Subgroups — C₃×#66069 #11755746

Order 960.

392 conjugacy classes of subgroups.

The subgroup lattice has 240 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×#4093	C₂	1	1	yes	maximal, characteristic
3	C₃×#4092	C₂	1	2	yes	maximal, complemented
4	C₃×#4090	C₂	1	1	yes	maximal, characteristic, complemented
5	C₃×#4044	C₂	1	2	yes	maximal, complemented
6	C₃×#4042	C₂	1	2	yes	maximal, complemented
7	C₃×#4040	C₂	1	2	yes	maximal
8	C₃×#4035	C₂	1	2	yes	maximal, complemented
9	C₃×#4022	C₂	1	2	yes	maximal, complemented
10	#66069	C₃	1	1	yes	maximal, characteristic, direct product factor, Hall
11	C₂×C₃×C₅×Q₈	C₂²	1	1	yes	characteristic
12	C₂×C₃×C₅×Q₈	C₂²	1	1	yes	characteristic
13	C₂×C₃×D₂₀	C₂²	1	1	yes	characteristic
14	C₂×C₃×C₄×D₅	C₂²	1	2	yes	—
15	C₂×C₃×Dic₁₀	C₂²	1	1	yes	characteristic
16	C₃×C₅×C₄⋊C₄	C₂²	1	2	yes	—
17	C₃×C₅×C₄⋊C₄	C₂²	1	2	yes	—
18	C₃×C₄²×C₅	C₂²	1	1	yes	characteristic
19	C₃×#685	C₂²	1	2	yes	—
20	C₃×#685	C₂²	1	2	yes	—
21	C₃×#685	C₂²	1	2	yes	—
22	C₃×#685	C₂²	1	2	yes	—
23	C₃×#685	C₂²	1	2	yes	—
24	C₃×C₄⋊Dic₅	C₂²	1	2	yes	—
25	C₃×#683	C₂²	1	2	yes	—
26	C₃×#683	C₂²	1	2	yes	—
27	C₃×#683	C₂²	1	2	yes	—
28	C₃×#683	C₂²	1	2	yes	—
29	C₃×#683	C₂²	1	2	yes	—
30	C₃×C₄×Dic₅	C₂²	1	1	yes	characteristic
31	C₃×C₄×Dic₅	C₂²	1	1	yes	characteristic
32	C₃×#563	—	5	1	no	maximal, Hall
33	C₅×C₄⋊Q₈	C₂×C₃	1	1	yes	characteristic, complemented
34	#4093	C₂×C₃	1	1	yes	characteristic
35	#4092	C₂×C₃	1	2	yes	complemented
36	#4090	C₂×C₃	1	1	yes	characteristic, complemented
37	#4044	C₂×C₃	1	2	yes	complemented
38	#4042	C₂×C₃	1	2	yes	complemented
39	#4040	C₂×C₃	1	2	yes	—
40	#4035	C₂×C₃	1	2	yes	complemented
41	#4022	C₂×C₃	1	2	yes	complemented
42	C₂²×C₃×D₅	C₂³	1	2	yes	—
43	C₃×C₅×Q₈	—	4	1	no	—
44	C₃×C₅×Q₈	—	4	1	no	—
45	C₂×C₃×C₄×C₅	C₂³	1	1	yes	characteristic
46	C₂×C₃×C₄×C₅	C₂³	1	1	yes	characteristic
47	C₂×C₃×C₄×C₅	C₂³	1	2	yes	—
48	C₂×C₃×C₄×C₅	C₂³	1	1	yes	characteristic
49	C₂×C₃×C₄×C₅	C₂³	1	2	yes	—
50	C₂×C₃×Dic₅	C₂³	1	2	yes	—
51	C₂×C₃×Dic₅	C₂³	1	2	yes	—
52	C₂×C₃×Dic₅	C₂³	1	2	yes	—
53	C₃×D₂₀	—	4	1	no	—
