Subgroups — C₃×#65897 #11755574

Order 960.

472 conjugacy classes of subgroups.

The subgroup lattice has 232 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₃×C₄×C₅×D₄	C₂	1	1	yes	maximal, characteristic, complemented
2	C₃×#4084	C₂	1	2	yes	maximal, complemented
3	C₃×#4076	C₂	1	2	yes	maximal, complemented
4	C₃×#4072	C₂	1	2	yes	maximal, complemented
5	C₃×#4038	C₂	1	2	yes	maximal, complemented
6	C₃×#4031	C₂	1	4	yes	maximal, complemented
7	C₃×C₄×D₂₀	C₂	1	1	yes	maximal, characteristic, complemented
8	C₃×#4015	C₂	1	1	yes	maximal, characteristic, complemented
9	#65897	C₃	1	1	yes	maximal, characteristic, direct product factor, Hall
10	C₂×C₃×C₅×D₄	C₂²	1	1	yes	characteristic
11	C₂²×C₃×C₄×C₅	C₂²	1	2	yes	—
12	C₂×C₃×C₂²⋊D₅	C₂²	1	4	yes	—
13	C₂×C₃×D₂₀	C₂²	1	1	yes	characteristic
14	C₂×C₃×C₄×D₅	C₂²	1	2	yes	—
15	C₂×C₃×Dic₁₀	C₂²	1	2	yes	—
16	C₃×C₅×C₄⋊C₄	C₂²	1	1	yes	characteristic
17	C₃×C₅×C₂²⋊C₄	C₂²	1	2	yes	—
18	C₃×C₄²×C₅	C₂²	1	1	yes	characteristic
19	C₃×C₂²⋊Dic₅	C₂²	1	4	yes	—
20	C₃×#685	C₂²	1	2	yes	—
21	C₃×#685	C₂²	1	4	yes	—
22	C₃×C₄⋊Dic₅	C₂²	1	2	yes	—
23	C₃×C₄⋊Dic₅	C₂²	1	2	yes	—
24	C₃×C₄⋊Dic₅	C₂²	1	1	yes	characteristic
25	C₃×C₄×Dic₅	C₂²	1	4	yes	—
26	C₃×#555	—	5	1	no	maximal, Hall
27	C₄×C₅×D₄	C₂×C₃	1	1	yes	characteristic, complemented
28	#4084	C₂×C₃	1	2	yes	complemented
29	#4076	C₂×C₃	1	2	yes	complemented
30	#4072	C₂×C₃	1	2	yes	complemented
31	#4038	C₂×C₃	1	2	yes	complemented
32	#4031	C₂×C₃	1	4	yes	complemented
33	C₄×D₂₀	C₂×C₃	1	1	yes	characteristic, complemented
34	#4015	C₂×C₃	1	1	yes	characteristic, complemented
35	C₂³×C₃×C₅	C₂³	1	2	yes	—
36	C₂²×C₃×D₅	C₂³	1	2	yes	—
37	C₃×C₅×D₄	—	2	2	no	—
38	C₂×C₃×C₄×C₅	—	2	4	no	—
39	C₂×C₃×C₄×C₅	C₂³	1	2	yes	—
40	C₂×C₃×C₄×C₅	C₂³	1	1	yes	characteristic
41	C₂×C₃×C₄×C₅	C₂³	1	1	yes	characteristic
42	C₂×C₃×C₄×C₅	C₂³	1	1	yes	characteristic
43	C₃×C₂²⋊D₅	—	4	4	no	—
44	C₂×C₃×Dic₅	C₂³	1	2	yes	—
45	C₂×C₃×Dic₅	C₂³	1	4	yes	—
46	C₃×D₂₀	—	2	2	no	—
47	C₃×C₄×D₅	—	2	4	no	—
48	C₃×Dic₁₀	—	4	2	no	—
49	C₃×C₄⋊Q₈	—	5	1	no	—
50	C₃×#124	—	5	4	no	—
51	C₃×#121	—	5	2	no	—
52	C₃×#121	—	5	2	no	—
53	C₃×C₄×D₄	—	5	1	no	—
54	C₃×C₄×D₄	D₅	1	1	yes	characteristic, complemented
55	C₃×#117	—	5	2	no	—
56	C₃×#117	—	5	2	no	—
57	C₂×C₅×D₄	C₂²×C₃	1	1	yes	characteristic
58	C₂²×C₄×C₅	C₂²×C₃	1	2	yes	—
59	C₂×C₂²⋊D₅	C₂²×C₃	1	4	yes	—
60	C₂×D₂₀	C₂²×C₃	1	1	yes	characteristic
61	C₂×C₄×D₅	C₂²×C₃	1	2	yes	—
62	C₂×Dic₁₀	C₂²×C₃	1	2	yes	—
63	C₅×C₄⋊C₄	C₂²×C₃	1	1	yes	characteristic
