yes	characteristic
48	C₃×#125	—	5	4	no	—
49	C₃×#125	—	5	2	no	—
50	C₃×C₄×Q₈	—	5	2	no	—
51	C₃×C₄×Q₈	—	5	2	no	—
52	C₃×C₄×Q₈	—	5	1	no	—
53	C₃×C₄×Q₈	—	5	1	no	—
54	C₂×C₅×Q₈	C₂²×C₃	1	2	yes	—
55	C₂×Dic₁₀	C₂²×C₃	1	1	yes	characteristic
56	C₂×Dic₁₀	C₂²×C₃	1	1	yes	characteristic
57	C₅×C₄⋊C₄	C₂²×C₃	1	2	yes	—
58	C₅×C₄⋊C₄	C₂²×C₃	1	2	yes	—
59	C₄²×C₅	C₂²×C₃	1	1	yes	characteristic
60	C₄⋊Dic₅	C₂²×C₃	1	1	yes	characteristic
61	C₄⋊Dic₅	C₂²×C₃	1	1	yes	characteristic
62	C₄⋊Dic₅	C₂²×C₃	1	4	yes	—
63	#683	C₂²×C₃	1	2	yes	—
64	#683	C₂²×C₃	1	2	yes	—
65	#683	C₂²×C₃	1	4	yes	—
66	#683	C₂²×C₃	1	4	yes	—
67	C₄×Dic₅	C₂²×C₃	1	2	yes	—
68	C₄×Dic₅	C₂²×C₃	1	2	yes	—
69	C₄×Dic₅	C₂²×C₃	1	4	yes	—
70	#557	—	5	1	no	Sylow
71	C₂²×C₃×C₅	C₂⁴	1	1	yes	characteristic
72	C₃×C₄×C₅	—	2	4	no	—
73	C₃×C₄×C₅	—	2	1	no	—
74	C₃×C₄×C₅	C₂×Q₈	1	2	yes	—
75	C₃×C₄×C₅	#41	1	2	yes	—
76	C₃×Dic₅	—	2	2	no	—
77	C₃×Dic₅	—	2	2	no	—
78	C₃×Dic₅	—	2	2	no	—
79	C₃×Dic₅	C₂×Q₈	1	4	yes	—
80	C₂×C₃×Q₈	—	5	1	no	—
81	C₂×C₃×Q₈	—	5	1	no	—
82	C₂×C₃×Q₈	C₂×D₅	1	2	yes	—
83	C₃×C₄⋊C₄	—	5	2	no	—
84	C₃×C₄⋊C₄	—	5	2	no	—
85	C₃×C₄⋊C₄	—	5	4	no	—
86	C₃×C₄⋊C₄	—	5	4	no	—
87	C₃×C₄⋊C₄	—	5	1	no	—
88	C₃×C₄⋊C₄	—	5	4	no	—
89	C₃×C₄⋊C₄	—	5	1	no	—
90	C₃×C₄⋊C₄	C₂×D₅	1	2	yes	—
91	C₃×C₄⋊C₄	C₂×D₅	1	2	yes	—
92	C₃×C₄²	—	5	4	no	—
93	C₃×C₄²	—	5	2	no	—
94	C₃×C₄²	—	5	2	no	—
95	C₃×C₄²	C₂×D₅	1	1	yes	characteristic
96	C₅×Q₈	—	2	4	no	—
97	C₂×C₄×C₅	C₂³×C₃	1	4	yes	—
98	C₂×C₄×C₅	C₂³×C₃	1	1	yes	characteristic
99	C₂×C₄×C₅	C₂³×C₃	1	1	yes	characteristic
100	C₂×C₄×C₅	C₂³×C₃	1	1	yes	characteristic
101	C₂×Dic₅	C₂³×C₃	1	2	yes	—
102	C₂×Dic₅	C₂³×C₃	1	2	yes	—
103	C₂×Dic₅	C₂³×C₃	1	2	yes	—
104	C₂×Dic₅	C₂³×C₃	1	2	yes	—
105	Dic₁₀	—	2	2	no	—
106	Dic₁₀	C₃×Q₈	1	4	yes	complemented
107	C₄⋊Q₈	—	5	2	no	—
108	C₄⋊Q₈	C₃×D₅	1	1	yes	characteristic
109	#125	—	5	4	no	—
110	#125	—	5	2	no	—
111	C₄×Q₈	—	5	2	no	—
112	C₄×Q₈	—	5	2	no	—
113	C₄×Q₈	—	5	1	no	—
114	C₄×Q₈	—	5	1	no	—
115	C₂×C₃×C₅	C₂²×Q₈	1	1	yes	characteristic
116	C₂×C₃×C₅	C₂×#41	1	1	yes	characteristic
117	C₂×C₃×C₅	2₋¹⁺⁴	1	1	yes	characteristic
118	C₃×Q₈	—	2	4	no	—
119	C₃×Q₈	—	5	4	no	—
120	C₃×Q₈	—	10	2	no	—
121	C₂×C₃×C₄	—	5	2	no	—
122	C₂×C₃×C₄	—	5	2	no	—
123	C₂×C₃×C₄	—	5	2	no	—
124	C₂×C₃×C₄	—	5	2	no	—
125	C₂×C₃×C₄	C₂²×D₅	1	4	yes	—
126	C₂×C₃×C₄	C₂²×D₅	1	1	yes	characteristic
127	C₂×C₃×C₄	C₂²×D₅	1	1	yes	characteristic
128	C₂×C₃×C₄	C₂²×D₅	1	1	yes	characteristic
129	C₂²×C₅	C₂⁴×C₃	1	1	yes	characteristic
130	C₄×C₅	—	2	4	no	—
131	C₄×C₅	—	2	1	no	—
132	C₄×C₅	C₂×C₃×Q₈	1	2	yes	—
133	C₄×C₅	C₃×#41	1	2	yes	—
134	Dic₅	—	2	2	no	—