54	C₃×C₄×D₅	—	4	2	no	—
55	C₃×Dic₁₀	—	4	1	no	—
56	C₃×C₄⋊Q₈	—	5	1	no	—
57	C₃×C₄⋊Q₈	D₅	1	1	yes	characteristic, complemented
58	C₃×#126	—	5	2	no	—
59	C₃×#126	—	5	2	no	—
60	C₃×#125	—	5	2	no	—
61	C₃×#124	—	5	1	no	—
62	C₃×#123	—	5	2	no	—
63	C₃×C₂²⋊Q₈	—	5	2	no	—
64	C₃×C₂²⋊Q₈	—	5	2	no	—
65	C₂×C₅×Q₈	C₂²×C₃	1	1	yes	characteristic
66	C₂×C₅×Q₈	C₂²×C₃	1	1	yes	characteristic
67	C₂×D₂₀	C₂²×C₃	1	1	yes	characteristic
68	C₂×C₄×D₅	C₂²×C₃	1	2	yes	—
69	C₂×Dic₁₀	C₂²×C₃	1	1	yes	characteristic
70	C₅×C₄⋊C₄	C₂²×C₃	1	2	yes	—
71	C₅×C₄⋊C₄	C₂²×C₃	1	2	yes	—
72	C₄²×C₅	C₂²×C₃	1	1	yes	characteristic
73	#685	C₂²×C₃	1	2	yes	—
74	#685	C₂²×C₃	1	2	yes	—
75	#685	C₂²×C₃	1	2	yes	—
76	#685	C₂²×C₃	1	2	yes	—
77	#685	C₂²×C₃	1	2	yes	—
78	C₄⋊Dic₅	C₂²×C₃	1	2	yes	—
79	#683	C₂²×C₃	1	2	yes	—
80	#683	C₂²×C₃	1	2	yes	—
81	#683	C₂²×C₃	1	2	yes	—
82	#683	C₂²×C₃	1	2	yes	—
83	#683	C₂²×C₃	1	2	yes	—
84	C₄×Dic₅	C₂²×C₃	1	1	yes	characteristic
85	C₄×Dic₅	C₂²×C₃	1	1	yes	characteristic
86	#563	—	5	1	no	Sylow
87	C₂²×C₃×C₅	C₂⁴	1	1	yes	characteristic
88	C₂×C₃×D₅	—	2	2	no	—
89	C₂×C₃×D₅	—	2	4	no	—
90	C₃×C₄×C₅	—	2	1	no	—
91	C₃×C₄×C₅	—	2	2	no	—
92	C₃×C₄×C₅	—	2	1	no	—
93	C₃×C₄×C₅	—	2	1	no	—
94	C₃×C₄×C₅	—	2	2	no	—
95	C₃×Dic₅	—	2	2	no	—
96	C₃×Dic₅	—	2	2	no	—
97	C₃×Dic₅	—	2	2	no	—
98	C₂×C₃×Q₈	—	5	1	no	—
99	C₂×C₃×Q₈	C₂×D₅	1	1	yes	characteristic
100	C₂×C₃×Q₈	C₂×D₅	1	1	yes	characteristic
101	C₂×C₃×D₄	—	5	1	no	—
102	C₂²×C₃×C₄	—	5	2	no	—
103	C₃×C₄⋊C₄	—	5	2	no	—
104	C₃×C₄⋊C₄	—	5	2	no	—
105	C₃×C₄⋊C₄	—	5	2	no	—
106	C₃×C₄⋊C₄	—	5	2	no	—
107	C₃×C₄⋊C₄	—	5	2	no	—
108	C₃×C₄⋊C₄	—	5	2	no	—
109	C₃×C₄⋊C₄	C₂×D₅	1	2	yes	—
110	C₃×C₄⋊C₄	C₂×D₅	1	2	yes	—
111	C₃×C₂²⋊C₄	—	5	2	no	—
112	C₃×C₂²⋊C₄	—	5	2	no	—
113	C₃×C₂²⋊C₄	—	5	2	no	—
114	C₃×C₂²⋊C₄	—	5	2	no	—
115	C₃×C₂²⋊C₄	—	5	2	no	—
116	C₃×C₄²	—	5	1	no	—
117	C₃×C₄²	—	5	1	no	—
118	C₃×C₄²	C₂×D₅	1	1	yes	characteristic
119	C₂²×D₅	C₂³×C₃	1	2	yes	—
120	C₅×Q₈	—	4	1	no	—
121	C₅×Q₈	—	4	1	no	—
122	C₂×C₄×C₅	C₂³×C₃	1	1	yes	characteristic
123	C₂×C₄×C₅	C₂³×C₃	1	1	yes	characteristic
124	C₂×C₄×C₅	C₂³×C₃	1	2	yes	—