64	C₅×C₂²⋊C₄	C₂²×C₃	1	2	yes	—
65	C₄²×C₅	C₂²×C₃	1	1	yes	characteristic
66	C₂²⋊Dic₅	C₂²×C₃	1	4	yes	—
67	#685	C₂²×C₃	1	2	yes	—
68	#685	C₂²×C₃	1	4	yes	—
69	C₄⋊Dic₅	C₂²×C₃	1	2	yes	—
70	C₄⋊Dic₅	C₂²×C₃	1	2	yes	—
71	C₄⋊Dic₅	C₂²×C₃	1	1	yes	characteristic
72	C₄×Dic₅	C₂²×C₃	1	4	yes	—
73	#555	—	5	1	no	Sylow
74	C₂²×C₃×C₅	—	2	2	no	—
75	C₂²×C₃×C₅	—	2	2	no	—
76	C₂²×C₃×C₅	—	2	2	no	—
77	C₂²×C₃×C₅	C₂⁴	1	1	yes	characteristic
78	C₂×C₃×D₅	—	2	2	no	—
79	C₂×C₃×D₅	—	2	2	no	—
80	C₂×C₃×D₅	—	2	2	no	—
81	C₃×C₄×C₅	—	2	2	no	—
82	C₃×C₄×C₅	—	2	1	no	—
83	C₃×C₄×C₅	#41	1	2	yes	—
84	C₃×C₄×C₅	#41	1	2	yes	—
85	C₃×Dic₅	—	2	2	no	—
86	C₃×Dic₅	—	2	4	no	—
87	C₂×C₃×Q₈	—	5	2	no	—
88	C₂×C₃×D₄	—	5	4	no	—
89	C₂×C₃×D₄	—	5	1	no	—
90	C₂×C₃×D₄	C₂×D₅	1	1	yes	characteristic
91	C₂²×C₃×C₄	—	5	2	no	—
92	C₂²×C₃×C₄	C₂×D₅	1	2	yes	—
93	C₃×C₄⋊C₄	—	5	1	no	—
94	C₃×C₄⋊C₄	—	5	2	no	—
95	C₃×C₄⋊C₄	—	5	2	no	—
96	C₃×C₄⋊C₄	C₂×D₅	1	1	yes	characteristic
97	C₃×C₂²⋊C₄	—	5	4	no	—
98	C₃×C₂²⋊C₄	—	5	4	no	—
99	C₃×C₂²⋊C₄	—	5	2	no	—
100	C₃×C₂²⋊C₄	C₂×D₅	1	2	yes	—
101	C₃×C₄²	—	5	4	no	—
102	C₃×C₄²	C₂×D₅	1	1	yes	characteristic
103	C₂³×C₅	C₂³×C₃	1	2	yes	—
104	C₂²×D₅	C₂³×C₃	1	2	yes	—
105	C₅×D₄	—	2	2	no	—
106	C₂×C₄×C₅	—	2	4	no	—
107	C₂×C₄×C₅	C₂³×C₃	1	2	yes	—
108	C₂×C₄×C₅	C₂³×C₃	1	1	yes	characteristic
109	C₂×C₄×C₅	C₂³×C₃	1	1	yes	characteristic
110	C₂×C₄×C₅	C₂³×C₃	1	1	yes	characteristic
111	C₂²⋊D₅	—	4	4	no	—
112	C₂×Dic₅	C₂³×C₃	1	2	yes	—
113	C₂×Dic₅	C₂³×C₃	1	4	yes	—
114	D₂₀	—	2	2	no	—
115	C₄×D₅	—	2	4	no	—
116	Dic₁₀	—	4	2	no	—
117	C₄⋊Q₈	—	5	1	no	—
118	#124	—	5	4	no	—
119	#121	—	5	2	no	—
120	#121	—	5	2	no	—
121	C₄×D₄	—	5	1	no	—
122	C₄×D₄	C₃×D₅	1	1	yes	characteristic, complemented
123	#117	—	5	2	no	—
124	#117	—	5	2	no	—
125	C₂×C₃×C₅	—	4	2	no	—
126	C₂×C₃×C₅	C₂×#41	1	1	yes	characteristic
127	C₂×C₃×C₅	C₂×#41	1	1	yes	characteristic
128	C₂×C₃×C₅	H(5,𝔽₂)	1	1	yes	characteristic
129	C₃×D₅	—	4	2	no	—
130	C₂³×C₃	—	5	2	no	—
131	C₂³×C₃	C₂²×D₅	1	2	yes	—
132	C₃×Q₈	—	20	2	no	—
133	C₃×D₄	—	2	2	no	—
134	C₃×D₄	—	10	2	no	—
135	C₃×D₄	—	20	4	no	—
136	C₂×C₃×C₄	—	2	4	no	—
137	C₂×C₃×C₄	—	5	2	no	—
138	C₂×C₃×C₄	—	5	4	no	—
139	C₂×C₃×C₄	—	10	4	no	—
140	C₂×C₃×C₄	C₂²×D₅	1	2	yes	—
141	C₂×C₃×C₄	C₂²×D₅	1	1	yes	characteristic
142	C₂×C₃×C₄	C₂²×D₅	1	1	yes	characteristic
143	C₂×C₃×C₄	C₂²×D₅	1	1	yes	characteristic
144	C₂²×C₅	—	2	2	no	—
145	C₂²×C₅	—	2	2	no	—
146	C₂²×C₅	—	2	2	no	—
147	C₂²×C₅	C₂⁴×C₃	1	1	yes	characteristic
148	C₂×D₅	—	2	2	no	—
149	C₂×D₅	—	2	2	no	—
