Subgroups — C₂×C₃×#4131 #11756963

Order 960.

532 conjugacy classes of subgroups.

The subgroup lattice has 200 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×C₃×#711	C₂	1	1	yes	maximal, characteristic
2	C₂×C₃×#704	C₂	1	2	yes	maximal, complemented
3	C₂²×C₃×#170	C₂	1	2	yes	maximal, complemented
4	C₃×#4131	C₂	1	8	yes	maximal, direct product factor
5	C₂×C₃×#700	C₂	1	1	yes	maximal, characteristic, complemented
6	C₂×C₃×D₅⋊C₈	C₂	1	1	yes	maximal, characteristic, complemented
7	C₂×#4131	C₃	1	1	yes	maximal, characteristic, direct product factor, Hall
8	C₂×C₃×C₅×D₄	C₄	1	1	yes	characteristic
9	C₂×C₃×C₂²⋊D₅	C₄	1	2	yes	—
10	C₂²×C₃×Dic₅	C₂²	1	2	yes	—
11	C₃×#711	C₄	1	4	yes	—
12	C₃×#711	C₂²	1	4	yes	—
13	C₂×C₃×C₄×D₅	C₂²	1	1	yes	characteristic
14	C₂×C₃×Dic₁₀	C₄	1	1	yes	characteristic
15	C₃×#704	C₂²	1	8	yes	complemented
16	C₂×C₃×#170	C₂²	1	1	yes	characteristic
17	C₂×C₃×#170	C₂²	1	1	yes	characteristic
18	C₂×C₃×#170	C₂²	1	2	yes	—
19	C₂×C₃×#170	C₂²	1	8	yes	complemented
20	C₃×#700	C₂²	1	4	yes	complemented
21	C₃×D₅⋊C₈	C₂²	1	4	yes	complemented
22	C₂×C₃×#131	—	5	1	no	maximal, Hall
23	C₂×#711	C₂×C₃	1	1	yes	characteristic
24	C₂×#704	C₂×C₃	1	2	yes	complemented
25	C₂²×#170	C₂×C₃	1	2	yes	complemented
26	#4131	C₂×C₃	1	8	yes	direct product factor
27	C₂×#700	C₂×C₃	1	1	yes	characteristic, complemented
28	C₂×D₅⋊C₈	C₂×C₃	1	1	yes	characteristic, complemented
29	C₂³×C₃×C₅	C₂×C₄	1	2	yes	—
30	C₂²×C₃×D₅	C₂×C₄	1	1	yes	characteristic
31	C₃×C₅×D₄	C₂×C₄	1	4	yes	—
32	C₂×C₃×C₄×C₅	C₂×C₄	1	1	yes	characteristic
33	C₃×C₂²⋊D₅	C₂×C₄	1	8	yes	—
34	C₂×C₃×Dic₅	C₂×C₄	1	2	yes	—
35	C₂×C₃×Dic₅	C₂×C₄	1	4	yes	—
36	C₂×C₃×Dic₅	C₂³	1	1	yes	characteristic
37	C₂×C₃×Dic₅	C₂³	1	4	yes	—
38	C₃×C₄×D₅	C₂×C₄	1	2	yes	—
39	C₃×C₄×D₅	C₂³	1	2	yes	—
40	C₃×Dic₁₀	C₂×C₄	1	4	yes	—
41	C₃×#170	C₂³	1	2	yes	—
42	C₃×#170	C₂³	1	2	yes	—
43	C₃×#170	C₂³	1	4	yes	—
44	C₂×C₃×#41	—	5	1	no	—
45	C₃×#131	—	5	8	no	—
46	C₂×C₃×M₁₆	—	5	1	no	—
47	C₂×C₃×M₁₆	—	5	2	no	—
48	C₂²×C₃×C₈	—	5	1	no	—
49	C₂²×C₃×C₈	—	5	2	no	—
50	C₂×C₅×D₄	C₃×C₄	1	1	yes	characteristic
51	C₂×C₂²⋊D₅	C₃×C₄	1	2	yes	—
52	C₂²×Dic₅	C₂²×C₃	1	2	yes	—
53	#711	C₃×C₄	1	4	yes	—
