Subgroups — C₃×#66004 #11755681

Order 960.

472 conjugacy classes of subgroups.

The subgroup lattice has 268 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₃×C₅×#124	C₂	1	1	yes	maximal, characteristic, complemented
2	C₃×#4093	C₂	1	1	yes	maximal, characteristic, complemented
3	C₃×#4090	C₂	1	1	yes	maximal, characteristic, complemented
4	C₃×#4085	C₂	1	2	yes	maximal, complemented
5	C₃×#4031	C₂	1	2	yes	maximal, complemented
6	C₃×#4030	C₂	1	2	yes	maximal, complemented
7	C₃×#4027	C₂	1	2	yes	maximal, complemented
8	C₃×#4023	C₂	1	2	yes	maximal, complemented
9	C₃×#4018	C₂	1	2	yes	maximal, complemented
10	#66004	C₃	1	1	yes	maximal, characteristic, direct product factor, Hall
11	C₂×C₃×C₅×Q₈	C₂²	1	1	yes	characteristic
12	C₂×C₃×C₅×D₄	C₂²	1	1	yes	characteristic
13	C₂×C₃×C₂²⋊D₅	C₂²	1	2	yes	—
14	C₂×C₃×C₂²⋊D₅	C₂²	1	2	yes	—
15	C₂²×C₃×Dic₅	C₂²	1	2	yes	—
16	C₂×C₃×D₂₀	C₂²	1	1	yes	characteristic
17	C₂×C₃×C₄×D₅	C₂²	1	2	yes	—
18	C₂×C₃×Dic₁₀	C₂²	1	1	yes	characteristic
19	C₃×C₅×C₂²⋊C₄	C₂²	1	2	yes	—
20	C₃×C₅×C₂²⋊C₄	C₂²	1	2	yes	—
21	C₃×C₄²×C₅	C₂²	1	1	yes	characteristic
22	C₃×C₂²⋊Dic₅	C₂²	1	2	yes	—
23	C₃×#685	C₂²	1	2	yes	—
24	C₃×#685	C₂²	1	2	yes	—
25	C₃×#685	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×C₄×Dic₅	C₂²	1	1	yes	characteristic
30	C₃×C₄×Dic₅	C₂²	1	1	yes	characteristic
31	C₃×C₄×Dic₅	C₂²	1	2	yes	—
32	C₃×#555	—	5	1	no	maximal, Hall
33	C₅×#124	C₂×C₃	1	1	yes	characteristic, complemented
34	#4093	C₂×C₃	1	1	yes	characteristic, complemented
35	#4090	C₂×C₃	1	1	yes	characteristic, complemented
36	#4085	C₂×C₃	1	2	yes	complemented
37	#4031	C₂×C₃	1	2	yes	complemented
38	#4030	C₂×C₃	1	2	yes	complemented
39	#4027	C₂×C₃	1	2	yes	complemented
40	#4023	C₂×C₃	1	2	yes	complemented
41	#4018	C₂×C₃	1	2	yes	complemented
42	C₂³×C₃×C₅	C₂³	1	2	yes	—
43	C₂²×C₃×D₅	C₂³	1	2	yes	—
44	C₃×C₅×Q₈	—	4	1	no	—
45	C₃×C₅×D₄	—	4	1	no	—
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	1	yes	characteristic
50	C₃×C₂²⋊D₅	—	2	4	no	—
51	C₃×C₂²⋊D₅	—	4	2	no	—
52	C₂×C₃×Dic₅	—	2	4	no	—
53	C₂×C₃×Dic₅	C₂³	1	2	yes	—
54	C₂×C₃×Dic₅	C₂³	1	2	yes	—
55	C₂×C₃×Dic₅	C₂³	1	2	yes	—
56	C₃×D₂₀	—	4	1	no	—
57	C₃×C₄×D₅	—	2	4	no	—
58	C₃×Dic₁₀	—	4	1	no	—
59	C₃×C₄⋊Q₈	—	5	1	no	—
60	C₃×#124	—	5	1	no	—
61	C₃×#124	—	5	2	no	—
62	C₃×#124	D₅	1	1	yes	characteristic, complemented
63	C₃×#121	—	5	2	no	—
64	C₃×#121	—	5	2	no	—
65	C₃×C₄×D₄	—	5	2	no	—
66	C₃×#117	—	5	2	no	—
67	C₃×#117	—	5	2	no	—
68	C₂×C₅×Q₈	C₂²×C₃	1	1	yes	characteristic
69	C₂×C₅×D₄	C₂²×C₃	1	1	yes	characteristic
70	C₂×C₂²⋊D₅	C₂²×C₃	1	2	yes	—
71	C₂×C₂²⋊D₅	C₂²×C₃	1	2	yes	—
72	C₂²×Dic₅	C₂²×C₃	1	2	yes	—
73	C₂×D₂₀	C₂²×C₃	1	1	yes	characteristic
74	C₂×C₄×D₅	C₂²×C₃	1	2	yes	—
