Subgroups — C₂³×C₃×C₅×C₈² #72186608

Order 7680.

16648 conjugacy classes of subgroups.

The subgroup lattice has 216 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂³×C₃×C₄×C₅×C₈	C₂	1	3	yes	maximal
2	C₂²×C₃×C₅×C₈²	C₂	1	28	yes	maximal, direct product factor
3	C₂³×C₅×C₈²	C₃	1	1	yes	maximal, characteristic, direct product factor, Hall
4	C₂³×C₃×C₈²	C₅	1	1	yes	maximal, characteristic, direct product factor, Hall
5	C₂⁴×C₃×C₅×C₈	C₄	1	6	yes	—
6	C₂³×C₃×C₄²×C₅	C₂²	1	1	yes	characteristic
7	C₂²×C₃×C₄×C₅×C₈	C₄	1	42	yes	—
8	C₂²×C₃×C₄×C₅×C₈	C₂²	1	42	yes	—
9	C₂×C₃×C₅×C₈²	C₂²	1	112	yes	direct product factor
10	C₂³×C₄×C₅×C₈	C₂×C₃	1	3	yes	—
11	C₂²×C₅×C₈²	C₂×C₃	1	28	yes	direct product factor
12	C₂⁴×C₃×C₄×C₅	C₂×C₄	1	3	yes	—
13	C₂³×C₃×C₅×C₈	C₈	1	96	yes	direct product factor
14	C₂³×C₃×C₅×C₈	C₂×C₄	1	84	yes	—
15	C₂²×C₃×C₄²×C₅	C₂×C₄	1	21	yes	—
16	C₂²×C₃×C₄²×C₅	C₂³	1	7	yes	—
17	C₂×C₃×C₄×C₅×C₈	C₂×C₄	1	252	yes	—
18	C₂×C₃×C₄×C₅×C₈	C₂³	1	84	yes	—
19	C₃×C₅×C₈²	C₂³	1	64	yes	direct product factor
20	C₂³×C₃×C₄×C₈	C₂×C₅	1	3	yes	—
21	C₂²×C₃×C₈²	C₂×C₅	1	28	yes	direct product factor
22	C₂⁴×C₅×C₈	C₃×C₄	1	6	yes	—
23	C₂³×C₄²×C₅	C₂²×C₃	1	1	yes	characteristic
24	C₂²×C₄×C₅×C₈	C₃×C₄	1	42	yes	—
25	C₂²×C₄×C₅×C₈	C₂²×C₃	1	42	yes	—
26	C₂×C₅×C₈²	C₂²×C₃	1	112	yes	direct product factor
27	C₂³×C₈²	C₃×C₅	1	1	yes	characteristic, direct product factor, Sylow
28	C₂⁵×C₃×C₅	C₄²	1	1	yes	characteristic
29	C₂³×C₃×C₄×C₅	C₄²	1	21	yes	—
30	C₂³×C₃×C₄×C₅	C₂×C₈	1	48	yes	—
31	C₂³×C₃×C₄×C₅	C₂²×C₄	1	21	yes	—
32	C₂²×C₃×C₅×C₈	C₂×C₈	1	672	yes	direct product factor
33	C₂²×C₃×C₅×C₈	C₂²×C₄	1	168	yes	—
34	C₂×C₃×C₄²×C₅	C₄²	1	42	yes	—
35	C₂×C₃×C₄²×C₅	C₂²×C₄	1	63	yes	—
36	C₂×C₃×C₄²×C₅	C₂⁴	1	7	yes	—
37	C₃×C₄×C₅×C₈	C₂²×C₄	1	168	yes	—
38	C₃×C₄×C₅×C₈	C₂⁴	1	24	yes	—
39	C₂⁴×C₃×C₈	C₄×C₅	1	6	yes	—