54	#711	C₂²×C₃	1	4	yes	—
55	C₂×C₄×D₅	C₂²×C₃	1	1	yes	characteristic
56	C₂×Dic₁₀	C₃×C₄	1	1	yes	characteristic
57	#704	C₂²×C₃	1	8	yes	complemented
58	C₂×#170	C₂²×C₃	1	1	yes	characteristic
59	C₂×#170	C₂²×C₃	1	1	yes	characteristic
60	C₂×#170	C₂²×C₃	1	2	yes	—
61	C₂×#170	C₂²×C₃	1	8	yes	complemented
62	#700	C₂²×C₃	1	4	yes	complemented
63	D₅⋊C₈	C₂²×C₃	1	4	yes	complemented
64	C₂×#131	—	5	1	no	Sylow
65	C₂²×C₃×C₅	—	2	4	no	—
66	C₂²×C₃×C₅	C₂²×C₄	1	4	yes	—
67	C₂²×C₃×C₅	C₂²×C₄	1	1	yes	characteristic
68	C₂×C₃×D₅	—	2	2	no	—
69	C₂×C₃×D₅	C₂²×C₄	1	2	yes	—
70	C₃×C₄×C₅	C₂²×C₄	1	2	yes	—
71	C₃×Dic₅	C₂²×C₄	1	4	yes	—
72	C₃×Dic₅	C₂²×C₄	1	1	yes	characteristic
73	C₃×Dic₅	C₂⁴	1	1	yes	characteristic
74	C₃×#41	—	5	4	no	—
75	C₃×#41	—	5	4	no	—
76	C₂×C₃×Q₈	—	5	1	no	—
77	C₂×C₃×D₄	—	5	2	no	—
78	C₂×C₃×D₄	AGL(1,𝔽₅)	1	1	yes	characteristic
79	C₂²×C₃×C₄	—	5	2	no	—
80	C₂²×C₃×C₄	—	5	1	no	—
81	C₃×M₁₆	—	5	4	no	—
82	C₃×M₁₆	—	5	8	no	—
83	C₂×C₃×C₈	—	5	4	no	—
84	C₂×C₃×C₈	—	5	1	no	—
85	C₂×C₃×C₈	—	5	1	no	—
86	C₂×C₃×C₈	—	5	2	no	—
87	C₂×C₃×C₈	—	5	8	no	—
88	C₂³×C₅	C₂×C₃×C₄	1	2	yes	—
89	C₂²×D₅	C₂×C₃×C₄	1	1	yes	characteristic
90	C₅×D₄	C₂×C₃×C₄	1	4	yes	—
91	C₂×C₄×C₅	C₂×C₃×C₄	1	1	yes	characteristic
92	C₂²⋊D₅	C₂×C₃×C₄	1	8	yes	—
93	C₂×Dic₅	C₂×C₃×C₄	1	2	yes	—
94	C₂×Dic₅	C₂×C₃×C₄	1	4	yes	—
95	C₂×Dic₅	C₂³×C₃	1	1	yes	characteristic
96	C₂×Dic₅	C₂³×C₃	1	4	yes	—
97	C₄×D₅	C₂×C₃×C₄	1	2	yes	—
98	C₄×D₅	C₂³×C₃	1	2	yes	—
99	Dic₁₀	C₂×C₃×C₄	1	4	yes	—
100	#170	C₂³×C₃	1	2	yes	—
101	#170	C₂³×C₃	1	2	yes	—
102	#170	C₂³×C₃	1	4	yes	—
103	C₂×#41	—	5	1	no	—
104	#131	—	5	8	no	—
105	C₂×M₁₆	—	5	1	no	—
106	C₂×M₁₆	—	5	2	no	—
107	C₂²×C₈	—	5	1	no	—
108	C₂²×C₈	—	5	2	no	—
109	C₂×C₃×C₅	—	2	4	no	—
110	C₂×C₃×C₅	#131	1	2	yes	complemented
111	C₂×C₃×C₅	C₂³×C₄	1	1	yes	characteristic
112	C₃×D₅	—	2	2	no	—
113	C₂³×C₃	—	5	1	no	—
114	C₂³×C₃	C₂×AGL(1,𝔽₅)	1	2	yes	—
115	C₃×Q₈	—	5	4	no	—
116	C₃×D₄	—	5	8	no	—
117	C₃×D₄	C₂×AGL(1,𝔽₅)	1	4	yes	—
118	C₂×C₃×C₄	—	5	4	no	—
119	C₂×C₃×C₄	—	5	4	no	—
120	C₂×C₃×C₄	—	5	2	no	—
121	C₂×C₃×C₄	—	5	1	no	—
122	C₂×C₃×C₄	—	5	2	no	—
123	C₂×C₃×C₄	—	5	2	no	—
124	C₂×C₃×C₄	C₂×AGL(1,𝔽₅)	1	1	yes	characteristic