75	C₂×Dic₁₀	C₂²×C₃	1	1	yes	characteristic
76	C₅×C₂²⋊C₄	C₂²×C₃	1	2	yes	—
77	C₅×C₂²⋊C₄	C₂²×C₃	1	2	yes	—
78	C₄²×C₅	C₂²×C₃	1	1	yes	characteristic
79	C₂²⋊Dic₅	C₂²×C₃	1	2	yes	—
80	#685	C₂²×C₃	1	2	yes	—
81	#685	C₂²×C₃	1	2	yes	—
82	#685	C₂²×C₃	1	2	yes	—
83	#683	C₂²×C₃	1	2	yes	—
84	#683	C₂²×C₃	1	2	yes	—
85	#683	C₂²×C₃	1	2	yes	—
86	C₄×Dic₅	C₂²×C₃	1	2	yes	—
87	C₄×Dic₅	C₂²×C₃	1	1	yes	characteristic
88	C₄×Dic₅	C₂²×C₃	1	1	yes	characteristic
89	#555	—	5	1	no	Sylow
90	C₂²×C₃×C₅	—	2	2	no	—
91	C₂²×C₃×C₅	—	2	2	no	—
92	C₂²×C₃×C₅	—	2	2	no	—
93	C₂²×C₃×C₅	C₂⁴	1	1	yes	characteristic
94	C₂×C₃×D₅	—	2	2	no	—
95	C₂×C₃×D₅	—	2	2	no	—
96	C₂×C₃×D₅	—	2	2	no	—
97	C₃×C₄×C₅	—	2	2	no	—
98	C₃×C₄×C₅	—	2	1	no	—
99	C₃×C₄×C₅	—	2	1	no	—
100	C₃×C₄×C₅	—	2	1	no	—
101	C₃×Dic₅	—	2	2	no	—
102	C₃×Dic₅	—	2	2	no	—
103	C₃×Dic₅	#41	1	4	yes	—
104	C₂×C₃×Q₈	—	5	1	no	—
105	C₂×C₃×Q₈	C₂×D₅	1	1	yes	characteristic
106	C₂×C₃×D₄	—	5	2	no	—
107	C₂×C₃×D₄	—	5	2	no	—
108	C₂×C₃×D₄	—	5	1	no	—
109	C₂×C₃×D₄	C₂×D₅	1	1	yes	characteristic
110	C₂²×C₃×C₄	—	5	2	no	—
111	C₂²×C₃×C₄	—	5	2	no	—
112	C₃×C₄⋊C₄	—	5	2	no	—
113	C₃×C₄⋊C₄	—	5	2	no	—
114	C₃×C₄⋊C₄	—	5	2	no	—
115	C₃×C₂²⋊C₄	—	5	2	no	—
116	C₃×C₂²⋊C₄	—	5	2	no	—
117	C₃×C₂²⋊C₄	—	5	2	no	—
118	C₃×C₂²⋊C₄	—	5	2	no	—
119	C₃×C₂²⋊C₄	C₂×D₅	1	2	yes	—
120	C₃×C₂²⋊C₄	C₂×D₅	1	2	yes	—
121	C₃×C₄²	—	5	1	no	—
122	C₃×C₄²	—	5	2	no	—
123	C₃×C₄²	—	5	1	no	—
124	C₃×C₄²	C₂×D₅	1	1	yes	characteristic
125	C₂³×C₅	C₂³×C₃	1	2	yes	—
126	C₂²×D₅	C₂³×C₃	1	2	yes	—
127	C₅×Q₈	—	4	1	no	—
128	C₅×D₄	—	4	1	no	—
129	C₂×C₄×C₅	C₂³×C₃	1	1	yes	characteristic
130	C₂×C₄×C₅	C₂³×C₃	1	2	yes	—
131	C₂×C₄×C₅	C₂³×C₃	1	1	yes	characteristic
132	C₂×C₄×C₅	C₂³×C₃	1	1	yes	characteristic
133	C₂²⋊D₅	—	2	4	no	—
134	C₂²⋊D₅	—	4	2	no	—
135	C₂×Dic₅	—	2	4	no	—
136	C₂×Dic₅	C₂³×C₃	1	2	yes	—
137	C₂×Dic₅	C₂³×C₃	1	2	yes	—
138	C₂×Dic₅	C₂³×C₃	1	2	yes	—
139	D₂₀	—	4	1	no	—
140	C₄×D₅	—	2	4	no	—
141	Dic₁₀	—	4	1	no	—
142	C₄⋊Q₈	—	5	1	no	—
143	#124	—	5	1	no	—
144	#124	—	5	2	no	—
145	#124	C₃×D₅	1	1	yes	characteristic, complemented
146	#121	—	5	2	no	—
147	#121	—	5	2	no	—
148	C₄×D₄	—	5	2	no	—
149	#117	—	5	2	no	—
150	#117	—	5	2	no	—
151	C₂×C₃×C₅	—	4	2	no	—
152	C₂×C₃×C₅	C₂×#41	1	2	yes	—
153	C₂×C₃×C₅	H(5,𝔽₂)	1	1	yes	characteristic
154	C₃×D₅	—	4	2	no	—
155	C₂³×C₃	—	5	2	no	—
156	C₂³×C₃	C₂²×D₅	1	2	yes	—
157	C₃×Q₈	—	4	1	no	—
158	C₃×Q₈	—	20	1	no	—
159	C₃×D₄	—	4	1	no	—
160	C₃×D₄	—	10	4	no	—
161	C₃×D₄	—	20	1	no	—
162	C₃×D₄	—	20	2	no	—
163	C₂×C₃×C₄	—	5	2	no	—
164	C₂×C₃×C₄	—	5	2	no	—
165	C₂×C₃×C₄	—	5	2	no	—
166	C₂×C₃×C₄	—	10	4	no	—
167	C₂×C₃×C₄	—	10	4	no	—
168	C₂×C₃×C₄	C₂²×D₅	1	2	yes	—
169	C₂×C₃×C₄	C₂²×D₅	1	1	yes	characteristic
170	C₂×C₃×C₄	C₂²×D₅	1	1	yes	characteristic