40	C₂³×C₃×C₄²	C₂²×C₅	1	1	yes	characteristic
41	C₂²×C₃×C₄×C₈	C₄×C₅	1	42	yes	—
42	C₂²×C₃×C₄×C₈	C₂²×C₅	1	42	yes	—
43	C₂×C₃×C₈²	C₂²×C₅	1	112	yes	direct product factor
44	C₂⁴×C₄×C₅	C₂×C₃×C₄	1	3	yes	—
45	C₂³×C₅×C₈	C₃×C₈	1	96	yes	direct product factor
46	C₂³×C₅×C₈	C₂×C₃×C₄	1	84	yes	—
47	C₂²×C₄²×C₅	C₂×C₃×C₄	1	21	yes	—
48	C₂²×C₄²×C₅	C₂³×C₃	1	7	yes	—
49	C₂×C₄×C₅×C₈	C₂×C₃×C₄	1	252	yes	—
50	C₂×C₄×C₅×C₈	C₂³×C₃	1	84	yes	—
51	C₅×C₈²	C₂³×C₃	1	64	yes	direct product factor
52	C₂³×C₄×C₈	C₂×C₃×C₅	1	3	yes	—
53	C₂²×C₈²	C₂×C₃×C₅	1	28	yes	direct product factor
54	C₂⁴×C₃×C₅	C₄×C₈	1	24	yes	—
55	C₂⁴×C₃×C₅	C₂×C₄²	1	7	yes	—
56	C₂²×C₃×C₄×C₅	C₄×C₈	1	168	yes	—
57	C₂²×C₃×C₄×C₅	C₂×C₄²	1	63	yes	—
58	C₂²×C₃×C₄×C₅	C₂²×C₈	1	168	yes	—
59	C₂²×C₃×C₄×C₅	C₂³×C₄	1	21	yes	—
60	C₂×C₃×C₅×C₈	C₂²×C₈	1	672	yes	direct product factor
61	C₂×C₃×C₅×C₈	C₂³×C₄	1	48	yes	—
62	C₃×C₄²×C₅	C₂×C₄²	1	42	yes	—
63	C₃×C₄²×C₅	C₂³×C₄	1	21	yes	—
64	C₃×C₄²×C₅	C₂⁵	1	1	yes	characteristic
65	C₂⁴×C₃×C₄	C₂×C₄×C₅	1	3	yes	—
66	C₂³×C₃×C₈	C₅×C₈	1	96	yes	direct product factor
67	C₂³×C₃×C₈	C₂×C₄×C₅	1	84	yes	—
68	C₂²×C₃×C₄²	C₂×C₄×C₅	1	21	yes	—
69	C₂²×C₃×C₄²	C₂³×C₅	1	7	yes	—
70	C₂×C₃×C₄×C₈	C₂×C₄×C₅	1	252	yes	—
71	C₂×C₃×C₄×C₈	C₂³×C₅	1	84	yes	—
72	C₃×C₈²	C₂³×C₅	1	64	yes	direct product factor
73	C₂⁵×C₅	C₃×C₄²	1	1	yes	characteristic
74	C₂³×C₄×C₅	C₃×C₄²	1	21	yes	—
75	C₂³×C₄×C₅	C₂×C₃×C₈	1	48	yes	—
76	C₂³×C₄×C₅	C₂²×C₃×C₄	1	21	yes	—
77	C₂²×C₅×C₈	C₂×C₃×C₈	1	672	yes	direct product factor
78	C₂²×C₅×C₈	C₂²×C₃×C₄	1	168	yes	—
79	C₂×C₄²×C₅	C₃×C₄²	1	42	yes	—
80	C₂×C₄²×C₅	C₂²×C₃×C₄	1	63	yes	—
81	C₂×C₄²×C₅	C₂⁴×C₃	1	7	yes	—
82	C₄×C₅×C₈	C₂²×C₃×C₄	1	168	yes	—
83	C₄×C₅×C₈	C₂⁴×C₃	1	24	yes	—
84	C₂⁴×C₈	C₃×C₄×C₅	1	6	yes	—
85	C₂³×C₄²	C₂²×C₃×C₅	1	1	yes	characteristic
86	C₂²×C₄×C₈	C₃×C₄×C₅	1	42	yes	—
87	C₂²×C₄×C₈	C₂²×C₃×C₅	1	42	yes	—
