Subgroups — C₂³×#4131 #13046027

Order 1280.

11422 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	C₂³×#711	C₂	1	1	yes	maximal, characteristic
2	C₂³×#704	C₂	1	2	yes	maximal, complemented
3	C₂⁴×#170	C₂	1	2	yes	maximal, complemented
4	C₂²×#4131	C₂	1	56	yes	maximal, direct product factor
5	C₂³×#700	C₂	1	1	yes	maximal, characteristic, complemented
6	C₂³×D₅⋊C₈	C₂	1	1	yes	maximal, characteristic, complemented
7	C₂³×C₅×D₄	C₄	1	1	yes	characteristic
8	C₂³×C₂²⋊D₅	C₄	1	2	yes	—
9	C₂⁴×Dic₅	C₂²	1	2	yes	—
10	C₂²×#711	C₄	1	28	yes	—
11	C₂²×#711	C₂²	1	28	yes	—
12	C₂³×C₄×D₅	C₂²	1	1	yes	characteristic
13	C₂³×Dic₁₀	C₄	1	1	yes	characteristic
14	C₂²×#704	C₂²	1	56	yes	complemented
15	C₂³×#170	C₂²	1	1	yes	characteristic
16	C₂³×#170	C₂²	1	1	yes	characteristic
17	C₂³×#170	C₂²	1	2	yes	—
18	C₂³×#170	C₂²	1	56	yes	complemented
19	C₂×#4131	C₂²	1	448	yes	direct product factor
20	C₂²×#700	C₂²	1	28	yes	complemented
21	C₂²×D₅⋊C₈	C₂²	1	28	yes	complemented
22	C₂³×#131	—	5	1	no	maximal, Sylow
23	C₂⁵×C₅	C₂×C₄	1	2	yes	—
24	C₂⁴×D₅	C₂×C₄	1	1	yes	characteristic
25	C₂²×C₅×D₄	C₂×C₄	1	28	yes	—
26	C₂³×C₄×C₅	C₂×C₄	1	1	yes	characteristic
27	C₂²×C₂²⋊D₅	C₂×C₄	1	56	yes	—
28	C₂³×Dic₅	C₂×C₄	1	2	yes	—
29	C₂³×Dic₅	C₂×C₄	1	28	yes	—
30	C₂³×Dic₅	C₂³	1	1	yes	characteristic
31	C₂³×Dic₅	C₂³	1	28	yes	—
32	C₂×#711	C₂×C₄	1	336	yes	—
33	C₂×#711	C₂³	1	112	yes	—
34	C₂²×C₄×D₅	C₂×C₄	1	14	yes	—
35	C₂²×C₄×D₅	C₂³	1	14	yes	—
36	C₂²×Dic₁₀	C₂×C₄	1	28	yes	—
37	C₂×#704	C₂³	1	224	yes	complemented
38	C₂²×#170	C₂³	1	14	yes	—
39	C₂²×#170	C₂³	1	14	yes	—
40	C₂²×#170	C₂³	1	28	yes	—
41	C₂²×#170	C₂³	1	224	yes	complemented
42	#4131	C₂³	1	512	yes	direct product factor
43	C₂×#700	C₂³	1	112	yes	complemented
44	C₂×D₅⋊C₈	C₂³	1	112	yes	complemented
45	C₂³×#41	—	5	1	no	—
46	C₂²×#131	—	5	56	no	—
47	C₂³×M₁₆	—	5	1	no	—
48	C₂³×M₁₆	—	5	2	no	—
49	C₂⁴×C₈	—	5	1	no	—
50	C₂⁴×C₈	—	5	2	no	—
51	C₂⁴×C₅	—	2	16	no	—
52	C₂⁴×C₅	C₂²×C₄	1	28	yes	—
53	C₂⁴×C₅	C₂²×C₄	1	1	yes	characteristic
54	C₂³×D₅	—	2	8	no	—
55	C₂³×D₅	C₂²×C₄	1	14	yes	—
56	C₂×C₅×D₄	C₂²×C₄	1	112	yes	—