125	C₂×C₄×C₅	C₂³×C₃	1	1	yes	characteristic
126	C₂×C₄×C₅	C₂³×C₃	1	2	yes	—
127	C₂×Dic₅	C₂³×C₃	1	2	yes	—
128	C₂×Dic₅	C₂³×C₃	1	2	yes	—
129	C₂×Dic₅	C₂³×C₃	1	2	yes	—
130	D₂₀	—	4	1	no	—
131	C₄×D₅	—	4	2	no	—
132	Dic₁₀	—	4	1	no	—
133	C₄⋊Q₈	—	5	1	no	—
134	C₄⋊Q₈	C₃×D₅	1	1	yes	characteristic, complemented
135	#126	—	5	2	no	—
136	#126	—	5	2	no	—
137	#125	—	5	2	no	—
138	#124	—	5	1	no	—
139	#123	—	5	2	no	—
140	C₂²⋊Q₈	—	5	2	no	—
141	C₂²⋊Q₈	—	5	2	no	—
142	C₂×C₃×C₅	H(5,𝔽₂)	1	1	yes	characteristic
143	C₂×C₃×C₅	2₋¹⁺⁴	1	2	yes	—
144	C₃×D₅	—	4	2	no	—
145	C₂³×C₃	—	5	2	no	—
146	C₃×Q₈	—	4	1	no	—
147	C₃×Q₈	—	4	1	no	—
148	C₃×Q₈	—	20	1	no	—
149	C₃×D₄	—	20	1	no	—
150	C₂×C₃×C₄	—	5	2	no	—
151	C₂×C₃×C₄	—	5	2	no	—
152	C₂×C₃×C₄	—	5	2	no	—
153	C₂×C₃×C₄	—	20	2	no	—
154	C₂×C₃×C₄	C₂²×D₅	1	1	yes	characteristic
155	C₂×C₃×C₄	C₂²×D₅	1	1	yes	characteristic
156	C₂×C₃×C₄	C₂²×D₅	1	2	yes	—
157	C₂×C₃×C₄	C₂²×D₅	1	1	yes	characteristic
158	C₂×C₃×C₄	C₂²×D₅	1	2	yes	—
159	C₂²×C₅	C₂⁴×C₃	1	1	yes	characteristic
160	C₂×D₅	—	2	2	no	—
161	C₂×D₅	—	2	4	no	—
162	C₄×C₅	—	2	1	no	—
163	C₄×C₅	—	2	2	no	—
164	C₄×C₅	—	2	1	no	—
165	C₄×C₅	—	2	1	no	—
166	C₄×C₅	—	2	2	no	—
167	Dic₅	—	2	2	no	—
168	Dic₅	—	2	2	no	—
169	Dic₅	—	2	2	no	—
170	C₂×Q₈	—	5	1	no	—
171	C₂×Q₈	C₂×C₃×D₅	1	1	yes	characteristic
172	C₂×Q₈	C₂×C₃×D₅	1	1	yes	characteristic
173	C₂×D₄	—	5	1	no	—
174	C₂²×C₄	—	5	2	no	—
175	C₄⋊C₄	—	5	2	no	—
176	C₄⋊C₄	—	5	2	no	—
177	C₄⋊C₄	—	5	2	no	—
178	C₄⋊C₄	—	5	2	no	—
179	C₄⋊C₄	—	5	2	no	—
180	C₄⋊C₄	—	5	2	no	—
181	C₄⋊C₄	C₂×C₃×D₅	1	2	yes	—
182	C₄⋊C₄	C₂×C₃×D₅	1	2	yes	—
183	C₂²⋊C₄	—	5	2	no	—
184	C₂²⋊C₄	—	5	2	no	—
185	C₂²⋊C₄	—	5	2	no	—
186	C₂²⋊C₄	—	5	2	no	—
187	C₂²⋊C₄	—	5	2	no	—
188	C₄²	—	5	1	no	—
189	C₄²	—	5	1	no	—
190	C₄²	C₂×C₃×D₅	1	1	yes	characteristic
191	C₃×C₅	#563	1	1	yes	characteristic, complemented, Hall
192	C₂²×C₃	—	10	2	no	—
193	C₂²×C₃	—	10	4	no	—
194	C₂²×C₃	C₂³×D₅	1	1	yes	characteristic
195	C₃×C₄	—	2	2	no	—
196	C₃×C₄	—	2	1	no	—
197	C₃×C₄	—	2	1	no	—
198	C₃×C₄	—	2	1	no	—
199	C₃×C₄	—	2	2	no	—
200	C₃×C₄	—	10	2	no	—
201	C₃×C₄	—	10	2	no	—