88	C₂×C₈²	C₂²×C₃×C₅	1	112	yes	direct product factor
89	C₂³×C₃×C₅	C₈²	1	64	yes	direct product factor
90	C₂³×C₃×C₅	C₂×C₄×C₈	1	84	yes	—
91	C₂³×C₃×C₅	C₂²×C₄²	1	7	yes	—
92	C₂×C₃×C₄×C₅	C₂×C₄×C₈	1	252	yes	—
93	C₂×C₃×C₄×C₅	C₂²×C₄²	1	21	yes	—
94	C₂×C₃×C₄×C₅	C₂³×C₈	1	84	yes	—
95	C₂×C₃×C₄×C₅	C₂⁴×C₄	1	3	yes	—
96	C₃×C₅×C₈	C₂³×C₈	1	96	yes	direct product factor
97	C₂⁵×C₃	C₄²×C₅	1	1	yes	characteristic
98	C₂³×C₃×C₄	C₄²×C₅	1	21	yes	—
99	C₂³×C₃×C₄	C₂×C₅×C₈	1	48	yes	—
100	C₂³×C₃×C₄	C₂²×C₄×C₅	1	21	yes	—
101	C₂²×C₃×C₈	C₂×C₅×C₈	1	672	yes	direct product factor
102	C₂²×C₃×C₈	C₂²×C₄×C₅	1	168	yes	—
103	C₂×C₃×C₄²	C₄²×C₅	1	42	yes	—
104	C₂×C₃×C₄²	C₂²×C₄×C₅	1	63	yes	—
105	C₂×C₃×C₄²	C₂⁴×C₅	1	7	yes	—
106	C₃×C₄×C₈	C₂²×C₄×C₅	1	168	yes	—
107	C₃×C₄×C₈	C₂⁴×C₅	1	24	yes	—
108	C₂⁴×C₅	C₃×C₄×C₈	1	24	yes	—
109	C₂⁴×C₅	C₂×C₃×C₄²	1	7	yes	—
110	C₂²×C₄×C₅	C₃×C₄×C₈	1	168	yes	—
111	C₂²×C₄×C₅	C₂×C₃×C₄²	1	63	yes	—
112	C₂²×C₄×C₅	C₂²×C₃×C₈	1	168	yes	—
113	C₂²×C₄×C₅	C₂³×C₃×C₄	1	21	yes	—
114	C₂×C₅×C₈	C₂²×C₃×C₈	1	672	yes	direct product factor
115	C₂×C₅×C₈	C₂³×C₃×C₄	1	48	yes	—
116	C₄²×C₅	C₂×C₃×C₄²	1	42	yes	—
117	C₄²×C₅	C₂³×C₃×C₄	1	21	yes	—
118	C₄²×C₅	C₂⁵×C₃	1	1	yes	characteristic
119	C₂⁴×C₄	C₂×C₃×C₄×C₅	1	3	yes	—
120	C₂³×C₈	C₃×C₅×C₈	1	96	yes	direct product factor
121	C₂³×C₈	C₂×C₃×C₄×C₅	1	84	yes	—
122	C₂²×C₄²	C₂×C₃×C₄×C₅	1	21	yes	—
123	C₂²×C₄²	C₂³×C₃×C₅	1	7	yes	—
124	C₂×C₄×C₈	C₂×C₃×C₄×C₅	1	252	yes	—
125	C₂×C₄×C₈	C₂³×C₃×C₅	1	84	yes	—
126	C₈²	C₂³×C₃×C₅	1	64	yes	direct product factor
127	C₂²×C₃×C₅	C₂×C₈²	1	112	yes	direct product factor
128	C₂²×C₃×C₅	C₂²×C₄×C₈	1	42	yes	—
129	C₂²×C₃×C₅	C₂³×C₄²	1	1	yes	characteristic
130	C₃×C₄×C₅	C₂²×C₄×C₈	1	42	yes	—
131	C₃×C₄×C₅	C₂⁴×C₈	1	6	yes	—
132	C₂⁴×C₃	C₄×C₅×C₈	1	24	yes	—
133	C₂⁴×C₃	C₂×C₄²×C₅	1	7	yes	—
134	C₂²×C₃×C₄	C₄×C₅×C₈	1	168	yes	—