57	C₂²×C₄×C₅	C₂²×C₄	1	14	yes	—
58	C₂×C₂²⋊D₅	C₂²×C₄	1	224	yes	—
59	C₂²×Dic₅	C₂²×C₄	1	28	yes	—
60	C₂²×Dic₅	C₂²×C₄	1	7	yes	—
61	C₂²×Dic₅	C₂²×C₄	1	168	yes	—
62	C₂²×Dic₅	C₂⁴	1	7	yes	—
63	C₂²×Dic₅	C₂⁴	1	56	yes	—
64	#711	C₂²×C₄	1	448	yes	—
65	#711	C₂⁴	1	64	yes	—
66	C₂×C₄×D₅	C₂²×C₄	1	84	yes	—
67	C₂×C₄×D₅	C₂⁴	1	28	yes	—
68	C₂×Dic₁₀	C₂²×C₄	1	112	yes	—
69	#704	C₂⁴	1	128	yes	complemented
70	C₂×#170	C₂⁴	1	28	yes	—
71	C₂×#170	C₂⁴	1	28	yes	—
72	C₂×#170	C₂⁴	1	56	yes	—
73	C₂×#170	C₂⁴	1	128	yes	complemented
74	#700	C₂⁴	1	64	yes	complemented
75	D₅⋊C₈	C₂⁴	1	64	yes	complemented
76	C₂²×#41	—	5	28	no	—
77	C₂²×#41	—	5	28	no	—
78	C₂³×Q₈	—	5	1	no	—
79	C₂³×D₄	—	5	2	no	—
80	C₂³×D₄	AGL(1,𝔽₅)	1	1	yes	characteristic
81	C₂⁴×C₄	—	5	2	no	—
82	C₂⁴×C₄	—	5	1	no	—
83	C₂×#131	—	5	448	no	—
84	C₂²×M₁₆	—	5	28	no	—
85	C₂²×M₁₆	—	5	56	no	—
86	C₂³×C₈	—	5	28	no	—
87	C₂³×C₈	—	5	1	no	—
88	C₂³×C₈	—	5	1	no	—
89	C₂³×C₈	—	5	2	no	—
90	C₂³×C₈	—	5	56	no	—
91	C₂³×C₅	—	2	112	no	—
92	C₂³×C₅	#131	1	8	yes	complemented
93	C₂³×C₅	C₂³×C₄	1	56	yes	—
94	C₂³×C₅	C₂³×C₄	1	7	yes	—
95	C₂²×D₅	—	2	56	no	—
96	C₂²×D₅	C₂³×C₄	1	28	yes	—
97	C₅×D₄	C₂³×C₄	1	64	yes	—
98	C₂×C₄×C₅	C₂³×C₄	1	28	yes	—
99	C₂²⋊D₅	C₂³×C₄	1	128	yes	—
100	C₂×Dic₅	C₂³×C₄	1	56	yes	—
101	C₂×Dic₅	C₂³×C₄	1	21	yes	—
102	C₂×Dic₅	C₂³×C₄	1	112	yes	—
103	C₂×Dic₅	C₂⁵	1	7	yes	—
104	C₂×Dic₅	C₂⁵	1	16	yes	—
105	C₄×D₅	C₂³×C₄	1	56	yes	—
106	C₄×D₅	C₂⁵	1	8	yes	—
107	Dic₁₀	C₂³×C₄	1	64	yes	—
108	#170	C₂⁵	1	8	yes	—
109	#170	C₂⁵	1	8	yes	—
110	#170	C₂⁵	1	16	yes	—
111	C₂⁵	—	5	1	no	—
112	C₂⁵	C₂×AGL(1,𝔽₅)	1	2	yes	—
113	C₂×#41	—	5	336	no	—
114	C₂×#41	—	5	112	no	—
115	C₂²×Q₈	—	5	28	no	—
116	C₂²×D₄	—	5	56	no	—
117	C₂²×D₄	C₂×AGL(1,𝔽₅)	1	28	yes	—
118	C₂³×C₄	—	5	28	no	—
119	C₂³×C₄	—	5	28	no	—
120	C₂³×C₄	—	5	1	no	—
121	C₂³×C₄	—	5	2	no	—
122	C₂³×C₄	—	5	14	no	—
123	C₂³×C₄	—	5	14	no	—
124	C₂³×C₄	C₂×AGL(1,𝔽₅)	1	1	yes	characteristic
125	#131	—	5	512	no	—
126	C₂×M₁₆	—	5	112	no	—
127	C₂×M₁₆	—	5	224	no	—
128	C₂²×C₈	—	5	112	no	—
129	C₂²×C₈	—	5	14	no	—
130	C₂²×C₈	—	5	14	no	—
131	C₂²×C₈	—	5	28	no	—