150	C₂×D₅	—	2	2	no	—
151	C₄×C₅	—	2	2	no	—
152	C₄×C₅	—	2	1	no	—
153	C₄×C₅	C₃×#41	1	2	yes	—
154	C₄×C₅	C₃×#41	1	2	yes	—
155	Dic₅	—	2	2	no	—
156	Dic₅	—	2	4	no	—
157	C₂×Q₈	—	5	2	no	—
158	C₂×D₄	—	5	4	no	—
159	C₂×D₄	—	5	1	no	—
160	C₂×D₄	C₂×C₃×D₅	1	1	yes	characteristic
161	C₂²×C₄	—	5	2	no	—
162	C₂²×C₄	C₂×C₃×D₅	1	2	yes	—
163	C₄⋊C₄	—	5	1	no	—
164	C₄⋊C₄	—	5	2	no	—
165	C₄⋊C₄	—	5	2	no	—
166	C₄⋊C₄	C₂×C₃×D₅	1	1	yes	characteristic
167	C₂²⋊C₄	—	5	4	no	—
168	C₂²⋊C₄	—	5	4	no	—
169	C₂²⋊C₄	—	5	2	no	—
170	C₂²⋊C₄	C₂×C₃×D₅	1	2	yes	—
171	C₄²	—	5	4	no	—
172	C₄²	C₂×C₃×D₅	1	1	yes	characteristic
173	C₃×C₅	#555	1	1	yes	characteristic, complemented, Hall
174	C₂²×C₃	—	2	2	no	—
175	C₂²×C₃	—	2	2	no	—
176	C₂²×C₃	—	2	2	no	—
177	C₂²×C₃	—	10	2	no	—
178	C₂²×C₃	—	10	2	no	—
179	C₂²×C₃	—	10	2	no	—
180	C₂²×C₃	C₂³×D₅	1	1	yes	characteristic
181	C₃×C₄	—	2	2	no	—
182	C₃×C₄	—	2	1	no	—
183	C₃×C₄	—	10	2	no	—
184	C₃×C₄	—	10	4	no	—
185	C₃×C₄	#709	1	2	yes	—
186	C₃×C₄	#711	1	2	yes	—
187	C₂×C₅	—	4	2	no	—
188	C₂×C₅	C₂×C₃×#41	1	1	yes	characteristic
189	C₂×C₅	C₂×C₃×#41	1	1	yes	characteristic
190	C₂×C₅	C₃×H(5,𝔽₂)	1	1	yes	characteristic
191	D₅	—	4	2	no	—
192	C₂³	—	5	2	no	—
193	C₂³	C₂²×C₃×D₅	1	2	yes	—
194	Q₈	—	20	2	no	—
195	D₄	—	2	2	no	—
196	D₄	—	10	2	no	—
197	D₄	—	20	4	no	—
198	C₂×C₄	—	2	4	no	—
199	C₂×C₄	—	5	2	no	—
200	C₂×C₄	—	5	4	no	—
201	C₂×C₄	—	10	4	no	—
202	C₂×C₄	C₂²×C₃×D₅	1	2	yes	—
203	C₂×C₄	C₂²×C₃×D₅	1	1	yes	characteristic
204	C₂×C₄	C₂²×C₃×D₅	1	1	yes	characteristic
205	C₂×C₄	C₂²×C₃×D₅	1	1	yes	characteristic
206	C₂×C₃	—	4	2	no	—
207	C₂×C₃	—	20	2	no	—
208	C₂×C₃	C₂×#709	1	1	yes	characteristic
209	C₂×C₃	C₂×#711	1	1	yes	characteristic
210	C₂×C₃	#4149	1	1	yes	characteristic
211	C₅	C₃×#555	1	1	yes	characteristic, complemented, Sylow
212	C₂²	—	2	2	no	—
213	C₂²	—	2	2	no	—
214	C₂²	—	2	2	no	—
215	C₂²	—	10	2	no	—
216	C₂²	—	10	2	no	—
217	C₂²	—	10	2	no	—
218	C₂²	C₂³×C₃×D₅	1	1	yes	characteristic
219	C₄	—	2	2	no	—
220	C₄	—	2	1	no	—
221	C₄	—	10	2	no	—
222	C₄	—	10	4	no	—
223	C₄	C₃×#709	1	2	yes	—
224	C₄	C₃×#711	1	2	yes	—
225	C₃	#65897	1	1	yes	characteristic, direct product factor, Sylow
226	C₂	—	4	2	no	—
227	C₂	—	20	2	no	—
228	C₂	C₂×C₃×#709	1	1	yes	characteristic
229	C₂	C₂×C₃×#711	1	1	yes	characteristic
230	C₂	C₃×#4149	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×C₄×C₅×D₄	C₂
socle	C₂²×C₃×C₅	C₂⁴
hypercenter	C₃×C₄×D₄	D₅
hypocenter	C₅	C₃×#555
LCS	G ▷ C₂²×C₅ ▷ C₅
UCS	C₁ ◁ C₂²×C₃ ◁ C₃×C₄×D₄