135	C₂²×C₃×C₄	C₂×C₄²×C₅	1	63	yes	—
136	C₂²×C₃×C₄	C₂²×C₅×C₈	1	168	yes	—
137	C₂²×C₃×C₄	C₂³×C₄×C₅	1	21	yes	—
138	C₂×C₃×C₈	C₂²×C₅×C₈	1	672	yes	direct product factor
139	C₂×C₃×C₈	C₂³×C₄×C₅	1	48	yes	—
140	C₃×C₄²	C₂×C₄²×C₅	1	42	yes	—
141	C₃×C₄²	C₂³×C₄×C₅	1	21	yes	—
142	C₃×C₄²	C₂⁵×C₅	1	1	yes	characteristic
143	C₂³×C₅	C₃×C₈²	1	64	yes	direct product factor
144	C₂³×C₅	C₂×C₃×C₄×C₈	1	84	yes	—
145	C₂³×C₅	C₂²×C₃×C₄²	1	7	yes	—
146	C₂×C₄×C₅	C₂×C₃×C₄×C₈	1	252	yes	—
147	C₂×C₄×C₅	C₂²×C₃×C₄²	1	21	yes	—
148	C₂×C₄×C₅	C₂³×C₃×C₈	1	84	yes	—
149	C₂×C₄×C₅	C₂⁴×C₃×C₄	1	3	yes	—
150	C₅×C₈	C₂³×C₃×C₈	1	96	yes	direct product factor
151	C₂⁵	C₃×C₄²×C₅	1	1	yes	characteristic
152	C₂³×C₄	C₃×C₄²×C₅	1	21	yes	—
153	C₂³×C₄	C₂×C₃×C₅×C₈	1	48	yes	—
154	C₂³×C₄	C₂²×C₃×C₄×C₅	1	21	yes	—
155	C₂²×C₈	C₂×C₃×C₅×C₈	1	672	yes	direct product factor
156	C₂²×C₈	C₂²×C₃×C₄×C₅	1	168	yes	—
157	C₂×C₄²	C₃×C₄²×C₅	1	42	yes	—
158	C₂×C₄²	C₂²×C₃×C₄×C₅	1	63	yes	—
159	C₂×C₄²	C₂⁴×C₃×C₅	1	7	yes	—
160	C₄×C₈	C₂²×C₃×C₄×C₅	1	168	yes	—
161	C₄×C₈	C₂⁴×C₃×C₅	1	24	yes	—
162	C₂×C₃×C₅	C₂²×C₈²	1	28	yes	direct product factor
163	C₂×C₃×C₅	C₂³×C₄×C₈	1	3	yes	—
164	C₂³×C₃	C₅×C₈²	1	64	yes	direct product factor
165	C₂³×C₃	C₂×C₄×C₅×C₈	1	84	yes	—
166	C₂³×C₃	C₂²×C₄²×C₅	1	7	yes	—
167	C₂×C₃×C₄	C₂×C₄×C₅×C₈	1	252	yes	—
168	C₂×C₃×C₄	C₂²×C₄²×C₅	1	21	yes	—
169	C₂×C₃×C₄	C₂³×C₅×C₈	1	84	yes	—
170	C₂×C₃×C₄	C₂⁴×C₄×C₅	1	3	yes	—
171	C₃×C₈	C₂³×C₅×C₈	1	96	yes	direct product factor
172	C₂²×C₅	C₂×C₃×C₈²	1	112	yes	direct product factor
173	C₂²×C₅	C₂²×C₃×C₄×C₈	1	42	yes	—
174	C₂²×C₅	C₂³×C₃×C₄²	1	1	yes	characteristic
175	C₄×C₅	C₂²×C₃×C₄×C₈	1	42	yes	—
176	C₄×C₅	C₂⁴×C₃×C₈	1	6	yes	—
177	C₂⁴	C₃×C₄×C₅×C₈	1	24	yes	—
178	C₂⁴	C₂×C₃×C₄²×C₅	1	7	yes	—
179	C₂²×C₄	C₃×C₄×C₅×C₈	1	168	yes	—
180	C₂²×C₄	C₂×C₃×C₄²×C₅	1	63	yes	—
