Subgroups — C₃×#66139 #11755816

Order 960.

668 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	C₂²×C₃×C₅×D₄	C₂	1	1	yes	maximal, characteristic, complemented
2	C₂×C₃×#709	C₂	1	1	yes	maximal, characteristic, complemented
3	C₃×#4099	C₂	1	4	yes	maximal, complemented
4	C₃×#4086	C₂	1	2	yes	maximal, complemented
5	C₃×#4084	C₂	1	4	yes	maximal, complemented
6	C₃×#4082	C₂	1	2	yes	maximal, complemented
7	C₃×#4072	C₂	1	1	yes	maximal, characteristic, complemented
8	#66139	C₃	1	1	yes	maximal, characteristic, direct product factor, Hall
9	C₂⁴×C₃×C₅	C₂²	1	2	yes	—
10	C₂×C₃×C₅×D₄	—	2	4	no	—
11	C₂×C₃×C₅×D₄	C₂²	1	2	yes	—
12	C₂×C₃×C₅×D₄	C₂²	1	2	yes	—
13	C₂²×C₃×C₄×C₅	C₂²	1	1	yes	characteristic
14	C₂×C₃×C₂²⋊D₅	C₂²	1	2	yes	—
15	C₂×C₃×C₂²⋊D₅	C₂²	1	8	yes	complemented
16	C₃×#709	—	2	4	no	—
17	C₂×C₃×D₂₀	C₂²	1	1	yes	characteristic, complemented
18	C₂×C₃×C₄×D₅	C₂²	1	2	yes	complemented
19	C₂×C₃×Dic₁₀	C₂²	1	1	yes	characteristic, complemented
20	C₃×C₂²⋊Dic₅	C₂²	1	2	yes	—
21	C₃×C₂²⋊Dic₅	C₂²	1	8	yes	complemented
22	C₃×C₄⋊Dic₅	C₂²	1	2	yes	complemented
23	C₃×C₄×Dic₅	C₂²	1	2	yes	complemented
24	C₃×#535	—	5	1	no	maximal, Hall
25	C₂²×C₅×D₄	C₂×C₃	1	1	yes	characteristic, complemented
26	C₂×#709	C₂×C₃	1	1	yes	characteristic, complemented
27	#4099	C₂×C₃	1	4	yes	complemented
28	#4086	C₂×C₃	1	2	yes	complemented
29	#4084	C₂×C₃	1	4	yes	complemented
30	#4082	C₂×C₃	1	2	yes	complemented
31	#4072	C₂×C₃	1	1	yes	characteristic, complemented
32	C₂³×C₃×C₅	—	2	4	no	—
33	C₂³×C₃×C₅	—	4	4	no	—
34	C₂³×C₃×C₅	C₂³	1	4	yes	—
35	C₂³×C₃×C₅	C₂³	1	1	yes	characteristic
36	C₂²×C₃×D₅	C₂³	1	2	yes	—
37	C₃×C₅×D₄	—	2	4	no	—
38	C₃×C₅×D₄	—	2	4	no	—
39	C₂×C₃×C₄×C₅	D₄	1	4	yes	complemented
40	C₂×C₃×C₄×C₅	C₂³	1	1	yes	characteristic
41	C₂×C₃×C₄×C₅	C₂³	1	1	yes	characteristic
42	C₃×C₂²⋊D₅	—	2	4	no	—
43	C₃×C₂²⋊D₅	—	4	8	no	—
44	C₂×C₃×Dic₅	C₂³	1	4	yes	—
45	C₂×C₃×Dic₅	C₂³	1	2	yes	—
46	C₃×D₂₀	—	2	2	no	—
47	C₃×C₄×D₅	—	2	4	no	—
48	C₃×Dic₁₀	—	2	2	no	—
49	C₂×C₃×#41	—	5	1	no	—
50	C₂²×C₃×D₄	D₅	1	1	yes	characteristic, complemented
51	C₃×#127	—	5	2	no	—
52	C₃×#124	—	5	2	no	—
53	C₃×#121	—	5	4	no	—
54	C₃×#120	—	5	4	no	—
55	C₃×#117	—	5	1	no	—
56	C₂⁴×C₅	C₂²×C₃	1	2	yes	—
57	C₂×C₅×D₄	—	2	4	no	—
58	C₂×C₅×D₄	C₂²×C₃	1	2	yes	—
59	C₂×C₅×D₄	C₂²×C₃	1	2	yes	—
60	C₂²×C₄×C₅	C₂²×C₃	1	1	yes	characteristic
61	C₂×C₂²⋊D₅	C₂²×C₃	1	2	yes	—