202	C₃×C₄	—	10	2	no	—
203	C₂×C₅	C₃×H(5,𝔽₂)	1	1	yes	characteristic
204	C₂×C₅	C₃×2₋¹⁺⁴	1	2	yes	—
205	D₅	—	4	2	no	—
206	C₂³	—	5	2	no	—
207	Q₈	—	4	1	no	—
208	Q₈	—	4	1	no	—
209	Q₈	—	20	1	no	—
210	D₄	—	20	1	no	—
211	C₂×C₄	—	5	2	no	—
212	C₂×C₄	—	5	2	no	—
213	C₂×C₄	—	5	2	no	—
214	C₂×C₄	—	20	2	no	—
215	C₂×C₄	C₂²×C₃×D₅	1	1	yes	characteristic
216	C₂×C₄	C₂²×C₃×D₅	1	1	yes	characteristic
217	C₂×C₄	C₂²×C₃×D₅	1	2	yes	—
218	C₂×C₄	C₂²×C₃×D₅	1	1	yes	characteristic
219	C₂×C₄	C₂²×C₃×D₅	1	2	yes	—
220	C₂×C₃	—	20	2	no	—
221	C₂×C₃	#4144	1	1	yes	characteristic
222	C₂×C₃	#4147	1	2	yes	—
223	C₅	C₃×#563	1	1	yes	characteristic, complemented, Sylow
224	C₂²	—	10	2	no	—
225	C₂²	—	10	4	no	—
226	C₂²	C₂³×C₃×D₅	1	1	yes	characteristic
227	C₄	—	2	2	no	—
228	C₄	—	2	1	no	—
229	C₄	—	2	1	no	—
230	C₄	—	2	1	no	—
231	C₄	—	2	2	no	—
232	C₄	—	10	2	no	—
233	C₄	—	10	2	no	—
234	C₄	—	10	2	no	—
235	C₃	#66069	1	1	yes	characteristic, direct product factor, Sylow
236	C₂	—	20	2	no	—
237	C₂	C₃×#4144	1	1	yes	characteristic
238	C₂	C₃×#4147	1	2	yes	—

	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	C₂²×C₃×C₅	C₂⁴
prime radical	C₂×C₃×D₂₀	C₂²
hypercenter	C₃×C₄⋊Q₈	D₅
hypocenter	C₅	C₃×#563
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
1. C₃×C₅×C₄⋊Q₈	C₂	semidirect
3. C₃×#4092	C₂	semidirect
4. C₃×#4090	C₂	semidirect
5. C₃×#4044	C₂	semidirect
6. C₃×#4042	C₂	semidirect
8. C₃×#4035	C₂	semidirect
9. C₃×#4022	C₂	semidirect
10. #66069	C₃	direct (N × H)
33. C₅×C₄⋊Q₈	C₂×C₃	semidirect
35. #4092	C₂×C₃	semidirect
36. #4090	C₂×C₃	semidirect
37. #4044	C₂×C₃	semidirect
38. #4042	C₂×C₃	semidirect
40. #4035	C₂×C₃	semidirect
41. #4022	C₂×C₃	semidirect
57. C₃×C₄⋊Q₈	D₅	semidirect
134. C₄⋊Q₈	C₃×D₅	semidirect
191. C₃×C₅	#563	semidirect
223. C₅	C₃×#563	semidirect
235. C₃	#66069	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×#4093	2
C₃×#4092	2
C₃×#4090	2
C₃×#4044	2
C₃×#4042	2
C₃×#4040	2
C₃×#4035	2
C₃×#4022	2
#66069	3
C₃×#563	5

Subgroups — C3×#66069 #11755746

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₃×#66069 #11755746