Subgroups — C₂×C₃×#4081 #11755807

Order 960.

684 conjugacy classes of subgroups.

The subgroup lattice has 208 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
2	C₂³×C₃×Dic₅	C₂	1	1	yes	maximal, characteristic, complemented
3	C₂×C₃×C₂²⋊Dic₅	C₂	1	1	yes	maximal, characteristic, complemented
4	C₂×C₃×C₂²⋊Dic₅	C₂	1	2	yes	maximal, complemented
5	C₃×#4081	C₂	1	8	yes	maximal, direct product factor
6	C₂×C₃×#683	C₂	1	2	yes	maximal, complemented
7	C₂×#4081	C₃	1	1	yes	maximal, characteristic, direct product factor, Hall
8	C₂⁴×C₃×C₅	C₂²	1	1	yes	characteristic
9	C₂⁴×C₃×C₅	C₂²	1	1	yes	characteristic
10	C₂×C₃×C₅×D₄	—	2	4	no	—
11	C₂×C₃×C₅×D₄	C₂²	1	4	yes	—
12	C₂²×C₃×C₄×C₅	C₂²	1	1	yes	characteristic
13	C₂²×C₃×Dic₅	—	2	4	no	—
14	C₂²×C₃×Dic₅	C₂²	1	2	yes	—
15	C₂²×C₃×Dic₅	C₂²	1	2	yes	—
16	C₂²×C₃×Dic₅	C₂²	1	4	yes	complemented
17	C₃×C₂²⋊Dic₅	C₂²	1	4	yes	complemented
18	C₃×C₂²⋊Dic₅	C₂²	1	8	yes	complemented
19	C₃×#683	C₂²	1	8	yes	complemented
20	C₂×C₃×#123	—	5	1	no	maximal, Hall
21	C₂²×C₅×D₄	C₂×C₃	1	1	yes	characteristic
22	C₂³×Dic₅	C₂×C₃	1	1	yes	characteristic, complemented
23	C₂×C₂²⋊Dic₅	C₂×C₃	1	1	yes	characteristic, complemented
24	C₂×C₂²⋊Dic₅	C₂×C₃	1	2	yes	complemented
25	#4081	C₂×C₃	1	8	yes	direct product factor
26	C₂×#683	C₂×C₃	1	2	yes	complemented
27	C₂³×C₃×C₅	—	2	4	no	—
28	C₂³×C₃×C₅	—	2	2	no	—
29	C₂³×C₃×C₅	—	2	4	no	—
30	C₂³×C₃×C₅	D₄	1	4	yes	—
31	C₂³×C₃×C₅	C₂³	1	2	yes	—
32	C₂³×C₃×C₅	C₂³	1	2	yes	—
33	C₂³×C₃×C₅	C₂³	1	1	yes	characteristic
34	C₃×C₅×D₄	—	2	8	no	—
35	C₂×C₃×C₄×C₅	—	2	2	no	—
36	C₂×C₃×C₄×C₅	C₂³	1	2	yes	—
37	C₂×C₃×Dic₅	—	2	4	no	—
38	C₂×C₃×Dic₅	—	2	8	no	—
39	C₂×C₃×Dic₅	—	2	4	no	—
40	C₂×C₃×Dic₅	C₂³	1	4	yes	—
41	C₂×C₃×Dic₅	C₂³	1	4	yes	—
42	C₂²×C₃×D₄	D₅	1	1	yes	characteristic
43	C₂³×C₃×C₄	—	5	1	no	—
44	C₃×#123	—	5	8	no	—
45	C₂×C₃×C₄⋊C₄	—	5	2	no	—
46	C₂×C₃×C₂²⋊C₄	—	5	1	no	—
47	C₂×C₃×C₂²⋊C₄	—	5	2	no	—
48	C₂⁴×C₅	C₂²×C₃	1	1	yes	characteristic
49	C₂⁴×C₅	C₂²×C₃	1	1	yes	characteristic
50	C₂×C₅×D₄	—	2	4	no	—
51	C₂×C₅×D₄	C₂²×C₃	1	4	yes	—
52	C₂²×C₄×C₅	C₂²×C₃	1	1	yes	characteristic
53	C₂²×Dic₅	—	2	4	no	—
54	C₂²×Dic₅	C₂²×C₃	1	2	yes	—
55	C₂²×Dic₅	C₂²×C₃	1	2	yes	—