132	C₂²×C₈	—	5	224	no	—
133	C₂²×C₅	—	2	112	no	—
134	C₂²×C₅	C₂×#131	1	28	yes	complemented
135	C₂²×C₅	C₂⁴×C₄	1	16	yes	—
136	C₂²×C₅	C₂⁴×C₄	1	7	yes	—
137	C₂×D₅	—	2	56	no	—
138	C₂×D₅	C₂⁴×C₄	1	8	yes	—
139	C₄×C₅	C₂⁴×C₄	1	8	yes	—
140	Dic₅	C₂⁴×C₄	1	16	yes	—
141	Dic₅	C₂⁴×C₄	1	7	yes	—
142	Dic₅	C₂⁶	1	1	yes	characteristic
143	C₂⁴	—	2	16	no	—
144	C₂⁴	—	5	14	no	—
145	C₂⁴	—	10	8	no	—
146	C₂⁴	C₂²×AGL(1,𝔽₅)	1	28	yes	—
147	C₂⁴	C₂²×AGL(1,𝔽₅)	1	1	yes	characteristic
148	#41	—	5	64	no	—
149	#41	—	5	448	no	—
150	C₂×Q₈	—	5	112	no	—
151	C₂×D₄	—	5	224	no	—
152	C₂×D₄	C₂²×AGL(1,𝔽₅)	1	112	yes	—
153	C₂²×C₄	—	5	7	no	—
154	C₂²×C₄	—	5	56	no	—
155	C₂²×C₄	—	5	7	no	—
156	C₂²×C₄	—	5	28	no	—
157	C₂²×C₄	—	5	168	no	—
158	C₂²×C₄	—	5	28	no	—
159	C₂²×C₄	—	5	84	no	—
160	C₂²×C₄	C₂²×AGL(1,𝔽₅)	1	14	yes	—
161	M₁₆	—	5	64	no	—
162	M₁₆	—	5	128	no	—
163	C₂×C₈	—	5	28	no	—
164	C₂×C₈	—	5	28	no	—
165	C₂×C₈	—	5	56	no	—
166	C₂×C₈	—	5	64	no	—
167	C₂×C₈	—	5	128	no	—
168	C₂×C₅	—	2	16	no	—
169	C₂×C₅	C₂²×#131	1	14	yes	complemented
170	C₂×C₅	C₂⁵×C₄	1	1	yes	characteristic
171	D₅	—	2	8	no	—
172	C₂³	—	2	112	no	—
173	C₂³	—	5	28	no	—
174	C₂³	—	10	56	no	—
175	C₂³	#4131	1	8	yes	direct product factor
176	C₂³	C₂³×AGL(1,𝔽₅)	1	56	yes	—
177	C₂³	C₂³×AGL(1,𝔽₅)	1	7	yes	—
178	Q₈	—	5	64	no	—
179	D₄	—	5	128	no	—
180	D₄	C₂³×AGL(1,𝔽₅)	1	64	yes	—
181	C₂×C₄	—	5	7	no	—
182	C₂×C₄	—	5	16	no	—
183	C₂×C₄	—	5	21	no	—
184	C₂×C₄	—	5	56	no	—
185	C₂×C₄	—	5	112	no	—
186	C₂×C₄	—	5	8	no	—
187	C₂×C₄	—	5	56	no	—
188	C₂×C₄	C₂³×AGL(1,𝔽₅)	1	28	yes	—
189	C₈	—	5	8	no	—
190	C₈	—	5	8	no	—
191	C₈	—	5	16	no	—
192	C₅	C₂³×#131	1	1	yes	characteristic, complemented, Sylow
193	C₂²	—	2	112	no	—
194	C₂²	—	5	8	no	—
195	C₂²	—	10	56	no	—
196	C₂²	C₂×#4131	1	28	yes	direct product factor
197	C₂²	C₂⁴×AGL(1,𝔽₅)	1	16	yes	—
198	C₂²	C₂⁴×AGL(1,𝔽₅)	1	7	yes	—
199	C₄	—	5	1	no	—
200	C₄	—	5	7	no	—
201	C₄	—	5	16	no	—
202	C₄	C₂⁴×AGL(1,𝔽₅)	1	8	yes	—
203	C₂	—	2	16	no	—
204	C₂	—	10	8	no	—
205	C₂	C₂²×#4131	1	14	yes	direct product factor
206	C₂	C₂⁵×AGL(1,𝔽₅)	1	1	yes	characteristic