171	C₂×C₃×C₄	C₂²×D₅	1	1	yes	characteristic
172	C₂²×C₅	—	2	2	no	—
173	C₂²×C₅	—	2	2	no	—
174	C₂²×C₅	—	2	2	no	—
175	C₂²×C₅	C₂⁴×C₃	1	1	yes	characteristic
176	C₂×D₅	—	2	2	no	—
177	C₂×D₅	—	2	2	no	—
178	C₂×D₅	—	2	2	no	—
179	C₄×C₅	—	2	2	no	—
180	C₄×C₅	—	2	1	no	—
181	C₄×C₅	—	2	1	no	—
182	C₄×C₅	—	2	1	no	—
183	Dic₅	—	2	2	no	—
184	Dic₅	—	2	2	no	—
185	Dic₅	C₃×#41	1	4	yes	—
186	C₂×Q₈	—	5	1	no	—
187	C₂×Q₈	C₂×C₃×D₅	1	1	yes	characteristic
188	C₂×D₄	—	5	2	no	—
189	C₂×D₄	—	5	2	no	—
190	C₂×D₄	—	5	1	no	—
191	C₂×D₄	C₂×C₃×D₅	1	1	yes	characteristic
192	C₂²×C₄	—	5	2	no	—
193	C₂²×C₄	—	5	2	no	—
194	C₄⋊C₄	—	5	2	no	—
195	C₄⋊C₄	—	5	2	no	—
196	C₄⋊C₄	—	5	2	no	—
197	C₂²⋊C₄	—	5	2	no	—
198	C₂²⋊C₄	—	5	2	no	—
199	C₂²⋊C₄	—	5	2	no	—
200	C₂²⋊C₄	—	5	2	no	—
201	C₂²⋊C₄	C₂×C₃×D₅	1	2	yes	—
202	C₂²⋊C₄	C₂×C₃×D₅	1	2	yes	—
203	C₄²	—	5	1	no	—
204	C₄²	—	5	2	no	—
205	C₄²	—	5	1	no	—
206	C₄²	C₂×C₃×D₅	1	1	yes	characteristic
207	C₃×C₅	#555	1	1	yes	characteristic, complemented, Hall
208	C₂²×C₃	—	2	2	no	—
209	C₂²×C₃	—	2	2	no	—
210	C₂²×C₃	—	2	2	no	—
211	C₂²×C₃	—	10	2	no	—
212	C₂²×C₃	—	10	2	no	—
213	C₂²×C₃	—	10	2	no	—
214	C₂²×C₃	C₂³×D₅	1	1	yes	characteristic
215	C₃×C₄	—	2	1	no	—
216	C₃×C₄	—	2	1	no	—
217	C₃×C₄	—	2	1	no	—
218	C₃×C₄	—	2	2	no	—
219	C₃×C₄	—	5	4	no	—
220	C₃×C₄	—	10	2	no	—
221	C₃×C₄	—	10	2	no	—
222	C₂×C₅	—	4	2	no	—
223	C₂×C₅	C₂×C₃×#41	1	2	yes	—
224	C₂×C₅	C₃×H(5,𝔽₂)	1	1	yes	characteristic
225	D₅	—	4	2	no	—
226	C₂³	—	5	2	no	—
227	C₂³	C₂²×C₃×D₅	1	2	yes	—
228	Q₈	—	4	1	no	—
229	Q₈	—	20	1	no	—
230	D₄	—	4	1	no	—
231	D₄	—	10	4	no	—
232	D₄	—	20	1	no	—
233	D₄	—	20	2	no	—
234	C₂×C₄	—	5	2	no	—
235	C₂×C₄	—	5	2	no	—
236	C₂×C₄	—	5	2	no	—
237	C₂×C₄	—	10	4	no	—
238	C₂×C₄	—	10	4	no	—
239	C₂×C₄	C₂²×C₃×D₅	1	2	yes	—
240	C₂×C₄	C₂²×C₃×D₅	1	1	yes	characteristic
241	C₂×C₄	C₂²×C₃×D₅	1	1	yes	characteristic
242	C₂×C₄	C₂²×C₃×D₅	1	1	yes	characteristic
243	C₂×C₃	—	4	2	no	—
244	C₂×C₃	—	20	2	no	—
245	C₂×C₃	#4144	1	1	yes	characteristic
246	C₂×C₃	D₅×#41	1	2	yes	—
247	C₅	C₃×#555	1	1	yes	characteristic, complemented, Sylow
248	C₂²	—	2	2	no	—
249	C₂²	—	2	2	no	—
250	C₂²	—	2	2	no	—
251	C₂²	—	10	2	no	—
252	C₂²	—	10	2	no	—
253	C₂²	—	10	2	no	—
254	C₂²	C₂³×C₃×D₅	1	1	yes	characteristic
255	C₄	—	2	1	no	—
256	C₄	—	2	1	no	—
257	C₄	—	2	1	no	—
258	C₄	—	2	2	no	—
259	C₄	—	5	4	no	—
260	C₄	—	10	2	no	—
261	C₄	—	10	2	no	—
262	C₃	#66004	1	1	yes	characteristic, direct product factor, Sylow
263	C₂	—	4	2	no	—
264	C₂	—	20	2	no	—
265	C₂	C₃×#4144	1	1	yes	characteristic
266	C₂	C₃×D₅×#41	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×#124	C₂
socle	C₂²×C₃×C₅	C₂⁴
hypercenter	C₃×#124	D₅
hypocenter	C₅	C₃×#555
LCS	G ▷ C₂²×C₅ ▷ C₅
UCS	C₁ ◁ C₂²×C₃ ◁ C₃×#124