56	C₂²×Dic₅	C₂²×C₃	1	4	yes	complemented
57	C₂²⋊Dic₅	C₂²×C₃	1	4	yes	complemented
58	C₂²⋊Dic₅	C₂²×C₃	1	8	yes	complemented
59	#683	C₂²×C₃	1	8	yes	complemented
60	C₂×#123	—	5	1	no	Sylow
61	C₂²×C₃×C₅	—	2	8	no	—
62	C₂²×C₃×C₅	—	2	4	no	—
63	C₂²×C₃×C₅	—	2	2	no	—
64	C₂²×C₃×C₅	—	2	4	no	—
65	C₂²×C₃×C₅	—	4	4	no	—
66	C₂²×C₃×C₅	C₂×D₄	1	4	yes	—
67	C₂²×C₃×C₅	C₂×D₄	1	2	yes	—
68	C₂²×C₃×C₅	#41	1	4	yes	—
69	C₂²×C₃×C₅	C₂⁴	1	1	yes	characteristic
70	C₃×C₄×C₅	—	2	2	no	—
71	C₃×Dic₅	—	2	4	no	—
72	C₃×Dic₅	—	2	4	no	—
73	C₂⁴×C₃	C₂×D₅	1	1	yes	characteristic
74	C₂⁴×C₃	C₂×D₅	1	1	yes	characteristic
75	C₂×C₃×D₄	—	2	4	no	—
76	C₂×C₃×D₄	C₂×D₅	1	4	yes	—
77	C₂²×C₃×C₄	—	5	4	no	—
78	C₂²×C₃×C₄	—	5	2	no	—
79	C₂²×C₃×C₄	—	5	2	no	—
80	C₂²×C₃×C₄	—	10	4	no	—
81	C₂²×C₃×C₄	C₂×D₅	1	1	yes	characteristic
82	C₃×C₄⋊C₄	—	5	8	no	—
83	C₃×C₂²⋊C₄	—	5	4	no	—
84	C₃×C₂²⋊C₄	—	5	8	no	—
85	C₂³×C₅	—	2	4	no	—
86	C₂³×C₅	—	2	2	no	—
87	C₂³×C₅	—	2	4	no	—
88	C₂³×C₅	C₃×D₄	1	4	yes	—
89	C₂³×C₅	C₂³×C₃	1	2	yes	—
90	C₂³×C₅	C₂³×C₃	1	2	yes	—
91	C₂³×C₅	C₂³×C₃	1	1	yes	characteristic
92	C₅×D₄	—	2	8	no	—
93	C₂×C₄×C₅	—	2	2	no	—
94	C₂×C₄×C₅	C₂³×C₃	1	2	yes	—
95	C₂×Dic₅	—	2	4	no	—
96	C₂×Dic₅	—	2	8	no	—
97	C₂×Dic₅	—	2	4	no	—
98	C₂×Dic₅	C₂³×C₃	1	4	yes	—
99	C₂×Dic₅	C₂³×C₃	1	4	yes	—
100	C₂²×D₄	C₃×D₅	1	1	yes	characteristic
101	C₂³×C₄	—	5	1	no	—
102	#123	—	5	8	no	—
103	C₂×C₄⋊C₄	—	5	2	no	—
104	C₂×C₂²⋊C₄	—	5	1	no	—
105	C₂×C₂²⋊C₄	—	5	2	no	—
106	C₂×C₃×C₅	—	2	4	no	—
107	C₂×C₃×C₅	—	4	2	no	—
108	C₂×C₃×C₅	#123	1	4	yes	complemented
109	C₂×C₃×C₅	C₂²×D₄	1	1	yes	characteristic
110	C₂×C₃×C₅	C₂×#41	1	2	yes	—
111	C₂³×C₃	—	2	4	no	—
112	C₂³×C₃	—	2	4	no	—
113	C₂³×C₃	—	2	2	no	—
114	C₂³×C₃	C₂²⋊D₅	1	4	yes	—
115	C₂³×C₃	C₂²×D₅	1	2	yes	—
116	C₂³×C₃	C₂²×D₅	1	2	yes	—
117	C₂³×C₃	C₂²×D₅	1	1	yes	characteristic
118	C₃×D₄	—	2	8	no	—
119	C₂×C₃×C₄	—	2	2	no	—
120	C₂×C₃×C₄	—	5	4	no	—
121	C₂×C₃×C₄	—	5	4	no	—
122	C₂×C₃×C₄	—	10	4	no	—
123	C₂×C₃×C₄	—	10	8	no	—
124	C₂×C₃×C₄	—	10	4	no	—
125	C₂×C₃×C₄	C₂²×D₅	1	2	yes	—
126	C₂²×C₅	—	2	8	no	—
127	C₂²×C₅	—	2	4	no	—
128	C₂²×C₅	—	2	2	no	—
129	C₂²×C₅	—	2	4	no	—
130	C₂²×C₅	—	4	4	no	—
131	C₂²×C₅	C₂×C₃×D₄	1	4	yes	—
132	C₂²×C₅	C₂×C₃×D₄	1	2	yes	—