	subgroup	G/H
center	C₂⁴	C₂²×AGL(1,𝔽₅)
commutator	C₂×C₅	C₂⁵×C₄
Frattini	C₂	C₂⁵×AGL(1,𝔽₅)
Fitting	C₂³×C₅×D₄	C₄
socle	C₂⁴×C₅	C₂²×C₄
prime radical	C₂³×#711	C₂
cosocle	Dic₅	C₂⁶
hypercenter	C₂³×D₄	AGL(1,𝔽₅)
hypocenter	C₅	C₂³×#131
LCS	G ▷ C₂×C₅ ▷ C₅
UCS	C₁ ◁ C₂⁴ ◁ C₂³×D₄

Complements

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

N	H ≅ G/N	type
2. C₂³×#704	C₂	semidirect
3. C₂⁴×#170	C₂	semidirect
4. C₂²×#4131	C₂	direct (N × H)
5. C₂³×#700	C₂	semidirect
6. C₂³×D₅⋊C₈	C₂	semidirect
14. C₂²×#704	C₂²	semidirect
18. C₂³×#170	C₂²	semidirect
19. C₂×#4131	C₂²	direct (N × H)
20. C₂²×#700	C₂²	semidirect
21. C₂²×D₅⋊C₈	C₂²	semidirect
37. C₂×#704	C₂³	semidirect
41. C₂²×#170	C₂³	semidirect
42. #4131	C₂³	direct (N × H)
43. C₂×#700	C₂³	semidirect
44. C₂×D₅⋊C₈	C₂³	semidirect
69. #704	C₂⁴	semidirect
73. C₂×#170	C₂⁴	semidirect
74. #700	C₂⁴	semidirect
75. D₅⋊C₈	C₂⁴	semidirect
92. C₂³×C₅	#131	semidirect
134. C₂²×C₅	C₂×#131	semidirect
169. C₂×C₅	C₂²×#131	semidirect
175. C₂³	#4131	direct (N × H)
192. C₅	C₂³×#131	semidirect
196. C₂²	C₂×#4131	direct (N × H)
205. C₂	C₂²×#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
C₂³×#711	2
C₂³×#704	2
C₂⁴×#170	2
C₂²×#4131	2
C₂³×#700	2
C₂³×D₅⋊C₈	2
C₂³×#131	5

Subgroups — C23×#4131 #13046027

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₂³×#4131 #13046027