181	C₂²×C₄	C₂²×C₃×C₅×C₈	1	168	yes	—
182	C₂²×C₄	C₂³×C₃×C₄×C₅	1	21	yes	—
183	C₂×C₈	C₂²×C₃×C₅×C₈	1	672	yes	direct product factor
184	C₂×C₈	C₂³×C₃×C₄×C₅	1	48	yes	—
185	C₄²	C₂×C₃×C₄²×C₅	1	42	yes	—
186	C₄²	C₂³×C₃×C₄×C₅	1	21	yes	—
187	C₄²	C₂⁵×C₃×C₅	1	1	yes	characteristic
188	C₃×C₅	C₂³×C₈²	1	1	yes	characteristic, direct product factor, Hall
189	C₂²×C₃	C₂×C₅×C₈²	1	112	yes	direct product factor
190	C₂²×C₃	C₂²×C₄×C₅×C₈	1	42	yes	—
191	C₂²×C₃	C₂³×C₄²×C₅	1	1	yes	characteristic
192	C₃×C₄	C₂²×C₄×C₅×C₈	1	42	yes	—
193	C₃×C₄	C₂⁴×C₅×C₈	1	6	yes	—
194	C₂×C₅	C₂²×C₃×C₈²	1	28	yes	direct product factor
195	C₂×C₅	C₂³×C₃×C₄×C₈	1	3	yes	—
196	C₂³	C₃×C₅×C₈²	1	64	yes	direct product factor
197	C₂³	C₂×C₃×C₄×C₅×C₈	1	84	yes	—
198	C₂³	C₂²×C₃×C₄²×C₅	1	7	yes	—
199	C₂×C₄	C₂×C₃×C₄×C₅×C₈	1	252	yes	—
200	C₂×C₄	C₂²×C₃×C₄²×C₅	1	21	yes	—
201	C₂×C₄	C₂³×C₃×C₅×C₈	1	84	yes	—
202	C₂×C₄	C₂⁴×C₃×C₄×C₅	1	3	yes	—
203	C₈	C₂³×C₃×C₅×C₈	1	96	yes	direct product factor
204	C₂×C₃	C₂²×C₅×C₈²	1	28	yes	direct product factor
205	C₂×C₃	C₂³×C₄×C₅×C₈	1	3	yes	—
206	C₅	C₂³×C₃×C₈²	1	1	yes	characteristic, direct product factor, Sylow
207	C₂²	C₂×C₃×C₅×C₈²	1	112	yes	direct product factor
208	C₂²	C₂²×C₃×C₄×C₅×C₈	1	42	yes	—
209	C₂²	C₂³×C₃×C₄²×C₅	1	1	yes	characteristic
210	C₄	C₂²×C₃×C₄×C₅×C₈	1	42	yes	—
211	C₄	C₂⁴×C₃×C₅×C₈	1	6	yes	—
212	C₃	C₂³×C₅×C₈²	1	1	yes	characteristic, direct product factor, Sylow
213	C₂	C₂²×C₃×C₅×C₈²	1	28	yes	direct product factor
214	C₂	C₂³×C₃×C₄×C₅×C₈	1	3	yes	—

	subgroup	G/H
center	G	C₁
commutator	C₁	G
Frattini, cosocle	C₄²	C₂⁵×C₃×C₅
socle, prime radical	C₂⁵×C₃×C₅	C₄²
LCS	G ▷ C₁
UCS	C₁ ◁ G

Complements

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

N	H ≅ G/N	type
2. C₂²×C₃×C₅×C₈²	C₂	direct (N × H)
3. C₂³×C₅×C₈²	C₃	direct (N × H)
4. C₂³×C₃×C₈²	C₅	direct (N × H)