135	Dic₅	—	2	2	no	—
136	Dic₅	—	2	2	no	—
137	Dic₅	C₂×C₃×Q₈	1	4	yes	—
138	C₂×Q₈	—	5	1	no	—
139	C₂×Q₈	—	5	1	no	—
140	C₂×Q₈	C₂×C₃×D₅	1	2	yes	—
141	C₄⋊C₄	—	5	1	no	—
142	C₄⋊C₄	—	5	2	no	—
143	C₄⋊C₄	—	5	2	no	—
144	C₄⋊C₄	—	5	4	no	—
145	C₄⋊C₄	—	5	1	no	—
146	C₄⋊C₄	—	5	4	no	—
147	C₄⋊C₄	—	5	4	no	—
148	C₄⋊C₄	C₂×C₃×D₅	1	2	yes	—
149	C₄⋊C₄	C₂×C₃×D₅	1	2	yes	—
150	C₄²	—	5	4	no	—
151	C₄²	—	5	2	no	—
152	C₄²	—	5	2	no	—
153	C₄²	C₂×C₃×D₅	1	1	yes	characteristic
154	C₃×C₅	#557	1	1	yes	characteristic, complemented, Hall
155	C₂²×C₃	C₂³×D₅	1	1	yes	characteristic
156	C₃×C₄	—	2	4	no	—
157	C₃×C₄	—	2	1	no	—
158	C₃×C₄	—	5	4	no	—
159	C₃×C₄	—	10	2	no	—
160	C₃×C₄	—	10	2	no	—
161	C₃×C₄	—	10	2	no	—
162	C₃×C₄	#711	1	2	yes	—
163	C₃×C₄	Q₈×D₅	1	2	yes	—
164	C₂×C₅	C₂²×C₃×Q₈	1	1	yes	characteristic
165	C₂×C₅	C₂×C₃×#41	1	1	yes	characteristic
166	C₂×C₅	C₃×2₋¹⁺⁴	1	1	yes	characteristic
167	Q₈	—	2	4	no	—
168	Q₈	—	5	4	no	—
169	Q₈	—	10	2	no	—
170	C₂×C₄	—	5	2	no	—
171	C₂×C₄	—	5	2	no	—
172	C₂×C₄	—	5	2	no	—
173	C₂×C₄	—	5	2	no	—
174	C₂×C₄	C₂²×C₃×D₅	1	4	yes	—
175	C₂×C₄	C₂²×C₃×D₅	1	1	yes	characteristic
176	C₂×C₄	C₂²×C₃×D₅	1	1	yes	characteristic
177	C₂×C₄	C₂²×C₃×D₅	1	1	yes	characteristic
178	C₂×C₃	C₂×#711	1	1	yes	characteristic
179	C₂×C₃	C₂×Q₈×D₅	1	1	yes	characteristic
180	C₂×C₃	#4147	1	1	yes	characteristic
181	C₅	C₃×#557	1	1	yes	characteristic, complemented, Sylow
182	C₂²	C₂³×C₃×D₅	1	1	yes	characteristic
183	C₄	—	2	4	no	—
184	C₄	—	2	1	no	—
185	C₄	—	5	4	no	—
186	C₄	—	10	2	no	—
187	C₄	—	10	2	no	—
188	C₄	—	10	2	no	—
189	C₄	C₃×#711	1	2	yes	—
190	C₄	C₃×Q₈×D₅	1	2	yes	—
191	C₃	#66055	1	1	yes	characteristic, direct product factor, Sylow
192	C₂	C₂×C₃×#711	1	1	yes	characteristic
193	C₂	C₂×C₃×Q₈×D₅	1	1	yes	characteristic
194	C₂	C₃×#4147	1	1	yes	characteristic

	subgroup	G/H
center	C₂²×C₃	C₂³×D₅
commutator, cosocle	C₂²×C₅	C₂⁴×C₃
Frattini	C₂²	C₂³×C₃×D₅
Fitting	C₃×C₅×C₄⋊Q₈	C₂
socle, prime radical	C₂²×C₃×C₅	C₂⁴
hypercenter	C₃×C₄⋊Q₈	D₅
hypocenter	C₅	C₃×#557
LCS	G ▷ C₂²×C₅ ▷ C₅
UCS	C₁ ◁ C₂²×C₃ ◁ C₃×C₄⋊Q₈

Complements

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

N	H ≅ G/N	type
9. #66055	C₃	direct (N × H)
45. C₃×Dic₁₀	Q₈	semidirect
106. Dic₁₀	C₃×Q₈	semidirect
154. C₃×C₅	#557	semidirect
181. C₅	C₃×#557	semidirect
191. C₃	#66055	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×Q₈×Dic₅	2
C₃×#4090	2
C₃×#4036	2
C₃×#4035	2
C₃×#4033	2
C₃×C₄×Dic₁₀	2
#66055	3
C₃×#557	5

Subgroups — C3×#66055 #11755732

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₃×#66055 #11755732