62	C₂×C₂²⋊D₅	C₂²×C₃	1	8	yes	complemented
63	#709	—	2	4	no	—
64	C₂×D₂₀	C₂²×C₃	1	1	yes	characteristic, complemented
65	C₂×C₄×D₅	C₂²×C₃	1	2	yes	complemented
66	C₂×Dic₁₀	C₂²×C₃	1	1	yes	characteristic, complemented
67	C₂²⋊Dic₅	C₂²×C₃	1	2	yes	—
68	C₂²⋊Dic₅	C₂²×C₃	1	8	yes	complemented
69	C₄⋊Dic₅	C₂²×C₃	1	2	yes	complemented
70	C₄×Dic₅	C₂²×C₃	1	2	yes	complemented
71	#535	—	5	1	no	Sylow
72	C₂²×C₃×C₅	—	2	2	no	—
73	C₂²×C₃×C₅	—	2	4	no	—
74	C₂²×C₃×C₅	—	2	8	no	—
75	C₂²×C₃×C₅	—	4	8	no	—
76	C₂²×C₃×C₅	C₂×D₄	1	2	yes	—
77	C₂²×C₃×C₅	C₂⁴	1	1	yes	characteristic
78	C₂×C₃×D₅	—	2	2	no	—
79	C₂×C₃×D₅	—	2	4	no	—
80	C₃×C₄×C₅	C₂×D₄	1	2	yes	—
81	C₃×C₄×C₅	C₂×D₄	1	2	yes	—
82	C₃×Dic₅	—	2	2	no	—
83	C₃×Dic₅	—	2	4	no	—
84	C₂⁴×C₃	C₂×D₅	1	2	yes	—
85	C₃×#41	—	10	4	no	—
86	C₂×C₃×Q₈	—	5	1	no	—
87	C₂×C₃×D₄	—	2	4	no	—
88	C₂×C₃×D₄	—	5	2	no	—
89	C₂×C₃×D₄	—	5	8	no	—
90	C₂×C₃×D₄	—	5	1	no	—
91	C₂×C₃×D₄	C₂×D₅	1	2	yes	—
92	C₂×C₃×D₄	C₂×D₅	1	2	yes	—
93	C₂²×C₃×C₄	—	5	2	no	—
94	C₂²×C₃×C₄	C₂×D₅	1	1	yes	characteristic
95	C₃×C₄⋊C₄	—	5	2	no	—
96	C₃×C₂²⋊C₄	—	5	2	no	—
97	C₃×C₂²⋊C₄	—	5	8	no	—
98	C₃×C₄²	—	5	2	no	—
99	C₂³×C₅	—	2	4	no	—
100	C₂³×C₅	—	4	4	no	—
101	C₂³×C₅	C₂³×C₃	1	4	yes	—
102	C₂³×C₅	C₂³×C₃	1	1	yes	characteristic
103	C₂²×D₅	C₂³×C₃	1	2	yes	—
104	C₅×D₄	—	2	4	no	—
105	C₅×D₄	—	2	4	no	—
106	C₂×C₄×C₅	C₃×D₄	1	4	yes	complemented
107	C₂×C₄×C₅	C₂³×C₃	1	1	yes	characteristic
108	C₂×C₄×C₅	C₂³×C₃	1	1	yes	characteristic
109	C₂²⋊D₅	—	2	4	no	—
110	C₂²⋊D₅	—	4	8	no	—
111	C₂×Dic₅	C₂³×C₃	1	4	yes	—
112	C₂×Dic₅	C₂³×C₃	1	2	yes	—
113	D₂₀	—	2	2	no	—
114	C₄×D₅	—	2	4	no	—
115	Dic₁₀	—	2	2	no	—
116	C₂×#41	—	5	1	no	—
117	C₂²×D₄	C₃×D₅	1	1	yes	characteristic, complemented
118	#127	—	5	2	no	—
119	#124	—	5	2	no	—
120	#121	—	5	4	no	—
121	#120	—	5	4	no	—
122	#117	—	5	1	no	—
123	C₂×C₃×C₅	—	2	2	no	—
124	C₂×C₃×C₅	—	4	4	no	—
125	C₂×C₃×C₅	C₂²×D₄	1	1	yes	characteristic
126	C₂×C₃×C₅	H(5,𝔽₂)	1	2	yes	—
127	C₃×D₅	—	4	2	no	—
128	C₂³×C₃	—	2	4	no	—
129	C₂³×C₃	—	4	4	no	—
130	C₂³×C₃	—	5	2	no	—
131	C₂³×C₃	C₂²×D₅	1	4	yes	—
132	C₂³×C₃	C₂²×D₅	1	1	yes	characteristic
133	C₃×Q₈	—	10	2	no	—
134	C₃×D₄	—	2	4	no	—
135	C₃×D₄	—	2	4	no	—
136	C₃×D₄	—	10	4	no	—
137	C₃×D₄	—	10	2	no	—
138	C₃×D₄	—	20	8	no	—
139	C₂×C₃×C₄	—	5	4	no	—
140	C₂×C₃×C₄	—	5	2	no	—
141	C₂×C₃×C₄	—	10	4	no	—
142	C₂×C₃×C₄	C₂²⋊D₅	1	4	yes	complemented
143	C₂×C₃×C₄	C₂²×D₅	1	1	yes	characteristic