9. C₂×C₃×C₅×C₈²	C₂²	direct (N × H)
11. C₂²×C₅×C₈²	C₂×C₃	direct (N × H)
13. C₂³×C₃×C₅×C₈	C₈	direct (N × H)
19. C₃×C₅×C₈²	C₂³	direct (N × H)
21. C₂²×C₃×C₈²	C₂×C₅	direct (N × H)
26. C₂×C₅×C₈²	C₂²×C₃	direct (N × H)
27. C₂³×C₈²	C₃×C₅	direct (N × H)
32. C₂²×C₃×C₅×C₈	C₂×C₈	direct (N × H)
43. C₂×C₃×C₈²	C₂²×C₅	direct (N × H)
45. C₂³×C₅×C₈	C₃×C₈	direct (N × H)
51. C₅×C₈²	C₂³×C₃	direct (N × H)
53. C₂²×C₈²	C₂×C₃×C₅	direct (N × H)
60. C₂×C₃×C₅×C₈	C₂²×C₈	direct (N × H)
66. C₂³×C₃×C₈	C₅×C₈	direct (N × H)
72. C₃×C₈²	C₂³×C₅	direct (N × H)
77. C₂²×C₅×C₈	C₂×C₃×C₈	direct (N × H)
88. C₂×C₈²	C₂²×C₃×C₅	direct (N × H)
89. C₂³×C₃×C₅	C₈²	direct (N × H)
96. C₃×C₅×C₈	C₂³×C₈	direct (N × H)
101. C₂²×C₃×C₈	C₂×C₅×C₈	direct (N × H)
114. C₂×C₅×C₈	C₂²×C₃×C₈	direct (N × H)
120. C₂³×C₈	C₃×C₅×C₈	direct (N × H)
126. C₈²	C₂³×C₃×C₅	direct (N × H)
127. C₂²×C₃×C₅	C₂×C₈²	direct (N × H)
138. C₂×C₃×C₈	C₂²×C₅×C₈	direct (N × H)
143. C₂³×C₅	C₃×C₈²	direct (N × H)
150. C₅×C₈	C₂³×C₃×C₈	direct (N × H)
155. C₂²×C₈	C₂×C₃×C₅×C₈	direct (N × H)
162. C₂×C₃×C₅	C₂²×C₈²	direct (N × H)
164. C₂³×C₃	C₅×C₈²	direct (N × H)
171. C₃×C₈	C₂³×C₅×C₈	direct (N × H)
172. C₂²×C₅	C₂×C₃×C₈²	direct (N × H)
183. C₂×C₈	C₂²×C₃×C₅×C₈	direct (N × H)
188. C₃×C₅	C₂³×C₈²	direct (N × H)
189. C₂²×C₃	C₂×C₅×C₈²	direct (N × H)
194. C₂×C₅	C₂²×C₃×C₈²	direct (N × H)
196. C₂³	C₃×C₅×C₈²	direct (N × H)
203. C₈	C₂³×C₃×C₅×C₈	direct (N × H)
204. C₂×C₃	C₂²×C₅×C₈²	direct (N × H)
206. C₅	C₂³×C₃×C₈²	direct (N × H)
207. C₂²	C₂×C₃×C₅×C₈²	direct (N × H)
212. C₃	C₂³×C₅×C₈²	direct (N × H)
213. C₂	C₂²×C₃×C₅×C₈²	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×C₃×C₄×C₅×C₈	2
C₂²×C₃×C₅×C₈²	2
C₂³×C₅×C₈²	3
C₂³×C₃×C₈²	5

Subgroups — C23×C3×C5×C82 #72186608

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₂³×C₃×C₅×C₈² #72186608