125	C₃×C₈	—	5	2	no	—
126	C₃×C₈	—	5	2	no	—
127	C₃×C₈	—	5	4	no	—
128	C₂²×C₅	—	2	4	no	—
129	C₂²×C₅	C₂²×C₃×C₄	1	4	yes	—
130	C₂²×C₅	C₂²×C₃×C₄	1	1	yes	characteristic
131	C₂×D₅	—	2	2	no	—
132	C₂×D₅	C₂²×C₃×C₄	1	2	yes	—
133	C₄×C₅	C₂²×C₃×C₄	1	2	yes	—
134	Dic₅	C₂²×C₃×C₄	1	4	yes	—
135	Dic₅	C₂²×C₃×C₄	1	1	yes	characteristic
136	Dic₅	C₂⁴×C₃	1	1	yes	characteristic
137	#41	—	5	4	no	—
138	#41	—	5	4	no	—
139	C₂×Q₈	—	5	1	no	—
140	C₂×D₄	—	5	2	no	—
141	C₂×D₄	C₃×AGL(1,𝔽₅)	1	1	yes	characteristic
142	C₂²×C₄	—	5	2	no	—
143	C₂²×C₄	—	5	1	no	—
144	M₁₆	—	5	4	no	—
145	M₁₆	—	5	8	no	—
146	C₂×C₈	—	5	1	no	—
147	C₂×C₈	—	5	1	no	—
148	C₂×C₈	—	5	2	no	—
149	C₂×C₈	—	5	4	no	—
150	C₂×C₈	—	5	8	no	—
151	C₃×C₅	C₂×#131	1	1	yes	characteristic, complemented, Hall
152	C₂²×C₃	—	2	4	no	—
153	C₂²×C₃	—	5	2	no	—
154	C₂²×C₃	—	10	2	no	—
155	C₂²×C₃	C₂²×AGL(1,𝔽₅)	1	4	yes	—
156	C₂²×C₃	C₂²×AGL(1,𝔽₅)	1	1	yes	characteristic
157	C₃×C₄	—	5	1	no	—
158	C₃×C₄	—	5	4	no	—
159	C₃×C₄	—	5	1	no	—
160	C₃×C₄	C₂²×AGL(1,𝔽₅)	1	2	yes	—
161	C₂×C₅	—	2	4	no	—
162	C₂×C₅	C₃×#131	1	2	yes	complemented
163	C₂×C₅	C₂³×C₃×C₄	1	1	yes	characteristic
164	D₅	—	2	2	no	—
165	C₂³	—	5	1	no	—
166	C₂³	C₂×C₃×AGL(1,𝔽₅)	1	2	yes	—
167	Q₈	—	5	4	no	—
168	D₄	—	5	8	no	—
169	D₄	C₂×C₃×AGL(1,𝔽₅)	1	4	yes	—
170	C₂×C₄	—	5	1	no	—
171	C₂×C₄	—	5	4	no	—
172	C₂×C₄	—	5	2	no	—
173	C₂×C₄	—	5	4	no	—
174	C₂×C₄	—	5	2	no	—
175	C₂×C₄	—	5	2	no	—
176	C₂×C₄	C₂×C₃×AGL(1,𝔽₅)	1	1	yes	characteristic
177	C₈	—	5	2	no	—
178	C₈	—	5	2	no	—
179	C₈	—	5	4	no	—
180	C₂×C₃	—	2	4	no	—
181	C₂×C₃	—	10	2	no	—
182	C₂×C₃	#4131	1	2	yes	direct product factor
183	C₂×C₃	C₂³×AGL(1,𝔽₅)	1	1	yes	characteristic
184	C₅	C₂×C₃×#131	1	1	yes	characteristic, complemented, Sylow
185	C₂²	—	2	4	no	—
186	C₂²	—	5	2	no	—
187	C₂²	—	10	2	no	—
188	C₂²	C₂²×C₃×AGL(1,𝔽₅)	1	4	yes	—
189	C₂²	C₂²×C₃×AGL(1,𝔽₅)	1	1	yes	characteristic
190	C₄	—	5	1	no	—
191	C₄	—	5	1	no	—
192	C₄	—	5	4	no	—
193	C₄	C₂²×C₃×AGL(1,𝔽₅)	1	2	yes	—
194	C₃	C₂×#4131	1	1	yes	characteristic, direct product factor, Sylow
195	C₂	—	2	4	no	—
196	C₂	—	10	2	no	—
197	C₂	C₃×#4131	1	2	yes	direct product factor
198	C₂	C₂³×C₃×AGL(1,𝔽₅)	1	1	yes	characteristic