144	C₂×C₃×C₄	C₂²×D₅	1	1	yes	characteristic
145	C₂²×C₅	—	2	2	no	—
146	C₂²×C₅	—	2	4	no	—
147	C₂²×C₅	—	2	8	no	—
148	C₂²×C₅	—	4	8	no	—
149	C₂²×C₅	C₂×C₃×D₄	1	2	yes	—
150	C₂²×C₅	C₂⁴×C₃	1	1	yes	characteristic
151	C₂×D₅	—	2	2	no	—
152	C₂×D₅	—	2	4	no	—
153	C₄×C₅	C₂×C₃×D₄	1	2	yes	—
154	C₄×C₅	C₂×C₃×D₄	1	2	yes	—
155	Dic₅	—	2	2	no	—
156	Dic₅	—	2	4	no	—
157	C₂⁴	C₂×C₃×D₅	1	2	yes	—
158	#41	—	10	4	no	—
159	C₂×Q₈	—	5	1	no	—
160	C₂×D₄	—	2	4	no	—
161	C₂×D₄	—	5	2	no	—
162	C₂×D₄	—	5	8	no	—
163	C₂×D₄	—	5	1	no	—
164	C₂×D₄	C₂×C₃×D₅	1	2	yes	—
165	C₂×D₄	C₂×C₃×D₅	1	2	yes	—
166	C₂²×C₄	—	5	2	no	—
167	C₂²×C₄	C₂×C₃×D₅	1	1	yes	characteristic
168	C₄⋊C₄	—	5	2	no	—
169	C₂²⋊C₄	—	5	2	no	—
170	C₂²⋊C₄	—	5	8	no	—
171	C₄²	—	5	2	no	—
172	C₃×C₅	#535	1	1	yes	characteristic, complemented, Hall
173	C₂²×C₃	—	2	2	no	—
174	C₂²×C₃	—	2	8	no	—
175	C₂²×C₃	—	2	4	no	—
176	C₂²×C₃	—	4	8	no	—
177	C₂²×C₃	—	10	4	no	—
178	C₂²×C₃	—	10	2	no	—
179	C₂²×C₃	C₂×C₂²⋊D₅	1	2	yes	—
180	C₂²×C₃	C₂³×D₅	1	1	yes	characteristic
181	C₃×C₄	—	10	4	no	—
182	C₃×C₄	—	10	2	no	—
183	C₃×C₄	C₂×C₂²⋊D₅	1	2	yes	—
184	C₃×C₄	C₂×C₂²⋊D₅	1	2	yes	—
185	C₂×C₅	—	2	2	no	—
186	C₂×C₅	—	4	4	no	—
187	C₂×C₅	C₂²×C₃×D₄	1	1	yes	characteristic
188	C₂×C₅	C₃×H(5,𝔽₂)	1	2	yes	—
189	D₅	—	4	2	no	—
190	C₂³	—	2	4	no	—
191	C₂³	—	4	4	no	—
192	C₂³	—	5	2	no	—
193	C₂³	C₂²×C₃×D₅	1	4	yes	—
194	C₂³	C₂²×C₃×D₅	1	1	yes	characteristic
195	Q₈	—	10	2	no	—
196	D₄	—	2	4	no	—
197	D₄	—	2	4	no	—
198	D₄	—	10	4	no	—
199	D₄	—	10	2	no	—
200	D₄	—	20	8	no	—
201	C₂×C₄	—	5	4	no	—
202	C₂×C₄	—	5	2	no	—
203	C₂×C₄	—	10	4	no	—
204	C₂×C₄	C₃×C₂²⋊D₅	1	4	yes	complemented
205	C₂×C₄	C₂²×C₃×D₅	1	1	yes	characteristic
206	C₂×C₄	C₂²×C₃×D₅	1	1	yes	characteristic
207	C₂×C₃	—	2	2	no	—
208	C₂×C₃	—	4	4	no	—
209	C₂×C₃	—	20	2	no	—
210	C₂×C₃	#4144	1	2	yes	—
211	C₂×C₃	C₂²×C₂²⋊D₅	1	1	yes	characteristic
212	C₅	C₃×#535	1	1	yes	characteristic, complemented, Sylow
213	C₂²	—	2	2	no	—
214	C₂²	—	2	8	no	—
215	C₂²	—	2	4	no	—
216	C₂²	—	4	8	no	—
217	C₂²	—	10	4	no	—
218	C₂²	—	10	2	no	—
219	C₂²	C₂×C₃×C₂²⋊D₅	1	2	yes	—
220	C₂²	C₂³×C₃×D₅	1	1	yes	characteristic
221	C₄	—	10	4	no	—
222	C₄	—	10	2	no	—
223	C₄	C₂×C₃×C₂²⋊D₅	1	2	yes	—
224	C₄	C₂×C₃×C₂²⋊D₅	1	2	yes	—
225	C₃	#66139	1	1	yes	characteristic, direct product factor, Sylow
226	C₂	—	2	2	no	—
227	C₂	—	4	4	no	—
228	C₂	—	20	2	no	—
229	C₂	C₃×#4144	1	2	yes	—
230	C₂	C₂²×C₃×C₂²⋊D₅	1	1	yes	characteristic