133	C₂²×C₅	C₃×#41	1	4	yes	—
134	C₂²×C₅	C₂⁴×C₃	1	1	yes	characteristic
135	C₄×C₅	—	2	2	no	—
136	Dic₅	—	2	4	no	—
137	Dic₅	—	2	4	no	—
138	C₂⁴	C₂×C₃×D₅	1	1	yes	characteristic
139	C₂⁴	C₂×C₃×D₅	1	1	yes	characteristic
140	C₂×D₄	—	2	4	no	—
141	C₂×D₄	C₂×C₃×D₅	1	4	yes	—
142	C₂²×C₄	—	5	2	no	—
143	C₂²×C₄	—	5	2	no	—
144	C₂²×C₄	—	5	4	no	—
145	C₂²×C₄	—	10	4	no	—
146	C₂²×C₄	C₂×C₃×D₅	1	1	yes	characteristic
147	C₄⋊C₄	—	5	8	no	—
148	C₂²⋊C₄	—	5	4	no	—
149	C₂²⋊C₄	—	5	8	no	—
150	C₃×C₅	C₂×#123	1	1	yes	characteristic, complemented, Hall
151	C₂²×C₃	—	2	4	no	—
152	C₂²×C₃	—	2	8	no	—
153	C₂²×C₃	—	2	4	no	—
154	C₂²×C₃	—	2	2	no	—
155	C₂²×C₃	—	4	4	no	—
156	C₂²×C₃	#711	1	4	yes	—
157	C₂²×C₃	C₂×C₂²⋊D₅	1	4	yes	—
158	C₂²×C₃	C₂×C₂²⋊D₅	1	2	yes	—
159	C₂²×C₃	C₂³×D₅	1	1	yes	characteristic
160	C₃×C₄	—	2	2	no	—
161	C₃×C₄	—	10	4	no	—
162	C₃×C₄	—	10	4	no	—
163	C₂×C₅	—	2	4	no	—
164	C₂×C₅	—	4	2	no	—
165	C₂×C₅	C₃×#123	1	4	yes	complemented
166	C₂×C₅	C₂²×C₃×D₄	1	1	yes	characteristic
167	C₂×C₅	C₂×C₃×#41	1	2	yes	—
168	C₂³	—	2	4	no	—
169	C₂³	—	2	4	no	—
170	C₂³	—	2	2	no	—
171	C₂³	C₃×C₂²⋊D₅	1	4	yes	—
172	C₂³	C₂²×C₃×D₅	1	2	yes	—
173	C₂³	C₂²×C₃×D₅	1	2	yes	—
174	C₂³	C₂²×C₃×D₅	1	1	yes	characteristic
175	D₄	—	2	8	no	—
176	C₂×C₄	—	2	2	no	—
177	C₂×C₄	—	5	4	no	—
178	C₂×C₄	—	5	4	no	—
179	C₂×C₄	—	10	4	no	—
180	C₂×C₄	—	10	8	no	—
181	C₂×C₄	—	10	4	no	—
182	C₂×C₄	C₂²×C₃×D₅	1	2	yes	—
183	C₂×C₃	—	2	4	no	—
184	C₂×C₃	—	4	2	no	—
185	C₂×C₃	#4081	1	4	yes	direct product factor
186	C₂×C₃	C₂×#711	1	2	yes	—
187	C₂×C₃	C₂²×C₂²⋊D₅	1	1	yes	characteristic
188	C₅	C₂×C₃×#123	1	1	yes	characteristic, complemented, Sylow
189	C₂²	—	2	8	no	—
190	C₂²	—	2	4	no	—
191	C₂²	—	2	4	no	—
192	C₂²	—	2	2	no	—
193	C₂²	—	4	4	no	—
194	C₂²	C₃×#711	1	4	yes	—
195	C₂²	C₂×C₃×C₂²⋊D₅	1	4	yes	—
196	C₂²	C₂×C₃×C₂²⋊D₅	1	2	yes	—
197	C₂²	C₂³×C₃×D₅	1	1	yes	characteristic
198	C₄	—	2	2	no	—
199	C₄	—	10	4	no	—
200	C₄	—	10	4	no	—
201	C₃	C₂×#4081	1	1	yes	characteristic, direct product factor, Sylow
202	C₂	—	2	4	no	—
203	C₂	—	4	2	no	—
204	C₂	C₃×#4081	1	4	yes	direct product factor
205	C₂	C₂×C₃×#711	1	2	yes	—
206	C₂	C₂²×C₃×C₂²⋊D₅	1	1	yes	characteristic