Complements

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

N	H ≅ G/N	type
1. C₃×C₄×C₅×D₄	C₂	semidirect
2. C₃×#4084	C₂	semidirect
3. C₃×#4076	C₂	semidirect
4. C₃×#4072	C₂	semidirect
5. C₃×#4038	C₂	semidirect
6. C₃×#4031	C₂	semidirect
7. C₃×C₄×D₂₀	C₂	semidirect
8. C₃×#4015	C₂	semidirect
9. #65897	C₃	direct (N × H)
27. C₄×C₅×D₄	C₂×C₃	semidirect
28. #4084	C₂×C₃	semidirect
29. #4076	C₂×C₃	semidirect
30. #4072	C₂×C₃	semidirect
31. #4038	C₂×C₃	semidirect
32. #4031	C₂×C₃	semidirect
33. C₄×D₂₀	C₂×C₃	semidirect
34. #4015	C₂×C₃	semidirect
54. C₃×C₄×D₄	D₅	semidirect
122. C₄×D₄	C₃×D₅	semidirect
173. C₃×C₅	#555	semidirect
211. C₅	C₃×#555	semidirect
225. C₃	#65897	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×C₄×C₅×D₄	2
C₃×#4084	2
C₃×#4076	2
C₃×#4072	2
C₃×#4038	2
C₃×#4031	2
C₃×C₄×D₂₀	2
C₃×#4015	2
#65897	3
C₃×#555	5

Subgroups — C3×#65897 #11755574

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₃×#65897 #11755574