	subgroup	G/H
center	C₂²×C₃	C₂³×D₅
commutator, cosocle	C₂²×C₅	C₂⁴×C₃
Frattini	C₂²	C₂³×C₃×D₅
Fitting	C₂²×C₃×C₅×D₄	C₂
socle	C₂²×C₃×C₅	C₂⁴
hypercenter	C₂²×C₃×D₄	D₅
hypocenter	C₅	C₃×#535
LCS	G ▷ C₂²×C₅ ▷ C₅
UCS	C₁ ◁ C₂²×C₃ ◁ C₂²×C₃×D₄

Complements

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

N	H ≅ G/N	type
1. C₂²×C₃×C₅×D₄	C₂	semidirect
2. C₂×C₃×#709	C₂	semidirect
3. C₃×#4099	C₂	semidirect
4. C₃×#4086	C₂	semidirect
5. C₃×#4084	C₂	semidirect
6. C₃×#4082	C₂	semidirect
7. C₃×#4072	C₂	semidirect
8. #66139	C₃	direct (N × H)
15. C₂×C₃×C₂²⋊D₅	C₂²	semidirect
17. C₂×C₃×D₂₀	C₂²	semidirect
18. C₂×C₃×C₄×D₅	C₂²	semidirect
19. C₂×C₃×Dic₁₀	C₂²	semidirect
21. C₃×C₂²⋊Dic₅	C₂²	semidirect
22. C₃×C₄⋊Dic₅	C₂²	semidirect
23. C₃×C₄×Dic₅	C₂²	semidirect
25. C₂²×C₅×D₄	C₂×C₃	semidirect
26. C₂×#709	C₂×C₃	semidirect
27. #4099	C₂×C₃	semidirect
28. #4086	C₂×C₃	semidirect
29. #4084	C₂×C₃	semidirect
30. #4082	C₂×C₃	semidirect
31. #4072	C₂×C₃	semidirect
39. C₂×C₃×C₄×C₅	D₄	semidirect
50. C₂²×C₃×D₄	D₅	semidirect
62. C₂×C₂²⋊D₅	C₂²×C₃	semidirect
64. C₂×D₂₀	C₂²×C₃	semidirect
65. C₂×C₄×D₅	C₂²×C₃	semidirect
66. C₂×Dic₁₀	C₂²×C₃	semidirect
68. C₂²⋊Dic₅	C₂²×C₃	semidirect
69. C₄⋊Dic₅	C₂²×C₃	semidirect
70. C₄×Dic₅	C₂²×C₃	semidirect
106. C₂×C₄×C₅	C₃×D₄	semidirect
117. C₂²×D₄	C₃×D₅	semidirect
142. C₂×C₃×C₄	C₂²⋊D₅	semidirect
172. C₃×C₅	#535	semidirect
204. C₂×C₄	C₃×C₂²⋊D₅	semidirect
212. C₅	C₃×#535	semidirect
225. C₃	#66139	direct (N × H)

Maximal subgroups

By isomorphism type — maximal subgroups of the same type appear once, whether or not they are normal.

subgroup	index
C₂²×C₃×C₅×D₄	2
C₂×C₃×#709	2
C₃×#4099	2
C₃×#4086	2
C₃×#4084	2
C₃×#4082	2
C₃×#4072	2
#66139	3
C₃×#535	5

Subgroups — C3×#66139 #11755816

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₃×#66139 #11755816