	subgroup	G/H
center	C₂³×C₃	C₂²×D₅
commutator, cosocle	C₂²×C₅	C₂⁴×C₃
Frattini	C₂²	C₂³×C₃×D₅
Fitting, prime radical	C₂²×C₃×C₅×D₄	C₂
socle	C₂³×C₃×C₅	C₂³
hypercenter	C₂²×C₃×D₄	D₅
hypocenter	C₅	C₂×C₃×#123
LCS	G ▷ C₂²×C₅ ▷ C₅
UCS	C₁ ◁ C₂³×C₃ ◁ C₂²×C₃×D₄

Complements

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

N	H ≅ G/N	type
2. C₂³×C₃×Dic₅	C₂	semidirect
3. C₂×C₃×C₂²⋊Dic₅	C₂	semidirect
4. C₂×C₃×C₂²⋊Dic₅	C₂	semidirect
5. C₃×#4081	C₂	direct (N × H)
6. C₂×C₃×#683	C₂	semidirect
7. C₂×#4081	C₃	direct (N × H)
16. C₂²×C₃×Dic₅	C₂²	semidirect
17. C₃×C₂²⋊Dic₅	C₂²	semidirect
18. C₃×C₂²⋊Dic₅	C₂²	semidirect
19. C₃×#683	C₂²	semidirect
22. C₂³×Dic₅	C₂×C₃	semidirect
23. C₂×C₂²⋊Dic₅	C₂×C₃	semidirect
24. C₂×C₂²⋊Dic₅	C₂×C₃	semidirect
25. #4081	C₂×C₃	direct (N × H)
26. C₂×#683	C₂×C₃	semidirect
56. C₂²×Dic₅	C₂²×C₃	semidirect
57. C₂²⋊Dic₅	C₂²×C₃	semidirect
58. C₂²⋊Dic₅	C₂²×C₃	semidirect
59. #683	C₂²×C₃	semidirect
108. C₂×C₃×C₅	#123	semidirect
150. C₃×C₅	C₂×#123	semidirect
165. C₂×C₅	C₃×#123	semidirect
185. C₂×C₃	#4081	direct (N × H)
188. C₅	C₂×C₃×#123	semidirect
201. C₃	C₂×#4081	direct (N × H)
204. C₂	C₃×#4081	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
C₂³×C₃×Dic₅	2
C₂×C₃×C₂²⋊Dic₅	2
C₃×#4081	2
C₂×C₃×#683	2
C₂×#4081	3
C₂×C₃×#123	5

Subgroups — C2×C3×#4081 #11755807

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₂×C₃×#4081 #11755807