Complements

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

N	H ≅ G/N	type
1. C₃×C₅×#124	C₂	semidirect
2. C₃×#4093	C₂	semidirect
3. C₃×#4090	C₂	semidirect
4. C₃×#4085	C₂	semidirect
5. C₃×#4031	C₂	semidirect
6. C₃×#4030	C₂	semidirect
7. C₃×#4027	C₂	semidirect
8. C₃×#4023	C₂	semidirect
9. C₃×#4018	C₂	semidirect
10. #66004	C₃	direct (N × H)
33. C₅×#124	C₂×C₃	semidirect
34. #4093	C₂×C₃	semidirect
35. #4090	C₂×C₃	semidirect
36. #4085	C₂×C₃	semidirect
37. #4031	C₂×C₃	semidirect
38. #4030	C₂×C₃	semidirect
39. #4027	C₂×C₃	semidirect
40. #4023	C₂×C₃	semidirect
41. #4018	C₂×C₃	semidirect
62. C₃×#124	D₅	semidirect
145. #124	C₃×D₅	semidirect
207. C₃×C₅	#555	semidirect
247. C₅	C₃×#555	semidirect
262. C₃	#66004	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×#124	2
C₃×#4093	2
C₃×#4090	2
C₃×#4085	2
C₃×#4031	2
C₃×#4030	2
C₃×#4027	2
C₃×#4023	2
C₃×#4018	2
#66004	3
C₃×#555	5

Subgroups — C3×#66004 #11755681

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₃×#66004 #11755681