	subgroup	G/H
center	C₂²×C₃	C₂²×AGL(1,𝔽₅)
commutator	C₂×C₅	C₂³×C₃×C₄
Frattini	C₂	C₂³×C₃×AGL(1,𝔽₅)
Fitting	C₂×C₃×C₅×D₄	C₄
socle	C₂²×C₃×C₅	C₂²×C₄
prime radical	C₂×C₃×#711	C₂
cosocle	Dic₅	C₂⁴×C₃
hypercenter	C₂×C₃×D₄	AGL(1,𝔽₅)
hypocenter	C₅	C₂×C₃×#131
LCS	G ▷ C₂×C₅ ▷ C₅
UCS	C₁ ◁ C₂²×C₃ ◁ C₂×C₃×D₄

Complements

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

N	H ≅ G/N	type
2. C₂×C₃×#704	C₂	semidirect
3. C₂²×C₃×#170	C₂	semidirect
4. C₃×#4131	C₂	direct (N × H)
5. C₂×C₃×#700	C₂	semidirect
6. C₂×C₃×D₅⋊C₈	C₂	semidirect
7. C₂×#4131	C₃	direct (N × H)
15. C₃×#704	C₂²	semidirect
19. C₂×C₃×#170	C₂²	semidirect
20. C₃×#700	C₂²	semidirect
21. C₃×D₅⋊C₈	C₂²	semidirect
24. C₂×#704	C₂×C₃	semidirect
25. C₂²×#170	C₂×C₃	semidirect
26. #4131	C₂×C₃	direct (N × H)
27. C₂×#700	C₂×C₃	semidirect
28. C₂×D₅⋊C₈	C₂×C₃	semidirect
57. #704	C₂²×C₃	semidirect
61. C₂×#170	C₂²×C₃	semidirect
62. #700	C₂²×C₃	semidirect
63. D₅⋊C₈	C₂²×C₃	semidirect
110. C₂×C₃×C₅	#131	semidirect
151. C₃×C₅	C₂×#131	semidirect
162. C₂×C₅	C₃×#131	semidirect
182. C₂×C₃	#4131	direct (N × H)
184. C₅	C₂×C₃×#131	semidirect
194. C₃	C₂×#4131	direct (N × H)
197. C₂	C₃×#4131	direct (N × H)

Maximal subgroups

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

subgroup	index
C₂×C₃×#711	2
C₂×C₃×#704	2
C₂²×C₃×#170	2
C₃×#4131	2
C₂×C₃×#700	2
C₂×C₃×D₅⋊C₈	2
C₂×#4131	3
C₂×C₃×#131	5

Subgroups — C2×C3×#4131 #11756963

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₂×C₃×#4131 #11756963